Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano: Application to Biology, Physics, Material Science, Mechanics, Structural and Processing Engineering [1 ed.] 9781402050619, 1-4020-5061-5

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MULTISCALING IN MOLECULAR AND CONTINUUM MECHANICS: INTERACTION OF TIME AND SIZE FROM MACRO TO NANO

Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano Application to biology, physics, material science, mechanics, structural and processing engineering

Edited by

G.C. SIH Lehigh University, Bethlehem, Pennsylvania, USA and East China University of Science and Technology, Shanghai, China

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 ISBN-13 ISBN-10 ISBN-13

1-4020-5061-5 (HB) 978-1-4020-5061-9 (HB) 1-4020-5062-3 (e-book) 978-1-4020-5008-4 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved © 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

Aims and Scope of the book The multi-disciplinary character of the book has been encouraged by the current trend of development of science and technology for seeking a common base of understanding. Multiscaling in time and size, from fast to slow and small to large can assist the micromanipulation of atoms and molecules to tailor make structureand bio-materials. The excitement lies in the likelihood that organic and inorganic matters might follow the same basic laws. The hope to discover the common denominator for all things has been the driving force for research in the past and will be in the future. The eighteen articles contained in this volume are evidence of the excitement that has generated in the application of fundamental science and high technology. That includes biology, physics, material science, mechanics, structural and processing engineering. Orders of magnitude improvement in the resolution of electron microscopes and accuracies of the electronic computers will no doubt continue to generate new discoveries and pleasant surprises. The theme of the 21st century research has been directed towards understanding the microscopic factors that control the macroscopic material degradation and biological malfunction. Chemical decomposition reacting at 10– 9 sec for cyclic nitramines can trigger detonation of macro-size solid propellants. Such a phenomenon could not have been known without examining the intermediate products of the gas phase reaction which are µm in size. Similar mechanisms for intergranular nano-meter size defects discovered by high resolution TEM and SEM may also explain the initiation of stress corrosion cracking. Psychosurgery via the use of brain implants is finding ways to send signals to neurons for curing blindness and mental disorders. Different fields from material to life science are finding solutions to their long waited answers by reaching down to the lower size scale. The contributions of this volume are aimed to add to the progress of micromanipulation by relating events from a wide range of size and time scale although there remains to be done in multiscaling where equilibrium at a larger scale may involve non-equilibrium at a lower scale. That is macroscopic homogeneity may entail microscopic heterogeneity. Contained in this volume are approaches involving the analytical, experimental and physical modeling, each serving a specific objective. The ultimate effort is to find a common ground of understanding. To this end, this book has made a step closer to the goal.

Contents Contributors

ix

Foreword

xv

“Deborah numbers”, coupling multiple space and time scales and governing damage evolution to failure Y.L. Bai, H.Y. Wang, M.F. Xia, F.J. Ke A multi-scale formulation for modeling of wrinkling formation in polycrystalline materials J.S. Chen, S. Mehraeen A multiscale field theory: Nano/micro materials Y.P. Chen, J.D. Lee, Y.J. Lei, L.M. Xiong Combined loading rate and specimen size effects on the material properties Z. Chen, Y. Gan, L.M. Shen

1

11

23

67

85

Discrete-to-continuum scale bridging J. Fish Micromechanics and multiscale mechanics of carbon nanotubes-reinforced composites X.Q. Feng, D.L. Shi, Y.G. Huang, K.C. Hwang

103

Multi-scale analytical methods for complex flows in process engineering: Retrospect and prospect W. Ge, F.G. Chen, G.Z. Zhou, J.G. Li

141

Multiscaling effects in low alloy TRIP steels G.N. Haidemenopoulos, A.I. Katsamas, N. Aravas

161

Ductile Cr-Alloys with solute and precipitate softening S. Hao, J. Weertman

179

vii

viii

Contents

A multi-scale approach to crack growth R. Jones, S. Barter, L. Molent, S. Pitt

197

Continuum-based and cluster models for nanomaterials D. Qian, K. Nagarajan, S.R. Mannava, V.K. Vasudevan

241

Segmented multiscale approach by microscoping and telescoping in material science G.C. Sih

259

Mode I segmented crack model: Macro/symmetry, micro/ anti-symmetry and dislocation/skew-symmetry G.C. Sih, X.S. Tang

291

Tensegrity architecture and the mammalian cell cytoskeleton D. Stamenoviü, N. Wang, D.E. Ingber Mode II segmented crack model: Macro/skew-symmetry, micro/anti-symmetry and dislocation/skew-symmetry X.S. Tang, G.C. Sih Microstructure and microhardness in surface-nanocrystalline Al-alloy material

321

339

369

Y.G. Wei, X.L. Wu, C. Zhu, M.H. Zhao Grain boundary effects on fatigue damage and material properties: Macro- and micro-considerations Z.F. Zhang, Z.G. Wang

389

Coupling and communicating between atomistic and continuum simulation methodologies J.A. Zimmerman, P.A. Klein, E.B. Webb III

439

Author Index

457

Subject Index

459

Contributors Aravas N

Bai YL Barter SA Chen FG

Chen JS

Department of Mechanical and Industrial Engineering, University of Thessaly, Volos, Greece, Email: Aravas@ mie.uth.gr LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China, Email: [email protected] Defence Science and Technology Organisation Victoria 3207Australia, Email: [email protected] Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100080, China, Email: fgchen@ home.ipe.ac.cn Civil & Environmental Engineering Department, University of California, Los Angeles, CA 90095 USA, Email: [email protected]

Department of Mechanical and Aerospace Engineering, The George Washington University Washington, DC 20052, USA, Email: [email protected] Chen Z Department of Civil and Environmental Engineering, University of Missouri-Columbia, Columbia, MO 652112200, USA, Email: [email protected] Feng XQ Department of Engineering Mechanics, Tsinghua University Beijing 100084, China, Email: fengxq@ tsinghua.edu.cn Fish J Rensselaer Polytechnic Institute, Troy NY 12180, USA, Email: [email protected] Gan Y Department of Civil and Environmental Engineering, University of Missouri-Columbia, Columbia, MO 652112200, USA, Email: [email protected] Ge W Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100080, China, Email: wge@home. ipe.ac.cn Haidemenopoulos GN Department of Mechanical and Industrial Engineering, University of Thessaly, Volos, 38334, Greece, Email: [email protected]

Chen YP

ix

x

Hao S

Huang YG

Hwang KC

Ingber DE

Jones R

Ke FJ

Katsamas AI

Klein PA Lee JD

Lei YJ

Li JG

Contributors

Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA, Email: suhao@ northwestern.edu Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign Urbana, IL61801 USA, Email: [email protected] Department of Engineering Mechanics Tsinghua University Beijing 100084 P.R. China, Email: huangkz@ tsinghua.edu.cn Departments of Pathology and Surgery, Children’s Hospital and Harvard Medical School, Boston, Massachusetts, USA, Email: [email protected] Mechanical Engineering, Monash University Melbourne, Melbourne Australia, Email: [email protected]. edu.au Department of Applied Physics, Beijing University of Aeronautics and Astronautics, Beijing 100080, China, Email: [email protected] Department of Mechanical and Industrial Engineering, University of Thessaly, Volos, 38334, Greece, Email: [email protected] Franklin Templeton Investments, San Mateo, CA 94403 USA, [email protected] Department of Mechanical and Aerospace Engineering, The George Washington University Washington, DC 20052, USA, Email: [email protected] Y.P. Department of Mechanical and Aerospace Engineering, The George Washington University Washington, DC 20052, USA, Email: [email protected] Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100080, China, Email: [email protected]. ac.cn

Contributors

Mannava SR

Mehraeen S

Molent L Nagarajan K

Pitt S Qian D

Shen LM

Shi DL SihGC

Stamenoviü D Tang XS

Vasudevan VK

Wang HY

xi

Department of Chemical and Materials Engineering, University of Cincinnati, Cincinnati, OH 45221-0012, USA, Email: [email protected] Civil & Environmental Engineering Department, University of California, Los Angeles, CA 90095, USA, Email: [email protected] Defence Science and Technology Organisation, Victoria 3207, Australia, Email: [email protected] Department of Mechanical, Industrial and Nuclear Engineering, University of Cincinnati, Cincinnati, OH 45221-0072, USA, Email: [email protected] Defence Science and Technology Organisation, Victoria 3207, Australia, Email: [email protected] Department of Mechanical, Industrial and Nuclear Engineering, University of Cincinnati, Cincinnati, OH 45221-0072, USA, Email: [email protected] Department of Civil and Environmental Engineering University of Missouri-Columbia, Columbia, MO 652112200, USA, Email: [email protected] Department of mechanics, Shanghai University, Shanghai 200444 China, Email: [email protected] School of Mechanical Engineering, East China University of Science and Technology, Shanghai 200237,China. Department of Mechanical Engineering and Mechanic, Lehigh University, Bethlehem, PA 18015, USA, Email: [email protected] and [email protected] Department of Biomedical Engineering, Boston University, Email: [email protected] School of Bridge and Structure Engineering, Changsha University of Science and Technology, Changsha, Hunan 410076, China, Email: [email protected] Department of Chemical and Materials Engineering, University of Cincinnati, Cincinnati, OH 45221-0012, USA, Email: [email protected] LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China, Email: why@lnm. imech.ac.cn

xii

Contributors

Wang N

Physiology Program, Harvard School of Public Health, Harvard University, Cambridge, Mass. USA, Email: [email protected]

Wang ZG

Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang, 110016, China, Email: zhgwang@ imr.ac.cn LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China, Email: Ywei@LNM. imech.ac.cn LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China, Email: Wu21@LNM. imech.ac.cn Sandia National Laboratories, Albuquerque, NM 87185 USA, Email: [email protected] Department of Material Science & Engineering, Northwestern University, Evanston, IL 60208, USA, Email: [email protected] Department of Physics, Peking University, Beijing 100871, China, Email: [email protected]

Wei YG

Wu XL

Webb EB III Weertman J

Xia MF Xiong LM

Zhang ZF

Zhao MH

Zhou GZ

Department of Mechanical and Aerospace Engineering, The George Washington University Washington, DC 20052, USA, Email: [email protected] Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang, 110016, China, Email: zhfzhang@ imr.ac.cn LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China, Email: Mzhao@LNM. imech.ac.cn Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100080, China, Email: gzzhou@home. ipe.ac.cn

Contributors

xiii

Zhu C

LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China, Email: Zhu@LNM. imech.ac.cn

Zimmerman JA

Sandia National Laboratories, Livermore, CA 94551 USA, Email: [email protected]

Foreword This volume on multiscaling has been motivated by the advancement of nano-technology in the past four decades. In particular, nano-electronics has paved the way to show that the behavior of nano-size bodies are not only different from macro-size bodies but they do not obey the same physical laws. There appears to be a mesoscopic region which separates the laws of quantum physics and continuum mechanics. A gap has been left in the full range of scaling from macro to nano. Micro-manipulation can be made more effective if the atomic and molecular scale activities can be identified more precisely with the use specific objectives. In this respect, material science has already benefited by positioning and structuring of nanometer-scale particles to arrive at the desired macroscopic material properties. The idea has been implemented to tailor-make structural materials for the Boeing 787 to better accommodate non-uniform stress and strain at different locations of the aircraft. Explored are also the possibility of coaxing DNA-based organisms such as viruses to improve performance of batteries, solar cells, fabrics, paints and other kinds of materials. The potential of assembling bio-molecules to build electronic components is also in the planning. The manipulation of molecules and atoms has been regarded as a common base for both material and life science. Quantum and continuum mechanics are being applied side by side for exploring the behavior of small and large objects moving at fast and slow speed. The establishment of a common basis of understanding for all sciences and technology has encouraged the contribution of the 18 articles in this volume in addition to the need for mutliscaling in time and size. Although the various disciplines covered seem to differ at first sight, their aims are directed to associate macroscopic behavior with the atomic and subatomic particles. Such a trend is likely to be the rule rather than the exception in the 21st century as new discoveries in science are made at an alarming pace. Dividing the matter into smaller and smaller entities has been the bias in the development of Western science for unraveling the secret of nature. When the size scale becomes increasingly small, quantum mechanics was devised to describe the behavior of electrons that cannot be explained by the laws of gravitation in general relativity. The confrontation between these two equally acceptable approaches, however, continues while awaiting for the discoveries of new laws and theories for their recon-citation. Off hand, there are issues that are not unrelated to the development of multiscaling models, a topic selected for discussion in this volume. Physical laws and theories are intended to establish organizational structure and order in nature such that they can be used to widen the range of existing knowledge and make new discoveries. Telescoping and microscoping involve the observation of events at increasingly large scales and decreasingly small scale. Up to this date, the progress seems to have been rested at both ends of the scale range, the subatomic and galactic scale. One of the immediate issues is concerned with space-time dimensionality. This entails the transition of two-dimensional surfaces to the three-dimensional volumes or vise versa. That is to consider the increase or decrease of dimensionality as a matter of perspective. The underlying implication xv

xvi

Foreword

is associated with the possible equivalency of particles and fields, the dualism of which has been taken for granted as a tradition. Entrenched in the indoctrinated discipline of particle mechanics is that the mathematical limit for an infinite numbers of particles would make up the continuum. Such an idealization has no room in subatomic physics since the vast space of the universe is likely to remain unknown. Even less is the number of particles required to fill a finite volume of space if the size of the smallest particle remains not known. Idealized mathematical models, however, do serve the purpose for testing and conceiving physical ideas. Conversion of volume integrals to surface integrals and surface integrals to contour integrals made, respectively, by applying the divergence and Green theorem do suggest that higher-dimensional and lower-dimensional realm are related. Classical mechanics defines tractions in two-dimensions while body force would require the third dimension. The trade off between the surface and volume energy density is also well known in the Gibb’s theory of crystal nucleation for determining the size of grains in poly-crystals. As a matter of fact, why shouldn’t surface- and volume-based quantities be related? The only reason is that by tradition classical continuum mechanics has chosen to develop theories by disconnecting surface from the bulk. That is by taking the limit of ∆V/∆A→0, an expediency that has severely restricted the use of continuum theories for small bodies with large surface to volume ratios. This happens to fall in the size range of nano-electronics. The aforementioned limiting process also entails the indiscriminate use of the bulk or equilibrium material properties. Although the approximation does no harm to large structures but it introduces serious errors for describing the properties of small devices that are predominantly in the state of non-equilibrium. Quantum mechanics was developed precisely to account for the non-equilibrating behavior of the atoms and electrons that are constantly moving. Incidentally, the use of equilibrium theory of dislocation at the atomic scale is an over simplification. It is also surprising that non-equilibrium continuum mechanics theory has not received the attention it deserves. The existence of non-equilibrium solution in continuum mechanics can be and has been proved in a space of isoenergy density although the solution may not be unique on account of the uncertainties involved of enforcing two identical boundary conditions in experiments at the atomic and /or molecular scale. For an open thermodynamic system, the air molecules next to the system will not occupy unique positions in two different experiments. Non-equilibrium mechanics theories encounter the same uncertainties as in quantum mechanics except that they are stated in different terms. One basic feature is that space, time and temperature would all be coupled and they tend to interact simultaneously. This can be made possible by keeping dV/dA finite in the theory such that the surface and volume density would be related. Constitutive relations are no longer pre-assumed throughout the continuum but rather it is derived for each point and instant of time simulating the true nature of non-homogeneity that is intrinsic of the non-equilibrium process. Incidentally, the rate of change of volume with surface dV/dA can also be quantized and expressed in terms of the de Broglie wavelength in quantum mechanics.

Foreword

xvii

Scaling models attempt to circumvent some of the complexities encountered in non-equilibrium theories. This may be accomplished by segmentation of the scale range from macro to nano and from secs to femto secs. Each divided segments can be made sufficiently small to apply the condition of equilibrium. Singularity representation can be used to describe the combined effects of material, geometry and applied load which is well known in equilibrium mechanics. Stress singularity order serves as the scaling factor. The inverse square root of the distance from the crack tip, for example, can be regarded as a scaling factor between the local and global stresses and geometric quantities with reference to the crack. This factor would change for a concentrated load that yields an inverse of the distance from the singular point. This corresponds to the kernel of the Cauchy singular integral equation. The spirit of developing multiscale space-time models is to find simpler ways of addressing complex physical phenomena without violating the axioms of mechanics. Large scale computations will not automatically distinguish the non-equilibrium character of the atoms and /or the electrons from the noise contributed from numerical inaccuracies, especially when the predictions are based on using equilibrium theories. Such a practice is not uncommon in the application of finite elements or similar techniques. The holographic paradigm that has been receiving attention in modern physics may also benefit the development of multiscaling models. Physicists are faced with the same conceptual difficulties when the scale range is extended to the limits where continuum ceases to apply while particles step in. Under certain situations, physicists have argued that subatomic particles such as electrons are able to instantaneously communicate with each other regardless of the distance separating them. Each particle seems to know what the other is doing. This property is known to prevail in a hologram where every part contains all the information possessed by the whole. The whole in every part feature is similar to the brain memories which are not confined to a specific location but are dispersed throughout the brain. For the time being, at least an explanation can be offered for the coexistence of particles on the surface and the continuum enclosed by the surface. The holographic theory is that clouds of quarks and gluons on the two-dimensional surface can describe three-dimensional objects in the enclosed volume. Such a correlation plays the central role in the development of the string theory. Strings lying on a two-dimensional surface but with different thicknesses could be directed to the interior volume to form three-dimensional objects. The dualism of particle and continuum is thus made possible by the equivalence of particles on the surface to the object in the interior space. The interplay of surface and volume appears to be fundamental for the argument involving the co-existence of particles and fields. Even though the influence of dV/dA on multiscaling remains to be found, there are ample evidence from the past and the present to show that surface and volume effects cannot be separated. To iterate, equilibrium theories are vulnerable when applied to explain non-equilibrium phenomena. The dormancy effect of empirically designed miniaturized machines is the results of neglecting the dV/dA effects. That is frictional forces may no longer be independent of the surface area

xviii

Foreword

as postulates in the classical law of friction. The cooling and heating behavior of uniaxial specimens that undergo non-equilibrium, has also been a fact ignored for decades. Open thermodynamic system data which are not in equilibrium have been misused for closed thermodynamic system theories. Such inconsistencies cannot be left untold in the text books and classrooms if modern science and technology are expected to advance in the future. The articles on multiscaling in this volume hopefully will provide insights to the need for distinguishing the fundamental difference between the behavior of large and small bodies. Because of the unique character of the articles in this volume, the contributors were given the choice to present their works at the 16th European Conference of Fracture on Failure Analysis of Nano and Engineering Materials and Structures. One of the messages of this volume is that fundamental science and applied engineering need much closer collaboration. This includes the manufacturing of nano-size components. The editor is indebted to the contributors for completing their work on time. The assistance of Anita Ren and Leslie Li for revising the art work and formatting the articles are much appreciated. Shanghai, China May, 2006

G.C. Sih Editor

“Deborah numbers”, coupling multiple space and time scales and governing damage evolution to failure Y.L. Baia, H.Y. Wanga *, M.F. Xiab,a, F.J. Kec,a a

LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China b Department of Physics, Peking University, Beijing 100871, China c Department of Applied Physics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China

Abstract Two different spatial levels are involved concerning damage accumulation to eventual failure. This can entail sample size A (∼ cm) to characteristic microdamage size c*(∼µm). Associated are three physical processes with three different rates, namely macroscopic elastic wave velocity a, nucleation and growth rates of microdamage n N* and V*. It is found that the trans-scale length ratio c*/L does not directly affect the process. Instead, * * * two independent dimensionless numbers: the trans-scale one De = ac AV including the * * *5 * V including mesoscopic parameters only, play length ratio and the intrinsic one D = n N c the key role in the process of damage accumulation to failure. The above implies that there are three time scales involved in the process: the macroscopic imposed time scale tim = A /a and two meso-scopic time scales, nucleation and growth of damage, t N =1 n *N c*4 and tV=c*/V*. Clearly, the dimensionless number De*=tV/tim refers to the ratio of microdamage growth time scale over the macroscopically imposed time scale. So, analogous to the definition of Deborah number as the ratio of relaxation time over external one in rheology. Let De be the imposed Deborah number while De* represents the competition and coupling between the microdamage growth and the macroscopically imposed wave loading. In stress-wave induced tensile failure (spallation) De* < 1, this means that microdamage has enough time to grow during the macroscopic wave loading. Thus, the microdamage growth appears to be the predominate mechanism governing the failure. Moreover, the dimensionless number D* = tV/tN characterizes the ratio of two intrinsic mesoscopic time scales: growth over nucleation. Similarly let D * be the “intrinsic Deborah number”. Both time scales are relevant to intrinsic relaxation rather than imposed one. Furthermore, the intrinsic Deborah number D* implies a certain characteristic damage. In particular, it is derived that D* is a proper indicator of macroscopic critical damage to damage localization, like D* ∼ (10–3~10–2) in spallation. More importantly, we found that this small intrinsic Deborah number D* indicates the energy partition of microdamage dissipation over bulk plastic work. This explains why spallation can not be formulated by macroscopic energy criterion and must be treated by multi-scale analysis.

(

(

)

)

Keywords: Deborah numbers; Trans-scale coupling; Damage evolution; Failure.

*

Corresponding author. E-mail address: [email protected] (H.Y. Wang).

1 G.C. Sih (ed.), Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano, 1–10. ” 2007 Springer.

2

Y.L. Bai et al.

1. Introduction From damage accumulation to failure, there prevails an important time-dependent phenomenon involving spallation in which failure occurs under transient loading like nano- to micro-seconds. Experimental observations suggest a time-integral criterion for spallation [1], ( ı ı* – 1) Ȟ × ǻt = K

(1)

where σ and σ* are stress and a stress threshold respectively, ǻt is the load duration, υ ν and K are two parameters. This criterion indicates that the critical stress to spallation is no longer a material constant, but a variable depending on its loading duration. Furthermore, since the power exponent υ in the criterion is usually neither 1 nor 2, the criterion implies neither momentum nor energy criteria macroscopically [1-3]. Comprehensive and critical reviews on spallation have been made [4-6]. It is stressed that “the continuum models based on the statistical nucleation and growth of brittle and ductile fracture appear to be an attractive approach, especially with a framework which provides some forms of a continuum cumulativedamage description of the evolving fracture state” [6]. Recently, the work in [7] suggests that “dynamic failure by the growth and coalescence of grain-boundary microcracks involve the cooperative interactions of propagating cracks. Insight into such processes is required from the perspective of stochastic mechanics and from computer simulations of the debonding of assemblages of grains”. It follows that spallation is a typical process with coupled multiple space and time scales. At least, there are two length scales: the sample size at macroscopic level and the microdamage size at mesoscopic level. On the other hand, there are three time scales: the stress wave loading duration macroscopically, the mesoscopic nucleation time and growth time of microdamage. So, spallation represents an illustrative example with multiple space and time scales. 2. Statistical Microdamage Mechanics The general evolution equation of microdamage number density is [8] ∂n + ∂t

I

∂ ( n ⋅ Pi )

i =1

∂p i

¦

= nN

(2)

where t is time, nN is the nucleation rate of microdamage number density. Pi = p i , “.” denotes the rate of variable p i , which represents the state of microdamage. After taking the phase variable p as the current size of microdamage c, i.e. p = c, we have obtained a general solution to microdamage number density n(t,c; σ), [9-10]

“Deborah numbers”, multiple space and time scales

­ c n N (c0 ;ı) ° ³ 0 V(c,c ;ı) dc0, c c f,0 °¯ ³c0f V(c,c0 ;ı) 0

3

(3)

where co is the nucleation size of microdamage and the time-dependent feature is expressed by the moving front of microdamage cof or cf in terms of the integral c

t=

³

c0f

c

f dc dc =³ V(c,co ;ı) c0 V(c,co ;ı)

(4)

More importantly, there are two mesoscopic rate processes of microdamage involved in solution (3): nN(c0;σ) is the nucleation rate of microdamage number density and V=V(c, c0; σ) is the growth rate of microdamage. The relation between continuum damage D and the number density of microdamage n is ∞

D(t, x) = ³ n(t, x, c) ⋅ τ ⋅ dc 0

(5)

where τ ∼ c3 is the failure volume of an individual microdamage with size c, [11-12]. Then, the statistical evolution equation of microdamage number density in Eq. (2) can be converted to the continuum damage field equation by integration under proper boundary conditions and its the one-dimensional Lagrangian form is [11-12]. ∂D ρ ∂v +D =f ∂T ρ0 ∂X

=³

∞

0

∞ cf ­ n N (c0 ; σ) ³ τ ′ (c)dcdc0 ½ ³ ° ° 0 c0 n N (c; σ)τ(c)dc ⋅ ®1 + ¾ ∞ n (c; ) (c)dc σ τ ° ° N ³0 ¯ ¿

(6)

where f is the dynamic function of damage, which represents the statistical average effects of nucleation and growth of microdamage on continuum damage evolution and IJ′=d IJ dc . Combining the damage field relation in Eq. (6) with the following conventional field equations of continuum, momentum and energy, ∂ε ∂v = ∂T ∂X ρ0

∂v ∂σ = ∂T ∂X

(7)

(8)

4

ρ0 c v

Y.L. Bai et al.

∂θ ∂ediss ∂2θ = +λ 2 ∂T ∂T ∂X

(9)

a coupled trans-scale framework is formed for the damage evolution [13-14], where ρ0 is density of intact material, λ is heat conductivity, θ is temperature and ediss is the energy dissipated in the material element. 3. Deborah Numbers, Coupling Multiple Space and Time Scales In accord with Π theorem in dimensional analysis, 10 parameters involved in spallation (see Table 1), can form 6 independent dimensionless numbers. However, when Eqs. (6-9) are non-dimensionalized it is found that there are 5 independent quantities only.

macroscopic

mesoscopic

Table 1. 8 parameters involved in spallation parameter symbol Sample size A Density ρ Elastic wave speed a Characteristic stress σY Impact velocity vf Heat conductivity λ Heat capacity cV Characteristic nucleation rate of microdamage nN* Characteristic growth rate of microdamage. V* Characteristic size of microdamage c*

dimension L ML– 3 LT –1 ML– 1 T –2 LT –1 MLT –3θ–1 L2T –2 θ – 1 L–4 T –1 LT –1 L

In the 5 dimensionless numbers, there are 3 conventional macroscopic ones: the well-known Mach number, damage number and Fourier number, related to continuum, momentum and energy equations respectively. Consider the other two ac* trans-scale dimensionless numbers: the imposed Deborah number De* = * and AV * *5 *

the intrinsic Deborah number D =

nN c . V*

For a group of Al alloy spallation tests, A ∼ (5-10)mm, vf ∼ 102 ms–1ˈ c* ∼ 4.27⋅10–6 m, V* ∼ 5.96 ms–1 and nN* ∼ 5.22⋅10 4 mm–3µm–1µs–1, derivation leads to De* ∼ 1 and D *∼ (10–2 ∼ 10–3) [10-12]. Noticeably, the ratio of two length scales on meso- and macro-levels c*/ A does not ac* appear here. Actually, the imposed Deborah number De* = * is a combination of AV two ratios: the size scale ratio c*/ A and the ratio of two velocities V*/a. Also, the imposed Deborah number De* is a unique trans-scale dimensionless parameter, since elastic wave speed a and sample size A are macroscopic parameters whereas microdamage size c* and microdamage growth rate V* are mesoscopic ones. This is

“Deborah numbers”, multiple space and time scales

5

very different from all other dimensionless parameters. On the other hand, De* = tV/tim refers to the ratio of microdamage growth time scale tV = c*/V* over the macroscopically imposed time scale tim = A /a. This is why we call it Deborah number.

n N*c*5 The intrinsic Deborah number D = characterizes the rate ratio of the V* * *

two intrinsic mesoscopic processes: nucleation over growth. Actually, D = tv/tN, −1 where tV = c*/V* and t N = ( n*N c*4 ) are the growth and nucleation time scales respectively. Note that the characteristic nucleation time tN is not the time needed for an individual microdamage to nucleate, but the time to form a characteristic nucle * means the damage fraction merely due to nucleaated damage fraction, since D N tion in unit time.

t *N = 1 = 1

n N * c* 4 (N N *

= 1 [(n N * c* )c * 3 ]

= 1 c* 3 ) DN *

(10)

Above all, in the case study of spallation, there are three time scales: the macroscopic imposed time scale tim= A /a∼10–6 s and two meso-scopic time scales, growth −1 − time scale tV = c*/V*∼10–6 s and the nucleation time scale t N = ( n*N C*4 ) ∼10 3 s. * –2 –3 These lead to De ∼1 and D * ∼ (10 ∼ 10 ) too. However, this is not the whole story of Deborah numbers. 4. The Significance of Deborah Numbers in Failure 4.1 Imposed Deborah number governs the failure process Note that the imposed Deborah number De*= tV/tim represents the competition and coupling between the microdamage growth and the macroscopically imposed wave loading. Also, in the concerned spallation case De* ∼ 1, this indicates that microdamage has enough time to grow during the macroscopic wave loading and then microdamage growth may be the predominate mechanism governing spallation. 4.2 Intrinsic Deborah number signifies characteristic damage and damage localization Turn to the intrinsic Deborah number D*. Firstly, D * implies a certain characteristic damage, since Eq. (5) can be written as

6

Y.L. Bai et al. ∞

D = ³ n(c) ⋅ τ ⋅ dc = 0

n*N c*5 V*

³

∞

0

n N dc0 ³

cf c0

c3 dc V

(11)

All variables with bar above denote non-dimensionalized and normalized ones, hence the preceding dimensionless combination, i.e. the intrinsic Deborah number D * , represents the magnitude of damage D. In the concerned spallation, this means a characteristic damage is of D * ∼ (10–2 ∼ 10–3) . Also, D* is a proper indicator of critical damage to localization. Damage localization can be formulated as follows[15]

[

∂

)∂∂XD ) ] ∂T ∂D ∂X

) )

∂D ) ) ≥ ∂T

(12)

D

It has been derived that provided the dynamic function of damage can be expressed by

f = f (D, σ)

(13)

Under a certain approximation the damage localization condition (12) can be expressed by the following inequality [15],

fD ≥

f D

(14)

∂f . Obviously, this condition (14) represents an intrinsic feature ∂D

Where f D =

and irrelevant to any specific conditions of tests. Combining the damage localization condition in Eq. (14), the definition of dynamic function of damage in Eq. (6) and the expression of damage in Eq. (5), a critical damage to localization can be derived:

* *5

n c Dc ≈ N * ⋅ V

³

∞

0

τ (cf )n N (c0 )dc0 ⋅ ³

∞

0

³

∞

0

τ (cf )n N (c0 ) dc0 V(c0 , cf )

(15)

τ′(cf )n N (c0 )dc0

*

= D ⋅ O(1)

This once more is the intrinsic Deborah number D*. As above, the dimensionless combination preceding the normalized integrals indicates the magnitude of the critical damage to localization. Compare the obtained

7

“Deborah numbers”, multiple space and time scales

n N*c*5 result with experiments. As noted before, experiments gave D = ∼ (10–2 ∼ * V –3 *

10 ). According to the localization condition in Eq. (14), there results in the critical damage to localization Dc ∼ 4 * 10–3 [12]. Clearly, the intrinsic Deborah number D* does characterizes the magnitude of the critical damage Dc. Some simulations (Fig. 1) demonstrate that the intrinsic Deborah number does signify a certain characteristics in damage localization. In the numerical study, a fixed De = De * /D * = 65.9 is taken. The most localized distribution occurs in the case of (De * = 0.151 equivalent to D * = 0.0023), whilst the case (De * = 0.415 equivalent to D * = 0.0063) demonstrates hardly localized distribution. This is in agreement with the analysis that lower D * indicates lower threshold of critical damage to localization. In Fig. 1, A is the plate thickness (sample size). 4.3 Intrinsic Deborah number signifies partition of energy dissipation The energy partition will be clarified by examining the energy equation [16-17]. After splitting the dissipation term into damage and plastic ones, Eq. (9) may be rewritten as ρ0 c V

∂θ ∂ε p ∂Σ ∂ 2θ =σ +γ +λ 2 ∂T ∂T ∂T ∂X

(16)

where Σ and γ are the total surface area of microdamage and corresponding equivalent surface energy in unit volume respectively. Certainly, the partition of plastic dissipation (the first term on the right hand side) and the damage one (the second term) are of our interest. Again, the dimensionless energy equation is: ∂θ ∂ε p ∂Σ ∂2 θ =σ + D* +Ψ 2 ∂T ∂T ∂T ∂Y

(17)

As before, all variables with bars are dimensionless and normalized, i.e. in O(1). The last term indicates the heat transfer and Ψ is a dimensionless number relevant to Fourier number. For aluminum, λ=238W/m⋅K, ρ0~2700Kg/m 3, cV~903J/kg⋅K, a~5000m/s, Ψ =

λ ρ0 v V κ a λ = = = 10 −3 i @ 1 ª § ∂Gu>i @ ·   : :  : D T d d P d « « » ¨ ¸  ¨ ¸ ijkl ijkl ji ³ AY «:³ ∂Yj ³ AY «:³ ¨ ∂Yj ¸ »» d: ¨ ∂Yl ¹¸ :X :X © ∂Xl ¹ ¬ Y ¼» ¬ Y© ¼

(7) where A Y is the area (or volume) of the unit cell in undeformed configuration, and ȍ Y is domain of unit cell. The other equation yields the macroscopic equilibrium equation:

ª ∂δu [0] § ∂∆u [k0] ∂∆u[k1] · º i D T + ³Ω «« Ω³ ∂X j ( ijkl ijkl ) ¨¨ ∂X l + ∂Yl ¸¸dΩ »» dΩ © ¹ ¼ X ¬ Y § ∂δu[i0] · [0] = ³ δu [i0] bi dΩ + ³ δu[i0]h i dΩ − ³ ¨ P dΩ ¨ ∂X ¸¸ ji j ¹ ΩX ΩX © Γ hX

1 AY

(8)

in which the body force is assumed constant over the unit cell. Introducing

(

u [i1] ( X,Y ) = Į kli ( Y ) ∂u[k0] ∂X l

)

in Eq. (7). This leads to

0 1 >1@ º § 1 ª § ∂Gu>i @ · ∂Dmnk  Y · °½ ∂'u>m@ °­ 1 ∂Gui D T dY d P d  G G  :  : « ¨ ¸  ® ¾ ¨ ¸ ijkl ijkl km nl ji ³ AY :³ ∂Yj ³ AY «:³ ¨ ∂Yj ¸ »» d: ∂Yl ¹ °¿ ∂Xn © :X ° :X ¹ Y ¯ ¬ Y© ¼ (9) Since the length scale of microstructure is considerably smaller than the macroscopic length scale, the residual due to nonlinearity in the microscopic equation in Eq. (9) can be ignored and it yields

15

Wrinkling formation in polycrystalline materials

∂α mnk ( Y ) ∂δu[i1] ∂δu[i1] ³ ∂Yj ( Dijkl + Tijkl ) ∂Yl dY = −Ω³ ∂Yj ( Dijmn + Tijmn ) dY ΩY Y

(10)

Note that microstructure deformation ui must be used to calculate Dijkl and

Tijkl in Eq. (10). By substituting u [i ] ( X,Y ) =Į kli ( Y ) ∂u[k ] ∂X l

into Eq. (8), ( ) the following macroscopic (homogenized) governing equation is obtained 1

∂Gu>i @ ∂'u>m@ D T  ³: ∂Xj  ijmn ijmn ∂Xn d: X 0

0

0

§ ∂Gu>i0@ · >0@ >0@ >0@ u b d u h d G : G : ³: i i ³h i i :³ ¨¨ ∂Xj ¸¸ Pji d: *X ¹ X X© (11)

where Dijmn is the homogenized material response tensor, and Tijmn is the homogenized geometric response tensor.

Dijmn = Tijmn

1 AY

1 = AY

³D

ΩY

ijkl

§ ∂α mnk ( Y ) · ¨ δkm δ nl + ¸ dΩ ∂Yl © ¹

§ ∂α mnk ( Y ) · ³Ω Tijkl ¨© δkmδnl + ∂Yl ¸¹ dΩ Y

(12)

[ 0]

from which coarse scale solution ǻu i is obtained. It can be easily identified that both the homogenized material response tensor Dijmn and the homogenized geometric response tensor Tijmn do not possess major symmetry. 3. Consistent Homogenization for Large Deformation Problems To recover symmetry property in the material and geometric response tensors, [0] we first define the point-wise macroscopic strain energy density W as the average of strain energy density in the unit cell as

W[0] =

1 AY

³ W ( F) dΩ

ΩY

(13)

Fij = ∂u i ∂X j where F and W are the microscopic deformation gradient and strain energy [0] density in the unit cell, respectively, and W is the point-wise macroscopic

16

J.S. Chen and S. Mehraeen

strain energy density. Consequently, the macroscopic first Piola-Kirchhoff stress tensor is obtained by

Pij[0] =

1 ∂W[0] = [0] AY ∂Fji

∂W

³ ∂F

[0] ji

ΩY

dΩ =

1 AY

1 ∂W ∂Fmn dΩ = Pnm K nmijdΩ [0] A Y ³Y mn ∂Fji

³ ∂F

ΩY

(14) and

∆Pij[0] =

§ 1 ∂ 2 W[0] [0] F ∆ = ¨ lk ¨ AY ∂Fji[0]∂Flk[0] ©

· [0] ∂ 2 W ∂Fqp ∂Fsr [0] [0] dY ¸ ³Ω ∂Fqp∂Fsr ∂Fji[0] ∂Flk[0] ¸ ∆Flk = Aijkl ∆Flk Y ¹

(15)

where

K nmij =

∂Fmn ∂ = [0] [0] ∂Fji ∂Fji

( (δ

mk

)

δ nl + ∂α kli ( Y ) ∂Yn ) Fkl[0] = δ jm δin + η jimn ,

ηklin = ∂α kli ( Y ) ∂Yn A[0] ijkl =

1 AY

³

(16)

A pqrs K pqijK rskldY, A pqrs =

ΩY

∂2W = C2pntr Fqn Fst + Spr δsq ∂Fqp ∂Fsr

(17)

Note that Eq. (15) demonstrates the major symmetry property of the first elastic[0] ity tensor at the global level A ijkl . Furthermore, unlike most asymptotic expansion based methods [2,11,15], the second elasticity tensor in above proposed method also possesses major symmetry property. Based on the incremental macroscopic equilibrium Eq. (11) and the above consistent homogenization procedures, the homogenized incremental equilibrium equation is obtained as

³

ΩX

∂δu [i ] [0] ∂∆u [m ] A ijmn dΩ = ∂X j ∂X n 0

0

³

ΩX

δu [i ] bi dΩ + 0

³

Γ hX

δu [i ] h i dΩ − 0

³

ΩX

∂δu [i ] [0] Pji dΩ, ∂Yj 0

(18) 4. Modeling of Wrinkling Formation in Polycrystalline Material In this example, consider the hemming process of a plane strain AA 1050 sheet metal with width AH = 0.022 m and thickness AE = 0.001m s hown in Fig. 2. According to the experimental setup shown in Fig. 2(a), the sheet metal is fixed

17

Wrinkling formation in polycrystalline materials

on the left end, positioned in between two frictionless blocks, and is bent by a rigid die with interface coefficient of friction µ = 0.1 pushing downward. The microstructure in a unit cell box size = 1µm × 1µm is shown in figure 2(c). The material properties of AA 1050 are listed in Table 1. Note that 10 GPa ≤ E 0 ≤ 100 GPa are generated randomly for each grain in unit cell 69 GPa. The hardening rule is given by with homogenized E = p p p n , where e is the effective plastic strain, K is the K e = K .008 + e material strength, and n is the strain hardening coefficient defined in Table 1.

( )

(

)

Table 1. General mechanical properties of AA 1050 Stain Hardening Coefficient Material Strength

(n)

(K)

158.8 MPa 125 MPa

Yield Stress Average Elasticity Modulus Poisson’s Ratio

0.295

(E)

69 GPa 0.3

The macroscopic large deformation analysis is conducted in 100 load steps. Figure 3 compares the vertical force-displacement predicted by single-scale and multi-scale approaches, as well as the experimental data [16]. The result shows that when wrinkling forms in the multi-scale calculation, the local instability leads to a reduction of the global stiffness when compared to the solution of the single-scale homogenization approach where the surface wrinkling is not captured. To observe the more pronounce wrinkling formation, the sheet metal is further bent numerically as shown in Fig. 4, and the comparison with non interactive single-scale homogenization and interactive multi-scale approach clearly demonstrates how the conventional non interactive single-scale homogenization approach is incapable of capturing the localized winkling response.

18

J.S. Chen and S. Mehraeen

Fig. 2. (a) Hemming process experimental setup, (b) FE discretization of the plane strain sheet metal, (c) microstructures in the unit cell with size = 1µm × 1µm (averaged grain size = 0.15µm).

Wrinkling formation in polycrystalline materials

19

Fig. 3. Comparison of vertical load-displacement curves of single-scale and multi-scale predictions, as well as experimental data [0].

Fig. 4. Comparison of large degree bending deformation predicted by non interactive single-scale homogenization and interactive multi-scale homogenization.

5. Conclusion The asymptotic method in conjunction with a consistent homogenization approach has been proposed for multi-scale modeling of wrinkling formation. The

20

J.S. Chen and S. Mehraeen

variational multi-scale governing equation and the scale-coupling relation have been formulated based on a total Lagrangian formulation. By introducing the asymptotic form of the material displacement into the test and trial functions in the variational equation, the resulting leading order equations yield the scale-coupling equation and the coarse scale governing equation. A Galerkin weak form was employed to solve the scale-coupling function numerically. It was found that the classical homogenization leads to homogenized tangent operator that does not possess major symmetry. A consistent homogenization method has been proposed by utilizing an averaged macroscopic strain energy density computed over the microstructure. It was demonstrated that homogenized formulation yields a tangent operator with major symmetry, and the proposed method achieves quadratic rate of convergence. Unlike the other methods [5, 15, 16], the proposed method allows consideration of full nonlinearity without approximation. The ability of the proposed multi-scale method to capture wrinkling modes by solving the microscopic solution has been demonstrated. The stage at which material tends to generate wrinkles has also been identified. In the numerical examples, surface wrinkling formation of a polycrystalline material has been presented. It has been shown that the bridging of microstructure information and macroscopic material behavior allows the prediction of local instability in the wrinkling formation. References [1] Segedin RH, Collins IF, Segedin CM, The elastic wrinkling of rectangular sheets, Int. J. Mech. Sci., 30 (1988) 719-732. [2] Kim JB, Yoon JW, Yang DY, Barlat F, Investigation into wrinkling behavior in the elliptical cup deep drawing process by finite element analysis using bifurcation theory, J. Material Processing Technology, 111 (2001) 170-174. [3] Groenewold J, Wrinkling of plates coupled with soft elastic media, Physica A, 298 (2001) 32-45. [4] Hatchinson JW, Plastic buckling, Advances in applied mechanics, 14 (1974) 67-144. [5] De Magalhaes Correia JP, Ferron G, Moreira LP, Analytical and numerical investigation of wrinkling for deep-drawn anisotropic metal sheets, Int. J. Mech. Sciences, 45 (2003) 1167-1180. [6] Stanuszek M, FE analysis of large deformation of membranes with wrinkling, Finite Elem. Anal. Design, 39 (2004) 599-618. [7] Dawe DJ, Yuan WX, Overall and local buckling of sandwich plates with laminated faceplates, part I: analysis, Comput. Methods Appl. Mech. Engrg., 190 (2001) 5197-5213. [8] Padovan J, Tabaddor F, Gent A, Surface wrinkles and local bifurcations in elastomeric components: seasls and gaskets, Part I: theory, Finite Elements Anal. Design, 9 (1991) 193-209. [9] Yang H, Lin Y, Wrinkling analysis for forming limit of tube bending processes, J. Mat. Processing Technology, 152 (2004) 363-369. [10] Epstein M, Forcinito MA, Anisotropic membrane wrinkling: theory and analysis, Int. J. Solids Struc., 38 (2001) 5253-5272. [11] Fish J, Shek K, Pandheeradi M, Shepherd MS, Computational plasticity for composite structures based on mathematical homogenization: theory and practice, Comput. Methods Appl. Mech. Engrg., 148 (1997) 53-73. [12] Smit RJM, Brekelmans WAM, Meijer HEH, Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling, Comput. Methods Appl. Mech. Engrg., 155 (1998) 181-192.

Wrinkling formation in polycrystalline materials

21

[13] Cricri G, Luciano R, Micro- and macro-failure models of heterogeneous media with micro-structure, Simulation Modeling Practice and Theory, 11 (2003) 433-448. [14] Takano N, Ohnishi Y, Zako M, Nishiyabu K, Microstructure-based deep-drawing simulation of knitted fabric reinforced themoplastics by homogenization theory, Int. J. Solids Struc., 38 (2001) 6333-6356. [15] Takano N, Ohnishi Y, Zako M, Nishiyabu K, The formulation of homogenization method applied to large deformation problem for composite materials, Int. J. Solids Struc., 37 (2000) 6517-6535. [16] Muderrisoglu A, Murata M, Ahmetoglu MA, Kinzel G, Bending, flanging and hemming of aluminum sheet-an experimental study, J. Mat. Proc. Tech. 59 (1996) 10-17.

A multiscale field theory: Nano/micro materials Y.P. Chen, J.D. Lee*, Y.J. Lei, L.M. Xiong Department of Mechanical and Aerospace Engineering The George Washington University, Washington, DC 20052, USA

Abstract A multiscale field theory is proposed for the application of nano/micro materials. Field representation of the conservation equations, stress, strains and stress-strain relation are formulated. Preliminary numerical examples from the new theory are presented.

1. Introduction 1.1 Atomic view The atomic view of a crystal is considered as a periodic arrangement of local atomic bonding units, Fig. 1. Each lattice point defines the location of the center of the unit. The space lattice is macroscopically homogeneous. Embedded in each lattice point is a group of bonded atoms, the smallest structural unit of the crystal. The structure of the unit together with the network of lattice points determines the crystal structure and hence the physical properties of the material.

Fig. 1. Atomic view of crystal structure. (a) Space lattice; (b) Crystal structure = lattice + local atomic bonding units.

Fig. 2. Typical motion for 2 atoms in a unit cell. LA: longitudinal acoustic, LO: longitudinal optical, TA: transverse acoustic, LO: transverse optical.

*

Corresponding author. E-mail address: [email protected] (J.D. Lee). 23

G.C. Sih (ed.), Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano, 23–65. ” 2007 Springer.

24

Y.. P. Chen et al.

For crystals that have more than one atom in the unit cell, the elastic distortions give rise to wave propagation of two types, as shown in Fig. 2. In the acoustic type (LA and TA), all the atoms in the unit cell move essentially in the same phase, sulting in the deformation of the lattice, as shown in Fig. 3. In the optical type (LO and TO), the atoms move within the unit cell, leave the lattice unchanged, and give rise to the internal deformations, as shown in Fig. 4.

Fig. 3. Acoustic vibrations of atoms give rise to lattice deformation. (a) Atomic displacements associated with LA, (b) Atomic displacements associated with TA [1].

Fig. 4. Optical vibrations of atoms give rise to internal deformation. (a) An TO mode of SrTiO3, (b) An TO model of silicon [2].

In real material response, atomic vibrations usually include simultaneous lattice deformation and internal deformation. The displacement of the α-th atom in the k-th unit cell, u(k,α), can be decomposed into lattice displacement u(k) and internal displacement ȟ(x, Į)

u(k,Į) = u(k)+ȟ(k,Į)

(1)

The analysis of phonon dispersion relation shows that, for a unit cell with ν atoms, there will be 3 acoustic and 3(ν-1) optical vibrational modes, and hence 3 lattice displacements and 3(ν-1) internal displacement patterns. There are two length/time scales associated with the atomic displacement in Eq. (1). The lattice deformation u(k) is homogeneous up to the point of structural instability (phase transformation). It is in the low and audible frequency region, and its length scale can be from

A multiscale field theory: Nano/micro materials

25

sub-nano to macroscopic. The internal displacement ȟ(x,Į) measures the displacement of atoms relative to the lattice, contributes to the inhomogeneous deformation. It is in the high frequency region, typically in the infrared, and its length scale is less than a nanometer. 1.2 Molecular dynamics viewpoint Consider a unit cell k in a multi-element crystal (k = 1, 2, 3,…n). Each unit cell is composed of ν atoms with mass m Į , position R kĮ and velocity V kĮ , (α = 1, 2…ν), cf. Fig. 5. The mass m, coordinate R k and velocity V k at the center of the unit cell, may be called unit cell average, can be obtained as: ν

m=

¦m

α

(2)

α =1

k

R =

1 m

k

V =

1 m

ν

¦m R α

kα

,

(3)

.

(4)

α =1 ν

¦m V α

αk

α=1

Fig. 5. Illustration of atomic and unit cell positions.

Denote the relative positions and velocities between atoms and the center of the unit cell as kĮ

kĮ

k

kĮ

kĮ

k

ǻr = R – R , ǻv =V –V ,

(5)

one has kĮ

k

R = R +ǻr V

kĮ

k

kĮ

kĮ

= V +ǻv .

(6) (7)

26

Y.. P. Chen et al.

and the total displacement of the α-th atom in k-th unit cell as

u (k, α ) = u (k) + ȟ (k, α )

(8)

The total atomic displacement of atom (k, α) is again expressed as a sum of a homogeneous lattice deformation u(k) and an inhomogeneous internal deformation ξ (k,α). 1.3 Breakdown of classical continuum mechanics Classic continuum mechanics views a crystal as a homogeneous and continuous medium rather than a periodic array of atomic bonding units. The basic structural unit of the crystal is taken without structure and is idealized as point mass. The optical vibrations or internal deformations within the unit are thus ignored. If the internal deformation has to be taken into account, as is necessary in nano/micro physics, classical continuum mechanics is no longer valid.

Fig. 6. Stresses in ABO3 ferroelectrics. (a) The smallest ABO3 structural unit. (b) Lattice points and the unit cell (2D illustration). (c) Stress evaluated at lattice position with the volume of a unit cell; (d) Moment stress resulting from the resultant moment of interatomic forces over the unit cell.

Consider ABO3 ferroelectrics as an example. There are five atoms in a primitive unit cell to form the smallest structural unit Fig. 6(a). This unit cell (a mass point in classical continuum theory) gives the smallest allowable volume in which the continuum hypothesis is not violated. However, at ferroelectric phase, for this mathematical infinitesimal, the vector sum of all interatomic force will not be passing through the mass center. This will results in a surface couple on the surface of this mathematical infinitesimal in addition to the body couple. As a consequence, a higher order moment stress, m, as in Fig. 6(d), is resulted. The existence of m is true for a deformed material that has more than one atom in the unit cell. It is also true for a material with defects. Whenever there is a microstructure, there will be an inhomogeneous deformation, and there will be a moment stress m. The role of inhomogeneous deformation becomes more significant as the dimension of the material approaches the dimension of the microstructure. For material without inhomogeneity, e.g., a monatomic crystal without defects, t = s and m = 0. As the length scale goes up to macroscopic, the volume associated with the macro-stress t becomes larger; both t – s and m are then negligible.

A multiscale field theory: Nano/micro materials

27

The analysis of phonon dispersion relation shows that the classical continuum mechanics is the long wavelength and low frequency limit of acoustic phonons. It achieves a correspondence with an atomic model only for acoustic vibrations near zero-frequency and zero-wave-vector. 1.4 An overview of multiscale material modeling The term “multiscale material modeling” refers to theory and simulation of material properties and behavior across length and time scales from microscopic to macroscopic. The relevance is fundamentally predicated on the belief that such modeling will bring about better understanding of material behavior and properties, which in turn is essential in understanding, control and accelerating development in new nano/micro-systems. An early multiscale modeling technique is combining several modeling techniques, each one being suited for a particular length and time scale, such as coupled tight-binding (TB)-molecular dynamics (MD)-finite element (FE) model or MD-FE model. This concurrent multiscale approach is well demonstrated through fracture mechanics simulation with MD or TB/MD for the crack tip region and FE for the region away from the crack tip [3-6]. Such hybrid models directly link atomic regions to continuum regions and require that the atomic and continuum regions be specified at the start of the computation. Although, after so many years, it is still attracting extensive attentions and efforts, this approach is only suited for problems in which length scales can be pre-defined by regions. Another method is an overlapping array of successively coarser modeling techniques, where at each plateau the parameters of the coarse description are based on the finer description. This method is also referred as “up scaling” or “bottom up”. The general subsequence is from quantum mechanical first principles calculations to MD simulation to mesoscopic micro-structural approach to continuum FE modeling. The problem with this sequential approach is that the method of coarsening the description from atomistic to micro-structural or from micro-structural to continuum is not so obvious as it is in going from electrons to atoms [7, 8]. The coarse-graining or homogenization process involved when going from discrete to continuum has posed great challenges. In most of cases, atomistic effects get lost during the processes of coarse graining or homogenization. The mixed atomistic-continuum or quasi-continuum theory [9-10] is also an “up scaling” approach. The basic idea is that every point in a continuum corresponds to a large region on the atomic scale. The constitutive relations are obtained based on interatomic potential. The Cauchy-Born hypothesis for homogeneous problems is assumed in the formulation. An inner displacement ξ is introduced to account for inner motion and is obtained by minimizing the strain energy W. The energy minimizing gives, ξ = ξ (F) where F is the deformation gradient, and ˆ (F) W = W( F, ȟ ) = W( F, ȟ( F )) ≡ W

(9)

28

Y.. P. Chen et al.

This then falls within the framework of classical continuum mechanics, governed by the classical balance laws. As a consequence of the energy minimization, the system in question is in equilibrium, and the model is valid for zero-temperature problems. Despite this limitation, different variants of this theory have been developed and documented over a series of publications with various applications [9-15]. It is also worthwhile to mention the well-established “top down” approach: Micromorphic Theory [16], which encompasses many popular microcontinuum theories including microstructure theory [17], micropolar theory [18], Cosserat theory [19], and couple stress theory [20]. It has been demonstrated that micromorphic theory overlaps quite a large application region of microscopic theories [21-26] and naturally spans a large time and length scales. The material parameters can be obtained through phonon dispersion relations [27-28]. However, for complex crystals it is very difficult for Micromorphic theory to match its internal displacement patterns to the eigenvectors of some optical phonons. On the other hand, with present-day computers, atomistic MD simulation can study systems of billions of particles. Such study already simulates some features of macroscopic physics and spans multi length scales. For small systems, MD can also take into account effects of thermal fluctuations to study finite-temperature properties. However, the state of the art MD simulation is not applicable for systems with simultaneously large length and time scales yet. MD simulation is in many respects very similar to real experiments. It consists of three principal steps: (1) construction of a model, (2) calculation of atomic trajectories, and (3) analysis of those trajectories to obtain physical properties. To obtain an observable quantity in a MD simulation, one must first of all be able to express this observable as a function of the positions and momenta of the particles in the system. For any field theory, there are two essential parts: conservation laws and constitutive relations. The constitutive relations define the correlations of physical quantities. The conservation laws that govern the dynamic behaviors of the system are the time evolution laws of conserved properties. It is seen that the analytical expressions of observable quantities in N-body MD system in fact contain entire information of (1) the correlations between these quantities and (2) the time evolutions of those quantities, which are able to define a field theory completely. There thus arises the question: can we take another path, start from microscopic many-body dynamics, proceed directly with the atomistic expressions of physical quantities and their averages to formulate a field theory that can work as an alternative to, but computationally much more efficient than MD simulations in studying statistical and finite temperature properties of materials in nano/microsystems? This work aims at formulation of a multiscale field theory. In the following sections, the atomistic definitions of physical quantities will be presented in Section 2; field representation of the balance laws for the new field theory are formulated and obtained in Section 3; stress and strain relation is derived in Section 4; a numerical example of phonon dispersion relation is presented in section 5, this paper ends with a summary and discussions in the Section 6.

29

A multiscale field theory: Nano/micro materials

2. Atomistic Definitions of Physical Quantities 2.1 Instantaneous physical quantities Macroscopic quantities are generally described by continuous (or piecewise-continuous) functions of physical space coordinates x and of time t. They are fields in physical space-time. Microscopic dynamic quantities, on the other hand, are functions of phase-space coordinates (r, p), i.e., the positions and momenta of atoms, cf. Eqs. (2) to (5):

{ p = {m V

r = R kα = R k + ∆r kα k = 1, 2,3,...n, α

kα

α = 1, 2,3,..ν

= m α V k + m α ∆v kα k = 1, 2,3...n,

}

α = 1, 2,3,...ν

}

(10)

where the superscript kα refers to the α-th atom in the k-th unit cell. Consider a one-particle dynamic function a(R kĮ ,V kĮ ) . The corresponding local density at a given point x in physical space can be represented by A(R kα , V kα ; x) = a(R kα , V kα )δ (R kα − x)

(11)

Here, the į-function, į(R kĮ − x) , is a localization function and provides the link between phase space and physical space descriptions. It can be a Dirac į-function, or a distribution function. For Dirac į-function [29], there results , x =R kα δ ( x − R kα ) = ­∞ ® kα

(12)

¯0 , x ≠ R

This means that, in a discontinuous atomic description, there can be a contribution to this function only if an atom happens to be located at x, that is, if RkĮ = x. In the distribution or weighting function approach, the localization function is a non-negative function that has a finite size and finite value [30-32], is peaked at x = R kĮ and tends to zero as x– R kĮ becomes large, e.g., Gaussian distribution function

(

2

)

δ ( x − R kα ) = π −3 / 2 exp − x − R kα / l 2 /l 3

(13)

where l characterizes the length of the region of a lattice point or of an atom. Both Dirac į-function and the distribution function shall satisfy

³ į(x– R

kĮ

)d 3 x = 1

(14)

V

Note that the above procedure is a standard treatment in statistical mechanics to define a mapping of the phase space into the physical space.

30

Y.. P. Chen et al.

Fig. 7. Field representation of the positions of the unit cell and the atoms relative to the unit cell.

This paper intends to employ a different field description that specifies the positions of the unit cell and of the atoms relative to the unit cell, cf. Fig. 7, similar to the MD representation in Fig. 5 for a multi-element crystal. Distinguishing from the standard treatment in statistical mechanics, in the rest of this paper, x is employed to represent the continuous collections of lattice points, corresponding to the phase space coordinates, R k ( k=1,2,3,...n ); and yĮ to represent the α-th atomic position relative to the lattice point x, corresponding to ǻr kĮ , cf. Fig. 7 and Fig. 5. Therefore, this localization function that define the mapping of the phase space into the physical space has following form į(R k + ǻr kĮ − x-yĮ) = į(R k −x) į(ǻr k Į−y Į)

(15)

with which, the correspondence between a lattice point x in physical space to the position of the center of k-th unit cell in phase space, R k , is then established, and the position of α-th atom associated with lattice point x, yĮ , shall be on one-to-one correspondence with, ǻr kĮ , the relative position of α-th atom in the unit cell k. That is, for any given physical point x at an instantaneous time, a unit cell can be found with its center R k located at this point, and a physical space description of the relative position of the α-th atom, yĮ , can be determined. The local density of any measurable phase-space function A(r,p) can then be defined as n

A(x, y α ) = ¦ A(r,p)δ(R k − x)δ(∆r kα − y α ) ≡ A α (x)

(17)

k =1

with normalization conditions

³ į(R –x)d x = 1 k

3

(k = 1, 2, 3, ...n)

(18)

V

where V is the volume of the whole system. Equation (18) implies that over the entire physical space all the unit cells, (k = 1, 2,3,..., n) , can be found. Then, for each

31

A multiscale field theory: Nano/micro materials

unit cell k , the second δ -function, δ (∆r kα − yα ) , identifies yα to be ∆r kα , i.e.,

­ 1 if ξ = α and ∆r kξ = y α δ( ∆r − y ) = ® α kξ ¯ 0 if ξ ≠ α or ∆r ≠ y kξ

α

(19)

It follows

į(ǻr

– yĮ ) =

kĮ

Ȟ

¦

į(ǻr kȟ – y Į )

(20)

ȟ=1

and

³ δ(R

k

− x)δ(∆r kα − y α )d3 x = 1 ( k = 1, 2, 3, ...n) (α = 1, 2, ..ν)

(21)

V

2.2 Averaged field variables To obtain an observable quantity in a MD simulation, one must first of all be able to express this observable as a function of the positions and momenta of the particles in the system. However, a measured value of A , called A m , is not obtained from an experiment performed at an instant; rather the experiment requires a finite duration. During that measuring period individual atoms evolve through many values of positions and momenta. Therefore the measured value A m is generally the phase function A(r, p) averaged over a time interval ∆t ǻt

A m (t) =

1 A(r(t+IJ),p(t+IJ))dIJ ǻt ³0

(22)

In equilibrium MD, it is assumed that thist time-interval average reliably approxi1 mates the time average A , A = Lim ³ A(r ( τ), p( τ))dτ , which would be obtained t →∞

t

0

from a measurement performed over an essentially infinite duration, i.e., Am = A

(23)

In statistical mechanics a macroscopic quantity is defined as the ensemble average of an instantaneous dynamical function:

32

Y.. P. Chen et al.

A ≡ ³ ³ A(r,p)f (r,p, t )drdp

(24)

p r

where f is the normalized probability density function, i.e., ³³ f (r, p, t)drdp = 1 . Eq. (24) distinguishes molecular dynamics from statistical mechanics. Statistical mechanics avoids time average by replacing it with an ensemble average, which is originally motivated by the inability to actually compute the phase-space trajectory of a real system containing huge number of molecules. When one departs from equilibrium very little theoretical guidance of statistical mechanics is available, and MD begins to play the role of an experimental tool. Most current MD applications involve systems that are either in equilibrium or in some time-independent stationary state; where individual results are subject to fluctuation, it is the well-defined averages over sufficiently long time intervals that are of interest. Extending MD to open systems, where coupling to the external world is of a more general kind, introduces many new problems. Not only are open systems out of thermodynamic equilibrium, but also in many cases they are spatially inhomogeneous and time-dependent. To smooth out the results and to obtain results close to experiments, measurements of physical quantities are necessary to be collected and averaged over finite time duration. Therefore, in deriving the field description of atomic quantities and balance equations, it is the time-interval averaged quantities that will be used, and the time-interval averaged (at time t in the interval ∆t) local density function reads α

A (x, t ) = A

α

α

≡ Am =

∆t

1

n

α k kα ³ ¦ A(r (t + τ), p(t + τ))δ(R − x)δ(∆r − y )dτ ∆t

(25)

0 k =1

The fundamental physical quantities considered in this paper are mass, momentum, atomic force, momentum flux, total and internal energy, heat flux and temperature. 2.3 Mass density Define the local mass density of α-th atom as a time-interval averaged quantity

ρ α (x, t) = ρ(x, y α , t) =

n

¦m

α

δ(R k − x)δ(∆r kα − y α )

(26)

k =1

The total mass of the system is given by § v · M = ³ ¨ ¦ ρα (x, t) ¸d 3 x = ¹ V © α=1

§

n

v

k =1

α=1

³ ¨© ¦ ¦ m

V

α

v · δ(R k − x)δ(∆r kα − y α ) ¸d 3 x = n ¦ m α = nm α=1 ¹

(27)

A multiscale field theory: Nano/micro materials

33

Here, the definition of mass densities is similar to that of [33], who defined the total mass density of a system involving ν different components, each with mass density ρ α , as ν

ρ = ¦ ρα .

(28)

α=1

2.4 Linear momentum density The linear momentum measures the flow of mass. The link between the atomic measure of the flow of mass and the field description of momentum density is achieved through the localization function and time interval averaging: ρ α ( v + ∆v α ) =

n

¦m

α

(V k + ∆v kα )δ(R k − x)δ(∆r kα − y α )

(29)

k =1

where v=x and ǻvĮ = y Į are the time-interval averaged velocity of the mass center of a unit cell and the velocity of α-th atom relative to the center of the unit cell, respectively. 2.5 Atomic forces It is assumed that the interatomic force can be derived from interatomic potential. For whether the interaction is through two or three-body potential, one always has the force acting on the atom i as fi = −

∂U , ∂R i

(30)

and the mutual interaction force between atom i and atom j can be obtained as f ij = −

∂U ∂U = = −f ji i j j i ∂(R − R ) ∂(R − R )

(31)

where U is the total potential energy of the system, f ij the interatomic force, and R i − R j the relative separation vector between the two atoms i and j. Generally, forces acting on an atom can be divided into three kinds. kĮ

f1l ȕ : interatomic force between (k, α) and (l, β) atoms in two different unit cells, kĮ

lȕ

with f1l ȕ = – f1kĮ. kĮ

f 2 ȕ : interatomic force between (k, α) and (k, β) atoms in the same unit cell, with kĮ

ȕ

f 2 ȕ = – f 2k Į. f 3kĮ : body force on atom (k, α) due to the external fields.

34

Y.. P. Chen et al.

The total force acting on an atom (k, α) can be written as Ȟ

n

Ȟ

kĮ

FkĮ = ¦¦ f1lȕ +¦ f 2 ȕ +f 3kĮ . l=1 ȕ=1

kĮ

(32)

ȕ=1

The body force density due to external field is fα ≡

n

¦f

kα 3

δ(R k − x)δ(∆r kα − y α )

(33)

k =1

and body couple density is Lα =

n

¦f

kα

⊗ (R k + ∆r kα )δ(R k − x)δ( ∆r kα − y α ) = f α ⊗ x + l α

3

(34)

k =1

n

¦f

where l α =

kα 3

⊗ ∆r kα δ(R k − x)δ( ∆r kα − y α ) .

k =1

Assume that the total internal potential energy is Uint , with the force-potential function relationship Eqs. (30, 31) the internal forces density due to atomic interaction can be expressed as n

ν

n

kα

ν

¦ (¦¦ f1lβ + ¦ f2 β )δ(R k − x)δ(∆r kα − y α )

f intα (x) ≡

k =1

l =1 β=1

β=1

∂U int δ(R k − x)δ(∆r kξ − y α ) kξ − R lη )

ν

n

kα

¦ ¦ ∂(R

=−

k,l =1 ξ , η=1

∂U int δ(R k − x)δ(∆r kξ − y α ) kξ − R kη )

ν

n

(35)

¦ ¦ ∂ (R

−

k =1 ξ , η=1

kȟ

lȘ

k ȟȘ

k ȟȘ

From Eq. (31), one has f l1Ș = –f1kȟ and f 2 = – f 2 . Utilizing the interchange of indices kξ and lη, it is seen that Į f int (x)=

1 2

Ȟ

n

k,l=1 ȟ,Ș=1

1 + 2

kȟ lȘ 1

¦¦f n

Ȟ

¦¦f k=1 ȟ,Ș=1

[į(R k  x)į(ǻr kȟ  yĮ )  į(R l  x)į(ǻr lȘ  y Į )]

(36) k ȟȘ 2

Į

Į

į(R  x)[į(ǻr  y )  į(ǻr  y )] k

kȟ

kȘ

Since the formulation involves many-body interactions, it is understood that the summation over k and l, does not include the case k = l, and the summation over α

A multiscale field theory: Nano/micro materials

35

and β does not include the case α = β. In this work, velocity-dependent interactions, such as interaction with magnetic fields, are not considered, and hence the forces depend only on atomic positions. However, the results can be generalized to include such cases. 2.6 Momentum flux density It is well accepted that the momentum flux in an N-body dynamics system can be divided into two parts: kinetic and potential parts [22-24, 34, 35]. The kinetic part of the momentum flux is the flow of momentum due to atomic motion, which, in the co-moving coordinate system, is s kin = −p ⊗ p / m .

(37)

By virtue of the possible macroscopic motion of the material body, the velocity that contributes to momentum flux is the difference between the instantaneous velocity and the stream velocity (the ensemble or time average of the velocity), i.e.,  kĮ =V kĮ  V kĮ = V k Į  (v+ǻ v Į ) V

(38)

This velocity difference, V kĮ , measures the fluctuations of atoms relative to the local equilibrium and is related to the thermal motion of atoms. In the field representation, the kinetic part of local density of momentum flux at Į-th atomic position embedded in lattice point x is α skin =−

n

¦m

α

 kα ⊗ V  kα δ(R k − x)δ(∆r kα − y α ) V

(39)

k

Fig. 8. Flow of momentum due to interatomic force.

The potential flow of momentum occurs through the mechanism of the interparticle forces (cf. Fig. 8). For a pair of particles α and β that lie on different side of a surface that intersects the line connecting the two particles at x + y α , the pair force kȟ

lȘ

f lȘ = f kȟ gives the rate at which momentum being transported from particle kξ to

particle lη. For each such pair the direction of this transport is along the direction of R k ȟ  R lȘ . So, the potential contribution to the momentum flux is kξ

s pot = −(R kξ − R lη ) ⊗ f lη

(39)

36

Y.. P. Chen et al.

which is continuous along the line connecting the two particles. Notice that δ(R k − x)δ( ∆r kξ − y α ) − δ(R l − x)δ(∆r lη − y α ) 1

=³ 0

d ª δ ( R k λ + R l (1 − λ ) − x ) δ ( ∆r kξ λ + ∆r lη (1 − λ ) − y α ) º dλ ¼ dλ ¬

(40)

With the consideration of all interatomic forces that pass through the atomic site (x, yĮ ) , the local density of the momentum transport at (x, y α ) due to atomic interaction may be expressed as 1

α =− spot

kξ n ν 1 kξ lη lβ k l kξ lη α d ( ) λ − ⊗ R R f ¦¦ 1 δ ( R λ + R (1 − λ ) − x ) δ ( ∆r λ + ∆r (1 − λ ) − y ) 2 ³0 k,l =1 ξ, η=1

1

−

ν n 1 kξ dλ ¦ ¦ (Rkξ − Rkη ) ⊗ f 2 η δ(R k − x)δ(∆r kξ λ + ∆r kη (1 − λ) − y α ) ³ 2 0 k =1 ξ,η=1

(41) The continuum counterpart of momentum flux density is the stress tensor. However, it is seen that the mathematical infinitesimal volume that does not violate the continuum assumption is the volume ǻV defining the density of lattice points, which is the volume of a unit cell. The vector sum of all the atomic forces within this volume may not pass through the mass center of the ǻV. The continuum definition of stress is, therefore, not the momentum flux density; for crystal with more than one atom in the unit cell, the continuum stress is only the homogenous part of the momentum flux summing over a volume at least of a unit cell, and it may not be symmetric. The total momentum flux is, therefore, better represented upon decomposition of a homogeneous part that is caused by lattice motion and deformation and is related to continuum stress, and an inhomogeneous part caused by internal (relative) atomic motion and deformation. They take the following forms: 1) The homogeneous kinetic part, α tkin =−

n

¦m

α

 k ⊗V  kα δ(R k − x)δ(∆r kα − y α ) V

(42)

k =1

2) The inhomogeneous kinetic part, α IJ kin =−

n

¦m k =1

α

 kα δ(R k − x)δ(∆r kα − y α ) ∆v kα ⊗ V

(43)

37

A multiscale field theory: Nano/micro materials

3) The homogeneous potential part, 1

n kξ ν 1 dλ ¦ ¦ (R k −R l ) ⊗ f1lη δ ( R k λ + R l (1 − λ) − x ) δ ( ∆r kξ λ + ∆r lη (1 − λ) − y α ) ³ 2 0 k,l =1 ξ, η=1

α tpot =−

(44) 4) The inhomogeneous potential part, 1

kξ n ν 1 kξ lη lβ d ( ) λ ∆ −∆ ⊗ δ ( R k λ + Rl (1 − λ) − x ) δ ( ∆r kξ λ + ∆r lη (1 − λ) − y α ) r r f ¦ ¦ 1 2 ³0 k,l =1 ξ,η=1

α =− IJpot

1

n ν 1 kξ dλ¦ ¦ (∆r kξ −∆r kη ) ⊗ f2 η δ(R k − x)δ ( ∆r kξ λ + ∆r kη (1 − λ) − y α ) ³ 2 0 k =1 ξ,η=1

−

(45) with d ªδ ( R k λ + R l (1 − λ ) − x ) δ ( ∆r kξ λ + ∆r lη (1 − λ) − y α ) º ¼ dλ ¬

(

= −∇ x ⋅ (R k − R l )δ ( R k λ + R l (1 − λ ) − x ) δ ( ∆r kξ λ + ∆r lη (1 − λ ) − y α )

)

(

−∇ y α ⋅ ( ∆r kξ − ∆r lη )δ ( R k λ + R l (1 − λ) − x ) δ ( ∆r kξ λ + ∆r lη (1 − λ ) − y α )

(46)

)

it is seen from Eqs. (36, 44, 45) that the divergences of the potential momentum fluxes are related to the internal forces by α α ∇ x ⋅ tpot + ∇ yα ⋅ IJ pot = f intα =

n

ν

n

kα lη 1

¦¦ (¦ f k =1 η=1

kα

+ f 2 η )δ(R k − x)δ(∆r kα − y α )

(47)

l =1

2.7 Total energy density and internal energy density The total energy of atom α in a microscopic N-body dynamics system is the sum of kinetic and potential energies. In continuum theory, the local energy density is usually defined as energy per unit mass. This implies ρα E α =

n

¦[ k =1

1 2

m α (V kα ) 2 + U kα ]δ(R k − x)δ( ∆r kα − y α )

(48)

The local density of internal energy, which is the state function of thermodynamics, can be expressed as the sum of thermal energy and potential energy ρα ε α =

n

¦[ k =1

1 2

 kα )2 + U kα ]δ(R k − x)δ(∆r kα − y α ) mα (V

(49)

38

Y.. P. Chen et al.

Rewriting the total energy density as n

¦ ª¬

ρα E α =

k =1

1 2

 kα )2 + 2V  kα ⋅ ( v + ∆v α ) + ( v + ∆v α )2 } + U kα ºδ(R k − x)δ(∆r kα − y α ) mα {(V ¼

(50) there results the macroscopic relation of densities of the total energy, the internal energy and the kinetic energy as, ρα E α = ρα ε α + 12 ρα ( v + ∆v α ) 2

(51)

2.8 Heat flux

The flow of energy by atomic motion, for all particles in the volume ǻV , gives the kinetic contributions to the energy flux. It comes from the rate at which the local energy E i of atom i moves with the local atomic velocity pi /mi , Q ikin = −

pi i E mi

(52)

The potential contribution to the energy flow occurs whenever two moving particles interact in such a way that one particle transfers a part of their joint energy to the other particle. It comes from the rate at which energy is transported through the action of interparticle forces: atom i is doing work on atom j, multiplied by the distance R i -R j over which this energy is transferred. 1 pi pj Q pot = − (R i − R j )( + ) × f ij 2 2mi 2m j

(53)

Noting that heat flux is the conductive flow of internal energy per unit time and area [2, 36], the local density function of kinetic and potential heat fluxes, therefore, are expressed as Qαkin = −

n

¦ V k =1

1

Qαpot = −

(54)

kξ ν n 1  kξ ⋅ f lη δ Rk λ + Rl (1 − λ) − x δ ∆rkξλ + ∆rlη (1 − λ) − yα dλ ¦ ¦ (Rkξ −Rlη )V 1 ³ 2 0 k,l=1 ξ,η=1 1

−

 kα )2 + U kα ]δ(R k − x)δ(∆r kα − y α ) [ mα (V

kα 1 2

(

) (

ν n ξ 1  kξ ⋅ f kη δ(Rk − x)δ ∆rkξλ + ∆r kη (1 − λ) − yα λ d (Rkξ − Rkη )V ¦ ¦ 2 ³ 2 0 k =1 ξ,η=1

(

)

)

(55) Clearly there are homogeneous and inhomogeneous parts. Similar to the decomposition of momentum flux density, let the heat flux density be decomposed as

39

A multiscale field theory: Nano/micro materials

1) The homogeneous kinetic part of heat flux, α qkin =−

n

¦ V [ k =1

k 1 2

 kα )2 + Ukα ]δ(R k − x)δ(∆r kα − y α ) mα ( V

(56)

2) The inhomogeneous kinetic part of heat flux, α jkin =−

n

¦ ∆v k =1

 kα ) 2 + U kα ]δ(R k − x)δ(∆r kα − y α ) [ mα (V

kα 1 2

(57)

3) The homogeneous potential part of heat flux, 1

α qpot =−

kξ n ν 1 k l  kξ k l kξ lη α lη d ( ) λ R − R V ⋅ f ¦¦ 1 δ R λ + R (1 − λ) − x δ ∆r λ + ∆r (1 − λ) − y 2 ³0 k,l =1 ξ,η=1

(

) (

)

(58) 4) The inhomogeneous potential part of heat flux, 1

α jpot =−

kξ n ν 1  kξ ⋅ f lη δ Rk λ + Rl (1 −λ) − x δ ∆rkξλ + ∆rlη (1 −λ) − yα dλ ¦ ¦ (∆rkξ −∆rlη )V 1 ³ 2 0 k,l=1 ξ,η=1

(

1

−

)(

n ν ξ 1  kξ ⋅ f kη δ(Rk − x)δ ∆rkξ λ + ∆rkη (1 − λ) − yα λ d (∆rkξ −∆rkη )V ¦ ¦ 2 ³ 2 0 k =1 ξ,η=1

(

)

)

(59) It is seen that the inhomogeneous heat flux is closely associated with the inhomogeneous stress and may represent the thermal currents that flow back and forth during vibration between stress inhomogeneities. With the definition of the potential parts of momentum fluxes and the identity of į function, it is straightforward to prove that the divergences of potential heat fluxes have the following characteristics. α α α α ∇ x ⋅ [q pot + tpot ⋅ ( v + ∆v α )] + ∇ y α ⋅ [ jpot + IJ pot ⋅ ( v + ∆v α )] kα 1 n ν (V kα + V lη ) ⋅ f1lη δ(R k − x)δ( ∆r kα − y α ) ¦ ¦ 2 k ,l =1 η=1

= +

(60)

1 n ν kα ¦¦ (V kα + V kη ) ⋅ f2 η δ(R k − x)δ(∆r kα − y α ) 2 k =1 η=1

2.9 Temperature The temperature T for the microscopic N-body system is also an average quantity. It can be most simply expressed in terms of thermal energy by the mean-squared velocity, relative to the local stream velocity [35], as

40

Y.. P. Chen et al.

T α ( x) =

∆V 3k B

∆V = 3k B

n

¦m

α

 kα ) 2 δ( R k − x)δ( ∆r kα − y α ) (V

k =1 n

¦m

α

kα 2

k

(V ) δ(R − x)δ(∆r

k =1

kα

mα −y ) − ( v + ∆v α ) 2 3k B

(61)

α

where k B is Boltzmann constant; V kα are the velocity differences or the fluctuations of atoms, and ǻV is the volume that define the density of lattice points, i.e., the volume of a unit cell. 2.10 Polarization Dielectrics and ferroelectrics are relating to various electrical phenomena and polarization of material. Classically, polarization P is defined as the dipole moment per unit volume [37] P=

e ¦ Zn R n ǻV

(62)

where R n is the position vector of charge Z n , e is the unit of charge, ∆V is the volume of unit cell. In an ionic crystal there are two sources of polarization. One is due to the electronic polarization of the ions. The other is due to the displacements of the ions. Polarization only makes sense when defined in terms of the displacements of ions summed over a neutral stochiometric unit, the smallest of which in a crystal is a primitive unit cell. In atomic-level computation, most common descriptions are through lattice dynamical rigid-ion model [38] or shell model [39, 40], cf. Fig. 9. In the rigid-ion model, the ionic charges are approximately by point charges centered at the nuclei. The ions are not polarizable. While in shell model, by assuming that each ion consists of a spherical charged shell isotropically bound to the rigid ion-core by a spring and can be displaced relative to the core, both the ionic and electronic polarizabilities can be taken into account. Correspondingly, the induced polarization for the rigid-ion model has the form as P(k)=

e Ȟ ¦ ZĮ u(k,Į) ǻV Į=1

(63)

and for the shell model as P (k) =

e ν ¦ [Zα u(k , α) + YαȢ(k , α)] ∆V α=1

(64)

41

A multiscale field theory: Nano/micro materials

In field representation, for the rigid-ion model, one has n

ν

P = P (x) = e¦¦ Zα u( k , α )δ(R k − x)

(65)

k =1 α=1

and for the shell model n

ν

P = P (x) = e¦¦ [Zα u( k , α ) + Yα Ȣ (k , a)]δ(R k − x)

(66)

k =1 α=1

Here, Z and Y are the core charge and shell charge respectively; u(k, α) is the core displacement with respect to the reference state, u(k, α ) = R kα − R krefα ; and Ȣ (k, α) is the shell to core relative displacement of atom (k, α) . The spontaneous polarization is thus P = P ( x) =

e n ν ¦¦ Zα R krefα δ(R k − x) ∆V k =1 α=1

(67)

where R krefĮ is the atomic position at reference temperature T, and is hence a function of T.

Fig. 9. The shell model in which the ion consists of an electronic shell and a rigid ion-core.

Eqs. (63) to (66) are definitions of induced polarization at constant temperature used in atomic-level computation, while the effect of temperature on polarization is accounted through Eq. (67). Despite its crudeness, the rigid-ion models are remarkably successful and have been fitted empirically to describe the alkali halides. They give fair estimates of cohesive energies and elastic properties, and when used in molecular dynamics simulations they reproduce thermodynamic and structural quantities in the molten state. However, they seriously misrepresent the dielectric properties of the crystals, and consequently give poor descriptions of lattice vibrational spectra. Allowance for ionic polarizability leads to effective dynamic charge. The shell model can result in correctly both static and high frequency dielectric constants for many ionic materials, and reproduce phonon spectra in a good agreement with the experimental measurement.

42

Y.. P. Chen et al.

3. Field Description of Conservation Laws 3.1 Time evolution of physical quantities As mentioned above, an observable quantity in a MD simulation is supposed to be a function of the positions and momenta of the particles in the system, cf. Eq. (17). With ∇ R δ(R k − x) = −∇ x δ(R k − x)

(68)

∇ ∆r δ(∆r kα − y α ) = −∇ y δ(∆r kα − y α )

(69)

k

kα

α

and in the general case that phase space function A does not involve field quantities, the time evolution of its local density function can be expressed as ∂A α ∂t

n

x,y α

 δ(R k − x)δ(∆r kα − y α ) = ¦A k =1

n

−∇ x ⋅ (¦ V k ⊗ Aδ(R k − x)δ(∆r kα − y α ))

(70)

k =1

n

−∇ yα ⋅ (¦ ∆v kα ⊗ Aδ(R k − x)δ(∆r kα − y α )) k =1

For the time-interval averaged (at time t in the interval ∆t) field quantity A Į , cf. Eq. (25). One has ∂A α ∂t

x,y

α

=

n

ν

¦ ¦ δ(R

k

k ,l =1 γ =1

−∇ x ⋅

n

¦V

− x)δ(∆r kα − y α )(V lγ ⋅ ∇ Rlγ + m1γ F lγ ⋅∇ Vlγ )A k

⊗ Aδ(R k − x)δ(∆r kα − y α )

(71)

k =1

−∇ yα ⋅

n

¦ ∆v

kα

⊗ Aδ(R k − x)δ(∆r kα − y α )

k =1

Eqs. (70) and (71) are the time evolution laws for instantaneous quantity A Į and averaged field quantity A Į , respectively. When A Į is a conserved property, it results in the local conservation laws that govern the time evolution of A Į and A Į , respectively. 3.2 Atomistic formulation of the balance laws A thermodynamic theory of irreversible processes starts with a set of general balance equations that govern the time evolution of the system. It is the objective of this paper to establish differential balance equations for a thermodynamic system on the same foundation of molecular dynamics: the classical N-body dynamics.

43

A multiscale field theory: Nano/micro materials

Those balance equations will follow exactly the time evolution laws that exist in a molecular dynamics simulation, where the atomic motion is fully described, the inhomogeneous internal motion is not ignored, and the smallest particles are atoms. n

¦m

With ρ α =

α

δ(R k − x)δ(∆r kα − y α ) , the time evolution of mass density

k =1

can be obtained as ∂ρα = −∇x ⋅ ∂t

n

¦ m V δ(R α

k

k

k =1

− x)δ( ∆r kα − y α ) − ∇yα ⋅

n

¦ m ∆v α

kα

δ(R k − x)δ( ∆r kα − y α )

k =1

(72) From the definition of linear momentum, one can immediately find ∂ρ α + ∇ x ⋅ ( ρα v) + ∇ y α ⋅ ( ρα ∆v α ) = 0 ∂t

(73)

or dρ α + ρα (∇ x ⋅ v + ∇ y α ⋅ ∆v α ) = 0 dt

(74)

For cell-average mass density ρ =

n

¦ mδ(R

k

− x) , it is readily to obtain

k =1

∂ρ + ∇ x ⋅ ( ρv) = 0 , ∂t

dρ + ρ∇ x ⋅ v = 0 dt

or

(75)

This is identical to the continuity equation in macroscopic physics. Turning to the balance of linear momemtum, consider using the field representation of local linear momentum density given by Eq. (20) which can be put into Eq. (71) to yield ∂ρ α ( v + ∆v α ) = ∂t

n

ν

¦ ¦ δ(R k − x)δ(∆r kα − y α )

k,l =1 γ =1

−∇ x ⋅

n

¦m

α

F lγ ⋅∇ lγ (m α V kα ) mγ V

V k ⊗ V kα δ( R k − x)δ( ∆r kα − y α )

k =1

−∇ yα ⋅

n

¦m k =1

α

∆v kα ⊗ V kα δ( R k − x)δ(∆r kα − y α )

(76)

44

Y.. P. Chen et al.

With the divergence of momentum flux, Eq. (47), it is found that n

¦F

kα

k =1

α α δ(R k − x)δ(∆r kα − y α ) = ∇ x ⋅ tpot + ∇ y α ⋅ IJ pot +fα

(77)

Since n

¦m

α

 k δ(R k − x)δ(∆r kα − y α ) = V

k =1

n

¦m

α

∆v kα δ(R k − x)δ(∆r kα − y α ) = 0

(78)

k =1

there results ∇x ⋅

n

¦m

α

V k ⊗ V kα δ(R k − x)δ(∆r kα − y α )

k =1

n

¦m

= ∇x ⋅

α

 k ⊗V  kα + v ⊗ ( v + ∆v α )]δ(R k − x)δ(∆r kα − y α ) [V

(79)

k =1

α = ∇ x ⋅ [− tkin + ρ α v ⊗ ( v + ∆v α )]

and ∇ yα ⋅

n

¦m

α

∆v kα ⊗ V kα δ(R k − x)δ(∆r kα − y α )

k =1

= ∇yα ⋅

n

¦m

α

 kα + ∆v α ⊗ ( v + ∆v α )]δ(R k − x)δ(∆r kα − y α ) [∆v kα ⊗ V

(80)

k =1

α = ∇ y α ⋅ [− IJ kin + ρα ∆v α ⊗ ( v + ∆v α )]

Į Į Į Į + tpot and IJ Į = IJkin + IJpot , the time evoluCombining Eqs. (77-80) and with t Į = tkin tion of linear momentum is obtained as

∂ α ( ρ ( v + ∆v α )) = ∇ x ⋅ [ t α − ρ α v ⊗ ( v + ∆v α )] + ∇ yα ⋅ [ IJ α − ρ α ∆v α ⊗ ( v + ∆v α )] + f α ∂t

(81) or ρα

d ( v + ∆v α ) = ∇ x ⋅ t α + ∇ y α ⋅ IJ α + f α dt

For cell-average linear momentum density, ρv ≡ evolution is obtained as

(82) n

¦ mV δ(R k =1

k

k

− x) , the time

45

A multiscale field theory: Nano/micro materials

∂ ( ρv ) = ∇ x ⋅ [ t − ρv ⊗ v ] + f , ∂t ν

ν

α=1

α=1

or

ρ

d v = ∇x ⋅ t + f . dt

(83)

t = ¦ t α and f = ¦ f α , are the cell averages of homogeneous momen-

where

tum flux density and body force density, respectively. Eq. (83) is identical with conservation law of linear momentum in macroscopic continuum mechanics. This is because the cell-average linear momentum is a homogeneous quantity. Consideration of the balance of moment of momentum or the angular momentum density gives ρα ȥ α ≡

n

¦m

α

V kα × R k α δ ( R k − x ) δ ( ∆ r kα − y α ) = ρ α ( v + ∆ v α ) × ( x + y α )

(84)

k =1

Substituting it into Eq. (71), there results ∂ α α (ρ ȥ ) = ∂t

n

¦m

α

V kα × V kα δ(R k − x)δ(∆r kα − y α )

k =1 n

¦

+

F kα × R kα δ(R k − x)δ(∆r kα − y α )

k =1

−∇ x ⋅

n

¦ (V

k

 kα + v + ∆v α ) × R kα δ(R k − x)δ(∆r kα − y α ) + v) ⊗ m α (V

k =1

−∇ yα ⋅

n

¦ (∆v

kα

 kα + v + ∆v α ) × R kα δ(R k − x)δ(∆r kα − y α ) + ∆v α ) ⊗ m α ( V

k =1

≡ A+B+C+D

(85)

with A=

n

¦ (m V Į

× V ka )δ(R k − x)δ(∆r kα − y α ) = 0

ka

(86)

k =1

B=

n

k =1

=

n

ν

kα lη 1

¦ (¦¦ f n

l =1 η=1 n

ν

kα lη 1

¦ (¦¦ f k =1

l =1 η=1

ν

× R kα + ¦ f 2 η × R kα )δ(R k − x)δ(∆r kα − y α ) + Lα kα

η=1

ν

+ ¦ f 2 η )δ(R k − x)δ(∆r kα − y α ) × (x + y α ) + Lα kα

(87)

η=1

α α ) × (x + y α ) + Lα = (∇ x ⋅ tpot + ∇ y α ⋅ IJ pot

ª n  k ⊗V  ka δ(R k − x)δ(∆r kα − y α ) × (x + y α ) º − ∇ ⋅ ( v ⊗ ρ α ȥ α ) C = −∇ x ⋅ « ¦ m α V x » ¬ k =1 ¼ α = −∇ x ⋅ [ v ⊗ ρα ȥ α − tkin × (x + y α )]

(88)

46

Y.. P. Chen et al.

ª D = −∇ y α ⋅ « ∆v α ⊗ ρ α ȥ α + ¬ = −∇ y α ⋅ (∆v α ⊗ ρα ȥ α )

n

¦ ∆v k =1

kα

º ⊗ m α V kα × R kα δ(R k − x)δ(∆r kα − y α ) » ¼

§ n  kα + v + ∆v α )δ(R k − x)δ(∆r kα − y α ) × (x + y α ) ·¸ −∇ y α ⋅ ¨ ¦ ∆v kα ⊗ m α (V © k =1 ¹ α α α α α = −∇ y α ⋅ [∆v ⊗ ρ ȥ − IJ kin × (x + y )]

(89)

Combining A, B, C, and D gives ∂ α α α α (ρ ȥ ) = −∇ x ⋅ [ v ⊗ ρ α ȥ α − tkin × (x + y α )] − ∇ y α ⋅ [∆v α ⊗ ρ α ȥ α − IJ kin × (x + y α )] ∂t (90) α α + (∇ x ⋅ tpot + ∇ yα ⋅ IJpot ) × (x + y α ) + Lα

or ρα

dȥ α α α α α ) × ( x + y α ) + Lα = ∇ x ⋅ [ tkin × (x + y α )] + ∇ yα ⋅ [ IJ kin × (x + y α )] + (∇ x ⋅ tpot + ∇ yα ⋅ IJ pot dt

(91) Notice that the time evolution of angular momentum can also be expressed from its field definition as ρα

dȥ α d( v + ∆v α ) = ρα × ( x + y α ) = (∇ x ⋅ t α + ∇ y α ⋅ IJ α + f α ) × ( x + y α ) dt dt

(92)

Since (1) t Į + IJ Į is symmetric and (2) x and yĮ are mutually independent within any unit cell, the balance law of angular momentum is shown to be identically satisfied. Let ρi iji = ρi ( v + ∆vi ) ⊗ y i . This definition is then similar to the generalized spin in Micromorphic theory [16], i.e., ν

ν

α=1

α=1

ρij = ¦ ρ α ( v + ∆v α ) ⊗ y α = ¦ ρα ∆v α ⊗ y α

(93)

The time evolution is then ν ν ∂ (ρij) + ∇ x ⋅ ( v ⊗ ρij − ¦ t α ⊗ y α ) + ¦ ∇ yα ⋅ (∆v α ⊗ ρ α ij α − IJ α ⊗ y α ) ∂t α =1 α =1 ν

= − IJ + ¦ ρ α ∆v α ⊗ ∆v α + l α =1

(94)

47

A multiscale field theory: Nano/micro materials

Assuming ∆v kα = Ȧ ⋅ ∆r kα , the above equation becomes identical to the balance equation of generalized spin in Micromorphic theory. The conservation of energy further can be applied by writing the local total energy n

¦[

density as ρ α E α =

k =1

1 2

m α ( V kα ) 2 + U kα ]δ(R k − x)δ( ∆r kα − y α ) . Hence, Eq. (71)

may be used to give ∂ α α (ρ E ) = −∇ x ⋅ ∂t

n

¦V k =1

n

¦ ∆v

−∇ y α ⋅

k =1

n

¦F

+

[ m α (V kα )2 + U kα ]δ(R k − x)δ(∆r kα − y α )

k 1 2

kα

kα 1 2

[ m α (V kα ) 2 + U kα ]δ(R k − x)δ(∆r kα − y α )

⋅V kα δ(R k − x)δ(∆r kα − y α )

(95)

k =1

ν

n

¦ ¦ (V

+

mγ

k,m =1 γ =1

⋅∇ Rmγ ) U kα δ(R k − x)δ(∆r kα − y α )

≡ A+ B+C+ D

A, B, C and D can be further derived as n

 kα )2 + V  kα ⋅ (v + ∆vα ) + 1 mα (v + ∆vα )2 + Ukα ] δ(Rk − x)δ(∆rkα − yα ) (Vk − v + v)[ 12 mα (V ¦ 2 k =1

A = −∇x ⋅

1 § α · α = −∇x ⋅ ¨ −qkin − tkin ⋅ (v + ∆vα ) + v[ρα ε α + ρα (v + ∆vα )2 ]¸ 2 © ¹

(96) n

¦ (∆v

B = −∇yα ⋅

kα

k =1

 ka )2 + V  ka ⋅ ( v + ∆vα ) + 1 mα ( v + ∆v α )2 + Ukα ] δ(R k − x)δ(∆r ka − y α ) − ∆vα + ∆vα )[ 12 mα (V 2

1 § α · = −∇ yα ⋅ ¨ − jkin − IJ αkin ⋅ ( v + ∆vα ) + ∆vα [ρα ε α + ρα ( v + ∆vα )2 ] ¸ 2 © ¹

(97) C=

ν

n

¦¦ V

kα

k= 1 β= 1

kα lβ 1

n

⋅ (¦f

kα

+ f2 β )δ(Rk − x)δ(∆r kα − yα ) +

l= 1

n

¦V

kα

⋅ f3δ(Rk − x)δ(∆r kα − yα )

k= 1

(98) D=

n

1 2

=− −

ν

n

¦¦¦ (V

lγ

⋅∇ Rlγ )U kα δ(R k − x)δ(∆r kα − y α )

k =1 l =1 γ =1 n

n

ν

1 2

¦¦¦ (V

1 2

¦¦ (V

kα

kα

− V lγ ) ⋅ f1lγ δ(R k − x)δ(∆r kα − y α )

k =1 l =1 γ =1 n

ν

k =1 γ =1

kα

kα

− V kγ ) ⋅ f 2 γ δ(R k − x)δ(∆r kα − y α )

(99)

48

Y.. P. Chen et al.

With the divergence of heat flux in. Eq. (60), it is seen that C+D =

kα 1 n ν kα lη k kα lη ( ) V V f + ⋅ − yα ) ¦¦ 1 δ ( R − x ) δ ( ∆r 2 k,l =1 η=1

+

1 n ν kα (V kα + V kη ) ⋅ f 2 η δ( R k − x)δ( ∆r kα − y α ) ¦¦ 2 k =1 η=1

= ∇ x ⋅ [q +

n

α pot

+ t ⋅ ( v + ∆v )] + ∇ yα ⋅ [ j + IJ

¦ (V

α pot

kα

α

α pot

α pot

(100) α

⋅ ( v + ∆v )]

− v − ∆v α ) ⋅ f3 δ(R k − x)δ( ∆r kα − y α ) / ∆V α + ( v + ∆v α ) ⋅ f α

k =1

In case the external field is not velocity-dependent, the sum of A, B, C, and D reads ∂ (ρ α E α ) = ∇ x ⋅ ( q α − v ρ α ε α + t α ⋅ ( v + ∆ v α ) ) + ∇ y α ⋅ ( j α − ∆v α ρ α ε α + IJ α ⋅ ( v + ∆v α ) ) ∂t 1 1 −∇ x ⋅ { ρα v( v + ∆v α )2 } − ∇ yα ⋅ { ρ α ∆v α ( v + ∆v α ) 2 } + ( v + ∆v α ) ⋅ f α 2 2

(101) From the conservation equation of mass and balance equation of linear momentum, the total energy equation can be rewritten in terms of internal energy as ∂ α α ∂ (ρ E ) = (ρ α ε α + 12 ρα ( v + ∆v α ) 2 ) ∂t ∂t ∂ §1 · §1 · = (ρ α ε α ) − ∇ x ⋅ ¨ ρ α v ( v + ∆v α ) 2 ¸ − ∇ y α ⋅ ¨ ρα ∆v α ( v + ∆v α ) 2 ¸ ∂t ©2 ¹ ©2 ¹

(

+ ( v + ∆v α ) ⋅ ∇ x ⋅ t α + ∇ yα ⋅ IJ α + f α

(102)

)

Finally, the time evolution of internal energy is obtained as ∂ α α (ρ ε ) +∇x ⋅ ( −qα + vρα ε α ) + ∇y ⋅ ( − j α + ∆vα ρα ε α ) = t α : ∇x (v + ∆vα ) + IJα : ∇y (v + ∆vα ) ∂t (103) or α

ρα

dε α =∇ x ⋅ q α +∇ y α ⋅ j α + t α : ∇ x (v + ∆v α ) + IJ α : ∇ y α (v + ∆v α ) dt

α

(104)

49

A multiscale field theory: Nano/micro materials

where t α : ∇ x ( v + ∆v α ) ≡ tijα

∂ (v j + ∆v αj ) ∂x i

,

IJ α : ∇ yĮ ( v + ∆v α ) ≡ τijα

∂ (v j + ∆v αj ) ∂yiα

(105)

The time evolution of cell-average energy can be found and will be different from the macroscopic equation of conservation of energy. This indicates the macroscopic form of conservation of energy equation no longer holds at nano/micro scale; the energy density of a unit cell is not a homogeneous quantity; and the contribution of the internal motion and deformation of atoms to the evolution of energy density cannot be ignored. 4 Field Description of Stress-strain Relations 4.1 Interatomic potential and interatomic forces Results of atomic-level molecular dynamics simulation depend critically on the interatomic forces. A key issue in atomic-level simulations is therefore the choice of a suitable potential energy function or interatomic force. For the sake of simplicity, this work considers only systems with central force pair potential. Assuming the separate distance of two atoms is d ij and the total potential energy of the system, U, is a function of the atomic positions only, one has n

U = ¦ U(d ij )

(106)

i≠ j

Denote G(dij ) ≡

1 ∂U d ij ∂d ij

(107)

the interatomic force between atom i and j can be obtained as f ij = −

∂U ∂U dij = − ij ij = −G(dij )dij ij ∂d ∂d d

(108)

In the notation of this work, the vectorial relative displacement between atom (k,Į) and atom (l,ȕ) is R kα − R lβ = (R okα − R loβ ) + u(k) − u(l) + ȟ (k, α) − ȟ (l, β)

(109)

Here, R okα and R olβ are the position vector of atom (k,Į) and atom (l,ȕ) at ground state, respectively; u(k) and u(l) are the displacements of the centers of the k-th and the l-th unit cells, respectively; ȟ(k, α) and ȟ(l, β) are the displacements of

50

Y.. P. Chen et al.

atoms (k,α) and (l,β) relative to their unit cell centers, i.e., lattice points, respectively. 4.2 Momentum flux density The temperature in an N-body dynamics systems is generally defined as ∆V 3k B

Tα =

n

¦m

α

 kα )2 δ(R k − x)δ(∆r kα − y α ) (V

(110)

k =1

It is seen that the kinetic parts of momentum flux in eqs. (2-33, 2-34), which are caused by the thermal motion of atoms, are related to temperature. They depend only on the magnitude of the fluctuations of atoms. This implies Į Į tkin + IJkin = − ȖT Į I

(111)

or Į tkin = − Ȗ1 T Į I

(112)

Į IJkin = − Ȗ2TĮ I

(113)

with Ȗ1 +Ȗ 2 = Ȗ Ȗ=

(114)

kB ǻV

(115)

With the identity of δ-function (see Appendix) 1

³ δ[R λ + R (1 − λ) − x]δ[∆r k

l

kξ

λ + ∆r lη (1 − λ) − y α ]dλ

0

∞

1 = ¦ [(R k − R l ) ⋅ ∇ x + (∆r kξ − ∆r lη ) ⋅∇ y α ]m −1 δ(R k − x)δ(∆r kξ − y α ) m =1 m!

(116)

it is seen from Eqs. (44) and (45) that the potential parts of momentum fluxes are functions of a series of high order gradients, with the zero-th order terms as

(t )

α 0 pot

=−

kξ 1 n ν (R k −R l ) ⊗ f1lη δ(R k − x)δ(∆r kξ − y α ) ¦ ¦ 2 k,l =1 ξ,η=1

(117)

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A multiscale field theory: Nano/micro materials

(IJ )

0 α pot

=−

kξ 1 n ν kξ lη k kξ lβ ( ) ∆ −∆ ⊗ − yα ) r r f ¦¦ 1 δ( R − x)δ( ∆r 2 k,l =1 ξ, η=1

1 n ν kξ − ¦ ¦ (∆r kξ −∆r kη ) ⊗ f2 η δ(R k − x)δ(∆r kξ − y α ) 2 k =1 ξ, η=1

(118)

Notice that the sum of zero-th momentum flux, t and τ, is the atomic virial stress. 0 α 0 With the expressions of interatomic forces, ( tpot ) and ( IJpotα ) can be expressed as

(t

(x, t) ) = 0

α pot

1 2(∆V) 2

υ

³ ¦ G(d

αβ

)(x − x′) ⊗ d αβ dV(x′)

(119)

G(d αβ )y αβ′ ⊗ d αβ dV(x′) +

1 υ ¦ G(yαβ )y αβ ⊗ y αβ (120) ∆V β =1

β=1

V( x ′ )

(IJ

α pot

(x, t) ) = 0

υ

1 2(∆V) 2

³ ¦ β =1

V( x ′ )

and the first order terms as § 1 1 ( t (x, t) ) = 4 ∇ x ⋅ ¨¨ (∆V) 2 ¨ ©

· ¸ G(d )(x − x′) ⊗ d dV(x′) ¸ ³ (x − x′)¦ β=1 ¸ V( x ′ ) ¹ § · υ 1 ¨ 1 ¸ αβ′ αβ αβ ′ ′ y G(d )(x − x ) ⊗ d dV(x ) ¸ + ∇ yα ⋅ ¨ ¦ 2 ³ 4 ¨ (∆V) V( x′) β=1 ¸ © ¹

1

α pot

§ 1 1 ( IJ (x, t) ) = 4 ∇x ⋅ ¨¨ (∆V) 2 ¨ © 1

α pot

1 + ∇ yα 4

υ

³ V( x ′ )

§ ¨ 1 ⋅¨ 2 ¨ (∆V) ©

αβ

αβ

(121)

· υ ¸ αβ αβ′ αβ ′ ′ (x − x )¦ G(d )y ⊗ d dV(x ) ¸ β =1 ¸ ¹

· ¸ αβ′ αβ αβ′ αβ ′ y y d x G(d ) dV( ) ⊗ ³ ¦ ¸ β =1 ¸ V( x ′ ) ¹ υ

(122)

§ 1 υ αβ · 1 y G(yαβ )y αβ ⊗ y αβ ¸ + ∇ yα ⋅ ¨ ¦ 4 V ∆ β =1 © ¹

where d αβ (x, x′) ≡ xo + u(x) + y αo + ȟ (x, α) − [x′o + u(x′) + y βo + ȟ (x′, β)] y

αβ

≡ y − y = y + ȟ (x) − [y + ȟ (x)] α

β

α o

α

β o

β

y ′ ≡ y − y ′ = y + ȟ (x) − [y + ȟ (x′)] αβ

α

β

α o

α

β o

β

(123) (124) (125)

52

Y.. P. Chen et al.

d αβ ≡ d αβ

(126)

yαβ ≡ y αβ

(127)

Eqs. (119) to (122) are the zero-th and the first order nonlinear nonlocal constitutive relations for the potential momentum flux density. The independent variables are the lattice displacement u(x , t ) and the relative atomic displacements ȟ(x, α, t )  ȟ α (x, t) . 4.3 Linear local momentum flux density To derive the linear constitutive relations for the potential momentum flux density, one may make the assumption of infinitesimal deformation, i.e., d αβ − d αo β → 0 , and hence the internal atomic force density can be written as fintα (x) ≈

1 (ǻV)2

ν

³ ¦ (c

αβ o

V(x ′ ) β=1

)

(x0 , x′0 ) + c1αβ (x0 , x′0 ) ( d αβ − d oαβ ) dV(x′)

(128)

αβ ′ where d αβ 0 = d ( x 0 , x 0 ) are separate vector between two atoms at ground state, αβ αβ αβ co (x0 , x′0 ) = c1 d o , and c1αβ (x0 , x′0 ) are the interatomic force constants, can be computed from quantum mechanical calculations, and are functions of the distance between as well as the types of atom α and atom β in question. For the sake of sim′ = cαβ ′ plicity, one may denote c1αβ′ = c1αβ (x 0 , x′0 ) , c1αβ = c1αβ (x 0 , x 0 ) and cαβ o o (x0 , x0 ) αβ and cαβ 0 = c0 (x0 , x0 ) . Using the linearized atomic force, Eqs. (119) to (122) become

(t

α pot

1 2(∆V) 2

(x, t) ) = 0

³ ¦ (x − x′) ⊗ ( c υ

β=1

)

′ + c1αβ′ ( d αβ − d αβ ′ o ) dV( x )

αβ o

(129)

V( x′ )

(t

α pot

1 ∇x ⋅ ³ 4(∆V)2 V( x ′ )

(x, t) ) = 1

+

(IJ

α pot

(x, t) ) = 0

υ

¦ β=1

1 ∇α⋅ 4(∆V) 2 y V(³x′ )

1 2(∆V) 2

υ

³ ¦ β=1

)

(

′ + c1αβ′ ( dαβ − d αβ ′ (x − x′)(x − x′) ⊗ cαβ o o ) dV( x ) υ

¦y

)

(

′ (x − x′) ⊗ cαβ ′ + c1αβ′ ( d αβ − dαβ ′ o o ) dV( x )

αβ

β=1

(130)

)

(

′ + c1αβ′ ( d αβ − d αβ ′ y αβ′ ⊗ cαβ o o ) dV( x )

(131)

V( x ′ )

1 υ αβ + ¦ y ⊗ coαβ + c1αβ ( y αβ − y oαβ ) 2∆V β=1

(

)

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A multiscale field theory: Nano/micro materials

(IJ

α pot

(x, t) ) = 1

)

(

υ 1 ′ + c1αβ′ ( d αβ − d αβ ′ ∇ x ⋅ ³ (x − x′)¦ y αβ′ ⊗ cαβ o o ) dV( x ) 2 4(∆V) β =1 V( x′ )

)

(

+

υ 1 ∇ yα ⋅ ³ ¦ y αβ′ y αβ′ ⊗ coαβ′ + c1αβ′ ( d αβ − d oαβ ) dV(x′) 2 4(∆V) V( x ′ ) β =1

+

υ 1 ∇ yα ⋅ ¦ y αβ y αβ ⊗ coαβ + c1αβ ( y αβ −y oαβ ) 4∆V β =1

(

(132)

)

Note that material properties should make the ground state stresses vanish, i.e., α α (0) = 0 . Hence, the expressions of potential momentum flux at ground tpot (0) = IJ pot state α (0) = tpot

1 2(∆V) 2

υ

³ ¦ (x

o

− x′o ) ⊗ coαβ′dV(x′)

(133)

β=1

V( x ′ )

α IJ pot (0) =

1 2(ǻV)2

ν

³ ¦y

αβ o

⊗ coαβ′dV(x′) +

β=1

1 ν αβ ¦ y o ⊗ coαβ ∆V β=1

(134)

V( x ′ )

′ αβ and the ground can be used for validation, once the material parameters cαβ 0 , c0 αβ state structural parameters x o and y o are obtained from quantum mechanical calculations. If the nonlocal effect is further neglected, i.e., consider only the interactions between unit cells in a close neighborhood are considered, there results

u(x) − u(x′) ≈ u, x ⋅ (xo − x′0 )

(135)

ȟ (x, β) − ȟ (x′, β) ≈ ȟ ⋅ (xo − x′0 )

(136)

ȟ (x, α) − ȟ (x, β)  Ȗ ⋅ (y − y )

(137)

β ,x

αβ

α o

β o

and hence x − x′ − (xo − x′o ) = u(x) − u(x′) ≈ u, x ⋅ (x o − x′0 )

(138)

y α − y β − (y 0α − y β0 ) = ȟ (x, α) − ȟ (x, β)  Ȗ αβ ⋅ (y αo − y βo )

(139)

y α − y ′β − (y α0 − y ′0β ) = ȟ (x, α) − ȟ (x′, β) = ȟ (x, α) − ȟ (x, β) + ȟ (x, β) − ȟ (x′, β) ≈ Ȗ αβ ⋅ (y αo − y βo ) + ȟ β, x ⋅ (xo − x′0 ) β αβ ′ ⋅ (y oα − y βo ) d αβ − d αβ 0 ≈ (u ,x + ȟ , x ) ⋅ ( x o − x 0 ) + Ȗ

(140) (141)

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Y.. P. Chen et al.

The zero-th order linear local potential momentum flux density can be then expressed as υ

t α (x, t) = −γ1T α I + ¦ (A1 : u, x + A 2 : Ȗ αβ + A 3 : ȟ β, x )

(142)

β=1

υ

IJ α (x, t) = −γ 2 T α I + ¦ (B1 : u, x + B 2 : Ȗ αβ + B3 : ȟ ,βx )

(143)

β=1

d Įȕ o ,

It is noticed that

x o − x ′o

Į ȕ and yĮȕ o =y o −y o

are material constants;

d , x − x ′ and y − y are up to first order in u ,x , ȟ , and Ȗ Įȕ . With Eqs. (138) to Įȕ

Į

ȕ ,x

ȕ

(141), it is found that the first order terms of momentum flux density are the strain gradient terms. Therefore, Eqs. (142) and (143) represent the linear local form of zero-th order homogeneous and inhomogeneous momentum flux density, respectively, and the sum of the two is the field representation of atomic virial stress. One can also write them in the tensor notation as: υ

2 2 α ε pq + A 3mnpq ε3pq ) tmn (x, t) = −γ1 δmn T α + ¦ (A1mnpq ε1pq + A mnpq

(144)

β=1 υ

2 2 α εpq + B3mnpq ε 3pq ) (x, t) = −γ 2 δ mn T α + ¦ (B1mnpq ε1pq + Bmnpq τmn

(145)

β=1

where ε1pq = u p,q ,

2 αβ εpq = γ pq ,

ε3pq = ξβp,q

(146)

and A1mnpq , A 2mnpq , A 3mnpq , B1mnpq , B2mnpq , B3mnpq are material constants, which can be expressed in terms of ground state structural parameters, x o − x ′0 and yĮo − yȕo , and material parameters, c1Įȕ . 4.4 Strain measures The field representation of momentum flux involves temperature, lattice deformation and relative atomic deformation. The linear local forms of momentum flux are expressed in terms of temperature and u ,x (x) , ȟ ,βx (x) , and Ȗ Įȕ (x) . Here, u ,x (x) , ȟ ,βx (x) , and Ȗ Įȕ (x) may be called the lattice strain, relative atomic strain, and atomic-bond strain, respectively. They are the lattice deformation gradient, relative atomic deformation gradient and relative atomic-bond stretch, and can be illustrated through Fig. 10.

55

A multiscale field theory: Nano/micro materials

Fig. 10. Illustration of strain measures in the formulated new field theory.

5. Numerical Examples 5.1 Numerical implementation Balance equations and constitutive relations are the two essential parts of a continuum field theory, with which the solution of a continuum problem under an external field can be uniquely and completely determined. Decomposing the atomic displacements into lattice and internal deformation, it is found that the atomic stress-strain relation as well as the balance equations can be fully described in terms of filed variables, of which the independent variables are temperature, the lattice displacement and the internal displacements. Among the balance equations, the balance equation of linear momentum is given by Eq. (82). Recalling the general procedure of numerical implementation of a continuum theory, one substitutes the constitutive relations into the balance equations to obtain a set of partial differential equations, the nodal force – nodal displacement relations, to solve for the nodal displacements. However, in this work, the force-displacement relation that is used to derive the constitutive relations can be directly derived from the interatomic potential, they can, therefore, be put directly into the balance equations. With the relation of momentum flux density and internal force density that can be found in Eq. (47), it is found that ρα

d ( v + ∆v α ) + γ1∇ x ⋅ T α = fintα + f α dt

(147)

For constant temperature and small deformation, the governing equation of a close system then becomes ρα

d 1 ( v + ∆v α ) = dt (∆V)2

ν

³ ¦ (c

V( x ′ ) β=1

αβ o

)

+ c1αβ ( d αβ − d oαβ ) dV(x′)

(148)

which is a linear nonlocal differential equation corresponding to the linear nonlocal stress-strain relation obtained in Section 4.

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Y.. P. Chen et al.

5.2 Phonon dispersion relations There are a number of material features, such as material hardness and symmetry, can be explained by atomic structure. There are, however, a large number of technically important properties that can only be understood on the basis of lattice dynamics. These include: temperature effect, energy dissipation, sound propagation, phase transition, thermal conductivity, piezoelectricity, thermo-mechanicalelectromagnetic coupling properties, etc. The atomic motions, revealed by those features, are not random. In fact they are determined by the forces that atoms exert on each other, and most readily described in terms of traveling waves. Those waves are the normal modes of vibration of the system. The quantum of energy in an elastic wave is called a phonon. The frequency-wave vector relationship of phonons is called phonon dispersion relation, which can be determined through experimental measurements, or through first principles approaches or phenomenological models. A delicate difference in phonon dispersion relations may indicate a delicate difference in the crystal structure. Through phonon dispersion relations, the dynamic characteristics of an atomic system can be represented [2], the applicability of a phenomenological model can be examined [21], interatomic force constants can be computed [41], various involved material constants can be determined [27]. In this work, phonon dispersion relations for ferroelectric crystalline material BiScO3 have been calculated based on the formulated field theory for constant room temperature, in which the interatomic potential and material parameters are obtained through fitting the energy surface to quantum mechanical calculations. Fig. 11(a) are the calculated phonon dispersion relations by the field theory, while Fig. 11(b) by an atomic-level molecular dynamics code GULP, a program for performing atomistic simulations on 3D periodic molecular solids and ionic materials – [42]. Here, the unit for frequency is cm 1, the speed of light divided by meter; the reduced wave vector is the wave vector normalized by the first reciprocal lattice vector lying along the direction of the wave vector. The results by the two methods are both displayed from the reduced wave vector q → 0 to q = 1. The reason that the phonon curves cannot be displayed at q = 0 is referred to [43]. Compared the results of the two methods, one can see that they are in almost exact agreement.

A multiscale field theory: Nano/micro materials

Fig. 11. Phonon dispersion relation of ferroelectric crystalline material BiSrO3 .

57

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Y.. P. Chen et al.

6. Summary and Discussions 6.1 About the conservation equations By decomposing atomic displacements, momentum and heat fluxes into homogeneous and inhomogeneous parts, the field representation of conservation laws at atomic scale has been formulated. The mathematical representations for conservation of mass, balance of linear momentum and conservation of energy are obtained as follows dρ α + ρα (∇ x ⋅ v + ∇ y α ⋅ ∆v α ) = 0 dt ρα

d ( v + ∆v α ) = ∇ x ⋅ t α + ∇ y α ⋅ IJ α + f α dt

ρα

dε α =∇ x ⋅ q α + ∇ y α ⋅ j α + t α : ∇ x (v + ∆v α ) + IJ α : ∇ y α (v + ∆v α ) dt

(74)

(82)

(104)

and the balance law of angular momentum at atomic scale is identically satisfied. It is seen that (1) The field representations of conservation equations were formulated within the framework of atomic N-body dynamics. They are the exact time evolution laws of conserved quantities in MD simulations. (2) Recall that in Micromorphic theory (Eringen and Suhubi [1964], Eringen [1999]), the balance laws for mass, linear momentum, generalized spin, and energy were obtained as dρ + ρ∇ x ⋅ v = 0 dt

(149)

ρ

dv = ∇x ⋅ t + f dt

(150)

ρ

dij = ∇ x ⋅ m + Ȧ ⋅ ρ i ⋅ Ȧ T + ( t − s )T + l dt

(151)

ρ

dε = t : ∇ v + m # ∇ Ȧ + Ȧ : ( s − t )T + ∇ ⋅ q dt

(152)

where ϕ is generalized spin, ω the gyration tensor and l external couple. Assuming that the inner atomic structure is a continuum and thus ∆v kα = Ȧ ⋅ ∆r kα , one will

A multiscale field theory: Nano/micro materials

59

find that the obtained balance laws in this paper can be reduced to the balance laws in Micromorphic theory upon such continuum assumption and cell averaging [22-24]. Note that because the atomic motion and deformation as well as momentum and heat fluxes are decomposed into homogeneous and inhomogeneous parts, the higher order moment stress is avoided in this paper. Also, if the atomic structure of the primitive unit cell and also the relative motion and deformation within this cell are ignored, i.e., the structural unit of the crystal is considered as point mass, the obtained balance equations can then be reduced to that of continuum mechanics. (3) For single component system the obtained balance laws are identical with that obtained by Irvine and Kirkwood in [29]. The averaged quantities are time-interval averages. If one uses ensemble averages, it is straightforward to prove that Liouville’s theorem would result in the same form as in Eq. (71) for the time evolutions in equilibrium statistical mechanics; the Boltzmann transport equation as well as BBGKY theory would also yield the same for conserved properties in non-equilibrium statistical mechanics [22-24, 33]. (4) While in molecular dynamics simulations, some physical phenomena may depend on the initial condition of the molecular dynamics model; it is noticed that the obtained mathematical representation of conservation laws is independent of the initial conditions. (5) The formulation has proved that for multi-element system the local conservation equations at atomic scale differ from that at macroscopic scale and the contribution of the internal motion and deformation of atoms cannot be ignored. 6.2 About the stress and stress-strain relations The field representation of momentum flux density is derived within the framework of atomic N-body dynamics. Three strain measures and the momentum flux density–strain relations are obtained. Major considerations and conclusions regarding the formulation may be summarized as follows: (1) The obtained momentum flux exactly represents the momentum flux in an atomic N-body dynamics model. Both the atomic-level momentum flux and the atomic displacements can be fully represented in term of field variables: temperature, lattice deformation and relative atomic deformation. All material constants involved can be obtained through the atomistic formulation. (2) This work has shown that the stress in the conventional continuum description is not the momentum flux density in an atomic N-body dynamics model; it is only the homogeneous part of momentum flux density summing over at least the volume of a primitive unit cell. Decomposing the momentum flux into homogeneous and inhomogeneous parts, one can establish the connection between the atomic momentum flux density and the continuum stress. (3) The formulations have shown that momentum flux density – strain relation, which may be referred to as atomic stress-strain relation, is nonlinear and nonlocal in displacements, and involves higher order gradients. In the case only the average

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stress of a the specimen is considered, or in the case of a homogeneous deformation, the gradient terms shall be disappeared, and the zero-th order term solution is equivalent to the complete series solution, i.e., it is identical to the virial theorem. (4) The three strain measures are obtained for the linear local constitutive relation. One may prove that the nonlinear nonlocal relation can also be expressed in terms of the temperature and the three strain measures. 6.3 Physical picture of the new field theory The atomistic expressions of physical quantities can be obtained from atomistic N-body dynamics. Those formulations actually contain all the information of the correlations and time evolutions of those quantities. The main goal of MD simulation is to understand physical phenomena and the underlying mechanism by numerical analysis of those quantities, the correlations and the time evolution. It is the suggestion of this work that a field-theory-based simulation can also achieve this objective.

Fig. 12. Physical Picture of the Multiscale Field Theory.

The physical picture of the new field theory is an embedded field theory, as shown in Fig. 12. Each point x in the field corresponds to a lattice point R k in the atomic model. Embedded in each point x is an inner structure with discrete ν atoms. The position of α-th atom in the k-th unit cell is R kα = R k + ∆r kα in phase space, x+y Į in physical space. The displacement of point x, u(x), gives rise to the homogeneous and continuous lattice deformation with length scale from nano to macroscopic. The displacements of embedded atom α, ȟ(x,Į) (α =1, 2,...ν), result in relative atomic deformation within the inner structure, describe the inhomogeneous and non-continuum atomic behavior, and the length scale is less than a nanometer. The total atomic displacement is u(x) +ȟ(x,Į) in physical space, corresponding to the u(k)+ȟ(k,Į) in the discrete atomic model in phase space, with the time scale of u(x) at audible frequency region and ȟ(x,Į) at infrared.

A multiscale field theory: Nano/micro materials

61

The only difference between the MD and that of the new embedded field theory is that the lattice deformation is assumed to be continuous with respect to x in the field theory. This assumption requires that the smallest ǻx , i.e., mesh size, in the simulation should be no less than the lattice spacing, i.e., the size of a unit cell. In the case ǻx equals lattice spacing, the formulated theory has same number of degrees of freedom, and scales the same as MD. However, in the computer simulation of the field theory, constitutive relations replace the need to evaluate the interatomic forces at each instantaneous time step. This will significantly reduce the computational cost. Also, the procedure of taking time average or ensemble average in molecular dynamics or statistical mechanics simulation is eliminated. Adaptive mesh can be used in regions of different concerns. The obtained field theory is thus computationally much more efficient than MD, and will be applicable to both large length and time scale phenomena. 6.4 Comparison with other theories A comparison on the geometric models, the basic assumptions, and the applicability and limitations of classical continuum mechanics, Micromorphic theory, the formulated new field theory and MD are summarized in Table. 1. Table 1. Comparison of classical continuum theory, Micromorphic theory, the new theory and MD.

Classic continuum mechanics views a crystal as a homogeneous and continuous medium. The basic structural unit of the crystal is taken without structure and is idealized as point mass, and the internal deformation is ignored. Its application is thus limited to homogeneous or macroscopic problems.

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In Micromorphic theory, a material body is envisioned as a continuous collection of deformable particles; each particle has finite size and 9 internal degrees of freedom describing the stretches and rotations of the particle. Compared with classical continuum mechanics, Micromorphic theory extends the application region of a continuum theory. However, the assumption of a continuous structure and deformation of the particle makes it difficult to describe complex crystalline materials. The formulated theory views a crystalline material as a continuous collection of lattice points, embedded with each lattice point is an inner structure, a group of bonding atoms. This dramatically expands further the application region of Micromorphic theory. While it retains most of the features of an atomic many-body dynamics model, the new field theory is computationally much more efficient than molecular dynamics simulations. Acknowledgement The support to this work by National Science Foundation under Award Numbers CMS-0301539 and CMS-0428419 is gratefully acknowledged. Appendix Define δ(λ; k, l, ξ, η, α) ≡ δ[ R k λ + R l (1 − λ ) − x]δ[ ∆r kξ λ + ∆r lη (1 − λ ) − y α ]  δ(λ )

(A1)

∆ (k, ξ, α) ≡ δ[ R k − x]δ[ ∆r kξ − y α ]  ∆ = δ(1) ,

(A2)

A ≡ (R k − R l ) ⋅∇ x

(A3)

B ≡ (∆r kξ − ∆r lη ) ⋅∇ y α ⋅

It is readily to verify that dδ = −(A + B)δ , dλ dδ = −(A + B)δdλ 1

³λ 0

(A4)

1

1

1

dδ 1 dλ = ³ λdδ = λδ 0 − ³ δdλ = δ(1) − ³ δdλ , dλ 0 0 0

1

1

0

0

³ δdλ = ∆ + (A + B)³ λδdλ ,

(A5)

(A6)

63

A multiscale field theory: Nano/micro materials 1

1

0

0

n n −1 ³ λ dδ = ∆ − n ³ λ δdλ .

(A7)

It leads to 1

³λ

1

n −1

δdλ =

0

∆ 1 + (A + B) ³ λ n δdλ . n n 0

(A8)

One may then prove that 1

1

0

0

³ δdλ = ∆ + (A + B)³ λδdλ 1

∆ 1 = ∆ + (A + B){ + (A + B) ³ λ 2 δdλ} 2 2 0 1

1 1 = ∆ + (A + B)∆ + (A + B) 2 ³ λ 2 δdλ , 2 2 0

(A9)

= ⋅⋅⋅⋅ ∞

1 (A + B) n −1 ∆ n =1 n!

=¦

i.e., 1

³ δ[R λ + R (1 − λ) − x]δ[∆r k

l

kξ

λ + ∆r lη (1 − λ) − y α ]dλ

0

∞

1 = ¦ [(R k − R l ) ⋅ ∇ x + (∆r kξ − ∆r lη ) ⋅∇ y α ]m −1 δ(R k − x)δ(∆r kξ − y α ) m =1 m!

(A10)

and 1

∞

1

³ δ ( R λ + R (1 − λ) − x ) dλ = ¦ m! ( (R k

l

m =1

0

k

− R l ) ⋅∇ x )

m −1

δ( R k − x )

(A11)

References [1] [2] [3] [4]

Dove MT, Introduction to Lattice Dynamics, Cambridge University Press, Cambridge, 1993. Cochran W (1973), “The dynamics of atoms in crystals”, Edward Arnold Limited, London. Rudd RE, Broughton JQ (1998), “Coarse-grained molecular dynamics and the atomic limit of finite elements”, Phys. Rev. B 58, R5893-6. Rudd RE, Broughton JQ (1999), “Atomistic simulation of MEMS resonators through the coupling of length scales”, J. Model. and Sim. of Microsys. 1, 26.

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[30] Hardy RJ, “Formulas for determine local properties in molecular dynamics simulations: shock waves”, J. Chem. Phys. 76, 622-628 (1982). [31] Hardy RJ, Formulas for determining local properties in molecular-dynamics simulations: Shock waves, J. Chem. Phys. 76(1) (1982) 622-628. [32] Ranninger J, Lattice thermal conductivity, Physical Review, 140 (1965) A2031-A2046. [33] Kreuzer HJ, Nonequilibrium thermodynamics and its statistical foundations, Clarendon Press, (1981). [34] Hoover WG, Molecular Dynamics, Springer-Verlag, Berlin 1986. [35] Hoover WG, Computational Statistical Mechanics, Elsevier, Amsterdam 1991. [36] Huang K, Statistical Mechanics, Willey, New York, 1967. [37] Kittel C, Introduction to Solid State Physics, John Wiley & Sons Inc, 1967. [38] Cochran W (1966), “Theory of phonon dispersion curves”, Phonons in perfect lattice and in lattice with point imperfections, edited by R. W. H. Stevenson, Plenum Press, New York, pp. 53-72. [39] Cochran W, Cowley RA (1967), Encyclopedia of Physics, 25/2a, Springer Berlin, Heidelberg, New York. [40] Dick BG, Overhauser AW (1958), “Theory of the dielectric constant of alkali halide crystal”, Phys. Rev. 112, 90-103. [41] Ghosez P, First principles study of the dielectric and dynamical properties of barium titanate, Ph.D dissertation, Universite Catholique Louvain 1997. [42] Gale JD, GULP – a computer program for the symmetry adapted simulation of solids, JCS Faraday Trans., 93, 629 (1997). [43] Gonze X, Lee C, Dynamic matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory, Physical Review B, 55, 10355 (1997).

Combined loading rate and specimen size effects on the material properties Z. Chen *, Y. Gan, L.M. Shen Department of Civil and Environmental Engineering University of Missouri-Columbia, Columbia, MO 65211-2200, USA

Abstract The current interests in developing multiscale model-based simulation procedures have brought about the challenging tasks of bridging different spatial and temporal scales within a unified framework. However, the research focus has usually been on the scale effect in the spatial domain with the loading rate being assumed to be quasi-static. Although material properties are rate-dependent in nature, little has been done in understanding combined loading rate and specimen size effects on the material properties at different scales. On the other hand, the length and time scales that can be probed by the molecular level simulations are still fairly limited due to the limitation of existing computational capability. Based on the experimental and computational capabilities available, therefore, attempts have been made recently to formulate a hyper-surface in both spatial and temporal domains to predict combined size and rate effects on the mechanical properties of engineering materials. It appears from the preliminary results, with the use of tungsten and diamond specimens, that the proposed procedure might provide an effective means to bridge different spatial and temporal scales in a unified multiscale modeling framework, and facilitate the application of nanoscale research results to engineering practice. To provide a foundation for the future study of the combined rate and size effects on fracture mechanics, the recent research results are presented in this chapter. Keywords: Size effect; Rate effect; Multiscale simulation; Hypersurface.

1. Introduction As can be seen from the open literature, much research has been conducted to investigate the rate-dependence and size-dependence of material properties, respectively. However, the focus has usually been on the scale effect in the spatial domain with the loading rate being assumed to be quasi-static, as shown by the representative references [1-5, among others]. The recent interests in developing multiscale model-based simulation procedures [6-8, among others] have brought about the challenging tasks of bridging different spatial and temporal scales within a unified framework. Although material properties are rate-dependent in nature,

*

Corresponding author. E-mail address: [email protected] (Z. Chen).

67 G.C. Sih (ed.), Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano, 67–84. ” 2007 Springer.

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little has been done in understanding combined loading rate and specimen size effects on the material properties at different scales. On the other hand, the molecular level investigations have been conducted recently to explore the rate effect, the size effect, and the mechanical responses of thin films [9-13, among others]. However, the length and time scales that can be probed by the molecular level simulations are still fairly limited due to the limitation of computational capability. A non-trivial question therefore arises: Can the current experimental facilities allow us to verify the molecular level simulation results? As can be found from the open literature related to the impact mechanics and shock physics [14-18, among others], not only the loading rate but also the specimen size used in the current molecular dynamics (MD) simulation can not be handled by the existing experimental techniques. Usually, a specimen of finite size is employed in the bar and plate impact experiments to investigate the rate-dependent mechanical properties under the loading rate which is well below that used in the MD simulation reported so far. Based on the experimental and computational capabilities available, hence, an attempt has been made recently to formulate a hyper-surface in both spatial and temporal domains to predict combined size and rate effects on the mechanical response of tungsten [19]. It appears from the preliminary results that the proposed procedure might provide an effective means to bridge different spatial and temporal scales in a unified multiscale modeling framework, and facilitate the application of nanoscale research results to engineering practice. To better understand combined size and rate effects, and to further demonstrate the features of the proposed hyper-surface, both tungsten and diamond specimens of various sizes under various loading rates are considered here with certain loading paths. Especially, the relationship between the model parameters and the Weibull probability theory is explored for diamond specimens to provide a foundation for integrated analytical, experimental and numerical study in the future. 2. Formulation of a Hyper-Surface in Spatial and Temporal Domains As shown in an asymptotic scaling analysis without considering the rate effect [1], the relationship between the nominal strength ı N and different sizes D of geometrically similar structures exhibits a two-sided asymptotic support, namely,

­σ0 = σ u D ≤ D u ° ° °log σ0 = log σm + ( log σu − log σm ) ⋅ ° ° ª § π log D − log Du · º ° «1 − sin ¨ ¸» ® ¬ © 2 log D m − log Du ¹ ¼ ° ° D u < D < Dm ° ° °σ0 = σ m D m ≤ D ° ¯

(1)

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the small-size asymptotic limit and large-size asymptotic limit, in the logD-logı N space. Hence, a simple set of equations could be chosen to represent the size effect on the quasi-static strength ı 0 in the spatial domain while ı u is the ideal ultimate strength as they are shown in Eqs. (1). Note that D u is the specimen size and D m is the minimum macro-scale size beyond which the strength ı m becomes size-independent. As can be seen from Eq. (1), the slope of the size-dependent portion with D u < D < D m is given by

d ( logı 0 ) = d ( logD ) § ʌ logD − logD u · ʌ ( logı u − logı m ) cos ¨ − ¸ 2 ( logD m − logD u ) © 2 logD m − logD u ¹

(2)

which can be normalized to be

d ( logı 0 )

( logı u − logı m ) = d ( logD ) ( logD m − logD u ) /

§ ʌ logD − logD u · ʌ − cos ¨ ¸ 2 © 2 logDm − logD u ¹

(3)

As can be found from Eq. (3), the normalized slope could be fully determined with the small-size asymptotic limit and large-size asymptotic limit, and is in the range ʌ between − and zero. Although the question on a reasonable small-size asymp2 totic limit remains open [1], the proposed formulation might provide a simple means to characterize the size-dependence of certain materials under quasi-static loading conditions. To describe the dependence of the material strength on the strain rate, a simple model proposed by Cowper and Symonds [20] is adopted as follows:

ı ( İ ) ıo

1/q

§ İ p · =1+ ¨ ¸ © İ r ¹

(4)

in which İ denotes the plastic strain rate, ı o is the quasi-static strength, and İ r and q are two model parameters that can be determined with two experimental p

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data points of ı ( İ ) . The size effect on the model parameters are not considered in the original model. As compared with the plastic strain, the elastic strain can be p neglected so that İ = İ could be assumed. As shown in [10, 11], however, there exists a critical strain rate for single-crystal metals, below which the material strength becomes rate-independent. The critical strain rate is increased with the decrease of the specimen size. In other words, the model parameters İ r and q in Eq. (4) could be assumed to be size-dependent if this model is used to predict the rate-dependent strength with the strain rate above the critical strain rate. Thus, a three-dimensional hyper-surface with respect to both spatial and temporal domains could be formulated to describe combined size and rate effects on the material strength, as below. Based on Eq. (4), the hyper-surface is assumed to have a two-sided asymptotic support, namely, the small-size asymptotic limit and large-size asymptotic limit, in

ı − ıo İ space. The small-size asymptotic limit and large-size − log ıo İ r asymptotic limit are represented by ( İ rs ,ı os ,q s ) and ( İ rl ,ı ol ,q l ) , respectively,

the log

which can be determined by continuum and molecular level studies. The size dependence of model parameters İ r and q can therefore be described as follows:

§ ʌ logD − logD rs · logİ r − logİ rl =1 − sin ¨ ¸ logİ rs − logİ rl © 2 logD rl − logDrs ¹

(5.1)

§ ʌ logD − logD rs · q − ql =1 − sin ¨ ¸ qs − ql © 2 logD rl − logD rs ¹

(5.2)

for D rs < D < D rl with D rs and D rl being the specimen sizes at the small-size asymptotic limit and large-size asymptotic limit, respectively. For D ≥ D rl , we have İ r = İ rl and q = q l , while for D ≤ D rs , we have İ r = İ rs and q = qs. It follows from Eqs. (4) and (5) that the three-dimensional hyper-surface takes the form of

ª § 1 ·º   r ( D)) ¸»  ) = ı o ( D ) «1+10** ¨¨ logİ-logİ ı ( İ,D ( ¸ © q (D) ¹ ¼» ¬«

(6)

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or

 ) = logı o ( D ) logı ( İ,D ª § 1 ·º + log «1+10** ¨¨ logİ − logİ r ( D ) ) ¸¸ » ( © q (D) ¹ ¼» ¬«

(7)

in the logİ − logD − logı space. In Eq. (6) or (7), İ r ( D ) and q ( D ) are determined from Eq. (5), and ı 0 ( D ) is defined by Eq. (1) because the size-dependent quasi-static strength is rate-independent. To characterize the hyper-surface, integrated molecular and continuum investigations must be performed to determine the small-size asymptotic limit and large-size asymptotic limit, as demonstrated in the next section. 3. Demonstration of the Proposed Procedure 3.1 Tungsten specimens Based on the recent research results on multiscale model-based simulation of tungsten (W) thin film delamination from silicon substrate [13, 21], combined loading rate and specimen size effects on the mechanical properties of W are explored with the use of the proposed hyper-surface. The MD simulation is performed first to find the small-size asymptotic limit, and the existing continuum level data are employed to determine the large-size asymptotic limit. To determine the small-size asymptotic limit, the computational set-up for the MD simulation of single-crystal W specimen under uniaxial tensile loading is shown in Fig. 1. The three-dimensional simulation super-cell consists of two parts. One part is referred to as the active zone in which the atoms move according to the interactions among the neighboring atoms; the other part, wrapped by the boxes as presented in Fig. 1, is referred to as the boundary zone where the atoms are assigned a fixed rigid body velocity. The dimension of active zone is indicated by H, D and W, while the size of each boundary zone in the z-direction is 2ao with ao being the lattice parameter of W. The crystal orientation of (x[1 0 0], y[0 1 0], z[0 0 1]) is considered with the periodic boundary conditions (PBCs) being imposed along both the x- and y-directions. In the simulation, all atoms are initially placed at their equilibrium positions at the temperature of 298 K. Those atoms in the boundary zone are then fixed. After the system has equilibrated for a certain period, constant velocities with the same magnitude and opposite direction are assigned to the atoms in the top and bottom boundary zones, respectively, to simulate a displacement-controlled uniaxial tensile loading in the z-direction. A velocity scaling technique is employed to maintain a constant temperature of 298 K. The Embedded Atom Method (EAM) [22] is used to model the interatomic potential among W atoms. The corresponding model

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parameters for W are based on the work in [23]. The method applied to integrate the equations of motion is the 6-value Gear predictor-corrector algorithm. The largest time step size that can keep the total system energy remaining constant in the adiabatic simulation for the motion of W atoms with the EAM potential is used in the numerical study. A time step size of 0.5 fs is chosen after performing several adiabatic simulations for the W atoms at different initial temperatures up to 2000 K.

Fig. 1. Computational model of single crystal W block under uniaxial tensile loading.

Stress calculation in the MD simulations has been the focus of many investigations over the past years. In this study, the formulations employed to calculate atomic-level stresses are motivated by the work in [24]. At each atom, the local stress tensor is defined to be

1 ȕi = − Ωi

Nn

¦f

ij

⊗ rij

(8)

j>i

in which i refers to the atom considered and j refers to the neighboring atom, rij is the position vector between atoms i and j, Nn is the number of neighboring atoms

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surrounding atom i, Ωi is the volume of atom i, and fij is the force vector on atom i due to atom j. The global continuum stress tensor is then determined by *

1 N ı = * ¦ ȕi N i

(9)

where N* represents the total number of atoms in a representative volume of continuum. Since the W block is under very large deformation, true strain ε is used in this

§ L· ¸ where Lo and L are the original and deformed lengths of © L0 ¹

study with İ = ln ¨

the specimen, respectively. To study the effects of specimen size and strain rate on the mechanical responses of single crystal W block under uniaxial tensile loading, seven MD simulation cases are designed, as specified in Table 1. By introducing different number of defects in Simulation 2, the effect of the vacancies in the W specimen on the stress-strain relation is also investigated to better understand the size effect. 3.1.1 The effect of specimen size To study the effect of specimen size on the tensile deformation of single crystal W, Simulations 1-5 are performed. Fig. 2 shows the corresponding stress-strain curves of different W blocks under tensile loading. Table 1. Description of the MD simulation cases. Specimen Peak Number Initial Size Stress of Active StrainD×W×H Atoms Rate [s−1] [GPa] [nm3] 1.27 × 1.27 384 2 × 109 39.0 1 × 3.83 1.59 × 1.59 39.1 2 750 2 × 109 × 4.78 2.54 × 2.54 28.6 3 3,072 2 × 109 × 7.65 3.19 × 3.19 4 24.8 6,000 2 × 109 × 9.56 6.38 × 6.38 48,000 2 × 109 5 18.5 × 19.13 3.19 × 3.19 22.2 6 6,000 2 × 108 × 9.56 3.19 × 3.19 6,000 2 × 1010 7 28.5 × 9.56

Simulation #

As can be seen from the figure, the initial elastic modulus of W is almost independent on the specimen size. However, the peak stress increases as the specimen

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size decreases, which is mainly due to the fact that larger specimens offer more spaces for dislocation to occur. Although there exist the constraints due to the boundary zones and the PBCs used in the simulations, the single crystal W specimen is a discrete system with atoms distributed only at certain positions. The bonds among atoms are the relatively weak parts while the individual atoms are the strong parts of the system. The increase of specimen size increases the number of relatively weak bonds, which in turn offers more possibilities for bond breaking and dislocation to occur, and thus reduces the strength of the specimen.

Fig. 2. The size effect on the stress-strain curves of W under tensile strain rate of 2 × 109 s−1 .

Fig. 3. The effect of vacancies distributed in the same x-y plane on the stress-strain curves.

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3.1.2 The effect of defects To verify the above claim that the decrease of the W strength with the increase of specimen size is mainly due to the fact that larger specimens provide more spaces for dislocation to occur, the effect of artificially introduced defects in a single crystal W block on the stress-strain relation is investigated. Different numbers of vacancies are introduced into Simulation 2 with other simulation conditions being kept the same. The stress-strain curves of W blocks with zero, one, two and four vacancies distributed in the same x-y plane which is initially located 2.152 nm away from the top end of the active zone are presented in Fig. 3. As can be seen from the figure, all the stress-strain curves are initially the same until failure occurs. The strength of W decreases as the number of vacancies increases, since more vacancies offer more possibilities for bond breaking and dislocation to occur. However, the rate of decrease in strength slows down when vacancies become saturated. To study the effect of the vacancy distribution in the z-direction on the stress-strain relation, four cases with zero, one, two and four vacancies, respectively, located at the center of x-y plane but different positions in the z-direction, are considered with other simulation conditions being kept the same as those in Simulation 2. The corresponding stress-strain curves are similar to those as shown in Fig. 3. Hence, it appears that the increase of the number of defects regardless of the distribution position in a single crystal W block would increase the possibility for dislocation to occur and thus reduce the strength of W, which is similar to the effect of increasing specimen size on the strength of W. Therefore, it is reasonable to conclude that larger single-crystal specimens provide more spaces for dislocation to occur which in turn reduces the strength of W. 3.1.3 The effect of strain rate To investigate the effect of the tensile strain rate on the stress-strain relation of a single-crystal W specimen at the atomic level, Simulations 4, 6 and 7 are performed with the initial strain rate being 2 × 109 s−1, 2 × 108 s−1 and 2 × 1010 s−1, respectively. Fig. 4 illustrates the corresponding stress-strain curves. As can be observed from the figure, the initial elastic modulus of W is insensitive to the strain rate, but the peak stress increases with the strain rate. The dependence of peak stress on the strain rate is mainly due to the dynamic wave effect that impedes the motion of dislocations [12]. Note that the rate-dependence of peak stress becomes weak with the decrease of the strain rate. In other words, there is a critical strain rate for single-crystal metals, below which the material strength becomes rate-insensitive. As shown in [10, 11], the critical strain rate is increased with the decrease of the specimen size. Hence, the size-dependent quasi-static strength, as described by Eq. (1), could be estimated with the current MD simulation capability if the specimen size is at the nanoscale, which makes it feasible to evaluate the small-size asymptotic limit.

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3.1.4 Determination of the hyper-surface for tungsten Based on the work in [10, 11], it can be assumed that the strength of W obtained at the strain rate of 2 × 108 s–1 is closed to the quasi-static strength at the given specimen size used in Simulation 6. As can be observed from Fig. 4, the strength of W decreases from 24.8 GPa at the rate of 2 × 109 s–1 in Simulation 4 to 22.2 GPa at the rate of 2 × 108 s–1 in Simulation 6, with a factor of 22.2/24.8 = 0.895. For the sake of simplicity, therefore, all the strengths obtained in Simulations 1-5 with the strain rate of 2 × 109 s–1 are multiplied by a factor of 0.895 to get the approximate quasi-static strengths at different spatial scales. Since the specimen sizes used in Simulations 1-3 are smaller than those used in Simulations 4-7, this treatment should result in a reasonable ideal strength because the critical strain rate is increased with the decrease of the specimen size. Indentation tests on single crystal W have shown that the material hardness depends on the specimen size but the indents with diagonals being longer than about 100 µm cease to display any hardness-dependence on the size [25]. Since the material hardness is directly related to the strength, Dm = 100 µm and σm = 1.5 GPa can be used in Eq. (1). As can be seen, σu = 34.9 GPa at Du =1.6 nm could be chosen for Eq. (1), which is in a good agreement with the ideal tensile strength of 29.5 GPa as reported in [26] using pseudopotential density functional theory. Thus, the multiscale quasi-static strength of W, as described by Eq. (1), could be shown in Fig. 5. To determine the hyper-surface in both spatial and temporal domains, as described by Eqs. (1), (5) and (7), the small-size asymptotic limit and large-size asymptotic limit, namely, ( İ rs ,ı os ,q s ) and ( İ rl ,ı ol ,q l ) , must be obtained via integrated molecular and continuum level studies. As shown in the experimental study at continuum level [27], the strength of W is sensitive to the strain rate so that two data points, (2.0 GPa, 2000.0 s–1) and (1.6 GPa, 1.0 s–1), could be taken to get

İ rl = 3.6 ×105 s−1, q l = 4.7

at Drl = 100 µm

based on Eq. (4), and with the use of ıol = ı m = 1.5 GPa. Based on the MD simulation results presented above, the small-size asymptotic limit can be found via Eq. (4) and ı os = 22.2 GPa at ε = 2.0 ×108 s–1and D = 3.2 nm as follows:

İ rs = 2.0 ×1011 s−1, qs = 2.6

at Drs = 3.2 nm

for which the two data points, (24.8 GPa, 2.0 × 109 s–1) and (28.5 GPa, 2.0 × 1010 s–1) at D = 3.2 nm, have been employed. As a result, the hyper-surface for W can be determined as shown in Fig. 6.

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Fig. 4. The effect of strain rate on the stress-strain curve of single-crystal W under tension.

Fig. 5. Size-dependent quasi-static strength of tungsten.

As can be seen, combined rate and size effects on the strength of W could be estimated with a two-sided asymptotic support. Although more data points are needed from multiscale studies to refine the hyper-surface, the proposed procedure might provide an effective means to bridge different spatial and temporal scales in a unified multiscale modeling framework. It should be indicated that the combined rate and size effects on the mechanical responses of tungsten are different under various loading paths. To demonstrate this point, preliminary MD simulations have been performed to explore the size effect

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on the mechanical response of single-crystal W under pre-compressed shear loading, as shown in Fig. 7 for the computational set-up. Three specimens of different sizes are first compressed to 0, 0.33, 0.5 and 0.7 of the corresponding compressive strength, respectively, and are then tested under shear loading. As can be observed from Figs. 8-10, the degree of strength oscillation is increased with the decrease of specimen size before fracture occurs, which is different from the uniaxial loading case as shown in Fig. 2 under the same strain rate. The shear strength dependence on the pre-compression level is also oscillating. It appears from the preliminary study that the rearrangement of lattice structure under pre-compressed shear loading is more pronounced than that under uniaxial tensile loading. However, both loading paths exhibit definite size effects on the strength of W.

Fig. 6. The hyper-surface for tungsten in both spatial and temporal domains.

3.2 Diamond specimens To better understand the mechanical response of ultrananocrystalline diamond (UNCD) and its grain boundary mechanism, a numerical study has been recently performed for the specimen size and rate effects on the mechanical properties of single crystal diamond and UNCD films under uniaxial and shear loading paths, respectively [28]. To be compared with the UNCD films, single crystal diamond blocks of various sizes under tensile loading in the ¢100² direction and shear loading with the {100}¢110² slip at different rates are investigated via the MD simulation. Based on the MD simulation results and experimental data available, the parameters for determining hyper-surface of diamond can be found as below.

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Fig. 7. A pre-compressed tungsten single crystal with orientation of (x [100], y [010], z [001]) under simple shear strain rate of 2 × 109 s−1 (PBCs being applied along the x-direction).

Fig. 8. The effect of pressure on the W single crystals with size 1: 5ao × 10ao × 5ao.

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Fig. 9. The effect of pressure on the W single crystals with size 2: 5ao × 20ao × 10ao.

Fig. 10. The effect of pressure on the W single crystals with size 3: 5ao × 40ao × 20ao.

The work of B. Peng and H.D. Espinosa has shown that the strength of UNCD thin films can be predicted using the Weibull statistics [29]. For specimens with the same shape and size, the cumulative failure probability of UNCD thin films, Pf , can be expressed as

ª § σ ·m º Pf = 1 − exp « − ¨ max ¸ » «¬ © σc ¹ »¼

(10)

where m is the Weibull modulus and ı c is the Weibull characteristic strength. The two parameters, m and ı c , can be obtained from the experimentally determined probability of failure stress and the measured failure stress σ max . If N specimens are ranked in ascending order of their measured failure stresses, the

81

Combined loading rate and specimen size effects on the material properties

i-1 2 experimental failure probability of ith specimen is defined to be . The failure N stress of UNCD thin films is defined to be the value of ı max corresponding to Pf = 5%. It is observed that the failure stress of sample with a dimension of width = 40 µm, length = 400 µm and thickness = 1.0 µm is 2.88 GPa, which is in a good agreement with the theoretical strength of diamond, 2.8 GPa [30]. Therefore, it is reasonable to adopt D m as 400µm and ı m as 2.8 GPa for Eq. (1). On the other hand, according to the MD simulation results presented in Table 2, Du and ı u can be aptly chosen to be 1.1 nm and 248.8 GPa, respectively. Continuum-level experiments have indicated that the strength of diamond is rate-dependent [31]. Then, two data points, (9.1 GPa, 3.6 ×10− 4 s–1) and (17.9 GPa, 203 s–1), are chosen to get

İ rl = 2 × 10 −9 , ql = 15

at Drl = 400 µm

via Eq. (4) and ı ol = 2.8 GPa. Similarly, based on MD simulation results and Eq. (4), (270 GPa, 2 ×10 8 s–1) and (282.4 GPa, 2 × 10 9 s–1) could be used to find

İ rs = 4.5 × 10 13 , qs = 5

at Drs = 1.1 nm

where ı os is taken to be 248.8GPa at ε = 4 ×10 7 s–1 and D = 1.1 nm. Thus, with the use of Eq. (1), Eq. (5) and Eq. (7), the hyper-surface of diamond can be formulated, as shown in Fig. 11. Table 2. Tensile strength of single crystal diamond based on MD simulations. Specimen Size D×W×H [nm3] 1.1 × 1.1 × 3.2 1.4 × 1.4 × 4.3 2.1 × 2.1 × 6.3 2.8 × 2.8 × 8.5 3.2 × 3.2 × 9.6 4.3 × 4.3 × 12.8

Strain rate

Strain rate

Strain rate

Strain rate

1× 1010 [s–1]

2 × 10 [s–1]

9

2 × 10 8 [s–1]

4 × 10 [s–1]

290.9 [GPa] 289.2 [GPa] 273.8 [GPa] 226.0 [GPa] 187.0 [GPa] 177.9 [GPa]

282.4 [GPa] 282.0 [GPa] 204.7 [GPa] 179.4 [GPa] 178.5 [GPa] 178.0 [GPa]

270.0 [GPa] 200.7 [GPa] 177.9 [GPa]

248.8 [GPa] 178.9 [GPa]

-

-

-

-

-

-

7

-

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Fig. 11. The hyper-surface for diamond in both spatial and temporal domains.

4. Conclusions Recent research results are presented in this chapter to investigate the effects of combined loading rates and specimen sizes, and to formulate a hyper-surface in both spatial and temporal domains to predict the combined size and rate effects on the mechanical properties of engineering materials. With a two-sided asymptotic support, the proposed hyper-surface could be determined via integrated molecular and continuum level investigations. To demonstrate the features of the proposed procedure, tungsten and diamond specimens of various sizes under different loading rates are considered with certain loading paths. Due to the simplicity in the current formulation, the loading-path dependence can not be predicted by the proposed hyper-surface. However, it appears from the preliminary results that the proposed procedure might provide an effective means to bridge different spatial and temporal scales in a unified multiscale modeling framework, and facilitate the application of nanoscale research results to engineering practice. An integrated experimental, analytical and computational effort is required to further improve the hyper-surface so that it could be applied to general cases. Acknowledgements This work was sponsored in part by the NSF-NIRT program under Grant No. 0304472, the Overseas Young Investigator Award from the National Natural Science Foundation of China (NSFC) under Grant No. 10228206, the Program for Changjiang Scholars and Innovative Research Teams (PCSIRT) in the NSFC under Grant No. 10421002, the National Key Basic Research Special Foundation of China under Grant No. 2005CB321704, and by Sandia National Laboratories (SNL). Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. The

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authors are grateful to the NIRT members: Profs. Espinosa, Hersam, Belytschko and Schatz at Northwestern University, and Prof. Auciello at the University of Illinois at Chicago, as well as the collaborators at Argonne National Laboratory: Drs. Carlisle and Zapol, and at SNL: Dr. Fang, for joint discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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[23] Zhang B, Ouyang Y, Theoretical Calculation of Thermodynamics Data for BCC Binary Alloys with the Embedded-Atom Method. Phys. Rev. B. 48:3022-3029, 1993. [24] Zhou M, A New Look at the Atomic Level Virial Stress: on Continuum-Molecular System Equivalence. Proc. Roy. Soc. A. 459:2347-2392, 2003. [25] Hutchinson JW, Plasticity at the Micron Scale. Int. J. Solids Struct. 37:225-238, 2000. [26] Roundy D, Krenn CR, Cohen ML, Morris JW, Ideal Strength of BCC Tungsten. Phil. Mag. A. 81:1725-1747, 2001. [27] Zurek AK, Follansbee PS, Kapoor D, Strain Rate and Temperature Effects in Tungsten and Tungsten Alloys. High Strain Rate Behavior of Refractory Metals and Alloys, Edited by Asfahani, R., Chen, E. and Crowson, A. The Minerals, Metals and Materials Society, 179-191,1992. [28] Shen L, Chen Z, A Numerical Study of the Size and Rate Effects on the Mechanical Response of Single Crystal Diamond and UNCD Films. To appear in International Journal of Damage Mechanics, 2005. [29] Peng B, Espinosa HD, Fracture Size effect in Ultrananocrystalline Diamond – Weibull Theory Applicability. Proceedings of IMECE’04, 2004 ASME International Mechanical Engineering Congress, California, November 13-19, 2004. [30] Field JE, The Properties of Diamond. London: London Academic, 1992. [31] Levitt CM, Nabarro FRN, The impact Strength of Diamond Under Different Rates of Strain. Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences 1966, 293(1433): 259-274.

Discrete-to-continuum scale bridging J. Fish* Rensselaer Polytechnic Institute, Troy NY 12180, USA

Abstract The book chapter describes several information-passing and concurrent discrete-to-continuum scale bridging approaches. In the concurrent approach both, the discrete and continuum scales are simultaneously resolved, whereas in the information-passing schemes, the discrete scale is modelled and its gross response is infused into the continuum scale. Most of the information-passing approaches provide sublinear computational complexity, (i.e., scales sublinearly with the cost of solving a fine scale problem). Among the information-passing bridging techniques, we present the Generalized Mathematical Homogenization (GMH) theory, which constructs an equivalent continuum description directly from molecular dynamics (MD) equations; the Multiscale Enrichment based on Partition of Unity (MEPU) method, which gives rise to the enriched coarse grained formulation, the Heterogeneous Multiscale Method (HMM), which provides equivalent coarse scale integrands; the Variational Multiscale Method (VMS), which can be viewed as an equivalent coarse scale element builder; the Coarse-Grained Molecular Dynamics (CGMD), which derives effective Hamiltonian for the coarse-grained problem; the Discontinuous Galerkin Method, which constructs discontinuous enrichment; the Equation-Free Method (EFM), which makes no assumption on the response of the coarse scale problem; and the Kinetic Monte Carlo (KMC)-based methods, which bridge diverse time scales by calibrating certain KMC parameters from molecular dynamics or quantum mechanics calculations. The second part of the book chapter focuses on multiscale systems, whose response depends inherently on physics at multiple scales, such as turbulence, crack propagation, friction, and problems involving nano-like devices. For these types of problems, multiple scales have to be simultaneously resolved in different portions of the problem domain. Among the Concurrent Multiscale techniques, we describe Domain Bridging (DBCM), Local Enrichment (DBCM) and Multigrid (MGCM) based concurrent multiscale methods. A space-time variant of the MGCM for bridging discrete scales with either coarse grained discrete or continuum scales is presented. The method consists of the wave-form relaxation scheme aimed at capturing the high frequency response of the atomistic vibrations and the coarse scale space-time solution (explicit or implicit) intended to resolve the coarse scale features of the discrete medium. Keywords: Multiscale; Concurrent; Information-passing; Multigrid; Homogenization; Enrichment; Atomistic; Bridging.

1. Introduction: The Tale of Two Engines Consider two engines depicted in Fig. 1. The size of the jet engine shown in Fig. 1a is of order of meters; continuum description by means of partial differential *

Corresponding author. E-mail address: [email protected] (J. Fish). 85

G.C. Sih (ed.), Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano, 85–102. ” 2007 Springer.

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equations prevails, and consequently, it is appropriate to use discretization methods such as finite elements or finite differences. On the other hand, the rotary motor in Fig. 1b has a diameter of 30 nm. It drives Salmonella and E. coli bacteria by rotating at around 20,000 rpm, at energy consumption of around 10 –16 W and with energy conversion efficiency close to 100% [1]. This remarkably efficient nano engine does not obey continuum principles because it is simply too small; continuum description does not account for predominant surface effects that would result in too stiff behavior. On the other hand, a brute force approach of modelling the rotary motor entirely on atomistic scale would necessitate billions of unknowns. This suggests a multiscale computational paradigm where important atomistic features could be captured at a fraction of computational cost required by atomistic simulation of the entire system [2]. Barriers In devising such a rigorous discrete-to-continuum scale-bridging framework, one of the main barriers is increased uncertainty/complexity introduced by discrete scales as illustrated in Fig. 2.

Fig. 1. (a) macro-engine, (b) nano-engine.

Fig. 2. Reduced precision due to increase in uncertainty and/or complexity.

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As a guiding principle for assessing the need for finer scales, it is appropriate to recall the statement made by Einstein, who stated that “the model used should be the simplest one possible, but not simpler.” The optimal multiscale model has to be carefully weighted on case-by-case basis. For example, in case of metal matrix composites (MMC) with almost periodic arrangement of fibers, introducing finer scales might be advantageous since the bulk material typically does not follow normality rules and developing a phenomenological coarse scale constitutive model might be challenging at best. The behavior of each phase is well understood and obtaining the overall response of the material from its fine scale constituents can be obtained using homogenization. On the other hand, in brittle ceramics composites (CMC), the microcracks are often randomly distributed and characterization of their interface properties is difficult. In this case, the use of fine scale models may not be desirable. For a complementary reading we refer to an excellent review article [3] and a comprehensive study on adaptive control of multiscale models [4]. 1.2 Qualification of multiscale methods A modelling and simulation approach is termed multiscale if it is capable of resolving certain quantities of interest with a significantly lower cost than solving the corresponding fine-scale system. Schematically, a multiscale method has to satisfy the so-called Accuracy and Cost Requirements (ACR) test:

Error in quantities of interest < tol Cost of multiscale solver = 1 Cost of fine scale solver This book chapter focuses on two categories of multiscale approaches: information-passing and concurrent. In the information-passing multiscale approach (see Section 2), the discrete scale is modelled and its gross response is infused into the continuum scale, whereas in the concurrent approach (see Section 3), both, the discrete and continuum scales are simultaneously resolved. Loosely speaking, the information-passing multiscale approach is likely to pass the ACR test provided that: (i) the quantities of interest are limited to or defined only on the coarse scale (provided that this quantities are computable from the fine scale), and (ii) special features of the fine scale problem, such as scale separation and self-similarity, are taken advantage of. On the other hand, for the concurrent multiscale approach to pass the ACR test, the following conditions must be satisfied: (i) the interface between the fine and coarse scales should be properly engineered (see Section 3), and

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(iia) the information-passing multiscale approach of choice should serve as an adequate mechanism for capturing the lower frequency response of the fine-scale system, or alternatively, (iib) the fine scale model should be limited to a small part of the computational domain. The ACR condition (iia) represents a stronger requirement typically satisfied by the multigrid-based concurrent methods (see Section 3.3), but not by the domain bridging (Section 3.1) or the local enrichment (Section 3.2) concurrent methods. It is important to note that even though concurrent approaches may pass the ACR test, their computational cost will typically exceed that of the information-passing methods. Nevertheless, they offer a distinct advantage over the information-passing methods by virtue of being able to resolve fine scale details in critical regions. Therefore, concurrent multiscale approaches are typically pursued when the fine scale information is either necessary, or if not resolved, may pollute significant errors on the coarse scale information of interest. 2. Information-passing Multiscale Methods In the information-passing multiscale methods, calculations at finer scales, and of high-computational complexity, are used to evaluate certain quantities for use in a more approximate or phenomenological computational methodology at a longer length/time scale. This type of scale bridging is also known as sequential, serial or parameter-passing. For nonlinear problems fine and coarse scale models are two-way coupled, i.e., the information continuously flows between the scales. In this section we present several information-passing bridging technique including: the Generalized Mathematical Homogenization (GMH) theory [5, 6], which constructs an equivalent continuum description directly from molecular dynamics (MD) equations; the Quasicontinuum method [7], which can be viewed as an engineering counterpart of the mathematical homogenization; the Multiscale Enrichment based on the Partition of Unity (MEPU) method [8], which gives rise to the enriched coarse grained formulation; the Heterogeneous Multiscale Method (HMM) [9], which provides equivalent coarse scale integrands; the Variational Multiscale Method (VMS) [10], which can be viewed as an equivalent coarse scale element builder; the Coarse-Grained Molecular Dynamics (CGMD) [11], which derives effective Hamiltonian for the coarse-grained problem; the Discontinuous Galerkin (DG) Method [12],which constructs discontinuous enrichment; the Equation-Free Method (EFM)[13], which makes no assumption on the response of the coarse scale problem; and the Kinetic Monte Carlo (KMC)-based methods, which bridge diverse time scales by calibrating certain KMC parameters from molecular dynamics or quantum mechanics calculations. 2.1 Generalized mathematical homogenization (GMH) theory In the GMH approach a multiple scale space-time asymptotic expansion is employed to approximate the displacement field u(x,y,IJ,t,t1 ,t 2 ) = u 0 + İu1…

(1)

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where x is a differentiable continuum coordinate; y = x/İ the discrete coordinate denoting position of atoms in a unit cell and 0 < e = 1 ; IJ the fast time coordinate, which tracks vibration of atoms for finite temperature applications; t the usual time coordinate; t1 ,t 2 the slow time coordinates, which from the physics point of view capture dispersion effects, whereas from the mathematics point of view eliminate secularity of asymptotic expansions. We first outline the O(1) GMH theory without consideration of slow and fast time scales. The primary objective of GMH is to construct continuum equations directly from Molecular Dynamics (MD) equations i (x,y) − m(y)u

¦

j (j ≠ i )

f ij ( rij ) = 0

(2)

where f ij , rij are the interatomic force and the radius vector between atoms i and j , respectively; and m i is the mass of atom i. In Eq. (2) for simplicity pairwise interatomic potential is considered, which may be inadequate for solids. Expanding rij = İrij0 + İ2 rij1 + … in asymptotic sequence and f ij in Taylor’s series expansion around the leading order term İrij0 yields a set of coupled continuum-atomistic governing equations  0 (x, t)  ‹x ¸ T = 0 Su

T=

1 œ œ rij  fij(rij ) 22 i j(vi)

œ f {(F(u ) ¸  ¡¢R 0

j(v i)

ij

ij

+ F (u1 (x, y j )  u1 (x, y i ))¯° = 0 ±

}

where S is the density; 2 the volume of the atomistic unit cell; ‹x ¸ ( ) the divergence operator; F(u 0 ) the coarse scale deformation gradient; R ij the initial separation between atoms i and j . It can be seen that the Cauchy stress ı derived from the O(1) GMH theory coincides with the mechanical term in the virial stress formula (see Irving and Kirkwood [14] and Hardy [15] among others). For more details on the O(1) GMH theory and its numerical implementation see [16]. The motivation for introduction of slow time scales was given in [17], where it has been shown that in absence of slow time scales the scaling parameter İ= O t becomes time dependent, where t is the normalized time coordinate. Consequently, as t → ∞ the asymptotic expansion becomes no longer uniformly valid.

( )

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Higher order GMH theory incorporating slow time scales leads to the nonlocal continuum description [5, 18, 19]. Alternatively, a close form solution for slow time scales can be obtained leading to an algebraic system of equations with a single time scale [6]. Extension of the GMH theory to finite temperatures using fast time scales has been given in [20]. Figure 3 compares the Generalized Mathematical Homogenization, with the spatial homogenization approach developed in [21] and molecular dynamics simulation for wave propagation in a layered lattice structure (see [6] for details). It can be seen that the GMH theory offers a comparable accuracy to MD simulation despite significant cost reduction. To this end we note that for the GMH approach to be valid both the temporal and spatial scales have to be separable. For instance, if the essential events of the faster fine scale model occur on the same time scales as the details of processes computed using the slower coarse model, then the time scales cannot be separated. Likewise, if the wavelength of the travelling signal is of the order of magnitude of the fine scale features, then the spatial scales cannot be separated. 2.2 Quasicontinuum Quasicontinuum is a continuum description where constitutive equations are constructed directly from atomistics rather than from a phenomenological constitutive model. The atomistically informed constitutive model is adequate as long as continuum fields are slowly varying over a unit cell domain. In its original form [7] the quasicontinuum method was formulated for simple Bravais crystals assuming uniform deformation of atoms. In a more general case with heterogeneous interatomic potentials, a unit cell problem has to be solved instead [22]. In this more general scenario, the quasicontinuum resembles GMH and as such it can be viewed as an engineering counterpart of the mathematical theory. Note that both the “engineering” and mathematical homogenization methods involve solution of an atomistic unit cell problem and subsequently feeding the continuum problem with effective properties. 2.3 Multiscale enrichment based on the partition of unity Multiscale Enrichment based on Partition of Unity (MEPU) [8] is a synthesis of the generalized mathematical homogenization [6] and Partition of Unity [23, 24, 25] methods. MEPU can be used to enrich the coarse scale continuum description or the coarse-grained discrete formulations. It is primarily intended to extend the range of applicability of the mathematical homogenization theory to problems where scale separation may not be valid, such as in the case of nonperiodic solutions or problems where the coarse solution may rapidly vary over the domain of the unit cell domain. MEPU belongs to the category of methods employing hierarchical decomposition of the approximation space in the form of u = uc + uf

(3)

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where uc and uf are the coarse and fine scale solutions, respectively. Note that in GMH uc = u0 (x) and uf x Fu1 (x, y) . In MEPU, on the other hand, u=

¦ N ( x ) d + ¦ H( x ) N( x ) a

(4)

Horizontal displacement (angstrom)

where N are the coarse scale element shape functions; H( x ) the influence function obtained from the unit cell solution; d, a are the nodal and enrichment degrees-of-freedom, respectively. The influence functions can be either discrete (obtained from the atomistic unit cell) or continuous. MEPU allows consideration of nonperiodic fields by associating different unit cells with different gauss point in the coarse scale elements. To reduce the computational cost, homogenization-like integration scheme is devised. By this approach the value of a function at a gauss point of a coarse scale element is replaced by an average computed over a unit cell domain cantered at a gauss point. It has been proved that the accuracy of the homogenization-like integration scheme is of order O(1/ n) where n is the number of unit cells in the coarse scale element domain. Figure 4 depicts a molecular model of a polymer subjected to uniform macroscopic fields. The polymer has been modelled using a single MEPU element with nine degrees of freedom per node [26]. The error in the L2 norm of displacements was 2% compared to the 9% using quasicontinuum method. 0.1 GMH

0.08

Classical homogenization

0.06 0.04 0.02 0 −0.02 −0.04 −0.06 0

5

10

15

20

Time (picosecond)

Fig. 3. Comparison of GMH with classical (spatial) homogenization and molecular dynamics (MD) simulations.

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Fig. 4. MD model of a polymer.

Coarse scale elements Interface

Fig. 5. VMS for enriching coarse grained models.

2.4 Variational multiscale method The Variational Multiscale Method (VMS) was originally developed for enriching continuum solutions with fine scale continuum description. Most common implementation of the method assumes the fine scale enrichment u f to be a residual free bubble vanishing on the coarse scale element boundaries. By this assumption the enrichment functions can be condensed out on the element level to give effective coarse scale elements as opposed to effective fields or material properties in the GMH approach. Alternatively, a better accuracy can be obtained by enforcing enrichment functions to vanish on the element boundaries in the weak sense. VMS can be easily extended to enriching coarse grained descriptions, such as for instance quasicontinuum. In this scenario the coarse grained description which amounts to interpolating the solution between the representative atoms, (element nodes in Fig. 5) can be enriched using the kinematics of individual atoms in the areas where such enrichment in necessary. Since positions of atoms may not coincide with the coarse scale element boundaries, homogeneous boundary condition of atoms residing in the close vicinity to the element boundaries can be enforced (weakly on in the strong sense) as shown in Fig. 5.

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2.5 Heterogeneous multiscale method The basic idea of the heterogeneous multiscale method (HMM) is to approximate the coarse scale integrands by data computed from the auxiliary fine scale problem [9]. The auxiliary fine scale problem is an atomistic cell subjected to boundary conditions extracted from the coarse scale solution. HMM can be viewed as a methodology for constructing effective integrands based on the fine scale data as opposed effective properties in GMH or effective elements in VMS. 2.6 Coarse-grained molecular dynamics The Coarse-Grained Molecular Dynamics (CGMD) [2, 27] method constructs a coarse grained Hamilton’s equations from MD equations under fixed thermodynamic conditions. The representative atoms, (similar to those employed in the quasicontinuum method) are enforced to preserve an average position and momenta of the fine scale atoms u c (t) = Ru f ( t)

(5)

pc (t) = Rpf (t)

where R is weighting or restriction operator, pc, p f are the momenta at the coarse and fine scales, respectively. The coarse-grained Hamiltonian, H c , is defined as the classical canonical ensemble average of the molecular dynamics Hamiltonian H f in the displacement-momenta space subjected to the restriction constraint (5) HC =

1 + PH f dp f du f Z³

(6)

where P = exp( − H f/k BT) is the probability function, T the temperature, H the fine scale Hamiltonian, k the Boltzmann constant, + enforces constraints in (5) and Z is the partition function. f

B

2.7 Discontinuous Galerkin (DG) method The so-called Multiscale finite element method (MsFEM) developed by Hou [12] belongs to the category of Discontinuous Galerkin methods. In the MsFEM the displacement field is approximated as u= u 0 (x)+ İ H (x)İ 0 (x) , which closely resembles the method of mathematical homogenization, but introduces no multiple spatial coordinates and thus results in C−1 continuous approximation of the solution. The oversampling idea of Babuska [28] is used to control the errors resulting from the discontinuity.

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2.8 Equation free method In the Equation Free Method (EFM) the fine scale problem is evolved at some sampling points in the coarse scale domain. These sampling points are represented by an atomistic unit cell. Unlike in the aforementioned information-passing methods the coarse problem is assumed to be unknown in EFM. Once the solution in two subsequent time steps on the fine scale is computed and then restricted to the coarse scale u c (t) =Ruf ( t) and u c (t+dt)=Ruc (t+dt) then the coarse scale solution at t ++t (+t  Et) is obtained by projective integration or extrapolation in time domain. The fine-to-coarse scale operators are well defined, but the definition of the information flow from the coarse to the fine scale remains to be the main challenge. This type of formulation may be attractive for complex bio-systems whose coarse scale behavior is often unknown. It is instructive to point out that the proper orthogonal decomposition (POD) [29] widely used in fluid dynamics community is closely related to EFM. POD generates a set of “snapshots” on the fine scale to create a reduced-space basis, onto which the fine scale equations are projected to predict the evolution on the coarse scale. 2.9 KMC-based information-passing methods Most of the aforementioned multiscale approaches with the exception of EFM and GMH deal with bridging the spatial scales. Linking diverse time scales is even more challenging. For instance, to capture the dynamics of atomistic vibrations, the time step should be of the order of femtoseconds, whereas the residence time of an adatom between hops is of the order of microseconds. This ‘time gap’ can be addressed using Kinetic Monte Carlo (KMC) based methods. The basic idea of this approach is summarized below. Let the probability of finding a system in state ı i at time t to be denoted as P ( ı i ,t ) and the rate of transitions from ıi to ı i+1 to be W(ıi ,ıi +1 ). KMC is an algorithm that solves for the probability function P ( ı i ,t ) such that ∂P ∂t

=

¦ ( P(σ , t)W ( σ , σ ) − P(σ i

i

i +1

i +1

, t)W ( σ , σ i +1

i

))

The rate of transition can be expressed as a product of an attempt rate and the probability of success per attempt, which is taken as an exponential of the energy barrier to the process. The attempt rate for event i is defined as ri = µ i exp(− E i / k B T)

(7)

where µi is a frequency prefactor expressed in terms of a vibrational frequency for surface processes [30, 31, 32] and Ei is the free energy barrier. KMC falls into

Discrete-to-continuum scale bridging

95

the category of information-passing methods because the frequency prefactor and the energy barrier can be calculated from molecular dynamics or quantum mechanics simulations. 3. Concurrent Multiscale Methods In this Section we present a class of multiscale approaches for systems, whose behavior depends on physics at multiple scales. Examples of problems falling into this category are turbulence, crack propagation, friction, and problems involving nano like devices are prime examples. In fracture, the crack tip bond breaking can be described with a quantum-mechanical model of bonding, while the rest of the sample is described with empirical potentials. In friction, it might be necessary to describe the surface interaction using quantum-chemical approaches while using continuum elasticity to simulate the contact forces. For these types of problems, multiple scales have to be simultaneously resolved in different portions of the problem domain. Multiscale methods based on the concurrent resolution of multiple scales are often coined as embedded, concurrent, integrated or hand-shaking multiscale methods. Various domain bridging methods [33, 34, 35, 36], multigrid methods [37, 38, 39, 40, 41, 42, 43] and local enrichment methods [44, 45, 46, 47, 48] are used to communicate the information between the subdomains represented by different mathematical models. An important aspect of concurrent methods is matching condition at the interface between different mathematical models. For instance, at the MD/continuum interface, MD generates phonons which are not represented in the continuum region and hence might be reflected at the continuum/MD interface. Formulation of absorbing interfaces include damping [33], Langevin equation [48], precomputing exact absorbing boundary conditions for harmonic potentials [49], approximating exact absorbing boundary conditions and calibrating coefficients to minimize reflection [50], matching the properties of continuum and MD at the interface [51], and bridging domain method [35, 52]. By refining the finite element mesh to atomistic scale at the interface [33], the issue of phonon reflection can be circumvented. A review of various interface formulations can be found in Curtin and Miller [34]. For concurrent bridging between discrete dislocations and continuum region we refer to [53]. Another important aspect of concurrent bridging is building temporal interfaces. For example, in a typical atomistic-continuum problem the time scale for integrating MD equations is dictated by the interatomic spacing and highly heterogeneous interatomic connections. At the continuum scale, the time step could be much larger primarily because the stiff connections have been homogenized out and the spacing between the discrete points (for instance, FE nodes) could be substantially larger. Temporal interfaces can be built using various multi-time-step methods [54] and local enrichment functions in time domain [55], whereas the space-time interfaces can be constructed using the space-time Discontinuous Galerkin (DG) method [56]. The multi-step technique developed in [35] preserves stability of time integrators and at the same time minimizes spurious reflections from the interfaces.

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In the following we outline the basic ideas of the domain bridging (Section 3.1), local enrichment (Section 3.2), and multigrid (Section 3.3) based concurrent multiscale methods common to several aforementioned approaches. We emphasize that the above multiscale methods are concerned with a concurrent bridging of dissimilar mathematical models representing different scales, as opposed to the classical domain decomposition, multigrid and enrichment methods, which are primarily concerned with efficient solution of a single scale mathematically similar models.

Γ = Ω ∩Ω C

Ω

Γ ⊂ Ωf

f

Ω

ΩC

ΩC ∂Γ

f

f

(b)

(a)

Fig. 6. (a) coexisting domains and (b) overlapping domains.

In the domain bridging based concurrent multiscale (DBCM) approach, the fine Ω f and coarse ΩC scale subdomains could be either overlapping or coexistent as shown in Fig. 6. The interface, Γ , could be the same or lower dimensional manifold. In the multigrid based concurrent multiscale (MGCM) and the local enrichment based concurrent multiscale (LECM) approaches the subdomains are coexisting (Fig. 6a). The interface, Γ ⊂ Ω f , is a subdomain in Ω f defined to be in the close vicinity to the boundary ∂Γ as shown in Fig. 6a. It could be the same or lower dimensional manifold, in which case Γ = ∂Γ . 3.1 Domain bridging based concurrent multiscale method Consider a conservative system consisting of continuum (c) – discrete (f) subdomains (Fig. 6b). Let p ,u be the momenta and displacements of atom i, and u (x),u (x) be the velocities and displacements of the continuum. The Hamiltonian that weakly satisfies compatibility between the discrete and continuum regions at the interface Γ is defined as f i

C

f i

C

ª 1  CT  C C C H = ³ αC «¬ 2 ρu u + w u ΩC

( )

(

+ ³ λ (x) ªδ uC (x) − u f i ¬ Γ

ª pf ⋅ pf º º C dΩ + ¦ αf « i i + w f u f , u f » »¼ i j f i,j≠i »¼ ¬« 2m

(

)

)º¼ dΓ

where Į + Į = 1 is enforced so that the energy would not count twice; Į =1 on Ω − Γ in case of overlapping subdomains and Ω − Ω in case of coexisting subdomains; in both cases α = 1 on Ω − Γ . δ is a delta function; the atomistic C

f

C

C

C

f

f

f

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Discrete-to-continuum scale bridging

region is defined over x if ∈ Ωf ; w C (x),w f (u if ,u fj ) are the continuum internal energy and the pairwise interaction of the atoms (for simplicity considered here), respectively. In [35] the overlapping subdomains were considered with ĮC ,Į f to vary linearly on Γ and compatibility enforced in a strong form. In [33] an overlapping subdomains were considered with ĮC = Įf = 0.5. The Arlequin method [52] was applied to bridge between continuum scales only. The above formulation can be extended to more than two scales. For instance, in [33] the computational domain was decomposed into three parts: a continuum region discretized with finite elements, an atomistic region modelled by molecular dynamics, and a quantum mechanical region where the tight binding model is used to model bond breaking. 3.2 Local enrichment based concurrent multiscale method The Local Enrichment based Concurrent Multiscale (LECM) method employs hierarchical decomposition of the form given in Eq. (3). For the enrichment scheme to be qualified in the LECM category, the enrichment function must have local supports and has to be complete, i.e., in the limit as the approximation of u f is enriched it converges to the fine scale description. For instance, in one of the VMS variants [10], in the limit as u f is locally enriched the method does not converge to the fine scale description because there is no refinement on coarse scale element boundaries. Examples of complete enrichment spaces can be found in [8, 47, 44, 45], but such complete enrichment spaces cannot be locally condensed out. The resulting Hamiltonian in LECM is given by

(

)

(

T 1 H = ¦ w f uC (x ) + αu f + ¦ m u C (x ) + αu f i i i i i⊂ Ω i⊂Ω 2

) (u C (xi ) + αu fi )

+ ¦ λ (x )u f i i i⊂ Γ

where α = 1 on Ω and α = 0 elsewhere. In the above u is a coarse grained model, such as quasicontinuum, defined on the entire problem domain Ω = Ω . w f is a pairwise interaction of the atoms. LECM offers the advantage inherent to hierarchical methods, including the ease of solution enrichment and a posteriori model error estimation. But unlike DBCM, which uses dissimilar mathematical models, LECM is limited to similar mathematical models at different scales. f

C

C

3.3 Multigrid based concurrent multiscale method The motivation for use of multigrid ideas for multiscale problems was given in [42, 43]. To convey the basic ideas, consider a one-dimensional two-scale elliptic problem

§ K(x) du · + b(x) = 0 x ∈ (0, L) u(0) = u(L) = 0 ¨ ¸ dx © dx ¹ d

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J. Fish

with oscillatory periodic piecewise constant coefficients K1 ,K 2 and 0.5 volume fraction. The above equation is discretized with 2 (m-1) elements – each element possessing constant coefficients. The eigenvalues can be computed in a closed form:  4K λ = k

h

sin

2

 4K

§k π · ¨ ¸ © 2m ¹

1 + 1 − q sin

2

,

§k π · ¨ ¸ © 2m ¹

λ

2m−k

=

h

sin

2

§k π · ¨ ¸ © 2m ¹

1 − 1 − q sin

2

§k π · ¨ ¸ © 2m ¹

where 1 ≤ k < m ; 2h the unit cell size; K the overall coefficients and q the ratio between geometric and arithmetic averages of the coefficients given as:  = K

2K1 K 2 K1 + K 2

,

q =

K1 K 2

(K

1

+ K2 ) / 2

30

20

Explicit

CPU 10

0

Multigrid

0.1 Time Integration

0.2

Fig. 7. Comparison of Explicit and multigrid methods.

Note that in many applications of interest and in particular those described at the atomistic scale K1  K 2 or K 2  K1 or 0 < q  1 . Consequently, the eigenvalues are clustered at the two ends of the spectrum, with one half being O(1) and the other half being O(1/q) . More importantly, the O(1) eigenvalues are identical to those obtained by the problem with homogenized coefficients. This character of the spectrum suggests a computational strategy based on the philosophy of multilevel methods. In such a multilevel strategy a smoother is designated to capture the higher frequency response of the fine scale model represented by a linear combination of the O(1/q) eigenmodes. The auxiliary coarse model is then engineered to effectively capture the remaining lower frequency response of the fine scale problem. For a periodic heterogeneous medium, such an auxiliary coarse model coincides with the boundary value problem with homogenized coefficients as evidenced by the identical eigenvalues. The resulting multiscale prolongation Q operator is given by Q = QC + Qf

(8)

Discrete-to-continuum scale bridging

99

where Q C ,Q f are the classical (smooth) prolongation and the fine scale correction obtained from the discretization of the influence functions, respectively. The rate of convergence of the multigrid process for the two-scale problem is governed by [42, 43]: ei +1 = q ei / ( 4 + q )

(9)

where ei+1 is the norm of error in iteration i+1. For example, if either K1 /K 2 or K 2 /K1 , is 100, then the two-scale process converges in three iterations up to the tolerance of 10 −5 . In principle, any information-passing approach described in Section 2 can be used as an auxiliary coarse model to capture the lower frequency response of the fine scale problem. The multiscale prolongation depends on the choice of the information-passing approach. To this end we describe the application of multigrid ideas to bridging diverse time scales. The space-time variational multigrid method developed in [39] is aimed at bridging between atomistic scale and either coarse grained discrete or continuum scales. The method consists of the wave-form relaxation scheme aimed at capturing the high frequency response of the atomistic vibrations and the coarse scale solution in space and time intended to resolve smooth features of the discrete medium. The waveform relaxation [57, 58] decomposes the system into very small subsystem (for instance, atom-by-atom decomposition) which can be integrated in parallel and take advantage of unstructured time integrators. The waveform relaxation can be also interpreted as atom-by-atom minimization of MD Hamiltonian. Then a coarse model correction at a certain time step u f +QeC can be calculated from the Hamilton principle on the subspace of coarse scale functions T 1 H ( ec ) = ¦ w f ( uf + Qec ) + ¦ m ( u f + Qec ) ( u f + Qec ) → min ec 2

The method has been used to simulate a polymer structure shown in Fig. 4 and compared to the classical explicit integration. Significant speed-ups have been observed. The coarse scale model was constructed using aggregation method [59], where each polymer chain was defined as an aggregate. The rate of convergence on a model problem with harmonic potentials has been studied in [38]. A multiscale filter [60, 61, 62] is needed for indefinite systems, such as those involving chemical reactions or breakage of interatomic connections due to mechanical loads. For related wave-form relaxation multigrid methods see [63].

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Acknowledgement The financial supports of National Science Foundation under grants CMS0310596, 0303902, 0408359 and Sandia contract DE-ACD4-94AL85000, ONR contract N00014-97-1-0687 are gratefully acknowledged. References [1] Namba K, Revealing the mystery of the bacterial flagellum – A self-assembling nanomachine with fine switching capability. Japan Nanonet Bulletin, 2004; 11th Issue February 5. [2] Rudd RE, Broughton, JQ, Coarse-grained molecular dynamics and the atomic limit of finite elements. Physical Review B 1998; 58(10): 93-97. [3] Curtin WA, Miller RE, Atomistic/continuum coupling in computational materials science. Modelling Simul. Mater. Sci. Eng. 2003; 11: R33-R68. [4] Tinsley J, Oden S, Prudhomme A, RomkesBauman P, Multi-Scale Modelling of Physical Phenomena: Adaptive Control of Models. ICES Report 05-13. Austin, 02/28/05. [5] Fish J, Schwob C, Towards Constitutive Model Based on Atomistics. Journal of Multiscale Computational Engineering 2003; 1: 43-56. [6] Chen W, Fish J, A Generalized Space-Time Mathematical Homogenization Theory for Bridging Atomistic and Continuum Scales. International Journal for Numerical Methods in Engineering 2005. In print. [7] Tadmor EB, Ortiz M, Phillips R, Quasicontinuum analysis of defects in crystals. Phil. Mag. A 1996; 73: 1529. [8] Fish J, Yuan Z, Multiscale Enrichment based on Partition of Unity. International Journal for Numerical Methods in Engineering. 2005; 62(10): 1341-1359. [9] EW, Engquist B, The heterogeneous multi-scale methods. Comm. Math. Sci. 2002; 1: 87-132. [10] Hughes TJR, Multiscale Phenomena; Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of the stabilized methods. Comp. Meth. Appl. Mech. Engng. 1995; 127: 387-401. [11] Rudd RE, Broughton, JQ, Coarse-grained molecular dynamics and the atomic limit of finite element. Phys. Rev. B 1998; 58(10): R5893. [12] Hou TY, Wu X, A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 1997; 134: 169-189. [13] Kevrekidis IG, et al., Equation-free coarse-grained multiscale computation: enabling microscopic simulators to perform system-level tasks. Communications in Mathematical Sciences 2003; 1(4): 715-762. [14] Irving JH, Kirkwood JG, The statistical mechanical theory of transport processes, IV. The equations of hydrodynamics. J. Chem. Phys. 1950; 18: 817-829. [15] Hardy RJ, Formulas for determining local properties in molecular dynamics simulations: shock waves. J. Chem. Phys. 1982; 76: 622-628. [16] Chen W, Fish J, Mathematical Homogenization Perspective of Virial Stress. International Journal for Numerical Methods in Engineering 2005. [17] Fish J, Chen W, Higher-Order Homogenization of Initial/Boundary-Value Problem. Journal of Engineering Mechanics 2001; 127(12): 1223-1230. [18] Fish J, Chen W, Nagai G, Nonlocal dispersive model for wave propagation in heterogeneous media. Part 1: One-Dimensional Case. International Journal for Numerical Methods in Engineering 2002; 54: 331-346. [19] Fish J, Chen W, Nagai G, Nonlocal dispersive model for wave propagation in heterogeneous media. Part 2: Multi-Dimensional Case. International Journal for Numerical Methods in Engineering 2002; 54: 347-363.

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[20] Fish J, Tang Y, Space-Time Generalized Mathematical Homogenization of Atomistic Media at Finite temperatures, to appear in International Journal for Multiscale Computational Engineering, 2005. [21] Chung PW, Computational method for atomistic homogenization of nano-patterned point defect structures. Int. J. Numer. Meth. Engng 2004; 60: 833-859. [22] Tadmor EB, Smith GS, Bernstein N, Kaxiras E, Mixed finite element and atomistic formulation for complex crystals. Physical Review B 1999; 59: 1. [23] Babuska G, Caloz, Osborn JE, Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM. J. Numer. Anal 1994; 4: 945-981. [24] Melenk JM, Babuska I, The Partition of Unity Finite Element Method. Basic Theory and Applications. Comp Methods in Appl. Mech and Engng 1996; 139: 289-314. [25] Moës N, Dolbow J, Belytschko T, A finite element method for crack growth without remeshing. Int. J. Num. Meth. Engnr 1999; 46:131-150. [26] Fish J, Yuan Z, Multiscale Enrichment based on Partition of Unity for non-periodic fields. International Journal for Numerical Methods in Engineering 2005; To appear. [27] Rudd RE, Coarse-Grained Molecular Dynamics for Computer Modeling of Nanomechanical Systems. International Journal for Multiscale Computational Engineering 2004; 2(2): 203-220. [28] Babuska I, Osborn J, Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Numer. Anal 1983; 20:510-536. [29] Lumley JL, The Structures of Inhomogeneous Turbulent Flow. Atmospheric Turbulence and Radio Wave Propagation 1967; 166-178. [30] Grujicic M, Cao G, Joseph PF, Multiscale Modelling of Delamination and Fracture of Polycrystalline Lamellar γ − TiAl + α 2 − Ti 3 A l Alloys. International Journal of Multiscale Computational Engineering 2003; 1: 1-22. [31] Picu C, A Nonlocal Formulation of Rubber Elasticity. International Journal of Multiscale Computational Engineering 2003; 1: 23-32. [32] Johnson HT, Bose R, Goldberg BB, Robinson HD, Effects of Externally Applied Stress on the Properties of Quantum Dot Nanostructures. International Journal of Multiscale Computational Engineering 2003; 1: 33-42. [33] Broughton JQ, Abraham FF, Berstein N, Kaxiras E, Concurrent coupling of length scales: Methodology and application. Phys. Review B 1999; 60: 2391-2403. [34] Miller RE, Direct Coupling of Atomistic and Continuum Mechanics in Computational Material Science. International Journal of Multiscale Computational Engineering 2003; 1: 57-72. [35] Belytschko T, Xiao S, Coupling Methods for Continuum Model with Molecular Model. International Journal of Multiscale Computational Engineering 2003; 1: 115-126. [36] Wagner GJ, Karpov EG, Liu WK, Molecular Dynamics Boundary Conditions for Regular Crystal Lattices. Computer Method in Applied Mechanics and Engineering 2004; 193: 1579-1601. [37] Datta DK, Picu RC, Shephard MS, Composite Grid Atomistic Continuum Method, An adaptive approach to bridge continuum with atomistic analysis. International Journal for Multiscale Computational Engineering 2004; 2(3):401-419. [38] Fish J, Chen W, Discrete-to-Continuum Bridging Based on Multigrid Principles. Comp. Meth. Appl. Mech. Engng 2004; 193: 1693-1711. [39] Waisman H, Fish J, Space-Time multigrid for bridging discrete scales. International Journal for Numerical Methods in Engineering 2005. To appear. [40] Knapek S, Matrix-Dependent Multigrid-Homogenization for Diffusion Problems. SIAM J. Sci. Comput 1999; 20: 515-533. [41] Moulton JD, Dendy JE, Hyman JM, The Black Box Multigrid Numerical Homogenization Algorithm. J. Comput. Phys 1998; 141: 1-29.

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[42] Fish J, Belsky V, Multigrid method for a periodic heterogeneous medium. Part I: Convergence studies for one-dimensional case. Comp. Meth. Appl. Mech. Engng 1995; 126: 1-16. [43] Fish J, Belsky V, Multigrid method for a periodic heterogeneous medium. Part 2: Multiscale modeling and quality control in multidimensional case. Comp. Meth. Appl. Mech. Engng 1995; 126: 17-38. [44] Fish J, Markolefas S, The s-version of the finite element method for multilayer laminates. International Journal for Numerical Methods in Engineering 1992; 33(5): 1081-1105. [45] Fish J, Markolefas S, Guttal R, Nayak P, On adaptive multilevel superposition of finite element meshes. Applied Numerical Mathematics 1994; 14: 135-164. [46] Moës N, Dolbow J, Belytschko T, A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 1999; 46(5): 131-150. [47] Strouboulis T, Babuska I, Copps K, The generalized finite element method. Computer Methods in Applied Mechanics and Engineering 2001; 190: 4081-4193. [48] Wagner GJ, Liu WK, Coupling of atomic and continuum simulations using a bridging scale decomposition. J. Comput. Phys 2003; 190: 249-274. [49] Cai W, de Koning M, Bulatov V, Yip S, Minimizing boundary reflections in coupled domain simulations. Phys. Rev. Lett. 2000; 85: 3213-3216. [50] EW, Huang Z, Matching conditions in atomistic-continuum modelling of materials. Phys. Rev. Lett. 2001; 87: 135501. [51] Muralidharan K, Deymier PA, Simmons JH, A Concurrent multiscale finite difference time/ domain/molecular dynamics method for bridging an elastic continuum to an atomic system. Modelling Simul. Mater. Sci. Eng. 2003; 11: 487-501. [52] Dhia HB, Multiscale mechanical problems: the Arlequin method. CR Acad. Sc. Paris 1998; 326 Ser-II b: 899-904. [53] Shilkrot LE, Miller RE, Curtin WA, Coupled atomistic and discrete dislocation plasticity. Phys. Rev. Lett. 2002; 89: 025501-1–025501-4. [54] Belytschko T, Smolinski P, Liu WK, Multi-stepping implicit-explicit procedures in transient analysis. In Proceedings of the International Conference on Innovative Methods for Nonlinear Problems. Pineridge Press International Limited, Swansea, U.K., 1984. [55] Bottasso CL, Multiscale temporal integration. Computer Methods in Applied Mechanics and Engineering 2002; 191: 2815-2830. [56] Yin L, Acharya A, Sobh N, Haber R, Tortorelli D, A Spacetime Discontinuous Galerkin Method for Elastodynamic Analysis. In: Cockburn B., Karniadakis G, Shu C, eds. Discontinuous Galerkin Methods Theory. Computation and Applications, Springer, 2000. [57] Miekkala U, Nevanlinna O, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 1987; 8(4): 459-482. [58] Giladi E, Keller HB, Space time domain decomposition for parabolic problems. Numerische Mathematik, 2002; 93(2): 279-313. [59] Fish J, Belsky V, Generalized Aggregation Multilevel Solver. International Journal for Numerical Methods in Engineering 1997; 40: 4341-4361. [60] Fish J, Qu Y, Global Basis Two-Level Method for Indefinite Systems. Part 1: Conver-gence Studies. International Journal for Numerical Methods in Engineering 2002; 49: 439-460. [61] Qu Y, Fish J, Global Basis Two-Level Method for Indefinite Systems. Part 2: Computational Issues. International Journal for Numerical Methods in Engineering 2002; 49: 461-478. [62] Waisman H, Fish J, Tuminaro RS, Shadid J, The Generalized Global-Basis (GGB) Method. International Journal for Numerical Methods in Engineering 2004; 61(8): 1243-1269. [63] Horton G, Vandewalle S, A space-time multigrid method for parabolic partial differential equations. SIAM Journal on Scientific and Statistical Computing 1995; 16(4): 848-864.

Micromechanics and multiscale mechanics of carbon nanotubes-reinforced composites X.Q. Fenga, *, D.L. Shib, Y.G. Huang c, K.C. Hwanga a

c

Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P. R. China b Department of Mechanics, Shanghai University, Shanghai 200444, P. R. China Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Abstract Owing to their superior mechanical and physical properties, carbon nanotubes (CNTs) seem to hold a great promise as an ideal reinforcing material for composites of high-strength and low-density. In the present paper, the stiffening and strengthening physical mechanisms of CNTs in polymer matrix are investigated theoretically by using micromechanics and multiscale mechanics methods. First, the stiffening effect of CNTs in composites is quantitatively examined by micromechanics methods. Second, a hybrid atomistic/continuum mechanics method is established in the present paper to study the deformation and fracture behaviors of CNTs in composites due to the enormous difference in the scales and mechanisms involved in this issue. A unit cell containing a CNT embedded in a matrix is divided in three regions, which are simulated by the atomic-potential method, the quasi-continuum method based on the modified Cauchy-Born rule, and the classical continuum mechanics, respectively. This method can not only predict the formation of Stone-Wales defects, but also simulate the subsequent deformation and fracture process of CNTs embedded in composites. The present study elucidates some key factors (e.g., waviness, agglomeration, residual stress, and interphase) that influence the mechanical properties of CNT-reinforced composites, and therefore may be useful for improving and tailoring their mechanical properties. Keywords: Carbon nanotube; Composite; Constitutive relation; Fracture; Stone-Wales transformation; Multiscale mechanics method; Micromechanics.

1. Introduction Since their discovery in 1991 [1], carbon nanotubes (CNTs) have attracted tremendous research interests due to their extraordinary mechanical, physical and chemical properties [2–5]. Much attention has been paid in the past decade to reveal the physical, chemical and other properties of CNTs and to explore their various potential applications in, e.g., novel nano-materials, devices and systems, and ultra-strong composite materials.

*

Corresponding author. E-mail address: [email protected] (X.Q. Feng). 103

G.C. Sih (ed.), Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano, 103–139. ” 2007 Springer.

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Both theoretical analysis [6–14] and experimental measurements [15–20] evidence that both single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) have Young’s moduli about 1 TPa under uniaxial tension. The Young’s modulus of CNT is a statistically averaged parameter of bonds and defects, and therefore, the results of theoretical calculations and experimental measurements of Young’s modulus generally have a reasonable agreement. However, the tensile strength and breaking strain are sensitive to the local deformation, types and orientations of defects, constraint conditions and some other factors. So the results of theoretical and experimental studies on CNTs strength and fracture often show a pronounced dispersion and inconsistency. The deformation and fracture of CNTs is studied by molecular dynamics simulation in [21]. They found that Stone-Wales transformation is a typical mechanism for fracture nucleation of CNTs and the fracture strains are in the range of 30–55%, depending on temperature. A hybrid atomistic/continuum model was developed to study the mechanical properties of CNTs and predicted an axial breaking strain of about 52%, which corresponds to the onset of deformation bifurcation [22, 23]. Other atomistic studies also predicted that the tensile fracture strains of CNTs could reach up to 30% [24–26]. However, experimental measurements [19, 20, 27, 28] shown that the fracture strains of CNTs are usually less than 13%. Someone [29, 30] conducted atomistic studies on failure of CNTs, and found that the discrepancies between the aforementioned atomistic and experimental studies can be attributed to the non-physical cutoff function in Brenner’s interatomic potential [31] for carbon. Using a modified Morse potential, they predicted the failure strains of CNTs in the ranges of 10–16% and 16–24%, respectively. In spite of the big dispersion between experimental results and theoretical predictions, it is well accepted that the mechanical properties of CNTs are much better than all the commercial fibers. Such superior properties make CNTs seems to be a very promising candidate as ideal reinforcing fibers for producing advanced composites with high strength and low density, which are of paramount interest in astronautical, aeronautical and other industries. Composites of CNTs dispersed in metallic or polymeric matrices have attracted a considerable attention in recent years [32, 33]. Different kinds of CNT-reinforced composites have been synthesized with enhanced properties. For example, a MWCNT reinforced polystyrene with good dispersion and CNT-matrix adhesion was reported in [34]. Using only 0.5% CNT reinforcement, the elastic modulus and tensile strength were improved about 40% and 25% over those of the matrix, respectively. It was found in [35] that by adding SWCNTs in isotropic petroleum pitch matrices, the tensile strength, elastic modulus, and electrical conductivity of the composite with 5 wt% content of purified SWCNTs were enhanced by about 90%, 150%, and 340%, respectively. It was reported in [36] that the incorporation of 1 wt% MWCNTs reinforcement produced a remarkable increase in the tensile strength and elastic modulus for non-drawn UHMWPE composite films of 49.7% and 38% and, more interestingly, enhanced significantly both the ductility and the strain energy absorption before fracture. However, a big difference still exists between these practical improvements and the expectations predicted from theoretical analysis, and many other studies demonstrated only modest improvement in the strength and stiffness after

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CNTs are incorporated into polymers [37–40]. Therefore, there is still a long way to go to make CNT-reinforced composites with superior comprehensive properties and to achieve their extensive applications in industry. Therefore, it is of great interest to investigate experimentally and theoretically the deformation and fracture behaviors of CNT-reinforced composites at nano, micro to macro scales and to examine the factors that influence their mechanical properties [41–56]. The excellent interfaces between CNTs and polymer matrix are reported in [43–45]. It was found in [34] that CNT breaking is a preferred process during cracking of CNT-reinforced film. In situ TEM observation indicated [46] that the external force can be effectively transferred to nanotubes and the embedded CNTs delay polymer film cracking via nanotube stretching and pulling out. Recently, three reinforcing mechanisms of CNTs have been observed: crack deflection at the CNT/matrix interface, crack bridging by CNTs, and CNT pullout on the fracture surfaces [47]. Based on shell theory, the buckling of a double-walled CNT embedded in an elastic matrix under axial compression was studied in [48]. The effective mechanical properties of CNT-reinforced composites have also been evaluated by using a nanoscale representative volume element (RVE) based on continuum mechanics and using the finite element method (FEM) [49, 50]. The nature of the interaction of CNTs and matrix was elucidated via molecular dynamics simulations [51]. A method for linking atomistic simulations of nano-structured materials to continuum models of the corresponding bulk materials was presented in [52], which provided a constitutive modeling of nanotube-reinforced polymer composites. The classical continuum mechanics provides a straightforward approach for estimating the macroscopically averaged or effective properties of deformation and fracture, but cannot predict defect nucleation and fracture at nanoscale. Atomistic studies such as molecular dynamic simulation have been playing an increasingly significant role in studying the physical process occurred in nanosystems like CNTs. However, atomistic studies are generally too computationally intensive for fracture problems of CNTs reinforced composites. In this chapter, micromechanics and multiscale mechanics models are developed to investigate the mechanical properties of CNTs and their composites. The micromechanics models provide important property-microstructure relations for CNT-reinforced composites, while hybrid continuum/atomistic model, which uses the atomistic model for atoms near the defect and the continuum model for atom far away from the defect, allows simulation of fracture behavior of CNTs embedded in matrix. 2. Stiffening Mechanisms of CNTs in Composites The overall effective elastic properties of composites depend upon the properties of individual constituents (e.g., matrix and reinforcing phases), the statistically averaged parameters of microstructures (e.g., the orientations, locations and sizes of inclusions or fibers, and the connectivity of the phases) as well as the interactions of phases. Various micromechanics schemes (e.g., self-consistent method and differential method) have been established to calculate the effective elastic moduli of

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heterogeneous materials. In this section, the influences of some important factors (waviness, agglomeration, and interphase properties) on the stiffening effect of CNTs in composites are investigated [53, 54]. 2.1 Waviness effect Experimental observations have shown that most CNTs in composites exist in a curved state. This is partially because of the very low bending stiffness and the nanometer diameter of CNTs. The finite element method has been used in [55] to study numerically the effect of CNT waviness on the elastic properties of composites. Here an analytical method is presented to calculate the effective elastic moduli of composites containing curved CNTs. Consider a three-dimensional unit cell of CNT composites, as shown in Fig. 1(a) and (b), where a curved CNT is modeled as a helical spring, with D being the spring diameter, ϑ the spiral angle, and ϕ the polar angle. The waviness of the CNT is quantified by the spiral angles ϑ. For instance, ϑ = 0 corresponds to a straight CNT, while ϑ = π 2 corresponds to a circular CNT. The length L of the curved CNT is related to these parameters by

L=

ϕD . 2 cos ϑ

(1)

The Mori-Tanaka method [57] is employed to estimate the stiffening effect of curved CNTs. Fig. 1 (a) shows a curved CNT embedded in a matrix subjected to the average matrix stress ı m in the far field, as assumed by the Mori-Tanaka method. The CNT is curved around the x 3 -axis of the global system o-x1x 2 x 3 . To estimate the stiffening effect of such a curved CNT, the RVE is divided into slices of infinitesimal thickness normal to the x 3 -axis, as shown in Fig. 1(c). The strain in the infinitesimal CNT in Fig. 1(c) is approximated by that in a long and straight CNT of the same orientation embedded in the matrix, as shown in Fig. 1(d). The CNT is along the x′2 -axis in the local coordinate system ( o-x1′ x ′2 x ′3 ), with Euler angles ϑ and ϕ , where ϑ is the angle between x 3 and x′3 , and ϕ is the angle between x1 and x1′ . It is noted that the local axis x′2 of the CNT and the x 3 axis around which the CNT is curved have a fixed angle ϑ . Therefore, the average strain in the curved CNT is obtained by integrating with respect to the angle ϕ . As shown in Fig. 1(c), the strain İ r ( ϑ, ϕ ) of an infinitesimal segment in the curved CNT is related to the stress ı m by

İ r ( ϑ, ϕ ) = A ( ϑ, ϕ ) :İ m = A ( ϑ, ϕ ) : C-1m :ı m ,

(2)

where A ( ϑ, ϕ ) is the strain concentration tensor. For a curved CNT, the average strain İr (ϑ) can be obtained from the integration of İ r ( ϑ, ϕ ) as

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σm x ′2 x′1

x 3′

J

x3

ϕ

x2 x1 a) Curve CNT in RVE

(b ) Infinitesimal thk. Slice

x3 ′

σr σ

x3

x2 ′ σ m

J

m

x2 ′

x1′ x3

x h′

(c) Finite thk. Slice

x2

ϕ x1

x1′

(d) Axes orientation

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İr (ϑ) =

1 ϕL

³

ϕL 0

ª¬ A ( ϑ, ϕ ) : C m−1 º¼ dϕ : ı m ,

(3)

where ϕL is the total polar angle along the CNT. Similarly, the average stress ı r (ϑ) in a curved CNT is given by

ı r ( ϑ) =

1 ϕL

³

ϕL 0

ª¬C r ( ϑ, ϕ ) : İ r ( ϑ, ϕ ) º¼ dϕ

1 ª ϕL Cr ( ϑ, ϕ ) : A ( ϑ, ϕ ) : Cm−1º dϕ : ım . = ³ « »¼ 0 ϕL ¬

(4)

The average stress and strain tensors in the composite can then be written in terms of ı m as

ı ( ϑ) = c r ır ( ϑ) + c m ı m ( ϑ ) ª c ϕL º = « r ³ ( Cr ( ϑ, ϕ ) : A ( ϑ, ϕ ) : C−m1 ) dϕ + cm I » : ım , ¬ ϕL 0 ¼ İ ( ϑ) = c r İr ( ϑ) + c m İm ( ϑ) ªc =« r ¬ ϕL

³ ( A ( ϑ, ϕ ) : C ) dϕ + c ϕL

0

−1 m

m

(5)

º C m−1 » : ı m . ¼

The elimination of ı m in Eq. (5) gives the tensor of effective elastic moduli of the composite reinforced by curved CNTs as

ªc C=« r ¬ ϕL

³

ªc :« r ¬ ϕL

ϕL

0

³

º −1 ¬ªCr (ϑ, ϕ) : A(ϑ, ϕ) : C m ¼º dϕ + cm I » ¼

ϕL

0

−1

º ª¬ A(ϑ, ϕ) : C −m1 ¼º dϕ + c m C −m1 » . ¼

(6)

If ϑ = 0 , Eq. (6) reduces to the effective stiffness tensor of composites reinforced by straight CNTs. The CNTs are assumed to be transversely isotropic elastic. Their elastic stiffness tensor is written as

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­ı11 ½ ª n r °ı ° « ° 22 ° « l r °°ı33 °° « l r ® ¾= « °ı 23 ° « 0 °ı13 ° « 0 ° ° « ¯°ı12 ¿° «¬ 0

(7)

lr k r +m r k r− m r 0 0 0

lr k r− m r k r +m r 0 0 0

0 0 0 mr 0 0

0 0 0 0 pr 0

0 º ­ İ11 ½ ° ° 0 »» ° İ 22 ° 0 » °° İ33 °° »® ¾, 0 » °2İ 23 ° 0 » °2İ13 ° »° ° p r »¼ ¯°2İ12 ¿°

where n r , lr , k r , m r and p r are Hill’s elastic constants. Specifically, k r is the plane-strain bulk modulus normal to the axial direction, n r is the uniaxial tension modulus in the axial direction, lr is the associated cross modulus, m r and p r are the shear moduli in planes normal and parallel to the CNT direction, respectively. Take the representative elastic constants of CNTs and matrix as E m = 1.9 GPa , Ȟ m = 0.3, k r = 30 GPa, m r = 1 GPa, l r = 10 GPa, and n r = 450 GPa [58]. Figure 2 shows the effective elastic modulus of the composite in the CNT axial direction (x 3 ) versus the volume fraction of aligned, curved CNTs in a polystyrene matrix for several spiral angles ϑ . It is observed that the modulus E 3 in the CNT axial direction decreases rapidly as the waviness increases. For example, E 3 at ϑ = 60 D is less than one half of that for straight CNTs ( ϑ = 90 D ). Figure 3 shows the effective elastic modulus E 1 (= E 2 ) of the composite normal to the CNT axial direction. It is seen that contrary to the axial moduli in Fig. 2, the lateral moduli increase with the waviness, even though the increase is rather small when ϑ changes from 90 D to 60 D . Therefore, one can conclude that the CNT waviness has little effect on the lateral moduli unless the spiral angle becomes very small (close to zero). 2.2 Agglomeration effect Since CNTs have low bending stiffness and high aspect ratio, they are easy to agglomerate in a polymer matrix. A micromechanics model is developed to study the influence of the agglomeration of CNTs on the effective elastic moduli of CNT-reinforced composites. The spatial distribution of CNTs in the matrix is assumed to be nonuniform such that some local regions have a concentration of CNTs different from the average volume fraction in the material. Without loss of generality, these regions with concentrated CNTs are assumed to have spherical shapes, and are considered as “inclusions” with different elastic properties from the surrounding material, as shown in Fig. 4. The total volume Vr of CNTs in the

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RVE V can be divided into the following two parts

Vr =Vrinclusion +Vrm ,

Fig. 2. CNT waviness on effective modulus in longitudinal direction.

Fig. 3. CNT waviness on effective modulus in transversal direction.

(8)

Micromechanics and multiscale mechanics of carbon nanotubes-reinforced composites

111

Fig. 4. Eshelby model of agglomeration of CNTs. inclusion

m

where Vr and Vr denote the volumes of CNTs dispersed in the inclusions (concentrated regions) and in the matrix, respectively. Introduce two parameters ξ and ζ to describe the agglomeration of CNTs

V ξ = inclusion , V

Vrinclusion ζ= , Vr

( 0 ≤ ξ, ζ ≤ 1)

(9)

where Vinclusion is the volume of the sphere inclusions in the RVE. ȟ denotes the volume fraction of inclusions with respect to the total volume V of the RVE. When ȟ = 1, nanotubes are uniformly dispersed in the matrix, and with the decrease of ȟ , the agglomeration degree of CNTs is more severe. The parameter ȗ denotes the volume ratio of nanotubes that are dispersed in inclusions and the total volume of the nanotubes. When ȗ = 1 , all the nanotubes are located in the sphere areas. In the case where all nanotubes are dispersed uniformly, one has that ȟ = ȗ . The average volume fraction cr of CNTs in the composite is

cr =

Vr . V

(10)

inclusion

Using Eqs. (8)–(10), the volume fractions of CNTs in the inclusions c r m in the matrix c r are expressed, respectively, as

and

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inclusion r

Vrinclusion ζcr , = = Vinclusion ξ

c (1 − ζ ) Vrm = r cmr = . V − Vinclusion 1− ξ

(11)

Thus, consider the CNT-reinforced composite as a system consisting of inclusions of sphere shape embedded in a hybrid matrix. Both the matrix and the inclusions contain CNTs. First, the effective elastic stiffness of the inclusions and the matrix are estimated respectively, and then calculate the overall property of the whole composite system. With the Mori-Tanaka method, the effective bulk moduli K in and K out and the effective shear moduli G in and G out of the inclusions and the matrix are given, respectively, by

 įr  3K m Į r cr ȗ , 3  ȟ  c r ȗ+c r ȗĮ r c r  į r  3K m Į r 1  ȗ K out = K m + , 3 >1  ȟ  c r (1  ȗ)+c r (1  ȗ)Į r @ c ȗ  Șr  2G mȕ r G in = G m + r , 2  ȟ  c r ȗ+c r ȗȕ r c r 1  ȗ  Șr  2G mȕ r G out = G m + . 2 ª¬1 ȟ c r 1  ȗ +c r 1  ȗ ȕ r º¼ K in = K m +

(12)

Finally, the effective bulk modulus K and the effective shear modulus G of the composite are derived from the Mori-Tanaka method as

ª º §K · ȟ ¨ in  1¸ « » K out ¹ © « », K = K out 1+ « § K in ·»  1¸ » « 1+Į 1 ȟ ¨ © K out ¹ ¼» ¬« ª º §G · ȟ ¨ in  1¸ « » © G out ¹ », G = G out «1+ « § G in ·»  1¸ » « 1+ȕ 1  ȟ ¨ © G out ¹ ¼» ¬« with Į = 1+ Ȟ out

 3  3Ȟ out

and ȕ =  8  10Ȟ out 15  15Ȟ out .

(13)

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Consider first the extreme case of agglomeration where all CNTs are concentrated in spherical subregions, i.e., ȗ = 1. Take the representative values of CNTs and matrix as above. Under different average contents cr of CNTs in the composite, the effective Young’s modulus is plotted in Fig. 5 with respect to the agglomeration parameter ȟ . When the CNTs are uniformly dispersed in the composite, i.e., ȟ = 1, the effective Young’s modulus has its maximum value. With the decrease of agglomeration parameter ȟ , the effective stiffness decreases very rapidly. In more general cases, both the parameters ȟ and ȗ are required to describe the agglomeration of CNTs. With ȟ = 0.2 and ȟ = 0.5, the effective Young’s moduli of composites are shown in Fig. 6. It is seen that with the increase of ȗ , the effective Young’s modulus decreases rapidly too.

, Young s modulus E (GPa)

40

cr = 0.05 = 0.1 = 0.2 = 0.4

30

20

10

0 0.4

0.5

0.6

0.7

0.8

0.9

1.0

Agglomeration parameter ξ Fig. 5. CNT agglomeration on effective modulus with ȗ = 1.

2.3 Interface effect As is well known, interfaces often play a significant role in mechanical properties of nanostructural materials. For CNT-reinforced composites, the high surface area of CNTs creates a large interfacial region. Nevertheless, there are seldom theoretical analyses about the interface effect on the properties of composites because of the nanoscale size of CNTs. Using an equivalent continuum modeling method, a CNT and its interface with matrix has been modeled in [52] as an effective continuum fiber. Here, the three-phase unit cell model is used to describe the microstructure of CNT composite.

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40

, Young s modulus E (GPa)

cr = 0.05 = 0.10 = 0.20

30

20

10

0 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Agglomeration parameter ζ (a) ξ = 0.2

40

, Young s modulus E (GPa)

cr= 0.05 = 0.10 = 0.20

30

20

10

0 0.5

0.6

0.7

0.8

0.9

1.0

Agglomeration parameter ζ (b) ξ = 0.5 Fig. 6. Effective modulus of CNT-reinforced composite with agglomeration effect.

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Fig. 7. Three-phase unit cell model for interface effect.

As shown in Fig. 7, rr , rs and rm denote the outer radii of CNT, interface and matrix phases, respectively. Thus, the volume fractions of the interface and matrix phases, cs and cm , can be written as

cs = ( Į 21) cr ,

cm = 1  Į 2c r ,

(14)

where Į = rs rr . An interphase layer with a finite thickness ( rs -rr ) is assumed between the CNT and the matrix phase to characterize their adhesion properties. The average strain and stress of composites can be written as

ı = ¦ ci ı i = ª¬ cr ( Cr :A r :C m−1 ) + cs ( Cs :As :C m−1 ) + c m I º¼ : ı m , i

İ = ¦ ci İ i = ( cr A r + cs A s+ c m I ) : C m−1:ı m .

(15)

i

Then, the effective stiffness tensor of the composite can be derived as −1

ª º ª º C= « ¦ ci Ci :Ai » : « ¦ ci Ai » , ¬ i ¼ ¬ i ¼ −1

where A i = ª¬I + S:Cm :(Ci Cm ) º¼

(i = r,s, m) −1

.

(16)

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To examine the effect of interface adhesion (or interphase) properties on CNT-reinforced composites, it is assumed the elastic modulus of the interphase in a reasonable range. The interface phase is considered to be isotropic, with the Young’s modulus E s= ȕE m , where ȕ is a nondimensional coefficient. Using the parameters of CNTs and matrix as in Section 2.1, the longitudinal tensile moduli of composites reinforced by aligned CNTs with c r = 0 .1 are plotted in Fig. 8. As the value of ȕ increases from 0.3 to 10, the effective axial tensile moduli increase, but less then 10%. Therefore, the interphase properties have an insignificant influence on the effective stiffness of CNTs-reinforced composites, though interfaces often play an important role in the strength and failure behaviours of composites. The above discussions show that the waviness and agglomeration of CNTs in composites may significantly reduce the stiffening effect of CNTs while the influence of the interphase properties the effective elastic maduli is generally insignificant. The present study not only gives some important relations between the effective elastic properties and the microstructural features of CNT-reinforced composites, but also may provide a guide for improving and tailoring the mechanical properties. The obtained results indicate that to reach superior mechanical properties of CNT-based composites, the CNTs should be controlled to have a straight shape and to be dispersed uniformly within the materials. These high requirements are by no means easy to be satisfied, but considerable developments have been made in this field by researchers of materials science. 60

Longitudinal modulus (GPa)

β = 0.3

55

=1 =5 = 10

50

45

40 1.0

1.2 1.4 1.6 1.8 Nondimensional interphase thickness α Fig. 8. Longitudinal moduli for

cr = 0.1.

2.0

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3. Atomistic-Based Continuum Theory for SWCNTs The deformation and fracture of CNTs were studied [59–61] by molecular dynamics simulation under the condition of high strain rates and high temperature. At the beginning of tension, a CNT undergoes uniform elastic deformation of hexagonal lattice. When the tensile strain reaches a critical value, Stone-Wales transformation occurs, causing the formation of 5-7-7-5 defects. A combined atomistic/ continuum mechanics method [62, 63] will be used in the sequel to investigate the Stone-Wales transformation and uniaxial tension fracture of CNTs under low temperature and quasi-static condition. 3.1 An empirical interatomic potential for carbon The empirical interatomic potential established in [64] for carbon has often been used in MD simulations of CNTs and is also employed here. The energy V stored in the atomic bond between atoms i and j is given by

V(rij ) =VR (rij )-BijVA (rij ),

(17)

where rij is the distance between atoms i and j; VR and VA are the repulsive and attractive pair terms given respectively by

VR (r) =

D(e) − e S-1

D(e)S − VA (r) = e S-1

2Sȕ(r-R (e) )

fc (r),

2 ȕ(r-R (e) ) S

(18)

fc (r),

which depend only on the distance of the two atoms. The parameters D (e) , S, ȕ (e) have been determined from the known physical properties of carbon, and R graphite and diamond as D (e) = 6.0 eV, S = 1.22 , ȕ = 21 nm −1, and R (e) = 0.1390 nm . The function f c( r) is a smooth cut-off function limiting the interaction range between two carbon atoms as

­1 ° ª ʌ(r-R (1) ) º ½° ° 1 ­° f c (r) = ® ®1+ cos « (2) (1) » ¾ ¬ R -R ¼ ¿° ° 2 ¯° ° ¯0

( r ≤ R (1) ) (R (1) < r ≤ R (2) ) , (r > R (2) )

(19)

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with R (1) = 0.17 nm and R (2) = 0.2 nm being the effective range of the cut-off function. The term Bij in Eq. (17) represents a multi-body coupling effect, i.e., the contribution of other atoms to the i-j bond, and is given by

Bij =

1 ( Bij +B ji ) , 2

(20)

while −į

ª º Bij = «1 + ¦ G(ș ijk )f c (rik ) » , ¬ k ( ≠i,j) ¼ ª º c02 c02 « », G(ș ijk ) = a 0 1+ 2 « d 0 d 2 + (1+cosș )2» 0 ijk ¼ ¬

(21)

with į = 0.500 , a 0 = 0.00020813 , c0 = 330 , and d 0 = 3.5 . 3.2 Carbon nanotubes prior to deformation A carbon nanotube can be considered as a rolled graphite sheet. Usually, it is considered that the distribution of carbon atoms on a graphite sheet is regular hexagon. In carbon nanotubes, however, a carbon atom and its three nearest-neighbor atoms form a “pyramid”, and the lengths of C-C bond become direction dependent. To determine the bond lengths in different directions, choose a respective volume element, including a representative atom A and its nearest-neighbor atoms B, C and JJG D, as shown JJG in Fig. 9(a). As conventionally defined in the literature, the vector BC and DC can be denoted as the base vectors a1 and a 2 . The positions of all the atoms can be expressed in terms of five independent parameters, a1 , a 2 , a 3 , a 4 and a 5 . Then the structure parameters of a CNT can be expressed as

C h = na1 +ma 2 ,

(

)

Ch = C h ⋅ C h = n 2 a12 +m 2 a 22 +nm a12 +a 22 -a 32 , 2

dt =

Ch ʌ

(

2

2

2

)

2na1 +m a1 +a 2 -a 3 C h ⋅ a1 = cos −1 , a1 C h 2a1 C h

ș = cos −1 ,

(22)

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Fig. 9. Volume Element: (a) Planar coordinates, (b) Cylindrical coordinates.

where Ch is the chiral vector of the CNT, d t the diameter, and ș the chiral angle. To characterize the mapping between the CNT and the “unrolled” plane, refer to a cylindrical coordinate system ( Z, Θ, R ) , as shown in Fig. 9(b). In this system, the positions of atoms A, B, C and D can be written as [62]

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ZA = a 4sin ( ∠ABC+ș ) , Θ A =

2a 4 cos ( ∠ABC + ș ) d , RA = t , dt 2

dt , 2 2a cosș d ZC = a1sinș, ΘC = 1 , RC = t , dt 2 ZB = 0, Θ B = 0, R B =

ZD = a 3 sin ( ∠DBC+ș ) , Θ D =

(23)

2a 3cos ( ∠DBC-ș ) dt

, RD =

dt . 2

Before the CNT deforms, the distance between two atoms can be written as

rIJ( ) = 0

d t2 2 ª¬1 − cos ( ΘJ − ΘI ) º¼ + ( ZJ -ZI ) . 2

(24)

With Eqs. (17) and (24), one can get the energy of the RVE as a function of five parameters a1 , a 2 , a 3 , a 4 and a 5 as

V ( a1 ,a 2 ,a 3 ,a 4 ,a 5 ) = V ( rAB ) +V ( rAC ) +V ( rAD ) .

(25)

The equilibrium condition requires that the derivatives of the energy V with respect to a I ( I = 1, 2, " , 5 ) vanish, that is,

∂V =0 ∂a I

( I = 1, 2, 3, 4,5 ) .

(26)

Then the positions of the all atoms in the RVE can be determined by solving the equations in Eq. (26). 3.3 Continuum description of deformed carbon nanotubes The Cauchy-Born rule [65] is an extensively adopted kinematic assumption for linking the deformation of an atomic system to that of an equivalent continuum. This rule states that for an atomic structure subjected to a homogeneous deformation, each atom moves according to a single mapping from the initial to the deformed configurations. It is important to point out that the Cauchy-Born rule holds only for centrosymmetric lattice structures. Therefore, it must be modified when used to simulate the deformation of CNTs with hexagonal lattice structure, which does not possess a centrosymmetry. By modifying the Cauchy-Born rule, an atomic potential-based continuum method was developed in [22, 23] for considering the uniform deformation of a CNT.

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The deformation gradient F = ∂x ∂X characterizes the deformation of a material point in the continuum analysis, where the material point represents many atoms that undergo locally uniform deformation, X and x denote the positions of the material point prior to and after deformation, respectively. For a CNT subject to tension, the deformed CNT remains to have a circular cross section, and can therefore be ‘‘unrolled’’ to a plane. Accordingly, the deformation gradient F is intrinsically two-dimensional. This mapping from the ‘‘unrolled’’ plane to a cylindrical surface, which is similar to the method introduced in [62, 63, 66], ensures that the Cauchy–Born rule is applicable to a curved surface. (0) Let rij denote the position vector from atom i to atom j prior to deformation. For a material point subject to the deformation gradient F , the position vector (0) (0) after the deformation. Therefore, the energy stored in rij becomes rij= F ⋅ rij the atomic bonds obtained from the empirical interatomic potential depends on F . Using the Cauchy–Born rule, which equates the strain energy at the continuum level to the energy stored in atomic bonds, one obtains the strain energy density W as a function of the deformation gradient F , i.e., W = W( F) . Such an approach, however, is limited to materials with a centrosymmetric atomic structure. In order to generalize the Cauchy–Born rule for the considered problem, a CNT prior to deformation is composed of two triangular sub-lattices (marked by open and solid circles, respectively), each sublattice possessing a centrosymmetry, as shown in Fig. 10 (a). Once the deformation is imposed, the Cauchy–Born rule discussed above can be applied to each sub-lattice, but the two sub-lattices may undergo a relative shift vector Ȣ [22, 23] and shown in Fig. 10 (b). This shift vector Ȣ plays the role of relaxing the atoms between two sub-lattices in order to ensure the equilibrium of atoms. The position vector rij between atoms i and j from two different sub-lattices then becomes

rij =F ⋅ rij(0) + Ȣ ,

(27)

and their distance is

rij = rij = Ȣ ⋅ Ȣ + 2Ȣ ⋅ F ⋅ rij(0) + rij(0) ⋅ FT ⋅ F ⋅ rij(0) .

(28)

The energy stored in the atomic bonds obtained from the interatomic potential in Eq. (17) now depends on both F and Ȣ . Thus, the Cauchy–Born rule gives the strain energy density W in the continuum analysis in terms of F and Ȣ , i.e., W= W( F , Ȣ) . The shift vector Ȣ is determined by energy minimization, which is equivalent to the equilibrium condition of atoms, i.e.,

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(a) Decomposition

Ȣ

(b) Shift vector Fig. 10. (a) Decomposition of hexagonal lattice for triangular sub-lattices, (b) Shift vector Ȣ between two sub-lattices to equilibrate atoms.

∂W = 0. ∂Ȣ

(29)

This is an implicit equation to determine the shift vector Ȣ in terms of F , i.e.,

Ȣ = Ȣ(F) . The strain energy density then becomes

W = W ( F,Ȣ ( F ) ) .

(30)

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The second Piola–Kirchhoff stress T is the work conjugate of the Lagrangian T strain E = 12 (F ⋅ F-I) , i.e.,

T = ∂W ∂E

(31)

where FT is the transpose of F , and I is the second-order identity tensor. The above method is first used to provide the tensile constitutive curves of CNTs without defect. Several representative stress-strain curves are plotted in Fig. 11(a). The chiral angles and diameters of CNTs influence the tensile behavior. When the deformation is small, the stress–strain relation is linear, and the Young’s modulus is determined in the range of 1.2–1.3 TPa, which agrees with experimental measurements and other numerical results. As the tensile strain increases, the stress-strain curves become nonlinear until the final rupture. The tensile stress-strain curves obtained from our simulation fit well with the experimental results in [18], as shown in Fig.11 (b). 3.4 Defect nucleation due to Stone-Wales transformation Figs. 12 (a) and (b) illustrate our hybrid continuum/atomistic model to study the Stone-Wales transformation in a CNT. Only the “unrolled” honeycomb lattice is shown, but all the calculations are done in the cylindrical configuration of CNTs. As the strain in the CNT reaches a critical value, a carbon bond rotates 90o, leading to the so-called Stone-Wales transformation or 5-7-7-5 ring pair consisting of two pentagons and two heptagons. The atoms in the CNT can then be divided to two groups, as shown in Fig. 12(b). The atoms with a local region Ωd around the defect are marked by open circles, while the others are marked by solid circles. The atoms in the latter group are thought to undergo relatively uniform deformation since the influencing scope of the bond rotation is limited to a local region around the defect according to the Saint-Venant principle. Therefore, the positions of these atoms (far away from the defect) can be determined by the atomistic-based continuum theory. The coordinates of the atoms near the defect are determined by direct molecular mechanics calculations by minimizing the energy of the system. Such an approach involves both continuum and atomistic calculations, and is therefore called a hybrid continuum/atomistic model in the sequel. Use the above hybrid continuum/atomistic model, one can calculate the energy E perfect of the system without the defect (Fig. 12a) and the energy E defect with a hypothesized Stone-Wales defect (Fig. 12b). Both E perfect and E defect depend on the deformation gradient F in the far field. E defect is larger than E perfect at smaller strains, indicating that no Stone-Wales transformation has occurred. As the strain reaches a critical value, E defect may become lower than E perfect (i.e., E defect < E perfect ). Then the Stone-Wales transformation becomes energetically favorable, and defect nucleation may occur. It should be mentioned that the

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adopted criterion of defect nucleation, E defect < E perfect , has not accounted for the kinetic process or energy barrier of defect formation. 200 (17,0) (10,10) (7,0) (5,5)

Stress σ (GPa)

150

100

50

0 0.00

0.03

0.06

0.09

0.12

0.15

0.18

Strain ε (a) Numerical

80

Stress σ (GPa)

60

(17,0) (10,10) (7,0) (5,5) experimental resluts [19]

40

20

0 0.00

0.01

0.02

0.03

0.04

0.05

0.06

Strain ε (b) Comparison with test Fig. 11. Tensile stress-strain curves (a) Numerical simulation (b) Comparison with results of Yu et al.

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Fig. 12. Hybrid continuum/atomistic model for Stone-Wales transformation: (a) Before transformation (b) After transformation.

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In order to ensure a high accuracy of the hybrid continuum/atomistic model, the region ȍ d should contain a sufficient number of atoms. Compare the numerical results for four sizes of ȍ d including 42, 80, 130, and 192 atoms, which correspond to the nearest one, two, three and four atom layers around the defect, respectively. The calculated critical strains of defect nucleation for two representative CNTs, (10, 10) and (17, 0), are given in Fig. 13 with respect to the size of ȍ d , where s stands for the number of atom layers around the defect. It is found that the obtained results converge to constants depending upon the chiral angle, and the estimations of the critical strains and energy from three and four layers of atoms are close to each other. Therefore, the ȍ d containing three layers of atoms around the defect seems suitable to achieve a balance of the accuracy of results and the simplicity of calculation. 0.20

critical strain εdefect

0.18

(17,0) (10,10)

0.16 0.14 0.12 0.10 0.08 0.06

1

2

3

4

Number of atom layers s Fig. 13. Critical Strains for different sizes of

ȍd .

3.5 Results and analysis Our calculation results of the critical strains for some representative CNTs are shown in Fig. 14. It is clearly seen that the critical strains are sensitive to the chiral angle but not on the diameter of CNTs. The Stone-Wales transformation is more easier to appear in armchair CNTs than in zigzag CNTs. The critical strains are about 7.6% and 13.2% for armchair and zigzag CNTs, respectively. These values are larger than the MD results in [60], because the simulated temperature in [60] is up to 1800–2000 K, while in our calculation, the temperature is 0 K. The difference of temperature induces different values of the critical strains of defect nucleation,

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but the relative changing tendencies with the chiral angle are identical at low and high temperatures. Both Nardelli et al.’s MD simulation and our results based on the hybrid atomistic/continuum method show that the critical strains of zigzag CNTs are nearly twice those of armchair CNTs.

Fig. 14. Critical strains of defect nucleation for representative CNTs.

Fig. 15. Tension fracture of a zigzag carbon nanotube (17, 0). (a) Uniform deformation without defect (b) Stable deformation with defect (c) Fracture.

The combined atomistic/continuum method allows also simulation the fracture behaviors of CNTs after defect nucleation. The deformation and fracture of a zigzag CNT (10, 10) and an armchair CNT (17, 0) are shown in Figs. 15(a–c) and 16(a–c), respectively. The changing curves of energy stored in the CNTs with the increasing tensile strain are given in Fig. 17. It is seen that the deformation process

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before final rupture can be divided into three stages. Firstly, when the tensile strain ε is lower than the critical strain of defect nucleation, the deformation of the CNT is completely elastic and uniform. Secondly, once İ reaches the critical strain, the Stone-Wales transformation occurs, causing formation of 5-7-7-5 defects. In spite of the Stone-Wales transformation, the deformation of the CNT is still stable, though non-uniform. Thirdly, when the tensile strain reaches the breaking strain (about 18.8% for zigzag CNTs and 16.3% for armchair CNTs), the deformation becomes highly non-uniform, and the system energy becomes unstable. Our calculations indicate that all CNTs fracture or cleavage in a brittle manner. In other words, at low temperature, the CNTs will fracture quickly without evident increase of the tensile strain after it reaches the breaking strain. Further analysis of CNT fracture will be published in [67].

Fig. 16. Tension fracture of armchair carbon nanotube (10, 10).: (a) Uniform deformation without defect, (b) Stable deformation with defect and (c) Fracture.

Fig. 17. Changing curve of energy.

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4. Multiscale Mechanics Method for CNTs in Composites Due to the enormous difference in scales involved in correlating the macroscopic properties with the microscopic physical mechanisms of nanostructured materials, multiscale analysis is necessary [68, 69]. Now the above hybrid atomistic/ continuum mechanics method is extended to study the deformation and fracture behaviors of CNTs embedded in composites [70].

Fig. 18. Multiscale model for CNT-reinforced composite.

4.1 Calculation model Consider a straight SWCNT embedded in a matrix and subjected to a uniform stress in the far field, as shown in Fig. 18. Assume that the interface between the CNT and matrix is perfectly bonded and that there is no residual strain in the composite. In our multiscale method, the unit cell in Fig. 18 is divided into three zones, A, B and C, which are dealt with in different manners according to their deformation features and the numbers of atoms in them: (i) A local subregion A is specified in the CNT, where Stone-Wales transformation and fracture will be assumed to initiate. The process of defect nucleation and evolution depends strongly on the directions and lengths of individual C-C bonds in this local region. Therefore, an atomistic method based on the Tersoff-Brenner potential is employed to determine the positions of all the atoms in this region by minimizing the total energy of the atomic system either before or after the defect nucleation. (ii) The region B covers all the atoms outside the local region A of the CNT. The positions of the atoms in B are insensitive to the defect and almost identical to

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those in a uniformly deformed, defectless CNT, provided that the size of A is large enough. The region B remains approximately the hexagonal lattice structure. Therefore, the positions of atoms in B can be calculated by using the continuum medium method based on the modified Cauchy-Born rule. (iii) Generally, the polymer matrix around the CNT has an atomistic system much larger than the CNT, and generally does not have a regular lattice. Therefore, the polymer matrix region C in the unit cell is simulated by a continuum medium. Here, the interaction of CNTs and the matrix is assumed to be perfect without considering the interphase between the matrix and CNTs. This multiscale method combines the advantages of both continuum mechanics and atomic-potential methods. The classical continuum mechanics provides a straightforward approach for estimating the macroscopically averaged or effective properties of deformation and fracture of composites, while the direct atomistic simulation allows prediction of defect nucleation and fracture at nanoscale. It will be shown that the present multiscale method is efficient for simulating the nanoscale fracture process occurred in a composite. 4.2 Effect of CNTs interaction The interaction of CNTs in a composite may influence their deformation and fracture behaviors. Many methods have been established in micromechanics for considering the interaction effect. To calculate the deformation gradient F in a 0 CNT as a function of the applied stress ı , the Mori-Tanaka method is adopted because of its simplicity and accuracy even at a high volume fraction of the reinforcing phase. For convenience, the CNTs are considered as straight fibers embedded in the composite. To estimate the interaction effect of distributed inclusions in composites, the Mori-Tanaka method assumes that each inclusion is placed in an infinite pristine 0 matrix and that the far-field stress ı is replaced by the average stress ı m in the matrix. ı m and ı r (average stress in the CNTs) can be written as

(

)

ı m = ı 0 + ı = Cm : İ 0 + İ ,

(

)

(

)

 ′ = Cm : İ 0 +İ+İ  ′– İ* ,  ′=Cr : İ 0 +İ+İ ı r = ı 0 +ı+ı

(32)

where ı denotes the average perturbed stress in the matrix due to the presence of * CNTs, ı ′ is the difference in the average stresses of the two phases, and İ serves as the Eshelby eigenstrain. With the average theorems of stresses ı 0 = (1 – cr ) ım + cr ır and Eshelby’s relation İ ′ = S:İ * , one gets

Micromechanics and multiscale mechanics of carbon nanotubes-reinforced composites

İ =  c  İc  İ* =  c  S  I :İ* .

131

(33)

From Eqs. (32) and (33), the average strain of the CNTs İ r and Eshelby eigen* strain İ reads

İ r = İ 0 + İ + İ′ = İ 0 + ª¬c r I+ (1 cr ) Sº¼: İ* ,

^

İ =   Cr  Cm : ª¬c r I+ 1  c r Sº¼ +Cm *

`

1

(34)

:  Cr  Cm :İ . 0

Then one can get the relation with İ r and İ . The CNTs are assumed to be transversely isotropic elastic. Their elastic stiffness tensor can be written as Eq. (7). The elastic tensor of the matrix, which is assumed isotropic, can be written as 0

Lmijkl = Ȝ m įijį kl + µ m ( įik į jl + įil į jk ) ,

(35)

where Ȝ m and µ m are the Lame constants. From Eq. (34), one gets ª D1 «1 « «¬ 1

1 D2 D3

1 º ­İ10 ½ ª B1 ° ° D3 »» ®İ 02 ¾ + «« B4 D 2 »¼ ¯°İ 30 ¿° «¬ B7

B2 B5 B8

B3 º ­ İ1* ½ ° ° B6 »» ®İ*2 ¾ = 0, B9 »¼ ¯° İ*3 ¿°

(36)

where

D1 =  n r  Ȝ m  2µ m  lr  Ȝ m , D 2 =  k r +m r  Ȝ m  2µ m  lr  Ȝ m , D3 =  k r  m r  Ȝ m  l r  Ȝ m , D 4 =  Ȝ m +2µ m  lr  Ȝ m ,

D5 = Ȝ m  l r  Ȝ m ,

B1 = c r D1 +D 4 + Ȟ m 1  c r 1  Ȟ m , B2 = B3 =c r +D5 + 1  c r  2  2Ȟ m , B4 = B7 =c r +D5 + Ȟ m 1  c r  D 2 +D3  2  2Ȟ m , B5 = B9 =c r D 2 +D 4 + 1  c r ª¬ 4Ȟ m D3  D3 +D 2  5  4Ȟ m º¼  8  8Ȟ m , B6 = B8 =c r D3 +D5 + 1  c r ª¬ 4Ȟ m D 2  D 2 +D3  5  4Ȟ m º¼  8  8Ȟ m , and Ȟ m is the Poisson’s ratio of the matrix.

(37)

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In the case of uniaxial tension along the axial direction ( x1 ) of the CNT (Fig. 0 0 0 18), one has İ 2 = İ 3 =-Ȟ m İ1 . Then, it is easy to solve from Eq. (36) that the eigen* 0 strains İ in the CNT can be written in terms of the applied strain İ as

İ1* =ǻ1İ10 , İ*2 =ǻ 2 İ10 , İ*3 =ǻ 3İ10 ,

(38)

where ∆1 , ∆ 2 , and ∆ 3 are functions of Bi and Di . From Eq. (34), one gets the strain of the CNT as a function of the applied strain as

İ1CNT = (1+cr ǻ1 ) İ10 , ª Ȟ (1 − c r ) ( 5 − 4Ȟ m )(1 − cr ) º ǻ °­ İ CNT = ®− Ȟ m + m ǻ1 + «c r + » 2 2 2 (1 − Ȟ m ) 8 (1 − Ȟ m ) ¬ ¼ ¯°

( 4Ȟ m − 1)(1 − cr ) ǻ ½° İ 0 , 3¾ 1 8 (1 − Ȟ m ) °¿ ­° Ȟ (1 − c r ) ( 4Ȟ m − 1)(1 − cr ) ǻ İ 3CNT = ® − Ȟ m + m ǻ1 + 2 2 (1 − Ȟ m ) 8 (1 − Ȟ m ) °¯ +

(39)

ª ( 5 − 4Ȟ m )(1 − cr ) º ǻ ½° İ 0 . + «cr + » 3¾ 1 8 (1 − Ȟ m ) ¬ ¼ ¿° Finally, the ratio between the circumferential and longitudinal tensile strains of the CNT is obtained as

İ CNT 2(3) İ

CNT 1

=

( cr − 1) E m Ȟ m +lr ( Ȟ m +1) [ −1+cr (2Ȟ m − 1)] . E m (1 − c r ) +2k r ª¬1+Ȟ m − cr ( 2Ȟ m 2 +Ȟ m − 1) º¼

(40)

The above result from the Mori-Tanaka, i.e. Eq. (40), is introduced in the hybrid atomistic/continuum method as the boundary condition of CNTs to account for the constraint effect of the matrix as well as the interaction effect on the deformation and fracture behaviors of the CNTs. The elastic constants of CNTs can be calculated by Section 2. Thus, the defect nucleation and fracture process of CNTs in composite can be simulated by the present multiscale mechanics method. In what follows, for illustration, the special case of uniaxial tension will be considered. Other loadings such as twisting, shearing or complex loading can also be analyzed similarly.

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Strain energy change ∆E (eV)

1.0

CNT not in composite CNT in composite

0.8

(10,10)

0.6

0.4

0.2

0.078 0.075

0.0 0.060

0.065

0.070

0.075

0.080

Tensile strain ε (a) (10,10) CNT

Strain energy change ∆E (eV)

0.5 CNT not in composite CNT in composite

0.4

(17,0)

0.3

0.2

0.1

0.136

0.132 0.0 0.120

0.125

0.130

0.135

0.140

Strain ε (b) (17,0) CNT Fig. 19. Averaged energy with increasing tensile strain: (a) (10, 10) CNT, and (b) (17, 0) CNT.

4.3 Results and analysis

First, compare the critical strains of Stone-Wales defect nucleation of two identical CNTs, one embedded in a composite and the other not. An armchair CNT (10,

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10) and a zigzag CNT (17, 0) are chosen as examples, whose diameters are almost the same. The elastic constants of the matrix are taken as the same as in Section 2.1. For the (10, 10) and (17, 0) CNTs, the changing curves of the averaged energy difference ǻE=E defect − E perfect over the region A are respectively given in Fig. 19 (a) and (b) with respect to the tensile strain. The solid and the dashed lines correspond to the CNTs that are embedded and not embedded in the composite, respectively. When ǻE reduces to zero, the Stone-Wales defect becomes favorable energetically. The critical strain of defect nucleation of the (10, 10) CNT is 7.6%, and becomes 7.8% after it is embedded in the composite. Similarly, the critical strain of the (17, 0) CNT increases from 13.2% to 13.6% after it is put into the composite. Due to the constraint effect of matrix, therefore, Stone-Wales defects become more difficult to form in CNTs embedded in composites, though the change in the critical strain İ SW is relatively small. The present method can not only predict the critical strain of Stone-Wales transformation, but also study the consequent defect evolution and fracture process of the CNT embedded in a composite. After the nucleation of Stone-Wales defects, the deformation of the CNT is still stable until the breaking of C-C bonds. Simulated the deformation and fracture process after defect nucleation, one will find that the deformation will become unstable and localized when the applied strain reaches a critical strain of breaking, İ break . With the localization of deformation, some C-C bonds will become so long that their forces will become very weak according to the Tersoff-Brenner potential. It is generally thought that a C-C bond will break when the distance of the two atoms reaches 0.2 nm. Our simulations indicate that both the armchair and zigzag CNTs will break in the direction nearly normal to the tensile direction, as shown in the insets of Fig. 20. The changing curves of energy with defect evolution and fracture are plotted in Fig. 20. The solid line corresponds to the CNTs embedded in the composite, and the dashed line corresponds to those not embedded. Corresponding to the deformation localization and breaking of a CNT, there is a rapid jump in the energy of the system, indicating the transformation from stable to unstable deformation. It is also found that when a CNT is placed in a composite, its critical strain of breaking will decrease though its critical strain of defect nucleation increases. For example, the critical strain of fracture of the (10, 10) CNT is about 16.3%, and reduces to 15.8% after it is embedded in the composite. Similarly, the critical strain of breaking of the (17, 0) CNT decreases from 18.8% to 17.1% after it is put into the composite. This might also be attributed to the constraint effect of the matrix. A CNT embedded in a composite is less effective to release the energy, and becomes more easily to fracture in comparison with that not embedded in the composite if the matrix is assumed not to fracture before CNTs. Finally, how the stiffness of matrix influences the critical strains of fracture of CNTs will be examined. For (10, 10) and (17, 0) CNTs, the critical strains of fracture are shown in Fig. 21 as a function of the Young’s modulus of matrix. It is found that the breaking strain of CNTs decreases with the increases in the stiffness of matrix.

Micromechanics and multiscale mechanics of carbon nanotubes-reinforced composites

Strain energy E (eV/atom)

0.35 CNT in composite CNT not in composite 0.30

(10,10)

0.25

0.158 0.20 0.150

0.163

0.155

0.160

0.165

0.170

0.175

Tensile strain ε (a) (10,10) CNT

Strain energy E (eV/atom)

0.40

0.35

CNT in composite CNT not in composite

(17, 0)

0.30

0.25

0.171 0.20 0.16

0.17

0.188

0.18

0.19

0.20

Tensile strain ε (b) (17, 0) CNT Fig. 20. Changing curves of energy: (a) (10, 10) CNT, and (b) (17, 0) CNT.

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5. Conclusions

Critical strain εbreak

In the present paper, the stiffening and strengthening physical mechanisms of CNTs in polymer matrix are investigated theoretically by using micromechanics and multiscale mechanics methods. The effect of waviness, agglomeration and interfaces of CNTs are examined. It is established that the waviness and agglomeration may significantly reduce the stiffening effect of CNTs. A combined continuum/micromechanics/atomistic method is developed to simulation the fracture process of CNTs in a polymer. The critical tensile strains of defect nucleation and final fracture are given, which show a strong dependence on the chiral angle. The present study not only provides the important relationship between the effective properties (both elastic modulus and strength) and the microstructure parameters of CNT-reinforced composites, but also may be useful for improving and tailoring their mechanical properties. 0.20

(10,10) (17,0)

0.19

νm= 0.2

0.18

0.17

0.16

0.15

0

20

40

60

80

100

Modulus Em (GPa) Fig. 21. Breaking strains of (10, 10) and (17, 0) CNTs with Young’s modulus of matrix.

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Multi-scale analytical methods for complex flows in process engineering: Retrospect and prospect W. Ge*, F.G. Chen, G.Z. Zhou, J. Li Institute of Process Engineering, Chinese Academy of Sciences, P. O. Box 353, Beijing 100080, China

Abstract The difficulties of scaling in process engineering lie in the multi-scaling of dynamic structure, the behavior of which becomes progressively involved with increasing scales. Multi-scale methodology looms naturally in practice that reveals the complexity of nature. Traditional hierarchical coarse-graining in continuum approaches has faced fundamental closure problems for the highly non-equilibrium multiphase flows in process engineering. An alternative approach called analytical (variational) multi-scale methodology has been proposed in [1]. It was based on the pioneering work on gas-solid multiphase flow [2, 3]. The later development can be found in [4, 5]. The inter-scale correlation was considered as a result of the compromise among the dominant mechanisms underlying the complex flow behavior. Mathematically, the dominant mechanisms can be expressed as extremum tendencies. And the compromise can be expressed as a mutually constrained extremum that constitutes the stability condition. This leads to a closed multi-scale model with dynamic equations. This methodology is presented in this work. The coverage is by no means complete, but it aims to be informative to those with different backgrounds. Implications of the approach to the general area of complex systems and nonlinear science are emphasized and presented to encourage the discovery of new possibilities that otherwise may have been hidden and escape the attention of other researchers. The review in [6] elucidates the methodology and its applications in more details. Keywords: Coarse-graining; Complex system; Compromise; Continuum approach; Extremum tendency; Multi-scale; Particle method; Variation.

1. Introduction The principal mission of the process industry is to make use of materials and/or energies of different forms in application. Involved are the chemical industry, power industry, food processing, mining and metallurgy, just to name a few. They lay the foundation for economy development even in the modern society of telecommunication. Loosely speaking, process industry constitutes the interface

*

Corresponding author. E-mail address: [email protected] (W. Ge). 141

G.C. Sih (ed.), Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano, 141–160. ” 2007 Springer.

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between the human society and the natural environment. Natural resources are proceessed to meet human needs, some of which are consumed directly and some are fed to other manufacturing industries as raw materials. Almost every activity in the human society is in need of material and/or energy in one way or another. In the final analysis, the net (irreversible) effect can be associated with the amount of entropy as a measure of the order or disorder inflicted to the society as a result of un balance between its supply and demand. Thermodynamically put, supplying negative entropy (information), can serve as the essential function of the process industry. It is understandable that these processes are typically non-equilibrium because of the entropy flux and production involved. Information technology may reduce the need for physical activities that require person-to-person contacts. This creates new possibilities. A virtual international conference on the Internet may produce much less entropy than a person-to-person meeting which can generate entropies associated with traveling. The development of telecommunication has also enhanced the knowledge of the universe. Otherwise, the searching for other intelligent lives in the universe would have produced much more entropy. What has transpired by Star Trek is no longer unthinkable in view of the development of “teletransportation” even though it applies only a few atoms at the present. In the same vein, the process industry is expected to progress and expand in the future even though the original concept dates back to the industrial revolution. On the other hand, the relentless pursuit for higher productivity and efficiency has been intensifying improvements for processing. This may involve the building of supercritical power plants for higher conversion ratio of heat to electricity and the invention of micro heat exchangers to achieve extremely high heat transfer capacity in unit volume. From a thermodynamic view point, intensification will generally deviate processing away from equilibrium conditions even further. That is, they will more often operate under nonlinear non-equilibrium conditions. The exponential relations between temperature and reaction rate, as well as the power-law dependence of hydraulic resistance on flow rate in turbulent pipes are among the most widely encountered nonlinearities. They are reflected by the non-equilibrium behavior in processing. The presence of phase change and phase separation, catastrophe and multiplicity has been the rule rather than the exception in the processing industry. It follows that characterizing and quantifying these processes require effective theoretical models in modern science, especially when nonlinearity, complex systems and multi-scale structures are involved. Awareness of the need for modern science in processing is gaining ground. In retrospect, the lack of theoretical guidelines and simulation tools in the scaling-up and scaling-down of laboratory processes has long been a bottleneck in industrial application. Despite of what has ben said, traditional mono-scale approaches still prevail in process engineering. Reliance on the validity of quasi-equilibrium without paying detail attention to scaling can still be found. This is because of the incompatibility between the scale limitation of the methods and the actually size range of the industrial facilities. Much of this can be eliminated by appropriate modeling and the use of virtual computational simulation techniques.

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Although multi-scaling has been used widely in other fields since the 1990’s, the advent of this approach to process engineering is quite limited. According to [1], there are three mainstreams in multi-scale simulation: descriptive, with descriptions on different scales for different spatio-temporal portions of the system integrated into one simulation; correlative, with smaller scale simulations providing constitutive correlations for simulations on the next larger scale; and analytical (variational), with equations on different scales correlated by stability conditions to give a lumped description of the system. Also in this order, the emphasis of the three approaches shifts from numerical techniques reproducing the complex phenomena to theoretical insights of the underlying mechanism. The second approach has a relatively long history. Its idea has been well exemplified by the turbulent models which are coarse-grained presentations of the Navier-Stokes equation, such as Reynolds-stress models (RANS, e.g. [7]) and large eddy simulations (LES, ref. [8]). However, the first approach is practiced more in recent years, such as the integration of molecular dynamics simulations into finite element numerical calculations to describe fracture developments in material mechanics [9,10] and in micro-flows [11,12]. The third approach is still in its infancy but it is believed to be a more profound solution considering its close relation to the study of complex systems. For particle-solid flows, the energy minimization multi-scale (EMMS) model [2, 3] has been a typical embodiment of this approach, and will be introduced as the theme of this chapter. As the EMMS model is proposed to solve the closure problems in correlative multi-scale approaches for particle-fluid flows, it will start with a brief discussion on this issue. 2. Hierarchical Coarse-graining in Particle-fluid Flows For single phase turbulence, hierarchical coarse-graining provides the most straightforward way towards a multi-scale approach to the hydrodynamics of particle-fluid two-phase flows. When phase-specific material properties are constant, the coarse-grained mass and momentum balance equations are written as:

∂ ˆ =0 (α k ρ k ) + ∇ ⋅ α k ρ k U k ∂t 1 ∂ n ˆ ) + ∇ ⋅ (α ρ U ˆ J k ⋅ dAik , (α k ρ k U k k k k U k ) = α k ρk g k + ∇ ⋅ α k P k + V ³Ai ∂t

(1) (2)

where α, ρ, U, g, P and J denote phase fraction, density, velocity, bulk force intensity, stress tensor and momentum flux, respectively, with subscript k being the phase index (“f” for fluid, “p” for particle and “i” for interface), V the control volume and dAi the outward surface element on the phase interface. Noting that, for constant phase-specific properties, phase-specific average and density weighted average, denoted by the superscripts “=” and “^” respectively, are actually identical, and the different notations are kept for comparison with other formulations. The problem is whether all these averaged terms can be expressed explicitly in the coarse-grained variables involved in the equations, namely α and U, otherwise

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the equation set is incomplete. The answer turns out to be nontrivial even for the simplest item. For the fluid phase, the stress tensor P k can be derived from the constitutive laws for a Newtonian fluid that

Pf = µ f (∇U f +∇U f T ) − (p f − ηf ∇ ⋅ U f )I ,

(3)

where µf and ηf are the shear and secondary viscosities, respectively. In our case, since ρf is constant and hence ∇ ⋅ U f = 0 , it can be simplified to

Pf = µ f (∇U f +∇U f T ) − p f I .

(4)

Averaging of this expression results

Pf =

T Pf µ f (∇U f +∇U f ) − pf I . = αf αf

(5)

Therefore, the problem is how to express ∇U f in terms of the coarse-grained variables, for which it should be noticed that

∇U f = ∇ U f +

1 U f dA i . V ³Ai

(6)

It is evident if each component of the tensor is written out and the differentiation rules for functions with discontinuities are applied [13]. To simplify the second term on the right hand side, only the case of rigid particles is considered, for which Uf on Ai can be decomposed into a translational part and a rotational part, that is,

U fi = U fc + Ȧ p × R ci ,

(7)

where subscript “c” denotes particle center of mass. So that

³

Ai

U f dAi = ³ U fc dAi + ³ Ȧ p × R ci dAi , Ai

Ai

(8)

As Ai is a closed surface and Ufc is independent of dAi, the first item on the right hand side is zero. For the second term, it can be written out explicitly as (for simplicity, the phase index for the components are omitted):

§ ωy rz − ωz ry · ¨ ¸ ³Ai ¨ ωz rx − ωx rz ¸ ( dA x ¨ω r −ω r ¸ y x ¹ © xy

dA y

dA z ) .

(9)

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A general expression for its components is

³

Ai

ωi rjdAi k = ωi ³ rjdAi k

(10)

Ai

Once again, as Ai is a closed surface, any value of rj corresponds to a pair of dAik’s with different signs if j k, which means the term is zero in that case (ref. Fig. 1). If j = k, the integration equals to the volume above the plane rk = C mk minus that below the plane as seen in Fig. 1, which equals to twice of the volume between the planes rk = Cvk and rk = Cmk, denoted as Vck here. Now in summary,

§ 0 2(1 − α f ) ¨ 1 ij = ³ U f dA i = ¦ ¨ ωz Vcx p A V i ¦ p Vp ¨ −ω V © y cx

−ωz Vcy 0 ωx Vcy

ωy Vcz · ¸ −ωx Vcz ¸ , 0 ¸¹

(11)

where subscript p is an index of the particles in the control volume V. It can be proved similarly that for two-dimensional cases, ω = ωz, and

ij=

§ 0 1 2(1 − ε) U f dA i = ω¨ ¦ ³ p V Ai ¦ p Vp © Vcx

− Vcy · ¸. 0 ¹

(12)

Obviously, for homogeneous solid particles, ij = 0 , and this is in fact a precondition for most formulations of particle-fluid flows, but for porous particles which are very common in process industry, ij ≠ 0 in general and

Fig. 1. Surface integration on an irregular rigid particle.

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Pf =

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* µf ∇ U f +ij − pf I , αf

(

)

(13)

where for a matrix M, M*=M+MT. Finding a theoretical expression for ij or merely measuring its values in flows may prove to be very difficult, because it depends not only on the structure of the particles, but also on their orientations in the flow, which is closely related to the fluid flow field below the scale of averaging and is not likely determined without introducing some empiricism.Unfortunately, what has been discussed is comparatively the simplest item to be expressed. The difficulties with other terms are much more complicated. For P p , the most theoretical way so far is to treat the particle phase as some kind of gas and derive the constitutive laws from its kinetic theory [14]. However, even for real gases, the kinetic theory is only well established for simple molecules under moderate conditions where the quasi-equilibrium assumption is well satisfied. For solid particles, their geometrical irregularity, inelasticity and most importantly, their strong interactions with the interstitial fluid which results the instability of particle suspension and highly non-uniform concentration distribution even on small scales, as well as many other factors, all drive them far from equilibrium. This is understandable in view of the flow velocities of the solids which are of the same magnitude or even lower than their fluctuation velocities [15], which means the flow of solids can be regarded as highly subsonic or supern introduces the nonlinearity that has been the major sonic. The inertial term U k Uk difficulty faced in the study of turbulence, no strict closure of this term has been found so far, and the chance for particle-fluid flows is even remoter. For the inter-phase friction term, that is, the last term of right hand side of Eq. (2), empiric correlations are used even for uniform suspension [14], and the effect of heterogeneity below the averaging scale has been totally overlooked until recently [16-18], though as will be demonstrated later, this can lead to errors in orders. So far, it can be justified to say that the coarse-graining route to correlative multi-scale models still have a formidably long way ahead, and the framework described above, which are usually called “two-fluid models”, are essentially mono-scale approaches yet, since the heterogeneities below the averaging scales have not been quantified or even considered effectively. Maybe a common difficulty associated with the quantification of smaller scales is the lack of scale separation between the hydrodynamic, kinetic and molecular behaviors in such flows, or in other words, the strong coupling between these scales. Cross-scale linkage thus becomes the central issue and the grand challenge for a multi-scale model of particle-fluid flows. The analytical approach is actually advantageous in this aspect. 3. Analytical Methodology The basic idea of analytical multi-scale approach is that, instead of finding the expressions explicitly, variational criteria are proposed to introduce additional constraints to the incomplete equation set. In this sense, it may also be named “variational” multi-scale approach. However, for most variational approaches in practice

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(ref. e.g. [19]), the equations are all known, at least theoretically, but some of the equations are then equivalently presented by variational criteria so as to employ an optimization scheme incorporated into the numerical solution of partial differential equations. Many simple equation sets, such as those for incompressible viscous flows and perfect fluid flows have their variational counterparts [19], but usually the latter is not advantageous for numerical solutions. For analytical multi-scale approaches, however, the emphasis is on how to propose varitional criteria from physical basis. The variational criteria may serve as short-cuts to the hidden dynamical equations, but they may work even when corresponding analytical equations are intrinsically nonexistent. Such variational criteria are usually of thermodynamic nature, with the second law of thermodynamics (that is, the maximization of entropy in isolated systems) being a typical example. Although its relation with micro-scale dynamics, especially the paradox of irreversibility, is still controversial, it is clearly not the consequence of the Newton’s law of motion only, but also the properties of the interactive mass points in the systems, such as the potential curves for their interactions and their initial conditions. That is variational criteria are deductive rather than inductive. The basic idea is to establish a genuine analytical multi-scale model for complex systems can be summarized as the follows: Physically, the compromising between dominant mechanisms resulting in the correlation between scales which shapes the multi-scale structure. Mathematically, the dominant mechanisms are expressed as extremum tendencies and their compromising are expressed as a mutually constrained extremum, which constitutes the stability condition and leads to a closed multi-scale model with dynamic equations. 3.1 Introduction to the EMMS model This idea is well presented by, and has actually stem from the energy minimization multi-scale (EMMS) model [2, 3] proposed for concurrent-up particle-fluid two-phase flow in the 1980’s. As shown in Fig. 2, the original model has based on a rather simplified physical picture. The flow was considered as one-dimensional, fully developed and free of wall effect. The suspension was assumed to separate distinctly into a particle-laden mixture called the dense “phase” (denoted by the subscript “d”) and a fluid-dominated mixture called the dilute “phase” (denoted by the subscript “l”). The flow was thought to be uniform and steady within each “phase”, which was then uniquely specified by the particle and fluid velocities (Vp and Vf), and voidages (that is, αf) in each phase, together with the volume fraction of the dense “phase” (f) and its “diameter” (l) since it was assumed to occur as spherical clusters. Six dynamic equations are introduced in the model. Firstly, all effective particle weight in the dilute “phase” is balanced by fluid drag, that is,

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1 − α fl π 3 dp 6

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CD0 (

U rl d p

α

νf 4.7 fl

)

π 21 d p ρf U rl2 = (1 − α fl )(ρp − ρf )g 4 2

(14)

Fig. 2 Physical picture of the EMMS model.

where Url=Ufl-Uplαfl/(1-αfl)=(Vfl-Vpl)αfl is the so-called superficial slip velocity of the dilute “phase” and CD0 is the drag coefficient function (The term “superficial velocity” denoted by U will be used frequently here, which is the imaginary flow velocity for the whole cross-section under the same flowrate, therefore, Ufl=Vflαfl, Upl=Vpl(1-αfl)=Vplαpl, Ufd=Vfdαfd and Upd=Vpd(1-αfd)ҏ=Vpdҏαpd). Secondly, effective particle weight in the dense “phase” is partially supported by the fluid flow inside, and the rest is supported by the bypassing dilute “phase” fluid flow, termed here as “inter-phase” (dilute-to-dense phase, also denoted by the subscript “i”) drag, that is,

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U ri l ) 1 − α fd νf π 21 νf π 2 1 2 d U l ρf U 2ri = (1 − α fd )(ρp − ρf )g (15) ρ + p f rd 4.7 π 3 π 4 2 4 2 α fd dp l3f 4.7 6 6 where Urd=Ufd-Updαfd/(1-αfd) and Uri=(1-f)(Ufl-Updαfl/(1-αfd)), which are dense phase and “interphse” slip velocities, respectively. Thirdly, the pressure drop in the dense “phase” is balanced by that of the dilute “phase” plus the “inter-phase”, that is, C D0 (

1 − α fl π 3 dp 6

CD 0 (

U rl d p

νf α fl4.7

1 − α fd = π 3 dp 6

CD 0 (

)

α

C D0 (

π 21 f d p ρf U 2rl + 4 2 1− f

U rd d p νf

)

)

4.7 fd

U ri l ) νf π 2 1 l ρ U2 π 3 4.7 4 2 f ri lf 6

CD0 (

(16)

π 21 d p ρf U 2rd 4 2

Besides the force balances, the continuity of the fluid and the particles requires (17)

U f = fU fd + (1 − f )U fl , U p = fU pd + (1 − f )U pl ,

(18)

where Uf and Up are the given mean superficial velocity of the fluid and the particles. Finally, a semi-empiric correlation for cluster diameter (l) is proposed (ρp − ρf )gU p

− Nst,mf ρp (1 − ε max ) l , = dp Nst − N st,mf

(19)

where by definition Nst=Wst/(1-αf) αp with αf=ҏfαfd+(1-f) αfl being the mean voidage, and αmax is the maximum αf exists for heterogeneous particle-fluid flow which is close to unity. The volume-specific energy consumption for suspending and transporting solids Wst =

1 − α fd π 3 dp 6

CD0 (

U rd d p νf

α fd4.7

)

π 21 d p ρf U rd2 U fd f + 4 2

U rl d p ª º U l CD0 ( ri ) «1 − α C D 0 ( ν ) π 1 » 1 ν π fl f f « d 2p ρf U rl2 + f l 2 ρf U ri2 » U fl (1 − f ) 4.7 π 3 4.7 4 2 α fl 4 2 « πd 3 » lf «6 p » 6 ¬ ¼

(20)

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The subscript “mf ” denotes the state of minimum fluidization where packed particles are just fully suspended. It can be derived from the foregoing equations that

N st,mf =

ρ p − ρf ρp

(U f,mf +

Upα f,mf 1 − α f,mf

)g

(21)

The constraints of the model consist of Eqs. (14)-(19), which involve 8 independent variables: Ufl, Upl, Ufd, Upd, afl, afd, f and l. To close the model, the stability condition NstÆmin is proposed to express the compromise between the tendency of the fluid to pass through the particle layer with least resistance, that is WstÆmin, and the tendency of the particle to maintain least gravitational potential, that is αf Æmin [20, 21]. Physically, such a compromise results in the aggregation of both phases to display a multi-scale heterogeneous structure. 3.2 Validation and Application The EMMS model has been checked against a wide range of experimental data and in situ measurements on commercial equipments, such as circulating fluidized bed reactors and boilers (ref. [1, 3-6, 20-22] and the references therein). Considering the simplicity of the model and the uncertainties in both modeling and measurements, the general agreement found in quite different systems under various conditions has been a convincible validation of the model. Maybe, its theoretical rationality and significance has been best demonstrated by its ability to provide a physical mapping of fluidization regimes which are partitioned by different saltations and bifurcations [20, 21]. Especially, an elegant interpretation has been found to shed some light on the long-lasting controversy about “choking” in vertical gas-solid pipe flows [21]. The highly heterogeneous regime of “fast fluidization” and the nearly homogeneous regime of “dilute transport” were found to have the same Nst under critical pairs of Up and Uf, so there can be jumps between these bi-stable states. The characteristic flow variables at choking, which have poorly predicted with even wrong orders of magnitude has now reduced to meeting engineering requirements and has found industrial applications (see chapter 8 of [6]). The calculation is now available on the Internet (http://www.ipe.ac.cn/csms). It is worthy of noting that this big picture of the nonlinearity and multiplicity in particle-fluid flows has not been well captured even for the most detailed mono-scale or correlative multi-scale models previously. In recent years, the EMMS has been incorporated into the framework of two-fluid models presented in Sec. 2 [18, 22], so as to establish a correlativeanalytical multi-scale method. The aggregation of particles may result in a dramatic drop in inter-phase drag as compared to uniform suspension. Numerical simulations with two-fluid models can not capture this effect below the element scale, which attributes greatly to their low accuracy. On the other hand, though the EMMS model has described this heterogeneity properly, it was originally applied on the vessel scale where its input of particle and fluid velocities are known as

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operational parameters. The integration employs two-fluid models to provide the velocities on the element scale to the EMMS model, and the EMMS model returns its calculated inter-phase drag coefficient to the two-fluid model for the next time step. In fact, a software interface has been developed to dock with mainstream commercial softwares for computational fluid dynamics (CFD). Such integration can improve the accuracy of two-fluid models greatly, in terms of solids concentration, solids circulation rate and the resolution for multi-scale heterogeneities [18, 22]. This progress has paved way for widespread applications of the EMMS model to practical facilities with complex configurations and operation modes, which provides new justifications of the model in turn. The existence of the stability condition has also been evidenced on the micro-scale with pseudo-particle modeling [23, 24], an ab initio method for simulating hydrodynamic behavior, which discretizes the apparently continuous phase into interactive fictitious particles. Different from other particle methods that are numerical discretization of continuum equations, pseudo-particle modeling resembles the physical processes of the simulated media on micro-scales. For this reason, it can deal with very complex systems at great ease. Moreover, its errors are mainly from the mapping of physical properties rather than numerical calculations, which can be analyzed and controlled much more easily. The evolution of mini particle-fluid systems were simulated, and the temporal variations of Nst in the systems were monitored (intrusively!) with machine accuracy, which was found to decrease monotonously to asymptotic values under different conditions, though with slight fluctuations [4, 5, 25]. On the other hand, common features have also been found in the compromise of different mechanisms, especially through the simulations with pseudo-particle modeling [4, 5, 25]. That is, each dominant mechanism can only realize its extremum tendency locally and instantaneously [1, 4, 5]. In fact, this is the way of spatial and temporal compromising on this scale, and this fluctuating establishes a spatio-temporal coupling which results in the meso-scale structures. As can be found in experiments as well as in simulations, the particles tend to aggregate into cluster of certain preferred shapes on the meso-scale, such as the U shape. The EMMS model has actually based its physical picture on this scale, which suggests that the model is only applicable when the space volume and time interval considered are large enough for the dominant mechanisms to find a compromise, and hence, the stability conditions become effective. In general, it requires that the statistical fluctuations associated with Nst and the relaxation time for the compromising process are small compared with this spatio-temporal domain, which are usually satisfied by the element sizes and time steps used for two-fluid models. On even larger scales, boundary conditions (such as walls, inlets and outlets) takes effect. The stability condition is still applicable, but the compromising process are expressed as the minimization of the averaged Nst over the whole cross-section, which leads to the radial distribution of this heterogeneity. As a result of this meso-scale compromising, the fluctuations of the averaged Nst are further smeared out on the global scale and the minimization tendency is best seen on the macro-scale.

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3.3 Extension and generalization The strategy of analytical multi-scale model as demonstrated by the EMMS model has been applied successfully for quite different systems. In single-phase pipe flow, the effect of viscosity to achieve minimum viscous dissipation and the effect of inertia to achieve maximum total dissipation are the pair of dominant mechanisms shaping the flow profile at different Reynolds numbers which reflects the relative dominance between them. At infinitely low Reynolds number, the flow is dominated exclusively by viscosity, resulting in the purely laminar regime characterized by the parabolic velocity profile. Conversely, at infinitely high Reynolds number, the flow is dominated exclusively by inertia, leading to a flat profile. Between these two extremes is the turbulent regime which has been shown quantitatively [26] that the compromising of the two mechanisms results in the well-known power-law profiles. The EMMS model has been extended to calculate the flow patterns of gas-solid-liquid three-phase fluidization [27], in which the liquid phase and the solids were considered to be a mixture and compromises with the gas phase. The predicted bubble sizes and phase fractions are in reasonable agreement with experimental data. A more sophisticated model has been proposed recently to predict the bubble size in bubble columns [28]. Compromise in other situations, such as that between the hydrophilic and lipophilic interactions in emulsion systems [29], diffusion and reaction processes in chemical reaction systems, is also being studied now. These case-studies are well demonstrating the ubiquity of compromising between dominant mechanisms in complex systems and the generality of the strategy embodied in the EMMS model. Due to the diversity of dominant mechanisms for different systems, the possibility of finding a general stability criterion is very remote, though it can be conceptually formulated into a multi-objective variational problem [1]. However, it seems promising that the physical researches to identify the dominant mechanism in different systems and mathematical works to solve the multi-objective variational problem may lead to the establishment a general methodology for analytical multi-scale modeling. 4. Prospects Although the EMMS model and its extensions have been successful as analytical multi-scale methods for engineering purpose, the theoretical foundation of the methods is yet to be consolidated. So far, the stability conditions have been proposed heuristically as prevision and validated by experimental data and known theoretical results. It is, therefore, highly desirable to deduce them from first principals, and in fact, a lot of fundamental problems will be faced in this process. As mentioned above, variational principals have been established in hydrodynamics under certain conditions [19], among which three cases are of close relation to the analytical multi-scale models discussed in this chapter, they are [19]:

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1) The Helmholtz-Korteweg principle for incompressible Stokesian flow which asserts that the steady state flow field has the minimum viscous dissipation among all flow fields having the same flow velocities on the boundaries and satisfying the continuity conditions. 2) For the suspension of particles and drops in slow flows, the total rate of dissipation (viscous and frictional (on the wall)) and potential energy (gravitational and interfacial) buildup for the steady motion is the minimum among all flow fields having the same flow velocities on the boundaries and satisfying the continuity conditions. 3) The Bateman-Kelvin principle for compressible inviscid steady flow which asserts that the enthalpy minus the internal energy plus twice of the kinetic energy is minimized among all velocity fields that satisfy the equation of state and the continuity conditions in a constant entropy process. The first principle is consistent with the effect of viscosity assumed by the analytical multi-scale approach to single-phase pipe flow. However, the effect of inertia to maximize total dissipation is not apparent from the third principle, but it must be noted that this effect is only proposed for single-phase pipe flow and the existence of a laminar bottom in the boundary layer is considered where the third principle is not applicable [26], so it may not conflict with the third principle though their relationship is unclear. Physically, it is understandable that viscous and inertial effects will compromise in any real flow, as expressed by the full Navier-Stokes equation. However, a general variational principle has found to be nonexistent for this equation (see Chapter 8 of [19] for an early review). Therefore, variational principles for turbulent flows should be derived with considerations to specific conditions, say, geometry, Reynolds number, etc. The second principle is consistent with the EMMS model in that both have expressed the compromise between the minimization of viscous dissipation and potential energy production. However, the second principle is based on steady motion and transient configuration of the phase interfaces, while the EMMS model is for a spatio-temporal domain of certain size, within which local accelerations and interface motions present all along. Besides that, the EMMS model is also not restricted to flows with negligible inertial effect. One way to generalize the second principle is to assume that the evolution of the system is always nearly steady, that is, the acceleration term in the momentum balance equation is negligible anytime anywhere in the flow field, as compared to other terms. In this case, the second principle is approximately applicable to each conformation passed in the evolution, and the actual spatio-temporal distribution of the fluid and particle velocities should accumulate least dissipation among all admissible distributions along the same conformational path (note that the potential energy depends on the conformation only). Self-organization of multi-scale structures in complex systems often involves a fast process with strong acceleration and, after the force balance has been well established, a slow but long-lasting process during which the structures undergo significant changes, such as glass transition. Weak but long-range interactions are expected to take effect at this stage. As shown

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in Fig. 10 of [25], particle-fluid systems and gas-liquid systems can display such characters also. Therefore, the quasi-steady assumption made here should be appropriate and very helpful to multi-scale methods. A possible application of this extension is to improve the accuracy of particle-trajectory models, an alternative of two-fluid models with the difference of tracking the motion of each particle explicitly. So far, arbitrary methods are used in simulations to interpolate the fluid velocities obtained from the computational grid onto particle centers [30]. Now, exact interpolation rules can be derived from the second principle using the particle positions and velocities known from previous time steps. Equal pressure drop along the direction of cell averaged slip velocity under given flow rate may work as an effective simplified rule. As both the potential energy and the actual dissipation rate are conformation-dependent only for a given system, they will approach to or fluctuate around bounded values in long term. A somewhat surprising outcome of this property is that for infinitely long time, the average potential increment is zero while average energy dissipation rate is positive, so whether their sum can be minimized is, after all, completely determined by the dissipation process. This conclusion is, however, misleading. The compromise process actually takes place from the initial state to the BEGINNING of the steady state (or the average state if the flow field is always fluctuating). The relative dominance of the two mechanisms: dissipation and potential is likely characterized by their values at these two ends and/or the changes between them. A general form may looks like φtc+ψ Æ min,

(22)

where φ, ψ and t are the characteristic values of dissipation rate, potential energy and evolution time, respectively. The additive form is certainly suggested by the second principle. It is intuitively reasonable to speculate that

φ = φ∞ ;

∞

t c = ³ (φ − φ∞ )dt φ∞ ; 0

ψ = ψ∞ − ψ0 ,

(23)

though many others can be tested. The coupling and hence the compromise between a dissipation (irreversible) process and a conservative (reversible) potential change is very common in various non-equilibrium systems, such as particle-fluid systems where the potential is mainly gravitational and gas liquid systems where surface tension also contributes to the potential (as shown in Fig. 3), especially at high surface-to-volume ratio. For compressible and non-Newtonian flows, the potential will also be stored in elastic deformation. Therefore Eq. (22) should be of great importance and wide applications if proved. However, the nonlinear inertial term in the Navier-Stokes equation has not been considered in Eq. (22), so it is still not general enough to recovery the EMMS model. Actually, the Navier-Stokes equation presents another paradigm of comprising, where thermodynamically linear processes (dissipative and irreversible)

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are coupled into a framework of nonlinear dynamical process (basically conservative and reversible), which results great complexity. As mentioned above, general variational criterion has found to be nonexistent for such systems. In the typical example of turbulence, the intrinsic transport coefficients such as viscosity, conductivity and diffusivity, are almost constant usually, so locally, the thermodynamic behavior is linearly non-equilibrium only. The nonlinearity is introduced by the inertia of the fluid as a dynamic contribution which results in convection. Therefore, nonlinear non-equilibrium thermodynamics is only relevant to the statistical behaviors of turbulence, where linear relations may reestablish, such as the constant effective viscosity in an isotropic homogeneous turbulence.

The liquid molecules are simulated by the shifted and truncated Lennard-Jones potential which takes σl=1, εl=1, rcl=4 and mass ml=4. For gas molecules simulated with pseudo-particle modeling, σg=0.5, m=1. The interaction between gas and liquid particles uses the repulsive terms only with σlg=1.25, εlg=0.5 and rclg=1.5. The time step is 0.01 and the flow field is 140 by 180. The liquid in the control layer (140 by 20) keeps constant flow velocity Vc=0.20 and temerature kTc=2.89. The adaptive bulk force G applies uniformly on the gas phase only, which always balances the drag FD from the liquid phase. Starting from equilibrium at t=3600 (when Vc is suddenly applied to the whole liquid phase), the compromise results in the increase of both dissipation rate φ=FDVc and interface potential ψs at first, and then a “tug of war” between the variables, that is, from a lose-lose game to a series of win-lose games, and finally approaches fluctuating “steady” state which seems to be a win-win game.

Fig. 3. Compromise of least dissipation and least potential increment in a gas-liquid flow with pseudo-particle modeling and molecular dynamics simulation.

It is interesting to note that, back to the molecular scale, the interactions which give linear transport laws are again highly nonlinear (just thinking of the famous Lennard-Jones potential). The alternation between linearity and nonlinearity along

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with increasing or decreasing scales should be inherently related to the compromising process, though its physical implications and possible applications are unclear yet. The coordination of local and global behavior may present another remarkable “ability” of complex systems to reach optimal comprise between dominant mechanisms. In many cases, the compromise is not reached at some locations and time intervals, so as to reach a better global compromise [4, 5, 25]. For instance, the particles on the surface of a cluster in particle-fluid flow are continuously stripped way so as to maintain a low-resistance state of the cluster in terms of shape and velocity distribution. The energy dissipation and (gravitational) potential increase are both large on the surface but low for the whole cluster in comparison with a homogeneous suspension. Such phenomena are also very common in more complex systems such as the human society where the individuals may sacrifice their interest for the overall welfare of a group or the whole society. Therefore, exploration to the origin of this “intelligence” may tough the essence of the compromising and complex systems. However, complex systems are not always “clever” in that strengthening the constraints for one dominant mechanism does not necessarily enhance the expression of this mechanism, nor does strengthening both constraints enhance the extent of compromising. Escape panic is a typical example [31], as well as traffic jams in many cases, in that stronger drive to move may result in blocking and hence much slow motion, so “haste makes waste”. For vertical particle-fluid flow in a thin pipe, increasing gravity and fluid flowrate may lead to slugging (with horizontal layer of particles and fluid alternatively which blocks the pipe), which is not favorable to the movements of both phases. For a wider pipe, however, particle clusters or fluid bubbles may form as a result of the “clever” self-organization of the system, so that both movement tendencies are better realized. When the system will be “clever”, or when the compromising will turn out to be a “win-win”, “win-lose” or “lose-lose” game is probably another general and fundamental problem for further study. Moreover, the study is of practical importance for the design and control of complex systems, which may provide a behavioral “phase diagram” of the system. From a statistical view, the relative number of microscopic state favorable to one or both movements and the probability to access them will be decisive factors to the realization of corresponding movement tendencies. The former is more dependent on the configuration of the system, say the pipe diameter for the particle-fluid flow, and the latter also depends on the speed to reach a stable state and hence the intensity of the constraints exerted, say the gravity and fluid flowrate in the pipe. The important problems for analytical multi-scale methods discussed above are not likely solved in theoretical studies only. Computer simulation, as it is now for most studies, may serve as a powerful probe. Particle methods such as pseudo-particle modeling [23, 24] that have been used to validate the EMMS model (see Sec. 3) are favorable approaches for this purpose, owing to their apparent physical reliability and computational simplicity [32]. Though particle methods are more costly in general, they are almost the only choice for the description of micro-scale and sometimes meso-scale (such as fine powders and polymers) phenomena, since

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nomena, since no continuum model naturally exists. And of course, micro-scale description is fundamental to the study of multi-scale approaches. The particle models can be classified into hard- and soft-particle models in general. The former keeps exact energy conservation but is computationally complicated and not very flexible in simulating complex fluids, while the latter is just on the contrary. Pseudo-particle modeling has been an attempt to combine the virtues of both models while avoiding their pitfalls [23, 24]. In this approach, the so-called pseudo-particles (PPs) undergo free flights and transient collisions in the style of hard-particle models, but the collisions are synchronized at the end of each time step for overlapping particles. It provides a simple particle-model that is conservative, time-driven and variable in size. However, pseudo-particle modeling has, so far, considered repulsive interactions between particles only. Therefore, it is not an adequate model for liquids and gases near the critical point, where the attractive forces between the molecules becomes important for displaying surface tension, tensile elasticity and instabilities. A simple remedy is that, for two pseudo-particles leaving each other, the magnitude of their relative velocity must exceed a certain threshold for the two particles to separate beyond a certain distance. The idea comes from the perception that soft-particles must overcome an attractive potential barrier to leave apart infinitively, which requires a certain amount of relative velocity just after the repulsive potential is fully released. As the repulsive potential in soft-particle models is replaced by an infinitely high step potential in hard-particle models, the attractive tail of the potential can be squeezed similarly to an impulse in the potential curve, which has a finite integrated value. In implementation, the effect of this potential barrier takes place at the end of the free flight period in each time step when the particles have already passed the location of the impulse. When a particle interacts with multiple neighbors, a predetermined order is applied to maintain the homogeneity and isotropy of the interactions. Test simulations seem to have reached a state of gas-liquid coexistence. For more complex fluid such as polymers, a group of pseudo-particles can also be linked in this way so as to present their chained molecules. An interesting recent discovery is that, for the particles used in macro-scale pseudo-particle modeling [32-34] and smoothed particle hydrodynamics (see [35] for a historical review), the viscoelasticity they display in simulations, which has been regarded a source of numerical error, are found to reproduce reasonably Bingham type non-Newtonian flow behavior under certain conditions, as illustrated in Fig. 4. Besides the practical significance of suggesting a simple way to develop particle model for complex fluids, the physical implication that the finite size effect of coarse-grained fluid particles bears similarities to the kinetics of complex molecules may be of greater interest to the analytical multi-scale methods. The ordered distribution of the particle in Fig. 4 suggests that the nonlinear properties are related to the self-organization of the particles. Together with the “long time tail” shown in pseudo-particle modeling [24], they have revealed more complex picture of the particle motion, they are definitely not stochastic!

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In summary, multi-scale modeling and simulations are still at its early stage. However, it has demonstrated advantages in both accuracy and computational cost over traditional mono-scale approaches for both theoretical explorations and practical applications. It is promising that with the developments in physics and chemistry, and the perfection of related mathematical and computational tools, the multi-scale method can be developed to satisfy the requirement for quantitative scaling of chemical processes, or even virtual processes. And this will, at the same time, contributes substantially to the progress of complexity science. 0.8

Effective viscosity (-)

0.7

0.6

Sheared side

0.5

Fixed wall

0.4

0.3

0.2 0

2

4

6

8

10

-5

Shear rate ( 10 ) The scheme is described by J.P. Morris, P.J. Fox, Y. Zhu, Modeling low Reynolds number incompressible flows using SPH, Journal of Computational Physics 136 (1997) 214-216. The parameters are normalized by the mass and interactive distance of the smoothed particles and the numerical time step, with the fluid density, viscosity and pressure being 12.83, 0.2566 and 1.668x105, respectively. Snapshots of particles shown here seems to suggest that ordering of the particle under stronger shearing, especially near the sheared side, may be the reason, similar to shear thinning in complex fluids.

Fig. 4. Non-Newton behavior of the Bingham type displayed in Couette flows simulated with smoothed particle hydrodynamics.

Acknowledgements The authors are grateful to the financial support from Natural Science Foundation of China (NSFC) under the grants 20336040 and 20221603; and Chinese Academy of Sciences under the grant KJCX-SW-L08. References [1] Li J, Kwauk M, Exploring Complex systems in chemical engineering – the multi-scale methodology. Chemical Engineering Science 58 (2003) 521-535. [2] Li J, Multi-Scale Modeling and Method of Energy Minimization for Particle-Fluid Two-Phase Flow, Doctor thesis, Institute of Chemical Metallurgy, Chinese Academy of Sciences, Beijing, 1987.

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[3] Li J, Kwauk M, Particle-fluid two-phase flow energy-minimization multi-scale method, Metallurgical Industry Press, Beijing, 1994. [4] Li J, Zhang J, Ge W, Liu X, Multi-scale methodology for complex systems, Chemical Engineering Science 59 (2004) 1687-1700. [5] Li J, Ge W, Zhang J, Li J, Multi-scale compromise and multi-level correlation in complex systems, Trans IChemE Part A, Chemical Engineering Research and Design 83 (2005) 574-582. [6] Li J, Ouyang J, Gao S, Ge W, Yang N, Song W, Multi-Scale Simulation of Particle-Fluid Complex Systems, Science Press, Beijing, 2005. (In Chinese) [7] Launder BE, Reece GJ, Rodi W, Progress in development of a Reynolds-stress turbulence closure, Journal of Fluid Mechanics 68 part 3 (1975) 537-566. [8] Lesieur M, Metais O, New trends in large-eddy simulations of turbulence, Annual Review of Fluid Mechanics 28 (1996) 45-82. [9] EW, Huang Z, Matching conditions in atomistic-continuum modeling of materials, Physical Review Letters 85 (2001) 135501-1-135501-4. [10] Curtin WA, Miller RE, Atomistic/continuum coupling in computational materials science, Modelling and Simulation in Materials Science and Engineering 11 (2003) R33-R68. [11] O’Connell ST, Thompson PA, Molecular dynamics-continuum hybrid computations: a tool for studying complex fluid flows. Physical Review E 52 (1995) 5792-5795. [12] Garcia AL, Bell JB, Crutchfield WY, Adaptive mesh and algorithm refinement using direct simulation Monte Carlo, Journal of Computational Physics 154 (1999) 121-134. [13] Jackson R, Locally averaged equations of motion for a mixture of identical spherical particles and a Newtonian fluid, Chemical Engineering Science 52 (1997) 2457-2469. [14] Gidaspow D, Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions, Academic Press, San Diego, 1994. [15] Carlos CR, Richardson JF, Solids movement in liquid fluidised beds-I particle velocity distribution, Chemical Engineering Science 23 (1968) 813-824. [16] Zhang DZ, VanderHyden WB, The effects of mesoscale structures on the macroscopic momentum equations for two-phase flows. International Journal of Multiphase Flow 28 (2002) 805-822. [17] Agrawal K, Loezos PN, Syamlal M, Sundaresan S, The role of meso-scale structures in rapid gas-solid flows. Journal of Fluid Mechanics 445 (2001) 151-185. [18] Yang N, Wang W, Ge W, Li J, CFD simulation of concurrent-up gas-solid flow in circulating fluidized beds with structure-dependent drag coefficient, Chemical Engineering Journal 96 (2003) 71-80. [19] Finlayson BA, The Method of Weighted Residuals and Variational Principles, with Application in Fluid Mechanics, Heat and Mass Transfer, Academic Press, New York, 1972. [20] Li J, Wen X, Ge W, Cui H, Ren J, Dissipative structure in concurrent-up gas-solid flow, Chemical Engineering Science 53 (1998) 3367-3379. [21] Ge W, Li J, Physical mapping of fluidization regimes ˉ the EMMS approach, Chemical Engineering Science 57 (2002) 3993-4004. [22] Yang N, Wang W, Ge W, Wang L, Li J, Simulation of heterogeneous structure in a circulating fluidized bed riser by combining the two-fluid model with the EMMS approach, Industrial and Engineering Chemistry Research 43 (2004) 5548-5561. [23] Ge W, Li J, Pseudo-particle approach to hydrodynamics of gas/solid two-phase flow, in: J. Li, M. Kwauk (Eds.), Proceedings of the 5th International Conference on Circulating Fluidized Bed, Science Press, Beijing, 1996, pp. 260-265. [24] Ge W, Li J, Macro-scale phenomena reproduced in microscopic systems–pseudo-particle modeling of fluidization, Chemical Engineering Science 58 (2003) 1565-1585. [25] Zhang J, Ge W, Li J, Simulation of heterogeneous structures and analysis of energy consumption in particle-fluid systems with pseudo-particle modeling, Chemical Engineering Science, 60 (2005) 3091-3099. [26] Li J, Zhang Z, Ge W, Sun Q, Yuan J, A simple variational criterion for turbulent flow in pipe, Chemical Engineering Science 54 (1999) 1151-1154.

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Multiscaling effects in low alloy TRIP steels G.N. Haidemenopoulos*, A.I. Katsamas, N. Aravas Department of Mechanical and Industrial Engineering, University of Thessaly, Volos, Greece, 38334

Abstract Low-alloy TRIP steels are a new class of steels with excellent combinations of strength and formability, which offer a unique field for the study of multiscale effects in materials, in the sense that experimental observations and models referring to different scale levels have to be combined, for the understanding and the design of these steels. In the present work, models involving multiscale physical quantities are reported, which regard prediction of the stability of retained austenite and of the kinetics of its mechanically-induced transformation to martensite, optimization of the heat-treatment stages necessary for austenite stabilization in the microstructure, as well as prediction of the mechanical behaviour of these steels under deformation. Austenite stability depends on chemical composition, austenite particle size, strength of the matrix and stress state, i.e. on factors ranging from the nano- to the macro-scale. The stability of retained austenite against mechanically-induced transformation to martensite is characterized by the M sı temperature, which can be derived as a function of the aforementioned multiscale factors by an appropriate model presented in this work. The kinetics of the mechanically-induced transformation of retained austenite to martensite are also dependent on multiscale factors, such as the population density of martensitic nucleation sites, the retained austenite particle size and the macroscopic level of plastic deformation. In the present work, a model describing the kinetics of this mechanically-induced transformation as a function of these factors is presented. Furthermore, the mechanical behaviour of TRIP steels also depends on the amount of retained austenite present in the microstructure, which is determined by the combinations of temperature and temporal duration of the heat-treatment stages undergone by the steel. Optimum amounts of retained austenite require optimization of the heat-treatment conditions. A physical model is presented in this work, which is based on the interactions between bainite and austenite during the heat-treatment of TRIP steels, which allows for the selection of treatment conditions leading to the maximization of retained austenite in the final microstructure. Finally, a constitutive micromechanical model is presented, which describes the mechanical behaviour of TRIP steels under deformation, taking into account the evolution of the microstructure during plastic deformation. This model is then used for the calculation of forming limit diagrams (FLD) for these complex steels, thus allowing for the optimization of stretch-forming and deep-drawing operations.

*

Corresponding author. E-mail address: [email protected] (G.N. Haidemenopoulo).

161 G.C. Sih (ed.), Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano, 161–178. ” 2007 Springer.

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1. Introduction Low-alloy TRIP steels are a relatively new class of steels that exhibit excellent combinations of strength and cold-formability, making them particularly suitable for sheet-forming applications in the automotive industry. These steels possess a multiphase microstructure containing ferrite, bainite and retained austenite. During cold-forming operations, such as stretch-forming and deep-drawing, the retained austenite transforms to martensite under the action of the applied stresses and strains. This mechanically-induced martensitic transformation of the retained austenite is responsible for the TRansformation-Induced Plasticity (TRIP) effects found in these materials. These effects include significant improvements in ductility and formability. Low-alloy TRIP steels offer a unique example for the study of multiscale effects in materials, in the sense that experimental observations and models, derived at different scale levels, can be combined for the understanding and design of these materials. The objective of this work is to shed light on the interplay between different scale levels, from the atomic scale to the macro or continuum mechanics level. 2. Multiscale Effects in Austenite Stability 2.1 The scales of stability

Fig. 1. The scale of the factors influencing austenite stability.

TRIP effects in the steels under consideration are influenced significantly by austenite stability. If austenite stability is low, then the austenite will transform very early during the forming operation, without beneficial effects in formability.

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On the other hand, if austenite stability is too high, then the austenite might not transform at all during the forming operation. Thus, an optimum level of stability is required in order to maximize formability. Austenite stability depends on the following factors: (a) chemical composition, (b) austenite particle size, (c) matrix strength and (d) stress state. These factors belong to different scale levels, as depicted in Fig. 1. Chemical composition refers to the atomic scale. In order to increase stability, the austenite should be enriched with carbon or other stabilizing elements by atomic diffusion. Austenite particle size refers to the micro-scale or the so-called microstructure scale. In order to increase stability, the austenite particle should be made finer, in order to reduce the probability of finding heterogeneous nucleation sites in the particle for triggering the martensitic transformation. The strength of the matrix also refers to the micro-scale. High strength destabilizes the austenite due to the higher mechanical driving force contribution to the total driving force for the martensitic transformation. In addition, it is well established that strength is controlled by microstructure via the various strengthening mechanisms. Finally, the stress state refers to the macro-scale. Its effect comes about due to the interaction of the transformational volume change with stress triaxiality. High stress triaxiality destabilizes the austenite. For example, consider an austenite particle, with a certain size and chemical composition, which could resist transformation in uniaxial tension. The same particle would transform readily in the triaxial stress field ahead of a crack tip. This is of particular importance when designing TRIP steels for forming applications, due to the complex stress states found in these processes. The austenite stability should be tuned for maximum TRIP interactions for the particular stress state. It is apparent that a model for the stability of austenite, accounting for all these factors, is needed. Since the parameters of the model come from different size scales, the model itself should possess a multiscale character, in the sense that it will incorporate these parameters in the same equation. 2.2 Modelling of stability

Fig. 2. Temperature for the start of martensitic transformation as a function of applied stress.

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The first step towards developing a model for the stability of austenite against mechanically-induced transformation is to identify a single parameter to characterize stability. In this work, the M sı temperature has been chosen for this task, in the same way that the Ms temperature is used to characterize the stability of austenite against transformation on cooling. The Msı temperature can be defined by considering the mechanically-induced martensitic transformation of austenite in Fig. 2. Spontaneous transformation occurs on cooling the austenite to the Ms temperature. This transformation is triggered at pre-existing nucleation sites in the austenite. The same sites can operate also above the M s temperature under the action of an externally applied stress. The higher the temperature, the higher the required stress. This stress-assisted transformation is denoted by line AC. At point C and at the Msı temperature, the applied stress reaches the yield stress of the austenite. Above Msı new potent nucleation sites, which are produced by the plastic deformation of austenite, trigger the strain-induced transformation. Thus the M sı temperature defines a boundary between the temperature regimes where separate modes of transformation dominate: below Msı the transformation is stress-assisted and above Msı the transformation is strain-induced. Near the Msı temperature both modes operate. Due to transformation plasticity the observed yield stress follows the stress for stress-assisted transformation below the Msı . A reversal in the temperature dependence of the flow stress provides a convenient determination of the Msı temperature. Actually this technique has been used successfully for the determination of the Msı temperature in steels containing austenitic dispersions, either as retained austenite in martensitic steels [1], or more recently for retained austenite in low-alloy TRIP steels [2]. Having defined the Msı temperature as the single parameter characterizing stability, the model objective is to develop an expression for the Msı as a function of the four stability parameters defined in the previous section, i.e.,

M sı = Msı (Xi ,Vp ,ı y ,ı h /ı)

(1)

In Eq. (1), Xi denotes the mole fraction of alloying elements (i.e. the chemical composition) in austenite, Vp is the mean austenite particle volume, ıy is the matrix yield strength and ı h /ı is the ratio of the hydrostatic to the von Mises equivalent stress, which characterizes the triaxiality of the stress state. The model details are given elsewhere [3], so only the key points will be presented here. The model is based on the fact that, for the case of stress-assisted transformation, the applied elastic stress aids the transformation kinetics by modifying the effective potency distribution of pre-existing nucleation sites. According to [4], heterogeneous martensitic nucleation can proceed by the dissociation of an existing defect, which has the form of a dislocation array on an existing grain boundary or interphase boundary. The dissociation of such a defect produces a fault structure or martensitic embryo with a thickness of n crystal planes, which possesses a fault energy Ȗf(n) per unit area:

Multiscaling effects in low alloy TRIP steels

165

Ȗ f (n) = n ȡ ( ǻG ch +Estr +Wf ) +2 Ȗ s

(2)

In Eq. (2), ǻGch is the chemical driving force for martensitic transformation per unit volume, Ȗs is the specific fault/matrix interfacial energy, ȡ is the density of atoms in the fault plane, Estr is the elastic strain energy per unit volume associated with distortions in the fault interface plane and Wf is the frictional work of interfacial motion, which occurs during the dissociation process. Spontaneous martensitic nucleation occurs when Ȗf (n) ≤ 0. In this case, the dissociation is barrierless and occurs at a critical value of the driving force. Based on the above, the potency of a nucleation site can be defined by the thickness (n) of the nucleus that can be produced from the defect by barrierless dissociation. The critical n for nucleation follows from Eq. (2) as:

n= −

2 Ȗs ȡ× ( ǻG ch +Estr +Wf )

(3)

Based on the above, it was derived in [5] that the cumulative defect-potency distribution from the Cech and Turnbull small-particle experiments in Fe–30%Ni alloys [6] as:

N v (n) = N ov exp( − a n)

(4)

In Eq. (4), Nv(n) is the number density of sites of sufficient potency to nucleate o martensite, N v is the total number of nucleation sites of all potencies, and Į is a constant. The effect of stress on the potency distribution can be found by adding a mechanical contribution term, ǻGı, to the chemical driving force of Eq. (3) to obtain:

ª º 2 a × Ȗs N v (ı) = Nvo × exp « » ¬ ȡ × ( ǻG ch +ǻG ı +E str +Wf ) ¼

(5)

The term ǻGı of Eq. (5) is equal to:

ǻG ı = ı

∂ ( ǻG ) ∂ı

(6)

In Eq. (6), ǻG is the overall Gibbs free energy change for the austenite to martensite transformation and ı the applied stress. The stress-assisted transformation of a dispersion of austenite particles of average particle volume Vp is controlled by the potency distribution of Eq. (5). The fraction of particles f to

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transform is equal to the probability of finding at least one nucleation site in the particle, assuming that a single nucleation event transforms the entire austenite particle to martensite. This probability is:

f =1 − exp ( − N v × Vp )

(7)

The transformation stress ı = ıt, at which the martensitic transformation is triggered, can be found by combining Eq. (5), (6) and (7) as:

­ ½ ° ° 2 a ⋅ γs 1 ° ° σt = − ∆G ch − E str − Wf ¾ ® ∂ ( ∆G ) ° ª ln(1 − f ) º ° ln ρ ⋅ «− o » ° ∂σ ° «¬ N v ⋅ Vp »¼ ¯ ¿

(8)

In the above equation the chemical driving force term is temperature and composition dependent. For the system Fe-C-Mn this term has a general form:

ǻG ch = f1 (X C ,X Mn ,T)

(9)

The frictional work term is also a function of chemical composition:

Wf = f 2 (X C ,X Mn )

(10)

The mechanical driving force contribution term is a function of stress-state:

∂ (ǻG ) = f3 (ı h /ı) ∂ı

(11)

The detailed analytical expression for the functions in Eqs. (9) to (11) are given in [3]. The Msı temperature can be found by letting the transformation stress to be equal to the yield stress (ıy) in Eq. (8) and solve for the temperature, since the term ǻGch is temperature dependent. This operation yields a general expression for the Msı temperature as:

­° ½° c1 + σy [d1 + d 2 (σh / σ)]¾ Msσ = (a1 + a 2 XC )−1 ®b1 + b2 XC + b3XMn + b4 XC XMn + ln ( c2 / Vp ) ¯° ¿° (12) Constants ( a i ,bi ,ci ,d i ) in Eq. (12) depend on the specific steel composition and,

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167

as stated earlier, are given in [3]. The key point is, however, that Eq. (12) provides an analytical expression for the Msı temperature as a function of the chemical composition of austenite, the austenite particle size, the yield strength and the stress state. It is interesting that parameters from three size scales (atomic, micro and macro) enter this equation. 3. Kinetics of the Mechanically-induced Transformation

As already explained with the aid of Fig. 2, transformation-induced plasticity can occur by two distinct mechanisms, i.e. stress-assisted and strain-induced transformation of retained austenite to martensite. In the stress-assisted regime martensite forms on pre-existing nucleation sites, whereas in the strain-induced regime new and more potent nucleation sites are created by plastic deformation of austenite. As the steel is stressed and deformed, retained austenite will transform to martensite by the simultaneous operation of both mechanisms. The stress-assisted mechanism prevails at stresses lower than the yield-strength of austenite, whereas the strain-induced mechanism prevails after the yield-strength has been surpassed. The total volume fraction of retained austenite transforming to martensite (f) can be expressed in the following form:

f = f(ı)stress + f(ı, İ)strain

(13)

In Eq. (13), f(ı)stress and f(ı,İ)strain denote the contributions of the stress-assisted and strain-induced mechanisms, respectively. As already mentioned, the formation of a martensitic nucleus can occur by the dissociation of an existing defect, which serves as a nucleation site for the transformation [4]. Dissociation of such a defect creates a fault structure or martensitic embryo, the growth of which is determined by the energy change accompanying the dissociation. The energy per unit area of an embryo with a thickness of (n) crystal planes, Ȗf(n), was given by Eq. (2). The potency of a nucleation site (defect) can be expressed in terms of the thickness n (in number of crystal planes) of the nucleus, which can be formed from barrierless dissociation of the defect. The critical value of n, for nucleation at a given chemical driving force per unit volume, was given by Eq. (3), whereas the cumulative defect-potency distribution was described by Eq. (4). In the case of transformation-induced plasticity, the potency distribution of pre-existing nucleation sites in the stress-assisted region is given by: ª 2Į×Ȗs º » «¬ A ×ρ »¼

NVstress = NoV × exp «

(14)

is the density of pre-existing where A = ǻGch + (1/3)ǻGı,max + Estr + Wf. Here Nstress v o nucleation sites and N v the density of pre-existing nucleation sites of all

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potencies. Eq. (14) contains the term ǻGı,max, which stands for the stress contribution to the driving force. This term has been determined in [7]:

ǻG ı,max = −

(

1 ı Ȗo2 +İo2 +ı×İ o 2

)

(15)

In Eq. (15), Ȗo and İo are the transformation shear and normal strains, respectively. A similar expression can be employed to describe the potency distribution of the new, strain-induced nucleation sites: ª 2Į × Ȗs º » ¬« B × ρ ¼»

Nstrain =NoV (İ) × exp « V

(16)

where B = ǻGch+ǻGı,max+Estr+Wf. All parameters in Eq. (16) represent the same physical quantities as explained for Eq. (14). It should, however, be emphasized that in contrast to Eq. (14), the total strain-induced nucleation sites density up to plastic strain İ, N ov (İ) , is in this case a function of plastic strain İ. The overall density of nucleation sites is equal to the sum of Eqs. (14) and (16):

N v =Nstress +Nstrain =Nov × exp ( Įstress × nstress ) + Nov (İ) × exp (− Įstrain × nstrain ) v v

(17)

Parameters nstress and nstrain are given by: §

2Ȗs · ¸ ¸ © A ×ρ ¹

n stress = − ¨¨

(18)

for the stress-assisted and, §

2γ s · ¸ ¸ © B ⋅ρ ¹

n strain = − ¨¨

(19)

for the strained-induced regimes of the transformation, respectively. Assuming that nucleating defects are randomly distributed, the volume fraction of retained austenite transforming to martensite will be given by Eq. (7). Subsequently, combining Eqs. (17) to (19), Eq. (7) gives: ­

ª

§ 2Ȗs ×Į

f=1− exp °®−Vp × «« Nov × exp ¨ ° ¯

«¬

¨ ©

stress

A ×ρ

§ 2Ȗs ×Į · strain ¸ +Nov (İ) × exp ¨ ¸ ¨ B ×ρ ¹ ©

· º ½° ¸» ¾ ¸» ° ¹ »¼ ¿

(20)

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Multiscaling effects in low alloy TRIP steels

An expression for the total strain-induced nucleation sites density up to plastic strain İ, N ov (İ) , has been proposed in [8]:

Nov (İ)=N × ª«1 − exp ( −k × İ n )º» ¬

(21)

¼

In Eq. (21), N represents the overall number of nucleation sites that can be induced by plastic strain, while k and n are dimensionless constants. Introducing Eq. (21) into Eq. (20), the later becomes: ­

ª

§ 2γ s ⋅α

f = 1 − exp ®°−Vp ⋅ «« Nov ⋅ exp ¨ ° ¯

«¬

¨ ©

stress

A ⋅ρ

· ¸ + N ⋅ ª1 − exp ¸ ¬« ¹

§ 2γ s ⋅α

( −k ⋅εn )»º ⋅ exp ¨ ¼

¨ ©

strain

B ⋅ρ

·º ½ ¸ » ¾° ¸» ¹ »¼ °¿

(22) Eq. (22) establishes a relation between the vol. fraction of retained austenite transforming to martensite (f) and true plastic strain (İ). It is, therefore, possible to predict the kinetics of the mechanically-induced transformation as a function of plastic strain. Fig. 3 depicts calculated (lines) and experimentally measured (symbols) ‘‘f–İ’’ kinetics for two typical low-alloy TRIP steels. The nominal chemical composition of steel ‘‘TRIP 1’’ was 0.20C–1.40Mn–0.50Si–0.70Al (in % mass), whereas steel ‘‘TRIP 2’’ had the same composition with a small addition of 0.03% mass Nb. As shown, model results display good agreement with experimental measurements. It should also be noted that a significant amount of retained austenite is transforming in the elastic region, due to the stress-assisted mechanism.

Fig. 3. Calculated and experimental results of vol. fraction retained austenite transformed to martensite as a function of plastic strain.

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Fig. 4. Typical heat-treatment processing of cold-rolled low-alloy TRIP steels.

4. Modelling of Heat-treatment for Austenite Stabilization

As stated in the introduction, TRIP effects and the associated formability enhancement depend directly on the amount and stability of retained austenite. A large research effort over the last years has focused on establishing the suitable heat-treatments for obtaining stable retained austenite in the microstructure. A typical heat-treatment for the production of a TRIP steel is shown in Fig. 4. It involves intercritical annealing to produce a ferrite-austenite mixture, followed by cooling to an intermediate temperature and holding for the isothermal transformation of austenite to bainite. During the bainitic transformation, carbon is rejected from the forming bainite to the austenite. This carbon stabilizes the remaining austenite against martensitic transformation on cooling to room temperature. The microstructure of a TRIP steel contains typically 50% ferrite, 40% bainite and 10% retained austenite. A research effort is underway to model the heat-treatment of Fig. 4, in order to calculate the amount and carbon enrichment of retained austenite and preliminary results will be presented here. Calculations involve physical modeling, as well as computational alloy thermodynamics and kinetics. The first stage involves the simulation of intercritical annealing. Thermo-Calc [9] and DICTRA [10] softwares have been employed to calculate the volume fraction and composition of austenite that forms during the intercritical annealing, by solving the 1-D moving boundary diffusion problem in the two phase field of ferrite-austenite [11]. The results of the simulation serve as the starting point for the simulation of the bainitic transformation of austenite and the associated stabilization during the second step of the heat-treatment. The model is based on the physical characteristics of the bainite transformation and on the fact that every bainitic ferrite platelet that forms contributes to austenite stabilization by carbon rejection to the remaining austenite. The model adopts a concept, originally introduced in [12], according to which each bainitic ferrite platelet (sub-unit) nucleates and instantaneously grows to a finite volume, which is controlled by the accommodation of plastic deformation in the surrounding austenite. From this

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mechanism it follows that once a bainitic ferrite sub-unit has formed, it begins to reject the excess carbon to the adjacent austenite layers. A key assumption here is that no carbide precipitation takes place. However, this assumption is reasonable for the case of TRIP steels, which contain Si and/or Al in excess of 1% mass [13]. In this way transient carbon profiles develop in the austenite, which tend to homogenize with time (Fig. 5). Taking into consideration that there exists a Ȗ minimum carbon content, w C, , which is necessary for the stabilization and min retention of austenite after quenching, it follows that each sub-unit contributes to the retention of a potential volume of austenite, characterized by the thickness s Ȗ R of the austenite layer. This thickness varies with time, as it depends on the position of the transient carbon concentration profile in austenite. This way the total amount of retained austenite can be calculated by: (a) considering the evolution of the population of bainite sub-units with transformation time and assuming that bainitic ferrite grows according to Johnson-Mehl-Avrami-Kolmogorov (JMAK) kinetics and (b) considering the population of “effective” sub-units, i.e. those sub-units, which contribute to austenite stabilization at any given point during the transformation. Details of the model are given in [14].

Fig. 5. Transient C-concentration profiles in austenite adjacent to bainitic ferrite sub-units and the corresponding thickness of the stabilized austenite layer ( s Ȗ ). R

Comparisons of experimental results (symbols) with model predictions (lines) are given in Fig. 6 for a Fe-0.2C-1.5Mn-1.5Si (% mass) TRIP steel. Intercritical

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annealing was performed at 800oC, while bainitic transformation was performed at 350, 400 and 450oC. The retained austenite was measured by the saturation magnetization technique [15]. As shown, the model provides very good results and can be used for the design of heat-treatments, aiming at obtaining the maximum amount of retained austenite in the final microstructure of TRIP steels.

Fig. 6. Experimental and calculated results for the evolution of retained austenite, as a function of temperature and holding time in the bainitic treatment stage.

5. Constitutive Modelling of TRIP Steels 5.1 Description of the model

Based on micromechanical considerations, a constitutive model was developed in [16] for the mechanical behavior of TRIP steels. In particular, based on previous work in [17] for dual-phase steels and in [16] developed constitutive equations for four-phase TRIP steels, which consist of a ferritic matrix with dispersed bainite and austenite, the later transforming gradually to martensite as the material deforms plastically. In the following a brief description of the constitutive model is reported. Details of the model can be found in [16]. The total deformation rate D is written as the sum of an elastic, a viscoplastic and a transformational part:

D = De + Dp + DTRIP

(23)

The elastic properties of all phases are essentially the same and the TRIP steel can be viewed as homogeneous in the elastic region. Standard isotropic linear hypoelasticity is assumed and the constitutive equation for De is written as:

173

Multiscaling effects in low alloy TRIP steels ∇

De = Me : ı , M e =

1 1 K+ J, 2µ 3κ

1 J = įį , 3 K = I − J,

(24) ∇

In Eq. (24), ı is the Jaumman or co-rotational stress rate, µ and ț are the elastic shear and bulk moduli, į and I are the second- and symmetric fourth-order identity tensors with Cartesian components įij (the Kronecker delta) and Iijkl = įik į jl + įil į jk /2 . The TRIP steel is considered as a four-phase composite material in which the isotropic, viscoplastic phases are distributed statistically uniformly and isotropically. The constitutive equation for Dp is determined by using the homogenization procedure developed in [18], who developed a variational procedure for the determination of the effective properties of such nonlinear composites; it was shown in [19] that a certain variation of the “secant method” (the so-called “modified secant method”) is identical to the variational procedure of Ponte Castañeda. The resulting constitutive equation is of the form [16]:

(

)

1 D p =İ p ( ı ) N İ p = ı ș hom , 3 3 N= s, 2ı

(25)

In Eq. (25), s is the stress deviator and șhom is determined by the aforementioned homogenization procedure and depends on σ , the volume fractions and the individual properties of the four phases involved. The transformation part DTRIP is written in the form:

∆ · σ § DTRIP = f ¨ A N + v į ¸ , A ( σeq ) = A0 + A1 * , 3 ¹ sα ©

(26)

In Eq. (26), f is the volume fraction of the martensite, a superposed dot denotes the material time derivative, ǻv is the relative volume change associated with the transformation and takes values in the range 0.02 to 0.05 in austenitic steels depending upon alloy composition, A 0 , A1 are dimensionless constants and s*α is a reference austenite stress. The model is completed by the evolution equations a 3 4 of the volume fraction of the individual phases. Let f,c ( ),c ( ),c ( ) be the volume fractions of martensite, austenite, bainite, and ferrite respectively. The corresponding evolution equations are of the form [16]:

(

)

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(

)

p a Martensite: f = c(a) A f İ ( ) +Bf Ȉ ,

(

(27)

)

Retained Austenite: c (a) =- ª1- c(3) +c(4) ǻ v º f, ¬ ¼ (3) (3)   Bainite: c =-c ǻ f,

(28)

Ferrite: c =-c ǻ v f,

(30)

v

(4)

(4)

(29)

where Ȉ=ı h / ı is the “triaxiality” of the stress state, ıh = ıkk/3 is the hydrostatic stress, and

(

)

r-1 p a A f İ ( ) ,Ȉ,T =Į ȕ0 r (1-fsb ) ( fsb ) P , ȕ 0 =C ȣm /ȣI , 2 ª § ′ º 1 g −g· » « P (g) = exp − ¨ ¸ dg′ , g ( Θ, Σ ) = g 0 − g1 Θ + g 2 Σ, « 2 ¨© sg ¸¹ » 2π s g −∞ ¬ ¼ σ p( a ) T − M s,ut , fsb §¨© ε p( a ) ·¸¹ = 1 − e −α ε , Θ (T ) = σ M d,ut − M s,ut g

1

(

Bf ε

p( a )

(31)

³

)

, Σ,T =

g2 2π s g

β0 ( f sb )

Bf = 0 if Σ ≤ 0 .

r

2 ª § · º − 1 g g exp « − ¨ ¸ » if Σ > 0, « 2 ¨© sg ¸¹ » ¬ ¼

(32)

(33)

(34) (35)

In the equations above, ȣm is the average volume per martensitic unit, υ I is the average volume of a shear-band intersection, parameter Į represents the rate of shear band formation at low strains, C is a geometric constant, the exponent r models the orientation of shear-bands (r = 2 for random orientation, r = 4 for ı are the initially parallel shear bands), T is the absolute temperature, M d,ut , M s,ut absolute characteristic Md and M sı temperatures for uniaxial tension. This constitutive model is used in the next section to predict the form of forming limit diagrams for TRIP steels. 5.2 Forming limit diagrams

A sheet made of TRIP steel is considered, which is deformed uniformly on its plane in such a way that the in-plane principal strain increments increase in proportion. The possibility of the formation of a neck in the form of a narrow straight band is studied (Fig. 7), and the corresponding forming limit diagram is constructed. The possibility of the formation of a neck as shown in Fig. 7 is examined; both inside and outside the band a state of uniform plane stress is

Multiscaling effects in low alloy TRIP steels

175

assumed. Since the in-plane displacements are continuous, their spatial derivatives parallel to the band remain uniform.

Fig. 7. Narrow band in biaxially stretched sheet.

The only discontinuities in the displacement gradient are restricted kinematically to the form [20]:

ª ∂u α º « » = G α Nβ , ¬« ∂Xβ ¼»

(36)

In Eq. (36), X is the position vector of a material point in the undeformed configuration, [ ] denotes the difference (jump) of the field within the band and the field outside the band, N is the unit vector normal to the band in the undeformed configuration (Fig. 7), and G is the jump in the spatial normal ª ∂u º derivative of the displacement u, i.e., G = « ⋅ N » . The vector G takes a ∂ X ¬ ¼ constant value within the band and depends on the imposed uniform deformation gradient outside the band; a method for the determination of G is discussed in the following. In view of Eq. (36), the in-plane components of the uniform deformation gradient F inside the band take the form: b FĮȕ = FĮȕ +G Į Nȕ

(37)

where superscript b denotes quantities within the band, and quantities with no superscript correspond to the uniform field outside the band. Using the equilibrium equations across the band together with the boundary conditions, two equations are derived of the form [16]:

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A Įȕ G ȕ = bĮ

(38)

In Eq. (38), AĮȕ and bĮ depend on the stress-state, the constitutive parameters, the orientation N, and the initial thicknesses Hb and H inside and outside the band, respectively. The last equation defines the evolution of G as the sheet is stretched. In a perfect sheet before necking occurs, the right hand side of Eq. (38) vanishes, and the deformation remains homogeneous G = 0 until at some stage the determinant of the coefficient matrix [A] vanishes, and this is the condition of a local necking bifurcation. In Eq. (38), Greek indices take values in the range (1, 2), where X1 – X2 is the plane of the sheet, and the “summation convention” on repeated indices is used. The approach of in [21], known as the “M-K” model, in which the sheet is assumed to contain a small initial inhomogeneity (imperfect sheet) and necking results from a gradual localization of the strains at the inhomogeneity was adopted. The inhomogeneity is in the form of straight narrow band (neck) of reduced thickness Hb < H (Fig. 7). Both inside and outside the band a state of uniform plane stress is assumed, and the analysis consists in determining the uniform state of deformation inside the band that is consistent kinematically and statically with the prescribed uniform state outside the band. In the presence of an initial thickness imperfection, the right hand side of Eq. (38) does not vanish, and these equations provide a system that defines the two unknowns G 1 and G 2 . Given the initial sheet thickness inside and outside the band and the imposed uniform deformation history outside the band, Eq. (38) are solved incrementally for ∆G = G ∆t to obtain the deformation history inside the band. Localization is said to occur when the ratio of some scalar measure of the amount of incremental straining inside the band to the corresponding value outside the band becomes unbounded. In the present calculations, an initial thickness imperfection is introduced and the deformation gradient outside the band F is prescribed in such a way, that the corresponding principal logarithmic strains İ1 and İ2 outside the band increase in proportion, i.e. dİ1/dİ2=İ1/İ2=ȡ=constant. The uniform solution outside the band is determined incrementally. At the end of every increment, Eq. (38) is used to determine ǻG and this defines the corresponding deformation gradient inside the band Fb. Then, the uniform solution inside the band is determined. Necking localization is assumed to occur, when the ratio of some scalar measure of the amount of incremental straining inside the band to the corresponding value outside the band becomes very large; in particular, the calculations are terminated when either one of the two conditions ∆G1 / ∆λ1 > 30 or ∆G 2 / ∆λ 1 > 30 is satisfied. The material constants used in the calculations are reported in [16]. The initial volume fractions of the four phases in the TRIP steel are assumed to be f 0= 0.017 , a 3 4 c( ) = 0.103 , c( ) = 0.38 and c( ) = 0.50 . For comparison purposes, a separate set of calculations is carried out for a non-transforming steel that consists of the three

(

)

Multiscaling effects in low alloy TRIP steels

177

phases, i.e., retained austenite, bainite and ferrite with constant volume fractions a 3 4 f 0 = 0 , c( ) = 0.12 , c( ) = 0.38 and c( ) = 0.50 . In all cases, a constant strain 4 1 – – rate outside the band İ 1 =10 sec is imposed. Fig. 8 shows forming limit curves obtained for imposed proportional straining ȡ for two different values of the initial thickness imperfection, namely Hb/H = 0.999 and Hb/H= 0.99. The two solid curves correspond to the TRIP steel, whereas the dashed curves are for the non-transforming steel. The TRIP phenomenon increases the necking localization strains. In particular, for an initial thickness imperfection of Hb/H = 0.99 and ȡ =0 cr (plane strain), the critical strain İ11 increases from 0.2145 for the non-transforming steel to 0.2541 for the TRIP steel; the corresponding values of cr İ11 for Hb/H = 0.999 and ȡ = 0 are 0.3179 for the non-transforming steel and 0.3567 for the TRIP steel. A comparison of the model predictions with available experimental data is also presented in Fig. 8. An “Erichsen” universal sheet metal testing machine was employed for the experiments. A hemispherical punch with a diameter of 50 mm was used and the punch velocity was set to 1 mm/sec. The agreement between the model prediction and the experimental data is reasonable.

Fig. 8. Forming limit curves for two different values of initial thickness inhomogeneities Hb/H. Solid lines correspond to the TRIP steel, whereas the dashed lines are for a non-transforming steel. Dark triangles are experimental data.

References [1] Haidemenopoulos GN, Grujicic M, Olson GB, Cohen M, “ Transformation Microyielding of Retained Austenite”, Acta Metall., Vol. 37, No. 6, pp. 1677-1682, 1989. [2] Vasilakos AN, Papamantellos K, Haidemenopoulos GN, Bleck W, “Experimental Determination of the Stability of Retained Austenite in Low-Alloy TRIP Steels”, Steel Res., Vol. 70, No. 11, pp. 466-471, 1999. [3] Haidemenopoulos GN, Vasilakos AN, “Modelling of Stability in Low Alloy Triple-Phase Steels”, Steel Res., Vol. 67, No. 11, pp. 513-519, 1996. [4] Olson GB, Cohen M, “A General Mechanism of Martensitic Nucleation: Part I. General Concepts and the FCC→HCP Transformation”, Metall. Trans. A, Vol. 7A, pp. 1897-1904, 1976.

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[5] Cohen M, Olson GB, Japan. Suppl. Trans. JIM 17, p. 93, 1976. [6] Cech RE, Turnbull D, “Heterogeneous Nucleation of the Martensite Transformation”, Trans. AIME, Vol. 206, pp. 124-132, 1956. [7] Patel JR, Cohen M, “ Criterion for the action of Applied Stress in the Martensitic Transformation ” , Acta Metall., Vol. 1, pp. 531-538, 1953. [8] Kuroda Y, “Kinetics of Deformation-Induced Transformation of Dispersed Austenite in two Alloy Systems ” , M.Sc. Thesis, Dept. of Materials Science & Engineering, M.I.T., Boston, Massachusetts, U.S.A., 1987. [9] Sundman B, Jansson B, Andersson J-O, “The Thermo-Calc Databank System ” , Calphad, Vol. 9, pp. 153-199, 1985. [10] Borgenstam A, Engström A, Höglund L, Ågren J, “DICTRA, a Tool for Simulation of Diffusional Transformations in Alloys ” , J. Phas. Equil., Vol. 21, pp. 269-280, 2000. [11] Katsamas AI, Vasilakos AN, Haidemenopoulos GN, “ Simulation of Intercritical Annealing in Low-Alloy TRIP Steels ” , Steel Res., Vol. 71, No. 9, pp. 351-356, 2000. [12] Matsuda H, Bhadeshia HKDH, “Kinetics of the Bainite Transformation ” , Proc. R. Soc. Lond. A, Vol. 460, pp. 1707-1722, 2004. [13] Traint S, Pichler A, Hauzenberger K, Stiaszny P, Werner E, “ Influence of Silicon, Aluminium, Phosphorous and Copper on the Phase Transformations of Low Alloyed TRIP Steels ” , Proc. Intern. Conf. on TRIP-Aided High Strength Ferrous Alloys, Vol. 1, pp. 121-128, 2002, Ghent, Belgium. [14] ECSC 7210-PR/370 project: “ Control and Exploitation of the Bake-Hardening Effect in Multi-Phase High-Strength Steels ” , Final Report, to be published by the European Commission. [15] Wirthl E, Pichler A, Angerer R, Stiaszny P, Hauzenberger K, Titovets Y.F. and Hackl M., “Determination of the Volume Amount of Retained Austenite and Ferrite in Small Specimens by Magnetic Measurements”, Proc. Intern. Conf. on TRIP-Aided High Strength Ferrous Alloys, Vol. 1, pp. 61-64, 2002, Ghent, Belgium. [16] Papatriantafillou I, Agoras M, Aravas N, Haidemenopoulos G, “Constitutive Modeling and Finite Element Methods for TRIP Steels”, Comput. Methods Appl. Mech. Engrg., to appear, 2006. [17] Stringfellow RG, Parks DM, Olson GB, “A Constitutive Model for Transformation Plasticity Accompanying Strain-Induced Martensitic Transformation in Metastable Austenitic Steels”, Acta Metall. Mater., Vol. 40, pp. 1703-1716, 1992. [18] Ponte Castañeda P, “The effective mechanical properties of nonlinear isotropic solids”, J. Mech. Phys. Solids 39, pp. 1757-1788, 1992. [19] Suquet P, “Overall properties of nonlinear composites: A modified secant moduli theory and its link with Ponte Castañeda’s nonlinear variational procedure”, C. R. Acad. Sci. Paris, II 320, pp. 563-571, 1995. [20] Hadamard J, “ Leçons sur la propagation des ondes et les équations de l’ hydrodynamique ” , Libraire Scientifique, Hermann, Paris, 1903. [21] Marciniac Z, Kuczynski K, “Limit strains in the process of stretch forming sheet metal”, Int. J. Mech. Sciences, Vol. 9, pp. 609-620, 1967.

Ductile Cr-Alloys with solute and precipitate softening S. Haoa, *, J. Weertmanb a, b

Department of Material Science & Engineering a Department of Mechanical Engineering Northwestern University, Evanston, IL 60208, U S A

Abstract A dislocation kinetics-based analysis has been performed to investigate toughening mechanisms of alloys. It is concluded that strength and toughening are determined by the combination of short range material adhesion and long range interaction between different phases, where the former refers to Peierls-Nabarro energy barrier and coherence embedded solute elements and the latter refers to the double kink formation of dislocation loops and associated pattern of slip line between dispersed solute atoms, precipitates, and second phase particles. A strategy for toughening alloys is proposed that contains two key elements: 1. Alloy softening through smeared out Peierls-Nabarro stress barriers. 2. Process-structure optimization to obtain desirable grain size, coherency and spacing between solutes, precipitates, and second phase of particles. Keywords: Dislocation; Double-kink formation; Ductile fracture; Alloy softening; Cr-based alloy; Peierls-Nabarro stress.

1. Introduction Cr is a promising base for alloy system due to its high melting temperature, high thermal conductivity with moderate strength, low density and low cost. On the other hand, a significant drawback of Cr, as compared with, for example, Fe-Ni-based alloys, is its high ductile-to-brittle transition temperature (about 150oC for unalloyed Cr). It presents a purely brittle behavior at room temperature that hampers effective applications for many engineering purposes. Although many research reports have been published about this class of alloys in recent years, a challenge remains in finding a Cr-based system with acceptable ductility. This work is a part of a broad study [1] for developing a new class of Cr-based alloys with high strength and high toughness. The alloy softening mechanism has been examined in [2-5] as way of achieving increased toughening.

*

Corresponding author. E-mail address: [email protected] (S. Hao). 179

G.C. Sih (ed.), Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano, 179–196. ” 2007 Springer.

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In contrast to cleavage fracture, ductility is deformation of metal by easy movement of crystal dislocations across certain slip systems. The barrier against easy movement of a crystal dislocation is a large amplitude Peierls-Nabarro periodic stress. The surmounting of the Peierls-Nabarro barrier by a straight dislocation occurs by the intermediate step of throwing a double kink over the barrier [6]. In order to identify the double kink formation in semi-conduct materials [7, 8] that, studied the theoretical solution for the motion of dislocation line under an applied stress. The work in [5] has investigated the dislocation double kink formation in Mg-based alloys with Li addition [5, 9] and with iron addition in [10]. The analysis below is an investigation of the condition for double kink formation in a metal with misfit second phase particles which combines a semi-analytical dislocation solution and the results of ab initio computations [11, 12]. This paper is organized as follows: Section 2 introduces the models and associated governing equations. The section that follows introduces a micromechanical analysis of particle misfit. Section 4 presents results of a semi-analytical solution of double kink formation. More discussions are given in Section 5 which leads to Section 6 with the conclusions.

Fig. 1. Dislocation double kink formation model.

2. Models and Governing Equations 2.1 The model Consider a metal with dispersed precipitates or solute atoms (see Fig. 1). Fig. 1(a) shows a dislocation loop in a sea of particles. Fig. 1(b) pictures a local double kink and Fig. 1(c) is a schematic plot of the effect of misfit atom on the periodic Peierls-Nabarro energy. (The derivative of the energy gives the periodic Peierls-Nabarro stress or force.) Obviously, the type of an actual dislocation motion is determined by alloy’s micro-nano structure and the interaction between dislocations and particles. Since a curved dislocation loop is formed by a summation of many kinks, its propagation can be characterized by one double kink formation as illustrated in Fig. 1(b). The double kink formation along the dislocation loop essentially governs the ductility of the alloy. Let the Peierls-Nabarro stress magnitude in a crystal be given by [13]:

Ductile Cr-Alloys with solute and precipitate softening

ı Peierls = b

K § ʌa' · exp ¨ − K¸ 2c' © c' ¹

181

(1)

where b is Burger vector, K is a constant determined by the second order derivative of interatomic potential, aƍ is the spacing between adjacent slip system, cƍ the lattice constant along the dislocation motion direction and µ' is shear stiffness. Note too that the following expression of Peierls-Nabarro potential E(y) has been proposed [13]:

ı c'b § 2ʌy · E ( y ) =E 0–E p cos ¨ ¸ ; Ep = 2ʌ © c' ¹ Peierls

(2)

where E 0 and E p are material constants. The corresponding Peierl-Nabarro stress is given by

τ xz b =

∂E ∂y

(3)

For a single crystal with simple cubic structure, in the three dimensional case, the Potential (2) can be written as

§ 2ʌx · § 2ʌy · § 2ʌz · E ( x,y,z ) =E0 – Ep cos ¨ ¸ cos ¨ ¸ cos ¨ ¸ © c' ¹ © c' ¹ © c' ¹

(4)

which can be expressed in the form of a Bloch’s-like potential:

­ ª 2ʌxi º ½ ­ ª 2ʌyi º ½ ­ ª 2ʌzi º ½ E ( x,y,z ) =E 0 –Ep Re ®exp « ¾ Re ®exp « ¾ Re ®exp « ¾ » » ¬ c' ¼ ¿ ¯ ¬ c' ¼ ¿ ¯ ¬ c' »¼ ¿ ¯

(5)

In this analysis, it is presumed that the inhomogeneities at the two ends of the dislocation line lie along x-axis with the same y and z coordinate in the cubic crystal. 2.2 Governing equation A dislocation line ȥ in a three-dimensional Cartesian coordinate system can be expressed as:

ȥ:

­x=x ° ®y = y ( x ) °z = z ( x ) ¯

(6)

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As described in [2], the double kink formation in the model of Fig. 1(b) is equivalent to finding the equilibrium dislocation configuration with the following energy minimum: 2 2 z y ª º dy · § dz · § »dx I ( ȥ ) = ³ « E ( x,y,z ) 1+ ¨ ¸ + ¨ ¸ − ³ ³ ( ı misf + ı appl bdydz ) xy dx ¹ © dx ¹ 0 0 » © −L « ¬ ¼ L

(7)

and

įI = 0

(8)

Also the dislocation line satisfies the boundary conditions, e.g. y ( ±L ) = 0 , z ( ±L ) = 0 , y ' ( ± L ) = c y , z ' ( ±L ) = cz

(9)

In Eqs. (7-9) L is the distance between two inhomogenieties which pin the dislocation line; c y and c z are constants which are determined the coherency of the inhomogenieties and are adjustable by the variational operation of Eq. (8); ı misf , ı appl are the misfit-induced stress and external applied stress, respectively; xy no other stress component is enforced on this field. Also we assume that the stress misf appl fields ı , ı xy are passive, so an integral of the blanket of Eq. (7) with respect to dydz along a closed route is path-independent. 2.3 Procedure of analysis The governing Eqs. (7-8) will be solved to obtain the equilibrium configuration toward double kink formation for the misfit center by alternating the PeierlsNabarro Potential and misfit-induced stress (see Fig. 2). The solution provides quantitative information about the effects of alloying and optimized spacing between particles, which will be used for assisting to establish quantitative correlations among alloying, micro/namo structures, and properties in metallurgical process design.

Fig. 2. The problem associated with Eqs. (7-9) for the misfit center located at (0, 0, h).

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183

3. Misfit Analysis The primary task in solving Eqs. (7-9) is to find the misfit induced stress field

ı misf when alloys additions are added to the Cr matrix. In this analysis the

Eshelby’s eigen-strain method [14] has been applied to obtain theoretical solution misf of ı , in which the matrix is assumed infinitely large as compared with the particles. The small particles are treated as second phase inclusions with different elastic constants than the matrix and details of the particles are ignored (see Fig. 3).

Fig. 3. Analytical model for a matrix with misfit particle/cluster based eigen-strain analysis.

3.1 Misfit in material properties In this case the problem associated with the model in Fig. 3 can be stated as (Eshelby [11]): for an infinite isotropic elastic body “I” with bulk modulus K1 A and Poisson’s ratio v1 under a remote uniform applied stress ıij ; then, the inclusion “II”, which is embedded in “I” with bulk modulus K 2 and Poisson’s * ratio v 2 , will be in the state characterized by an eigen-strain tensor İ ij and an A * corresponding stress tensor ıij . Under an uniform applied stress ıii , the İ ij and ıij , respectively, are determined by following:

İ*ii =

3 (1–v1 ) I ª¬( K 2–K1 ) İ iiA º¼ İiiA =Ciijj ı Ajj – – 4v 2 K 1+v K ( 1 ) 1 ( 1) 2

(10)

And

ıii = ( 2µ1+ Ȝ1 ) ( İ iiA + Siimn İ*mn − İ*ii )

(11)

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where Sijkl are Eshelby’s tensor:

Sijijįij =

7–5v1 5v1–1 4 –5v1 , Sijkm įijį km = , Sijijİ ijk = 15 (1– v1) 15 (1– v1 ) 15 (1–v1 )

(12)

The bulk modulus is correlated to Young’s modulus E by:

K=

E 3 (1 – 2v)

Fig. 4. A model for volumetric misfit analysis.

3.2 Volumetric misfit By assuming the inclusion to be a sphere that is larger than the void existing in material “I”, then the problem can be described by the model in Fig. 4 stated as: for the infinite isotropic elastic body “I” with a spherical void with the radius r1 , an inclusion “II” with the radius r2 is embedded in “I” and r2 ≠ r1 . Hence, misfit-induce diameter change occurs for both the void in “I” and the inclusion, + which are denoted as dr and dr , respectively, in Fig. 4. The interfacial traction, denoted as p (pressure), is determined by the following relation: p=

4µ1µ 2 ( r2 − r1 )

(13)

r1 ( 2µ 2 +µ1 ( ț 2−1) )

where

ț=

vE E Ȝ+3µ , µ= , Ȝ= 2 (1+v) Ȝ+ µ (1−2v) (1+ v)

The corresponding stresses components are

σϑϑ =

p § r1 · ¨ ¸ 2© r ¹

3

§r · ı rr = − p ¨ 1 ¸ ©r¹

3

(14)

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Ductile Cr-Alloys with solute and precipitate softening

4. Results 4.1 Materials constants Some materials constants of the alloys correlated Cr are listed in Table I: Crystal structure at room temperature

Lattice (nm)

Cr Re Ni V

HCP/BCC HCP FCC BCC

0.407/0.291 0.446 0.352 0.303

297 463 200 128

v Poisson ratio 0.21 0.3 0.31 0.37

Fe

BCC

0.287

210

0.297

constant

E (GPa)

Ep/b (J/M2) (BCC)1.44 0.349

4.2 Misfit induced stress Displayed in Fig. 5 are the increments of the amplitude of stress caused by elastic constants misfit, which shows the misfit stress increases when either Young’s modulus or Poisson’s ratio in the inclusion, which is termed “misfit center” in the figure, is larger than that in the matrix. Fig. 6 shows the stress distribution of the volumetric misfit induced stress. It demonstrates a significant stress concentration near the inclusion whose amplitude is proportional to the size of the misfit center and is of the order of Young’s modulus.

Fig. 5. The increment of stress amplitude due to elastic constants misfit.

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Fig. 6. Distribution of the volumetric misfit induced stress, where b is the burgers vector of the matrix and bx is that of the misfit center.

4.3 Results of double kink formation The governing Eqs (7) can be expanded into as following L z y ª º I ( ȥ ) = ³ « E ( x,y,z ) Q ( x,y,z ) − ³ ³ ( ı misf +ı appl xy ) bdydz »dx « −L ¬ 0 0 ¼»

(15)

where 2

§ dy · § dz · G ( x,y,z ) = 1+ ¨ ¸ + ¨ ¸ © dx ¹ © dx ¹

2

2

2 2 2 2 1 ª§ dy · § dz · º 3 ª§ dy · § dz · º =1+ «¨ ¸ + ¨ ¸ » + «¨ ¸ + ¨ ¸ » + ... 2 ¬«© dx ¹ © dx ¹ ¼» 2 ¬«© dx ¹ © dx ¹ ¼»

(16)

Eq. (15) has been solved numerically by taking the first two terms of (16) and

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Ductile Cr-Alloys with solute and precipitate softening

letting L = 20b and assuming the sum of the derivative terms in Eq. (16) is less than unity. Also, the following dimensionless constants are introduced in the analysis:

ER =

Ep E0 , Eı = 2 2 bµ bµ

(17)

where E 0 and E p are the constants in the Peierls-Nabarro potential (1,2); and

IJij IJij E v b M E = x , M ȣ = x , M V = x , IJij = Peierls , IJijµ = µ ı E v b A

A

(18)

Here the subscription “x” indicates the materials constants of the misfit center. In (18) the parameter M V represents the degree of volumetric misfit. Displayed in Figs. 7-9 are the results when the misfit center lies in the {x, y} plane, i.e, h = 0 in Fig. 2. Figure 7 shows how the dislocation line “climbs” the Peierls-Nabarro barrier as it decreases or the applied shear stress increases when a very “soft” misfit center is at the middle (which acts as a “void”).

Fig. 7. The dislocation line “climbs” up when applied stress increases.

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The obtained equilibrium configurations of dislocation lines when a “strong” misfit center exists (due the volumetric misfit) are presented in Figs. 8(a) and 8(b). These are obtained by varying the Peierls-Nabarro stress barrier when the applied stress is fixed. Similarly to Fig. 7, the dislocation lines are pinned by the misfit center but two equilibrium solutions can be found for each given materials misfit and applied stress. The first one shows the same trend as that in Fig. 7 but with different scales whereas the second one presents a “Orowan” curve for precipitate strengthening.

Fig. 8. Two kinds of equilibrium configurations of the dislocation lines when strong misfit center presents.

Ductile Cr-Alloys with solute and precipitate softening

189

Fig. 9. Two equilibrium configurations of dislocation line when misfit center is away from {x,y} plane.

Fig. 9 presents the solutions when h =10b, i.e. the misfit center is away from the {x,y} plane, see Fig. 2; where the negative and positive misfit refer to the case M V 1 , respectively. Although it looks like the negative volumetric misfit is beneficial, the physical meaning in this diagram is still needs to be clarified. Recall the analysis presented in the Section 3, i.e. the misfit analysis from Fig. 6. A small lattice constant misfit will cause huge stress, of the order of Young’s modulus. Therefore, a debonding between misfit center and matrix or a dislocation inside the misfit particle may take place. Plotted in Fig. 10 is a graphical analysis, using (10-14), to compute the misfit stress when the displacement caused by the debonding/misfit particle dislocation appears (=1b, 2b, 3b,…), which demonstrates that such a location-induced displacement can reduce the misfit-induced stress significantly. Logically, the debonding/misfit particle dislocation works as a dislocation core which may track a dislocation line to reduce the misfit stress, so as to reduce the system energy significantly to reach an equilibrium configuration. By assuming the case of dislocation line touching misfit center to be a permissible solution, the solutions obtained in this analysis demonstrates that the dislocation line always towards to the misfit center at equilibrium states. Fig. 11 shows a set of 3D numerical results of Eqs. (7-9) by varying the distance “h”, which presents a “helix-like” topology. This is a favorable dislocation mode and more discussion will be given in the next section.

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Fig. 10. The variation of misfit-induced stress when a decohesion/dislocation induced displacement, denoted as “ ǻu dis ” in the figure, takes place.

5. Discussions The analysis in this study reveals that a volumetric and elastic modulus misfit center may result two different dislocation equilibrium configurations, depending upon the position: (1) When all particles lie in the same slip plane, the major function of these misfit centers is to pin the dislocation line which leads to a precipitation strengthening

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Fig. 11. Two views of the equilibrium configurations by varying the distance to {x, y} plane, presuming the misfit stress reduction when the dislocation line touches the misfit center.

(2) The misfit centers locate on different slip plane may result a helix equilibrium configurations of dislocation line when dislocation is easily to be activated in a misfit center. In the study of Mg-based alloys with Li addition, it was indicated in [3, 7] that two dislocation motions: basal slip and prismatic slip, may occur in the crystal with close-packed structure such as HCP and BCC. The above mentioned two modes are correlated these two motions. The precipitate strengthening is to increase the resistance of basal slip. This is because a basal slip takes place along the slip planes with the minimum energy barrier, which is similar to the slip systems in FCC; hence, the dislocation front has the trend to move forward sweeping over large area which leads to fail of the material. On the other hand, the zigzag path of a prismatic slip, as the “helic” path demonstrated in Fig. 11, will dissipate more energy than straight slip while induce additional resistance to dislocation motion. A high strength alloy should avoid massive basal slip since it represents plastic deformation without hardening. The enhanced high toughness of an alloy requires the tolerance to the localized helix slip system that is able to smooth out the energy concentration around precipitates and defects which are usually the source of brittle fracture. Considering a metal specimen under external load, the corresponding equivalent appl appl and İ , respectively. The applied load stress and strain are denoted as ı causes dislocation loops and plastic deformation within the metal, the corinst responding stress is termed “intrinsic strengh”, denoted as ı th in this work, which reflects the capacity of the metal against deformation. Therefore a criterion can be written for the failure of the metal as below:

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t ı appl ≥ ıins th

(19)

Based on the analysis introduced in the previous sections, an estimate of the inst intrinsic strength ı th for an alloy with solute/precipitate strengthening/softening, has been suggested as following: matrix ıinst + ǻı sol th = ı th th

(20)

where ı th is the yield strength of the matrix, which is essentially determined sol by the Peierls-Nabarro stress of the material; and ǻı th represents the contribution of precipitate strengthening/solute softening, which is a function of appl applied strain İ : matrix

ǻı sol th =

8ȕ ­ ț § 1-v · matrix appl 2 rclust 2 ½ 2 particle ı mis ) ( rclust ) L¾ ®( ǻı th ¨ ¸ ( ı th İ ) ⋅ Lb ¯ ȕ 2 © 2-v ¹ b ¿

(21)

where the coefficient ȕ =1 for in-plane dislocation line and ȕ ≈ 1.4 for spirical dislocation line, e.g, that in Fig. 11. The average space between particles is L and b is the Burger’s vector. Note that the average diameter of the second phase particle matrix is the flow strength of matrix, which is determined by is rclust and ı th Peierls-Nabarro barrier. Moreover, in the expression

ǻı partilce = ı partilce ı thmatrix th th

(22)

the quantity ı th has the following interpretation: (1) for hard particle, it is the coherent stress at the interface between precipitate particle and matrix, ≈ ı solut ; where ısolut is the (2) for solute atom or lose-connected cluster, ı partilce th th th bulk flow strength of the solute atoms. (3) for soft particle, it’s the yield stress of the particles partilce

The misfit-induced stress, applied equivalent strain and softening coefficient to be appl mis calibrated by more computation are denoted, respectively by ı , İ and ț . The Eq. (21) actually provides a prediction of strain softening curve through integrating the information of matrix Peierls-Nabarro stress, strength, average size and spacing of the second phase particle, the coherence and misfit between these two phases, as well as types of the associated dislocation loops. Displayed in Fig. 12 is a set of predicted alloy’s strain softening curves. In this diagram the size and

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spacing of the second phase particles are fixed ( rclust /b =9 , L/b= 60 ) whereas the Peierls-Nabarro stress of matrix, i.e. ı matrix varies with the fixed ratio: th particle matrix ǻı th /ı th = 0.9 . Since an idealized ductile alloy should possesses an infinite long yielding plateau at the strengthened level, as depicted by the straight line on the top in this diagram, a fast drop of stress with minimum applied strain refers to a brittle behavior of an alloy. As expected, Fig. 12 shows that reducing Peierls-Nabarro stress leads to an increase of ductility, demonstrated by the decrease of strain softening speed; meanwhile the gain of ductility is traded off by a deduction of strength. However, by carefully selection of second phase additions (small precipitate particles or solute atoms) with appropriated size and distribution, it may result in desirable dislocation pattern, such as the spiral loops, with minimum misfit stress. Under this condition, a considerable gain of ductility can be achieved without changing Peierls-Nabarro stress, as plotted in Fig. 13. In this diagram it is presumed no misfit stress. By increasing the density, i.e. decreasing the spacing, of hard secondary phase particle/cluster, it shows a trend of the improvement in both strength and ductility. When the ratio L/b is greater than 60, surely the trend will be continuing. However, the volume fraction of precipitates becomes high which may result other structural and deformation pattern. Hence, the prediction plotted in Fig. 13, in conjunction with the results of misfit analysis demonstrated in Fig. 10, leads to the conclusion that the non-coplanar second phase-particles/precipitate with an average diameter 5-10b and spacing L =20-40nm is an optimized structure to achieve both high-strength and high ductility in an alloy. 6. Conclusions The formation of a double kink along a dislocation line is governed by the competition between the driving force that induced by applied stress and misfit induced stress, and the material resistance that is represented by Peierls-Nabarro energy barrier. The reduction of the effective Peierls-Nabarro stress by the stress field of solute atoms and precipitate particles leads to massive dislocation double-kink formation and possible softening behavior in stress-strain curve of a metal. Based on the theory in [2,6] and the expremental result [9], this analysis concludes that the capability to form double kinks is essentially determined by the heterogeneity of metal, through the misfit between matrix and second phase particle/precipitate/solute atoms and their geometric distribution. According to the degree of misfit, two classes of second phases are identified: termed strong and weak misfit centers, respectively; where the former refers to the case that second phase particle has a larger lattice or elastic constant whereas the latter has smaller lattice constant or lower material constants. The amplitude of the misfit-induced stress caused by volumetric (lattice) mismatch, which has the magnitude of Young’s modulus, is about one order higher than that caused by elastic constants mismatch. A strong misfit center leads to higher driving force for double kink formation.

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The size and spacing of the second phase particles also play an important role in double kinks formation and enhanced materials plasticity. This is because the curvature and length of dislocation segments determine the corresponding yield stress. The analysis indicates that the non-coplanar second phase-particle/ precipitates with an average diameter 5-10b and spacing L/b of order 40 will maximize the strengthening effect.

Fig. 12. A diagram of the relationship between Peierls-Nabarro energy barrier and strain softening.

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Fig. 13. The effects of the spacing of second phase particles and dislocation loops.

Acknowledgements The authors gratefully acknowledge the support of NSF. References [1] Fine ME, Weertman J, Freeman A, Cr-Based Alloys, a NSF-sponsored project, Northwestern University, 2004. [2] Weertman J, Dislocation Model of Low-Temperature Creep. Journal of Applied Physics, 1958. 29(12): pp. 1685-1689. [3] Arsenault RJ, Solid solution strenghtening and weakening of bcc solid solutions. Acta Metallurgica, 1969. 17(10): pp.1291-1297. [4] Fine ME, Tongen A, Gagliano, Dinteraction of coherent nanoscale preciptates with screw dislocations to lower the Peierls stress in low carbon steels. In Electron Microscopy: Its Role in Materials Science (The Mike Meshii Symposiun). Edited by J.R. Weertman, M. Fine, K. Faber, W. King and P. Liaw. TMS (The Minerals, Metlas and Materials Society) Warrendale, PA 2003, pp. 229-234. [5] Urakami A, Fine ME, Influence of misfit on formation of helical dislocations. Scripta Metallurgica, 1970. 4(9): pp. 667-672. [6] Weertman J, Mason’s dislocation relaxation mechanism. Physical Review, 1956, 101(4): pp. 1429-1430.

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[7] Celli V, et al., Theory of Dislocation Mobility in Semiconductors. Physical Review, 1963. 131(1): pp. 58-&. [8] Guyot P, Dorn JE, A Critical Review of Peierls Mechanism. Canadian Journal of Physics, 1967. 45(2P3): pp. 983-&. [9] Urakami A, Ph D Thesis, Advsor: M.E. Fine, Effect of Li additions on mechanical properties of Mg based single crystals in basal and prinsmatic sliping, in Dept. Mat. Sci. Northwestern University, Evanston, 1971. [10] Sato A, Ph D Thesis: Effect of electron irridation on the strength of iron single crystals, (Advsor: M. Meshii) Northwestern University: Evanston, 1972. [11] Freeman AJ, Full-Potential Linear Argument Plane Wave Code ( FLAPW) 1991. [12] Hao S, Moran B, Liu WK, Olson GB, A Hierarchical Multi-Physics Model for Design of High Toughness Steels. J. Compute-Aided Materials Design, 2003. 10(2): pp. 99-142. [13] Weertman J, Weertman JR, Elementary Dislocation Theory, 1992, Oxford University Press, Oxford. [14] Eshelby JD, Elastic inclusions and inhomogeneities, in Progress in Solid Mechanics, I.N. Sneddon, Hill, R., Editor. 1961, North-Holland: Amsterdam. pp. 89-140.

A multi-scale approach to crack growth R. Jones*, S. Barter, L. Molent, S. Pitt DSTO Centre of Expertise for Structural Mechanics, Department of Mechanical Engineering, Monash University, P.O. Box 31, Monash University, Victoria, 3800, Australia

Abstract This work first examines the fatigue crack growth histories (from microns to millimetres) of a range of test specimens and service loaded components and concludes that in most cases rack growth follows the generalised Frost and Dugdale crack growth law, i.e. as a first approximation there is a linear relationship between the log of the crack length or depth and the service history (number of cycles). It is then shown how the Frost and Dugdale crack growth law, incomplete self-similarity, the two parameter crack growth model, and fractal fatigue concepts are related. Also shown are how this law can be used to predict crack growth from sub microns to mm’s in a range of full-scale aircraft fatigue tests and coupon tests, including aircraft fuselage lap-joints. Keywords: Crack growth; Short cracks; Fractals; Experimental crack growth; Fatigue prediction.

1. Introduction Whilst there is little experimental data relevant to nanocrystalline metals, it has been postulated that dislocation slip in metal grains leads to void formation and Stress Sg(ksi)

A 1 2 3 4

14

B

12

11 12 13 14 15 16 17 18 19 20

B 1 2 3 4 5 6

7 8 9 10

7 2 3 4

5

11 12 13 14 15 16 17 18 19 20 21

1 2 3 4 5 6

7 8 9 10 11

Creak length a

Fig. 1. Striations formed on an aluminium alloy 2024 fatigue crack surface as shown by a replica in a transmission electron microscope 0. The insert indicates the loading used to produce the striations. *

Corresponding author. E-mail address: [email protected] (R. Jones) 197

G.C. Sih (ed.), Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano, 197–240. ” 2007 Springer.

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micro-cracking 0. Thus some of the principals shown to apply to fatigue crack growth of sub-millimetre materials are probably relevant to sub-micron-structures. Indeed ductile striations, considered to be the product of slip during fatigue crack growth can be observed to step forward distances per load cycle between 10 and 100 nanometres (the limit of observation for such features). These striations show no significant difference to those associated with step forward distances several orders of magnitude greater. Examples of striations in this size range are shown in Fig. 1, from 0. The science of fatigue crack growth has traditionally revolved around the relationship between the stress intensity factor range, ∆K and the crack growth rate, da/dN. The first paper making this correlation was published in 1961 0. The K-value was used from the analysis of the stress field around the tip of a crack as proposed in 0. Here the results of a series of constant-amplitude crack growth tests were used to express the crack growth rate da/dN (where N is the number of fatigue cycles, and a is the crack depth, or length, at time N) as a function of ∆K on a log-log scale. Plotting the data in this fashion revealed a region of growth where a linear relation between log(da/dN) and log(∆K) appeared to exist. This led to the well-known Paris equation, viz:

da/dN = C∆Km

(1)

where C and m are experimentally obtained constants. These constants are not easily associated with any physical property of the material. For many commonly used structural materials the value of m has been found to lie between 2.5 to 4.0. The equation is a simple model (curve fit) of the central region (region II or the ‘Paris’ region) of results of fatigue crack growth experiments 0. This law has continued to be modified to account for a variety of real life observations including, stress ratio R = (σmax/σmin) and crack closure effects [5, 7], and dependency on the peak stress intensity factor (Kmax) [8, 9], etc. However, in recent years a number of modifications have been questioned [8, 9]. In the mid 1970’s, it was shown in [10] that fatigue crack growth laws determined for macroscopic crack growth data, using methods such as those outlined in ASTM standard 647, could not be used to predict the growth of small sub-millimetre cracks, and that the constants in the crack growth law were a function of the size of the crack. It was also revealed that this inconsistency was not due to crack-tip plasticity effects. This work presents an explicit expression for this crack length dependency, and reveals how it is related to incomplete self-similarity, the two parameter crack growth model [8, 9] and to fractal fatigue concepts. At this stage it is important to note that the Paris equation was not the first law proposed to describe crack growth. The first law (according to the work in 0) can be attributed to an early work [12] of the Australian Defence Science and Technology Organisation (DSTO). Subsequently, the observation of self-similar crack growth [12] was used in [11]. Reported in this work is that crack growth under constant amplitude loading could be described via a simple log-linear relationship, viz:

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Ln ( a ) = ȜN + Ln(a o )

(2)

or a = a o eȜN

which gives as the crack growth rate equation: da/dN = λa

(3)

where λ, which we define as the “growth acceleration rate” constant, is a parameter that is material, geometry and load dependent, N is the “fatigue life”, and a0 is the initial crack-like flaw size (depth of the crack at the start of loading). When the far field stress and crack geometry remain relatively constant in a homogeneous (Many of the variations in the growth of very small fatigue cracks may be the result of local inhomogeneities in the material through which the crack is travelling. These effects can strongly influence the results of fatigue crack growth experiments, particularly those dealing with short crack growth, potentially leading to incorrect conclusions.) material then: In sequel, Eq. (2) shall

be known as the Frost Dugdale law.

Ȝ =f ( ı )

(4)

For constant amplitude loading, it was found [11, 13] that λ could be expressed as:

Ȝ= ij ( ǻı )

3

(5)

where φ only depends on the nature of the loading, and the geometry of the structure. For physically short cracks, it was further shown in [14-22] that a near linear relationship between ln(a) and N can be used. In this context, researchers at DSTO have for many years observed, in a wealth of experimental data [23-36], that exponential growth rates are often found for most of the life of cracks grown in service or from full-scale fatigue tests, and coupon tests under service spectra, down to cracks in the order of a few microns in depth. Examples include the aluminium alloy 7050 crack growth data used in the F/A-18 aircraft (see for example the quantitative fractography (QF) results in Fig. 2), and steels used in the F111, and the Aermacchi aircraft (Fig. 3). In these fig.s, it can be seen that cracks grown under typical complex (variableamplitude) can be approximated by Eq. (2). With respect to MEMS technology it is now known that, although bulk silicon does not exhibit a significant susceptibility to cyclic fatigue, micron-scale structures made from silicon films are vulnerable to fatigue in ambient air environments. As outlined in [37-39] metal-like stress-life (S-N) curves were produced from fatigue tests on silicon specimens. Of particular interest is the fact that the crack growth histories presented in [37, 38] also appear to conform to the work in [11], see Fig. 4. The same conclusion was highlighted in the compendium [23] for F/A-18 fatigue crack growth data0. This compendium consists of QF fracture

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surface measurements along with the relevant fatigue crack information, including initiating flaw size and type of test (mechanical, environmental or chemical). This work, which examined more than 300 different cracks in various full-scale fatigue tests and the associated coupon test programs, revealed a near log-linear relationship holds for crack growth from starting lengths of near microns up to lengths of 5 mm’s or more. A more general and comprehensive review of the applicability of Eq. (2) to represent crack growth is given in a general review of natural (Natural cracks may be considered to be those that grow from a natural or artificial discontinuity in a fashion where the far field loading is consistent throughout the cracks life and not manipulated in order to alter the crack tip K at different crack depths.) crack growth data in 0, and a re-

view of crack growth in the fuselage lap joints in commercial transport aircraft 0 . While this log-linear relationship appears to be a reasonable approximation for most of the life of many typical cracks, growth rate acceleration towards the end of life can cause an upward trend in the crack growth acceleration rate. Fractographic observations of many cracks indicates that this appears to be accounted for by the onset of static fracture modes 0 such as: inclusion fracture and ultimately tearing, changes in geometry and the coalescence of other cracks growing nearby to form a larger crack. Where load shedding is not involved this deviation from exponential growth frequently only accounts for a small fraction of the total crack life. Small variation may also occur when the crack is very small.

Fig. 2. Graph of QF crack depth against flight hours for cracks in aluminium alloy 7050-T7451 test coupons loaded with two F/A-18 usage spectra, from 0.

As a consequence of the relationship given in Eq. (2) and numerous crack growth measurements taken of coupons loaded at several different stress levels,

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Fig. 3. Graph of QF crack depth against ‘Programs’ of the FALSTAFF spectrum for cracking in Aermacchi centre section lower boom for two different configurations of boom (-O4a, -O4c); Material was 4340 steel 0.

Fig. 4. Crack growth at the nano-scale in 2 µm-thick polysilicon, adapted from [37, 38].

DSTO have shown that Eq. (5) holds for the aluminum alloy (AL) 7050-T7451. This is illustrated in Fig. 5 where the slopes of the crack growth curves from 0, as determined by QF, have been plotted against applied notch stress. In this fatigue test program up to five low Kt AL7050 specimen sets were tested at up to four

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stress levels each. Up to five complex service load histories typical of fighter aircraft spectra were tested, two of which are shown in Fig. 2. The effective crack like nature of the discontinuity from which a crack may grow, for a specific surface finish, has been established by back projecting from the measured results of test, or service cracks to time zero using Eq. (2). In this fashion an initial crack-like size of 10 microns has been found to be typical for the materials and surface treatments used in 036].

Fig. 5. Crack growth slopes versus notch stress for two spectra, from 0.

The data presented in Fig. 5, along with that presented in [13, 40, 45] suggests that by using the cubic relationship we can estimate the growth rates at one stress level (σ2) from those of another stress level (σ1) given the initial crack size a0 for the new crack and the value of λ for the original crack growth, viz:

a 2 = ao 2 e

(ı ı ) Ȝ N 3

2

1

1

(6)

Several predictions using Eq. (6) are presented in the following sections. 1.1 Cracks as fractals It has been shown that fracture surfaces can be considered as an invasive fractal set 0. Indeed, It was stated that:

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“When a piece of metal is fractured either by tensile or impact loading the facture surface that is formed is rough and irregular. Its shape is affected by the metal’s microstructure (such as grains, inclusions, and precipitates where characteristic length is large relative to the atomic scale), as well as by ‘macrostructural’ influences (such as the size, the shape of the specimen, and the notch from which the fracture begins). However, repeated observation at various magnifications also reveal a variety of additional structures that fall between ‘micro’ and ‘macro’ and have not yet been described satisfactorily in a systematic manner. The experiments reported here reveal the existence of broad and clearly distinct zone of intermediate scales in which the fracture is modelled very well by a fractal surface.” This concept, i.e. of a fracture surface as a fractal, has been further developed in [47, 48]. Used was made of the renormalisation techniques to develop a growth law for an invasive lacunar fractal, viz. da/dN = C(a)(∆K)p = C1aφ(∆K)p

(7)

This law is dimensionally identical to the classical Paris law except that the coefficient C is afubnction of a whereas in the Paris law it is assumed to be a material constant. It was also revealed 0 that such a crack growth law also corresponds to incomplete self-similarity, or self-similarity of the second kind. These findings together with the realisation that “in the threshold regime, there is something missing either in the model…”, see 0, led to the conjecture [40, 41] that in the low ∆K region, i.e. Paris Region I, the crack growth rate can be expressed in the form: da/dN = C ( a/a*)(1-m*/2) (∆Keff) m*

(8)

where C, a*, and m* are constants, and ∆Keff is an “effective”, or “equivalent”, stress intensity factor range, as used in either [7] or [8]. It is clear that this relationship follows the form proposed in [47, 48] and therefore has an underlying physical basis. Modelled in [10] is a notch [50] with a small but finite tip radius ȡ > 0 to represent the crack, see Fig. 6, which as per the Neuber micro-support concept 0 remains open during crack growth. The Smith-Watson-Topper (SWT) fatigue damage parameter 0 was used to determine failure of the ligament immediately in front of the notch, to obtain a crack growth law of the form first proposed in 0, viz:

da = C ª¬ (ǻK + ) (1-p) K pmax,appl º¼ dN

Ȗ

(9)

Here ∆K+ corresponds to the tensile part of the load cycle, i.e. the tensile part of the stress intensity range. After rearranging the expressions for C given in 0, it is found that C can be expressed in the form: C = ȡ1-γ/2 C (10) where C is independent of ȡ.

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Fig. 6. Crack with a finite tip radius, from 0.

It was revealed in [50] that:

(ȥ ) ȡ= *

y,1

2ʌ

2

§ ǻK th · ¨ a ¸ © ǻı th ¹

2

(11)

where ∆ıҊath is the actual threshold stress range over the first elementary block in front of the crack tip, ȥy,1 is a constant, and ∆Kth is the threshold stress intensity range. The formulae presented in Section 2 of the NASGRO users manual 0 can now be used to relate the threshold stress intensity range to the crack length, viz: ǻKth = ϖ √(a/(a+a0))

(12)

where ϖ is a function of the R ratio, constraint state and prior load history, see 0. Thus for physically small cracks a 20%). This result was consistent with our experimental observations on fibroblasts and endothelial cells [77] and results of others [40]. In the case of our affine model, however, we compared model predictions primarily with data obtained from measurements on airway smooth muscle cells where applied strains were smaller than 20%. Thus, it is believed that omitting the contribution of intermediate filaments as a tension-bearing structure did not substantially influence quantitative prediction of this model. One limitation of the affine approach is the assumption that local strains follow global strains. Thus this approach is known to lead to overestimate of elastic moduli [cf. 60]. Since it is not very likely that local strains of the CSK follow global strains applied to the cell, one should expect that the affine model predicts higher values of the elastic moduli than the measured ones. This can, in part explain, the quantitative discrepancy between the model predictions and the experimental data (Fig. 6). Furthermore, this model cannot predict long-distance propagation of forces in the cytoplasm that has been observed in living cells [28, 46], since the model presumes a continuum behavior which, in turn, implies that local loads produce only local deformations (in continuum mechanics this is known as the principle of local action). Nevertheless, the affine model has been successful in describing and predicting a number of essential mechanical properties of living cells such as shape stability, the contribution of microtubules cell stiffness, and the load shift between the CSK and the ECM [61]. 4. Other Models of Cellular Mechanics There are other models of the CSK in the literature, most notably models based on a cortical network [9, 14], open cell foam [55], a tensed cable network [62], and percolation [21]. While these models have been successful at explaining some particular aspects of cellular mechanics, they fail short of describing many other mechanical behaviors that are important for cell function. In particular, the models based on the open cell foam network and the percolation network do not take into account the effect of cytoskeletal prestress on cell deformability, or the contribution of microtubules to cell mechanical behaviors. The cortical network model, the open cell foam model, and the percolation model also ignore the contribution of the ECM to cellular mechanics, and the cortical network model cannot explain the observed transmission of mechanical signals from cell surface to the nucleus as well as to basal FAs [28, 46, 76]. On the other hand, all of these features (and many others) can be explained by the cellular tensegrity model [34, 35, 64]. Moreover, none of the other models provide a mechanism to explain how mechanical stresses applied to the cell surface result in force-dependent changes in biochemistry at discrete sites inside the cell (e.g., FAs, nuclear membrane, microtubules), whereas tensegrity can [36]. Thus, we believe that the cellular tensegrity model represents a good platform for further research on the cell structure-function relationship and mechanotransduction.

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It is important to clarify that according to a mathematical definition of tensegrity that is based on considerations of structural stability [cf. 8], the cortical membrane model and the tensed cable network model also fall in the category of tensegrity structures. They differ only by the manner in which they balance the prestress. However, in the structural mechanics literature, this difference is used to make a distinction between various types of prestressed structures and consequently, tensed cable nets and tensegrity architecture are considered as two distinct types of structures [73]. Although this may be understandable from a theoretical modeling standpoint, a self-stabilized cable net (i.e., the one that is not attached to an external world) cannot be ‘tensed’ unless it contains at least one compression element that balances these internal forces; hence, the more general definition of tensegrity may be relevant for describing real, three-dimensional structures in the living world. 5. Tensegrity and Cytoskeletal Rheology and Future Directions During the past decade, biomechanical studies of the cell have been focused on its rheological behavior. This is important since the CSK is a dynamic system which undergoes continuous remodeling and in its natural habitat it is exposed to dynamic loads. Rheological studies on various cell types and with various techniques yielded two distinct features: 1) that cell mechanical behavior conforms to a weak power law dependence on both frequency of loading and time of loading; and 2) that this power-law dependence is influenced by cytoskeletal prestress [1, 18, 53, 67]. In particular, it has been observed that the power-law dependence is inversely related to cytoskeletal prestress. Since the power-law behavior is directly related to deformability, (i.e., when a power-law exponent approaches zero or unity we have a solid-like or fluid-like behaviors, respectively), then the observations suggest that cells use mechanical prestress to regulate their transition between a solid-like and a malleable behaviors [67]. This was quite a surprising finding since a standard paradigm was that this transition is regulated by chemical mechanisms that govern polymerization and depolymerization of the CSK [68]. A number of empirical and semi-empirical mathematically sophisticated models have been offered to explain these rheological behaviors of living cells [6, 15, 18, 44]. All these models could provide explanations and descriptions for the power-law behavior. However, none can explain the observed dependence of the cell rheological behavior on cytoskeletal prestress. Importantly, these models have no structural correlates in living cells, thus cannot predict how specific cytoskeletal structural alterations (e.g., reorientation and rearrangement) might be related to cellular mechanical behaviors. To address this problem, we proposed a viscoelastic model based on tensegrity [69]; in a tensegrity model of the type shown in Fig. 1, elastic cables were replaced by simple Voigt spring-dashpot units. It is shown that this model can account for the prestress-dependent rheological behavior of the cell. However, in order to explain the power-law behavior observed in cells, it was necessary to assume ad hoc a very high degree of non-homogeneity between structural element properties in order to provide a wide spectrum of time constants that leads to the power-law.

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To provide a mechanistic explanation for the observed rheological behavior of living cells, initiated recently is an investigation that would link the cytoskeletal prestress to molecular dynamics of polymers of the CSK. The rationale is as follows. The rheological behavior of the CSK must necessarily reflect dynamics of polymer chains of the CSK. The dynamics of long chain molecules are characterized by thermally driven fluctuations that are governed by power law-like rheological behavior [cf. 11, 33]. However in the living CSK, polymer chains are under tension due to prestress forces. This tension, in turn, should impact cytoskeletal polymer dynamics in such a way that thermally-driven fluctuations diminish with increasing tension. This would push the cytoskeletal rheology closer to the solid-like behavior and thus, the power-law dependence should diminish. Our preliminary statistical models of fluctuating polymer chains under sustained tension yielded behaviors that are qualitatively consistent with the observations in living cells. This leads us to believe that this approach provides a good physical basis to explain how cytoskeletal prestress may affect the rheology of molecules within the CSK, and how these molecular scale features feed back to alter the mechanical properties of the entire cell through the unifying mechanism of cellular tensegrity. It is well known that living cells exhibit significant regional differences in mechanical stiffness [24]. However, since the whole cell responds to an external mechanical stimulus as an integrated unit, these local units within the cytoplasm must be mechanically connected via the CSK, possibly via the prestress-bearing elements. A more comprehensive analytical tensegrity model needs to be developed in order to capture the behavior of mechanical heterogeneity and anisotropy observed in living cells [28, 30]. 6. Summary In this work, it is shown that the tensegrity model is a useful framework for studying mechanics and mechanotransduction of living adherent cells. The model identifies mechanical prestress borne by the CSK as a key determinant of shape stability within living cells and tissues. It also shows how mechanical interactions between the CSK and ECM come into play in the control of various cellular functions. Furthermore, the model provides a way to channel mechanical forces in distinct patterns, to shift them between different load-bearing elements in the CSK and ECM, and to focus them on particular sites where biochemical remodeling may take place. If successful, this approach may show the extent to which prestress plays a unifying role in terms of both determining cell rheological behavior, and orchestrating mechanical and chemical responses within living cells. Moreover, it will elucidate potential mechanisms that link cell rheology to the mechanical prestress of the CSK, from the level of molecular dynamics and biochemical remodeling events, to the level of whole cell mechanics.

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Mode II segmented crack model: Macro/skew-symmetry, micro/anti-symmetry and dislocation/skew-symmetry a

X.S. Tanga, c, G.C. Siha, b ∗

School of Bridge and Structural Engineering, Changsha University of Science and Technology, Changsha, Hunan 410076, China b Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, Pennsylvania USA c School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai, China

Abstract Material non-homogeneity can vary with different degrees of severity depending on the size scale at which the observation is being made. For a polycrystalline material, the state of affairs within a grain can differ widely from those in a cluster of grains. The situation is further complicated by the presence of defects and imperfections which may grow under load while the stress symmetry conditions with reference to the defects in the form of a line or area can also change with size scale. Macro/symmetry for a line defect or crack dominated by applied load may no longer prevail when viewed at the microscopic scale where the material structure can influence the symmetric. In the work to follow, in-plane macro-shear load is considered such that the corresponding macro-stress field would be skew-symmetric with reference to a line defect if the grain size is small in comparison with the continuum element. When the grain structure comes into play, the line defect may no longer be in a state of pure in-plane shear. Micro-normal and micro-shear stresses may both be present on the micro-crack rendering a state of mixed mode micro-crack extension. This effect will be considered in addition to the generation of edge dislocations from the end of a micro-crack. According to the classical theory of dislocations, edge dislocations pertain only to the skew-symmetric stress field. The transitions from mode II macro/ skew-symmetry to mixed mode micro/anti-symmetry and finally to dislocation/skew-symmetry is considered in a line defect model using the continuum mechanics approach while realizing the physical process from macro to atomic is one of non-equilibrium in addition to the effect of material non-homogeneity where the properties of the bulk and those of the local region can differ. Segmentation of the scale range is made to alleviate the use of non-equilibrium behavior, the complexities of which would be beyond the scope of this investigation. Discontinuities of the volume energy densities are thus introduced at the scale crossing junctions. Their severity can be adjusted by the prevailing material and geometric parameters that are also affected by the applied macroscopic load. The presented analytical model is useful for making sensitivity analyses involving the influence of atomic and microscopic effects on the macroscopic behavior. As it is to be expected, the results depend on a combination of load, geometry and material at the different scales.

∗ Corresponding author.

E-mail address: [email protected] (G.C. Sih). 339 G.C. Sih (ed.), Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano, 339–367. ” 2007 Springer.

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Unlike previous models, the method of segmentation and connection by using the concept of scale multipliers in addition to displacement and stress compatibility has made possible the obtainment of closed form local solutions where stress singularities would dominate the energy transfer process. This technique has been used for problems of material damage. It is likely that the same method can be applied to examine the multiscaling behavior of problems in life science. Keywords: Segmented scaling; Macro-crack; Micro-crack; Edge dislocations; Restraining zones; Scale multipliers; Energy density discontinuities; Macroscopic tension; Microscopic mixed mode; Multiscaling model.

1. Introduction When the size and time scales are changed, the difference between organic and inorganic matters becomes smaller and smaller. It is not inconceivable that the gap between life and material science can also be decreased. Such a thinking has no doubt been instilled by the increasing capacity of the electronic computer and the challenge to consider physical events on a common basis. The behaviour of the electrons has replaced the atoms which were regarded not long ago as the smallest constituents of matter. The development of multiscaling models is to keep abreast of the results at the different scales and even more important to learn how differences detected at the lower scale can affect those at the higher scale, particularly when the tools of particle mechanics have been used at both the cosmic and atomic scales. Ambitious as it may be, science probes into the very large and the very small in order to gain physical and analytical understanding of what can be observed. Hence, multiscaling is intended to include all scales and to achieve consistency [1] as a necessary requirement ahead of righteousness that seems to depend on the progress of scientific discovery. Based on the material science models advanced to-day, diversified views on the definition of strength, toughness and damage can be found. For the most part, however, they all fall into the pattern that the constituents of the matter tends to play a role in the process of breaking up a specimen or increasing the distance between atoms. It is the relation between these events at the different scales that has been stifling the understanding of material behaviour. That is to make use specific materials from the atomic, if not even smaller size scales. All of this is well said and yet the progress has been slow because material inhomogeneity seems to defy a universal interpretation of material integrity. Putting aside all of the concerns mentioned, the learning process may not be able to follow leaps and jumps in the development of science and technology. Instead of achieving generality, the spirit of the present work has been finding a possible way to piece together the disconnected crack and dislocation models as a start knowing well that the same model will have to include the disturbance at the subatomic scale as well [2, 3]. The objective therefore is to bring the results from the macro to the atomic scale with emphasis placed on the mode II macro/skew-symmetric shear load in contrast to the mode I macro/symmetric normal load [4]. Up to this stage, it is not all together clear how the interatomic force model of two atoms in tension is related to the edge dislocation in contrast to tension and shear at the macroscopic scale. It is obvious that these are topics for future studies.

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2. Description of the Physical Problem and Mathematical Model Cracking is the macroscopic description of a solid creating free surfaces although it can also occur at the microscopic scale where the surfaces may or may not be free of mechanical tractions. The texture of the surface can vary depending on the rate at which the surfaces are being pulled apart. It can be smooth or rugged. This is indicative of whether the energy dissipation process is dominated by volume change or shape change referred to, respectively, as dilatational or distortional [5, 6]. The terms tension and shear are also used in the global sense referring to the applied loading. For a real material such as polycrystalline metals, the situation can be more complex because of the presence of the grain boundaries whose properties are not exactly known. The force or energy required for creating free surfaces between the grain boundaries and across the grains are different, not to mention the environment conditions. Moreover, modeling of the physical process is not altogether straight forward because surface creation may not be a continuous. A grain or cluster of grains ahead of a macro-crack may fracture before the finite ligament would give in. This is in fact the situation because of inhomogeneity of the grains at the microscopic scale. Hence, the crack may change directions like a saw tooth as illustrated in Figs. 1 and 2. When the defect size changes scale, physical and/or mathematical discontinuities may be introduced if sudden changes are permitted for the sake of ignoring the fine details. The restraining zones invoked in Fig. 2 are to account for the macro- to micro-transition and micro- to dislocation-transition. They were first use for developing dual scale models [7, 8].

Fig. 1. Physical model of macro-micro-dislocation damage.

Fig. 2. Analytical model displaying transition zones.

Depicted in Fig. 3 is the in-plane applied shear stress τ∞ of the triple scale damage model where the micro-crack and dislocation are contained in the segment with width w. The details of the restraining zones 1 and 2 corresponding, respectively, to the macro-micro and micro-dislocation transitions are as shown in Fig. 4. Note that skew-symmetry at the macro-scale is not carried into the

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micro-scale where anti-symmetry can prevail because the micro-crack does not follow a straight path. It can zig-zag following the irregularity of the grains. The effect is most intense in the segment denoted as restraining zone 2 whereby σ2 and τ2 are applied. Beyond this zone, n edge dislocations along rb are generated owing to the sliding motion. The number of dislocations is denoted by n and b stands for the magnitude of the Burger’s vector.

Fig. 3. In-plane shear of macro-crack with micro/dislocation ahead.

Fig. 4. Notations for macro-micro-dislocation model with one-half symmetry.

3. Macroscopic Stress Field The in-plane shear load gives rise to a stress field that is skew-symmetrical with respect to the x1-axis. Neglecting the non-singular terms of the stresses and requiring the displacement to be single-valued, the asymptotic expressions of the macro-stresses can be written as [9]

Mode II segmented crack model

macro σ11 =

343

3θ − K macro θ θ II sin ( 2 + cos cos ) 2 2 2 2πrmacro

macro σ 22 =

3θ K macro θ θ II sin cos cos 2 2 2 2πrmacro

macro σ12 =

K macro 3θ θ θ II cos (1 − sin sin ) 2 2 2 2πrmacro

(1)

The notations for the (r, θ) coordinates can be found in Fig. 3. A unique character of Eqs. (1) is that the load and geometry effects are controlled by the macro-stress macro which can be divided into three parts intensity factor K II

K macro = K ∞II + K (II1) + K (II2 ) , K macro =0 II I

(2)

It is known from [9] that

K ∞II = τ ∞ πc

(3)

where c is the half macro-crack length. As in the case of mode I crack extension [9], the mathematical results for mode II are similar. Hence,

K (II1) = −

2 2 Λ 1τ1 πc , K (II2 ) = − Λ 2 τ 2 πc π π

(4)

in which

Λ 1 = sin −1

a+d f +h a f − sin −1 , Λ 2 = sin −1 − sin −1 c c c c

(5)

It is seen that Λ1 reflects the macro-effect and that owing to the macro-micro transition while Λ 2 accounts for the micro-effect and that due to the microdislocation transition. Also note from Fig. 4 that

f = a + d + g , c = a + d + g + h + rb = f + h + rb

(6)

Substituting Eqs. (3) and (4) into Eq. (2), it is found that

K macro = Λ macro τ1 πc II

(7)

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where

Λ macro =

τ 2 π τ∞ ( − Λ1 − 2 Λ 2 ) τ1 π 2 τ1

(8)

Λmacro is a scale parameter and should be always positive. Numerical values of the macro-stress intensity factor K macro will be obtained II for a = 10mm, g = 1mm, h/g = 0.01, rb /h = 1 and τ∞ = 10MPa while the ratios τ1/τ∞ and τ2/τ1 will be varied one at a time. With τ2/τ1 = 5, curves for τ1/τ∞ = 1, 2, 3 and 4 are displayed in Fig. 5 for K macro against the length ratio d/a. It is anticipated that II high applied stress τ∞ or low stress ratio τ1/τ∞ tends to raise the amplitude of K macro . The curves all decrease in magnitude as d/a is increased. The rate of II macro decrease is less pronounced for large applied stress τ∞. The variations of K II with the length ratio d/a for τ1/τ∞ = 2 and τ2/τ1 = 1, 5 and 10 can be found in Fig. 6. The crack-tip macro-stress intensity tends to increase with decreasing ratio of τ2/τ1. This corresponds to large restraining macro-stress in comparison with the restraining micro-stress. Setting both the ratios of τ1/τ∞ and τ2/τ1 at 2 and 5, Fig. 7 macro with d/a for h/g = 0.01, 0.03 and 0.05. The shows the decay of K II micro-length ratio h/g does have an effect on the macro-stress intensity factor. Large h/g tends to damp the macro-crack tip intensity an effect that is not unexpected. Variations of the dislocation length ratio rb /h = 1, 2 and 3 have a much less effect on K macro . This is shown in Fig. 8 with τ2/τ1 = 5. This means that II macro-stress is not so sensitive to changes of the dislocation length ratio.

Fig. 5. Macro-stress intensity factor

K macro versus normalized length d/a for different ratio τ1/τ∞. II

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Mode II segmented crack model

Fig. 6. Macro-stress intensity factor

K macro II

versus normalized length d/a for different ratio

K macro II

versus normalized length d/a for different ratio

τ2/τ1.

Fig. 7. Macro-stress intensity factor h/g.

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Fig. 8. Macro-stress intensity factor rb/h.

K macro II

versus normalized length d/a for different ratio

4. Microscopic Stress Field In view of the argument that was concerned with the anti-symmetry of the micro-crack tip stress field, both the mode I and mode II micro-stress intensity micro micro will be present as shown below: and K II factors K I micro σ11 =

θ§ θ K micro 3θ · K micro 3θ θ θ I II cos ¨1 − sin sin ¸ − sin ( 2 + cos cos ) 2 2 2 2 2 2 2 πrmicro 2πrmicro © ¹

σ micro = 22

θ§ θ K micro 3θ · I cos ¨1 + sin sin ¸ + 2¹ 2© 2 2 πrmicro

micro σ12 =

θ θ K micro 3θ I + sin cos cos 2 2 2 2 πrmicro

θ θ K micro 3θ II sin cos cos 2 2 2 2 πrmicro

(9)

θ θ K micro 3θ II cos (1 − sin sin ) 2 2 2 2 πrmicro

4.1 Mixed mode micro-stress intensity factors micro

will contain the parameter Λmicro The micro-stress intensity factor K II correcting for the presence of the micro-dislocation transition zone. Following the argument in [4] for mode I loading, the expression

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Mode II segmented crack model

K micro = Λ micro τ 2 πe II

(10)

can be written such that

Λ micro = 1 −

h g

(11)

The presence of the micro-stress intensity factor K micro in Eqs. (9) is reflected via I

K micro = I

2 σ 2 Λ 2 πc π

(12)

where Λ2 is given by the second of Eqs. (5). Displayed in Figs. 9, 10 and 11 are the micro-stress intensity factor in Eq. (10) as a function of the ratio h/g when the applied shear τ∞ is fixed at 10MPa. With fixed τ2/τ1 = 5 and τ1/τ∞ = 1, 2, 3 and 4, Fig. 9 shows that K micro stays nearly constant for II micro as the shear stress ratio τ1/τ∞ each ratio τ1/τ∞ with increasing amplitude of K II is increased. The same conclusion can be drawn from the results in Fig. 10 where τ2/τ1 takes the values of 1, 5 and 10 with τ1/τ∞ = 2. Increase of the macro-stress restrain amplitude ratio τ2/τ1 raises the micro-stress crack tip intensity. Again, micro changes of the dislocation length ratio have no appreciable effects on K II . This can be seen from the overlapping curves in Fig. 11.

Fig. 9. Micro-stress intensity factor applied shear stress τ1/τ∞.

K micro II

versus normalized length h/g for different remotely

348

Fig. 10. Micro-stress intensity factor

X.S. Tang and G.C. Sih

K micro versus II

normalized length h/g for different ratio

τ2/τ1.

Fig. 11. Micro-stress intensity factor rb /h.

K micro II

versus normalized length h/g for different ratio

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4.2 Macro-micro transition via scale multiplier A connection between the macro- and micro-stress intensity factors can be made by application of the scale multiplier α1 as

K micro = α1 K macro II II

(13)

where α1 is a positive and dimensionless factor. Eqs. (7) and (10) can be inserted to Eq. (13) to give

α1 =

Λ micro τ 2 Λ macro τ1

e c

(14)

Equation (14) shows how macro- and micro-effects compete with one another as they appear in the form of ratios. Numerical computations of Eq. (14) are made for the different macro- and micro-parameters. Plotted is the scale multiplier α1 as a function of the ratio d/a. The four curves in Fig. 12 correspond to τ1/τ∞ = 1, 2, 3 and 4 with τ2/τ1 = 5. The other geometric parameters are fixed to the same values as used earlier. The multiplier factor remain nearly constant for large applied shear stress with τ1/τ∞ = 1 and 2. When τ1/τ∞ is increased to 3 and 4 meaning that τ∞ is decreased, then the multiplier will rise quickly with d/a. Fig. 13 exhibits the results for τ1/τ∞ = 2 and τ2/τ1 = 1, 5 and 10. Only the curve with τ2/τ1 = 10 increased slightly with d/a while the remaining two curves stayed constant. The effects of h/g and rb /h are shown, respectively, in Fig. 14 for rb /h = 1 and Fig. 15 for h/g = 0.01. Both ratios are seen to have very small effects on α1.

Fig. 12. Scale multiplier α1 versus normalized length d/a for different ratio τ1/τ∞.

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Fig. 13. Scale multiplier α1 versus normalized length d/a for different ratio τ2/τ1.

Fig. 14. Scale multiplier α1 versus normalized length d/a for different ratio h/g.

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351

Fig. 15. Scale multiplier α1 versus normalized length d/a for different ratio rb /h.

5. Edge Dislocation Stress Field It is well known that the elastic stress field produced by n edge dislocations is [10]

σ

disl 11

K disl sin θ( 2 + cos 2θ) II =− rdisl (1 − ν disl ) 2π

σ disl 22 =

K disl sin θ cos 2θ II rdisl (1 − ν disl ) 2π

disl σ12 =

K disl cos θ cos 2θ II rdisl (1 − ν disl ) 2π

(15)

in which Edisl is the elastic modulus, νdisl is the Poisson’s ratio, b is the length of the Burger’s vector, n is the number of edge dislocations in the segment rb. The dislocation-stress intensity factor can be defined as

K disl II =

µ nb E disl nb = disl 2(1 + ν disl ) 2 π 2π

(16)

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X.S. Tang and G.C. Sih

in which n is still unknown. Eqs. (10) and (16) can be connected as micro K disl II = α 2 h K II

(17)

It can be derived from Eqs. (10), (16) and (17) that

α2 =

E disl nb 2π(1 + ν disl )Λ micro τ 2 2h e

(18)

The results of Eq. (18) depends on a knowledge of the number of dislocations that will be made available by using the crack sliding displacements that will be determined in the work to follow. For the moment, assume that Eq. (32) is known so that a discussion of Eq. (18) can be made. The micro/dislocation transition depends on the use of the material constants Emacro = 200GPa, Emicro / Emacro = Edisl /Emicro = 5 and νmacro = νmicro = νdisl = 0.3. Moreover, with d/a = 0.1, h/g = 0.01 and τ2/τ1 = 5, a plot of the multiplier α2 versus rb/h is given in Fig. 16 for τ1/τ∞ = 1, 2, 3 and 4. All of the curves rise with increasing ratio of the dislocation segment. The most pronounced rise is for large applied shear where τ1/τ∞ = 1. This is expected. Similar plots for varying τ2/τ1 = 1, 5 and 10 with τ1/τ∞ = 2 are shown in Fig. 17. Small macro-stress restrain for τ2/τ1 = 1 gave large increase in α2 while the other two curves for τ2/τ1 = 5 and 10 remained nearly constant. Changes for d/a = 0.05, 0.10 and 0.15 do not seem to disturb the multiplier. This is shown in Fig. 18. By fixing d/a = 0.1 and changing the normalized micro-stress restraining zone ratio as h/g = 0.01, 0.03 and 0.05, three curves in Fig. 19 are obtained. They are sensitive to change in h/g. Large h/g increases α2 more.

Fig. 16. Micro/dislocation scale multiplier α2 versus normalized length rb /h for different ratio τ1/τ∞.

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Fig. 17. Micro/dislocation scale multiplier α2 versus normalized length rb /h for different ratio τ2/τ1.

Fig. 18. Micro/dislocation scale multiplier α2 versus normalized length rb /h for different ratio d/a.

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Fig. 19. Micro/dislocation scale multiplier α2 versus normalized length rb /h for different ratio h/g.

Holding the applied shear constant at τ∞ = 10MPa, the dislocation stress intensity factor in Eq. (16) will be examined for the same material constants used before. disl With d/a = 0.1, h/g = 0.01 and τ2/τ1 = 5, Fig. 20 shows that K II versus rb /h curves are not sensitive to changes of τ1/τ∞ = 1, 2, 3 and 4. The same is seen in Fig. 21 for τ2/τ1 = 1, 5 and 10 with τ1/τ∞ = 2 and Fig. 22 for d/a = 0.05, 0.10 and 0.15 and τ2/τ1 = 5. That is the results do not change with running parameters. The ratio of h/g, disl however, does affect the K II appreciably for d/a = 0.1. This is shown in Fig. 23 where the largest increase of K disl II corresponds to high h/g of 0.05. 6. Crack Sliding Displacement CODII and Edge Dislocation Number n The crack opening or sliding displacements CODI and CODII are defined as

COD I = u 2+ − u 2− , COD II = u1+ − u1−

(19)

CODI is produced by the micro-stress σ2

4(1 − ν 2micro )σ2 [2Λ 2 c 2 − x12 + (f + h)X f +h CODI = πE micro − x1Yf +h − f X f + x1Yf ]

(20)

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Mode II segmented crack model

Fig. 20. Dislocation-stress intensity factor

K disl II

versus normalized length rb /h for different

K disl II

versus normalized length rb /h for different

ratio τ1/τ∞.

Fig. 21. Dislocation-stress intensity factor ratio τ2/τ1.

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X.S. Tang and G.C. Sih

Fig. 22. Dislocation-stress intensity factor ratio d/a.

K disl II

versus normalized length rb /h for different

Fig. 23. Dislocation-stress intensity factor ratio h/g.

K disl II

versus normalized length rb /h for different

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Mode II segmented crack model

The crack sliding displacement CODII can be divided into three parts

COD II = COD ∞II + COD (II1) + COD (II2 )

(21)

where COD∞II , COD (II1) and COD (II2 ) are, respectively, produced by the remotely applied shear stress τ∞, the restraining stresses τ1 and τ2. It has been known that

COD ∞II =

4(1 − ν 2macro ) τ ∞ E macro

CODII(1) =

4(1 − ν2micro )τ1 [−2Λ1 c2 − x12 − (a + d)Xa+d + x1Ya+d + aXa − x1Ya ] πEmicro

c 2 − x 12

(22)

(23)

CODII( 2) =

2 micro

4(1 − ν )τ2 [−2Λ2 c2 − x12 − (f + h)Xf +h + x1Yf +h + fXf − x1Yf ] πEmicro (24)

in which

X a = ln

c 2 − x 12 + c 2 − a 2 c 2 − x 12 − c 2 − a 2

, Ya = ln

a c 2 − x 12 + x 1 c 2 − a 2 a c 2 − x 12 − x 1 c 2 − a 2

(25)

The other quantities of X and Y with subscripts a+d, f and f+h have the same functional forms as the quantities in Eqs. (25). They can be obtained by replacing the subscripts a in Eqs (25) by a+d, f and f+h. Eqs. (22), (23) and (24) can be substituted into Eq. (21) to give

COD II = c 2 − x 12 [

4(1 − ν 2macro ) τ ∞ 8(1 − ν 2micro ) − ( Λ 1τ1 + Λ 2 τ 2 )] πE micro E macro

+

4(1 − ν 2micro )τ1 [ −(a + d )X a +d + x 1Ya +d + aX a − x 1 Ya ] πE micro

+

4(1 − ν 2micro )τ 2 [−(f + h ) X f +h + x 1 Yf + h + fX f − x 1Yf ] πE micro

(26)

in which x1≠a, a+d, f and f+h. Special considerations are given for evaluating the CODs at x1=a, a+d, f and f+h, At these locations the expressions of CODII are given by

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COD II

X.S. Tang and G.C. Sih

x1 = a

= { c 2 − x 12 [

4(1 − ν 2macro ) τ ∞ 8(1 − ν 2micro ) − ( Λ 1τ1 + Λ 2 τ 2 )] πE micro E macro

4(1 − ν 2micro ) τ1 + [−(a + d ) X a +d + x 1 Ya + d ] πE micro 4(1 − ν 2micro ) τ 2 + [ −(f + h )X f + h + x 1Yf + h + fX f − x 1 Yf ]} x1 =a πE micro (27)

COD II

x1 = a + d

= { c 2 − x 12 [

2 macro

2 micro

4(1 − ν )τ ∞ 8(1 − ν ) − ( Λ 1τ1 + Λ 2 τ 2 )] E macro πE micro

+

4(1 − ν 2micro ) τ1 [aX a − x 1 Ya ] πE micro

+

4(1 − ν 2micro ) τ 2 [−(f + h ) X f + h + x 1 Yf + h + fX f − x 1 Yf ]} x1 =a +d πE micro (28)

COD II

x1 = f

= { c 2 − x 12 [

2 macro

2 micro

4(1 − ν )τ ∞ 8(1 − ν ) − ( Λ 1τ1 + Λ 2 τ 2 )] πE micro E macro

+

4(1 − ν 2micro ) τ1 [ −(a + d ) X a +d + x 1 Ya +d + aX a − x 1 Ya ] πE micro

+

4(1 − ν 2micro ) τ 2 [−(f + h ) X f + h + x 1 Yf + h ]} x1 =f πE micro (29)

COD II

x1 = f + h

= { c 2 − x 12 [ +

2 macro

2 micro

4(1 − ν )τ ∞ 8(1 − ν ) ( Λ 1τ1 + Λ 2 τ 2 )] − E macro πE micro

4(1 − ν 2micro )τ1 [ −(a + d )X a +d + x1 Ya +d + aX a − x 1 Ya ] πE micro

4(1 − ν 2micro )τ 2 + [fX f − x 1Yf ]} x1 =f + h πE micro (30)

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359

6.1 Number of edge dislocations The displacement continuity condition at the location x1 = f+h is

nb = CODII

x1 =f +h

(31)

where f = a+d+g as shown in Fig. 4. Since all terms in Eq. (30) are now known, it can be substituted into eq. (31) to give

nb = { c 2 − x 12 [ +

4(1 − ν 2macro )τ ∞ 8(1 − ν 2micro ) − ( Λ 1 τ1 + Λ 2 τ 2 )] πE micro E macro

4(1 − ν 2micro ) τ1 [−(a + d ) X a +d + x 1 Ya +d + aX a − x 1 Ya ] πE micro

(32)

4(1 − ν 2micro ) τ 2 + [fX f − x 1 Yf ]} x1 =f + h πE micro Now the edge dislocation number n can be determined from Eq. (32). Let Emacro = 200GPa, Emicro /Emacro = 5 and νmacro = νmicro = 0.3 together with d/a = 0.1, h/g = 0.01, τ1/τ∞ = 2 and τ2/τ1 = 5, Fig. 24 shows that the number of edge dislocations generated increase with the applied shear stress τ∞ = 10, 20 and 30 MPa and the ratio of the dislocation length rb /h. At τ∞ = 30 MPa and rb /h = 4, the number of edge dislocations is found to be n = 5,200. If the applied shear stress is fixed at τ∞ = 10MPa, the number of dislocations are found to vary as the ratio τ1/τ∞ = 1, 2, 3 and 4. It is shown in Fig. 25 that for τ1/τ∞ = 1 and rb /h = 4 about 17,000 dislocations are generated. The number of dislocations emitted is not sensitive to changes of τ2/τ1 and d/a and hence their numerical values will not be shown. Changes in the ratio h/g will alter n as a function of rb /h. This is illustrated in Fig. 26 for h/g = 0.01, 0.03 and 0.05. It is seen that the ratio h/g has a pronounced influence on the dislocation number n. 6.2 Numerical results of crack sliding displacements Variations of the macro/micro crack sliding displacement CODII with the coordinate x1 are displayed in Fig. 27 for Emacro = 200GPa, Emicro /Emacro = 5 and νmacro = νmicro = 0.3. The four curves are obtained by making use of Eqs. (26) to (30) for τ1/τ∞ = 1, 2, 3 and 4 with τ∞ = 10MPa, d/a = 0.1, h/g = 0.01 and τ2/τ1 = 5. Referring to Fig. 4, it can be seen that the micro-crack tip is located at x1 = f+h = 12.1 mm. Wiggles in the CODII are seen for x1 between 9 to 11 mm. This is caused by the large restraining stress in zone 1 in Fig. 4. As the ratio τ1/τ∞ = 1, the CODII curve is nearly smooth. For τ1/τ∞ = 2 and τ∞ = 10 MPa, it is found that the ratio τ2/τ1 does not affect the results for CODII. The macro-stress restraining zone ratio d/a does

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influence the crack sliding displacement. This shown by the three curves in Fig. 28 for d/a = 0.05, 0.10 and 0.15. The wiggle in the curve diminishes with decreasing d/a ratio.

Fig. 24. Edge dislocation number n versus normalized length rb /h for different applied shear stress τ∞.

Fig. 25. Edge dislocation number n versus normalized length rb /h for different ratio τ1/τ∞.

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361

Fig. 26. Edge dislocation number n versus normalized length rb /h for different ratio h/g.

7. Discontinuities of dW/dV at the Junctions of Scale Crossings As mentioned earlier, discontinuities in the volume energy density function were chosen to allow the use of equilibrium theories with each segmented scale range. The significance of the discontinuity requires special attention that would not be considered here. As an average, the location r/a = 10– 4 is chosen as the connection – between the macroscopic and microscopic zones. The location r/a = 10 7 is chosen as the connection between the microscopic and atomic zones.

Fig. 27. Macro/micro crack sliding displacement CODII versus coordinate x1 for different ratio τ1/τ∞.

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X.S. Tang and G.C. Sih

Fig. 28. Macro/micro/dislocation sliding displacement CODII versus coordinate x1 for different ratio d/a.

7.1. Macro-volume energy density function The macro-volume energy density function for plane-strain is

§ dW· ¸ ¨ © dV ¹

macro

=

1 + νmacro macro 2 2 macro 2 macro 2 macro [(σ11 ) + (σ22 ) − νmacro(σ11 ) ] + σmacro 22 ) + 2(σ12 2Emacro (33)

It should be emphasized that only plane strain prevails near the crack front since Eq. (1) apply to the limit rmacro→0 which is sufficiently small when compared to any finite plate thickness. Inserting Eqs. (1) into Eq. (33), there results

§ dW · ¨ ¸ © dV ¹

macro

=

1 + ν macro ( K macro )2 II E macro 2πrmacro

(34)

Further substitution of Eq. (7) into Eq. (34) gives

§ dW · ¨ ¸ © dV ¹

macro

(1 + ν macro )( Λ macro τ1 ) 2 c = 2 E macro rmacro

(35)

The familiar 1/rmacro singularity for the energy density function is seen from Eq. (35).

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Mode II segmented crack model

7.2 Micro-volume energy density function The micro-volume energy density can be computed by substituting Eq. (9) into

§ dW · ¨ ¸ © dV ¹

micro

=

1 + νmicro micro 2 2 micro 2 2 micro [(σ11 ) + (σmicro + σmicro 22 ) + 2(σ12 ) ] 22 ) − ν micro(σ11 2Emicro (36)

This leads to the micro-volume energy density

§ dW · ¸ ¨ © dV ¹

micro

=

( K micro ) 2 ( K micro )2 1 + ν micro + II ] [(1 − 2ν micro ) I 2πrmicro 2πrmicro E micro

(37)

Both Eqs. (10) and (12) can be put into Eq. (37) to give

§ dW · ¸ ¨ © dV ¹

micro

=

(1 + ν micro )c 4 e [ 2 (1 − 2ν micro )( Λ 2 σ 2 ) 2 + ( Λ micro τ 2 ) 2 ] c 2 E micro rmicro π

(38)

Macro-effect is also reflected by Eq. (38) via the parameter c. 7.3. Dislocation-volume energy density function The dislocation-volume energy density can also be written as

§ dW · ¨ ¸ © dV ¹

disl

=

1 + ν disl disl 2 2 disl disl 2 disl 2 (σ11 ) + ( σ disl 22 ) − ν( σ11 + σ 22 ) + 2( σ12 ) 2 E disl

[

]

(39)

Using the dislocation stresses in Eq. (15), it can be shown from Eq. (39) that

§ dW · ¨ ¸ © dV ¹

disl

E disl ( nb) 2 1 = 2 2 2 16π (1 + ν disl )(1 − ν disl ) rdisl

(40)

The edge dislocation number n is given by Eq. (32). 7.4. Log-log plots of volume energy density function for the full scale range Summarized in Figs. 29 to 32 inclusive are the variations of the volume energy density function for the full scale range from the dislocation to the macroscopic – – range. As mentioned earlier, the discontinuities are preset at r/a = 10 4 and 10 7 although the choice is arbitrary. It is possible to vary these locations to obtain the desired accuracy for the particular scale range of interest.

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X.S. Tang and G.C. Sih

Assuming that the material constants are known to be Emacro = 200GPa, Emicro/Emacro = Edisl/Emicro = 5 and νmacro = νmicro = νdisl = 0.3. The geometric parameters are set at d/a = 0.1, h/g = 0.01 and rb/h = 1 while σ2 = τ2, τ∞ = 10MPa and τ2/τ1 = 5. Referring to the log-log plot in Fig. 29, it can be seen from the full scale range of the radial decay of the volume energy density function that dW/dV decreases gradually with the distance r as it should be. For τ1/τ∞ = 1, 2, 3 and 4, the (dW/dV)macro increases with decreasing ratio of τ1/τ∞ from 4 to 1. The (dW/dV)micro is seen to have the opposite effect. When the ratio d/a is altered from 0.05 to 0.15 with τ∞ = 10MPa and τ1/τ∞ = 2, Fig. 30 shows that the variations of macroscopic length ratio d/a has little or no effect on (dW/dV)micro and (dW/dV)disl. On the other hand, Fig. 31 shows that the variations of the microscopic length ratio h/g = 0.01, 0.03 and 0.05 do not exert must influence on (dW/dV)macro. Similarly, the dislocation length ratio rb/h is varied and it affects only the (dW/dV)disl while its influence on (dW/dV)macro and (dW/dV)micro is not visible. 8. Conclusions and Future Considerations While the macro/micro/dislocation model presented is only an approximate model of line defect under mode II macro-shear applied load, it does reveal the complexities of multiscaling when the interaction of load, geometric and material parameters at the different scales are considered. In retrospect, simplicity of the continuum mechanics model cannot be denied although it cannot be justified in general, particularly when material inhomogeneity and/or anisotropy come into play in the crack tip region. In such situations, it has been shown that the application of fracture criterion concepts based on homogeneity and isotropy can lead to absurd conclusions. Refer to negative energy release rates found for piezoelectric materials [11, 12].

Fig. 29. Volume energy densities (dW/dV)macro, (dW/dV)micro and (dW/dV)disl versus distance r for different ratio τ1/τ∞.

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365

Fig. 30. Volume energy densities (dW/dV)macro, (dW/dV)micro and (dW/dV)disl versus distance r for different ratio d/a.

Fig. 31. Volume energy densities (dW/dV)macro, (dW/dV)micro and (dW/dV)disl versus distance r for different ratio h/g.

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Fig. 32. Volume energy densities (dW/dV)macro, (dW/dV)micro and (dW/dV)disl versus distance r for different ratio rb/h.

It is the aim of this research effort to further extend this model to the subatomic scale where the behavior of the electrons at the outer periphery of the atomic cloud is considered [13] important for studying the macroscopic properties of metals and non-metals. The excitation of the electrons from other energy sources can alter the macro-strength and/or macro-toughness of metals. These effects can be associated with chemical decomposition. Intergranular stress corrosion cracking and instability behavior of solid rocket propellant are cases in point although direct evidence from observation at the subatomic scale remains to be shown. Nevertheless, analytical models can start at the molecular scale. References [1] Sih GC, Survive with the time o’clock of nature, in Sih GC and Nobile L, Restoration, Recycling and Rejuvenation Technology and Architecture Application, Aracne Editrice S.r.l. (2004) 3-22. [2] Sih GC, Crack tip system for environment assisted failure of nuclear reactor alloys: multiscaling from atomic to macro via mesos, J. of Pressure Equipment and Systems, 3 (2005) 1-25. [3] Sih GC, Multiscale cracking aggravated by environment effects ofelectron behavior and local chemistry, in: G. C. Sih, S. T. Tu and Z. D. Wang, Multiscale Damage Related to Environment Assisted Cracking, East China Univer. Sci. and Tech. Press (2005) 1-11. [4] Sih GC, The role of surface and volume energy in the mechanisms of fracture, in: V. Balakrishnan and C. E. Bottani (eds.), Mechanical Properties and Behavior of Solids: Plastic Instability, World Scientific, Singapore (1985) 396-461. [5] Sih GC, Chen EP, Dilatational and distorsional behavior of cracks in magnetoelectroelastic materials, J. of Theoretical and Applied Fracture Mechanics, 40(1) (2003) 1-21.

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[6] Sih GC, Tang XS, Dual scaling damage model associated with weak singularity for macroscopic crack possessing a micro/mesoscopic notch tip, J. of Theoretical and Applied Fracture Mechanics, 42(1) (2004) 1-24. [7] Sih GC, Tang XS, Simultaneity of multiscaling for macro-meso-micro damage model represented by strong singularities, J. of Theoretical and Applied Fracture Mechanics, 42(3) (2004) 199-225. [8] Sih GC, Mechanics of fracture initiation and propagation, Kluwer Academic Publishers, The Netherlands, 1991. [9] Sih GC, Tang XS, Mode I segmented crack model: macro/symmetry, micro/anti-symmetry and dislocation/skew-symmetry, in: G. C. Sih (Ed.), Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano, Springer (2006) 291-319. [10] Friedel J, Dislocations, Pergamon Press, Oxford, 1964. [11] Spyropoulos CP, Energy release rate and path independent integral study for piezoelectric material with crack, International Journal of Solids and Structures 41 (2004) 907-921. [12] Lin S, Narita F, Shindo Y, Comparison of energy release rate and energy density criteria for a piezoelectric layered composite with a crack normal to interface, J. of Theor. and Appl. Fract. Mech. 39(3) (2003) 229-243. [13] Sih GC, Signatures of rapid movement of electrons in the valence band region: interdependence of position, time and temperature, J. of Theoretical and Applied Fracture Mechanics, 45(1) (2005) 1-12.

Microstructure and microhardness in surface-nanocrystalline Al-alloy material Y.G. Wei*, X.L. Wu, C. Zhu, M.H. Zhao LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China

Abstract Discussed and analyzed are the surface-nanocrystallization of Al-alloy material manufactured by using the ultrasonic shot peening method. The transmission electron microscope (TEM) is used to examine the microstructure features and nanocrystalline mechanisms, such as, the formed microbands, the divided subbands, the infinitesimally divided and formed nanometer-sized grains and grain boundaries, etc. Based on the microscale observation and measurement, the mechanical behaviors of the surface-nanocrystalline Al-alloy material are investigated experimentally at submicron scale by means of the nano-indentation test. The load- and hardness-indent depth curves are measured. The grain size and its nonuniform effects are investigated. In the theoretical modeling, based on the microstructure characteristics and the experimental features, a dislocation pile-up model considering grain size effect based on the Mott theory is presented and used. The experimental hardness-indent depth curves which display the strong size, geometry and nonuniformity effects are successfully modeled and predicted. Keywords: Surface-nanocrystalline Al-alloy; Microstructure; TEM observation; Nanoindentation test; Size effect.

1. Introduction Recent studies have shown that the high strength nano-structured materials can be fabricated by using some special techniques. The severe plastic deformation (SPD) method can be used to fabricate the nanocrystalline (NC) materials [1-4]. In addition, the mechanical behaviors of the conventional materials can be improved by using some NC methods. The surface-nanocrystalline (SNC) technique can make the nanocrystallization within a material surface layer such that material behaviors can be improved considerably [5, 6]. The adopted SPD methods include: the large torsion method [1], the large pressing method [4], the ultrasonic shot peening (USP) method [5, 6], etc. The microstructure features of both the nano-structured bulk materials and the SNC materials have been widely investigated in [1-6]. Recent investigations have displayed that the SNC materials have the regular microstructures overall or locally within the SNC surface layer. The

*

Corresponding author. E-mail address: [email protected] (Y.G. Wei). 369

G.C. Sih (ed.), Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano, 369–387. ” 2007 Springer.

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variation of the representative cell size of the microstructures is from tens to hundreds of nanometers, even to microns from the NC surface to the interior of the material inside. The mechanical behavior of the NC materials has also been studied. Besides the works in [1-6], a wide characteristics of the NC materials have been investigated. They include the grain boundary behavior and plasticity in the NC Ni [7], the compressive behaviors of the NC Al-alloy [8], the surface roughness effect on the hardness of the NC Al-alloy [9], the grain rotation model of a 9-grain cluster mechanism for the NC copper [10], the strain-rate sensitivity in the NC Ni [11], the formed nanometer crystal grain due to indented for a bulk amorphous metal alloy [12], the tensile behaviors of the NC electrodeposited Ni [13] and the high tensile ductility of the NC copper [14]. Below the micron scale, the materials tend to display the strong size effects. Research along this line has been related to the nanoindentation tests for single crystal or coarse-grained metals [15-19]. The results show that indent depth decreases. The measured hardness curve displays an increasing trend, i.e., the size effect which has been explained by the strain gradient theories [20-25], the dislocation density theory [16, 9, 26], as well as the discrete dislocation theory [27]. The modeling and simulation results were consistent with the experimentally measured results. However, the NC materials are very complicated and quite different from the single crystal or coarse-grained materials. Besides the size effect, the influence of both the crystal grain size and grain shape distributions should affect the material mechanical behavior. This is because the NC material grain size is so small that it is comparable to the material length scale or the strain gradient sensitive zone size. Previous research on the nano-polycrystal Al and the thin film/substrate system [28], both the crystal grain size and the shape distribution effects were referred to as the “geometrical effect” so as to distinguish therm from the size effect described by the microscale parameter of the strain gradient theories. The microstructure cell model and the strain gradient plasticity theory have been studied with reference to the size and geometrical effect. The predicted and experimental results have been studied with reference to the effects of grain size and the microstructure characteristics in relation to the microscale parameter of the strain gradient theory [28]. Similar model has been used to investigate the size, geometry and nonuniformity effects in the SNC materials [29]. In what follows, the microstructure features in the SNC Al-alloy LC4 are first considered ked by using the USP method through transmission electron microscopy (TEM) and high resolution TEM (HRTEM) observation and measurement. Next, the mechanics behavior of the SNC Al-alloy material will be studied experimentally and theoretically based on the nanoindentation experiments. The specimens are designed and prepared according to the microstructure features of the NC material. The load- and hardness-depth curves are measured and analyzed. Using the Mott dislocation pile-up theory [30], a dislocation mechanism considering the grain boundary constraint effect will be presented and used to model the nano-indentation experiments for NC Al-alloy material as in [31]. Here, the attention will be focused on the microstructure observation, measurement and experiments. In addition, the material hardness curves will be predicted by using the model based on the Mott dislocation pile-up theory. Finally, through experimental research and the theoretical simulation and analysis for the SNC Al-alloy material, limitations and further work will also be discussed.

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2. SNC Materials and Procedures The procedures of the experimental specimen preparations are similar to those shown in [6] for another material, Al-alloy 7075. A brief description of the procedures is given as follows. 2.1 Material The experimental material was a high purity Al-alloy LC4, with a composition (wt pct) of 2.5Mg, 5.6Zn, 0.4 Mn and 1.8Cu, balance Al. A commercially available plate was cut into pieces with 100 × 100 × 6 mm3 in dimension. A smooth surface finish was attained on the faces by polishing on 800-1200 grade SiC papers. Microscopic examination revealed an initial grain size of the order of ~80 microns. 2.2 The USP technique The principle of the USP technique was described in [5]. A high-energy ultrasonic generator of high frequency (45 kHz) is used to vibrate the reflecting chamber by application of the stainless steel shots of 7.45mm diameter. The shots then performed repetitive, high-speed, and multi-directional impact onto the surface of the materials. Severe plastic strains were imparted into the surface by striking. The USP processing was conducted under vacuum at room temperature for 15 minutes. Through the USP technique treatment, the nano-scale crystal grains were formed near the NC surface, and the grain sizes change with the distance away from the NC (striking) surface in a gradient law, as described in the sketch figure, Fig. 1. Near the NC surface, nanometer-sized grains are formed. They also occur at the distance far away from the NC surface, for instance distance D > 80 micron. The grain size keeps the original coarse grain size. Between them, the grain sizes change with the distance away from the NC surface. 2.3 Microstructure examination The JEM-2000FXII transmission electron microscope (TEM) operated at 200 kV was used for examination of the general microstructure features at low magnifications. A JEM-2010FEF high-spatial-resolution analytical electron microscope (HRTEM) is used for high magnifications and lattice image observations in grains with a focused beam having a diameter of ~1 nm. Lattice images were taken at close to the optimum defocus conditions, typically at a magnification of 500,000X, with the selected grain oriented close to for lattice imaging. Thin TEM films were prepared by the steps: (1) sticking a castolite plate 2 mm thick on the peened surface, (2) cutting a bar 3 mm in diameter with the peened layer located at the middle, (3) cutting discs 30 micron thick using a diamond saw normal to the long axis of the bar and (4) dimpling and argon-ion-beam thinning to perforation at room temperature. This method allows the inspection of a well-identified depth of the processed surface layer. The grain size measurements were made directly from

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TEM photomicrographs and the reported values are the averages from 40-60 individual measurements. Because of the elongated nature of the subgrains, the datum points were separately presented for measurements of the average of the short axis, the average of the long axis, and the average from randomly selected directions, whereas measurements for the grains were taken consistently along randomly selected directions.

Fig. 1. Sketch of the surface-nanocrystallization principle for a coarse-grained metal undergoing the ultrasonic shot peening (USP) technique.

Fig. 2. Statistically averaging grain sizes changing with the distance away from the nanocrystalline(NC) surface of Al-alloy material.

3. Tem Observation and Measurement for SNC Al-alloy 3.1 Grain size distribution Fig. 2 shows the relationship of the averaged grain size changing with distance away from the NC surface. According to the TEM observation and measurementˈ the original coarse grain size is kept at approximately 80 micron for distance larger than this. There the nanocrystallization effect seems to be negligible. From Fig. 2, the grains are nanocrystallized remarkably and the grain size attains the nanometer scale (grain size ı as . In particular, the saturation stress of the bicrystal RB increases with increasing strain amplitude, showing no plateau feature in the applied strain range. This difference in saturation stresses between the bicrystals CB and RB is believed to be purely caused by the large-angle GB, This will be discussed subsequently. The curves of the saturation resolved shear stress IJ as vs. saturation plastic resolved shear strain Ȗ pl of the crystals (G1, G2) and the bicrystals (CB, RB) are plotted in Fig. 11(b). The plateau saturation resolved shear stresses IJ as of the crystals (G1, G2) and the bicrystal RB nearly maintain the constant value in the range of 29-31MPa. This is consistent with the results of the copper single crystals oriented for single-slip [15, 16]. But the actual grown bicrystal RB does not display a plateau in its CSSC and shows a higher saturation stress (31.4 – 34.0 MPa) than that (29.4 – 30.0 MPa) of the bicrystal CB. This cyclic stress-strain behavior of the bicrystal RB should be attributed to the existence of the large-angle GB. 3.2 Surface slip morphologies Optical microscopy shows that only the primary slip system B4 (111) [101] was activated within the component crystals (G1, G2) of the bicrystal CB under all applied

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strain amplitudes. This indicates that the cyclic plastic strain of the bicrystal CB was carried by the primary slip bands within the two component crystals (G1, G2). However, the secondary slip besides the primary slip was observed within the component crystal G2 near the GB in the bicrystal RB, as shown in Figs. 12(a) and (b). The primary slip lines are regularly modulated by the secondary slip lines and no other slip systems are involved. The primary slip lines and secondary slip lines emerge alternately near the GB to form a GB affected zone (GBAZ) with a cross-weaved structure, which is similar to the deformation bands (DBs) in appearance (see Fig. 12). This phenomenon can be associated with the incompatibility of stress-strain at the GB [3-6]. Meanwhile, The width or volume fraction of the GBAZ was found to depend on the applied strain amplitude, as shown in Figs. 13(a) and (b). The measured mean width WGB and volume fraction VGB of the secondary slip region are listed in Table 2. It is obvious that the width WGB and volume fraction VGB of the GBAZ increase and the interactions of primary slip with secondary slip become more serious with increasing strain amplitude. In general, the macroscopic strain compatibility conditions at a GB plane can be fulfilled if the total number of independently operating slip systems in both neighboring crystals is four [1, 38]. However, it is interesting to note that the total number of activated slip systems is only 3 beside the GB in the present real bicrystal RB. 3.3 Orientation factor of copper bicrystals The Schmid factor Ω is widely used to calculate the resolved shear stress of single crystals from the imposed axial stress. For polycrystals, the Taylor factor (M = 3.06) or the Sachs factor (M = 2.24) is often employed [39-43]. However, the orientation factor of a bicrystal was seldom discussed before. Hu et al. [44, 45] has selected the mean value of the Schmid factors of two component crystals to illustrate the CSSCs of copper bicrystals. But the physical meaning and crystallographic basis for this selection are not clear. For a bicrystal CB with two individual single grains G1 and G2 combined in parallel, as shown in Fig. 14, the two component crystals will deform independently owing to the absence of GB. As the bicrystal CB is cyclically saturated, the force balance yields as

PasCB = PasG1 + PasG2

(10)

or G1 G2 ı CB as A CB = ı as A G1 + ı as A G2 , CB

G1

G2

(11)

where Pas , Pas and Pas are the loads applied to the bicrystal CB and the CB G1 G2 component crystals G1 and G2; ı as , ı as and ı as are the axial saturation

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stresses applied to the bicrystal CB and the component crystals G1 and G2; ȍ B , ȍ G1 and ȍ G2 are the orientation factors of the bicrystal CB and the component crystals G1 and G2; A CB , A G1 and A G2 are the areas of the bicrystal CB and G2 CB G1 the component crystals G1 and G2. Substituting IJ as , IJ as and IJ as (Eq. (1), Eq. (3) and Eq. (5)), the Schmid factors ( Ω B , ΩG1 and ΩG 2 ) of two component crystals (G1, G2) and the combined bicrystal CB into Eq. (11), the results are

IJCB IJG1 IJ G2 as A CB = as A G1 + as A G2 ȍB ȍ G1 ȍ G2

(12)

Fig. 12. Slip morphology in the vicinity of GBs in the bicrystal RB fatigued at εa = 3.0 × 10–3.

Fig. 13. GB affected zone (GBAZ) in the bicrystal RB. (a) εa = 1.5 × 10 –3 and (b) εa = 2.5 × 10–3.

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Fig. 14. Diagrammatic sketch of stress distribution in a combined bicrystal CB.

Fig. 15. CSSCs of the [679] //[145] copper bicrystals RB and CB along with the [135] //[135],

[135] //[ 235] and [235] //[ 235] copper bicrystals.

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The saturation resolved shear stress IJ as of the single copper crystal oriented for single slip maintained a constant value in the range of 28-30MPa in the region B of its CSSC and is independent of crystal orientations [16]. Therefore PSB

G1 G2 PSB IJCB as = IJ as = IJ as = IJ as

(13)

Substituting Eq. (19) into Eq. (12), it is found that

V V 1 = G1 + G2 ȍB ȍ G1 ȍ G2

(14)

Thus, the orientation factor ΩB of the bicrystal can be expressed as

§V V · ȍ B = ¨ G1 + G2 ¸ © ȍ G1 ȍ G2 ¹

−1

(15)

where VG1 and VG2 are volume fractions of the component crystals G1 and G2 in the bicrystal CB, respectively.For the bicrystals CB and RB, the values VG1 = VG2 = 0.5, ΩG1 = 0.35 and ΩG2 = 0.47 are used. Substituting the values of VG1, VG2, ΩG1 and ΩG2 into Eq. (15), the results is ΩB = 0.40. Essentially, the orientation factor Ω B of the bicrystal can be regarded as a geometrical transformation factor, by which the saturation resolved shear stresses along the primary slip system of the component crystals in a bicrystal can be well estimated. 3.4 Comparison of CSSCs in copper bicrystals To have a better understanding of the effect of GBs on cyclic deformation behavior, the CSSCs of the copper bicrystals CB and RB as well as [135] //[135] , [135] //[ 235] and [235] //[ 235] copper bicrystals [44, 45] are shown in Fig.15. For comparison, all the saturation resolved shear stresses are calculated by using the orientation factor Ω B of the bicrystal in terms of Eq. (11). It can be seen that nearly all the CSSCs show a plateau region except the bicrystal RB, however, the plateau saturation resolved shear stresses strongly depend on the types of the bicrystals. For the [135] //[135] bicrystal [45] and the bicrystal CB, their plateau saturation resolved shear stresses are nearly the same, namely 29-30MPa, and hence basically equal to that (28-30MPa) of copper single crystals [15, 16]. This indicates that the strengthening effect of the GB in the co-axial [135]//[135]

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bicrystal can be negligible. For the [135] //[ 235] and [235] //[ 235] copper bicrystals [48], their plateau resolved shear stresses (about 32.5MPa and 36MPa) are different and obviously higher than that (28-30MPa) of copper single crystals. This demonstrates that the strengthening effect of the large-angle GBs in the two bicrystals is significantly improved by changing the combination of different component crystals and the GB properties. However, the saturation resolved shear stress of the bicrystal RB is always higher than that (29-30MPa) of the bicrystal CB and increases with increasing strain amplitude without showing a plateau region. It can be attributed to the effect of the GBAZ produced by GB and will be explained as follows. 3.5 Strengthening Effect of Large-angle GBs The effect of large-angle GB on the flow stress has been extensively discussed for bicrystals with GBs parallel to the stress axis under uniaxial loading [1-6, 10-14]. The works in[10] have given a stress relation for iso-axial β-brass bicrystals as

ı T = ı B +VGB ( ı GB - ı B ) ,

(16)

where σGB was defined as the average stress in the GBAZ. It can be determined from the applied stress σT, the flow stress σB of single crystal and the volume fraction VGB in GBAZ. The increase in flow stress of the bicrystal is mainly attributed to the existence of the GBAZ. Found in [3, 4] was the increase in the flow stresses of some bicrystals by the presence of Σ7 and Σ21 coincidence GBs. As shown in Fig. 12 and 13, there is a GBAZ in the grown bicrystal RB subjected to cyclic deformation. However, there is no such GBAZ in the combined bicrystal CB. Apparently, the difference in saturation stresses between the bicrystals CB and RB can be attributed to the higher mean stress within the GBAZ. As the bicrystals (CB and RB) were cyclically saturated, similar to Eq. (16), their axial saturation stresses will obey the following relation, GB CB ı asRB = ı CB as +VGB ( ı as -ı as ) .

(17)

Fig. 16(a) shows that G1 G2 ı CB as = ı as VG1 +ı as VG2 .

(18)

As listed in Table 2 and shown in Fig. 16(b), the stress difference ǻı as in axial saturation stresses between the bicrystals CB and RB can be simply calculated by the following relation B

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ǻı asB = ı asRB -ı CB as .

(19)

If ǻı as is attributed to the higher mean stress ı as in the GBAZ, by combining GB Eq. (17) – (19), ı as in the GBAZ can be expressed as B

ı GB as =

GB

ı asRB -ı asCB ǻı B + ı asCB = as +ı CB as VGB VGB

(20)

The calculated results of ǻı as and ı as at different strain amplitudes for the GB bicrystals CB and RB are listed in Table 3. It can be seen that the mean stress ı as CB RB in the GBAZ is much higher than the mean stresses ( ı as , ı as ) of the [679] //[145] bicrystals CB and RB, and decreases with increasing strain amplitude. GB It is generally recognized that the stress ı as in the GBAZ is not a constant value GB and decreases apart from GB [10,12]. Actually, the mean stress ı as in GBAZ can be expressed more precisely as B

ı =ı GB as

CB as

³ +

VGB

0

ǻı Bas ( V ) dV VGB

GB

,

(21)

where ǻı as (V) is a function of the position and decreases apart from the GB, as shown in Fig. 16(b). Thus, it may be responsible for the reduction of the higher GB mean stress ı as in the GBAZ. B

Table 3. The average stresses in the GBAZ for the bicrystal RB. Strain amplitude 0.1% 0.15% 0.2% 0.25%

0.3%

∆σ %DV (MPa)

4.9

6.2

7.3

8.7

9.8

σ *DV% (MPa)

184.6

177.3

174.1

171.5

168.5

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Fig. 16. Illustration of axial saturation stress distribution in the [679] //[145] bicrystals CB and RB. (a) The combined bicrystal CB; (b) really grown bicrystal RB.

From the viewpoint of plastic strain localization, the cyclic plastic strain of a bicrystal is mainly carried by PSBs within the component crystals G1 and G2 according to the two-phase model [17, 18]. As shown in Fig. 17(a) and by using Eq. (3a) and Eq. (11), the saturation resolved shear stress of the combined bicrystal CB can be described as −1

IJ

CB as

=IJ

PSB as

=ı

CB as

§ VG1 VG2 · + ¨ ¸ . © ȍ G1 ȍ G2 ¹

(22)

When PSBs meet a GB in a real grown bicrystal RB, an additional shear stress should be applied to PSBs due to the constraint of GB, as shown in Fig. 17(b). Consequently, the bicrystal RB should have a higher saturation stress than the bicrystal CB owing to the effect of GB. The saturation resolved shear stress τ asRB of the bicrystal RB can be calculated from Eq. (22) as

IJ

RB as

=ı

RB as

§ VG1 VG2 · + ¨ ¸ © ȍ G1 ȍ G2 ¹

−1

(23)

Combining Eqs. (22) and (23), the additional GB resistance ∆τ asGB to PSBs can be introduced as

Grain boundary effects on fatigue damage and material properties

ǻIJ

GB as

=IJ

RB as

−IJ

PSB as

= (ı

RB as

§V V · − ı ) ¨ G1 + G2 ¸ © ȍ G1 ȍ G2 ¹

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−1

CB as

(24)

The calculated additional GB resistance ǻIJ as at different strain amplitudes is GB listed in Table 3. It can be seen that ǻIJ as is also increased with increasing strain amplitude. From the analysis and calculations above, it seems that the existence of a GBAZ with a higher stress may be responsible for the increase in the cyclic saturation stress and the disappearance of the plateau region in the CSSC of the really grown bicrystal RB. GB

4. Effects of Crystallographic Orientations and Perpendicular GBs on CSSCs 4.1 Comparison of CSSCs for copper bicrystals with perpendicular GBs The cyclic deformation behavior of different copper bicrystals with a perpendicular GB has been systematically investigated [46-49], including [123] ⊥ [335] , [134] ⊥ [134] , [345] ⊥ [117] and [5913] ⊥ [579] copper bicrystals. The component crystal orientations of these bicrystals are shown in Fig. 6(b). The results show that these copper bicrystals have quite different CSSCs, as shown in Fig.18. The [345] ⊥ [117] bicrystal [46] has higher cyclic saturation stress than the other bicrystals. At lower axial plastic strain range from 1.2 × 10–4 to 6.3 × 10–4, the cyclic saturation stress of [345] ⊥ [117] bicrystal rapidly increases with increasing plastic strain amplitude. In axial plastic strain range from 6.3 × 10–4 to 1.62 × 10–3, the saturation stress slowly increases from 93 to 98 MPa, showing a pseudo-plateau region. At higher plastic strain range, the saturation stress can reach a very high value of about 120 MPa. For the [123] ⊥ [335] copper bicrystal, its CSSC displays a plateau region in the lower axial plastic strain range from 2.1 × 10–4 to 1.1 × 10–3 [48]. The axial saturation stress in the plateau region is about 65 – 68 MPa. When the axial plastic strain amplitude is higher than 1.1 × 10–3, its cyclic saturation stress increases monotonically with increasing the plastic strain amplitude. There are two plateau regions in the CSSC of the [5913] ⊥ [579] copper bicrystal. The axial saturation stress is about 62 to 64 MPa in the lower plateau region from axial plastic strain range of 1.8 × 10–4 to 1.35 × 10–3 [49]. While in higher axial plastic strain range from 1.7 × 10–3 to 2.56 × 10–3, the axial saturation stress increases to 70 – 71 MPa. Similar to the copper single crystal oriented for single slip, the [134] ⊥ [134] copper bicrystal [50] shows a clear plateau region over a very wide axial plastic strain range from 9 × 10–5 to 3.3 × 10–3. The plateau axial

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saturation stress is about 61 – 62 MPa in the [134] ⊥ [134] bicrystal. This indicates that the copper bicrystals with a perpendicular GB can display quite different CSSCs, which strongly depend on the crystallographic orientations of the component crystals.

Fig. 17. Illustration of saturation resolved shear stress distribution in the [679] //[145] bicrystals CB and RB. (a) The combined bicrystal CB; (b) The really grown bicrystal RB.

Fig. 18. The CSSCs of [134] ⊥ [134] , [5913] ⊥ [579] , [123] ⊥ [335] and [345] ⊥ [117] copper bicrystals.

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Fig. 19. Surface slip morphology of the [5913] ⊥ [5 79] copper bicrystal cycled at different strain amplitudes and cycles. (a) εPl = 0.76 × 10–3; (b) εPl = 2.30 × 10–3.

4.2 Plastic deformation features of the bicrystals with perpendicular GBs It is well known that the CSSCs of copper single crystals oriented for double slip and multiple slip may or may not show a plateau region [50-54].The slip morphology features in the copper bicrystals have been shown in [49-52]. For [134] ⊥[134] and [5913] ⊥ [579] copper bicrystals, only the primary slip bnads on the component crystal surfaces were found for all of the applied strain amplitudes. Fig. 19(a) shows the slip morphology near the grain boundary in the [5913] ⊥ [579] copper bicrystal. Obviously, there is GBAZ, which is quite different from that near the parallel GB, Figs. 12 and 13. At higher strain amplitude, the slip morphology of the [5913] ⊥ [579] bicrystal is shown in Fig. 19(b). There is still no GBAZ and only the primary slip bands were activated on both component crystals G1 [5913] and G2 [579] . In particular, the plastic strains carried by two component crystals

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G1 and G2 display obvious difference as shown in Fig. 19(b). The crystal orientations are shown to play a decisive role in the CSSCs of the bicrystal with a perpendicular GB. Consequently, the difference in the CSSCs of the copper bicrystals can be explained from the combination of component crystals in the bicrystals. The work in [46] attempted to compare the CSSCs between [345] ⊥ [117] copper bicrystals and polycrystals by defining an orientation factor (ΩB) for the bicrystal:

ȍB =

ȍ G1 + ȍ G2 . 2

(25)

Calculated was the saturation resolved shear stress (τas) of the copper bicrystal by the following formula

IJas = ı as × ȍ B = ı as ×

ȍ G1 + ȍ G2 2

(26)

For the bicrystal with a perpendicular GB, the axial stresses (σas) applied on each component crystal are equal during cyclic deformation. The resolved shear stresses G1 G2 ( IJas , IJ as ) applied on the primary slip systems of each component crystal will be not the same if their Schmid factors (ΩG1, ΩG2) are different. As a matter of fact, G1 G2 the saturation resolved shear stresses ( IJas , IJ as ) on the primary slip systems can be calculated by using the following formulae separately, i.e.

IJG1 as = ı as × ȍ G1

(27)

IJG2 as = ı as × ȍ G2

(28)

Thus, the calculated resolved shear stress (τas) of copper bicrystal by using Eq. (26), in fact, does not represent the true resolved shear stress in any primary slip systems B G1 of the component crystals. When a bicrystal was cyclically saturated, İ pl , İ pl G2 and İ pl , which are the axial plastic strain amplitudes applied to the bicrystal and to the component crystals G1 and G2, will have the following relation

İ plB = İ plG1VG1 + İ G2 pl VG2 .

(29)

where VG1 and VG2 are the volume fractions of the crystals G1 and G2. Taking VG1 = VG2 = 0.5, it can be shown that

Grain boundary effects on fatigue damage and material properties G2 2İ Bpl = İ G1 pl + İ pl

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(30)

As reported in [55], according to the slip morphology in Fig. 19(b) and Eqs. (27) and (28), it can be concluded that the component crystal G1 with a higher Schmid factor will be subjected to higher resolved shear stress and carry more plastic strain than the component crystal G2 with a lower Schmid factor, i.e. G1 İ G2 pl < İ pl .

(31)

Combining Eq. (30) with Eq. (31), İ pl , İ pl and İ pl should obey the following relation, B

G1

G2

G1 B İ G2 pl < İ pl < İ pl .

(32)

Therefore, the plastic strain İ pl carried by the crystal G1 with a higher Schmid G2 fator should be higher than that İ pl by the crystal G2 with a smaller Schmid factor. Figure 19(b) provides a powerful evidence for this relation. Obviously, there exists inhomogeneous plastic strain in such bicrystals. G1

4.3 Effect of crystallographic orientations on CSSCs The bicrystals discussed above contain at least one component crystal oriented for single slip. By using Eqs. (27) and (28), the cyclic saturation resolved shear stresses in the primary slip systems of the single slip oriented component crystals in these bicrystals were calculated at the applied plastic strain amplitudes. The calculated saturation resolved shear stress is plotted as a function of the applied axial plastic strain amplitude, as shown in Fig. 20.The resolved shear stresses in the plateau region basically maintain a constant value of about 28-31MPa for [5913] , [123] and [134] component crystals. However, the component crystal [579] and [345] showed a lower plateau region and an intermediate pseudo-plateau region. Apparently, the plateau (or pseudo-plateau) behavior in the cyclically deformed copper bicrystals should be attributed to the cyclic saturation of the single slip oriented component crystals at the region B in their CSSCs. From the results above, we can classify these bicrystals into two types, i.e. (i) Single-single combined bicrystals, such as [134] ⊥ [134] and [5913] ⊥ [579] bicrystals. (ii) Single-double combined bicrystals, such as [123] ⊥ [335] and [345] ⊥ [117] bicrystals.

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For the first type of bicrystals, if the Schmid factors of two component crystals are identical, such as [134] ⊥ [134] bicrystal, the resolved shear stresses and plastic strain applied on each component crystal will have no difference. Therefore, the CSSC of the bicrystal should display one plateau region over a wide plastic strain range as that of a single slip oriented single crystal (see Fig. 18 and Fig. 20). This indicates that the perpendicular GB in such bicrystal does not affect its CSSC. When the Schmid factors of two component crystals oriented for single slip are different, such as [5913] ⊥ [579] bicrystal, the plastic strain and resolved shear stresses on the primary slip systems of each component crystal will be different during cyclic deformation. At lower plastic strain range, most part of the plastic strain will be carried by the soft component crystal [5913] with a higher Schmid factor (0.452). As a result, when the bicrystal was cyclically saturated, the plastic strain carried by the soft component crystal [5913] will be situated in the region B of its CSSC. But the hard component crystal [579] with a lower Schmid factor (0.406) is actually deformed at the plastic stain range below region B of its CSSC. As shown in Fig. 19, the resolved shear stresses of the crystals [5913] and [579] are 28-29MPa and 25-26MPa, respectively, in the lower plateau region. This indicates that the occurrence of the lower plateau region of CSSC in [5913] ⊥ [579] bicrystal is due to the cyclic saturation of the soft component crystal [579] at the region B of its CSSC. With increasing plastic strain amplitude, the plastic strain carried by the soft component crystal [5913] will reach the region C of its CSSC with higher cyclic saturation stress. As shown in Fig. 20, the resolved shear stresses of two component crystals are 31-32MPa and 28-29MPa, respectively in the upper plateau region. In this case, the plastic strain carried by the hard component crystal [579] should have reached the region B of its CSSC. Dislocation observation on the cyclically saturated [5913] ⊥ [579] copper bicrystal [52] supported the hypothesis above. Therefore, the appearance of double plateau region in the CSSC of the [5913] ⊥ [579] copper bicrystal can be schematically illustrated as in Fig. 21(a). The argument above can also be applied to the second type bicrystals. If the Schmid factor of the single-slip oriented component crystal is higher than that of the double-slip oriented component crystal, such as [123] ⊥ [335] bicrystal, the plastic strain and resolved shear stresses applied on the primary slip systems of each component crystal will be also different. In the lower axial plastic strain range from 2.1×10 − 4 to 1.1×10 −3 , as the bicrystal cyclically saturated, the soft [123] component crystal with a relatively higher Schmid factor of 0.467 will be cyclically deformed within the region B of its CSSC with a saturation resolved shear stress of about 28-30MPa (Fig. 19). However, the hard component crystal [335] with a relatively low Schmid factor of 0.38 is subjected to a cyclic saturation below

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415

region B. As a result, the CSSC of [123] ⊥ [335] bicrystal exhibits a plateau region at lower plastic strain range. At higher strain range, the plastic strain carried by [123] component crystal will reach the region C of its CSSC. Meantime, the cyclic saturation stress of the component crystal [335] oriented double slip may increase with increasing the plastic strain amplitude, which is similar to [001], [117] and [111] copper single crystals [53-56]. Consequently, the [123] ⊥ [335] bicrystal did not show an upper plateau region as the [5913] ⊥ [579] bicrystal at higher plastic strain range. Fig. 21(b) schematically illustrates the formation of one plateau region in the CSSC of [123] ⊥ [335] bicrystal.

Fig. 20. The curves of the resolved shear stress of single slip oriented component crystals vs plastic strain amplitude in the [134] ⊥ [134] , [5913] ⊥ [579] , [123] ⊥ [335]

and

[345] ⊥ [117] copper bicrystals.

For the [345] ⊥ [117] bicrystal, the Schmid factor (0.432) of the double-slip oriented component crystal [117] is higher than that (0.390) of the single-slip oriented component crystal [345] . At lower plastic strain range, the plastic strain carried by the crystal [117] should be higher than that carried by the crystal [345] . TEM observations in the [345] ⊥ [117] bicrystal [46] revealed that the cyclic saturation dislocations were characterized by loop patch (or vein) structures in [345] component crystal cycled at lower plastic strain range. Therefore, the plastic strain

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carried by [345] component crystal should be in the region A of its CSSC. Meanwhile, the cyclic saturation stress of [117] copper single crystal increased with increasing strain amplitude at a wide plastic strain range [51]. As a result, the saturation stress of the [345] ⊥ [117] bicrystal should increase with increasing plastic strain amplitude in lower plastic strain range, as shown in Fig. 17. As the axial plastic strain was increased to 6.3 × 10–4, the plastic strain carried by the crystal [345] may reach the region B in its CSSC and resulted in a plateau region in its CSSC. However, the calculated resolved shear stress of the [345] component crystal in [345] ⊥ [117] bicrystal is obviously higher than 28-30MPa and its CSSC only displayed a pseudo-plateau region, as shown in Fig. 18. This may be associated with the operation of multiple slip systems in [345] ⊥ [117] bicrystal [46]. As a result, the [345] ⊥ [117] bicrystal displayed a higher saturation resolved shear stress at intermediate plastic strain range than the other bicrystals. As increasing plastic strain amplitude, the saturation stress of the [345] ⊥ [117] bicrystal will increase again because the plastic strain carried by [345] component crystal may reach the region C in its CSSC. This process can be schematically illustrated as in Fig. 21(c). 5. Fatigue Cracking Mechanisms 5.1 Fatigue cracking behavior of bicrystals containing type I GBs During cyclic deformation, [123] ⊥ [335] , [5913] ⊥ [579] and [014] ⊥ [115] copper bicrystals with a perpendicular GB exhibit rapid initial cyclic hardening and consequent cyclic saturation behavior [51,52,59]. Surface observations reveal that many PSBs are activated and terminate at the GBs for all the bicrystals. Figs. 22(a) and 22(b) show typical interactions of slip bands with a perpendicular GB in those bicrystals. In general, primary slip bands can transfer through the whole component grains and sometimes some secondary slip bands occur near the GBs, depending on the orientations of the component grains. If cyclic deformation continues to apply to the bicrystal specimens, after a long cyclic saturation, a rapid cyclic softening corresponding to fatigue crack initiation can be found. Surface observations show that all the fatigue cracks nucleated and propagated along the GB in all the copper bicrystals above. Figure 23(a) shows a typical intergranular fatigue cracking of the bicrystals with a perpendicular GB. It is noted that there are only primary slip bands near the GB even the intergranular crack has opened a larger displacement (about 40 µm). It seems that the primary slip bands should control the GB fatigue cracking process of the bicrystals. With further cyclic deformation, fatigue crack will continuously propagate along the GB, until the occurrence of intergranular fatigue fracture. If the observations are focused on the whole surface of the bicrystals,

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however, PSBs never become the sites of fatigue cracking after intergranular fracture of the bicrystals. These results are consistent with the observations in [117] ⊥ [345] , [134] ⊥ [134] , [001] ⊥ [149] and [149] ⊥ [149] copper bicrystals [46, 47, 55]. In those copper bicrystals, the secondary slip bands were activated near the GBs due to double- or multi-slip orientations of the component crystals. Investigated in [57, 58] are the fatigue cracking behavior of the bicrystals with a perpendicular GB. It was found that fatigue cracks often nucleate and propagate along the GB during cyclic deformation even though some GBs have a low Σ value, such as Σ3, Σ9 and Σ41 GBs. For the copper bicrystal with a GB tilting to the stress axis, we also found that GB cracking is the unique cracking mode under low or high strain amplitude, as shown in Fig. 23(b), which is identical with the bicrystals with a perpendicular GB. From the results above, it can be concluded that intergranular fatigue cracking is the dominant damage mode even though the orientations of the component grains and the GB structures are obviously different in the copper bicrystals with a perpendicular or tilting GB during cyclic deformation. In other words, the existence of a GB will significantly decrease the fatigue lives of the bicrystals due to early occurrence of intergranular fatigue cracking [59]. On the other hand, when cyclic deformation were performed on [679] //[145] , [3610] //[ 457] and [4916] //[ 4927] copper bicrystals with a parallel GB, it is found that strain incompatibility in the vicinity of the GB always become more serious than that in the bicrystals with a perpendicular GB [60-62]. Figs. 12 and 13 show typical deformation morphology near the GB in a fatigued [679] //[145] copper bicrystal. The deformation morphology strongly depends on the applied strain amplitude [60-62]. Normally, secondary slip bands were activated near the GB, forming a GBAZ. With increasing strain amplitude, the width of GBAZ also increases and the degree of plastic strain incompatibility within the GBAZ become serious, which often leads to GB strengthening effect and disappearance of the plateau in its CSSC. For [4916] //[ 4927] copper bicrystal, when it is deformed at low strain amplitude, only secondary slip bands appear near the GB and primary slip bands can not reach the GB [61]. At higher strain amplitude, both primary and secondary bands can reach the GB and strain incompatibility often become serious, as shown in Fig. 24. For [135] //[135] , [135] //[235] and [235] //[ 235] copper bicrystals with a low Σ GB parallel to the stress axis, Hu et al. [45] also observed obvious secondary slip near the GB. The strain incompatibility near the GBs had been widely observed in different bicrystals subjected to unaxial deformation [1-5], however, strain incompatibility induced by cyclic deformation seems to be more serious and complicated [60-62]. With continuous cyclic deformation, it is observed that fatigue cracks first nucleated along the GB in all the bicrystals above even though the GB is parallel to the stress axis. As an example, fatigue cracking processes of [4916] //[ 4927] bicrystal are elucidated and shown in Fig. 25(a)-(c). When the bicrystal is cyclically deformed at low strain amplitude (εpl = 5 × 10–4), fatigue crack nucleation

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seems to be rather difficult. After longer cycles (about 5 × 104), some cracks begin to initiate at the intersection sites of PSBs with GB, as shown in Fig. 25(a). With further cyclic deformation, the number of fatigue cracks increases and links with each other along the GB, in the end, forming a long intergranular crack. At higher strain amplitude (εpl = 2 × 10–3), as cyclic number is higher than 104, obvious cracks nucleating along GB can be clearly seen (Fig. 25(b)). The fatigue crack had propagated to be rather long, finally can extend to the whole gauge region of the bicrystal specimen. In particular, the intergranular cracking path displays a zigzag feature at some region owing to the interactions of PSBs with GB. As cyclic number is high enough (N >105), GB cracks become very wide, as shown in Fig. 25(c), a rift with a width of 10 µm can be seen. However, no apparent fatigue cracking along those PSBs is found on the component crystal surface except some extrusions on the surfaces. For [679] //[145] and [3610] //[ 457] copper bicrystals, intergranular fatigue cracking processes are nearly the same as [4916] //[ 4927] bicrystal. Besides, GB fatigue cracks are also observed to preferentially nucleate in [135] //[135], [135] //[235] and [235] //[ 235] copper bicrystals with a parallel Σ3 or Σ13a GB during cyclic deformation [45]. These results further indicate that large-angle GBs in the bicrystals are preferential sites leading to fatigue cracking even though they are parallel to the stress axis. From the results above, it can be concluded that the GB cracking is always the preferential damage mode in copper bicrystals, no matter whether they are perpendicular, tilting or parallel to the stress axis. In other words, for all the type I GBs in copper bicrystals, the preferential intergranular fatigue cracking behavior is independent of the GB structure and the interaction angle between the GB plane and the stress axis. 5.2 Fatigue cracking behavior of the crystals containing type II GBs For some copper columnar crystals containing low-angle GBs, it is found that all the fatigued specimens display initial cyclic hardening and saturation behavior. The cyclic saturation resolved shear stress ranges from 29.0 – 29.6 MPa in applied strain range of 7 × 10–4 to 4.7 × 10–3 [66], which is similar to that of copper single crystals oriented for single slip [15, 16]. This indicates that the low-angle GBs do not show perceivable effect on its cyclic stress-strain response. The result is quite different from that of [679] //[145] , [3610] //[ 457] and [4916] //[ 4927] bicrystals with a large-angle GB, which often plays an obvious strengthening role in the bicrystals and results in a increase in the saturation stress. Surface observations show that all the slip bands beside the low-angle GBs have good one-to-one relationship as shown in Figs. 26(a) and 26(b). They show that PSBs had transferred through the low-angle GB continuously. Meanwhile, no secondary slip bands are activated, which further demonstrates good plastic strain compatibility near the low-angle GBs. The transmission of slip bands across low-angle GBs was also observed in deformed titanium [64]. Therefore, the interactions of slip bands with

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low-angle GBs should be more compatible in comparison with those near the large-angle GBs. With further cyclic deformation, it is found that the fatigue cracks are difficult to initiate in the columnar crystals. However, as cyclic number is high enough, fatigue crack always preferentially nucleates along PSBs, rather than along low-angle GBs, independent of the interaction angle between the GB plane and the stress axis. Figure 27 shows a typical fatigue crack initiating along PSBs, in the end, leading to transgranular fatigue fracture. With increasing strain amplitude and cyclic number, fatigue cracks invariably nucleate along PSBs, however, intergranular crack is never observed along low-angle GBs during cyclic deformation of the columnar copper crystals. From the present results, it can be concluded that fatigue crack initiation and fracture along PSBs is the dominant damage mode in columnar copper crystals with low-angle GBs, which is similar to copper single crystals [65-67], but is against with copper bicrystals with large-angle GBs.

Fig. 21. Diagrammatic sketch of the CSSCs for the copper bicrystals. (a) [5913] , [579] and

[5913] ⊥ [579] copper bicrystal; (b) [123] , [335] and [123] ⊥ [335] copper bicrystal; (c) [345] , [117] and [345] ⊥ [117] copper bicrystal.

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Fig. 22. Surface slip morphologies of the copper bicrystals with a perpendicular large-angle GB. (a) [5913] ⊥ [579] copper bicrystal; (b) [123] ⊥ [335] copper bicrystal.

Fig. 23. Intergranular fatigue cracking in the bicrystal with a large-angle perpendicular (a) and tilting to the stress axis (b).

5.3 Fatigue cracking behavior of the crystals containing type III GB For the [41520] /[1827] copper bicrystal, there exists a coplanar slip system between the two component crystals, as shown in Fig. 9(a) and (b), which is similar to that in the columnar crystals. It is expected that the Σ19b GB will show also intrinsically strong resistance to intergranular cracking as the low-angle GBs. Cyclic deformation was applied to the bicrystal in the applied axial plastic strain range of 1.5 × 10–4 – 2.13 × 10–3. It is found that all the bicrystal specimens exhibited a cyclic saturation behavior with nearly the same axis stress of 61.6 – 63.5 MPa [68].

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If taking the Schmid factors (ΩG1 = 0.47 and ΩG2 = 0.49) of the two grains into account, the saturation resolved shear stresses τ G1 and τ G 2 applied on the primary slip system of the two grains will be in the range of 28.9 – 29.8 MPa and 30.2 – 31.1 MPa, respectively, which is approximately equal to the saturation resolved shear stress (28-30 MPa) of the copper single crystal oriented for single-slip. This result indicates that the cyclic stress-strain response of the [41520] /[1827] bicrystal is also similar to copper single crystals oriented for single-slip [15,16] or the columnar crystals. Therefore, it can be concluded that the large-angle Σ19b GB did not produce an obvious strengthening effect on the cyclic stress-strain response of the [41520] /[1827] bicrystal. After cyclic deformation, it is observed that the common primary slip bands are activated on the whole surfaces of the two crystals, including the vicinity of GB. Figs. 28(a) and 28(b) give typical slip morphologies near the GB on the two surfaces of the bicrystal specimen. It is clear that all the slip bands have a good one-to-one relationship across GB, and no secondary slip bands can be seen. This indicates that the common primary slip bands had transferred through the Σ19b GB without interruption during cyclic deformation, which should be attributed to its special crystallographic relationship of the bicrystal, as shown in Figs. 9(a) and 9(b). The present results shed light on the similar effect of the Σ19b GB on the stress-strain response and the transmission of PSB across the GB as to the low-angle GBs in the columnar crystals.

Fig. 24. Slip morphologies beside the GB in the [ 4916] //[ 4927] copper bicrystal.

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Fig. 25. Fatigue cracking along the GB parallel to the stress axis in the [ 4916] //[ 4927] copper – bicrystal deformed at low and high strain amplitudes for different cycles. (a) εpl = 5 × 10 4, 5 –3 4 –3 5 N = 2 × 10 ; (b) εpl = 2 × 10 , N = 2 × 10 ; (c) εpl = 2 × 10 , N = 2 × 10 .

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Fig. 26. Surface slip morphologies of the columnar copper crystals containing low-angle GBs. All the slip bands have a good one-to-one relationship across the low-angle GBs.

Fig. 27. Fatigue cracking along the PSB in columnar copper crystals containing low-angle GBs.

When cyclic deformation continues to apply on the bicrystal specimens, however, fatigue crack always initiates at the GB at first, as shown in Fig. 29(a). Beside intergranular fatigue crack, PSBs still have a one-to-one relationship and secondary slip bands are not activated. So the effect of secondary slip bands on the GB cracking can be excluded. With further cyclic deformation, the intergranular fatigue crack gradually propagated along the GB, finally leading to intergranular fatigue fracture along the whole GB, as shown in Fig. 29(b). The present results prove that although surface slip feature of the bicrystal is close to those of a fatigued copper single crystal or columnar crystals containing low-angle GBs, and the GB does not affect the continuity of surface PSBs, eventually fatigue cracking along the GB is still inevitable. Therefore, intergranular fatigue cracking characteristics of the

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[41520] /[1827] bicrystal is similar to the large-angle type I GBs, but is contrary to

the low-angle GBs. Although there are similar effects of the Σ19b GB and the low-angle GBs on the stress-strain response and slip morphologies, fatigue cracking mechanism display quite different features between them, which is an interesting phenomenon that will be further discussed through the observations of dislocations near the GBs. 5.4 Intergranular fatigue cracking mechanisms Typical fatigue cracking mechanisms for f.c.c. metals are either along PSBs or along GBs or both of them in single-, bi- or poly-crystalline materials [65-80]. Up to now, the widely accepted models for the two typical fatigue cracking are extrusion-intrusion mechanism [74-76] and PSB-GB (or piling-up of dislocations) mechanism [77-80]. Both of the two models are based on the movements of dislocations in fatigued crystals, therefore, it is necessary to further observe the dislocation arrangements near the GBs for better understanding the difference in fatigue cracking mechanisms along the three types of GBs. Since SEM-ECC technique provide a convenient and applicable way to obtain the dislocation arrangements on a large area and some special sites, all the fatigued specimens have been observed especially near the GBs. Fig. 30(a) shows a typical dislocation arrangements near a large-angle GB perpendicular to the stress axis. It can be seen that some white bands should correspond to PSBs and they can reach GB. When PSBs are close to the GB, their end becomes sharp and irregular due to the blocking effect of the adjacent grain. The dislocation observations further demonstrate that PSBs can not transfer through a large-angle GB, which is consistent with the surface observations, as shown in Figs. 22 and 24. When the large-angle GB is parallel to the stress axis, it is found that all the PSBs become irregular near the GB due to the serious plastic strain incompatibility, as shown in Fig. 30(b). In the upper grain, no PSBs can be observed and a labyrinth structure appears in the GBAZ due to the interactions of primary and secondary slip bands (see Fig. 24). From the observations in the vicinity of the large-angle GBs, it can be concluded that the dislocation arrangements beside the type I GB are discontinuous, leading to the piling-up of dislocations at large-angle GBs, which can be attributed to the difference in orientations of the adjacent grains and the blocking effect of GB on slip bands. By means of the TEM technique. Similar phenomenon was also found in [49,68] for [135] //[135] and [235] //[ 235] copper bicrystals with a parallel GB and a [117] ⊥ [345] copper bicrystal with a perpendicular GB. Since piling-up of dislocations at large-angle GBs has been widely observed and accepted in fatigued polycrystals, it is natural that the crystallographic relationship between GB and slip systems of the two neighboring grains can be illustrated as in Fig. 31. The activated PSBs can reach the GB, but cannot pass through it because there is no a coplanar slip system. It has been well known that most plastic strains are carried by PSBs during cyclic deformation of f.c.c. metals [17,18]. PSBs may become main carrier and channel transporting residual dislocations and vacancies

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from the interior of grains into GBs. As the residual dislocations and vacancies within a large-angle GB are accumulated to be high enough in density, intergranular cracking along GB will occur under external cyclic stress. Consequently, it can be concluded that the essence of intergranular fatigue cracking should be attributed to the reversal interactions of PSBs with GB, or the accumulation of dislocations and vacancies within the GB. In other words, the PSB-GB mechanism will dominate the intergranular fatigue cracking of all the bicrystals with a large-angel GB, independent of the interaction angles between GB plane and the stress axis. From the common crystallographic relationship between the GB and the slip systems in Fig. 30, it is natural that the interaction angle cannot change the essence of accumulation of dislocations within the GB. The only case is that the processes of intergranular fatigue cracking become relatively difficult when the GB plane is parallel to the stress axis. Therefore, regardless of the interaction angle between the GB plane and the stress axis, all type I GBs are the preferential sites for the nucleation of fatigue crack, which can be attributed to the accumulation of dislocations within the GBs.

Fig. 28. Surface slip morphologies of the [ 41520] /[1827] copper bicrystal on (a) the top surface and (b) lateral surface of the bicrystal specimen.

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Fig. 29. Fatigue cracking along the Σ19b GB in the [ 41520] /[1827] copper bicrystal.

The SEM-ECC technique has clearly shown the dislocation patterns near the low-angle GBs in Fig. 32(a) and ladder-like PSBs transfer through a low-angle GB continuously. The dislocation arrangement near low-angle GB is consistent with the surface slip morphologies in Fig. 26. To further verify the continuity of dislocation arrangements near the low-angle GBs, a more direct method is to observe the dislocation arrangements on the common slip planes. The observation results show that the dislocation patterns near the low-angle GB are still continuous, as shown in Fig. 32(b). Since the slip direction of a PSB is perpendicular to the dislocation walls, as indicated by the arrows, the slip directions of the two neighboring grains should be nearly the same. The present observation gives powerful evidence that both ladder-like PSBs (see Fig. 32(a)) and parallel dislocation walls (see Fig. 32(b)) are continuous across the low-angle GB. It can be concluded that cyclic stress-strain response, surface slip morphology and dislocations are not affected by the low-angle GBs during cyclic deformation. Therefore, the columnar crystals can be regarded as a single crystal; furthermore, its fatigue cracking mechanism can be in analogy with the single crystals. It has been known that the irreversibility of slip within PSBs results in the roughness on the crystal surfaces, which manifests itself in the form of the extrusions and intrusions at the PSB-matrix interfaces. The

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interfaces between PSBs and matrix are the preferential sites for the nucleation of fatigue cracks in copper single crystals [19, 65-67]. The fatigue cracking mechanism of single crystals has been explained by the surface roughness model based on the dislocation annihilation within PSBs [74], as illustrated in Fig. 33(a). Since low-angle GBs in columnar crystals do not block the transmission of the dislocations within PSBs, the interactions of the dislocation walls within a PSB lamina can be illustrated as in Fig. 33(b). The parallel dislocation walls can be transported from one grain into the adjacent grain freely without piling-up of dislocations at the low-angle GBs. Therefore, the fatigue cracking mechanism of the columnar crystals will be identical with that of copper single crystals, which can well explain the PSB cracking in the columnar crystal, as illustrated in Fig. 33(c).

Fig. 30. Dislocation arrangements near the large-angle GB. (a) [5913] ⊥ [579] copper bicrystal with a perpendicular GB: (b) [ 4916] //[ 4927] copper bicrystal with a parallel GB.

For the [41520] /[1827] bicrystal, the SEM-ECC observations also clearly reveal the fatigued dislocation arrangements near the Σ19b GB. Figure 34(a) shows the typical interaction of the Σ19b GB with the dislocations within PSBs on the specimen surface. The ladder-like PSBs within one grain do not transfer through the GB continuously, but terminate at the Σ19b GB. However, the dislocation arrangements within PSBs in another grain are not clear due to the difference in the orientation of the two grains. By careful observation, one can see that there seems to exist piling-up of dislocations at the Σ19b GB, as indicated by the arrows, indicating that the surface slip bands with one-to-one relationship are actually discontinuous. After

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that, the bicrystal specimen was also cut along the common primary slip plane as in the columnar crystals. In this case, the dislocation arrangements beside the Σ19b GB are also discontinuous and a dislocation-affected zone (DAZ) appears, as shown in Fig. 34(b). The DAZ is about 10 µm in width and is very similar to the dislocation-free-zone (DFZ) observed by TEM technique in the [117] ⊥ [345] bicrystal and polycrystals [32, 46, 81]. The observations above are not consistent with the surface slip bands in Fig. 11, indicating that, actually, the ladder-like PSBs cannot completely transfer through the Σ19b GB over the whole specimen despite the two grains in the bicrystal having a coplanar slip system. Therefore, it can be concluded that the ladder-like PSBs beside the Σ19b GB actually are discontinuous, which is acceptable since the slip directions b1 and b2 of the two component grains are obviously different, as shown in Fig. 9. From the dislocation observations, the difference in the fatigue cracking mechanisms between the Σ19b GB and the low-angle GBs can be explained by their different crystallographic features. As seen in Fig. 9, the primary slip planes of the two component grains are coplanar, whereas, the slip directions b1 and b2 have a large angle of 13.8°. Based on the surface slip morphology and dislocation arrangements near the Σ19b GB, the interactions of PSBs with the Σ19b GB can be schematically illustrated in Fig. 35(a) and (b). During cyclic deformation of the [41520] /[1827] bicrystal, the PSBs in the two component grains will impinge at the GB. If the Burgers Vectors b1 and b2 of the two PSBs are identical, the dislocations carried by each PSB should easily transfer through the Σ19b GB and move into the adjacent grains, as in the columnar crystals containing low-angle GBs. Because there is an angle of 13.8° between the slip directions b1 and b2, actually the dislocations carried by PSBs cannot fully transfer through the Σ19b GB, which has been proved by the observations in Fig. 34. With further cyclic deformation, parts of dislocations carried by PSBs will terminate at the GB, leading to the pilling-up of dislocations. The DAZ near the Σ19b GB in this bicrystal should be a direct evidence for the piling-up of dislocations. Therefore, the intergranular fatigue cracking mechanism of the special [41520] /[1827] bicrystal can be still attributed to the piling-up mechanism of dislocations, which is in essence consistent with the damage mechanism of the large-angle Type I GBs.

Fig. 31. Illustration of the interactions of the adjacent slip planes with a large-angle GB.

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Fig. 32. Dislocation arrangements near the low-angle GB on the surface (a) and on the common primary slip plane (b) in the columnar crystals.

Based on the results and discussion above, it is apparent that the fatigue damage of the f.c.c. crystals always originates from PSBs or GBs. In essence, the activation of a PSB should correspond to the onset of the micro-damage induced by cyclic deformation. However, the final nucleation of a fatigue crack will depend on the movement processes of the dislocations carried by PSBs. When the dislocations within PSB move from the interior of a grain, they will be transported into the crystal surface if there is no blocking effect of GB in the path of the dislocations. In this situation, the PSB cracking mechanism will dominate the fatigue damage processes and fatigue crack in general originates from the surface roughness. But in most case, the dislocations carried by PSBs will always meet a GB, so they will face two choices: (1) fully transmission through the GB; (2) blocking and piling-up at the GB. From our observations, it has been confirmed that only those low-angle GBs can be fully transferred through by the dislocations. Therefore, the dislocations can be moved into the adjacent grains, furthermore reach the surface, and lead to the PSB cracking. Although some investigators had given some criterions for the transmission of dislocation across a GB under unaxial loading, however, the transmission of dislocations across a large-angle GB is rather difficult. Even for the special [41520]/[1827 ] copper bicrystal with a coplanar slip system, the piling-up of dislocations is also observed near the GB, as shown in Fig. 33. It is indicated that the piling-up of dislocations near a large-angle GB is a common phenomenon, which will be responsible for the nucleation of GB fatigue crack. Probably, in some special conditions, such as the criterions in the literature [82-85], some dislocations might transfer through a large-angle GB under unaxial loading. However, it should be pointed out that the interactions of dislocations with a GB are very complicated under cyclic loading. Even partial dislocation is kept at GB, the accumulation of

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the residual dislocations can contribute to the piling-up of dislocations at the GB, and the nucleation of intergranular fatigue cracking. From this viewpoint, therefore, the interactions of PSBs with GBs will strongly affect the fatigue cracking mechanism of f.c.c. metals. According to the GBCD model, there might exist some special GBs, which are insensitive to cracking under unaxial loading, such as the GBs with a low value or good match at the boundary. But the adjacent PSBs are in general not coplanar beside the GB. For example, [41520]/[1827 ] bicrystal has a coplanar slip system under the present stress condition. But if the [41520] / [1827] bicrystal is deformed by other stress axis, the primary slip systems of the two component grains will be changed. Although the GB is still Σ19b in type, whereas, the activated PSBs will be not coplanar: Therefore, the intergranular fatigue cracking mechanism of the [41520]/[1827] bicrystal will be similar to those of the bicrystals with a large-angle type I GB. The similar situation should take place in the bicrystals containing the Σ3, Σ5, Σ7, Σ9GBs and so on. Therefore, the GB structure itself cannot dominate the fatigue cracking mechanism and the interactions of PSBs with GBs should be more important for the fatigue crack nucleation. However, for the low-angle GBs, all the slip systems beside the GBs have a coplanar relationship, independent of the stress axis, which results in the non-cracking behavior of the low-angle GBs. Since intergranular fatigue cracking is attributed to the impingement of PSB to GBs and piling-up of dislocations. We can design other special bicrystals, which is not satisfied with the conditions of piling-up of dislocations at the GB. Furthermore, intergranular fatigue cracking mechanism can be well understood. One of the ideas is to avoid the impingement of PSBs to GB. According to the principle, the unique case is that both of primary slip planes of the two component grains are parallel to the GB plane in a bicrystal with a tilting GB, as illustrated in Fig. 36(a). Meanwhile, it should make the two component crystals oriented as typical single slip so that the impingement of the secondary slip bands to the GB can be avoided during cyclic deformation. In this case, the activated primary slip bands will not impinge to the GB, however, the fatigue crack mechanism in such a bicrystal is still a maze. The second way is to make the slip planes of two component grains are still coplanar, as in the [ 41520] /[1827] bicrystal. But the slip directions of the two crystals are also the same as illustrated in Fig. 36(b). Since the [41520]/[1827] bicrystal is created by rotating 46.2° from one component crystal around the common rotation [111] axes, the new designed bicrystal can be produced by rotating 60° around the common axis of [111] axes, accordingly a Σ3 GB will form between the two crystals. As a result, the GB in the specially designed bicrystal should belong to the second type with a coplanar slip system and identical slip directions, whereas, it is a large-angle one. The identical slip directions between the two component crystals might result in the full passing through of the dislocations across the GB and the piling-up of dislocations at the GB will disappear during cyclic deformation. Furthermore fatigue crack might nucleate along the PSBs and the special designed Σ3 GB will become intrinsically strong to fatigue cracking as in the low-angle GBs. However, experimental work should be further carried out.

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Fig. 33. (a) Illustrations of fatigue cracking along the PSBs in copper single crystals; (b) Illustrations of transmission of dislocation walls across low-angle GBs; (c) Illustrations of fatigue cracking along the PSBs in copper columnar crystal.

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Fig. 34. Dislocation arrangements near the Σ19b GB of the [ 41520] /[1827] copper bicrystal on the specimen surface (a) and on the common primary slip plane (b).

Fig. 35. (a) Illustrations of the interactions of the surface PSBs with the Σ19b GB; (b) Illustrations of the interactions of the dislocation walls with the Σ19b GB on the common primary slip plane in the [ 41520] /[1827] copper bicrystal.

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Fig. 36. Illustrations of crystallographic relationships in two special bicrystals: (a) both of the primary slip planes of two component crystals in the bicrystal are parallel to the GB plane; (b) the primary slip systems of two component crystals are coplanar and have the identical slip directions.

6. Conclusions The effects of GBs and crystallographic orientations on the macro-scale cyclic stress-strain response and fatigue damage mechanisms in micro-scale were systematically investigated and summarized. The following conclusions can be drawn: 1)

2)

3)

−1

By introducing an orientation factor Ω B = ( VG1/ȍ G1 +VG2 /ȍ G2 ) , the saturation resolved shear stress of the bicrystals can be well calculated. It is found that the large-angle GBs can play a different strengthening effect on the copper bicrystals when the GB plane is parallel to the loading direction. The surface observations revealed a GBAZ in the bicrystals and the width WGB and volume fraction VGB of the GBAZ increased with increasing strain amplitude. The strengthening effect of the large-angle GBs can be attributed to the addiB tional GB resistance ǻIJ as to PSBs. When the large-angle GBs are perpendicular to the loading direction, the copper bicrystals [123] ⊥ [335] , [134] ⊥ [134] , [5913] ⊥ [579] and [345] ⊥ [117] displayed quite different CSSCs. It is found that the component crystal orientations play a decisive effect on the CSSCs of those bicrystals. In combination with the cyclic deformation behavior of copper single crystals oriented for single slip, it is suggested that the occurrence of the plateau or pseudo-plateau of their CSSCs is mainly resulted from the cyclic saturation of the single-slip-oriented component crystals at the region B of their CSSCs. For all the large-angle GBs in copper bicrystals, fatigue cracks always prefer to nucleate at the GBs, rather than along PSBs. The GB cracking is independent of the interaction angles between the GB plane and the stress axis.

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The intergranular fatigue cracking can be well explained by the PSB-GB damage (or piling-up of dislocations) mechanism. For the columnar copper crystals containing low-angle GBs, fatigue cracks would initiate along PSBs rather than along low-angle GBs. The PSB cracking mechanism in essence is identical with that in fatigued copper single crystals. The main reason for the non-cracking behavior of the low-angle GBs can be explained that the dislocations carried by PSBs have been fully transported from one grain to adjacent grain through their coplanar slip system. In other words, the low-angle GBs do not block the transmission of dislocations carried by PSBs, therefore, no piling-up of dislocations exists at the low-angle GB. For a special [41520] /[1827] copper bicrystal with a coplanar slip system, surface PSBs are continuous across the Σ19b GB after cyclic deformation. However, the GB is still the preferential site for the nucleation of fatigue crack even though the two component crystals have a coplanar slip system. The fatigue cracking mechanism can be attributed to the difference in the slip directions of the adjacent grains. Therefore, the dislocations carried by PSBs can not be fully moved into the adjacent grain through the coplanar slip. The partial dislocations are kept at the GB and form a dislocation-affected-zone (DAZ), which is a direct evidence for the piling-up of dislocations at the Σ19b GB. It is suggested that intergranular fatigue cracking strongly depends on the interactions of PSBs with GBs, rather than the GB structure itself. The nucleation of fatigue cracks along a PSB or a GB is a competitive process, depending on the movement processes of dislocations carried by PSB. The fatigue damage of the ductile f.c.c. crystals originates from the activation of dislocations carried by slip bands. When the dislocations can be freely transported to the surface by slip bands, the gradual surface toughness will become the origin of fatigue crack, in most case, resulting in PSB cracking. When dislocations carried by PSBs cannot transfer through a GB, they will pile-up at the GB, furthermore result in the intergranular fatigue cracking due to the accumulation of the residual dislocations.

Acknowledgements The authors would like to thank Gao W, Hu YM, Jia WP, Laird C, Li GY, Li SX, Li XW, Lukas P, Mughrabi H, Su HH, Wang ZR, Wu SD, Wu XM, and Yao G for their assistance to crystal growth, fatigue tests, SEM observations and stimulating discussions. This work was financially supported by National Natural Science Fund of China (NSFC) under grant No. 50571104, the National Basic Research Program of China (No. 2004CB619306), “Hundreds of Talents Project” and the Special Fund for the National 100 Excellent Ph. D Thesis provided by the Chinese Academy of Sciences.

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Coupling and communicating between atomistic and continuum simulation methodologies J.A. Zimmermana, *, P.A. Kleinb, E.B. Webb IIIc a

Sandia National Laboratories, P.O. Box 969 - MS 9042, Livermore, CA 94551 USA b Franklin Templeton Investments, 1 Franklin Pkwy, San Mateo, CA 94403 USA c Sandia National Laboratories, P.O. Box 5800 - MS 1411, Albuquerque, NM 87185 USA

Abstract A primary objective of modern materials modeling is to predict the material response and failure governed by deformation mechanisms, and to assess the mechanical reliability of components. These material deformation mechanisms operate at specific length scales, which vary from nanometers to microns. Multi-scale materials simulations have been the focus of many studies using techniques such as atomistic simulation and finite element (FE) analysis. This chapter will review a recently developed formulation for coupling atomistic and continuum mechanical simulation methods for quasistatic analysis. The formulation assumes a FE mesh covers all parts of the computational domain, while atomistic crystals are introduced only in regions of interest. This formulation allows the geometry of the mesh and crystal to overlap arbitrarily, and uses interpolation and projection operators to link the kinematics of each region, which in-turn are used to formulate a system potential energy from which coupled equilibrium equations are derived. A hyperelastic FE formulation is used to compute the deformation of the defect-free continuum using the Cauchy-Born rule, and a correction is introduced in the overlap region to minimize fictitious boundary effects. For both direct coupling of atomistic and continuum simulation approaches, as well as informational coupling methods ubiquitously used in multi-scale modeling research, it is essential that proper definitions for continuum quantities exist that can be evaluated within an atomistic framework. Continuum variables that are important during thermo-mechanical deformation processes include stress, temperature and heat flux. In this chapter, expressions are reviewed for mechanical stress, temperature and heat flux in atomistic systems derived from a continuum formalism developed by R.J. Hardy. Also presented is an extension of Hardy’s technique to include time averaging of stress, temperature and heat flux measures for finite temperature systems. Molecular dynamics simulations will be presented that characterize the consistency of these expressions with continuum models of heat transport. In addition, spatial and time averaging simulation studies have been performed to determine the limits at which fluctuations do not overwhelm the expectation values of continuum properties. Keywords: Atomistic simulation; Continuum mechanics; Stress; Temperature; Coupling.

*

Corresponding author. E-mail address: [email protected] (J.A. Zimmerman). 439

G.C. Sih (ed.), Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano, 439–455. ” 2007 Springer.

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1. Introduction Multi-scale materials simulations have been the focus of many studies using techniques such as atomistic simulation and finite element (FE) analysis. Continuum mechanical modeling efforts have evolved beyond using ad-hoc failure criteria to include cohesive approaches for surface separation and damage accumulation models for bulk material degradation. However, these techniques only capture anticipated deformation phenomena. Atomistic simulation procedures, such as molecular statics (MS) and dynamics (MD), use simple inter-atomic potentials as the underlying constitutive relation between material particles and allow the derived forces to govern the basic physics of the system’s response to an applied load. Atomistic simulation has the ability to display competing mechanisms of material deformation, such as fracture, dislocation nucleation and propagation, and void nucleation, growth and coalescence. However, limits of computational power prohibit analysis of micro-scale systems using only atomistic simulation, even in large-scale, parallel calculations. Early work in [1] and collaborators created the methodology, FEAt, that combines FE analysis with atomistic modeling. More recently, several methods have been introduced, including the Quasicontinuum method (QC) [2], Coarse-Grained Molecular Dynamics [3], Molecular-Atomistic-Ab Initio Dynamics [4] and Kaxiras, and the Bridging Scale Decomposition method [5]. These coupling methods have been used successfully to simulate phenomena such as crackgrain boundary interactions, dislocation nucleation from nanoindentation and the dynamic fracture of silicon. However, the weaknesses of these methods show that more consideration is needed in developing a coupled atomistic-continuum approach. Specifically, a rigorous methodology for partitioning potential energy between atomistic bonds and continuum strain energy within the overlapping regions needs to be developed. Improper partitioning of potential energy leads to fictitious, or “ghost” forces acting on atoms and nodes within the overlap region. A recent review [6] details the origins and effects of ghost forces that arise due to use of the QC method. This article also discusses newer approaches in [7] and [8] that attempt to overcome this problem. This chapter will review a recently developed formulation [9] for coupling atomistic and continuum mechanical simulation methods for quasistatic analysis. The formulation assumes a FE mesh covers all parts of the computational domain, while atomistic crystals are introduced only in regions of interest. Moreover, the formulation allows the geometry of the mesh and crystal to overlap arbitrarily. Interpolation and projection operators are used to link the kinematics of each region, which are then used to formulate a system potential energy, and from there, coupled equilibrium equations. A hyperelastic FE formulation is used to compute the deformation of the defect-free continuum using the Cauchy-Born rule. A correction to the Cauchy-Born rule is introduced in the overlap region to minimize fictitious boundary effects. For both direct coupling of atomistic and continuum simulation approaches, as well as informational coupling methods ubiquitously used in multiscale modeling research, it is essential that proper definitions for continuum quantities exist that

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can be evaluated within an atomistic framework. Connections between continuum variables and microscopic quantities originate from long-wavelength elasticity theory or long-time, equilibrium ensemble averages giving rise to macroscopic balance equations. The instantaneous atomic contributions to these averages do not have the same physical interpretation as the corresponding “point-wise” continuum quantities. In this work, expressions are reviewed for mechanical stress, temperature and heat flux in atomistic systems derived from a formalism developed in [10-12]. Hardy’s expressions are evaluated at a fixed spatial point and use a localization function to dictate how nearby atoms contribute to the continuum properties at that point, thereby performing a local spatial averaging. Also presented is an extension of Hardy’s technique to include time averaging of stress, temperature and heat flux measures for finite temperature systems. Molecular dynamics simulations will be presented that characterize the consistency of these expressions with continuum models of heat transport. In addition, spatial and time averaging simulation studies have been performed to determine the limits at which fluctuations do not overwhelm the expectation values of continuum properties. 2. Coupled Atomistic-continuum Simulations for Arbitrary Overlapping Domains This section will summarize the recently developed formulation in [9] for coupling atomistic and continuum mechanical simulations methods for quasistatic analysis. Expanded details on both the theory and computational aspects of the formulations, as well as many example problems, can be found in [9]. 2.1 Kinematics The coupled atomistic-continuum system is shown in Fig. 1. A FE mesh covers all parts of the computational domain, while only limited regions of interest, such as crack tips or other defects, are also covered with an atomic crystal. Let the at-

Fig. 1. Patch of a coupled atomistic-continuum system. The set of FE nodes N is shown as †, the set of nodes Nˆ

Aˆ

is shown as z.

is shown as „, the set of atoms A is shown as |, and the set of atoms

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  (Į) T omistic displacements in the system be written as Q ¬ªq ¼º , where Į  A ,  and A is the set of all atoms. Let the nodal displacements be written as    (a ) T U ª¬u º¼ , where a  N , and N is the set of all FE nodes. Greek symbols denote atom indices, while lower case Roman symbols denote node indices. In order to satisfy continuity of the displacement field across the atomistic-continuum boundary, the continuum displacement field prescribes the motion of some of the atoms. For simplicity, these atoms are called “ghost atoms”. This



T

ˆ ªq º , where subset of atomistic displacements Q will be denoted as Q ¬ ¼ D  Aˆ , while the complement which contains the unprescribed atomistic dis(Į) T ª placements (for free atoms) will be denoted as Q q  ¬ º¼ , where D  A , (D)

Aˆ ‰ A = A and Aˆ ˆ A = ‡ . Analogously, the underlying lattice prescribes the motion of some FE nodes. These displacements will be denoted as T ˆ ª u (a) º , where a  Nˆ , and the unprescribed nodal displacements will U ¬ ¼  T ª u (a) º , where a  N , Nˆ ‰ N = N , and be denoted as U

¬

¼

Nˆ ˆ N = ‡ . One can interpolate the continuum displacement field to the locaĮ a Į a Į  N tion of any atom as u X X u , where X is the unaN

 ¦



a

deformed position of atom Į and N is the FE shape function associated with node a. The FE shape functions typically have compact support, so the sum shown Į above involves only the nodes whose support includes X . Generally, one can consider the atomistic and continuum displacement fields to be related as

­°Q ½° ­°U °½ ­Qc½ N ®ˆ¾ ® ˆ ¾® ¾ ¯°U ¿° ¯ 0 ¿ ¯°Q ¿°

(1)

where (2) The sub-matrices of N contain shape functions as defined by the interpolation stated above. Q c is introduced since the FE shape functions N QU and N QUˆ are generally too coarse to represent the atomistic displacement exactly. Since Q and Uˆ may have arbitrary dimensions, and it is expected that the 1 number of free atoms to exceed the number of prescribed nodes, N QUˆ will not be ˆ are chosen to minimize the error defined, and so U

e Qc ˜ Q c

¦ ¬ªq

DA

(D)





 ¦ aN N  a X D u a º ¼

2

(3)

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This error is minimized by the solution (4) (5) where (6) (7) (8) (9) and . Klein and Zimmerman have also shown that a moving least squares (MLS) interpolation can be substituted for the projection operation, thereby with a matrix of shape functions possessing unique propreplacing the term BUQ ˆ erties, and resulting in a set of equivalent relations to replace Eqs. (4) – (9). Details on this substitution and the subsequent development can be found in [9]. 2.2 Coupled equilibrium equations The solution for the unprescribed displacements Q and U can be determined by developing equilibrium equations derived by formulating the total potential energy of the coupled atomistic-continuum system, expressed as

  U, Uˆ  Q, U  F ˜ Q  F

ˆ  Q, U  3  Q, U 3 Q Q, Q 3U

Q

U ˜U

(10)

where 3 Q represents the potential energy in the bonds of the crystal, 3 U is the strain energy density integrated over the continuum, and F Q and F U are external forces acting on the atoms and nodes, respectively. Static equilibrium equations are derived via chain rule differentiation of Eq. (10) and when combined with Eqs. (4) and (5) can be expressed as

RQ =

∂Π Q ∂Q

+ BQQ ˆ T

∂Π Q ˆ ∂Q

+ B UQ ˆ T

∂Π U ˆ ∂U

− FQ = 0

(11)

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RU =

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∂Π U ∂U

+ B UU ˆ T

∂Π U ˆ ∂U

+ BQU ˆ T

∂Π Q ˆ ∂Q

− FU = 0

(12)

Solutions to (11) and (12) can be obtained by linearizing the equations about Q and U to derive the components of the symmetric tangent matrix, as is shown in [9], and then using this tangent matrix within a Newton solution scheme. Alternatively, a preconditioned conjugate gradient algorithm [13] can be used to directly solve Eqs. (11) and (12). 2.3 Correction to the Cauchy-Born rule Although atomistic and continuum degrees of freedom are now coupled both kinematically and through equations of equilibrium, the specific form of the total potential has not yet been given. The contribution embodied within 3 Q is computed from a sum of bond energies in the crystal. 3 U is computed using the Cauchy-Born rule [14,15], which accurately describes the long wavelength behavior of the lattice. An important detail is correcting for the overlap of the continuum and the underlying crystal. Within this overlapping region, the weighting of the contributions to potential energy from the bonds and finite elements needs to be determined such that the total energy for the coupled system is consistent with the result one would obtain from a full crystal, regardless of the location and orientation of the underlying crystal with respect to the overlaying FE mesh. It is immediately apparent that the weighting of the bonds between free and ghost atoms must always be unity to accurately calculate the energy per atom among free atoms. Contributions to potential energy from elements containing both free nodes and ghost atoms must be modified from the conventional Cauchy-Born rule to maintain the correct strain energy density for an equivalent continuum. For )d: where ) )C, X is the strain energy density, such elements, 3 U a function the left Cauchy-Green tensor C for a deformed material point located at X in the undeformed configuration. For a crystal subject to pair interactions,

³

ri



1 nb ȡi  X ji r i , where M is the inter-atomic potential, ¦ i V0 R i ˜ C ˜ R i , and R  i is the vector representing bond (i) in the unde-

)  C, X

formed configuration. Here, (i) refers to not just a single bond, but rather all bonds having the same orientation and length in the undeformed configuration. The spatially varying bond density contributions, 0 d ȡ i  X d 1 , are introduced so that the calculation of strain energy density only accounts for the “missing” energy that is not represented by actual bonds between atoms within the system. The functions ȡ i  X are only unknown within the overlap region. For regions of the domain completely covered by the underlying crystal, ȡ  i  X 0 , whereas ȡ i  X 1 for elements with no underlying crystal.

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The functions ȡ i  X are determined by enforcing a condition of homogeneous deformation given the appropriate boundary conditions. This is accomplished by minimizing the fictitious forces on both free and prescribed nodes within the overlap region for the coupled system. The development presented in [9] shows that for the choice of pair potentials, this amounts to minimizing the quantity

P(i)

2· 2 1 §¨ (a ) ˆ (a ) º ¸ ª ª º R f R f ˜  ˜ ¦ ¦ ¬ i (i) ¼ ¸ (i) ¼ 2 ¨ aN (i) ˆN ¬ i a N ( i ) ˆNˆ © ¹

(13)

where

fˆ(i)(a )

R i

E:X

¦

E

N

 (a ) : 0(i)

a

 X  E

ª º wN  a d:  » « ³ « :(0a( i)) wX » 1 R R … > @i « » a V0 « U wN d: » «  ³( a ) i wX » ¬« :0 ( i ) ¼»

∑

f (i)(a) = fˆ(i)(a) +

ˆ b∈( Ν(i) ∩ N)

fˆ(i)(b) B(a) (X (b) )

(14)

(15)

:0ai denotes the region of the domain that supports node a, is associated with   a denotes the region of the bonds of type (i), and for which ȡ i  X =1 , ȍ 0 i

domain that supports node a, is associated with bonds of type (i), and for which    a , and 0 d ȡi  X d 1 , N (i) denotes the subset of N that lie in the region ȍ 0 i a

the term B

 X b

corresponding with nodes refers to the element of B UU ˆ

a  N and b  Nˆ . As discussed in [9], an additional term may be required in Eq. (13) to ensure solutions for the functions ȡ i  X , as the number of unknowns may exceed the number of independent equations. Techniques such as Tikhonov regularization [16] may be used. 2.4 Example calculations One-dimensional example. Key features of this coupling technique are demonstrated by examining a one-dimension chain of atoms subject to multiple neighbor interactions. For this example, a Lennard-Jones potential [17, 18] that has been

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truncated [19] to a fifth nearest neighbor interaction range is used. For this calculation, the chain has been given free boundary conditions on the atoms at either end and the system is relaxed using a conjugate gradient, energy minimization algorithm. The system contains 30 atoms, 6 nodes and 5 elements, giving a ratio of element size to atomic spacing of 6:1. Atomic displacements for this system are shown in Fig. 2.

Fig. 2. Displacements for the relaxation of a free, one-dimensional atomic chain. Each curve represents a different number of layers of free atoms used at the chain’s surfaces.

It is observed that as successively more free atoms are used at the outer layers, the displacement field converges to that of a pure atomistic system, with convergence achieved when 6 or more atomic layers are used, a distance just beyond the potential interaction range of the outermost atoms. Two-dimensional example. An analogous example in two dimensions is also examined. Figure 3(a) shows a system composed of a hexagonal lattice with free surfaces overlapped by a quadrilateral mesh. Atoms within the outer layer of elements are free while all other elements contain ghost atoms. For this example, a Lennard-Jones potential [17, 18] is used that has been shifted and truncated to have a range out to an atom’s third nearest neighbors. The system relaxes outward, as shown in Fig. 3(b). Agreement in relaxation displacements is observed to be very good, although not perfect due to the severe inhomogeneous deformation at the lattice’s corners.

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Fig. 3. (a) A two-dimensional, hexagonal lattice with free surfaces composed of free (light gray) and ghost (dark gray) atoms. The overlapping quadrilateral mesh is also shown. (b) The relaxed configuration of (a) for the coupled system (dark gray) and a system treated purely with atomistics (light gray). Displacements are magnified by a factor of 200.

3. Evaluation of Thermo-mechanical Continuum Variables within Atomistic Simulation In addition to the mechanisms for coupling between atomistic and continuum simulation methods, for either direct coupling as discussed in section 2 or in a hierarchical, indirect coupling, it is important to have a common language between the two methods. Results obtained with one methodology need to be interpreted with regard to the other in order to provide insight on the phenomena simulated. In this section, expressions for mechanical stress, temperature and heat flux in atomistic systems derived from a continuum formalism developed in [10-12] are reviewed and examined. The material summarized in this section can be found in

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more detail in the articles by Zimmerman et al. [20] and Zimmerman and Webb [21], and in the technical report by Aubry et al. [22]. 3.1 Hardy’s formalism Hardy’s formalism begins with the spatial forms of the balance of mass, linear momentum and energy for a dynamic continuum [23]. Hardy then defines the continuum fields of mass ( ȡ ), momentum (p) and energy (e) densities in terms of Į Į atomic positions ( x ) and velocities ( v ) through use of localization functions, ȥ:

ȡ  x,t = ¦ Į=1 m Į ȥ  x Į  x

(16)

p  x,t = ¦ Į=1 m Į vĮ ȥ  x Į  x

(17)

2 N ­1 ½ e  x,t = ¦ Į=1 ® m Į  v Į +ij Į ¾ ȥ  x Į  x ¯2 ¿

(18)

N

N

Į

D

Here, I represents the potential energy attributed to atom Į , m represents the atom’s mass, and the localization function ȥ is a weighting factor for averaging the properties of the atoms, allowing each atom to contribute to a continuum property at a fixed spatial point x at time t. ȥ has units of inverse volume and \ z 0 only in some characteristic volume surrounding the spatial point x. This is the volume associated with the material point occupying the point x at time t, and its size influences the smoothness of the resulting continuum fields. More detailed information regarding the use and properties of these localization functions can be found in [10, 12, 20, 22]. Using these density functions within the balance equations, Hardy developed expressions for stress ( V ) and heat flux ( q ) at a spatial point: N ­1 N ½ DE DE DE ° 2 ¦ D 1 ¦ EzD x … F B  x ° ® ¾ ° ¦ N m D u D … u D \  x D  x ° D 1 ¯ ¿

V  x, t q  x, t



º 1 N ª N wIE x DE … x DE DE B  x » ˜ u D ¦ DE DE D 1 « ¦ EzD 2 x wx ¬ ¼

2 N ­1 ½  ¦ D 1 ® m D  u D  ID ¾ u D \  x D  x ¯2 ¿

(19)

(20)

Coupling and communicating between atomistic and continuum simulation methodologies Įȕ

Į

DE

ȕ

DE

449

Įȕ

Here, x =x  x , x x , F is the inter-atomic force exerted on atom Į by atom ȕ , uD { vD  v where v { p ȡ , and BĮȕ is the bond function

defined by the expression B

Įȕ

 x = ³0 ȥ  Ȝx Įȕ +x ȕ  x dȜ . When 1

ȥ is defined

as a constant for the entire volume of the system, Eq. (19) exactly reproduces the virial theorem, while Eq. (20) yields an analogous expression traditionally used for heat flux. Hardy and co-workers have also presented [24] an expression for temperature within this same continuum mechanical framework:

1 T  x,t = 3k B

¦

mĮ  u Į ȥ  x Į  x 2

N Į=1

¦

ȥ  xĮ  x Į=1 N

(21)

3.2 Evaluation of stress Simulations presented in this sub-section were performed using the embedded atom potential for Cu in [25]. Unless otherwise noted, ȥ is a radial step function of size R c and periodic boundary conditions were used. Figure 4 shows stress evaluated within a cube containing 4,000 atoms, equilibrated to zero pressure at a temperature of 300K. Stress is evaluated at two distinct spatial points located at positions (A) {11.3 Å, 15.5 Å, 7.0 Å} and (B) {–11.7 Å, –12.5 Å, –14.8 Å}, where the bounds of the system are ±18.075 Å in each direction. Temperature control is accomplished by exerting an atomic drag force proportional to both the difference between the current and desired temperatures of the system and the atom’s velocity [26].

Fig. 4. Hardy stress at 300K and zero pressure. Stress is evaluated at two distinct spatial points, A and B.

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It is observed that the curves shown for the two points differ significantly from each other. However, for R c > 6 Å, the variations in each curve (~0.005 eV/Å3), and the differences between the two curves, are very small and the value of stress is close to zero. It can be shown that averaging the stress expression over multiple time steps results in values that display smaller variations [21, 22]. Figure 5 shows time averaged curves for the Hardy stress for point A of Fig 4. Stress is averaged over 1, 10, 100 and 1000 MD time steps of 2 fs. It is observed that for averaging of 100 time steps or greater, variations decrease in magnitude severely for values of R c as low as 5 Å. Just as the value of R c — the size of the characteristic volume for spatial averaging — is required to be larger than some minimum value in order to achieve a sufficient level of accuracy at the continuum scale, so too should the time averaging window size be larger than some minimum period. This minimum period is a correlation time [26], and may be influenced by phonon lifetimes particular to the materials and temperatures simulated.

Fig. 5. Hardy stress at 300K and zero pressure averaged over 1, 10, 100 and 1000 time steps ( 't = 2 fs).

3.2 Transient and steady state heat flow The expressions for both temperature and heat flux were evaluated for MD simulations of a rectangular cross-section rod with dimensions 1453.102 Å × 36.329 Å × 36.326 Å containing 160,000 atoms. The crystal’s initial temperature was 300K, and a constant heat flux boundary condition [22] was implemented in cross sectional slabs of thickness 36.325 Å in the x (long) direction. One boundary condition control slab was used in the middle of the crystal to remove energy from the system, while another was used at the ends of the crystal (encompassing the periodic boundary) to insert energy. Fig 6 displays the crystal depicted by a random selection of 1600 spatial points, each shaded according to its value of temperature.

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Fig. 6. Hardy temperature for a random selection of spatial points. Simulation times are shown.

A value of R c = 10 Å and a normalized quartic localization function were used. Figure 6 shows a complex transient behavior that evolves to an apparent steady state within ~ 200 ps. The corresponding time frame in the figure qualitatively exhibits a nearly linear distribution of temperature between the slabs at the middle and ends at which the constant heat flux boundary condition is active. The transient and steady state behaviors of Hardy’s temperature and heat flux expressions can be better understood by examining their distributions along the long direction of the rod. Cross-sectional bins of width 36.325 Å were used to average a subset of the spatial point values. Figs. 7 and 8 show the temperature (T) and heat flux in the x direction (qx), respectively. Each figure displays both the instantaneous distribution and a time averaged distribution for several instances in time, where the time averaging period is 40 ps.

Fig. 7. Distribution of temperature (in units of K) for binned regions along the length of the crystal (in units of Å) shown in Fig. 6. Both instantaneous (dotted) and time averaged (solid) values are plotted.

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These distributions exhibit many features. It is observed that a steady state response is not achieved until at least 400 ps have elapsed. The spatial fluctuations are nearly absent for the time-averaged distributions, enabling one to easily extract thermal properties of the system. For example, the regions of crystal between the boundary condition slabs display a temperature gradient of 0.179 ± 0.002 K/Å in Fig. 7 and a heat flux of 0.20866 ± 0.01 nano-Watts/Å2 in Fig. 8. This value of heat flux is estimated after over 800,000 steps of simulated time, i.e. 800 ps, and is the average of heat flux magnitudes for the two halves of the crystal. This estimate is very close to, and within its uncertainty of, the prescribed heat flux of 0.20924 nano-Watts/Å2. Combining these two simulated “measurements” results in quantifying the thermal conductivity of the system, assuming Fourier’s law of heat conduction as the long-time behavior of the system. Thermal conductivity is estimated to be a value of 11.65 ± 0.69 W/(m•K), an uncertainty of approximately 6%. More detail about these simulations can be found in [22].

Fig. 8. Distribution of heat flux component qx (in units of nano-Watts/Å2) for binned regions along the length of the crystal (in units of Å) shown in Fig 6. Both instantaneous (dotted) and time averaged (solid) values are plotted.

It is of interest to compare how various local averaging techniques for heat flux compare. For example, the expression Eq. (20), can be compared with an analogous expression taken from the virial theorem that consists of the sum of discrete, atomic contributions,

qt

­ 1 NV ª NI DE ½ ¦ ¦ F … x DE ¼º ˜ u D °° 1 °° 2 D 1 ¬ EzD ® ¾ 2 N ­1 V° ½  ¦ D V1 ® m D  u D  ID ¾ u D ° °¯ ¯2 ¿ °¿

(22)

where V is the volume of a subset of the system that contains N V atoms, and N I is the number of atoms interacting with atom Į . Use of this expression to

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quantify heat flux is done in two ways: either fully including interactions that cross the boundary of an analysis region, N I = N , or excluding those interactions, N I = N V . Note how this compares with the use of the bond function in Eq. (20). Evaluation of Eq. (22) in analysis regions outside the heat insertion/extraction regions reveals that steady state is reached after approximately 700 ps of simulated time. The time period between 760 and 800 ps is used to time average Eq. (22), and the resulting spatial distribution for qx is plotted in Fig. 9.

Fig. 9. Distribution of time averaged qx (in units of nano-Watts/Å2) calculated using the Hardy (solid), a discrete expression that fully includes interactions that cross the surface of the analysis volume (dashed) and a discrete expression that excludes those interactions (dotted).

Not surprisingly, including all interactions that cross the analysis bounds in Eq. (22) over-estimates the expectation value whereas excluding them under-estimates it. The Hardy method, on the other hand, lies in between these two and is closest to the expectation value. This result demonstrates the superiority of the Hardy method over the other methods for computing q. This is not surprising given the Hardy method’s more physically robust methodology of accounting for interactions that cross the surface of an analysis region. As was demonstrated in [20], a similar result was seen for computing local averages of stress. Performing a similar analysis as before results in an estimate of thermal conductivity of 11.2 ± 0.04 W/(m•K), close to the estimate made using Hardy’s expressions.

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Acknowledgement The authors gratefully acknowledge input provided by discussions with N.S. Weingarten at the Catholic University of America, R.J. Hardy at the University of Nebraska at Lincoln, and J.J. Hoyt, R.E. Jones, D.J. Bammann, G.J. Wagner, C.J. Kimmer and S. Aubry at Sandia National Laboratories. Sandia is a multiprogram laboratory operated by the Sandia Corporation, a Lockheed Martin company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. References [1] Kohlhoff S, Schmauder S, A new method for coupled elastic-atomistic modelling. In: Vitek V, Srolovitz D, editors, Atomistic Simulation of Materials: Beyond Pair Potentials, Plenum Press, New York, 1989, pp. 411-418. [2] Tadmor E, Ortiz M, Phillips R, Quasicontinuum analysis of defects in solids, Phil Mag A 1996; 73(6): 1529-1563. [3] Rudd R, Broughton J, Coarse-grained molecular dynamics and the atomic limit of finite elements, Phys Rev B 1998; 58; R5893-R5896. [4] Broughton J, Abraham F, Bernstein N, Kaxiras E, Concurrent coupling of length scales: Methodology and application, Phys Rev B 1999; 60; 2391-2403. [5] Wagner GJ, Liu WK, Coupling of atomistic and continuum simulations using a bridging scale decomposition, J Comp Phys 2003; 190; 249-274. [6] Curtin WA, Miller RA, Atomistic/continuum coupling in computational materials science, Modelling Simul Mater Sci Eng 2003; 11; R33-R68. [7] Shenoy VB, Miller R, Tadmor EB, Rodney D, Phillips R, Ortiz M, An adaptive finite element approach to atomic-scale mechanics – the quasicontinuum method, J Mech Phys Solids 1999; 47; 611-642. [8] Knap J, Ortiz M, An analysis of the quasicontinuum method, J Mech Phys Solids 2001; 49; 1899-1923. [9] Klein PA, Zimmerman JA, Coupled atomistic-continuum simulations using arbitrary overlapping domains, J Comp Phys 2006; In Press. [10] Hardy RJ, Formulas for determining local properties in molecular-dynamics simulations: Shock waves, J Chem Phys 1982; 76(1); 622-628. [11] Hardy RJ, Karo AM, Stress and energy flux in the vicinity of a shock front. In: Shock Compression of Condensed Matter, Proc. Amer Phys Soc Topical Conference, 1990, pp. 161-164. [12] Hardy RJ, Root S, Swanson DR, Continuum properties from molecular simulations. In: 12th International Conference of the APS Topical Group on Shock Compression of Condensed Matter , Pt. 1 of AIP Conference Proceedings, 2002; 620, pp. 363-366. [13] Schewchuk JR, An introduction to the conjugate gradient method without the agonizing pain, on-line tutorial 1994; URL: http://www.cs.cmu.edu/~quake-papers/painless-conjugategradient.pdf [14] Huang K, On the atomic theory of elasticity, Proc Roy Soc London A 1950; 203; 178-194. [15] Born M, Huang K, Dynamical Theory of Crystal Lattices. Oxford: Clarendon Press, 1956. [16] Engl H, Hanke M, Neubauer A, Regularization of Inverse Problems, Dordrecht: Kluwer Academic Publishers, 1996. [17] Lennard-Jones JE, The determination of molecular fields I. From the variation of the viscosity of a gas with temperature, Proc Roy Soc London A, 1924; 106; 441. [18] Lennard-Jones JE, The determination of molecular fields II. From the equation of state of a gas, Proc Roy Soc London A, 1924; 106; 463.

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[19] Haile JM, Molecular Dynamics Simulation: Elementary Methods. New York: Wiley, 1992. [20] Zimmerman JA, Webb III EB, Hoyt JJ, Jones RE, Klein PA, Bammann, DJ, Calculation of stress in atomistic simulation, Modelling Simul Mater Sci Eng, 2004; 12; S319-S332. [21] Zimmerman JA, Webb III EB, Evaluating Thermo-Mechanical Continuum Variables in Atomistic Simulation. In: Proceedings of the Sixth International Conference for Mesomehanics, 2004, pp. 530-537. [22] Aubry S, Bammann DJ, Hoyt JJ, Jones RE, Kimmer CJ, Klein PA, Wagner GJ, Webb III EB, Zimmerman JA, A Robust, Coupled Approach for Atomistic-Continuum Simulation. Technical Report SAND2004-4778, Sandia National Laboratories, September 2004. [23] Malvern LE, Introduction to the Mechanics of a Continuous Medium. New Jersey: Prentice-Hall, Inc., 1969. [24] Root S, Hardy RJ, Swanson DR, Continuum predictions from molecular dynamics simulations: Shock waves, J Chem Phys 2003; 18(7); 3161-3165. [25] Foiles SM, Baskes MI, Daw MS, Embedded-atom-method functions for the FCC metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys, Phys Rev B 1986; 33; 7983-7991. [26] Allen MP, Tildesley DJ, Computer Simulation of Liquids. Oxford: Clarendon Press, 1987.

Author Index

Aravas, N. 161, 178

Lee, J.D. 23, 64 Lei, Y.J. 23

Bai, Y.L. 1, 10

Li, J.G. 141, 159

Barter, S.A. 197, 237 Mannava, S.R. 241 Chen, J.S. 11, 83

Mehraeen, S. 11, 83

Chen, F.G. 141, 160

Molent, L. 197, 237

Chen, Y.P. 23, 64 Chen, Z. 67, 83

Nagarajan, K. 241

Feng, X.Q. 103, 137

Pitt, S. 197, 238

Fish, J. 20, 85, 100 Qian, D. 241, 257 Gan, Y. 67, 83 Ge, W. 141, 159

Shen, L.M. 67, 83 Shi, D.L. 103, 139

Haidemenopoulos, G.N. 161, 178

Sih, G.C. 259, 287, 291, 339, 366

Hao, S. 179, 196

Stamenoviü, D. 321, 335

Huang, Y.G. 103, 137 Hwang, K.C. 103, 137

Tang, X.S. 287, 291, 319, 339, 367

Ingber, D.E. 321, 335

Vasudevan, V.K. 241

Jones, R. 197, 238

Wang, H.Y. 1, 10 Wang, N. 321, 338

Katsamas, A.I. 161, 178

Wang, Z.G. 389, 436

Ke, F.J. 1, 10

Webb, E.B. III 439, 455

Klein, P.A. 439

Weertman, J. 179, 195, 435

457

458

Wei, Y.G. 369, 386 Wu, X.L. 369, 386 Xia, M.F. 1, 10 Xiong, L.M. 23

Author Index

Zhang, Z.F. 389, 436 Zhao, M.H. 369, 386 Zhou, G.Z. 141 Zimmerman, J.A. 439, 454

Subject Index Diamond 78 Discontinuities 314 Discrete Model 251 Dislocation 273, 303, 309, 351, 359, 380 Displacement field Macroscopic 267, 295 Microscopic 263, 293

Agglomeration 109 Aluminium 7050-T7451 213 Alloy material 369, 372 Atomic 23, 28 Atomic chains 253 Atomistic-continuum 439, 441 Austenite stability 162

Eigenfunction 265 Eigenvalue 271 Energy density 276, 295, 300, 303, 362 Energy dissipation 7 Environmental effects 222

Barriers 86 Black hole 285, 286 Boeing 767 and 757 218 Carbon 103 Cauchy-Born rule 444 Cellular mechanics 332 Complex loading 214 Conservation laws 42 Constitutive relation 172 Copper bicrystals Fatigue cracking 416, 418, 420 Intergranular 424 Normal to grain 377 Orientation 402 Plastic deformation 411 Classical continuum mechanics 26 Corner crack 281 Crack opening displacement 303, 354 Crack growth 197, 205, 232, 235 Cr-Alloys 179 Crystalline solids 243, 244 Cyclic stress-strain 397 Cytoskeletal rheology 333

F4 218 F16 217 F111 231 F/A18 217, 228 FAA panel 211 FCC structure 254 Failure 5 Fatigue crack growth 205 Fatigue life 228, 231 Fluid flow 143 Fractals 202 Galerkin method 93 Grain boundary 389 Grain size 372 Hardy’s formalism 448 Heat flow 450 Heat flux 38 Homogeneity 15 Hyperelastic continuum 249 Hyper-surface 68, 76

Deborah number 1, 4 Defect 75, 123 Density Energy 276, 296, 300, 303, 362 Internal energy 37 Linear momentum 33 Mass 32 Momentum flux 35, 50, 52

Interface effect 113 Interatomic forces 49 Interatomic potential 49, 117

459

460

Lap joint 210, 232, 235 Limit diagrams 174 Loading type 264 Macro-crack 304 Mass 32 Mechanical transformation 167 Micro-crack 268 Microdamage 2 Microhardness 377, 384 Microstructure 371 Microtubes 332 Milky way 285 Mirage 216 Misfit 183, 185 Molecular dynamics 25, 93 Momentum 33, 50, 52 Multiscale 23, 87, 88, 95, 96, 97, 129, 259 Nanomaterials 241 Nanotubes 103, 118, 120 Newtonian force 275 Overload 213 Phonon dispersion 56 Polarization 40 Polycrystal 11, 16 Processing 141 Quasicontinuum 90 Reinforced composites 103 Restraining stress 293

Subject Index

Scale invariant 275 Scale multiplier 298 Scale shifting 276 Segmented model 278 Singularity 265, 268, 273 Size effect 73 Slip morphologies 401 Space and time 4 Statistical mechanics 2 6train rate 75 Stress field Macroscopic 269, 297, 346 Microscopic 266, 295, 342 Stress intensity factors Macro 295 Micro 269, 270 Mode I and II 267, 346 Stress-Strain 49, 59 Surface crack 282 TEM 372 Temperature 39 Tensegrity 321, 325, 333 Thermo-mechanical 447 Time and space 4 TRIP steels 162, 172 Tungsten 71 USAF panel 212 Variational method 92 Waviness 106 Wrinkling 11