Continuum Micromechanics: Theory and Application to Multiscale Tectonics 3031233123, 9783031233128

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Springer Geophysics

Dazhi Jiang

Continuum Micromechanics Theory and Application to Multiscale Tectonics

Springer Geophysics

The Springer Geophysics series seeks to publish a broad portfolio of scientific books, aiming at researchers, students, and everyone interested in geophysics. The series includes peer-reviewed monographs, edited volumes, textbooks, and conference proceedings. It covers the entire research area including, but not limited to, applied geophysics, computational geophysics, electrical and electromagnetic geophysics, geodesy, geodynamics, geomagnetism, gravity, lithosphere research, paleomagnetism, planetology, tectonophysics, thermal geophysics, and seismology.

Dazhi Jiang

Continuum Micromechanics Theory and Application to Multiscale Tectonics

Dazhi Jiang Department of Earth Sciences Western University London, ON, Canada

This work was supported by Natural Sciences and Engineering Research Council of Canada ISSN 2364-9119 ISSN 2364-9127 (electronic) Springer Geophysics ISBN 978-3-031-23312-8 ISBN 978-3-031-23313-5 (eBook) https://doi.org/10.1007/978-3-031-23313-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The book reflects my learning and teaching journey as I apply the micromechanical approach to rock deformation. As a graduate student and then a postdoctoral fellow in the 1990s, my interest in structural geology was in the kinematics of flow and finite deformation and the application of theoretical understandings to geological structures observed from the microscale (under a microscope) to the regional scale of a field area. However, as I learned more, the limitations of the kinematic approach became ever more apparent. The kinematic approach starts with an a priori velocity field, which in most, if not all, cases is homogeneous and steady state. It then investigates the evolution of finite deformation resulting from the velocity field. The advantage of the approach is that it can readily deal with large finite strains. Considering large finite strains is necessary to understand the evolution of structures such as sheath folds in ductile shear zones. But it is almost impossible to use the kinematic approach to deal with heterogeneous deformation, which is ubiquitous in nature. The difficulty is that even considering an apparently simple heterogeneous velocity field may violate the strain compatibility condition, and heterogeneous flows are also generally non-steady. Many qualitative kinematic models involving spatial variation of deformation proposed in the literature do not render finite strain solutions because they violate the strain compatibility requirement. The kinematic approach cannot go much further than homogeneous and steady-state deformations. To address the heterogeneous (and generally non-steady) deformation of rocks, one needs to follow a complete mechanics approach to derive kinematics from the full set of mechanical principles. But there are many challenges to the mechanics’ approach. To apply to understanding geological structures and fabrics, one must consider 3D deformations to large finite strains. Whereas 2D mechanical problems are relatively easier to handle, and so are many 3D mechanical problems with small strains, solving 3D mechanical problems with large finite strains is not trivial and is usually computationally costly. Perhaps the mechanics approach’s biggest challenge is the multiscale nature of rock deformation. Rocks are heterogeneous on a wide range of observation scales. Our observations are commonly made on relatively small scales, from outcrops to v

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under microscopes. And they are always from rheologically distinct elements, such as a granite pluton embedded within a tectonic unit or from within a ductile shear zone in an orogenic belt. The characteristics of such structures and fabrics are not directly related to the macroscale deformation at the scale of the tectonic unit or orogenic belt in question. Instead, they are related to the microscale deformation within the granite pluton or the shear zone. Therefore, rocks’ rheologically heterogeneous composition necessitates considering multiscale deformations. A singlescaled mechanics approach is insufficient to understand the richness of geological structures. Micromechanics provides an effective multiscale approach for studying the multiscale deformation and the associated development of structures and fabrics in Earth’s lithosphere. Micromechanics is the most recent development and active branch of continuum mechanics since Eshelby (1957, 1959). The literature is rich, diverse, and also quite formidable to geoscience students. Available textbooks on the subject are written for materials science students. The purpose of this book is to present the most fundamental principles of continuum micromechanics to geoscience students. The book is particularly intended for geoscientists interested in applying micromechanics to analyze and model fabric development. The book contains several application examples. Many Mathcad worksheets and MATLAB programs are provided with the text to facilitate using the multiscale approach presented in this book. As a piece of advice to readers who may find the math tools used in this book challenging, my math background was two semesters of calculus (taken during my undergraduate program) and one semester of linear algebra (taken during my master’s program). Many Canadian undergraduate geoscience programs require two calculus courses. Some require a linear algebra course as well. You should be okay with the math if you have a similar math background as I do. Cartesian tensors used in this book are just applications of linear algebra. With modern mathematics applications (e.g., Mathcad and MATLAB), tensor operations are much simpler than my graduate student time. I have done most derivations in the book with the symbolic engine of Mathcad, which is like doing math with pen and paper. The symbolic derivation is more efficient and makes fewer mistakes than I would with pen and paper. Therefore, do not let the formality of tensor notations turn you down. You will get used to them soon as I did, and mathematics applications like Mathcad and MATLAB handle them most elegantly. My experience is that math is better learned and improved as one applies it to exciting geology problems. If you wish to refresh your mathematics, I recommend the synoptic book by R. Shankar (1995) entitled Basic Training in Mathematics: A Fitness Program for Science Students, published by Plenum Press. An outline of the book is as follows. Chapter 1 gives a brief background and summarizes the mathematic notations used in the book. Chapter 2 begins with using unit vectors to define the orientation of lines in space and moves to using Euler angles or equivalents to define the orientation of 3D elements. The close relation between rotation and direction is established. The distribution of orientations in 3D space is discussed, and methods to generate a set of orientation data that follow some

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prescribed distributions are presented. The treatment in this chapter uses the mathematics applications MATLAB and Mathcad. Chapter 3 begins with a review of basic concepts of stress, strain, and elasticity. The general Hooke’s law for anisotropic materials and related stiffness matrix are presented. The chapter concludes with remarks on the multiscale nature of stress and strain and the concept of effective stiffness for real materials. Chapters 4 and 5 summarize the kinematics of finite deformation and flow. Chapter 6 applies the kinematic theory presented in Chaps. 4 and 5 to ideal high-strain zones. Where possible, analytical expressions between flow field parameters and finite strain geometry are presented for plane-straining general shearing, monoclinic transpression, and triclinic transpression. The chapter then discusses kinematically permissible combinations of simple and pure shearing for the kinematic modeling of high-strain zones. It concludes with remarks on the limitations of the kinematic approach. Chapter 7 presents constitutive equations and particularly Newtonian viscosity and power-law viscous behaviors. Chapters 8 and 9 focus on the motion of rigid objects in viscous flows based on Jeffery’s (1922) work. Analytical solutions for spheroids in monoclinic flows are given in Chap. 8. Numerical methods to solve the differential equation for rigid body rotation are explained, which will be used in later chapters. Chapter 9 focuses on prolate objects in simple shearing flows. Forces on such objects and the resultant deformation are analyzed. The classical Eshelby’s inclusion/inhomogeneity solutions for elastic and Newtonian viscous materials are presented in Chap. 10. The solutions are extended to anisotropic viscous inclusion and inhomogeneities in Chap. 11. Integral forms for viscous Green functions and their derivatives are provided together with associated Mathcad worksheets. Chapter 12 demonstrates that the same Green function approach can treat 2D inclusion and inhomogeneity problems as 3D problems. And the Green function method is simpler and numerically more efficient than the previous complex variable formulation. Chapter 13 applies the partitioning equations developed in Chaps. 10 and 11 to determine the effective stiffnesses of heterogeneous materials. Chapters 14, 15, and 16 demonstrate how the Eshelby method can be applied to geological problems. In addition, analytical expressions presented in this chapter can serve to verify numerical codes. Finally, Chap. 17 extends Eshelby’s solutions to nonlinear viscous materials and gives a selfconsistent formulation to solve the multiscale deformation problem of heterogeneous materials. Where applicable, at the end of a chapter, Mathcad worksheets and/or MATLAB programs related to the chapter are listed and the readers may obtain these programs from the online resource of this book. There are many possible choices of powerful mathematics application software to solve micromechanical problems. Using Mathcad is like doing math with a pen and paper. The algorithm for solving a problem is so explicit in a Mathcad worksheet that one who has not used the program before would be able to follow. It is my habit to develop and implement the algorithm for a problem in Mathcad. The explicit nature of Mathcad worksheets makes communicating problem-solving strategies in my research group easy. Once the algorithm is fully developed and optimized, it can be implemented in MATLAB

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if the latter makes a substantial difference in the computation time required for the problem. Numerical solution of micromechanics problems involving extremely elongated or flat heterogeneities requires intensive computation. The heavy weight-lifting computation of Eshelby tensors for flat or elongate elements is implemented in C and called in MATLAB. Regardless of the mathematic applications, the algorithms are clear from the enclosed programs. References Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond Ser A-Math Phys Sci 241(1226):376– 396. https://doi.org/10.1098/rspa.1957.0133 Eshelby JD (1959) The elastic field outside an ellipsoidal inclusion. Proc R Soc Lond Ser A-Math Phys Sci 252(1271):561–569. https://doi.org/10.1098/rspa. 1959.0173 Jeffery GB (1922) The motion of ellipsoidal particles in a viscous fluid. Proc R Soc Lond Ser A-Containing Papers of a Mathematical and Physical Character 102(715):161–179. https://doi.org/10.1098/rspa.1922.0078 Shankar R (1995) Basic training in mathematics: a fitness program for science students. Springer Science & Business Media London, ON, Canada October 2022

Dazhi Jiang

Acknowledgments

The book originates from my lecture notes of a graduate course at Western University, Canada, and several invited summer short courses for graduate students and junior faculty at the University of Chinese Academy of Sciences, Northwest University, and the Hefei University of Technology China. Financial and logistic support from these institutions is greatly appreciated. Funding from Ontario’s eCampus Initiative for a project to create an online course on “Multiscale Analysis and Modeling in Structural Geology” covered some teaching relief for me to revise and update the book’s content. The Natural Sciences and Engineering Research Council of Canada (NSERC), the Canada Foundation for Innovation, and the National Natural Science Foundation of China (NSFC) provided funding for my research program, the results of which have been used in this book. This support is gratefully acknowledged. I am grateful to many people who taught me geology in the classroom, lab, and field and helped me with my career. My mentors, Paul Williams and Joe White, not only shared their knowledge and ideas with me generously, but they and their families helped my wife and me settle in Canada. Our time at the University of New Brunswick, Canada, was full of joyful memories. Paul’s love of structural geology and critical attitude toward science are a lifelong inspiration for me. I am grateful to Win Means for his support and encouragement in the early stage of my career. My undergraduate and graduate-time classmate Shoufa Lin has been a collaborator and long-time friend with whom I share ideas and seek insights. I am indebted to Profs. Guowei Zhang, Yunpeng Dong, and Anlin Guo (Northwest University, China), Profs. Guang Zhu and Chuanzhong Song (Hefei University of Technology, China), Prof. Quanlin Hou (University of Chinese Academy of Science), and Prof. Scott R. Paterson (University of Southern California) for opportunities of scientific collaboration and exchange of ideas therewith. I thank Dr. R.A. Lebensohn (Los Alamos National Laboratory, USA) for generously sharing the VPSC code. I have enjoyed reading so many landmark geology books. I am grateful to Dr. Eileen McLellan (former Associate Professor at the University of Maryland ix

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and now a lead senior scientist at the US Environmental Defense Fund). Shortly after I arrived at Maryland as an Assistant Professor and before she left Maryland to work on Capitol Hill as a Congressional Science Fellow, Eileen kindly let me take any books I liked from her bookshelf. Many of the books were out of print at the time. While I developed the theoretical equations, constructed the numerical algorithms, and implemented them in Mathcad, my graduate students, postdoctoral fellows, and visiting scientists over the past decade in Western’s Laboratory for Structural Geology and Tectonics have translated the Mathcad worksheets into MATLAB programs in their applications. They have refined some algorithms. Specifically, Dr. Mengmeng Qu developed a MATLAB program to compute Eshelby tensors for elongated or flattened ellipsoids. Mengmeng was also responsible for coupling MOPLA with the VPSC code. Dr. Lucy X. Lu was responsible for translating MOPLA from a Mathcad worksheet to a MATLAB package that is computationally more efficient and powerful. Lucy was also a principal player in the development of other MATLAB programs. In collaboration with Lucy, Dr. Ankit Bhandari developed MATLAB programs for the exterior fields around an ellipsoidal heterogeneity. Dr. Takayuki Miyoshi made the VPSC program run in our lab. The book has been improved from comments and feedback from Drs. Yvette Kuiper, Changcheng Li, Biwei Xiang, Yin Chen, and Rui Yang. These people have applied the theory and numerical methods presented in this book to their research. My current student Bohan Zhu helped run many numerical experiments, format some figures, and check many references. Finally, I thank my wife Changyun (Cathy) and children Eric and Lillian for their love and support, without which this book would not have been here.

List of Computer Programs and Movies

Mathcad Worksheets The worksheets can be run in Mathcad. You may also just read the PDF of the sheets to understand the algorithms used in solving various problems. This will help you understand the MATALAB codes below as similar algorithms have been used. Chapter 1 CommonTensorOperations.mcdx: This worksheet includes the most commonly used functions, such as tensor contraction, fourth-order tensor inversion, and mapping of a fourth-order tensor to a matrix. ViscousCommonTensorOperations.mcdx: The same as the above worksheet except that deviatoric subspace is considered. Fourth-order inversion is applicable to incompressible materials. One of the above two worksheets must be included at the beginning of many problem-solving worksheets. Chapter 2 3D_Fabric.mcdx: This worksheet presents the algorithm for the generation and plot of linear and 3D orientations. Chapter 3 StiffnessCompliance.mcdx: This worksheet computes the elastic compliance tensor from the elastic stiffness tensor. The input elastic stiffnesses are in Voigt matrix notation. AnisotropicElasticGreenFunction.mcdx: This sheet computes the Green function tensor and the first derivative of the Green function for an infinite anisotropic elastic solid.

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Chapter 4 PolarDecomposition.mcdx: This worksheet calculates the right and left stretch tensors and the rotation tensor for a given position gradient tensor. Chapter 6 L2F.mcdx: This sheet computes the corresponding position gradient tensor from a given constant velocity gradient tensor. TabularZoneModel.mcdx: This sheet computes the finite strain geometry (principal strains and orientations of principal strain axes) in a homogeneous tabular zone. Chapter 8 RigidEllipsoidSingle.mcdx: This sheet calculates the rotation path of a single rigid ellipsoid in a given viscous flow, defined by a constant velocity gradient tensor. The angular velocity is from Jeffery (1922). The Runge–Kutta and Rodriges methods are applied. The MATLAB program RigidEllipsoidSingle.m is based on the same algorithm. RigidEllipsoidGroup.mcdx: This sheet calculates the evolution of shape-preferred orientations of a group of non-interacting rigid ellipsoids. SpheroidAnalytic.mcdx: This worksheet is based on analytical solutions of the rotation path of a given prolate or oblate object. Chapter 9 Revolution.mcdx: This worksheet is based on Eq. (9.8). Chapter 10 IsotropicElasticEshelbyTensor.mcdx: This worksheet is for the calculation of interior Eshelby tensors (S and Π) in an isotropic elastic medium. AnisotropicElasticEshelbyTensor.mcdx: This worksheet is for the calculation of interior Eshelby tensors (S and Π) in a general anisotropic elastic medium. The isotropic case is contained as a special case. EshelbySingleEllipsoid.mcdx: This worksheet computes the evolution of the shape and orientation of a deformable ellipsoid in viscous flow. The model is for the more general situation of power-law viscous rheology of the system (ellipsoid and the matrix). For Newtonian systems, the stress exponents of the matrix and ellipsoid are set to 1. The extension of the Eshelby theory to power-law viscous materials is presented in Chap. 17. Chapter 11 ViscousGreenFunctions.mcdx: Integral form Green functions for the velocity and pressure in anisotropic viscous materials. ViscousInteriorEshelbyTensor.mcdx: Computes the Eshelby tensors (S and Π) for an ellipsoid in a linear anisotropic viscous medium. IsotropicExteriorEshelbyTensor.mcdx: Computes the Eshelby tensors (S and Π) outside an ellipsoid in a linear isotropic viscous medium.

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AnisotropicExteriorEshelbyTensor.mcdx: Computes the Eshelby tensors (S and Π) outside an ellipsoid in a linear anisotropic viscous medium. Chapter 13 EffectiveElasticStiffness.mcdx: This worksheet computes the effective elastic stiffness tensor (fourth-order) for a polycrystal aggregate from constituent stiffnesses and microstructures (shapes, orientations) using the self-consistent method. The example input parameters in the worksheet are specifically for a polycrystal quartzite made of n-set = 400 equant α-quartz grains. The crystal orientations are uniform in 3D. One can easily extend the computation to the effective stiffness tensor of a polyphase aggregate with or without preferred orientation. Chapter 14 ElasticStressPartitioning.mcdx: This worksheet calculates the Cauchy stress tensor in a prolate inclusion from a given remote stress tensor and the elastic constants of the matrix and the inclusion (both isotropic). ProlateStress.mcdx: This worksheet evaluates Eq. (14.43). It calculates the three deviatoric stresses in a prolate object lying on the VNS in simple shearing.

MATLAB Programs and Packages In the following, a MATLAB Program shall mean a single MATLAB-runnable file whereby input variables and output results are all contained in that file. A MATLAB Package, on the other hand, consists of a set of MATLAB files to solve a problem. The functions are usually divided into Main Functions and Supporting Functions. The use of each package is explained by a few examples. In the MOPLA package, some intensive computation (Eshelby tensors for extremely elongate or flat RDEs) is done through C++ and called in MATLAB. Please read the Readme.PDF included in each Program or Package and run the examples provided. Chapter 8 RigidEllipsoidSingle: Like the counterpart Mathcad worksheet RigidEllipsoidSingle. mcdx above, this MATLAB Package computes and plots the rotation path of a single rigid ellipsoid in a given viscous flow, defined by a constant velocity gradient tensor. Chapter 12 PressureFieldEllipse.m: This MATLAB Program computes and plots the pressure field around an ellipse in plane-straining flow (Eq. 12.15). Chapter 13 Homogenization: This package uses the three homogenization methods (Eshelby, Mori-Tanaka, and self-consistent) covered in Chap. 13 to get the effective rheology of the macroscale material.

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List of Computer Programs and Movies

Chapter 16 FlankingStructure: This package simulates the development of flanking structures around an ellipsoidal heterogeneity (the cutting element). Chapter 17 MOPLA: This package solves homogenization and partitioning equations for powerlaw viscous materials simultaneously. The output results are the effective rheology of the HEM, the shapes and orientations of all RDEs making up the HEM, and the partitioned flow field and stress tensor in each RDE. The homogenization computation in this package differs from the Homogenization package of Chap. 13 in that the power-law rheology of RDEs is considered. Therefore, if the HEM is made of power-law viscous phases, homogenization should be performed using MOPLA.

Movies A few movies of flanking structure evolution simulated in Chap. 16 are provided here.

Contents

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Background, Mathematic Preliminaries and Notations . . . . . . . . . . 1.1 The Necessity of a Multiscale Approach . . . . . . . . . . . . . . . . . 1.2 A Micromechanical Approach . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Mathematic Preliminaries, Notation, and Convention . . . . . . . . 1.3.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Coordinate System Transformation of Cartesian Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Matrix Exponentiation . . . . . . . . . . . . . . . . . . . . . . . 1.3.6 Differential Operators, Convention, and Related Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Fourth-Order Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orientation of Fabric Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Orientation of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Orientation of a 3D Object . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Spherical Angles (θ, ϕ, ϑ) . . . . . . . . . . . . . . . . . . . . . 2.2.2 Euler Angles (α, β, γ) . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Direction of a3 and a Rotation Angle (ξ, φ, γ) . . . . . . . 2.3 The Invariant Area Element on a Unit Sphere . . . . . . . . . . . . . 2.4 Rotation around a Coordinate Axis . . . . . . . . . . . . . . . . . . . . . 2.5 Rotation around a General Axis . . . . . . . . . . . . . . . . . . . . . . . 2.6 Euler Angles as a Set of Rotations . . . . . . . . . . . . . . . . . . . . . 2.7 Generation of a Population of Lines Following a Specified Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 A Set of Lines with Uniform Distribution . . . . . . . . . . 2.7.2 A Set of Lines Having a Gaussian Point Maximum . . .

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2.7.3 Line Set Forming a Small or Great Circle Girdle . . . Generation of a Population of 3D Fabric Elements Following a Specified Distribution . . . . . . . . . . . . . . . . . . . . 2.8.1 A Set of 3D Orientations Forming a Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 3D Objects Having Preferred Orientations . . . . . . . . 2.9 Misorientation Between Two Objects . . . . . . . . . . . . . . . . . . 2.10 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Stress, Strain, and Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Sign Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Strain of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Displacement Field and Strain Tensor in a Continuous Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 The Eulerian Displacement Gradients, Infinitesimal Strain, and Infinitesimal Rotation . . . . . . . . . . . . . . . . . . . . . . 3.7 Hooke’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 In Isotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 In Anisotropic Materials . . . . . . . . . . . . . . . . . . . . . . 3.8 Matrix Expression of the Elastic Stiffness Components . . . . . . 3.9 Boundary Value Problems of Linear Elasticity . . . . . . . . . . . . 3.10 Multiscale Stress and Strain in Real Materials . . . . . . . . . . . . . 3.11 The Effective Rheology on the Macroscale . . . . . . . . . . . . . . . 3.12 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Green Function for an Infinite Anisotropic Elastic Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 63 64 64

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Deformation: Strain and Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Position Gradient Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Pure Strain Deformation . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Simple Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Pure Strain Followed by Simple Shear . . . . . . . . . . . . 4.2 Polar Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Finite Strain Tensors, Stretch, and Shear of Lines . . . . . . . . . . 4.3.1 Stretch of Material Lines . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Shear Between a Pair of Initially Orthogonal Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Shear Strain and Shear Direction of a Material Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Other Useful Decompositions of Finite Deformation . . . . . . . . 4.5 Infinitesimal Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.6 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5

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Flow: Strain Rate and Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Material and Spatial Coordinates . . . . . . . . . . . . . . . . . . . . . . 5.2 Displacement Field, Velocity Field, Spatial and Material Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Strain Rate Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Velocity Gradient Tensor, Strain Rate, and Vorticity . . . . . . . . 5.5 Some Simple Flow Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Flow Described in Different Reference Frames . . . . . . . . . . . . 5.7 Decomposition of Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Some Examples of Vorticity Decomposition . . . . . . . . . . . . . . 5.9 Flow Field Arising from Multiple Slip Systems . . . . . . . . . . . . 5.10 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 115 116 117 119 121 126 128 129 134 136 136

Flow and Finite Deformation in Tabular Zones . . . . . . . . . . . . . . . . 6.1 Flow Apophyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Relationship Between Flow and Finite Deformation . . . . . 6.3 Kinematic Models for Homogeneous Tabular Zones . . . . . . . . 6.3.1 Simple Shearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Monoclinic General Shearing . . . . . . . . . . . . . . . . . . 6.4 Kinematics of Combining Simple and Pure Shearing Components in Tabular Zones . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Superposition of Simple Shearing Motions . . . . . . . . . 6.4.2 Kinematically Permissible Superposition of Simple and Pure Shearing Components in Constructing the Macroscale Flow in Tabular Zone . . . . . . . . . . . . . . . 6.4.3 Spatial Variation of Simple and Pure Shearing Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Limitations of Kinematic Modeling . . . . . . . . . . . . . . . . . . . . 6.6 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 139 141 144 146 147

Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Newtonian Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Power Law Viscous Behavior . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Flow Laws for Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Wet Quartzite Flow Laws from Experiments . . . . . . . . . . . . . . 7.5 Tangent Viscous Stiffness and Linearization . . . . . . . . . . . . . . 7.6 Plasticity as a Limit Behavior of Power Law Viscosity . . . . . . 7.7 Anisotropic Secant Compliance Tensor for a Polycrystal Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Boundary and Initial-Value Problems of Viscosity . . . . . . . . . . 7.9 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 164 167 169 171 173

154 155

157 158 160 160 161

174 176 177 177

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10

Contents

Rotation of Rigid Objects in Homogeneous Flows . . . . . . . . . . . . . . 8.1 Rotation and Coordinate System Transformation Revisited . . . 8.2 From Angular Velocity to Finite Rotation . . . . . . . . . . . . . . . . 8.2.1 Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 The Relationship Between Angular Velocity and Finite Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Angular Velocity of a Rigid Object in Slow Viscous Flows the Jeffery Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Analytical Solutions for Spheroids in Monoclinic Flows . . . . . 8.4.1 Equations Governing the Motion of Spheroidal Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Solutions of Spheroidal Objects in Monoclinic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 The Behavior of Rigid Spheroids in Monoclinic Flows . . . . . . 8.6 Numerical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 The Runge-Kutta Method . . . . . . . . . . . . . . . . . . . . . 8.6.2 Rodrigues Rotation Approximation . . . . . . . . . . . . . . 8.6.3 Runge-Kutta-Rodrigues Approximation . . . . . . . . . . . 8.6.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181 181 182 182

Further Analysis of Spheroids in Simple Shearing Flows . . . . . . . . 9.1 Jeffery Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Revolution Around the Distinct Axis . . . . . . . . . . . . . . . . . . . 9.3 Rotation of a Population of Rigid Spheroids . . . . . . . . . . . . . . 9.4 Forces Acting on a Prolate Object in Simple Shearing . . . . . . . 9.5 Deformation of a Prolate Object in Simple Shearing Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 205 206 209 211

Eshelby’s Inclusion and Inhomogeneity Problem . . . . . . . . . . . . . . 10.1 Eshelby’s Elastic Inclusion/Inhomogeneity Problem . . . . . . . . 10.2 Eshelby Tensors and the Auxiliary Interaction Tensor . . . . . . . 10.3 Extension to Newtonian Viscous Materials . . . . . . . . . . . . . . . 10.4 Expressions of Eshelby Tensors for Linear Isotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Eshelby Tensor Expressions for Penny-Shaped, Rod-Like, and Spherical Objects in Isotropic Elastic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Eshelby Tensor Expressions for Penny-Shaped, Rod-Like, and Spherical Bodies in Isotropic Newtonian Materials . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Strain and Rotation of a Deformable Ellipsoid . . . . . . . . . . . . .

221 222 227 229

184 185 186 186 189 191 199 199 200 201 201 202 203

213 218 219 219

230

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10.6 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 11

12

13

Viscous Inclusions in Anisotropic Materials . . . . . . . . . . . . . . . . . . 11.1 Limitation of the Penalty Approach . . . . . . . . . . . . . . . . . . . . 11.2 Green Functions for Viscous Materials . . . . . . . . . . . . . . . . . . 11.3 Viscous Eshelby Tensors and Auxiliary Quantities . . . . . . . . . 11.4 Formal Solutions for the Interior and Exterior Fields . . . . . . . . 11.5 Isotropic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 Viscous Tensor Identities . . . . . . . . . . . . . . . . . . . . . 11.5.2 An Explicit Approach to Evaluate Exterior Solutions for Isotropic Systems . . . . . . . . . . . . . . . . . 11.6 Some Analytic Solutions for Isotropic Systems . . . . . . . . . . . . 11.6.1 Kinematics of an Ellipsoid in 3D Flows . . . . . . . . . . . 11.6.2 Deviatoric Stresses and Pressure . . . . . . . . . . . . . . . . 11.6.3 An Ellipse in 2D Flows . . . . . . . . . . . . . . . . . . . . . . 11.7 Equations for Ellipsoid Rotation in Anisotropic Viscous Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.1 Angular Velocity Equations . . . . . . . . . . . . . . . . . . . 11.7.2 Shear Spin When the Inclusion Is Instantaneously a Spheroid or Sphere . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral Expressions of Gij,Gij, l, and Hi . . . . . . . . . . . . . . . . . Relations Among Tensor Quantities for Isotropic Incompressible Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivation of Equations for Incompressible Isotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Dimensional Inclusion Problems . . . . . . . . . . . . . . . . . . . . . . . 12.1 Previous Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Generalized Plane Flows in Anisotropic Viscous Materials . . . 12.3 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Application to Materials with a Planar Anisotropy . . . . . . . . . . 12.5 Pressure Field Around a 3D Inclusion in a Viscous Matrix with Planar Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245 245 247 252 255 256 256 259 260 260 262 262 266 266 267 267 268 269 269 273 275 279 283 284 285 285 287 295 297 297 297

Effective Stiffnesses of Heterogeneous Materials . . . . . . . . . . . . . . . 299 13.1 Scales, Micro- and Macroscale Fields . . . . . . . . . . . . . . . . . . . 299 13.2 Macroscale Averages and Hill’s Lemma . . . . . . . . . . . . . . . . . 301

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13.3

Determination of Effective Stiffnesses for Linear Materials: Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Determination of Effective Stiffnesses for Linear Materials: Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Examples for Computing Effective Elastic Stiffness Tensor of Polycrystal Aggregates from Single Crystal Stiffness Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Comparison of the Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7 Expressions for Effective Stiffness of Multiphase Composites from Noninteracting Approximation . . . . . . . . . . . 13.8 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

15

Application Example 1: An Elastic Prolate Object in a Viscous Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 An Elastic Prolate Inclusion in an Elastic Matrix . . . . . . . . . . . 14.2 Comparison with the Fiber-Loading Theory . . . . . . . . . . . . . . 14.3 An Elastic Prolate in a Newtonian Viscous Matrix . . . . . . . . . . 14.3.1 Equations for Stress and Strain in the Prolate Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 14.3.2 Solution of Jd - S dσ þ Sσ = Σ When Σ t db Is Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 Comparison with Earlier Work on Simple Shearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 14.3.4 Solution of Jd - S dσ þ Sσ = Σ for a Rod-Like t db Prolate Object on the Vorticity Normal Section in a Simple Shearing Flow . . . . . . . . . . . . . . . . . . . . 14.4 Application to Microboudinage . . . . . . . . . . . . . . . . . . . . . . . 14.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Solutions of an Elastic Flat Oblate Body in a Newtonian Viscous Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application Example 2: A Penny-Shaped Viscous Inclusion in an Elastic Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 The Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Solution for the Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 The Deviatoric Stress in the Inclusion . . . . . . . . . . . . . . . . . . . 15.4 The Pressure Field in the Inclusion . . . . . . . . . . . . . . . . . . . . . 15.5 Analysis of the Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.1 Stress Relaxation and Creep in the Viscous Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.2 Dynamic Pressure in the Viscous Inclusion . . . . . . . . 15.6 Discussion and Geological Implications . . . . . . . . . . . . . . . . .

302 305

313 315 317 320 321 323 324 325 329 329 331 337

339 340 344 344 345 347 349 350 353 355 356 359 359 361 362

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15.7 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 16

17

Application Example 3: Deformation Around a Heterogeneity— Flanking Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 The Motion of Material Particles Around an Ellipsoid from the Exterior Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Macroscale Flows and Model Geometry . . . . . . . . . . . . . . . . . 16.3 Modeling Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . . . . 16.5 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Conversion Between Cartesian and Ellipsoidal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalization of Eshelby’s Formalism and a Self-Consistent Model for Multiscale Rock Deformation . . . . . . . . . . . . . . . . . . . . . 17.1 Nonlinear Rheology and Partitioning Equations . . . . . . . . . . . . 17.2 Non-ellipsoidal Shape of Rheological Distinct Elements . . . . . 17.3 Inclusions in a Finite Space . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Interface Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Heterogeneous Matrix and Homogenization . . . . . . . . . . . . . . 17.6 A Self-Consistent Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 17.7 Multiscale Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.8 Behaviors of Fabric Elements . . . . . . . . . . . . . . . . . . . . . . . . . 17.8.1 Finite Strains in RDEs . . . . . . . . . . . . . . . . . . . . . . . 17.8.2 Rigid or Deformable Fabric Elements in RDEs . . . . . . 17.8.3 Crystal Lattice Rotation . . . . . . . . . . . . . . . . . . . . . . 17.8.4 Empirical Behaviors of Fabric Elements . . . . . . . . . . . 17.9 A Self-Consistent Multiscale Model for the Deformation of Earth’s Heterogeneous Lithosphere . . . . . . . . . . . . . . . . . . . . . 17.10 A Continuum Micromechanics-Based Multiscale Approach . . . 17.11 Notes and Key References . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

371 372 377 379 382 384 385 386 389 389 394 395 395 398 402 404 405 405 406 407 407 408 410 411 412

Chapter 1

Background, Mathematic Preliminaries and Notations

Abstract This Chapter gives a brief background on the need for a multiscale approach in structural geology and tectonics. A brief review of matrix algebra, vectors, and cartesian tensors is presented. Mathematic notations and conventions commonly used in the presentation of mechanical equations are explained. Mathematic operations involving fourth-order tensors are described and the method, using orthonormal second-order tensor bases, to get the inverse of a symmetric fourthorder tensor (such as the elastic stiffness tensor) is introduced. The related mathematic operations are summarized in a Mathcad worksheet.

1.1

The Necessity of a Multiscale Approach

Earth’s dynamic history has left abundant “patterns” in the lithosphere, which we generally call geological structures and fabrics. The term structure is often used to refer to specific features in Earth’s lithosphere like folds, faults, ductile shear zones, and so on that we can observe and characterize individually. The term fabric refers to patterns defined by many repetitive (penetrative) elements observed in a volume of rock. Fabric-defining elements can be shapes of mineral grains and geological bodies, interfaces and boundaries, and other heterogeneities in rocks. We broadly refer to fabrics defined by these as shape fabrics or shape-preferred orientations (SPOs). Foliations and lineations in rocks are examples of shape fabrics. Fabricdefining elements can also be lattice orientations of the constituent mineral grains in a rock volume. In such cases, the fabrics are called lattice-preferred orientations (LPOs) or crystallographic-preferred orientations (CPOs). An example is quartz caxis preferred orientations in a quartzofeldspathic mylonite. The presence of an LPO may not be observed directly but is detected indirectly, arising from certain anisotropic (direction-dependent) physical properties such as seismic velocity, electric

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-23313-5_1. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Jiang, Continuum Micromechanics, Springer Geophysics, https://doi.org/10.1007/978-3-031-23313-5_1

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1 Background, Mathematic Preliminaries and Notations

conductivity, magnetic susceptibility, and so forth. In some situations, the underlying element(s) giving rise to the observed anisotropies may even be unknown. Most structures and fabrics are products of deformation. By studying them, geologists hope to understand the ancient deformation as much as possible. This includes deformation conditions, mechanisms, rheological behaviors of rocks at the time of deformation, kinematics, dynamics, and boundary conditions (load and movement history). Structures and fabrics occur over a vast range of characteristic lengths, from crystal lattice spacing (e.g., Burger’s vector of a dislocation) to the size of lithospheric plates. Structural Geology is a science that mainly focuses on the scale range most easily accessible—typically from outcrops to hand samples and under petrological microscopes. A structural geologist spends a lot of time in the field doing geological mapping to collect outcrop and hand-sample scale structural data from rock exposures and try to identify the geometrical (spatial) and overprinting (temporal) relations of the structures over a region. The work is often further carried to the scale under petrological microscopes. Features observed under petrological microscopes on oriented thin sections are commonly called microstructures. SPOs under microscopes can often be measured directly with some image analysis software. LPOs such as quartz c-axis fabric used to be measured with a Universal Stage mounted on a petrological microscope or by X-ray diffraction. The modern technique is the Electron Backscatter Diffraction (EBSD) technique—a device installed on a Scanning Electron Microscope (SEM). Crystal lattice-scale features such as dislocations are observed under Transmission Electron Microscopes (TEM). We shall call structures and fabrics in the Structural Geology scale range minor structures for easy reference. Minor structures are abundant in rocks and are most commonly observed and characterized. High-precision GPS-enabled digital mapping techniques and modern computer-aided microstructural analysis such as EBSD have allowed structural data to be obtained more efficiently. Tectonics deals with regional to global-scale features and processes. As the scale length of a structural element increases, it becomes increasingly harder to determine its 3D geometry because geologists inevitably must work with discontinuous outcrops of a 2D nature. An important motivation for studying minor structures (Structural Geology) has been to gain clues about Earth’s tectonic-scale deformation and rheology. It is not trivial to relate minor structures to tectonic scale processes because the lithosphere is made of rheologically heterogeneous elements over a wide range of observation scales. One needs a multiscale approach to bridge the scale gap between Structural Geology and Tectonics. For many decades, geologists have relied on principles of continuum mechanics to understand minor structures and relate them to tectonics. The application of continuum mechanics has led to significant advances in structural geology since Ramsay (1967) and Hobbs et al. (1976). This is reflected in many landmark textbooks (Twiss and Moores 1992; Johnson and Fletcher 1994; Passchier and Trouw 1995; Pollard and Fletcher 2005; Pollard and Martel 2020) and a large number of research papers in the Earth and planetary science literature. This

1.1

The Necessity of a Multiscale Approach

3

approach is robust in analyzing structures of a single or limited range of characteristic length scales (Fig. 1.1). However, the standard continuum mechanics (Spencer 1980; Truesdell 1991)) cannot effectively address the heterogeneous deformation in Earth’s lithosphere, where one needs to consider rheological heterogeneities spanning a wide range of characteristic length scales (Fig. 1.2). This inability is due to the fact that the classical approach does not contain material parameters with length dimensions that can capture the characteristic scales of rheological heterogeneities. Rheological heterogeneities are ubiquitous in the lithosphere and can cause significant fluctuations in flow field from one rheologically distinct unit to another—a phenomenon called flow partitioning (Lister and Williams 1983). Consequently, in the deformation of a large region such as a tectonic-scale deformation zone, it is the local flow field fluctuations that are responsible for small-scale features like stretching lineations, kinematic indicators, and quartz crystallographic preferred orientation fabrics. A single-scale approach to relate these small-scale structures to large-scale boundary conditions is unrealistic. Many people have recognized the scale problem discussed here (e.g., Lister and Williams 1979, 1983; Ishii 1992; Jiang 1994a, b; Jiang and White 1995; Jiang and Williams 1999; Hudleston 1999; Goodwin and Tikoff 2002; Jones et al. 2005). Geologists distinguish different scale quantities routinely using terms like ‘regional’ (or ‘bulk’) stress vs ‘local’ stress, bulk flow vs local flow and so on. They also frequently resort to “flow partitioning” to explain heterogeneities of natural structures and fabrics (e.g., Kilian et al. 2011; Carreras et al. 2013). It is perhaps safe to say that very few geologists would deny the significance of flow partitioning in lithospheric deformation. Yet, how to address the problem remains elusive. Some simple analyses are based on geometric and kinematic grounds (e.g., Jiang 1994a, b; Jiang and White 1995; Jiang and Williams 1999; Hudleston 1999; Jones et al. 2004; Passchier et al. 2005). There are also 2D mechanical analyses (e.g., Ishii 1992; Griera et al. 2011; Griera et al. 2013; Dabrowski and Schmid 2011). But there is still no rigorous and generally applicable means to handle flow partitioning. In many works, “partitioning” remains a qualitative concept used in various contexts: sometimes to criticize simple-minded extrapolations from small-scale observations to big-scale processes. At other times, without much harm, “partitioning” is used as a convenient means to “explain” observations that do not fit the predictions of simple single-scaled models. Some earth scientists have taken the ubiquitous “flow partitioning” in nature to support their notion, often expressed informally, that small-scale structures are too complicated to be useful. While this notion is not acceptable (Ramsay and Huber 1987, p.vi), it demonstrates the need for a more rigorous consideration of heterogeneities at various scales.

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1

Background, Mathematic Preliminaries and Notations

Fig. 1.1 A Ramsay and Graham type ductile shear zone in the Roses granodiorite near Cadaqués, Spain. The deformation and fabric development in such a shear zone is a single-scale problem. The shape preferred orientations of feldspar grains and deformed quartz aggregates define the schistosity, the deflection of which across the shear zone is related to the boundary displacement. Larger (such as crustal-scale) shear zones are usually more complex as they are made of rheologically heterogeneous elements which cause deformation field partitioning. Partitioned deformation fields are responsible for fabric development (Lister and Williams 1983; Jiang 1994a, 1994b; Jiang and White 1995). A multiscale approach is needed to connect observations on various scales to the boundary conditions (Jiang and Bentley 2012; Jiang 2014)

1.2

A Micromechanical Approach

We generally call the substances that make up a deforming body materials. A material has no boundaries or shapes but has intrinsic properties such as density, shear modulus, viscosity, and thermal conductivity. An object is a physical body having a geometrical shape and size. An object is made of material(s) whose deformation depends on its shape, size, boundary conditions, and material properties. The standard continuum mechanics treats a material as a uniform continuous mass but does not explicitly address any internal details within the material, such as the presence of grain boundaries, inclusions, and heterogeneities. We shall collectively refer to the constituent heterogeneous elements making up a deforming body as Rheologically Distinct Elements (RDEs). With RDEs ignored, the standard continuum mechanics uses a single-scale characterization of the material. The mechanical fields (stress, strain, strain rate, etc.) derived from this single-scale approach reflect the ‘homogenized’ fields over a large volume element (so that RDEs become inconspicuous)—the macroscale fields. However, as pointed out

1.2 A Micromechanical Approach

5

Fig. 1.2 A large shear zone spanning many discontinuous outcrops, such as the Sierra Crest Shear zone in California, cannot be treated as a single-scale problem because outcrop-observed minor structures have been developed in many rheologically distinct elements that have undergone distinct deformation histories. In the multiscale approach that will be discussed in this book, the plate-scale boundary conditions establish a macroscale deformation field which is partitioned into many mesoscale rheologically distinct elements. The partitioned deformation fields in these elements are used to investigate the minor structural development. Micromechanics-based partitioning equations will govern how macroscale deformation is partitioned. Modified after Bentley (2004)

6

1 Background, Mathematic Preliminaries and Notations

above, rocks are made of heterogeneous materials, and our observations are inevitably made from individual RDEs. Using the single-scale macroscale fields is insufficient to understand the observed microscale features. Micromechanics, on the other hand, addresses the internal heterogeneities explicitly. In micromechanics, the materials making up the individual RDEs are each treated as a continuum via continuum mechanics. Micromechanics (Mura 1987; Nemat-Nasser and Hori 1999; Qu and Cherkaoui 2006) is a new and fast-growing branch of continuum mechanics. It deals with the deformation of heterogeneous materials at both the global (sample) scale and the scale of the constituent heterogeneities. Micromechanics originated from the milestone work of Eshelby (1957, 1959, 1961) on the interaction between an elastic inclusion (or inhomogeneity) with the surrounding infinite elastic matrix. His novel method of solving the problem, now commonly referred to as Eshelby’s inclusion/inhomogeneity solutions (e.g., Mura 1987, p. 74; Qu and Cherkaoui 2006, p. 77; Li and Wang 2008, p. 94) or more simply the Eshelby formalism (e.g., Lebensohn and Tomé 1993), was first extended to Newtonian viscous materials by Bilby et al. (1975) and Bilby and Kolbuszewski (1977) and then to non-Newtonian materials using the linearization approach (Molinari et al. 1987; Lebensohn and Tomé 1993; Ponte Castañeda 1996; Masson et al., 2000). The extension to power-law materials led to the self-consistent viscoplastic (VPSC) formulations (Lebensohn and Tomé 1993; Lebensohn et al., 2011) for simulating lattice preferred orientation fabrics in crystalline aggregates. Castelnau et al. (2010), in their multiscale investigation of the anisotropic rheology of olivine polycrystals and Earth’s upper mantle dynamics, and Montagnat et al. (2014), in their recent simulation of ice deformation, gave a succinct review of the first-order, second-order, and full-field viscoplastic approaches. The VPSC theory and code have been used by many geoscientists to investigate texture development in crustal and mantle rocks (e.g., Wenk et al. 1989; Tommasi et al. 2000; Lebensohn et al. 2003; Wenk et al. 2009; Keller and Stipp 2011) for over two decades. More recently, Griera et al. (2011) and Griera et al. (2013) have used the full-field viscoplastic formulation based on the fast Fourier transformation (Lebensohn 2001) to simulate the rotation of rigid porphyroclasts and strain localization near the clasts. The extended Eshelby formalism for power-law viscous materials can also be adapted to address the general problem of flow partitioning (Jiang and Bentley 2012; Jiang 2012, 2013, 2014). With the idea of “inhomogeneities within inhomogeneities,” one can apply the extended Eshelby formalism to multi-hierarchical levels of flow partitioning in materials containing rheologically heterogeneous elements of varying characteristic lengths. This has led to the development of a self-consistent MultiOrder Power Law Approach (self-consistent MOPLA, Jiang 2014). MultiOrder means that the approach considers rheological elements of multiple characteristic lengths, from large (low-order) elements to small (fabric-defining, high-order) elements. And power law designates the fact that the model considers the non-Newtonian rheological behavior of natural rock deformation. Theoretical advances in micromechanics have also been promoted by powerful and user-friendly mathematics applications like MATLAB and Mathcad, which

1.3

Mathematic Preliminaries, Notation, and Convention

7

make numerical and theoretical (symbolic) analysis accessible (Jiang 2007a, 2007b, 2012, 2013, 2014). All this has enabled us to analyze and simulate multiscale 3D deformations of rocks to large finite strains and to investigate the associated fabric development.

1.3

Mathematic Preliminaries, Notation, and Convention

In this section, we briefly review the concepts of vectors, matrices, tensors and the standard mathematic notations that will be used in this book. Do not worry if it takes some time for you to get used to some of these notations (like the Einstein summation convention). The tensor/matrix notations are necessary to present mechanical equations. They are also convenient in the numerical solutions of mechanical equations using mathematics applications like MATLAB and Mathcad. Therefore, it is beneficial that you will become familiar with the notations.

1.3.1

Vectors

A vector is a physical quantity that has both magnitude and direction. It is often represented by an oriented arrow. Changing the sign of a vector reverses its direction. Multiplying it by a positive number changes its magnitude. Two vectors are equal if they have equal magnitude and direction. Vectors are added by putting them in tail-to-head strings. The vector sum is a vector from the tail of the first vector in the string to the head of the last one (Fig. 1.3). The Cartesian components of a vector are its projection onto a Cartesian coordinate system which has orthogonal coordinate axes. Another way to look at the Cartesian components is to regard a vector as composed of a linear combination of 3 orthogonal base unit vectors (Fig. 1.4). v = v1 e 1 þ v 2 e 2 þ v 3 e 3 =

X vi ei

ð1:1Þ

i

Clearly, a vector’s Cartesian components depend on the base vector set. If another set of mutually orthogonal unit base vectors e'i is used, the corresponding components v'i will be different as: v=

X X vi ei = v ′ ie ′ i i

i

ð1:2Þ

8

1

Background, Mathematic Preliminaries and Notations

Fig. 1.3 Vectors and vector addition. Note a vector is represented by an arrow in 3D space. Vectors v, w, and z are not necessarily in the same plane. To analyze vectors, we express vectors in terms of their Cartesian components

Fig. 1.4 Vectors in 2D and 3D space and their Cartesian components in terms of base unit vectors

Further, we do not always need to explicitly write out the unit base vectors. We often write a vector just as vi with the free index i varying between 1 and n, where n is the dimension of space and, in most cases, n = 3. To simplify mathematic expressions, the Einstein summation convention is used. An index repeated in a product implies a summation over that index from 1 to n for an n-dimensional problem. With this convention, the summation symbol in Eqs. (1.1) and (1.2) is omitted, and Eq. (1.2) is simply written as: v = vi ei = v ′ i e ′ i

ð1:3Þ

An index not repeated is a free index, meaning a list of components or expressions. v3, if v is a quantity in 3D space. Therefore, from now on, vi will stand for v1, v2, and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The magnitude (length) of a vector v is jvj = v1 2 þ v2 2 þ v3 2 which, using the summation convention, can be expressed simply as: pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi jvj = vi vi = vj vj = vk vk Note it makes no difference whether the repeated index is i, j, k, or anything else in a summation expression because it merely stands for summation with respect to the index. For this reason, a repeated index is called a dummy index in contrast to a free index. We will use the Kronecker delta frequently, which is defined as:

1.3

Mathematic Preliminaries, Notation, and Convention

 δij =

1 0

if i = j if i ≠ j

9

ð1:4Þ

The dot product (also called scalar product, inner product) of two vectors u and v is a scalar, defined by: u  v = jujjvj cos θ where θ is the angle between u and v that satisfies 0 ≤ θ ≤ π. For a set of Cartesian unit vectors, ei, δij = ei  ej. Also, note the identity δikvk = δimvm = vi, which is the index change for vector components i. In terms of Cartesian components, the dot product is:     u  v = ðui ei Þ  vj ej = ui vj ei  ej = ui vj δij = ui vi

ð1:5Þ

uv Therefore, the angle between two vectors can be obtained by cos θ = jujjvj = uiffivpi ffiffiffiffiffi pffiffiffiffiffi b b b b u u v v = u  v where u and v are unit vectors parallel to uand v respectively. i i

i i

We also use the permutation symbol: 8 >

: 0

if ijk is an even permutation of 123 if ijk is an odd permutation of 123 if 2 or 3 indices are equal

ð1:6Þ

Another way to understand this definition is that starting with ε123 = 1 any time two adjacent indices swap positions, a - 1 is multiplied. For example, ε213 = (1)ε123 = - 1 and ε231 = (-1)ε213 = (-1)2ε123 = 1 so forth. The cross product (or vector product) of two vectors u and v is a new vector whose length is |u||v| sin θ and whose direction is perpendicular to the plane of u and v, toward a right-handed screw from u to v (Fig. 1.5). In terms of Cartesian components, the cross product is: u × v = εijk uj vk ei = ε1jk uj vk e1 þ ε2jk uj vk e2 þ ε3jk uj vk e3 = ðu2 v3 - u3 v2 Þe1 þ ðu3 v1 - u1 v3 Þe2 þ ðu1 v2 - u2 v1 Þe3

1.3.2

ð1:7Þ

Matrices

A matrix is a rectangular table of numbers or mathematic expressions called elements of the matrix. Examples of matrices are:

10

1

Background, Mathematic Preliminaries and Notations

Fig. 1.5 The cross (or vector) product of two vectors is a new vector

0

3

B @ 10 12

2:7 - 0:125 3:14

0

1

0 - 0:3 B 2 C B C B C 2:7 A, ð2:7 4 2Þ, B C, @ 17 @ -5A 8 0 7 1

1

13

2

1

C 0A 9

The size of a matrix is specified by its row number and column number. Matrix A below has m rows and n columns and is called an m by n or m × n matrix. 0

A11

B A21 B B A = B :: B @ :: Am1

A12

...

A22 ::

... ::

:: Am2

:: ...

A1n

1

A2n C C :: C C C :: A Amn

The element at row i and column j is denoted by Aij. 0

1 7 B B = @ - 0:2 2 π 2:7

3 13 5

1 25 C 4 A,B31 = π, B24 = 4 pffiffiffi 2

We often use a bold font to refer to a matrix, such as A and B above. We also use Aij and Bij, with free indices varying in their range, to refer to matrices if no ambiguities result. The transpose of matrix A is a new matrix, denoted AT, whose row i elements are the column i elements of A, i.e., (AT)ij = Aji. In other words, the rows and columns are swapped in matrix transpose. If a matrix has n rows and n columns, it is called a square matrix of order n, such as:

1.3

Mathematic Preliminaries, Notation, and Convention

0

11

A11 BA B 21 B @ A31

A12 A22

A13 A23

A32

A33

1 A14 A24 C C C A34 A

A41

A42

A43

A44

The A11, A22, A33, and A44 are called the diagonal elements. The sum of them is called the trace of the matrix and is denoted by: traceðAÞ = Aii If A is a matrix and c is a scalar, then the product cA is the matrix obtained by multiplying every component of A by c. An example is: 0

1

B A = @ - 0:2 π

7 2 2:7

3

25

1

0

C 13 4 A; pffiffiffi 5 2

2

14

6

B 2A = @ - 0:4 4 2π 5:4

26 10

50

1

C 8 A pffiffiffi 2 2

If A is a m × r matrix and B a r × n matrix, then the product AB is a m × n matrix whose ij-element is obtained in the following way: ðABÞij = Aik Bkj The repeated index k in the above expression is a dummy index, meaning that r P Aik Bkj ). The index not summation over k in the range of 1 to r (i.e., Aik Bkj  k=1

repeated here are i and j. They are free indices standing for all elements of the product matrix. Free indices in an equation must be consistent. As the Cartesian components of a vector are often written in a column matrix format, it is more convenient in computer programing to write the cross product of two vectors in the form of a matrix product. The Cartesian components of u × v can be expressed by the following matrix product: 0

0

B ðu × vÞi = U ik vk = @ u3 - u2

- u3 0 u1

u2

10

v1

1

CB C - u1 A @ v 2 A 0 v3

ð1:8Þ

It is easy to confirm Eq. (1.8). The matrix elements Uij and vector components ui are related by the following:

12

1

Background, Mathematic Preliminaries and Notations

ui = - εijk U jk 1 U ij = - εkij uk 2

ð1:9Þ

A square matrix like Uij with all diagonal elements being zero and Uij = - Uji is called a skew-symmetric or anti-symmetric matrix. An anti-symmetric 3 × 3 matrix has only three independent elements like a 3D vector. If A is a square matrix and a square matrix B exists such that AikBkj = BikAkj = δij, then B is the inverse of A and is written as B = A-1. The matrix Iij = δij is called the identity matrix. It is so called because for any square matrix A, IA = AI = A. A square matrix A has an associate determinant, denoted by det A, which is a scalar. For a 2 × 2matrix, its determinant is simply calculated according to the following rule:  det

a

b

c

d

 = ad - bc

For a 3 × 3 matrix, the determinant is calculated by the following rule:

where the red products are positive and green products negative, the above expression can be written using the permutation symbol and summation convention as: detA = εijk εrst Air Ajs Akt

ð1:10Þ

The cross product u × v of Eqs. (1.7) and (1.8) can be expressed, using determinant, as: e1

e2

u × v = j u1 v1

u2 v2

e3

u2 u3 j = j v2 v3

u3 v3

je1 - j

u1

u3

v1

v3

je2 þ j

u1

u2

v1

v2

je3

ð1:11Þ

If A is a square matrix, x is a column matrix (vector). The multiplication, Ax, yields a new vector which can be viewed as the “transformed” (by A) vector. It is of particular interest in mechanics to find those vectors that, after transformation by

1.3

Mathematic Preliminaries, Notation, and Convention

13

A, remain in the same orientation as their original vectors. Such vectors are called eigenvectors of the matrix. Eigenvectors for A can be found by solving the following Equation: Ax = λx,or ðA - λIÞx = 0 where λ is a scaler called the eigenvalue corresponding to the eigenvector. A square 3 × 3 matrix can have up to three distinct eigenvalues and associated eigenvectors.

1.3.3

Tensors

A vector v = viei is a first-order tensor which is a linear combination of the base unit vectors ei. A second-order tensor T can be formed by the tensorial (or dyadic) product of two vectors (first-order tensors) u and v: T = u  v = ui vj ei  ej = T ij ei  ej

ð1:12Þ

The Cartesian components Tij = uivj form a square matrix. The bases for T are second-order terms ei  ej. Physical quantities like stress, strain, and displacement gradients are all secondorder tensors. Like with vectors, we do not always need to write out the base tensors ei  ej explicitly. We often denote a tensor just by its Cartesian components, which, for second-order tensors, are square matrices. It should be clear that Cartesian components of a second-order tensor are expressed as a square matrix. The inverse is not generally true—not all matrices are tensor components. High-order tensors can be constructed similarly by the tensorial production of more vectors. A third-order tensor is a dyadic product of three vectors, u  v  w, which has 27 Cartesian components uivjwk and a fourth-order tensor has 81 Cartesian components. High-dimension arrays which have more than two subscript indices are used to present the Cartesian components of high-order tensors such as Aijk and Cijkl for third and fourth-order tensor components. Tensor operations are done with their Cartesian components and, for secondorder tensors, are similar to matrix operations outlined above. Tensor operations involving higher-order arrays can be tedious, but modern mathematics applications like MATLAB and Mathcad can handle such operations readily. We will use the inner product of tensors frequently. Tensor inner product operation is commonly referred to as contraction. If A = Aijei  ej, B = Bijei  ej, and b = biei, the single-index contraction of the two tensors is defined as follows:

14

1

A  b = Aik bk ei  ek  ek = Aik bk ei b  A = bk Aki ek  ek  ei = bk Aki ei

Background, Mathematic Preliminaries and Notations

) 1st order with components : Aik bk

ð1:13Þ

) 1st order with components : bk Aki ð1:14Þ

A  B = Aik Bkj ei  ek  ek  ej = Aik Bkj ei  ej ) 2nd order with components : Aik Bkj B  A = Bik Akj ei  ej

) 2nd order with components : Bik Akj

ð1:15Þ ð1:16Þ

where the underlined dot products of base vectors in Eqs. (1.13), (1.14), and (1.15) all become one. Double index contraction is encountered when high-order tensors. For example, the stress and strain in linear elasticity are both second-order tensors. They are related, for general anisotropic materials, by the following general Hooke’s law: σ ij = C ijkl ekl

ð1:17Þ

where σ ij and ekl are respectively second-order stress and strain tensors, Cijkl is the fourth-order elastic stiffness tensor. This relationship is written in tensor notation as: σ=C : e

ð1:18Þ

where the colon “:” stands for double index contraction. Some examples of double-index contraction operations are as follows. If C = Cijklei  ej  ek  el and S = Sijklei  ej  ek  el are fourth-order tensors, then: C : A = C ijkl Akl ei  ej  ek  el : ek  el = Cijkl Akl ei  ej ) 2nd order A : C = Akl Cklij ek  el : ek  el  ei  ej = Akl C klij ei  ej ) 2nd order C : S = C ijmn Smnkl ei  ej em  en : em  en  ek  el = C ijmn Smnkl ei  ej  ek  el

)

4th order

A : B = Aij Bij ei  ej : ei  ej = Aij Bij ) 0nd order ðscalarÞ

ð1:19Þ

where the underlined parts all become one by double dot production. The single index contractions in Eqs. (1.13), (1.15), and (1.16) are the same as matrix multiplication if bi is taken as a column matrix. The dot is often omitted when the single index contraction produces the same components as the matrix product. Therefore, A  b, A  B, and B  A in Eqs. (1.13), (1.15), and (1.16) are simply written as Ab, AB, and BA respectively. However, the operation in Eq. (1.14) cannot be a matrix multiplication and the dot in b  A cannot be omitted. This book uses a combination of Cartesian tensor notation and a standard tensor/ matrix notation.

1.3

Mathematic Preliminaries, Notation, and Convention

1.3.4

15

Coordinate System Transformation of Cartesian Tensors

We pointed out above that the Cartesian components of a vector are different in different coordinate systems. Tensors are dyadic product of vectors. The Cartesian components for a second or higher order tensor also depend on the coordinate system’s base vectors. In mechanics and fabric analysis, it is often necessary to convert the components of a vector or tensor expressed in one coordinate system to corresponding components in another coordinate system. This is called the coordinate system transformation of a tensor. Let us begin with the transformation of a vector (first order tensor). As pointed out above Eq. (1.3), vector v has two distinct sets of Cartesian components in two coordinate systems: v = v i ei = v ′ i e ′ i To relate v'i and vi, define the coordinate system transformation matrix Qij as: Qij = e ′ i  ej

ð1:20Þ

Qij are the direction cosines of e'i relative to the coordinate system defined by ei and e'i = Qikek. Thus, the ith row of matrix Qij are the Cartesian components of e'i in the coordinate system defined by ej. In other words, Qij is simply formed by stacking the three sets of Cartesian components of e'i (i = 1, 2, 3) relative to the first coordinate system. The components transform according to the following rule: v ′ i = Qik vk

ð1:21Þ

This expression is the rule of coordinate transformation for a vector (first order tensor). Because e'i  e'j = δij, it is trivial to show that QikQjk = δij. The transformation matrix Qij has the following property: QQT = QT Q = I detðQÞ = 1

ð1:22Þ

Qij is thus a proper orthogonal matrix (Spencer 1980, p. 6). It is trivial, with Eq. (1.21) to confirm that the transformation keeps the length of v invariant:

16

1

Background, Mathematic Preliminaries and Notations

pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi jvj = vi vi = v ′ i v ′ i Now consider the transformation of the components of a second-order tensor T = u  v. As T = u  v = uivjei  ej = u'iv'je'i  e'j, the Cartesian components are Tij = uivj and T'ij = u'iv'j respectively. How are T'ij related to Tij? According to Eq. (1.21), u'i = Qimumand v'j = Qjnvn. We thus have the following transformation expression: T ′ ij = u ′ i v ′ j = Qim Qjn um vn = Qim Qjn T mn

ð1:23Þ

This is the tensor transformation rule for second-order tensors, which can also be written in a matrix operation as: T 0 = QTQT

ð1:24Þ

The Q-term occurs two times in the transformation of a second-order tensor. For a kth-order tensor, the coordinate transformation formula is as follows, where the Q-term occurs k times: Ψ ′ i1 i2 ⋯ik = Qi1 m1 Qi2 m2 ⋯Qik mk Ψi1 i2 ⋯ik

ð1:25Þ

The reason that physical quantities are expressed as tensors is that physical laws are independent of coordinate systems. Mathematic equations for physical laws are of the same forms regardless of coordinate systems if physical quantities laws are written in terms of tensors. For example, the form of Hooke’s law (Eqs. (1.17) or (1.18)) should be the same regardless of the coordinate system. In a different coordinate system, the stress tensor components become σ'ij = QimQjnσ mn, the strain tensor components e'ij = QimQjnemn, and the elastic stiffness tensor components become C'ijkl = QimQjnQksQltCmnst. One still has σ'ij = C'ijkle'kl. The tensor form (Eq. (1.18)) is invariant. The transformation matrix having the property of Eq. (1.22) is called an orthogonal matrix. If1α, β, and γ are three mutually perpendicular unit vectors, then 0 α1 α2 α3 B C @ β1 β2 β3 A is an orthogonal matrix. Because of the property QikQjk = δij, it γ1 γ2 γ3 follows that δij = αi αj þ βi βj þ γ i γ j

ð1:26Þ

1.3

Mathematic Preliminaries, Notation, and Convention

1.3.5

17

Matrix Exponentiation

In the analysis of homogeneous finite deformation problems, one often encounters the solution of the following set of linear differential equations with initial conditions: 8 < dx = = Ax dt : x ð 0Þ = x0

ð1:27Þ

where A is a constant matrix independent of t and x. This problem has a unique solution according to the Picard-Lindelöf theorem (Ref). The solution can be expressed in terms of the matrix exponential by: x = eAt x0 = exp ðAt Þx0

ð1:28Þ

The exponential of a matrix is defined as: eA = I þ A þ

A2 A3 þ þ⋯ 2! 3!

ð1:29Þ

A special case of Eq. (1.29) is where the matrix A = Ω and Ω is anti-symmetric 1 P Ωn (Ωij = - Ωji). In such a case, exp ðΩÞ = n! has a closed expression as shown n=0

below.

0

1

0 - ω3 ω2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C B b where ω = ω1 2 þ ω2 2 þ ω3 2 . Being Let Ω = @ ω3 0 - ω1 A = ωΩ - ω2 ω1 0 skew-symmetric, the powers of Ω have the following property: Ω0 = I

b Ω1 = ωΩ b2 Ω2 = ω2 Ω

b Ω3 = - ω3 Ω 4 4 b2 Ω = -ω Ω b Ω5 = ω5 Ω

ð1:30Þ

⋯⋯ Submitting these expressions into exp ðΩÞ =

1 P n=0

Ωn n!

and regrouping yields:

18

1

Background, Mathematic Preliminaries and Notations

b2 b b2 b Ω 2Ω 3Ω 4Ω exp ðΩÞ = I þ ω þ ω -ω ⋯ -ω 2! 4! 3!  1!  3 5 2 ω2 ω ω4 ω 6 ω ω b b þ -⋯ þ Ω þ ⋯ =I þ Ω 3! 5! 2! 4! 6! 1!

ð1:31Þ

As the two parentheses in Eq. (1.31) are the expansions of sinω and 1 - cos ω respectively, we therefore have: b þ ð1 - cos ωÞΩ b exp ðΩÞ = I þ sin ωΩ

2

ð1:32Þ

This expression is the well-known Rodrigues relation for rotation (e.g., Başar and Weichert 2000, p. 31). The following is a list of some matrix exponential identities. It is important to note that not all rules for scalar exponential apply to matrix exponential, mainly because matrix multiplication is generally not commutable: d  At  e = AeAt dt e0 = I where 0 is the zero matrix BeAt ≠ eAt B unless AB = BA eAt eBt ≠ eðAþBÞt unless AB = BA eAt eAs = eAðtþsÞ only if At and As commute  At  - 1 = e - At e eAt = eλ1 t I þ

e λ1 t - e λ 2 t ðA - λ1 I Þ A is 2 λ1 - λ2

× 2, λ1 and λ2 are eigenvalues of A, both real and λ1 ≠ λ2 eAt = eλ1 t I þ teλ1 t ðA - λ1 I Þ A is 2 × 2 λ1 = λ2 and real  sin βt At αt ðA - αI Þ A is 2 × 2,λ1 ,λ2 = α ± iβ,β > 0 e = e cos βtI þ β

1.3.6

ð1:33Þ

Differential Operators, Convention, and Related Theorems

We will use the following shorthand called the comma convention to write the derivative with respect to spatial coordinates:

1.3

Mathematic Preliminaries, Notation, and Convention

A,j =

19

∂A ∂xj

ð1:34Þ

where A can be a scalar, vector, or tensor of any order. ∂G For example, the derivatives of a second-order tensor Gij are written as ∂xkij = Gij ,k 2

∂ G

, ∂xk ∂xij l = Gij ,kl and so forth. Note that the summation convention in Gij,j means the following: Gij ,j = Gi1 ,1 þ Gi2 ,2 þ Gi3 ,3 =

∂Gi1 ∂Gi2 ∂Gi3 þ þ : ∂x1 ∂x2 ∂x3

∂ The differential operator ∇ = e1 ∂x∂1 þ e2 ∂x∂2 þ e3 ∂x∂3 = ei ∂x , called the “del” operai

∂ϕ is a tor, is commonly encountered in field analysis. For a scalar field ϕ, ∇ϕ = ei ∂x i ∂v

vector field of the gradient of ϕ. For a vector field v(x1, x2, x3), ∇v = ∂xji ei  ej is a i second order tensor called the gradients tensor of v, ∇  v = ∂v = vi ,i is a scalar ∂xi e1 e2 e3 ∂ ∂ ∂ j is a ∂vk called the divergence of v(x1, x2, x3), and ∇ × v = εijk ∂xj ei = j ∂x1 ∂x2 ∂x3 v1 v2 v3 vector called the curl of v. Gauss’s divergence theorem (e.g., Korn and Korn 1968; p. 5.61) states that the volume integration of the divergence of v in a closed region is equal to the flux of v throughout the boundary surface of the region. Mathematically, the theorem is the following expression: Z

Z ∇  v dV = Ω

Z v  dS

Z vi ,i dV =

or Ω

∂Ω

vi ni dS

ð1:35Þ

∂Ω

where Ω is the volume and ∂Ω its boundary surface. The general Gauss’s theorem states that if A and its first derivatives A,i are continuous and single-valued in a given volume Ω whose boundary surface is S and outward normal is ni, then: Z

Z A,i dV =

Ω

Ani dS

ð1:36Þ

S

The Stokes’ curl theorem states that the total flux of the curl of v over surface S is equal to the circulation of v along the closed boundary line of S. Mathematically, it is the following relation:

20

1

Z

I ð∇ × vÞ  dS =

S

Background, Mathematic Preliminaries and Notations

Z v  dx

C

I εjik vi,j nk dS =

or

vi dxi

ð1:37Þ

C

S

where nk is the unit vector normal to S and C the boundary of S. The general Stokes’ theorem is the following relation, replacing vi in the above Equation by a general tensor field A: Z

I εjik A,j nk dS =

1.4

Adxi

ð1:38Þ

C

S

Fourth-Order Tensors

In micromechanics, fourth-order tensors are commonly used in mathematical equations. We summarize the basic relations and operation rules below that are used later in this book. The following fourth-order identities are introduced for the convenience of presentation (Jiang 2014, 2016): J ijkl = δik δjl 1 Jm ijkl = 3 δij δkl  1  1 J sijkl = J ijkl þ J jikl = δik δjl þ δjk δil 2 2 J dijkl = J sijkl - J m ijkl  1 a J ijkl = δik δjl - δjk δil 2

ð1:39Þ

Where m 1 Jm ijkl is called the fourth-order mean identity, as J ijkl akl = 3 ða11 þ a22 þ a33 Þδij for any given second-order tensor. J sijkl and J aijkl are the fourth-order symmetric and anti-symmetric identities respec    tively, as J sijkl akl = 12 aij þ aji and J aijkl akl = 12 aij - aji . J dijkl is the fourth-order deviatoric identity, as J dijkl akl = aij - 13 ða11 þ a22 þ a33 Þδij. A fourth-order isotropic tensor can be written as (Hashin 1988): A = 3αJm þ 2βJd

ð1:40Þ

Therefore, the elastic stiffness tensor for an isotropic body can be expressed as a fourth-order tensor:

1.4

Fourth-Order Tensors

21

C = 3KJm þ 2μJd

ð1:41Þ

where K is the bulk modulus and μ the shear modulus. The inverse of A, in the sense of A-1 : A = A : A-1 = Js is: A-1 =

1 m 1 J þ Jd 3α 2β

ð1:42Þ

The elastic compliance tensor for an isotropic elastic body is thus: M = C-1 =

1 m 1 J þ Jd 3K 2μ

ð1:43Þ

If B = 3aJm + 2bJd, then AB = BA = 9aαJm + 4bβJd. In general, anisotropic materials, the elastic stiffness tensor is not isotropic and finding its inverse is much more complicated. However, if a fourth-order tensor has the following symmetry: Aijkl = Ajikl = Aijlk

ð1:44Þ

and a unique fourth-order tensor B exists, which also has the above symmetry and A : B = B : A = Js

ð1:45Þ

then B is the inverse of A and is denoted by B = A-1. If A-1 exists, then A is said to be non-singular. The most efficient approach to finding the inverse of a fourth-order tensor having the symmetry of Eq. (1.44) is presented by Lebensohn et al. (1998), Tomé (1998), and Dinçkal (2012). The approach introduces a set of 6 second-order orthonormal base tensors bλ (λ = 1, 2, . . .6), as 0 1 -1 0 0   1 1 B C b1 = pffiffiffi 3δ3i δ3j - δij = pffiffiffi @ 0 -1 0A 6 6 2 0 0 1 0 0 0 -1  1  1 B C b2 = pffiffiffi δij - δ3i δ3j - 2δ1i δ1j = pffiffiffi @ 0 -1 0A 2 2 0 0 0 0 1 0 0 0  1  1 B C 3 b = pffiffiffi δ2i δ3j þ δ3i δ2j = pffiffiffi @ 0 0 1 A 2 2 0 1 0

22

1

Background, Mathematic Preliminaries and Notations

0

0 0  1 B 1  4 b = pffiffiffi δ3i δ1j þ δ1i δ3j = pffiffiffi @ 0 0 2 2 1 0 0 0 1  1  1 B 5 b = pffiffiffi δ1i δ2j þ δ2i δ1j = pffiffiffi @ 1 0 2 2 0 0 1 0 1 0 0 1 1 B C b6 = pffiffiffi δij = pffiffiffi @ 0 1 0 A 3 3 0 0 1

1 1 C 0A 0 0

1

C 0A 0

This set of base tensors is orthonormal in that bλij bηij = = δλη . A fourth-order tensor A having the symmetry of Eq. (1.44) relates two secondorder symmetric tensors α and β (αij = αji, βij = βji) by: α = A : β or αij = Aijkl βkl

ð1:46Þ

The inverse relationship of Eqs. (1.46), if exists (A being non-singular), is: -1 αkl β = A - 1 : α or βij = Aijkl

ð1:47Þ

The procedure to obtain A-1 is as follows using the orthonormal bλ(λ = 1, 2, . . .6). First, the second-order symmetric tensors α and β are mapped to 6-dimensional vectors by: b βλ = βij bλij ðλ = 1, 2, . . . 6Þ αλ = αij bλij ,b

ð1:48Þ

b are as follows, expressed in a row matrix: The explicit components of α 

2α33 - α11 - α22 pffiffiffi 6

α22 - α11 pffiffiffi 2

pffiffiffi 2α23

pffiffiffi 2α13

pffiffiffi 2α12

α11 þ α22 þ α33 pffiffiffi 3



The explicit components of b β are of similar forms. The 6-dimensional vector components and the second-order tensor components are related by: αλ bλij , βij = b βλ bλij αij = b The relationship of Eq. (1.46) is converted to:

ð1:49Þ

1.4

Fourth-Order Tensors

23

b βη ðλ, η = 1, 2, . . . 6Þ αλ = Aλη b

ð1:50Þ

where Aλη is a 6 × 6 matrix representation of the fourth-order tensor A. The mutual transformations between Aλη and Aijkl are as follows: (

Aλη = bλij bηkl Aijkl Aijkl =

ð aÞ

= Aλη bλij bηkl

ð bÞ

ðλ, η = 1, 2, . . . 6Þ

ð1:51Þ

-1 -1 b The relation of Eq. (1.47) can also be expressed in the form b βλ = Aλη αη . Now Aλη can be obtained from Aλη readily by the standard matrix inversion of Aλη. We -1 back to complete the fourth-order tensor inversion by converting the matrix Aλη the fourth-order tensor components using Eq. (1.51) to get: -1 -1 = bλij bηkl Aλη Aijkl

ð1:52Þ

For incompressible materials such as viscous materials, the viscous stiffness tensor relates the deviatoric stresses (5 independent components) to the incompressible strain rates (also 5 independent components). Therefore, the viscous stiffness tensor is singular in the full 6-dimensional Cauchy space. In the deviatoric subspace, a fourth-order tensor relates two second-order tensors in the following form: α0=A0 : β0

ð1:53Þ

where α' and β' are deviatoric (α'kk = 0 and β'kk = 0). The inverse relation is: β 0 = A 0-1 : α 0

ð1:54Þ

A 0 : A 0 - 1 = A 0 - 1 : A 0 = Jd

ð1:55Þ

such that

To find A'-1, we can use the same approach as finding A-1 above, except that the β ′6 = range for λ and η is from 1 to 5. This is because, being deviatoric, b α ′6 =b = βp′ffiffi3kk  0 . Therefore, we only need five orthonormal second-order base tensors bλ(λ = 1, 2, ⋯5) to map deviatoric α' and β' into 5-dimensional vectors. Accordingly, the stiffness tensor A' is converted to a 5 × 5 matrix which can be inverted and then converted back to the fourth-order tensor. αp′ffiffikk 3

24

1.5

1

Background, Mathematic Preliminaries and Notations

Notes and Key References

The algorithm for common tensor operations including fourth-order tensor inversion is presented in the Mathcad worksheet named “CommonTensorOperations.mcdx”. Shankar (1995) is an excellent textbook to refresh one’s mathematics. Spencer (1980) is a succinct textbook on continuum mechanics. Chapters 2 and 3 of Spencer (1980) concisely review matrix algebra, vectors, and cartesian tensors. Jiang and Bentley (2012) and Jiang (2014) discuss the significance of a multiscale approach for natural deformation in more detail.

References Başar Y, Weichert D (2000) Nonlinear continuum mechanics of solids: fundamental mathematical and physical concepts. Springer Science & Business Media Bentley C (2004) Rock Fabric Analysis of the Sierra Crest Shear Zone System, California: implications for Crustal-Scale Transpressional Shear Zones. MSc Thesis, University of Maryland Bilby BA, Eshelby JD, Kundu AK (1975) Change of shape of a viscous ellipsoidal region embedded in a slowly deforming matrix having a different viscosity. Tectonophysics 28(4): 265–274. https://doi.org/10.1016/0040-1951(75)90041-4 Bilby BA, Kolbuszewski ML (1977) Finite deformation of an inhomogeneity in 2-dimensional slow viscous incompressible-flow. Proc R Soc Lond Ser A-Math Phys Eng Sci 355(1682): 335–353. https://doi.org/10.1098/rspa.1977.0101 Carreras J, Cosgrove JW, Druguet E (2013) Strain partitioning in banded and/or anisotropic rocks: implications for inferring tectonic regimes. J Struct Geol 50:7–21. https://doi.org/10.1016/j.jsg. 2012.12.003 Castelnau O, Cordier P, Lebensohn RA, Merkel S, Raterron P (2010) Microstructures and rheology of the Earth’s upper mantle inferred from a multiscale approach. Comptes Rendus Physique 11(3–4):304–315. https://doi.org/10.1016/j.crhy.2010.07.011 Dabrowski M, Schmid DW (2011) A rigid circular inclusion in an anisotropic host subject to simple shear. J Struct Geol 33(7):1169–1177. https://doi.org/10.1016/j.jsg.2011.05.003 Dinçkal Ç (2012) On the properties of anisotropic engineering materials based upon orthonormal representations. IAENG Int J Appl Math 42(3) Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond Ser A-Math Phys Sci 241(1226):376–396. https://doi.org/10.1098/ rspa.1957.0133 Eshelby JD (1959) The elastic field outside an ellipsoidal inclusion. Proc R Soc Lond Ser A-Math Phys Sci 252(1271):561–569. https://doi.org/10.1098/rspa.1959.0173 Eshelby JD (1961) Elastic inclusions and inhomogeneities. Progr Solid Mech 2:89–104 Goodwin LB, Tikoff B (2002) Competency contrast, kinematics, and the development of foliations and lineations in the crust. J Struct Geol 24(6–7):1065–1085. https://doi.org/10.1016/S01918141(01)00092-X Griera A, Bons PD, Jessell MW, Lebensohn RA, Evans L, Gomez-Rivas E (2011) Strain localization and porphyroclast rotation. Geology 39(3):275–278. https://doi.org/10.1130/G31549.1 Griera A, Llorens MG, Gomez-Rivas E, Bons PD, Jessell MW, Evans LA, Lebensohn RA (2013) Numerical modelling of porphyroclast and porphyroblast rotation in anisotropic rocks. Tectonophysics 587:4–29. https://doi.org/10.1016/j.tecto.2012.10.008

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Hashin Z (1988) The differential scheme and its application to cracked materials. J Mech Phys Solids 36(6):719–734 Hobbs BE, Means WD, Williams PF (1976) An outline of structural geology. Wiley Hudleston PJ (1999) Strain compatibility and shear zones: is there a problem? J Struct Geol 21(8–9):923–932. https://doi.org/10.1016/S0191-8141(99)00060-7 Ishii K (1992) Partitioning of noncoaxiality in deforming layered rock masses. Tectonophysics 210(1–2):33–43. https://doi.org/10.1016/0040-1951(92)90126-Q Jiang D (1994a) Vorticity determination, distribution, partitioning and the heterogeneity and nonsteadiness of natural deformations. J Struct Geol 16(1):121–130. https://doi.org/10.1016/ 0191-8141(94)90023-X Jiang D (1994b) Flow variation in layered rocks subjected to bulk flow of various kinematics vorticities – theory and geological implications. J Struct Geol 16(8):1159–1172. https://doi.org/ 10.1016/0191-8141(94)90059-0 Jiang D (2007a) Numerical modeling of the motion of rigid ellipsoidal objects in slow viscous flows: a new approach. J Struct Geol 29(2):189–200. https://doi.org/10.1016/j.jsg.2006.09.010 Jiang D (2007b) Numerical modeling of the motion of deformable ellipsoidal objects in slow viscous flows. J Struct Geol 29(3):435–452. https://doi.org/10.1016/j.jsg.2006.09.009 Jiang D (2012) A general approach for modeling the motion of rigid and deformable ellipsoids in ductile flows. Comput Geosci 38(1):52–61. https://doi.org/10.1016/j.cageo.2011.05.002 Jiang D (2013) The motion of deformable ellipsoids in power-law viscous materials: formulation and numerical implementation of a micromechanical approach applicable to flow partitioning and heterogeneous deformation in Earth’s lithosphere. J Struct Geol 50:22–34. https://doi.org/ 10.1016/j.jsg.2012.06.011 Jiang D (2014) Structural geology meets micromechanics: a self-consistent model for the multiscale deformation and fabric development in Earth’s ductile lithosphere. J Struct Geol 68:247–272. https://doi.org/10.1016/j.jsg.2014.05.020 Jiang D (2016) Viscous inclusions in anisotropic materials: theoretical development and perspective applications. Tectonophysics 693:116–142. https://doi.org/10.1016/j.tecto.2016.10.012 Jiang D, Bentley C (2012) A micromechanical approach for simulating multiscale fabrics in largescale high-strain zones: theory and application. J Geophys Res Solid Earth 117. https://doi.org/ 10.1029/2012JB009327 Jiang D, White JC (1995) Kinematics of rock flow and the interpretation of geological structures, with particular reference to shear zones. J Struct Geol 17(9):1249–1265. https://doi.org/10. 1016/0191-8141(95)00026-A Jiang D, Williams PF (1999) A fundamental problem with the kinematic interpretation of geological structures. J Struct Geol 21(8–9):933–937. https://doi.org/10.1016/S0191-8141(99)00068-1 Johnson AM, Fletcher RC (1994) Folding of viscous layers: mechanical analysis and interpretation of structures in deformed rock. Columbia University Press Jones RR, Holdsworth RE, Clegg P, McCaffrey K, Tavarnelli E (2004) Inclined transpression. J Struct Geol 26(8):1531–1548. https://doi.org/10.1016/j.jsg.2004.01.004 Jones RR, Holdsworth RE, McCaffrey KJW, Clegg P, Tavarnelli E (2005) Scale dependence, strain compatibility and heterogeneity of three-dimensional deformation during mountain building: a discussion. J Struct Geol 27(7):1190–1204. https://doi.org/10.1016/j.jsg.2005.04.001 Keller LM, Stipp M (2011) The single-slip hypothesis revisited: crystal-preferred orientations of sheared quartz aggregates with increasing strain in nature and numerical simulation. J Struct Geol 33(10):1491–1500. https://doi.org/10.1016/j.jsg.2011.07.008 Kilian R, Heilbronner R, Stunitz H (2011) Quartz microstructures and crystallographic preferred orientation: which shear sense do they indicate? J Struct Geol 33(10):1446–1466. https://doi. org/10.1016/j.jsg.2011.08.005 Korn GA, Korn TM (1968) Mathematical handbook for scientists and engineers. McGraw-Hill, New York, NY Lebensohn RA (2001) N-site modeling of a 3D viscoplastic polycrystal using Fast Fourier Transform. Acta Mater 49(14):2723–2737

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Lebensohn RA, Dawson PR, Kern HM, Wenk HR (2003) Heterogeneous deformation and texture development in halite polycrystals: comparison of different modeling approaches and experimental data. Tectonophysics 370(1–4):287–311. https://doi.org/10.1016/S0040-1951(03) 00192-6 Lebensohn RA, Ponte Castañeda P, Brenner R, Castelnau O (2011) Full-field vs. homogenization methods to predict Microstructure–Property relations for polycrystalline materials. In: Ghosh S, Dimiduk D (eds) Computational methods for microstructure-property relationships. Springer, Boston, MA, pp 393–441. https://doi.org/10.1007/978-1-4419-0643-4_11 Lebensohn RA, Tomé CN (1993) A self-consistent anisotropic approach for the simulation of plastic-deformation and texture development of polycrystals – application to zirconium alloys. Acta Metall Mater 41(9):2611–2624. https://doi.org/10.1016/0956-7151(93)90130-K Lebensohn RA, Turner PA, Signorelli JW, Canova GR, Tomé CN (1998) Calculation of intergranular stresses based on a large-strain viscoplastic self-consistent polycrystal model. Model Simul Mater Sci Eng 6(4):447–465. https://doi.org/10.1088/0965-0393/6/4/011 Li S, Wang G (2008) Introduction to Micromechanics and Nanomechanics. World Scientific Lister GS, Williams PF (1979) Fabric development in shear zones – theoretical controls and observed phenomena. J Struct Geol 1(4):283–297. https://doi.org/10.1016/0191-8141(79) 90003-8 Lister GS, Williams PF (1983) The partitioning of deformation in flowing rock masses. Tectonophysics 92(1-3):1–33. https://doi.org/10.1016/0040-1951(83)90083-5 Masson R, Bornert M, Suquet P, Zaoui A (2000) An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals. J Mech Phys Solids 48(6-7): 1203–1227. https://doi.org/10.1016/S0022-5096(99)00071-X Molinari A, Canova GR, Ahzi S (1987) A self-consistent approach of the large deformation polycrystal viscoplasticity. Acta Metall 35(12):2983–2994. https://doi.org/10.1016/0001-6160 (87)90297-5 Montagnat M, Castelnau O, Bons P, Faria S, Gagliardini O et al (2014) Multiscale modeling of ice deformation behavior. J Struct Geol 61:78–108. https://doi.org/10.1016/j.jsg.2013.05.002 Mura T (1987) Micromechanics of defects in solids. Martinus Nijhoff Nemat-Nasser S, Hori M (1999) Micromechanics: Overall Properties of Heterogeneous Materials. Elsevier Science Passchier CW, Mancktelow NS, Grasemann B (2005) Flow perturbations: a tool to study and characterize heterogeneous deformation. J Struct Geol 27(6):1011–1026. https://doi.org/10. 1016/j.jsg.2005.01.016 Passchier CW, Trouw RA (1995) Microtectonics. Springer Pollard DD, Fletcher RC (2005) Fundamentals of structural geology. Cambridge University Press Pollard DD, Martel SJ (2020) structural geology: a quantitative introduction. Cambridge University Press Ponte Castañeda P (1996) Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J Mech Phys Solids 44(6):827–862. https://doi.org/10.1016/ 0022-5096(96)00015-4 Qu J, Cherkaoui M (2006) Fundamentals of micromechanics of solids. Wiley Ramsay JG (1967) Folding and fracturing of rocks. McGraw Hill Ramsay JG, Huber MI (1987) The techniques of modern structural geology, vol 2. Folds and fractures, Academic Press Shankar R (1995) Basic training in mathematics: a fitness program for science students. Springer Science & Business Media Spencer AJM (1980) Continuum mechanics. Longman Tommasi A, Mainprice D, Canova G, Chastel Y (2000) Viscoplastic self-consistent and equilibrium-based modeling of olivine lattice preferred orientations: implications for the upper mantle seismic anisotropy. J Geophys Res Solid Earth 105(B4):7893–7908. https://doi. org/10.1029/1999JB900411

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Tomé CN (1998) Tensor properties of textured polycrystals. In: Kocks UF, Tomé CN, Wenk HR, Beaudoin AJ, Mecking H (eds) 1998. Cambridge University Press, Texture and anisotropy: preferred orientations in polycrystals and their effect on materials properties Truesdell C (1991) A first course in rational continuum mechanics. 1. General concepts. Academic Press Twiss RJ, Moores EM (1992) Structural geology. W.H. Freeman Wenk HR, Armann M, Burlini L, Kunze K, Bortolotti M (2009) Large strain shearing of halite: experimental and theoretical evidence for dynamic texture changes. Earth Planet Sci Lett 280(1–4):205–210. https://doi.org/10.1016/j.epsl.2009.01.036 Wenk HR, Canova G, Molinari A, Kocks UF (1989) Viscoplastic modeling of texture development in quartzite. J Geophys Res Solid Earth Planets 94(B12):17895–17906. https://doi.org/10.1029/ JB094iB12p17895

Chapter 2

Orientation of Fabric Elements

Abstract This Chapter uses a unit vector to mathematically present the orientation of a linear or planar fabric element (such as a fold hingeline, a lineation, or the pole to a foliation) and a set of three mutually orthogonal unit vectors to present the orientation of a 3D fabric element such as the orientation of a porphyroclastic feldspar in a mylonite sample. A unit vector has two degrees of freedom (plunge direction and plunge angle) and we use two spherical angles to characterize its orientation. A 3D orientation requires three angles to define. We use the orientation matrix Q to define a 3D orientation. In a mylonite for instance, each feldspar grain corresponds to a Q and the distribution of Q of all feldspar grains making up the sample defines the preferred orientation of feldspars in that sample. During progressive deformation, Q is a function of time. We mathematically characterize the orientation and rotation of line elements and 3D fabric elements. We also explore how to generate linear or 3D orientation sets to follow a certain given distribution in 3D space in this Chapter.

2.1

Orientation of a Line

When dealing with the orientation of a line in space, it is most convenient to represent it by a unit vector (Fisher et al. 1987, Fig. 2.1). In a Cartesian coordinate system x1x2x3 with the three base unit vectors being respectively e1 e2, and e3 (Fig. 2.1), a unit vector u = u1e1 + u2e2 + u3e3 completely defines a line’s direction. The Cartesian components can be further expressed in terms of two spherical angles θ and ϕ, with θ being the angle between the projection of u on the x1x2-plane and the positive x1-axis and ϕ is the angle between u and the positive x3-axis. The angle θ is very much like the trend of a line in geology if x1x2x3 is the geographic system (Fig. 2.1b). The angle ϕ is related to, but not the same as, the plunge angle in

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-23313-5_2. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Jiang, Continuum Micromechanics, Springer Geophysics, https://doi.org/10.1007/978-3-031-23313-5_2

29

30

2 Orientation of Fabric Elements

geology. In the geographic system where x3 points vertically down, ϕ is the complementary angle of the plunge if u3 ≥ 0 (ϕ≤π2). Otherwise, when u3 < 0 (ϕ>π2 ), we can call ϕ - π2 the ‘elevation’ angle. The range for θ is between 0 and 2π(we use radians generally in this book), and that for ϕ between 0 and π. The reason for using ϕ, instead of the plunge angle, is that ϕvaries monotonically to cover all possible directions of a unit vector, whether it points to the lower or upper hemisphere. In most circumstances in geology, only the orientation of a fabric element matters, not its direction. One can use either u or -u to represent the orientation in such cases. For instance, a fold hingeline or the pole to a foliation can always be represented by a downward pointing vector. Only one half-space (usually the lower hemisphere) is sufficient to cover all possible orientations. Where the direction of a line is important (e.g., the direction of glacier striation, the younging direction of a strata etc.), u and -u represent two opposite directions that must be distinguished. We must use half spaces (upper and lower hemispheres) to represent such data. The Cartesian components of u in terms of θ and ϕ are expressed in a column matrix below: 0

1 0 1 cos θ sin ϕ u1 B C B C u = @ u2 A = @ sin θ sin ϕ A u3

ð2:1Þ

cos ϕ

The following inverse relations allow for θ and ϕ to be calculated where the unit vector components are known: 8 π signðu2 Þ , u1 = 0 > > 2 > > > >   > < u arctan 2 , u1 > 0 θðuÞ = u1 > > >   > > > > : π þ arctan u2 , u1 < 0 u1 ϕðuÞ = arccos ðu3 Þ

ð2:2Þ

The orientation of a line is commonly plotted on a stereographic (also called an equal-angle) projection or an equal-area projection. Both projections are like a 2D polar coordinate system. To plot a unit vector u, one needs its polar angle θ with respect to a reference direction (north in the geographic coordinate system, for instance) and the radial length r from the center origin, which is determined by the angle ϕ (Fig. 2.1). The angle θ is already known and is the same for both equal-angle and equal-area projections. The radial lengths are different depending on the projection used. For the stereographic (equal-angle) projection, the polar coordinate rs can be calculated according to the following relation (Hobbs et al. 1976, p. 487):

2.1

Orientation of a Line

31

Fig. 2.1 (a) In a Cartesian coordinate system, the direction of a line is represented by a unit vector u, which in turn is defined by two spherical angles θ and ϕ. (b) In the geographic coordinate system, θ is the trend of u and ϕ is the angle the vector makes with the plumb line. Where ϕ ≤ π2 , the complementary angle π2 - ϕ is the familiar plunge angle. Where ϕ > π2 , the line points upward (having a negative plunge angle), and the angle ϕ - π2 can be called the ‘elevation angle’. (c) and (d) stereographic or equal-area projection of u associated with (a) and (b), respectively. For stereographic projections, the radial coordinate rs=op is calculated using Eq. (2.3). For equal-area projection, the radial coordinate rEA is calculated using Eq. (2.4). Both hemispheres are needed to cover the whole range 0 ≤ ϕ ≤ π. We usually use different symbols to distinguish plots in the X3-positive and X3-negative hemispheres

8   > ϕ > > < tan 2 ,   rs = > ϕ > > : cot 2 ,

π 2

in x3 ‐positive hemisphere

π ϕ> 2

in x3 ‐negative hemisphere

ϕ≤

, equal angle

ð2:3Þ

For the equal-area projection, the polar coordinate rEA is calculated by (Hobbs et al. 1976, p. 496):

32

2

8   pffiffiffi > ϕ > > 2 sin , < 2   r EA = pffiffiffi > ϕ > > : 2 cos 2 ,

ϕ≤

π 2

Orientation of Fabric Elements

in x3 ‐positive hemisphere

π ϕ> in x3 ‐negative hemisphere 2

, equal area

ð2:4Þ

If ϕ varies between 0 and π, both hemispheres are needed for plotting. Plots in the x3positive hemisphere can be distinguished from plots in the x3-negative hemisphere using different symbols. If only orientations (not directions) matter, one can always choose one hemisphere, such as the x3-positive hemisphere, to present the data. In such cases, simply replace u by -u if u points to the x3-negative hemisphere (i.e., u3 1 must be replaced by 2 - ηi. The required set {ϕ} is then obtained by taking the inverse cosine of {η}, i.e., ϕi = cos-1ηi. With both sets {θ} and {ϕ} generated, the elements of set {u} are obtained accordingly using ui = 0

1 cos θi sin ϕi B C @ sin θi sin ϕi A . Figure 2.11a is the equal-area plot of 500 lines with the mean cos ϕi vertical direction and standard deviation of 0.05 generated by the above procedure. Applying a rigid rotation of Eq. (2.26) to every element in the above set then produces the desired set (Fig. 2.11b and c).

2.7.3

Line Set Forming a Small or Great Circle Girdle

The above procedure can be extended to the generation of a set of lines forming a small circle girdle or great circle girdle distributions. Again, we need only consider the special situation where the small/great circle axis is vertical first and then transform the set to any required orientation by using Eq. (2.26). Suppose the mean angular radius of the small circle is K from the cone axis, and there is no additional preferred orientation within the girdle. Where the cone axis is vertical, normal distribution of unit vectors around the mean small circle is defined by a population of unit vectors whose corresponding set {ϕ} satisfy the condition that the set {cosϕ} represents a normal distribution around the mean value cosK with a specified standard deviation σ measuring the degree of concentration. We thus proceed as follows: First, generate an auxiliary set {η} with Gaussian normal distribution with the required mean (cosK ) and standard deviation (σ). The set {ϕ} is obtained by taking the inverse cosine of every element of {η}, i.e., ϕi = cos-1ηi. As the small circle cone axis is vertical and there is no additional preferred orientation other than the girdle, {θ} is uniform in the range of 0 and 2π. With both sets {θ}

2.7

Generation of a Population of Lines Following a Specified Distribution

49

Fig. 2.11 A set of 500 lines having Gaussian point maximum distribution. (a) Point maximum center parallel to the vertical axis. (b) Point maximum center at θ = 60∘ and ϕ = 40∘. (c) Point maximum center at θ = 60∘ and ϕ = 80∘. The standard deviation is 0.05 = cos 18∘

and {ϕ} generated, the elements of set {u} are obtained accordingly using 0

1 cos θi sin ϕi B C ui = @ sin θi sin ϕi A. cos ϕi Figure 2.12a is an equal area plot, in the x3- positive hemisphere, of a set of 500 lines forming a small circle girdle distribution with K = 40∘ and standard deviation 0.05 = cos 18∘. The girdle axis is vertical. A rigid body rotation can move the girdle axis to any desired orientation (Figs. 2.12b and c). The above method also applies to great circle girdles (K = π2) (Fig. 2.12d).

50

2

Orientation of Fabric Elements

Fig. 2.12 A set of 500 lines forming a small circle girdle with standard deviation 0.05 = cos 18∘. (a) K = 40∘, small circle girdle center vertical. (b) Same small circle girdle with center at θ = 60∘ and ϕ = 40∘. (c) Same small circle girdle with center at θ = 60∘ and ϕ = 80∘. (d) K = 90∘ and the distribution is along a great circle girdle with pole at θ = 60∘ and ϕ = 40∘

2.8

Generation of a Population of 3D Fabric Elements Following a Specified Distribution

As the orientation of a 3D object is defined by its Q matrix, a population of 3D objects with uniform distribution orientations in space means that the set {Q} is isotropic, having no preferred orientations. A simple preferred orientation may be that {Q} tends to concentrate around a mean Qm, for instance. It is important to note that the three axes that define a 3D object in space cannot be considered separately. Therefore, one cannot consider the distribution of a1, a2, and a3as separate lines when preferred 3D orientations are analyzed. How do we generate a set of {Q} that represents a given distribution? How do we generate a {Q} that has a mean of Qm and a standard deviation of Qσ? How to mathematically formulate the standard deviation of a Q distribution? These are subjects covered in the literature on directional statistics (Mardia and Jupp 2000; Banerjee et al. 2005; Sra 2012). We will only attempt to consider some simple situations of Q distribution here.

2.8

Generation of a Population of 3D Fabric Elements Following a. . .

2.8.1

51

A Set of 3D Orientations Forming a Uniform Distribution

A uniform distribution of Q is easy to visualize and is often the initial condition in most fabric modelling (Lister and Hobbs 1980; Lebensohn and Tomé 1993; Yang et al. 2019; Bhandari and Jiang 2021). For instance, the grain shape and lattice orientations in an undeformed granite are most likely uniform. It is relatively easy to generate a population of uniformly distributed Q using Euler angles (α,β, γ). Because the volume-like element dα  d cos β  dγ is invariant (Eq. (2.14)), one can generate a set of (α,β, γ) such that α and γ are both uniformly varying in the range of 0 to 2πand cosβ is uniformly varying in the range of -1 and 1. The associated {Q} is uniform in space. Alternatively, one can use the Euler angles (ξ, φ, γ). Generate a set of them such that ξ, cosφ, and γ all vary uniformly within their respective ranges of 0 to 2π, -1 to 1, and -π to π. Using the spherical angles (θ, ϕ, ϑ) to generate a uniform distribution {Q} is trickier as the invariant measure is too complicated. Jiang (2007a) used the following procedure. Because the associated a2 axes have no preferred orientations on the a2a3 plane, the ‘pitch’ angle ψ of a2 must vary uniformly between -π and π (Fig. 2.13). As ϑis related to θ, ϕ, and ψ by the following relation, ϑ can be obtained accordingly: ϑ=

π þ θ þ tan - 1 ð tan ψ cos φÞ 2

ð2:27Þ

Therefore, the procedure is as follows. First, generate a population of uniformly distributed lines for the a1 which form the (θ, ϕ) set. Also, generate a set of uniform numbers between -π and π for {ψ}. Using Eq. (2.27) to get the associated elements for {ϑ} and combining the set of {θ, ϕ} with {ϑ}forms the required population {θ, ϕ, ϑ}. Fig. 2.13 The relationship between ϑ, θ, and ϕ through ψ, which is the ‘pitch’ of a2on the a2a3 plane

52

2.8.2

2

Orientation of Fabric Elements

3D Objects Having Preferred Orientations

Let us adopt the Euler angle set (ξ, φ, γ) to express the Q matrix. Recall that the spherical angles (ξ, φ) represent direction of a3 axis. The corresponding Q matrix is given in Eq. (2.11) reproduced below: Qðξ, φ, γ Þ 0 1 - sin ξ cos γ - cos ξ cos φ sin γ cos ξ cos γ - sin ξ cos φ sin γ sin φ sin γ B C C =B @ sin ξ sin γ - cos ξ cos φ cos γ - sin ξ cos φ cos γ - cos ξ sin γ sin φ cos γ A cos ξ sin φ sin ξ sin φ cos φ A simple Gaussian point maximum of a3 around x3- axis can be generated by requiring uniform ξ between 0 and 2π, uniform γ between -π and π, and a normal distribution of cosφ with the mean of one and standard deviation σ. Such a point maxima of a3 and uniform a1 (and a2) (Fig. 2.14a, b, c) can be rotated to any required initial orientation of a3 maximum by applying the rotation of Eq. (2.16 and 2.26) (Fig. 2.14d, e, f).

Fig. 2.14 Equal-area projection of 500 3D orientations with uniform a1 axes (a, d) and a2 axes (b, e). The a3 axes (c, f) form a Gaussian maximum. (a), (b), and (c) are for the a3 maximum parallel to the vertical axis. (d), (e), and (f) are when the a3 maximum is at ξm = 20∘ and φm = 50∘. The standard deviation is 0.05(18∘)

2.8 Generation of a Population of 3D Fabric Elements Following a. . .

53

Fig. 2.15 Equal-area projection of 500 3D orientations with the a3 axes (c and f) forming a small circle girdle with κ = 40∘ and a1 axes (a and d) and a2 axes (b and e) uniformly distributed on the broad girdle normal to a3. The upper row plots, (a), (b), and (c), are for the situation a3 cone axis is parallel to the vertical axis. The bottom row plots, (d), (e), and (f), are when the cone axis is at ξm = 20∘ and φm = 70∘. The standard deviation for the concentration along the small circle is 0.05 (18∘)

A similar approach can be used for the a3 axes to have a small (or great) circle girdle distribution. In such a case, one only needs to give a cone angle (κ) for the small circle. Figure 2.15 is for κ = 40∘. The distribution of a1 axes (and therefore a2 axes) is uniform with respect to θ in (Figs. 2.14a and b and 2.15a and b). To consider an additional Gaussian concentration for a1 axes (and therefore a2 axes), we can replace the uniform distribution of ξ by a Gaussian normal distribution. To do so, we first generate a set {ξn} with the required normal distribution. We then rotate every orientation in the initial {Qu}, where subscript u stands for uniform distribution in ξ, by the following rotation around x3 axis:   Ri = R3 ξin - ξiu Qin = Qiu RiT

ð2:28Þ

where ξiu is the ith element in the initial uniform distribution and ξin is the corresponding element in the normal distribution set {ξn}. The above operation converts all elements in the initial {Qu}, which represent no preferred orientation

54

2

Orientation of Fabric Elements

Fig. 2.16 (a), (b), and (c) are the same as (a), (b), and (c) of Fig. 2.14 except that ξhas normal distribution around mean 0 with standard deviation of 15∘. Both a1 and a2 axes form point maxima now. The bottom row (d), (e), (f) are corresponding plots of (a), (b), and (c) when the a3 maximum axis is rotated to ξm = 25∘ and φm = 75∘.

of a1 (and therefore a2) axes in the broad girdle normal the a3 cone axis (Figs. 2.15a, b, d, and e), to corresponding elements in {Qn} which have the required normal distribution in ξ. Because the operation is a rigid rotation, all three axes of every fabric element are affected by the same rotation, and their mutual orthogonality is maintained. Figure 2.16 is produced by adding the normal distribution in ξ with mean 0 and standard deviation of 15∘ for the a1 axes (and a2 axes) to the dataset of Fig. 2.14. Similarly, a Gaussian normal distribution in ξ can also be applied to Fig. 2.15, where the a3 follow a small circle girdle distribution. The resulting plots are in Fig. 2.17.

2.9

Misorientation Between Two Objects

The orientation of a 3D object is defined by Q. The difference in orientation ΔQ (misorientation) between two objects, Q1 and Q2 is measured by the rotation that the moves Q1 to Q2 and ΔQ = Q1TQ2. The magnitude of misorientation between  two Q’s is measured by a rotation of angle ω = cos - 1 12 ½traceðΔQÞ - 1 around an axis that can be obtained using Eq. (2.25). Note that R = QT.

2.10

Notes and Key References

55

Fig. 2.17 The upper row (a, b, and c) is the same as (a), (b), and (c) of Fig. 2.15 except that θ has normal distribution around mean 0 with standard deviation of 15∘. Both a1 and a2 axes have a concentration in a broad small circle. The bottom row (d, e, f) is when the a3 maximum is at θm = 25∘ and ϕm = 75∘

Note Q2 - Q1 is not a rotation because it is not a proper orthogonal matrix (that is, (Q2 - Q1)(Q2 - Q1)T ≠ I). Multiplication among Q’s produces proper orthogonal matrices, but addition or subtraction does not.

2.10

Notes and Key References

The Mathcad worksheet named “3D Fabric.mcdx” presents the algorithm for the generation and plot of linear and 3D orientations. Chapter 1 of Ramsay (1967) presents a statistical analysis of lines and planes (as poles) using stereonet and unit vectors. Woodcock (1977), Fisher et al. (1987), and Vollmer (1990, 1995) present statistical methods and computer programs to analyze the orientation of liner data. Davis and Titus (2017) give a review of 3D orientation data which is complementary to the analysis of this Chapter. More advanced mathematical treatment of orientation distribution in 3D space is presented by Bunge (1982) and Mardia and Jupp (2000).

56

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Orientation of Fabric Elements

References Banerjee A, Dhillon IS, Ghosh J, Sra S (2005) Clustering on the unit hypersphere using von MisesFisher distributions. J Mach Learn Res 6:1345–1382 Başar Y, Weichert D (2000) Nonlinear continuum mechanics of solids: fundamental mathematical and physical concepts. Springer Science & Business Media Bhandari A, Jiang D (2021) A multiscale numerical modeling investigation on the significance of flow partitioning for the development of quartz c-axis fabrics. J Geophys Res Solid Earth 126(2). https://doi.org/10.1029/2020JB021040 Bunge HJ (1982) Texture analysis in materials science: mathematical methods (English edition, trans: Morris PA). Butterworths Davis JR, Titus SJ (2017) Modern methods of analysis for three-dimensional orientational data. J Struct Geol 96:65–89. https://doi.org/10.1016/j.jsg.2017.01.002 Fisher NI, Lewis T, Embleton BJJ (1987) Statistical analysis of spherical data. Cambridge University Press Freeman B (1985) The motion of rigid ellipsoidal particles in slow flows. Tectonophysics 113(1–2): 163–183. https://doi.org/10.1016/0040-1951(85)90115-5 Goldstein H (1980) Classical mechanics, 2nd edn. Addison-Wesley Hauser W (1965) Introduction to the principles of mechanics. Addison-Wesley Hobbs BE, Means WD, Williams PF (1976) An outline of structural geology. Wiley Ježek J (1994) Software for modeling the motion of rigid triaxial ellipsoidal particles in viscous flow. Comput Geosci 20(3):409–424 Jiang D (2007a) Numerical modeling of the motion of rigid ellipsoidal objects in slow viscous flows: a new approach. J Struct Geol 29(2):189–200. https://doi.org/10.1016/j.jsg.2006.09.010 Jiang D (2007b) Numerical modeling of the motion of deformable ellipsoidal objects in slow viscous flows. J Struct Geol 29(3):435–452. https://doi.org/10.1016/j.jsg.2006.09.009 Korn GA, Korn TM (1968) Mathematical handbook for scientists and engineers. McGraw-Hill, New York, NY Lebensohn RA, Tomé CN (1993) A self-consistent anisotropic approach for the simulation of plastic-deformation and texture development of polycrystals – application to zirconium alloys. Acta Metall Mater 41(9):2611–2624. https://doi.org/10.1016/0956-7151(93)90130-K Lister GS, Hobbs BE (1980) The simulation of fabric development during plastic-deformation and its application to quartzite – the influence of deformation history. J Struct Geol 2(3):355–370. https://doi.org/10.1016/0191-8141(80)90023-1 Mardia KV, Jupp PE (2000) Directional statistics. Wiley, London Ramsay JG (1967) Folding and fracturing of rocks. McGraw Hill Spencer AJM (1980) Continuum mechanics. Longman Sra S (2012) A short note on parameter approximation for von Mises-Fisher distributions: and a fast implementation of I-s(x). Comput Stat 27(1):177–190. https://doi.org/10.1007/s00180-0110232-x Vollmer FW (1990) An application of eigenvalue methods to structural domain analysis. Geol Soc Am Bull 102(6):786–791 Vollmer FW (1995) C program for automatic contouring of spherical orientation data using a modified Kamb method. Comput Geosci 21(1):31–49 Woodcock NH (1977) Specification of fabric shapes using an eigenvalue method. Geol Soc Am Bull 88(9):1231–1236. https://doi.org/10.1130/0016-7606(1977)882.0. CO;2 Yang R, Jiang D, Lu LX (2019) Constrictional strain and linear fabrics as a result of deformation partitioning: a multiscale modeling investigation and tectonic significance. Tectonics 38(8): 2829–2849. https://doi.org/10.1029/2019TC005490

Chapter 3

Stress, Strain, and Elasticity

Abstract This Chapter reviews the concepts of stress and strain in the context of linear elasticity first. Although linear elasticity is concerned with the infinitesimal strain tensor, more general strain tensors are introduced from displacement field gradients in preparation for finite deformation study. Anisotropic elasticity is discussed with elastic stiffness and elastic compliance expressed both in the standard Vogit matrix notation and in terms of fourth-order tensors. The method introduced in chapter one to inverse fourth-order symmetric tensors is used to calculate the elastic compliance tensor from the stiffness tensor or vice versa. The significance of multiscale stress and strain in heterogeneous materials is emphasized. Following a brief presentation on the relationship between stresses (and strains) of different scales, the concept of effective rheology of a heterogeneous material on a representative volume element is introduced. Simplistic homogenization through the Vogit and Ruess averages is introduced. The Green function approach to solve boundaryvalue elastic problems is presented. Elastic Green function and its first derivative for an infinitely extending anisotropic elastic material are derived in the appendix. These expressions are required in elastic micromechanics and the approach will be further developed for viscous micromechanics in later Chapters.

3.1

Stress

When a volume of a solid body is free from any load at its boundaries and has no strain anywhere internally, it may be in a stress-free state. In a stress-free state, the only actions among the constituent material particles are the “bonding forces” that hold the particles together to form the solid continuum. In this state, material particles have equilibrated interatomic spacing. The geometrical assemblage of all particle positions corresponds to the stress-free configuration. A perfect stress-free

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-23313-5_3. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Jiang, Continuum Micromechanics, Springer Geophysics, https://doi.org/10.1007/978-3-031-23313-5_3

57

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3 Stress, Strain, and Elasticity

state is an idealization. Atomic scale defects (vacancies, impurities, dislocations, etc.) are inevitable even in “perfect” crystals, which will cause local disturbances (e.g., lattice strain) in the configuration from a truly stress-free state. On the macroscale, atomic scale interactions are cancelled out. For all practical purposes, one can regard a solid body not undergoing deformation and without external load as the reference stress-free state. Now imagine that an external load acts on the body, and the body undergoes a configuration change (the spatial assemblage among constituent particles changes). We call such configuration change deformation generally. In the deformed configuration, the constituent particles are forced to move closer or farther apart. This induces additional actions beyond the reference stress-free state, which generates stress in the material. Figure 3.1 illustrates the origin of stress in a condensed material schematically. The potential energy (U ) from the interaction between interatomic particles is a function of inter-particle spacing (r) (Fig. 3.1a). The characteristic equilibrium interparticle spacing (r0) corresponds to the minimum U at temperature 0∘K. This minimum U measures the bond strength. At temperatures above 0∘K, the interparticle spacing fluctuates around r0. Figure 3.1b shows the corresponding interparticle force (F ¼ ∂U ). The inter-particle force vanished (F ¼ 0) if r ¼ r0. F is ∂r attractive if r > r0 and repulsive if r < r0. In Figures 3.1c–f, material particles are shown as small orange balls, and the force between two nearby particles is represented by an elastic spring. In the equilibrium state (Fig. 3.1c), the springs are neither compressed nor extended. There is no net force and, therefore, no stress across the AA section. Under an external vertical tension, the spacing between horizontal layers is increased to r0 + δr, which induces attractive force across the horizontal section AA and, therefore, tensile stress on AA (Fig. 3.1d). Under vertical compression, on the other hand (Fig. 3.1e), the spacing between horizontal layers is reduced to r0  δr. This causes repulsive force across section AA, which results in compressive stress on AA. When the external load is a shear force parallel to the horizontal layers, some springs are extended, and others compressed. Angular distortion (shear angle ψ) is induced across section AA. The net result is shear stress on section AA. Notice that as long as the change in spacing (δr) is small relative to r0 (i.e., δr/r0 being small), F is approximately linearly related to δr (green line segment in Fig. 3.1b). Hooke’s law of linear elasticity captures this linear relation. To define the stress at a material point in a continuum, imagine a surface element passing through the point. The intensity of the action across the surface element is called the tractiont, between the two parts separated by the surface, which is defined as: t ¼ lim

ΔF

ΔA!0 ΔA

ð3:1Þ

where ΔF is the force due to configuration change and ΔAis the area of action. The traction has the dimension of force per unit area. The traction depends on the orientation of the surface element. We need to consider the tractions on three planes

3.1

Stress

59

Fig. 3.1 Schematic diagram explaining the origin of stress in a condensed material. (a) The potential energy (U ) from the interaction between neighboring particles as a function of interparticle spacing (r). The equilibrium inter-particle spacing is r0. (b) The inter-particle force as a function of r. (c) The stress-free state corresponding to r ¼ r0. (d) Tensile stress on horizontal section AA is produced as a result of an external vertical tension. (e) Compressive stress on AA results from vertical compression. (f) Shear stress is generated in response to an external shear load parallel to the horizontal layers. Based on Putnis (1992, pp. 255–256), Batchelor (2000, p. 3), and Shortley and Williams (1971, p. 236)

to completely describe the stress at a point, and it is most convenient to use three mutually orthogonal planes in a Cartesian coordinate system. The traction acting on the x2x3 plane is named t1 because it is on the plane whose normal is the x1 axis (or e1). Similarly, t2 and t3 are tractions on the x1x3 and x1x2 planes, respectively.

60

3

Stress, Strain, and Elasticity

Each traction ti(i ¼ 1, 2, 3) is a vector, ti ¼ σ i1e1 + σ i2e2 + σ i3e3, simply written as: ti ¼ σ ij ej

ð3:2Þ

Therefore, the stress at a point is completely defined by the matrix σ ij. Of course, the Cartesian components σ ij depend on the coordinate system used. The matrix σ ij are Cartesian components of the second-order stress tensor, denoted by σ, called the Cauchy stress tensor. The stress tensor has nine components. But the balance of angular momentum requires that the stress tensor be symmetric, i.e., σ ij ¼ σ ji. This means that the stress tensor has six independent components. If the stress tensor σ ij at a point is known, the traction on any plane passing the point can be calculated. On a general plane with its normal unit vector n (Fig. 3.2b), the traction (a vector) is simply: t ¼ n  σ ¼ σn, or in components: t i ¼ σ ik nk

ð3:3Þ

Note that n  σ is tensor inner product notation whereas σn is the matrix notation (Chap. 1). The normal stress acting on the plane is the projection of t onto the plane’s normal direction: σ n ¼ ðn  σÞ  n ¼ σ ij ni nj

ð3:4Þ

The shear stress parallel to the plane is a vector: τ ¼ t  σ n n ¼ σn  σ kl nk nl n ¼ ðσ  σ kl nk nl IÞn

ð3:5Þ

which in component terms is: τi ¼ ðσ im  σ kl nk nl δim Þnm ¼ σ im nm  σ kl nk nl ni

ð3:6Þ

The magnitude of the total shear stress τ on the plane is: τ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi t 2  σ 2n ¼ ðσ2 Þij ni nj  σ ij ni nj

ð3:7Þ

In the practice of reconstructing the stress field from the fault-slip data, it is assumed that the slip direction on a fault surface is parallel to the resolved shear stress τ (e.g., Lisle 2013). In theories of plasticity, a slip plane and slip direction together form a slip system. To activate a slip system, the resolved shear stress on the slip plane and

3.1

Stress

61

Fig. 3.2 (a) The stress at a point is defined by the tractions on three coordinate planes, ti (i ¼ 1, 2, 3), each being a vector with 3 Cartesian components (σi1, σi2, σi3). σii (i ¼ 1, 2, 3) are normal stresses on respective coordinate planes. σij (i6¼j) are shear stresses. The total shear stress on the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi plane normal to x1 is τ ¼ σ 212 þ σ 213 . Therefore, the stress at a point is completely defined by the matrix σij. (b): The traction and stress components on an arbitrary plane with plane-normal defined by unit vector n (yellow) can be obtained once σij are known

in the slip direction must reach a critical value, commonly referred to as the critical resolved shear stress (CRSS). Denote the slip direction by unit vector s. The resolved shear stress τr is: τr ¼ σ ij ni sj

ð3:8Þ

For a viscous material, the resolved shear stress determines the rate of slip accommodated by that slip system. In continuum mechanics, it is common and useful to regard the Cauchy stress tensor as the sum of an isotropic part, called the mean stress, and a part called the deviatoric stress tensor: 1 σ ij ¼ σ kk δij þ σ ′ ij ¼ σ m δij þ σ ′ ij 3

ð3:9Þ

  1 1 σ m ¼ σ kk ¼ trace σ ij 3 3

ð3:10Þ

where

is the mean stress (a scalar) and

62

3

σ ′ ij ¼ σ ij  σ m δij

Stress, Strain, and Elasticity

ð3:11Þ

is the deviatoric stress tensor. In the theory of elasticity, the mean stress causes the volume strain, and the deviatoric stress tensor is responsible for the shape change (shear strains of the body). Viscous materials are generally assumed to be incompressible. The constitutive equations relate the deviatoric stresses to the strain rates of the material. The simplest stress state is a hydrostatic or lithostatic state with all shear stress components vanishing, and the normal stresses are equal to the pressure: σ ij ¼ pδij

ð3:12Þ

This is the stress state in water (hydrostatic), inviscid fluids, or viscous fluids at rest. In Earth’s lithosphere at isostatic equilibrium and not subjected to any tectonic load, the stress state may approximate such a state (called the lithostatic state). In a lithostatic situation, the pressure increases with depth byp ¼ ρgz where zis the depth, ρ is the average density of the overburden rocks, and gis the acceleration due to gravity. The common practice of converting pressure estimates from metamorphic petrology to paleodepths is based on the lithostatic stress state assumption. Another simple state of stress is in the laminar simple shearing flow of an isotropic viscous fluid. Suppose the shear plane is x1x3 and shear direction along x1- axis. The deviatoric stresses are: σ'11 ¼ σ'22 ¼ σ'33 ¼ σ 13 ¼ σ 31 ¼ σ 23 ¼ σ 32 ¼ 0 and σ 12 ¼ σ 21 6¼ 0. The deviatoric principal stress directions are: p1ffiffi2  ðe1 þ e2 Þ,e3 ,and p1ffiffi2 ðe1  e2 Þ corresponding to principal stresses: σ 1 ¼ σ 12, σ 2 ¼ 0, and σ 3 ¼  σ 12, where e1, e2, and e3 are base unit vectors parallel to x1-, x2-, and x3- axes, respectively. As a second-order symmetric tensor, the stress tensor has up to three distinct eigenvalues and eigenvectors. The eigenvalues are the magnitudes of principal stresses and the corresponding eigenvectors are the directions of the principal stress axes. The eigenvalues can be found by solving the following Equation for σ (see Chap. 1):   det σ ij  σδij ¼ 0 which gives a cubic equation: σ3  I 1 σ2  I 2σ  I 3 ¼ 0 where

3.2

Equilibrium Equations

63

I 1 ¼ σ 11 þ σ 22 þ σ 33 I 2 ¼ σ 212 þ σ 213 þ σ 223  ðσ 11 σ 22 þ σ 22 σ 33 þ σ 33 σ 11 Þ   I 3 ¼ det σ ij

ð3:13Þ

are the three invariants of the Cauchy stress tensor. For the deviatoric stress tensor, the first invariant is I'1  0 and the second invariant is: 1 I ′ 2 ¼ σ ′ ij σ ′ ij 2

ð3:14Þ

The square root of I'2 is called the effective shear stress (Ranalli 1995, p. 76): σE ¼

 1=2 1= 1 σ ′ ij σ ′ ij ¼ ð I '2 Þ 2 2

ð3:15Þ

The effective shear stress measures the stress magnitude of a deviatoric stress tensor. The magnitude of stress is also often described by the principal stress differences, σ 1  σ 3 and σ 1  σ 2, commonly known as the differential stresses. In most laboratory experiments on creep deformation of rocks, the loading condition is such that σ 2 ¼ σ 3 and is equal to the confining pressure. In such situations, the principal stress difference σ 1  σ 3, or the differential stress, is equivalent to σ E. In a simple shear experiment, it can be demonstrated that the shear stress on the shear plane is equal to σ E.

3.2

Equilibrium Equations

At static equilibrium, every point in the elastic body is stationary. Applying Newton’s second law to an arbitrary element in the elastic body leads to the following equations for the stress (Ranalli 1995, pp. 23–25): 8 ∂σ 11 ∂σ 21 ∂σ 31 > > þ þ þ f1 ¼ 0 > > ∂x1 ∂x2 ∂x3 > > < ∂σ 12 ∂σ 22 ∂σ 32 þ þ þ f2 ¼ 0 > ∂x1 ∂x2 ∂x3 > > > > > : ∂σ 13 þ ∂σ 23 þ ∂σ 33 þ f 3 ¼ 0 ∂x1 ∂x2 ∂x3 which in index notation is often written as:

64

3

Stress, Strain, and Elasticity

∂σ ij þ f j ¼ 0,or σ ij,i þ f j ¼ 0 ∂xi

ð3:16Þ ∂σ

where fj is the body force per unit volume. The shorthand σ ij,i ¼ ∂xiji follows the comma convention for differentiation with respect to the Cartesian coordinates (see Chap. 1). If the only body force is gravity, then f1 ¼ f2 ¼ 0 and f3 ¼ ρg if the x3 coordinate axis is vertically down.

3.3

Sign Conventions

In the mechanics and materials science literature, tensile stress is positive and compressive stress negative. The compressive stress positive convention is commonly used in Structural Geology, Tectonics, and some geodynamics literature (e.g., Hobbs et al. 1976; Turcotte and Schubert 1982). This is because the stress state inside the Earth is most commonly compressive. However, this convention means that the stress sign and strain sign are opposite, as extension strain is always taken as positive. Using the compressive-stress-positive convention also means that mathematic equations and numerical codes developed in the large mechanics and materials science literature must be changed before they can be used for geology problems. The tensile-stress-positive convention is used in this book to be consistent with the mechanics and materials science literature where most micromechanics works are published.

3.4

Strain of a Line

Before introducing the strain as a tensor quantity, let us review a few strain definitions used in engineering and structural geology (e.g., Hobbs et al. 1976, pp. 33–34; Jaeger 1964, p. 22). The elongation (or simply strain) e of a line element is defined as: e¼

Δl l

ð3:17Þ

where Δl is the change in length and l is the length of the line segment taken at a reference state. The reference state is the one with respect to which strain is measured. It can be the undeformed state or any intermediate state during a deformation. The length l of a line segment changes during deformation. Therefore, it is important to clarify the reference state in a strain definition. However, if the strain is small (like < M ijkl ¼ J ijkl þ J ijkl E E ) > : C ijkl ¼ E J d þ E J m 1 þ ν ijkl 1  2ν ijkl M ijkl ¼

ð3:48Þ

As the shear modulus is related to Young’s modulus and Poisson’s ratio by E ¼ 2μ(1+ ν), the isotropic elastic stiffness and compliance tensors are also expressed as: 8   1 1  2ν m > J ijkl þ J dijkl < M ijkl ¼ 2μ 1 þ ν   > : C ¼ 2μ 1 þ ν J m þ J d ijkl ijkl 1  2ν ijkl

ð3:49Þ

Because the bulk modulus relates the volume strain and the mean stress by 13 σ kk ¼ Kεkk , one can easily demonstrate using Eq. (1.41) or Eq. (3.48) that K and the tensor components Cijkl are related by (Walpole 1984): 1 K ¼ C iijj 9

ð3:50Þ

For isotropic materials, the shear modulus μ is related to the tensor components Cijkl by: μ¼

1 ðC  3KÞ 10 ijij

ð3:51Þ

The above presentation for linear elasticity can be applied, en masse, to Newtonian viscous fluids where the stress is linearly related to the strain rate by the viscosity. In

3.8

Matrix Expression of the Elastic Stiffness Components

77

an isotropic Newtonian material, it has a bulk viscosity and a shear viscosity. In general anisotropic Newtonian materials, the viscosity is a fourth-order tensor called the viscous stiffness tensor which has the major symmetry as the elastic stiffness tensor. However, viscous materials are commonly assumed incompressible, and the deviatoric stresses are related to the incompressible strain rates. We will discuss viscous behavior later in the book.

3.8

Matrix Expression of the Elastic Stiffness Components

It is customary to write the components of elastic stiffness tensor in a matrix format – the so-called contracted notation (Voigt 1910; Lekhnitskii 1950). The Voigt notation is still the de facto standard for writing elastic constants and Hooke’s law in most texts (Mura 1987; Nemat-Nasser and Hori 1999). In this notation, the ij pair is contracted to a single index α and the kl pair to a single index β according to the following rule (Mura 1987, p. 502; Nemat-Nasser and Hori 1999, p. 76; Ting 1996, pp. 35–37): 0

Cαβ

C1111

BC B 2211 B B C3311 ¼B BC B 2311 B @ C3111 C1211

C 1122

C1133

C 1123

C 1131

C 2222

C2233

C 2223

C 2231

C 3322 C 2322

C3333 C2333

C 3323 C 2323

C 3331 C 2331

C 3122 C 1222

C3133 C1233

C 3123 C 1223

C 3131 C 1231

C1112

1

C2212 C C C C3312 C C C2312 C C C C3112 A

ð3:52Þ

C1212

The stress tensor and strain tensors are mapped to 6-component vectors with 0

σ 11

1

Bσ C B 22 C B C B σ 33 C B C; b σ¼B C B σ 23 C B C @ σ 13 A σ 12

0

ε11

1

B ε C B 22 C B C B ε33 C C bε ¼ B B 2ε C B 23 C B C @ 2ε13 A 2ε12

The above rule of mapping is summarized by Ting (1996, p. 36) as:

ð3:53Þ

78

3

( α¼ (

Stress, Strain, and Elasticity

if i ¼ j

i,

9  ði þ jÞ, if i 6¼ j

β¼

if k ¼ l

k,

ð3:54Þ

no summation

9  ðk þ lÞ, if k 6¼ l bα ¼ σ ij , bβ ¼ ð2  δkl Þεkl , Cαβ ¼ C ijkl Hooke’s law can be written in a matrix notation: bα ¼ Cαβbβ

ðα, β ¼ 1, 2, : . . . 6Þ

ð3:55Þ

Table 3.1 shows elastic stiffnesses for quartz at various temperatures from Ohno et al. (2006). The α-quartz has trigonal symmetry. With the c-axis parallel to coordinate x3-axis, the six independent elastic stiffnesses fill up the Voigt matrix as follows (Ting 1996, p. 45): 0

Cαβ

B B B B B ¼B B B B @

C11

1

C12

C 13

C14

0

0

C11

C 13 C 33

C 14 0

0 0

0 0

C44

0 C44

0 C 14 C 11  C12 2

sym

C C C C C C C C C A

ð3:56Þ

The β-quartz has hexagonal symmetry, and the five independent elastic stiffnesses fill the Voigt matrix as follows: 0

Cαβ

B B B B B ¼B B B B @

C11

C 12 C 11

sym

C 13 C 13

0 0

0 0

0 0

C 33

0

0

0

C44

0 C 44

0 C14 C 11  C12 2

1 C C C C C C C C C A

ð3:57Þ

For a complete list of relations for materials of different symmetries between independent elastic stiffnesses and the Voigt matrix, the reader may refer to Ting (1996). As an example, from Table 3.1, the Vogit matrix for the elastic stiffness of αquartz at 19∘C is:

3.8

Matrix Expression of the Elastic Stiffness Components

79

Table 3.1 Elastic Stiffnesses of Quartz (GPa) (From Ohno et al. 2006, with permission for reuse from Springer Nature)

80

3

0

Cαβ

88:2 6:5 B 6:5 88:2 B B B 12:4 12:4 ¼B B 18:8 18:8 B B @ 0 0 0

Stress, Strain, and Elasticity

18:8 18:8

0 0

107:2

0

0

0 0

58:5 0

0 58:5

C C C 0 C CGPa 0 C C C 18:8 A

0

0

18:8

40:85

0

0 0

1

12:4 12:4

ð3:58Þ

Using the relation of Eq. (3.52), one can get the fourth-order elastic stiffness tensor readily. In a nested matrix format [Cijkl ¼ (Cij)kl], it is: 2 0

88:2 0

6 B 6 B 0 6 @ 6 6 60 0 6 0 6 6B 6 40:85 Cijkl ¼ 6 B 6@ 6 6 18:8 6 0 6 0 6 6 B 6 B 18:8 4 @ 58:5

0

1 0

C 6:5 18:8 C A 18:8 12:4 1 40:85 18:8 C 0 0 C A 0 0 1 18:8 58:5 C 0 0 C A 0 0

0

40:85 18:8

1 0

B C B 40:85 0 0 C @ A 18:8 0 0 0 1 6:5 0 0 B C B 0 88:2 18:8 C @ A 0 18:8 12:4 0 1 0 18:8 0 B C B 0 18:8 58:5 C @ A 0 58:5 0

0 18:8 58:5

1 3

B C 7 B 18:8 0 7 0 C @ A 7 7 7 58:5 0 0 0 17 7 0 18:8 0 7 B C7 B 0 18:8 58:5 C 7 GPa @ A7 7 7 0 58:5 0 7 0 17 7 12:4 0 0 7 B C7 B 0 12:4 0 C 7 @ A5 0 0 107:2 ð3:59Þ

Hooke’s law can also be written in a reverse form of Eq. (3.55): bα ¼ Mαβbβ

ðα, β ¼ 1, 2, : . . . 6Þ

ð3:60Þ

where Mαβ is the inverse matrix of Cαβ (i.e., Mαγ Cγβ ¼ Cαγ Mγβ ¼ δαβ ). The elastic compliance matrix corresponding to Eq. (3.58) is easily obtained by matrix inversion: Mαβ ¼ C1 αβ 0 12:627 1:737 1:26 B 1:737 12:627 1:26 B B B 1:26 1:26 9:62 ¼B B 4:616 4:616 0 B B @ 0 0 0 0 0 0   3 1  10 GPa

4:616

0

4:616 0

0 0

20:061

0

0 0

20:061 9:232

0

1

C C C C C 0 C C C 9:232 A 0 0

28:729 ð3:61Þ

The transformation from the elastic compliance matrix Mαβ to the elastic compliance tensor Mijkl does not follow the rule of Eq. (3.52) because the stress tensor does not

3.8

Matrix Expression of the Elastic Stiffness Components

81

map to the stress vector in the same way as the strain tensor is mapped to the strain vector Eq. (3.53). The following rules must be applied to convert the matrix components to the tensor ones (Bacon et al. 1980; Ting 1996, p. 37): 0

Mαβ

M 1111

B M B 2211 B B M 3311 ¼B B 2M 2311 B B @ 2M 3111 2M 1211

M 1122

M 1133

2M 1123

2M 1131

M 2222

M 2233

2M 2223

2M 2231

M 3322 2M 2322

M 3333 2M 2333

2M 3323 4M 2323

2M 3331 4M 2331

2M 3122 2M 1222

2M 3133 2M 1233

4M 3123 4M 1223

4M 3131 4M 1231

2M 1112

1

2M 2212 C C C 2M 3312 C C 4M 2312 C C C 4M 3112 A 4M 1212

which can be summarized as: M ijkl ¼ Mαβ if both α and β  3 2M ijkl Mαβ if either α or β  3 4M ijkl ¼ Mαβ if both α and β > 3

ð3:62Þ

One can use the rules above to transform the matrix of Eq. (3.61) to the tensor components Mijkl. However, it is a rather cumbersome method. A much more efficient approach to use a set of orthonormal second-order tensor base tensors (Lebensohn et al. 1998) as presented in Chap. 1 Eqs. (1.44) to (1.51). The algorithm is presented in a Mathcad worksheet named StiffnessCompliance.mcdx. The inverse of Cijkl of Eq. (3.60) is obtained immediately using the worksheet: M ijkl ¼ C 1 ijkl 20

12:627

0

0

1 0

C 6B C 6B 0 1:737 2:308 C 6B A 6@ 6 6 0 2:308 1:26 6 60 1 6 0 7:182 2:308 6 6B C 6B C 0 0 C ¼ 6 B 7:182 6@ A 6 6 2:308 0 0 6 60 1 6 0 2:308 5:015 6 6B C 6B C 6 B 2:308 0 0 C 4@ A 5:015 0 0   103 GPa1

0

7:182 2:308

1 0

C B C B 0 0 C B 7:182 A @ 2:308 0 0 1 0 1:737 0 0 C B C B 12:627 2:308 C B 0 A @ 0 2:308 1:26 1 0 2:308 0 0 C B C B 2:308 5:015 C B 0 A @ 0 5:015 0

0

2:308 5:015

13

C7 B C B 0 0 C7 B 2:308 7 A7 @ 7 7 5:015 0 0 7 1 7 0 7 2:308 0 0 7 C 7 B C 7 B 2:308 5:015 C 7 B 0 A 7 @ 7 7 0 5:015 0 7 1 7 0 7 1:26 0 0 7 C 7 B C 7 B 1:26 0 C 7 B 0 A 5 @ 0 0 9:62 ð3:63Þ

One can confirm that the tensor components in Eq. (3.63) are consistent with Eq. (3.61) considering the rules of Eq. (3.62).

82

3.9

3

Stress, Strain, and Elasticity

Boundary Value Problems of Linear Elasticity

Where an elastic body is subjected to given boundary conditions, which can be displacements, boundary loads (stresses), or a mixed boundary condition (displacements for part of the boundary and loads for the remaining boundary), the effort to find out the complete elastic state of the body, is called the solution of a boundary value elastic problem. Commonly, a boundary value problem is reduced to the solution for the displacement field. Combining Eqs. (3.16), (3.33), and (3.41), and also considering the symmetry of Cijkl (e.g., Qu and Cherkaoui 2006, p. 35), one gets the following Equation for the displacement field: C ijkl uk,li þ f j ¼ 0

ð3:64Þ

This fundamental Equation in elasticity is called the Navier equation. The Navier equation and proper boundary conditions constitute a well-posed boundary-value problem of linear elasticity. In most cases, numerical methods are needed to obtain solutions to the displacement field. Once the displacement field is known, the strain field is obtained using Eq. (3.33) and the stress field using Eq. (3.41). Because the Navier equation is linear, if uαk is the solution for given body force f αj (i.e., C ijkl uαk,li þ f αj ¼ 0) and uβk is the solution for f βj (i.e., Cijkl uβk,li þ f βj ¼ 0), then the a and b  are constants) isthe solution for the linear combination auαk þ buβk (where   β β α α α body force af j þ bf j (i.e., Cijkl auk,li þ buk,li þ af j þ bf βj ¼ 0 . Therefore, if we can find the solution for a family of f kj ðxÞ, we can write the solution to any linear combinations f kj ðxÞ immediately. This leads to the powerful concept and formal approach to solving the Navier equation by the Green function method. As the Green function method is pivotal to Eshelby’s (1957, 1959) solution of an elastic inclusion within an infinite elastic medium and the development of micromechanics, this method is outlined below. If the displacement at x, ui(x), due to a point force F ¼ (F1, F2, F3) acting at another point x' is expressed as:   ui ðxÞ ¼ Gik x, x 0 F k

ð3:65Þ

where Gij(x, x') is called the elastic Green function. Gij(x, x') is the i-component of displacement at x caused by a unit point force in j-direction at x'. How to find the Green function is a separate problem. But once the Green function is known, the displacement field due to any body force distribution can be obtained by integrating the contribution of all body force, i.e., by the following convolution relation:

3.9

Boundary Value Problems of Linear Elasticity

Z ui ð xÞ ¼

83

      Gij x, x 0 f j x 0 dV x 0

ð3:66Þ

Ω

where Ω is the volume containing all body force sources of the elastic solid. In the Appendix to this Chapter, expressions for the Green function and its derivatives in an infinitely-extending anisotropic elastic body are derived following Barnett (1972). The Green function for an infinite isotropic elastic body was first derived by Lord Kelvin in 1848 (Love 1927): 

Gij x, x

0



"  # ð xi  x i 0 Þ x j  xj 0 1 ð3  4νÞδij þ ¼ 16πμð1  νÞjx  x 0 j jx  x 0 j2

ð3:67Þ

In an infinite elastic body, the Green function depends on the relative location, hence Gij(x, x') ¼ Gij(x  x') and Eq. (3.65) becomes:   uk ðxÞ ¼ Gkm x  x 0 F m

ð3:68Þ

The point force Fk can be written as a body force density using the Dirac delta function (not to be confused with the Kronecker’s delta). The three-dimensional Dirac delta function is defined as: (

8 > > > > >
> 0 > δðx  x ÞdV ðxÞ ¼ > > : 0

x 6¼ x 0 x 6¼ x 0

Ω

if x 0 2 Ω

ð3:69Þ

if x 02 =Ω

which enables us to write the point force as a body force field function:   f j ð xÞ ¼ F j δ x  x 0

ð3:70Þ

The displacement field due to a point force Eq. (3.68) must satisfy the Navier Eq. (3.64), which leads to:     Cijkl Gkm,li x  x 0 F m þ δ x  x 0 F j ¼ 0     ) C ijkl Gkm,li x  x 0 F m þ δjm δ x  x 0 F m ¼ 0

ð3:71Þ

Because Eq. (3.71) is valid for an arbitrary point force load, we have:     Cijkl Gkm,li x  x 0 þ δjm δ x  x 0 ¼ 0

ð3:72Þ

The Green function in an anisotropic infinite elastic body is found by solving this partial differential equation set. In the Appendix of this Chapter, the Green function Gij and its derivative Gij, l are obtained in integral formalism. This approach will be followed up later when we deal

84

3 Stress, Strain, and Elasticity

with anisotropic viscous materials. A Mathcad worksheet is provided online that is based on the algorithm.

3.10

Multiscale Stress and Strain in Real Materials

In continuum mechanics, a field quantity like the stress tensor is defined at every “point” in the medium. The spatial variation of the quantity defines the field, like the stress field. However, real materials like rocks are not continuous in the mathematical sense. Take the density of granite, for example. To get the local density, one takes a sample of the granite at the locality, and the density is the mass of the sample divided by the volume of the sample, that is: ρ¼

Δm ΔV

ð3:73Þ

where ρ is density, Δm the mass, and ΔV the volume of the sample. This is the average density over the volume, of course. To get the density at the locality x, one should use a small sample volume. In a mathematic sense, the density truly at a point is the following limit: Δm ΔV!0 ΔV

ρðxÞ ¼ lim

ð3:74Þ

The same idea applies to other quantities, such as the stress, at a point and hence the limit in Eq. (3.1). However, in reality, the sample size cannot be vanishingly small. There is a critical volume Vc below which the density one gets might be the density of a single mineral grain like quartz, plagioclase, or even a void space (Fig. 3.7). The continuum assumption for granite breaks down if ΔV < Vc. The local density is always the averaged density over a Representative Volume Element (RVE) greater than Vc: ρ ¼ lim

ΔV!V c

Δm ΔV

ð3:75Þ

All other continuum concepts (stress, strain, vorticity, displacement etc.) for real materials must also be associated with a critical RVE. The critical RVE is the physically sound maximum resolution for a field quantity at a point. The size of the RVE, however, depends on the heterogeneity of the material and the nature of the problem under investigation. In the study of natural rock deformation, it is often necessary to consider distinct characteristic length scales separated by orders of magnitude in dimension. Multiscale and hence multiple RVEs are considered. In structural geology and tectonic studies, one often speaks of “regional” and “local” stress and strain fields. The two scales are different by many (at least three)

3.10

Multiscale Stress and Strain in Real Materials

85

Fig. 3.7 The density of rock like granite as a function of the volume used to measure the density. At the scales below the critical volume Vc, the density fluctuates due to grain and sub-grain scale heterogeneities and cannot represent the granite density. The ‘local’ density at a point must be measured at the finite Vc. The continuum concept for granite breaks down below Vc. See text for detail. Modified after Batchelor (2000) and Ranalli (1995)

orders of magnitudes. It is critical that one appreciates this scale gap and understands how fields of multiscale are related. For the convenience of presentation, let us consider microscale and macroscale stress and strain fields. Denote the microscale stress and strain fields by lowercase σ and ε, and macroscale counterparts by uppercase Σ and Ε. The macroscale RVE on which Σ and Ε are defined contains a large assemblage of microscale elements. Σ(X) is the average of σ(x) over the macroscale RVE centered on X. The relation is: 1 ΣðXÞ ¼ V

Z σðxÞdV RVE

ΕðXÞ ¼

1 V

Z

ð3:76Þ εðxÞdV

RVE

The integration Eq. (3.89) can be expressed as the volume-weighted average of stress (or strain) of all heterogeneous microscale elements contained in the macroscale RVE: Σ¼

N X i

vi σi ¼ hσi and E ¼

N X

vi εi ¼ h εi

ð3:77Þ

i

where vi is the volume fraction of the ith element, σi and εi the average stress and strain respectively within the microscale element, and N is the total number of microscale elements in the macroscale RVE. In micromechanics, the inverse relationships of Eq. (3.89) are as follows:

86

3

ε ð xÞ ¼ A ð x Þ : Ε σðxÞ ¼ BðxÞ : Σ

Stress, Strain, and Elasticity

ð3:78Þ

where A(x) and B(x) are, respectively, the strain partitioning tensor and the stress partitioning tensor. Therefore, in real materials, physical quantities like stress and strain are still defined at every point, but the point quantity is defined as the average over an appropriate RVE centered at the point. Geologists distinguish different scale quantities by using terms like ‘regional’ (or ‘bulk’) stress vs ‘local’ stress. Σ can be regarded as regional stress, whereas σ is local stress. By averaging, Σ is much smoother and simpler than σ. It captures a larger scale, longer wavelength pattern of the stress field but lacks the details to understand small-scale structures and fabrics. A geodynamic finite element model with mesh sizes of tens of kilometer gives Σ and Ε that do not govern how smallscale structures observed in a single outcrop have developed. The latter is related to σ and ε which are distinct and cannot be derived from Σ and Ε. One must investigate the partitioning function A(x) and B(x) in Eq. (3.78) to get σ and ε. That requires a micromechanics-based multiscale investigation. The clarification on the relations between multiscale stress and strain fields is important as there has been some confusion on the relation between local vs regional stresses (e.g., Tikoff and Wojtal 1999). If one chooses a sufficiently large RVE, the macroscale averaged stress and strain will undoubtedly conform to the plate boundary conditions. The fact that the World Stress Map (Zoback 1992) and the Global Strain Rate Map (Kreemer et al. 2014) show that regional stress and strain fields are aligned with plate motions is a natural expectation of the macroscale mechanical behavior of the elastoviscous lithospheric plates. The multiscale nature of stress is also significant in evaluating the practice of reconstructing the paleo-stress field from fault-slip data. The method assumes that all faults from which fault slip data are obtained were formed in a homogeneous and isotropic elastic medium and under a single-phase and a single-scale stress field (the regional stress Σ). To collect a sufficient number of fault slips, slip data from many faults and slip surfaces are compiled. A more plausible situation is that the slips were due to a mix of multiscale stresses and multi-phase deformation increments, considering the ubiquitous rheological heterogeneities in rocks. Large spatial variations in principal stress axes determined from earthquake moment (Rebai et al. 1992) highlight the heterogeneity and multiscale nature of the stress field in Earth’s lithosphere.

3.11

3.11

The Effective Rheology on the Macroscale

87

The Effective Rheology on the Macroscale

Rocks are heterogeneous materials. Suppose we know the rheological behaviors of the microscale elements making up the assemblage on the macroscale RVE. What is the overall or ‘bulk’ rheological behavior of the material? We will explore this subject in more detail later in this book but can discuss some general principles here. Consider a macroscale RVE containing N heterogeneous elements. Let us assume that the rheological behavior in each element is linearly elastic, as expressed below: σ αij ¼ Cαijkl εαkl

ð3:79Þ

where superscript α stands for the α-th element, and the elastic stiffness C αijkl is known. D E The bulk or macroscale stress and strain are, respectively Σij ¼ σ αij and Εij ¼ D E εαij Eq. (3.77). We would like to know the effective behavior of the macroscale material, which can be expressed as: Σij ¼ C ijkl E kl

ð3:80Þ

where Cijkl is the overall stiffness or effective stiffness of the macroscale material. The method to obtain C ijkl from known C αijkl (α ¼ 1, 2, ⋯N ) is called homogenization in micromechanics. From Eq. (3.79), we take the volume-weighted average and get: D E D E Σij ¼ σ αij ¼ Cαijkl εαkl

ð3:81Þ

We can go no further at this stage unless we make some simplistic assumptions. Suppose we assume that the strains in all microscale elements are the same. That is εαij  Eij, which is the uniform strain model or the Taylor model. Equation (3.81) is simplified to D E Σij ¼ C αijkl Ekl

)

D E Cijkl ¼ Cαijkl

ð3:82Þ

The overall stiffness of the effective material is the volume-weighted average of the constituent stiffnesses. This is the well-known Voigt average for effective stiffness. On the other hand, if we use the inverse relation εαij ¼ M αijkl σ αkl and assume that the microscale stresses are all equal σ αij  Σij , the so-called Sachs model, then we get:

88

3

Stress, Strain, and Elasticity

D E D E D E Eij ¼ εαij ¼ M αijkl σ αkl ¼ M αijkl Σkl

ð3:83Þ

D E1 C ijkl ¼ M αijkl

ð3:84Þ

and in turn:

which is the well-known Reuss (1929) average for effective stiffness. Of course, neither the uniform strain model nor the uniform stress model is realistic. Hill (1952) has proved that the actual effective stiffness lies between the Reuss and Voigt averages, regardless of the microstructure. Therefore, the Voigt average provides the upper bound estimate and the Reuss average the lower bound estimate for the effective material. Unfortunately, the two bounds are typically far apart. A more rigorous homogenization is to use the partitioning relations in Eq. (3.78). As the microscale fields can be expressed in the following forms: εαij ¼ Aαijkl Ekl σ αij ¼ Bαijkl Σkl

ð aÞ ðbÞ

ð3:85Þ

Submitting Eq. (3.85) into Eq. (3.81) yields: D E D E Σij ¼ Cαijkl εαkl ¼ Cαijmn Aαmnkl E kl Therefore, an expression for the bulk stiffness tensor is D E Cijkl ¼ Cαijmn Aαmnkl

ð3:86Þ

The strain partitioning tensor for the microscale elements Aαmnkl will depend on the shapes and orientations of the elements. How the partitioning tensors are obtained will be discussed later in the book.

3.12

Notes and Key References

The following two Mathcad worksheets associated with this Chapter are provided: StiffnessCompliance.mcdx: computes the elastic compliance (fourth-order) tensor from an input elastic stiffness coefficients. The input elastic stiffness coefficients are in the Voigt matrix notation. AnisotropicElasticGreenFunction.mcdx expresses the elastic Green function tensor (and the its first derivative) for an infinite anisotropic elastic solid. The

Appendix: Green Function for an Infinite Anisotropic Elastic Body

89

corresponding derivation for the integral expressions for the Green function and its derivative are in the Appendix of this Chapter. All structural geology textbooks include an introduction to the stress and strain. The more advanced discussion in this Chapter is necessary for the content of this book. Means (1976), Ranalli (1995), and Pollard and Martel (2020) give comprehensive and accessible treatment for stress, strain, and elasticity. The readers are also referred to classic continuum mechanics books by Fung (1965) and Spencer (1980) for additional information and alternative explanation of the stress and strain concepts.

Appendix: Green Function for an Infinite Anisotropic Elastic Body Integral expressions for the Green function Gij and its derivative Gij, l for a homogeneous (Cijkl being constant everywhere) and infinite medium are given by Barnett (1972) and Mura (1987, pp. 33–34). Barnet’s procedure of getting the expressions is summarized below. This method will form the basis for getting corresponding Green functions and their derivatives for incompressible viscous materials in Chap. 11. We must solve the PDE in Eq. (3.64) to find the Green function. Because in an infinite medium, Gij(x, x') ¼ Gij(x  x'), we can consider the case of x' ¼ 0 without loss of generality. Equation (3.72) becomes CijklGkm, li(x) + δjmδ(x) ¼ 0. Because Cijkl is symmetric with respect to the swap of the i and j indices, this Equation can be written as: C ijkl Gkm,lj ðxÞ þ δim δðxÞ ¼ 0

ð3:87Þ

This Equation is solved by the Fourier transform method (Barnett 1972; Mura 1987). The Fourier transform of Gij(x) is defined as: Z1 gij ðKÞ ¼

Gij ðxÞ exp ðiK  xÞdx

ð3:88Þ

1

and the inverse Fourier transform is: 1 Gij ðxÞ ¼ ð2π Þ3

Z1 gij ðKÞ exp ðiK  xÞdK 1

Direct differentiation on Eq. (3.89) yields:

ð3:89Þ

90

3

1 Gkm,lj ðxÞ ¼ ð2π Þ3 1 ¼ ð2π Þ3

Z1

Stress, Strain, and Elasticity

2

gkm ðKÞ 1 Z1

∂ ½ exp ðiK  xÞdK ∂xj ∂xl ð3:90Þ

K j K l gkm ðKÞ exp ðiK  xÞdK 1

On the other hand: 1 δ ð xÞ ¼ ð2π Þ3

Z1 exp ðiK  xÞdK

ð3:91Þ

1

Inserting Eqs. (3.90) and (3.91) into Eq. (3.87) leads to: 1 ð2π Þ3

Z1

 Cijkl K j K l gkm ðKÞ þ δim exp ðiK  xÞdK ¼ 0

ð3:92Þ

1

The requirement that this Equation holds for any x requires: Cijkl K j K l gkm ðkÞ ¼ δim

ð3:93Þ

To separate orientation dependence from magnitude dependence, define K ¼ |K| and z¼K K . Thus, z is the unit vector parallel to K and K ¼ Kz. Equation (3.93) is expressed as: C ijkl zj zl K 2 gkm ðKÞ ¼ δim

ð3:94Þ

The term Cijklzjzl is called the second-order Christoffel stiffness tensor (Barnett 1972). Denoting it by Aik ¼ Cijklzjzl, Eq. (3.94) becomes K2Aikgkm(K) ¼ δim, and: gij ðKÞ ¼

A1 ij K2

ð3:95Þ

Equation (3.95) is the Green function in Fourier space. It has a relatively simple form. It depends on the magnitude of the wave vector K by K12 and the orientation of the wave vector by A1 ij ðzÞ. The Green function in real space Gij(x) is the inverse Fourier transform of gij(K):

Appendix: Green Function for an Infinite Anisotropic Elastic Body

1 Gij ðxÞ ¼ 3 8π

Z1

A1 ij ðzÞ K2

1

91

exp ðiK  xÞdK

ð3:96Þ

With Euler’s formula, the expression becomes: 1 Gij ðxÞ ¼ 3 8π

Z1

A1 ij ðzÞ K2

1

cos ðKz  xÞdK

ð3:97Þ

Only the real part of the integral is concerned as Gij is real. The expression for Gij, l can be obtained by differentiating Eq. (3.97) with respect to xl: Gij,l ðxÞ ¼ 

1 8π 3

Z1 1

zl A1 ij ðzÞ sin ðKz  xÞdK K

ð3:98Þ

Let us simplify Eqs. (3.97) and (3.98). Note that the integrations are over the entire 3D Fourier space. That is, ðÞdK ¼

R1 1

dK 1

R1 1

dK 2

R1 1

R1 1



ðÞdK 3. With the following set of variable substitution:

b x ¼ xr and K ¼ kr , and dK ¼ dk r3 where r ¼ |x|, Eqs. (3.97) and (3.98) are reduced to: 1 Gij ðxÞ ¼ 3 8π r

Z1

A1 ij ðzÞ

1

1 Gij,l ðxÞ ¼  3 2 8π r

Z1 1

k2

cos ðkz  b xÞdk

zl A1 ij ðzÞ sin ðkz  b xÞdk k

ð3:99Þ

ð3:100Þ

We use a spherical coordinate system (Fig. 3.8) aligned with b x to evaluate these integrals: dk ¼ k2 sin σdkdσdψ zb x ¼ cos σ

ð3:101Þ

where ψ is the polar angle in the plane perpendicular to b x (i.e., z  b x ¼ 0) with respect to any reference direction α (Fig. 3.8) With Eqs. (3.101), (A.33) and (A.34) become, respectively:

92

3

Stress, Strain, and Elasticity

Fig. 3.8 Stereographic projection showing how, for any given x, unit vector z in the plane normal to x (i.e., the b x  z ¼ 0 plane) can be expressed in terms of any two mutually orthogonal unit vectors (α and β) on the b x  z ¼ 0 plane by z ¼ cos ψα + sin ψβ following Synge (1957) and Barnett (1972). Shown here is one possible pair of α and β with α parallel to the strike of the x-normal plane and β the dipline of the plane. If b x is expressed in terms of spherical angles θ and φ by b x¼ 0

cos θ sin φ

1

0

 sin θ

1

0

 cos θ cos φ

1

B C B C B C @ sin θ sin φ A , then α ¼ @ cos θ A , and β ¼ @  sin θ cos φ A . Green function Gij and its cos φ 0 sin φ x  z ¼ 0 plane derivative Gij, l are expressed as contour integrals along a unit circle on the b

1 Gij ðxÞ ¼ 3 8π r

8 Z2π 1 > > A1 Gij ðxÞ ¼ 2 > ij ðzÞjσ¼π2 dψ > > 8π r
Z2π  >  > 1 > 1 > ð x Þ ¼ b x A þ z M dψ G > l ij l ij > : ij,l 8π 2 r 2 σ¼π2

ð3:112Þ ð bÞ

0



 1 where M ij ¼ A1 xq þ zqbxp . iα Ajβ C αpβq zpb For a general anisotropic elastic medium, these integrals can only be evaluated numerically. A Mathcad worksheet named “General GreenFunctions” presents the algorithm to evaluate Eq. (3.112). In an isotropic medium, Cijkl is simple Eq. (3.48). We can easily obtain the Christoffel tensor and its inverse, which are:  zi zj  Aij ¼ μ δij þ 1  2ν   zi zj 1 > : A1 ij ¼ μ δij  2ð1  νÞ 8 >
0 (a thinning zone) and synthetic otherwise where cosα = Wk (Bobyarchick 1986) (Fig. 5.1). The two eigenvectors are directions of zero angular velocity. Material lines parallel to a flow apophysis are irrotational. A material line not parallel to a flow apophysis will rotate toward or away from the apophysis, depending on whether the apophysis is a stable or unstable apophysis. In a thinning plane-strain general shearing flow, the shear direction is the stable apophysis and the one antithetic to the shear direction is the unstable apophysis. (d) Vortex flow and the reference frame dependence of vorticity We now examine a vortex flow field to highlight the significant point of the frame dependence of vorticity.

5.5

Some Simple Flow Fields

123

Fig. 5.1 In an isochoric plane-straining general shearing flow. There are two flow apophyses in the VNS. In a thinning zone flow case shown here, the stable apophysis (solid line) is parallel to the shear direction and the unstable one (dashed line) is at angle α = cos-1Wk antithetic to the shear. The two apophyses define forward rotation sectors (FWR) and backward rotation (BWR) sectors. A material line whose orientation lies in the FWR sector rotates with the vorticity and away from the unstable apophysis toward the stable apophysis. A material line in the BWR sector rotates against vorticity toward the stable apophysis. For this reason, the stable apophysis is also called a fabric attractor (Passchier 1997)

In the neighbourhood of a vortex line lying along the x3 axis, the particles move in circles about the vortex line. Suppose the angular speed is inversely proportional to the squared distance from the vortex line (Truesdell 1991, p. 120). The velocity field can be expressed, most simply, in a cylindrical coordinate system (Fig. 5.2) as: dx dθ κ dr = 2 , = 0, 3 = 0 dt dt r dt

ð5:24Þ

where κ is a constant, θ and r are polar coordinates. Converted into a Cartesian system, the velocity field is: v1 = -

κx2 κx , v2 = 2 1 2 , v3 = 0 x21 þ x22 x1 þ x2

which gives the following quantities for the flow field:

ð5:25Þ

124

5

Flow: Strain Rate and Vorticity

Fig. 5.2 Particle path, maker deformation, and spin of the instantaneous stretching axis (ISA) in a vortex flow. In a vortex flow, all particles follow circular paths around the vortex axis. A planar marker containing particles a, b, c, . . . g (shown in this section view as a straight line) indicated becomes a spiral after a period of deformation with particles moved to A, B, C, . . . G. The instantaneous stretching axis at any point is always at a fixed 45° with respect to the radial line passing that point. As the particle moves in a circular path, the ISA attached to it spins at an angular velocity equal to the particle’s angular velocity with respect to the vortex center, that is, dθ dt . As a result, the local material element has vorticity relative to the local ISA. The magnitude of this vorticity is equal to the spin of the ISA


 κ x22 - x21 2κx1 x2 2
2 2 B
2 x1 þ x22 B x1 þ x22 B
 2 2 L=D=B 2κx1 x2 B κ x2 - x 1 B
2 -
2 2 2 2 @ x þx x1 þ x22 1 2 0 0 pffiffiffi 2κ ,kW k = 0 = 2 x1 þ x22 0

1 0C C C C, W = 0,kDk C 0C A 0 ð5:26Þ

The vorticity vanishes and Wk = 0 everywhere in the flow field. We know that planestraining pure shearing has Wk = 0. Does this mean that the vortex flow has the same characteristic as a pure shearing flow? Why does the flow have no vorticity, but all material particles move in circular paths? One can gain some insights by applying Stokes’ curl theorem (e.g., Korn and Korn 1968, p. 5.62) (see Jiang 1994a, b):

5.5

Some Simple Flow Fields

125

Z

I curlv  dS =

v  dx

ð5:27Þ

C

S

where C is a closed loop and S is any surface with C as its boundary. The curlv term H is the vorticity vector Eq. (5.18). It is easy to confirm with Eq. (5.25) that v  dx = 0 for every closed loop unless the loop contains the vortex C

line. Therefore, theHfluid is indeed irrotational everywhere. For a loop that does contain the vortex, v  dx  2πκ. Thus, all the ‘vorticity’ for this flow is concenC

trated on the vortex line, the singularity of the flow field. This flow is isochoric and plane-straining everywhere because the three principal κ κ _ 2 = - x2 þx _ 1 , and ε_ 3 = 0 . The strain rates are respectively: ε_ 1 = x2 þx 2 , ε 2 = -ε 1

2

1

2

orientations of the principal strain rate axes are as follows: the ε_ 1 and ε_ 2 axes are in the x1x2 plane with ε_ 1 axis at θ - 45∘ relative to the x1 axis and ε_ 2 axis at θ + 45∘ relative to the x1 axis (Fig. 5.2). As a material element moves circularly at angular κ _ 1 and ε_ 2 axes also rotate at the same angular speed. speed dθ dt = r2 , the local ε Therefore, unlike the simple shearing and pure shearing flows that we considered above, where the principal strain rate axes are fixed, in the vortex flow case, the principal strain rate axes rotate with the fluid element. Consequently, although the local material element has no vorticity relative to the coordinate system x1x2, the element does have vorticity relative to the local principal strain rate axes because these axes themselves rotate. We denote this vorticity by W for the tensor notation and w for the vector notation. w and rotation of the local fluid element cancel each other so that the fluid is everywhere irrotational. Therefore, the magnitude of w is of the strain rate axes, we have jwj = 2κ twice the r2 . With Eq. (5.21), this angular pffiffiffispeed κ gives W = 2 r2 . If W is used, the associated kinematic vorticity number is W k = 1. As the flow field is isochoric plane straining, the flow field for every material element is the same as a simple shearing flow! It is not difficult to visualize that the flow field is simple shearing everywhere, with the local shear plane being the tangential plane of the cylindrical surface passing that point (Fig. 5.2). For structure and fabric development, it is W (or w)—the vorticity measured with respect to the local principal strain rate axes of ISA—that is relevant. In a flow field where the ISA is fixed to the coordinate system in which the flow field is defined, W = W. In the vortex flow example, the ISA spins and the distinction between W and W becomes significant. We consider the frame dependence of flow fields below to elucidate this concept further and develop equations to calculate W.

126

5.6

5

Flow: Strain Rate and Vorticity

Flow Described in Different Reference Frames

The vortex flow example highlights the dependence of a flow field on the observer’s frame of reference. In studying structural and fabric development in rocks, one wishes to use “objective” or frame-independent characteristics of a flow field. We therefore investigate the frame dependence of flow description rigorously in mathematic terms. First, it is necessary to distinguish a coordinate system from a reference frame. A reference frame is a set of physical points considered fixed in space, with respect to which motion is measured. A motion makes sense only when it is referred to a reference frame. A coordinate system is a mathematical scheme through which the position of a point in space is uniquely defined by a set of real numbers (coordinates). In a reference frame, one can, and is often necessary to, use different coordinate systems to solve a problem. With a coordinate system, the position (coordinates), the change of position, and the rate of change of position, etc. of a particle can be expressed mathematically. Suppose Alice and Bob both use Earth as the reference frame to describe the motion of an object. Alice chooses a geographic Cartesian coordinate system with the origin fixed at a point on Earth’s surface, the x1 axis horizontal and pointing north, the x2 axis horizontal and pointing east, and the x3 axis point down vertically. Bob uses another coordinate system x1 x2 x3 that differs from Alice’s by a transformation matrix Qij and xi = Qik xk . Alice’s description will differ in form from Bob’s description of the motion, but the differences are completely resolved by the tensor transformation between their coordinate systems. We say that the description of a motion is invariant in different coordinate systems (of the same reference frame). Now suppose Cathy uses the cruise ship she is on as her reference frame. She uses a Cartesian coordinate system x1'x2'x3' fixed to the ship to describe the motion. The difference between her description of the motion and either Alice’s or Bob’s cannot be resolved by the tensor transformation between coordinate systems because the x1'x2'x3' is attached to another reference frame. The description of a motion is NOT invariant in different reference frames. Let us denote two reference frames by ℝ and ℝ'. Let a Cartesian coordinate system fixed to ℝ be x1x2x3 [three base unit vectors being ei (i = 1, 2, 3)] and another Cartesian coordinate system fixed to ℝ' be x1'x2'x3' [three base unit vectors being e'i (i = 1, 2, 3)]. Let us consider the consequence of ℝ' rotating with respect to ℝ. The kinematic consequence of relative translation between the two systems is trivial and irrelevant in flow field analysis. First, the coordinate transformation is still the same form but Q now is a function of time: x 0 = Qx; x = QT x 0

ð5:28Þ

5.6

Flow Described in Different Reference Frames

127 0

0 dx The velocity of a particle viewed in ℝ is v = dx dt and viewed in ℝ' is v = dt . The relation between the two velocities is found by differentiating Eq. (5.28) with respect to time:

dx 0 _ þ Q dx = Qx dt dt

ð5:29Þ

_ is due to the reference frame change. If the ℝ and ℝ' were fixed to each other The Qx _ (Q = 0 ), Eq. (5.29) is the well-known coordinate system transformation for the velocity in the same reference frame. The right-hand side of Eq. (5.29) can also be expressed in terms of e'i using Eq. (5.28) to give the following expression: _ T x 0 þ QLQT x 0 ,in x 0 ‐coordinates v 0 = QQ

ð5:30Þ

T In getting Eq. (5.30), v = dx dt is expressed by v = Lx = LQ x'. From Eq. (5.30), we get the transformation relation for the velocity gradients tensor across reference frames:

_ T þ QLQT L 0 = QQ _ þ QT L 0 Q L = - QT Q

ð aÞ ðbÞ

ð5:31Þ

Equations (5.31) establish the transformation rules for the velocity gradient tensor between reference frames. Again, if the ℝ and ℝ' were fixed to each other, the first term on the right of both Eqs. (5.31) and (5.31) would vanish, and the Equations would reduce to the familiar forms of coordinate system transformation for L. Applying the Euler-Cauchy-Stokes decomposition Eq. (5.16) to Eq. (5.31) yields: D 0 = QDQT _ T þ QWQT W 0 = QQ

ð aÞ ð bÞ

ð5:32Þ

Therefore, in a reference frame change, the strain rate tensor follows exactly the coordinate system transformation, unaffected by a change in the reference frame. _ T ) besides the coordinate system The vorticity tensor has an additional term (QQ transformation term. In other words, the two observers agree on the strain rate, but their vorticities differ by a term depending on the relative rotation of their respective reference frames. The frame dependence of the vorticity also explains why the vorticity term never enters any constitutive equations. A constitutive equation must always satisfy the reference-frame invariance (Spencer 1980). _ T at a point should be irrelevant for structural and fabric developThe term QQ ment, as it arises entirely due to the choice of the reference frame. But this raises the question of what reference frame one should choose in fabric analysis. How do the two observers tell whose frame of reference is better? What is an objective frame of reference?

128

5.7

5

Flow: Strain Rate and Vorticity

Decomposition of Vorticity

The fact that the vorticity of a flow is reference-frame dependent implies that we should define an objective (frame-independent) measurement for vorticity. One such measure is W (or w) that we introduced above, which is the vorticity of the flowing material relative to the local principal strain rate axes. In a classic paper on vorticity, Means et al. (1980) explain the significance of W and its distinction from W thoroughly. They name W the internal vorticity (Means et al. 1980). Astarita (1979) provides a complete formulation for W to be computed for any flow. Owing to Astarita (1979) and Means et al. (1980), the vorticity of a flow field can be decomposed into internal vorticity, which is the vorticity of the local material relative to the local principal strain rate axes, and a spin which is the angular velocity of the local principal strain rate axes. The decomposition is mathematically expressed as follows: W=W þ Ω

ð5:33Þ

where Ω is the spin tensor. Astarita (1979) defines Ω by the following equation: Ddi = Ωdi Dt

ð5:34Þ

where di (i = 1,2,3) are the three distinct eigenvectors of D. In the event two of the eigenvalues of D are identical, say the two corresponding to d1 and d2, the definition of Eq. (5.34) is complete by setting Ω12 = W12 (Drouot 1976; Astarita 1979). The situation where all three eigenvalues of D are identical is trivial because the flow is then a pure volume change plus a rigid-body rotation (all vorticity is spin Ωij = Wij). Both W and Ω depend on the reference frame but W does not. In any given flow field, a numerical method of decomposition is given by Jiang (2010) which is summarized below. First, denote the three principal strain rate axes at time t and t' by di and d'i (i = 1,2,3), respectively. d'i can always be regarded as a rotation Ψ, of di: d'i = Ψdi ,di = ΨT d 0 i The rotation matrix Ψ can be constructed by Ψ = Λ'ΛT, where Λ is a matrix formed by augmenting the three unit vectors di, column by column, and Λ' is formed in the same way from d'i. The time rate of the rotation of the principal axes is then found by: ðΨ - IÞ Ddi d' - di = lim i0 = lim 0 di Dt t0 →t t - t t0 →t t - t Comparing the above expression with Eq. (5.34) yields:

ð5:35Þ

5.8

Some Examples of Vorticity Decomposition

129

Ω = lim

Δt → 0

Ψ-I Δt

ð5:36Þ

where Δt = t' - t. As the backward rotation from d'i to di is described by ΨT, we thus have: ΨT - I Δt Δt → 0

- Ω = lim

ð5:37Þ

Combining Eqs. (5.36) and (5.37) leads to: Ψ - ΨT Δt → 0 2Δt

Ω = lim

ð5:38Þ

To sum up, the procedure for obtaining Ω at time t is as follows: First, calculate the eigenvectors of D at the given time t and t + Δt. Second, construct Ψ using the eigenvectors at both time t and t + Δt. And finally, use Eq. (5.38) to get Ω by making Δt as small as is necessary so that the accuracy for Ω is satisfactory. A dimensionless number like the one in Eq. (5.20), called the internal vorticity number, can be defined (Means et al. 1980) which measures the instantaneous degree of non-coaxially of a flow field: Wk =

W kD k

ð5:39Þ

W k does not change with reference frames. It is, therefore, a fundamental characteristic of a flow field. We use a few examples to show how the above vorticity decomposition can be carried out and W k obtained.

5.8

Some Examples of Vorticity Decomposition

(a) Spinning pure shearing. Imagine a competent layer undergoing folding by the so-called tangential longitudinal strain mode (layer a in Fig. 5.3). Suppose the coordinate system x1'x2'x3', attached to frame R', co-rotates with the layer at its angular speed ω_ about x3' and the coordinate system x1x2x3, attached to frame R, is stationary. Also, note that at t=0 (when the layer is not folded), the two coordinate systems are parallel to each other. The flow field for the tangential longitudinal strain in x1'x2'x3' is plane-straining pure shearing: v10 = ε_ x 0 1 ,v20 = - ε_ x 0 2 ,v30 = 0 , which gives the velocity gradient tensor as:

130

5

0

ε_ B 0 L =@0 0

Flow: Strain Rate and Vorticity

1 0 0 C - ε_ 0 A: 0 0

This flow has no internal vorticity and is coaxial. For the sake of exercise, let us see what the flow field is like in the stationary x1x2x3? It will undoubtedly have vorticity. How would one still recognize that it is a coaxial flow if the flow were given in the stationary system and therefore has vorticity? First, the coordinate transformation matrix between xi and xi' is a function of time: 0

_ Þ cos ðωt B _ Þ Qðt Þ = @ - sin ðωt 0

_ Þ sin ðωt _ Þ cos ðωt 0

1 0 C 0A

ð5:40Þ

1

With Eqs. (5.31), it is easy to find the velocity gradient tensor in the x1x2x3 system: 0

_ Þ ε_ cos ð2ωt B _ Þ þ ω_ L = @ ε_ sin ð2ωt 0

_ Þ - ω_ ε_ sin ð2ωt _ Þ - ε_ cos ð2ωt 0

0

1

C 0A 0

The strain rate and vorticity tensors are, respectively: 0

_ Þ ε_ cos ð2ωt B _ Þ D = @ ε_ sin ð2ωt 0

1 0 _ Þ ε_ sin ð2ωt 0 0 C B _ Þ 0 A,W = @ ω_ - ε_ cos ð2ωt 0 0 0

1 - ω_ 0 C 0 0A 0 0

Let us now use the procedure outlined above to decompose the vorticity to see if it has any internal vorticity. The three eigenvalues of D can be obtained and they are respectively ε_ , 0, and - ε_ . The corresponding eigenvectors at the time t and t' are found and written in a matrix form: 0

_ Þ 0 cos ðωt B _ Þ 0 Λ = @ sin ðωt 0 1 Ψ is constructed:

0 1 _ Þ - sin ðωt _ 0Þ cos ðωt C 0 B _ Þ A,Λ = @ sin ðωt cos ðωt _ 0Þ 0 0

0 0 1

1 _ 0Þ - sin ðωt C _ 0Þ A cos ðωt 0

5.8

Some Examples of Vorticity Decomposition

131

0

1 - sin ðω_ ðt 0 - t ÞÞ 0 C cos ðω_ ðt 0 - t ÞÞ 0 A 0 1

cos ðω_ ðt 0 - t ÞÞ B Ψ = Λ 0 ΛT = @ sin ðω_ ðt 0 - t ÞÞ 0

ð5:41Þ

The spin can be obtained: 1 19 0 0 - ω_ 0 0 -1 0 > = 2 sin ðω_ ðt - t ÞÞ B Ψ-Ψ B C C 0 0A = lim Ω = lim @ 1 0 0 A = @ ω_ 0 - tÞ 0 - tÞ 0 0 2 t 2 ð t ð > > t →t t → t: ; 0 0 0 0 0 0 8 >
> > dt = - ΘQ
ΘðQ, aÞ = QWQT þ BðaÞ
QDQT angular velocity equation ðbÞ > > : Q ðt 0 Þ = Q 0 initial condition ð cÞ ð8:14Þ where ½BðaÞij =

a2j - a2i a2j þa2i

is a 3 × 3 matrix function of the ellipsoid shape (ai). The ! vectorize operator A
B produces  !a matrix by multiplying corresponding elements of the matrices A and B, i.e., A
B = Aij
Bij, without summation with respect to the ij

repeated indices.

8.4

Analytical Solutions for Spheroids in Monoclinic Flows

Analytical solutions to Eq. (8.14) have been obtained for special situations. Jeffery (1922) himself provided analytical solutions for spheroids in simple shearing flows. Goldsmith and Mason (1967) reviewed the work on the rotation of rigid spheroids in simple shearing. A thorough treatment for the analytical solution of spheroids in monoclinic flows is given below, following Ježek et al. (1996) and Jiang (2007).

8.4.1

Equations Governing the Motion of Spheroidal Objects

The shape of an ellipsoid is described by the matrix ½BðaÞij =

a2j - a2i a2j þa2i

(Eqs. 8.13 and

8.14). A single parameter B is sufficient for spheroidal objects to define their shape. We shall denote the distinct semi-axis of a spheroid by a1. The three semi-axes are

8.4

Analytical Solutions for Spheroids in Monoclinic Flows

187

Fig. 8.2 The orientation of a spheroid in space can be defined by its distinct axis (a1) direction (θ and ϕ). The directions of a2 and a3 axes are respectively (θ þ π2,π2) and (θ + π,π2 - ϕ).This leads to the transformation matrix of Eq. (8.16)

a1 > a2 = a3 for a prolate object and a1 < a2 = a3 for an oblate object. B is defined below: B=

r2p - 1 a21 - a22 , = a21 þ a22 r2p þ 1

rp =

a1 a2

ð8:15Þ

Thus, for spheroids, |B| < 1 and 0 < rp < 1. B > 0 and rp > 1 are for prolate objects; B < 0 and rp < 1 are for oblate objects. B = - 1 (rp = 0) and B = 1 (rp = 1) are, respectively, for material planes and material lines. Because the orientation of a spheroid is defined by its distinct a1 axis, its orientation matrix Q is, in turn, also defined by the spherical angles (θ, ϕ) of a1 axis (Fig. 8.2): 0 B Q=@

cos θ sin ϕ

sin θ sin ϕ

- sin θ - cos θ cos ϕ

cos θ - sin θ cos ϕ

cos ϕ

1

C 0 A sin ϕ

ð8:16Þ

Differentiating Eq. (8.16) with respect to t gives: 0

- sin θ sin ϕθ_ þ cos θ cos ϕϕ_ B _ Q=@ - cos θθ_ sin θ cos ϕθ_ þ cos θ sin ϕϕ_

cos θ sin ϕθ_ þ sin θ cos ϕϕ_ - sin θθ_

- cos θ cos ϕθ_ þ sin θ sin ϕϕ_

- sin ϕϕ_

1

C A 0 _ cos ϕϕ ð8:17Þ

Submitting Eq. (8.17) into Eq. (8.12) yields Θ13 = θ_ sin ϕ , Θ32 = θ_ cos ϕ , and Θ21 = - ϕ_ . Therefore, differential Eq. (8.12) is reduced to the following two equations for θ and ϕ:

188

8

8 > Θ21 > > < sin ϕ θ_ = > > Θ32 > : cos ϕ _ϕ = Θ13

Rotation of Rigid Objects in Homogeneous Flows

ϕ ≠ 0 or π otherwise

ð8:18Þ

On the other hand, the angular velocity tensor for a spheroid can be obtained from Eq. (8.13): e 13 - BD e 13 Θ13 = W e 32 Θ32 = W

ð8:19Þ

e 21 þ BD e 12 Θ21 = W e ij = Qim Qjn W mn , are: The vorticity components in the ellipsoid axis coordinates, W e 32 = W 21 cos ϕ þ sin ϕðW 32 cos θ þ W 13 sin θÞ, W e 13 = W 13 cos θ - W 32 sin θ W e 21 = W 21 sin ϕ - cos ϕðW 32 cos θ þ W 13 sin θÞ W

ð8:20Þ

The strain rate components in Eq. (8.19) are obtained by tensor transformation e ij = Qim Qjn Dmn : D h i e 12 = sinϕ D22 -D11 sin2θþD12 cos2θ þ ðD23 cosθ-D13 sinθÞcosϕ D 2   D -D cos 2 θ-D22 sin 2 θ-D12 sin2θ 33 11 e 13 = sin2ϕ- ðD13 cosθþD23 sinθÞ cos2ϕ D 2 ð8:21Þ Substituting Eqs. (8.20) and (8.21) into (8.18) yields the following set of differential equations for the rotation of a spheroid in a general flow field:

8.4

Analytical Solutions for Spheroids in Monoclinic Flows

189

8 W 21 - cotϕ½ðW 32 -BD23 Þcosθþ ðW 13 þBD13 Þsinθ > > dθ >   > = > < dt - B D11 -D22 sin2θ-D cos2θ 12 2   > 2 2 > cosθ-W W > 13 32 sinθ-B D33 -D11 cos θ-D22 sin θ-D12 sin2θ sinϕcosϕ dϕ > > : dt = - ðD13 cosθþD23 sinθÞcos2ϕ ð8:22Þ

8.4.2

Solutions of Spheroidal Objects in Monoclinic Flows

Although simpler than the general Eqs. (8.14), (8.22) for rigid spheroids still cannot be solved analytically in general. Analytical solutions for Eq. (8.22) are possible only in special flows where many components of Dij and Wij vanish (Jeffery 1922; Ježek et al. 1996; Jiang 2007). Equation set (8.22) can be further simplified to allow for analytic solutions if the velocity field gradient tensor has a form like (Fig. 8.3):

Fig. 8.3 (a) Coordinate system used for the solution of a rigid spheroid in homogeneous monoclinic flows. The shear plane is parallel to the yz-plane, and the shear direction is along the y-axis. Shear sense is sinistral. The vorticity vector is parallel to the z-axis. The θ angle is the trend of the object’s distinct axis, measured with respect to the x-axis. The ϕ angle (not shown in this view) is between the object’s distinct axis and the z-axis. The strain rates along the coordinate axes are ε_ x, ε_ y, and ε_ z (not shown). The xy-plane is the VNS. (b) Mohr circle presentation for the flow on the VNS. The sectional vorticity number (Eq. 8.24) is Ws and Ws = cos 2θR

190

8

Rotation of Rigid Objects in Homogeneous Flows

0

ε_ x B L = @ γ_

0 ε_ y

0

0

1 0 C 0A ε_ z

ð8:23Þ

Such flows have a monoclinic symmetry, and the vorticity vector is parallel to the z-axis. Submitting components in the matrix of Eq. (8.23) into Eq. (8.22), reorganizing, and assuming volume-constant deformation (i.e., ε_ x þ ε_ y þ ε_ z = 0 ) yields: 8

 > dθ γ_ B > > cos 2ðθ þ θR Þ = 1þ < dt 2 Ws

 > dϕ B_γ sin 2ϕ 1 > > sin 2ðθ þ θR Þ - 3Z : dt = 4 Ws

ð aÞ ð8:24Þ ðbÞ

h i - 12 2 where W s = γ_ ε_ x - ε_ y þ γ_ 2 is the sectional vorticity number on the vorticitynormal section (Fig. 8.3, Li and Jiang 2011), θR = ± 0.5cos-1Ws (the positive and negative signs are for ε_ x - ε_ y > 0 and ε_ x - ε_ y < 0 situations respectively), and Z = ε_γ_z. Using a variable substitution γ_ dt = dγ and denoting Λ = WBs , Eq. (8.24) is further simplified to: 8 dθ 1 > ½Λ þ cos 2ðθ þ θR Þ = < dγ 2Λ sin 2ϕ > dϕ : ½ sin 2ðθ þ θR Þ - 3W s Z  = dγ 4Λ Equation (8.25) can be reorganized as

Rθ θ0

dθ Λþ cos 2ðθþθR Þ

ð aÞ ð bÞ =

Rγ 0

dγ 2Λ

ð8:25Þ

which can be inte-

grated (Ryshik and Gradstein 1963, p. 103). Equation (8.25) can be reorganized as Rγ h 3BZ sin 2ðθþθR Þi Rϕ 2dϕ dγ = - 2Λ þ = Λþ cosdθ2ðθþθR Þ from Eq. (8.25) dγ, which with 2Λ sin 2ϕ 2Λ

ϕ0

0

can also be integrated. The analytical solutions to Eq. (8.25) are as follows:

8.5

The Behavior of Rigid Spheroids in Monoclinic Flows

191

tan ðθ þ θR Þ 8 "pffiffiffiffiffiffiffiffiffiffiffiffiffi # ( pffiffiffiffiffiffiffiffiffiffiffiffiffi) 2 > 1 tan ð θ þ θ Þ Λ Λ þ 1 γ Λ2 - 1 > 0 R 1 > pffiffiffiffiffiffiffiffiffiffiffiffiffi tan tan > þ if Λ2 > 1ðaÞ > 2 > Λ þ 1 2Λ > Λ -1 > > > > > > tan ðθ0 þ θR Þ þ γ if Λ = 1ðbÞ > > > > -1 < ð cot ðθ0 þ θR Þ - γ Þ if Λ = - 1ðcÞ = ! r ffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffi p > > > 1þΛ γ 1 - Λ2 > > tanh tan ðθ0 þ θR Þ þ > > 1-Λ 2Λ > > > > if Λ2 < 1ðdÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffi! r ffiffiffiffiffiffiffiffiffiffiffi ffi > > 2 > γ 1-Λ 1-Λ > > tan ðθ0 þ θR Þ tanh >1 þ : 2Λ 1þΛ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   Λ þ cos 2ðθ þ θ Þ 3BZγ 0 R tan ϕ = tan ϕ0 exp j j 2 Λ þ cos 2ðθ þ θR Þ

8.5

ð8:26Þ ð8:27Þ

The Behavior of Rigid Spheroids in Monoclinic Flows

∘ By analyzing Eq. (8.25), one can see that dϕ dγ = 0 if ϕ = 0, ϕ = 90 , or sin2(θ + θ R) 3WsZ = 0, and dθ dγ = 0, if Λ + cos 2(θ + θR) = 0. These are special orientations of the object. With Eqs. (8.26) and (8.27), we can thoroughly examine the behavior of spheroids in monoclinic flows. The rotational behavior for rigid spheroids in monoclinic flows is rich, depending on their shapes, initial orientations, and flow field properties.

Behavior 1: Permanently Rotating Objects (Λ2 > 1) In the situation of Λ2 > 1 (i.e., |B| < Ws), θ changes with increasing γ according to Eq. (8.26): Λþ1 tan ðθ þ θR Þ = pffiffiffiffiffiffiffiffiffiffiffiffiffi tan Λ2 - 1

( tan

-1

# "pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi) Λ2 - 1 tan ðθ0 þ θR Þ γ Λ2 - 1 þ 2Λ Λþ1

 pffiffiffiffiffiffiffiffiffiffiffiffiffi - 1 . In terms of a finite strain which has a periodicity of T = 4πΛ γ_ Λ2 - 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi measure, this periodicity can be expressed as γ P = T γ_ = 4πΛ= Λ2 - 1. The object’s distinct axis returns to the same γ P position for every increment of the deforming fluid. The corresponding ϕ changes with strain γ according to Eq. (8.27). The behavior is more complicated depending on the BZ term.

192

8

Rotation of Rigid Objects in Homogeneous Flows

For a plane-straining flow (Z = 0), tan ϕ = tan ϕ0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos 2ðθ0 þθR Þ jΛþ Λþ cos 2ðθþθR Þ j. ϕ also has the

same periodicity as θ. Thus, the rotational path for a spheroid in a plane-straining flow is an ellipse called the Jeffery orbit. The principal directions of the orbit ellipse are defined by the two θ values when ϕ reaches the maximum and minimum values, respectively. The principal long axis of the orbit of a prolate object is inclined at θR = ± 0.5cos-1Wsrelative to the shear direction (y-axis). The positive and negative signs are for ε_ x - ε_ y > 0 and ε_ x - ε_ y < 0 situations, respectively. In a simple shearing flow, for instance, θR = 0 and the long axis of the orbit for a prolate object is parallel to the shear direction (Fig. 8.4a). An oblate object having rp reciprocal to a prolate object has identically-shaped Jeffery orbits but with the principal long dimension rotated 90∘ to the prolate orbit (Fig. 8.4b, c, d). Objects of the same rp but different initial orientations rotate on different orbits.   term in Eq. (8.27) decreases (if BZ > 0) In a flow field with Z ≠ 0, the exp - 3BZγ 2 or increases (if BZ < 0) exponentially. The ϕ angle will thus drift to the vorticity axis (where BZ > 0) or to the VNS (ϕ=90∘, where BZ < 0). Therefore, the rotation paths become elliptical spirals, either converging to the vorticity axis (if BZ > 0) or diverging to the vorticity normal section (if BZ < 0) (Fig. 8.4c, f–h). For instance, the distinct axis of a prolate object in a Z > 0 field will follow an elliptical spiral path converging toward the vorticity axis (Fig. 8.4c), whereas an oblate object in the same flow will have its distinct axis following a spiral path toward the VNS (Fig. 8.4f). In either case, the object will rotate permanently even though the ϕ angle has reached the ultimate value. A prolate object parallel to the vorticity vector rotates at a steady angular velocity equal to 0:5_γ; an oblate object (with |B| < Ws) with its distinct axis lying on the vorticity normal section rotates at a pulsating angular velocity with the fluid vorticity. Behavior 2: Objects Having Special Paths (Λ2 = 1) In this case, the θ evolves according to the second and third equations in Eq. (8.26): tan(θ + θR) = tan (θ0 + θR) + γ for prolate objects and cot(θ + θR) = cot (θ0 + θR) - γ for oblate objects. The associated ϕ evolves according to Eq. (8.26) which in the current situation is reduced to:  tan ϕ = exp tan ϕ0 prolate objectsðaÞ  tan ϕ = exp tan ϕ0 oblate objectsðbÞ

 sin ðθ - θ0 Þ cos ðθ0 þ θR Þ 3BZ j j, 2 cos ðθ þ θR Þ cos ðθ0 þ θR Þ cos ðθ þ θR Þ  sin ðθ - θ0 Þ sin ðθ0 þ θR Þ 3BZ j j, 2 sin ðθ0 þ θR Þ sin ðθ þ θR Þ sin ðθ þ θR Þ

ð8:28Þ

For a prolate object in plane-straining flows (Z = 0 of Eq. (8.28)), the rotation paths cos ðθ0 þθR Þ tan ϕ are defined by tan ϕ0 = j cos ðθþθR Þ j, which are great circles with the same strike of θ þ θR = ± π2 . In a thinning zone flow, the common strike line is 0.5cos-1Ws antithetic to the shear plane (Fig. 8.5a, b). For an oblate object in plane-straining sin ðθ0 þθR Þ tan ϕ flows, the path is defined by tan ϕ = j sin ðθþθR Þ j, which are the same great circles but 0

8.5 The Behavior of Rigid Spheroids in Monoclinic Flows

193

Fig. 8.4 Rotation paths for spheroids in monoclinic thinning zone flows with Λ2 > 1. The shear direction is north-south, and the sense of shear is sinistral. (a) Ws = 1, B = 0.8, and Z = 0. Prolate objects follow elliptical Jeffery orbits with the long dimension of the orbits parallel to the shear direction in simple shearing (Ws = 1). (b) Ws = 0.866, B = 0.8, and Z = 0. Prolate objects follow Jeffery orbits with the long dimension of the orbits inclined antithetic (0.5cos-10.866 = 15∘) to the shear direction. (c) Ws = 0.866, B = 0.8, and Z = 0.5. Prolate objects rotate to the vorticity vector. (d–f) are for oblate objects. (d) Ws = 1, B = - 0.8, and Z = 0. (e) Ws = 0.866, B = - 0.8, and Z = 0. (f) Ws = 0.866, B = - 0.8, and Z = 0.5. The distinct axis rotates to the VNS and becomes stable at 75∘ (=90∘ - 0.5cos-10.866) synthetic to the shear direction. (g) The same as (c) except Z = - 0.5. The prolate objects rotate toward the VNS and become stable at 15∘ antithetic to the shear direction. (h) The same as (f) except Z = - 0.5. The oblate’s distinct axis rotates toward the vorticity vector and continues to spin with the vorticity

194

8

Rotation of Rigid Objects in Homogeneous Flows

Fig. 8.5 Rotation paths for spheroids in monoclinic thinning zone flows with Λ2 = 1. (a) Ws = 1, B = 1, and Z = 0. (b) Ws = 0.866, B = 0.866, and Z = 0. (c) Ws = 0.866, B = 0.866, and Z = 0.5. (d) W s = 1, B = - 1, and Z = 0. (e) Ws = 0.866, B = - 0.866, and Z = 0. (f) Ws = 0.866, B = - 0.866, and Z = 0.5. (g) The same as (c) except Z = - 0.5. (h) The same as (f) except Z = - 0.5. In (a, b, d, and e), the rotation paths follow great circles. In (c), the prolate object rotates toward the vorticity vector and continues to spin with vorticity. In (f), the oblate object axis rotates to the VNS toward a transit position at 75∘ (= 90∘ - 0.5cos 1 0.866) synthetic to the shear direction. In (g) the prolate object rotates toward the VNS and stops at 0.5cos -10.866 = 15∘ antithetic to the shear. In (h) the oblate object’s distinct axis eventually aligns with the vorticity vector and continues to spin with vorticity -1

with a common strike of θ + θR = 0 or π (or π2 - cos 2 W s syn-thetic to the shear plane) (Fig. 8.5d, e). That is, oblate paths are orthogonal to their counterpart prolate paths.

8.5

The Behavior of Rigid Spheroids in Monoclinic Flows

195

In thinning zones (Jiang and Williams 1998) with stretching along the vorticity vector (Z > 0), a prolate object rotates with vorticity toward but never passes the trending of 0.5cos-1Ws antithetic to the shear direction. At the same time, the object axis gets progressively closer to the vorticity vector (Fig. 8.5c). Ultimately the prolate axis becomes parallel to the vorticity axis, and the object continues rotating around its own distinct axis at a rate of 0:5_γ . An oblate object’s distinct axis rotates toward the VNS at a trend of 0.5cos-1Ws off the shear plane normal (x-axis) in the sense of vorticity (Fig. 8.5f). This is a quasi-stable position. The oblate’s distinct axis will remain on the VNS (the xy-plane) but rotate at a pulsating rate with the vorticity. In thinning zones with shortening along the vorticity vector (Z < 0), prolate objects rotate away from the vorticity vector toward a quasi-stable position trending at 0.5cos-1Ws antithetic to the shear plane on the VNS (Fig. 8.5g). Oblate objects rotate toward aligning their distinct axis with the vorticity vector and continue to spin with vorticity (Fig. 8.5h). Behavior 3: Objects Having Stable Positions (Λ2 < 1) In the situation of Λ2 < 1 (i.e., |B| > Ws), θ changes with increasing γ according to the fourth Equation in Eq. (8.26): qffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffi2ffi γ 1-Λ tan ðθ0 þ θR Þ þ 11þΛ tanh -Λ 2Λ qffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffi2ffi tan ðθ þ θR Þ = γ 1-Λ -Λ 1 þ 11þΛ tan ðθ0 þ θR Þ tanh 2Λ As strain increases, the tanh

 pffiffiffiffiffiffiffiffiffi2ffi γ 1-Λ 2Λ

term tends to be 1 for prolate objects and -1

for oblate objects. Thus, the ultimate angle θ1 is given by the following expression: qffiffiffiffiffiffiffiffi tan ðθ0 þ θR Þ ± 11þΛ -Λ qffiffiffiffiffiffiffiffi tan ðθ1 þ θR Þ = -Λ 1 ± 11þΛ tan ðθ0 þ θR Þ

ð8:29Þ

where the positive and negative signs are for prolate and oblate objects, respectively. qffiffiffiffiffiffiffiffi -Λ = tan θc , Eq. (8.29) can be Denoting θc = 0.5cos-1Λ and, therefore 11þΛ simplified to: tan ðθ1 þ θR Þ =

tan ðθ0 þ θR Þ ± cot θc = ± cot θc 1 ± tan θc tan ðθ0 þ θR Þ

ð8:30Þ

π þ nπ 2

ð8:31Þ

The above Equation implies: θ1 þ θR ± θc = ± where n = 0, 1, 2, . . . .

196

8

Rotation of Rigid Objects in Homogeneous Flows

Therefore, a prolate object in a flow field with ε_ x - ε_ y < 0 (such as in a monoclinic thinning zone, Jiang and Williams 1998) will reach an ultimate trending (note that θR = - 0.5cos-1Ws in this case):   π 1 π W 1 cos - 1 W s - cos - 1 Λ þ = cos - 1 W s - cos - 1 j s j þ ð8:32Þ B 2 2 2 2   This trend is 12 cos - 1 W s - cos - 1 jWBs j antithetic to the shear plane along the y-axis in our analysis (Fig. 8.3). On the other hand, an oblate object in the same flow field with ε_ x - ε_ y < 0 will reach a final trend: θ1 =

θ1 =

  π 1 1 W cos - 1 W s þ cos - 1 Λ - = cos - 1 W s - cos - 1 j s j B 2 2 2

ð8:33Þ

which is orthogonal to the stable orientation of its prolate counterpart, of course. To see the limit of Eq. (8.27) as strain increases in the situation Λ2 < 1, we must express γ in the Equation in terms of θ. Taking the logarithm of Eq. (8.27) yields: ln

Because

γ 2Λ

=

Rθ θ0

Λ þ cos 2ðθ0 þ θR Þ tan ϕ 3BZγ 1 =þ ln j j tan ϕ0 2 2 Λ þ cos 2ðθ þ θR Þ

dθ Λþ cos 2ðθþθR Þ

ð8:34Þ

from Eq. (8.25), under the condition of Λ2 < 1, this

gives the following expression:  2Λ γ = pffiffiffiffiffiffiffiffiffiffiffiffiffi tanh - 1 ½ tan θc tan ðθ þ θR Þ - tanh - 1 ½ tan θc tan ðθ0 þ θR Þ 2 1-Λ   qffiffiffiffiffiffiffi With the identity exp tanh - 1 ðxÞ = 11þx - x , the above relation can be written in the following form: Λ γ = pffiffiffiffiffiffiffiffiffiffiffiffiffi ln 1 - Λ2



cos ðθ þ θR - θc Þ cos ðθ0 þ θR þ θc Þ cos ðθ þ θR þ θc Þ cos ðθ0 þ θR - θc Þ

 ð8:35Þ

Inserting Eq. (8.35) into (8.35) and organizing gives: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #( )ffi u" χ -1 u cos ðθ0 þ θR - θc Þχþ1 ð þ θ Þ cos θ þ θ R c tan ϕ = tan ϕ0 tj j cos ðθ0 þ θR þ θc Þχ - 1 cos ðθ þ θR - θc Þχþ1 s ffi ffiffiffiffiffiffiffiffiffi where χ = p3ZW . 2

1-Λ

ð8:36Þ

8.5

The Behavior of Rigid Spheroids in Monoclinic Flows

197

The explicit Eq. (8.36) allows for investigating the ultimate value ϕ1 associated with θ1. For plane-straining flows (χ = 0), from Eq. (8.36) tanϕ1 is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos ðθ0 þ θR - θc Þ cos ðθ0 þ θR þ θc Þ tan ϕ1 = tan ϕ0 j j cos ðθ1 þ θR þ θc Þ cos ðθ1 þ θR - θc Þ

ð8:37Þ

Combining Eq. (8.31) into (8.37), we have ϕ1 = π2. Therefore, the distinct axis will ultimately be on the VNS, trending as θ1 given by Eq. (8.32) if it is a prolate object or trending as θ1 given by Eq. (8.33) if it is an oblate object. For 3D monoclinic flows (χ ≠ 0), the final ϕ value has three possible outcomes depending on χ. For prolate objects, as θ1 þ θR þ θc = π2 þ nπ (Eqs. 8.30, 8.36) gives the following results for the ultimate ϕ value:

tan ϕ1 =

8 > > > < > > > :

0 cos ðθ0 þ θR - θc Þ tan ϕ0 j j sin 2θc π 2

if χ > 1 if χ = 1

ð8:38Þ

if χ < 1

For oblate objects, as θ1 þ θR - θc = - π2 þ nπ (Eq. 8.30), the ultimate ϕ value is obtained from Eq. (8.36): 8 > > >
> > π : 2

if χ < - 1 if χ = - 1

ð8:39Þ

if χ > - 1

Thus, the distinct axis may eventually be parallel to the vorticity, on the VNS, or at an inclined angle determined by the shape and initial orientation of the object and the flow field property. In the event the distinct axis reaches the vorticity vector eventually (ϕ1=0), it continues to spin around its distinct axis with the vorticity of the flow field. This contrasts with the situations where the distinct axis ultimately reaches the vorticitynormal section or the inclined orientation (if χ = ± 1) when the object is truly stable. The behaviors and rotation paths for situations Λ2 < 1 are shown in Fig. 8.6 for prolate objects, Fig. 8.7 for oblate objects, and Fig. 8.8 in the special case where χ = ± 1.

198

8

Rotation of Rigid Objects in Homogeneous Flows

Fig. 8.6 Rotation paths for prolate objects in monoclinic thinning zone flows with Λ2 < 1. (a) Ws = 0.866, B = 1, and Z = 0. (b) Ws = 0.866, B = 0.9, and Z = 0. (c) Ws = 0.866, B = 0.9, and Z = 0.5. (d) Ws = 0.866, B = 0.9, and Z = - 0.5. In both (a, b), the object reaches a stable position  on the VNS at 0:5 cos - 1 W s - cos - 1 WBs antithetic to the shear plane. In (c), the object rotates with vorticity and  toward parallel with the  vorticity vector. In (d), the object rotates toward the VNS and stops at 0:5 cos - 1 W s - cos - 1 WBs antithetic to the shear direction. The object rotates with vorticity if lying in the forward rotation sector against vorticity if lying in the backward rotation sector

Fig. 8.7 Rotation paths for oblate objects in monoclinic thinning zone flows with Λ2 < 1. (a) Ws = 0.866,  B = -0.9, and Z = 0. The distinct axis rotates toward the stable position at

s 90∘ - 0:5 cos - 1 W s - cos - 1 W jBj synthetic to the shear direction on the VNS. (b) Ws = 0.866,

B = - 0.9, and Z = 0.5. The distinct axis also rotates toward the stable position as in (a), but via different paths. (c) Ws = 0.866, B = - 0.9, and Z = - 0.5. The distinct axis rotates toward the vorticity direction and continues to spin with vorticity

8.6

Numerical Approach

199

Fig. 8.8 In the special case where Λ2 < 1 and χ = ± 1, the object’s distinct axis rotates to a final stable position determined by the initial conditions of the object and the flow field parameters. (a) Ws = 0.866, B = 0.9, Z = 0.106, and thus χ = 1. The object’s long axis rotates toward a plane striking at 0:5 cos - 1 W s - cos - 1 WBs antithetic to the shear direction. The ultimate plunge angle of the axis is determined by Eq. (8.38). (b) Ws = 0.866, B =  0.9, Z = - 0.106, and therefore s χ = - 1. The distinct axis rotates toward a plane striking 0:5 cos - 1 W s - cos - 1 W jBj counter-

clockwise to the shear plane normal. The ultimate plunge angle is determined by Eq. (8.39)

8.6

Numerical Approach

In the general situation where the object is not a spheroid and/or the flow field is not monoclinic, no analytical solutions for rigid ellipsoid rotation in ductile flows are possible. The problem can be solved numerically. Traditionally, the orientation of a rigid ellipsoid is defined by a set of three Euler angles (such as the set of α, β, and γ in Chap. 2). The differential equations for the angular velocity of the ellipsoid are then three coupled non-linear differential equations for the Euler angles. A far more efficient approach to solve for rigid ellipsoid rotation is to solve the single differential equation of Q from Eq. (8.14). The initial orientation of the object is still defined by a set of Euler angles, but they give the initial orientation Q(0) (Eq. 8.14). From the many numerical methods that exist, we describe the Runge-Kutta fourth-order method, the Rodrigues method, and the Runge-Kutta-Rodrigues method to solve Eq. (8.14) below.

8.6.1

The Runge-Kutta Method

The Runge-Kutta fourth-order method is an accurate and flexible method based on a Taylor series approximation. For the initial value problem of Eq. (8.14), the RungeKutta algorithm for determining the approximation of Qn + 1 from the state of Qn is:

200

8

Rotation of Rigid Objects in Homogeneous Flows

1 Qnþ1 = Qn þ ðK1 þ 2K2 þ 2K3 þ K4 Þ 6

ð8:40Þ

where Ki (i = 1, 2, 3, 4) are calculated by the following procedure: K1 = - δtΘðQn ÞQn   δt c ← I - ΘðQn Þ Q 2 n   d ← I - ΘðQn Þδt Qn   1 K2 = - δtΘðcÞ Qn þ K1 2   1 K3 = - δtΘðcÞ Qn þ K2 2 K4 = - δtΘðd ÞðQn þ K3 Þ

ð8:41Þ

where δt is the step time for computation. It is the time increment between Qn and Qn + 1 states. The Runge-Kutta method is a very accurate method; the local  error indetermining  Qn + 1 from Qn is equivalent to the fifth order of δt Θðt n Þ , where Θðt n Þ is an appropriate norm, such as the Euclidean norm, of Θðt n Þ. Thus, the choice of δt must ensure that infinitesimal rotation, i.e.,   each step of computation represents  an  δt Θðt n Þ < < 1 , so that the fifth order of δt Θðt n Þ is a tolerable error for computation.

8.6.2

Rodrigues Rotation Approximation

Another numerical approach to solve the set of Eq. (8.14) is to use the Rodrigues rotation (Başar and Weichert 2000, p. 31). If the angular velocity tensor Θ is assumed to be constant within a time increment δt, the solution to Eq. (8.14) is: Qðt þ δt Þ = exp ðΩÞQðt Þ

ð8:42Þ

where Ω = - ΘðQn Þδt is an anti-symmetric tensor, and therefore, the Rodrigues relation (Eq. 1.32) can be used. The approximation of Qn + 1 from the state Qn is:    b þ ð1 - cos ωÞ
ω b 2
Qn ð8:43Þ Qnþ1 = exp - ΘðQn Þδt
Qn = I þ sin ω
ω where ω = δt

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 b 2 Θij Θij and ω =

- Θδt ω .

This approximation is computationally more efficient than the Runge-Kutta method, but it is not as accurate (Jiang 2013). It is found that in simple shearing

8.6

Numerical Approach

201

flows, the time step must be about half of the one used for the Runge-Kutta method if a similar level of accuracy is desired.

8.6.3

Runge-Kutta-Rodrigues Approximation

One can use the Rodrigues relation in the incremental calculation of the Runge-Kutta method to achieve an extremely accurate, but of course, computational more costly, approximation. As Eq. (8.14) can be written as: d lndt Q = - Θ. Note that Θ is a function of Q and t. One can construct a Runge-Kutta approximation with the increments at the middle and end points of a time step calculated with the Rodrigues relation. Specifically, the angular velocity tensor is calculated using Eq. (8.43): Qnþ1 = exp ðθδt Þ
Qn

ð8:44Þ

where the exponential term is defined by Rodrigues relation in Eq. (1.32) and θ=

    1 θ1 þ Qn 2 q2 T θ2 q2 þ q3 T θ3 q3 þ q4 T θ4 q4 Qn T 6

ð8:45Þ

The procedure to calculate the terms in the above expression is as follows: θ1 = - ΘðQn Þ, angular velocity at start   θ δt Qn , one orientation at middle time q1 = exp 1 2 one angular velocity at middle point θ2 = - Θðq1 Þ,   θ δt Qn , one orientation at middle point q2 = exp 2 2 θ3 = - Θðq2 Þ,

ð8:46Þ

another angular velocity at middle point

q3 = exp ðθ3 δt ÞQn , orientation at end point angular velocity at end θ4 = - Θðq3 Þ q4 = exp ðθ4 δt ÞQn orientation at end

8.6.4

Implementation

It is straightforward to implement the above algorithms in a mathematics application. A Mathcad worksheet called “RigidEllipsoidSingle.mcdx” and a MATLAB program called “RigidEllipsoidSingle.m” are based on the algorithm presented above. The use

202

8 Rotation of Rigid Objects in Homogeneous Flows

of the programs is self-explanatory in the worksheet and through annotations in the MATLAB code. To use the Mathcad worksheet, first provide the input variables, which include the following: • The three semi-axis lengths (a1,a2, and a3) of the ellipsoid as a 3 × 1 column matrix • The initial orientation of the ellipsoid, (θ, ϕ, ϑ) as a 3 × 1 column matrix • The flow velocity gradient tensor, L, as a 3 × 3 matrix • The step length δt for computation. • The total steps of computation, STEPS. The total actual time duration of deformation is therefore STEPS∙δt. • The number of steps of computation, mm, between outputs. As it is unnecessary to output the results of every computational step, the parameter mm specifies the number of steps of calculation between output states. For example, for a simulation with the total STEPS set at 5000, if one wants an output set for the ellipsoid state every 50 steps of calculations, mm is set to 50, and the total number of output sets will be 100 (STEPS/mm). The choice of the step length δt will determine the accuracy of the computation. One must choose a δt small enough so that each step of calculation corresponds to an infinitesimal rotation. In the Runge-Kutta method, the  local error  in determining Q(tn + 1) from Q(tn) is equivalent to the fifth order of δt Θðt n Þ , where kAk is an appropriate normffi of the tensor A. One simple choice is the Euclidean norm qffiffiffiffiffiffiffiffiffiffiffiffi kAkE = 12 Aij Aij. Numerical experiments indicate that for the worksheets provided,   if δt ΘE < 0:05, the local error is so small that further reducing the time step leads to no significant difference in the result after many thousand steps of computation equivalent to a shear strain of 100 for a progressive simple shear case.

8.7

Notes and Key References

The following three Mathcad worksheets are associated with this Chapter: RigidEllipsoidSingle.mcdx, as the name indicates, calculates the rotation path of a single rigid ellipsoid in a given viscous flow, defined by a constant velocity gradient tensor. The angular velocity is from Jeffery (1922). The MATLAB program RigidEllipsoidSingle.m is nased on the same algorithm. RigidEllipsoidGroup.mcdx calculates the evolution of shape-preferred orientations of a group of non-interacting rigid ellipsoids. SpheroidAnalytic.mcdx is based on analytical solutions of the rotation path of a given prolate or oblate object. The classic work of Jeffery (1922) is a thorough investigation of the kinematics and dynamics of a single rigid ellipsoid in a Newtonian viscous fluid. The angular velocity equation (Eq. 37 of Jeffery 1922) is used in this Chapter. Both theoretical

References

203

consideration (Bretherton 1962; Willis 1977) and experimental investigations (Arbaret et al. 2001; Ferguson 1979; Ghosh and Ramberg 1976; Goldsmith and Mason 1967; Taylor 1923; Trevelyan and Mason 1951) have shown that Jeffery’s angular equations are applicable to rigid non-ellipsoidal objects, if a correction factor is used for the object’s shape. Ježek et al. (1996) solved rigid spheroid rotation in monoclinic flows. Ježek (1994) published a FORTRAN program for the rotation of rigid ellipsoids. Passchier (1987) analyzed the stable positions of rigid spheroids in noncoaxial flows. Jiang (2007) published Mathcad worksheets for modeling the rotation of rigid ellipsoids. Rigid object rotation was described by three coupled differential equations of three Euler (or spherical) angles. Jiang (2012) reduces the coupled equations to a single differential equation for Q and simplifies the algorithm to solve for the motion of rigid (and deformable) ellipsoids.

References Arbaret L, Mancktelow NS, Burg JP (2001) Effect of shape and orientation on rigid particle rotation and matrix deformation in simple shear flow. J Struct Geol 23(1):113–125. https://doi.org/10. 1016/S0191-8141(00)00067-5 Başar Y, Weichert D (2000) Nonlinear continuum mechanics of solids: fundamental mathematical and physical concepts. Springer Science & Business Media Bretherton FP (1962) The motion of rigid particles in a shear flow at low Reynolds number. J Fluid Mech 14(2):284–304. https://doi.org/10.1017/S002211206200124X Ferguson CC (1979) Rotations of elongate rigid particles in slow non-Newtonian flows. Tectonophysics 60(3–4):247–262. https://doi.org/10.1016/0040-1951(79)90162-8 Ghosh SK, Ramberg H (1976) Reorientation of inclusions by combination of pure shear and simple shear. Tectonophysics 34(1–2):1–70. https://doi.org/10.1016/0040-1951(76)90176-1 Goldsmith HL, Mason SG (1967) The microrheology of dispersions, Rheology: theory and applications, vol 4. Academic Press, New York, pp 5–250 Jeffery GB (1922) The motion of ellipsoidal particles in a viscous fluid. Proc R Soc Lond Ser A-Containing Papers of a Mathematical and Physical Character 102(715):161–179. https://doi. org/10.1098/rspa.1922.0078 Ježek J (1994) Software for modeling the motion of rigid triaxial ellipsoidal particles in viscous flow. Comput Geosci 20(3):409–424 Ježek J, Schulmann K, Segeth K (1996) Fabric evolution of rigid inclusions during mixed coaxial and simple shear flows. Tectonophysics 257(2–4):203–221. https://doi.org/10.1016/0040-1951 (95)00133-6 Jiang D (2007) Numerical modeling of the motion of rigid ellipsoidal objects in slow viscous flows: a new approach. J Struct Geol 29(2):189–200. https://doi.org/10.1016/j.jsg.2006.09.010 Jiang D (2012) A general approach for modeling the motion of rigid and deformable ellipsoids in ductile flows. Comput Geosci 38(1):52–61. https://doi.org/10.1016/j.cageo.2011.05.002 Jiang D (2013) The motion of deformable ellipsoids in power-law viscous materials: formulation and numerical implementation of a micromechanical approach applicable to flow partitioning and heterogeneous deformation in Earth’s lithosphere. J Struct Geol 50:22–34. https://doi.org/ 10.1016/j.jsg.2012.06.011 Jiang D, Williams PF (1998) High-strain zones: a unified model. J Struct Geol 20(8):1105–1120. https://doi.org/10.1016/S0191-8141(98)00025-X Li C, Jiang D (2011) A critique of vorticity analysis using rigid clasts. J Struct Geol 33(3):203–219. https://doi.org/10.1016/j.jsg.2010.09.001

204

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Rotation of Rigid Objects in Homogeneous Flows

Passchier CW (1987) Stable positions of rigid objects in noncoaxial flow – a study in vorticity analysis. J Struct Geol 9(5–6):679–690. https://doi.org/10.1016/0191-8141(87)90152-0 Ryshik IM, Gradshteĭn IS (1963) Tables of series, products, and integrals. Pleumm Taylor GI (1923) The motion of ellipsoidal particles in a viscous fluid. Proc R Soc Lond Ser A-Containing Papers of a Mathematical and Physical Character 103(720):58–61. https://doi.org/ 10.1098/rspa.1923.0040 Trevelyan BJ, Mason SG (1951) Particle motions in sheared suspensions. 1. Rotations J Coll Sci 6(4):354–367. https://doi.org/10.1016/0095-8522(51)90005-0 Willis DG (1977) Kinematic model of preferred orientation. Geol Soc Am Bull 88(6):883–894. https://doi.org/10.1130/0016-7606(1977)882.0.CO;2

Chapter 9

Further Analysis of Spheroids in Simple Shearing Flows

Abstract In Chap. 8, we investigated the kinematics of rigid ellipsoids in slow flows, particularly spheroids in monoclinic flows. We focus on spheroids in simple shearing flows in this Chapter. We develop analytic expressions for the stress in a prolate object embedded in an isotropic Newtonian material under simple shearing. Simple shearing flows are the most studied flow type in materials science and geology literature. The results of this Chapter will be used later to verify a more general investigation of an elastic prolate object in a Newtonian viscous flow.

9.1

Jeffery Orbits

The history of θ is given by Eq. (8.26). For simple shearing specifically, the expression is simplified to:
   γrp tan θ0 þ 2 tan θ = rp tan tan - 1 rp rp þ 1

ð9:1Þ

The period of rotation is:    2π  rp þ rp - 1 or γ p = γ_ T = 2π rp þ rp - 1 γ_ γ p ≈ 2πrp , for rp >> 1 2π , for rp 1), ϕ1 is the maximum value and ϕ2 the minimum. For an oblate object, it is the opposite. It varies between 0 and 1. C = 0 corresponds to the situation where the distinct axis is parallel to the vorticity. The object spins around with the vorticity axis at a steady angular velocity of 0:5_γ . C = 1 is where the distinct axis lies on the VNS. The object rotates around vorticity, with the a1 axis lying on the VNS. The object does not spin around its distinct axis. 0 < C < 1 defines various elliptical orbits. For this reason, C is referred to as the Jeffery orbit constant, or simply the orbit constant. For a prolate object, the long dimension of the orbit ellipses is parallel to the shear direction. For an oblate object, the long dimension of the orbit ellipses is perpendicular to the shear direction (Fig. 9.1).

9.2

Revolution Around the Distinct Axis

The rotation of an object in space is the change of its orientation with time, which is the Jeffery orbits for spheroids. Revolution (Jiang and Williams 2004; Jiang 2007), on the other hand, is the accumulated amount of rotation a principal axis has revolved around itself. Inclusion trail microstructures encapsulated within syn-kinematic porphyroblasts, such as snowball garnets (Williams and Jiang 1999), result from revolution (Fig. 9.2).

9.2

Revolution Around the Distinct Axis

207

Fig. 9.1 (a) The coordinate system used. The shear direction is parallel to the x2 axis. (b) Jeffery orbits for the rotation of a prolate object with rp = 3 in a simple shearing flow. The rotation paths for a spheroid are ellipses called its Jeffery orbit. Objects of the same rp but with different initial orientations follow different orbits characterized by their orbit constant 0 ≤ C ≤ 1. The distinct axis is parallel to the vorticity (the x3 axis) when C = 0 and on the VNS (the x1x2 plane) when C = 1. Jeffery orbits for an oblate object of aspect ratio rp are 90∘ rotation of those for the prolate object with aspect ratio 1/rp

Fig. 9.2 Curved inclusion trails in a porphyroblastic garnet from the Tay Nappe recorded the revolution of the garnet as it overgrows a foliation. Notice the continuity of the ilmenite from the foliation outside the garnet to the interior of the garnet. The rotation is counter-clockwise and is around 220∘. The thin section was kindly provided by the late Prof. Peter Stringer of the Univerity of New Brunswick

The angular velocity around the distinct a1 axis of a spheroid is given by e 32 (Eq. 8.19). Because the simple shearing velocity gradient tensor is Θ32 = W 0 1 0 0 0 B C (Fig. 9.3) Lij = @ γ_ 0 0 A and Qij is given in Eq. (8.16), it is trivial with 0 0 0 e ij = Qim Qjn W mn , to get: W

208

9

a

Further Analysis of Spheroids in Simple Shearing Flows

b

C=0 360 360 C=0.5 330 330 C=0 300 300 C=0.5 =2 =5 r r C=1 p p 270 270 240 240 C=1 Rv210 Rv 210 180 180 150 150 120 120 C=5 90 90 C=5 60 60 30 30 C=10 C=10 0 0 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 J J

Fig. 9.3 Revolution (Rv) in degree as a function of orbit constant for two prolate objects with rp = 2 (a) and rp = 5 (b), respectively. Where C = 0, the object spins like a spherical object and revolution increases linearly according to Rv = 0.5γ. Revolution decreases as the object axis is tilted from the vorticity. This effect is pronounced as the object is more elongated (greater rp)

e 32 = γ_ cos ϕ Θ32 = W 2

ð9:5Þ

Combining Eqs. (9.4) and (9.4) yields the following expression for cosϕ (Forgacs and Mason 1959): 1 cos ϕ = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi



   γrp γr p 1 þ C 2 cos 2 r2 p þ1 þ 1 þ rp 2 C 2 sin 2 r2 p þ1

ð9:6Þ

Therefore, the revelation around the a1 axis is the time integration of Eq. (9.5): Zt Rv = 0

γ_ 1 cos ϕdt = 2 2

Zγ 0

dγ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h

i   γrp 1 þ C 2 1 þ rp 2 - 1 sin 2 r2 p þ1

ð9:7Þ

In the C = 0 case (the a1 axis parallel to the vorticity), Θ32 jC = 0 = 0:5_γ and Rv = 0.5γ. In the case of C = 1, the object has zero angular velocity of spin (Θ32 jC = 1 = 0) and Rv = 0. At the intermediate orbit constant 0 < C < 1, the amount of revolution around a1 achieved by the object at a given shear strain can be obtained by numerical evaluation of Eq. (9.7). Figure 9.3 shows the dependence of revolution on the shape and initial orientation, which determines the orbit constant. Rγp For one complete rotation (γ = γ p), the revolution is Rv = 12 cos ϕdγ which can 0

be expressed as Anczurow and Mason (1967):

9.3

Rotation of a Population of Rigid Spheroids

209

8  2  2 rp þ 1 > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffi E ðζ 1 Þ, > > 2   < rp C þ 1   Rv γ p = 2 rp 2 þ 1 > > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eðζ 2 Þ, > : rp C 2 r p 2 þ 1

for rp < 1 for rp > 1

ð9:8Þ

π

R2

dx ffi is the complete elliptic integral of the first kind, ζ 1 = where Eðζ Þ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 - ζ 2 sin 2 x 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi ffi C 2 ð1 - r p 2 Þ rp 2 - 1 C and ζ . = 2 2 2 2 r C þ1 C þr p p

9.3

Rotation of a Population of Rigid Spheroids

In the ideal situation of a viscous fluid with dilute rigid spheroid particles under simple shearing, each particle rotates on its Jeffery orbit continuously until it interacts with another particle and then separates. The interaction causes the particle to shift its orbit. A group of similar particles may define a preferred orientation, but because of continuous rotation, the preferred orientation will oscillate. For a group of identical particles with the same rp, the preferred orientation will oscillate with a periodicity of T. If the initial orientation distribution is uniform in space, then the distribution density function is pðθ0 , ϕ0 Þ = sin4πϕ0 . The density distribution function at a future shear strain is (Okagawa et al. 1974): pðθ, ϕÞ = 4π



sin ϕ cos 2 ϕ

þ χ 2 sin 2 ϕ

32

ð9:9Þ

where 1 χ 2 = χ 1 sin 2 θ þ χ 2 sin 2θ þ χ 3 cos 2 θ 2     1 4πγ -2 -2 χ1 = 1 þ rp þ 1 - r p cos 2 γp  -1  4πγ χ 2 = rp - rp sin γp     1 4πγ χ3 = 1 þ rp 2 þ 1 - rp 2 cos 2 γp

ð9:10Þ

From these expressions, the density distribution functions for material lines (rp → 1) and material planes (rp → 0) can be readily obtained. For material lines, it is easy to show that χ 2 = 1 - γ sin 2θ, and the density distribution function is:

210

9

Further Analysis of Spheroids in Simple Shearing Flows

Fig. 9.4 Evolution of orientations of a population of 1000 prolate objects in a simple shearing flow. γ γ 3γ All objects have the same rp = 5. (a) Initial uniform distribution. (b) γ = 8p . (c) γ = 4p . (d) γ = 8p . γp γ (e) γ = 2 . The preferred orientation fabric intensifies and reaches the maximum at strain γ = 4p . Then the fabric intensity decreases with increasing strain and returns to the uniform distribution at γ γ = 2p

pðθ, ϕÞ =

h

sin ϕ

i32

ðMaterial linesÞ

i32

ðMaterial planesÞ ð9:12Þ

4π cos 2 ϕ þ ð1 - γ sin 2θÞ2 sin 2 ϕ

ð9:11Þ

For material planes, χ 2 = 1 + γ sin 2θ, and: pðθ, ϕÞ =

h 4π

sin ϕ cos 2 ϕ

2

þ ð1 þ γ sin 2θÞ sin ϕ 2

Figure 9.4 shows the evolution of orientations of a group of 1000 prolate particles, all with rp = 5 and with initially uniform distribution in space (Fig. 9.4a). A preferred orientation develops as the shear strain increases. Figure 9.4b shows a girdle at a small angle antithetic to the shear direction (vertical in the plot). The girdle also has a point maximum on the VNS. The girdle rotates and the fabric intensifies and towards parallel with the shear direction. The strongest concentration of the orientation is at the shear strain γ = 0.25γ p (Fig. 9.4c). The girdle rotates past the shear direction and becomes synthetic to the shear direction and then weakens with further increase of shear strain. The orientations return to uniform distribution at γ = 0.5γ p. Thus, the

9.4

Forces Acting on a Prolate Object in Simple Shearing

211

preferred orientations oscillate every 0.5γ p. Note although it takes γ = γ p for every object to return to its original orientation, because of symmetry, it only takes γ = 0.5γ p for the fabric to return to the original uniform distribution. Fabrics made of identically-shaped rigid spheroid particles are not possible in natural mylonite. But the point that fabrics defined by rigid elements can oscillate is clear from this example. This implies that the intensity of a fabric defined by rigid elements is not a monotonic function of the strain magnitude.

9.4

Forces Acting on a Prolate Object in Simple Shearing

The traction on the spheroid surface exerted by the embedding fluid results in stress and strain within the object. Jeffery (1922, Eq. 34 there) presents equations for the traction at a point on the ellipsoid’s surface. Based on Jeffery’s equations, Goldsmith and Mason (1967) obtain the following expressions for the forces acting on half the ellipsoid surface parallel to the object’s distinct axis (the a1 direction): 

P1 = πτa22

 rp 2 - 1 R   sin 2θ sin ϕ 2 2rp - R 2rp 2 þ 1

ðaÞ

rp 2 - 1  rp 2 þ 1 ð3R - 2Þ

ð bÞ

rp 2 - 1  rp 2 þ 1 ð3R - 2Þ

ð cÞ

2

P2 = πτa22 sin 2θ sin 2ϕ 

P3 = 2πτa22 cos 2θ sin ϕ 

where τ is the shear stress of the simple shearing flow, R =

R1 0

ð9:13Þ

pffiffiffiffiffiffiffiffi ffi dλ which 2

rp ð1þλÞ2

rp þλ

can be expressed analytically (Okagawa et al. 1974): 8 rp 2 rp cosh - 1 rp > > for rp > 1, prolate objects >  3 , > rp 2 - 1 > 2-1 2 > r p > > < 2 -1 R = rp cos rp - rp , for rp < 1, oblate objects  3 > > 1 - rp 2 2 2 > 1 r > p > > > > : 2, for rp = 1, sphere objects 3

ðaÞ ð bÞ

ð9:14Þ

ðcÞ

The corresponding stresses on the object’s principal section normal to the a1 direction are: σ 11 =

P1 P P , σ 12 = 22 , σ 13 = 32 πa22 πa2 πa2

ð9:15Þ

212

9

Further Analysis of Spheroids in Simple Shearing Flows

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Making use of the formula cosh - 1 rp = ln rp þ rp 2 - 1 and neglecting terms smaller than rp-2 ln rp when rp >> 1 in Eq. (9.14), Eq. (9.13) can be simplified to (Goldsmith and Mason 1967): P1 ≈

πa21 τ sin 2θ sin 2 ϕ   2 ln 2rp - 1:5

ð9:16Þ

which can also be expressed using C to express ϕ (Eq. (9.4): πa21 τC 2 sin 2θ   for rp >> 1: P1 ≈  2 ln 2rp - 1:5 C2 þ cos 2 θ For any given C, the force along the a1 direction of a prolate object (P1) is tensile for 0 > 1

100 90 80 70 60 V11 W 50 40 30 20 10 0 0

2

4

6

8

10 12 rp

14

16

18

20

9.5

Deformation of a Prolate Object in Simple Shearing Flows

9.5

213

Deformation of a Prolate Object in Simple Shearing Flows

As Fig. 9.4 shows, the maximum longitudinal stress in a prolate object increases rapidly with rp. For an elongated object with rp about 4, the stress is already over an order of magnitude greater than the flow shear stress. One can imagine that for any given flow shear stress τ, a prolate object has a critical rp above which |σ| exceeds the tensile strength, and the object will develop tensile fractures (boudinage), or |σ| exceeds the threshold for buckling instability to develop. We analyze the two situations where the object is on the VNS and undergoes the maximum compressive stress at θ = - π4 and maximum tension θ = π4, respectively. Let us use an object coordinate system xyz which is related to the (external) coordinate system x1x2x3 by Fig. 9.6. Because we are considering the stress and strain when the object’s long axis is on the VNS, the z-axis is always parallel to the x3-axis. The coordinate transformation matrix is: 0 1 0 cos θ x B C B @ y A = @ - sin θ 0

z

10 1 0 x1 CB C 0 A @ x2 A

sin θ cos θ 0

1

x3

Because this is a plane-straining problem, we only need to consider the 2D deformation in the VNS, and the transformation is:
 x y


=

cos θ

sin θ

- sin θ

cos θ




x1



x2


)

Qij =

cos θ

sin θ

- sin θ

cos θ

 ð9:18Þ

The stress field of simple shearing Σij, in the x1x2x3, and Σobj ij in the xyz systems, are respectively: Σij = Σobj ij

0

τ

τ

0

!

= Qim Qjn Σij =

ð aÞ τ sin 2θ

τ cos 2θ

τ cos 2θ

- τ sin 2θ

!

ð9:19Þ ð bÞ

We regard an elastic rod-like object as a beam and apply the well-known Euler’s beam equation (Timoshenko and Gere 1961) to analyze its buckling instability: EI

d2 w þ P1 ðxÞw = 0 dx2

ð9:20Þ

214

9

Further Analysis of Spheroids in Simple Shearing Flows

Fig. 9.6 (a) External coordinate system (x1x2x3) and object geometry with the long axis on the VNS. The object’s coordinate system is xyz. (b) The origin of the object coordinate system is at the center of the object. Thus, the object domain is -a1 ≤ x ≤ a1. Because the stress inside an ellipsoid embedded in a uniform infinite matrix is constant (Eshelby 1957), the longitudinal force varies along x because the cross-section area of the object drops to 0 as x increases from 0 to x = a1

where E is Young’s modulus, I the minimum area moment of inertia of the object, πa4 and w is the deflection. For a rod-like body, I = 4 2 (Timoshenko and Gere 1961, p. 6). From Eq. (9.16), on the VNS P1(x) is:  2    2 πΣobj πτ sin 2θ a21 - x2 11 a1 - x     P 1 ð xÞ = = 2 ln 2rp - 1:5 2 ln 2rp - 1:5

ð9:21Þ

This relation also follows from Eshelby’s (1957) conclusion that the stress field in an ellipsoid embedded in an infinite uniform material subject to homogeneous deformation remotely is uniform. Uniform stresses along the a1 direction imply that P1(x) follows Eq. (9.21)  because  the sectional area of the object drops according to the relation of πa22 1 - x2 =a21 (Fig. 9.6b). Combining Eqs. (9.20) and (9.21) yields the following differential equation for the buckling of the prolate object:   d2 w þ c a21 - x2 w = 0 dx2 where

ð9:22Þ

9.5

Deformation of a Prolate Object in Simple Shearing Flows

c=

215

2τ πτ   = 4  Ea2 ln 2rp - 1:5 2EI ln 2rp - 1:5

ð9:23Þ

Equation (9.22) has been solved in terms of Hermite polynomials (Forgacs and Mason 1959). 1 With variable substitution z = c4 x, Eq. (9.22) becomes:  d 2 w pffiffiffi 2 þ ca1 - z 2 w = 0 2 dz

ð9:24Þ

z2

With a further substitution of w = ye - 2 , Eq. (9.24) becomes:  d2 y dy pffiffiffi - 2z þ ca21 - 1 y = 0 dz dz2

ð9:25Þ

pffiffiffi 2 ca1 - 1 = 2n

ð9:26Þ

Provided that

Equation (9.25) is a Hermite differential equation and has the solution (e.g., Jeffrey 1995, p. 291): y = H n ð z Þ = ð - 1Þ n e z

2

d n - z2 e dzn

ð9:27Þ

where Hn denotes the Hermite function of order n. The n = 0 case corresponds to the solution of y = H0(z) = 1 representing the first mode of buckling of the object. The solution is thus: w = w0 e -

ffi

p 2 cx 2

þC

where w0 is the amplitude of the deflection, which cannot be determined by the linear analysis here (Timoshenko et al. 1961, p. 48; Turcotte and Schubert 1982, p. 119). w0can have any value as long as the deflection is small, so that Eq. (9.20) remains valid. The integral constant C is determined using the pinned end condition w(a1) = w(-a1) = 0. The solution becomes:
pffi 2 pffi 2  ca w - c2x - 21 = e -e w0

ð9:28Þ 2

One can confirm with Eq. (9.28) that the other end condition, ddxw2 jx = ± a1 = 0 is met pffiffiffi due to ca21 - 1 = 0, as from Eq. (9.28):

216

9

Further Analysis of Spheroids in Simple Shearing Flows

Fig. 9.7 Predicted buckling profile (normalized deflection vs. normalized length) for a rod with hinged end conditions in a simple shearing flow

pffiffiffi pffiffiffi  d2 w = - w0 c 1 - cx2 e j 2 x = ± a1 dx This solution corresponds to

ffi

p 2 cx 2

jx = ± a1 = 0

pffiffiffi 2 ca1 - 1 = 0 and implies E

2τrp 4

ð ln 2rp - 1:5Þ

= 1. Therefore,

the corresponding shear critical stress is: τcrit: =

  E ln 2rp - 1:5 2rp 4

ð9:29Þ

Submitting Eq. (9.29) into Eq. (9.23) to get the expression for c at the critical shear and then replacing c in Eq. (9.28), we finally get the following expression for the buckling state of the object:
b = exp w

-



bx2 1 - exp 2 2

ð9:30Þ

b = ww0 and bx = ax1 . where w Figure 9.7 is a plot of Eq. (9.30) for - 1 ≤ bx ≤ 1. This is the smallest critical stress for the rod to buckle. Substituting n = 1, 2, ⋯ into Eq. (9.26), we obtain the corresponding critical shear stresses: τcrit = 9E ð ln 2rp - 1:5Þ 25E ð ln 2rp - 1:5Þ , τcrit = . . . for other buckling modes. These other 2r4 2r4 p

p

forms of buckling require much higher compressive force and are produced only when the rod is prevented from lateral deflection (to realize the first mode of buckling) by other mineral grains. The critical stress required to buckle a prolate object drops rapidly with increasing : rp. For rp around 5, the critical buckling shear stress is about τcrit. ¼ 6.4 × 10-4E. If

9.5

Deformation of a Prolate Object in Simple Shearing Flows

217

Fig. 9.8 Critical flow shear stress (normalized against the object Young’s modulus) required to buckle the object. The prediction is valid for rp > 5

we take the estimate of Young’s modulus for tourmaline as 201.5 GPa (Ozkan : 1979), then τcrit. ¼ 129 MPa. Taking the estimate of Young’s modulus for feldspar as : 70 GPa (Ji and Li 2021), then τcrit. ¼ 45 MPa. Folds in elongate minerals are relatively rare in mylonites. In contrast, microboudins of these minerals are common. The stress condition for tensile fractures perpendicular to the object’s long axis is: 

τr2p

2 ln 2rp - 1:5

 =T

)

τcrit: =

  2T ln 2rp - 1:5 r2p

ð9:31Þ

where T is the tensile strength of the object. This relation is plotted as the dashed curve in Fig. 9.8. The tensile strengths for minerals like tourmaline in natural deformation pressure and temperature conditions : are poorly known. Ji and Li (2021) suggest the empirical relation T ¼ 7.2 × 10-4E between Young’s modulus and tensile strength for geological materials. Using this empirical estimate and taking rp around 5, the critical shear stress for : : boudin fracturing is about τcrit. ¼ 9.3 MPa for tourmaline and τcrit. ¼ 3.2 MPa for feldspar. This are very small shear stresses. But the estimate is not inconsistent with the observations that all tourmaline grains are fractured regardless of their aspect ratios in the Gaoligong metamorphic belt (Yunnan Province, China) (Li and Ji 2020). Natural microboudins are observed in grains with rp as low as 1. In this case, the above approximation based on rp >> 1 cannot be used. We can use the exact result for ℜ in Eq. (9.14) to write the fracturing condition:

218

9

Further Analysis of Spheroids in Simple Shearing Flows

Fig. 9.9 Critical flow shear stress (normalized against the object tensile strength) required to fracture the object. The dashed line is based on the rp > > 1 approximation (Eq. 9.31). The solid line is based on the exact relation (Eq. 9.32)

"

 # 2rp 2 - R 2rp 2 þ 1   τcrit: = T , rp 2 - 1 R

where R =

rp cosh - 1 rp rp 2  3 ð9:32Þ rp 2 - 1 rp 2 - 1 2

The above relation is plotted as the solid curve in Fig. 9.9. The maximum possible : τcrit. as rp = 1 is τcrit. = 0.4T, which means τcrit. ¼ 2.88 × 10-4E assuming the empirical relation of Ji and Li (2021). This corresponds to ~58 MPa for a spherical tourmaline grain to fracture and 20 MPa for a spherical feldspar grain to fracture, assuming the same estimates for Young’s moduli of these minerals as stated above. If all tourmaline grains are fractured, these estimates provide a low limit estimate of the actual flow shear stress at the time of deformation.

9.6

Concluding Remarks

The stress and strain analysis in prolate objects above is based on Jeffery’s (1922) equations on the traction the fluid exerts on the object. The results are quite different from the fiber-loading theory or shear-lag model that is used by many authors to understand microboudinage structures in mylonites (e.g., Masuda and Kuriyama 1988; Masuda et al. 1989; Ji and Zhao 1994; Zhao and Ji 1997). We will have a more thorough investigation of the deformation of prolate objects in viscous flows in Chap. 14.

References

9.7

219

Notes and Key References

Mathcad worksheet Revolution.mcdx is based on Eq. (9.8). Goldsmith and Mason (1967) is a comprehensive and exhaustive review of the kinematics and dynamics of rigid (mostly prolate) objects in simple shearing flow of Newtonian viscous fluids. The paper contains many analytical expressions for the kinematics and stress field in elongate prolates (rods) and flat oblates (discs). Some of their equations are recast in Sect. 9.4 here. In a more general analysis in Chap. 14, the stress and strain in an elastic prolate object will be investigated and the results in this Chapter will be used for verification.

References Anczurow E, Mason SG (1967) Kinetics of flowing dispersions. 2. Equilibrium orientations of rods and discs (theoretical). J Colloid Interface Sci 23(4):522–532. https://doi.org/10.1016/00219797(67)90199-3 Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond Ser A-Math Phys Sci 241(1226):376–396. https://doi.org/10.1098/ rspa.1957.0133 Forgacs OL, Mason SG (1959) Particle motions in sheared suspensions: IX. Spin and deformation of threadlike particles. J Colloid Sci 14(5):457–472. https://doi.org/10.1016/0095-8522(59) 90012-1 Goldsmith HL, Mason SG (1967) The microrheology of dispersions, Rheology: theory and applications, vol 4. Academic Press, New York, pp 5–250 Jeffery GB (1922) The motion of ellipsoidal particles in a viscous fluid. Proc R Soc Lond Ser A-Containing Papers of a Mathematical and Physical Character 102(715):161–179. https://doi. org/10.1098/rspa.1922.0078 Jeffrey A (1995) Handbook of mathematical formulas and integrals. Academic Press Ji S, Li L (2021) Feldspar microboudinage paleopiezometer and its applications to estimating differential stress magnitudes in the continental middle crust (examples from west Yunnan, China). Tectonophysics 805. https://doi.org/10.1016/j.tecto.2021.228778 Ji S, Zhao P (1994) Strength of 2-phase rocks – a model-based on fiber-loading theory. J Struct Geol 16(2):253–262 Jiang D (2007) Numerical modeling of the motion of rigid ellipsoidal objects in slow viscous flows: a new approach. J Struct Geol 29(2):189–200. https://doi.org/10.1016/j.jsg.2006.09.010 Jiang D, Williams PF (2004) Reference frame, angular momentum, and porphyroblast rotation. J Struct Geol 26(12):2211–2224. https://doi.org/10.1016/j.jsg.2004.06.012 Li L, Ji S (2020) On microboudin paleopiezometers and their applications to constrain stress variations in tectonites. J Struct Geol 130. https://doi.org/10.1016/j.jsg.2019.103928 Masuda T, Kuriyama M (1988) Successive mid-point fracturing during microboudinage – an estimate of the stress-strain relation during a natural deformation. Tectonophysics 147(3-4): 171–177. https://doi.org/10.1016/0040-1951(88)90185-0 Masuda T, Shibutani T, Igarashi T, Kuriyama M (1989) Microboudin structure of piedmontite in quartz schists – a proposal for a new indicator of relative paleodifferential stress. Tectonophysics 163(1–2):169–180. https://doi.org/10.1016/0040-1951(89)90124-8 Okagawa A, Cox RG, Mason SG (1974) Particle behavior in shear and electric-fields. 6. Microrheology of rigid spheroids. J Colloid Interface Sci 47(2):536–567. https://doi.org/10. 1016/0021-9797(74)90286-0

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9

Further Analysis of Spheroids in Simple Shearing Flows

Ozkan H (1979) Elastic-constants of tourmaline. J Appl Phys 50(9):6006–6008 Timoshenko S, Gere JM (1961) Theory of elastic stability. McGraw-Hill Turcotte DL, Schubert G (1982) Geodynamics: application of continuum physics to geological problems. Wiley, New York Williams PF, Jiang D (1999) Rotating garnets. J Metamorph Geol 17(4):367–378. https://doi.org/ 10.1046/j.1525-1314.1999.00203.x Zhao P, Ji S (1997) Refinements of shear-lag model and its applications. Tectonophysics 279(1–4): 37–53. https://doi.org/10.1016/S0040-1951(97)00129-7

Chapter 10

Eshelby’s Inclusion and Inhomogeneity Problem

Abstract Following the last Chapter, one naturally wonders how a non-rigid inclusion would behave in a viscous flow. A non-rigid inclusion changes both shape and orientation during deformation. We discuss the powerful method of John Douglas Eshelby (1916–1981) in this Chapter which solves the problem of a deformable ellipsoid. The point force and equivalent inclusion method of Eshelby’s can be applied to many other problems, such as deformation in and around dykes, faults, and shear zones. Eshelby (Proc R Soc Lond Ser 241(1226):376–396, 1957) solved what is now known as Eshelby’s inclusion and inhomogeneity problem. That work and Eshelby (Proc R Soc Lond Ser A-Math Phys Sci 252(1271):561–569, 1959) pioneered an elegant approach to deal with the interaction between an isolated elastic region and the infinite elastic medium. The approach has led to the development of a new and active branch in continuum mechanics called Micromechanics (Mura, Micromechanics of defects in solids. Martinus Nijhoff, 1987; Nemat-Nasser and Hori, Micromechanics: overall properties of heterogeneous materials. Elsevier Science, 1999; Qu and Cherkaoui, Fundamentals of micromechanics of solids. Wiley, 2006). Eshelby’s approach and the more general micromechanics have also been extended, first to isotropic Newtonian viscous materials by Bilby et al. (Tectonophysics 28(4):265–274, 1975) and Bilby and Kolbuszewski (Proc R Soc Lond Ser A-Math Phys Eng Sci 355(1682):335–353, 1977), and then to general power-law and anisotropic materials by Molinari et al. (Acta Metall 35(12): 2983–2994, 1987) and Lebensohn and Tomé (Acta Metall Mater 41(9): 2611–2624, 1993a; Philos Mag A-Phys Condensed Matter Struct Defects Mech Prop 67(1):187–206, 1993b). Earth’s lithosphere is made of rheologically heterogeneous materials on a wide range of observation scales. One major issue in structural geology, tectonics, and the rheology of Earth in general, has been how to tackle the deformation of such a body effectively. Micromechanics provides the fundamental ingredients for dealing with the heterogeneous deformation and accompanying fabric development in Earth’s

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-23313-5_10. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Jiang, Continuum Micromechanics, Springer Geophysics, https://doi.org/10.1007/978-3-031-23313-5_10

221

222

10 Eshelby’s Inclusion and Inhomogeneity Problem

lithosphere. We will briefly work through Eshelby’s approach in this Chapter to understand its essential ingredients in order to apply them to corresponding viscous problems.

10.1

Eshelby’s Elastic Inclusion/Inhomogeneity Problem

Eshelby (1957) considered what is now known as his classical inclusion/inhomogeneity problem: What is the elastic field of an infinite uniform elastic solid caused by an internal “inclusion” (Ω in Fig. 10.1a) undergoing a homogeneous strain field, which Eshelby (1957, 1959) called the “transformation strain”, if the inclusion were cut out from the elastic medium (Fig. 10.1b, c)? Mura (1987) referred to transformation strains generally as eigenstrains which will be used here and denoted by e*. Eshelby (1957, 1959, 1961) solved the inclusion problem through a series of transformations illustrated in Fig. 10.1. The inclusion is first cut out to undergo the eigenstrain to reach its stress-free state (Ω
in Fig. 10.1c). The strain of the inclusion at this state is e
ij, but the stress is zero (it is important to note that the eigenstrain state corresponds to the stress-free state). The infinite body with a hole remains stress and strain free (Fig. 10.1b). Applying traction on the inclusion surface equal to t i = - Cijkl e
kl nj, where nj is the outward unit normal of the surface element dS, will bring the transformed inclusion back to the original form (shape and size) (the Ω state in Fig. 10.1c) so that it can be placed back to the void (Fig. 10.1d). The strain of inclusion at this state is zero. The stress of inclusion is - C ijkl e
kl . The stress in the matrix is still zero at this state. Put the inclusion back into the void and let the contact between the inclusion and matrix be perfectly glued. There is no change in stress and strain in the inclusion, and the matrix is still stress and strain free. The system is nevertheless different from the original problem in that it is acted upon by external forces - C ijkl e
kl nj dS. The next step is to remove the traction forces, which amounts to applying a body force bi = - t i = C ijkl e
kl nj dS over the inclusion-matrix contact. This brings the system back to the original inclusion problem. The elastic field caused by this body force in the inclusion and in the matrix is the solution to the problem. Through the above series of transformations, the inclusion problem is reduced to the elasticity in an infinite body caused by a body force distribution over the inclusion surface. As the inclusion is now constrained by the surrounding matrix medium and, therefore, cannot recover to the stress-free eigenstrain state, the interaction between the matrix and the inclusion (caused by the body force bi) leads to a constrained state (the Ωc state in Fig. 10.1c). Eshelby (1957, p. 384, his Eq. 3.5) proved an amazing result that if the inclusion shape is an ellipsoid, the final constrained state of stress and strain in the inclusion is uniform. Specifically, the constrained strain ec and the constrained rotation ωc in the ellipsoidal inclusion are related to e
by two 4th-order tensors, S and II:

10.1

Eshelby’s Elastic Inclusion/Inhomogeneity Problem

223

Fig. 10.1 The classic Eshelby inclusion problem. (a) A homogeneous solid with elastic stiffness Cm everywhere is stress and strain free. A region Ω, called inclusion, undergoes “transformation” (shape change). (b) If the inclusion region were cut out, the remaining solid body would remain stress- and strain-free. (c) The “transformation” amounts to a uniform eigenstrain field (e
) that changes Ω to Ω*. In the Ω* (eigenstrain) state, the inclusion is stress-free. Surface traction is applied to the boundary of the inclusion to bring it back to its original form. The stress of the inclusion in this state is -Cm : e
. (d) The shape-restored inclusion is placed back to the viod in (b). The interface is perfected glued. (e) The interaction between the matrix and the inclusion leads to the equilibrated constrained state of the inclusion Ωc which differs from the initial shape by a strain field ec and a rotation ωc. If the inclusion is ellipsoidal, ec and a ωc are uniform inside the inclusion and are related to the eigenstrain by two Eshelby tensors. The matrix outside the inclusion is also in a state of strain and rotation [ec(x) and ωc(x)]. This strain and rotation field dies out far away from the inclusion

ec = S : e


ð aÞ

ωc = Π : e


ð bÞ

ð10:1Þ

Tensors S and Π are known as the symmetric and anti-symmetric Eshelby tensors, respectively. We will discuss how Eshelby tensors can be obtained for linear elastic and linear viscous materials. It can be shown that S has minor symmetry (Sijkl = S jikl = Sijlk, Eq. 1.44), but in general, does not have major symmetry

224

10 Eshelby’s Inclusion and Inhomogeneity Problem

(Eq. 3.46), i.e., S jikl ≠ Sklij. Π is anti-symmetric in that Πijkl = Πijlk = - Πjikl = Πjilk. Because the stress and strain fields in the ellipsoidal inclusion is uniform, both S and Π are constant inside the ellipsoid. The stress inside the inclusion is σc = Cm : (ec - e
). The matrix outside the inclusion is also in a state of stress and strain, which varies in space and dies out far away from the inclusion. Eshelby (1959) showed that the strain and rotation outside the inclusion can also be related to the eigenstrain by what are now known as two exterior-point Eshelby tensors, SEijkl ðxÞ and ΠijklE(x): eC ðxÞ = SE ðxÞ : e
ω ð xÞ = Π ð xÞ : e C

E

ð aÞ


ð bÞ

ð10:2Þ

Eshelby (1957, 1959) called a region, linearly elastic but having different elastic stiffnesses from the embedding matrix medium, an inhomogeneity. To solve for the stress-strain field in an isolated inhomogeneity embedded in an infinite homogeneous elastic body under uniform elastic stress (strain) at infinity, Eshelby (1957) had the brilliant idea that an inhomogeneity can always be replaced uniquely by an “equivalent inclusion” of the same shape and with the proper eigenstrain field so that the stress state inside the inclusion is the same as when the inhomogeneity is present (Fig. 10.2). As long as the inhomogeneity is ellipsoidal, the stress and strain field inside it is uniform. The eigenstrain for the equivalent inclusion is found using the principle of superposition (Fung 1965, p. 3) which applies to all linear elastic materials. Let the uniform elastic strain at infinity be denoted by eM. If there were no inhomogeneity, the strain field would be uniform everywhere in the elastic medium. One can regard the strain in the inhomogeneity e, which is unknown and to be solved, as the superposition of eM and ec: e = ec þ eM

ð10:3Þ

where ec is caused by an eigenstrain e* of the equivalent inclusion. The ec term in Eq. (10.3) can be expressed as ec = S : e
(Eq. 1.2). The requirement that the equivalent inclusion produces the same stress field as the inhomogeneity in the ellipsoidal region means that:
 C1 : e = CM : eC þ eM - e


ð10:4Þ

where CM and C1 are the elastic stiffnesses of the matrix and the inclusion, respectively. On the right-hand side of Eq. (10.4), the eigenstrain e
is subtracted from the total strain because the eigenstrain state (the Ω
state in Fig. 10.1c) corresponds to the inclusion’s stress-free state. Combining Eqs. (10.1), (10.3) and (10.4), we get:

10.1

Eshelby’s Elastic Inclusion/Inhomogeneity Problem

225

Fig. 10.2 Schematic illustration of Eshelby’s (1957) approach of solving the inhomogeneity problem by the equivalent inclusion approach. (a–c) The inclusion solutions where the strain and stress state in the inclusion are related to the eigenstrain. (d) The inhomogeneity (heterogeneous inclusion) problem: What is the stress and strain in a heterogeneous inclusion (elastic stiffness C1) embedded in a homogeneous medium (elastic stiffness Cm) subjected to uniform strain (eM) at infinity? (e) This problem is solved by adding a uniform strain (eM) to the inclusion problem (c) and solving for an adequate eigenstrain so that the stress field in the inclusion is identical to that in the inhomogeneity. This leads to Eq. (10.4) and then the partitioning Eq. (10.5) for strain and rotation. Such inclusion is called equivalent inclusion. The eigenstrain for the equivalent inclusion is e
= S1 : (e - eM). See text for more details

 -1 M e = JS - S þ S : CM- 1 : C1 :e
 -1 M M ω = ω þ Π : S : e - e = ωM þ Π : S - 1 : ee

ð aÞ ð bÞ

ð10:5Þ

where ee = e - eM stands for the difference between the strain in the inhomogeneity and the strain in the remote medium. Henceforth, the tilde on a symbol shall stand for the difference between its value in the inhomogeneity and in the remote medium. The exterior strain field can also be solved similarly by using the equivalent inclusion approach. Letting e(x) = eC(x) + eM first and then using Eq. (10.2), we get:

226

10 Eshelby’s Inclusion and Inhomogeneity Problem

eeðxÞ = SE ðxÞ : S - 1 : ee e ðxÞ = ΠE ðxÞ : S - 1 : ee ω

ð aÞ

ð10:6Þ

ð bÞ

where, as noted above, eeðxÞ = eðxÞ - eM . We shall analyze both the interior and exterior fields using Eqs. (10.5) and (10.6), respectively, in more detail later. Let us now focus on the interior field using Eq. (10.5). First, Eq. (10.5) relate the remote uniform strain to the uniform strain inside the ellipsoid. This can be expressed explicitly by rewriting Eq. (10.5) in the following set: e = A : eM ω=ω þ Π : S M

σ = C1 : A :

-1

CM- 1




: A-J

S



ð aÞ :e

M

: Σ=B : Σ

ð bÞ

ð10:7Þ

ð cÞ


 - 1 is called the strain where the fourth-order tensor A = JS þ S : CM- 1 : C1 - JS partitioning tensor and the fourth-order tensor B = C1 : A : CM- 1 the stress partitioning tensor. Eq. (10.7) is obtained by applying Hooke’s law to Eq. (10.7). Accordingly, Eq. (10.7) will be called partitioning equations. The relation between the inclusion field and the matrix field is also described, equivalently, by the following interaction equations which are obtained by inserting Eqs. (1.2) and (10.3) into Eq. (10.4) to replace the eigenstrain term and then rearranging: e = - H : ee σ _ ee = - H : σ e

ð10:8Þ


-1 _ e = σ - σM , H = CM : (S-1 - JS), and H = H - 1 = S - 1 - JS : CM- 1 . where σ H is sometimes called Hill’s constraint tensor (Qu and Cherkaoui 2006, pp. 90, 315) _ and its inverse H is known as the interaction tensor (Lebensohn and Tomé 1993a, b). Similarly, the partitioning equation for rotation (Eq. 10.5) is often written as: e = Π : S - 1 : ee ω

ð10:9Þ

e = ω - ωM . where ω In essence, Eshelby’s approach to solving the problem of an inhomogeneity in a uniform elastic body subjected to far-field homogeneous deformation is to treat the problem as a superposition of a homogeneous remote field and a perturbation field caused by an equivalent inclusion. Because an inhomogeneity can always be regarded as an equivalent inclusion, we will use the general term “inclusion” hereafter for both the inclusion and inhomogeneity in Eshelby’s sense. Where a distinction is required, we shall call an inhomogeneity a heterogeneous inclusion.

10.2

Eshelby Tensors and the Auxiliary Interaction Tensor

10.2

227

Eshelby Tensors and the Auxiliary Interaction Tensor

In the above formal solutions for the inclusion (and inhomogeneity) problems, two Eshelby tensors were introduced. To explain how they are defined and computed, we need to solve the boundary-value elasticity problem for inclusions using the Green function method. The elastic inclusion problem is posed completely by the following set of equations: ð aÞ Cijkl uck,lj þ bi = 0 (
Cijkl ekl nj on inclusion surface bi = 0 elsewhere ðcÞ uci ðxÞjjxj → 1 = 0

ð bÞ

ð10:10Þ

where uci ðxÞ is the displacement field for the constrained state of the inclusion. Equation (10.10) is the equilibrium equation, Eq. (10.10) is the body force distribution corresponding to the problem, and Eq. (10.10) states the boundary condition. The solution to Eq. (10.10) can be expressed by means of the Green function method (Eq. 3.66): Z uci ðxÞ =



 Gir x, x 0 C rsmn e
mn ns dS x 0

ð10:11Þ

∂Ω

where Gij(x, x') is the Green function and ∂Ω the surface of the inclusion. Because in an infinite body, Gij(x, x') = Gij(x - x'), the above Equation can be expressed in the following form: Z uci

=




Gir x - x

0



Crsmn e
mn ns dS = C rsmn e
mn

∂Ω

Z


 Gir x - x 0 ns dS

ð10:12Þ

∂Ω

Therefore, the displacement gradients are: 2 uci,j =

∂ 6 4C rsmn e
mn ∂xj

Z ∂Ω

3 Z

 7 Gir x - x 0 ns dS5 = Crsmn e
mn Gir,j x - x 0 ns dS ð10:13Þ ∂Ω

Using the Gauss divergence theorem (Eq. 2.28), Eq. (10.13) can also be written as a volume integral over the ellipsoid inclusion:

228

10 Eshelby’s Inclusion and Inhomogeneity Problem

uci,j = - C rsmn e
mn

Z


 Gir,js x - x 0 dV

ð10:14Þ

Ω

As ecij =

1 2

    uci,j þ ucj,i and ωcij = 12 uci,j - ucj,i , the two Eshelby tensors defined in

Eq. (10.2) are: Sijmn = -

Z

1 C 2 rsmn

Ω

1 Πijmn = - Crsmn 2

Z




 Gir,js þ Gjr,is dV

ð aÞ ð10:15Þ


 Gir,js - Gjr,is dV

ð bÞ

Ω

Define an auxiliary 4th-order tensor Tijkl, sometimes called the Green interaction tensor (Lebensohn et al. 1998): Z T ijkl ðxÞ = -

Gik,jl





 x - x dV x 0 = 0



Ω

Z



 Gik,l x - x 0 nj dS x 0

ð10:16Þ

∂Ω

with which one can demonstrate that the two Eshelby tensors are, respectively: S = Js : T : C Π = Ja : T : C

ð10:17Þ

For points inside an ellipsoidal inclusion, Tijkl is a constant tensor, and so are Sijkl and Πijkl. For points outside the inclusion (in the matrix), these tensors vary in space. In the following, we shall use Tijkl, Sijkl and Πijkl for interior points and T Eijkl ðxÞ, SEijkl ðxÞ, and ΠEijkl ðxÞ for exterior points. Because the Green function for an infinite isotropic elastic material is known in analytic form (Eq. 3.67), with Eqs. (10.16) and (10.17), it is possible to express the interior-point Eshelby tensors explicitly or in terms of elliptic integrals (Eshelby 1957; Mura 1987). For anisotropic elastic materials, the interior-point Tijkl can be expressed by the following integral (Lebensohn et al. 1998; Jiang 2014, 2016): aaa T ijkl = 1 2 3 4π

Z2π Zπ 0

0

Aik- 1 zj zl ρ - 3 sin ϕdϕdθ

ð10:18Þ

10.3

Extension to Newtonian Viscous Materials

229

0

cos θ sin ϕ

1

B C where ai (i = 1, 2, 3)are the three semi-axes of the ellipsoid, z = @ sin θ sin ϕ A , cos ϕ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ρ = ða1 z1 Þ2 þ ða2 z2 Þ2 þ ða3 z3 Þ2 , and Aik = Cijklzjzl is called the Christoffel stiffness tensor (e.g., Barnett 1972; Walker 1993). Because Cijkl has major symmetry (3.46), Aik is clearly symmetric. It follows from Eq. (10.18) that Tijkl has the following symmetry: T ijkl = T kjil = T ilkj

ð10:19Þ

The two exterior point Eshelby tensors, SE(x) and ΠE(x), can be calculated numerically from Green functions (Jiang 2016) using Eqs. (10.16) and (10.17). The evaluation of Eqs. (10.16) and (10.18) requires numerical computation. For isotropic elastic materials, more efficient methods have been developed by Ju and Sun (1999), Meng et al. (2012), and Jiang (2016) to calculate SE(x) and ΠE(x). More details will be presented in the next Chapter.

10.3

Extension to Newtonian Viscous Materials

Because linear elastic theories and linear viscous theories are formally equivalent (Spencer 1980, pp. 110–118), Eshelby’s elastic inclusion solutions can be extended to Newtonian viscous materials readily. Bilby et al. (1975) and Bilby and Kolbuszewski (1977) have applied Eshelby’s elastic inclusion solutions to isotropic Newtonian viscous materials. They replaced the elastic strain tensor with the viscous strain rate tensor and the elastic infinitesimal rotation tensor with the vorticity tensor in all elastic equations and set Poisson’s ratio to 0.5 to achieve the viscous incompressible condition. By this approach, Eq. (10.7) are rewritten, for general Newtonian materials, as: ε=A : E w = W þ Π : S - 1 : ðε - E Þ

ðaÞ ð bÞ

σ=B : Σ 
 - 1 A = Jd þ S : Cm- 1 : Ci - Jd

ð cÞ

B = C1 : A : CM- 1

ð10:20Þ ð dÞ

ð eÞ

where lowercase ε, w, and σ are, respectively, strain rate, vorticity, and stress tensors in the inclusion; uppercase E, W, and Σ are corresponding tensors in the remote matrix; C now stands for the viscous stiffnesses (viscosity) tensor with subscript “i” and “m” representing the inclusion and matrix respectively; S and Π are viscous Eshelby tensors for incompressible Newtonian materials. The two fourth-order

230

10 Eshelby’s Inclusion and Inhomogeneity Problem

tensors, A and B are, respectively, the strain-rate and stress partitioning tensors. Because we are dealing with the deviatoric subspace for incompressible materials, Jd is used instead of Js. Bilby et al. (1975) and Bilby and Kolbuszewski (1977) used elastic Eshelby tensors for isotropic materials for incompressible viscous Eshelby tensors by setting the Poisson’s ratio to 0.5. This approach is called the penalty approach (more details below). Like the elastic stiffnesses, the viscous stiffnesses tensor C and compliances tensor M (=C-1) are fourth-order tensor quantities. Just as elastic C and M can be expressed in terms of two scalar moduli when the material is isotropic, for isotropic Newtonian materials, the viscous C and M can be expressed as (Eqs. (1.41) and (1.43); Qu and Cherkaoui 2006): C = 3ηb Jm þ 2ηs Jd 1 m 1 d M= J þ J 3ηb 2ηs

ð10:21Þ

where ηb and ηs are the bulk and shear viscosities, analogous to elastic bulk and shear moduli, respectively. In the deviatoric subspace for incompressible materials where ηb → 1, M = 2η1 Jd and C = 2ηsJd. Viscous stiffness and compliance tensors are s inverses of each other in the sense of C : M = M : C = Jd. ηi ηi Using Eq. (10.21), the Cm- 1 : Ci term in Eq. (10.20) reduces to ηmb Jm þ ηms Jd (note ηi

ηi

b

s

Jm : Jd = Jd : Jm = 0). If ηmb and ηms are equal to a common viscosity ratio r, then s b
 Cm- 1 : Ci = r Jm þ Jd = rJs and Eq. (10.20) becomes: ε = ½Js þ ðr - 1ÞS - 1 : E

ð10:22Þ

Substituting this expression into Eq. (10.20), the vorticity equation of Bilby and Kolbuszewski (1977, Eq. 2 there) can be obtained.

10.4

Expressions of Eshelby Tensors for Linear Isotropic Materials

Eshelby tensors in linear anisotropic viscous materials will be discussed in Chap. 11. For isotropic linear viscous materials, the Eshelby tensors can be obtained using corresponding expressions for linear elastic materials which are already known (Eshelby 1957; Mura 1987, p. 77):

10.4

Expressions of Eshelby Tensors for Linear Isotropic Materials

231

3 1 - 2ν a2  þ  8π ð1 - νÞ i ii 8π ð1 - νÞ i 3 1 - 2ν Seliijj = a2   8π ð1 - νÞ j ij 8π ð1 - νÞ i    3 1 - 2ν
Selijij = a2i þ a2j ij þ i þ  j 16π ð1 - νÞ 16π ð1 - νÞ   j i Πelijij = Πelijji = 8π Πeljiij = Πeljiji = - Πelijij Seliiii =

ð10:23Þ

where ai (i = 1, 2, 3) are the lengths of ellipsoid semi-axes and the -terms can be defined in terms of two elliptic functions (Mura 1987, p. 78): Z F ðθ, kÞ =

θ

dw

1 1 - k sin 2 w 2 Z θ
1 E ðθ, kÞ = 1 - k2 sin 2 w 2 dw 0




2

ð10:24Þ

0

where θ = arcsin

rffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 - a2 1 - ða3 =a1 Þ2 , k = a12 - a22 . Specifically, the -terms for six 1

3

different situations are as follows: Triaxial ellipsoids (a1 > a2 > a3): 4πa1 a2 a3 1 =

1=2 ½F ðθ, k Þ - E ðθ, kÞ 2 a1 - a22 a21 - a23 "
# 1=2 a2 a21 - a23 4πa1 a2 a3 3 =
- Eðθ, k Þ 
1=2 a1 a3 a 2 - a2 a2 - a2 2

3

1

ð10:25Þ

3

2 = 4π - 1 - 3 and the remaining -terms follow from the following formula (Eshelby 1957): j - i  ij =  3 a2i - a2j 4π i1 þ i2 þ i3 = 2 3ai

ð aÞ ð10:26Þ ð bÞ

It should be pointed out that, in Mura (1987) and Qu and Cherkaoui (2006), the  - definition of ij is ij = a2j - ai2 , which leads to dissimilar expressions for ii and ij . i

j

That is, Eq. (10.26) will be of the form 311 þ 12 þ 13 =

4π 3a21

etc. Eshelby’s

expressions are used here, as in Jiang (2007b, 2012), because they retain the

232

10 Eshelby’s Inclusion and Inhomogeneity Problem

symmetry for all ij terms (including i = j terms) and also make the expressions in Eq. (10.23) symmetric. Oblate spheroids (a1 = a2 > a3): "



1=2 # a23 2πa21 a3 a3 - 1 a3 1- 2 1 = 2 =
3=2 cos a1 a1 a1 a21 - a23 3 = 4π - 21  - 13 = 23 =
12 32  3 a 3 - a1 π 3 11 = 22 = 2 - 13 a1 4  12 = 11 3 4π 33 = 2 - 213 3a3

ð aÞ ð bÞ ð cÞ ð dÞ ð eÞ ðf Þ ð10:27Þ

Penny-shaped inclusions (a1 = a2 > > a3): In Chap. 9, we used rp to characterize the shape of spheroids. rp is the ratio of the spheroid’s distinct semiaxis length to the semiaxis length on the section normal to the distinct axis. Because we order the three semiaxes of an ellipsoid such that a1 ≥ a2 ≥ a3 to express Eshelby tensors, rp = aa31 < 1 for oblate objects, and rp = a1 a3 > 1 for prolate objects. We shall use re instead of rp in this subsequent Chapters for the axial ratio. The subscript “e” stands for “equivalent”. This is because, in the application of micromechanics, it is always the axial ratio of its “equivalent ellipsoid” for a real object that should be used. For penny-shaped inclusions, re < < 1. The following formulae are obtained by Taylor expansion of Eq. (10.27) at re = 0. Therefore, they are valid approximations only for penny-shaped inclusions (re < < 1). Specifically, the approximations neglect terms smaller than re2:

10.4

Expressions of Eshelby Tensors for Linear Isotropic Materials

233

1 = 2 ≈ πre ðπ - 4re Þ
 3 ≈ 2π 2 - πre þ 4re 2 π ð4 - 3πre þ 16re 2 Þ 3a21

πr 3π - 16re 11 = 22 ≈ 2e 4 a1

 πr 3π - 16re 12 = 11 ≈ 2e 3 12 a 1

2π 2 2 33 ≈ 2 4 þ 3πr 16r e e 3a1 re 2 13 = 23 ≈

ð10:28Þ

Prolate spheroids (a1 > a2 = a3): " #

1=2 2πa1 a23 a1 a21 a -1 - cosh - 1 1 2 = 3 =
3=2 a 2 a3 2 2 a 3 3 a1 - a3 1 = 4π - 22  - 12 = 13 =
22 12  3 a 1 - a2 4π 11 = 2 - 212 3a1 π 3 22 = 33 = 2 - 13 a2 4  23 = 22 3

ð aÞ ð bÞ ð cÞ ð dÞ

ð10:29Þ

ð eÞ ðf Þ

Rod-shaped inclusions (a1 > > a2 = a3): The following formulae are obtained by Taylor expansion of Eq. (10.29) at re1 = 0. They are valid for elongated prolate objects where re > > 1, specifically, where terms smaller than re-2 are neglected.

4π ð ln 2re - 1Þ 1 - ln 2re , =  ≈ 2π 1 þ 2 3 r2e r2e

4π ð3 ln 2re - 4Þ 2π 4 - 3 ln 2re , 11 ≈ 12 = 13 ≈ 2 1 þ r2e 3a1 3a21 r2e 

1 1 4 - 3 ln 2re 22 = 33 ≈ π 2 - 2 1 þ r2e a3 2a1 

π 1 1 4 - 3 ln 2re 23 ≈ 1 þ 3 a23 2a21 r2e 1 ≈

Spheres (a1 = a2 = a3):

ð10:30Þ

234

10 Eshelby’s Inclusion and Inhomogeneity Problem

1 = 2 = 3 =

4π 4π  , = 22 = 33 = 2 , 12 = 13 = 23 = 11 3 11 3 5a1

ð10:31Þ

With the above expressions for  -terms, the analytical expressions for Eshelby tensors in isotropic elastic materials can be obtained using Eq. (10.23). To obtain Eshelby tensor components for isotropic incompressible Newtonian materials, Bilby et al. (1975) and Bilby and Kolbuszewski (1977) assumed that Sijkl = Selijkl jν = 0:5and Πijkl = Πelijkl jν = 0:5 (where Sijkl and Πijkl without superscript are for viscous materials and those with superscript “el” are for elastic materials). The non-zero components of Selijkl jν = 0:5 and Πelijkl jν = 0:5 are, from Eq. (10.23):   3 2 3 2 3 ai ii ,Seliijj jν = 0:5 = aj ij , Selijij jν = 0:5 = a2i þ a2j ij 4π 4π 8π j - i el el el el Πijij jν = 0:5 = Πijji jν = 0:5 = , Πjiij jν = 0:5 = Πjiji jν = 0:5 = - Πelijij jν = 0:5 8π Seliiii jν = 0:5 =

ð10:32Þ

It will be shown in the next Chapter that viscous Πijkl are indeed equal to their elastic counterparts when the Poisson’s ratio is set to 0.5, i.e., Πijkl = Πelijkl jν = 0:5 . However, in general Sijkl ≠ Selijkl jν = 0:5 . Instead, they are related by: Sijkl = Selijkl jν = 0:5 þ Λij δkl

ð10:33Þ

where Λij is the second order tensor for the pressure field in incompressible viscous materials that will be introduced in Chap. 11. For isotropic materials, the following relation holds:   1 Λij = - δmn Selijmn jν = 0:5 3 (  1
- Selii11 þ Selii22 þ Selii33 jν = 0:5 = 3 0

i=j i≠j

ð10:34Þ

and thus, Eq. (10.33) can be written as: Sijkl = Selijmn jν = 0:5 J dmnkl

ð10:35Þ

From Eq. (10.35), one has: Sijkl Akl  Selijkl jν = 0:5 Akl

if tr ðAÞ = 0

ð10:36Þ

Therefore, for isotropic viscous materials, although Sijkl ≠ Selijkl jν = 0:5 , using the penalty approach (assuming Sijkl = Selijkl jν = 0:5 ) does not lead to any error in the calculated deviatoric stress and strain rate tensors of the inclusion, because

10.4

Expressions of Eshelby Tensors for Linear Isotropic Materials

235

deviatoric stress and strain tensors are traceless. For general anisotropic viscous materials, however, Eqs. (10.34)–(10.36) do not hold and Sijkl ≠ Selijkl jν = 0:5 is important (see Chap. 11). With Eqs. (10.25)–(10.34), viscous Eshelby tensors for isotropic incompressible materials can be calculated for ellipsoids of any shape.

10.4.1

Eshelby Tensor Expressions for Penny-Shaped, Rod-Like, and Spherical Objects in Isotropic Elastic Materials

For penny-shaped oblate objects, the elastic Eshelby tensor components are obtained by combining Eqs. (10.23) and (10.28). 0

13 - 8ν πr 32ð1 - νÞ e

B B B el 8ν - 1 B b πr Sij = Seliijj = B 32ð1 - νÞ e B B @ ν 4ν þ 1 πr 1 - ν 8ð1 - νÞ e 7 - 8ν Sel1212 = πr 32ð1 - νÞ e   1 ν-2π Sel1313 = Sel2323 = 1þ re 2 1-ν4

8ν - 1 πr 32ð1 - νÞ e

2ν - 1 πr 8ð 1 - ν Þ e

13 - 8ν πr 32ð1 - νÞ e

2ν - 1 πr 8ð 1 - ν Þ e

ν 4ν þ 1 πr 1 - ν 8ð 1 - ν Þ e

1þ

2ν - 1 πr 4ð 1 - ν Þ e

1 C C C C C C C A

ð10:37Þ

bij = Aiijj . This shorthand notation will be where we have used simplified notation A used frequently hereafter. 0

9πre B 16 B B 3πre el b Sij jv = 0:5 = B B 16 B @ 3πre 14 3πre el S1212 jv = 0:5 = 16

3πre 16 9πre 16 3πre 14

Sel1313 jv = 0:5 = Sel2323 jv = 0:5 =

1 0C C C 0C C C A 1

ð10:38Þ

  1 3πre 14 2

The elastic Eshelby tensor components for rod-like prolate objects are obtained by combining Eqs. (10.23) and (10.28):

ð10:39Þ

1 ð3ln2re -4Þþ ð1-2νÞð ln2re -1Þ 1-2ð1-2νÞð ln2re -1Þ 1-2ð1-2νÞð ln2re -1Þ C B 2ð1-νÞr2e 4ð1-νÞr2e 4ð1-νÞr2e C B

 C  B C el B 4-3ln2r ð 1-2ν Þ ð 1ln2r Þ 4 ð 1-2ν Þ ð 1ln2r Þ-3 1þ4 ð 1-2ν Þ ð 1ln2r Þ 1 1 1 e e e e C b 5-4νþ 4ν-1þ2ν Sij ¼ B 2 2 2 C B 4ð1-νÞ 8 ð 1-ν Þ 8 ð 1-ν Þ r 2r 2r e e e C B B 

C

@ 3-2ð1þvÞln2re þ2v 1þ4ð1-2νÞð1- ln2re Þ 4ð1-2νÞð1- ln2re Þ-3 A 1 1 1 4ν-15-4νþ þ2v 2 2 4ð1-νÞ 8ð1-νÞ 8ð1-νÞ re 2re 2r2e  2þv- ð1þvÞln2re 1 Sel1212 ¼Sel1313 ≈ þ ð1-νÞ 4ð1-νÞ r2e



1 1 1-2ν 1- ln2re 1- 2 þ 1þ Sel2323 ¼ 8ð1-νÞ 4ð1-νÞ 2re r2e

0

236 10 Eshelby’s Inclusion and Inhomogeneity Problem

10.4

Expressions of Eshelby Tensors for Linear Isotropic Materials

0

3 ln 2re - 4 r2e

B B

B 1 4 - 3 ln 2r e B þ 1 =B r2e B2 v = 0:5 B

@ 1 4 - 3 ln 2re þ 1 2 r2e

el b Sij



1 2r2e

3 14 1 14

1 2r2e 1 2r2e



237

1 2r2e

1

C

C 1 1 C 1- 2 C 4 2re C C

C 3 1 A 1- 2 4 2re

ð10:40Þ

For spheres, combining Eqs. (10.23) and (10.28) yields an isotropic fourth-order tensor that can be expressed as (Qu and Cherkaoui 2006, p. 293): Sel = 3αJm þ 2βJd

ð10:41Þ

3ðKþ2μÞ K 4 - 5ν = 9ð1þν where α = 3Kþ4μ 1 - νÞ and β = 5ð3Kþ4μÞ = 15ð1 - νÞ, withK, μ, and ν being respectively the bulk modulus, shear modulus, and Poisson’s ratio.

10.4.2

Eshelby Tensor Expressions for Penny-Shaped, Rod-Like, and Spherical Bodies in Isotropic Newtonian Materials

The corresponding expressions for viscous Eshelby tensors Sijkl and Λij, for pennyshaped, rod-like, and spherical objects are obtained by using Eqs. (10.32)–(10.34). The results are as follows: Penny-shapes spheroids: πr Λ11 = Λ22 = - e 4 πre Λ33 = - 1 þ 2 0 1 5 -1 -4 C r ð3π - 16re Þ B b B -1 Sij = e 5 -4C @ A 48 -4 -4 8  1
S1313 = S2323 = 4 - 3πre þ 20re 2 8 re ð3π - 16re Þ S1212 = 16

ð10:42Þ

ð10:43Þ

238

10 Eshelby’s Inclusion and Inhomogeneity Problem

Rod-like objects Λ11 =

1 - ln 2re r2e

Λ22 = Λ33 = 0 2 ln 2r - 3 3 - 2 ln 2re e r2e 2r2e B B B 3 - 2 ln 2r 1 7 - 4 ln 2re B e b Sij = B 2 4 2r 8r2e B e B @ 3 - 2 ln 2r 1 4 ln 2re - 5 e - þ 4 8r2e 2r2e

1 5 - 3 ln 2re S1212 = S1313 ≈ 1þ 4 r2e

1 1 S2323 ≈ 1- 2 4 2re

ln 2re - 1 - r2e 2r2e 3 - 2 ln 2re 2r2e

ð10:44Þ 1

C C 1 4 ln 2re - 5 C C - þ C 4 8r2e C C 1 7 - 4 ln 2re A 4 8r2e

ð aÞ

ð bÞ ð cÞ ð10:45Þ

The expression of Eq. (10.45) is slightly simplified to the following form: 0

Η 2

B Η B 1 B Η 1 ð1 þ Η Þ - 2 b Sij = B 2 4 B 8re B @ Η 1 1 ð Η - 1Þ þ 2 2 4 8re

Η 2

1

C C 1 1 C ð Η - 1Þ þ 2 C 4 8re C C 1 1 A ð1 þ ΗÞ - 2 4 8re

ð10:46Þ

where: Η=

2 ln 2re - 3 r2e

ð10:47Þ

For spherical inclusions, it is easy to get from Eq. (10.41) that Sel jν = 0:5 = Jm þ 25 Jd from which we get: Λij = S=

2 d J 5

1 δ 3 ij

ð10:48Þ

10.5

10.5

Strain and Rotation of a Deformable Ellipsoid

239

Strain and Rotation of a Deformable Ellipsoid

The partitioning equations are pivotal to micromechanics. We shall discuss the application of partitioning equations in the following Chapters. In the remaining of this Chapter, we use Eq. (10.20) for the motion of a viscous ellipsoid embedded in a slow viscous flow. Unlike a rigid object whose motion is defined by its angular velocity only, the motion of a deformable ellipsoid is defined by the angular velocity of its instantaneous semi-axes and the shape change rate of the ellipsoid. One needs to track the orientation of the coordinate system always parallel with the three semi-axes of the ellipsoid, as well as the length change of the three semi-axes. The instantaneous angular velocity of the semi-axes of a deformable ellipsoid in a viscous flow is determined by two factors. One is the vorticity w in the ellipsoid (Eq. 10.20). This determines the angular velocity of the material making up the ellipsoid. The other factor is the migration of the ellipsoid’s semi-axes through the material due to the shape change of the ellipsoid. Note the instantaneous principal strain rate axes in the ellipsoid domain (principal directions of ε, Eq. (10.20) are generally not coincident with the semi-axes of the ellipsoid, and this causes the migration of the semi-axes through the material. This angular velocity of this migration is described by a term called shear spin, denoted by ws, defined as (Goddard and Miller 1967; Bilby and Kolbuszewski 1977; Jiang 2012, 2016): 8 2 2 < ai þ a j ε s wij = a2i - a2j ij : wij

ai ≠ a j

no sum

ð10:49Þ

ai = aj

where εij and wij are respectively the strain rate and vorticity tensors of the ellipsoid expressed in the ellipsoid’s coordinate system (Eq. 10.20). The definition of wsij = wij at ai = aj is to avoid the geometric singularity where the semi-axes cannot be defined in the event the ellipsoid goes through a spheroidal or spherical shape (see more details in Jiang 2016). The total angular velocity tensor for an ellipsoid’s semi-axes, written in the ellipsoid’s coordinate system, Θij , is thus (e.g., Kocks et al. 1998; p. 395): Θij = wij - wsij

ð10:50Þ

The evolution of the ellipsoid is governed by the following two coupled equations (Jiang 2012):

240

10 Eshelby’s Inclusion and Inhomogeneity Problem

8 >
: dQ = - ΘQ dt

ð10:51Þ

0

1 a1 B C where a = @ a2 A, bε is the diagonal matrix made of the diagonal components of εij, a3
 i.e., bε = diagonal εij . Equation set (10.51) can be solved numerically. A fourth-order Runge-Kutta method (Jiang 2007b, 2012) or the Rodrigues rotation (Jiang 2013, 2014) can be used. One efficient numerical approximation using the Rodrigues rotation can be expressed as: (

anþ1 = exp ðbεδt Þan   Qnþ1 = exp - ΘðQn , an Þδt  Qn

ð10:52Þ

  where exp - ΘðQn , an Þδt is defined by Rodrigues relation in Eq. (8.43). A coupled fourth-order Runge-Kutta-Rodrigues method is as follows: (

  anþ1 = exp b hδt an Qnþ1 = exp ðθδt Þ  Qn

ð10:53Þ

where the exp(θδt) term is defined by the Rodrigues relation in Eq. (8.43); b h and θ are calculated by the following expressions: 8 n  
  o 1 b >

: θ= θ1 þ Qn 2 k2 T θ2 k2 þ k3 T θ3 k3 þ k4 T θ4 k4 Qn T 6

ð10:54Þ

The terms in Eq. (10.54) are calculated from the following procedure: θ1 = -ΘðQn , an Þ,h1 =εðQn , an Þ, b h1 =diagðh1 Þ

k1 = exp



b θ1 δt h δt Qn , α1 = exp 1 an 2 2

θ2 = - Θðk1 , α1 Þ, h2 = εðk1 , α1 Þ,b h2 = diagðh2 Þ

k2 = exp



b θ2 δt h δt Qn , α2 = exp 2 an 2 2

angular velocity and strain rate at the start

orientation and shape at halftime angular velocity and strain rate at half point

orientation and shape at the midpoint

10.5

Strain and Rotation of a Deformable Ellipsoid

θ = - Θðk2 , α2 Þ, h3 = εðk2 , α2 Þ,b h3 = diagðh3 Þ



another set of angular velocity and strain rate at the middle point

 k3 = exp ðθ3 δt ÞQn , α3 = exp b h3 δt an

θ4 = - Θðk3 , α3 Þ, h4 = εðk3 , α3 Þ, b h4 = diagðh4 Þ

k4 = exp ðθ4 δt ÞQn

241

orientation and shape at the end angular velocity and strain rate at the end

orientation at the end

The Mathcad worksheet named EshelbySingleEllipsoid.mcdx models the motion of an isotropic power-law viscous ellipsoid in an isotropic power-law medium. How the Eshelby solutions may be extended to power-law viscous materials will be discussed later. For a Newtonian ellipsoid in a Newtonian matrix, simply set the power-law stress exponents for the matrix and ellipsoid to one. The worksheet requires the following input parameters to evaluate: L: a 3 × 3 matrix for the velocity gradient tensor of the flow, such as L = 0

0 1 B @0 0

1 0 C 0 A for a dextral simple shearing flow parallel to the horizontal x-axis.

0 0 0 y0: a vector of 8 components for the initial state of the system, such as y0 = (120  deg 60  deg 25  deg 5 3 1 20 0.5)T, where the first three components are the spherical angles for the ellipsoid’s initial orientation (Chap. 3, Jiang 2007a, b), the next three components (5, 3, 1) are the initial lengths of the three semi-axes, the 7th component (20) is the initial ratio of the ellipsoid’s effective viscosity to that of the matrix at the bulk strain rate state. The 8th and last component (0.5) is the second invariant of the given flow, which is 0.5. δt: the time step for computation, such as δt = 0.01. The choice of δt will determine the accuracy of the computation, as discussed in Chap. 9. The selection must ensure that Lδt represents an infinitesimal increment of deformation. The selection of δt depends on the numerical method used. STEPS: the total number of steps of calculation. mm: an integer parameter specifying the number of steps of calculation between output results. Nm: the stress exponent for the matrix. Nm ≥ 1 and Nm = 1 if the matrix is Newtonian. Nc: the stress exponent for the ellipsoid. Nc ≥ 1 and Nc = 1 if the matrix is Newtonian. The orientation evolution of the ellipsoid is represented by the equal area projected paths for three semi-axes. The shape evolution is represented by the length vs steps plot for each semiaxis. The finite strain history in the ellipsoid is also tracked with principal finite strain axes plotted as equal-area paths and principal stretches as a function of steps.

242

10.6

10 Eshelby’s Inclusion and Inhomogeneity Problem

Notes and Key References

The following two Mathcad worksheets are associated with this Chapter: AnisotropicElasticEshelbyTensor.mcdx: This worksheet computes the Eshelby tensors (S and Π) in a general anisotropic elastic medium. The computation of Eshelby tensors for isotropic elastic medium is included within this worksheet. EshelbySingleEllipsoid.mcdx: See the explanation in the last section. This worksheet models the evolution of the shape and orientation of a deformable ellipsoid in viscous flow. The worksheet is written for the more general situation of powerlaw viscous rheology of the system (ellipsoid and the matrix). For a Newtonian system, the stress exponents of the matrix and ellipsoid are set to 1. The extension of the Eshelby theory to power-law viscous materials is presented in Chap. 17. Mura (1987) is perhaps the most authoritative and comprehensive text on the micromechanics of elastic materials. Qu and Cherkaoui (2006) is an introductory text on micromechanics that is quite accessible to geoscientists. Bilby et al. (1975) was the first paper published in an Earth Science journal on the deformation of viscous ellipsoids in viscous flow. It is also the first paper to apply Eshelby’s (1957) elastic inclusion/inhomogeneity solution to isotropic Newtonian materials. Freeman (1987) developed a numerical method to investigate the motion of deformable ellipsoids using Eshelby’s equations. Jiang (2007b, 2012) developed new algorithms to model the motion of deformable ellipsoids in viscous flows. Expressions for Eshelby tensor (S) components in isotropic elastic materials are given in many books on elastic micromechanics (e.g., Mura 1987; Qu and Cherkaoui 2006). The expressions in this Chapter for penny-shaped and rod-shaped objects are accurate to a higher order so that they can be used in geology problems where the aspect ratios of a deforming inclusion do not deviate extremely from 1.

References Barnett DM (1972) The precise evaluation of derivatives of the anisotropic elastic Green’s functions. Phys Status Solidi B 49(2):741–748. https://doi.org/10.1002/pssb.2220490238 Bilby BA, Eshelby JD, Kundu AK (1975) Change of shape of a viscous ellipsoidal region embedded in a slowly deforming matrix having a different viscosity. Tectonophysics 28(4): 265–274. https://doi.org/10.1016/0040-1951(75)90041-4 Bilby BA, Kolbuszewski ML (1977) Finite deformation of an inhomogeneity in 2-dimensional slow viscous incompressible-flow. Proc R Soc Lond Ser A-Math Phys Eng Sci 355(1682): 335–353. https://doi.org/10.1098/rspa.1977.0101 Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond Ser A-Math Phys Sci 241(1226):376–396. https://doi.org/10.1098/ rspa.1957.0133 Eshelby JD (1959) The elastic field outside an ellipsoidal inclusion. Proc R Soc Lond Ser A-Math Phys Sci 252(1271):561–569. https://doi.org/10.1098/rspa.1959.0173 Eshelby JD (1961) Elastic inclusions and inhomogeneities. Progr Solid Mech 2:89–104

References

243

Freeman B (1987) The behavior of deformable ellipsoidal particles in 3-dimensional slow flows – implications for geological strain analysis. Tectonophysics 132(4):297–309. https://doi.org/10. 1016/0040-1951(87)90349-0 Fung YC (1965) Foundations of solid mechanics. Prentice-Hall, New Jersey Goddard JD, Miller C (1967) Nonlinear effects in rheology of dilute suspensions. J Fluid Mech 28: 657–673. https://doi.org/10.1017/S0022112067002381 Jiang D (2007a) Numerical modeling of the motion of rigid ellipsoidal objects in slow viscous flows: a new approach. J Struct Geol 29(2):189–200. https://doi.org/10.1016/j.jsg.2006.09.010 Jiang D (2007b) Numerical modeling of the motion of deformable ellipsoidal objects in slow viscous flows. J Struct Geol 29(3):435–452. https://doi.org/10.1016/j.jsg.2006.09.009 Jiang D (2012) A general approach for modeling the motion of rigid and deformable ellipsoids in ductile flows. Comput Geosci 38(1):52–61. https://doi.org/10.1016/j.cageo.2011.05.002 Jiang D (2013) The motion of deformable ellipsoids in power-law viscous materials: formulation and numerical implementation of a micromechanical approach applicable to flow partitioning and heterogeneous deformation in Earth’s lithosphere. J Struct Geol 50:22–34. https://doi.org/ 10.1016/j.jsg.2012.06.011 Jiang D (2014) Structural geology meets micromechanics: a self-consistent model for the multiscale deformation and fabric development in Earth’s ductile lithosphere. J Struct Geol 68:247–272. https://doi.org/10.1016/j.jsg.2014.05.020 Jiang D (2016) Viscous inclusions in anisotropic materials: theoretical development and perspective applications. Tectonophysics 693:116–142. https://doi.org/10.1016/j.tecto.2016.10.012 Ju J, Sun L (1999) A novel formulation for the exterior-point Eshelby’s tensor of an ellipsoidal inclusion. J Appl Mech-Trans Asme 66(2):570–574. https://doi.org/10.1115/1.2791090 Kocks UF, Tomé CN, Wenk HR, Beaudoin AJ, Mecking H (1998) Texture and anisotropy: preferred orientations in polycrystals and their effect on materials properties. Cambridge University Press Lebensohn RA, Tomé CN (1993a) A self-consistent anisotropic approach for the simulation of plastic-deformation and texture development of polycrystals – application to zirconium alloys. Acta Metall Mater 41(9):2611–2624. https://doi.org/10.1016/0956-7151(93)90130-K Lebensohn RA, Tomé CN (1993b) A study of the stress state associated with twin nucleation and propagation in anisotropic materials. Philos Mag A-Phys Condensed Matter Struct Defects Mech Prop 67(1):187–206. https://doi.org/10.1080/01418619308207151 Lebensohn RA, Turner PA, Signorelli JW, Canova GR, Tomé CN (1998) Calculation of intergranular stresses based on a large-strain viscoplastic self-consistent polycrystal model. Model Simul Mater Sci Eng 6(4):447–465. https://doi.org/10.1088/0965-0393/6/4/011 Meng C, Heltsley W, Pollard DD (2012) Evaluation of the Eshelby solution for the ellipsoidal inclusion and heterogeneity. Comput Geosci 40:40–48. https://doi.org/10.1016/j.cageo.2011. 07.008 Mura T (1987) Micromechanics of defects in solids. Martinus Nijhoff Qu J, Cherkaoui M (2006) Fundamentals of micromechanics of solids. Wiley Spencer AJM (1980) Continuum mechanics. Longman Walker KP (1993) Fourier integral-representation of the green-function for an anisotropic elastic half-space. Proc R Soc Lond Ser A-Math Phys Eng Sci 443(1918):367–389. https://doi.org/10. 1098/rspa.1993.0151

Chapter 11

Viscous Inclusions in Anisotropic Materials

Abstract This Chapter applies Eshelby’s Green function and point-force approach to inclusion/inhomogeneity problems in incompressible anisotropic viscous materials. In Chap. 10, we presented the solutions for the classic inclusion/inhomogeneity problem (Eshelby, Proc R Soc Lond Ser A- 241(1226):376–396, 1957; Eshelby, Proc R Soc Lond Ser A- 252(1271):561–569, 1959) of an elastic inclusion/inhomogeneity in an isotropic elastic medium. The solutions have been extended to isotropic Newtonian viscous materials (Bilby et al., Tectonophysics 28(4):265–274, 1975; Bilby and Kolbuszewski, Proc R Soc Lond Ser A- 355(1682):335–353, 1977) by the so-called “penalty approach” whereby the Poisson’s ratio is set to 0.5 in corresponding elastic expressions in order to get equivalent viscous expressions. Since Eshelby’s approach has been applied to general anisotropic linearly elastic material (Mura, Micromechanics of defects in solids. Martinus Nijhoff, 1987), it may appear that the “penalty approach” could be extended to anisotropic Newtonian viscous materials as well. Unfortunately, this is not the case. The incompressible viscous inclusion/inhomogeneity problems must be solved anew, which is the subject of this Chapter. We present the formulations of Molinari et al. (Acta Metall 35(12):2983–2994, 1987) and Lebensohn et al. (Model Simul Mater Sci Eng 6(4): 447–465, 1998) and develop more explicit formal expressions for the viscous inclusion problem in this Chapter.

11.1

Limitation of the Penalty Approach

In principle, one may utilize a large modus operandi ‘bulk viscosity’ (K ) to make an incompressible viscous material slightly compressible so that corresponding elastic solutions can be used en masse for viscous problems. By making K sufficiently large, one expects to get satisfactorily accurate approximations for an incompressible viscous problem. Hutchinson (1976), Bilby et al. (1975), Bilby and Kolbuszewski

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-23313-5_11. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Jiang, Continuum Micromechanics, Springer Geophysics, https://doi.org/10.1007/978-3-031-23313-5_11

245

246

11

Viscous Inclusions in Anisotropic Materials

(1977), and Lebensohn and Tomé 1993), among others, have used the penalty approach. This approach is rather easy to construct and implement, as shown below. One can express the viscous stiffness and compliance tensors as: e ijkl = 3KJ m þ C ijkl C ijkl 1 e ijkl = M J m þ M ijkl 3K ijkl

ð aÞ ð bÞ

ð11:1Þ

where Cijkl and Mijkl are viscous stiffness and compliance tensors relating deviatoric e ijkl and M e ijkl are the hypothetical stiffness and stresses and strain rates, respectively, C compliance tensors used in the penalty approach. Although Cijkl and Mijkl are invertible only in the deviatoric space (i.e., Cijmn M mnkl = M ijmn C mnkl = J dijkl ) due to e ijkl and M e ijkl are invertible in the Cauchy space (i.e., incompressibility, C s e ijpq M e pqkl = J ). e pqkl = M e ijpq C C ijkl Because a coordinate system transformation does not affect the first term (with Jm ijkl) of either Eq. (11.1), adding the fictitious bulk viscosity term would appear to be e ijkl and M e ijkl replacing their quite a simple effort in numerical computations. With C elastic counterparts, the viscous equilibrium equations are identical in form to the static elastic Navier equations and all elastic solutions would be applicable to corresponding viscous problems. In practice, the above penalty approach is known to cause serious problems in numerical calculations. Unlike isotropic situations, solutions for anisotropic materials are generally not expressible in closed forms. Consequently, the penalty approach cannot be carried out by taking a limit operation as K → 1. Instead, a large K must be used in numerical calculations to approximate incompressibility. A sufficiently large K relative to the norm of Cijkl to approximate incompressibility will, because of the finite precision of the computer, deteriorate the numerical e ijkl and accumulation of round-off errors calculations due to ill-conditioning of C (e.g., Pelletier et al. 1989). This difficulty limits the applicability of the penalty approach where the material is anisotropic. A practical and certainly more efficient approach is to regard the viscous incompressibility as an additional kinematic constraint in the set of equations. Constitutive equations only relate the deviatoric stress tensor to the strain rate tensor through viscosity. The total Cauchy stress tensor is the sum of the deviatoric stress tensor and the pressure. The latter is also to be solved from the (new) equilibrium equation. The equilibrium equations for a viscous material become (e.g., Lebensohn et al. 1998): Cijkl uck,lj - p,j δij þ f i = 0 and the kinematic constraint of incompressibility requires:

ð11:2Þ

11.2

Green Functions for Viscous Materials

247

uck,k = 0

ð11:3Þ

Equation (11.2) is called Stokes’s Equation for slow viscous motions, and Eq. (11.3) is commonly referred to as the continuity equation. For the problem of a viscous inclusion with eigenstrain, Eqs. (11.2) and (11.3) together substitute the equivalent Eq. (10.10a) for a corresponding elastic inclusion with eigenstrains. Like the elastic inclusion problem stated in Eq. (10.10), the viscous eigenstrain is also replaced by a body force distribution over the inclusion-matrix interface:  fi =

C ijkl εkl nj

on inclusion surface

0

elsewhere

ð11:4Þ

which is similar to Eq. (10.10b). The following boundary condition (similar to Eq. 10.10c) completes the statement of the viscous inclusion problem: uci ðxÞ = 0 as jxj → 1

11.2

ð11:5Þ

Green Functions for Viscous Materials

We can use the same Green function method (Eshelby 1957, 1959; Barnett 1972; Bacon et al. 1980) to solve the viscous inclusion problem. To do so, we need to find expressions for viscous Green functions first. In the following, “viscous materials” shall always mean the most general “incompressible anisotropic viscous materials.” In solving linear elasticity problems (Chaps. 4 and 10), a single Green function for displacement was used. To solve incompressible viscous inclusion problems, Molinari et al. (1987) and Lebensohn et al. (1998) used two Green functions: Gij(x, x') for velocity, which is analogous but not the same as the elastic Green function for displacement, and Hi(x, x') for pressure. The Green function for velocity, Gij(x, x'), represents the velocity at x, ui(x), caused by a unit point force at location x'. The Green function for pressure, Hi(x, x'), represents the pressure at x, p(x), caused by a unit point force at x'. In terms of these two Green functions, the velocity and pressure fields are then expressed by the following convolution relations:

248

11

Z ui ðxÞ= Ω

Z pðxÞ=

Viscous Inclusions in Anisotropic Materials

      Gij x, x 0 f j x 0 dV x 0

ð aÞ

      H i x, x 0 f i x 0 dV x 0

ð bÞ

ð11:6Þ

Ω

where f(x') is the body force distribution in the deforming body and Ω is the volume containing all body force sources. Following the treatment of an elastic inclusion problem in the last Chapter, an ellipsoid inclusion with eigenstrain rate εkl can be replaced by a fictitious layer of body force f(x') = Cijklεklnj(x')dS(x') and a heterogeneous inclusion (Eshelby’s inhomogeneity) is then regarded as an equivalent homogeneous inclusion with the appropriate eigenstrain rate (Eshelby 1957). As pointed out in Chaps. 4 and 10, in an infinite medium Gij(x, x') = Gij(x - x') and, similarly, Hi(x, x') = Hi(x - x'). For simplicity of presentation, the Green functions are written as Gij(x) and Hi(x) for below, setting the source point at the origin. Following Chap. 4, the Green functions must satisfy the following auxiliary set of linear partial differential equations (Molinari et al. 1987, Eq. 27 there): 

C ijkl Gkm,jl ðxÞ - H m,i ðxÞ þ δim δðxÞ = 0 Gkm,k ðxÞ = 0

ð11:7Þ

with boundary conditions Gkm = 0,H m = 0 at infinity

ð11:8Þ

where δ(x) is the Dirac delta function. To solve for the two Green functions, the Fourier transform method is used as in Chap. 3. Applying Fourier transform to the set of Eq. (11.7) converts it to the following algebraic system: 

Aik K 2 gkm - zi iKhm = δim zk K 2 gkm = 0

ð11:9Þ

11.2

Green Functions for Viscous Materials

249

where 8 Aik = C ijkl zj zl > > > > 1 Z > > > > > Gkm ðxÞ exp ðiK  xÞdx < gkm ðKÞ =

ð11:10Þ

-1 > > > Z1 > > > > h ðKÞ = H m ðxÞ exp ðiK  xÞdx > > : m -1

The symmetric Aik is called the viscous Christoffel stiffness tensor, similar to the elastic Christoffel stiffness tensor (e.g., Barnett 1972; Walker 1993), gkm(K) and hm(K) are the two Green functions in Fourier space, pffiffiffiffiffiffiffiffiK the Fourier wave vector, ), and i = - 1 (not to be confused with a z unit vector parallel to K (i.e., z = K K tensor index i). The linear system of Eq. (11.9) can be written in the following form (Lebensohn et al. 1998): 0

A11 BA B 21 B @ A31 z1

A12 A22

A13 A23

A32 z2

A33 z3

10 2 z1 K g11 B K2g z2 C CB 21 CB z3 A@ K 2 g31 0 - iKh1

1 0 1 K 2 g13 C B 2 K g23 C B 0 C=B K 2 g33 A @ 0

1 0 0 1 0C C C 0 1A

- iKh3

0 0

K 2 g12 K 2 g22 K 2 g32 - iKh2

0

which is written in the form of a block matrix below (Jiang 2016): 

A

z

zT

0





K 2g - iKhT

=

  I

ð11:11Þ

0

Let us define: _

A ζT

ζ λ

!

 =

A

z

zT

0

-1 ð11:12Þ

The solution of Equation set (11.11) is then 

K2g - iKhT



_

=

A ζT

ζ λ

!  I 0

_

=

A ζT

! ð11:13Þ

Therefore, the Green functions in Fourier space, gij and hi are expressed, respectively, as:

250

11

Viscous Inclusions in Anisotropic Materials

_

Aij ðzÞ K2 iζ hi ðKÞ = i K gij ðKÞ =

ð aÞ

ð11:14Þ

ð bÞ

_

_

Note that A, ζ, and λ are all functions of z (Eq. 11.12). Note also that Aij ðzÞ is a 3 × 3 matrix distinct from Aij- 1 ðzÞ. The Green functions in real space are inverse Fourier transforms of Eq. (11.14). In the Appendix to this Chapter, expressions of Gij, Gij, l, and Hi are obtained (Eqs. 11.78a, b, c). The results are summarized below: 1 Gij ðxÞ = 2 4π r

Zπ

_

Aij dψ

ðaÞ

0

1 H i ð xÞ = - 2 2 4π r

Zπ F i dψ

ðbÞ

_ _ - xbl Aij þ zl M ij dψ

ð cÞ

ð11:15Þ

0

Gij,l ðxÞ =

1 4π 2 r 2

Zπ  0

where r = |x| (not to be confused with the viscosity ratio r), bxi = xri ,  _   _ F i = Ain Υ mn ζ m þ λAim þ ζ i ζ m bxm ,with Υ αβ = C αpβq zpbxq þ zqbxp _ _ _ _ _ M ij = Aim Ajn Υ mn þ Aim ζ j þ Ajm ζ i bxm _

_

Note that like Aij , Fi and M ij are both functions of z. The integrations of Eq. (11.15) are line integrals along a complete unit circle in the b x  z = 0 plane. To perform the integrations, z in the b x  z = 0 plane can be expressed in terms of two mutually orthogonal unit vectors [α and β, Synge 1957; Barnett 1972] and angle ψ as shown in the stereographic projection in Fig. 11.1. Both α and β are expressed in terms of b x, and ψ is the integration variable.   _ For isotropic viscous materials, it can be shown that Aij = 1η δij - zi zj , F i = - bxi,   _ and M ij = 1η zibxj þ zjbxi . And explicit integration of Eq. (11.15) gives the following well-known results for isotropic materials (e.g., Bilby et al. 1975; Mura 1987, p. 22; Buryachenko 2007, pp. 53, 56; Yin et al. 2014):

11.2

Green Functions for Viscous Materials

251

Fig. 11.1 Stereographic projection showing how, for any given x, unit vector z in the plane normal x to x (i.e., the b x  z = 0 plane, b x = jxj ) can be expressed in terms of any two mutually orthogonal unit vectors (α and β) on the b x  z = 0 plane by z = cos ψα + sin ψβ following Synge (1957) and Barnett (1972). Shown here is one possible pair of α and β. α is parallel to the ‘strike’ of the xnormal 0 plane and β the x is expressed spherical angles θ and φ 1 discipline of 0the plane.1If b 0 in terms of 1 cos θ sin φ - sin θ - cos θ cos φ B C B C B C by b x = @ sin θ sin φ A, then α = @ cos θ A , and β = @ - sin θ cos φ A. Green functions and cos φ 0 sin φ their derivatives (Gij, Gij, l, and Hi) for anisotropic viscous materials are expressed as contour integrals along a unit circle on the b x  z = 0 plane

 1  δ þ bxibxj 8πηr ij  1  - δijbxl þ δjlbxi þ δilbxj - 3bxibxjbxl Giso ij,l ðxÞ = 8πηr2 bxi H iso i ð xÞ = 4πr 2 Giso ij ðxÞ =

ð aÞ ð bÞ

ð11:16Þ

ðcÞ

where η is the Newtonian viscosity and the superscript “iso” stands for isotropic viscous materials. One can also get Eq. (11.16) by setting the Poisson’s ratio term to 0.5 in Eq. (3.67). More explicit forms than Eq. (11.15) for Green functions and their derivatives are not available for viscous materials, although some closed-form expressions for elastic Gij and its derivatives have been found for transverse isotropic, orthotropic, and cubic materials and on the symmetry plane of monoclinic materials (e.g., Ting and Lee 1997 and references therein; Buryachenko 2007). However, they cannot be used directly for incompressible viscous materials.

252

11

Viscous Inclusions in Anisotropic Materials

Although in integral forms, Eq. (11.15) can be evaluated numerically using standard quadrature and with mathematics applications such as MATHCAD or MATLAB (Jiang 2014, 2016; Qu 2018; Qu et al. 2016). A Mathcad worksheet is provided for evaluating Eq. (11.15).

11.3

Viscous Eshelby Tensors and Auxiliary Quantities

With the Green function Gij, viscous Eshelby tensors can be defined like their elastic counterparts. The deviatoric stress, strain rate, and vorticity tensors and the pressure inside and outside the inclusion can be solved accordingly. The confined velocity and pressure field due to a layer of body force on the inclusion-matrix interface C ijkl εkl nj is: Z

      Gir x - x 0 C rjkl εkl nj x 0 dS x 0 = C rjkl εkl

uci = ∂Ω

Z

p ð xÞ =



Z

    Gir x - x 0 nj dS x 0

∂Ω

 0

    H i x - x Cijkl εkl nj x 0 dS x 0 = C ijkl εkl

∂Ω

Z

ð aÞ

      H i x - x 0 nj x 0 dS x 0

ð bÞ

∂Ω

ð11:17Þ The following two auxiliary tensors are introduced to facilitate the calculation of velocity and pressure (Lebensohn et al. 1998, Eq. B14 there; Molinari et al. 1997, Eq. 31 there): Z Λij ðxÞ = -

    H i,j x - x 0 dV x 0 =

Ω

Z

T ijkl ðxÞ = -

Gik,jl Ω



 0

Z

      H i x - x 0 nj x 0 dS x 0

∂Ω

  x - x dV x 0 =

Z

      Gik,l x - x 0 nj x 0 dS x 0

ð aÞ ð bÞ

∂Ω

ð11:18Þ Λij is a second-order tensor for pressure calculation and Tijkl is analogous to its elastic counterpart. The second equal signs in Eq. (11.18) are due to the application of Gauss’s divergence theorem (e.g., Korn and Korn 1968, p. 5.6.1). To express Λij(x) and Tijkl(x) more explicitly, introduce the following substitution:

11.3

Viscous Eshelby Tensors and Auxiliary Quantities

253

Fig. 11.2 Integration over the ellipsoidal surface is realized with integration with respect to integration over θ and ϕ

0

a1 cos θ sin ϕ

1

B C x 0 = @ a2 sin θ sin ϕ A, a3 cos ϕ

0

cos θ sin ϕ

1

B C ξ = @ sin θ sin ϕ A cos ϕ

to get (Fig. 11.2): 1 0 1 0 - a1 sin θ sin ϕ a1 cos θ cos ϕ    C B C B     ∂x 0 ∂x 0 C C B n x 0 dS x 0 = dϕ × dθ = B @ a2 sin θ cos ϕ A × @ a2 cos θ sin ϕ Adϕd θ ∂ϕ ∂θ - a3 sin ϕ 0 20 1 3 a2 a3 cos θ sin ϕ 6B C 7 B C 7 =6 4@ a1 a3 sin θ sin ϕ A sin ϕ5dϕdθ a1 a1 cos ϕ 

ð11:19Þ Equation (11.18) can be expressed as:

254

11

Z2π Zπ Λij ðxÞ = a1 a2 a3 0

Viscous Inclusions in Anisotropic Materials

H i ðx - x 0 Þξj sin ϕdϕdθ aj

ð aÞ

0

Z2π Zπ T ijkl ðxÞ = a1 a2 a3 0

0

Gik,l ðx - x Þξj sin ϕdϕdθ aj

ð11:20Þ ð bÞ

0

For the interior fields, Λij and Tijkl are constant tensors. They can be further expressed, respectively, as the following integrals (Lebensohn et al. 1998; Weinberger et al. 2005; Jiang 2016):

Λij = -

a1 a 2 a 3 4π

Z2π Zπ 0

T ijkl =

a1 a2 a3 4π

ð aÞ

0

Z2π Zπ 0

ζ i zj ρ - 3 sin ϕdϕdθ

ð11:21Þ _

Aik zj zl ρ - 3 sin ϕdϕdθ

ð bÞ

0

where ai (i = 1, 2, 3) are the three semi-axes of the ellipsoid, ρ = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða1 z1 Þ2 þ ða2 z2 Þ2 þ ða3 z3 Þ2 , and unit vector z is expressed in terms of θ and ϕ 0 1 cos θ sin ϕ B C by z = @ sin θ sin ϕ A . cos ϕ By definition of Eq. (11.12), we have ziζ i = 1. Combining this with Eq. (11.21) R2π Rπ - 3 R2π a2 a3 implies that Λkk = - a14π ρ sin ϕdϕdθ . Applying the identity  Rπ

0

ρ

-3

sin ϕdϕdθ =

0

4π a1 a2 a3

0

0

due to Michelitsch et al. (2003), we get the following

identity: Λkk  - 1

ðfor all viscous materialsÞ

ð11:22Þ

Whereas from Eq. (11.21), it is clear that Λij ≠ Λji

ð11:23Þ

In general, for isotropic materials, it can be shown that ζ i = zi and the integrand in Eq. (11.21) is odd with respect to z if i ≠ j. This means that Λij is always diagonal:

11.4

Formal Solutions for the Interior and Exterior Fields

255

Λij = 0 ði ≠ jÞ ðfor isotropic materialsÞ

ð11:24Þ

The two viscous Eshelby tensors, S and II, are defined as follows (Jiang 2013): S = Jd : T : C and Π = Ja : T : C

ð11:25Þ

which are identical to the definition for elastic Eshelby tensors except that Jd is used instead of Js. The above expressions are also equivalent to the following expressions used in Appendix 2 of this Chapter: S = Ts : C and Π = Ta : C     where T sijkl = 14 T ijkl þ T jikl þ T ijlk þ T jilk and T aijkl = 14 T ijkl - T jikl þ T ijlk - T jilk (Lebensohn et al. 1998). Because of the Eshelby uniformity (stress and strain field being uniform in an ellipsoid inclusion), both Λij and Tijkl are constant tensors for points within the ellipsoid. The mechanical fields in the matrix (exterior points), ΛijE(x), SijklE(x), and ΠijklE(x) vary with space. To evaluate tensors SE(x) and ΠE(x), we need to get TijklE(x) from Eq. (11.20).

11.4

Formal Solutions for the Interior and Exterior Fields

p= From Eq. (11.17), the pressure field inside the inclusion due to eigenstrain is e Λ : C : S - 1 : eε . Together with interaction Eq. (10.8), the mechanical fields inside the ellipsoid are related to the remote fields by the following set of equations:  -1 eε = Jd - S - 1 e : C-1 : σ -1

e = Π : S : eε w e p = Λ : C : S - 1 : eε

ð aÞ ð bÞ ð cÞ

ð11:26Þ

This set is collectively called “the solutions to an interior-point Eshelby problem” or simply “the interior solutions.” Of course, such solutions are formal—to get the actual value of pressure, for instance, one must evaluate the equations numerically in most cases. Similarly, the exterior point fields are also expressed by a set of interaction equations as follows:

256

11

Viscous Inclusions in Anisotropic Materials

eεðxÞ = SE ðxÞ : S - 1 : eε e ðxÞ = ΠE ðxÞ : S - 1 : eε w e pðxÞ = Λ ðxÞ : C : S E

-1

ð11:27Þ : eε

This set is often referred to as “the solutions to an exterior-point Eshelby problem” or simply “the exterior solutions.” Again, these are formal expressions; numerical evaluation of the fields is required to get the actual results, which generally requires numerical computation.

11.5

Isotropic Systems

If the matrix and the inclusion are both isotropic, the calculation of the viscous tensors can be greatly simplified. Many analytical expressions can be derived from the formal expressions.

11.5.1

Viscous Tensor Identities

In Chap. 10, we presented expressions (Eqs. 10.32 and 10.33) to calculate the isotropic viscous Eshelby tensors for interior points from isotropic elastic Eshelby tensor expressions with the Poisson’s ratio term set to 0.5. In Appendix 2 to this Chapter, we give the derivation for that expression. It is shown in Appendix 2 that for isotropic materials, the symmetric elastic Eshelby tensor components with Poisson’s ratio set at 0.5, Selijkl jν = 0:5 are related to the symmetric viscous Eshelby tensor components Sijkl in the following manner: First, Selijkl jν = 0:5 = Sijkl = 0 except for components with repeated indices (i.e., Siijj and Sijij including i = j). Second, Sijij = Selijij jν = 0:5 (if i ≠ j ). Third, Siijj (including i = j) are related to Seliijj jν = 0:5 by the following relations:  1  el S þ Selii22 þ Selii33 jν = 0:5 3 ii11 Siijj = Seliijj jν = 0:5 þ Λii

Λii = -

ðno sumÞ ðno sumÞ

ð11:28Þ

The anti-symmetric Eshelby tensors for isotropic viscous materials are equal to the corresponding components for elastic materials with the Poisson’s ratio set at 0.5: Πijkl  Πelijkl jν = 0:5

ð11:29Þ

11.5

Isotropic Systems

257

For the exterior fields, the tensors ΛijE(x), SijklE(x), and ΠijklE(x) are all functions of the position x. These tensors are to be calculated numerically using integral expressions in Eqs. (11.15, 11.18) together with Green function expressions in Eq. (11.15). As the Green functions are in explicit analytic forms for isotropic materials, there has been great success in the effort to express ΛijE(x), SijklE(x), and ΠijklE(x) more explicitly. By the definition of Eqs. (10.15) and (11.18), it is straightforward, albeit a bit tedious, to get the following integral expressions for incompressible isotropic materials: SEijkl ðxÞ

1 ¼ 4π

Z Ω

( 1 r3

" 3 - δij δkl þ δikbxjbxl þ δjkbxibxl þ δilbxjbxk þ δjlbxibxk 2



þ 2 δijbxkbxl þ δklbxibxj ΠEijkl ðxÞ =

3 8π

Z Ω

# 

) - 15bxibxjbxkbxl dV

 1 δjkbxibxl þ δjlbxibxk - δikbxjbxl - δilbxjbxk dV 3 r ΛEij ðxÞ =

ð11:30Þ

1 4π

Z Ω

 1 3bxibxj - δij dV r3

ð11:31Þ

ð11:32Þ

xi - x 0 i jx - x 0 j, and Ω as in previous equations stand for the volume 2 2 2 x0 x0 x0 the ellipsoid, specifically a12 þ a22 þ a32 ≤ 1. 1 2 3 One notes from Eq. (11.30) that SEijkl = SEklij , that is, it has major symmetry.

where r = jx - x 0 j, bxi = of

Because the symmetric Eshelby tensor for isotropic elastic materials does not have major symmetry, the major symmetry of SEijkl is clearly a consequence of the incompressibility condition. It is an open question whether SEijkl has major symmetry for all incompressible materials or just for incompressible and isotropic materials. It turns out that the major symmetry of SEijkl for isotropic viscous materials makes the computation of viscous tensors much simpler, as will be shown below. To evaluate Eq. (11.30), we use the following two identities:

  bxibxj 1 δij ∂ bxi = þ 3 r 3 ∂x 0 j r 2 r3

   bxibxjbxkbxl 1 1  ∂ bxibxjbxk b b b b b b x x x x x x = δ þ δ þ δ þ jl i k kl i j 5 r3 il j k r3 r2 ∂x 0 l to convert the following volume integrals to surface integrals by use of the Gauss theorem:

258

11

Z Ω

Z Ω

Viscous Inclusions in Anisotropic Materials

 bxibxj 1 dV = δij I 0 þ Φij 3 r3

ð aÞ



bxibxjbxkbxl 1  dV = δil Φjk þ δjl Φik þ δkl Φij þ Ψijkl 3 5 r

ð bÞ

Z

I0 = Ω

dV ,Φ = r 3 ij

Z

bxi nj dS,and Ψijkl = r2

∂Ω

Z

bxibxjbxk nl dS r2

ð11:33Þ

ð cÞ

∂Ω

Substituting these expressions into Eqs. (11.30)–(11.32) and using variable substitutions to convert the integrations over an ellipsoid surface to a unit sphere, we get the following expressions: SEijkl ðxÞ =

i h  1 1 ðaÞ δij Φkl þ δik Φjl þ δjk Φil - δil Φjk - δjl Φik - 3Ψijkl 4π 2  1  ð bÞ δ Φ þ δjl Φik - δik Φjl - δil Φjk ΠEijkl ðxÞ = 8π jk il 1 ΛEij ðxÞ = Φ ðcÞ 4π ij ð11:34Þ

where Φij and Ψijkl can be expressed explicitly as: Z2π Zπ

ðxi - ai zi Þ sin ϕ h i3=2 z j 2 ðx1 - a1 z1 Þ þ ðx2 - a2 z2 Þ2 þ ðx3 - a3 z3 Þ2 dϕdθ 0 0 aj   Z2π Zπ ðxi - ai zi Þ xj - aj zj ðxk - ak zk Þ sin ϕ Ψijkl ðxÞ = a1 a2 a3 h i5= 2 2 2 2 zl x ð a z Þ þ ð x a z Þ þ ð x a z Þ dϕdθ 1 1 1 2 2 2 3 3 3 0 0 al ð11:35Þ

Φij ðxÞ = a1 a2 a3

Also, note the identity from the above expression: Φij = Ψααij

ð11:36Þ

With Eq. (11.35), solving the exterior fields (equivalent to obtaining SEijkl , ΠEijkl , and ΛEij) boils down to the evaluation of Φij and Ψijkl, which can be performed efficiently and accurately with built-in quadrature in MATLAB and Mathcad or with a combination of product Gaussian quadrature and Lebedev quadrature (Qu et al. 2016).

11.5

Isotropic Systems

11.5.2

259

An Explicit Approach to Evaluate Exterior Solutions for Isotropic Systems

With the above exploration, the solution to the exterior viscous problems in isotropic materials can be further simplified. First, Eq. (11.35) implies that Φij = Φji and Φkk = 0. Therefore, ΛEij ðxÞ is both symmetric and traceless for isotropic materials, that is: ΛEij = ΛEji

ð11:37Þ

ΛEkk = 0

Second, from Eqs. (11.34) and (11.35), one can confirm that SEααkl = 0. This, together with the major symmetry of SEijkl means: SEααkl = SEijαα = SEiiαα = 0 ðsum applies to repeated Greek letters onlyÞ

ð11:38Þ

Third, Ju and Sun (1999) develop an explicit approach to calculate the exterior-point symmetric Eshelby tensor for isotropic elastic materials [their Gijkl ðxÞ]. Comparing their Eq. (4) (see also Li and Wang 2008, p. 105) with Eq. (11.30) above, we get the following relation: SEijkl ðxÞ = Gijkl ðxÞjν = 0:5 þ ΛEij ðxÞδkl

ð11:39Þ

which is a generalized relation of Eq. (11.87) we get in Appendices. With Eqs. (11.37)–(11.39), we are now able to calculate all three viscous tensors, ΛEij ðxÞ, SEijkl ðxÞ, and ΠEijkl ðxÞ, required for the solution of the exterior field at any point, from the single tensor Gijkl ðxÞ by the following relations: 8 1 > ði ≠ jÞ < - Gijαα ðxÞjν = 0:5 , 3 E Λij ðxÞ =

> 1 : ði = jÞ - Gijαα ðxÞ - Gααij ðxÞ jν = 0:5 , 3 except i = j = k = l SEijkl ðxÞ = Gijkl ðxÞjν = 0:5 þ ΛEij ðxÞδkl h i SEiiii ðxÞ = - SEiijj ðxÞ þ SEiikk ðxÞ , ði ≠ j, i ≠ k, and j ≠ k Þ no sum  1 ΠEijkl ðxÞ = δjk ΛEil þ δjl ΛEik - δik ΛEjl - δil ΛEjk 2

ð aÞ ðbÞ ð11:40Þ ðcÞ ðdÞ

As the explicit computation for Gijkl ðxÞ is given in Ju and Sun (1999), the set of Eq. (11.40) gives a complete quasi-analytical solution for viscous exterior fields (stress, strain rate, and velocity field).

260

11

Viscous Inclusions in Anisotropic Materials

The elastic exterior Eshelby tensor Gijkl ðxÞ of Ju and Sun (1999) allows for the solution of the stress and strain fields but not the displacement field because the antisymmetric Eshelby tensor is not determined. Equation (11.40) cannot be applied to linear elastic materials because the latter are compressible. Meng et al. (2012) develop a complete algorithm for the exterior elastic field, closing many unfinished derivatives in Mura (1987, Eq. 11.41 there). The explicit expressions given in Meng et al. (2012) lead to a complete solution to the displacement field (Eq. 21 of Meng et al. 2012) rather than just the elastic stress and strain fields.

11.6

Some Analytic Solutions for Isotropic Systems

It is shown in this section that explicit analytic expressions for the interior fields of all-isotropic (both inclusion and matrix are isotropic) systems presented in Chap. 10 can be derived directly, as special cases, from the general anisotropic formulation.

11.6.1

Kinematics of an Ellipsoid in 3D Flows

From Eq. (11.26), one can get the following explicit strain rate and vorticity partitioning equations (see also Jiang 2013):   - 1 :E ε = Jd þ S : C - 1 : C1 - Jd w=W þ Π : S

-1

: ðε - EÞ

ð bÞ

ð aÞ

ð11:41Þ

where, as in Chap. 10, C1 stands for viscous stiffness tensor of the inclusion and C the viscous stiffness of the matrix. When both the matrix and the inclusion are isotropic, the C-1 : C1 term in Eq. (11.41) is reduced to rJd, where r (not to be confused with r in the Green functions) is the scalar ratio of inclusion viscosity to the matrix viscosity. In such cases, Eq. (11.41) is simplified to (see Jiang (2013, Eq. 11 there)):

-1 :E εjisotropic = Jd þ ðr - 1ÞS

ð11:42Þ

This is a system of five linear equations with five unknowns and can be solved explicitly (11.85 and 11.86, Appendices 2 and 3) to give the following expressions:

11.6

Some Analytic Solutions for Isotropic Systems

261

8 1 > ε11 jisotropic = f½1 þ ðr -1ÞðS2222 -S2233 ÞE 11 - ðr -1ÞðS1122 -S1133 ÞE 22 g > > Δ > < 1 ε22 jisotropic = fðr -1ÞðS2233 -S2211 ÞE11 þ ½1 þ ðr -1ÞðS1111 -S1133 ÞE 22 g > > Δ > > : ε33 jisotropic = - ðε11 þ ε22 Þ E ij εij jisotropic = , i≠j ðno sumÞ ð bÞ 1 þ 2ðr -1ÞSijij

ð aÞ

ð11:43Þ where Δ = [1 + (r - 1)(S1111 - S1133)][1 + (r - 1)(S2222 - S2233)] - (r 1)2(S1122 - S1133)(S2211 - S2233). These results were first given by Freeman (1987) with corrections by Jiang (2007). All viscous Sijkl in the above equations can be obtained from related elastic Eshelby tensors, as shown in Chap. 10. In Appendix 3 to this Chapter, it is shown that (Eq. 11.111): e ij jisotropic = w

a2i - a2j a2i þ a2j

eεij

ðno sumÞ

ð11:44Þ

which combined with Eq. (11.43) gives: e ij jisotropic = w

a2i - a2j a2i

þ

!

a2j

2ðr - 1ÞSijij E 1 þ 2ðr - 1ÞSijij ij

ðno sumÞ

ð11:45Þ

The shear spin (Eq. 10.49) is: wij jisotropic = s

a2i þ a2j a2i

- a2j

! E ij 1 þ 2ðr - 1ÞSijij

ðno sumÞ,

ð11:46Þ

The angular velocity of an isotropic ellipsoid embedded in an isotropic matrix is, therefore: " Θij jisotropic = W ij - 2ðr - 1ÞSijij 

E ij 1 þ 2ðr - 1ÞSijij

a2i - a2j a2i þ a2j

! þ

a2i þ a2j

!#

a2i - a2j

ðno sumÞ

Jeffery’s result is reproduced by taking the limit as r → 1 of Eq. (11.47):

ð11:47Þ

262

11

Viscous Inclusions in Anisotropic Materials

a2i - a2j

!

E ij Θij j rigid body = W ij - 2 ai þ a2j in isotr:matrix

ð11:48Þ

which is identical to Eq. (8.13). The angular velocity of an incompressible inviscid ellipsoid in an isotropic matrix is obtained by setting r = 0 in Eq. (11.47): E ij Θij j isotropic = W ij þ 1 - 2Sijij inviscid

2Sijij

a2i - a2j a2i þ a2j

-

a2i þ a2j a2i - a2j

! ð11:49Þ

Mulchrone (2007) derives a 2D version of this relation.

11.6.2

Deviatoric Stresses and Pressure

The deviatoric stresses inside the inclusion can be calculated from its strain rate tensor (obtained above) and the viscosity of the inclusion σ dij = 2ηc εij . The pressure in the inclusion can be calculated using Eq. (11.26). Because Λij is diagonal (Eq. 11.24) and all Sijkl are zero except for terms with repeated indices for isotropic materials, the expression for the pressure is simplified to: -1 eεJj e p = 2ηm ΛiI SiIjJ

ð11:50Þ

where ηm is the viscosity of the matrix, and the tilde stands for the difference of the quantity from the far-field matrix value. Summation applies to repeated lowercase indices, while uppercase indices take their lowercase values but are not summed. This is the convention used in Mura (1987).

11.6.3

An Ellipse in 2D Flows

Many early investigations on inclusion problems were on 2D plane-strain deformations. It is shown in this section that the 2D results can be reproduced readily from the general 3D formulation. An isotropic viscous ellipse in a 2D plane-straining flow can be regarded as an elliptic cylinder with its axis perpendicular to the plane of the flow. Elastic Eshelby tensors for isotropic elliptic cylinders are already known (Jaswon and Bhargava 1961; Mura 1987, p. 80). With the Poisson’s ratio set to 0.5, the non-vanishing components are:

11.6

Some Analytic Solutions for Isotropic Systems

263

2R þ 1 1 1 , Sel1122 = , Sel1133 = Rþ1 ð R þ 1Þ 2 ðR þ 1Þ2 R2 2R þ R2 R , Sel2222 = , Sel2233 = Sel2211 = 2 2 R þ 1 ð R þ 1Þ ðR þ 1Þ 2 R þ1 Sel1212 = 2 ð R þ 1Þ 2 Sel1111 =

ð11:51Þ

where R = ab (a and b being the two semi-axes of the elliptic section with a ≥ b). The corresponding viscous Λij can be calculated using Eq. (11.28) and non-vanishing Eshelby tensor components can be calculated accordingly. The results are as follows: Λ11 = -

1 R , Λ22 = Rþ1 Rþ1

S1111 = S2222 = - S1122 = - S2211 =

R R2 þ 1 ; S = 1212 2 ð R þ 1Þ 2 ð R þ 1Þ 2

ð11:52Þ

Introducing a fourth-order deviatoric identity tensor, analogous to J dijkl in 3D (Jiang 2016): J 2d ijkl =

 1 1 δik δjl þ δjk δil - δij δkl 2 2

-1 The non-vanishing components of Sijmn for elliptic cylinders under plane-straining -1 Smnkl = J 2D flows can be obtained using the identity Sijmn ijkl. There are only two distinct non-vanishing components:

-1 -1 -1 -1 = S2222 = - S1122 = - S2211 = S1111 -1 S1212

ð R þ 1Þ 2  =  2 2 R þ1

ðR þ 1Þ2 4R

ð aÞ ð bÞ

ð11:53Þ

With the above, we can express the interior strain rate and vorticity fields of the inclusion in terms of any given far-field plane-straining flow. If the far-field plane-straining flow is a 2D general shear with vorticity parallel to x3, submitting Eq. (11.52) into Eqs. (11.43) and (11.45) gives the following relations for the strain rates and vorticity of the ellipse, equivalent to the results of Bilby and Kolbuszewski (1977):

264

11

Viscous Inclusions in Anisotropic Materials

ðR þ 1Þ2 E11 R2 þ 2rR þ 1 ð R þ 1Þ 2 ε2D E 12 12 = rR2 þ 2R þ r  2  ð1 - r Þ R - 1 2D e 12 = E 12 w rR2 þ 2R þ r

2D ε2D 11 = - ε11 =

ð aÞ ð bÞ

ð11:54Þ

ð cÞ

Submitting Eq. (11.53) into Eq. (11.47) yields the angular velocity of the ellipse (Note for the elliptic cylinder considered here, the non-vanishing components are 2D 2D Θ12 = - Θ21 ): 2D Θ12 jisotropic

 2   R þ 1 r ðR - 1Þ þ 2R = W 12 þ E 12 R-1 rR2 þ 2R þ r

ð11:55Þ

which is equivalent to the result of Bilby and Kolbuszewski (1977, their Eq. 15). The deviatoric stresses and pressure inside the inclusion can also be calculated readily. As Λii and Sijkl are given in Eq. (11.52), the pressure in the ellipse, using Eq. (11.26), is:   2ηm Ε11 ð1 - rÞ R2 - 1 e p= R2 þ 2rR þ 1

ð11:56Þ

which is equivalent to Schmid and Podladchikov (2003, their Eq. 67). The viscous fields outside the ellipse vary in space. The pressure field can be obtained from the work of Jaswon and Bhargava (1961). Combining their Eqs. (13) and (21), applying the principle of superposition (Fung 1965, p. 3), and replacing the eigenstrain term in their equations with an equivalent inclusion, one gets the following explicit expression for the pressure (Jiang 2016): e pðξ, ηÞ =   1-

8ηm RðR þ 1Þ R-1     eε12 eε11 sin 2η sinh 2ξ cosh 2ξ - cos 2η R2 þ 1 cosh 2ξ - cos 2η 2R

ð11:57Þ

where (ξ, η) are exterior points in confocal elliptic coordinates. The confocal elliptic coordinates are related to the Cartesian coordinates by: pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 - b2 cosh ξ cos η pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 = a2 - b2 sinh ξ sin η x1 =

The Cartesian coordinate axes are parallel to the ellipse’s semi-axes and the coordinate origin is at the center of the ellipse.

11.6

Some Analytic Solutions for Isotropic Systems

265

Fig. 11.3 Pressure deviation field around a strong ellipse with r = 100 and R = 2, computed from Eq. (11.57). (a) In a pure shearing flow (vertical shortening and horizontal extension). (b) In a top to the right simple shearing flow. The pressure is normalized against the second invariant of the matrix deviatoric stress tensor. The pressure inside the inclusion is uniform and can be calculated using Eq. (11.56). It is 0.73 for (a) and 0 for (b)

One can recast Eq. (11.57) in the more familiar polar coordinates (ρ, θ) which are related to the Cartesian coordinates by x1 = ρ cos θ and x2 = ρ sin θ to get: e pðρ, θÞ = - 8ηm

  RðR þ 1Þ eε eε ½1 - Aðρ, θÞ 11 - Bðρ, θÞ 2 12 R-1 2R R þ1

ð11:58Þ

where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F þ ð1 - h cos 2θÞ pffiffiffi Aðρ, θÞ = 2F pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F - ð1 - h cos 2θÞ pffiffiffi Bðρ, θÞ = ± 2F pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F = 1 þ h2 - 2h cos 2θ h=

a 2 - b2 ρ2

B(ρ, θ) takes a positive sign when θ is in the first and 3rd quadrants and a negative sign when it is in the second and fourth quadrants. eε11 and eε12 in the above expression can also be expressed in terms of the far-field strain rates using Eq. (11.54). Figure 11.3 is a plot based on Eq. (11.57).

266

11

11.7

Viscous Inclusions in Anisotropic Materials

Equations for Ellipsoid Rotation in Anisotropic Viscous Materials

11.7.1

Angular Velocity Equations

The angular velocity tensor of an ellipsoid, expressed in the ellipsoid’s coordinate system, Θij , is given in Eq. (10.50). Combining Eq. (11.26), the angular velocity equation for the general case of a deformable anisotropic inclusion in a dissimilar anisotropic matrix is: -1 eεkl - wsij Θij = W ij þ Πijmn Smnkl

ð11:59Þ

For a rigid inclusion, εkl  0 (equivalent to eεkl  - Εkl ) and there is no shear spin wsij  0, the above expression is reduced to the following expression: -1 = W ij - Πijmn Smnkl Ekl

Θij j rigid in

ð11:60Þ

anisotropic This is the expression for a rigid ellipsoid in an anisotropic viscous matrix. Of a2 - a2 course, it is reduced to Jeffery’s result, Θij j rigid in = W ij - aI2 þa2J Εij , (Eq. 8.13), I

J

isotropic when the matrix is isotropic, as demonstrated in Appendix 3. Another limiting case is an inviscid ellipsoid in an anisotropic matrix. In this case, e = - Σ ). Submitting this the deviatoric stresses in the inclusion vanish (i.e., σ condition into Eqs. (11.43)–(11.47) and rearranging, one gets:  -1  -1 : Ε,oreεj general = S - 1 - Jd :Ε εj general = Jd - S inviscid



e j general = Π : Jd - S w

-1

ð aÞ

inviscid :Ε

ð bÞ

ð11:61Þ

inviscid

  -1 a2 þ a2J d  d J - S klmn Emn J Θij j general = W ij þ Πijkl - I2 ijkl 2 aI - aJ

ðcÞ

inviscid where in Eq. (11.61) repeated lowercase indices are summed up from 1 to 3, while uppercase indices which take on the same numbers as the lowercase ones are not summed up. This notation was already used in Eq. (11.50).

11.8

Summary

11.7.2

267

Shear Spin When the Inclusion Is Instantaneously a Spheroid or Sphere

The vorticity w and the strain rate ε in the inclusion are both always well defined, whether the inclusion is a triaxial ellipsoid, spheroid, or sphere. The set of principal strain rate axes, the three eigenvectors of ε, generally has an angular velocity which is called the spin component of the vorticity w (Eq. 5.33) (Astarita 1979; Means et al. 1980). The spin component of vorticity must not be confused with the shear spin ws (Eq. 10.49). The latter describes the phenomenon of a deforming ellipsoid whose shape principal axes (not the principal axes of the strain rate tensor in the ellipsoid!) swipe through the material as the ellipsoid changes shape (e.g., Goddard and Miller 1967; Bilby and Kolbuszewski 1977). It is called a ‘shear spin’ by Kocks et al. (1998, p. 385) in recognition that ws arises from shearing between the instantaneous shape principal axes of the ellipsoid: wsij = 0 if εij is diagonal (shearing components vanish). In a rigid object, the shear spin is zero. The shear spin must be considered for a deformable inclusion. However, the definition of wsij is singular if the inclusion is a spheroid (ai = aj ). This is a geometric singularity, occurring because of the simple fact that a perfect circle has no definable principal axes. The vorticity in the ellipsoid has no singularity. This geometric singularity must be considered in the numerical modeling of deformable ellipsoids as some deformable ellipsoids may go through a spheroidal or even spherical state during deformation in rare cases and that can cause errors in the modeling. Jiang (2007) considers all possible singular cases that may be encountered in the numerical calculation of isotropic inclusion problems. Jiang (2012) simplifies the singularity handling by setting wsij = wij when ai = aj as in Eq. (10.49). A more detailed explanation of this handling of the geometric singularity is in Jiang (2016). It turns out that my handling of the geometric singularity for ws is like the definition of the spin component of vorticity in the event the strain rate tensor has repeated eigenvalues. Where two of the eigenvalues of D are identical, say the two corresponding to d1 and d2, the definition of Eq. (5.34) is completed by setting Ω12 = W12 (Drouot 1976; Astarita 1979).

11.8

Summary

The solution of the instantaneous mechanical state and time-evolution of a solitary ellipsoid in an infinite Newtonian fluid boils down to obtaining three viscous tensors, Λij, Sijkl, and Πijkl, for the interior fields, and ΛEij ðxÞ, SEijkl ðxÞ, and ΠEijkl ðxÞ for the exterior fields. For isotropic viscous systems, these tensors can be expressed explicitly in terms of elliptic integrals or closed expressions that can be evaluated efficiently by numerical methods. Because of incompressibility, the three exterior field related tensors, ΛEij ðxÞ, SEijkl ðxÞ, and ΠEijkl ðxÞ, can be readily calculated from the single

268

11

Viscous Inclusions in Anisotropic Materials

tensor Gijkl ðxÞ for which quasi-analytical evaluation is available from Ju and Sun (1999). For anisotropic viscous systems, the viscous tensors (Λij, Sijkl, Πijkl, ΛEij ðxÞ, SEijkl ðxÞ , and ΠEijkl ðxÞ) are expressed in integral forms which can be evaluated numerically. The strain rate and vorticity of the flow field inside the ellipsoid as well as the angular velocity of the ellipsoid semi-axes are obtained once the interior fields are available. This enables the solution of Eq. (10.51) to give the progressive evolution of the ellipsoid in a Newtonian fluid. The general equations for an ellipsoid in an anisotropic incompressible Newtonian fluid are reduced to special case equations where the ellipsoid is rigid or the matrix is isotropic. The general equations for 3D deformation are also reduced to well-known equations for 2D deformations. For example, Eq. (11.48) is the angular velocity of an ellipsoid in the most general case. In the special case of a rigid ellipsoid in anisotropic fluid, the Equation is reduced to Eq. (11.49). In the further special case of a rigid ellipsoid in an isotropic fluid, the Equation is further reduced to the Jeffery equation (Eq. 8.13).

11.9

Notes and Key References

The following Mathcad worksheets are associated with this Chapter: ViscousGreenFunctions.mcdx: Integral form Green functions for the velocity and pressure in anisotropic viscous materials. ViscousInteriorEshelbyTensor.mcdx: Computes the Eshelby tensors (S and Π) for an ellipsoid in a linear anisotropic viscous medium. IsotropicExteriorEshelbyTensor.mcdx: Computes the Eshelby tensors (S and Π) outside an ellipsoid in a linear isotropic viscous medium. AnisotropicExteriorEshelbyTensor1.mcdx, AnisotropicExteriorEshelbyTensor2. mcdx, and AnisotropicExteriorEshelbyTensor3.mcdx: These three worksheets compute the Eshelby tensors (S and Π) outside an ellipsoid in a linear anisotropic viscous medium. The three sheets differ from one another slightly in the quadrature method used to evaluate the tensor T. Bilby et al. (1975) and Bilby and Kolbuszewski (1977) were the first work on deformable viscous ellipses in an isotropic Newtonian fluid. Solving 3D anisotropic viscous inclusion/inhomogeneity problems by the Green function method was pioneered by Molinari et al. (1987) and further developed by Lebensohn and Tomé (1993) and Lebensohn et al. (1998). Jiang (2016) derives integral expressions for the viscous Green functions and their first-order derivatives to be used in the calculation of viscous Eshelby tensors. The Appendix to this Chapter is an updated version of the derivation. Jiang (2016) also extends Ju and Sun’s (1999) explicit calculation of exterior Eshelby tensor for isotropic elastic materials to isotropic

Appendices

269

viscous materials. This extension is summarized in Eq. (11.40), upon which the Mathcad worksheet IsotropicExteriorEshelby Tensor.mcdx is based.

Appendices Integral Expressions of Gij,Gij, l, and Hi Integral expressions for Gij and Gij, l for anisotropic elastic materials are given in the Appendix to Chap. 4, Barnett (1972) and Mura (1987, pp. 33–34). The same procedure is used here to obtain integral expressions of Gij, Gij, l, and Hi for incompressible viscous materials. The two Green functions in real space are inverse Fourier transforms of the expressions in Eq. (11.14): 1 Gij ðxÞ = 3 8π

H i ð xÞ =

Z1

_

-1

1 8π 3

Aij ðzÞ exp ð - iK  xÞdK K2

ð11:62Þ

iζ i exp ð - iK  xÞdK K

ð11:63Þ

Z1 -1

where z = K K and the integrations are over the entire infinite 3D Fourier space. With Euler’s formula, the expressions become: 1 Gij ðxÞ = 3 8π

Z1 -1

1 H i ð xÞ = 3 8π

Z1 -1

_

Aij ðzÞ cos ðKz  xÞdK K2 ζi sin ðKz  xÞdK K

ð11:64Þ

ð11:65Þ

Here only the real parts of the integrals are concerned as Gij and Hi are real. The expression for Gij, l can be obtained by differentiating (11.64) with respect to xl: 1 Gij,l ðxÞ = - 3 8π

Z1 -1

_

zl Aij ðzÞ sin ðKz  xÞdK K

With the following set of variable substitutions:

ð11:66Þ

270

11

Viscous Inclusions in Anisotropic Materials

x k dk ,K = ,and therefore dK = 3 r r r

b x=

ð11:67Þ

Equations (11.64–11.66) are reduced to: 1 Gij ðxÞ = 3 8π r

Z1

_

Aij ðzÞ cos ðkz  b xÞdk k2

-1

Z1

1 H i ð xÞ = 3 2 8π r 1 Gij,l ðxÞ = - 3 2 8π r

-1

Z1 -1

ð11:68aÞ

ζi sin ðkz  b xÞdk k

ð11:68bÞ

_

zl Aij ðzÞ sin ðkz  b xÞdk k

ð11:68cÞ

A spherical coordinate system aligned with b x is used to evaluate these integrals, whereby: d 3 k = k2 sin σdkdσdψ,z  b x = cos σ

ð11:69Þ

where σ is the angle between z and b x, ψ is the angle between the projection of z on the plane perpendicular to b x (i.e., z  b x = 0) and any fixed reference direction in the zb x = 0 plane (Fig. 11.1). With (11.69), the three integrals in (11.68a, b, and c) become: 1 Gij ðxÞ = 3 8π r

8 Z2π J 2D 11kl > > B > > cijkl = 2ηn @ > < 1 2D J 12kl m > > > > J 2D > 1 11kl > > s = ijkl : 2ηn mJ 2D

12kl

1 2D 1 J m 12kl C A J 2D 22kl ! mJ 2D 12kl

ð12:6Þ

J 2D 22kl

where the lowercase cijkl and sijkl are 2D viscous stiffness and compliance tensors, respectively, with all indices varying from 1 to 2, J 2D ijkl is the fourth order deviatoric 1 identity tensor for plane-straining deformations defined as: J 2D ijkl = 2    1 ηn δik δjl þ δjk δil - 2 δij δkl (i, j, k, l = 1, 2), and m = η is a measure of the strength s of anisotropy (Treagus 2002; Fletcher 2004, 2009; Chen et al. 2014) which also corresponds to a similar concept in transverse isotropic elastic materials (Brace 1965). Equation (12.6) is a block matrix with each component itself being a 2 × 2 matrix. In the ellipse’s coordinate system, which is at an angle ϕ relative to the anisotropy plane (Fig. 12.1), the stiffness tensor components can  be obtained by tensor  transcos ϕ sin ϕ . The formation c'ijkl = QiaQjbQkcQldcabcd, where Qij = - sin ϕ cos ϕ non-vanishing components of c'ijkl are:

12.4

Application to Materials with a Planar Anisotropy

289

8   i ηn h 1 1 > 0 0 0 > c = c = c = 1 þ þ 1 cos 4ϕ 1111 2222 1122 > > 2 m m > <  ηn  1 0 0 c 1112 = - c 1222 = - 1 sin 4ϕ > 2 m > > h   i > η 1 1 > : - 1 cos 4ϕ c 0 1212 = n 1 þ þ 2 m m

ð12:7Þ

Other non-vanishing components are obtained by permutation of the above ones through the minor and major symmetry of c'ijkl. Those that cannot be obtained by permutation are all zero. Because of the symmetry of Eq. (12.6), replacing 2ηn by 2η1 and m1 by m in n Eq. (12.7), we immediately get the corresponding non-vanishing compliance components: 8 1 > > ½ðm þ 1Þ þ ð1 - mÞ cos 4ϕ s 0 1111 = s 0 2222 = - s 0 1122 = > > 8η > n > < 1 ðm - 1Þ sin 4ϕ s 0 1112 = - s 0 1222 = 8η > n > > > > 1 > : s 0 1212 = ½ðm þ 1Þ þ ðm - 1Þ cos 4ϕ 8ηn

ð12:8Þ

which are equivalent to Fletcher (2009, Eq. 5 there). Again, all other non-vanishing components are obtained by permutation of the above ones (e.g., s'1121 = s'2111 = s'1211 = s'1112). From Eq. (12.7), the Christoffel stiffness tensor (Eq. 11.10) components can be calculated: 8

η > A11 = n sin 2 ð2ϕ þ ψ Þ þ m cos 2 ð2ϕ þ ψ Þ > > m > < η A12 = A21 = n ð1 - mÞ sin ð4ϕ þ 2ψ Þ 2m > > >

> : A22 = ηn m sin 2 ð2ϕ þ ψ Þ þ cos 2 ð2ϕ þ ψ Þ m

ð12:9Þ

_

which in turn leads to the following expressions for Aik and ζ i (defined in Eq. 11.12) through matrix inversion: 8 > > > > > >
50%. The self-consistent method yields higher effective viscosity, closer to the upper-bound Voigt average than other methods. The differential and Mori-Tanaka

316

13

Effective Stiffnesses of Heterogeneous Materials

Fig. 13.3 Effective viscosity calculated according to different models as a function of the concentration of the second phase. The horizontal axis is the concentration of the second phase, and the vertical axis is the ratio of the effective viscosity to the matrix viscosity. Second phase inhomogeneities are all spheres with a viscosity contrast r = 20 to the matrix. SC self-consistent, DF1 differential method with the strain-rate partitioning tensor based on the Eshelby method, DF2 differential method with the strain rate partitioning tensor based on the Mori-Tanaka method, MT Mori-Tanaka

methods give effective viscosity between the self-consistent estimates and the Reuss bound. These results are consistent with investigations on elastic materials (e.g., Mura 1987; Nemat-Nasser and Hori 1999; Qu and Cherkaoui 2006). Handy (1990, 1994) has suggested that for two minerals of contrasting stiffness, the microstructure may be described as a load-bearing framework (LBF) if the stronger mineral supports stress around pockets of a weaker mineral, or as an interconnected weak layer (IWL) if the weaker mineral forms a matrix around microboudins of the stronger minearl. In the LBF case, both minerals have more or less the same strain, and the effective viscosity is be close to the upper bound Taylor model (uniform strain). The IWL is more like the Sachs model (uniform stress) with effective viscosity close to the Reuss lower bound. As a rock is deformed, the self-consistent method is more applicable at lower strain states. At higher strains, a connected weaker phase may be formed, consisting of dynamically recrystallized grains, brittle fractured phases, or newly formed fine-grained minerals. The self-consistent method may overestimate the effective viscosity in this case. The Mori-Tanaka method may be more reasonable (Figs. 13.3 and 13.4).

13.7

Expressions for Effective Stiffness of Multiphase Composites. . .

317

Fig. 13.4 Rheological weakening is associated with the formation of a connected weak matrix during the mylonitization of rocks in natural shear zones. (a) In an undeformed granite, the polycrystal aggregate behaves close to the equal-strain model. The self-consistent method may be valid for estimating the effective viscosity. (b) In a granitic mylonite, a connected matrix is made mainly of dynamically recrystallized quartz. The Mori-Tanaka method is more appropriate for estimating its effective viscosity

13.7

Expressions for Effective Stiffness of Multiphase Composites from Noninteracting Approximation

The examples of two-phase composite materials given in Sect. 13.4 are for cases where the second phase heterogeneities are all spherical. Another simple situation is for the second phase to be made of identically shaped and aligned ellipsoids so that the Eshelby tensors for the heterogeneities are all the same. Analytical expressions for the effective stiffness of an isotropic elastic solid containing cracks, pennyshaped, or rod-shaped second phase heterogeneities are derived in many papers (e.g., Budiansky and O’Connell 1976; Hashin 1988; Weng 1990; Ju and Chen 1994). In the case of an isotropic elastic solid containing randomly oriented elliptical cracks which have different size but identical ratio of minor to major axes a/b, the small rack density result for the effective bulk and shear moduli are (Hashin 1988): K = K 0 ð1 - κ 1 α Þ μ = μ0 ð1 - κ 2 αÞ where

ð13:48Þ

318

13

Effective Stiffnesses of Heterogeneous Materials

P π ab2 α= 2E ðkÞV

ð13:49Þ

is called the crack density parameter and the sum stands for summation over all cracks. The two parameters κ1 and κ 2 are defined below:   16 1 - v20 9ð1 - 2v0 Þ 

32 3 1 1 κ2 = ð 1 - v0 Þ 1 þ þ 45 4 1 þ βv0 1 þ ρv0 κ1 =

ð13:50Þ

where β=

k21 F ðkÞ - E ðkÞ k 2 E ðk Þ

ρ=

k21 Eðk Þ - F ðkÞ k 2 E ðk Þ

k 2 = 1 - b2 =a2 ,

ð13:51Þ k21 = b2 =a2

Here F(k) and E(k) are the complete elliptic integrals of the first and second kind, respectively, (e.g., Mura 1987, p. 83): Zπ=2 E ðk Þ =



 1 - k 2 sin 2 ϕ dϕ

0

Zπ=2 F ðk Þ =

ð13:52Þ 

1 - k2 sin 2 ϕ

 - 1=2

dϕ

0

Using the differential method, Hashin (1988) obtained the following expressions for penny-shaped cracks (a = b), originally given by Zimmerman (1985): 1-v 5 v0 15 45 1þv 5 3-v þ þ ln ln þ ln ln v 64 1 - v0 128 1 þ v0 128 3 - v0 8 10=9

 v E 3 - v0 1=9 = v0 E0 3-v α=

μ 1 þ v0 E = 1 þ v E0 μ0 P 3 a where α = V .

ð13:53Þ

13.7

Expressions for Effective Stiffness of Multiphase Composites. . .

319

In the so-called noninteracting approximation where the averaged perturbations in a heterogeneity due to other surrounding heterogeneities are neglected, Ju and Chen (1994) have obtained analytical results identical to the classical variational bounds (Hashin and Shtrikman 1963). For the effective stiffness for a multiphase composite made of a matrix phase and inhomogeneities of identically shaped and unidirectionally aligned ellipsoids, Ju and Chen (1994) obtained the following expression: h i C = C0 Jx þ ΖðJx - SΖÞ - 1

ð13:54Þ

where Ζ=

i-1 X h cr S þ ð C r - C 0 Þ - 1 C 0

ð13:55Þ

r

and Jx = Js and Jd, respectively, for linear elastic and Newtonian viscous materials. For a two-phase composite with heterogeneities unidirectionally aligned and identically shaped, Eq. (13.54) is reduced to (dropping the subscript for heterogeneities):  C = C0 :

h

J þ c ðC1 - C0 Þ x

-1

: C0 þ ð1 - cÞS

i - 1

ð13:56Þ

Equation (13.56) is valid for isotropic or anisotropic C0 and C1. This expression is identidal to the tensorial expression for effective stiffness given by Willis (1977), which is Walpole’s (1966a, b) generalization of Hashin and Shtrikman’s (1963) result. Furthermore, Eq. (13.56) is identical to Zhao et al. (1989) based on the MoriTanaka method. In the event all heterogeneities are spherical and both the matrix and heterogeneities are isotropic elastic, the effective bulk and shear moduli are (Ju and Chen 1994):   3ð1 - v0 ÞðK 1 - K 0 Þc K = K0 1 þ ð aÞ 3ð1 - v0 ÞK 0 þ ð1 - cÞð1 þ v0 ÞðK 1 - K 0 Þ   15ð1 - v0 Þðμ1 - μ0 Þc ð bÞ μ = μ0 1 þ 15ð1 - v0 Þμ0 þ ð1 - cÞð8 - 10v0 Þðμ1 - μ0 Þ

ð13:57Þ

Equation (13.57) are the lower (or upper) bounds, if the matrix is the softer (or harder) phase, expressions derived by Hashin and Shtrikman (1963). They are also the same as the results of Weng (1984, 1990) based on the Mori-Tanaka method. In the case of a multi-phase composition with the same S (meaning all heterogeneities are identically shaped and unidirectionally aligned), Weng’s (1990) analysis

320

13

Effective Stiffnesses of Heterogeneous Materials

based on the Mori-Tanaka method led to the same expression as Eq. (13.56) (Ju and Chen 1994). In elastic solids with parallel penny-shaped microcracks, (C1 - C0)-1 : C0 = s J , c = 4πr3e α with α defined above denoting the microcrack density. The noninteracting approximation of Ju and Chen (1994) (Eq. 13.56) leads to the following tensorial expression for the effective elastic stiffness:

 4π C = 1 þ α C0 : Ξ 3 where Ξ = lim

h

1 re → 0 re

ðS - Js Þ - 1 -

ð13:58Þ

i-1

4π 3 αS

The components of Eshelby tensor for penny-shaped objects are given in Chap. 10 (Eq. 10.37). By substituting the components of Sinto the above expression and taking the limit, one gets the (transversely isotropic) effective compliance tensor of the solid with parallel microcracks are (the microcracks are all normal to the x3 axis):  16  1 - v20 α 3 16ð1 - v0 Þ 4μ0 M 1313 = 1 þ α = 4μ0 M 2323 = 4μ0 M 3131 = 4μ0 M 3232 3ð 2 - v0 Þ E 0 M 3333 = 1 þ

ð13:59Þ

The remaining components of M are identical to those of the isotropic matrix compliance. If the microcracks are sealed by veins or dykes, then C1 ≠ 0. Equation (1.56) together Eq. (10.37) are to be used to compute the effective stiffness. Where both the matrix and the sealed cracks are isotropic, and assuming that KK 10 = μμ1 = r, it can be 0 shown that Eq. (13.56) becomes: o  n C = 3K 0 Jm þ 2μ0 Jd Js þ cðr- 1Þ½Js þ ð1 - cÞðr - 1ÞS - 1

ð13:60Þ

The homogenization is discussed in the context of linear materials in this chapter. We will revisit the subject of homogenization when we extend the Eshelby formalism is extended to non-linear viscous materials in Chap. 17.

13.8

Notes and Key References

Mathcad worksheet EffectiveElasticStiffness.mcdx computes the effective elastic stiffness tensor (fourth-order) for a polycrystal aggregate from constituent stiffnesses and microstructures (shapes, orientations) using the self-consistent method. The

References

321

example input parameters in the worksheet are specifically for a polycrystal quartzite made of 400 equant α-quartz grains. The crystal orientations are uniform in 3D. The MATLAB software Homogenization implements the various homogenization methods numerically. Chapter 7 of Mura (1987) is on effective elastic properties. Nemat-Nasser and Hori (1999) is a thorough treatment on the effective properties of heterogeneous materials. Qu and Cherkaoui (2006) given an accessible presentation of determining effective elastic properties of heterogeneous materials. Chapter 12 of Karato (2008) presents a succinct introduction to calculate the effective properties of mainly two-phase materials. Effective strength of multiphase rocks is also investigated phenomenologically by Ji and Zhao (1993), Ji et al. (2001), and Ji (2004).

References Aboudi J, Arnold SM, Bednarcyk BA (2013) Micromechanics of composite materials: a generalized multiscale analysis approach. Butterworth-Heinemann Budiansky B, O’Connell RJ (1976) Elastic moduli of a cracked solid. Int J Solids Struct 12:81–97 Carmichael RS (1989) Practical handbook of physical properties of rocks and minerals. CRC Press, Boca Raton Handy MR (1990) The solid state flow of poly mineralic rocks. J Geophys Res 95:8647–8661 Handy MR (1994) Flow laws for rocks containing 2 nonlinear viscous phases – a phenomenological approach. J Struct Geol 16(3):287–301. https://doi.org/10.1016/0191-8141(94)90035-3 Hashin Z (1988) The differential scheme and its application to cracked materials. J Mech Phys Solids 36(6):719–734 Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11:127–140 Hobbs BE, Means WD, Williams PF (1976) An outline of structural geology. Wiley Jeffery GB (1922) The motion of ellipsoidal particles in a viscous fluid. Proc R Soc Lond Ser A-Containing Papers of a Mathematical and Physical Character 102(715):161–179. https://doi. org/10.1098/rspa.1922.0078 Ji S (2004) A generalized mixture rule for estimating the viscosity of solid-liquid suspensions and mechanical properties of polyphase rocks and composite materials. J Geophys Res Solid Earth 109:B10207. https://doi.org/10.1029/2004JB003124 Ji S, Wang Z, Wirth R (2001) Bulk flow strength of forsterite–enstatite composites as a function of forsterite content. Tectonophysics 341(1):69–93. https://doi.org/10.1016/S0040-1951(01) 00191-3 Ji S, Zhao P (1993) Location of tensile fracture within rigid-brittle inclusions in a ductile flowing matrix. Tectonophysics 220(1–4):23–31. https://doi.org/10.1016/0040-1951(93)90221-5 Ji S, Li L, Motra HB, Wuttke F, Sun S, Michibayashi K, Salisbury MH (2018) Poisson’s ratio and auxetic properties of natural rocks. J Geophys Res Solid Earth 123:1161–1185. https://doi.org/ 10.1002/2017JB014606 Jiang D (2014) Structural geology meets micromechanics: a self-consistent model for the multiscale deformation and fabric development in Earth’s ductile lithosphere. J Struct Geol 68:247–272. https://doi.org/10.1016/j.jsg.2014.05.020 Ju JW, Chen TM (1994) Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidal inhomogeneities. Acta Mech 103:103–121 Karato S-I (2008) Deformation of earth materials: an introduction to the rheology of solid earth. Cambridge University Press

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Kimizuka H, Ogata S, Li J, Shibutani Y (2007) Complete set of elastic constants of α-quartz at high pressure: a first-principle study. Phys Rev B 75:054109. https://doi.org/10.1103/PhysRevB.75. 054109 Li S, Wang G (2008) Introduction to Micromechanics and Nanomechanics. World Scientific Mura T (1987) Micromechanics of defects in solids. Martinus Nijhoff Murshed MR, Ranganathan SI (2017) Hill–Mandel condition and bounds on lower symmetry elastic crystals. Mech Res Commun 81:7–10. https://doi.org/10.1016/j.mechrescom.2017.01. 005 Nemat-Nasser S, Hori M (1999) Micromechanics: Overall Properties of Heterogeneous Materials. Elsevier Science Ohno I (1995) Temperature variation of elastic properties of α-quartz up to the α-β transition. J Phys Earth 43:157–169 Ohno I, Harada K, Yoshitomi C (2006) Temperature variation of elastic constants of quartz across the alpha-beta transition. Phys Chem Miner 33(1):1–9. https://doi.org/10.1007/s00269-0050008-3 Qu J, Cherkaoui M (2006) Fundamentals of micromechanics of solids. Wiley Taylor GI (1932) The viscosity of a fluid containing small drops of another fluid. Proc R Soc A138:41–48. https://doi.org/10.1098/rspa.1932.0169 Turner FJ, Weiss LE (1963) Structural analysis of metamorphic tectonites. McGraw-Hill Vand V (1948) Viscosity of solutions and suspensions. 1. Theory. J Phys Coll Chem 52(2): 277–299. https://doi.org/10.1021/j150458a001 Walpole LJ (1966a) On bounds for overall elastic moduli of inhomogeneous systems – I. J Mech Phys Solids 14:151–162 Walpole LJ (1966b) On bounds for overall elastic moduli of inhomogeneous systems – II. J Mech Phys Solids 14:289–301 Weng GJ (1984) Some elastic properties of reinforced solids, with special reference to isotropic ones containing spherical inclusions. Int J Eng Sci 22:845–856 Weng GJ (1990) The theoretical connection between Mori-Tanaka’ theory and the HashinShtrikman-Walpole bounds. Int J Eng Sci 28:1111–1120 Wenk H-R (1998) Plasticity modeling in minerals and rocks. In: Kocks UF, Tome CN, Wenk H-R (eds) Texture and anisotropy: preferred orientations in polycrystals and their effect on materials properties. Cambridge University Press, pp 560–596 Willis JR (1977) Bounds and self-consistent estimates for the overall properties of anisotropic composites. J Mech Phys Solids 25:185–202 Zhao YH, Tandon GP, Weng GJ (1989) Elastic moduli for a class of porous materials. Acta Mech 76:105–131 Zimmerman RW (1985) The effect of microcracks on the elastic moduli of brittle materials. J Mater Sci Lett 4:1457–1460

Chapter 14

Application Example 1: An Elastic Prolate Object in a Viscous Matrix

Abstract We analyzed the longitudinal stress in an elastic prolate object in Newtonian viscous fluid under simple shearing flow in Chap. 9. That analysis was based on Jeffery’s (Proc R Soc Lond Ser A-Containing Papers of a Mathematical and Physical Character 102(715):161–179, 1922) original equations on the traction forces the viscous fluid exerts on the object’s surface [Forgacs and Mason (J Colloid Sci 14(5):457–472, 1959); Goldsmith and Mason (The microrheology of dispersions, Rheology: theory and applications. Academic Press, 1967); Okagawa et al. (J Colloid Interface Sci 47(2):536–567, 1974)]. Having presented Eshelby’s solutions for the inclusion/inhomogeneity problems, we carry out a more thorough investigation of the mechanical interactions between a prolate object and the embedding matrix in this chapter. We shall consider the following two situations: The first situation is where both the prolate object (inclusion hereafter) and the matrix are linearly elastic, but the inclusion is stronger. The one-dimensional shear-lag model or fiber-loading model commonly used in the analysis of microboudinage development is based on an elastic-elastic interaction. The second situation is where the inclusion is elastic, but the embedding matrix is Newtonian viscous. This scenario is perhaps closer to the case of a strong rod-like mineral in a flowing mylonite matrix. For simplicity, we assume that both the matrix and the inclusion are isotropic. The first situation is a direct application of Eshelby’s (Proc R Soc Lond Ser A-Math Phys Sci 241(1226):376–396, 1957) inclusion/inhomogeneity solutions for an elastic system presented in Chap. 10. Although Eshelby’s solutions do not apply to the second situation directly, we will solve the problem using the equivalentinclusion method of Eshelby’s illustrated in Chap. 10. In what follows, we first find the expressions for the mechanical fields in a prolate object embedded in an elastic or viscous matrix. We then use the expressions to investigate the deformation of prolate objects and compare the results with other studies.

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-23313-5_14. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Jiang, Continuum Micromechanics, Springer Geophysics, https://doi.org/10.1007/978-3-031-23313-5_14

323

324

14.1

14 Application Example 1: An Elastic Prolate Object in a Viscous Matrix

An Elastic Prolate Inclusion in an Elastic Matrix

Eshelby’s solutions (Eq. 10.7) apply directly to the case of a prolate rheologically distinct elastic inclusion embedded in an elastic matrix. The stress and strain fields in the inclusion are related to the far-field stress and strain by the following relations: ε=A : E

ð aÞ

σ=B : Σ

ð bÞ

ð14:1Þ

Where, as in Eq. (10.7), the lowercase ε and σ are respectively the elastic strain and stress tensors in the inclusion, corresponding uppercase symbols are counterparts in the far-field matrix, A and B are the strain and stress partitioning tensors, respectively, defined as: 
 - 1 A = JS þ S : Cm- 1 : Ce - JS
-1 B = Ce : Cm : A - 1

ðaÞ ðbÞ

ð14:2Þ

where Ce and Cm are elastic stiffness tensors of the (ellipsoid) inclusion and the matrix, respectively. Since for isotropic elastic materials, the stiffness tensor can be expressed as (Chap. 1): Cx = 3K x Jm þ 2μx Jd 1 m 1 d C - 1x = J þ J 3K x 2μx

ð14:3Þ

where the subscript x = e for the inclusion and x = m the matrix. Elastic stiffness of isotropic materials is also commonly expressed in terms of Young’s modulus (E) and Poisson’s ratio (ν), and: E x = 2μx ð1 þ νx Þ,

Kx =

Ex , 3ð1 - 2νx Þ

μx =

3ð1 - 2νx Þ K 2ð1 þ νx Þ x

ð14:4Þ

The stress partitioning tensor for a prolate object is a function of five independent variables, four elastic constants plus the object’s aspect ratio (re), such as B(re, Ke, Km, μe, μm) or B(re, Ee, Em, νe, νm). Because the stress (and strain) partitioning tensor is related only to the ratios among the elastic constants, not their absolute values, we can use dimensionless variables by setting, say, Km = 1. Denoting R = KKme , it follows from Eq. (14.4) that:

14.2

Comparison with the Fiber-Loading Theory

K m = 1,

μm =

3ð1 - 2νm Þ , 2ð1 þ νm Þ

E m = 3ð1 - 2νm Þ,

325

K e = R,

μe =

3ð1 - 2νe Þ R 2ð1 þ νe Þ

Ee = 3ð1 - 2νe Þ

In the literature on microboudinage, the ratio of Young’s moduli of the object to the matrix Ee/Em is often used. It is related to other parameters by: μ ð1 þ νe Þ K ð1 - 2νe Þ Ee = e = e Em μm ð1 þ νm Þ K m ð1 - 2νm Þ

ð14:5Þ

One can see that the ratio Ee/Em is coupled with R and Poisson’s ratios. To facilitate a comparison of the investigation here with the literature, we shall set νm = νe = ν so that Eq. (14.5) becomes: μ Ee K = e = e =R E m μm K m

ð14:6Þ

The Poisson’s ratio for rocks can vary between 0.1 and greater than 0.4, but a typical value is 0.25 (Pollard and Martel 2020). It will be shown below that the stress partitioning tensor depends very weakly on Poisson’s ratio. Therefore, the stress partitioning tensor shall be a function of three dimensionless variables, re, R, and ν. The stress tensor in the inclusion can be related to any matrix stress tensor through: σ = Bðre , R, νÞ : Σ B = Ce ½Cm þ ðR - 1ÞCm S - 1

ð aÞ ð bÞ

ð14:7Þ

As analytical expressions for S are known and given in Chap. 10, it is straightforward to compute the stress partitioning tensor B(re, R, ν) for any prolate objects. The stress and strain tensors can thus be obtained. The explicit analytic expression for B(re, R, ν) is too cumbersome, but the calculation is easily done in MATHCAD or MATLAB. A Mathcad worksheet named “Prolate Analysis.mcdx” is provided online for the calculation. In the following section, we analyze the stress and strain in prolate objects and compare our analysis with the fiber-loading and shear-lag models, which are commonly used to understand the deformation of elongated objects embedded in a weaker matrix.

14.2

Comparison with the Fiber-Loading Theory

The deformation of a strong elongated object in a mechanically weaker matrix has been understood with the fiber-loading theory in the field of mechanics of composite materials. The original fiber-loading theory (Kelly 1973; Lloyd et al. 1982; Masuda

326

14 Application Example 1: An Elastic Prolate Object in a Viscous Matrix

and Kuriyama 1988) considered the case of a strong fiber enclosed in an elastic medium subjected to bulk fibre-parallel uniaxial tension. The one-dimensional theory predicts that the tensile stress in the fiber varies from zero at the ends and increases to a maximum value at the midpoint. The maximum tensile stress at the midpoint is given by:   σ max E 1 = e 1σ0 Em cosh ðβre Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Em β=u t R E e ð1 þ νm Þ ln 0 r0

ð aÞ ð bÞ

ð14:8Þ

where σ 0 is the bulk uniaxial tensile stress, R0 the mean distance between adjacent fibers and r0 is fiber radius. In the case of dilute fiber concentrations (β → 0) where inter-fiber interactions can be ignored, Eq. (14.8) gives a meaningless result σ max= 0, which would imply that an isolated fiber is stress-free! Equation (14.8) also does not make sense when the fiber is the same material as the matrix, as noticed by Ji and Zhao (1994) and Zhao and Ji (1997). These authors developed the following refined relation:  

Ee Em 1 σ max = 1þ -1 σ0 Em Ee cosh ðβre Þ

ð14:9Þ

Because the shear-lag model is based on a uniaxial extension setup, it is not clear how the model can be applied to a general stress state. The one-dimensional quantity σ 0 in Eq. (14.9) was referred to as the “bulk stress” (e.g., Masuda and Kuriyama 1988; Masuda et al. 1989). With no further justification, this group has also regarded σ 0 as “the far-field differential stress” later (Masuda et al. 2003, 2008; Matsumura et al. 2017). Li and Ji (2020) questioned regarding σ 0 as the far-field differential stress. The analysis here can give the full stress state in the fiber under any triaxial bulk stress state. But to compare the analysis here with the shear-lag model prediction, we choose a uniaxial stress state with Σ11 = σ 0 and all other stress components being set to zero. Figure 14.1 compares the shear-lag model predictions with those from the current analysis. The triaxial stress state of the fiber-matrix system does affect the resulting tensile stress in the fiber, as the analysis here shows. Figure 14.2 compares the tensile stresses in the fiber resulting from the one-dimensional bulk stress (as the shear-lag model), a plane-stress state, and a uniaxial extension with confining pressure. In the latter two cases, σ 0 is taken to be the differential stress. As shown in Fig. 14.1, the shear-lag model predicts a much higher tensile stress than the analysis here. The tensile stress rapidly reaches the upper bound stress as re > 5. The present analysis predicts fiber tensile stress between the lower bound

14.2

Comparison with the Fiber-Loading Theory

327

Fig. 14.1 Tensile stress, normalized against the uniaxial tensile stress, versus the aspect ratio predicted by this analysis and the shear-lag model. The elastic contrast R = 50. The solid black curve is based on Eq. (14.9). The dashed black curve is based on Eq. (14.8). The results from this study are plotted for three Poisson’s ratios, 0.1 (green), 0.25 (purple), and 0.4 (red), respectively. Solid curves are based on exact expressions for prolate objects. Corresponding dashed curves are based on rod approximations (re > > 1). The rod approximation is acceptable for re > 5. The shearlag model predicts upper-bound stress even in moderately elongated objects re < 10. The analysis here predicts intermediate stress between the lower bound (uniform stress or Reuss model) and upper bound (uniform strain or Voigt model). The shear-lag model likely overestimates the stress in the fiber significantly

(uniform stress model) and the upper bound. Therefore, the shear-lag model likely overestimates the stress in the fiber significantly. The assumption of ellipsoid shape in the analysis here means that the stress obtained should be taken as the average stress in the inclusion. If the object is a rod-like object with a constant cross section area, then the stress within the object will vary along the length of the object and the tensile stress will be the maximum at the midpoint. In this case, the midpoint tensile stress will be greater than the predictions here, but it is still likely far below the shear-lag model prediction. Figure 14.1 also suggests that Poisson’s ratio plays a minor role in the stress magnitude in the object. Microboudinage structures are commonly reported in elongated and rheologically strong minerals in mylonites (Fig. 14.3). The aspect ratio can vary significantly from

328

14 Application Example 1: An Elastic Prolate Object in a Viscous Matrix

Fig. 14.2 Tensile stress in the fiber is dependent on the triaxial stress state of the fiber-matrix system. The purple line is for the 1D extension as in Fig. 14.1, and σ 0 for this case is the bulk tensile stress. The red line is for a plane-stress state with the extension parallel to the fiber and σ 0 is the principal stress difference. The green line is for a uniaxial extension stress state with confining pressure and σ 0 is the difference between the bulk extension stress and the confining pressure. The Poisson’s ratio is 0.25 for all calculations

re < 2 to re > 25 (Masuda and Kimura 2004). Masuda and coworkers have applied the shear-lag model to the development of microboudins. They argued that the microboudinage formation is due to elastic interaction between the boudinaged mineral and the embedding matrix. The ductile flow of the matrix is only responsible for the separation of the boudins, not the initiation of the boudin fractures. If this is true, the stress relevant to boudin development is a transient state of stress, not the long-term flow stress associated with the dominant dislocation creep mechanism during mylonitization. This proposal is inconsistent with the observations that microboudins are all developed in mylonites. Further, as the shear lag model predicts unrealistically high stress in the fiber, it is possibly not a physically sound model. In the following sections, we analyze the mechanical interactions between the elastic prolate object with the flow ductile matrix, furthering the preliminary analysis in Chap. 9 based on Jeffery’s equations for the traction forces on the surface of a rigid object.

14.3

An Elastic Prolate in a Newtonian Viscous Matrix

329

Fig. 14.3 Photomicrographs of typical microboudinage structures tourmaline (a), amphibole (b), orthopyroxene (c), and feldspar (d). (a) From Cenozoic granitic mylonites in the Chongshan metamorphic zone, Yunnan, China (Reflected polarized light). (b) From amphibole-bearing granitic protomylonite from Yunkai Mountain, Guangdong, China. (c) Peridotite mylonite from the Yushugou massif, Tianshan, Xinjiang, China. (d) Feldspar porphyroclasts from the same shear zone as (a). Abbreviations: F feldspar, Hb hornblende, Ol olivine, Opx orthopyroxene, Q quartz, T tourmaline (Photos courtesy of S. Ji)

14.3

An Elastic Prolate in a Newtonian Viscous Matrix

In this section, we derive governing equations for the mechanical interaction between a viscous matrix and an elastic heterogeneity. We then solve the equations for the case of rod-like heterogeneity to get the complete stress and strain field in the heterogeneity.

14.3.1

Equations for Stress and Strain in the Prolate Object

In an all-elastic (or all-viscous) system, the interaction equation:
 σ - Σ = Cm : Jx - S - 1 : ðε - EÞ

ð14:10Þ

For an all-elastic problem, the lowercase σ and ε are the Cauchy stress and elastic strain tensors for the inclusion and the uppercase Σ and E are the Cauchy stress and

330

14 Application Example 1: An Elastic Prolate Object in a Viscous Matrix

elastic strain tensors of the far-field matrix. Cm is the elastic stiffness tensor of the matrix, and S is the elastic Eshelby tensor. For elastic problems, Jx = Js. For an all-viscous problem, σ and Σ are the deviatoric stresses of the inclusion and the matrix, respectively. ε and E are strain rate tensors of the inclusion and the far-field matrix, respectively. Cm is the viscous stiffness tensor of the matrix and S is the viscous Eshelby tensor. For viscous problems, Jx = Jd. We are interested in a mixed rheology problem where the inclusion is linearly elastic, but the matrix is Newtonian viscous. Equation (14.10) cannot be applied to such mixed rheology directly, but we can apply the idea of the equivalent-inclusion method of Eshelby (1957) to solve the problem. Let us consider the viscous version of Eq. (14.10) but replace the inclusion with an equivalent elastic inclusion and require that the instantaneous stress field in the elastic inclusion be the same as if the inclusion were viscous. With such a fictitious equivalent viscous inclusion, Eq. (14.10) becomes:
 σ - Σ = C m : Jd - S - 1 :



Ce- 1 :

dσ - Cm- 1 : Σ dt

ð14:11Þ

Note the stress tensor Σ is deviatoric because the matrix is incompressible. The stress tensor σ in the above equation is the deviatoric part of the stress in the elastic prolate object. S is the viscous Eshelby tensor. For the isotropic matrix and inclusion that we consider here, the above equation is simplified to:
 σ - Σ = 2η Jd - S - 1 :



1 dσ Σ 2μ dt 2η

ð14:12Þ

where η and μ are, respectively, the matrix viscosity and the inclusion’s shear modulus. Equation (14.12) can be rearranged into a simpler form:
d  dσ þ Sσ = Σ J -S dbt

ð14:13Þ

where bt = μη t is a dimensionless time, which can be expressed as: bt = t , tr

tr =

η μ

ð14:14Þ

where tr is the relaxation time of the deforming elastic-viscous system. Note that there is a subtle difference between the relaxation time of a viscoelastic material (Turcotte and Schubert 1982, p. 337–338), which is the viscosity of the material divided by its own shear modulus, from the system relaxation time here. The relaxation time here measures the system behavior of an elastic heterogeneity

14.3

An Elastic Prolate in a Newtonian Viscous Matrix

331

(with shear modulus μ) interacting with the embedding viscous matrix (with viscosity η). Neither the heterogeneity nor the embedding matrix is viscoelastic. This point is significant in the next chapter when we consider the viscous “shear zone” interacting with the country rock. The mean stress σ m = 13 σ kk in the object is to be obtained from the following equation which can be derived from Eq. (11.26c): -1 - σm - P = Λij C ijmn Smnkl



dεkl - E kl dt



ð14:15Þ

We solve Eqs. (14.13) and (14.15) below to get the Cauchy stress in the prolate object.

14.3.2


 Solution of Jd - S dσ þ Sσ = Σ When Σ Is Constant t db

From Chap. 10, it is clear that elastic and viscous Eshelby tensor S in an isotropic medium has the following property. All components are zero except those with repeated indices Siiii and Sijij (where i ≠ j and summation is not implied for repeated indices). Other fourth-order tensors associated with isotropic materials, including the stiffness (and compliance) tensors (C and M), the strain (or strain rate) partitioning tensor A, and the stress partitioning tensor B, also have this property. We shall bij, B bij and so forth). Sij (and accordingly, A denote Siiii and Siijj by the simpler notation b The computations of normal stresses and normal strains (or strain rates) are only related to these components. The computations of shear stresses and shear strains (or strain rates) are only related to the Sijij (i ≠ j) components. Therefore, in isotropic materials, the normal stresses (σ 11, σ 22, σ 33) can be solved separately from the shear stresses σ ij.

Solutions for Shear Stresses For the three independent shear stresses, Eq. (14.13) is reduced to the following three differential equations, each for a separate shear stress component: 2Sijij dσ ij Σij þ σ ij = 1 - 2Sijij 1 - 2Sijij dbt

ði ≠ j,

no sum for repeated indicesÞ

ð14:16Þ

The viscous Eshelby tensor expressions for  a prolate object are already  known (Eq. 10.45). Specifically S1212 = S1313 ≈ 14 1 þ 5 - 3r2ln 2re and S2323 ≈ 14 1 - 2r12 . e

e

Equation (14.16) can be explicitly solved (e.g., Jeffery 1995, p. 317) to give the following expressions:

332

14 Application Example 1: An Elastic Prolate Object in a Viscous Matrix

 2A Σ 1 - exp σ 12 = A - 1 12  2A σ 13 = Σ 1 - exp A - 1 13 
 σ 23 = 2Σ23 1 - exp - bt where A =

 A - 1b t Aþ1

 A - 1b t Aþ1

ð14:17Þ

r2e 3 ln 2re - 5.

Solutions for Normal Stresses For the three deviatoric normal stresses, only b Sij components are relevant. We write Eq. (14.13) in a matrix form below: 

 h i-1 h d i-1 d dσ b þ b J ij - b Sij ½σ = b Sij J ij - b Sij ½Σ ð14:18Þ dbt 3 3 3 2 2 2 σ 11 Σ11 σ 11 h i 7 7 7 6 6 6 where dσ = d 4 σ 22 5, ½σ = 4 σ 22 5, and ½Σ = 4 Σ22 5. t t db db σ 33 σ 33 Σ33 All expressions for isotropic viscous b Sij are given in Chap. 10. As pointed out earlier, we are dealing with deviatoric stresses (σ 11 + σ 22 + σ 33 = Σ11 + Σ22 + Σ33  0). The 3 × 3 matrices in Eq. (14.18) are singular. Only two of the three equations in the linear system of Eq. (14.18) are independent. Such deviatoric linear systems are commonly encountered in incompressible viscous problems. We use the following method to solve them. First, we reduce a deviatoric system like: 2

M11 6 M 4 21

M12 M22

32 3 2 3 Σ11 M13 σ 11 76 7 6 7 M23 54 σ 22 5 = 4 Σ22 5

M31

M32

M33

σ 33

ð14:19Þ

Σ33

to the following 2 × 2 system: 

M11 - M13

M12 - M13

M21 - M23

M22 - M23



σ 11 σ 22



 =

Σ11



Σ22

which can be solved readily. σ 33 is then obtained from σ 33 = - (σ 11 + σ 22). And the solution of Eq. (14.19) can be expressed as:

14.3

An Elastic Prolate in a Newtonian Viscous Matrix

333

σ = M - 1Σ

ð14:20Þ

where: 0

M22 - M23

M13 - M12

B Mij- 1 = ΔB @ M23 - M21

M11 - M13

M21 - M22

M12 - M11

and

0

1

C 0C A 0

ð14:21Þ

Δ = ½ðM11 - M13 ÞðM22 - M23 Þ - ðM12 - M13 ÞðM21 - M23 Þ - 1 :

-1 This method will be used when we need to get b Sij from b Sij . With this method, it is straightforward to show that Eq. (14.18) can be reduced to the following two linear differential equations:

8 1 dσ 11 2ℬ > > þ ð aÞ σ 11 = Σ < 1 þ 2ℬ 1 þ 2ℬ 11 b dt ð ℬ - 1Þ > dσ 22 ð1 þ ℬ Þð1 - ℬ Þ > σ 11 þ σ 22 = Σ þ 2Σ22 ðbÞ : ℬ ð1 þ 2ℬ Þ ð1 þ 2ℬ Þ 11 dbt

ð14:22Þ

2

where ℬ = 3ð2 ln r2re e - 3Þ. Equation (14.22) is solved in two different situations. In the first situation, Σ11 and Σ22 do not vary with time. This is the case if the macroscale stress field is constant and the prolate object does not rotate in the external stress field. In the second situation, Σ11 and Σ22 are functions of time, which can arise from the change of the macroscale stress field with time, rotation of the object in a fixed stress field (like the rotation in a simple shearing flow), or both. In the first situation, the solution to Eq. (14.22) is relatively simple (e.g., Jeffrey 1995, p. 317). The analytical solution to Eq. (14.22) is:  σ 11 = 2ℬΣ11 1 - exp



bt 1 þ 2ℬ

 ð14:23Þ

Submitting the above expression into Eq. (14.22) leads to the following solution for σ 22: σ 22 = 2Σ22 ð1 - e - t Þ  

þ ð1 - ℬ ÞΣ11 1 þ ℬ - 1 e - t - ℬ - 1 þ 1 exp -

bt 1 þ 2ℬ

 ð14:24Þ

The expression for σ 33 follows from the deviatoric condition σ 33 = - (σ 11 + σ 22).

334

14 Application Example 1: An Elastic Prolate Object in a Viscous Matrix

Solutions for the Mean Stress -1 The mean stress σ m is solved using the interaction Eq. (14.15), which requires Sijkl and Λij. The non-vanishing components of Λij for prolate objects are already given in Chap. 10 (Eq. 10.44):

Λ11 =

1 - ln 2re ; r2e

Λ22 = Λ33 =

ln 2re - 1 - r2e 2r2e

ð14:25Þ

Sijkl for prolate objects are also given in Chap. 10 (Eq. 10.45). The expression for -1 b Sij can be obtained easily with the method presented above (Eq. 14.21): 0

1

2r2e - 1 4r2e

B B B 2 -1 8r ℬ 2r2 - 1 B 1 b Sij jrod = 2e - e 2 B 2re - 1 B 4ℬ 8re B @ 2r2 - 1 1 - e 2 4ℬ 8re

0 1 2ℬ -

1 2ℬ

0C C C C 0C C C A 0

ð14:26Þ

Submitting Eqs. (14.25) and (14.26) into Eq. (14.15), expanding the expression to span over all non-vanishing terms, and simplifying, one eventually gets: - σ m - P = 2η



3 - 3 ln 2re þ r2e dε11 - E 11 dt 3ð2 ln 2re - 3Þ

ð14:27Þ

From Eq. (14.23), we have: 

 bt ℬ ε11 = Σ11 1 - exp μ 1 þ 2ℬ

ð14:28Þ

and dε11 μ dε11 ℬΣ11 = exp = dt η dbt ηð1 þ 2ℬ Þ



bt 1 þ 2ℬ

ð14:29Þ

Submitting this into Eq. (14.27) yields the expression for the mean stress: σm = - P þ



 bt 3 - 3 ln 2re þ r2e 2ℬ exp Σ11 11 þ 2ℬ ð1 þ 2ℬ Þ 3ð2 ln 2re - 3Þ

ð14:30Þ

Therefore, we have obtained the full Cauchy stress tensor in the prolate object. To sum up, the complete solution to the stresses in the prolate object is:

14.3

An Elastic Prolate in a Newtonian Viscous Matrix

335

Fig. 14.4 Deviatoric stresses in a prolate object as a function of the dimensionless time bt . (a) Normal stresses (σ 11 and σ 22) for various shapes of the object. The bulk stress state is a uniaxial extension along the prolate long axis with Σ11 > 0, and Σ22 = Σ33 = - 0.5Σ11. The stresses are normalized against the bulk differential stress (1.5Σ11). (b) Shear stresses normalized against their bulk shear stress counterparts. Orange color for Σσ1212 , with uniform dashed line, solid line and intermittent dashed lines for re = 5, 10, and 15, respectively. Green color for Σσ 2323 which is equal to σ 13 σ 23 σ 13 Σ13 . Σ23 (and Σ13 ) is approximately the same for re = 5, 10, and 15



 8 bt > > σ =2ℬΣ 1exp > 11 11 > 1 þ 2ℬ > > > 

 > > 

-1 bt > -1 -t > -t > σ =2Σ ð 1-e Þ þ ð 1-ℬ ÞΣ 1 þ ℬ e þ 1 exp ℬ 22 22 11 > > 1 þ 2ℬ > > > < σ = - ðσ þ σ Þ 33 11 22 

 > 2A A -1b > > σ 12 =σ 13 = Σ 1- exp t ≈2Σ12 ð1-e -t Þ > > A -1 12 A þ1 > > > > σ =2Σ ð1-e -t Þ > > 23 23 > > 

 > > bt 3-3ln2re þ r2e > 2ℬ > : σ m = -P þ 1exp Σ11 1 þ 2ℬ 3ð2ln2re -3Þ ð1 þ 2ℬ Þ

ðaÞ ðbÞ ðcÞ ðdÞ ðeÞ ðf Þ

ð14:31Þ

The corresponding elastic strain tensor for the object follows immediately by using Hooke’s law. Figure 14.4 shows the increase of deviatoric stresses in the prolate object with time. The normal deviatoric stresses increase with a characteristic time ~ð1 þ 2ℬ Þ μη, much greater than the system relaxation time, toward their asymptotic ultimate values dependent on the shape of the object (Fig. 14.4a). In contrast, the shear stresses reach their ultimate magnitudes, which are about two times their bulk stress

336

14 Application Example 1: An Elastic Prolate Object in a Viscous Matrix

Fig. 14.5 Mean stress in a prolate object as a function of the dimensionless time bt for various shapes of the object. The bulk stress state is the same as in Fig. 14.4. The mean stress is normalized against the differential stress. Hydrostatic pressure is zero

counterparts, much more rapidly, with a characteristic time equal to the system relaxation time μη (Fig. 14.4b). The asymptotic deviatoric stress magnitudes are: 8 σ 11 j1 = 2ℬΣ11 > > > > > σ 22 j1 = 2Σ22 þ ð1 - ℬ ÞΣ11 ≈ ð1 - ℬ ÞΣ11 > > > > < σ 33 j = - 2Σ22 - ð1 þ ℬ ÞΣ11 ≈ - ð1 þ ℬ ÞΣ11 1 σ j > 12 1 = 2Σ12 > > > > > σ 13 j1 = 2Σ13 > > > : σ 23 j1 = 2Σ23

ð14:32Þ

Whereas the normal stresses are amplified significantly depending on their shape aspect ratios, the ultimate shear stresses in the object are two times the background magnitudes. Therefore, the stress state in a prolate object is always such that the principal stresses are nearly parallel to the object semiaxes, regardless of the object’s orientation with respect to the bulk stress field. Because of this, we shall ignore the shear stresses and call σ 11, σ 22, and σ 33 respectively σ 1 σ 2, and σ 3. In fact, the stress state in the object is similar to a uniaxial tension or compression because σ 11|1 = 2ℬΣ11, σ 22|1 ≈ σ 33|1 ≈ - ℬΣ11. The differential stress in the object is σ 1 - σ 2 ≈ σ 1 - σ 3 ≈ 3ℬΣ11. This stress state does not mean, however, that one can use a one-dimensional stress analysis, like the shear-lag model, to consider σ 1 only, because σ 2 ≈ σ 3 are significant (Fig. 14.4a).

14.3

An Elastic Prolate in a Newtonian Viscous Matrix

337

The mean stress in a prolate object increases, like the normal deviatoric stresses, with a characteristic time ~ð1 þ 2ℬ Þ μη , toward the ultimate mean magnitude (Fig. 14.5): σ m j1 = - P þ

14.3.3

3 - 3 ln 2re þ r2e Σ 3ð2 ln 2re - 3Þ 11

ð14:33Þ

Comparison with Earlier Work on Simple Shearing

Goldsmith and Mason (1967, p. 138) obtained the following approximate solution for the total Cauchy normal stress in a rod-like rigid object in a simple shearing flow: σ1 =

r2e Σ 2 ln 2re - 3 11

ð14:34Þ

The results of this analysis are valid for any bulk flow type. For a rigid object in a viscous flow, bt = 1. The ultimate Cauchy normal stresses along the three axes of the prolate object are: 1 - ln 2re þ r2e Σ11 2 ln 2re - 3 ln 2re - 2 σ 2 = σ 22 j1 = 2Σ22 þ Σ 2 ln 2re - 3 11 4 - 3 ln 2re σ 3 = σ 33 j1 = - 2Σ22 þ Σ 2 ln 2re - 3 11 σ 1 = σ 11 j1 =

ð aÞ ð bÞ

ð14:35Þ

ð cÞ

In the above equation, we have left out the uniform hydrostatic pressure term in the matrix. In the case of a simple shearing flow, Σ11 = τ sin 2θsin2ϕ. Equation (14.35) becomes:


 1 - ln 2re þ r2e τ sin 2θ sin 2 ϕ σ1 = 2 ln 2re - 3

ð14:36Þ

In Chap. 9, we obtained the normal stress on the section normal to the prolate long axis (Eqs. 9.15 and 9.16): σ1 =

r2e τ sin 2θ sin 2 ϕ 2 ln 2re - 3

ð14:37Þ

338

14 Application Example 1: An Elastic Prolate Object in a Viscous Matrix

Fig. 14.6 Comparison of the solution of this study with that of Goldsmith and Mason (1967) for a rigid rod object in a simple shearing flow. The vertical axis is the axial stress parallel to the object’s long axis normalized against the simple shear flow stress. The horizontal axis is the object’s aspect ratio. The solid black line is based on exact prolate objects. The dashed red line is based on the approximate solution of Goldsmith and Mason (1967, p. 138). The solid green line is based on the approximate solution of this study. Approximate solutions are acceptable for re > 5

Figure 14.6 compares Eq. (14.36) with Eq. (14.37). Both solutions are excellent approximations when re> 5. The present solution is a better approximation. In a simple shear flow field, the maximum tensor stress in the object is reached when Σ11 = τ and Σ22 = - τ, which is when the object’s long axis is 45∘ antithetic to the shear direction on the VNS. In that position, the three axial stresses in the prolate object are: 1 - ln 2re þ r2e τ 2 ln 2re - 3

4 - 3 ln 2re τ σ2 = 2 ln 2re - 3

ln 2re þ 1 σ3 = τ 2 ln 2re - 3 σ1 =

ð14:38Þ

14.3

An Elastic Prolate in a Newtonian Viscous Matrix

14.3.4

339


 Solution of Jd - S dσ þ Sσ = Σ for a Rod-Like t db Prolate Object on the Vorticity Normal Section in a Simple Shearing Flow

The above analysis is valid when Σij in the differential Eq. (14.13) is a constant quantity. This requires that both the macroscale stress field is constant with time and that the prolate object does not rotate in the stress field. A prolate object rotates continuously in a progressive non-coaxial deformation such as a steady state simple shearing flow. Therefore, Σij is a function of the object’s θ and ϕ. In such cases, one must combine Eq. (14.13) with the rotation equations (Chaps. 8 and 9) to solve for the stress and strain history in a prolate object. Such an investigation will require a numerical approach. We consider the simpler situation where the distinct axis of a prolate object lies on the VNS and the flow is simple shearing. In this case, Σij is a function of θ only (Fig. 9.1a and Eq. 9.19b): Σij = τ

sin 2θ

cos 2θ

cos 2θ

- sin 2θ

ð14:39Þ

where τ is the shear stress of the fluid. 2 As we have dθ dt = γ_ cos θ from Eq. (8.24), we get the following relation: dθ η_γ τ = cos 2 θ = cos 2 θ μ μ dbt

ð14:40Þ

which enables us to make the variable substitution to Eqs. (14.16) and (14.22) from bt to θ. Together with Eq. (14.39) to express Σij, one gets the following set of differential equations for the stresses in the object: μ sec 2 θ dσ 11 4μℬ σ = þ tan θ dθ τð1 þ 2ℬ Þ 11 1 þ 2ℬ    dσ 22 μ 1 þ ℬ μ ð1 - ℬ Þ sec 2 θσ 11 - 6τ tan θ þ sec 2 θσ 22 = ℬ dθ τ 1 þ 2ℬ τ
 2Aμ dσ 12 μ A - 1 2 - sec 2 θ sec 2 θσ 12 = þ dθ τAþ1 Aþ1

ð aÞ ðbÞ ð14:41Þ ð cÞ

Note that σ 13 = σ 23 = 0 because Σ13 = Σ23 = 0 for a prolate on the VNS. The solutions to the above three equations can be expressed in the following forms:

340

8 > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > :

14 Application Example 1: An Elastic Prolate Object in a Viscous Matrix

8

< Z θ



σ 11 =

σ 22 =



4μℬ μtanθ exp 1 þ 2ℬ τð1 þ 2ℬ Þ :





θ0

8

< Z θ

1þℬ μ μtanθ exp 1 þ 2ℬ τ τ :

exp

θ0

9 = μtanζ exp tanζdζ τð1 þ 2ℬ Þ ;

μtanζ τ







9  = ð1-ℬ Þ sec 2 ζσ 11 -6τ tanζ dζ ℬ ;

8 9 

< Zθ 

 = 
2Aμ A -1 μ A -1 μ exp exp σ 12 = tanθ tanζ 2- sec 2 ζ dζ A þ1 A þ1 τ A þ1 τ : ; θ0

ð14:42Þ

Expressed in normalized terms, the deviatoric stresses in the object are: 8 8 9

< Z θ

> = > > tanζ σ tanθ ℬ 11 > > ða Þ tanζdζ = exp exp > > γ c ð1þ2ℬ Þ γ c ð1þ2ℬ Þ : ; > τ γ c ð1þ2ℬ Þ > > θ 0 > > 8 9 > >  =

< Zθ

 >  < tanζ ð1-ℬ Þ 2 σ 11 σ 22 1 1þℬ tanθ exp = -6tanζ dζ ðbÞ sec ζ exp γc γc : τ γ c 1þ2ℬ τ ℬ ; > > > θ0 > > > 8 9 > >



 < Zθ   > = >
 > A -1 tanζ σ 2A A -1 tanθ 1 12 > > = 2- sec 2 ζ dζ exp ðc Þ exp > > A þ1 γ c A þ1 γ c : ; : τ γ c A þ1 θ0

ð14:43Þ

where γ c = μτ . The mean stress can be obtained with Eq. (14.27) by first getting the expression of ε11 = σ2μ11 from Eq. (14.42) and then in turn dεdt11 . The result is: 3 - 3 ln 2re þ r2e  ðσ 11 þ Σ11 Þ σm =
3 2 ln 2rp - 3 þ 2r2e

14.4

ð14:44Þ

Application to Microboudinage

When the differential stress in the prolate object is tensile and reaches the object’s tensile strength, tensile fracturing perpendicular to the object’s long axis will occur. When the differential stress is compressive and reaches the buckling stress threshold, the object’s bending may occur. Starting with the simple scenario, the asymptotic differential stress in the object is, from Eq. (14.32):

14.4

Application to Microboudinage

341

σ 1 - σ 2 ≈ σ 1 - σ 3 ≈ 3ℬΣ11 =

r2e Σ 2 ln 2re - 3 11

ð14:45Þ

In the case of a simple shearing flow, Σ11 = τ sin 2θsin2ϕ and the above equation reproduces the expression of Goldsmith and Mason (1967). The fracturing condition is, from Eq. (9.17): r2e T = 2 ln 2re - 3 Σ11

ð14:46Þ

where T is the tensile strength of the prolate object. The buckling condition is, from Eq. (9.29): 4r4e E = 2 ln 2re - 3 Σ11

ð14:47Þ

In a simple shearing flow, Σ11 is the maximum compressive stress Σ11 = - τ realized when the prolate object is aligned at θ = 45∘ synthetic to the shear on the VNS. Σ11 is the maximum tensile stress Σ11 = τ when the prolate object is aligned at θ = 45∘ antithetic to the shear on the VNS. In the latter case, Eq. (14.46) can be reformulated into a piezometer for the flow shear stress of the matrix: τcrit: 1 = T 3ℬ

ð14:48Þ

Figure 14.7 plots the relation of Eq. (14.48). The dashed purple line separates the 1 space into two domains. In the domain above the line (τcrit: T > 3ℬ, shaded), the object is expected to be fractured, whereas, below the line, it should still be intact. Thus if microbodins are developed in more than one mineral in a mylonite zone, one may potentially use this diagram to constrain both the flow shear stress and the tensile strengths of boudinaged minerals in a self-consistent way. For example, if we have different minerals (A, B, and C) with microboudinage structures, If the critical aspect 1 , ratio can be determined for each phase, then we get three equations, (Tτa = 3ℬ a τ 1 τ 1 T b = 3ℬ b , and T c = 3ℬ c ) for four unknowns (τ, Ta, Tb, Tc). If at least one mineral’s tensile strength is known in addition, then the remaining three unknowns can be solved. There are still many puzzles regarding microboudins from natural mylonites. While it is commonly assumed that tensile fractures develop in sequence by successive midpoint fracturing (e.g., Masuda et al. 1989), microstructures of microboudins (e.g., Fig. 14.3) do not support this assumption. The fractures appear to have developed randomly in the object, and the final fractured pieces can often have aspect ratios below one. Some insights may be gained by considering the visco-elastic interaction between the matrix and the object. This requires that we consider the time effect rather than

342

14 Application Example 1: An Elastic Prolate Object in a Viscous Matrix

Fig. 14.7 The relation between the flow shear stress, normalized against the tensile strength, and the aspect ratio of the prolate objects. The solid purple line is based on the exact prolate solution that is applicable for all re ≥ 1. The thick dashed line is based on the approximation for rod-like objects 1 , shaded) and (Eq. 14.48). The solid purple line divides the space into a boudinaged domain (Tτ > 3ℬ 1 an intact domain (Tτ < 3ℬ ). Minerals with different tensile strengths in the same mylonite zone are expected to have different critical aspect ratios, which may be used to constrain both the flow stress and the minerals’ tensile strengths self-consistently

just the asymptotic stresses. Because of viscoelastic interaction, the stress loading and unloading in the prolate object are not instantaneous (Eqs. (14.31) and (14.44). This implies that the current stress state in the object depends on its past history and that fracturing of the object does not immediately relax the stress in the object. Multiple parallel tensile fractures can develop quasi-simultaneously in a criticallystress prolate object. The location of the fractures is likely controlled by geometric irregularities and mechanical flaws rather than at the midpoint. Figure 14.8 plots the differential stress history for prolate objects (all with re = 10) having different elastic stiffness and initial orientations. The plots are based on Eq. (14.43). The stress history depends on the dimensionless parameter γ c. If γ c~0.002 or less (Fig. 14.8a), which implies that the prolate object is highly stiff, the stress state in the object is like that in a rigid object. The interaction between the viscous matrix and the object is nearly instantaneous. On the VNS, the

14.4

Application to Microboudinage

343

Fig. 14.8 Differential stress histories in prolate objects (re = 10) in a simple shearing flow. The object is on the VNS and the differential stress is normalized against the flow shear stress. The instantaneous state of differential stress depends on the object’s initial orientation and elastic stiffness. The beginning point of each curve is the initial θ of the object. (a) In a highly stiff object (γ c = 0.002), the stress state is like what is expected within a rigid object. (b) In a more compliant object (γ c = 0.01), the differential stress state has more memories of the initial conditions. (c) When γ c = 0.05, the peak differential stresses and the compression to tension transition are skewed significantly toward θ = 90∘. See text for more details

differential stress reaches its maximum compression at θ = - 45∘ and its maximum tension at θ = 45∘. And θ = 0∘ is the compression to tension transition orientation. Regardless of the initial orientation of the object, its differential stress attains the ultimate state rapidly. As the object becomes more compliant (Fig. 14.8b, c), it has more memory of its initial condition. The peak differential stresses (maximum compression, maximum tension) are skewed toward higher θ, and the compression to tension transition is at θ > 0∘. In the case of γ c = 0.05, if the object’s initial stressfree state is at θ = - 30∘ (the red line), it attains the maximum compression at about θ ≈ - 15∘. The compression to tension transition is at θ ≈ 15∘ and the maximum tension is at θ ≈ 70∘. Suppose this object fractures at θ = 70∘. The fractured pieces have reduced aspect ratios and also inherited stress as their initial stress states. Their stress state will evolve on different paths. Therefore, even in a simple shearing flow field, microboudins are expected to develop in the range 45∘ < θ < 90∘. The more elastically compliant the object, the closer the microbodins are parallel to the shear direction at the time of their formation.

344

14.5

14 Application Example 1: An Elastic Prolate Object in a Viscous Matrix

Concluding Remarks

We have applied Eshelby’s solutions and the equivalent inclusion approach to the mechanical problem of an elastic prolate object in an elastic and viscous matrix. Microboudins in natural mylonites are better explained by viscoelastic interactions between the ductile flow matrix and the prolate-like mineral grain. Proposals based on elastic-elastic interactions are not supported by the microstructures of microboudins. Furthermore, the one-dimensional shear-lag model predicts upperbound stress and is likely unrealistic. The complete mechanical analysis based on Eshelby’s approach allows the full stress and strain states and the stress history in an elastic prolate object to be obtained. The primary intention of this chapter is to demonstrate how Eshelby’s theory and the equivalent inclusion idea can be applied to solve practical geology problems. It is not meant to investigate the development of microboudins exhaustively. We have limited ourselves to linear rheology. Natural mylonites demonstrate dislocation creep as their dominant deformation mechanism, suggesting power-law viscous rheology. We shall discuss the extension of Eshelby’s linear theory to power-law viscous rheology in Chap. 17. Many natural examples of microboudinage structures (and folds) involve platelike elements in addition to rod-like ones investigated above. Plate-like elements can be treated as flat ellipsoids or penny-shaped ellipsoids. The approach of this chapter can be easily applied to such inclusions. In the Appendix of this chapter, the complete solutions for the stress and strain field in a penny-shaped elastic object embedded in a Newtonian viscous matrix are given, with derivation steps omitted.

14.6

Notes and Key References

The following Mathcad worksheet presents the solution to the stress field in an elastic prolate object in a viscous fluid in simple shearing: ElasticStressPartitioning.mcdx: This worksheet calculates the Cauchy stress tensor in a prolate inclusion from a given remote stress tensor and the elastic constants of the matrix and the inclusion (both isotropic). ProlateStress.mcdx: This worksheet evaluate Eq. (14.43). It calculates the three deviatoric stresses in a prolate object. The differential stress histories (Fig. 14.8) are obtained using this worksheet. Goldsmith and Mason (1967) is a comprehensive review of the rotation and stress in rigid rods and discs in simple shearing flow. The recast of some of Jeffery’s (1922) equations in Goldsmith and Mason (1967) is very helpful. The paper also has many analytical expressions for the kinematics and stress field in rods and discs, including the one reproduced as Eq. (14.49) here which is used to verify the more general

Appendix: Solutions of an Elastic Flat Oblate Body in a Newtonian. . .

345

analytical solutions in this chapter. The one-dimensional elastic fiber-loading theory of Kelly (1973) was applied to the formation of boudin structures by Lloyd et al. (1982). Masuda and coworkers used and developed the theory to the development of microboudinage structures (Masuda and Kuriyama 1988; Masuda et al. 1989, 2003; Matsumura et al. 2017). Ji and coworkers (Ji and Li 2021; Ji and Zhao 1993, 1994; Zhao and Ji 1997) have further refined and extended the fiber-loading theory. Their papers have documented many examples of microboudinage structures in natural mylonite. The readers are referred to these original papers for more details.

Appendix: Solutions of an Elastic Flat Oblate Body in a Newtonian Viscous Matrix The approach of this chapter can be extended easily to the analysis of a flat oblate elastic object in a viscous matrix. The solutions are presented here. The procedure is omitted. Note that for flat oblate bodies, the distinct axis is normal to the principal plane of the object and re≫ 1. The deviatoric stresses in the body when the macroscale stresses are constant in the object’s coordinate system:


 2 2ðΣ11 - Σ22 Þα2 bt þ ðΣ11 þ Σ22 Þα1 bt σ 11 bt = 3πre


 2 ðΣ11 þ Σ22 Þα1 bt - 2ðΣ11 - Σ22 Þα2 bt σ 22 bt = 3πre
 4Σ33
 α bt σ 33 bt = 3πre 1
 8Σ σ 12 ≈ 12 α2 bt 3πre
 4Σ13 α3 bt σ 13 = 4 - 3πre
 4Σ23 α3 bt σ 23 = 4 - 3πre

ð aÞ ð bÞ ð cÞ ð dÞ

ð14:49Þ

ð eÞ ðf Þ

where:

3πre b t 4 - 3πre


 3πre b t α2 bt = 1 - exp 8 - 3πre


 4 - 3πre b t α3 bt = 1 - exp 3πre
 α1 bt = 1 - exp



-

ð14:50Þ

346

14 Application Example 1: An Elastic Prolate Object in a Viscous Matrix

The mean stress is: σm = - P -

4Σ33
b α t - Σ33 3πre 1

ð14:51Þ

For extremely flat bodies (re< 0.01) and assuming P = 0 in the matrix, the above results can be simplified to the following:
 2ð 1 - B Þ σ 11 bt = f2ðΣ11 - Σ22 Þ þ ðΣ11 þ Σ22 Þð1 þ BÞg 3πre
 2ð 1 - B Þ σ 22 bt = fðΣ11 þ Σ22 Þð1 þ BÞ - 2ðΣ11 - Σ22 Þg 3πre
 4Σ
 σ 33 bt = 33 1 - B2 3πre 8Σ12 σ 12 = ð1 - BÞ 3πre 

 4 b σ 13 = Σ13 1 - exp t 3πre 

 4 b t σ 23 = Σ23 1 - exp 3πre  4Σ
: σ m ¼ - 33 1 - B2 - Σ33 3πre
3πr  where B = exp - 8 e bt The Cauchy stresses are:
 2 ð 1 - BÞ σ c 11 bt = f2ðΣ11 - Σ22 Þ - 3Σ33 ð1 þ BÞg 3πre
 2ð1 - BÞ ½2ðΣ11 - Σ22 Þ þ 3Σ33 ð1 þ BÞ σ c 22 bt = 3πre
 σ c 33 bt = - Σ33

ð14:52Þ

ð14:53Þ

In a simple shearing flow, if a flat oblate object is oriented with its distinct axis on the VNS, the bulk stresses are Σ11 = τ, Σ33 = - τ, and Σ22 = 0. The principal Cauchy stresses in the object are: 2ð1 - BÞð5 þ 3BÞ τ 3πre 2ð1 - BÞð1 þ 3BÞ σc 2 = τ 3πre σ c 33 = 0 σc 1 =

ð14:54Þ

References

347

And the principal stress difference is: σc1 - σc 2 =

8ð1 - BÞτ 3πre

ð14:55Þ

References Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond Ser A-Math Phys Sci 241(1226):376–396. https://doi.org/10.1098/ rspa.1957.0133 Goldsmith HL, Mason SG (1967) The microrheology of dispersions, Rheology: theory and applications, vol 4. Academic Press, New York, pp 5–250 Jeffery GB (1922) The motion of ellipsoidal particles in a viscous fluid. Proc R Soc Lond Ser A-Containing Papers of a Mathematical and Physical Character 102(715):161–179. https://doi. org/10.1098/rspa.1922.0078 Jeffrey A (1995) Handbook of mathematical formulas and integrals. Academic Press Ji S, Li L (2021) Feldspar microboudinage paleopiezometer and its applications to estimating differential stress magnitudes in the continental middle crust (examples from west Yunnan, China). Tectonophysics 805. https://doi.org/10.1016/j.tecto.2021.228778 Ji S, Zhao P (1993) Location of tensile fracture within rigid-brittle inclusions in a ductile flowing matrix. Tectonophysics 220(1–4):23–31. https://doi.org/10.1016/0040-1951(93)90221-5 Ji S, Zhao P (1994) Strength of 2-phase rocks – a model-based on fiber-loading theory. J Struct Geol 16(2):253–262 Kelly A (1973) Strong solids. Clarendon Press Li L, Ji S (2020) On microboudin paleopiezometers and their applications to constrain stress variations in tectonites. J Struct Geol 130. https://doi.org/10.1016/j.jsg.2019.103928 Lloyd GE, Ferguson CC, Reading K (1982) A stress-transfer model for the development of extension fracture boudinage. J Struct Geol 4(3):355–372. https://doi.org/10.1016/0191-8141 (82)90019-0 Masuda T, Kimura N (2004) Can a Newtonian viscous-matrix model be applied to microboudinage of columnar grains in quartzose tectonites? J Struct Geol 26(10):1749–1754. https://doi.org/10. 1016/j.jsg.2004.02.008 Masuda T, Kimura N, Hara Y (2003) Progress in microboudin method for palaeo-stress analysis of metamorphic tectonites: application of mathematically refined expression. Tectonophysics 364(1–2):1–8. https://doi.org/10.1016/S0040-1951(03)00045-3 Masuda T, Kuriyama M (1988) Successive mid-point fracturing during microboudinage – an estimate of the stress-strain relation during a natural deformation. Tectonophysics 147(3-4): 171–177. https://doi.org/10.1016/0040-1951(88)90185-0 Masuda T, Nakayama S, Kimura N, Okamoto A (2008) Magnitude of σ 1, σ 1, and σ 3 at mid-crustal levels in an orogenic belt: microboudin method applied to an impure metachert from Turkey. Tectonophysics 460(1–4):230–236. https://doi.org/10.1016/j.tecto.2008.08.025 Masuda T, Shibutani T, Igarashi T, Kuriyama M (1989) Microboudin structure of piedmontite in quartz schists – a proposal for a new indicator of relative paleodifferential stress. Tectonophysics 163(1–2):169–180. https://doi.org/10.1016/0040-1951(89)90124-8

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14 Application Example 1: An Elastic Prolate Object in a Viscous Matrix

Matsumura T, Kuwatani T, Masuda T (2017) The relationship between the proportion of microboudinaged columnar grains and far-field differential stress: a numerical model for analyzing paleodifferential stress. J Mineral Petrol Sci 112(1):25–30. https://doi.org/10.2465/ jmps.160711 Pollard DD, Martel SJ (2020) Structural geology: a quantitative introduction. Cambridge University Press Turcotte DL, Schubert G (1982) Geodynamics: application of continuum physics to geological problems. Wiley, New York Zhao P, Ji S (1997) Refinements of shear-lag model and its applications. Tectonophysics 279(1–4): 37–53. https://doi.org/10.1016/S0040-1951(97)00129-7

Chapter 15

Application Example 2: A Penny-Shaped Viscous Inclusion in an Elastic Matrix

Abstract Ductile shear zones are common features in Earth’s lithosphere. They vary from submeter to plate scales. As rheologically distinct elements in the lithosphere, they undoubtedly affect the rheology of the lithosphere. Shear zones also serve as conduits for the movement of fluids and melt and are often sites of economically significant minerals. In this chapter, we use Eshelby’s equivalentinclusion approach to investigate the stress field and progressive deformation in a viscous shear zone embedded in the lithosphere. The situation that the lithosphere is also viscous but with a higher viscous stiffness is trivial, as the problem of viscous inhomogeneity in a viscous medium is solved in Chap. 11. We consider the situation of a viscous shear zone enclosed in an elastic lithosphere instead. Past studies of ductile shear zones often treat them in isolation. In kinematic models (Chap. 6), the only connection between a shear zone and its country rocks is the boundary displacement or velocity condition, chosen to maintain compatibility between the country rock and the deforming zone. In mechanical models, shear zones are often assumed to be infinitely-extending zones of rheologically weaker rocks between parallel walls subjected to a boundary displacement or traction condition (Lockett and Kusznir 1982; Turcotte and Schubert 1982, p. 375–378; Robin and Cruden 1994). However, shear zones in Earth’s ductile lithosphere are finite in size. Small ductile shear zones, in particular, are commonly entirely enclosed in the lithosphere (Fig. 15.1). It is inappropriate to use a constant boundary condition for such a zone because the mechanical interaction between the zone and the country rock is dynamic during deformation. In this chapter, the ductile shear zone is regarded as a rheologically-distinct element (RDE, a heterogeneity) embedded in the lithospheric country rock. Microstructures of mylonites from natural ductile shear zones suggest that the dominant mechanism of deformation is dislocation creep which is associated with power-law viscous rheology (Tullis 2002; Kohlstedt et al. 1995). We regard ductile shear zones as Newtonian viscous in this chapter. The analysis here may be extended to power-law viscous rheology using the extended Eshelby theory to be elucidated in Chap. 17. The simplest approach is to regard a ductile shear zone as a viscous inclusion in a viscous but stronger matrix because the partitioning Eq. (10.20) or interaction Eq. (11.24) can be applied directly. Here, we consider a more challenging scenario © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Jiang, Continuum Micromechanics, Springer Geophysics, https://doi.org/10.1007/978-3-031-23313-5_15

349

350

15

Application Example 2: A Penny-Shaped Viscous Inclusion in an Elastic Matrix

Fig. 15.1 A small-scale ductile shear zone in metagabbro, Castell Odair, N. Uist. (Photo Courtesy of J.C. White). See Ramsay and Graham (1970) and Ramsay and Huber (1983, p. 36) for a description. Such shear zones are common in basement rocks (Ramsay 1980; Ramsay and Alison 1979). They are entirely constrained by the country rock, which is often undeformed. We regard such zones as penny-shaped Eshelby inclusions (a1 = a2 > > a3, re = aa31 < < 1) in an elastic matrix in this chapter

where the country rock is elastic. Most shear zones in the crust have country rocks that are either undeformed or much less deformed. The elasticity of the country rock must have been mechanically significant during shear zone evolution.

15.1

The Equations

In Chap. 14, we considered an elastic prolate object in a viscous matrix. In the Appendix to Chap. 14, we presented the solutions for a penny-shaped elastic object in a viscous matrix. Now, we consider the inverse scenario—a viscous penny-shaped inclusion in an elastic matrix (Fig. 15.1). Let us start with the elastic interaction equation:
 σ - Σ = Cm : Js - SðelÞ - 1 : ðε - EÞ

ð15:1Þ

where the lowercase σ and ε are the stress and strain tensors in a penny-shaped elastic inclusion (to be replaced by a viscous one soon), the uppercase Σ and E are the stress and strain tensors of the matrix, Cm is the elastic stiffness tensor of the

15.1

The Equations

351

matrix, and S is the elastic Eshelby tensor. Note Eq. (15.1) is in the full Cauchy space. Apply the equivalent inclusion approach by replacing the inclusion with a viscous heterogeneity and require that the stress in the viscous heterogeneity satisfies Eq. (15.1). We assume that both the viscous inclusion and the matrix are isotropic. Equation (15.1) becomes:  
 dε K m μ d ε - SðelÞ - 1 ε þ SðelÞ - 1 E = J þ J dt κ η

ð15:2Þ

where Km and μm are, respectively, the bulk and shear moduli of the matrix; κ and η, respectively, the bulk and shear viscosities of the inclusion. K We assume that the viscous inclusion is
incompressible, which h i κ = 0.  implies that ðelÞ - 1

ðelÞ - 1

Equation (15.2) becomes dtij = μη εij - Sijkl - 13 δij Sααkl ðεkl - Ekl Þ . Using μ the dimensionless time bt = η t as in the last chapter, the differential equation finally becomes: dε


 dεij 1 ðelÞ - 1 ðelÞ - 1 = εij - Sijkl - δij Sααkl ðεkl - Ekl Þ 3 dbt

ð15:3Þ

The Eshelby tensor components for a penny-shaped inclusion are expressed in Eq. (10.37) and are reproduced below: 0

13 - 8ν πr 32ð1 - νÞ e

8ν - 1 πr 32ð1 - νÞ e

2ν - 1 πr 8ð 1 - ν Þ e

1

B C B C B C el 8ν 1 13 8ν 2ν 1 B b πre πre πre C Sij = B C 32ð1 - νÞ 32ð1 - νÞ 8ð 1 - ν Þ B C B C ð15:4Þ @ ν A 4ν þ 1 ν 4ν þ 1 2ν - 1 πre πre 1 þ πre 1 - ν 8ð1 - νÞ 1 - ν 8ð 1 - ν Þ 4ð 1 - ν Þ
 7 8ν 1 ν 2 π Sel1212 = πr , Sel1313 = Sel2323 = 1þ r 2 1-ν4 e 32ð1 - νÞ e The inverse Eshelby tensor components can be obtained by inversion of Eq. (15.4) (see Appendices to Chap. 11 for details):

352 S^el-1 ij

15

Application Example 2: A Penny-Shaped Viscous Inclusion in an Elastic Matrix

0

1 15ð1-2νÞπre þ4ð17ν-13Þ ð2ν-1Þð1þ16νÞπre þ4ð13ν-16ν2 -1Þ 1-2ν C B πre ð8ν-7Þ πre ð8ν-7Þ B C B ð2ν-1Þð1þ16νÞπre þ4ð13ν-16ν2 -1Þ C 15ð1-2νÞπre þ4ð17ν-13Þ B =NB 1-2ν C C ð 8ν-7 Þ ð 8ν-7 Þ πr πr B C e e @ A ð4νþ1Þπre -8ν ð4νþ1Þπre -8ν 3 πre πre ð15:5Þ

and ðelÞ - 1

S1212 = ðelÞ - 1

1 ðelÞ 4S1212 ðelÞ - 1

=

S1313 = S2323 =

8ð1 - νÞ , ð7 - 8νÞπre 2ð1 - νÞ 4ð1 - νÞ þ ðν - 2Þπre

ð15:6Þ

where N = ð1þνÞ½ð2ν -11-Þπrν e þð3 - 4νÞ. In addition, the following quantities follow from Eq. (15.5) and will be used in the subsequent derivations:   8 8ð1 - 2νÞ > -1 -1 < Selαα11 = Selαα22 = N 8ν - 1 þ πre > : el - 1 Sαα33 = N ð5 - 4νÞ   16ð1 - 2νÞ el - 1 Sααββ = N 3ð4ν þ 1Þ þ πre

ðaÞ ð15:7Þ ðbÞ

1 el- 1 1 Z ij = Seliijj - 3 Sααjj δii 0 1 - ½ð32ν - 19Þπre þ ð50-64νÞ ½ð16ν -5Þπre þ ð22 -32νÞ -1 B C πre ð8ν -7Þ πre ð8ν -7Þ B C B C 2 B ½ð16ν -5Þπre þ ð22- 32νÞ - ½ð32ν- 19Þπre þ ð50- 64νÞ C = ð1 þ νÞN B -1 C 3 B C πre ð8ν -7Þ πre ð8ν -7Þ B C @ A 2ðπre - 2Þ 2ðπre -2Þ 2 πre πre ð15:8Þ

15.2

Solution for the Strain

15.2

353

Solution for the Strain

For shear strains εij (i ≠ j), Eq. (15.3) becomes

dεij

 þ 2S - 1 ijij - 1 εij = 2S - 1 ijij Eij

db t (with no sum over repeated indices), which in the explicit form are the following two equations:

dε12 þ β1 ε12 = β1 E 12 dbt dεα3 þ β2 εα3 = Eα3 , α = 1,2 dbt

ð15:9Þ

where β1 =

16ð1 - νÞ ð7 - 8νÞπre

and

β2 =

ð2 - νÞπre ð2 - νÞπre ≈ 4ð1 - νÞ þ ðν - 2Þπre 4ð 1 - ν Þ

ð15:10Þ

Their solutions are:
  ε12 bt = E 12 1 - e - β1bt   E
εα3 bt = α3 1 - e - β2bt , β2

ð15:11Þ

α = 1,2,

For normal strains, Eq. (15.3) can be reorganized to: 0

ε11

1

0

ε11

1

0

ε11 - E11

1

dB C B C B C @ ε22 A = @ ε22 A - Z ik @ ε22 - E22 A dbt ε33 ε33 ε33 - E33

ð15:12Þ

-1 δii is given in Eq. (15.8). where Z ij = Seliijj- 1 - 13 Selααjj To solve Eq. (15.12), first, reorganize it into the following form:

0

ε11

1

0

ε11

1

0

E11

1

dB C C C B B @ ε22 A þ ðZ - IÞ@ ε22 A = Z @ E22 A dbt ε33 ε33 E33 As the above system is deviatoric because ε11 + ε22 + ε33  0, the following sub-system is solved

354

15

Application Example 2: A Penny-Shaped Viscous Inclusion in an Elastic Matrix

        d ε11 ε11 ε11 E 11 e e þZ =Z ε22 ε22 E 22 d^t ε22

ð15:13Þ

ÞN 2ð1 - νÞ where ℛ = 23 ð7ð1þν - 8νÞπre ≈ 3ð7 - 8νÞð3 - 4νÞπre . Equation (15.13) becomes the following two equations:

dε11 þ ½ℛð50-64νÞ-1ε11 -ℛð22-32νÞε22 =ℛð50-64νÞE11 -ℛð22-32νÞE 22 dbt dε22 -ℛð22-32νÞε11 þ ½ℛð50-64νÞ-1ε22 = -ℛð22-32νÞE11 þℛð50-64νÞE22 dbt which are reorganized into the following two equations: d ðε11 þ ε22 Þ þ ½4ð7 - 8νÞℛ - 1ðε11 þ ε22 Þ = 4ð7 - 8νÞℛðE11 þ E 22 Þ dbt ð15:14Þ d ðε11 - ε22 Þ þ ½24ð3 - 4νÞℛ - 1ðε11 - ε22 Þ = 24ð3 - 4νÞℛðE11 - E 22 Þ dbt The above system can be readily solved to yield: 
 4ð7 - 8νÞℛðE 11 þ E 22 Þ
1 - e - β3bt ≈ ðE 11 þ E22 Þ 1 - e - β3bt β3 
 24ð3 - 4νÞℛðE 11 - E22 Þ
1 - e - β4bt ≈ ðE 11 - E 22 Þ 1 - e - β4bt ε11 - ε22 = β4 ð15:15Þ ε11 þ ε22 =

where 8ð1 - νÞ 3ð3 - 4νÞπre 16ð1 - νÞ β4 = 24ð3 - 4νÞℛ - 1 ≈ = β1 ð7 - 8νÞπre β3 = 4ð7 - 8νÞℛ - 1 ≈

ð15:16Þ

From Eqs. (15.15) and (15.16), we finally get the following results: ε11 = E 11 - E 11 h1 - E22 h2 ε22 = E 22 - E 11 h2 - E22 h1
 ε33 = - ðE 11 þ E22 Þ 1 - e - β3bt where h1 =

t t e - β3b þe - β4b and 2

h2 =

t t e - β3b - e - β4b , 2

ð15:17Þ

15.3

15.3

The Deviatoric Stress in the Inclusion

355

The Deviatoric Stress in the Inclusion

It is straightforward to get the deviatoric stresses in the inclusion, having obtained the expressions for the viscous strains, because: σ ij = 2η

dεij dbt dεij dεij = 2η = 2μ dt dt b dt dbt

ð15:18Þ

The dimensionless strain rates are easily calculated from Eqs. (15.9) and (15.17): β e - β3bt þ β4 e - β4bt dε11 = E 11 3 2 dbt β e - β3bt - β4 e - β4bt dε22 = E 11 3 2 dbt

! þ E22

β3 e - β3bt - β4 e - β4bt 2

þ E 22

β3 e - β3bt þ β4 e - β4bt 2

!

! !

dε33 = - β3 ðE 11 þ E22 Þe - β3bt dbt dε12 = β1 E 12 e - β1bt dbt dεα3 = E α3 e - β2bt ðα = 1, 2Þ dbt The deviatoric stresses follow from the above two equations: ! ! 8 - β3b t - β4b t - β3b t - β4b t > β e þ β e β e β e > 3 4 3 4 > þ 2μE 22 > σ 11 = 2μE 11 > 2 2 > > > > ! ! > > > - β3b t - β4b t - β3b t - β4b t > β e β e β e þ β e > 3 4 3 4 < σ 22 = 2μE 11 þ 2μE22 2 2 > > > > > σ 33 = - 2μβ3 ðE 11 þ E22 Þe - β3bt > > > > > > σ 12 = 2μβ1 E 12 e - β1bt > > > : σ = 2μE e - β2bt ðα = 1, 2Þ α3

α3

We can make use of the relations E m = relations in terms of matrix stresses:

Σm 3K

and E ij =

Σm 3K δij

Σd

þ 2μij to express the above

356

15

Application Example 2: A Penny-Shaped Viscous Inclusion in an Elastic Matrix

8 > σ 11 = > > > > > > > > > σ = > < 22

 β3
4μ β  Σm - Σd33 e - β3bt þ 1 Σd11 - Σd22 e - β1bt 2 3K 2  β3
4μ β1  d d - β3b t Σm - Σ33 e Σ11 - Σd22 e - β1bt 2 3K 2
 4μ d - β3b t σ 33 = - β3 Σ - Σ33 e > > 3K m > > > > > > σ 12 = Σ12 e - β1bt > > > : σ α3 = Σα3 e - β2bt ðα = 1, 2Þ

15.4

ð aÞ ð bÞ ð cÞ

ð15:19Þ

ð dÞ ðeÞ

The Pressure Field in the Inclusion

To solve for the pressure of this inclusion, we must consider the interaction of the spherical part (volume strains) of the strain tensors. We shall regard the viscous inclusion as being compressible with a bulk viscosity κ and investigate the asymptotic behavior of the pressure as κ approaches infinity. This is the penalty approach (Chap. 10). But since both the inclusion and the matrix are isotropic and analytical expressions are sought here, the approach is effective as shown below. Taking the trace of Eq. (15.2), we get: K dεkk = dt κ

 -1  -1  -1 -1 -1 -1 ε11 þ Sαα11 ε22 þ Sαα33 ε33 þ Sαα11 E11 þ Sαα11 E22 þ Sαα33 E33  εkk - Sαα11 ð15:20Þ Note in the above equation, εkk ≠ 0 because we assume that the viscous inclusion is compressible. Regarding the total strain tensor as a sum of deviatoric and spherical parts, i.e., εij = εdij þ 13 εkk δij and E ij = Edij þ 13 E kk δij Equation (15.20) can be written in the following form: h dεkk K 1 1  -1 -1 -1 -1 -1 þ Sαα33 þ Sαα22 þ Sαα33 = ε - S - 1 þ Sαα22 εkk þ Sαα11 E kk dt κ kk 3 αα11 3  -1 d  -1 d -1 d -1 d -1 d -1 d ε22 þ Sαα33 ε33 þ Sαα11 E11 þ Sαα22 E22 þ Sαα33 E33  - Sαα11 ε11 þ Sαα22 which can be reorganized and simplified to the following form: i h K dεkk 1 -1 1 -1 εkk þ Sααββ E kk þ G = εkk - Sααββ dt κ 3 3 where

ð15:21Þ

15.4

The Pressure Field in the Inclusion

357

-1  d  d  d  -1 -1 G = - Sαα11 ε11 - E d11 þ Sαα22 ε22 - E d22 þ Sαα33 ε33 - E d33

ð15:22Þ

Note the solutions for the strain field (Eq. 15.17) are for the deviatoric strains of the inclusion, although the superscript “d” was omitted there because of dealing with incompressible viscous inclusion. As we are considering compressibility in this section, it is important to remember that the total viscous strain εij is not traceless. _ Introducing another dimensionless time t = Kκ t , which is distinct from bt = μη t introduced in Eq. (15.3), Eq. (15.21) becomes:
 1 -1 1 -1 dεkk Sααββ - 1 εkk = Sααββ E kk þ G _ þ 3 3 dt _

Note t =

K κ

ð15:23Þ

t is related to bt = μη t by: _

t=

Kη b t μκ

ð15:24Þ

-1 (for i = 1, 2, 3) are given in Eq. (15.7), the following terms can As the required Sααii be calculated from Eq. (15.17):

εd11 - Ed11 = Em - E 11 h1 - E 22 h2 εd22 - Ed22 = Em - E 11 h2 - E 22 h1 εd33 - Ed33 = - 2E m þ e - β3bt ðE11 þ E 22 Þ The expression for G is, from Eq. (15.22), as follows:  h i 4 2E m - e - β3bt ðE 11 þ E 22 Þ G = 2N ð1 - 2νÞ 3 πre -1 - 1, Eq. (15.23) becomes: Denoting S = 13 Selααββ

i h
 _ dεkk - U t ðE 11 þ E 22 Þ _ þ Sεkk = 3ðS þ 1ÞE m - P 2E m - exp dt

ð15:25Þ

where 

 8ð1 - νÞð1 - 2νÞ 4 -3 ≈ πre πre ð1 þ νÞð3 - 4νÞ 8ð1 - νÞ μκ μκβ3 U= ≈ Kη 3ð3 - 4νÞπre ηK P = 2N ð1 - 2νÞ

ð15:26Þ

358

15

Application Example 2: A Penny-Shaped Viscous Inclusion in an Elastic Matrix

The solution to this equation is (Jeffery 1995, p. 317–318), assuming the initial condition of εkk|t = 0 = 0:


i _ ½3ðS þ 1Þ - 2P E m h 1 - exp - S t S h

i _ _ P ðE 11 þ E22 Þ þ exp - U t - exp - S t g S-U εkk =

ð15:27Þ

Σd

Σm Σm and Eij = 3K δij þ 2μij to replace the strain terms in Use the relations Em = 3K Eq. (15.27). The above equation is simplified to the following expression:

 
i _ ½3ðS þ 1Þ - 2P Σm h 2P 2Σm Σd33 εkk = 1 - exp - S t 3SK 2μ S - U 3K h

 i _ _  exp - S t - exp - U t ð15:28Þ Because p = - κ dεdtkk = - K dε_kk , the explicit expression for the pressure is obtained dt by combining Eqs. (15.25) and (15.28). The result is as follows:  
 _ p 2P PS 2 KΣd33 exp - S t = - ð S þ 1Þ þ Σm 3 S - U 3 2μΣm    PU 2 KΣd33 exp - β3bt þ S - U 3 2μΣm

ð15:29Þ

In arriving at this analytical expression for the pressure, we have used a characteristic relaxation time in Eq. (15.23): tK =

κ ; K

_

t = t=t K

ð15:30Þ

For a truly incompressible viscous inclusion (rigid with respect to volume strain), tK = 1. To get the expression for the limit case of an incompressible inclusion, take _ the limit of Eq. (15.29) as κ → 1. Note that U → 1 and t → 0 as κ → 1. The pressure in the viscous inclusion is thus:      p 2P K Σd33 2 = - ð S þ 1Þ þ þP exp - β3bt Σm 3 2μ Σm 3 The first term on the right-hand side of the above equation is the long-term value. The second term is related to deviatoric stress σ 33 (Eq. 15.19). Combining Eq. (15.19) and the above equation leads to:

15.5

Analysis of the Solutions

359



 2P K σ 33 Σm þ P p = - ð S þ 1Þ þ 3 2μ β3

ð15:31Þ

the long-term pressure in the inclusion is p = As

σ 33 is transient,  2P - ð S þ 1Þ þ 3 Σ m .

15.5 15.5.1

Analysis of the Solutions Stress Relaxation and Creep in the Viscous Inclusion

In a viscoelastic system, such as the one considered here, the stress and strain response of the deforming inclusion is not instantaneous as in a perfectly elastic system. The viscoelastic interaction between the viscous inclusion and the elastic matrix has a characteristic relaxation time for the system (Eq. 15.3): tμ =

η μ

ð15:32Þ

which is a measure of the system response time. tμ = 0 represents instantaneous response (pure elastic behavior) whereas tμ = 1 implies pure viscous behavior. The stress and strain variation with time of the inclusion (Eqs. 15.11, 15.17, and t t t 15.19) is controlled by three distinct characteristic relaxation times: βμ , βμ , and βμ 1 2 3 (note that β4 ≈ β1). The three normal stresses and normal strains as well as the shear strain ε12 are t t associated with βμ and βμ . The shear stresses and shear strains (σ α3 and εα3) are 1 3 t associated with βμ . β1 and β3 are inversely proportional to re while β2 is proportional 2 t t t to re. This means that βμ is greater than βμ and βμ by a factor of re-2. Therefore, the 2 1 3 creep and relaxation histories for σ α3 and εα3 are fundamentally different from the histories of other stress and strain components. If we regard the penny-shaped viscous inclusion as a ductile shear zone (Fig. 10.1), the principal plane normal to the distinct axis is the ‘shear plane’ for the shear zone. σ α3 and εα3 are respectively the (shear-zone parallel) shear stresses and shear strains. We will use the conventional engineering shear strain γ = ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ε23ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ε13 þ p and combined shear stress τ = σ 13 2 þ σ 23 2 below. The counterpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 parts Γ = 2 E 13 þ E 23 and Τ = Σ13 2 þ Σ23 2 are corresponding quantities in the country rock. The three normal strains (Eq. 15.17) and the shear strain ε12 and associated stresses are related to the stretching of the shear zone itself. To illustrate the characteristic stress and strain histories, let us consider a country rock with shear modulus μ = 10 GPa and Poisson’s ratio ν = 0.25, enclosing a viscous penny-shaped shear zone with viscosity η = 1020Pas and re = 0.01. This gives tμ ≈ 317 years, β1 ≈ 76, β2 ≈ 0.018, and β3 ≈ 32. The three characteristic

360

15

Application Example 2: A Penny-Shaped Viscous Inclusion in an Elastic Matrix

Fig. 15.2 Viscous strain history in the shear zone. (a) The shear strain ε12 increases rapidly to its
  asymptotic bulk shear strain E12 rapidly in about 10 years, according to ε12 bt = E12 1 - e - β1bt . (b) The shear-zone parallel shear strain increases to a much higher asymptotic value >50 times the bulk shear strain. And the shearing is active for >20 thousand years. Note the difference in scale between (a) and (b) t

t

t

response times are respectively βμ ≈ 4 years, βμ ≈ 10 years, and βμ ≈ 17,610 years. 1 3 2 The shear zone parallel stress and strain persist about four orders of magnitude (π 2re-2) longer than that the other stress and strain components. Figure 15.2 plots the strain accumulation history for ε12 and γ. Σm , It is of interest to note that even in a lithostatic stress state, E11 = E22 = E33 = 3K a penny-shaped viscous inclusion undergoes deviatoric stresses. From Eq. (15.19), one finds that in a lithostatic country rock stress state, the stresses in the inclusion are: 2μ Σ β e - β3bt 3K m 3 4μ σ 33 = Σ β e - β3bt 3K m 3 σ 11 = σ 22 =

Thus, in a compressive lithosphere Σm < 0. Deviatoric tensile stress is produced due to the bulk mean compressive stress. In general, the deviatoric stress σ 33 may be of the same or opposite sign to the macroscale Σd33 . As is clear from Eq. (15.19)),  σ = β Σd - 4μ Σ e - β3bt , if, say, Σd < 0 (compressive), σ may be tensile, if 33

3

33

3K

m

33

33

4μ Σd33 - 3K Σm > 0. All stress and strain components are transient except the shear zone parallel τ and γ. The shear zone parallel shear stress starts at the bulk shear stress value pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (Τ = Σ13 2 þ Σ23 2 ) and relaxes according to τ = Τe - β2bt . A smaller re corresponds

15.5

Analysis of the Solutions

361

to a longer lifespan of the shear zone activity. For an infinitely extending shear zone (re = 0), τ  Τ, which is a well-known result. A viscous shear zone is a domain of shear strain concentration. For the parameters used above, the ultimate shear strain in the shear zone domain is ~56 times the bulk shear strain. The deviatoric stresses in the shear zone are always below those in the country rock. All deviatoric stresses are relaxed in a few decades in the above example except τ = Τe - β2bt which is also always below the shear stress in the country rock. The result that a viscous shear zone in an elastic country rock has a limited lifespan is the consequence of the mechanical interaction between the zone and the elastic medium. Localized shear in the zone causes elastic strains in the vicinity of the shear zone—the exterior-point Eshelby’s problem (Chap. 11). Unless the elastic strains in the matrix near the shear zone are released, continued shearing in the shear zone is prevented. There are many plausible mechanisms for exterior elastic strains to be relaxed. Increasing the length of a shear zone and connecting neighbouring zones will drop re and prolong the lifespan of a shear zone. Fracturing or slipping on existing faults in the elastic medium reduces the elastic strains by converting accumulated elastic strains to permanent offsets.

15.5.2

Dynamic Pressure in the Viscous Inclusion

The pressure in the viscous shear zone is given by Eq. (15.31). Submitting the intermediate parameters into the expression and reorganizing lead to:  1-ν b p bt = 3 - 4ν    d   Σ33 2ð1 - 2νÞ 4ν þ 1 8 exp - β3bt sgn ðΣm Þ  þ 1þν 1 þ ν 3πre Σm where b p= n

The

ð15:33Þ

p jΣm j.

initial h d Σ33 8 - 4νþ1 þ 1þν 3πre Σm -

ν b transient pressure is pð0Þ = 31--4ν  io 2ð1 - 2νÞ sgn ðΣm Þ which is relaxed to the asymptotic value 1þν t

rapidly, according to the relaxation time βμ : 3

b pjlong term = -

ð1 - νÞð4ν þ 1Þ sgn ðΣm Þ ð1 þ νÞð3 - 4νÞ

ð15:34Þ

This relation shows that b pjlong term ≤ 1 for ν ≤ 0.5, and the equal sign holds when ν = 0.5. Therefore, the viscous shear zone’s long-term pressure is always below the country rock’s mean stress. In the typical range of 0.1 < ν < 0.4, 0:44
> σ 11 = > > > > > > > > > > > σ 22 = > > < σ 33 = > > > > > > > σ 12 = > > > > > > > > : σ 3α =

2r ½ð16 - 3ξÞΣ11 þ ξΣ22  ð8 - ξÞð4 - ξÞ 2r ½ð16 - 3ξÞΣ22 þ ξΣ11  ð8 - ξÞð4 - ξÞ 4rΣ33 4-ξ 8rΣ12 8-ξ 4rΣ3α α = 1, 2 4r - ξ

ð15:35Þ

where r is the viscosity ratio of the shear zone to the country rock and ξ = 3πre(1 r). Because the shear zone is much weaker than the country rock, therefore r ≫ 1 and ξ ≈ 3πre < < 1. Equation (15.35) can be represented by the approximate relations σ ij ≈ rΣij for all components except σ 3α ≈ Σ3α. Thus, all stresses in the shear zone are lower than the bulk stress field in the country rock by a factor of r, except for the shear stress parallel to the shear zone, which is approximately equal to the country rock value. The pressure difference of the shear zone from the country rock can be obtained using Eq. (11.26c). With expressions for Λij given in Eq. (10.42) and Eshelby tensor components given in Eq. (10.43), it can be shown that the pressure difference is: e p= -

2ð1 - r ÞΣ33 ð1 - r ÞΣ33 ≈≈ - 0:5Σ33 2 ð4 - ξ Þ

ð15:36Þ

If we regard a shear zone as a 2D ellipse (or a 3D elliptical cylinder), we get from Eq. (11.56) the following relation: e p= -

ð1 - r Þð1 - re 2 ÞΣ33 ≈ - Σ33 1 þ 2rre þ re 2

ð15:37Þ

The above two expressions imply that the shear zone is over-pressured (e p > 0) if the p < 0) if Σ33 is tensile. deviatoric stress Σ33 is compressive and under-pressured (e Therefore, both the viscous-elastic and the all-viscous model predict the shear zone is a domain of concentrated shear strain but a domain of lower deviatoric stresses. Both models predict that the only significant stress in the shear zone is the zone-parallel shear stress. In the viscous-elastic model, the zone-parallel shear stress

364

15

Application Example 2: A Penny-Shaped Viscous Inclusion in an Elastic Matrix

is equal to the bulk field value and relaxes slowly (Eq. 15.19). In the all-viscous model, the zone-parallel shear stress is equal to the bulk field value. These predictions are consistent with the conventional wisdom that ductile shear zones are domains of weakened rocks (White et al. 1980). The two models have contrasting predictions of the pressure within the shear zone. In the all-viscous model, the shear zone is over- or under-pressured by 0.5|Σ33| to |Σ33|, whereas in the viscous-elastic model, the long-term pressure inside the shear zone is below the mean stress in the elastic country rock (Eq. 15.34 and Fig. 15.3). It is worth pointing out that all rocks are viscoelastic rheologically. Whether it behaves more like a viscous fluid, or an elastic solid depends on the time scale of the problem and the relaxation time of the material (its viscosity divided by shear modulus, for an isotropic material). Where a viscous shear zone is bonded by a country rock with a viscosity many orders of magnitude higher than the zone (an assumption of the all-viscous model to account for the strain localization into shear zones), the relaxation time of the country rock must also be orders of magnitude higher than that of the shear zone domain. This is because the shear modulus of rocks varies by at most two orders of magnitude (e.g., Pollard and Fletcher 2005, p. 319–322). This means that, for the time scale relevant to the shear zone deformation, the country rock is essentially elastic. Therefore, our viscoelastic model, considering a viscous shear zone interacting with an elastic matrix, may better capture the physics of natural shear zone deformation. Many studies have established that ductile shear zones are preferential conduits for fluid flow and melt segregations (McCaig 1987; Kisters et al. 2000; Lin and Corfu 2002; Weinberg et al. 2015) (Fig. 15.4). Fluid flow into active shear zones leads to hydrating mineral reactions and is also important for facilitating metamorphic reactions that do not involve hydration (Pennacchioni 1996; Austrheim and Engvik 1997). Variation in minor-element chemistry (Marquer et al. 1994; Rolland et al. 2003) and stable isotopes (Kerrich et al. 1984) also support fluid flow into shear zones. These lines of evidence are consistent with the prediction of our viscoelastic model that ductile shear zones are lower pressure domains into which fluids and melts flow. Because the all-viscous model predicts an overpressure in a viscous shear zone that is thinned (Σ33 compressive) which is inconsistent with the observations that ductile shear zones are sinks and conduits of fluids and melts, Mancktelow (2006) argues that a ductile shear zone might be better represented by a plastic rheology. Cyclic brittle failure is invoked to explain fluid flow into ductile shear zones. The mechanisms by which repeated brittle failure takes place in a mylonite zone are not clear. The fault-valve model (Sibson et al. 1988) attributes cyclic brittle failure at the brittle/ductile transition zone to seismicity in reverse fault systems. But fluid and melt flow into the shear zone is observed in all tectonic settings and at all tectonic levels. Our viscous-elastic model predicts that the shear zones are sinks of fluids and melts during its active slip, except for transient disturbances. Of course, our model does not exclude brittle failures of the shear zone. The results of this chapter may also shed some light on axial plane veins, styloliteparallel veins, and veins parallel to crenulation cleavages. The occurrence of such

15.6

Discussion and Geological Implications

365

Fig. 15.4 (a) Simplified geologic map of a striped outcrop at the High Rock Island gold deposit. The shear zone is a brittle-ductile dextral strike-slip zone. The zone of alteration coincides with the shear zone approximately. The main quartz vein occurs in the center of the zone. (b) A field photo of the central part of the map area. Modified after Lin and Corfu (2002). Photo courtesy of S. Lin

veins has been puzzling as the orientation is far from the tensile fracture direction where veins are supposed to form. A fundamental mechanism in developing crenulation cleavage is the dissolution and precipitation mechanism (Williams 1972, 1990) driven by gradients in the chemical potential of minerals like quartz (Fletcher 1977; Gray and Durney 1979). This mechanism does not produce foliation-parallel veins. Some of such veins were formed initially in other orientations, likely along tensile fractures, and subsequently rotated toward parallelism with the foliation (Fig. 15.5) (e.g., Hanmer and Passchier 1991; Williams and Jiang 2005; Druguet

366

15

Application Example 2: A Penny-Shaped Viscous Inclusion in an Elastic Matrix

Fig. 15.5 Axial plane veins and leucosomes as a result of progressive deformation. (a) A quartz vein (Q) is folded and transposed toward parallelism (arrows) with a differentiated crenulation cleavage (parallel to the pencil) in the argillite of the Gowganda Formation of the Huronian Supergroup, near Whitefish Fall, Ontario. (b) Layered migmatites with leucosomes parallel to the D1 transposition foliation in the Grenville Front Tectonic Zone near Markstay, Ontario. See Li (2012) for more details on geology

2019). Others were formed parallel to the foliation (Figs. 15.4 and 15.6). Many publications have attributed them to brittle failure (e.g., Lucas and St-Onge 1995; Vernon and Paterson 2001), whereby the veins emplace a pre-existing foliation with a much low tensile strength. An alternative interpretation is that axial plane crenulation foliations, once developed, can become zones of viscous shear, and fluid/melt flows into these zones due to the visco-elastic mechanical interaction between the shear zones and the bounding rocks without brittle failure (Fig. 15.6).

15.7

Notes and Key References

The kinematic modeling of shear zones is reviewed in Chap. 6. Turcotte and Schubert (1982, p. 375–378) give an analytical solution for the velocity field in a one-dimensional shear zone. Lockett and Kusznir (1982) solved the thermomechanical differential equations governing deformation in viscous shear zones for both constant velocity and constant stress boundary conditions. The model of Sibson et al. (1988) explains how fluid-pressure-activated valves can produce repeated tension veins in the brittle-ductile transition regime. Mancktelow (2006) stated that if ductile shear zones are assumed viscous, the pressure in them is expected to be higher than the country rock, which would be at odds with the observations that shear zones are conduits of melt and metamorphic fluids. He argued that plastic rheology must be significant, a conclusion that seems inconsistent with the dominant dislocation creep mechanism in mylonites. The analysis in this Chapter shows that if the elasticity of the country rock is considered, viscous shear zones are low-pressure domains.

References

367

Fig. 15.6 (a) A field photo showing quartz beads or pseudo boudins parallel to the foliation in the contact zone between a Nipissing diabase and the Gowganda Formation argillites near Whitefish Fall, Ontario. The diabase (~2.2 Ga) is undeformed away from the contact. The Gowganda argillite bedding is folded with an axial plane cleavage parallel to the contact. (b) A field of mylonitic granite from the Grenville Front Tectonic Zone, about 2 km south of Coniston, east of Sudbury, Ontario. The photo section is perpendicular to the mylonitic C-foliation and parallel to the down-dip stretching lineation. The sense of shear is south-side up thrusting. A quartz vein is parallel to the foliation. (c) A photomicrograph (crossed polarization) of the quartz vein. The thin section was cut parallel to the lineation and perpendicular to the C-foliation (horizontal in the photo). The quartz vein is completely dynamically recrystallized, like the host granite. The deformation temperature is constrained between 450 and 500 °C. See Chang (2022) for more details

References Austrheim H, Engvik AK (1997) Fluid transport, deformation and metamorphism at depth in a collision zone. In: Jamtveit B, Yardley BWD (eds) Fluid flow and transport in rocks. Springer, Dordrecht, pp 123–137. https://doi.org/10.1007/978-94-009-1533-6_7

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Chang X (2022) Deformation conditions of quartz-rich mylonites of the Grenville front tectonic zone and application to the crustal strength. MSc Thesis, Western University. https://ir.lib.uwo. ca/etd/8626 Druguet E (2019) Deciphering the presence of axial-planar veins in tectonites. Geosci Front 10(6): 2101–2115. https://doi.org/10.1016/j.gsf.2019.02.005 Fletcher RC (1977) Quantitative theory for metamorphic differentiation in development of crenulation cleavage. Geology 5(3):185–187. https://doi.org/10.1130/0091-7613(1977)52.0.CO;2 Gray DR, Durney DW (1979) Crenulation cleavage differentiation – implications of solutiondeposition processes. J Struct Geol 1(1):73–80. https://doi.org/10.1016/0191-8141(79)90023-3 Hanmer S, Passchier CW (1991) Shear-sense indicators: a review. Geol Surv Can Pap:90–17 Jeffrey A (1995) Handbook of mathematical formulas and integrals. Academic Press Jiang D, Bhandari A (2018) Pressure variations among rheologically heterogeneous elements in Earth’s lithosphere: a micromechanics investigation. Earth Planet Sci Lett 498:397–407. https:// doi.org/10.1016/j.epsl.2018.07.010 Kerrich R, Latour TE, Willmore L (1984) Fluid participation in deep fault zones – evidence from geological, geochemical, and o-18/o-16 relations. J Geophys Res 89(NB6):4331–4343. https:// doi.org/10.1029/JB089iB06p04331 Kisters AFM, Kolb J, Meyer FM, Hoernes S (2000) Hydrologic segmentation of high-temperature shear zones: structural, geochemical and isotopic evidence from auriferous mylonites of the Renco mine, Zimbabwe. J Struct Geol 22(6):811–829. https://doi.org/10.1016/S0191-8141(00) 00006-7 Kohlstedt DL, Evans B, Mackwell SJ (1995) Strength of the lithosphere – constraints imposed by laboratory experiments. J Geophys Res Solid Earth 100(B9):17587–17602. https://doi.org/10. 1029/95JB01460 Li C (2012) An investigation of deformation structures and their tectonic significance across the Grenville front tectonic zone in the vicinity of Sudbury, Ontario, Canada. Electronic Thesis and Dissertation Repository. 464. https://ir.lib.uwo.ca/etd/46 Lin S, Corfu F (2002) Structural setting and geochronology of auriferous quartz veins at the High Rock Island gold deposit, northwestern Superior Province, Manitoba, Canada. Econ Geol Bull Soc Econ Geol 97(1):43–57. https://doi.org/10.2113/97.1.43 Lockett JM, Kusznir NJ (1982) Ductile shear zones: some aspects of constant slip velocity and constant shear stress models. Geophys J Int 69(2):477–494. https://doi.org/10.1111/j.1365246X.1982.tb04961.x Lucas SB, Stonge MR (1995) Syntectonic magmatism and the development of compositional layering, Ungava Orogen (Northern Quebec, Canada). J Struct Geol 17(4):475–491. https:// doi.org/10.1016/0191-8141(94)00076-C Mancktelow NS (2006) How ductile are ductile shear zones? Geology 34(5):345–348. https://doi. org/10.1130/G22260.1 Marquer D, Petrucci E, Iacumin P (1994) Fluid advection in shear zones – evidence from geological and geochemical relationships in the aiguilles-rouges massif (Western Alps, Switzerland). Schweiz Mineral Petrogr Mitt 74(1):137–148 McCaig AM (1987) Deformation and fluid – rock interaction in metasomatic dilatant shear bands. Tectonophysics 135(1–3):121–132. https://doi.org/10.1016/0040-1951(87)90156-9 Pennacchioni G (1996) Progressive eclogitization under fluid-present conditions of pre-Alpine mafic granulites in the Austroalpine Mt Emilius Klippe (Italian western Alps). J Struct Geol 18(5):549–561. https://doi.org/10.1016/S0191-8141(96)80023-X Pollard DD, Fletcher RC (2005) Fundamentals of structural geology. Cambridge University Press Ramsay JG (1980) Shear zone geometry – a review. J Struct Geol 2(1–2):83–99. https://doi.org/10. 1016/0191-8141(80)90038-3 Ramsay JG, Allison I (1979) Structural analysis of shear zones in an Alpinised Hercynian granite (Maggia Lappen, Pennine Zone, Central Alps). Schweiz Mineral Petrogr Mitt 59:251–279

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Chapter 16

Application Example 3: Deformation Around a Heterogeneity—Flanking Structures

Abstract In Chaps. 14 and 15, we applied Eshelby’s interior solutions and the equivalent inclusion approach. In this chapter, we investigate the development of flanking structures as an example to use Eshelby’s exterior solutions. Flanking structures [Fig. 16.1; Passchier (J Struct Geol 23(6–7):951–962, 2001)] are deflections of linear and planar fabric elements around a cross-cutting element such as a fault, vein, dyke, or any other heterogeneity. Many authors have investigated the development of flanking structures by a combination of field, numerical modeling, analogue modeling, and analytical approaches [Exner and Dabrowski (J Struct Geol 32(12):2009–2021, 2010); Exner et al. (J Struct Geol 26(12): 2191–2201, 2004); Grasemann et al. (J Struct Geol 25(1):19–34, 2003), (J Struct Geol 33(11):1650–1661, 2011); Grasemann and Stuwe (J Struct Geol 23(4): 715–724, 2001); Kocher and Mancktelow (J Struct Geol 27(8):1346–1354, 2005), (J Struct Geol 28(7):1139–1145, 2006); Wiesmayr and Brasemann (J Struct Geol 27(2):249–264, 2005)]. These studies have greatly improved the understanding of flanking structures. The main conclusions are that the final geometrical features of flanking structures depend on the initial conditions of the cutting element (shape, orientation, relative rheology) and the characteristics of the flow field. One must therefore be cautious when interpreting the kinematics of flanking structures (e.g., Grasemann et al. (J Struct Geol 25(1):19–34, 2003); Wiesmayr and Brasemann (J Struct Geol 27(2):249–264, 2005)]. Because flanking structures result from the progressive deformation around a cutting element heterogeneity, we can regard the cutting element as an Eshelby ellipsoid and apply the exterior solutions (Chap. 11) to the development of flanking structures. Compared to previous investigations, the application of Eshelby’s exterior solutions has the following advantages. First, there is no limit to the flow type and the initial geometrical conditions of the cutting element (any ellipsoidal shape, arbitrary orientation with respect to the host element and the bulk flow field). Second, there is no limit to the amount of strain that can be reached because the approach does not rely on meshing. Third, the approach can handle anisotropic

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-23313-5_16. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Jiang, Continuum Micromechanics, Springer Geophysics, https://doi.org/10.1007/978-3-031-23313-5_16

371

16 Application Example 3: Deformation Around a Heterogeneity—Flanking. . .

372

viscosity readily, although at a more expensive computational cost (Jiang and Bhandari 2018; Bhandari 2021). Furthermore, although the approach is based on linear viscous materials so far (Chaps. 10 and 11), it can be extended to power-law viscous materials by a linearization approach [Jiang (J Struct Geol 68:247–272, 2014), (Tectonophysics 693:116–142, 2016); Chap. 17]. This chapter first summarizes the principal equations related to the exterior mechanical fields in the vicinity of a heterogeneous ellipsoid. An outline of the algorithm for the numerical modeling of flanking structures is then presented. The algorithm is implemented in MATLAB. The MATLAB programs are provided in the online resource of the book. The program is applicable to anisotropic viscous materials under any flow field. As examples of using the numerical approach and the MATLAB program, we present numerical results for the development of flanking structures in plane-strain general shearing flows. The cutting element and the host medium are assumed to be isotropic and linear viscous materials. Furthermore, for simplicity, we limit ourselves to cutting elements with one principal axis parallel to the flow vorticity so that the whole deforming system has monoclinic symmetry.

16.1

The Motion of Material Particles Around an Ellipsoid from the Exterior Solutions

To simulate the development of flanking structures, one must track the motion of the cutting element (Fig. 16.1), which is approximated by an ellipsoidal heterogeneity here, and the motion of particles outside but close to the element. The former is defined by the interior solutions and the latter by the exterior solutions.

a

b

flanking fold CE

Flanking Shear Bands

Flanking Folds

synthetic displacement: s-Type

external HE

m

HE

HE

CE

m CE

internal HE

no displacement: n-Type flanking shear band

CE

antithetic displacement: a-Type

external HE

internal HE

Fig. 16.1 Flanking structure terminology after Passchier (2001). (a) Schematic drawing of two types of flanking structures: flanking folds and flanking shear bands formed from deflection of the host fabric element (HE—here a layering) close to the cross-cutting element (CE—here a vein). The HE further away from the CE is undisturbed, suggesting that the exterior fields around the CE heterogeneity were responsible for the development of flanking structures. (b) Terminology of flanking structures used in this chapter. After Passchier (2001), with permission for reuse from Elsevier

16.1

The Motion of Material Particles Around an Ellipsoid from the. . .

373

The formal solutions for the interior and exterior stress and flow fields are given by the following set (Eqs. 11.26 and 11.27; Jiang 2016): ( (


-1 : C - 1 : ð σ - ΣÞ ε = E þ Jd - S - 1 w = W þ Π : S - 1 : ðε - EÞ εðxÞ = E þ SE ðxÞ : S - 1 : ðε - EÞ wðxÞ = W þ ΠE ðxÞ : S - 1 : ðε - EÞ

interior fields

ð aÞ ð16:1Þ

exterior fields

ð bÞ

Figure 16.2 illustrates the symbols. To evaluate the two exterior Eshelby tensors SE(x) and ΠE(x), we need to get TijklE(x) from the following relation (Eq. 11.20): Z2π Zπ T ijkl ðxÞ = a1 a2 a3 0

Gik,l ðx - x 0 Þξj sin ϕdϕdθ aj

ð16:2Þ

0

where Gik, l(x - x') is the derivative of the Green function for velocity x 0 = 0

a1 cos θ sin ϕ

1

0

cos θ sin ϕ

1

C B C B @ a2 sin θ sin ϕ A and ξ = @ sin θ sin ϕ A. cos ϕ a3 cos ϕ

Fig. 16.2 Eshelby’s solutions (Chap. 11) relate the mechanical fields inside a Rheologically Distinct Element (RDE) to the remote fields by Eq. (16.1). The mechanical fields outside the RDE are related to the remote fields by Eq. (16.1). The cutting element of a flanking structure is modeled as an ellipsoidal RDE in this chapter. Modified after Jiang and Bhandari (2018)

374

16 Application Example 3: Deformation Around a Heterogeneity—Flanking. . .

For general anisotropic Newtonian materials, the Green tensor derivative Gik,l(x - x') is given in the integral form in Eq. (11.15):

Gik,l





1 x-x = 2 4π ðx - x 0 Þ2 0

Zπ 

 _ _ - bxl Aik þ zl M ik dψ

ð16:3Þ

0

where ψ is defined in Eq. (11.15) and Fig. 11.1.   _ _ Let the integrand be denoted by ½gðψ Þikl = - bxl Aik þ zl M ik . The above integral can be evaluated numerically using the Gauss-Legendre quadrature: Gik,l ðx, θ, ϕÞ =

nψ X 1 wm ½gðψ m Þikl 8π ½r ðx, θ, ϕÞ2 m = 1

ð16:4Þ

where r ðx, θ, ϕÞ = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx1 - a1 cos θ sin ϕÞ2 þ ðx2 - a2 sin θ sin ϕÞ2 þ ðx3 - a3 cos ϕÞ2 , wm are the Gauss-Legendre weights and ψ m are the Gauss-Legendre grid nodes, nψ is the  _ _ number of nodes. Note ½gðψ Þikl = - bxl Aik þ zl M ik is also a function of ϕ and θ. We use the product Gauss-Legendre quadrature as Jiang (2013, 2014) to evaluate Tijkl(x) in Eq. (16.2). Combined with Eq. (16.4), we have the following quadrature expression: T Eijkl ðxÞ =

nψ n n πa1 a2 a3 X X X wp wq wm sin ϕq ξj ½gðψ m Þikl 16 q = 1 p = 1 m = 1
r x, θp , ϕ 2 aj q

ð16:5Þ

where (θp, ϕq)are Gauss-Legendre grid nodes, wp are the Gauss-Legendre weights and n is the number of nodes and weights. The total number of grid nodes used to evaluate Eq. (16.5) is thus n2 × nψ . Although not needed for the modeling of flanking structures in this chapter, ΛEij ðxÞ (Eq. 11.20) for the exterior pressure field (Jiang and Bhandari 2018) can be numerically evaluated similarly by the following quadrature: ΛEij ðxÞ =

nψ n n πa1 a2 a3 X X X wp wq wm sin ϕq ξj ½F ðψ m Þi 16 q = 1 p = 1 m = 1
r x, θp , ϕ 2 aj q

ð16:6Þ

where Fi as defined in Eq. (11.15b) is a function of x, ϕ and θ. The number of grid points to use to evaluate Eqs. (16.5) and (16.6) is unknown a priori. Our approach is to prescribe a tolerance and evaluate the equation iteratively, starting with a lower number of nodes and progressively increasing the node number until the current output converges with the previous one within the prescribed

16.1

The Motion of Material Particles Around an Ellipsoid from the. . .

375

tolerance. We note, as expected that exterior points closer to the inclusion-medium interface require more Gaussian nodes for accurate evaluation. We vary the number of Gaussian nodes according to the position from the inclusion-medium contact. We use the parallel computing toolbox in MATLAB to increase the computation speed. Once TE(x) is computed, SE(x) and ΠE(x) are computed readily with Eq. (11.25). For isotropic viscous materials, the rheological contrast between the cutting element and the matrix is reduced to viscosity ratio r, and SE(x) and ΠE(x) can be obtained with quasi-analytical precision efficiently using the closed-form expressions given in Eq. (11.40). Once SE(x) and ΠE(x) are obtained, the strain rate and vorticity tensors are known, and associated velocity gradient tensors are, respectively: L=ε þ w

interior points

LðxÞ = εðxÞ þ wðxÞ

exterior points ðbÞ

ð aÞ

ð16:7Þ

The incremental displacement of a material point on a marker (such as foliation) is determined from its instantaneous velocity, which, at the location x, is given by the following integral:   vi ð x Þ = v i x 0 þ

Zx Lik ðxÞdxk xo

along any path from location x0, where the velocity vi(x0) is known, to the location x. It is the most straightforward to select a point on the inclusion’s surface as x0 because the velocity there is known to be vo i = vi ðx0 Þ = Lik x0k . The velocity at x is thus: Zx vi ðxÞ = Lik x0k

þ

Lik ðxÞdxk

ð16:8Þ

xo

The evaluation of this equation is discretized in the interval (xo, x) as follows: v i ð xÞ = vo i þ

n X j=0

   Lik xj xj - xj - 1 k

ð16:9Þ

Note that repeated k implies summation. While one can choose any point on the inclusion surface as x0 and the integration path can also be arbitrary, computationally, the most efficient path is selected using the confocal elliptic (in 2D case) coordinates (ξ, η) (Fig. 16.3) or the confocal ellipsoidal (in 3D case) coordinates (φ, λ, h) (Fig. 16.4). In the 2D case, the elliptic coordinates for the Cartesian location x are (ξ, η). We choose the x0 such that its elliptical coordinates (ξ0, η0) satisfy ηo = η. The path integral of Eq. (16.8) is reduced

376

16 Application Example 3: Deformation Around a Heterogeneity—Flanking. . .

Fig. 16.3 (a) The cutting element modeled an ellipsoid in plane-straining general shear flows. The longest axis of the ellipsoid is parallel to the flow vorticity (the Z-axis). The plane-straining flow is examined in the XY plane. The shear direction is parallel to the X-axis. (b) The confocal elliptic coordinate system. The red elliptic curves are the variation ofξ. Supposea and c are respectively the  a long and short semi-axes of the ellipse. ξ = cosh - 1 pffiffiffiffiffiffiffiffiffiffi is on the ellipse surface. The dashed 2 2 a -c

blue curves are variation of η from 0 to 2π (the numbers shown are in degrees). The green line shows the path following a constant η axis connecting xo and x. Modified after Bhandari (2021)

to the integration concerning ξ only. The integral path is divided into n increments along the η axis such that the coordinates of grid points are (ξi, η) where i = 1, 2⋯n (Fig. 16.3b). The elliptic coordinates are converted back to Cartesian coordinates for evaluating the integral. Similarly, in the general 3D case, we choose xo such that its ellipsoidal coordinates (φo, λo, ho) satisfy φo = φ, λo = λ. The path from xo to x can then be divided into increments along the h axis such that the grid points are (φ, λ, hi) where i = 1, 2⋯n (Fig. 16.4b). The grid points are converted back to Cartesian coordinates for evaluating the integral. The conversion between the Cartesian and Ellipsoidal coordinate system requires numerical evaluation. The details are presented in the Appendix to this chapter. Once the instantaneous velocity at a point is determined from the above procedure, the incremental displacement is determined for the particle and its position is updated according to: xðt þ Δt Þ = xðt Þ þ vΔt

ð16:10Þ

The system state, including the shape, orientation of the inclusion, and all marker points coordinates, is updated. The procedure to update the shape and orientation of an ellipsoid is already described in Chap. 11. When the system update is completed, the next time step calculation follows. The process continues until a prescribed bulk finite strain or boundary displacement is reached.

16.2

Macroscale Flows and Model Geometry

377

Fig. 16.4 A cutting ellipsoidal element in a general 3D flow. (a) The cutting element surrounded intersecting a red planar marker (the host element). (b) Cartesian coordinate system XYZ and the confocal ellipsoidal coordinate system φ, λ, hi. The integration of Eq. (16.8) is evaluated along a constant φ and constant λ path (green). The conversion between the Cartesian and ellipsoidal coordinates is given in the Appendix to this chapter. Modified after Bhandari (2021)

The above procedure is implemented in a MATLAB program available in the Appendix.

16.2

Macroscale Flows and Model Geometry

The development of flanking structures is a 3D problem. The CE ellipsoidal body can intersect the host element (a layering, for instance) in different orientations. The CE and the HE may be arbitrarily orientated in a given flow field. Our observed sections are usually also arbitrarily oriented. Below, we shall consider the simple situations where the flow field is plane-straining general shearing, and one semi-axis of the cutting element ellipsoid is parallel to the vorticity. In such cases, both the flow and the deforming system are monoclinic. The developing flanking structure can be examined on the section normal to the vorticity and passing through the center of the cutting element. Let the shear plane be the XY plane and the shear direction the X-direction (Fig. 16.5a). A plane-straining steady-state general shearing flow is defined by the following velocity gradient tensor: sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 - W 2s B± 1 B 2 W 2s γ_ 0 B B C - ε_ 0 A = γ_ B B 0 B 0 0 @ 0

0

ε_

B L=@0 0

0

1 0C C C sffiffiffiffiffiffiffiffiffiffiffiffiffiffi C C 2 1 1 - Ws C ∓ 0C 2 2 A Ws 0 0 1

ð16:11Þ

378

16 Application Example 3: Deformation Around a Heterogeneity—Flanking. . .

Fig. 16.5 Geometry of the problem investigated on the vorticity-normal section (VNS). The shape of the cutting element in this section is measured by the aspect ratio a2/a3. The orientation is measured by the angle θ between the a2axis and the X-axis

where Ws is the sectional kinematic vorticity number on the vorticity-normal section (Jiang and White 1995; Bhandari and Jiang 2021). Equation (16.11) describes a thinning zone where ε_ > 0 and a thickening zone where ε_ < 0 (Jiang and Williams 1998). strain and vorticity tensor are, respectively D = 12   The Tassociated   rate tensor  T 1 and W = 2 L - L . The second invariant of the strain rate tensor is L þq L ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε_ II =

1 2 Dij Dij

(Ranalli 1995, p. 77–78). If the flow is simple shearing (Ws = 1), we

use the shear strain γ to measure the progression of deformation. If the flow is a pure shearing or general shearing flow (0 ≤ Ws < 1), we use the following invariant strain to measure the advance of bulk deformation: εII = ε_ II t

ð16:12Þ

where t is the time of deformation. The invariant strain defined in Eq. (16.12) is similar to the so-called Von Mises equivalent strain rate multiplying time. The Von Mises equivalent strain rate used in qffiffipffiffiffiffiffiffiffiffiffiffiffi ffi VM 2 the materials science literature is often defined as ε_ eq = 3 Dij Dij (Tomé and pffiffi Lebensohn 2009). Therefore, εII = 23 ε_ VM eq t. As pointed out above, we let the a1-axis of the CE ellipsoid be parallel to the Z-axis, which is the flow vorticity. A very large a1 relative to a2 and a3 will thus approximate an elliptical cylinder, and the modeling results will be equivalent to those obtained from 2D elliptic cutting inclusions (e.g., Mulchrone and Walsh 2006). Because a1-axis is aligned with the Z-axis, the orientation of the CE is defined by θ—the angle of the a2-axis relative to the shear direction (Fig. 16.5), and the shape of the CE observed on the XY plane is defined by the aspect ratio a2/ a 3.

16.3

16.3

Modeling Results

379

Modeling Results

Passchier (2001) summarizes many potential mechanisms for the development of flanking structures. The modeling here is relevant to those flanking structures formed around a pre-existing rheologically distinct CE. In other words, the modeling applies to mechanisms II, III, III', and IV of Passchier (2001). The rheology of the medium embedding the CE is uniform, and the initial geometry of the HE is planar or linear without deflections. Passchier’s (2001) mechanism V (and V'), namely passive amplification of asymmetric or symmetric deflections in the HE close to CE, falls into two different scenarios. In the first, the CE itself is also passive, having no rheological contrast with the HE. These flanking structures develop purely kinematically and can be analyzed by following the kinematics of the progressive bulk deformation (Chaps. 4–6). In the second, the CE is a rheologically distinct element, and the HE has pre-existing deflections. To model flanking structures overprinting pre-existing deflections, one can input the initial coordinates defining the initial markers with deflections and then apply the exterior solutions to track the motion of the markers. The results presented below are all for the initial planner HE parallel to the shear plane. If the CE is much stiffer than the HE (r > > 1), it rotates in the bulk flow and deflects the HE close to it. There is an insignificant internal strain within the CE. As a result, n-type flanking folds (Fig. 16.6) develop, as Grasemann and Stüwe (2001) observed. Flanking folds develop if the CE rotates with vorticity. The development of n-type flanking shear band is limited to Ws < 1 and the CE initially in the

Fig. 16.6 Development of n-type flanking folds. Simulated flanking folds in simple shearing for a very stiff CE (r = 100) with an aspect ratio 100:1.5:0.1. Rows a, b, and c are for different initial orientations of the CE: a-row θ = 135∘, b-row θ = 90∘, and c-row θ = 45∘. N-type flanking folds are developed, as expected. Where the initial intersection of the CE and HE are oblique (a) and (c), the fold geometries appear s-type (a3 and a4) or a-type (c3 and c4) (Movies 16.1–16.3)

380

16 Application Example 3: Deformation Around a Heterogeneity—Flanking. . .

Fig. 16.7 Progressive development of flanking structures close to very stiff CE (r = 100) with an aspect ratio 100:1.5:0.1 and θ = 110∘ in pure shearing (vertical shortening and horizontal extension). An n-type flanking shear band is developed at a low strain state (a2). Thereafter, the structure evolves into an n-type flanking fold (Movie 16.4)

backward rotation sector. Figure 16.7 is for a very stiff CE (with r = 100) initially at θ = 110∘in a pure shearing flow. Except at low strains (Fig. 16.7a2) where n-type shear band may be recognized, the developing structures are all n-type flanking folds. As pointed out by Passchier (2001), where the initial intersection of the CE and HE are oblique (Fig. 16.6a, c), the n-type flanking folds appear s-type (Fig. 16.6a3, a4) or a-type (c3 and c4). A weaker CE accumulates more internal strain and causes a larger offset of the HE. A rich history is observed in the evolution of the related flanking structures. Figure 16.8 shows flanking folds around weak CEs (r = 0.01 for rows a, b, c, and r = 0.1 for row d) in a dextral simple shearing flow. Depending on the initial orientation of the CE, a-type flanking folds (Fig. 16.8a2, b2) or s-type flanking folds (Fig. 16.8c2) are developed initially. Earlier a-type flanking folds evolve into n-type flanking folds (Fig. 16.8a4, b4, c3) and then s-type flanking folds (Fig. 16.8a5, b5, c4, c5) as the CE rotates toward the shear direction with increasing bulk strain. The same evolution is observed as the CE becomes more viscous (r = 0.1, row d). The initial a-type flanking fold (Fig. 16.8d2, d3) evolves into the n-type flanking fold (Fig. 16.8d4) and continues toward the s-type (Fig. 16.6d5). The evolution of the flanking structure around an inviscid CE from an a-type to an s-type as strain increases was observed in analogue experiments by Exner et al. (2004). A weak CE in the backward rotation orientation in a thinning zone leads to the progressive development of the s-type flanking shear band (Fig. 16.9a1–a5). Other types of flanking shear bands (Fig. 16.1b) are more difficult to produce. Figure 16.9b1–b6 shows the evolution of flanking structures around a weak CE in a plane-straining thickening zone. An s-type flanking shear band is developed with a minor offset (Fig. 16.9b2). It evolves into a HE geometry with barely any deflection (Fig. 16.9b3) and then an a-type flanking shear band (b4). With continuing strain and rotation of CE with vorticity, the a-type flanking shear band evolves into the a-type flanking folds (Fig. 16.9b5, b6).

16.3

Modeling Results

381

Fig. 16.8 Simulated flanking folds in simple shearing. The CE aspect ratio is 100:1.5:0.1 (the long axis is perpendicular to the paper). Rows a, b, and c are for r = 0.01 but with the different initial orientations of the CE: a-row (a1–a5) θ = 135∘, b-row (b1–b5) θ = 90∘, c-row (c1–c5), θ = 45∘. A-type flanking folds are developed first in a and b because of the counter sense of offset across the CE. In c, because of CE’s initial orientation, no counter sense of offset occurred and s-type flanking folds are developed. As shear strain increases, the sense of offset across the CE becomes the same as the bulk sense of shear. This leads earlier a-type flanking folds in (a and b) to evolve into n-type (a4, b4) and eventually s-type flanking folds. Row d is for θ = 90∘ and r = 0.1. It shows a similar trend, except that the transition from a-type to s-type requires higher bulk strain (Movies 16.5–16.8)

Figure 16.10 shows simulation results of flanking fold development around a stiff CE in dextral simple shearing flow. The simulation is motivated by field structure (Fig. 16.10f) observed in the Cross Lake group metasandstone in the Cross Lake greenstone belt in the Canadian Superior Province in Manitoba. The HE is the bedding and is assumed to be parallel to the shear plane. The CE represents the quartz veins set at 30∘ synthetic to the shear plane initially, and the viscosity ratio is 100. Figure 16.9d at the shear strain of 2.1 reproduces the overall geometry of the field structure. A movie of this simulation is available from the online source of the book. Figure 16.11 shows simulation results of the development of flanking shear bands a very weak CE. The flow field is a thinning general shear vertical shortening, horizontal extension and dextral horizontal shear. The simulation is motivated by the field structure (Fig. 16.11f) from a mylonitic quartzite from the northern Cap de Creus shear belt. The HE is the horizontal layering which is the mylonitic foliation, and the CEs are the synthetic shear bands which are represented by a very weak (r = 0.01) flattened ellipsoid. For the parameters used in simulation, the CE rotates against vorticity for a minor 5∘ as the s-type flanking shear band develops.

382

16 Application Example 3: Deformation Around a Heterogeneity—Flanking. . .

Fig. 16.9 (a1–a5) Progressive development of s-type flanking shear band in plane-straining thinning general shearing (vertical thinning, horizontal extension, and top to right shearing) with Ws = 0.866, CE aspect ratio 100:1.5:0.03, initial θ = 160∘, and r = 0.01. (b1–b5): Rotation about 8 degrees. Everything the same as row a, except that the bulk flow is plane-straining thickening general shearing (horizontal shortening, vertical extension, and top to right shearing). S-type shear band forms initially. As the CE rotates with vorticity, the shear band becomes n-type (b3), and then s-type shear band (b4) and eventually becomes a-type flanking folds (b5 and b6) (Movies 16.9 and 16.10)

16.4

Summary and Concluding Remarks

The forward modeling method based on the exterior solutions is robust and can achieve any finite strain magnitude, in any given flow field, for any ellipsoidalshaped CE, with any relative viscosity. Our numerical modeling reproduces all types of observed flanking structures. The modeling already shows the richness of the development and evolution of flanking structures. Although well-developed s-type flanking shear bands and s-type flanking folds have simple kinematic interpretations and can serve as good shear sense indicators, other flanking structures do not have unique kinematic interpretations. They must be used carefully with other structures to render reliable kinematic interpretation. Passchier (2001) presented a Mohr circle method to estimate the sectional finite strain based on n-type flanking fold geometry, assuming no change in the initial angle between the CE and the HE (valid for a large r). Kocher and Mancktelow (2005) and Bhandari (2021) have developed a reverse-dynamic modelling method to extract quantitative information about the flow vorticity, finite strain and the CE’s relative viscosity from observed natural flanking structures. The forward modeling approach here allows iterative testing of initial conditions, bulk flow type, and finite strain magnitudes to compare observed flanking structures with the modeling results.

16.4

Summary and Concluding Remarks

383

Fig. 16.10 N-type flanking fold around an initially synthetical CE having the following initial conditions and flow field parameters. CE aspect ratio: 100:1.5:0.03 (long axis is perpendicular to the paper), initial θ = 150∘, the viscosity ratio r = 100, the flow vorticity Ws = 1, and the sense of shear is top to the right. The shear strain for each state is shown. (f) N-type flanking fold in the Cross Lake group sandstone from the Cross Lake Greenstone Belt, Manitoba, Canada. Note that because of the initial oblique intersection between the CE and the HE, the final folds in (d) and (e) appear like s-type flanking folds, although there is no offset (Movie 16.11)

As pointed out at the beginning of this chapter, the development of flanking structures is an intrinsically 3D problem. The treatment here is a simplistic one. The anisotropic rheology of the HE and non-linear rheology may be significant as well. The exterior solutions and the numerical implementation can handle anisotropic viscosity readily. The challenge lies in defining anisotropic viscosities to represent rock deformation and, more significantly, to track the evolution of rheological

384

16 Application Example 3: Deformation Around a Heterogeneity—Flanking. . .

Fig. 16.11 Flanking shear band around a synthetical CE having the following initial conditions and flow field parameters. CE aspect ratio: 100:1.5:0.03 (the long axis is perpendicular to the paper), initial θ = 160∘, the viscosity ratio r = 0.01, the flow vorticity Ws = 0.866, and the sense of shear is top to the right. The CE is nearly irrotational with ~5∘ backward rotation from (a) to (e). The second invariant strains, εII, from (a) to (e) are, respectively, 0, 0.15, 0.3, 0.45, and 0.6. (f) S-type flanking shear bands in mylonitic quartzite from the northern Cap de Creus shear belt. The field of view is about 15 cm (Movie 16.12)

anisotropies. Non-linear rheology will be dealt with in the chapter by the linearization approach.

16.5

Notes and Key References

The MATLAB package Flanking Structures is provided in the online resource of this book.

Appendix: Conversion Between Cartesian and Ellipsoidal Coordinates

385

The terminology and classification of flanking structures used in this chapter follow Passchier (2001). Exner and Dabrowski (2010) use the Eshelby solution to investigate the development of flanking structures in plane-straining progressive deformation. The cutting element is assumed to be an inviscid slit.

Appendix: Conversion Between Cartesian and Ellipsoidal Coordinates Suppose the ellipsoid inclusion’s three principal axes (a ≥ b ≥ c) are respectively parallel to the x-, y-, and z-axes. The Cartesian coordinates (X, Y, Z) can be calculated from the respective Ellipsoidal coordinates (φ, λ, h) using the following relation: X = (N + ℎ) cosφ cosλ Y = [N (1 - ee2) + ℎ] cosφsinλ Z = [N (1 - ex2) + ℎ] sinφ Where φ is the latitude and -π/2 ≤ φ ≤ + π/2 λ is the longitude and -π < λ ≤ +π) ℎ (-b ≤ ℎ < +1) is the (ellipsoidal) height (See Fig. 16.3b). a ffi is the radius of curvature in the prime vertical normal N = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 1 - e1 sin φ - e2 cos φ sin λ

section 2 2 2 e1 = a a-2 c and e22 =

a2 - b2 a2

are the eccentricities squared of the triaxial ellipsoid.

The above calculations are straightforward. To obtain Ellipsoidal coordinates (φ, λ, h) from the Cartesian coordinates (X, Y, Z), one needs to obtain the projection (x, y, z) of the point (X, Y, Z) onto this ellipsoid along the normal to this surface, also known as the foot point. This can be evaluated using Bisection method as in Eberly (2018). Once (x, y, z) is known, φ and λ can be evaluated as follows: φ

=

8 2 > > > > 6 > arctan 4 > > > < > > > > > > > > :

3      qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 - e22 z 7 2 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi if 1 e z ≤ 1 e 1 - e22 x2 þ y2 5 2 1    2 1 - e22 x2 þ y2 1 - e21 q 2 3  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 e 1 - e22 x2 þ y2 1 π 5   - arctan 4 otherwise 2 1 - e22 z

The conventions regarding the proper quadrant for φ are applied from the sign of Z.

16 Application Example 3: Deformation Around a Heterogeneity—Flanking. . .

386

λ=

8 > > > > > > > < > > > > > > > :

2 6 

3

y 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5  2 2 2 2 y 1 -2e2 x þ 1 - e2 x þ 3   1 - e22 x π 7 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi - 2 arctan 4 5   2 2 yþ 1 - e22 x2 þ y2 2 arctan 4

 2

if

y ≤ ð1 - e22 Þx

otherwise

The conventions regarding the proper quadrant for λ are applied from the signs of X and Y. For the height coordinate h, the Euclidean distance between the points (x, y, z) and (X, Y, Z) is used: h=

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð X - xÞ 2 þ ð Y - yÞ 2 þ ð Z - zÞ 2

References Bhandari A (2021) An investigation on flow field partitioning related to the rheological heterogeneities and its application to geological examples. PhD thesis, Western University. https://ir.lib. uwo.ca/etd/7763/ Bhandari A, Jiang D (2021) A multiscale numerical modeling investigation on the significance of flow partitioning for the development of quartz c-axis fabrics. J Geophys Res Solid Earth 126(2). https://doi.org/10.1029/2020JB021040 Eberly D (2018) Distance from a point to an ellipse, an ellipsoid, or a hyperellipsoid. https://www. geometrictools.com/Documentation/DistancePointEllipseEllipsoid.pdf Exner U, Dabrowski M (2010) Monoclinic and triclinic 3D flanking structures around elliptical cracks. J Struct Geol 32(12):2009–2021. https://doi.org/10.1016/j.jsg.2010.08.002 Exner U, Mancktelow NS, Grasemann B (2004) Progressive development of s-type flanking folds in simple shear. J Struct Geol 26(12):2191–2201. https://doi.org/10.1016/j.jsg.2004.06.002 Grasemann B, Stuwe K (2001) The development of flanking folds during simple shear and their use as kinematic indicators. J Struct Geol 23(4):715–724. https://doi.org/10.1016/S0191-8141(00) 00108-5 Jiang D (2013) The motion of deformable ellipsoids in power-law viscous materials: formulation and numerical implementation of a micromechanical approach applicable to flow partitioning and heterogeneous deformation in Earth’s lithosphere. J Struct Geol 50:22–34. https://doi.org/ 10.1016/j.jsg.2012.06.011 Jiang D (2014) Structural geology meets micromechanics: a self-consistent model for the multiscale deformation and fabric development in Earth’s ductile lithosphere. J Struct Geol 68:247–272. https://doi.org/10.1016/j.jsg.2014.05.020 Jiang D (2016) Viscous inclusions in anisotropic materials: theoretical development and perspective applications. Tectonophysics 693:116–142. https://doi.org/10.1016/j.tecto.2016.10.012 Jiang D, Bhandari A (2018) Pressure variations among rheologically heterogeneous elements in Earth’s lithosphere: a micromechanics investigation. Earth Planet Sci Lett 498:397–407. https:// doi.org/10.1016/j.epsl.2018.07.010

References

387

Jiang D, White JC (1995) Kinematics of rock flow and the interpretation of geological structures, with particular reference to shear zones. J Struct Geol 17(9):1249–1265. https://doi.org/10. 1016/0191-8141(95)00026-A Jiang D, Williams PF (1998) High-strain zones: a unified model. J Struct Geol 20(8):1105–1120. https://doi.org/10.1016/S0191-8141(98)00025-X Kocher T, Mancktelow NS (2005) Dynamic reverse modelling of flanking structures: a source of quantitative kinematic information. J Struct Geol 27(8):1346–1354. https://doi.org/10.1016/j. jsg.2005.05.007 Mulchrone KF, Walsh K (2006) The motion of a non-rigid ellipse in a general 2D deformation. J Struct Geol 28(3):392–407. https://doi.org/10.1016/j.jsg.2005.12.008 Passchier CW (2001) Flanking structures. J Struct Geol 23(6–7):951–962. https://doi.org/10.1016/ S0191-8141(00)00166-8 Ranalli G (1995) Rheology of the earth, 2nd edn. Chapman & Hall Tomé CN, Lebensohn RA (2009) Manual for code: visco-plastic self-consistent (VPSC), Version 7c

Chapter 17

Generalization of Eshelby’s Formalism and a Self-Consistent Model for Multiscale Rock Deformation

Abstract In Chaps. 10–12, we have obtained formal or explicit, where possible, solutions for a single ellipsoidal inclusion in an infinite homogeneous matrix of linear rheology (either elastic or Newtonian viscous). We have also presented methods to evaluate the formal solutions numerically. In Chaps. 14–16, we applied the theory to some geology problems, assuming that the rheology is linear and the problems can be approximated by “a single ellipsoid in an infinitely-extending homogeneous matrix.” These assumptions require more justification. This chapter discusses the efforts and advances in applying Eshelby’s formalism to “real” deformations of Earth’s lithosphere. We will first extend Eshelby’s solutions to nonlinear rheologies, particularly power-law viscous rheology, by the linearization method. We then formulate a self-consistent multiscale model based on extended formalism. Finally, we summarize how the self-consistent model may be applied to the modelling of multiscale structures in Earth’s lithosphere.

17.1

Nonlinear Rheology and Partitioning Equations

The exact solutions for Eshelby’s inclusion problems, as summarized in Eq. (11.26) for the interior points and Eq. (11.27) for exterior points, are a consequence of the linear rheology of the matrix. While the Earth’s brittle lithosphere is commonly treated as a linear elastic body, the ductile lithosphere is believed to be power-law viscous (Kohlstedt et al. 1995; Tullis 2002). There are no exact solutions like the above equations for an ellipsoidal inclusion in such nonlinear materials. The linearization approach (Molinari et al. 1987; Lebensohn and Tomé 1993a; Ponte Castañeda 1996; Masson et al. 2000) uses approximate linear rheology to replace the nonlinear rheology in the vicinity of a stress and strain (or strain rate) state. Equation (11.26) for the interior points and Eq. (11.27) are then used as approximate solutions in an infinitesimal stress and strain (or strain rate) increment. Large finite-

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-23313-5_17. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Jiang, Continuum Micromechanics, Springer Geophysics, https://doi.org/10.1007/978-3-031-23313-5_17

389

390

17

Generalization of Eshelby’s Formalism and a Self-Consistent Model. . .

strain deformation of a viscous material is the accumulation of infinitesimal increments. The mechanical fields for each increment are approximated by the solutions based on the linearization approach. Let us revisit the linearization concept introduced in Chap. 7 (Eqs. 7.8 and 7.26). Take a power-law viscous material as an example. The nonlinear constitutive equation is often written in a pseudo-linear form (e.g., Hutchinson 1976, p. 104, his Eq. 2.5): (

σðεÞ = Cð sec Þ ðεÞ : ε εðσÞ = Mð sec Þ ðσÞ : σ

ð17:1Þ

where C(sec) and M(sec) are the secant viscous stiffness and the secant viscous compliance tensors, respectively (e.g., Lebensohn and Tomé 1993a). C(sec) is also known as effective viscosity (e.g., Ranalli 1995, p. 78). The relations in the above equation are pseudo-linear because, as pointed out in Chap. 7, C(sec) and M(sec) are not constant coefficients. They depend on the state of deviatoric stress (or strain rate). The relations are nonlinear. One cannot apply C(sec) and M(sec) directly to Eshelby’s solutions because of nonlinear rheology. The linearization approach seeks the following linear-form relations: (

b : ε þ σ0 σ=C b : σ þ ε0 ε=M

ð17:2Þ

b and M b are called, respectively, the linearized stiffness to approximate Eq. (17.1). C and linearized compliance tensors, ε0 and σ0 are the associated back-extrapolated stress and strain rate terms. The two relations in Eq. (17.2) are equivalent if b : ε0 , and ε0 = - M b - 1 = M, b σ0 = - C b : σ0 . C b and A variety of linearization schemes have been proposed, differing in how M 0 0 b ε (or C and σ ) are constructed. The tangent linearization (Eq. 7.8) is based on the first-order Taylor expansion of Eq. (17.1) at a given state of stress and strain rate (σ0 and ε0):   ∂σ ∂σ ∂σ jε = ε0 : ðε - ε0 Þ = jε = ε0 : ε þ σðε0 Þ - jε = ε0 : ε0 ∂ε ∂ε ∂ε   ∂ε ∂ε ∂ε εðσÞ ≈ εðσ0 Þ þ jσ = σ0 : ðσ - σ0 Þ = jσ = σ0 : σ þ εðσ0 Þ - jσ = σ0 : σ0 ∂σ ∂σ ∂σ σðεÞ ≈ σðε0 Þ þ

which, for a power-law viscous material, can be written as:

17.1

Nonlinear Rheology and Partitioning Equations

391

1 σðεÞ≈ Cð sec Þ ðε0 Þ : ε þ σ0 =Cð tan Þ ðε0 Þ : ε þ σ0 ðaÞ n εðσÞ≈nMð sec Þ ðσ0 Þ : σþ ε0 =Mð tan Þ ðσ0 Þ : σ þ ε0 ðbÞ

ð17:3Þ

1 Cð tan Þ = j Cð sec Þ ; Mð tan Þ = nMð sec Þ n   1 0 σ = 1 - σðε0 Þ; ε0 = ð1 - nÞεðσ0 Þ n

ð17:4Þ

where

The superscript “tan” stands for tangent, n is the power-law stress exponent, σ0 and ε0 are the back-extrapolated stress and strain rate terms (Hutchinson 1976; Molinari et al. 1987; Lebensohn and Tomé 1993a). Submitting the linearized stiffness or compliance into Eqs. (11.26) and (11.27) makes them approximate solutions for the nonlinear viscous inclusion problem. For simple presentation, the partitioning Eq. (11.26) is recast into the form of interaction Eq. (10.8). With the linearized stiffness and compliance used, the interaction equations are as follows: (

_

eε = - H : σ e e σ = - H : eε

ð17:5Þ

 -1   _ b : S - 1 - Jd . b and H = C :M where H = S - 1 - Jd Equation (17.5) is the generalized Eshelby solutions for nonlinear materials. Submitting Eq. (17.4) into Eq. (17.5) and rearranging, we get the following partitioning equations for power-law materials (Jiang 2014, Eq. 10 there): (

σ=B : Σ

ε=A : E h  i - 1  -1 : C1 ð sec Þ - Jd : Jd þ ðnm - 1ÞS A = Jd þ S : nm Cm ð sec Þ B = nm C1 ð sec Þ : A : Cm ð sec Þ

-1

ð aÞ ð bÞ ð cÞ ð17:6Þ

where A and B are the strain-rate partitioning and stress partitioning tensors, respectively, nm is the power-law stress exponent of the matrix, and C1(sec) and Cm(sec) are the secant stiffnesses for the inclusion and the matrix, respectively. Equation (17.6) is the principal relation used by Xiang and Jiang (2013) and Chen et al. (2014). The tangent linearization (Eq. 17.4) is based on a first-order Taylor expansion of the nonlinear relation at a given stress (or strain rate) state. For a homogeneous stress

392

17

Generalization of Eshelby’s Formalism and a Self-Consistent Model. . .

(or strain rate) state, the linear relations can be made to approach the nonlinear relations (Eq. 17.1) to any prescribed precision as long as it is applied within a sufficiently small vicinity of the stress (or strain rate) state used for the Taylor expansion. The stress (or strain rate) field is uniform in the inhomogeneous ellipsoid. The stress (or strain rate) field is also homogeneous in the matrix far away from the inhomogeneity. However, the stress (or strain rate) in matrix material near the inhomogeneity is heterogeneous. This causes the viscous compliance to vary spatially. In other words, the matrix material there is not rheologically uniform. Therefore, tangent linearization is valid only when the local stress variation near the inhomogeneity is small. Gilormini and Germain (1987) and Molinari and Toth (1994) compared results based on finite element analysis with results calculated using the Eshelby-type solution based on tangent linearization. They concluded that the results from tangent linearization could be made to fit finite element results better if an empirical “effective” stress exponent neff is used instead of nM as the stress exponent of the matrix material and consistently 1 < neff < nM. Such an approach has been called the intermediate model (Lebensohn et al. 2004). Molinari and Toth (1994) obtained an empirical relation between neff and nM based on their analysis of 5n2 spherical inhomogeneities: neff ≈ 3n2 þ4nMM - 2 . According to this relation, the range M

for neff is pretty small (1 < neff < 1.67). Zhong (2012) compared results obtained from FLAC modelling of elliptical cylinders, in a matrix with nM = 4.7, with results calculated from Eshelby equations based on the tangent linearization. He found out that neff ≈ 1.5. By adopting the fitting parameter neff, the strain rate partitioning tensor in Eq. (17.6) becomes: h  i - 1  ð sec Þ ð sec Þ - 1 - JS : JS þ ðneff - 1ÞS A = JS þ S : neff MM : ME This empirical partitioning equation can represent different strain partitioning schemes proposed in the literature: the secant model (neff = 1), the tangent model (neff = nM), the intermediate model (1 < neff < nM), the homogeneous stress (Sachs) model (neff = 1), and the homogeneous strain rate (Taylor) model (neff = 0). The tangent linearization and its modified empirical intermediate linearization may apply well to materials made of single phases, such as quartzite with identical stress exponent. In such cases, this stress exponent of the matrix as a whole may be the same as its constituent phases (Hutchinson 1976). For a heterogeneous continuum made of poly-phases with different stress exponents, no simple relations are expected between the bulk stress exponent and those of the constituent phases. Other linearization schemes should be used. Many other schemes of linearization have been proposed (Molinari et al. 1987; Lebensohn and Tomé 1993a; Ponte Castañeda 1996; Masson et al. 2000). For any linearization (Eq. 17.2), the stress and strain rate relations in the inclusion and in the matrix are expressed, respectively:

17.1

Nonlinear Rheology and Partitioning Equations

(

b 1 : ε þ σ0 ; σ=C b m : E þ Σ0 ; Σ=C

b 1 : σ þ ε0 ε=M b m : Σ þ E0 E=M

393

for the inclusion for the inclusion

ð17:7Þ

where subscript “m” stands for the matrix and “1” for the inclusion. The corresponding expressions for the partitioning equations can be obtained from the interaction equations, and they are (Jiang 2014): b :Eþα b :Σþb b σ=B β; ε = A  -1   bm ; b= HþC b1 : HþC A _ -1 b1 b= HþM B

ð aÞ -1   b1 b= HþC : Σ 0 - σ0 ðbÞ ð17:8Þ α _  _ -1   bm ; b b1 : HþM : E0 - ε0 ðcÞ β= H þ M 

b and α b being fourth-order and b are the strain rate partitioning tensors with A where A b b b being fourth-order and b second order; B and β are the stress partitioning tensors, B α b β second order. Note that in a Newtonian viscous material, the stain rate partitioning (or the stress partitioning tensor) is characterized by a single fourth-order tensor (A or B). For a b nonlinear material, the partitioning equations contain both a fourth-order tensor (A b for stress) and a second-order partitioning tensor (b for strain rate and B α for strain rate and b β for stress). In the special case of tangent linearization, the partitioning equations can still be packed into a single fourth-order tensor (Eq. 17.5), because of the simple relations between secant and tangent stiffnesses or compliances (Eq. 17.4). The partitioning Eqs. (17.5) and (17.8) relate the microscale fields in a rheologically distinct ellipsoid to the macroscale fields in the matrix. They are the most significant relations for us to develop a multiscale model for the heterogeneous deformation of Earth’s lithosphere. In the simple situation of a solitary heterogeneous inclusion in an infinite and uniform but nevertheless, power law viscous material undergoing homogeneous b m , Σ0 , M b m , and E0) far-field flow (Σ and E being constant), the linearized terms (C are all independent of the stress and strain rate states of the inclusion. All three application examples in Chaps. 14–16 are concerned with an isolated microscale inclusion in a macroscale matrix. Therefore, the treatment there is still valid if the Newtonian viscosity of the matrix is replaced by a linearized viscosity such as the tangent viscosity. In a more general situation, the assumption of isolated microscale heterogeneity is unjustified if the volume fraction of the heterogeneities is more than dilute. In such cases, the “rheology of the matrix” is that of an effective medium that depends on the geometry, microstructures, and mechanical states of the constituent microscale elements. A homogenization approach is needed to get the effective stiffness (or compliance) for the effective medium. We will discuss this point later in this chapter.

394

17.2

17

Generalization of Eshelby’s Formalism and a Self-Consistent Model. . .

Non-ellipsoidal Shape of Rheological Distinct Elements

Rheological distinct elements (RDEs hereafter) are of irregular shapes. The simple reason for the common practice of regarding RDEs as ellipsoids in micromechanics (e.g., Mura 1987; Nemat-Nasser and Hori 1999; Li and Wang 2008) is because exact solutions for inclusion problems are limited to ellipsoids and cuboids in an infinite matrix. The Eshelby uniformity is limited only to ellipsoids (Eshelby 1961; Mura 1987). The Eshelby solutions (Eqs. 11.26 and 11.27) have been taken as approximations for the average fields in non-ellipsoidal inclusions. It is hard to justify this practice quantitatively (see Mancktelow 2011; Jiang 2013 and references therein for some discussions). Budiansky and Mangasarian (1960) may be the first to treat individual grains in a polycrystalline material as ellipsoidal inhomogeneities embedded in the polycrystalline medium which is taken to be mechanically homogeneous (Qu and Cherkaoui 2006). This treatment has been followed up in the various homogenization efforts to use the Eshelby formalism to obtain the effective mechanical properties of a material from the properties of the material’s constituent elements (Mura 1987, p. 421–433; Nemat-Nasser and Hori 1999, Part I; Molinari 2002). The practical success of homogenization lends indirect support to the ellipsoid shape assumption. In addition, dislocations and stacking faults in crystalline solids, cracks, weakened zones, and other discontinuities in Earth’s crust have also been treated as heterogeneous “ellipsoids” (e.g., Mura 1987, p. 15–20, 240–379; Rudnicki 1977; Healy et al. 2006; Exner and Dabrowski 2010). In the geology literature, it is common to treat rigid or deformable RDEs as ellipsoids (e.g., Ramsay 1967, p. 209–221; Dunnet 1969; Gay 1968a, b, c; Ghosh and Ramberg 1976; Ježek et al. 2002). Experiments (e.g., Ferguson 1979; Arbaret et al. 2001; Ghosh and Ramberg 1976) and theoretical considerations (Bretherton 1962; Willis 1977) have demonstrated that the rotational behavior of rigid non-ellipsoidal objects is indeed close to that of their best-fit (or equivalent) ellipsoids. Where the inclusion is not ellipsoidal, the Eshelby tensor is not constant in the inclusion. Zou et al. (2010) obtain some explicit expressions of the Eshelby Tensor Field (ETF) and its average for a wide variety of non-elliptical inclusions, including arbitrary polygons and those characterized by the finite Laurent series. They conclude that the ETF inside a non-elliptical inclusion can be quite non-uniform. However, using the elliptical approximation as the average of the ETF is valid for convex non-elliptical inclusions. The approximation becomes less acceptable for non-convex non-elliptical inclusions. This result should be considered when applying Eshelby’s inclusion solutions to geological problems. Where the ellipsoid assumption is not acceptable, one must take the numerical challenge to address the so-called Eshelby’s problem of non-ellipsoidal inclusions for which there has been some progress in isotropic elastic materials (e.g., Zou et al. 2010). We shall regard the ellipsoid shape assumption as valid for microscale heterogeneities in this chapter. This assumption enables the application of modern

17.4

Interface Properties

395

micromechanics to the heterogeneous deformation of Earth’s lithosphere. It is also worth pointing out that an ellipsoid can fit a wide range of natural shapes. By changing the relative length of its three semi-axes, an ellipsoid varies from rod-like (a1 → 1 , a2 = a3 → 0), to spherical (a1 ≈ a2 ≈ a3), to plane-like (a1 ≈ a2 ≪ a3 → 0), and many other shapes.

17.3

Inclusions in a Finite Space

Another issue to be addressed before applying Eshelby’s solutions to rock deformation is that the solutions are strictly only for a single ellipsoid in an infinite uniform matrix. This condition is never fully satisfied in lithosphere deformation. A smallscale Ramsay-type shear zone, far away from other shear zones, may well be regarded as an inclusion in an infinite matrix. A rigid porphyroclast in an ultramylonite matrix may also be regarded as an isolated inclusion. Crystal twins have also been treated as Eshelby heterogeneous inclusions in the host crystal (Lebensohn and Tomé 1993b; Jones et al. 2018). Central to these treatments is the observation that the average size of the RDEs in each problem is much smaller than the RVE appropriate for the problem. This is the condition of scale separation (Zaoui 2002; Jiang 2014). One must be mindful of the scales of RDEs and ensure that the condition of scale separation is met to regard the RDEs as isolated inclusions. Where the size of the inclusion phase becomes appreciable compared to its distance from the deformation boundary, boundary effects become significant. The inclusion’s translational and angular velocities are affected (e.g., Goldsmith and Mason 1967) by the boundary effects. This leads to the inclusion problems in a finite space which differ from the classic Eshelby inclusion problems but have been considered by Kinoshita and Mura (1984). Existing work is still limited to the spherical shape. Li et al. (2007) obtained closed-form solutions for the stress fields of a spherical inclusion concentrically placed in a finite spherical space. Zou et al. (2012) proposed a superposition framework to study inclusions in a finite space. Further progress to these finite space problems would enable better estimation of the mechanical properties of real materials. On the other hand, analogue experiments and numerical modelling have improved understanding of the rotational behaviors and surrounding flow patterns of rigid inclusions in confined simple shear (e.g., Marques et al. 2005a, b). In this chapter, we assume that the scale separation condition is satisfied.

17.4

Interface Properties

It is well known that the mechanical interactions between an inclusion and its embedding matrix are significantly influenced by the interface property. The Eshelby inclusion theory assumes perfect bonding at the interface. There is microstructural

396

17

Generalization of Eshelby’s Formalism and a Self-Consistent Model. . .

Fig. 17.1 Photomicrograph of a mylonite from the Okanagan shear zone in the Canadian Cordillera. The sense of shear is top to the left. Feldspar and micafish phorphyroclasts are at an angle ~30∘ antithetic to the shear direction. This geometry is universally observed from mylonites and suggests that the contact between the porphyroclasts and the matrix was slippery

evidence that the interface between heterogeneities and their matrix is not perfectly coherent. Rigid porphyroclasts (feldspar, micafish, etc.) in mylonites are most commonly at a small angle antithetic to the shear direction (Fig. 17.1). This observation suggests that interface slip may be common in natural shear zones (Mancktelow et al. 2002; Schmid and Podladchikov 2004, 2005; Liu 2009). There is also good evidence from analogue experiments for slippery interface (Ildeonse and Mancktelow 1993; Arbaret et al. 2001; Marques and Coelho 2001; Marques and Bose 2004; Marques et al. 2005c). Inclusions with weak interfaces constitute the so-called problems of coated inclusions (Buryachenko 2007, p. 78–79). The shear stress on the inclusion interface is reduced. The limiting end-member situation is a perfectly-slippery interface which supports no shear stress. This is the situation where the stress state in the inclusion is hydrostatic. The inclusion behaves like a rigid inviscid inclusion (Xu et al. 2011) or a rigid void (Schmid and Podladchikov 2004). Rigid in the sense that it does not change shape because of vanishing shear stresses; inviscid or void because its interaction with the matrix is like an inclusion with zero viscosity. The angular velocity of such a rigid void is given in Eq. (11.61). The shear spin must be considered for the rotational behavior of rigid voids, because a freely slipping interface, by definition, means that the particles in contact with the inclusion slide freely (Fig. 17.2) which is equivalent to allowing the shape principal axes of the inclusion to swipe away from their current positions. Equation (11.61) can be used to describe the angular velocity of a perfectly slippery ellipsoid in a general anisotropic

17.4

Interface Properties

397

Fig. 17.2 Schematic diagrams for an inclusion with a slippery interface contact with the matrix. The system can be regarded as a matrix with a void occupied by the inclusion. In state (a) the material point (p) on the inclusion side at the tip of the major semi-axis coincides with the material point (p') on the matrix side of the interface. The points p and p' are at the same location but are drawn slightly apart across the interface for easy visualization. In state (b), p' has moved along the hoop with respect to p because of interface slip. This means that the void’s principal axes have angular velocity to swipe through the material. Therefore, the shear spin effect must be considered in the angular velocity equation

viscous material, and Eq. (11.49) is for the special situation where the matrix is isotropic. Mulchrone (2007) derived a 2D version of Eq. (11.49). Johnson et al. (2009) have confirmed Eq. (11.61). A slippery interface is important not only for us to understand the microstructures of mylonites but also for the rheology of shear zones. We understand that the presence of strong particles in a weak viscous matrix will increase the viscous stiffness of the material (see Chap. 12). But if the rigid prophyroclasts are equivalent to rigid voids in the event of perfect slippery interfaces, their presence will weaken the material. The most plausible scenario in natural shear zones may be that the interfaces are partly slippery. Regardless, it poses a problem for homogenization. More work is needed to further clarify the degree of interface slip during mylonitization. In the meantime, we can only use a parametric approach to assign low viscosities to the inclusions to simulate their mechanical interaction with the matrix.

398

17.5

17

Generalization of Eshelby’s Formalism and a Self-Consistent Model. . .

Heterogeneous Matrix and Homogenization

With the discussion so far, the generalized Eshelby inclusion solutions are applicable to RDEs in a uniform matrix which can be power law viscous. Therefore, one can apply the solutions where the RDEs are dilute in a connected groundmass. This is like the situation where the Eshelby model is applicable to get the effective property of the bulk material (Chap. 13). As the concentration of RDEs increases, the presence RDEs affects the bulk property of the material. An RDE actually interacts with the Homogeneous Equivalent Medium (HEM) made of the matrix and all RDEs in the RVE. Where the RDE concentration is high enough, there may be no connected matrix phase. An RDE interacts with the HEM made of all RDEs of the RVE. The approach to deal with the heterogeneous matrix is to use the concept of HEM which is a hypothetical uniform medium. The rheological properties of HEM (viscous stiffness and compliances tensors) are obtained by the homogenization methods. Depending on the concentration of the RDEs, different homogenization methods are chosen (Chap. 13). Figure 17.3 illustrates the different scenarios. In the dilute RDE case, the HEM is the connected matrix (Fig. 17.3b) and no additional effort of homogenization is needed. In other cases, the interaction between an RDE with the heterogeneous material is idealized by the interaction of an ellipsoid with a hypothetical uniform HEM. We discussed the homogenization of linear materials in Chap. 13. Homogenization methods rely on strain (strain rate) and stress partitioning tensors. As we are dealing with power-law materials now, we must revisit the homogenization methods, using linearized stiffness and compliances instead. In a macroscale RVE containing many RDEs, with the tangent linearization, the microscale stress and strain rate fields in the kth RDE are related to the macroscale fields in the HEM by Eq. (17.5): ð aÞ σðkÞ = BðkÞ : Σ; εðkÞ = AðkÞ : E h  i - 1 h i AðkÞ = Jd þ SðkÞ : nm Cð sec Þ - 1 : CðkÞ ð sec Þ - Jd : Jd þ ðnm - 1ÞSðkÞ ð bÞ BðkÞ = nm CðkÞ ð sec Þ : AðkÞ : Cð sec Þ - 1

ð cÞ ð17:9Þ

where the superscript “k” stands for the kth RDE, nm is the power-law stress exponent of the HEM. For a general linearization, the partitioning equations are (Eq. 17.8):

17.5

Heterogeneous Matrix and Homogenization

399

Fig. 17.3 Three different scenarios of inclusion-matrix interaction. (a) An infinite composite medium subjected to a remote field (strain rate Em, vorticity Wm and deviatoric stress Σm) at infinity. The macroscale fields (E, W and Σ) at every point are defined over a suitable RVE. (b) The dilute RDE situation is where the volume fraction of the RDEs is so low that they are far apart from each other and are embedded in a connected groundmass phase. In this scenario, the HEM rheology is simply that of the connected phase. No homogenization is needed. (c) Where the RDE concentration is higher but the RDEs are still embedded in a connected phase (matrix-supported), the overall rheology of the composite cannot be represented by that of the connected phase (C0) but is represented by a HEM whose rheology is obtained by the Mori-Tanaka approach. (d) Where RDEs are so crowded that little or no connected matrix phase exists, the material is a poly-element composite. Each RDE is embedded a HEM representing the poly-element aggregates. In this scenario, a self-consistent approach is used to obtain the effective rheology of the HEM. (e) The interaction between an RDE and the heterogeneous material is idealized by the interaction of an ellipsoidal RDE with a hypothetical uniform HEM. Modified from Qu et al. (2016), with permission for reuse from Elsevier

ðk Þ b ðkÞ : E þ α b ðk Þ : Σ þ b b ; σð k Þ = B εð k Þ = A β    -1   -1  b ðkÞ = HðkÞ þ C b ðkÞ b ; α b ðk Þ b ðkÞ = HðkÞ þ C A : HðkÞ þ C : Σ0ðkÞ - σ0ðkÞ

-1

-1    _ ðk Þ _ ðk Þ _ ðk Þ ðk Þ ðk Þ ðk Þ ðk Þ b b b b b : H þM ; β = H þM : E0ðkÞ - ε0ðkÞ B = H þM

ð17:10Þ Where a connected matrix phase is present (Fig. 17.3c), the Mori-Tanaka (Mori and Tanaka 1973; Molinari and Mercier 2004; Mercier and Molinari 2009) homogenization is adopted to calculate the stiffness of the HEM. In this approach, the matrix b 0) is adopted as a reference medium to solve the partitioned fields (ε(k), w(k) phase (C (k) and σ ) for individual RDEs using Eq. (17.10).

ð aÞ ð bÞ ð cÞ

400

17

Generalization of Eshelby’s Formalism and a Self-Consistent Model. . .

A self-consistent homogenization is used to obtain the effective properties of the HEM where a connected matrix phase is not present (Fig. 17.3d). In the selfconsistent scheme, the HEM is the reference medium to solve the partitioned fields for individual RDEs using Eq. (17.10). Although the choices of the reference medium in the Mori-Tanaka and selfconsistent approaches are different, in both cases, each RDE is regarded as a heterogeneous inclusion embedded in the reference medium subjected to remote strain rate Εm (a priori different from E), which is the average strain rate in the reference medium. The macroscale response of the reference medium is represented by (Molinari 2002): Σm = C : Em þ Σ0 ,

Σ = C : E þ Σ0

ð17:11Þ

where Σm is the average stress in the reference medium resulting from Εm; Σ is the macroscale stress due to the imposed strain rate E. Specifically, the homogenization procedure is as follows. Consider an RVE made of N different microscale heterogeneous elements labelled by k (k = 0, 1, 2, . . . N ). k = 0 represents the connected matrix phase if present. The average strain rate ε(k) and average stress σ(k) in a given element are given by (Eq. 17.5): _ ðk Þ

εð k Þ - E m = - H

  : σðkÞ - Σm

ð17:12Þ

_ ðk Þ

where H is evaluated at Em. The strain rate partitioning equations follow from Eqs. (17.9) and (17.10) as: εðkÞ = AðkÞ : Em tangent linearization 8 < εðkÞ = A b ðkÞ : Em þ α b ðk Þ general linearization ðk Þ : ðkÞ b ðkÞ b σ = B : Σm þ β

ð aÞ ð bÞ

ð17:13Þ

Imposing the consistency condition hε(k)i = E on Eq. (17.13), where hi means volume average over the RVE, gives:

17.5

Heterogeneous Matrix and Homogenization

401

D E-1 Em = AðkÞ : E tangent linearization 8 D ðkÞ E - 1 D ðkÞ E - 1 D E > > b b b ðkÞ >Em = A : Ε- A : α > > > < > > > D ðkÞ E - 1 D ðkÞ E - 1 D ðkÞ E > > > b b : Σm = B : Σ- B : b β

ð aÞ

general linearization

ð bÞ

ð17:14Þ Combining the nonlinear response of each element Eq. (17.1), σ(k) = C(k)(sec) : ε(k), with Eqs. (17.13) and (17.14) yields σ(k) = C(k)(sec) : A(k) : hA(k)i-1 : E. Imposing the stress consistency condition hσ(k)i = Σ on this equation gives the following expression for the homogenized secant viscosity: ð sec Þ

C

D E D E-1 = CðkÞð sec Þ : AðkÞ : AðkÞ

ð17:15Þ

Combining the linearized response of each element Eq. (17.2), σðkÞ = b ðkÞ : εðkÞ þ σ0ðkÞ , with Eqs. (17.13) and (17.14) and then imposing the stress C D ðk Þ E b :A b ðk Þ : E m þ consistency condition hσ(k)i = Σ leads to Σ = C D ðkÞ E b :α b ðkÞ þ σ0ðkÞ . Substituting Eq. (17.14) into this equation yields the following C expressions for the linearized form of homogenized viscous stiffness and the backextrapolated term: 8 D ðk Þ E D ðkÞ E - 1 > > b ðkÞ : A b :A b 0 D b ðk Þ > b ðk Þ : A b :A b :Σ = C : α b ðkÞ þ σ0ðkÞ - C b ðk Þ : α

ð17:16Þ

A similar procedure can be followed with the stress partitioning equations from Eqs. (17.5) and (17.6) to get the following for homogenized compliances and the back-extrapolated term: D ðkÞð sec Þ E D E-1 ð sec Þ b M = M : BðkÞ : BðkÞ tangent linearization 8 D ðkÞð sec Þ ðkÞ E D ðkÞ E - 1 ð sec Þ > b b b > : B = M :B

> : E0 = M b :B b b :b b : B β þ ε0ðkÞ - M : b β

ðaÞ general linearization ðbÞ ð17:17Þ

402

17

Generalization of Eshelby’s Formalism and a Self-Consistent Model. . .

Equation (17.17) are the homogenization equations used in the self-consistent model of Jiang (2014). As pointed out above, the above homogenization equations are valid for both the Mori-Tanaka and the self-consistent schemes. These two schemes are distinguished by choice of the reference medium only.

17.6

A Self-Consistent Algorithm

The microscale deformation in a RDE can now be solved numerically by solving the partitioning and homogenization equations simultaneously following the following algorithm. First, the macroscale flow field is known and given by a velocity gradient tensor L strain rate and vorticity are known with  which implies that the  macroscale  E = 12 L þ LT and W = 12 L - LT . Second, because the deformation in each RDE is assumed uniform, tangent linearization applies perfectly to RDEs with M(k) (tan) = n(k)M(k)(sec) and CðkÞð tan Þ = n1ðkÞ CðkÞð sec Þ . As the rheological properties of the ð tan ÞðkÞ ð sec ÞðkÞ constituent RDEs are given, Minitial and Minitial are known, defined at a reference stress or strain rate state such as the homogeneous state of the macroscale strain rate E. We can get initial guesses of M and E0 using the volume-weighted (Taylor) average of the tangent compliances of all RDEs according to: D E D E ð tan ÞðkÞ ð sec ÞðkÞ Minitial = Minitial = nðkÞ Minitial

D E - 1 ð sec ÞðkÞ 0 S :E J - Minitial : Minitial E initial =

=

D E D E ð sec ÞðkÞ ð sec ÞðkÞ - 1 JS - nðkÞ Minitial : Minitial

ð17:18Þ

 :E

Third, the self-consistent computation is achieved by two nested iterative loops (Fig. 17.4). In the outer iterative loop, the Eshelby and interaction tensors (S(k) and _ ðk Þ

H ) for each RDE are calculated starting with the initial guessed M and E0 (Eq. 17.18). This leads to the launch of the inner iterative loop to calculate the stress partitioning tensors (B(k) and β(k)) for each RDE, starting with the initial M(sec)(k) (or M(tan)(k)and ε0(k)) defined at the homogeneous macroscale strain rate state. The partitioned stress in the RDE, σ(k), and the partitioned strain rate ε(k) can be calculated (Eq. 17.13) which are in turn used to update M(sec)(k) (or M(tan)(k) and ε0 (k) ), for the RDE for the next round of iteration. As the calculation for each RDE’s stress and strain rate is independent of other RDEs, parallel computing is used to speed up the computation. The inner iteration for an RDE terminates when the output compliances coincide with the input ones within a specified tolerance which is checked by ensuring that the stresses in the phase satisfy:

17.6

A Self-Consistent Algorithm

403

Fig. 17.4 Flow charter showing the nested loops to numerically solve for the microscale stress and strain rate fields and the macroscale rheology (homogenized stiffness and compliance tensors and back-extrapolated terms). See text for more details. Modified after Lu (2020)

404

17

Generalization of Eshelby’s Formalism and a Self-Consistent Model. . .

ðk Þ σ current - σðkÞ last < tolerance kσðkÞ last k The computation then moves to the next RDE. When all RDEs are computed, the computation uses the new set of M(tan)(k), ε0(k),B(k), and β(k) to update the HEM’s M and E0. A new round of outer iteration is executed starting with the new M and E0. The outer iteration continues until the input M and E0 coincide with their output values within a specified tolerance which is checked by ensuring both the macroscopic compliances and the macroscopic deviatoric stresses satisfy: Mcurrent - Mlast kΣ 0 current - Σ 0 last k < tolerance, < tolerance Mlast kΣ 0 last k When both loops are completed, the computation updates the state of the system for all RDEs, and the calculation at the microscale for one time step is completed. The microscale fields and the HEM rheology can be used to investigate the fabric development in the heterogeneity.

17.7

Multiscale Modelling

Once the computation for one time step is completed, there are two possible ways to proceed to investigate the development of fabrics in the microscale RDEs. The microscale stress and strain rate fields for those RDEs of interest can be exported as an output file which will be used as input in a separate program to simulate the development of fabrics in those RDEs. For instance, Bhandari and Jiang (2021) investigated the development of quartz CPOs as a result of partitioned flows. They coupled the self-consistent computation above with the visco-plastic self-consistent (VPSC) model of Lebensohn and Tomé (1993a). Alternatively, to investigate the development of microscale shape fabrics, it is possible to simulate microscale fabrics in RDEs in the same program with the self-consistent computation aforementioned. In this case, when the computation for the microscale fields is completed for one time step, the computation continues to track the behaviors of fabric elements within each RDE. The computation does not move to the next time step until all RDEs and their embedding fabric elements are updated. The multi-order computation continues until a prescribed boundary displacement, or a macroscale finite strain state is achieved. This self-consistent multi-order computation is called the MultiOrder Power Law Approach (MOPLA, Jiang and Bentley 2012). The Multi-Order aspect is already explained above, and Power-Law designates the rheology of the system. Yang et al. (2019) used MOPLA to investigate the development of L-tectonites in high-strain zones. Therefore, we can achieve multiscale modelling of rock deformation by either coupling different programs as in Bhandari and Jiang (2021) or carrying out

17.8

Behaviors of Fabric Elements

405

multi-order computation in a single program as in Yang et al. (2019). From a computational point of view, the latter is more efficient and makes the visualization of the modelling results much easier. Furthermore, it eliminates possible errors from combining software programs developed by different researchers and (or) computations executed on different hardware platforms. In the self-consistent MOPLA, as the macroscale deformation advances, shape and lattice-preferred orientation fabrics in RDEs evolve. The rheological properties of the RDEs vary with time due to their changing internal stress (and strain rate) state as well as fabric. Stain and alignment of RDEs also define shape preferred orientations. Both lead to evolving rheology and rheological anisotropy at the macroscale. The formulation so far captures some of these evolving properties in a self-consistent manner.

17.8

Behaviors of Fabric Elements

As outlined in the last section, once the microscale (or partitioned) flow field in an RDE is calculated, it can be used for investigating the evolution of microscale fabric elements contained in the RDE. The behavior of a fabric element is governed by its own dynamics. For LPOs, geologists have widely utilized the VPSC model (Lebensohn and Tomé 1993a), replacing the earlier Taylor-Bishop-Hill model (Lister and Hobbs 1980). For SPOs, geologists have regarded the fabric-defining elements as passive markers, rigid objects, and deformable objects. In reality, a combination of these elements defines many SPOs in rocks.

17.8.1

Finite Strains in RDEs

The March (1932) model regards a fabric element as a line (e.g., an acicular grain) or plane (e.g., a mica flake) element which rotates passively in a given flow field. Fabrics defined by such elements then follow the finite strains in the RDE. Preferred orientations defined by passive strain markers (e.g., oolites in limestone, Cloos 1947) also follow finite strain axes. Foliations and lineations in rocks are also commonly taken to be parallel to the finite strain axes. The finite strain in an RDE is most easy to track. As the partitioned flow field is known (L(k) = ε(k) + w(k)), the associated finite strain evolves by the following expression (Eq. 6.6): dFðkÞ = LðkÞ FðkÞ dt

ð17:19Þ

where F(k) is the position gradient tensor for the RDE. This equation can be solved numerically by: F(k)(t + δt) = (I + L(k)δt)F(k)(t) and F(k)(0) = I (Chap. 6).

406

Generalization of Eshelby’s Formalism and a Self-Consistent Model. . .

17

One uses Eq. (17.9) to track the finite strains in every RDE. Note that the finite strain in each RDE is distinct (Jiang and Bentley 2012; Jiang 2014).

17.8.2

Rigid or Deformable Fabric Elements in RDEs

If the fabric element is rigid, the behavior of the fabric element is: b2 - b2J ðkÞ ðk Þ ΘFij j rigid in = wij - 2I ε bI þ b2J ij isotropic ðk Þ

ðk Þ

ð aÞ

ðk Þ - 1 ðk Þ

= wij - Πijmn Smnkl εkl

ΘFij j rigid in

ð bÞ

ð17:20Þ

anisotropic where ΘFij is the angular velocity of the fabric element and bI (I = 1, 2, 3) are its three semi-axis lengths. Equation (17.20) is Jeffery’s (1922) result and Eq. (17.20) is based on Eq. (11.60) and Jiang (2016). An equivalent ellipsoid shape should be used where the rigid element is far from ellipsoidal. Slippery interface also alters the rotational behavior significantly. The angular velocity for a rigid ellipsoid with a perfect slippery interface is (Chap. 11 and Jiang 2016): ΘFij j perfect

ðk Þ = wij

þ

ðk Þ Πijkl

! -1 b2I þ b2J d  d ðk Þ kÞ - 2 J S εðmn J ijkl klmn bI - b2J

ð17:21Þ

slippery Where the fabric element can be regarded as a deformable ellipsoid, Eshelby’s formalism applies to their angular velocities and shape change: εF = AF : εðkÞ w =w F

ðk Þ

db F = bε b, dt

þΠ

ðk Þ F

:S

ðk Þ - 1



: ε -ε

dQ = - ΘF QF dt

F

ðk Þ



ð aÞ ð bÞ

ð17:22Þ

ð cÞ

where bεFij is a diagonal matrix made of the diagonal components of εFij , the strain rate of a microscale element. The angular velocity ΘF is calculated by using Eshelby’s formalism to the microscale-fabric system. Expressions for ΘF are given by Eq. (11.59).

17.8

Behaviors of Fabric Elements

17.8.3

407

Crystal Lattice Rotation

An RDE is usually a polycrystal rock made of monomineralic or polyphase minerals. The plastic deformation of the minerals leads to development of LPOs in additions to shape fabrics. The LPO is a major cause of the rheological anisotropy (e.g. Wenk 1998). The development of LPO within an RDE is clearly due to partitioned flow within the RDE rather than the macroscale flow. The angular velocity of the lattice of a mineral grain is: ΘLattice j in ij

ðk Þ

e ij = wij þ w

  1X s s bi nj - bsj nsi γ_ s 2 s

ð17:23Þ

RDE ðk Þ

e ij is the grain’s deviation of vorticity with respect to wij ; bsi , nsi , and γ_ s are where w respectively the slip direction unit vector, the slip plane-normal unit vector, and the shear rate of a slip system. The last term is the anti-symmetric component of the e ij term can be obtained using plastic deformation of the grain (see Eq. 5.46b). The w Eq. (17.22) by regarding the grain as a heterogeneous inclusion in the polycrystal RDE. The VPSC model (Lebensohn and Tomé 1993a) is widely utilized to model the development of LPOs. The partitioned flow field in an RDE can be used as input flow for the VPSC model so that LPO development in selected RDEs can be simulated (Bhandari and Jiang 2021).

17.8.4

Empirical Behaviors of Fabric Elements

Sometimes, the dynamic equations of microstructures are written in an empirical form. For an axisymmetric (not necessarily a spheroid) element with a distinct axis represented by a vector vF, Olbricht et al. (1982) gave the following Equation of evolution:   dvFi F α ðk Þ ðk Þ ðk Þ εst bvFsbvFt vFi vF = wim vFm þ G εim vFm Fþ1 Fþ1 i dt

ð17:24Þ

where b vF = jvvF j , and the coefficients G, α, and F are parameters depending on the properties of the element and the hosting RDE. By selecting G, α, and F, Eq. (17.24) is reduced to behaviors for particular fabric elements, including irregular rigid particles (Bretherton 1962; Willis 1977). A linear elastic dumbbell corresponds to G = 1, α = constant, and F = 0 (Olbricht et al. 1982), reducing the evolution  ðk Þ dvFi equation to dt = Lim - αδim vFm . The situation G = 1 indicates that the particle F

408

17

Generalization of Eshelby’s Formalism and a Self-Consistent Model. . .

would rotate and deform identically to a line element of the fluid for α = 0. Almost all rigid axisymmetric particles exhibit G < 1. Any one or a combination of the above evolution equations may be used to track the state of fabric elements contained in selected RDEs in a multiscale modelling investigation. When all fabric elements in selected RDEs are updated, the multiscale computation for one time step is completed. The computation now moves to the next time step. This procedure continues until a given amount of boundary displacement or macroscale finite strain state are reached. The model-predicted fabric patterns can then be compared with natural observations.

17.9

A Self-Consistent Multiscale Model for the Deformation of Earth’s Heterogeneous Lithosphere

As we pointed out in Chap. 1, geological observations are mostly made on a small scale and from rheologically heterogeneous elements. No simple, single-scaled relations exist between our observations and the large-scale boundary conditions we wish to unravel. The key to bridging the scale gap is the partitioning relations established in continuum micromechanics, specifically the generalized Eshelby formalism for power-law viscous materials. In the self-consistent MOPLA, we first solve for the partitioned flow fields in rheological heterogeneities and then use the partitioned flows to investigate the fabric development in the rheological heterogeneities. We use a hypothetical high-strain zone deformation to highlight our approach’s assumptions, principles and procedure. As shown in Fig. 17.5, the ‘regional’ deformation of the high-strain zone is the macroscale flow subjected to the tectonic-scale boundary conditions (Fig. 17.5a). We denote the characteristic length of the zone by D. Macroscale field quantities (‘regional’ stress, strain rate, and vorticity fields) are defined at every location in terms of a macroscale RVE (Fig. 17.5a) that is a large enough volume element, centered at the location and containing a representative assemblage of RDEs. The size of the RVE, D, represents the macroscale characteristic length. To capture the spatial variation of the macroscale deformation, D must be much smaller than D and the fluctuation length λ, of the boundary loading or velocity conditions, i.e., D ≫ D and D ≫ λ. In the deformed state, as observed in the natural high-strain zones (e.g., Williams and Jiang 2005), the RDEs in an RVE are typically made of transposed rheologically heterogeneous units shown schematically in Fig. 17.5b. The microscale deformation fields can be regarded as the ‘local’ field quantities (‘local’ stress, strain rate, and vorticity). They refer to field quantities in individual RDEs. RDEs can vary in size greatly, from a plutonic body in a tectonic zone to outcrop-scale lithological elements. In a single problem, one can always divide larger elements into smaller ones so that all RDEs in a macroscale RVE have similar sizes. The mean size of RDEs is denoted by d (Fig. 17.5b). We regard all RDEs as

17.9

A Self-Consistent Multiscale Model for the Deformation of Earth’s. . .

409

Fig. 17.5 The self-consistent MOPLA approach illustrated in a hypothetic high-strain zone situation. (a) A tectonic-scale high-strain zone with characteristic width D and boundary conditions fluctuating on characteristic length scale λ. (b) The RVE for the macroscale flow, centered at position point X, has the characteristic length D. D ≫ D and D ≫ λ. (c) The RVE contains a large assemblage of RDEs whose mean size is d (d ≫ D). Microscale (or partitioned) fields are fields within individual RDEs. (d) Each RDE is regarded as an ellipsoid embedded in and interacts with the uniform HEM whose rheology is obtained by homogenization over the RVE. (e), (f), and (g) are examples of outcrop scale structures, microstructures under the microscope, and LPOs, respectively, that one may observe within an RDE. The characteristic length for these fabrics is denoted by δ (δ ≫ d). The separation of scale is stated by Eq. (17.25). Modified after Jiang (2014), Qu (2018), and Bhandari (2021)

410

17

Generalization of Eshelby’s Formalism and a Self-Consistent Model. . .

ellipsoids and assume that d ≫ D. We further assume that d is small enough such that intra-RDE deformation fields are uniform. The fabric elements within RDEs have characteristic length denoted by δ. In the hypothetical example here, they are typical structural geology-scale features observed directly on outcrops (Fig. 17.5c, d). If the deformation within an RDE is homogeneous down to the microscopic scale, then δ may also correspond to the microstructural shape and lattice fabrics observed under microscopes (Fig. 17.5e, f). Because our direct observations are within RDEs, fabrics are products of the ‘local’ deformation fields—deformation within RDEs. We assume that δ ≫ d so that the presence of fabric elements does not violate the uniform intra-RDE deformation assumption. To sum up, the gaps separating various scales are stated by the following expression: δ < < d < < D < < D, D < < λ

ð17:25Þ

This is the condition of scale separation (Zaoui 2002; Qu and Cherkaoui 2006, p. 99–102; Nguyen et al. 2011) specifically for the hypothetical problem. This condition forms the basis for treating RDEs as Eshelby inhomogeneities in a macroscale deforming mass and regarding fabric elements as higher-order inclusions in partitioned flows in RDEs. It is worth emphasizing the necessity of a multiscale approach for a problem like Fig. 17.5. It is impractical to tackle a problem with such a wide range of scales in a single-scale numerical model like in a finite element analysis. As δ and D are separated by many orders of magnitude, to account for δ-scale features in a model extending to the D-scale, one would have to consider a ridiculously large number of elements that would require a prohibitive computational resource (c.f. Nemat-Nasser and Hori 1999, p. 574).

17.10

A Continuum Micromechanics-Based Multiscale Approach

At its core, micromechanics is about the mechanical interaction between a heterogeneous element (approximated by an ellipsoid heterogeneity) and the embedding material (simplified as a uniform HEM). This interaction is dynamic in a large-finite strain deformation, as the heterogeneity changes shape and orientation and the HEM’s rheology evolves with time. The formal interior solutions are often recast into interaction Eq. (17.5) and are sometimes referred to as “interaction laws” (Molinari 2002; Molinari and Mercier 2004). They form the basis for the partitioning and homogenization computations of heterogeneous materials. The exterior solutions are used for the investigation of deformation and fabric development around rheological heterogeneities. The flow in

17.11

Notes and Key References

411

a rheological heterogeneity is non-steady due to its continuous interaction with the embedding matrix. This non-steady flow gives rise to fabric development in rheological heterogeneities. Mechanical interactions imply that the deformation of any heterogeneous element cannot be studied in isolation. As demonstrated in Chap. 15, the total displacement along a ductile shear zone in an elastic medium is limited because the displacement causes elastic strain in the country rock. The activity, lifespan, and geometric evolution of a shear zone are inextricably linked to the deformation of the entire crust. Therefore, one should consider the whole orogenic system subjected to a far-field tectonic loading or displacement condition rather than treating a shear zone in isolation with ad hoc imposed boundary conditions. In this system approach, the deformation of a shear zone represents a partitioned field due to the rheological interaction between the shear zone and the embedding deforming lithosphere. Mechanical interactions also imply that one cannot use a single-scale extrapolation of the plate-scale boundary conditions, determined by relative plate motions that can be steady for many Ma’s (e.g., DeMets and Wilson 1997; DeMets and Traylen 2000; Bird 2003), to interpret outcrop-scale observations. Attia et al. (2022) demonstrate that observed fabrics in the central Sierra Nevada (eastern California, USA) are to be understood with partitioned flows. Arc activity did not control the location, intensity, or kinematics of intra-arc deformation. Instead, the arc lithosphere appears to have strengthened rheologically as the arc-orogenic flare-up proceeded. The rheological evolution of the arc lithosphere has played a key role in the spatiotemporal patterns of Late Cretaceous deformation observed across-strike of the entire Cordilleran margin. The multiscale concept of natural deformation has implications for the “rheology of the lithosphere.” As the stress and strain (strain rate) are multiscale concepts, the stress field and/or magnitude in a ductile shear zone (or any other rheologically distinct element) reflects a partitioned field. It does not represent the strength of the lithosphere on an RVE in which the shear zone is a heterogeneous element. Assuming that the shear zone stress represents the lithosphere strength amounts to assuming a uniform stress model (Platt and Behr 2011) for the heterogeneous lithosphere, which likely represents the lower bound for the lithosphere strength. The kinematic approach is unable to deal with heterogeneous and non-steady natural deformations ubiquitous in nature. The traditional continuum mechanical approach has significant difficulties achieving large finite strains for 3D deformations. It also cannot handle multiscale deformations well. The continuum micromechanics approach outlined in this book combines the benefits of the kinematic and mechanical analyses.

17.11

Notes and Key References

The self-consistent model presented in this chapter is implemented in the MATLAB software MOPLA that is provided in the online resource of this book.

412

17

Generalization of Eshelby’s Formalism and a Self-Consistent Model. . .

Appling Eshelby’s formalism in an “inclusion in inclusion” approach was first presented at the GSA Penrose conference on “Deformation localization in rocks: new advances” (Jiang and Bentley 2011). The approach has been fully developed since then (Jiang and Bentley 2012; Jiang 2014, 2016) and has been applied in several studies (Jiang and Bhandari 2018; Yang et al. 2019; Bhandari and Jiang 2021).

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