Multiscale Modelling of Plasticity and Fracture by Means of Dislocation Mechanics (CISM International Centre for Mechanical Sciences, 522) 3709102820, 9783709102824
The latest state of simulation techniques to model plasticity and fracture in crystalline materials on the nano- and mic
131 76 22MB
English Pages 401 Year 2010
Title Page
Copyright Page
Preface
Table of Contents
Atomistic Simulation Methods and their Application on Fracture
1 Introduction
2 Description of Interatomic Bonds
2.1 Quantum Mechanics Based Descriptions of the Atomic Bound- ing
2.2 Atomic Interaction Models, Potentials
3 The Molecular Dynamics Method
3.1 Force Calculation
3.2 Integrating the Equations of Motion
3.3 Relaxation Algorithms
3.4 Boundary and initial conditions
3.5 Stable Defects under Load
3.6 Visualisation and Analysis of Defects
4 Comcurrent Multiscale Methods
4.1 Introduction and Classification of Multiscale Methods
4.2 The Finite Element Atomistic (FEAt) Method
4.3 The Quasicontinuum-Method Based on the Cauchy Born Rule
4.4 The Fully Nonlocal Cluster-Based Quasicontinuum-Method
4.5 Other Concurrent Multiscale Methods
4.6 'Learn-On-The-Fly' - LOTF
5 Atomistic Aspects of Fracture
5.1 Lattice Trapping and the Directional Cleavage Anisotropy
5.2 Metastable Fracture Surfaces
6 Dynamics of Brittle Crack Propagation
7 Summary
Biblliography
Fundamental dislocation theory and 3D dislocation mechanics
1 Basic Dislocation Theory
1.1 Heuristic Dislocation Creation
1.2 Basic Dislocation Types
1.3 Moving Dislocations
1.4 Dislocations In Real Crystals
2 Curved Dislocations
2.1 Line Tension Model
2.2 Dislocation Self-Interaction
3 2-D Applications
3.1 Simulation Technique
3.2 Static Simulations Using the Line Tension Model
3.3 Simulation Using Dislocation Self Interaction: Particle Strengthening
3.4 Simulations of Thermally Activated Dislocation Glide
4 3-D aspects
4.1 Non-elastic 3-D effects
4.2 Computational Aspects for 3-D Simulations
Bibliography
Plasticity of moderately loaded cracks and the consequence of the discrete nature of plasticity to fatigue and fracture
1 Introduction
2 Stress field of a crack in a linear elastic material
3 Dislocation crack tip interaction
3.1 Linear elastic analysis of the stresses and deformations induced by an edge dislocation near a crack tip
3.2 Moderate cyclic loading of a crack
3.3 The involved length scales
4 Modelling of plasticity, crack propagation and fracture surface contact
4.1 The cyclic plastic deformation as a function of load amplitude and ber of cycles
4.2 The effect of boundaries
4.3 The threshold of cyclic plastic deformation and the effective threshold of stress intensity range
4.4 Other discrete discrete dislocation simulations of fatigue crack propagation
Bibliography
Discrete Dislocation Plasticity Analysis of Cracks and Fracture
1 Introducton
2 Elastic Models of Dislocations
2.1 General Idea
2.2 Edge Dislocations
3 Boundary Value Problems
4 Dislocation Dynamics
4.1 Annihilation
4.2 Frank-Read sources
5 Methodology for Crack Problems
6 Cracks in Single Crystals
6.1 Stationary Crack - Tip Fields
6.2 Crack Propagation under Monotonic Loading
7 Fatigue Crack Growth
8 Cracks in Polycrystals
Bibliography
Statistical physical approach to describe the collective properties of dislocations
1 Introduction
2 Kroner-Kosevich field theory of dislocations
2.1 Nye's dislocation density tensor
2.2 Internal stress generated by the dislocation system
2.3 Second order stress function tensor
2.4 2D problems
2.5 Time evolution of the dislocation density tensor
2.6 Time evolution of the displacement field
2.7 Problems related to coarse graining
3 Linking micro- to mesoscale for a 2D dislocation system
3.1 Discrete dislocation dynamics simulations in 2D
3.2 Continuum theories developed for other systems, analogies and differences
3.3 Hierarchy of the different oder density ffunctions
3.4 Evolution of the plastic shear
3.5 Self-consistent field approximation
3.6 Stability ananlysis
3.7 Numerical studies
3.8 The role of dislocation-dislocation correlation
3.9 Deformation of a constrained channel
3.10 Application to metal-matrix composite
3.11 Boltzmann thery of dislocation
3.12 Hydrodynamics approach proposed by Kratochvil and Sedlacek
4 Internal stress distribution generated by the dislocations
4.1 General considerations
4.2 Stress distribution at the dislocations
4.3 The mean values of distributions P(r) and Po(r)
4.4 Asymptomic properties of the stress distribution function
Bibliography
Basic ingredients, development of phenomenological models and practical use of crystal plasticity
1 Introduction
2 A thermodynamical approach to single crytal plasticity
2.1 General fframework
2.2 Derivation of single crystal models
2.3 Yield surfaces
2.4 Identification under tension and tension-shear loaadings
2.5 Slip system selection
2.6 Other crystal plasticity models
3 Finite element computations of single crystalline components
3.1 Algorithm for the numerical integration
3.2 Laboratory specimens
3.3 Turbine Blades
4 Finite Element Crystal Plasticity
5 Uniform field models
5.1 Yield surfaces
5.2 Scale transition rules
5.3 Complex paths
6 Conclusion and perspecives
Bibliography
Computational homogenization
1 Introduction
2 Underlying principles and assumptions
2.1 Scale separation
2.2 Local periodicity
2.3 Homogenization principles
2.4 Computational homogenization scheme
2.5 Kinematically driven multi-scale scheme
3 The micro-scale problem
3.1 The representative volume element
3.2 Micro-scale characterization & equilibrium
3.3 The macro-micro scale transition
3.4 Micro-scale RVE boundary comditions
4 The macro-scale problem
4.1 The micro-macro scale transition
4.2 Macroscopec stress tensors
5 Two-scale numerical solution strategy
5.1 RYE Boundary value problem
5.2 Extraction of the macroscopic stress
5.3 Extraction of the macroscopic tangent operator
5.4 Nested solution strategy
6 Example: two-scale coupled analysis in bending
7 The RVE in first-order computational homogenization
7.1 General concept of an RVE
7.2 Unit cells versus RVEs
8 Second-order computational homogenization
8.1 Principles
8.2 Two-scale higher-order kinematics
8.3 Extracting stress tensors
8.4 Two-scale computational solution strategy
8.5 Parallel solution of the multi-scale nested boundary value problens
9 Higher-order issues
9.1 First-order versus second-order
9.2 Full gradient versus couple stress
9.3 Geometrical size effects
9.4 Large macroscopic gradients
9.5 Macroscopic localization
9.6 The higher-order RVE
10 Conclusions
Bibliography
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~ SpringerWienNewYork
CISM COURSES AND LECTURES
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Executive Editor Paolo Serafini - Udine
The series presents lecture notes, monographs, edited works and proceedings in the field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences.
INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 522
MULTISCALE MODELLING OF PLASTICITY AND FRACTURE BY MEANS OF DISLOCATION MECHANICS
EDITED BY REINHARD PIPPAN AUSTRIAN ACADEMY OF SCIENCES, LEOBEN, AUSTRIA PETER GUMBSCH FRAUNHOFER-INSTITUT FÜR WERKSTOFFMECHANIK, IWM, FREIBURG, GERMANY
This volume contains 187 illustrations
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2010 by CISM, Udine Printed in Italy SPIN 80016150
All contributions have been typeset by the authors.
ISBN 978-3-7091-0282-4 SpringerWienNewYork
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8
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Atomistic Simulation Methods and their Application on Fracture
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10
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Atomistic Simulation Methods and their Application on Fracture
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Atomistic Simulation Methods and their Application on Fracture
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Atomistic Simulation Methods and their Application on Fracture
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16
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+ )# '1# ### )# '1+!# 1# # ! )# J!+ # +L .)# # #+ UA+ # +# ##! )# !+1 + !#+ + )# !G )+ ) + !#F#! )# 1 !+1 + 1#') ?)# #: UA # 13#L # #1 G 6! = UA+ + )# 3+1# !+1 + !#+ # # 1 !+1 + ) 11 3# +# !#% 3+L % # !++!# # + ?OOA +3# # U ! #!
Fundamental Dislocation Theory and 3D Dislocation Mechanics
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# >)# #> ?B = =A&G B # + )# $ # +L #1# )# 3 + 3 % )# + )# ! # ! #1 + % 3+' !+1 + ? #' !1# % % # ) ! +# )# !+1 +A ?B = 6!
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8 + 11#! )# #1#! )# #L + #%1 /## + 3+! ) )# !#F++ - + ! #!G )# #1#! )# # 31+1+#! )# ? A 3'+!# % )# '# # % !+1 + # 1 )# % # # + 1#') )+ !+1 +: 8 = AL )1! 1 # #! ) + )# !# ++ % 8 # 3!# 3+ )# 1+# # G )# # 8 + +!##!# % L ## G # 3#! ) 7 1 )# % # + F' # 7G )+ ) + 11#1 )# '1+!# 1#G ! !# )! # ###! +L % # 3# 7 11 % # +#!G + + +!#! 3+ 11 ! # + ?OTA /# )# 3 # '## 1 % 3: 8 = + 7 +
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'+G + + )# 31+ #! '1+!# 1# # L % # 3# + = ?N@ N@ OAG # #+ # + ?OTAL ,+ # )# '# # 1+# + )# '1+!# 1# + 3# +)#; % 3 )+ + # ## ) 1 )# 3# % # # 7 '+# +# 8 ! )# # ! +# !+1 + '1+!# ?7 ! 7 % # + ?OTAAL + ?O6A -# # # 7 + )# >'1+!# #3>G !#F#! )# '1+!# 1# ! )# '# # L # #G
Fundamental Dislocation Theory and 3D Dislocation Mechanics *+
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
Fundamental Dislocation Theory and 3D Dislocation Mechanics
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73
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% )# # # #+# + 1 #! +') % 3 )# # ? / NAG + ## ++# # 8 )+ ) ! +# + 3 # ++# J1#G +L#L % 3 )# #: 11#1 # !+1 + ##1 # ) )# +% )# )# )# 3# +'L 4+) #+)# +# #!G )# !+1 + # 11#! + 11#1G ! )# # ) )# L 4)# )# 3##G )# +)+1# # ) )# L + #G )# +' )1% 1# % + 11#1 #!'# !+1 + % 3 %11 1# % )# 1L # + ?QNA + )! ## 3#! ) )# !+1 + # ! )# '1+!# 1# + = ?N@ N@ OAL & # # !+1 + + + 3#!+3 # ! '1+!# 1#; % )+ # + 3 # !# +#! ) 8 = 0 A?Q4 AG )# # + )# !+ # ### )# !+1 +L ## G + #1 1 ?# J + OLSAG # !+1 + 3+ + +11 ! # + 11' )+ 1#L % # +!# #!'# !+1 + ? /# )#! + F' # ONAG )# 'J # # # = ? @ N@ NA ! = ? @ N@ NAG ! )# 31 '1+!# # 8 + 3 ) 3 # 31+ #!L + 1 G )# +' % )# +# J + !##! ! : - 0
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# #++' !+1 + + #! )# ST J1+# )# # - = - ; )# #G 8 = NL .)+ 3# ) #!'# !+1 + % +# +' #! % 3 !+1 + !+1#G +) ST '1# ### )# '1+!# 1# ! )# ) # # # +' )L
+1 + !+1# )# )# +3 # ) )# # ! +# )3'## # 7 : )# #!'# !+1 + # ! +# + +# !+ # +L .)+ 3 #/ )# !+1#G + +11 ! +# +L )+ ##G !+1# # +!# #! +33+1#G + )# 3+1# !+1 + )+ ) 11 #1# 1+ + L # +% )# 3#G )# + % !#% 3+ )# #!'# !+1 + # #1 # ) )# % )# 3 L )# +3 %# # % !+1# + ) )# !+1 + +1#! 3# ! # ) )# 3+ % '1+!# ! 1+3 ! F11 +)+1#L 1+3 # + # 3# +1 !+W+G ! )# # )+ +)+1+ + /+#+ 11 +)++#!L + # )# #!'# !+1 + # ) )# G )# !+1# #!# ! ++' % # )+ ) #3+ +# ++1 3
Fundamental Dislocation Theory and 3D Dislocation Mechanics
75
+#L ### 11#1 #!'# ! # !+1 + )# # + #1+ +# J +: )# 3++ % = ?N@ @ NAG = ? @ N@ NAG ! = ? @ N@ NAG = ?N@ @ NAG ) +#1! 8 = N ! 8 = NG ## +#1 L # # +) ## ' )'#G 11#1 # ! #!'# !+1 + # 3#! %11 + +!#L )+ #! # +') !+1 + +) 3+#! ) # + '## #!L # #1 G )+ 3# ) 3+#! ) # !+J 1 + +) '# # 3 # +!# #! #+' 3#! % #!'# ! # !+1 + +) '# # 3# = + ! = + G ## +#1 L 4+) )##G )# # # 7 % 3+#! ) # !+1 + 3 # +# 7 ! ?A = 7 ? A D 7 ? A
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
#! )# +# +!# % )# % #: % 3 # + ?SA + # ## ) )# 1 J+)+' # 3# + 1+! + 1 !+# % !+1 J , + 1' )# J+G 7 G !##! 1 )# !+ # = - D G )+ ) + 1 ++#L # #G 11 ## % )# !+1 + ?+) %,& G %,& = G F' # OQ?AA ! + +3'# # ?+) %-& = %,& G %-& = DA 3+#! +) + )# )1# 1# ?N@ @ AL .)# +'+1
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#! !+1 + ?# + OLQAG ! )# 3# 7 % )# 3## % # ) )# ?# + ?7AAL + # % 7 G )# 3# % )# !+1 + !! # # )# +' % 7 !##! 1 ) % 1 ) % ?# + ?7AAL F' # OQ?A )# J+)+' # 3# # +!+ #! ' #
L # % +') !+1 +G )# +# + ### )# !+1 + ! + +3'# + +11 '+# # + ?QOAG +) = N ! = QG )+ ) #1 8%-&%,& = 0A?S4?O 3AA ?+ ) ! )#G O99QAL .)# # 3# 7 # +31 #3#! 3+ 1J +1 )+% 1' )# !+ # + % )# ' #
L .)# # % % ## % # 3 # +!# #! +# % # ++!# # +3# )# # )# )# 3!1 + )# % 3 0 # G )# # )# 3# +1 + +F+#1 %L .)# )# # #3# + +# % # +J F+#1 +'+! 3# +1 ?0 AG )+ ) 11 #1+ ? 1+ A +: = = = = = = NL ,+3+1 )# # !+ #! #G )+ # )+##! !!+' 3# +1 +) +3'# !+1 + +) )# 3# 'J # # : %-& = D%,& L ) # )# !+1 + ## +# # )# 3# 3 % ## 8%-&%,& G +) +# #! +': .)# !+1 + + #J #11#! % 3 )# +'+! L 86;
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Fundamental Dislocation Theory and 3D Dislocation Mechanics
77
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Fundamental Dislocation Theory and 3D Dislocation Mechanics
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'+# 1#L .)+ #+ # ) ) J '#! # +!+ # +)# +#! % 3 )# 1 #G )# '1+!# 1# ! )# '1+!# !+ # +L J + 3+ % )# 3G +L#L )# 31 1# )#3G )#1 !+1 + # 3# )+ #+ #G +% )# '1+!# 1# # # 3)#! #3J # #L .)# #% #G 8 + % + % #3# #L # % % # # #! + 1 )# '1+!# 1# # 1#1 /#! ?F' # ORA ! )# #% # >3)>; )# &#+# 1 # + )## 1 + #'1+'+1# ##
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($ 8 #')#+' 3!#1L .)# ' # # + # )# )# !+1 + #/ ) ') )# 1# + )# 3+!!1#G +1 )# # 1# + %!L # # % G # % )# +1# G G ! # # /L + + / ) # '# ! +' !+1 + '1+!#G # # 1# 3 # %! )# !+1 + %# + ) # 3# )# L .)+ 3# ) OA # 1 )# # !#+ % 1#G G )+ ) + F#! ! # 3# #!G = - G )# # !## )# # 1+ # +'L .)+ + )# ) # !+ # ### 1# +% )# # #
'#! + # 1+ #L "#G )# % # # +1+ +3 + )# 1# + 1+#!: )# ++ 1 # 8 G )# % # 8 # !+1 + % )# 1#') ! )# # )# 1#G # '# ! = 8 L +11 G % 3 )# # # +1+ +3 ### )# 1# # / )G 1+/# + )# $ 3!#1 #G 8 = 8 = A ?$ AL 4+) )## +!# +G # + ?Q7A # #1#! % )# 1#') G 11#! )# +#!#1 1#'): = = ) )
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L .)+ 3# ) )# !+1 + )# 1 # %# 1# )# )# # #/G G +!+ #! #G )# )# !+1 + + # +'+! ! #3+ +')L .)+ #3)+ # )# +3 # % )# 1+# #+ 3!#1G # # )# 1+# ## ' + ##!#! # )# 1# % # ! G ## + +31J + + )+ ) )# 1+# #+ 3!#1 + #! 11L ## G + 3 # ##! '+ ) + )! ## 3#! ) + L % + % + % )# !+1 + ) # G %# ? 1 A )# 1+# #+ &?:A + ! #! + # + QLO + # #! +#! % ?:AL 4+) !
Fundamental Dislocation Theory and 3D Dislocation Mechanics
87
) % 3 # + ?Q8AG )# #')#+' #W# 8 % )# 1# F11 # +# : 8 =
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# +L .)# # 8 ?# + ?QTAAG ! )# % # U = 8 U 31 )# #'3# # U #1#! % 3 )+ 3+L .)+ )1! % 1+# ## ' L 11 + % + % )# !+1 J + ) # G +!+ #! )# '1# : ### '# J ! 1+# # L 3 ?:AG !+1 + #'3# % 1#') U ## # U ) #3 # )# #'3# ) ) + ## ' ? U A + 1# #!: 9? U A 9 ?RNA ?:A = ?:A = 9: 9: 0 )# # = UAU + 1#') # +F #G 11#! 1+# # + 1J ' )# 1+# ## ' L # # U + 3#! # # # )# #W# % 1#') )'# ) 1 #! ## # #! 8 + # + QLOL .)# # +# # )# #'3#G % )# 3 )+ + #J ##! )# #+') +' #'3# + # )# # + )# 3# !+ # + )# )# # 1# ! ## )# 3# #L F' # QN ) ## #'3# +) )# 3# 1#') U # 1#!L .)# ## 1+')1 +' # U %,& G U %+& G ! U %(,& #1+' % 3 +' L .)# # U #1 + % # + )# #'3#> #! +L ### )# #'J 3# ?NA ! ?DOAG )# 3 % % # + J!+ # + + U = U %+& AU U %(,& AU = %+& %(,& L 4+) )# !+ # ### )# J # + #+' )# #'3# 1#') U G )#
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Fundamental Dislocation Theory and 3D Dislocation Mechanics 7
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"# ) %-& + # + ?R6A ! + F' # QQ ) #'+# +'; )+ + # # )+ % # )# 1#% +!# % !+1 + #'3#G ! )# +') %,& L .)+ + +# +) # + ?ROAL # + F' # QQG ) % # # %,& ! %-& + + ++# J!+ # + '+ )# + 1#L .)+ + #! )# % ) ?:A )'# + +' )# '1# : ### '# ! 1+# # )'# % 3 1 D1 + )# 1#; )# +' % + 1 ) +# # )# !+1 + + # ) # G )+ ) ) )# 1# ## ' L .)# 1 !+W# # # ### F' # QQ?A ! QQ?A 1+# + )# 1+# #
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Fundamental Dislocation Theory and 3D Dislocation Mechanics
91
# # U #1 # %,& %-& G ! U: + - +# #! +) !3'#L .)# ?U AU:A # #! )# F !# ++# % ?:AL 4)+1# +' )# + 1#G )# 1+# # )'# U: = Q1 ?F' # QQ?AAL " +# +' )+ ! #1 +' 1 +1 ?1 NAG ! +' = 9 A9: ?# + ?RNAAG # '#:
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.)# /# + # + ?R8A + +1 +!#+ 1 +) )# 1+# #+ & + ! #! + # + ?RSAL # # # + ?R8A # #! +% + F' # QQ?AG % # & %,& ! & %-& #! )# 1# +#! %
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V. Mohles
4 ( - . .)# '## 1 #1+ # # +1+ +3 # +# 7 D = NG )# # 7 + '## 1 # # G + 13# % # ?% # # + 13#A ! !## )# ' !+# J # L 4+) )# !#F++ % + 7 + # 3 % !+1 #3# G = = ?9 A9 D 9 A9 A AQG ! /#> 1 7 = = G )# # # +1+ +3 # +# #1+ ### ### 13# % # ! )#
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# # !## )# S) ' !# # % #1+ G ! +#+> 33+ #+ # # 1 +!+ # + 1+#!L .)# # # !# +J + + # + #! #1+ 11 + + 3#!+3L )+G )# # % #1+ # +# + # 3 % )# )# 3!1 0 ! )# &+ + 3: = C