MULTISCALING IN MOLECULAR AND CONTINUUM

MECHANICS: INTERACTION OF TIME AND SIZE FROM

MACRO TO NANO

Multiscaling in Molecular and

Continuum Mechanics: Interaction of

Time and Size from Macro to Nano

Application to biology, physics, material science,

mechanics, structural and processing engineering

Edited by

G.C. SIH

Lehigh University, Bethlehem, Pennsylvania, USA

and

East China University of Science and Technology, Shanghai, China

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

ISBN-10 1-4020-5061-5 (HB)

ISBN-13 978-1-4020-5061-9 (HB)

ISBN-10 1-4020-5062-3 (e-book)

ISBN-13 978-1-4020-5008-4 (e-book)

Published by Springer,

P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

www.springer.com

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© 2007 Springer

The multi-disciplinary character of the book has been encouraged by the current trend of development of science and technology for seeking a common base of understanding. Multiscaling in time and size, from fast to slow and small to large can assist the micromanipulation of atoms and molecules to tailor make structure- and bio-materials. The excitement lies in the likelihood that organic and inorganic matters might follow the same basic laws. The hope to discover the common denominator for all things has been the driving force for research in the past and will be in the future. The eighteen articles contained in this volume are evidence of the excitement that has generated in the application of fundamental science and high technology. That includes biology, physics, material science, mechanics, structural and processing engineering. Orders of magnitude improvement in the resolution of electron microscopes and accuracies of the electronic computers will no doubt continue to generate new discoveries and pleasant surprises.

The theme of the 21st century research has been directed towards understanding the microscopic factors that control the macroscopic material degradation and

9 sec for cyclic nitramines can trigger detonation of macro-size solid propellants. Such a phenomenon could not have been known without examining the intermediate products of the gas phase reaction which are µm in size. Similar mechanisms for intergranular nano-meter size defects discovered by high resolution TEM and SEM may also explain the initiation of stress corrosion cracking. Psychosurgery via the use of brain implants is finding ways to send signals to neurons for curing blindness and mental disorders. Different fields from material to life science are finding solutions to their long waited answers by reaching down to the lower size scale. The contributions of this volume are aimed to add to the progress of micromanipulation by relating events from a wide range of size and time scale although there remains to be done in multiscaling where equilibrium at a larger scale may involve non-equilibrium at a lower scale. That is macroscopic homogeneity may entail microscopic heterogeneity. Contained in this volume are approaches involving the analytical, experimental and physical modeling, each serving a specific objective. The ultimate effort is to find a common ground of understanding. To this end, this book has made a step closer to the goal.

biological malfunction. Chemical decomposition reacting at 10–

Aims and Scope of the book

Contents

Contributors

Foreword

“Deborah numbers”, coupling multiple space and time scales and governing damage evolution to failure

Y.L. Bai, H.Y. Wang, M.F. Xia, F.J. Ke

in polycrystalline materials J.S. Chen, S. Mehraeen

A multiscale field theory: Nano/micro materials Y.P. Chen, J.D. Lee, Y.J. Lei, L.M. Xiong

Combined loading rate and specimen size effects on the

Z. Chen, Y. Gan, L.M. Shen

Discrete-to-continuum scale bridging J. Fish

Micromechanics and multiscale mechanics of carbon nanotubes-reinforced composites

X.Q. Feng, D.L. Shi, Y.G. Huang, K.C. Hwang

Multi-scale analytical methods for complex flows inprocess engineering: Retrospect and prospect

W. Ge, F.G. Chen, G.Z. Zhou, J.G. Li

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

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

1

11

23

67

85

103

141

161

179

material properties

A multi-scale formulation for modeling of wrinkling formation

ix

xv

vii

Contents

Y.G. Wei, X.L. Wu, C. Zhu, M.H. Zhao

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

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

Segmented multiscale approach by microscoping and telescoping in material science

G.C. Sih

Mode I segmented crack model: Macro/symmetry, micro/anti-symmetry and dislocation/skew-symmetry

G.C. Sih, X.S. Tang

Tensegrity architecture and the mammalian cell cytoskeleton D. Stamenovi , N. Wang, D.E. Ingber

Mode II segmented crack model: Macro/skew-symmetry,

X.S. Tang, G.C. Sih

Microstructure and microhardness in surface-nanocrystalline

Grain boundary effects on fatigue damage and material properties:

Z.F. Zhang, Z.G. Wang Macro- and micro-considerations

Coupling and communicating between atomistic and continuum

simulation methodologies J.A. Zimmerman, P.A. Klein, E.B. Webb III

Author Index

Subject Index

389

439

457

459

339

321

291

259

241

197

Al-alloy material 369

micro/anti-symmetry and dislocation/skew-symmetry

viii

Aravas N Department of Mechanical and Industrial Engineering,

University of Thessaly, Volos, Greece, Email: Aravas@

mie.uth.gr

Bai YL

Barter SA Defence Science and Technology Organisation Victoria

3207Australia, Email: simon.barter@dsto.defence.gov.au

Chen FG Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100080, China, Email: fgchen@

home.ipe.ac.cn

Chen Z Department of Civil and Environmental Engineering,

University of Missouri-Columbia, Columbia, MO 65211-

2200, USA, Email: ChenZh@missouri.edu

Feng XQ Department of Engineering Mechanics, Tsinghua

tsinghua.edu.cn

Fish J Rensselaer Polytechnic Institute, Troy NY 12180, USA,

Email: fish@rpi.edu

Gan Y Department of Civil and Environmental Engineering,

Ge W Institute of Process Engineering, Chinese Academy of

Sciences, Beijing 100080, China, Email: wge@home.

ipe.ac.cn

Haidemenopoulos GN Department of Mechanical and Industrial Engineering,

University of Thessaly, Volos, 38334, Greece, Email:

hgreg@mie.uth.gr

LNM, Institute of Mechanics, Chinese Academy of Sciences,

University of Missouri-Columbia, Columbia, MO 65211-

2200, USA, Email: yg8p2@izzou.edu

Beijing 100080, China, Email: Baiyl@lnm.imech.ac.cn

Contributors

University Beijing 100084, China, Email: fengxq@

ix

Chen JS Civil & Environmental Engineering Department, Univer-

sity of California, Los Angeles, CA 90095 USA, Email:

jschen@seas.ucla.edu

Chen YP Department of Mechanical and Aerospace Engineering,

The George Washington University Washington, DC

20052, USA, Email: ypchen@gwu.edu

Hao S Department of Mechanical Engineering, Northwestern

University, Evanston, IL 60208, USA, Email: suhao@

northwestern.edu

Huang YG Department of Mechanical and Industrial Engineering,

University of Illinois at Urbana-Champaign Urbana,

Hwang KC Department of Engineering Mechanics Tsinghua

University Beijing 100084 P.R. China, Email: huangkz@

tsinghua.edu.cn

Ingber DE Departments of Pathology and Surgery, Children’s

Hospital and Harvard Medical School, Boston,

Jones R Mechanical Engineering, Monash University Melbourne,

Melbourne Australia, Email: Rhys.Jones@eng.monash.

edu.au

Ke FJ Department of Applied Physics, Beijing University of

Aeronautics and Astronautics, Beijing 100080, China,

Email: Kefj@lnm.imech.ac.cn

Department of Mechanical and Industrial Engineering,

University of Thessaly, Volos, 38334, Greece, Email:

akatsam@mie.uth.gr

Klein PA Franklin Templeton Investments, San Mateo, CA 94403

USA, pklein@frk.com

Lee JD Department of Mechanical and Aerospace Engineering,

The George Washington University Washington, DC

20052, USA, Email: jdlee@gwu.edu

Lei YJ Y.P. Department of Mechanical and Aerospace

Engineering, The George Washington University

Washington, DC 20052, USA, Email: yajielei@gwu.edu

Li JG Institute of Process Engineering, Chinese Academy of

Sciences, Beijing 100080, China, Email: jhli@home.ipe.

ac.cn

Massachusetts, USA, Email: IngberDE@aol.com

Katsamas AI

IL61801 USA, Email: huang9@uiuc.edu

x Contributors

Contributors xi

Mehraeen S Civil & Environmental Engineering Department, Univer-

shafigh@math.ucla.edu

Nagarajan K Department of Mechanical, Industrial and Nuclear

Engineering, University of Cincinnati, Cincinnati, OH

45221-0072, USA, Email: nagaraka@email.uc.edu

Pitt S Defence Science and Technology Organisation, Victoria

3207, Australia, Email: susan.pitt@dsto.defence.gov.au

Qian D Department of Mechanical, Industrial and Nuclear

Engineering, University of Cincinnati, Cincinnati, OH

45221-0072, USA, Email: dong.qian@uc.edu

Shen LM Department of Civil and Environmental Engineering

University of Missouri-Columbia, Columbia, MO 65211-

2200, USA, Email: ShenL@missouri.edu

Shi DL Department of mechanics, Shanghai University, Shanghai

200444 China, Email: sdongli00@mails.tsinghua.edu.cn

SihGC School of Mechanical Engineering, East China Univer-

sity of Science and Technology, Shanghai 200237,China.

Department of Mechanical Engineering and Mechanic,

Lehigh University, Bethlehem, PA 18015, USA, Email:

gcs@ecust.edu.cn and gcs8866@yahoo.com

Stamenovi D Department of Biomedical Engineering, Boston Univ-

ersity, Email: dimitrij@bu.edu

Tang XS School of Bridge and Structure Engineering, Changsha

University of Science and Technology, Changsha, Hunan

Vasudevan VK Department of Chemical and Materials Engineering,

University of Cincinnati, Cincinnati, OH 45221-0012,

USA, Email: vijay.vasudevan@uc.edu

Wang HY LNM, Institute of Mechanics, Chinese Academy of

Sciences, Beijing 100080, China, Email: why@lnm.

imech.ac.cn

sity of California, Los Angeles, CA 90095, USA, Email:

410076, China, Email: tang_xs231@sohu.com

Molent L Defence Science and Technology Organisation, Victoria

3207, Australia, Email: lorrie.molent@dsto.defence.gov.au

Mannava SR Department of Chemical and Materials Engineering,

University of Cincinnati, Cincinnati, OH 45221-0012,

USA, Email: manny.mannava@uc.edu

xii Contributors

Wang N Physiology Program, Harvard School of Public Health,

Harvard University, Cambridge, Mass. USA, Email:

nwang@hsph.harvard.edu

LNM, Institute of Mechanics, Chinese Academy of

Sciences, Beijing 100080, China, Email: Ywei@LNM.

imech.ac.cn

Wu XL LNM, Institute of Mechanics, Chinese Academy of

imech.ac.cn

Webb EB III Sandia National Laboratories, Albuquerque, NM 87185

USA, Email: ebwebb@sandia.gov

Weertman J Department of Material Science & Engineering,

Northwestern University, Evanston, IL 60208, USA,

Email: j-weertman2@northwestern.edu

Xiong LM Department of Mechanical and Aerospace Engineering,

The George Washington University Washington, DC

20052, USA, Email: xlm@gwu.edu

Zhou GZ Institute of Process Engineering, Chinese Academy of

Sciences, Beijing 100080, China, Email: gzzhou@home.

ipe.ac.cn

Wei YG

Sciences, Beijing 100080, China, Email: Wu21@LNM.

Wang ZG Shenyang National Laboratory for Materials Science,

Institute of Metal Research, Chinese Academy of

Sciences, Shenyang, 110016, China, Email: zhgwang@

imr.ac.cn

Xia MF Department of Physics, Peking University, Beijing

100871, China, Email: xiam@lnm.imech.ac.cn

Zhang ZF Shenyang National Laboratory for Materials Science,

Institute of Metal Research, Chinese Academy of

Sciences, Shenyang, 110016, China, Email: zhfzhang@

imr.ac.cn

Zhao MH LNM, Institute of Mechanics, Chinese Academy of

Sciences, Beijing 100080, China, Email: Mzhao@LNM.

imech.ac.cn

Contributors xiii

Zimmerman JA Sandia National Laboratories, Livermore, CA 94551

USA, Email: jzimmer@sandia.gov

Zhu C LNM, Institute of Mechanics, Chinese Academy of

imech.ac.cn

Sciences, Beijing 100080, China, Email: Zhu@LNM.

This volume on multiscaling has been motivated by the advancement of nano-technology in the past four decades. In particular, nano-electronics has paved the way to show that the behavior of nano-size bodies are not only different from macro-size bodies but they do not obey the same physical laws. There appears to be a mesoscopic region which separates the laws of quantum physics and continuum mechanics. A gap has been left in the full range of scaling from macro to nano. Micro-manipulation can be made more effective if the atomic and molecular scale activities can be identified more precisely with the use specific objectives. In this respect, material science has already benefited by positioning and structuring of nanometer-scale particles to arrive at the desired macroscopic material properties. The idea has been implemented to tailor-make structural materials for the Boeing 787 to better accommodate non-uniform stress and strain at different locations of the aircraft. Explored are also the possibility of coaxing DNA-based organisms such as viruses to improve performance of batteries, solar cells, fabrics, paints and other kinds of materials. The potential of assembling bio-molecules to build electronic components is also in the planning. The manipulation of molecules and atoms has been regarded as a common base for both material and life science. Quantum and continuum mechanics are being applied side by side for exploring the behavior of small and large objects moving at fast and slow speed. The establishment of a common basis of understanding for all sciences and technology has encouraged the contribution of the 18 articles in this volume in addition to the need for mutliscaling in time and size. Although the various disciplines covered seem to differ at first sight, their aims are directed to associate macroscopic behavior with the atomic and subatomic particles. Such a trend is likely to be the rule rather than the exception in the 21st century as new discoveries in science are made at an alarming pace.

Dividing the matter into smaller and smaller entities has been the bias in the development of Western science for unraveling the secret of nature. When the size scale becomes increasingly small, quantum mechanics was devised to describe the behavior of electrons that cannot be explained by the laws of gravitation in general relativity. The confrontation between these two equally acceptable approaches, however, continues while awaiting for the discoveries of new laws and theories for their recon-citation. Off hand, there are issues that are not unrelated to the development of multiscaling models, a topic selected for discussion in this volume.

Physical laws and theories are intended to establish organizational structure and order in nature such that they can be used to widen the range of existing knowledge and make new discoveries. Telescoping and microscoping involve the observation of events at increasingly large scales and decreasingly small scale. Up to this date, the progress seems to have been rested at both ends of the scale range, the subatomic and galactic scale. One of the immediate issues is concerned with space-time dimensionality. This entails the transition of two-dimensional surfaces to the three-dimensional volumes or vise versa. That is to consider the increase or decrease of dimensionality as a matter of perspective. The underlying implication

Foreword

x v

Foreword

is associated with the possible equivalency of particles and fields, the dualism of which has been taken for granted as a tradition. Entrenched in the indoctrinated discipline of particle mechanics is that the mathematical limit for an infinite numbers of particles would make up the continuum. Such an idealization has no room in subatomic physics since the vast space of the universe is likely to remain unknown. Even less is the number of particles required to fill a finite volume of space if the size of the smallest particle remains not known. Idealized mathematical models, however, do serve the purpose for testing and conceiving physical ideas. Conversion of volume integrals to surface integrals and surface integrals to contour integrals made, respectively, by applying the divergence and Green theorem do suggest that higher-dimensional and lower-dimensional realm are related. Classical mechanics defines tractions in two-dimensions while body force would require the third dimension. The trade off between the surface and volume energy density is also well known in the Gibb’s theory of crystal nucleation for determining the size of grains in poly-crystals. As a matter of fact, why shouldn’t surface- and volume-based quantities be related? The only reason is that by tradition classical continuum mechanics has chosen to develop theories by disconnecting surface from the bulk. That is by taking the limit of ∆V/∆A→0, an expediency that has severely restricted the use of continuum theories for small bodies with large surface to volume ratios. This happens to fall in the size range of nano-electronics. The aforementioned limiting process also entails the indiscriminate use of the bulk or equilibrium material properties. Although the approximation does no harm to large structures but it introduces serious errors for describing the properties of small devices that are predominantly in the state of non-equilibrium. Quantum mechanics was developed precisely to account for the non-equilibrating behavior of the atoms and electrons that are constantly moving. Incidentally, the use of equilibrium theory of dislocation at the atomic scale is an over simplification. It is also surprising that non-equilibrium continuum mechanics theory has not received the attention it deserves. The existence of non-equilibrium solution in continuum mechanics can be and has been proved in a space of isoenergy density although the solution may not be unique on account of the uncertainties involved of enforcing

scale. For an open thermodynamic system, the air molecules next to the system will not occupy unique positions in two different experiments. Non-equilibrium mechanics theories encounter the same uncertainties as in quantum mechanics except that they are stated in different terms. One basic feature is that space, time and temperature would all be coupled and they tend to interact simultaneously. This can be made possible by keeping dV/dA finite in the theory such that the surface and volume density would be related. Constitutive relations are no longer pre-assumed throughout the continuum but rather it is derived for each point and instant of time simulating the true nature of non-homogeneity that is intrinsic of the non-equilibrium process. Incidentally, the rate of change of volume with surface dV/dA can also be quantized and expressed in terms of the de Broglie wavelength in quantum mechanics.

two identical boundary conditions in experiments at the atomic and /or molecular

xv i

Foreword

Scaling models attempt to circumvent some of the complexities encountered in non-equilibrium theories. This may be accomplished by segmentation of the scale range from macro to nano and from secs to femto secs. Each divided segments can be made sufficiently small to apply the condition of equilibrium. Singularity representation can be used to describe the combined effects of material, geometry and applied load which is well known in equilibrium mechanics. Stress singularity order serves as the scaling factor. The inverse square root of the distance from the crack tip, for example, can be regarded as a scaling factor between the local and global stresses and geometric quantities with reference to the crack. This factor would change for a concentrated load that yields an inverse of the distance from the singular point. This corresponds to the kernel of the Cauchy singular integral equation. The spirit of developing multiscale space-time models is to find simpler ways of addressing complex physical phenomena without violating the axioms of mechanics.

Large scale computations will not automatically distinguish the non-equilibrium

numerical inaccuracies, especially when the predictions are based on using equilibrium theories. Such a practice is not uncommon in the application of finite elements or similar techniques.

The holographic paradigm that has been receiving attention in modern physics may also benefit the development of multiscaling models. Physicists are faced with the same conceptual difficulties when the scale range is extended to the limits where continuum ceases to apply while particles step in. Under certain situations, physicists have argued that subatomic particles such as electrons are able to instantaneously communicate with each other regardless of the distance separating them. Each particle seems to know what the other is doing. This property is known to prevail in a hologram where every part contains all the information possessed by the whole. The whole in every part feature is similar to the brain memories which are not confined to a specific location but are dispersed throughout the brain. For the time being, at least an explanation can be offered for the coexistence of particles on the surface and the continuum enclosed by the surface. The holographic theory is that clouds of quarks and gluons on the two-dimensional surface can describe three-dimensional objects in the enclosed volume. Such a correlation plays the central role in the development of the string theory. Strings lying on a two-dimensional surface but with different thicknesses could be directed to the interior volume to form three-dimensional objects. The dualism of particle and continuum is thus made possible by the equivalence of particles on the surface to the object in the interior space. The interplay of surface and volume appears to be fundamental for the argument involving the co-existence of particles and fields. Even though the influence of dV/dA on multiscaling remains to be found, there are ample evidence from the past and the present to show that surface and volume effects cannot be separated. To iterate, equilibrium theories are vulnerable when applied to explain non-equilibrium phenomena. The dormancy effect of empi-rically designed miniaturized machines is the results of neglecting the dV/dA effects. That is frictional forces may no longer be independent of the surface area

character of the atoms and /or the electrons from the noise contributed from

xvii

Foreword

as postulates in the classical law of friction. The cooling and heating behavior of uniaxial specimens that undergo non-equilibrium, has also been a fact ignored for decades. Open thermodynamic system data which are not in equilibrium have been misused for closed thermodynamic system theories. Such inconsistencies cannot be left untold in the text books and classrooms if modern science and technology are expected to advance in the future. The articles on multiscaling in this volume hopefully will provide insights to the need for distinguishing the fundamental difference between the behavior of large and small bodies.

Because of the unique character of the articles in this volume, the contributors

One of the messages of this volume is that fundamental science and applied engineering need much closer collaboration. This includes the manufacturing of nano-size components. The editor is indebted to the contributors for completing their work on time. The assistance of Anita Ren and Leslie Li for revising the art work and formatting the articles are much appreciated.

Shanghai, China May, 2006 Editor

G. C. Sih

xviii

were given the choice to present their works at the 16th European Conference ofFracture on Failure Analysis of Nano and Engineering Materials and Structures.

Y.L. Baia, H.Y. Wanga *, M.F. Xiab,a, F.J. Kec,a

a LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, Chinab Department of Physics, Peking University, Beijing 100871, China

c Department of Applied Physics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China

Abstract

Two different spatial levels are involved concerning damage accumulation to eventual failure.

nucleation and growth rates of microdamage nN* and V*. It is found that the trans-scale length ratio c*/L does not directly affect the process. Instead,

two independent dimensionless numbers: the trans-scale one ( )* * *V including the * * *5 *

N c V including mesoscopic parameters only, play the key role in the process of damage accumulation to failure.

The above implies that there are three time scales involved in the process: the macroscopic imposed time scale tim = /a and two meso-scopic time scales, nucleation and growth of dam-age, ( )* *4

N Nt =1 n c and tV=c*/V*. Clearly, the dimensionless number De*=tV/tim refers to the ratio of microdamage growth time scale over the macroscopically imposed time scale. So, analogous to the definition of Deborah number as the ratio of relaxation time over external one in rheology. Let De be the imposed Deborah number while De represents the competition and coupling between the microdamage growth and the macroscopically imposed wave loading. In stress-wave induced tensile failure (spallation) De* < 1, this means that microdamage has enough time to grow during the macroscopic wave loading. Thus, the microdamage growth appears to be the predominate mechanism governing the failure.

Moreover, the dimensionless number D* = tV/tN characterizes the ratio of two intrinsic mesoscopic time scales: growth over nucleation. Similarly let D be the “intrinsic Deborah number”. Both time scales are relevant to intrinsic relaxation rather than imposed one. Further-more, the intrinsic Deborah number D* implies a certain characteristic damage. In particular, it is derived that D* is a proper indicator of macroscopic critical damage to damage localization, like D* ∼ (10–3~10–2) in spallation. More importantly, we found that this small intrinsic Deborah num-ber D* indicates the energy partition of microdamage dissipation over bulk plastic work. This explains why spallation can not be formulated by macroscopic energy criterion and must be treated by multi-scale analysis.

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

1

length ratio and the intrinsic one D = nDe = ac

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

* Corresponding author. E-mail address: why@lnm.imech.ac.cn (H.Y. Wang).

This can entail sample sizethree physical processes with three different rates, namely macroscopic elastic wave velocity a,

to failure and time scales and governing damage evolution“Deborah numbers”, coupling multiple space

(∼cm) to characteristic microdamage size c*(∼µm). Associated are

*

*

2007 Springer.

Y.L. Bai et al. 2

1. Introduction

From damage accumulation to failure, there prevails an important time-dependent phenomenon involving spallation in which failure occurs under transient loading like nano- to micro-seconds. Experimental observations suggest a time-integral criterion for spallation [1],

*( 1) × t = K (1)

where σ and σ* are stress and a stress threshold respectively, t is the load dura-tion, υ ν and K are two parameters. This criterion indicates that the critical stress to spallation is no longer a material constant, but a variable depending on its load-ing duration. Furthermore, since the power exponent υ in the criterion is usually neither 1 nor 2, the criterion implies neither momentum nor energy criteria macro-scopically [1-3]. Comprehensive and critical reviews on spallation have been made [4-6]. It is stressed that “the continuum models based on the statistical nucleation and growth of brittle and ductile fracture appear to be an attractive approach, espe-

grain-boundary microcracks involve the cooperative interactions of propagating cracks. Insight into such processes is required from the perspective of stochastic mechanics and from computer simulations of the debonding of assemblages of grains”.

It follows that spallation is a typical process with coupled multiple space and time scales. At least, there are two length scales: the sample size at macroscopic level and the microdamage size at mesoscopic level. On the other hand, there are three time scales: the stress wave loading duration macroscopically, the mesoscopic nucleation time and growth time of microdamage. So, spallation represents an illustrative example with multiple space and time scales.

2. Statistical Microdamage Mechanics

The general evolution equation of microdamage number density is [8]

( )Ii

Ni 1 i

n Pnn

t p=

∂ ⋅∂ + =∂ ∂

(2)

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

–

cially with a framework which provides some forms of a continuum cumulative-damage description of the evolving fracture state” [6]. Recently, the workin [7] suggests that “dynamic failure by the growth and coalescence of

3

0f

cN 0

00

cN 0

c0

n (c ; )

V(c,c ; )n(t,c; ) =

n (c ; )

V(c,c ; )

(3)

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

f

0f 0

c c

o oc c

dc dc t = =

V(c,c ; ) V(c,c ; ) (4)

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

The relation between continuum damage D and the number density of micro-damage n is

0D(t, ) n(t, , c) dc

∞= ⋅ τ ⋅x x (5)

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

f

0

0

c

N 0 00 c

N0

N0

D vD f

T X

n (c ; ) (c)dcdcn (c; ) (c)dc 1

n (c; ) (c)dc

∞

∞

∞

∂ ρ ∂+ =∂ ρ ∂

σ τ= σ τ ⋅ +

σ τ

(6)

where f is the dynamic function of damage, which represents the statistical average effects of nucleation and growth of microdamage on continuum damage evolution and ′conventional field equations of continuum, momentum and energy,

vT X

∂ε ∂=∂ ∂

(7)

0

v

T X

∂ ∂σρ =∂ ∂

(8)

′

0 f,0dc , c<c

dc , c>c0 f,0

“Deborah numbers”, multiple space and time scales

=d dc . Combining the damage field relation in Eq. (6) with the following

Y.L. Bai et al.

2diss

0 v 2

ec

T T X

∂∂θ ∂ θρ = + λ∂ ∂ ∂

(9)

a coupled trans-scale framework is formed for the damage evolution [13-14], where ρ0 is density of intact material, λ is heat conductivity, θ is temperature and ediss is the energy dissipated in the material element.

In accord with Π theorem in dimensional analysis, 10 parameters involved in spallation (see Table 1), can form 6 independent dimensionless numbers. However, when Eqs. (6-9) are non-dimensionalized it is found that there are 5 independent quantities only.

Table 1. 8 parameters involved in spallation parameter symbol dimension

Sample size LDensity ρ ML–3

Elastic wave speed a LT –1

Characteristic stress σY ML–1T–2

Impact velocity vf LT –1

Heat conductivity λ MLT –3θ–1

macroscopic

Heat capacity cV L2T–2θ 1

Characteristic nucleation rate of microdamage nN* L–4T–1

Characteristic growth rate of microdamage. V* LT –1mesoscopic Characteristic size of microdamage c* L

In the 5 dimensionless numbers, there are 3 conventional macroscopic ones: the well-known Mach number, damage number and Fourier number, related to contin-uum, momentum and energy equations respectively. Consider the other two

trans-scale dimensionless numbers: the imposed Deborah number*

**

acDe

V= and

the intrinsic Deborah number* *5

* N*

n cD

V= .

For a group of Al alloy spallation tests, ∼ (5-10)mm, vf ∼ 102 ms 1 c* ∼4.27⋅10–6 m, V* ∼ 5.96 ms–1 and nN

* ∼ 5.22⋅104 mm–3µm–1µs–1, derivation leads to De* ∼ 1 and D ∼ (10–2 ∼ 10–3) [10-12].

Noticeably, the ratio of two length scales on meso- and macro-levels c*/ does not

appear here. Actually, the imposed Deborah number *

**

acDe

V= is a combination of

two ratios: the size scale ratio c*/ and the ratio of two velocities V*/a. Also, the im-posed Deborah number De* is a unique trans-scale dimensionless parameter, since elastic wave speed a and sample size are macroscopic parameters whereas mi-crodamage size c* and microdamage growth rate V* are mesoscopic ones. This is

3. Deborah Numbers, Coupling Multiple Space and Time Scales

–

–

4

*

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

The intrinsic Deborah number* *5

* N*

n cD =

V characterizes the rate ratio of the

two intrinsic mesoscopic processes: nucleation over growth. Actually, D*= tv/tN,where tV = c*/V* and ( ) 1* *4

N Nt n c−

= are the growth and nucleation time scales re-spectively. Note that the characteristic nucleation time tN is not the time needed for an individual microdamage to nucleate, but the time to form a characteristic nucle-ated damage fraction, since *

ND means the damage fraction merely due to nuclea-tion in unit time.

*4 3N

N N

3N N

1 1tn c [(n c )c ]

1 1(N c ) D

= =

= =(10)

Above all, in the case study of spallation, there are three time scales: the macro-scopic imposed time scale tim= /a∼10–6 s and two meso-scopic time scales, growth time scale tV = c*/V*∼10–6 s and the nucleation time scale ( ) 1* *4

N Nt n C−

= ∼10 3 s.These lead to De* ∼1 and D ∼ (10–2 ∼ 10–3) too. However, this is not the whole story of Deborah numbers.

4. The Significance of Deborah Numbers in Failure

4.1 Imposed Deborah number governs the failure process

Note that the imposed Deborah number De*= tV/tim represents the competition and coupling between the microdamage growth and the macroscopically imposed wave loading. Also, in the concerned spallation case De*∼ 1, this indicates that mi-crodamage has enough time to grow during the macroscopic wave loading and then microdamage growth may be the predominate mechanism governing spallation.

*

istic damage, since Eq. (5) can be written as

−

and damage localization 4.2 Intrinsic Deborah number signifies characteristic damage

5“Deborah numbers”, multiple space and time scales

*Turn to the intrinsic Deborah number D . Firstly, D implies a certain character-

* * ***

* * *

*

Y.L. Bai et al. 6

f

0

* *5 3c

NN 0*0 0 c

n c cD n(c) dc n dc dc

VV

∞ ∞= ⋅ τ ⋅ =

(11)

All variables with bar above denote non-dimensionalized and normalized ones, hence the preceding dimensionless combination, i.e. the intrinsic Deborah number D , represents the magnitude of damage D. In the concerned spallation, this means a characteristic damage is of D ∼ (10–2 ∼ 10–3) .

Also, D* is a proper indicator of critical damage to localization. Damage local-ization can be formulated as follows[15]

DDX[ ] )T T

D DX

∂∂ ∂∂∂ ∂≥∂∂

(12)

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

f f (D, )= σ (13)

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

D

ff

D≥ (14)

Where D

ff

D

∂=∂

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

and irrelevant to any specific conditions of tests. Combining the damage localization condition in Eq. (14), the definition of dy-

namic function of damage in Eq. (6) and the expression of damage in Eq. (5), a critical damage to localization can be derived:

f N 0f N 0 0 0* *5 0 0

N 0 fc *

f N 0 00

*

(c )n (c )(c )n (c )dc dc

n c V(c , c )D

V (c )n (c )dc

D O(1)

∞ ∞

∞

ττ ⋅≈ ⋅

′τ

= ⋅

(15)

This once more is the intrinsic Deborah number D*.As above, the dimensionless combination preceding the normalized integrals in-

dicates the magnitude of the critical damage to localization. Compare the obtained

)))

))

*

*

result with experiments. As noted before, experiments gave * *5

* N*

n cD

V= ∼ (10–2 ∼

10–3). According to the localization condition in Eq. (14), there results in the criti-cal damage to localization Dc ∼ 4 10–3 [12]. Clearly, the intrinsic Deborah number D* does characterizes the magnitude of the critical damage Dc. Some simulations (Fig. 1) demonstrate that the intrinsic Deborah number does signify a certain char-acteristics in damage localization.

In the numerical study, a fixed De = De /D = 65.9 is taken. The most localized distribution occurs in the case of (De = 0.151 equivalent to D = 0.0023), whilst the case (De = 0.415 equivalent to D = 0.0063) demonstrates hardly localized distribution. This is in agreement with the analysis that lower D indicates lower threshold of critical damage to localization. In Fig. 1, is the plate thickness (sample size).

4.3 Intrinsic Deborah number signifies partition of energy dissipation

The energy partition will be clarified by examining the energy equation [16-17]. After splitting the dissipation term into damage and plastic ones, Eq. (9) may be rewritten as

p 2

0 V 2T T T X

∂θ ∂ε ∂Σ ∂ θσ + γ + λ∂ ∂ ∂ ∂

where Σ and γ are the total surface area of microdamage and corresponding equivalent surface energy in unit volume respectively. Certainly, the partition of plastic dissipation (the first term on the right hand side) and the damage one (the second term) are of our interest. Again, the dimensionless energy equation is:

p 2*

2D

T T T Y

∂θ ∂ε ∂Σ ∂ θ= σ + + Ψ∂ ∂ ∂ ∂

(17)

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

a~5000m/s, 30 V

0 V

10v a

1c a a

−λ ρλ κΨ = = = = <<ρ

. This implies that spal-

lation appears to be adiabatic. Now, the energy dissipation involved in spallation consists of two parts only:

plastic dissipation and damage dissipation. Examine the term of damage dissipation in more details. Approximately, γ can be estimated with (σYεY c ) and Σ with (nN c 4/V ) whereas the plastic dissipation with σYεY. therefore, the dimen-sionless and normalized combination preceding the damage dissipation term

7

ρ =c

“Deborah numbers”, multiple space and time scales

3

*

* *

* *

*

*

*

***

*

(16)

Y.L. Bai et al. 8

T

∂Σγ∂

is (σYε Y c ) nN c 4/V / σYε Y = nN c 5/V = D as shown in Eq. (17).

It becomes clear that the energy dissipation involved in spallation consists of two very different parts: plastic dissipation in O(1) whilst damage dissipation in O(D ), see Eq. (17). It can be said that the temperature rise in spallation is mainly due to plastic dissipation, since D <<1. Fig. 2 gives a numerical result of the energy parti-tion in spallation. One can notice that the damage dissipation varies around the magnitude of the intrinsic Deborah number D .

It is clear from Fig. 2(a) that the temperature increments induced by damage is much less that that induced by plastic dissipation. And the ratio of temperature in-crements due to damage over plastic dissipation varies around the value of D , as shown in Fig. 2(b).

5. Conclusion

Two independent dimensionless numbers: the trans-scale imposed Deborah

number *

**

acDe =

V and the intrinsic Deborah number

* *5* N

*

n cD =

Vgovern the

process of damage accumulation to failure.

The trans-scale imposed Deborah number*

**

acDe =

V represents the competition

and coupling between the microdamage growth and the macroscopically imposed wave loading. In spallation De* < 1 means that micro-damage has enough time to grow during the macroscopic wave loading. Thus, the microdamage growth ap-pears to be the predominate mechanism governing the failure.

Fig. 1. The spatial distribution of damage normalized by its maximum.

*******

*

*

*

*

*

*

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

De* = 0.415

0.277

0.207

0.166

0.151

M = 0.0305S = 0.153De = 65. 9

Nor

mal

ized

dam

age

dist

ribut

ion

D/D

max

Normalized coordiante x/l

.

Fig. 2. The spatial distribution of temperature rise.

The intrinsic Deborah number* *5

* N*

n cD =

V characterizes the ratio of micro-

damage growth rate over nucleation. The intrinsic Deborah number D* implies a certain characteristic damage and is a proper indicator of critical damage to dam-age localization. More importantly, this small intrinsic Deborah number D* ∼ (10–2

∼ 10–3) indicates the energy partition of microdamage dissipation over bulk plastic work in spallation.

9“Deborah numbers”, multiple space and time scales

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

Normalized coordinate x/l

Plastic Deformation Induced Damage Evolution Induced Conductivity Induced Total Temperature Increment

(a) Temperature increment

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

D* = 0.00944

Rat

io o

f tem

pera

ture

incr

emen

t *10

2

(Dam

age

indu

ced/

plas

tic d

efor

m. i

nduc

ed)

Normalized coordinate x/l

(b) Ratio of temp. increment

Tem

pera

ture

incr

emen

t∆θ

Y.L. Bai et al. 10

This work is granted by the National Natural Science Foundation of China (NSFC No. 10432050, 10432040, 10372012, 10302029, 10472118) and Chinese Academy of Sciences.

References

[[1] Tuler FR and Butcher BM. A criterion for the time dependence of dynamic fracture. Int J Fracture Mech 1968; 4(4): 431-437.

[2] Shen LT, Zhao SD, Bai YL and Luo LM. Experimental Study on the Criteria and Mecha-nism of Spallation in an Al Alloy. Int J Impact Eng 1992; 12(1): 9-19.

[3] Zhu ZX. private communication 1985. [4] Curran DR, Seaman L, Shockey DA. Dynamic failure of solids. Phys Rep 1987; 147:

253-388. [5] Meyers MA. Dynamic Behaviour of Materials. New York: John Wiley & Sons Inc., 1994. [6] Grady DE. The spall strength of condensed matter. J Mech Phys Solids 1988; 36(3):

353-384. [7] Clifton RJ. Response of materials under dynamic loading. Int J Solids Struct 2000; 37(1-2):

105-113. [8] Bai YL, Ke FJ, Xia MF. Formulation of statistical evolution of microcracks in solids. Acta

Mechanica Sinica 1991; 7: 59-66. [9] Ke FJ, Bai YL, and Xia MF. Evolution of ideal micro-crack system. Science in China Ser.

[10] Han WS, Xia MF, Shen LT and Bai YL. Statistical Formulation and Experimental Deter-mination of Growth Rate of Micrometer Cracks under Impact Loading. Int J Solids Struct 1997; 34: 2905-2925.

[11] Xia MF, Han WS, Ke FJ and Bai YL. Statistical meso-scopic damage mechanics and dam-

145-173. [12] Bai YL, Bai J, Li HL, Ke FJ, Xia MF. Damage evolution, localization and failure of solids

subjected to Impact Loading. Int J Impact Engng 2000; 24: 685-701. [13] Bai YL, Xia, MF, Wei YJ, Ke FJ. Non-Equilibrium Evolution of Collective Microdamage

and Its Coupling with Mesoscopic Heterogeneities and Stress Fluctuations. In: Horie Y, Davison L, Thadhani NN, editors. High Pressure Shock Compression of Solids VI – Old Paradigms and New Challenges. Springer-Verlag, 2002, p. 255-278.

[14] Bai YL, Xia MF, Ke FJ and Li HL (2002), Closed trans-scale statistical microdamage me-chanics, Acta Mech. Sinica, 18(1)1-17.

[15] Bai YL, Xia MF, Ke FJ, Li HL (1997). Damage field equation and criterion for damage localization. In: Dordrecht, Wang R, editors. Rheology of Bodies with Defects. Kluwer

damage evolution. 2nd int Conf. Structural Stability and Dynamics, 2002. [17] Wang HY, Bai YL and Wei YJ. Analysis of spallation based on trans-scale formulation of

damage evolution. Acta Mech Sinica 2004; 20(4): 400-407.

[16] Wang HY Bai YL, Wei YJ. Analysis of spallation based on trans-scale formulation of

A 1990; 33: 1447-1459.

Academic Publishers, 1997, p. 55-66.

age evolution induced catastrophe. Advances in Mechanics (in Chinese) 1995; 25: 1-40,

Acknowledgements

formation in polycrystalline materials

J.S. Chen*, S. Mehraeen 5713 Boelter Hall, Civil & Environmental Engineering Department

University of California, Los Angeles (UCLA), Los Angeles, CA 90095

Abstract

This work aims to develop multi-scale mathematical formulation and computational algorithm for modeling microstructure evolution and wrinkling formation in polycrystalline materials. A multi-scale variational formulation based on asymptotic expansion method, as well as associ-ated numerical methods are presented. Proposed is a consistent homogenization method that yields a tangent operator with major symmetry. The homogenized material response tensors vary in space in the macroscopic structures. This feature captures the local instability when the re-gional deformation gradient reaches certain ill-conditioned (non elliptic) state. Moreover, it is shown that the conventional non interactive single-scale homogenization approach is incapable of capturing the localized winkling response.

1. Introduction

The early work of wrinkling prediction was developed under the framework of a single-scale continuum mechanics, for example, by examining the eigen properties

shifts in definiteness of the tangent stiffness have been used as the indicator of

anisotropic membranes has been studied based on a saturation function as a wrin-

ignore microstructural effects on the onset of wrinkling formation. Approaches considering homogenization of heterogeneous materials have been

investigated by another group of researchers. An asymptotic expansion approach

* Corresponding author.

G.C. Sih (ed.), Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from

11

Macro to Nano, 11–21.

sub-structuring approach with B-spline finite strip discretization has been em-

of stiffness matrix based on elastic plate theory [2], elastoplasticity [9], and plate buckling on elastic foundation [8]. The onset of wrinkling has been analyzed using

ployed [2] to predict the buckling stresses of rectangular sandwich plates. The

wrinkles initiation [11]. The wrinkling wave function has been investigated based on thin-shell theory, forming theory, and the energy principle [17]. The wrinkling of

kle indicator [4]. However, the above continuum mechanics based methods entirely

the bifurcation functional [10] based on the Donnell-Mushtari-Vlasov shell theory[3], and by the sign of natural strain based on membrane theory [4]. A

E-mail address: jschen@seas.ucla.edu (J.S. Chen).

A multi-scale formulation for modeling of wrinkling

2007 Springer.

J.S. Chen and S. Mehraeen

macroscopic constitutive tensor with consideration of microstructural deformation. Applied in [15, 16] is the micro-structure based homogenization method to com-posites under manufacturing process.

Despite considering micro-macro relation in these methods, nonlinearity of macroscopic structure has not been approximated thoroughly. In this work, asymp-totic formulation with consideration of the fully geometric and material nonlineari-ties in micro- and macro-structures is formulated, and a fully coupled multi-scale formulation by introducing an averaging field theory is developed to construct macroscopic material constituent where microscopic deformation is linked to mac-roscopic deformation at macrostructure integration points. The outline of this paper is as follows. Section 2 discusses multi-scale governing equations for large defor-mation. In Section 3, consistent homogenization based on an averaged macroscopic strain energy density computed over the microstructure is presented. This approach yields a fully symmetric tangential stiffness. A numerical example demonstrating the effectiveness of proposed method is studied in Section 4, followed by conclud-ing remarks in Section 5.

2. Micro-macro Governing Equations

Two different coordinate systems are introduced as shown in Fig. 1, i.e., ( )1 2X , X and ( )1 2Y = Y , Y which denote the macroscopic and micro-

scopic coordinates in undeformed configuration, respectively. Correspondingly,

Fig. 1. Micro and macro coordinate systems.

12

stress field. In the context of nonlinear analysis, a macro failure surface was de-fined [1] as the locus of the points corresponding to the loss of positivity in the

has introduced in [6] for a small strain multi-scale elastoplastic analysis. The ex-tension to large strain multi-scale analysis was introduced in [13] via averaging RVE

of periodic microstructures.

macroscopic and microscopic coordinates in deformed configuration are denotedby x and y , respectively. Here we assume a body is constructed by the assembly

X =

Wrinkling formation in polycrystalline materials 13

( ) [ ] ( ) [ ] ( ) ( )0 1 2i i iu X,Y =u X + u X,Y +O (1)

where is asymptotic parameter which can be viewed as the scale ratio (i.e.,

i iY = X , i iy = x ), [ ] ( )0iu X and [ ] ( )1

iu X,Y are coarse and fine scale components of rate of displacement, respectively. Further considering that

[ ] ( )0iu X varies linearly with respect to X over the microstructure (unit cell), [ ] ( ) [ ]( ) [ ]( )0 0 0i i j j i j ju X u X X u X Y≈ ∂ ∂ = λ ∂ ∂ , and the coarse-fine scale relation

[ ] ( ) ( ) [ ]( )1 0i kli k lu X,Y = Y u X∂ ∂ , where ( )kli Yα is the coupling function, the

displacement measured in microstructure coordinates, scaled by 1/ λ of Eq.

( ) ( )( ) [ ]0i ik jl j kli klu Y = Y + Y F (2)

where [ ] [ ]0 0kl k lF u X= ∂ ∂ . Let the equilibrium equation be defined in the unde-

formed domain XΩ with boundary h gX X XΓ = Γ ∪ Γ as

ji,J i XP b 0 in+ = Ω (3)

hji j i X

gi i X

P N h on= Γ (4)

where hXΓ and g

XΓ are natural and essential boundaries in the undeformed con-figuration, respectively, ( ) ( ) j, J

. . X= ∂ ∂ , jiP is first Piola-Kirchhoff stress, ibis the body force defined in X , ih is the surface force divided by the unde-formed surface area, and jN is the surface normal defined on XΓ . The incre-mental variational equation is

( )h

X X XX

i,J ijkl ijkl k,L i i i i i,J jiu D T u d u b d u h d u P dΩ Ω ΩΓ

δ + ∆ Ω = δ Ω + δ Ω − δ Ω (5)

where 2ijkl im jlmn kn ijkl jl ikD F C F , T S= = δ , 2

jlmnC is second elasticity tensor,

mnE is Green-Lagrange strain, and ijS is the 2nd Piola-Kirchhoff stress. Intro-ducing scale decomposition to Eq. (5), there results

2.1 and holding X fixed, is obtained as

u g= Γon

Consider asymptotic expansion of displacement measured at the macroscopic coordinate:

J.S. Chen and S. Mehraeen

[ ] [ ]( )

[ ] [ ]

[ ] [ ]( ) [ ] [ ]( )[ ] [ ]

X

hX XX

0 1 0 1i i k k

ijkl ijklj j l l

0 10 1 0 1 i i

i i i i i i jij j

u u u uD T d

X Y X Y

u uu u b d u u h d P d

X Y

Ω

Ω ΩΓ

∂δ ∂δ ∂∆ ∂∆+ + + Ω∂ ∂ ∂ ∂

∂δ ∂δ= δ + λδ Ω + δ + λδ Ω − + Ω∂ ∂

(6)

where we used scale relation. By treating [ ]0iu and [ ]1

iu as independent varia-tions, taking average of Eq.(6) over the unit cell, and considering the limit of

0λ → , Eq. (6) gives rise to two decoupled equations. This provides the equation to solve for scale coupling function ( )kli Y , i.e.,

(7)

where YA is the area (or volume) of the unit cell in undeformed configuration, and Y is domain of unit cell. The other equation yields the macroscopic equi-librium equation:

[ ]( )

[ ] [ ]

[ ] [ ][ ]

[ ]

X Y

hX XX

0 0 1i k k

ijkl ijklY j l l

00 0 0i

i i i i jij

u u u1D T d d

A X X Y

uu b d u h d P d

X

Ω Ω

Ω ΩΓ

δ ∆ ∂∆+ + Ω Ω∂ ∂ ∂

∂δ= δ Ω + δ Ω − Ω∂

(8)

in which the body force is assumed constant over the unit cell. Introducing [ ] ( ) ( ) [ ]( )1 0i kli k lu X,Y = Y u X∂ ∂ in Eq. (7). This leads to

(9) Since the length scale of microstructure is considerably smaller than the macro-

scopic length scale, the residual due to nonlinearity in the microscopic equation in Eq. (9) can be ignored and it yields

14

X Y X Y

1 0 1 1i k k i

ijkl ijkl jiY j l l Y j

u u u u1 1D T d d P d dA Y X Y A Y

X Y X Y

1 0 1mnki m i

ijkl ijkl km nl jiY j l n Y j

Yu u u1 1D T dY d P d dA Y Y X A Y

∂ ∂ ∂ ∂∂ ∂ ∂ ∂

∂∂

∂∂∂∂∂

∂∂∂

Wrinkling formation in polycrystalline materials 15

[ ]( ) ( ) [ ]

( )Y Y

1 1mnki i

ijkl ijkl ijmn ijmnj l j

Yu uD T dY D T dY

Y Y YΩ Ω

∂α∂δ ∂δ+ = − +∂ ∂ ∂

(10)

Note that microstructure deformation iu must be used to calculate ijklD and

ijklT in Eq. (10). By substituting [ ] ( ) ( ) [ ]( )1 0i kli k lu X,Y = Y u X∂ ∂ into Eq. (8),

the following macroscopic (homogenized) governing equation is obtained

(11)

where ijmnD is the homogenized material response tensor, and ijmnT is the ho-mogenized geometric response tensor.

( )

( )Y

Y

mnkijmn ijkl km nl

Y l

mnkijmn ijkl km nl

Y l

Y1D D d

A Y

Y1T T d

A Y

Ω

Ω

∂α= δ δ + Ω

∂

∂α= δ δ + Ω

∂

(12)

from which coarse scale solution [ ]0iu is obtained. It can be easily identified that

both the homogenized material response tensor ijmnD and the homogenized geo-metric response tensor ijmnT do not possess major symmetry.

3. Consistent Homogenization for Large Deformation Problems

To recover symmetry property in the material and geometric response tensors, we first define the point-wise macroscopic strain energy density [0]W as the av-erage of strain energy density in the unit cell as

( )Y

[0]

Y

ij i j

1W W F d

A

F u X

Ω

= Ω

= ∂ ∂(13)

where F and W are the microscopic deformation gradient and strain energy

hX X XX

0 0 00 0 0i m i

ijmn ijmn i i i i jij n j

u u uD T d u b d u h d P dX X X

density in the unit cell, respectively, and [0]W is the point-wise macroscopic

∂∂∂∂∂∂