Challenges in computational materials science: Multiple ... · Challenges in computational materials science: Multiple scales, multi-physics and evolving discontinuities Rene´ de

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Computational Materials Science 43 (2008) 1–15

Challenges in computational materials science: Multiple scales,multi-physics and evolving discontinuities

Rene de Borst

Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 5600, 5600 MB Eindhoven, Netherlands

LaMCoS – UMR CNRS 5514, INSA de Lyon, 69621 Villeurbanne, France

Available online 4 September 2007

Abstract

Novel experimental possibilities together with improvements in computer hardware as well as new concepts in computational math-ematics and mechanics – in particular multiscale methods – are now, in principle, making it possible to derive and compute phenomenaand material parameters at a macroscopic level from processes that take place one to several scales below. Because of this quest to ana-lyse and quantify material behaviour at ever lower scales, more often (evolving) discontinuities have to be taken into account explicitly.Also, many applications require that one or more diffusion-like phenomena are considered in addition to a standard stress analysis.Accordingly, multiscale analysis, multi-physics and the ability to explicitly and accurately model evolving discontinuities are importantchallenges in computational science, and further progress on these topics is indispensible for an improved understanding of the behaviourand properties of materials. In this contribution we will give an impression of some developments.� 2007 Elsevier B.V. All rights reserved.

PACS: 02.70.Dh; 46.30.N2; 62.20.Fe; 62.40.+i

Keywords: Multiscale analysis; Multi-physics; Discontinuities; Finite element method; Fracture; Phase transformation; Porous medium

1. Introduction

The past two decades have seen a tremendous develop-ment in computational science. On one hand, computa-tional possibilities have vastly increased. Here, one canthink of the fast developments in computer hardware,where ever faster processors become available at an afford-able price. This, together with high-speed local area net-works have opened up the possibility of connectingnumerous low-cost personal computers. With standardisedsoftware protocols like MPI this has made massive parallelcomputing come within reach. At the same time, new con-cepts have been proposed and algorithms have been elabo-rated in computational mechanics and computationalmathematics that sometimes reduce computation timesby an order of magnitude, and for other cases enablecomputations that were not possible, or even conceivable,

0927-0256/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.commatsci.2007.07.022

E-mail address: [email protected]

hitherto. An example is the tracking of evolving disconti-nuities such as solid–solid phase boundaries, that movethrough the body during the deformation process. Widelyused discretisation concepts such as finite element methods,finite difference methods or finite volume methods havebeen designed to solve continuum problems and cannotreadily handle such evolving discontinuities inside bodies.On the other hand, major improvements in experimentaltechniques have made it possible to visualise physical pro-cesses and measure relevant parameters at fine scales thatwere not thought possible only a few decades ago. Exam-ples are (at different scales) Digital Image Correlation,Computer Tomography, Scanning Electron Microscopy,Atomic Force Microscopy, Nano-Indentation. Progresshas also been made at identifying parameters in constitu-tive models at different scales from measurements that stemfrom these advanced experiments [1,2].

From the technology side, demands have come for moreaccurate, more reliable and faster predictions and new

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2 R. de Borst / Computational Materials Science 43 (2008) 1–15

applications areas have arisen, for Micro-ElectricalMechanical Systems (MEMS), biomedical applications,new joining techniques such as friction-stir welding, andan increasing attention for durability and sustainability ofmaterials and systems. In virtually all cases, diffusion prob-lems have to be considered in addition to – and often evensimultaneous with – the solution of conventional stressproblems. Furthermore, the demand for new or improvedmaterials, e.g. thermal-barrier coating systems which enablehigher operating temperatures in engines with an ensuinghigher efficiency, has contributed to the drive towards thedevelopment of multiscale techniques. The latter class ofmethods aims at understanding the material behaviour ata lower level of observation, and with this understandingthese methods can be a major tool for developing and engi-neering new materials systems. When considering materialsat a lower length scale, the classical concept of a continuummore and more fades away. Where, at the macroscopic levelwe already have to take into account cracks, shear bands,Luders bands and Portevin-Le Chatelier bands, to mentionsome of the most commonly observed discontinuities, at alower level we encounter for example grain boundaries incrystalline materials, solid–solid phase boundaries as in aus-tenite–martensite transformations [3], and discrete disloca-tions [4,5]. Thus, the proper modelling of discontinuitiesplays a more and more important role in the analysis ofmaterial behaviour. As aforementioned, classical discretisa-tion methods are not well amenable for capturing disconti-nuities. Accordingly, next to multi-physics phenomena andmultiscale analysis, the proper capturing of discontinuitiesis a third major challenge in contemporary computationalmechanics of materials. A fourth challenge constitutes thefurther development of computational methods to assessthe probability of failure, which is driven by the require-ment that systems must become more reliable, and that itshould become possible to better assess their reliability [6].However, this fourth challenge will not be further pursuedin this contribution.

This contribution will strive to give an overview of somedevelopments in the three first-mentioned challenges –multi-physics, multiscale analysis, and the capturing ofevolving discontinuities – with particular reference to fail-ure of materials. We will start by a rather generic overviewof multiscale techniques, where a distinction will be madebetween upscaling and concurrent computing at multiplescales. Next, the capturing of discontinuities will beaddressed. Basically, two methods exist to handle them:by distributing them over a finite distance, or by treatingthem as true discontinuities. The first method has been asubject of much research in the past two decades and willbe discussed only briefly here. Recently, however, methodsthat treat discontinuities in a truly discrete manner haveemerged. They will be discussed and the most promisingof these techniques, which exploits the partition-of-unityproperty of finite element shape functions, will be examinedin more detail. Examples will be given of several types ofdiscontinuities, including fast crack propagation in hetero-

geneous materials and solid–solid phase transformations.The last section turns towards multi-physics. Here, all threechallenges come together, since we discuss how to analyseflow in cracks as well as in the surrounding porous mediumin a truly two-scale approach.

2. Multiscale methods

As briefly touched upon in the Introduction, there is animportant difference between upscaling methods and con-current multiscale computing. In the former class of meth-ods constitutive models at higher scales are constructedfrom observations and models at lower, more elementaryscales. By a sophisticated interaction between experimentalobservations at different scales and numerical solutions ofconstitutive models at increasingly larger scales, physi-cally-based models and their parameters can be derivedat the macroscopic scale. An example can by found in[7]. We consider also methods of computational homogeni-sation [8–10] to belong to this class.

In concurrent multiscale computing one strives to solvethe problem simultaneously at several scales by an a prioridecomposition. So far, almost only two-scale methods havebeen considered in practice, whereby the decomposition ofa problem is made into a coarse scale and a fine scale. Infact, this idea has, in an intuitive manner, been used inengineering for decades, if not for centuries. Also in com-putational science, large-scale problems have been solved,and local data, for instance displacements, forces or veloc-ities, have been used as boundary conditions for the resolu-tion of more detail in a part of the problem. Recent yearshave witnessed the development of multiscale methods incomputational science, which set out at coupling fine scalesand coarse scales in a more systematic manner. Consider-ing the generic nonlinear problem

AðuÞ ¼ B ð1Þwith A the nonlinear operator, u the unknown field and Bthe forcing term, we can transform this boundary-valueproblem into the following variational statement:

aðu; vÞ ¼ bðvÞ ð2Þ

with v the test function. A two-scale decomposition thenconsists of

u ¼ �u|{z}coarse scale

þ ~u|{z}fine scale

ð3Þ

Substitution into the variational statement (2) then yields:

að�u;�vÞ þ að~u;�vÞ ¼ bð�vÞ ð4aÞað�u;~vÞ þ að~u;~vÞ ¼ bð~vÞ ð4bÞ

Having made a rigorous decomposition of the probleminto fine scales and coarse scales, various approaches exist,which essentially differ in how to couple the fine scale to thecoarse scale, or, put differently, in finding a computation-ally inexpensive, but still accurate approximation toað~u;�vÞ. The Variational Multiscale Method proposed by

R. de Borst / Computational Materials Science 43 (2008) 1–15 3

Hughes and co-workers [11,12] is a most promising mem-ber of this family, but for instance, multigrid methodscan also be classified as multiscale methods. The same con-jecture can be substantiated for hp-adaptive finite elementmethods. Before discussing the different aspects that distin-guish multiscale methods in more detail, however, we wishto emphasise that they can be used in conjunction with var-ious discretisation methods, for instance, finite element, fi-nite difference or finite volume methods.

There are at least three features in multiscale methodswhich are crucial for the distinction between the differentapproaches: (i) the properties of the underlying physicalproblem, (ii) the processing with respect to the spatial scale,and (iii) the processing with respect to the temporal scale.Below we will discuss them in more detail.

Properties of the underlying problem. The underlyingproblem can be classified by questions like: is it a problemdescribed by different equations on different scales or mayone equation describe all relevant scales, how many scalelevels have to be considered, how strong is the couplingbetween the scales, is there a periodic microscale structure,and is the fine scale information required locally or glob-ally? An example of a problem where the same set of equa-tions describe the different scales is turbulence. Indeed, theNavier–Stokes equations are able to capture all relevantscales of Newtonian fluid flow, including the turbulent flowregime. On the other hand, molecular dynamics approachesutilise very different equations than those used by finite ele-ment methods for continua, and multiscale methods forcoupling them, e.g. the Bridging Domain Method [13], havea different character. Indeed, while the standard continuummodel has a local character, the potentials that are used inmolecular dynamics imply long-range forces, and thereforehave a nonlocal character. The coupling between bothdomains has to take this into account.

Spatial scale processing. The various approaches thataddress the processing of the spatial scales can be distin-guished by considering aspects like which kind of discreti-sation is applied (coarse to fine, fine to coarse, or separatedomains), how are the fine scales represented, how are theinter-element continuity on the fine-scale and on the large-scale levels taken care of, which are the assumptions on theboundaries of the fine-scales, how is information passedbetween the scales. Concerning the discretisation, thecoarse-to-fine-approach departs from a large-scale discret-isation. The fine-scale information is evaluated eitherlocally or globally in this case. The Discontinuous Enrich-ment Method [14] is believed to belong to this class. On theother hand, the fine-to-coarse approach departs from afine-scale discretisation from which the large-scale repre-sentation is derived, e.g., in multigrid approaches like Dis-crete-to-Continuum Bridging [15], which has been used tocouple fine-scale molecular dynamics to large-scale contin-uum mechanics. On the large-scale level, most methodsassume continuity between the individual elements,whereas on the small-scale level, for instance, the moleculardynamics approaches are discontinuous. Also for the com-

munication between the scales – a most essential part ofmultiscale methods – a wide variety of possibilities exists,ranging from variational projection as in the New Varia-tional Multiscale Method [16], to Lagrange multipliermethods as utilised in the Discontinuous EnrichmentMethod [14].

Temporal scale processing. With respect to the processingof the temporal scales, we distinguish between so-calledconcurrent, two-way coupled procedures and sequential,or serial procedures. In the latter case, at a given momentin time, the fine scales and the large scales can be treatedindependently. Another aspect is the space–time coupling:is one framework being used for the spatial dimensions aswell as for the time, or are they treated differently? Mostmethods discretise space and time in different ways, a nota-ble exception being the space–time finite element method.One can further distinguish between methods which usethe same time step on all scales and methods that use subcy-cling, as, e.g., is done in the Bridging Domain Method [13].

3. Discontinuities

When scaling down, discontinuities arise which need tobe modelled in an explicit manner. When the discontinuityhas a stationary character, such as in grain boundaries, thisis not so difficult, since it is possible to adapt the discretisa-tion such that the discontinuity, either in displacements orin displacement gradients, is modelled explicitly. An evolv-ing discontinuity, however, is more difficult to handle. Apossibility is to adapt the mesh upon every change in thetopology, as was done by Ingraffea and co-workers in thecontext of linear elastic fracture mechanics [17], and laterby Camacho and Ortiz [18] for cohesive fracture.

Another approach is to model discontinuities within theframework of continuum mechanics. A fundamental prob-lem is then that standard continuum models do not furnisha length scale which is indispensible for describing fracture,or, more precisely, they result in a zero length scale. Sincethe energy dissipated in the failure process is given per unitarea of material that has completely degraded, and since avanishing internal length scale implies that the area inwhich failure occurs goes to zero, the energy dissipated inthe failure process also tends to zero. Two approaches havebeen followed to avoid this physically unrealistic situation,namely via discretisation and via regularisation of the con-tinuum, see Fig. 1.

In the first approach, researchers have let the spacing ofthe discretisation take over the role of the missing internallength scale, so that the discontinuity in the left part ofFig. 1 is replaced by a displacement distribution as in theright-lower part of this figure, where w is the spacing ofthe discretisation. The idea is then to choose the discretisa-tion such that the spacing of the discretisation coincideswith the internal length scale that derives from the physicsof the process. Evidently, a good knowledge of the problemis required and solutions, including the proper choice forthe discretisation, are problem-dependent. Nevertheless,

Fig. 1. Application of regularisation and discretisation methods to adiscontinuity.


this approach has been used successfully to obtain insightin various issues in materials science [19–21].

Yet, this approach cannot be called a proper solution inthe sense that the mathematical setting of the initial valueproblem remains unchanged. Indeed, the introduction ofdegradation of the material properties in a standard,rate-independent continuum model – and therefore, theintroduction of a stress–strain curve with a descendingslope – can locally cause the governing differential equa-tions to change character. Without special provisions suchas the application of special interface conditions betweenboth domains at which different types of differential equa-tions hold, the initial value problem becomes ill-posed.Numerically, this has the consequence that the solutionbecomes totally dependent upon the discretisation [22,23].An example is shown in Figs. 2 and 3. It concerns a bar

Fig. 2. Top: Uniaxial bar subject to impact load. Bottom: Applied stressas function of time (left) and local stress–strain diagram (right).

0 10 20 30 40 50 60 70 80 90 100x [m]

1

2

3

4

5

6

stra

in [x

0.0

001]

Fig. 3. Strain profiles along the bar for 101 gr

composed of a porous, fluid-saturated material that isloaded by an impulse load at the left end [24]. Upon reflec-tion at the right boundary, the stress intensity doubles andthe stress in the solid exceeds the yield strength and enters alinear descending branch, Fig. 2. The results are shown inFig. 3 in terms of the strain profile along the bar at discretetime intervals. It is observed that a Dirac-like strain distri-bution develops immediately upon wave reflection, provingthat the governing equations change locally from hyper-bolic, as is normal in wave propagation, to elliptic, whichimplies that a standing wave develops. To furtherstrengthen this observation the analysis was repeated witha 25% refined mesh, which resulted in a marked increase ofthe localised strain (Fig. 3), and has been plotted on thesame scale as the results of the original discretisation. Weremark that in dynamic calculations of softening mediawithout regularisation, not only the spatial discretisationstrongly influences the results, but also the time discretisa-tion [24,25].

As a more rigorous solution, various regularisationmethods have been proposed, including nonlocal averag-ing, the addition of viscosity or rate dependency, or theinclusion of couple stresses or higher-order strain gradi-ents, see [26] for an overview. The effect of all these strate-gies is that the discontinuity shown in the left side of Fig. 1is transformed into the continuous displacement distribu-tion shown in the right-upper part of this figure. In contrastto discretisation strategies, the internal length scale w isnow set by the constitutive model for the solid material,and as soon as a sufficiently fine discretisation has beenadopted to properly capture this displacement distribution,the numerically calculated results only change in a sensethat is normally expected upon mesh refinement. It isemphasised that the above observations for discretisationand regularisation hold for any discretisation method,including finite element approaches, finite difference meth-ods, meshfree methods and finite volume methods [26].

In view of the fact that discretisation provides only apartial remedy to the ill-posedness of the underlying initialvalue problem, and that difficulties still persist with regular-isation strategies – notably the unresolved issue of addi-tional boundary conditions, the need to use very finemeshes in the zone of the regularised discontinuity, and

0 10 20 30 40 50 60 70 80 90 100x [m]

1

2

3

4

5

6

stra

in [x

0.0

001]

id points (left) and 126 grid points (right).


the need to determine additional material parameters fromtests that impose an inhomogeneous deformation field –have been a contributing factor to revisit the research into(more flexible) methods to capture arbitrary, evolving dis-continuities in a discrete sense.

At present, four such methods exist: zero-thicknessinterface elements, meshless or meshfree methods, the par-tition-of-unity method which exploits the partition-of-unity property of finite element shape functions [27] – alsoknown as the extended finite element method [28,29], anddiscontinuous Galerkin methods. Zero-thickness interfaceelements and the partition-of-unity method have becomethe most widely used methods in solid mechanics, andtherefore, we shall discuss them in some detail below.Meshfree methods were originally thought to behold agreat promise for fracture analyses due to the fact that thisclass of methods does not require meshing, and subsequentremeshing upon crack propagation, but the high costs, andespecially the difficulties to properly redefine the support ofa node when it is partially cut by a crack, have led to adecreasing interest [30–34]. However, they are of impor-tance, if only because out of the research into this classof methods, the analysis methods that exploit the parti-tion-of-unity property of finite element shape functionshave arisen, which are now believed to be the most viableoption for large-scale fracture analyses. Finally, the factthat discontinuous Galerkin methods [35] have been quitesuccessful in capturing shock waves in fluid flows, hasdrawn attention to their possible use in solid mechanics,especially for problems involving cracks [36], or for consti-tutive models that incorporate higher-order spatial gradi-ents [37]. In the former case, use of a discontinuousGalerkin formalism can be an alternative way to avoidtraction oscillations in the pre-cracking phase. In the lattercase, the fact that discontinuous Galerkin methods do notrequire inter-element continuity a priori, by-passes therequirement of C1-continuity on the damage or plastic mul-tiplier field which plague the implementation of many gra-dient models in continuous Galerkin finite elementmethods.

3.1. Zero-thickness interface elements

The classical way to represent discontinuities in solids isto introduce zero-thickness interface elements between twoneighbouring (solid) finite elements, e.g. Fig. 4 for a planar

Fig. 4. Planar interface element between two three-dimensional finiteelements.

interface element. The governing kinematic quantities ininterfaces are relative displacements: sunb, susb, sutb forthe normal and the two sliding modes, respectively. Whencollecting these relative displacements in a vector sub, theycan be related to the displacements at the upper (+) andlower sides (�) of the interface, u�n , uþn , u�s , uþs , u�t , uþt , by

sut ¼ Lu ð5Þ

with uT ¼ ðu�n ; . . . ; uþt Þ and L an operator matrix [38,39].The displacements contained in the array u are interpolatedin a standard manner, as

u ¼ Ha; H ¼ diag h h h h h h½ � ð6Þ

with h a 1 · N matrix containing the interpolation polyno-mials, and a the element nodal displacement array,a ¼ ða1

n; . . . ; aNn ; a

1s ; . . . ; aN

s ; a1t ; . . . ; aN

t ÞT, N being the total

number of nodes in the interface element. The relation be-tween nodal displacements and relative displacements forinterface elements is now derived from Eqs. (5) and (6) as:

sut ¼ LHa ð7Þ

where

LH ¼�h h 0 0 0 0

0 0 �h h 0 0

0 0 0 0 �h h

264

375 ð8Þ

which subsequently has to be transformed to the localcoordinate system of the integration point or node-set.

For analyses of fracture propagation that exploit inter-face elements, cohesive-zone models are used almost exclu-sively. In this class of fracture models, a discrete relation isadopted between the cohesive tractions at the discontinuitytd and the relative displacements:

td ¼ tdðsut; jÞ ð9Þ

with j a history parameter. After linearisation, necessary touse a tangential stiffness matrix in an incremental-iterativesolution procedure, one obtains:

dtd ¼ Tdsut ð10Þ

with the d-symbol denoting an infinitesimal increment, andT the material tangent stiffness matrix of the discrete trac-tion–separation law:

T ¼ otd

osutþ otd

ojoj

osutð11Þ

Fig. 5 shows some commonly used decohesion relations,one for ductile fracture (left), e.g. [40] and one for quasi-brittle fracture (right), e.g. [41]. For ductile fracture, themost important parameters of the cohesive zone model ap-pear to be the tensile strength ft and the work of separationor fracture energy Gc ([42]), which is the work needed tocreate a unit area of fully developed crack. It has thedimensions J/m2 and is formally defined as:

Gc ¼Z 1

sunt¼0

tn dsunt ð12Þ

f

tn

ft

n

u n

t

t

un

Fig. 5. Traction–displacement curves for ductile separation (left) andquasi-brittle separation (right).


with tn the normal traction across the fracture processzone. For more brittle decohesion relations as shown forinstance in the right part of Fig. 5, i.e., when the decohe-sion law stems from micro-cracking as in concrete orceramics, the shape of the traction–separation relationplays a much bigger role and is sometimes even moreimportant than the value of the tensile strength ft

([43,44]). Evidently, cohesive-zone models as defined aboveare equipped with an internal length scale, since the quo-tient Gc=E, with E a stiffness modulus for the surroundingcontinuum, has the dimension of length.

In cases where the direction of crack propagation isknown a priori, interface elements equipped with cohe-sive-zone models have been used with considerable success.Fig. 6 shows this for mixed-mode fracture in a single-edgenotched concrete beam. In this example the mesh has beendesigned such that the interface elements, which areequipped with a quasi-brittle cohesive-zone model, areexactly located at the position of the experimentallyobserved crack path [45].

To allow for a more arbitrary direction of crack propa-gation, Xu and Needleman [46] have inserted interface ele-ments equipped with a cohesive-zone model between allcontinuum elements. Although such analyses provide muchinsight, they suffer from a certain mesh bias, since the direc-tion of crack propagation is not entirely free, but isrestricted to inter-element boundaries. This has been dem-onstrated in [47], where the single-edge notched beam of

Fig. 6. Deformed configuration of a single-edge notched beam that resultsfrom an analysis where interface elements equipped with a quasi-brittlecohesive zone model have been placed a priori at the experimentallyknown crack path [45].

Fig. 6 has also been analysed, but now with a finite elementmodel in which interface elements equipped with a quasi-brittle decohesion relation were inserted between all contin-uum elements, Fig. 7.

Conventional interface elements have to be inserted inthe finite element mesh at the beginning of the computa-tion, and therefore, a finite stiffness must be assigned inthe pre-cracking phase with at least the diagonal elementsbeing non-zero. Prior to crack initiation, the stiffnessmatrix in the interface element therefore reads:

T ¼ diag½dn; ds; d t� ð13Þ

with dn the stiffness normal to the interface and ds and dt

the tangential stiffnesses. The undesired elastic deforma-tions can be largely suppressed by choosing a high valuefor the stiffness dn. However, this can lead to spurious trac-tion oscillations in the pre-cracking phase for high stiffnessvalues [38], which may cause incorrect crack patterns. Anexample of an oscillatory traction pattern ahead of a notchis given in Fig. 8. When analysing dynamic fracture, spuri-ous wave reflections can occur as a result of the introduc-tion of such artificially high stiffness values prior to theonset of delamination. Moreover, the necessity to alignthe mesh with the potential planes of delamination, re-stricts the modelling capabilities.

3.2. Partition-of-unity method

Out of the research into meshless methods, a methodhas emerged in which a discontinuity in the displacementfield is captured exactly. It has the added benefit that itcan be used at different scales, from microscopic to macro-scopic analyses. The method exploits the partition-of-unityproperty of finite element shape functions [27]. A collectionof functions Ni, associated with nodes i, form a partition ofunity if

Pni¼1Ni ¼ 1 with n the number of discrete nodal

points. For a set of functions Ni that satisfy this property,a field u can be interpolated as follows:

u ¼Xn

i¼1

Ni �ai þXm

j¼1

wj~aij

!ð14Þ

with �ai the ‘regular’ nodal degrees-of-freedom, wj the en-hanced basis terms, and aij the additional degrees-of-free-dom at node i which represent the amplitude of the jthenhanced basis term wj. In conventional finite elementnotation we thus interpolate a displacement field as:

u ¼ Nð�aþW~aÞ ð15Þ

where N contains the standard shape functions Ni, Wcontains the enhanced basis terms wj, and a� and ~a collectthe conventional and the additional nodal degrees-of-freedom, respectively. A displacement field that containsa single discontinuity, Fig. 9, can be represented bychoosing [28,29]:

W ¼HCdI ð16Þ

a b

c d

e f

Fig. 7. Crack patterns for different discretisations using interface elements between all solid elements. Only the part of the single-edge notched beam nearthe notch is shown [47].

100 mm

20 mm

250 mm

P=1000 N

Fig. 8. Left: Geometry of symmetric, notched three-point bending beam. Right: Traction profiles ahead of the notch using linear interface elements withGauss integration. Results are shown for different values of the ‘dummy’ stiffness D = dn in the pre-cracking phase [38].

Γd

ΓdΓp

Γq

ΓtΓu

utp p

n

Ωp

q

p

p

Fig. 9. Boundary conditions and internal discontinuity Cd for body X.


Substitution into Eq. (15) gives

u ¼ N�a|{z}�u

þHCdN~a|{z}

~u

ð17Þ

Identifying �u ¼ N�a and ~u ¼ N~a we observe that Eq. (17)exactly describes a displacement field that is crossed by asingle discontinuity, but is otherwise continuous. Accord-ingly, the partition-of-unity property of finite elementshape functions can be used in a straightforward fashionto incorporate discontinuities in a manner that preservestheir discontinuous character.

From Eq. (15) we infer that the partition-of-unityconcept can naturally be conceived as a multiscalemethod [11]. We can formally show this by decomposingu as

u ¼ �uþ ~u ð18Þ

with

�u ¼ N�a ð19Þ


representing the coarse scale and

~u ¼ NW~a ð20Þrepresenting the fine scale.

It is emphasised that in finite element methods thatexploit the partition-of-unity property of finite elementshape functions to model discontinuities, the additionaldegrees-of-freedom cannot be condensed at element level,because it is node-oriented and not element-oriented. It isthis property which makes it possible to represent a discon-tinuity such that it is continuous at inter-elementboundaries.

The partition-of-unity property of finite element shapefunctions is a powerful way to introduce cohesive surfacesin continuum finite elements [48–50]. Using the interpola-tion of Eq. (17) the relative displacement at the discontinu-ity Cd is obtained as:

sut ¼ ~ujx2Cdð21Þ

and the tractions at the discontinuity are derived from Eq.(9). A key feature of the method is the possibility of extend-ing a (cohesive) crack during the calculation in an arbitrarydirection, independent of the structure of the underlying fi-nite element mesh.

When the discontinuity coincides with an element edge,the traditional interface element formulation is retrieved[39,51]. But, even though formally the matrices can coin-cide for the partition-of-unity based method and for theconventional interface formulation, both concepts are quitedifferent. Indeed, simulations of delamination using thepartition-of-unity property of finite element shape func-tions offer advantages. Because the discontinuity does nothave to be inserted a priori, no (dummy) stiffness is neededin the elastic regime. Indeed, there does not have to be anelastic regime, since the discontinuity can be activated atthe onset of cracking. Consequently, the issue of spurioustraction oscillations in the elastic phase becomes irrelevant.Also, the lines of the potential delamination planes nolonger have to coincide with element boundaries, so thatunstructured meshes can be used. Moreover, the methodcan be generalised to large displacement gradients in astraightforward and consistent manner [52].

3.2.1. Crack propagation in heterogeneous media

The physics of crack initiation and crack growth in a het-erogeneous quasi-brittle material are illustrated in Fig. 10

Fig. 10. Experimentally observed ‘diffuse’ crack pat

[53]. The heterogeneity of the material, i.e., the presenceof particles of different sizes and stiffnesses leads to a com-plex stress field where existing cracks branch (‘a’ in Fig. 10)and new cracks nucleate (‘b’). Smeared crack models do notproperly capture these processes of crack initiation, growth,coalescence and branching, because essential features arelost in the smoothing process. This observation also par-tially holds for regularised smeared models that have beenenhanced with a capability to form true discontinuities inthe later stages of the deformation process [54–56].

More detail is preserved if the initiation, growth andeventual coalescence of the cracks are modelled at the mes-oscopic level of observation in Fig. 10 separately. Hithertothis could not be carried out, not only because of the highcomputational effort that this would require, but alsobecause a suitable numerical framework was lacking. Theexploitation of the partition-of-unity property of finite ele-ment shape functions can make such calculations feasible.As a further extension to a continuous cohesive crack thatruns through an existing finite element mesh without bias,one can define cohesive segments that can arise at arbitrarylocations and in arbitrary directions and allow for the res-olution of complex crack patterns including crack nucle-ation at multiple locations, followed by growth andcoalescence [57,58]. Although at present the goal of accu-rately simulating the complex cracking phenomena illus-trated in Fig. 10 has not yet been attained, significantprogress is being made.

A key feature of the cohesive segments approach is thepossible emergence of multiple cohesive segments in adomain. Consider a domain X which contains m disconti-nuities Cd,j, j = 1, . . . ,m. Each discontinuity splits thedomain in two parts, denoted as X�j and Xþj , such thatX�j [ Xþj ¼ X. Generalising Eq. (17), the displacement fieldcan be written as the sum of m + 1 continuous displace-ment fields �u and ~uj [59]:

u ¼ �uþXm

j¼1

HCd;j~uj ð22Þ

with HCd;j separating the continuous displacement fields �uand ~uj.

When the criterion for the initiation of decohesion is metin one of the integration points in the domain, a cohesivesegment is inserted through this integration point. The seg-ment is taken to extend throughout the element to which

tern [53] and possible numerical representation.


the integration point belongs and into the neighbouringelements, see Fig. 11a. The magnitude of the displacementjump is determined by a set of additional degrees of free-dom which are added to all nodes whose support is crossedby the cohesive segment. The nodes of the element bound-ary that is touched by one of the two tips of the cohesivesegment are not enhanced in order to ensure a zero openingat these tips [48]. Subsequently, the evolution of the separa-tion of the cohesive segment is governed by a decohesionconstitutive relation in the discontinuity. When the crite-rion for the initiation is met at one of the two tips, thecohesive segment is extended into a new element, as dem-onstrated in Fig. 11b. The extension is straight within theelement, but does not necessarily have to be aligned withprevious parts of the segment, so that curved crack pathscan be simulated.

Each cohesive segment is supported by its own set ofadditional degrees of freedom. When two segments meetwithin a single element, the nodes that support the elementare enhanced twice, once for each cohesive segment. In thesituation depicted in Fig. 12a, segment A is only extendeduntil it touches segment B, which can be regarded as a freeedge. This implies that there is no crack tip, so that all fournodes of the element are enhanced. A special case is shownin Fig. 12b. When two segments approach as shown in thisfigure, they are simply joined.

Fig. 11. (a) A single cohesive segment in a quadrilateral mesh. The segment pasThe solid nodes contain additional degrees of freedom that determine the magnare influenced by the cohesive segment. (b) A cohesive segment is extended intthat have just been added to support the extension of the cohesive segment.

a

Fig. 12. (a) Interaction of two cohesive segments. Segment A is extended (dashethere will be no crack tip for segment A. Consequently, all four nodes of thesegment A (denoted by the gray nodes). (b) Two segments are connected (das

Because the crack is not taken as a single entity a prioriin the cohesive segments approach, the method can equallynaturally simulate distributed cracking which frequentlyoccurs in a heterogeneous solid. Thus, the cohesive seg-ments approach embraces both extremes, distributedcracking with crack nucleation, growth and eventual coa-lescence at multiple locations as well as the initiation andpropagation of a single, dominant crack without requiringspecial assumptions. It is only needed to specify the condi-tions for crack nucleation and for the crack propagationdirection, and a decohesion relation at the crack.

The specimen in Fig. 13 has been subjected to a high-rate shear loading [60], and has dimensions L = 0.003 mand W = 0.0015 m, with an initial crack with lengtha = 0.0015 m. It is made of an isotropic linear elasticmaterial with Young’s modulus 3.24 · 109 N/m2, Poisson’sratio 0.35 and density q = 1190.0 kg/m3. The correspond-ing dilatational, shear and Rayleigh wave speeds arecd = 2090 m/s, cs = 1004 m/s and cR = 938 m/s, respec-tively. The ultimate normal traction of the material is equalto 100.0 · 106 N/m2 and the fracture toughness is 700 N/m.The lower part of the specimen is subjected to an impulseload in the positive x-direction which is modelled as a pre-scribed velocity with magnitude v0 = 20 m/s and a rise timetr = 0.1 ls. At sufficiently high rates of loading, fracture inthis configuration takes place by cleavage cracking at an

ses through an integration point (�) where the fracture criterion is violated.itude of the displacement jump. The gray shade denotes the elements that

o a new element (dashed line). The gray nodes contain degrees of freedom

b

d line) until it touches segment B. Since this can be regarded as a free edge,element will be enhanced in order to support the displacement jump of

hed line).

Fig. 13. Geometry and loading conditions of the specimen for thedynamic shear failure test.


angle of approximately 60� to 70� with respect to the initialcrack.

The specimen is discretised using bilinear displacementquadrilateral elements. The specific length of the smallestelements, which are located in the vicinity of the initialcrack tip, is le = 15.0 lm. Based on this element size, thetime increment was set to Dt = 1.0 · 10�10 s. At the startof the simulation, the model has 11587 degrees of freedom.The initial crack is modelled as a traction free discontinuityand therefore, the crack tip is sharp.

The extension of the crack during the simulation isshown in Fig. 14 [58]. The crack starts to propagate att = 3.01 ls at an angle of roughly 65�, which is in agree-ment with experimental observations [60]. At t = 3.7 lsthe propagation of the crack slows down and changes to

Fig. 14. Position of crack at various time steps. The crack has been visualisedduring post-processing [58].

an angle of 50�. This small deviation is caused by one ofthe reflections of the initial compressive stress wave whichtravels through the specimen in the horizontal direction atthe dilatational speed. At t = 3.7 ls this wave passes theregion just to the right of the original crack tip for the thirdtime. The additional compressive stress changes the stressstate at the current crack tip and therefore the angle ofthe principal stress. Once this wave has passed, the cracktrajectory continues at an angle of approximately 65�.

3.2.2. Solid–solid phase boundaries

While discontinuities like cracks involve a jump in thedisplacement field, other physical phenomena exist forwhich the displacements at the discontinuity remain contin-uous, but for which the displacement gradient experiencesa finite jump. Typical examples are solid–solid phaseboundaries, e.g. between martensite and austenite, orbetween blades in twinned martensite, Fig. 15 – left [3].In such cases, the interface conditions are characterised by

srut ¼ c� nCdð23Þ

with c a nonzero vector. Instead of taking the Heavisidefunction HCd

at the discontinuity Cd as the enrichmentfunction wj, the distance function DCd

is substituted forwj [61]. This function is continuous at the discontinuityCd, but its normal derivative is discontinuous and equalto HCd

:

nCd� rDCd

¼HCdð24Þ

by splitting the elements that contain a cohesive segment in two elements

Fig. 15. Left: Twinning in martensite as an example of solid–solid phase boundary. Right: Plate under uniaxial tension with different stiffness moduli.


thus meeting condition (23). The enriched displacementfield is subsequently obtained by substituting wj ¼ DCd

intoEq. (15), so that:

u ¼ N�a|{z}�u

þDCdN~a|{z}

~u

ð25Þ

A simple example of a weak discontinuity is given in theright part of Fig. 15, which shows a plate subjected to auniaxial stress. Because of the different values of Young’smodulus on the left and the right parts of the plate, therewill be a discontinuity in the displacement gradient. Instandard finite element analysis this discontinuity is cap-tured by letting it coincide with the boundaries of C0-con-tinuous finite elements. However, as shown in the right partof Fig. 15 the enriched displacement interpolation of Eq.(25) can capture this weak discontinuity in an exact mannerwithout the need to align the boundaries of finite elementswith the discontinuity. Indeed, this concept can be takenfurther to the limiting case that one of Young’s modulivanishes. Then, the boundary of a structure can be mod-elled without the need to align finite element boundarieswith the structure boundaries. This advantage can beexploited to model complex geometries and microstruc-tures while avoiding the need to carry out a complicatedmeshing of the structure [62].

While the example of Fig. 15 is simple and can be solvedusing a standard interpolation, this is no longer the case forevolving discontinuities as solid–solid phase boundaries.Level-set functions [63–65] are the most common approachto track the evolution of moving discontinuities. The idea isthat the position of the discontinuity Cd coincides with thezero level set of a smooth, scalar-valued function /,

Cd ¼ fx 2 X : /ðx; tÞ ¼ 0g ð26ÞThe distance function of Eq. (24) can be chosen as level-setfunction: / ¼ DCd

, and its evolution is then governed bythe Hamilton-Jacobi equation:

_DCdðx; tÞ þ V nkrDCd

ðx; tÞk ¼ 0 ð27Þwith Vn the normal component of the propagation velocityof the discontinuity Cd. This equation needs to be solvedfor every time step, and its solution gives the position ofthe discontinuity. Subsequently, the enriched displacementinterpolation of Eq. (25) can be used to carry out the stressanalysis on a fixed grid.

4. Two-scale modelling of fluid flow in porous media

In the previous section it has been argued that the par-tition-of-unity approach can be conceived naturally as avariational two-scale method. In this section we will dem-onstrate this for a two-phase medium, and show how amodel for flow inside the discontinuity – the fine scale –can be coupled naturally to the flow and deformation inthe surrounding porous medium – the large scale. In thissense, this section brings together the focal points of thisarticle: multiscale modelling, numerical description ofevolving discontinuities, and flow and deformation inmulti-phase media.

4.1. Governing equations for a fluid-saturated

porous medium

The bulk is considered as a two-phase medium subjectedto the restrictions of small displacement gradients andsmall variations in the concentrations. Furthermore, theassumptions are made that there is no mass transferbetween the constituents, that convective terms and thegravity acceleration can be neglected, and that the pro-cesses which we consider, occur isothermally and quasi-statically. With these assumptions, the balances of linearmomentum for the solid and the fluid phases read, e.g.[24,66]:

r � rp þ pp ¼ 0 ð28Þ

with rp the stress tensor of constituent p. As in the remain-der of this paper, p = s,f, with s and f denoting the solidand fluid phases, respectively. Further, pp is the source ofmomentum for constituent p from the other constituent,which takes into account the possible local drag interactionbetween the solid and the fluid. Evidently, the latter sourceterms must satisfy the momentum production constraintP

p¼s;f pp ¼ 0. Adding both momentum balances, takinginto account the momentum production constraint, anddefining r = rs + rf, one obtains the ‘standard’ equilibriumequation for the mixture:

r � r ¼ 0 ð29Þ

Under the same assumptions as for the balance of momen-tum, one can write the balance of mass for each phase as:


_qp þ qpr � vp ¼ 0 ð30Þ

with qp the apparent mass density and vp the absolutevelocity of constituent p. We multiply the mass balancefor each constituent p by its volume ratio np, add themand utilise the constraint

Pp¼s;f np ¼ 1 to give:

r � vs þ nfr � ðvf � vsÞ þ nsq�1s _qs þ nfq

�1f _qf ¼ 0 ð31Þ

The change in the mass density of the solid material is re-lated to its volume change by

ða� 1Þr � vs ¼ nsq�1s _qs ð32Þ

with Ks the bulk modulus of the solid material, Kt the over-all bulk modulus of the porous medium, and a = 1 � Kt/Ks

the Biot coefficient [67]. For the fluid phase, a phenomeno-logical relation is assumed between the incrementalchanges of the apparent fluid mass density and of the fluidpressure p [67]:

Q�1 dp ¼ nfq�1f dqf ð33Þ

with Q�1 = (a � nf)/Ks + nf/Kf the compressibility modu-lus, and Kf is the bulk modulus of the fluid. InsertingEqs. (32) and (33) into the balance of mass of the totalmedium, Eq. (31), gives:

ar � vs þ nfr � ðvf � vsÞ þ Q�1 _p ¼ 0 ð34Þ

4.2. Weak form and micro–macro coupling

To arrive at the weak form of the balance equations, wemultiply the momentum balance (29) and the mass balance(34) by kinetically admissible test functions for the dis-placements of the skeleton, g, and for the pressure, f. Sub-stitution into Eqs. (29) and (34), using Darcy’s relation

nfðvf � vsÞ ¼ �kfrp ð35Þ

with kf the permeability coefficient of the porous medium,integrating over the domain X and using the divergencetheorem leads to the corresponding weak forms:Z

Xðr � gÞ � rdXþ

ZCd

sg � rt � nCddX ¼

ZC

g � tp dX ð36Þ

and

�Z

Xafr � vs dXþ

ZX

kfrf � rp dX�Z

XfQ�1 _p dX

þZ

Cd

nCd� sfnfðvf � vsÞtdC ¼

ZC

fnC � qp dC ð37Þ

Because of the presence of a discontinuity inside the do-main X, the power of the external tractions on Cd andthe normal flux through the faces of the discontinuity areessential features of the weak formulation. Indeed, theseterms enable the momentum and mass couplings betweenthe discontinuity – the small scale – and the surroundingporous medium – the large scale.

The momentum coupling stems from the tractionsacross the faces of the discontinuity and the pressure

applied by the fluid in the discontinuity onto the faces ofthe discontinuity. We assume stress continuity from thecavity to the bulk, so that we have:

r � nCd¼ td � pnCd

ð38Þ

with td the cohesive tractions, which are given by Eq. (9).Therefore, the weak form of the balance of momentumbecomes:Z

Xðr � gÞ � rdXþ

ZCd

sgt � ðtd � pnCdÞdC ¼

ZC

g � tp dC

ð39Þ

Since the tractions have a unique value across the disconti-nuity, the pressure p must have the same value at both facesof the discontinuity, and, consequently, this must also holdfor the test function for the pressure, f. Accordingly, themass transfer coupling term for the water can be rewrittenas follows:

�Z

Xafr � vs dXþ

ZX

kfrf � rp dX�Z

XfQ�1 _p dX

þZ

Cd

fnCd� qd dC ¼

ZC

fnC � qp dC ð40Þ

where qd = nfsvf � vsb represents the fluid flux through thefaces of the discontinuity.

To quantify the influence of the ‘micro’-flow inside thediscontinuity on the ‘macro’-scale, we recall the balanceof mass, which, for the ‘micro’-flow in the cavity reads:

_qf þ qfr � v ¼ 0

subject to the assumptions of small changes in the concen-trations and that convective terms can be neglected, cf. Eq.(30). We assume that the first term can be neglected be-cause the problem is monophasic in the cavity and thevelocities are therefore much higher than in the porousmedium, although this assumption is not essential, cf.[68]. With this assumption, and focusing on a two-dimen-sional configuration, the mass balance inside the cavitysimplifies to:

ovoxþ ow

oy¼ 0 ð41Þ

with v ¼ v � tCdand w ¼ v � nCd

the tangential and normalcomponents of the fluid velocity in the discontinuity,respectively, see Fig. 16. Accordingly, the difference in thefluid velocity components that are normal to both crackfaces is given by

swft ¼ �Z h

�h

ovox

dy ð42Þ

To proceed, the velocity profile of the fluid flow inside thediscontinuity must be known. Different possibilities exist,but here we follow Ref. [69], in which a Newtonian fluidwas assumed. Together with the balance of momentumfor the fluid in the discontinuity and integrating from

nΓd

Γdt

2h

y

x

Fig. 16. Cavity geometry.


y = �h to y = h, Fig. 16, we obtain the following velocityprofile:

vðyÞ ¼ 1

2lopoxðy2 � h2Þ þ vf ð43Þ

with l the viscosity of the fluid. The essential boundaryv = vf has been applied at both faces of the cavity, andstems from the relative fluid velocity in the porous mediumat y = ±h:

vf ¼ ðvs � n�1f kfrpÞ � tCd

Substitution of Eq. (43) into Eq. (42) and again integratingwith respect to y then leads to:

swft ¼2

3lo

oxopox

h3

� �� 2h

ovf

oxð44Þ

This equation gives the amount of fluid attracted in thetangential fluid flow. It can be included in the weak formof the mass balance of the ‘macro’-flow to ensure the cou-pling between the ‘micro’-flow and the ‘macro’-flow. Sincethe difference in the normal velocity of both crack faces isgiven by swst ¼ 2 _h, the mass coupling term becomes:

nCd� qd ¼ nfswf � wst

¼ nf2h3

3lo

2pox2þ 2h2

lopox

ohox� 2h

ovf

ox� 2 _h

� �ð45Þ

As in the preceding section, the interpolation of each com-ponent of the displacement field u of the solid phase is en-riched with discontinuous functions, cf. Eq. (17). Withrespect to the interpolation of the pressure p we note thatthe fluid flow normal to the discontinuity is discontinuous.Since the fluid velocity is related to the pressure gradientvia Darcy’s law, the gradient of the pressure normal tothe discontinuity is therefore discontinuous across the dis-continuity. Accordingly, the enrichment of the interpola-tion of the pressure must be such that the pressure itselfis continuous, but has a discontinuous first spatial deriva-tive. The distance function DCd

defined in Eq. (24) satisfiesthis requirement, and accordingly, all nodes whose supportis cut by the discontinuity hold additional pressure degreesof freedom such that:

p ¼ H�pþDCdH~p ð46Þ

where H contain the shape functions Hi used as partition ofunity for the interpolation of the pressure field p, and �p and

~p are the nodal arrays assembling the amplitudes that cor-respond to the standard and enhanced interpolations of thepressure field.

The choices for Ni and Hi are driven by modellingrequirements. Indeed, the modelling of the fluid flow insidethe cavity needs the second derivatives of the pressure, seeEq. (45). Hence, the order of the finite element shape func-tions Hi has to be sufficiently high, otherwise the couplingbetween the fluid flow in the cavity and the bulk will not beachieved. Further, the order of the finite element shapefunctions Ni must be greater than or equal to the orderof Hi for consistency in the discrete balance of momentumequation.

In a Bubnov–Galerkin sense we choose the test func-tions for the displacements and the pressures, g and f,respectively, in the same space as the interpolation func-tions for the displacements and the pressure, and after sub-stitution into Eqs. (39), (40) we require that the resultingequations hold for all admissible discrete test functions.This gives:Z

XrNTrdX¼ �Fext ð47aÞZ

XHCdrNTrdXþFinter¼ ~Fext ð47bÞ

�Z

XaHTmT _us dXþ

ZX

kfrHTrpdX�Z

XQ�1HT _pdX¼ �Qext

ð47cÞ

�Z

XaDCd

HTmT _us dXþZ

XkfrðDCd

HÞTrpdX

�Z

XQ�1DCd

HT _pdXþQinter¼ ~Qext ð47dÞ

where, for two dimensions, m = [1, 1,0]. The external forceand flux vectors are given by

�Fext ¼Z

CNTtp dC; ~Fext ¼

ZCHCd

NTtp dC ð48Þ

�Qext ¼Z

CHTnTqp dC; ~Qext ¼

ZCDCd

HTnTqp dC ð49Þ

The interfacial force vector Finter is discretised from Eq.(39) to give:

Finter ¼Z

Cd

NTtddC�Z

Cd

NTnCdpdC ð50Þ

In a similar fashion, the interfacial flux vector can be de-rived as:

Qinter ¼Z

Cd

HTnTCd

qd dC ð51Þ

where

nTCd

qd ¼nf

12lðnT

Cd~uÞ3ðtT

Cdrr�pÞ þ nf

4lðnT

Cd~uÞ2ðtT

Cdr�pÞ

� ðnTCd

~uÞtTCdðnfð _�uþ _~u=2Þ � kfr�pÞ � nfðnT

Cd

_~uÞ ð52Þ


Subsequently, the semi-discrete set of Eqs. (47a)–(47d),(48)–(51) has to be integrated over time. Since the resultingset of discrete equations is nonlinear due to the cohesive-zone model for the interface tractions and the momentumand mass coupling terms, an iterative procedure has to beapplied. Details of the time integration and the linearisa-tion of the discrete equations that is necessary when apply-ing a Newton–Raphson iterative procedure are given in[69].

As an example the initial value problem of Fig. 17 isconsidered, where a set of 10 cracks has been generatedin a random manner [69]. The block is 10 · 10 m2, whilethe length of the cracks ranges from one to three meters,and the fault angle varies from a = �10� to a = 30�. Forsimplicity sake, the tractions across the cracks have beenset to zero, and are stationary. In order to take the stresssingularity at the crack tips into account, singular functionshave been added to the solution in those elements whose

.

q .

q.q n =

n =

p =

n =0

0

0

qo

Fig. 17. Fractured block with boundary conditions.

Fig. 18. L2-norm of the pressure gradient in fractured block.

support contains a crack tip [28,29]. Fig. 18 shows theinfluence of the cracks on the norm of the pressure gradi-ent. The global fluid flow is strongly affected by the‘micro’-flows inside the cracks. From Fig. 18 it is observedthat the main effect is due to the two longest cracks. Evi-dently, the effect of a fault on the ‘macro’-flow increaseswith its length because more fluid can flow inside the cav-ity. For this simulation, a discretisation has been used thatis composed of 30 · 30 quadrilateral elements with bilinearshape functions.

5. Concluding remarks

Some challenges in computational mechanics have beenaddressed in this contribution. In particular the emergingconcept of multiscale analysis, which appears to becomea new paradigm in computational science, the importanceof accounting for one or several diffusion-like phenomenain addition to a stress analyses for many contemporaryproblems of materials science, and the necessity to trackand compute evolving discontinuities, which appear at avariety of scales. Only some directions have been given,and not even in an exhaustive manner. Indeed, it cannotbe avoided that a discussion like this is biased and incom-plete. For instance, the discussion has been limited to con-tinuum mechanics concepts, and important developmentsin molecular dynamics, coupling atomistics to continua[70,71] or coupling atomistics to discrete dislocations [72]have not been included.

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Challenges in computational materials science: Multiple ... · Challenges in computational materials science: Multiple scales, multi-physics and evolving discontinuities Rene´ de