multi scale crack

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2007; 71:14661482Published online 12 February 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2001

A multiscale projection method for macro/microcrack simulations

Stefan Loehnert, and Ted Belytschko

Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road,

Evanston, IL 60208, U.S.A.

SUMMARY

We present a new multiscale method for crack simulations. This approach is based on a two-scale

decomposition of the displacements and a projection to the coarse scale by using coarse scale testfunctions. The extended finite element method (XFEM) is used to take into account macrocracks as wellas microcracks accurately. The transition of the field variables between the different scales and the roleof the microfield in the coarse scale formulation are emphasized. The method is designed so that the finescale computation can be done independently of the coarse scale computation, which is very efficient andideal for parallelization. Several examples involving microcracks and macrocracks are given. It is shownthat the effect of crack shielding and amplification for crack growth analyses can be captured efficiently.Copyright q 2007 John Wiley & Sons, Ltd.

Received 8 December 2006; Revised 20 December 2006; Accepted 22 December 2006

KEY WORDS: multiscale; cracks; XFEM

1. INTRODUCTION

Multiscale methods offer great promise in modelling the effects of microstructural features such as

microcracks. Different concepts of multiscale methods are the variational multiscale methods [1],the homogenized Dirichlet projection method [2, 3], the FE2 method [4], domain decompositionmethods [5 7], multiscale methods based on homogenization techniques [8], and other concurrentmethods, e.g. [9]. When local effects in certain subdomains of a large structure need to be modelled,multiscale methods are attractive since standard homogenization techniques, which in general are

very useful for the bulk of the structure, usually fail to predict localizing phenomena accurately.

Using a multiscale method in those domains overcomes this difficulty since it is possible to

Correspondence to: Stefan Loehnert, Department of Mechanical Engineering, Northwestern University, 2145 SheridanRoad, Evanston, IL 60208, U.S.A.

E-mail: [email protected]

Contract/grant sponsor: Army Research Office; contract/grant number: W911NF-05-1-0049

Copyright q 2007 John Wiley & Sons, Ltd.


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A MULTISCALE PROJECTION METHOD FOR MACRO/MICROCRACK SIMULATIONS 1467

predict the localizing phenomena with a detailed microstructural computation. A method using

superimposed meshes for the improvement of solutions in local areas applied to linear elastostatics

can be found in [10].Two of the most common localization phenomena which can be observed in a variety of materials

are cracks and shear bands. For the investigation of crack propagation, stress or strain criteria in thevicinity of the existing crack tip or the stress intensity factors of the crack tip are used to predict

the crack propagation and its direction. In many ceramic materials it can be observed that around

the crack tip of an existing macrocrack, microcracks nucleate and develop. These microcracks

have a significant influence on the stress field in the vicinity of the macrocrack tip. They can

result in crack shielding or crack amplification and thus greatly influence the propagation of the

macrocrack. A brute force, single-scale analysis of such problems would require an extreme local

adaptive mesh refinement in the vicinity of each macrocrack tip. Refining to the needed scale

around such a local area would lead to poorly conditioned equations and high computational cost

which make such an approach awkward.

This paper presents a computationally efficient method for treating subscale (i.e. microscale)

behaviour for cracked bodies. The approach is based on exploiting the separation of scales typical

of microscale/macroscale problems. As shown here, such a scale separation enables the problem to

be treated in two parts: a macroscale model which embodies the overall effects of the microcracks,

and a microscale model which models detailed interaction of cracks. The novel contribution of

this paper is that we show that the acceptable accuracy can be obtained by using the coarse mesh

test functions in conjunction with the effects of the microcracks.

Our method is formulated within the context of the extended finite element method (XFEM)

[11, 12] which makes it possible to perform crack simulations with an arbitrary number of randomlydistributed cracks without remeshing whenever a crack propagates. XFEM has been applied to large

arrays of cracks at a single scale by Budyn et al. [13] and in microcrack/macrocrack strategies byGuidault et al. [14]. The synthesis of XFEM with level sets we use here was proposed by Stolarskaet al. [15] and Belytschko et al. [16]. Improvements of XFEM for crack applications have beenproposed by Xiao and Karihaloo [17] and Laborde et al. [18]. Extensions to three dimensions arereported by Gravouil et al. [19, 20] and Areias and Belytschko [21]. Applications to shear bandsare given by Areias and Belytschko [22] and Samaniego and Belytschko [23]. Applications todislocations are given in Ventura et al. [24] and Gracie et al. [25]. Thus, the method proposedhere is also applicable to localized shear bands and dislocations.

In Section 2, the multiscale method for crack simulations is derived. In particular, this involves the

transition of the field variables between the different scales and consequently the decoupling of the

computations for different scales as well as the separation of the structural details for each scale. In

Section 3, the discretized equations and the proposed projection of the field variables are presented.

Section 4 shows the solution procedure. In Section 5, the multiscale method is applied to several

tests for which analytical results can be found in literature and to some crack shielding/amplification

problems involving a larger number of randomly distributed microcracks around the crack tips of

a macrocrack. Section 6 gives some conclusions and an outlook to future work.

2. THE MULTISCALE STRATEGY

For simplicity, here we restrict ourselves to two scales even though the multiscale strategy described

in the following can be extended to an arbitrary number of scales. We assume that the behaviour

Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 71:14661482

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1468 S. LOEHNERT AND T. BELYTSCHKO

Figure 1. Fine scale (microscale) and coarse scale (macroscale).

of the problem, either because of its geometry or the evolution of the cracks, exhibits two scalesof behaviour. We denote by l0 the scale of coarse scale features and by l1 the scale of fine

scale features. Furthermore, we assume that the geometry and response are characterized by scale

separation, i.e. that the fine scale features are significantly smaller than the coarse scale features,

i.e. l0l1.Let the domain of the structure under consideration be 0. We denote the boundary of 0

by *0 and subdivide it into complementary boundaries *0t and *0u where the tractions and

displacements, respectively, are prescribed. The domain 0 consists of parts where a fine scale

analysis is not necessary and other parts 1 0 where the fine scale behaviour has great influenceon the coarse scale behaviour (see Figure 1). It is assumed that along the boundaries *1, thefluctuations of the field variables due to the microstructure are negligible. This strongly affects

the choice of the area within which the detailed microstructure has to be resolved accurately. On

the coarsest scale, only cracks which are longer than a typical finite element size are explicitlyconsidered. In the fine scale model, the microcracks and all the discontinuities on coarser scales

are taken into account.

We consider a body subjected to body forces f so that the equilibrium equation is

div(r) + f= 0 (1)

In addition, the traction boundary conditions on *0t and the crack surfaces must be satisfied.

r n= t on *0t (2)

r n

=t

c

on i iD (3)where n is the unit normal to the surface pointing out from the body. The coarse scale model

includes macrocracks which correspond to discontinuities 0D in the displacement field, while

the microcracks in 1 corresponds to discontinuities 1D . For convenience, we let

iD =

iiD ,

i = 0, 1.


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The macro displacement (coarse scale displacement) is given by

u0(X)=u0C(X) + u0D(X) (4)

where u0C(X) is the continuous part of the macro displacement and u0D(X) is the discontinuouspart of the macro displacement. We require that u0C U0C and u0D U0D where

U0C = {u|u H1(0), u= u on *0u} (5)

U0D = {u|u H1(0\0D),u= 0 on *0u} (6)

In 1, the displacement field is given by

u1 =u0 + u1 (7)

where the micro displacement field u1

is given by

u1(X)= u1C(X) + u1D(X) (8)

where u1C and u1D are, respectively, continuous and discontinuous parts of the micro displacement

field, u1C U1C and u1D U1D and

U1C = {u|u H1(0),u= 0 on *1,u= 0 in 0\1} (9)

U1D = {u|u H1(0\{0D 1D}),u= 0 on *1,u= 0 in 0\1} (10)

We include both cohesive cracks and cracks with singular crack tips in the formulation, so that

crack bridging can be treated, cf. Bao and Suo [26].The weak form on 0 will be based on the macro displacement field, i.e. it will resolve

equilibrium on the coarse level. For this purpose, we employ as test functions the functions from

the macro displacement space U0C U0D. The weak form is obtained by multiplying the equilibriumequation by the test function g0(X)

0

div(r(u0 + u1)) g0 d=

0f g0 d+

*0t

t g0 d*+

0D

tc 'g0D( d (11)

where

g0(X)

=g0C(X)

+g0D(X) (12)

and g0C U0C, g

0D U

0D with

U0C = {g|g H1(0), g= 0 on *0u} (13)

U0D = {g|g H1(0\0D), g= 0 on *0u} (14)


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Integration by parts yields

0r(u0 + u1) : gradsym(g0) d

=

0f g0 d+

*0

t g0 d*+

0D

tc 'g0D( d g0C U

0C, g

0D U

0D (15)

where tc is the cohesive traction across the cracks.

Note that the microcrack cohesive tractions are eliminated from the equilibrium equation because

the macroscale test field is continuous across the microcracks, i.e. the discontinuities 1D. Thus, the

use of the coarse scale test functions eliminates the role of the cohesive tractions on the microcracks

from the coarse scale equilibrium equations. However, their effect is still included in the sense

that u1 depends on the microcracks. When the cracks are traction free, they do not appear in the

coarse scale work expression, but it is easy to verify that the EulerLagrange equations (15) are


=

0f g0 d+

*0

t g0 d* g0C U0C, g

0D U

0D (16)

with r n= 0 on 0D.For the purpose of obtaining the fine scale equilibrium equations, we define the micro test

functions

g1(X)=g1C(X) + g1D(X) (17)where g1

C U

1

Cand g1

D U

1

D. We then multiply the equilibrium equation by these test functions,

which yields, after integrating by parts, that


=

1f g1 d+

0D

tc 'g1D( d+

1D

tc 'g1D( d (18)

Note that cohesive forces of both the coarse scale and the fine scale cohesive cracks enter the fine

scale equations. As for the macroscale, when the cohesive forces vanish, the last terms do not

appear. In this case, the EulerLagrange equations are

1r(u0 + u1) : gradsym(g1) d=

1f g1 d (19)

with natural boundary conditions r n= 0 on 0D 1D, i.e. the tractions vanish on the macrocrackand microcrack surfaces. In the above, we did not account for crack closure, which would entail

either changing (15) and (18) to variational inequalities or checking for the absence of crack

closure after the computation. We did the latter.


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Figure 2. Coarse scale mesh and fine scale mesh.

3. DISCRETIZATION

The domain 0 is subdivided into elements 0e . The subdomain 1 is subdivided into elements

that are congruent to the elements 0e , so that for any element 0e on the coarse scale there exists a

set of elementsSe so that 0e =

Se

1 (see Figure 2). The fine scale and coarse scale domains

will be treated separately, with the fine scale solution only providing the fine scale stress field

to the coarse scale equations, see Equation (15). Instead of discretizing the field u1, we directly

discretize u1. Following Moes et al. [12] the displacement approximations can then be written inthe form

u0h =n0n

I=1N0I

u0I +

nenrj=1

fja0j I (20)

u1h =

n1nI=1

N1I

u

1I +

nenrj=1

fja1j I

(21)

where fj are the enrichment functions chosen to account for crack behaviour, N0

Ithe shape

functions for the coarse scale, N1I the shape functions for the fine scale, u0I and u

1I the standard

nodal displacement degrees of freedom for the coarse and fine scale, respectively, and a0j I and a1j I

are the coarse scale and fine scale enrichment parameters.

The nodes belonging to elements within which a crack ends are enriched with the crack tip

enrichment functions developed by Fleming et al. [27]

f1(r,)=

rsin

2

(22)

f2(r,)=

rcos

2

(23)


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f3(r,)=

rsin

2

sin() (24)

f4(r,)

=

rcos 2 sin() (25)where

r=(x xtip)2 + (y ytip)2 (26)

= arctan

y ytipx xtip

(27)

and is the angle of the crack with the x -axis at the crack tip which is located at (xtip, ytip). If the

element is completely intersected by a crack, the element nodes are enriched with the Heaviside

step function

f1(n)= H((n))=+1, (n)0

1, (n)


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with

giI =

giI

b

i

1I

...

binenrI

(33)

The symmetric part of the gradient of the test function is given by

gradsym(gih)=nin

I=1B

iI giI (34)

where BiI=

gradsym

N

iI. Accordingly, the discrete weak form of balance of momentum on the

coarse scale becomes

n0nI=1

(g0I)

T

0B

0I

T : r(u0h + u1h) d fint

0N

0I fh d

*0

N0I th d*

0D

N0I tch d

= 0 (35)

The integration of the nodal internal forces fint is carried out over the entire domain including

the domain 1. While in the domain 0

\

1 the stresses only depend on the displacement field

u0h since in

0\1 we have u1h 0, in the domain 1 the stresses depend on the displacementfield u1h = u0h + u1h which are computed from the fine scale model by solving the discretized formof (18)

n1nI=1

(g1I)T

1B

1I

T : r(u1h) d

1N

1I fh d

0D 1D

N1I tch d

= 0 (36)

for the displacement field u1h .

In solving the standard macroscale problem, standard Gauss quadrature methods are used in

0\1 except around the macrocrack, where element subdivision as described in Moes et al. [12] is

employed. In the macroscale problem, 0

1 is integrated by taking advantage of the congruence

of the two meshes. Thus, we employ

0e

() d= Se

1

() d (37)

where we use Gauss quadrature for each element Se. The subdivision of the microscaleelements around discontinuities is also carried over in the quadrature, though not indicated above.


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Figure 3. Enrichments along the boundary of the fine scale domain.

For the fine scale problem, the boundary conditions (i.e. continuity conditions) are

u1h =u0h on *1 (38)

Although the meshes are congruent, the enriched functions of the macro and micro fields differ.

Therefore, the continuity condition is enforced by a least-squares projection

1

I

N1I u1I

J

N0J u0J

M

N1M g1M

d= 0 (39)

Note that in (39) the integral is carried out over a subdomain 1 1 of the microstructure

contiguous to the interface. This is necessary to avoid non-uniquenesses of u1

h

which can occur

when the integral is performed only over the boundary of the microstructure. Figure 3 shows

the reason for such a situation. Here, due to the macrocrack, within the fine scale domain nodes

A and B on the boundary are enriched with the jump enrichment, although no crack intersects

the boundary between the two enriched nodes. On the boundary of the fine scale domain, the

standard degrees of freedom and the enriched degrees of freedom of those nodes are linearly

dependent. Therefore, it is necessary to integrate over the domain of the fine scale element C,

which takes into account the geometry of the crack within the microstructure, to eliminate linear

dependence. Accordingly, the domain 1

is chosen to be a strip of a width of one element along

the boundary.

This projection method requires solving another linear equation system. However, this system

is much smaller than the equation system for the microscale finite element problem. Additionally,

it is possible to apply a lumping strategy for the coefficient matrix such that this operation is stillquite cheap.

Close to the crack tip of a macrocrack, the displacement field has high gradients. For arbitrary

crack problems with microcracks around a macrocrack tip, the approximation of the displacement

field by the macroscopic ansatz may be relatively coarse. The macroscopic solution will then be

stiffer than the solution of the microstructural problem, not only because it cannot accurately capture

the discontinuities within the microstructure, which weaken the effective stiffness of the material,


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but also because it simply contains a much lower number of degrees of freedom. Therefore, to

avoid an excessively stiff boundary of the microstructure domain this domain should be big enough

so that the fluctuations due to the microstructure and the non-linearities due to the macrocrack are

relatively small on the boundary. This will be investigated in Section 5.

4. SOLUTION PROCEDURE

The solution procedure consists of a series of iterations in which the fine scale domain is solved

independently, followed by solution of the coarse scale equations for the entire domain. The

essence of the procedure for a linear elastic material without cohesive cracks is shown in Figure 4.

The method can easily be extended to non-linear material models and finite deformation theory

by replacing the solution step 4 by a Newton iteration step and using a geometrically consistent

linearization of the weak form and a consistent material tangent operator D in solution step 2.

However, for simplicity for the test cases investigated in Section 5 we restrict ourselves to isotropiclinear elasticity.

1. Initialize

k = 0, u0,kh

= 0, u1,kh

= 0,0,kh

= 0,1,kh

= 0

2. Solve macroscale iteration step

K0 u

0,kh

= f0

ext f0,k

int for u0,kh

with

K0

IJ =0B

0I

T D B

0J d and

f0,kint,I 0B0I

T : u0,kh + u1,kh d + 0N0I fh d

set u0,k+ 1h

= u0,kh

+ u0,kh

3. Project boundary conditions (continuity conditions) from the macroscale solution onto the

boundary of the microscale domain by solving

P u1,k+ 1h

= rkh for u

1,k+ 1h

on the boundary 1 with

PIJ =1

N1I

T N1

J d and rk

I= 1

N1I

M

N0M

u0,k+ 1

Md

4. Solve microscale iteration step

K1 u

1,k+ 1h

= f1,k+ 1ext f

1int for u

1,k+ 1h

with

K1IJ =1B1I

T D B1J d , f1int,I =1

N1I fh d and

f1,k+ 1

ext,I = K1

IJ u1,k+ 1

Jfor all nodes J on the boundary of the microscale domain

5. If|| u0,kh

|| > tol0 or ||u1,k+ 1h

u1,kh

|| > tol1 then set k k + 1 and goto step 2.

Figure 4. Solution procedure of the multiscale method.


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5. NUMERICAL STUDIES

In this section, we examine a number of numerical test cases to study the different parameters

influencing the results of the described multiscale method. In particular, those parameters are the

mesh resolution of the coarse and fine scale meshes and the domain size for the microstructure. Forall computations only square-shaped standard displacement elements are used. The stress intensity

factors, which are the quantities we investigate in all the numerical tests, are computed using a

domain form of the interaction integral as proposed in [11, 12, 29].

5.1. Influence of the coarse scale mesh resolution

The influence of the resolution of the coarse scale meshes on the result is investigated by considering

crack shielding of the problem shown in Figure 5. The body is loaded by prescribing mode 1

displacement boundary conditions along all edges. This example has been studied in Reference [30]using analytical methods. The geometry is shown in Figure 5. In the coarse scale problem only

the large edge crack is considered explicitly. The influence of the two small cracks (b = 2135 a) iscaptured only implicitly through the fine scale solution. To keep the influence of the size of the fine

scale domain (the darkly shaded area in Figure 5) negligible, the radius r around the macrocrack tip

within which all elements are defined as multiscale elements is chosen to be constant (r= 0.165a)for all mesh resolutions. Additionally, the fine scale mesh resolution is kept constant for all coarse

scale mesh resolutions. This means that mainly the number of coarse scale elements within the

fine scale domain will change in this convergence study. The number of fine scale elements

remains almost constant. We choose the coarse scale mesh to consist of 9 9, 15 15, 27 27,45 45 and 81 81 elements, respectively. The fine scale mesh is chosen to have a mesh sizeof h = 1

1215a.

Figure 5. Sketch of a crack shielding test: left, coarse scale mesh; and right, fine scale mesh.


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Figure 6. Stress intensity factor KI of the macrocrack tip depending on the coarse scale mesh resolution.

The results in Figure 6 show that there is hardly any influence of the coarse scale mesh

resolution on the resulting stress intensity factor of the macrocrack. Even for a very coarse mesh

resolution the values for the stress intensity factor deviate only in the third digit. For comparison

purposes, it should be noted that the stress intensity factor of the macrocrack in the absence ofthe microcracks is 1.0, so there is substantial crack shielding. The results shown in Figure 6

indicate that the far field solution is almost independent of the detailed microstructure in the

vicinity of the macrocrack tip and that our two-scale approach can effectively capture the effects

of the microstructure. It also shows that the size of the fine scale domain is large enough for this

test and that the projection of the displacement field on the boundary of the fine scale domain

works well.

5.2. Influence of the fine scale mesh resolution

For the investigation of the influence of the fine scale mesh resolution on the accuracy of the result

we use the same crack shielding example (see Figure 5) but with b =2

27 a. The mesh resolution ofthe coarse scale mesh is kept constant at 15 15 elements, and the radius r for the fine scale domainis set to r= 0.165a again. The fine scale element size varies between h = 145 a and h = 11215 a whichcorresponds to a range between 225 and 164 025 elements. The results for mode 1 stress intensity

factor of the macrocrack tip are shown in Figure 7. Even for the coarsest fine scale mesh the

results are quite good. The deviation of the coarse mesh solutions from the solution for the finest

mesh resolution is less than 5%.


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Figure 7. Stress intensity factor KI of the macrocrack tip depending on the fine scale mesh resolution.

5.3. The domain size of the microstructural domain

The size of the domain in which a microstructural computation needs to be performed has to be

chosen so that the fluctuations of the field variables on the boundary of this domain are negligible.

This means that the boundary of the fine scale domain has to be far away from the zone of influence

of the microstructure on the macrocrack. However, in order to reduce the computational cost, it

is desirable to reduce the size of the fine scale domain as much as possible without losing toomuch accuracy. Here, the domain is chosen to be of circular shape with the centre at the tip of the

macrocrack. To investigate the influence of the fine scale domain size, the radius of the circle is

changed. The geometry for this test again is given in Figure 5 with b = 2243

a. The coarse and fine

scale mesh element sizes are fixed. The coarse scale mesh consists of 81 81 elements andthe fine scale element size is set to h = 1

2187a. This rather fine discretization is chosen to

minimize the influence of the mesh resolution. Figure 8 shows mode 1 stress intensity factor

as a function of the normalized radius r/a. It can be seen that even for the smallest fine scale

domain, for which the radius r is not even twice as big as the length of the microcracks, the error

of the stress intensity factor is less than 1%. For a slightly bigger radius, the error even decreases

to less than 12. This shows that the fluctuations in the field variables due to the presence of the

microcracks rapidly decay with the distance to the microcracks.

5.4. Mixed mode multiple crack problem

We consider a crack problem with many microcracks and a macrocrack under mixed mode loading.

The purpose of this example is to show the applicability of the method to random microstructures

and arbitrary loading conditions. A sketch of the geometry and loading conditions is shown

in Figure 9. The 114 microcracks, which are randomly distributed around the crack tip of the


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Figure 8. Stress intensity factor KI of the macrocrack tip depending on the radius of the fine scale domain.

Figure 9. Sketch of a mixed mode multiple crack problem.

macrocrack, are of length 1600

a. The darkly shaded area in Figure 9 is chosen to be the fine scale

domain and is chosen to be a square of ( 349

a) ( 349

a). The macrocrack is at an angle of= 15.64and ends at the centre of the plate. On the coarse scale level a mesh of 49 49 elements is used,


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Figure 10. Deformed configuration of the fine scale domain.

Figure 11.

yy stress component of the fine scale domain.

while on the fine scale 261 261 elements are used to capture the quite non-smooth displacementand stress field in the vicinity of the macrocrack tip accurately. In Figure 10 is displayed the

deformed fine scale domain. One can see that the macro- and microcrack behaviour is captured

correctly in the whole domain and on the boundary of the fine scale domain. None of the mode 1

stress intensity factors of the microcracks is negative which indicates that crack closure does not


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occur. Figure 11 shows the yy component of the stresses in the fine scale domain. The mode 1

stress intensity factor for the macrocrack computed by the macro/micro model is KI = 19.836. Fora coarse scale mesh resolution of 147 147 elements, the stress intensity factor for the macrocrackis 19.924. In the absence of the microcracks, the stress intensity factor is KI

=28.163. We have

not been able to do a single-scale computation for this problem because the microcracks are veryshort compared to the scale of the structure.

6. CONCLUSIONS

A method has been presented for the multiscale analysis of bodies with extensive microstructure

on a scale significantly smaller than features of the macrostructure. This scale separation makes it

possible to solve the microproblem separately from the macroproblem, thus providing substantial

savings in computational cost. The scale separation is affected by using as test functions the coarse

scale functions on the macroscale, but retaining the microscale features in the stress field. In this

paper, the focus has been on microcrack/macrocrack interactions. As shown in the development,

only the cohesive tractions from the macrocracks appear in the coarse scale equations, whereas

the macrocrack and microcrack cohesive tractions appear in the fine scale equations. The proposed

multiscale method for crack simulations considerably improves the efficiency of simulations of

microcrack/macrocrack interaction problems.

We have employed the extended finite element method for crack modelling. This facilitates

the treatment of complex crack patterns. However, the method is applicable to other multiscale

approaches, such as combinations of molecular and continuum mechanics.

The numerical examples demonstrate the excellent convergence behaviour of the method in the

interplay of coarse scale and fine scale behaviour, namely, the stress intensity factor as influenced

by the microcracks. The effectiveness of the method has been demonstrated entirely in terms of

examples. At this time, we have no mathematical basis for how well and under what circumstances

the method works. However, it is apparent from the examples that when there is significantseparation of scales, a two-scale method such as proposed here is quite effective. Additionally, this

method is applicable to a large variety of other problems, such as dislocations and shear bands.

An advantage of this method is the ease of its implementation, for it requires little modification

of standard finite element software. The method can also easily be extended to three-dimensional

problems and other applications such as heterogeneities.

ACKNOWLEDGEMENTS

The support of the Army Research Office under grant W911NF-05-1-0049 is gratefully acknowledged.

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DOI: 10.1002/nme

multi scale crack

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