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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
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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
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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
0r(u0 + u1) : gradsym(g0) d
=
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
1r(u0 + u1) : gradsym(g1) d
=
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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