COMPUTATIONAL HOMOGENIZATION OF DISCRETE FRACTURE INFIBRE-EPOXY SYSTEMS

M.V. Cid Alfaro1, A.S.J. Suiker1,∗, C.V.Verhoosel1, R. de Borst2

1 Delft University of Technology2 Eindhoven University of Technology

ABSTRACT: In the present paper the effective mesoscale failure response of a fibre-epoxy sample is computed fromits complex microscale fracture behaviour. The mesoscale failure response is represented by a traction-separation curvederived from numerically homogenizing the fracture response of a periodic fibre-epoxy microstructure loaded underuniaxial tension. The traction-separation curve can be applied in material points of interface elements that are used forsimulating mode I mesoscopic fracture in macroscopic laminate failure problems. The effect of the size of the microscopicfibre-epoxy sample on the mesoscale failure response is examined, as well as the effect of local imperfections at fibre-epoxy interfaces.

KEYWORDS: Computational homogenization, discrete fracture, fibre-epoxy systems, GLARE.

1 INTRODUCTIONFinite element simulations serve as an important toolfor studying the complex failure behaviour of fibre-metallaminates (FMLs) and optimizing their performance in(macroscopic) engineering applications. However, the ex-ecution of a direct numerical simulation on the mechan-ical response of an engineering structure, where all thedetails of the underlying microstructure are incorporated,requires a very fine finite element mesh, leading to animpractical amount of computational time. A more effi-cient approach is to use material models that represent themechanical response of the underlying microstructure inan effective fashion. When the underlying microstructureis complex and is characterised by a non-linear constitu-tive behaviour, computational homogenization is a recom-mendable tool for this purpose. Computational homoge-nization is based on numerically averaging the generatedmicrostructural stress and strain over a representative vol-ume element (RVE), thereby implicitly (i.e., not in closed-form) establishing an effective, mesoscopic constitutiverelation between the average stress and strain, see for ex-ample [1].The overall failure behaviour of FMLs strongly dependson small-scale fracture events occurring within individ-ual fibre-epoxy (prepreg) layers, such as fibre decohesionand matrix cracking. Accordingly, the present paper fo-cuses upon the derivation of the effective mesoscale fail-ure response of a fibre-epoxy sample from its complexmicroscale fracture behaviour. The mesoscale failure re-sponse is represented by a traction-separation curve con-structed from numerically homogenizing the fracture re-

∗Corresponding author: Delft University of Technology, Faculty ofAerospace Engineering, Kluyverweg 1, 2629 HS, Delft, Netherlands,tel: +31 (0)15 2781629, e-mail: A.S.J.Suiker@tudelft.nl

sponse of a periodic fibre-epoxy microstructure loadedunder uniaxial tension. The traction-separation curve canbe applied in material points of interface elements that areused for simulating mesoscopic fracture in macroscopiclaminate failure problems. This modelling approach is anappealing and acceptable alternative to the (costly) directsimulation of microscale fracture in macroscopic prob-lems if the fluctuations of the microscale crack trajectoriesin the thickness direction of the laminate are much smallerthan the laminate thickness itself (i.e., a separation oflength scales can be warranted). The analysis of specificmacroscopic problems, however, falls beyond the scopeof the study in the present paper; the attention is directedhere to the numerical homogenization of the response of amicrostructural RVE to a mesoscopic traction-separationcurve, thereby considering the influence of microstruc-tural sample size and local imperfections. Although thenumerical examples treated focus upon uniaxial tension,the homogenization framework used is applicable to arbi-trary loading conditions.

2 MICRO- AND MESOSCALE MOD-ELLING

Consider a mesoscopic domain ΩM

⊂ R2 with an ex-

ternal boundary ΓM , see Figure 1. The external bound-ary is subjected to displacements u

M and tractions tM

at ΓMu and ΓM

t , respectively. The mesoscopic domainis crossed by an adhesive layer ΓM

coh. The response in amaterial point of the adhesive layer is connected to thelower-scale response of a heterogeneous, microscopic do-main Ωm

⊂ R2, see Figure 1. The microscopic do-

main is represented by a fibre-epoxy sample of width b.Since the thickness of the mesoscale adhesive layer ΓM

coh

DOI 10.1007/s12289-009-0593-7© Springer/ESAFORM 2009

Int J Mater Form (2009) Vol. 2 Suppl 1:931–934

1 2

34

Γm

B

Γm

R

Γm

T

Γm

L

Γm

coh

nR

nL

n

xm

1

xm

2 xm

xM

1

xM

2 xM

ΩM

Ωm

n

ΓM

u

ΓM

t

ΓM

coh

b

Figure 1: Mesoscopic domain ΩM and microscopic do-

main Ωm.

is zero, the thickness of the microscopic domain does notneed to be accounted for in the homogenization procedure.Further, fracture in the microscopic domain occurs alongthe cracking path Γm

coh. The boundary conditions at theouter edges Γm of the microscopic domain are assumedto be periodic, where the fluctuations at the top and bot-tom edges are restrained. For establishing an expressionfor the mesoscopic traction, tMi , in a material point at themesoscale cohesive interface ΓM

coh, the computational ho-mogenization scheme presented in [2] is applied. Thisscheme uses the averaging principle of Hill [3], whichstates that the spatial average of the virtual work at themicroscale, δWm, needs to be equal to the virtual workin a local material point of the mesoscale cohesive inter-face, δwM

coh. Accordingly, the following expression for themesoscopic traction is obtained [2]

tMi =1

b

∫

ΓmT

tTi dΓmT , (1)

where tTi is the microscopic traction at the top surfaceΓm

T of the microscopic domain Ωm. Within a finite ele-ment discretisation the integral term in the right-hand sideof Eq.(1) is approximated by the summation of the nodalforces at the top edge of the RVE.

3 FIBRE-EPOXY SPECIMEN SUB-JECTED TO UNIAXIAL TENSION

Although Eq.(1) can be applied to samples subjected toarbitrary combinations of tensile and shear loading, forsimplicity the examples studied focus upon uniaxial ten-sion. The interface damage model presented in [2] is usedto simulate matrix cracking and debonding between fibresand matrix.

3.1 GEOMETRY AND BOUNDARY CONDI-TIONS OF THE FIBRE-EPOXY SAMPLE

In order to check whether the sample size converges to arepresentative volume element, in the numerical analysestwo different specimen widths are considered, see Figure2. The samples have a fibre volume fraction of 0.3, andare subjected to uniaxial tension by prescribing the ver-tical displacement, u, at the top edge of the sample. Pe-riodic boundary conditions are prescribed at the left andright edges of the sample. The displacement at the bottom

edge of the sample is constrained in the vertical direction,and the displacement of the node at the bottom-left cor-ner of the specimen is also constrained in the horizontaldirection, see Figure 2.

The first sample studied is square-shaped, with a width(and height) equal to 0.125 mm. The second sampleis rectangular-shaped, where, in comparison with thesquare-shaped sample, the width is chosen two timeslarger, i.e., 0.250 mm, while the height is kept the same.Essentially, the chosen sample height is representative ofthe thickness of fibre-epoxy layers used in the fibre-metallaminate GLARE [4]. The typical diameter of the S2 glassfibres used in GLARE is 10 µm. As illustrated in Figure2, the internal material structure of the samples relates toa random fibre distribution that is geometrically periodicin the x1-direction.

u

x1

x2

0.125 mm

0.1

25

mm

(a) Square-shaped specimen.

u

x1

x2

0.1

25

mm

0.250 mm

(b) Rectangular-shaped specimen.

Figure 2:

10 µm.

3.2 FINITE ELEMENT MODEL

The finite element model used for the microscale compu-tations is plane-strain and thus is representative of fibreswith a relatively large (actually infinite) length. The fi-bre and the epoxy matrix are meshed with 6-node trian-gular elements equipped with a 7-point Gauss quadrature.Debonding between fibre and matrix and fracture withinthe epoxy are simulated by 6-node interface elements witha 3-point Newton-Cotes quadrature. The interface ele-ments describing fracture in the epoxy are placed in be-tween the triangular continuum elements constructing theepoxy matrix, an approach that was originally proposedby Xu and Needleman [5]. The material parameters ofthe S2 glass fibre, the FM94 epoxy and the interface be-tween fibres and epoxy are taken as reported in [2]. Thevalues of the ultimate normal and shear strengths of the(adhesive) fibre-epoxy interface are taken equal, namelytu1

= tu2

= tuadh = 50 MPa. In addition, the cohesivestrength of the epoxy material is set equal to the adhesivestrength of the fibre-epoxy interface, tucoh = tuadh = 50

MPa. In order to avoid numerical convergence problemsduring crack bifurcation, the interface damage model isexhibited with a small viscosity [2]. The number of con-tinuum elements and interface elements used in the finiteelement meshes of the samples in Figure 2 are listed inTable 1. Note from this table that the number of elements

Geometry and boundary conditions of fibre-epoxy samples with a fibre volume fraction of 0.3. Thediameter of the glass fibres is

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used for modelling the rectangular sample is about twotimes the number of elements used for the square sample.

Sample Continuum Fibre-epoxy Epoxyshape elements int. elems. int. elems.

Square 5194 772 5001Rectang. 10390 1549 10035

Table 1: Number of elements used for samples of differentsize, see Figure 2.

3.3 INFLUENCE OF SAMPLE SIZE

The failure responses of the samples with the two dif-ferent widths are compared to asses the convergence ofthe numerical result upon an increasing microstructuralsample volume. The mesoscopic traction-separation re-lation obtained after applying Eq.(1) to the numerical re-sults is shown in Figure 3. It can be observed that the

00 0.01 0.02 0.03

10

20

30

40

50

60

Nor

mal

trac

tion

tM 2[M

Pa]

Displacement jump Ju2KM [mm]

Square-shaped specimen

Rectangular-shaped specimen

Figure 3: Mesoscopic traction-separation response for the

bre volume fraction of 0.3, shown in Figure 2.

traction-separation responses of the square-shaped andrectangular-shaped samples show a strong resemblance interms of shape and characteristic properties, such as frac-ture strength and fracture toughness. This resemblance in-dicates that the square-shaped sample is sufficiently largefor being considered as an RVE. Essentially, the traction-separation responses closely follow the mode I traction-separation law of the epoxy material. Although not shownhere, this is due to the fact that the final failure crack de-velops mainly through the epoxy material and is mode Idominated (i.e., the orientation of the crack is more or lessperpendicular to the direction of the tensile loading). Be-cause the present analysis demonstrates that the square-shaped sample can be considered as a representative vol-ume element, this sample will be used for the forthcomingcomputations in this paper.

3.4 INFLUENCE OF IMPERFECTIONS ON FAIL-URE RESPONSE

For studying how the homogenized traction-separation re-sponse is influenced by imperfections at fibre-epoxy in-

terfaces, four different configurations are considered, seeFigure 4. The fibre volume fraction of the sample is 0.3and the imperfections are represented by a local absenceof the adhesive bonding strength at specific fibre-epoxy in-terfaces. As illustrated in Figure 4, the four configurationsconsidered are, respectively, characterised by: (i) A singleimperfection in the upper region of the sample (i.e., in theleft-top quadrant), (ii) A single imperfection in the lowerregion of the sample (i.e., in the right-bottom quadrant),(iii) Two imperfections in the upper and lower regions ofthe sample (i.e., a combination of configurations (i) and(ii)), and (iv) Two imperfections in the upper and lowerregions of the sample, where the upper imperfection coin-cides with that in configuration (i) and the lower imperfec-tion is placed somewhat higher than that of configuration(ii).

(i) (ii) (iii) (iv)

Figure 4:

of the sample is 0.3 and imperfections are indicated in red.

The mesoscopic traction-separation response for the dif-ferent configurations in Figure 4 is plotted in Figure 5. Forcomparison, the response of the specimen without imper-fections, as plotted in Figure 3, has also been included inthis figure. The corresponding fracture patterns are de-

0.01 0.02 0.0300

10

20

30

40

50

60

Nor

mal

trac

tion

tM 2[M

Pa]

Displacement jump Ju2KM [mm]

Without imperfection

Imperfection upper region (i)

Imperfection lower region (ii)

Two imperfections (iii)

Two imperfections (iv)

(ii),(iii)

(i),(ii),(iii)

Figure 5: Mesoscopic (mode I) traction-separation law

bre volume fraction of 0.3 (shown in Figure 2(a)), plottedfor a different number and positions of imperfections.

picted in Figures 6 to 9. In all configurations the imper-fections clearly act as nucleation sites for crack develop-ment. In addition, their location typically is included inthe geometry of the dominant crack that develops uponcomplete failure of the fibre-epoxy specimen. An excep-tion in this respect, however, is the imperfection locatedin the upper half of configuration (iii), see Figure 8; thisimperfection initially acts as a nucleation site for crack-ing but eventually is not included in the geometry of the

square-shaped and rectangular-shaped samples with a fi-

Four fibre-epoxy samples with different locations and/or number of imperfections. The fibre volume fraction

corresponding to the square-shaped specimen with a fi-

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dominant failure crack, due to a local unloading behaviourduring the loading process. This is an important differencewith configuration (iv), where both imperfections are in-cluded in the geometry of the dominant failure crack, seeFigure 9. The net result of this feature is that the totalcrack length of configuration (iv) is substantially largerthan that of configuration (iii), which thus requires the dis-sipation of more energy in order to fail the sample. Cor-respondingly, the effective fracture toughness of config-uration (iv) is larger than that of configuration (iii), seeFigure 5, and also is larger than that of the sample with-out imperfections. Hence, the presence of imperfectionsin the fibre-epoxy sample may have a positive effect on itsoverall failure characteristics.

0 10 20 30 40 50σ22 [MPa]

Figure 6: Axial normal stress σ22 depicted in the de-case (i) in Figure 4.

Ju2KM

=

0.000821, 0.00194 and 0.00599 mm, respectively.

0 10 20 30 40 50σ22 [MPa]

Figure 7: Axial normal stress σ22 depicted in the de-case (ii) in Figure 4.

Ju2KM

=

0.00082, 0.00185 and 0.00586 mm, respectively.

0 10 20 30 40 50σ22 [MPa]

Figure 8: Axial normal stress σ22 depicted in the de-case (iii) in Figure 4.

Ju2KM

=

0.000821, 0.00199 and 0.00589 mm, respectively.

0 10 20 30 40 50σ22 [MPa]

Figure 9: Axial normal stress σ22 depicted in the de-case (iv) in Figure

Ju2KM

=

0.000822, 0.00186 and 0.00804 mm, respectively.

4 CONCLUSIONSIn this paper the homogenization framework presented in[2] has been applied to link the microscale response ofa fibre-epoxy sample to a mesoscale traction-separationcurve that can be used for simulating the failure responseof a (material point in a) cohesive interface. The formula-tion is based on Hill’s averaging principle, which statesthat the spatial average of the virtual work at the mi-croscale needs to be equal to the virtual work in a localmaterial point of the mesoscale cohesive interface. Mi-croscale numerical simulations are performed on a fibre-epoxy sample subjected to uniaxial tension. The numeri-cal response is characterised by a failure pattern that de-velops mainly through the epoxy matrix, where the orien-tation of the dominant failure crack is approximately per-pendicular to the loading direction (i.e., mode I failure).It is demonstrated that imperfections between fibres andepoxy generally trigger crack nucleation, and their loca-tion typically is included in the geometry of the dominantcrack that develops upon complete failure of the sample.As a result of this behaviour, imperfections may increasethe length of the dominant failure crack (as compared tothe crack length for a sample without imperfections), andthus may enhance the effective fracture toughness of thesample.

REFERENCES[1] P.M. Suquet. Local and global aspects in the math-

ematical theory of plasticity. In A. Sawczuk andG. Bianchi, editors, Plasticity Today; Modelling,Methods and Applications, pages 279–310. ElsevierApplied Science Publishers, London, 1985.

[2] M.V. Cid Alfaro. Multiscale Analyses of Fibre MetalLaminates. PhD Dissertation, Delft University ofTechnology, 2008.

[3] R. Hill. The essential structure of constitutive lawsfor metal composites and polycrystals. J. Mech. Phys.Solids, 13:189–198, 1967.

[4] A. Vlot and J.W. Gunnink. Fibre-Metal Laminates -An Introduction. Kluwer Academic Publishers, 2001.

[5] X.P. Xu and A. Needleman. Numerical simulationsof fast crack growth in brittle solids. J. Mech. Phys.Solids, 42:1397–1407, 1994.

formed, cracked configuration, for the4. The deformed configuration are considered at

formed, cracked configuration, for The deformed configurations are considered at

formed, cracked configuration, for The deformed configuration are considered at

formed, cracked configuration, for The deformed configuration are considered at

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