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Cracks propagation and interaction in an orthotropic elastic material:Analytical and numerical methods

T. Sadowski a,*, L. Marsavina a,b, N. Peride c, E.-M. Craciun c

a Lublin University of Technology, Nadbystrzycka 40, 20-618 Lublin, Polandb Politehnica University of Timisoara, Department Strength of Materials, Blvd. M. Viteazu, Timisoara, 300222, Romaniasc ‘‘Ovidius” University of Constanta, Romania

a r t i c l e i n f o

Article history:Received 11 November 2008Received in revised form 1 May 2009Accepted 9 June 2009Available online 15 July 2009

Keywords:Orthotropic elastic materialPlemelj’s formulasCauchy integralsCrack closure integralFRANC2D

a b s t r a c t

An elastic orthotropic material containing a crack in Mode I is considered to formulate a new analyticalmodel. The boundary conditions for the crack existence in the material lead to the solution of the homo-geneous Riemann–Hilbert problems. The mathematical model was elaborated for a single and two collin-ear cracks of different lengths and distance for Mode I in order to investigate cracks interaction problem.Using the theory of Cauchy’s integral and the numerical analysis, the fields in the vicinity of the crack tipswere determined.

Finite Element Method was applied to compare the mathematical analytical solution and to determinethe fields in the vicinity of the crack tips. The critical values of applied stress which caused cracks prop-agation were evaluated. The interaction of cracks in an orthotropic aramid-epoxy material was studied indetails. Comparison of both approaches to crack propagation leads to the conclusion that the new analyt-ical model is correct and can be applied to more complex cracks geometries, including inclined cracks.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

The problem of a linear elastic solid with multiple cracks wasinvestigated by many authors [1–5]. Kachanov [1] proposed anapproximate method for the analysis of multiple cracks by assum-ing that traction in each crack can be represented as a sum of anuniform component and a non-uniform component, and the inter-action among the cracks is only due to the uniform components.These assumptions simplify considerably the mathematical ap-proach and allow for ‘closed-form’ solutions to be obtained forsome cases. However, it is noted that the assumptions may notbe valid when the cracks are very close. Therefore, an improvedmethod for elastic solids with closely spaced multiple cracks wasproposed by Li et al. [2]. Otsu et al. [3] solved a plane strain prob-lem for two bonded dissimilar planes containing a crack parallel tothe interface in each layer using Gauss–Chebyshev integral formu-lae for various material combinations and geometrical parameters.Gorbatikh et al. [4] developed a method that predicts Stress Inten-sity Factors (SIF) at tips of two-dimensional cracks at distances thatmay be substantially smaller in comparison to the crack lengths.

Numerical methods are often used to find the stress and dis-placement fields around multiple cracks and to estimate the frac-ture mechanics parameters. The Boundary Element Method wasused by Blackburn [5] in order to calculate the fracture parameters

for the two circular coplanar cracks. Laplace Finite Element Alter-nating Method was used by Chen et al. [6] for elastodynamic frac-ture analysis of multiple cracks.

We consider an orthotropic elastic material containing astraight crack of a length 2a situated in x1x3 plane, Fig. 1. Let ussuppose that the material is unbounded and the crack faces aresubjected to the constant normal stresses p.

The first aim is to create a new analytical model of crack in anorthotropic elastic material and to determine the elastic state cre-ated by cracks in the material using Guz’s representation theorem,[7]. The boundary conditions of our problem lead to homogeneousRiemann–Hilbert problems. Using the theory of Cauchy’s integral[7–12] and the numerical analysis we determine the fields in thevicinity of the crack tips.

The second aim of the paper is to validate the analytical solutionby classical numerical approach to determine the critical values ofthe stresses and direction of crack propagation. Illustrative exam-ples where done for an orthotropic aramid-epoxy material contain-ing two collinear straight cracks with different dimensions anddistance.

2. Guz’s representation theorem for displacement and stressfields

Let us consider an orthotropic, linear elastic material. The sym-metric planes of the material are taken as coordinate planes. We

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* Corresponding author. Tel.: +48 81 5384386; fax: +48 81 5384173.E-mail address: t.sadowski@pollub.pl (T. Sadowski).

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assume that the material is unbounded and contains a straightcrack of length 2a, as is shown in Fig. 1.

Let us suppose that the crack is subjected to a distributed exter-nal stress having a constant value p(x1) = p = const. > 0, see Fig. 1.Moreover, we assume also that the initial deformed equilibriumconfiguration of the body is locally stable.

As it was shown by Guz [7], the elastic state of the orthotropicmaterial can be expressed by two analytical complex potentialsWj(zj) defined in two complex planes zj, j = 1,2. We denote by u1,and u2 the displacement vector components, whereas by h11, h12,h21 and h22 the stress tensor components.

According to Guz’s representation formulae, we have:

u1 ¼ 2Refb1U1ðz1Þ þ b2U2ðz2Þg;u2 ¼ 2Refc1U1ðz1Þ þ c2U2ðz2Þg;

ð2:1Þ

h11 ¼ 2Refa1l21W1ðz1Þ þ a2l2

2W2ðz2Þg;h12 ¼ �2Refl1W1ðz1Þ þ l2W2ðz2Þg;

ð2:2Þ

h21 ¼ �2Refa1l1W1ðz1Þ þ a2l2W2ðz2Þg;h22 ¼ 2RefW1ðz1Þ þW2ðz2Þg:

ð2:3Þ

In the above relations

WjðzjÞ ¼ U0jðzjÞ ð2:4Þ

and

zj ¼ x1 þ ljx2; j ¼ 1;2:

The parameters aj, bj, cj, j = 1,2 have the following expressions:

aj ¼ ð-2112-1122l2j �-1111-1212Þ=ðBjl2

j Þ;bj ¼ �ð-1122 þ-1212Þ=Bj; ð2:5Þcj ¼ ð-2112l2

j þ-1111Þ=ðBjljÞ; ð2:6ÞBj ¼ -2222-2112l2

j þ-1111-2222 �-1122ð-1122 þ-1212Þ: ð2:7Þ

The instantaneous elasticities -klmn (k,l,m,n = 1, 2) can be ex-pressed through the elastic coefficients C11, C12, C22 and C66 ofthe material by the following relations:

-1111 ¼ C11; -1212 ¼ C66; -2222 ¼ C22;

-1221 ¼ C66; -1122 ¼ C12; -2112 ¼ C66:ð2:8Þ

In turn the elastic coefficients can be expressed using the engi-neering constants of the material and we have:

C11 ¼1� m23m32

E2E3H; C22 ¼

1� m13m31

E1E3H;

C12 ¼m12 þ m13m32

E1E3H; C66 ¼ G12 ð2:9Þ

with

H ¼ ð1� m12m21 � m23m32 � m31m13 � m21m32m13 � m12m23m31Þ=ðE1E2E3Þ:ð2:10Þ

In these relations E1, E2, E3 are Young’s moduli in the corre-sponding symmetry directions of the material, m12, . . . ,m32 are Pois-son’s ratios and G12, G23, G31 are the shear moduli in thecorresponding symmetry planes. The parameter lj can be obtaineddetermining the roots mj, j = 1,2 of the algebraic equation:

m2 þ 2Amþ B ¼ 0 ð2:11Þ

with

A ¼ ½-1111-2222 þ-1221-2112 � ð-1122 þ-1212Þ2�=ð-2222-2112Þ;B ¼ ð-1111-1221Þ=ð-2112-2222Þ: ð2:12Þ

As was shown in [7,8], the above Eq. (2.11) cannot have real roots ifthe initial deformed equilibrium configuration of the body is locallystable. The complex parameters lj can be calculated through theroots mj, j = 1,2 using equations:

l1 ¼ffiffiffiffiffim1p

; l2 ¼ �ffiffiffiffiffim2p

if Im mj–0; j ¼ 1;2 ð2:13Þ

and

l1 ¼ffiffiffiffiffim1p

; l2 ¼ �ffiffiffiffiffim2p

if Immj ¼ 0 and Remj < 0; j ¼ 1;2:

ð2:14Þ

It can be shown that the parameters lj (j = 1,2) satisfy the relations:

Imðl1l2Þ ¼ 0 and Reðl1 þ l2Þ ¼ 0: ð2:15Þx1

x3

x2

a-a -l2a l1a

p(x1) = p = const.

Fig. 2. Two collinear cracks in an orthotropic elastic material.

Table 1Elastic properties of the orthotropic material.

Elastic moduli (MPa) Poisson ratios

Elastic constants of the orthotropic materialE1 = 75,000 m12 = 0.22E2 = 5560 m13 = 0.2E3 = 5560 m23 = 0.2G12 = 2000

x1

x3

x2

p(x1) = p = const.

a-a

r

Fig. 1. Single crack in an orthotropic elastic material.

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In what follows we assume that parameters l1 and l2 are notequal, i.e.

l1–l2: ð2:16Þ

The above condition is satisfied for orthotropic materials as wellas for prestressed isotropic materials.

The expressions of the complex potentials Wj(zj), (j = 1,2) corre-sponding to our boundary value problem were determined in [7–9]and have the following forms:

W1ðz1Þ ¼a2l2KI

2Dffiffiffiffiffiffipap z1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z21 � a2

q � 1

0B@

1CA;

W2ðz2Þ ¼ �a1l1KI

2Dffiffiffiffiffiffipap z2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z22 � a2

q � 1

0B@

1CA: ð2:17Þ

In a small neighborhood of the tip a, letting

x1 ¼ aþ r sinu; x2 ¼ r sin u; ð2:18Þ

where r and u are polar coordinates, assuming that r is small incomparison of the half crack length a and taking into account theapproximationffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z2j � a2

q¼

ffiffiffiffiffiffiffiffi2arp

vjðuÞ; j ¼ 1;2: ð2:19Þ

we obtain the following asymptotic values of the complexpotentials:

W1ðz1Þ ¼a2l2KI

2Dffiffiffiffiffiffiffiffiffi2prp 1

v1ðuÞ; W2ðz2Þ ¼ �

a1l1KI

2Dffiffiffiffiffiffiffiffiffi2prp 1

v2ðuÞ; ð2:20Þ

where

D ¼ a2l2 � a1l1 ð2:21ÞvjðuÞ ¼ ðcos uþ lj sinuÞ; j ¼ 1;2 ð2:22Þ

and

KI ¼ pffiffiffiffiffiffipap

; ð2:23Þis the Stress Intensity Factor (SIF) corresponding to Mode I of frac-ture for an applied stress p > 0.

3. Two collinear unequal cracks

We consider an unbounded orthotropic material containing twocollinear cracks situated in the same plane, parallel to the oneorthotropic direction, Fig. 2. The plane containing the cracks is ta-ken as x1x3-plane, the x2-axis being perpendicular to the crackfaces. The produced plane state is a plane state relative to x1x2-plane. Hence, we can use the complex representation describedin the previous section to study our problem corresponding tothe Mode I in classical fracture mechanics.

Here a > 0 is a given positive constant and l1 and l2 are given po-sitive numbers satisfying the restriction 0 < l1, l2 < 1.

Let us assume that p(x1) = p = const. > 0. Using the Plemelj’sfunction theory and imposing the uniformity of the displacementfield, as well as implicitly of the complex potential W1(z1), weget the following equation for complex potential:

W1ðz1Þ ¼pa2l2

2Dz2

1 �Maz1 � a2Nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðz1 � a2Þðz1 þ l2aÞðz1 � l1aÞ

p � 1

!;

W2ðz2Þ ¼ �pa1l1

2Dz2

2 �Maz2 � a2Nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðz2 � a2Þðz2 þ l2aÞðz2 � l1aÞ

p � 1

!;

ð3:1Þ

where

M ¼ Mðl1; l2Þ ¼R2S0 � S2R0

R0S1 þ S0R1; N ¼ Nðl1; l2Þ ¼

R2S1 þ S2R1

R0S1 þ S0R1ð3:2Þ

and

Rn ¼Z 1

l1

tn

RðtÞdt; Sn ¼Z 1

l2

tn

SðtÞdt; n ¼ 0;1;2; ð3:3Þ

RðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� t2Þðt þ l2Þðt � l1Þ

q; SðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� t2Þðt � l2Þðt þ l1Þ

q:

ð3:4ÞIn order to evaluate the Energy Release Rate (ERR) correspond-

ing to a crack tip we have to know the asymptotic values of theabove complex potentials.

We start to obtain these asymptotic values near the crack tip aof the crack (l1a,a) by taking:x1 ¼ aþ r cos u; x2 ¼ r sinu; ð3:5Þwhere the polar coordinates r and u denote the radial distance fromthe crack tip and the angle between the radial line and the lineahead of the crack, respectively. Now, from (3.1) we obtain theasymptotic values of the complex potentials in a small neighbour-hood of the tip a:

W1ðz1Þ¼pa2l2m1

2D

ffiffiffiapffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2rv1ðuÞp

!; W2ðz2Þ¼�

pa1l1m1

2D

ffiffiffiapffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2rv1ðuÞp !

;

ð3:6Þ

U1ðz1Þ¼pa2l2m1

2D

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2arv1ðuÞ

q; U2ðz2Þ¼�

pa1l1m1

2D

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2arv2ðuÞ

q;

ð3:7Þwhere

m1 ¼ m1ðl1; l2Þ ¼l21 � l1M � Nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� l2

1Þðl1 þ l2Þq and

vjðuÞ ¼ cos uþ lj sin u; j ¼ 1;2: ð3:8Þ

x1

x2

p(x1) = p = const.

d3mm 10mm

O P Q R

Fig. 3. Two collinear cracks configurations used in the illustrative example.

Table 2Energy release rates (ERRs) at the crack tips.

Distance d (mm) Angle b (�) ERR (J/m)

GIO GIP GIQ GIR

4 0 93.95 103.34 263.78 254.8290 20.57 23.61 66.49 64.18

8 0 84.55 87.05 254.92 252.2490 20.84 21.99 66.00 65.92

16 0 78.99 79.63 251.47 249.5090 19.66 19.38 63.48 63.62

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According to Irwin’s formula for the energy release rate GI(a) corre-sponding to the propagation of the tip a of the crack (l1 a,a), we getfollowing value:

GIðaÞ ¼ pap2m21q: ð3:9Þ

In a similar way we get the corresponding energy release rates ofthe other tips:

GIðl1aÞ ¼ pap2m22q;

GIð�l2aÞ ¼ pap2m23q;

GIð�aÞ ¼ pap2m24q;

ð3:10Þ

where m2, m3 and m4 have similar forms as m1 from Eq. (3.8)1, see[9], and q depends on the elastic coefficients of the material asfollows:

q ¼ C11 þ C22

2C11ðC11 � C12Þ: ð3:11Þ

3.1. Illustrative example

To demonstrate the capacity of the method let us considerorthotropic material with the elastic constants presented in Table 1.

Moreover, the cracks geometry presented in Fig. 3, was taken forcalculation with three different values of d = 4 mm, 8 mm and16 mm to investigate cracks interaction. According to the theorypresented in Section 3, the calculations of ERR GI for cracks tips:O, P, Q, R were done and results are presented in Table 2. The calcu-lations were performed for two orientations in relation to the prin-cipal material direction x1, i.e. for inclination angle b = 0� and 90�.

4. Critical values of the stresses. Cracks interaction

In order to obtain the critical values of the applied stress forwhich one of the tips starts to propagate, we use Griffith’s energet-ical criterion.

According to Griffith’s criterion (see for instance [7]), the tip astarts to propagate if the following condition is fulfilled:GIðaÞ ¼ 2c: ð4:1Þ

Let pcrit(a) be the critical value of the applied stress for whichthe tip a starts to propagate. Then from (3.9) and (4.1) we get:

pcritðaÞ ¼ 2q

ffiffiffiffiffifficpa

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ l2Þð1� l1Þ

p1�M � N

; ð4:2Þ

where c is the surface fracture energy of the considered material.The other values corresponding to the propagation of the other

tips are consequently as follows:

pcritðl1aÞ ¼ q

ffiffiffiffiffiffi2cpa

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� l2

1Þðl1 þ l2Þq�l2

1 þ l1M � N;

pcritð�l2aÞ ¼ q

ffiffiffiffiffiffi2cpa

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� l2

2Þðl1 þ l2Þq�l2

2 þ l2M � N;

pcritð�aÞ ¼ 2q

ffiffiffiffiffifficpa

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� l2Þð1þ l1Þ

p1�M � N

:

ð4:3Þ

To study the interaction of the crack we introduce the followingdimensionless quantities:

v1ðl1; l2Þ ¼pcritðaÞ

v ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1þ l1Þð1� l2Þ

p1�M � N

;

v2ðl1; l2Þ ¼pcritðl1aÞ

v ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� l2

1Þðl1 þ l2Þq�l2

1 þ l1M � N;

v3ðl1; l2Þ ¼pcritð�l2aÞ

v ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� l2

1Þðl1 þ l2Þq�l2

2 � l2M � N;

v4ðl1; l2Þ ¼pcritð�aÞ

v ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1� l2Þð1þ l1Þ

p�1�M � N

;

v ¼ q

ffiffiffiffiffiffiffiffi2cpa0

s:

ð4:4Þ

300 mm

200

mm

3mm d 10

mm

O P Q R

t

Fig. 4. Two unequal collinear cracks model (d = 4 mm for closed cracks andd = 16 mm for far cracks).

Fig. 5. Numerical model and mesh detail around the two unequal collinear cracks.

Table 3Non-dimensional SIFs for homogeneous material.

SIFs (MPa mm0.5) Tip O Tip P Tip Q Tip R

FIO,FE FIP,FE FIP,AN FIQ,FE FIR,FE

d = 4 mm 1.125 1.180 1.172 1.036 1.018d = 8 mm 1.068 1.084 1.073 1.038 1.033d = 16 mm 1.033 1.035 1.031 1.050 1.050

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The analysis of the interaction of cracks requires the knowledgeof the order between the four stresses, for different values of thedimensionless parameters l1 and l2. The necessary order relationwill be determined if the values of the dimensionless quantitiesv1, v2, v3 and v4 are known. The tip which will propagate the firstis that one which corresponds to the minimum value of vj forj = 1,4.

Numerical analysis for the equal and unequal cracks led us tothe following:

� if the distance between the equal cracks is much smaller thantheir length, there exists a strong interaction between the cracksand they tend to unify;

� if the distance between the equal cracks is much greater thantheir length, there is a weak interaction between the cracks;

� the tips of the longer crack start to propagate simultaneouslyand the interaction between cracks is relatively weak whenthe distance between them is relatively large compared withthe length of the shorter crack;

� if the distance between the cracks is relatively small comparedwith the length of the short crack, the interaction between thecracks is strong. Therefore, the inner tips start to propagate first.

5. Numerical simulation

The Finite Element Method (FEM) was used to estimate the frac-ture parameters at the tip of the cracks placed in homogeneous andorthotropic materials. The Modified Crack Closure Integral (MCCI)was used to extract the SIFs from the numerical results, according

Fig. 6. Deformed meshes.

Fig. 7. Normal stress distribution r11 around the two cracks.

Fig. 8. Normal stress distribution r22 around the two cracks.

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with the methodology reviewed by Ingraffea [13] and imple-mented in FRANC2D software [14,15].

The MCCI method was used successfully for determination ofSIF’s and ERR in bi-material models with cracks approaching theinterface [16], and for kinked cracks from interface [17].

In order to validate the analytical predictions for the twounequal collinear cracks, three numerical models wereconsidered with different distances (d = 4 mm, 8 mm and16 mm) between the two cracks, Fig. 4. PMMA material wasfirst considered with Young’s modulus E = 3250 MPa and Pois-son ratio m = 0.4 in order to calibrate the numerical model ver-sus analytical solution of Yokobori et al. [18] based oncontinuous distribution of infinitesimal dislocations, includedin Murakami’s stress intensity factors handbook [19]. Planestrain conditions were considered.

The FRANC2D code was used for numerical simulation witheight node isoparametric elements, [14]. The mesh consists of2852 isoparametric elements connected in 8319 nodes. Eight sin-gular elements were placed around the crack tip as a commontechnique to model the crack tip stress singularity, Fig. 5. The mod-el was uniaxially loaded with a normal stress p = 10 MPa.

A convergence study was performed considering different meshdensities at the crack tip in order to obtain high accuracy of numer-ical results. Table 3 presents a comparison for non-dimensionalMode I SIFs, FI ¼ KI=ðp

ffiffiffiffiffiffipap

Þ obtained numerical FI,FE and analyticalFI,AN according with [18]. It should be mentioned that analyticalresults are provided only for the tip P.

The errors between numerical and analytical results at tip P arebelow 1%, which validates the numerical methodology.

The orthotropic material was defined in Table 1. It correspondsto aramid-epoxy composite material. Three orientations betweenglobal x1-axis and 1st principal material direction were consideredb = 0�, 45� and 90�.

Fig. 9. Tangential stress distribution r12 around the two cracks.

Table 4Energy release rates (ERRs) results for orthotropic material.

Distance between tips, d (mm) Angle b (�) Energy release rates (J/m)

GIO GIIO GIP GIIP GIQ GIIQ GIR GIIR

4 0 95.1 0 104.6 0 267.8 0 258.7 045 160.1 64.1 177.6 71.5 467.5 190.7 445.4 181.390 20.84 0 23.9 0 67.51 0 65.16 0

8 0 85.58 0 88.11 0 258.8 0 256.1 045 143.4 55.47 148.7 57.76 439.5 176.0 435 176.790 21.1 0 22.26 0 67.01 0 66.92 0

16 0 79.95 0 80.6 0 255.3 0 254.6 045 134.5 51.24 135.7 51.19 444.3 183.0 444.8 187.190 19.9 0 19.62 0 64.45 0 64.59 0

Fig. 10. Normal stress r22 versus distance between cracks.

692 T. Sadowski et al. / Computational Materials Science 46 (2009) 687–693

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The deformed mesh for the case when b = 0� and two distancesd = 4 mm and 16 mm are shown in Fig. 6. Figs. 7–9 show the stressdistributions around the two cracks with d = 4 mm and 16 mm andb = 0�.

It could be observed in Figs. 7–9 that for the case when d = 4mm the stress field around the two cracks is influenced by thepresence of the neighbour crack. For the case of far cracks(d = 16 mm) the stress field around crack tip is not influenced byeach other.

Analyzing the numerical results of ERR’s for the orthotropiccase, Table 4, it could be observed that a strong interaction for closecracks (d = 4 mm) which decreases for far cracks (d = 16 mm). Forb = 45� presence of mixed mode (I and II) at the crack tip is shown.For each considered distance between cracks, the maximum valuesof ERR’s were obtained for b = 45� and minimum value for b = 90�.The maximum ERR values appeared in the tips of longer crack. Forthe case of far cracks (d = 16 mm) equal values at the tips of thecracks were obtained, while for close cracks (d = 4 mm) the valueof the ERR at the inner tip of the longer crack reached maximum,which confirmed the analytical predictions.

Comparison to results of the analytical model, Table 2, leads tothe conclusion that differences of ERR in the corresponding cracktips are of order 1–1.5%.

Fig. 10 shows the asymptotic distribution of normal stress r22

versus distance between the two tips for the orthotropic case withb = 0�. It could be observed that normal stress r22 values are higherat the crack tips P and Q when the distance between cracks is 4 mmand decreases for 8 mm and 16 mm.

6. Final remarks

The paper proposes the new analytical model for a single andtwo collinear cracks in orthotropic elastic material using Guz’s rep-resentation theorem [7]. Using complex analysis the solution leadsto the homogeneous Riemann–Hilbert problem. It is original ap-proach for two collinear cracks problem, e.g. [20].

The capacity of the model was illustrated in case of two collin-ear cracks of different lengths and distance for Mode I. The classicalproblem of cracks interaction was investigated. The cracks interac-tion strongly depends on the crack dimensions and distance be-tween them. If the lengths of the cracks are much greater thanthe distance between them, the interaction between the cracks isstrong and the cracks tend to unify. The interaction between cracksis weak when the cracks lengths are much smaller than the dis-tance between them.

In case of equal cracks all the tips start to propagate simulta-neously and in case of unequal cracks the tips of the longer crackstart to propagate nearly in the same time and larger cracks spreadto smaller one.

The analytical model was compared with numerical calcula-tions done within FEM. The obtained results are very close. Exper-

imental evidences for considered orthotropic material behaviour(for different cracks configurations) with application of the digitalimage ARAMIS system will be reported in [21]. The obtained re-sults lead to the conclusion that proposed model can be appliedto more complex cracks geometries, including inclined cracks.

Acknowledgements

This work was supported by the Marie Curie Transfer of Knowl-edge Project MTKD-CT-2004-014058: Modern Composite Materi-als Applied in Aerospace, Civil and Mechanical Engineering:Theoretical Modelling and Experimental Verification.

Financial support of Structural Funds in the Operational Pro-gramme – Innovative Economy (IE OP) financed from the EuropeanRegional Development Fund – Project No. POIG.0101.02-00-015/08is gratefully acknowledged.

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