Modeling of edge cracks interaction

From experimental and theoretical investigations it is known that cracks are sensitive to geometry, e.g., to the inclination angle to the load. A small deviation of a crack from the normal direction to a tensile load causes mixed mode conditions near the crack tip which lead to deviation of the crack from its initial propagation direction. Besides, the presence of other cracks, inhomogeneities, surfaces and their interaction causes additional deformations and stresses which also have influence on the initiation of the crack propagation and on the direction of this propagation. The aim of this paper is to show the effects of the interaction of edge cracks on further crack formation. The main fracture characteristics, such as, stress intensity factors, fracture angles and critical loads are provided in this study. A series of illustrative examples is presented for different geometries of arbitrarily inclined edge cracks.


INTRODUCTION
urface cracking is observed in many engineering structures, e.g., aircraft structures, turbine blades, engine components and many others, see [1,2] for some examples and references.In everyday life we can see asphalt pavement cracking, called also as crocodile cracking [3].The structures are subjected to different mechanical and thermal loading as well have to resist high temperature, wear and aggressive environments.Cracks can initiate from initial defects or microcracks appear during manufacturing or service.An example of multiple surface cracking is the fracture of thermal barrier coatings (TBCs), where the upper layer is usually a ceramic -the brittle material.Investigations of thermal barrier coatings show that heating and then subsequent cooling of the coating causes the surface to experience a tensile stress leading to surface cracking [4].

S
There is abundant experimental results (e.g., [4][5][6]) showing that when TBCs are subjected to thermal shock, multiple cracks occur at the ceramic surface.Besides, the crack patterns strongly depend on the microstructure of the materials and on the type of loading.Numerous investigations are devoted to different types of fracture including surface fracture.In previous papers of the authors [7][8][9][10] the fracture of FGM/homogeneous bimaterials (an infinite medium) under thermal and mechanical loadings were investigated, besides, in [11] some results for edge cracks in FGM/homogeneous structures (a semi-infinite medium) were obtained in the frame of the approach used in [7][8][9][10].The results show that the fracture of materials (both composites and homogeneous) is significantly affected by a complex crack interaction mechanism, e.g., interacting cracks can enhance or suppress the propagation of each other.During further studying of the fracture of functionally graded coatings on a homogeneous substrate and the preparation of the results for the influence of material non-homogeneity on surface fracture it became clear that a classical problem for edge cracks interaction is still not well examined.Before presenting the results for more complicated cases of nonhomogeneous materials (FGMs, bimaterials, and others), modeling of the interaction of edge cracks should be done for a homogeneous medium.From experimental and theoretical investigations it is known that cracks are sensitive to geometry, e.g., to the inclination angle to the load.A small deviation of a crack from the normal direction to a tensile load causes mixed mode conditions near the crack which lead to deviation of the crack from its initial propagation direction.Besides, the presence of other cracks, inhomogeneities, surfaces and their interaction causes additional deformations and stresses which are also influenced on the initiation of the crack propagation and on the direction of this propagation.That is, the picture of the fracture with respect to the crack pattern for a system of arbitrary inclined edge cracks will be different from the picture of regularly distributed cracks, e.g, for periodically distributed equal (and non-equal) cracks, this case was often studied, see [12][13][14].The goal of this paper is to show the effects of the interaction of edge cracks on further fracture formation.The main fracture characteristics, such as, stress intensity factors, fracture angles and critical loads are provided for this study.A series of illustrative examples is presented for different geometries of arbitrarily inclined edge cracks.

PROBLEM FORMULATION AND ASSUMPTIONS
he geometry of the problem is presented in Fig. 1 a.A homogeneous half-medium contains pre-existing edge cracks inclined arbitrarily on angles βn to the surface.A Cartesian coordinate system (x, y) has x-axis along the boundary of the half-plane, and local coordinate systems (x k , y k ) are attached to each crack.The lengths of the cracks are 2an, and the midpoint coordinates are is imaginary unity).The homogeneous medium is subjected to tension p applied parallel to the free surface.The problem is solved by using the method of singular integral equations.The cracks are modeled by displacement jumps on the crack faces and unknown functions in this formulation are the derivatives of displacement jumps For arbitrary located cracks in a half-plane the system of singular integral equations is written as [15,16] An overbar   ... is the complex conjugate.N is number of cracks.The method of superposition was used in deriving of Eq. ( 2) where the loads at infinity are reduced to the corresponding loads on the crack faces.The functions p n in the right side of Eq. ( 2) are these loadings, and in the case of a homogeneous half-plane under tension p they are written as (see Fig. 1 a).If a non-homogeneous medium is considered, e.g., a functionally graded structure with continuous gradation of the thermo-mechanical properties with the coordinate y, and this structure is cooled, then tensile residual stresses are arising due to mismatch in the coefficients of thermal expansion [4,14].The influence of this inhomogeneity can be accounted via continuously varying residual stresses p* which are written as follows [14]: This function is added to the right side of Eq. ( 2).It should be noted, that in this case we also have the problem for a halfplane under tension.

NUMERICAL SOLUTION, STRESS INTENSITY FACTORS
he solution of singular integral equations (Eq.2) is obtained by a numerical method which is based on Gauss-Chebyshev quadrature.The method is similar to the method presented by Erdogan and Gupta [17], but we will follow the version formulated in [15,16].The equations (2) are rewritten in dimensionless form with the non-dimensionless coordinates For edge cracks the function possess a singularity less than at the edge point 1    and this condition is accounted as [15,16] In spite of the exact singularity at the edge points is not taking into account, the numerical results have shown good accuracy [15,16].Using Gauss's quadrature formulae for the regular and singular integrals the integral equations are reduced to the following system of NxM (N -number of cracks, M -number of nodes) algebraic equations with M is the total number of discrete points of the unknown functions within the interval (-1,1).Applying the conjugate operation to the system (6) additional NxM equations are obtained, i.e. 2NxM equations should be solved, where N is the number of cracks.Eq. ( 7) is obtained from the condition (5) and the interpolation formula for the functions Here T r are Chebyshev polynomials of the first kind.Inserting (8) into Eq.( 4) the derivative of displacement jumps on the crack lines are obtained and then the displacement jumps can be derived by integrating the function ( 4) with (8).
The stress intensity factors (SIFs) are calculated according to the following formula

CRITICAL LOADS, FRACTURE ANGLES
or general crack problems the stress intensity factors are both nonzero, i.e. mixed-mode conditions are in the vicinity of cracks.For this mixed-mode case the cracks deviate from their initial propagation direction.For the prediction of the crack growth and direction of this growth a fracture criterion should be applied.Using the maximum circumferential stress criterion (see [18] and for references [15,16]) the direction of the initial crack propagation (Fig. 1 b) is evaluated as and the critical stresses can be calculated from the expression Here KIc is the fracture toughness of the material.The critical stresses are given as Here and p0= KIc/(2πa) 1/2 is critical load for a single crack in a material with the fracture toughness KIc.
For the system of cracks the fracture starts from the crack tip where P cr is minimal, i.

RESULTS AND DISCUSSION
ome examples for edge crack interaction is investigated and presented here for homogeneous materials.The verification of the method and the numerical outcomes has been done in [11], where the results for some special cases were compared with the results for SIFs for a single inclined edge crack cited in [19] and with SIFs for periodic edge cracks cited in [20].The tensile loading p is applied parallel to the boundary and on the crack lines we have the loading Eq. ( 3).The nondimensional stress intensity factors Mode I and Mode II ( II I k , ) are defined by Eqs. ( 9) and ( 15).Non-dimensional k I for a single edge crack normal to the surface is equal to kI =1.12 and SIF kII is kII = 0.  10) and critical loads pcr by Eq. ( 12).The schemes of the direction of cracks propagation are shown in Fig. 11 for inclination angles β=90° and 60° and for different distances between the cracks.For a single edge crack with β=90° the fracture angle is equal to ϕ=0°.It is shown in Fig. 11 a the both cracks will change the direction of the propagation and the fracture angles are larger for closely located cracks for d=2 than for the distances d=4 and 6 between the cracks where the fracture angles are ϕ=12° and 8°.
The results for two unequal cracks are shown in Fig. 11 c.The crack 1 slightly deviates away from crack 2 if the distance is d=2 and will propagate straight for far distances between the cracks.At the same time the influence of the big crack on the propagation direction of the small is rather strong.Fig. 11 d show the fracture angles for the cracks inclined under the angle β=60°.For a single edge crack with β=60° the fracture angle is equal to 31°.The difference in the fracture angles due to their interaction is dependent on the distance between the cracks and the size of these cracks.As shown in Figs. 3, 6 and 9, there are crack configurations for which cracks are not deviating from their initial direction, i.e. ϕ =0.Tab. 1 presents such results, e.g., crack 1 has the fracture angle ϕ =0° for β 1,2 =103°, the crack 2 for this case has ϕ =30°.For close located cracks (for d=1) with β 1,2 =90° the crack interaction gives the fracture angles (ϕ 1 , ϕ 2 )=(0°, 50°), see column first in the Tab. 1.For a single edge crack for β=90° the crack has the zero fracture angle.

Three arbitrary inclined cracks
Consider the case for three arbitrary inclined edge cracks.Figs. ( 10)-( 12) are for the cracks with same sizes and Figs. ( 15)-( 22) for different sizes, i.e. a 1 =1, a 2 =0.5 and a 3 =0.5.Fig. 12 shows the SIFs kI and kII , Fig. 13 -the fracture angles, Figs. 14 -the critical loads as functions of inclination angle β=βn (n=1, 2, 3) for three interacting edge cracks of the same size.The results are presented for the crack 2 (the middle crack), for other cracks 1 and 3, the plots are similar to the previous case for two interacting cracks, Fig. 2. The distances between the cracks are equal (see Fig. 21 a) and the calculation is performed for d=2, 4 and 6.
The curves for kI for all three cracks are similar, but the values for kI are different kI(2) < kI(1,3) for all β and d.The influence of the distance on the k I is stronger for the crack 2, than for cracks 1 and 3.The SIF k II (2) is also less than k II for cracks 1 and 3 and have small dependence on the distance d.The strong shielding effect is observed for the middle crack 2.
The fracture angles ϕ are presented in Fig. 13 only for the crack 2, for cracks 1 and 3 they are nearly the same as in Fig. 3 for two cracks.Some selected cases for directions of crack propagation are shown in Fig. 21 a, b.The influence of the inclination angle on the fracture angles is evident as well as the influence of interaction between the cracks.Fig. 14 shows the non-dimensional critical loads for the three equal cracks, Fig. 14 a for 60°≤ β ≤120° and Fig. 14 b for 15°≤ β ≤90°.The critical loads p cr(1,3) < p cr (2) for all parameters, hence the outer cracks 1 and 3 will start to propagate first.Some results are presented in Figs. 15 -17 for cracks with different sizes a 1 =1 and a 2 = a 3 =0.5 and equally inclined to the surface.Fig. 15 shows SIFs for cracks 2 and 3, the curves for the crack 1 (the bigger crack) are similar to the curves for the crack 1 in Fig. 5 a, c. k I(2) < k I(3) and k I(2,3) < k I(1) for all parameters, that is, the maximum shielding effect is observed for the crack 2. The SIFs k II are small (close to zero) and their absolute values for cracks 2 and 3 are less than for crack 1.
The fracture angles ϕ are presented in Fig. 16 for the small cracks 2 and 3 and for the crack 1 the curves are similar to the curves in Fig. 6 a, c.Some schemes for the direction of the crack propagation are presented in Fig. 21 c, d.The non-dimensional critical loads for the three unequal cracks are presented in Fig. 17. p cr(2) >> p cr (1,3) for 60°≤ β ≤120° and for 15°≤ β ≤90° (d=2), besides p cr(2) > p cr(3) > p cr (1) .The larger crack will propagate first.The last case for the three edge cracks is presented in Figs.18 -22.The sizes of cracks are equal, the first crack is inclined with β 1 =90° and β 2,3 =β are varied.The SIFs are shown in Fig. 18.The dependence of k I and k II with changing β are similar as for two interacting cracks, Fig. 8, but the values of k I are smaller for the three crack case for all parameters.The fracture angles are presented in Fig. 19.The dependence of ϕ with β for crack 3 is similar to the two-crack case in Fig. 9 b.The fracture angles for the crack 1 (β1 =90°) are larger than for the crack 1 interacting with only one crack, Fig. 9 a, and they are much smaller than the values ϕ for cracks 2 and 3, as expected, because of a single crack with β1 =90° don't change the direction of propagation.Fig. 20 shows the non-dimensional critical loads for this case of interacting cracks, Fig. 20 a for the crack 1 and Fig. 20 b is for three cracks.The critical load for the middle crack 2 is much larger than for other cracks for all β values.For inclinations β 2,3 =β close to 90° (88°≤ β ≤102°), when the three cracks nearly parallel, p cr(1) = p cr(3) , so that the cracks 1 and 3 will start to propagate first, for other inclination angles the weaker is the crack 1.

Crack interaction effect
The shielding -amplification effects are observed in crack interaction problems, i.e. some geometries of the interacting cracks can enhance or suppress the propagation of each other.This problem has long history of discussion especially for macro-microcrack interaction problems [21].In a recent paper by Wang et al [22] the authors return to this problem.The shielding and amplification effects of transverse array of microcracks on a main crack were investigated using extended FEM with respect to SIFs.The SIFs at the main crack tip include not only the interaction between the main crack and each of small cracks, but also include mutual interaction of microcracks.It was pointed out that the interaction between microcracks can weaken their amplification-shielding effect on the main crack and this effect was demonstrated in [22].
In the present problem for edge cracks we can check whether this interaction effect presents or not.Consider the case of two edge cracks with sizes (a1, a2)=(1.0,0.5) and three edge cracks with sizes (a1, a2, a3)=(1.0,0.5, 0.5), in both cases the cracks have same inclination angle β to the surface βn=β.
Introduce the a notation Δ1 = kI (1) -k 0 for two cracks with distance between the cracks d=2 and for distance d=4 for 3 cracks with equal distances between the cracks d=2, where k I (n) is the SIF at the tip of large crack, k 0 is the SIF for a single edge crack.These values of Δ determine interaction effects between the cracks, i.e. the influence of small cracks on the large crack 1.
The value Δ 1 + Δ 2 -Δ 3 will give the magnitude of mutual interaction of small cracks (crack 2 and 3) and is the relative value of this small crack interaction.The data for SIFs are given in Tab. 2. SIF k 0 =0.92 for a single crack with inclination angle β=60° can be found in [19] in Tab. 4 or in [11] in Tab. 1.The SIF kI (1) for crack 1 as influenced by a crack 2 on the distance d=2 is 1.094 and this value is less than SIF k 0 =1.12 on the value |Δ1|=0.026(first column in Tab. 2) , and for the cracks on the distance d=4 the SIF kI (2) =1.071 is less than k 0 on |Δ 2 |=0.05 (second column in Tab. 2).The total interaction effect of the cracks 1 and 2 on the large crack 1 is |Δ 1 + Δ2|=0.076(if superposition of these interactions is assumed).The SIF kI for crack 1 under the influence of pair interacting cracks 2 and 3 is kI (3) =1.061 (distances between the cracks are d=2) and |Δ 3 |=0.06 is for this case.|Δ 3 | is less than |Δ 1 + Δ 2 | on 0.016.That is, the solution to the problem of three edge cracks gives interaction effect on 0.016 less than the superimposed effect of two separate cracks derived from the problems of two-crack interaction.It is an example of calculations for β=90°, for β=60° calculations are similar.Tab. 2 in the last column show that the interaction between small cracks (crack 2 and 3) weaken the shielding effect on 1.5% for the cracks inclined on β=90° and on 2.17% for the cracks with β=60°.It is rather small values, but, probably, for interaction of multiple cracks the effect will be stronger.

CONCLUSIONS
he effects of the interaction of edge cracks on further crack formation were studied with respect to main fracture characteristics, namely, stress intensity factors, fracture angles and critical loads.Some illustrative examples show the influence of inclination angles, distances between the cracks and sizes of the cracks on this interaction.The interaction of cracks leads to mixed mode conditions near the crack tips even for symmetric geometries and loading normal to the crack lines.As an illustration, a classical edge crack (i.e. the crack normal to the boundary and under tension normal to the crack line) in the presence of other crack is under mixed mode conditions and deviates from its initial propagation direction, albeit the other crack is small (Fig. 9 a and 11 c).The crack shielding takes place (as expected for this problem) for most parameters of the problem.The maximum magnitude of the shielding effect is observed for close located cracks and for a middle crack in the case of the interaction of three cracks.The influence of two interacting cracks on the third crack can weaken the shielding effect and it was shown for SIF Mode I which is dominant in this problem.The applied method of singular integral equations (which have been solved by well-known numerical method based on the quadratic formulas for integrals) in combination of a fracture criteria (the maximum hoop stress criteria has been used) is effective method for modeling of the crack interaction at the initial stage of their propagation.

ACKNOWLEDGEMENT
he support of the German Research Foundation under the grant Schm 746/139-1 is greatly acknowledged.


The kernels K nk (t,x) and L nk (t,x) are the same as for the system of cracks in an infinite plane, and the additional terms in Eqs.(A.1) and (A.2) are responsible for the influence of the edge of the half plane.αn is the inclination angle of n-th crack to the x-axis and

Figure 1 :
Figure 1: (a) Edge cracks inclined arbitrarily with an angle βn to the surface of the medium.an -a half length of n-th crack, zn 0 = xn 0 +i yn 0 -the crack midpoint coordinate.(b) The angle ϕ of crack deflection (the fracture angle).

T
Here [u n ] and [v n ] are shear and vertical displacement jumps, respectively, on the n-th crack line, modulus, E -Young's modulus,  -Poisson's ratio, a length of the k-th crack.The unknown function ) continuous function in the segment [-1,1]) and the weight function

2
The non-dimensional distances a d d /  between the cracks are d = 1, 2, 4, 6, k k a a max  and we remind that 2ak is the size of the k-th crack.After obtaining SIFs the fracture angles  are calculated by Eq. (

Figure 12 :Figure 13 :Figure 14 :
Figure 12: Stress intensity factors kI and kII as functions of the inclination angle β=βn of the edge cracks to the surface for different distances d between the cracks: (a) for crack 2 (60°≤ β ≤120°) (b) for crack 2 (15°≤ β ≤90°) (SIFs for cracks 1 and 3 are similar as in Figs. 2 a, b for two crack).Three equal edge cracks.

APPENDIXn
Eqs.(2) the kernels R nk (t,x) and S nk (t,x) are written as

Table 1 :
Some cases for crack configurations for which cracks are not deviating from their initial direction.

Table 2 :
SIFs kI at crack 1 interacting with small cracks and their interaction effects.