The treatment of crack propagation in inhomogeneous materials using the boundary element method A. Boussekine," L. Ulmet," S. Caperaa* " Laboratoire de Genie Civil, Universite de Limoges, 19300 Egletons, France * Ecole Nationale d'ingenieurs de Tarbes, 6500 Tarbes, France ABSTRACT The purpose of this paper consists in the implementation of a boundary element model for which the crack propagation is easily studied thanks to an automatic remeshing process. The traction singular quarter point element was used in order to improve the accuracy of the values of stress intensity factors. The Erdogan and Sih criterion allows us to determine the new growth direction after each step. Validity of this implement is demonstrated, with respect to experimental results, by the example of a holed plate. In this study, we can show the manner in which the crack may propagate when the distance between the initial crack's axis and the tangent hole is changed. Two distinct behaviours can be pointed out with respect to this distance : in some cases the crack may be simply deflected by the hole, and propagates across the plate until a complete fracture is reached. In other ones, the crack propagates toward and reaches the hole. In the framework of fatigue under cyclic loading, the manner in which growth rate is affected by the proximity of a hole is shown. INTRODUCTION The use of the Boundary Element Method for the solution of crack problems has been of interest to many investigators over recent years. However, for non-symetric problems where the two crack surfaces have the same geometrical coordinates, the direct application of this method leads to a singular system of equations. Some special techniques have been established to overcome this difficulty. The most general are the subregions method introduced by Blandford et al [1] and the dual bondary element method established by Portela et al [2]. In our study we have used the first method for the modelisation of the crack problems. We have shown [3] that the use of this method does not alter the automatization of crack propagation.
376 Localized Damage THE BOUNDARY ELEMENT METHOD The boundary integral equation relates displacements i% and t< on the boundary F of an elastic homogeneous isotropic body. It can be written as :, Q)uj(Q)dT(Q) = f 0y (P, Q)ti(Q)dT(Q) (1) Jr In which the tensors T^ and Uij are the components number j of the tractions and displacements at Q due to a unit i direction load at P. The term Cij represents the behaviour of the singularity in Ty. When P is on a smoothpart of T, Cij = &,/2. Uj and tj denote components of the boundary displacement vector and the traction vector respectively. The boundary is now divided into N isoparametric elements. By using a collocation method, the system of equations is produced on the form : M {«}=[«]«(2) The matrices [A] and [J3] contain coefficients of either Tij and U^. METHOD FOR THE CALCULATION OF STRESS INTENSITY FACTOR 'SIP' In order to evaluate correctly and accuratly the stress intensity factor, we have compared several methods like : the J Integral, Traction Singular Quarter Point Element (TSQP) and the Crack Opening Displacement method in which Kj is calculated for different formulations : Figure 1. Quarter point element /c = (3 4f/) for plane strain and =(3 %/)/(! + %/) for plane stress In figure 2, we denote as follows : Quarter point elements and one-point displacement formula Quarter point elements and two point displacement formula KIUIS Traction singular quarter-point elements and one-point displacement formula Traction singular quarter-point elements and two-point displacement formula Kit Traction singular quarter-point elements and nodal value formula The extrapolation method of the displacement a, I The crack size and the lenght of the element at crack tip This analysis is similar to that studied by Blandford et al [1] and Dominguez et al [4]. The error upon Kj obtained by the use of the j-integral, for several problems, is about 2%. We present below the curves of this error for the center crack plate. These results are compared with analytical solution given by Tada et al [5]
Different methods for SIF calculation Localized Damage 377 Figure 2 : Comparison of the different methods STRESS INTENSITY FACTOR CALCULATION USING TSQP In the aim of studying crack propagation in mixed mode, the domain is divided in two subdomains by mean of an internal boundary called Active interface. This technique posses a lot of advantages : on the one hand, it enables us to avoid a singular matrix, where two similar sets of equation are produced, on the other hand, it allows us to study an interfacial crack occuring between two different materials. Each propagation step is performed by removing an element from the interface to the crack faces, behind the crack tip. Such a procedure modifies only the mesh in the immediate vicinity of the crack head. In this case, it is possible to have an angular interfacial singularity. The cancel of this problem is realised by choosing a Active interface angle just at crack tip, as shown in figure 3 : angular singularity crack tip fictivc interface Figure 3 : Interfacial angular singularity
378 Localized Damage In this configuration, the stress intensity factors can be calculated as follows : The components of the stress vector </% r" are expressed by : r" = - cr(i sin 0 cos 0 + a'^ sin 9 cos 0 + a'^ cos 29 a" =(7(1 sin* 6 + 032 cos* 0-2<7Jg cos 0 sin 0 where <J^ are expressed by Irwin [6] as follows : (4) The substitution of the last equation in the first one, leads to : Where : =(/22 - /ii) sin 0cos0 + /«cos 20 =(<7u + /22)sin0cos0 + Pi2cos20 =/n sin* 0 -f /22 cos* 9-2/12 sin 9 cos 0 (5) =0ii sin* 9 + 022 cos* 9 2pi2 sin 9 cos 0 the vector {r",cr"} is related to the vector {ti,^} by : r" / cos a sinawti The use of the singular traction quarter point boundary element (TSQP) proposed by Cruse and Wilson [7], leads directly to stress intensity factors [4] : where t7, TI are the traction values at the crack tip, and / is the TSQP element length. and finally, the expression between the ti, tj and JC/* KII is : -sino
Localized Damage 379 For several examples, we have compared the values of Kj and KJJ for different angles of the Active interface with the reference value. This latter corresponds to the case for which the interface is on the crack's axis. These results show the strong sensibility of stress intensity factor values with the Active interface angle, when the absolute value of this angle is greater than 60.,. Kj Kjref, Kjj we denote e\ - - - and 63 = - angle o 0.00 26.56 45.00 56.31 63.43 79.69 100.30 el (%) 0.00 1.16 2.30 3.34 4.40 5.80 11.70 Active i crack state of reference Figure 4- Influence of thefictiveinterface angle for mode I angle 0 10 5-5 -10-15 -20-25 -30-35 -40-45 ei(%) 0.00 0.16 0.10-0.09-0.14-0.13 0.33 0.10-0.21 0.03 0.02 0.12 *,(%) 0-0.20-0.04-0.07-0.30-0.71-0.40-0.15 40.03-0.40-0.35-0.36 fictive intei crack state of reference Figure 5. Influence of the fictive interface angle for mixed mode ERDOGAN AND SIH CRITERION After the calculation of the failure parameters Kj and KU, it is possible to determine the direction for which the crack propagates. In this aim several criteria may be used. These criteria include the minimum strain energy theory [8], the maximum energy release rate theory [9], and the maximum principal stress (maximum tangential tensile stress) theory [10]. The last criterion shows better agreement with experimental data when it is compared to two other methods [11]. This criterion assumes that the crack propagates in the direction which is normal to the maximum circumferential stress o>0. This direction
380 Localized Damage is defined by an angle 0o» which is expressed as follows : 00 = arctant -- (8) The crack is usually supposed to be unstable, which enables the propagation. FATIGUE CRACK GROWTH The growth rate for mode / loading is determined by the use of Paris's law [12] as : da dn (9) in wich, a is the crack's length, N is the number of cycles, C and m are material properties and AHf/ is the range of the stress intensity factor. The majority of failure problems in engineering are of mixed mode type, where the crack does not propagate in the direction normal to the load's axis because of the lack of geometric symetry. Tanaka [13] generalised the Paris model to the mixed mode as follows : the "effective" range of K is defined by : NUMERICAL RESULTS The following example shows the manner by which our boundary element program carried-out the crack propagation. The results are presented in figure (6) for different distances d between initial crack's axis and the tangent hole. The propagation trajectory for each initial crack is presented in figure (7). The figure (8) shows the experimental propagation trajectory. This testing was carried out on plexiglass material. The distance d between the crack and the hole was taken equal to 1. Number of cycle according to the distance d - N (d-0,9) - N (d=0,8) N (d=0,7) N (d-0,6) - N (d=0,5) N (d=0,4) crack size Figure 6 : Cycle number versus crack's size
Localized Damage 381 irnmimmmt!mnu = i Hole radius R = 0.7 Distance d = 1---- > 0.3. Initial crack size a = 0.4 W = 4 mnrnmiiuihimiu = i Before propagation After propagation Figure 7 : Propagation trajectory Figure 8 : Experimental trajectory
382 Localized Damage CONCLUSION In this paper, we have shown by using Paris's law that the crack reveals three steps of propagation. In thefirstone, it propagates very quickly before the hole. When the crack is just under the hole, it slows down. In this stage, the crack will meet the hole if the distance d is small, else it continues its propagation with again acceleration until the failure occurs. The first part of this work points out the good accuracy of the results obtained by the traction singular quarter point element method (TSQP). The incremental propagation process, using this kind of element, is completely automatic in the computation and in the generation of the successive meshes. The remeshing scheme is simplified thanks to the opportunity of having an interfacial angle at crack tip. This angle doesn't affect the accuracy on stress intensity factor computation. REFERENCES [1] G.E.Blandford, A.R.Ingraffea and J.A.Liggett Two Dimensional Stress Intensity Factor Computation using the Boundary Element Method Int.J. numer.meth, Engng 17 387 404 (1981) [2] A.Portela,M.H.Aliabadi and D.P.Rooke The Dual Boundary Element Method : Effective Implementations for Crack Problems, Int.J.Numer.Meth.Engng 33 pp :1269-1288 (1992) [3] A.Boussekine, L.Ulmet, J.Rosier A Boundary Element Method for modelling crack propagation Mechanisms in Inhomogeneous Materials (Ed. C.A.Brebbia, J.Dominguei, F. Paris), Boundary Elements 14 vol 2pp.383-394,(1992) [4] J. Martinez and J.Domingue* Short Communication On The Use Of Quarter Point Boundary Element for Stress Intensity Factor Computation [5] H.Tada,P.C.Paris, G.R.Irwin Stress analysis of cracks handbook, Del research corporation, Hellertown,Pa,USA,1973 [6] G.R. Irwin Fracture in Encyclopaedia of Physics (Ed. S. Flugge), Vol VI, Springer-Verlag, 1958 [7] T. Cruse and R.B. Wilson Boundary Integral Equation Method for Elastic Fracture Mechanics AFSOR-TR-78-0355,10-11,1977 [8] G.C. Sih Strain energy density factor applied to mixed mode crack problem Int.J.Fracture 11, 305-321 (1974) [9] M. Hussain, S.L. Pu and J.H. Underwood Strain energy release rate for a crack under combined mode I and mode II Fract. Anal.,ASTM STP 560, 2-28 (1974) [10] F.Erdogan and G.C. Sih On the crack extension in plates under plane loading and transverse shear, J. Basic Engng Fracture Mech. 5,647-655 (1973) [11] D. Broek and J. Rice Fatigue crack growth properties of rail steels Battelle report to DOT/TC(1976) [12] P.O. Paris and F.Erdogan A critical analysis of crack propagation laws J.bas.Engng,Trans.ASME,Ser. D 85, 528-533 (1963) [IS] K. Tanaka Fatigue crack propagation from a crack inclined to the cyclic tensile axis, Engng.Fract. Mech.,Vol.6,PP 493-507. 1974