Numerical Analysis of Dynamic Crack Propagation: Generation and Prediction Studies

Results of “generation” (determination of dynamic stress-intensity factor variation with time, for a specified crack-propagation history) studies, as well as “prediction” (determination of crack-propagation history for specified dynamic fracture toughness vs crack-velocity relationships) studies of dynamic crack propagation in plane-stress/strain situations are presented and discussed in detail. These studies were conducted by using a transient finite element method wherein the propagating stress-singularities near the propagating crack-tip have been accounted for. Details of numerical procedures for both the generation and prediction calculations are succinctly described. In both the generation and prediction studies, the present numerical results are compared with available experimental data. It is found that the important problem of dynamic crack propagation prediction can be accurately handled with the present procedures. INTRODUCTION FOR DYNAMIC crack propagation in finite elastic bodies, the interaction with the crack-tip of stress waves reflected from the boundaries and/or emanated by the other moving crack-tip plays an important role in determining the intensity of the dynamic singular stress-field at the considered crack-tip. Because of the analytical intractability of such elasto-dynamic crack-propagation problems, computational techniques are mandatory. A critical appraisal of several such computational techniques was made by Kanninen[ll in 1978. Most of the finite element techniques reviewed in[ll use conventional assumed displacement finite elements near the crack-tip and hence do not account for the known crack-tip singularity. Moreover in these techniques, crack-propagation was simulated by the well-known “node release” technique, which, as discussed in[l], may not be sufficiently accurate. The literature on dynamic finite element methods for simulation of fast fracture, since the appearance of [I], has been reviewed inI2-41. In Refs.[24], the authors have presented a “translating-singularity” finite element procedure for simulation of fast crack propagation in finite bodies. In this procedure, a singular-element, wherein the analytical eigen functions for a propagating crack in an infinite domain were used as basis functions for assumed displacements, was used near the crack-tip. In simulating crack-propagation, this singular element was translated by an arbitrary amount AC in each time-increment At of the time-integration scheme. During this translation, the crack-tip retains a fixed location within the singular element; however, the regular isoparametric elements surrounding the moving singular element deform appropriately. It was shown[2-4] that the above finite element method, which was based[2,3] on an energy-consistent variational principle for bodies with changing internal boundaries, leads to a direct evaluation of dynamic K-factors for propagating cracks. Attempts at simplifying the above procedure, by employing alternatively, a singular element with only the well-known Williams’ eigen-functions for a stationary crack being used as element basis-functions, or distorted triangular isoparametric elements (the so called “quarter-point elements”), in place of the above described singular-element, were made in [51. However, all the examples presented in [2-51 fall into the category of “generation studies” in the sense decribed earlier. Specifically, results for finite-domain counterparts of the well-known analytical problems for infinite domains, solved by Broberg, Freund, Nillsson, Thau and Lui, Sih et al (as referenced in [2,3]), were presented in[2-51, to indicate the effects of finite boundaries, and stress-wave interactions, on dynamic crack-tip stress-intensity, in these problems. In the present paper, which emphasises the “propagation” or “prediction” problem, namely the determination of crack-tip propagation history in a plane stress/strain problem for a specified dynamic fracture-toughness vs crack-velocity relation, the following topics are discussed: (i) a synopsis of the mathematical formulation for analysis of the “generation” problem; (ii) description of the details of analysis of the “prediction” problem; (iii) detailed description and discussion of the numerical results of both the “generation” and “prediction” studies of wedge-loaded rectangular double cantilever, and tapered double cantilever, beam specimens for which experimental data has been reported by Kalthoff et al. [6,71 and independent numerical results have been reported by Kobayashi et al. [8], and Popelar and 304 T. NISHIOKA and S. N. ATLURI Gehlen[9]. The present paper ends with some conclusions and a discussion of the open questions in numerical analysis of fast crack propagation in realistic metallic structures. SYNOPSIS OF THE FORMULATION OF “GENERATION” PROBLEM Consider two instants of time t, and t2 = t, t At. Assuming, without loss of generality, that the crack propagation is in pure mode I, let the crack lengths at ti and t2 be C, and & = XI t AC, respectively. Let the displacements, strains, and stresses at t, and tz be, respectively, (~lj, e{j, and a$), and (~2, E& and ~3. The variables at time t, are presumed known. It has been shown[2,3f that the variational principle governing the dynamic crack propagation between t, and t2 can be written as: In the above, V, is the domain, and sc2 the external boundary where time-dependent tractions are prescribed, at time t2; fj are the prescribed tractions at time tl at s,, (= s,J as well as at AX+; ( 1’ indicates the upper haif of the crack face, which only is considered in the present mode I problem. It is seen that CT~V/ are the cohesive forces holding the crack-faces together at time t,. Thus, it is seen that the integrand (o~jv/)‘(Su,2)” in the last term of the rhs. of eqn (1) corresponds to the term of energy-release rate due to dynamic crack propagation. The eqn (1) may thus be viewed as a virtual energy-balance relation for dynamic crack-propagation, and hence the present numerical method based on eqn (I) is inherently energy-consistent. In eqn (l), (ii/, oij) are known, while (o& ef, and uf) are the variables. Now, eqn (1) is used to develop a finite element approximation at time tz. Thus, the domain V, is discretized into a finite number of elements, with a domain V, immediately-surro~ding the crack-tip being treated as the so-called “singular element”, and the domain VTV, being mapped by the welI-known, 8-noded, isoparametric elements. In the singular-element V,, the basis functions for assumed displacements are the crackvelocity dependent eigen-function solutions to the elasto-dynamic problem of crack-propagation in an infinite domain, as discussed in this paper. Note that at time fZ, in the present mode I problem, the crack tip is located at x1 = C, + AX and hence the singular-element is centered at x1 = X, + AC. In developing the equations for the finite element mesh at tZ, is is seen from eqn (1) that the variation of pij and u/ must be known in the finite element mesh at tp. However, oijy u/, and ti/ were solved for, in the finite element mesh at t,. In the mesh at t, the crack-tip was located at x1 = 2, and hence the crack element was centered at Xi. Thus between t, and tz (t, + At) the crack element is translated by an amount AZ. While the crack-element is translated, only the elements surrounding the moving crack-tip are distorted. Thus the finite element meshes at times tl and t2 differ only in the location of the crack-tip (and hence the crack-element) and the shapes of the immediately surrounding isoparametric elements. Thus, the known data at ci and uif in the mesh at t, is interpolated easily into corresponding data in the mesh at tz. Further details of the above translatingsingularity-element method of simulating dynamic crack propagation in arbitrary shaped finite bodies can be found in[2,3]. We now remark briefly on the basis functions for assumed displacments used in the singular element. Let .Q((Y = 1,2) be fixed rectangular coordinates in the plane of the present Zdimensional elastic body, with the crack-tip moving along the x, axis and x2 is normal to the crack-axis. We introduce a coordinate system (5, x,) which remains fixed w.r.t. the propagating crack-tip, such that 6 = xl vt, where v is, without loss of generality, the constant speed of crack-propagation. It can be shown12,33 that the elastodynamic equations, governing this problem, for the wave potentials 4 (dilatational) and II, (shear) are: [I (u/cd)21(~2~~~~2) + ( z~l~x2z) = (2u~c~)(~2~l~t~~) t (l~c~)(~z#~~t*) (2) and a similar equation for 4, except that cd in eqn (2) is to be replaced by c,, where cd and c,~ are the Numerical analysis of dynamic crack propagation 305 dilatational and shear wave speeds respectively. The “steady-state” eigen-function solution to the homogeneous part of eqn (2), namely, the solution which appears time-invariant to an observer moving with the cracktip, and satisfies the prescribed traction conditions on the crack face ([ < 0, x2 = + 0) can be derived easily, as indicated in [2] and elsewhere. We use these eigen function solutions for an infinite body, as basis functions for assumed displacements within the “crack-tip-singularity-element”. However to satisfy the full eqn (2), the undetermined coefficients, pi below, in the eigen function expansion are taken to be functions of time. Thus, within the singular element, U,(~,X2,f)=Uaj(5,X2,V)Pj(t) [a=1,2; i= 1,2..Nl (3) where uUj are the above described eigen-functions, and pj are undetermined parameters, which are to be determined from the finite element equations for the cracked body. As seen from eqn (3), the eigen functions U,j depend on the crack-tip velocity. In the present numerical approach, the crack-tip velocity is assumed to be constant within each time-increment At, say U, between t, and t, +At, and u2 between t2 and tZ+Af, etc. Thus, between t, and t, +At, the eigen-functions embedded in the singularity-element correspond to velocity U, and those between t2 and f2 t At correspond to velocity v2. Thus, the present finite element method is capable of handling non-uniform-velocity crack propagation. The total velocities and accelerations of a material particle in the singular element, within each time step, corresponding to eqn (3), can be written as: lit = Uaibj VU,j.@j (4)