Material and Geometry Effects on Stress Singularities in Anisotropic Heterogeneous Composites

The eigenvalue problem for a multi-material wedge composed of several anisotropic elastic sectors is formulated in a completely general manner, including the cases of degenerate and extra-degenerate material sectors, and various types of edge conditions for both open and closed wedges. Accurate elasticity solution of a multimaterial wedge may require many non-singular eigenfunctions in addition to the singular eigenfunctions. In the present work, over one hundred multi-material wedge configurations with various combinations of sector angles, material orientations and edge boundary conditions are examined, and their eigenvalues and eigenfunctions are determined by a combination of symbolic algebra and numerical root-finding. However, only the results of singular eigenvalues are shown in the paper. These results suggest certain material and geometrical aspects of the wedge as having particularly strong influence on the nature of singularity. INTRODUCTION Composite structures involving interfaces, joints, free edges and cracks generally develop singular elastic stress fields near the intersection of lines of material and geometrical discontinuity. Examples include interface cracks, transverse matrix cracks impinging upon an adjacent ply, lap and beveled adhesive joints, skinstiffener interfaces, and p ly drops in laminated structures. These localized regions of severe stress are possible sites of failure initiation and growth. The local geometry involves a multi-material wedge containing two or more anisotropic material sectors. These dissimilar sectors are bonded by radial interfaces which converge at the vertex of the wedge. A general analysis method has been developed to obtain accurate elasticity solutions of multimaterial wedges using a substructure approach [1, 2]. In ____________________________________________ * Professor; Associate Fellow, AIAA. Phone: (404) 894-2773; E-mail: wanlee.yin@ce.gatech.edu Copyright © 2002 by Wan-Lee Yin. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. this scheme, conventional finite element analysis of the global structure is performed to provide the traction boundary data on a path bordering the wedge. Accuracy of the data on this path may be ensured by requiring the path to be separated from the vertex by a distance of a few elements. An elasticity solution of the region interior to the path is obtained subsequently by constructing an eigenseries that matches the traction data at various points along the path using least square-error fitting. For bimaterial wedges associated with free edges or interface cracks, this method yields elasticity solutions that are in close agreement with existing numerical solution of a local model using singular finite elements. In the case of interface cracks, the energy release rate predicted by the dominant singularity of the elasticity solution is in excellent agreement with the result of the Jintegral evaluated along a remote boundary path. Further validation is shown by obtaining and comparing several eigenseries with different numbers of terms, by changing the collocation path, and by fitting the displacement data of the finite element solution, instead of the traction data. The stress field associated with an eigenfunction has the following form in polar coordinates Fij = C fij (2) r , (1) where 8 is a (real or complex) eigenvalue, and where the angle-dependent functions fij (2) in the successive sectors are related by the traction and displacement continuity conditions across the sector interfaces 2 = 2k . When Re[8] < 1, the stress field of the eigenfunction is singular. The severity of the dominant singularity (i.e., the eigenfunction with smallest Re[8]) depends not only on the power of singularity, Re[8]!1, but also on the multiplicative factor C, and the latter depends on mechanical and thermal loading. Much of the past research on multi-material wedges is focused on the calculation of the singular eigenvalues [46], with the apparent implication that the stronger the singularity (i.e., the smaller the eigenvalue), the more critical will be the stress state in the wedge. However, a 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con 22-25 April 2002, Denver, Colorado AIAA 2002-1760 Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. 2 wedge configuration that yields a stronger power of singularity (corresponding to a smaller value of Re[8]) may consistently give relatively small value of C under common types of loading. A question that does not suggest a simple answer is whether this poses a lesser or greater threat than another wedge having a weak order of singularity but yielding greater values of C under the same loads (which result in more severe stresses except in a neighborhood of minuscule, perhaps subatomic size, around the vertex). Presently, no adequate set of failure parameters and criteria is available for evaluating and discriminating among the competing stress singularities associated with the enormous variety of multi-material wedges with diverse geometry, edge conditions, and material combinations. The issue remains ambiguous even in the relatively simple case of bimaterial interface cracks, since the mode mixity, or the ratio of the stress intensity factors KI and KII , is not an intrinsic physical quantity under a given loading but depends on an arbitrary reference length scale . A useful evaluation of the criticality of the wedge stress canno t be based on the order of dominant singularity and the associated stress intensity factors or generalized stress intensity factor. As suggested by the elasticity solutions of a number of wedges, other singular eigenfunctions with different mode shapes may contribute just as significantly, in any region of physically meaningful size, under appropriate types of mechanical and thermal loads. Still, singular eigenvalues are important parameters because a strong order of singularity indicates the possibility of intense localized stress. They also may be obtained with little computational effort compared to the higher (nonsingular) eigenvalues. In this paper, the analytical formulation of the eigenvalue problem of anisotropic multi-material wedges is recapitulated, and extended to encompass explicitly general types of homogeneous edge conditions, including traction-free, rigid, sliding, floating and elastically supported edges. The richness and variety of multimaterial wedges is thereby substantially expanded, and the results point to the crucial importance of the combination of edge conditions on stress singularity. It can be shown that if a wedge is formed by one or two 180 sectors, then the characteristic equation can be solved exactly. The case includes mono-sector half planes with different boundary conditions on 2 = 0 and 2 = 180. It also includes the various cases of bimaterial interface cracks, some familiar (i.e., traction-free cracks) while others not. For other types of multi-material wedges, the eigenvalues must be determined numerically from the exceedingly lengthy characteristic equation that involves the sums and products of polynomial and exponential functions with various complex exponents. The task is achieved by using the Mathematica software, which includes and integrates the several capabilities indispensable for the task, i.e., symbolic algebra, numerical computation in complex numbers, and plotting and contour plotting of functions. Over one hundred bimaterial and tri-material wedges with various combinations of sector angles and edge conditions were examined in this study, and all real and complex eigenvalues with Re[8] < 5 are obtained for each wedge. However, only the results of the singu lar eigenvalues are reported in this paper because of their greater importance. The material in all sectors is taken to be the same strongly orthotropic unidirectional graphite/epoxy composite, but the orientation of its fiber axis vary from sector to sector, so that the effective elastic constants also vary significantly among the sectors. The reason for this choice is that previous studies have suggested the dominant influence of wedge geometry and edge conditions upon the singular eigenvalues, and the relatively milder effects due to the magnitude of elastic constants. By using the same orthotropic unidirectional composite in all sectors while varying the orientations, one can more easily compare the effects of varying sector angles, material orientations, and edge conditions upon the orders of singularity. Some general patterns of dependency may be discerned from the results obtained, and they are suggested as tentative conclusions. EIGENVALUES AND EIGENFUNCTIONS A general formulation of the eigenvalue problem of multi-material wedges in two-dimensional anisotropic elasticity has been presented in previous papers [2, 3]. The main results are recapitulated in this section. Let "ij (i, j = 1, . . . , 6) denote the anisotropic elastic compliance constants relating the strain components ,x, ,y, ,z , (yz , (xz , (xy to the stress components Fx , Fy , Fz , Jyz, Jxz, Jxy , and let $ij = "ij ! "i3 "j3 /"33 (for i, j 3). Then, for generalized plane deformations (i.e., deformations in which all strain components depend only on x and y), one has