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Boundary Element Applications to Polymer Fracture

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Boundary Element Applications to Polymer Fracture Transactions on Modelling and Simulation vol 35, © 2003 WIT Press, www.witpress.com, ISSN 1743-355X Boundary element applications to polymer fracture S. Syngellakis & J. Wu School of Engineering Sciences, University of Southampton, U.K. Abstract This paper reports on applications of the boundary element method (BEM) to polymer fracture. Both Laplace transformed and time domain two-dimensional analyses, based on linear viscoelasticity, are developed and applied to centre- cracked plates under tension in order to assess their relative accuracy and efficiency. The time dependence of stress intensity factors is assessed for various viscoelastic models as well as loading conditions. Various approaches for direct assessment of the energy release rate are proposed; its representation through path-independent J-type integrals is also explored. Systematic procedures for J- integral evaluation are developed requiring the derivation of additional boundary integral equations for the displacement gradient distributions in the solid domain. The BEM formulation is extended to the determination of the energy dissipation rate for incremental crack growth. Then, the application of conservation of energy, combined with the knowledge of the critical and current values of strain energy release rate, leads to the assessment of the crack growth rate. Numerical results are compared with other analytical solutions and some experimental measurements. The consistency between BEM predictions and other published results confirms the method as a valid modelling tool for polymer fracture characterisation and investigation under complex conditions. 1 Introduction The increasing use of polymers has prompted extensive research on their failure mechanisms. Polymer fracture, in particular, has been the subject of many theoretical studies concerned with the identification and determination of parameters governing crack initiation and growth. The finite element method has been mainly applied to simulations of fracture behaviour in polymer matrix Transactions on Modelling and Simulation vol 35, © 2003 WIT Press, www.witpress.com, ISSN 1743-355X 84 Boutrdary Elcmatr~~XXV composites [l]. Relatively few attempts have been made to predct material parameters characterising polymer fracture. The application of the boundary element method (BEM) in this area has been even more limited. An early such attempt was concerned the prediction of stress and displacement fields in the neighbourhood of a crack filled with failed, CO-calledcraze material [2]. More recently 131, a BEM formulation was applied to the evaluation of an expression for the strain energy release rate derived from a functional corresponding to the potential energy in elasticity. The present BEM analysis attempts to address a wide range of issues relating to the modelling of the problem. Linear viscoelasticity, the most commonly adopted model for polymers, is assumed. The correspondence principle is used to solve the quasi-static problem in the Laplace transform domain with the time- dependent responses determined by numerical inversion. This approach essentially uses the fundamental solutions or Green's functions of the corresponding elastic problems, which can be adapted to viscoelastic models of any complexity and can be chosen to satisfy the traction-free conditions over the crack surface. Solutions are alternatively obtained through time domain BEM formulations derived from viscoelastic reciprocity relations using fundamental solutions specific to the viscoelastic model used. Both formulations require particular attention to the boundary modelling around the crack tip so that the stresses in this region are approximated with reasonable accuracy. Both transformed and time domain analyses were initially applied to the prediction of near crack-tip stress and crack-opening displacement time histories in order to assess their relative accuracy and efficiency. The time dependence of stress intensity factors was assessed for various viscoelastic models as well as loading and support conditions. Various approaches were proposed and applied for determining the strain energy release rate or crack extension force under isothermal conditions. The representation and of the crack extension force through path-independent J-type integrals was examined. Systematic procedures for their evaluation were developed requiring the derivation of additional boundary integral equations for the displacement gradient distributions in the solid domain. The BEM formulation was extended to the determination of the rate of energy dissipation for incremental crack growth. Then, the application of global conservation of energy, combined with the knowledge of the critical and current values of strain energy release rate, led to an assessment of the crack growth rate. Numerical results were compared with other analytical solutions and experimental measurements. 2 Boundary element modelling 2.1 Laplace transform domain If the correspondence principle is applied to the linear viscoelastic problem, the relevant boundary integral equation in the Laplace transformed domain is written = I[B($)U; (S) - (s)~;(s)]~ + f,u;d~a (1) r a Transactions on Modelling and Simulation vol 35, © 2003 WIT Press, www.witpress.com, ISSN 1743-355X where G, = 0.54, in the case of a smooth boundary, 4 , p,, f, are the Laplace transforms of displacements, tractions and body forces, respectively, and S is the transform space variable; D and rrepresent, respectively, the domain and its boundary; (U;, p,; ) is the elastic fundamental solution for displacements and tractions in which however the elastic constants have been replaced by the corresponding functions in the transformed space according to - 5..= sC.. E V rlk, kl = ~tklEkl (2) where Ekl and Gilklare the Laplace transforms of stresses, strains and relaxation moduli, respectively. The summation convention over repeated indices has been adopted. The boundary element modelling and solution is performed in exactly the same manner as for the respective elasticity problems. Inversions of obtained Laplace transforms to time histories are carried out using established techniques [4l. 2.2 Time domain The boundary integral equation can be obtained directly from the reciprocal theorem of linear viscoelasticity [5]leading to ~,,u,(t)= (U, * dpi- pc *dui )U + fi * du;d~ (3) I J R r with the Stieltjes convolution of two functions g(t) and h(t) defined as [5] The time-dependent fundamental solution U$ (X-4,t) satisfies the field equation in an infinite domain with the body force given by fi* 46( x-S)H(t) (5) where 6, is the Kronecker delta, 6( X-5) the delta function and H(t) the Heaviside step function. Applying the correspondence principle, ut; (X-5,t) is obtained as the inverse Laplace transform of the corresponding elastic fundamental solution in the transformed space divided by the Laplace domain variable S. A fast general procedure was devised for obtaining time-dependent fundamental solutions standard linear solid (SLS) creep or relaxation models with a high number of springs and dashpots. The modelling in the time domain was based on a linear variation of the boundary variables within a time step. The introduction of a constant spatial model led to a system of equations yielding the solution at any time step in terms of the current boundary conditions and the solutions at all previous time steps. Transactions on Modelling and Simulation vol 35, © 2003 WIT Press, www.witpress.com, ISSN 1743-355X 86 Boutrdarv Elcmatr~~XXV 3 Fracture models Stress intensity factor 3.1 The two-dimensional visco-elastic problem considered is schematically drawn in Fig. 1 where, for simplicity, the cracked solid is assumed symmetric relative to the two co-ordinate axes xi (i = 1,2) parallel and perpendicular to the crack with origin at the centre of the crack. Time histories of the stress field D&) in the neighbourhood of the crack tip can be determined by either the Laplace transformed or time domain BEM formulations. The stress intensity factor can then be obtained using with the stress a22described relative to a polar frame of reference with origin at the crack tip. X2 J-integral path / Crack tip & : Figure 1 Centre-crack plate under tension. It is also possible to obtain the Laplace transform of the stress intensity factor K, (S) from a path-independent integral [6] 7 (S) = 1 - piq,,dr) r, where I; represents the integration path, a comma followed by an index indicates differentiation with respect to the corresponding co-ordinate and W =La.($lqj 2 " The relations between (S) and (S) are similar to those between J and KI in linear elastic fracture mechanics, namely Transactions on Modelling and Simulation vol 35, © 2003 WIT Press, www.witpress.com, ISSN 1743-355X where p is 1 for plane stress and l-SF for plane strain. 3.2 Strain energy release rate Crack growth occurs due to the application of either edge displacement Ci(t) or traction jji (t) loading histories. These loads produce stress and strain fields $(t) and ~~(t),respectively, with the total reversibly stored energy E and dissipated energy D, given by [7] f E t D, = jog(r).Cu (r)d~dr (6) OR where a dot above a symbol indicates differentiation with respect to time. Assuming small strain rates, that is, linear relations between strain and deformation rates, Eq. (6) can be transformed via the application of Green's theorem to t E D, = piiidrdr (7) + or The principle of conservation of energy requires that [X] W=E+D,+D,, (8) where W is the total work done by boundary tractions and D, the total surface energy. The crack is assumed to be stationary, that is, after a certain period of loading t = tl, it can be forced to grow instantaneously by a small amount 6a. A crack extension 6a means the removal of the tension pi(x,,O, t) for t > tl over this length along XI,changes in work and energy should therefore be evaluated with this tension reversed, that is, applied as compression. The derivation of an incremental relation similar to Eq. (8) is performed with reference to the geometry and loading of Fig. 1 in order to assess the corresponding rate of surface energy increase.
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