ON A PATH-INDEPENDENT INTEGRAL APPROACH TO FAST TEARING OF ELASTOMERS Gianluca Medri* MECCANICA 21 (1986), 47-50 SOMMARIO. In questo artieolo si discutono le possibilita offerte dall'impiego dell'integrale J e del relativo criterio di stabitit6 della cricca nello studio della frattura ((veloce>> di rnateriali elastomerici. SUMMARY. The feasibility of a path-independent integral approach to fracture mechanics of elastomers and the objectivity of the pertinent crack stability criterion are discussed with reference to tearing with negligible mecha- nical dissipations. LIST OF F FELED T T/ U W X X SYMBOLS AND ABBREVIATIONS Deformation gradient. Finite Element for Large Elastic Deformations. Strain energy release per unit crack growth (termed G in fracture mechanics of metals). First Piola-Kirchhoff stress tensor. Displacement vector. Stored energy function per unit of volume in the undeformed configuration. Spatial co-ordinates. Material co-ordinates. 1. INTRODUCTION The fracture mechanics approach to the strength properties of elastomeric materials is relevant in the study of tyre, belting and seal failure mechanism. In particular, crack growth is easily triggered by ciclic loadings in the presence of tensile stresses. The propagation may become catastrophic when the instability conditions are fulfilled [ 1 ]. Rivlin and Thomas [2] tried an energetic approach to these phenomena by introducing the <<tearing energy>> T. When using such an approach the viscous component of the mechanical behaviour of elastomers may be taken into account by means of a sort of phenomenologicat convention [1 ]. A way to insert <<relaxation>> into the study of fracture mechanics of elastomers, outflanking the <<troubles>> involved in 'time', has been introduced [3] [4] resorting to the concept of two networks of cross-links (from a former idea of Green and Tobolsky). Nevertheless, parameter T is more suitable, in its original meaning and definition, to compare the behaviour of diffe- rent elastomers in simple specimens than to evaluate the * Associate professor of Machine Design - Istituto di Meccanica applicata alle Macchine - University of Bologna, Italy. failure resistance of complex-shaped cracked rubber elements, because of computational difficulties. When the material is ((elastic>), at least from a practical point of view, the well-known path-independent integral proposed by Rice [5] is a valid energy-based tool to analyse a crack in a stressed continuum. Usually named J-integral, this is widely used in the non-linear fracture mechanics of metals. A stability criterion of the crack is based upon the hypothesis of the existence of a critical value of the J-integral (Je) [6]. Chang [7] proposed a path-independent integral with the same features as Rice's integral, which can be used in finite elasticity. Of course, the J-integral approach to the fracture of elastomers is subject to binding hypotheses and leads to more or less accurate results depending on the importance of the anelastic component of the behaviour of the material on hand [8] [9]. This paper reports the main results and thoughts of a preliminary study on the suitability of numerical methods, based on the J-integral, to evaluate the stability of cracks in rubber units in working conditions. 2. THE J-INTEGRAL Rice's classical integral was proposed for infinitesimal elasticity (i.e. with reference to the linear strain tensor) but without restrictions on the non-linearity of the constitutive equation of the material. Since constitutive equation of ductile metals may have a phenomenological non-linear <<elastic>>form (within the hypotheses of monotonous loading and fixed strain rate) the J-integral is used in the analysis of the fracture mechanics of those materials. On the other hand Rice's integral and similar formulations [10] cannot be used for elastomers as a rule. In fact, the large deformations involved (<<average strain>> is usually greater than 20%) oblige resorting to finite elasticity theories. In finite elasticity the form of the J-integral is [8] [9] i ~176 JLD : WdS/-tl. aX. dS (1) I where S. is the projection of S (a surface enclosing the crack tip in the undeformed configuration) on the coordinate plane perpendicular to the X i axis and t! = T! 9 n the surface traction vector per unit area of the undeformed surface. The JLD-integral is very similar to Chang's form [7] (x has been replaced by u = x-X) and the way to prove its <<path-independence>> is the same. Eq. (1) reduces to the well-known 21 (1986) 47