Comparison of Different Methods of Evaluation of Weight Functions for 2-d Mixed-mode Fracture Analyses

In this paper, two different techniques for mixed-mode type engineering fracture analysis, by using the weight-function method, are presented. These two methods are (a) the virtual crack extension technique (VCE), and (b) the equivalent domain integral method (EDI). For mixed-mode weight-function evaluation, a technique of using two reference load states, both of which cannot be either pure mode I or pure mode II, is combined with the above mentioned numerical schemes. The numerical results show that the weight functions calculated by these two methods coincide with each other well. With the proposed techniques, it is very straightforward to perform fracture analysis of isotropic, orthotropic and composite materials. 1. ~RODU~ION STRESS intensity factors K, and K,, are important parameters in solving fracture problem@ J. There are several numerical methods for evaluating the stress intensity factors. The hybrid finite element method[2-4], integral equation[5], and the alternating method[6] are some examples for such numerical schemes. In the practical application of fracture mechanics, the determination of weight functions is often more advantageous than the calculation of stress intensity factors alone: in life prediction studies it is indispensable to analyse the effects of cyclic load changes on a flawed structure. The use of weight functions can obviate the repeated computer calculations of stress intensity factors for given structural and crack geometries. With the weight function concept, the stress intensity factor is expressed as a sum of a worklike product between the applied loads and the values of weight functions at thier points of application. In a previous paper[q, the weight functions calculated by the alternating finite element method have been presented. In[7], only the untracked structure was modelled by the finite element method, while the crack was accounted for by considering the embedded crack in an infinite domain subjected to abitrary crack face tractions. If the material is nonhomogeneous, or if the crack exists in a complicated structure such as a bi-material plate or a stiffened plate, etc., the direct numerical modeling of the crack itself is unavoidable, in determining the weight functions. In this paper, two post processes (a) the virtual crack extension technique (VCE), and (b) the equivalent domain integral method (EDI) for fracture analysis are discussed for this purpose. In the following section, the weight function concept of elastic fracture mechanics is briefly reviewed. The virtual crack extension technique, combined with the fracture-mode d~omposition concept for energy release rate calculation, is presented. The equivalent domain integral method for weight function evaluation is then discussed. Numerical examples for certain illustrative problems, which encompass the above mentioned areas, are presented. 2._WEIGHT FUNCTIONS OF ELASTIC FRACTURE The concept of weight functions for elastic crack problems dates back to the work of Bueckner[8] and Bice{9] (see also Bortman and Banks-Sills(lO]). The “weight function” may generally be viewed as the appropriately normalized rate of change of displacements due to a unit change in the crack length for a reference state of loading. The practical importance of the concept of the weight functions lies in the fact that, if the weight functions are evaluated from a (perhaps simple) reference state of loading, then the stress intensity factors for any ~bit~ state of loading can be computed by using an integral of the worklike product between the applied