Applications of the concept of J-integrals for calculation of generalized forces

Abstract

The J-integral technique is used to assess the risk of defects in a particular structure by explicit computation of the arising generalized forces. Three different situations are considered: stress concentrators in cycloidal specimens, the interaction of a dislocation with a circular inclusion, and the interaction between two dislocations. Special emphasis is given to an effective but analytical calculation of the corresponding path integrals. IntroductionEver since the pioneering work of Eshelby and Rice the J-integral has been known to be an effective way of assessment of the risk of defects in a structure. In fact, it can be used to quantify various kinds of "defects" in a very broad sense of the word. The term "defect" can refer to the singular stress-strain field developing around the tip of a crack which illustrates the crucial importance of the concept of J-integrals in fracture mechanics. The defect can also be a stress concentrator like a notch or, on a more elementary scale, apply to imperfections in crystals, such as dislocations (e.g., Kienzler [1]).The purpose of this paper is, first, to illustrate the possibility to obtain closed-form solutions for the J-integral in the aforementioned situations by suitable choice of the integration path. Second, the attention is drawn to the potential of epicycloidal specimens for use in fracture mechanics. The J-integral will explicitly be evaluated for epicycloids of order 1 and 2 which are subjected to thermal loading by a central hot spot. Energy release rates will be computed for cusp-like notches and then be used to predict the stress intensity factors of cusp-like cracks in such specimens (Section 3.1). Third, for the case of dislocation problems, it will be shown that the evaluation of the path integral can be extremely facilitated if a certain limit procedure is applied (Section 3.2 and 3.3). Theoretical backgroundConsider a closed surface, ~ V, in three-dimensional space. In the static case under absence of body forces, for small deformations, and within the framework of deformation theory the J-integral is defined as follows (e.g., Rice [17], Buggisch, Gross and Kriiger [2], Kienzler and Kordisch [3], or W.H. Miiller and G. Kemmer Kienzler [1]):