In order to analyze the fracture problems of composite materials with complex interfaces, such as particle reinforced composite materials (PRCMs), we developed a new interaction integral method by which the stress intensity factors (SIFs) can be solved using an integral domain with arbitrarily complex interfaces. The interaction (energy) integral method([1])) was derived from the J-integral by considering a composition of two admissible states (the actual and auxiliary fields) to obtain mode I and mode II SIFs separately for homogeneous materials. Subsequently, the interaction integral method was successfully used to solve the crack problems in functionally graded materials (FGMs)([2]). Generally, the contour integral should be converted into an equivalent domain integral in numerical computations. Since divergence theorem can not be used in a domain with material interfaces, the material properties in the integral domain are assumed to be continuous in previous studies. In the authors' work([3]), the interaction integral was studied when the integral domain contains material interfaces. As shown in Figure I. the interaction integral is defined as l = lim(r -> 0)integral(Gamma)[1/2(sigma(aux)(jk) epsilon(jk) + sigma(jk)epsilon(aux)(jk))delta(li) - (sigma(aux)(ij) u(j.1) + sigma(ij)u(j.1)(aux))]n(i)d Gamma (1) Since the integral domain is divided by Gamma(interface) into two domain A(1) and A(2), the interaction integral should be written as I = lim(r -> 0) closed integral(partial derivative A1) + closed integral(partial derivative A2) [sigma(aux)(ij) u(j.1) + sigma(ij)u(j.1)(aux) - +1/2(sigma(aux)(ij) epsilon(jk) + sigma(ij) epsilon(aux)(jk))delta(li))m(i)q]d Gamma + l (2) where l* is a line integral along the interface and it can be proved that l* = 0. Then, applying divergence theorem in A(1) and A(2), we can obtained that I = integral(A) (sigma(aux)(if) u(j.1) + sigma(ij)u(j.1)(aux) -sigma(aux)(ik)epsilon(jk)delta(li))q(j)dA+integral(Alpha)sigma(ij)[S-ijkl(tip) - S-ijkl (x)]sigma(aux)(krl)qdA (3) Compared with the J-integral and the traditional interaction integral, the present interaction integral has two advantages. I) It does not involve any derivatives of material properties. Therefore, the interaction integral does not need the material properties to be differentiable. Since it may be difficult to obtain the derivatives of material properties or there are no derivatives in many actual cases. 2) The validity of the present interaction integral method is not affected by material interfaces in the integral domain. Namely, the present method does not require the material to be continuous and hence, it can be used to compute the Sit's of the composite material with complex interfaces effectively. For the interface crack problems and the three-dimensional crack problems in the composite materials with complex interfaces, the interaction integral method with the above two advantages have been obtained([4, 5]). Moreover, the method can deal with curved interface crack problems effectively. On account of these advantages, the interaction integral method has become a very promising technique in the fracture analysis of the composite materials with complex interfaces.
