13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- solids, the J-integral is equal to the total potential energy release rate which is expressed as [6] 1 , ( , , , , ) 2 MN M N J Y K K MN IIIIVV = = (18) where MN Y is the generalized Irwin matrix which depends on the material constants at the crack-tip location [6]. Applying Eq. (18) to the superimposed state S gives ( ) 1 ( )( ) 2 S aux aux MN M M N N J Y K K K K = + + . (19) By expanding Eq. (19) as ( )S aux J J J I = + + , we can obtain the interaction integral as aux MN M N I Y K K = . (20) By taking values of the vector [ aux II K , aux IK , aux IV K , aux VK ] to be [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0] and [0, 0, 0, 1], sequentially, Eq. (19) reduces to ( ) 11 12 14 15 II II I IV V I K Y KY K Y K Y = + + + , (21) ( ) 21 22 24 25 I II I IV V I K Y KY K Y K Y = + + + , (22) ( ) 41 42 44 45 IV II I IV V I K Y KY K Y K Y = + + + , (23) and ( ) 51 52 54 55 V II I IV V I K Y KY K Y K Y = + + + . (24) By simultaneously solving Eqs. (21)-(24), the IFs IK , II K , IV K and VK can be obtained. 3. Numerical implementation 3.1. Extended finite element method Figure 2. Finite element mesh of a plate with a crack and a particle [7] To compute the interaction integral, the values of the actual fields Iu , IJσ and IJε in the integral domain need to be obtained first. Generally, the numerical methods such as the finite element method (FEM), the extended finite element method (XFEM) and the element-free Galerkin method (EFGM) are adopted to compute these values. Here, the XFEM is used and the approximations of the expanded displacements are adopted as
RkJQdWJsaXNoZXIy MjM0NDE=