13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- problems. Then, the mixed-mode SIFs can be easily extracted from the Irwins’ relation after we get the values of the I-integral. The Newmark’s method of direct integration schemes is used in dynamic analysis finally. 2.3. Numerical Examples Young’s modulus and mass density vary exponentially, such that / E constent ρ≡ , as given by 0 1 2 exp( ) E E x y β β = + (18) 0 1 2 exp( ) x y ρ ρ β β = + (19) where 0E and 0ρ are Young’s modulus and mass density for initial values. 1β and 2β are the non-homogeneity parameters along the x- and y- directions, respectively. When 1β and 2β are equal to zero, the above two equations return to the case of homogeneous materials. A constant Poisson ratio of 0.3 is used during the whole simulation and the plane strain status is assumed. The geometry and the boundary conditions are illustrated in Fig. 3. The data used in the computation are: 20 L mm = , 40 D mm = , 2 4.8 a mm = , 0 199.992 E GPa = , 3 0 5000 / kg m ρ = , 7.34 / dC mm sμ = . The time is normalized with respect to the dilatational wave speed ( dC ), and the DSIFs are normalized with respect to 0 0 K a σ π = (20) where the 0σ is the magnitude of the applied stress and a is half of the total crack length. A time step is 0.1 t sμ Δ = . Fig. 3 Center cracked tension specimen: (a) non-homogeneous materials (b) exponentially graded materials in the y-direction In order to employ severe material gradations, relatively high β values are assigned: 1 0.1 β= and 2 0.1 β = . Here, the units corresponding to the material gradation parameters are millimeters. Material properties vary simultaneously along both the x- and y-directions according to Fig. 3(a). Fig. 4 shows DSIFs at the right crack tip calculated by the present I-integral and M-integral and the reference ones in the paper written by Song et al. [4]. It can be found that there is an excellent agreement between the present numerical results and the reference ones. It demonstrates that the present method is valid for the fracture problems of such materials. L D 2a L D 2a ( )a ( )b
RkJQdWJsaXNoZXIy MjM0NDE=