In-plane transient analysis of two dissimilar nonhomogeneous half-planes containing several interface cracks

Abstract

In this study, the linear elastic fracture mechanics theory is employed to calculate the dynamic stress intensity factors (DSIFs) of multiple cracks which are located at the interface between two dissimilar nonhomogeneous half-planes subjected to in-plane impacts. The change in the material properties is illustrated by an exponential law. First, the integral transforms and Volterra-type climb and glide edge dislocation at the interface of two dissimilar nonhomogeneous materials are used to obtain the solution. Then, using the distributed displacement technique, singular integral equations with Cauchy singularity are obtained. By solving these equations numerically in the Laplace domain, the dislocation density on the crack faces used to determine the DSIFs is obtained. Finally, numerical results exhibited graphically the effects of the gradient nonhomogeneous constant, crack length, the variation of time and the interaction between of cracks on the DSIFs.

Appendix

The parameters appearing in Eq. (10) are

$$\begin{aligned} A_{11}= & {} {[a_{24} (-b_{33} b_{41} +b_{31} b_{43} )+a_{23} (b_{34} b_{41} -b_{31} b_{44} )+a_{21} (-b_{34} b_{43} +b_{33} b_{44} )]} / {\Delta ,} \\ A_{12}= & {} {[b_{34} (a_{23} -b_{43} )+a_{24} (-b_{33} +b_{43} )+b_{44} (-a_{23} +b_{33} )]} / {\Delta ,} \\ A_{21}= & {} {[a_{14} (b_{33} b_{41} -b_{31} b_{43} )+a_{13} (-b_{34} b_{41} +b_{31} b_{44} )+a_{11} (b_{34} b_{43} -b_{33} b_{44} )]} / {\Delta ,} \\ A_{22}= & {} {[a_{14} (b_{33} -b_{43} )+b_{34} (-a_{13} +b_{43} )+b_{44} (a_{13} -b_{33} )]} / {\Delta ,} \\ B_{31}= & {} {[a_{14} (a_{23} b_{41} -a_{21} b_{43} )+a_{13} (-a_{24} b_{41} +a_{21} b_{44} )+a_{11} (a_{24} b_{43} -a_{23} b_{44} )]} / {\Delta ,} \\ B_{32}= & {} {[a_{24} (-a_{13} +b_{43} )+a_{14} (a_{23} -b_{43} )+b_{44} (a_{13} -a_{23} )]} / {\Delta ,} \\ B_{41}= & {} {[a_{14} (-a_{23} b_{31} +a_{21} b_{33} )+a_{13} (a_{24} b_{31} -a_{21} b_{34} )+a_{11} (-a_{24} b_{33} +a_{23} b_{34} )]} / {\Delta ,} \\ B_{42}= & {} {[a_{24} (a_{13} -b_{33} )+a_{14} (-a_{23} +b_{33} )+b_{44} (-a_{13} +a_{23} )]} / {\Delta ,} \\ \Delta= & {} a_{11} a_{24} b_{33} -a_{11} a_{23} b_{34} -a_{24} b_{33} b_{41} +a_{23} b_{34} b_{41} -a_{11} a_{24} b_{43} +a_{24} b_{31} b_{43} +a_{11} b_{34} b_{43} -a_{21} b_{34} b_{43} \\&+ a_{14} [a_{23} (b_{31} -b_{41} )+b_{33} (-a_{21} +b_{41} )+b_{43} (a_{21} -b_{31} )]+b_{44} [a_{23} (a_{11} -b_{31} )+b_{33} (-a_{11} +a_{21} )] \\&+ a_{13} [b_{34} (a_{21} -b_{41} )+a_{24} (-b_{31} +b_{41} )+b_{44} (-a_{21} +b_{31} )]. \end{aligned}$$