It has been a great challenge for many scientists and engineers to compute elastic–plastic solutions for dynamically loaded cracked structures due to the fact that the solutions are much more complicated and computationally time consuming than corresponding static problems. The path independent integral J ^ F , originally developed for a two-dimensional dynamically loaded stationary circular arc crack in a homogeneous and isotropic material, is evaluated in the present study by taking elastic–plastic material properties to quantify the crack problem. All the numerical results presented in this study were evaluated using general purpose finite element commercial software ANSYS together with the post-processing program (using FORTRAN) developed by the authors. Numerical results thus obtained are compared with good agreement within the limits of computational accuracy and the path invariant property of J ^ F is well preserved over the integration contours. arbitrary region around the process zone body force vector physical component of body force vector B i Rice's path independent integral crack driving force for circular arc crack under multiple loads positive unit outward normal physical component of positive unit outward normal n β plane polar coordinate radius of the crack traction vector physical component of the traction vector T i displacement vector covariant derivative of u i with respect to j physical component of u i , j mechanical strain energy density function contravariant stress tensor physical component of the stress tensor σ i j strain tensor covariant derivative of ε i j with respect to β physical component of ε i j , β thermal strain tensor initial strain tensor arbitrary curve surrounding A angle material density symbols not listed are defined as they appear in text. 
Elastic–plastic dynamic fracture analysis for stationary curved cracks