A two-crown finite element technique for the determination of tearing modulus

The importance of approach to the subject of crack instability for the design of structures containing cracks has increased considerably over the last few years. The tearing modulus theory recently enunciated by Paris and co-workers has emerged as one of the leading criterions for stable crack growth and for instability, and the estimation of T termed Tearing modulus in the theory has since been extensively investigated theoretically as well as experimentally. Analytical methods exist for calculating the tearing modulus of various crack configurations in simple-shaped structures under certain loading conditions. However, for arbitrary structures under general loading, more sophisticated calculation techniques are required. Extending the virtual crack extension method introduced independently by Hellen and Parks, a new numerical approach for calculating the tearing modulus is presented hereafter and put in a form suitable for the instability analysis of structures containing one single crack or several interacting cracks. As it is well-known that the calculation of the energy release rate in elasticity by the virtual crack extension method is related to a stiffness derivative to which only a small region around the crack tip has a contribution, the technique described in the paper shows that it would be reasonable to evaluate the tearing modulus, or rather, the second derivative of potential energy with respect to the crack length, by means of two stiffness derivative calculations in two crowns around the crack tip. In particular, when one crown is strictly included in another one, computation is largely curtailed at this point with some saving of computer time, but a very accurate value of tearing modulus is obtained. As an interesting consequence, an another expression of the tearing modulus is carried out. In Section 4: the classical tearing modulus is proved to be precisely equivalent to a line integral which is independent of integration path. Numerical example using the proposed method is given in the final section