International Journal of Fracture 35:295-310 (1987) © Martinus Nijhoff Publishers, Dordrecht Printed in the Netherlands 295 A general treatment of crack tip contour integrals B. MORAN and C.F. SHIH Division of Engineering, Brown University, Providence, RI 02912, USA Received 12 January 1987; accepted 19 May 1987 Abstract. Given a general balance statement we derive an expression for the associated crack tip flux integral. The conditions under which the integral is physically meaningful and yields a non-trivial result are outlined. To illus- trate the approach a number of well known integrals in use in fracture mechanics are derived. It is demonstrated that complementary analogues to these integrals can be derived in a similar fashion and a result indicating the equality of dual integrals under quite general conditions is presented. We discuss the domain integral method as an alternative means of representing crack tip integrals and we show that the method may be interpreted as a par- ticular form of Signorini's theorem of stress means. A discussion of some associated integral identities is presented. 1. Introduction In this paper we present a perspective on crack tip integrals from the point of view that these integrals can be simply derived from appropriate balance laws. The approach has its origin in a crack tip integral expression for the elastodynamic energy release rate proposed by Atkinson and Eshelby [1] and independently derived from the field equations by Kostrov and Nikitin [2] and Freund [3]. It has since been recognized that the result is also valid for general material response [4, 5]. We build upon the approach taken by Nakamura et al. [5] to obtain some general results. We begin with a general balance statement and derive an expression for the associated crack tip flux integral in Section 2. Then we examine the conditions under which this integral is physically significant (namely path-independent in the crack tip region) and yields a non-trivial result. In Section 3, the general result is specialized to particular balance laws and a number of crack tip integrals currently used in fracture analysis are derived. No a priori restrictions on material response and crack tip motion are required in the derivations. To illustrate the approach, the fundamental crack tip energy flux and energy release rate integrals are derived in this manner for both total and mechanical energy balance statements. We then derive a general dissipation integral from a variational form of momentum balance and show that it reduces to the C(t) integral (e.g., Bassani and McClintock [6]) for power law creeping solids. Also crack tip integrals for conduction/diffusion problems are discussed. In Section 4, we illustrate that the complementary counterparts of the above mentioned crack tip integrals can be derived using the same approach. For nonlinear elastic response, Bui [7] has shown that the energy and complementary energy release rates are equal. We establish in a direct manner that Bui's observation is valid for quite general material response and for all dual crack tip integrals under discussion. The implication is that no new information can be extracted from complementary integrals which is not already provided by their counterparts. However, in certain circumstances, the pair of integrals, referred to as dual integrals, may be employed to obtain a bound for the crack tip parameter.