On conservation integrals in micropolar elasticity

Two conservation laws of nonlinear micropolar elasticity (Jk = 0 and Lk = 0) are derived within the framework of Noether’s theorem on invariant variational principles, thereby extending the earlier authors’ results from the couple stress elasticity. Two non-conserved M -type integrals of linear micropolar elasticity are then derived and their values discussed. The comparison with related work is also given. x 1: INTRODUCTION Three conservation integrals of infinitesimal non-polar elasticity (Jk, Lk, and M ) were derived by employing Noether’s (1918) theorem on variational principles invariant under a group of infinitesimal transformations by Günther (1962), and Knowles and Sternberg (1972). When evaluated over a closed surface which does not embrace any singularity, these integrals give rise to conservation laws Jk = 0, Lk = 0, and M = 0. The law Jk = 0 applies to anisotropic nonlinear material, the law Lk = 0 to isotropic nonlinear material, and M = 0 to anisotropic linear material. If the surface embraces a singularity or inhomogeneity (defect), Eshelby (1951,1956) has shown that the value of Jk is not equal to zero but represents a configurational or energetic force on the embraced defect (vacancy, inclusion, dislocation). The path-independent J integral of plane fracture mechanics, independently introduced by Rice (1968), has proved to be of great practical importance in modern fracture mechanics, allowing the prediction of the behavior at the crack tip from the values of the remote field quantities. 1Tel: 858-534-3169; Fax: 858-534-5698; Email: [email protected] Budiansky and Rice (1973) interpreted the Lk and M integrals as the energetic forces (potential energy release rates) conjugate to rotation (by erosion/addition of material) and self-similar expansion (erosion) of the traction-free void. Freund (1978) used the M conservation law for certain plane elastic crack problems to calculate the elastic stress intensity factor without solving the corresponding boundary value problem. The reference to other related work can be found in the papers by Eshelby (1975) and Rice (1985). Noether’s theorem was further applied by Fletcher (1975) to obtain a class of conservation laws for linear elastodynamics. Jarić (1979) and Vukobrat and Jarić (1981) studied the conservation laws in thermoelasticity and linear theory of elastic dialectrics, and Vukobrat (1989) and Vukobrat and Kuzmanović (1992) in micropolar and nonlocal elastodynamics. Yang and Batra (1995) and Huang and Batra (1996) used Noether’s theorem to derive the conservation laws and energy-momentum tensors for piezoelectric materials and nonsimple dialectrics. Pucci and Saccomandi (1990) applied a version of Noether’s theorem to deduce the conservation laws of micropolar elasticity, but their analysis was unnecessarily restricted to linear constitutive equations. Nikitin and and Zubov (1998) derived the conservation laws for the Cosserat continuum under finite deformations. Lubarda and Markencoff (2000) modified a non-polar analysis of Knowles and Sternberg (op. cit.) and derived the conservation laws Jk = 0 and Lk = 0 for the couple stress elasticity. The results are here extended to the more general framework of the nonlinear micropolar elasticity, in which the local rotation of material elements is not constrained as in the couple stress elasticity, but independent of the displacement field. The derived conservation laws correspond to infinitesimal invariance of the strain energy relative to translational and rotational transformations of the position coordinates, and the displacement and rotation fields. It is then shown that the quadratic strain energy is not infinitesimally invariant under a self-similar scale change, which prevents