Energy theorems and the J-integral in dipolar gradient elasticity

Within the framework of Mindlin s dipolar gradient elasticity, general energy theorems are proved in this work. These are the theorem of minimum potential energy, the theorem of minimum complementary potential energy, a variational principle analogous to that of the Hellinger–Reissner principle in classical theory, two theorems analogous to those of Castigliano and Engesser in classical theory, a uniqueness theorem of the Kirchhoff–Neumann type, and a reciprocal theorem. These results can be of importance to computational methods for analyzing practical problems. In addition, the J-integral of fracture mechanics is derived within the same framework. The new form of the J-integral is identified with the energy release rate at the tip of a growing crack and its path-independence is proved. The theory of dipolar gradient elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Microstructure in linear elasticity. Arch. Rational Mech. Anal. 16, 51–78] and is appropriate to model materials with periodic structure. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain (as in classical elasticity) and the second gradient of the displacement (additional term). Specific cases of the general theory considered here are the well-known theory of couple-stress elasticity and the recently popularized theory of strain-gradient elasticity. The latter case is also treated in the present study.