Continua with microstructure modelled by the geometry of higher-order contact

In the paper, the Finslerian-geometry-oriented model of the continuum with microstructure is formulated within the frame of Newtonian±Eshelbian continuum mechanics, based on the information characterizing a structure-dependent evolution of state variables. In this approach, position- and direction-dependent deformation and strain measures are used to describe the motion of the continuum with microstructure at the macro- and microlevel. The variational arguments for a Lagrangian functional de®ned on the Finslerian bundle are used to derive dynamic balance laws, boundary and transversality conditions for macro- and microstresses of deformational and con®gurational type. The dissipation inequality for the thermo-inelastic deformation processes is formulated by the suciency condition of Weierstrass type for the action integral. The presented geometric technique is illustrated in the following examples. The damage tensor, identi®ed with a measure of reduction of load carrying area elements caused by the development of microcracks or microvoids, is de®ned on the tangent bundle using the lifting technique. The macro±micro constitutive equations and the associated phenomenological constitutive relations for the thermo-inelastic processes are derived in terms of the free energy functional and a dissipation potential. A strain-induced crack propagation criterion, de®ned by the di€erence between the strain energy release rate and the rate of the surface energy of the crack, is formulated for the kinking of cracks