Eulerian rates of elastic incompatibilities for crystal plasticity applied to size-dependent hardening in finite bending

By following the work in (Rubin and Bardella, 2024), this investigation develops measures of rates of elastic incompatibilities, denoted as R_ij, for crystal plasticity. This effort relies on Eulerian constitutive equations for finite-deformation anisotropic elastoplasticity governed by the evolution of microstructural material vectors. The rates R_ij depend on the crystallography as the latter enters the rate of plasticity L_p and the R_ij are obtained by evaluating the opposite of the current curl of L_p relative to the microstructural vectors. Each component of R_ij is invariant under superposed rigid body motions, such that it can be independently employed in the constitutive equations. In crystal plasticity, the adopted Eulerian framework allows for singling out in R_ij the contributions due to rates of densities of geometrically necessary dislocations and to the elastic distortion of the crystal lattice. In this work, R_ij are used to enhance the hardening, which is applied to the size-dependent material response of structurally thick circular sectors subjected to uniform large-deformation bending.