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.