On higher gradients in continuum-atomistic modelling

Continuum-atomistic modelling denotes a mixed approach combining the usual framework of continuum mechanics with atomistic features like e.g. interaction or rather pair potentials. Thereby, the kinematics are typically characterized by the socalled Cauchy–Born rule representing atomic distance vectors in the spatial configuration as an affine mapping of the atomic distance vectors in the material configuration in terms of the local deformation gradient. The application of the Cauchy–Born rule requires sufficiently homogeneous deformations of the underlying crystal. The model is no more valid if the deformation becomes inhomogeneous. Nevertheless the development of microstructures with inhomogeneous deformation is inevitable. In the present work, the Cauchy–Born rule is thus extended to capture inhomogeneous deformations by the incorporation of the second-order deformation gradient. The higher-order equilibrium equation as well as the appropriate boundary conditions are presented for the case of finite deformations. The constitutive law for the Piola–Kirchhoff stress and the additional higher-order stress are represented for the simplified case of pair potential-based energy density functions. Finally, a deformation inhomogeneity measure is introduced and studied for a particular non-homogeneous simple-shear like deformation.