On higher gradients in continuum-atomistic modelling
R. Sunyk and P. Steinmann
International Journal of Solids and Structures, 40, 6877-6896 (2003).
ABSTRACT
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 so-called 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.