Nonlocal Force Equilibrium Condition for Non-Simple Materials

For simple materials Noll’s principle of local action yields the stress tensor function to depend only on the local deformation gradient or its history (Noll in The Foundations of Mechanics and Thermodynamics, 1974, p. 20, Theorem 3). Consequently, the stress field is of class \(C^{1}\) and the standard force equilibrium condition exhibits the divergence of stress. Nonlocal models, e.g., couple stress theories, drop the principle of local action. They account for higher gradients in deformation and additional kinematical variables, respectively. Then, the stress tensor field in the contiguity of a continuum point may not be a linear function. In the context of power series expansion, higher order terms of stress appear in the representative volume element around the point. We axiomatically consider the stress field tensor and the body force vector as nonlinear functions of class \(C^{n}\), approximated via power series expansion of order \(m \leq n\) from the midpoint of a cubic representative volume element. Depending on the grade of approximation, the series expansion reproduces nonlinearities of the stress field in the cube and also on its surface. The proposed procedure yields a nonlocal force equilibrium condition extending the local condition by an additional term with internal length scale parameter. It evolves from integrating tractions on the surface of a finite region. Thus, we make no use of Green’s divergence theorem. Our approach is not restricted by material constitution. Thus, it is valid for solids and fluids. However, we limit our examples to solids, where an internal length scale arises from the inner structure of material. Additionally, the variational approach of a gradient elasticity model with explicit constitutive assumptions is under investigation. Latter leads to a similar force equilibrium condition, however, with a reversed sign for the proposed extension.