A model for a constrained, finitely deforming, elastic solid with rotation gradient dependent strain energy, and its specialization to von Karman plates and beams

摘要

The aim of this paper is to develop the governing equations for a fully constrained finitely deforming hyperelastic Cosserat continuum where the directors are constrained to rotate with the body rotation. This is the generalization of small deformation couple stress theories and would be useful for developing mathematical models for an elastic material with embedded stiff short fibers or inclusions (e.g., materials with carbon nanotubes or nematic elastomers, cellular materials with oriented hard phases, open cell foams, and other similar materials), that account for certain longer range interactions. The theory is developed as a limiting case of a regular Cosserat elastic material where the directors are allowed to rotate freely by considering the case of a high "rotational mismatch energy". The theory is developed using the formalism of Lagrangian mechanics, with the static case being based on Castigliano's first theorem. By considering the stretch U and the rotation R as additional independent variables and using the polar decomposition theorem as an additional constraint equation, we obtain the governing and as well as the boundary conditions for finite deformations. The resulting equations are further specialized for plane strain and axisymmetric finite deformations, deformations of beams and plates with small strain and moderate rotation, and for small deformation theories. We also show that the boundary conditions for this theory involve "surface tension" like terms due to the higher gradients in the strain energy function. For beams and plates, the rotational gradient dependent strain energy does not require additional variables (unlike Cosserat theories) and additional differential equations; nor do they raise the order of the differential equations, thus allowing us to include a material length scale dependent response at no extra "computational cost" even for finite deformation beam/plate theories