Mixed convolved Lagrange multiplier variational formulation for size-dependent elastodynamic couple stress response

To model the dynamic response of materials at the submicron scale, non-classical theories of continuum mechanics must be invoked to capture the often-expected size-dependent behavior, and the corresponding variational formulations must be established. In addressing this need, a novel mixed convolved action approach is developed here for consistent couple stress theory. This action uses mixed variables, including displacements and rotations, along with impulses of symmetric force stress and skew-symmetric couple stress. In addition, the skew-symmetric components of force stress appear as Lagrange multipliers to enforce the rotation-displacement constraint, thus allowing development of a C-0 variational statement. Meanwhile, the temporal convolution operator is introduced to permit proper accounting for the variations to maintain compatibility with the specified initial conditions. The resulting action functional is energy preserving for well-posed elastodynamic couple stress problems and leads directly to an innovative time-space finite element method. A low-order version of this finite element method is then applied to several two-dimensional problems to test the validity of the formulation and numerical implementation and to bring out a few interesting features of the underlying skew-symmetric couple stress theory under impulsive loading, including the dispersive nature of wave propagation. Note, however, that the formulation is not restricted to impulsive loading and, in fact, can be applied to problems with general time- and space-dependent loads.

Automated summary

In this paper, a novel mixed convolved action approach is developed for consistent couple stress theory, and it is applied to different problems to study the characteristics of impulsive loading under skew-symmetric couple stress theory.