A unifying treatise on variational principles for gradient and micromorphic continua

Abstract

The effective macroscopic behaviour of most materials depends significantly on the structure exhibited at the microscopic level. In recent decades, considerable effort has thus been directed towards the development of continuum theories accounting for the inherent microstructure of materials. A unified theory is, however, not yet available. A reliable and efficient use of microstructured materials would benefit from the former, which is also the foundation of any subsequent numerical analysis. Focusing on two kinds of microstructured materials (gradient continua, micromorphic continua), an attempt is made to identify their role and their potential with regard to numerical implementations in a generic theory. An analysis of both continua in the framework of various variational formulations based on the Dirichlet principle shows that they are closely related (which is intuitively not obvious): a gradient continuum can, for example, be modelled by applying a mixed variational principle to a micromorphic continuum. This has considerable consequences for the numerical implementation: the Euler–Lagrange equations describing the seemingly more intricate micromorphic continuum require only continuous approximations when implemented in a FEM code, while, in contrast, the Euler–Lagrange equations deduced for the gradient continuum require differentiability of the involved unknowns.