On the Material Setting of Gradient Hyperelasticity

Abstract

The behavior of most materials is influenced by inhomogeneously distributed microscale properties. A reliable and efficient use of microstructured materials in engineering, nanotechnology, biomedical application and earth sciences requires and justifies high efforts in their mathematical modeling and numerical treatment and has led to the development of extended continuum theories. In the field of computational mechanics, the material force method has been established to assess the configurational equilibrium of inhomogeneities, but tacitly assumes that the material behavior is described sufficiently accurately by its state variables and their first gradients. Since this restriction might be too severe, a material setting based continuum formulation for hyperelastic media involving second deformation gradients is investigated in this article. In doing so, we wish to transfer the universally accepted improvements and successes gained by considering higher gradient continua in the spatial setting to the material setting, which has been advocated as the intrinsic framework to describe material rearrangements of inhomogeneities at the microscopic level. Focusing on quasistatic conditions, the balance law of linear momentum is derived in a variational framework. The dual nature of the spatial and material setting of Boltzmann continua of grade 2 is highlighted. Based on an enhanced Eshelby stress tensor arising in the material-setting, a fracture criterion applicable to gradient continua is obtained that encompasses, in the limit of vanishing higher deformation gradients, the classical J integral of fracture mechanics.