Second-grade elasticity revisited

We present a compact, linearized theory for the quasi-static deformation of elastic materials whose stored energy depends on the first two gradients of the displacement (second-grade elastic materials). The theory targets two main issues: (1) the mechanical interpretation of the boundary conditions and (2) the analytical form and physical interpretation of the relevant stress fields in the sense of Cauchy. Since the pioneering works of Toupin and Mindlin et al. in the 1960's, a major difficulty has been the lack of a convincing mechanical interpretation of the boundary conditions, causing second-grade theories to be viewed as 'perturbations' of constitutive laws for simple (first-grade) materials. The first main contribution of this work is the provision of such an interpretation based on the concept of ortho-fiber. This approach enables us to circumvent some difficulties of a well-known 'reduction' of second-grade materials to continua with microstructure (in the sense of Mindlin) with internal constraints. A second main contribution is the deduction of the form of the linear and angular-momentum balance laws, and related stress fields in the sense of Cauchy, as they should appear in a consistent Newtonian formulation. The viewpoint expressed in this work is substantially different from the one in a well known and influential paper by Mindlin and Eshel in 1968, while affinities can be found with recent studies by dell'Isola et al. The merits of the new formulation and the associated numerical approach are demonstrated by stating and solving three example boundary value problems in isotropic elasticity. A general finite element discretization of the governing equations is presented, using C1-continuous interpolation, while the numerical results show excellent convergence even for relatively coarse meshes.