The paper briefly summarizes the theoretical derivation of the objective stress rates that are work-conjugate to various finite strain tensors, and then briefly reviews several practical examples demonstrating large errors that can be used by energy inconsistent stress rates. It is concluded that the software makers should switch to the Truesdell objective stress rate, which is work-conjugate t...

The Serre or Green and Naghdi equations are fully-nonlinear and weakly dispersive and have a built-in assumption of irrotationality. However, like the standard Boussinesq equations, also Serre’s equations are only valid for long waves in shallow waters. To allow applications in a greater range of h0/l, where h0 and l represent, respectively, depth and wavelength characteristics, a new set of ex...

In this paper, we introduce and analyze discontinuous Galerkin methods for a Naghdi type arch model. We prove that, when the numerical traces are properly chosen, the methods display optimal convergence uniformly with respect to the thickness of the arch. These methods are thus free from membrane and shear locking. We also prove that, when polynomials of degree k are used, all the numerical tra...

A nonlinearly elastic shell is modeled either by the nonlinear threedimensional shell model or by a nonlinear two-dimensional shell model. We show how such two-dimensional shell models can be derived from the minimization problem associated with the nonlinear three-dimensional shell model. For shells made of a Saint Venant-Kirchhoff material, we obtain in particular the nonlinear shell models o...

A nonlinear extension of the Caginalp phase field system is considered that takes thermal memory into account. The resulting model, which a first-order approximation thermodynamically consistent system, inspired by theories developed Green and Naghdi. Two equations, from dynamics universal balance law for internal energy, are written in terms variable (representing non-conserved order parameter...

We propose a system of first-order balance laws that describe the propagation internal solitary waves in multilayer stratified shallow water with non-hydrostatic pressure upper and lower layers. The construction this model is based on use additional variables, which make it possible to approximate Green–Naghdi-type dispersive equations by system. In Boussinesq approximation, governing allow one...

We consider unsteady undular bores for a pair of coupled equations of Boussinesq-type which contain the familiar fully nonlinear dissipationless shallow-water dynamics and the leading-order fully nonlinear dispersive terms. This system contains one horizontal space dimension and time and can be systematically derived from the full Euler equations for irrotational flows with a free surface using...