A new mixed formulation for Total Lagrangian fast solid dynamics has been recently proposed by some of the authors. This new methodology differs from the classic displacement-based formulation in that it employs a set of conservation laws where the linear momentum, the deformation gradient and the total energy of the system are the unknown conservation variables. Displacement-based methods present a series of shortcomings, such as reduced spacial accuracy for derived variables, namely stresses and strains. For low order discretisation schemes, this can produce an overly stiff behaviour of the solid under bending dominated scenarios. In addition, locking phenomena can also appear in the case of nearly incompressible materials. The introduction of the new mixed formulation can circumvent these deficiencies by employing a different set of unknown variables.
In the context of this new mixed formulation, the conservation laws form a system of hyperbolic equations where Computational Fluid Dynamics techniques can be used for numerical simulation. Among those, Finite Volume-based schemes have already been successfully implemented for 2-D/3-D applications. In particular, an adapted Jameson-Schmidt-Turkel (JST) scheme has been recently developed utilising fourth order and second order stabilisation terms. The fourth order term provides stabilisation against odd-even decoupling phenomena, whilst the second order term activates with steep gradients, capturing any discontinuity in the solution.

Aim of the paper is to introduce the mixed formulation in the context of Smooth Particle Hydrodynamics (SPH), also incorporating concepts from the JST Finite Volume methodology. In SPH, the second order (harmonic) and fourth order (biharmonic) operators can be readily obtained by closed-form differentiation of the interpolating kernel functions. Crucially, SPH derivatives are smooth even in the presence of very large gradients of deformations.
With the purpose of demonstrating the capabilities of the new methodology, applications of the mixed formulation will be presented and compared against alternative numerical methodologies for a series of benchmark tests.