Traumatic injuries to the central nervous system (brain and spinal cord) have recently been put under the spotlight because of their devastating socioeconomical cost. At the cellular scale, recent research efforts have focussed on primary injuries by making use of models aimed at simulating mechanical deformation induced axonal electrophysiological functional deficits. The overwhelming majority of these models only consider axonal stretching as a loading mode, while other modes of deformation such as crushing or mixed modes-highly relevant in spinal cord injury-are left unmodelled. To this end, we propose here a novel 3D finite element framework coupling mechanics and electrophysiology by considering the electrophysiological HodgkinHuxley and Cable Theory models as surface boundary conditions introduced directly in the weak form, hence eliminating the need to geometrically account for the membrane in its electrophysiological contribution. After validation against numerical and experimental results, the approach is leveraged to model an idealised axonal dislocation injury. The results show that the sole consideration of induced longitudinal stretch following transverse loading of a node of Ranvier is not necessarily enough to capture the extent of axonal electrophysiological deficit and that the non-axisymmetric loading of the node participates to a larger extent to the subsequent damage. On the contrary, * Corresponding authors Email addresses: man.kwong@eng.ox.ac.uk (Man Ting Kwong ), antoine.jerusalem@eng.ox.ac.uk (Antoine Jérusalem ) Preprint submitted to Computer Methods In Applied Mechanics And EngineeringJune 6, 2018 a similar transverse loading of internodal regions was not shown to significantly worsen with the additional consideration of the non-axisymmetric loading mode.

This paper presents a Discontinuous Galerkin scheme in order to solve the nonlinear elliptic partial differential equations of coupled electro-thermal problems.In this paper we discuss the fundamental equations for the transport of electricity and heat, in terms of macroscopic variables such as temperature and electric potential. A fully coupled nonlinear weak formulation for electro-thermal problems is developed based on continuum mechanics equations expressed in terms of energetically conjugated pair of fluxes and fields gradients. The weak form can thus be formulated as a Discontinuous Galerkin method.The existence and uniqueness of the weak form solution are proved. The numerical properties of the nonlinear elliptic problems i.e., consistency and stability, are demonstrated under specific conditions, i.e. use of high enough stabilization parameter and at least quadratic polynomial approximations. Moreover the prior error estimates in the H 1 -norm and in the L 2 -norm are shown to be optimal in the mesh size with the polynomial approximation degree.

A coupled Electro-Thermo-Mechanical Discontinuous Galerkin (DG) method is developed considering the non-linear interactions of electrical, thermal, and mechanical fields. In order to develop a stable discontinuous Galerkin formulation the governing equations are expressed in terms of energetically conjugated fields gradients and fluxes. Moreover, the DG method is formulated in finite deformations and finite fields variations. The multi-physics DG formulation is shown to satisfy the consistency condition, and the uniqueness and optimal convergence rate properties are derived under the assumption of small deformation. First the numerical properties are verified on a simple numerical example, and then the framework is applied to simulate the response of smart composite materials in which the shape memory effect of the matrix is triggered by the Joule effect.

Abstract. Discontinuous Galerkin (DG) methods are attractive tools to integrate several PDEs in engineering sciences, due to their high order accuracy and their high scalability in parallel simulations. The main interest of this work is to derive a constant and stable Discontinuous Galerkin method for two-way electro-thermal coupling analyses. A fully coupled nonlinear weak formulation for electro-thermal problems is developed based on continuum mechanics equations which are discretized using the Discontinuous Galerkin method. Toward this end, the weak form is written in terms of energetically conjugated fields gradients and fluxes. In order to validate the effectiveness of the formulation and illustrate the algorithmic properties, a numerical test for composite materials is performed. 2558

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