Symmetric stiffness matrices for isoparametric finite elements in nonlinear elasticity

The article illustrates a position-based finite element formulation, which greatly simplifies the statement of nonlinear elasticity problems. The formulation adopts as main unknowns the nodal positions in the current configuration instead of the nodal displacements. As a result, simple analytical expressions are obtained of the secant and tangent stiffness matrices for general isoparametric finite elements. Contrary to most formulations of the literature, the secant stiffness matrices turn out to be symmetric. Furthermore, any hyperelastic constitutive law can be easily implemented. Specialised expressions are deduced for the stiffness matrices of a two-node truss bar element and a three-node planar triangular element. The validity of the proposed approach is illustrated through the analysis of a steep von Mises truss and Cook’s membrane. For illustration, the de Saint Venant–Kirchhoff and neo-Hookean material models are considered.