A mixed finite element formulation for slightly compressible finite elasticity with stiff fibre reinforcement. Two fibre families. Uniaxial tension formulation

We propose a novel finite element based formulation for the solution of the static mechanical mixed boundary value problem of a finite elastic solid reinforced by two distinct, stiff fibre families. The fibre tensions, are assumed decoupled and uniaxial, at out-set. The associated energy conjugate fibre stretch rates are shown to be uniaxial by duality. The natively displacement dependent fibre tension-fibre stretch pairs are replaced by auxiliary independent variables. The complementary, displacement based, stresses and energy conjugate strain rates become tensionless and stretch-rate-less in the two fibre directions, respectively, by construction. An additively decoupled hyperelastic strain energy ansatz in terms of the fibre stretches and a novel apparently doubly stretchless Cauchy-Green tensor is used. The displacement based part of the formulation is set in an apparently inextensible fibre metric space. The proposed uniaxial fibre tension description is statically exact for the fully constrained problem, and the novel doubly stretchless Cauchy-Green tensor is conditionally kinematically admissible in its vicinity. The formulation is realised as a five-field mixed finite element method admitting separate higher order approximations in H1, for the displacement, and in L2, for the energy conjugate fibre tensions and stretches, respectively. The convergence and correctness of the implementation is verified by numerical and analytical examples.