Minimization of Shear Energy in Two Dimensional Continua with Two Orthogonal Families of Inextensible Fibers: The Case of Standard Bias Extension Test

In this paper we consider Pipkin-type bi-dimensional continua with two orthogonal families of inextensible fibers. We generalize the representation formula due to Rivlin (J. Ration. Mech. Anal. 4(6):951-974, 1955) valid for planar placement fields. They are the sum of two vector functions each of which depends on one real variable only, are piece-wise C2 and may exhibit jumps of their first gradients on some inextensible fibers. Subsequently we consider a deformation energy depending only on the shear deformation relative to the inextensible families of fibers. In the suitably introduced space of configurations representing considered constrained kinematics, we formulate the relative energy minimization problem for the standard bias extension test problem, i.e., elongation of specimens which (i) have the shape of a rectangle with one side exactly three times longer than the other; (ii) are subject to a relative displacement of shorter sides in the direction parallel to the longer one. By exploiting the material and geometric symmetries, we reduce the aforementioned minimization problem to the determination of a piece-wise real function defined in a real interval. A delicate calculation of the energy first variation produces a necessary stationarity condition: it consists of an integral equation which is to be satisfied by the unknown function. The crucial points of this deduction are represented by (i) the reduction of two-dimensional integrals to one-dimensional integrals by the Fubini Theorem and (ii) the determination of the set of admissible kinematical functions on the basis of imposed boundary conditions, which implies a further integral constraint condition on the unknown function. Therefore the energy minimization problem requires the introduction of a global Lagrange multiplier. The established integral equations are solved numerically with a scheme based on a contraction type iterative process. Finally, the equilibrium shapes of the specimen undergoing large deformations, as determined by the presented model, are shown and briefly discussed.