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This work presents a generalized Kirchhoff–Love shell theory that can explicitly capture fiber-induced anisotropy not only in stretching and out-of-plane bending, but also in in-plane bending. This setup is particularly suitable for heterogeneous and fibrous materials such as textiles, biomaterials, composites and pantographic structures. The presented theory is a direct extension of classical Kirchhoff–Love shell theory to incorporate the in-plane bending resistance of fibers. It also extends existing second-gradient Kirchhoff–Love shell theory for initially straight fibers to initially curved fibers. To describe the additional kinematics of multiple fiber families, a so-called in-plane curvature tensor—which is symmetric and of second order—is proposed. The effective stress tensor and the in-plane and out-of-plane moment tensors are then identified from the mechanical power balance. These tensors are all second order and symmetric in general. Constitutive equations for hyperelastic materials are derived from different expressions of the mechanical power balance. The weak form is also presented as it is required for computational shell formulations based on rotation-free finite element discretizations. The proposed theory is illustrated by several analytical examples.

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Polyconvexity is an important mathematical condition imposed on a strain energy function. In particular, it is sufficient for the ellipticity of the constitutive equation and for the material stability and becomes especially crucial in the context of nonlinear elasticity. In combination with another condition referred to as coercivity, polyconvexity ensures existence of the global minimizer of the total elastic energy which implies a solution of a boundary value problem. While a great variety of polyconvex energies are known for isotropic and have recently been proposed for anisotropic elastic solids, there are, to the best of our knowledge, no results on polyconvexity for electro- and magneto-elastic materials. In the present paper, we extend the notion of polyconvexity to the coupled electro- and magneto-elastic response and formulate polyconvex free energy functions for electro- and magneto-sensitive elastomers. In analogy to the purely elastic response, these free energy functions will ensure the positive features of the constitutive equations mentioned above, although a strict mathematical proof of this fact should be supplied later. The proposed model is able to describe the electro- and magnetostriction and demonstrates good agreement with the corresponding experimental data.

Strain-induced crystallization is a unique crystallization process taking place solely in polymers subjected to large deformations. It plays a major role for reinforcement and improvement of mechanical properties of polymers with a high regularity of the molecular structure. In this paper, we develop a micromechanical model for the strain-induced crystallization in filled rubbers. Accordingly, the strain-induced crystallization is considered as a process triggered by fully stretched and continued by semistretched polymer chains. The model extends the previously proposed network evolution model [Dargazany and Itskov, Int. J. Solids Struct. 46, 2967 (2009)] and can thus, in addition to the stress upturn and evolution of crystallinity, take into account several inelastic features of filled rubbers, such as the Mullins effect, permanent set, and induced anisotropy. Finally, the accuracy of the model is verified against different set of experimental data both with respect to the stress-strain and crystallization-strain relations. The model exhibits good agreement with the experimental results, which, besides its relative simplicity, makes it a good option for finite-element implementations.

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