Vu Ngoc Khiêm

A polyconvex anisotropic free energy function for electro- and magneto-rheological elastomers

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.

Multi-scale modeling of soft fibrous tissues based on proteoglycan mechanics

Collagen in the form of fibers or fibrils is an essential source of strength and structural integrity in most organs of the human body. Recently, with the help of complex experimental setups, a paradigm change concerning the mechanical contribution of proteoglycans (PGs) took place. Accordingly, PG connections protect the surrounding collagen fibrils from over-stretching rather than transmitting load between them. In this paper, we describe the reported PG mechanics and incorporate it into a multi-scale model of soft fibrous tissues. To this end, a nano-to-micro model of a single collagen fiber is developed by taking the entropic-energetic transition on the collagen molecule level into account. The microscopic damage occurring inside the collagen fiber is elucidated by sliding of PGs as well as by over-stretched collagen molecules. Predictions of this two-constituent-damage model are compared to experimental data available in the literature.

Electroelasticity of dielectric elastomers based on molecular chain statistics

The mechanical response of dielectric elastomers can be influenced or even controlled by an imposed electric field. It can, for example, cause mechanical stress or strain without any applied load; this phenomenon is referred to as electrostriction. There are many purely phenomenological hyperelastic models describing this electroactive response of dielectric elastomers. In this contribution, we propose an electromechanical constitutive model based on molecular chain statistics. The model considers polarization of single polymer chain segments and takes into account their directional distribution. The latter results from non-Gaussian chain statistics, taking finite extensibility of polymer chains into account. The resulting (one-dimensional) electric potential of a single polymer chain is further generalized to the (three-dimensional) network potential. To this end, we apply directional averaging on the basis of numerical integration over a unit sphere. In a special case of the eight-direction (Arruda–Boyce) model, directional averaging is obtained analytically. This results in an invariant-based electroelastic constitutive model of dielectric elastomers. The model includes a small number of physically interpretable material constants and demonstrates good agreement with experimental data, with respect to the electroactive response and electrostriction of dielectric elastomers.