A polyconvex framework for soft biological tissues. Adjustment to experimental data

Abstract

The main goal of this contribution is to provide a simple method for constructing transversely isotropic polyconvex functions suitable for the description of biological soft tissues. The advantage of our approach is that only a few parameters are necessary to approximate a variety of stress-strain curves and to satisfy the condition of a stress-free reference configuration a priori in the framework of polyconvexity. The proposed polyconvex stored energies are embedded into the concept of structural tensors and the representation theorems for isotropic tensor functions are utilized. As an example, the medial layer of a human abdominal aorta is investigated, modeled by some of the proposed polyconvex functions and compared with experimental data. Hereby, the economic fitting to experimental data, and hence the easy handling of the functions is shown.

A mechanochemical 3D continuum model for smooth muscle contraction under finite strains

“…It is worth noting that the restrictions I 4 , I 6 , J 4 , J 6 ≥ 0 render the free energies in Eqs. (31) and (32) polyconvex which implies Legendre-Hadamard ellipticity, see, e.g., Balzani et al [2]. Following Stålhand et al [21], take the function C in Eq.…”

“…By comparing to the expressions for η k in [21], it is concluded that the parameters D 1 to D 4 should be divided by the factor λ f + 1.12 where λ f is computed to be 1 by substituting λ x = λ z = 1 in Eq. (48) 2 and taking the square root. The parameters associated with the myosin heads, E 3 , E 4 , f 3 , and f 4 , must also be specified.…”

On a tensor cross product based formulation of large strain solid mechanics

“…An example of particular interest in many bioengineering applications is that of transversely isotropic materials [39], [40]. In such cases the strain energy w tr (I 1 , I 2 , I 3 , I 4 , I 5 ) can be expressed as a function of two further invariants, which with the current notation can be defined as…”

Relaxed Incremental Variational Formulation for Damage in Fiber‐Reinforced Materials

“…For the effective energy a polyconvex function is chosen in order to guarantee material stability in the undamaged regime, see [12]. A series of polyconvex functions for fiber-reinforced materials is given in [2]. The incremental stress potential for the one-dimensional fiber-reinforcement is considered as…”