Damage-Induced Stress-Softening Effects in Elastomeric and Biological Membranes

Abstract

Abstract. The two-dimensional finite strain constitutive model for membranes is presented; it incorporates stress-softening behaviour typically observed in elastomeric and natural or biologically-derived soft membranes subjected to severe deformations. It is assumed that the experimentally observed progressive degradation of a membrane stiffness under monotonous and cycling loading can macroscopically be modelled by a scalar damage variable. The evolution of this variable during the deformation process is specified by the kinetic law of damage growth, which together with the constitutive equation for the surface stress tensor and the damage criteria completely determines the presented constitutive model. It is shown that the general constitutive model can be specified for particular classes of problems under certain additional assumptions. In particular, a remarkable simplification of the model is achieved assuming that the state of strain at membrane points can be characterised by a single scalar variable, the so-called effective (equivalent) strain. This assumption is combined with the hypothesis of maximum strain according to which the stress softening in the membrane depends only on the maximum previous strain experienced during deformation history. Within these two hypotheses the progressive degradation of membrane stiffness is completely described by a softening function which determines the current value of damage variable in terms of maximum equivalent strain. Various specific forms of such a softening function as well as different definitions of the effective strain are considered.

Statics of hybrid structures composed of interacting membranes and strings

“…In [3][4][5] cables arranged along the boundary of membrane structure have only been considered with the additional assumption of perfect elasticity of the membrane and cables. A more general approach to the statics of hybrid structures comprised of interacting membranes and strings has been outlined in the author's earlier works [8,9]. The aim of this paper is to show that the equations governing large deformations of hybrid structures with membrane and string elements arranged in any way and of any material behavior can be derived directly from the basic principles of mechanics.…”

“…In the theory of smooth membranes, the integral balance laws may be replaced by partial differential equations (the local equilibrium equations) at interior points of the membrane whenever all fields are sufficiently smooth for the surface divergence theorem to be applicable (Gurtin and Murdoch [11]). The extension of that theorem to piecewise smooth surfaces and fields was presented in [8,9] and the main results are given in Appendix (see also Forte and Vianello [12] in the context of piecewise smooth boundaries of three dimensional bodies).…”

“…The equations (48) provide the most general form of the constitutive equations satisfying the principle of frame indifference. Specific forms of the elastic constitutive equations for elastomeric and biological membranes have been considered in [15] and for anisotropic fabric membranes in [9].…”

“…However, the modeling of elastomeric and biological membranes requires that inelastic effects, such as stress softening, permanent set and hysteresis, are also properly accounted for. These aspects of the constitutive modeling have been considered in [16] for smooth membranes and the same approach may be adopted for hybrid structures.…”