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.
Highlights
Membranes are widely used as structural elements in many engineering fields
In the theory of elastic membranes, the constitutive equations relating to surface stresses and strains are typically formulated in terms of a strain energy function which depends on the deformation tensor C alone
According to the general theory of irreversible processes, the complete constitutive model of an inelastic membrane requires that an elastic potential and the stress tensor S are given by appropriate constitutive equations in terms of C and a including possibly their surface and time derivatives
Summary
Membranes are widely used as structural elements in many engineering fields. Fabric (pneumatic and tension) structures, automobile airbags, parachutes, meteorological balloons are examples. A typical inelastic effect observed in the inflation and subsequent deflation of balloons is that of hysteresis accompanied by stress-softening (Fig 1). These effects are observed in biological membranes. The aim of this work is to develop a constitutive model which is capable to capture main effects of damage-induced stress-softening observed in polymeric and biological membranes. This goal is achieved through the use of certain concepts from the field of continuum damage mechanics. A simple description of stress-softening effects is derived through successive specification of the general constitutive equations
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