In this work, we rst recapitulate experimental ndings under dierent testing conditions available in the literature. Next, we establish a theoretical framework for the analysis of evolution of deformation into a highly localized pattern, where the theoretical approach with necessary adjustments to accommodate the constitutive behaviour of the solid polymer is applied in the course of development. A detailed theoretical analysis of critical stress conditions required for shear band initiation shows that in order to predict the material instability in accordance with experimental ndings it is necessary to introduce a multi-parameter type of constitutive relation.
To incorporate the strain rate dependence and strain memory properties in the constitutive formulation, we introduce integral hereditary operators of Volterra type in place of free material constants of a general hyper stress constitutive model. The corresponding constitutive equations, which are derived from the proposed form of a free energy functional, are based on a single-integral type representation of the response functionals inherent in the expression for free energy.
FoundationsA classical concept of an elastic material as a simple material is based on an assumption that the material behavior does not depend on a strain history. Such an assumption is usually augmented by the requirement of existence of a stress potential, which is a function of deformation gradient or some other possible strain measure. Stored energy described in the form of strain energy function serves as a starting point for modeling different types of material response. In the sequel, we describe in short terms the thermodynamic consideration of thermo-rheologically simple material and propose a constitutive model which can accomodate a broad range of thermo-hyperelastic as well as thermo-viscoelastic behavior.Consider a material body in an initial state of zero stress and strain which is referred to as a preferred natural configuration in a three dimensional Euclidean space IR 3 . The material elements or particles of a continuous medium, or body B, are denoted by the position vector X ∈ B in a fixed reference configuration with the mass density R . Sometimes it is preferable to use the current placement of a body as a natural reference configuration and define the past histories with respect to it. If x(t) denotes the current position of a particle, then the spatial position history of the particle can be expressed as x(τ) = χ(x(t), τ), where x(τ) denotes the location at time τ of the particle that will occupy the position x(t) at time t so that χ = χ t (x, τ) represents a backward path line of a particle τ ≤ t presently at x = χ t (x, t). Consequently, the relative deformation gradient and the relative right Cauchy-Green deformation tensor can be defined as F t (τ) = ∇ x χ t (x, τ), F ij = ∂χ i /∂x j , and
The modelThe functional relationship established in [9] σ(t) = G t τ=−∞ [C t (τ); B(t)], where G is an isotropic functional and B denotes the left Cauchy-Green strain tensor is a starting point in the discussion of our constitutive model. The dependence on B implies the existence of a natural reference state. From the relation above a single integral type of constitutive equation for a compressible isotropic material with fading memory can be derived which retains only linear terms in the strain history. The equation readsThe model consists of twelve relaxation functions χ A = χ A (I . In what follows we will present an extention of previously described isothermal relations into a temperature domain. The resulting relations can be considered as a special case of a generalized thermo-rheological model that is still of a single i...
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