Constitutive equations in finite viscoplasticity of semicrystalline polymers.
The paper deals with semicrystalline polymers at isothermal finite deformation, for which uniaxial tensile tests with different constant rates of deformation have been performed. In developing the model, the authors start from the multiplicative decomposition of the deformation gradient into its elastic and plastic (irreversible) parts, widely utilized in finite elasto-plasticity. Physical background of the intermediate (relaxed or isoclinic) configurations, with respect to which the material response is considered to be elastic, is related to the crystalline lattice of the material. In crystal plasticity theory, the rate of plastic deformation is generally prescribed by multislip in the appropriate slip systems. Here the authors identified the elastic parts to network stretch and slippage stretch, but in the model no slip systems, no yield condition or viscoplastic surface have been introduced. Moreover, the rate of plastic deformation, via formula (27), is considered to be proportional to a part of the deviator of Cauchy stress (but how can it be identified?), rotated with the elastic rotation. This feature is generally associated with the normality flow rule, when a quadratical yield condition is accepted. We leave away physical motivations in deriving the constitutive relations, and we draw out from the multitude of formulae presented in the paper, the two ones which can be considered to define the model: formula (37), which expresses the rate of left Cauchy-Green elastic tensor in terms of itself, of the velocity gradient, where the strain energy density and the scalar coefficient \(a(t)\) are considered to be given, and formula (35), the stress-strain relation which prescribes the deviatoric part of Cauchy stress in terms of the left Cauchy-Green tensors of the total deformation, as well as of the elastic deformation. Certain simplified versions, see formulae (42) and (43), provide the constitutive equations for the analysis of experimental data in three-dimensional mechanical tests, as the authors notified. But here the previous coefficient \(a(t)\), introduced in formula (27), has been replaced by the ratio of intensity of the rate of total deformation and that of the equivalent stress. In order to characterize the behaviour of constitutive equations (36) with (35), when the history of the deformation gradient is given on a certain time-interval during an homogeneous process, the authors solve first the differential equation system (36) for the left Cauchy-Green elastic tensor, when initial conditions are prescribed, and subsequently, the deviatoric part of Cauchy stress tensor can be defined via formula (35). Consequently, when the deformation for uniaxial tension and for shear stress, in (44) and (51), are given, then the left Cauchy-Green elastic tensor can be expressed in the form written in (46) and (57), if and only if they are solutions of the above-mentioned differential equation system (37) attached to the constitutive description. Finally, we mention that the equivalent stress, introduced via the formula for \(a(t)\), is considered to be one of the adjustable parameters of the model.
scientific article; zbMATH DE number 2067528