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A new class of quasi-linear models for describing the nonlinear viscoelastic response of materials. (English) Zbl 1291.74051

The authors construct a class of quasi-linear viscoelastic models for homogeneous isotropic viscoelastic bodies based on the casuality (forces (stresses) as the cause, and motions (strains) as the effect). The main idea of the authors is the following: the deformation gradient can be expressed in terms of a functional of the history of the loading. In this context, the authors suggest the relationship for nonlinear viscoelasticity in the form \[ B = f(T(0), t) + \int _0 ^t \frac{\partial f (T(s), t-s) }{\partial T(s)} \frac{dT(s)}{ds}ds, \] where \(B\) is the Cauchy-Green tensor, \(T\) is the stress tensor and \(f\) is an isotropic function. In the particular case of small strains, tensor \(B\) is replaced by the tensor of linearized strains \(\varepsilon\), and \(f\) is replaced by \(\hat{f}=\frac 12(f-1)\). The authors research the model in detail for the uniaxial case when \(\hat{f}(T(t),t)=\{G(T)\}J(t)\), where the ‘generalized creep function’ \(J(t)\) has an exponential expression, and the function \(G(T)\) has two forms (a second-order polynomial and a composition of exponential and linear fractional functions). The authors analyze the one-dimensional model for the cases of creep and relaxation (under the step input), and under the constant stress rate. They suggest a numerical iteration based on the Newton-Raphson method to describe the relaxation. The sufficiency of the supposed rheological model is verified by the comparison of calculated and experimental data of a biological material (human patellar tendons for two age groups) under a uniaxial tension with constant stress rate and constant strain rate, and under a cyclic loading with peak and valley strains equal to 2.4 and 1.1 percent, respectively. The authors find an agreement between the calculated and experimental data.

MSC:

74D10 Nonlinear constitutive equations for materials with memory
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