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We propose an anisotropic and nonlinear generalization of the Kelvin–Voigt viscoelastic model obtained considering the additive splitting of the Cauchy stress tensor in an elastic and a dissipative part. The former one corresponds to a fiber-reinforced hyperelastic material while the dissipative effect is described by the most general linear transverse-isotropic tensorial function of symmetric part of the velocity gradient. In a such a way we characterize the dissipative contribution via three viscoelastic moduli. We then show, by a detailed analysis of the simple shear quasistatic motion and the corresponding creep phenomena, that this motion may be used to determine experimentally the viscoelastic parameters. [ABSTRACT FROM AUTHOR]

Based on the classical theory of simple materials of differential type and the results on the analytical form of constitutive models consistent with the laws of thermodynamics, we introduce a very general response function for the Cauchy stress tensor of a dispersive hyperelastic solid. Next, by focusing on the propagation of localised waves in slightly dispersive quasi incompressible solids, we prove the existence of a rich variety of solitary wave solutions as well as kink wave solutions. Our analysis and results can be easily specialised to shape memory alloys. [Display omitted] • We develop a mathematical model for the stress tensor of dispersive hyperelastic solids within the theory of simple materials of differential type. • We study the wave propagation in dispersive hyperelastic solids. • In the small but finite regime, we find exact solutions representing solitary and kink waves. [ABSTRACT FROM AUTHOR]

One of the simplest problems in the classical Navier–Stokes theory that illustrates the development of boundary layers is the asymptotic suction problem (Schlichting (1960)). This problem has been studied within the context of many different non-Newtonian fluid models and in this note we consider it within the context of a new class of constitutive relations that has been put into place. Depending on the values of the parameters, this class of constitutive relations includes models for both shear thinning and shear thickening fluids, and truly implicit shear stress shear rate relationships. For any set of the material parameters we are able to find exact solutions to the governing equations, study their stability, show that suction has a stabilizing effect, and delineate the development of boundary layers. [ABSTRACT FROM AUTHOR]

The purpose of the present research is to investigate how the nonlinearity and the boundary conditions have the power to influence the coupling of the various modes of deformation when a plane strain deformation is superimposed on anti-plane shear deformation for a class of fiber-reinforced incompressible hyperelastic materials described by a strain energy density W. Attention is confined when W depends only on the first invariant of the strain tensor and on the square of the stretch in the direction of the fibers. We are able to write down the governing equations of equilibrium as a coupled system of three nonlinear partial differential equations for three displacement fields. Two displacements are the in-plane components and one displacement is the anti-plane state. The system that we are able to deduce in a compact form is always compatible at variance with the case in which the anti-plane shear problem is analyzed. As explicit example of our findings we study the problem of the helical shear and we investigate into details the coupling of its axial and azimuthal components. [ABSTRACT FROM AUTHOR]

Abstract: A material is of coaxial type if the Cauchy stress tensor and the strain tensor are coaxial for all deformations. Clearly a hyperelastic material is of coaxial type if and only if it is isotropic. Here we present a weaker definition of materials of coaxial type. Anisotropic materials may be of a coaxial type in a weak sense if for a given specific we have that . We denote these materials -coaxial. We show that for transverse isotropic materials weak coaxial constitutive equations may be characterized using universal relations. We discuss the impact of -coaxial materials in the modeling of soft tissues. We conclude that -coaxial materials are a strong evidence that in real world materials two anisotropic invariants are always necessary to model in a meaningful and correct way single fiber reinforced materials. [Copyright &y& Elsevier]

We investigate some possible nonlinear generalizations of the Kelvin—Voigt viscoelastic models. We use the usual idealization of creep and recovery experiments to discuss the mechanical significance of some constitutive requirements and we show that in the case of a shear-rate dependent viscosity localization of the solution is possible under a simple constitutive characterization. [ABSTRACT FROM PUBLISHER]

Acousto-elasticity is concerned with the propagation of small-amplitude waves in deformed solids. Results previously established for the incremental elastodynamics of exact non-linear elasticity are useful for the determination of third- and fourth-order elastic constants, especially in the case of incompressible isotropic soft solids, where the expressions are particularly simple. Specifically, it is simply a matter of expanding the expression for ρv2, where ρ is the mass density and v the wave speed, in terms of the elongation e of a block subject to a uniaxial tension. The analysis shows that in the resulting expression: ρv2=a+be+ce2, say, a depends linearly on μ; b on μ and A; and c on μ, A, and D, the respective second-, third, and fourth-order constants of incompressible elasticity, for bulk shear waves and for surface waves. [ABSTRACT FROM AUTHOR]

We consider deformations and motions that correspond to a non-uniform axial stretch superimposed to a general pseudoplane deformation of the first kind. We provide the general determining equation for this kind of solution when the constitutive equation for the stress tensor is derived from the elastic neo-Hookean strain-energy density. Then we provide some explicit solutions which are generalizations of the non-symmetric torsion of a slab and some finite-amplitude pseudo-planar elliptical motions. [ABSTRACT FROM PUBLISHER]

Investigates the helical shear problem for a circular cylindrical tube composed of an isotropic hyperelastic incompressible material. Constitutive models that account for hardening at large deformations; Characterization of the stress fields and displacements; Mechanical properties of elastomeric materials.

We consider shear wave propagation in soft viscoelastic solids of rate type. Based on objective stress rates, the constitutive model accounts for finite strain, incompressibility, as well as stress- and strain-rate viscoelasticity. The theory generalises the standard linear solid model to three-dimensional volume-preserving motions of large amplitude in a physically-consistent way. The nonlinear equations governing shear motion take the form of a one-dimensional hyperbolic system with relaxation. For specific objective rates of Cauchy stress (lower- and upper-convected derivatives), we study the propagation of acceleration waves and shock waves. Then we show that both smooth and discontinuous travelling wave solutions can be obtained analytically. We observe that the amplitude and velocity of steady shocks are very sensitive to variations of the stress relaxation time. Furthermore, the existence of steady shocks is conditional. Extension of these results to the case of multiple relaxation mechanisms and of the Jaumann stress rate is attempted. The analysis of simple shearing motions is more involved in these cases. • The modelling of dissipation in soft incompressible solids is discussed. • Finite-strain viscoelastic models with objective stress rates are presented. • Shear wave propagation is governed by a hyperbolic system of balance laws. • Travelling wave solutions are either smooth kinks or steady shocks. • Exact analytical results are obtained for the lower- and upper-convected rates. [ABSTRACT FROM AUTHOR]

We consider three-dimensional nonlinear viscoelastic models that account for both stress relaxation and creep/recovery phenomena. These models are based on different frame indifferent time derivatives: the Oldroyd (or upper-convected) derivative, the Jaumann (or co-rotational) derivative and the Cotter–Rivlin (or lower-convected) derivative. Under a simple tension creep process, these constitutive equations predict the same stress relaxation but lead to different situations. The models based on the Oldroyd and the lower-convected derivative require restrictions on the values of the material parameters as well as on the traction/compression stress. The model based on the Jaumann derivatives does not require any restriction. All the constitutive models examined are used to study the finite amplitude, horizontal oscillatory motion of a mass attached to a rate-type viscoelastic string. In this way we generalize the classical results by Beatty and Zhou (1991). [ABSTRACT FROM AUTHOR]