One-dimensional Viscoelasticity Research Articles

Stresses in Viscoelastic Half Space at Given on the Boundary Non-stationary Normal Displacement

Within the framework of linear model with coinciding volume and shear relaxation kernels non-stationary problem for viscoelastic half space with normal displacements given on its boundary is considered. Solution representation in the form of generalized convolution of corresponding plane elasticity theory problem solution and one-dimensional viscoelasticity theory problem solution is used. These solutions are written down as convolution of boundary conditions with the kernels - surface Green functions. Time Laplace transform is used for kernels constructing. Its inversion is carried out exactly using either analytical representations of the transforms (for plane problem of elasticity theory) or asymptotic method (for one-dimensional problem of viscoelasticity). Explicit formulas for normal and shear stresses on the boundary of viscoelastic half space are obtained using the structure of Green functions. Examples of calculation show that contrary to elastic medium discontinuities of the first kind don’t exist on the wave front.

До задачі розрахунку релаксації напружень у тонкостінних трубчастих елементах із лінійно- в’язкопружних матеріалів

The relaxation of isotropic homogeneous and non-aging linear-viscoelastic materials under conditions of complex stress state is considered. Thin-walled tubular specimens of High Density Polyethylene (HDPE) for creep under a single-axial stretching, with a pure twist and combined load tension and torsion are considered as base experiments, tests. The solution is obtained by generalizing the initial one-dimensional viscoelasticity model to a complex stressed state, constructed using the hypothesis of the proportionality of deviators. The heredity kernels are given by the Rabotnov’s fractional-exponential function. The dependence between the kernels of intensity and volumetric creep is established, which determine the scalar properties of linear viscoelastic materials in the conditions of a complex stressed state in the defining equations of the type of equations of small elastic-plastic deformations, and the kernels of longitudinal and transverse creep defining the hereditary properties of linear-viscoelastic materials under the conditions of the uniaxial tension. The problems of stress relaxation calculation of thin walled tubes under combined tension with torsion have been solved and experimentally approved.

Symmetry group classification for one-dimensional elastodynamics problems in nonlocal elasticity

The symmetry groups of one-dimensional elastodynamics problem of nonlocal elasticity are investigated and we get a classification for the problem. The determining equations of the system of Fredholm integro-differential equations corresponding to one-dimensional nonlocal elasticity equation are found and solved. We get the differential equations that include the kernel function and the independent term. The symmetry groups are determined using these functions. We compare the results of one-dimensional nonlocal elasticity with the results of the Voltera integro-differential equation corresponding to one-dimensional visco-elasticity equation in the conclusion section of the manuscript.

On a stochastic nonlinear equation in one-dimensional viscoelasticity

In this paper we discuss an initial-boundary value problem for a stochastic nonlinear equation arising in one-dimensional viscoelasticity. We propose to use a new direct method to obtain a solution. This method is expected to be applicable to a broad class of nonlinear stochastic partial differential equations.

On the dynamics of fine structure

We investigate models for the dynamical behavior of mechanical systems that dissipate energy as timet increases. We focus on models whose underlying potential energy functions do not attain a minimum, possessing minimizing sequences with finer and finer structure that converge weakly to nonminimizing states. In Model 1 the evolution is governed by a nonlinear partial differential equation closely related to that of one-dimensional viscoelasticity, the underlying static problem being of mixed type. In Model 2 the equation of motion is an integro—partial differential equation obtained from that in Model 1 by an averaging of the nonlinear term; the corresponding potential energy is nonlocal.

L2estimates for the perturbed phase transitions in unbounded one-dimensional nonlinear viscoelasticity

The motion of one-dimensional viscoelastic materials with a cubic stress-strain relation is considered in an unbounded region. Energy estimates are established for the perturbed travelling wave solutions including the dynamic phase transitions

Effectiveness and Robustness with Respect to Time Delays of Boundary Feedback Stabilization in One-Dimensional Viscoelasticity

The damping effect of a boundary feedback mechanism on torsional vibrations of a homogeneous viscoelastic rod is examined. A characteristic equation is derived for the oscillatory modes, and the solutions of this equation are studied by analytic and numerical methods, as functions of the feedback gain parameter and of a parameter for feedback delay. Results are compared to recent studies of elastic materials, where a feedback delay can cause exponential instability; here this phenomenon depends significantly on the short-time behavior of the viscoelastic memory kernel. Finally, an existence result is given, showing that the behavior of a weak solution corresponds in the expected way to the location of the characteristic roots.

Solutions to the Equations of One-Dimensional Viscoelasticity in BV

The initial value problem associated with the equations of one-dimensional viscoelasticity of the rate type is studied. The system of equations is linearized and the solutions to the resulting linear system are completely analyzed in the $L^1 $-setting by the method of Fourier transform. This enables us to establish the global existence of solutions to the original nonlinear system with small initial data in $L^1 \cap {\text{BV}}$. As a by-product, we also obtain precise results on asymptotic behavior of solutions.

The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity

It has been proved (Lax [I], MacCamy-Mizel [2]) that the equations of one-dimensional nonlinear elasticity do not admit, in general, smooth solutions in the large. It is expected though, that if the stress depends on the history of motion in an appropriate fashion, then smooth solutions exist. The simplest model of a solid with history dependence is provided by one-dimensional viscoelasticity, where the stress a is a function of the deformation gradient u, and its time derivative zi, . Greenberg, MacCamy and Mizel [3] have considered the semilinear case, cr(uZ , zi.) = ~J(zL~) + ti, where 93 is a strictly increasing function. They prove the existence of a unique solution which is asymptotically stable. In this paper we consider the traction boundary value problem in the general case where O(U, , ti,) may be nonlinear in both u, , zi, . The form of the dependence of CT(U~ , 2%) on ti, is restricted by the requirement that the viscosity be bounded away from zero. On the contrary, the dependence on U, is essentially unrestricted apart from certain requirements of boundedness. It turns out that the viscoelastic part dominates the elastic part and secures the existence of a unique solution in the large. This solution is smooth enough so that all derivatives entering the equation of motion are Holder continuous. The tools of the proof are certain “energy” estimates combined with known a priori bounds from the theory of parabolic equations, and the Leray-Schauder fixed point theorem. In the final part of the paper, we investigate the asymptotic stability of the solution. The mechanism which provokes the decay of the solution is induced by the viscosity. On the other hand, the number and the nature of all possible static configurations depend entirely on the elastic part of