Quasistatic Viscoelasticity Research Articles

A variational approach to frame-indifferent quasistatic viscoelasticity of rate type.

The three-dimensional dynamical model for nonlinear viscoelasticity of strain-rate type is investigated in a quasistatic setting under the assumption of higher-order regularity of the deformation, which in the literature is referred to as the case of non-simple materials. The existence of weak solutions is proven using a time-discretization technique while respecting the condition of dynamical frame indifference. Some observations on frame indifference for strain-rate-type stresses are made, and corrections are proposed for some related work in the literature. Finally, a counterexample is given to show that the assumed higher-order regularity is necessary in order to obtain the required compactness.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Quasistatic Viscoelasticity with Self-Contact at Large Strains

The frame-indifferent viscoelasticity in Kelvin-Voigt rheology at large strains is formulated in the reference configuration (i.e., using the Lagrangian approach) considering also the possible self-contact in the actual deformed configuration. Using the concept of 2nd-grade nonsimple materials, existence of certain weak solutions which are a.e. injective is shown by converging an approximate solution obtained by the implicit time discretisation.

Study of the Viscoelasticity of Chopped Aramid Fiber Reinforced Rubber Composite

The chopped aramid fiber reinforced rubber composite (AFRC) has been widely used in the tire treads for its excellent characteristics. The viscoelasticity which is an important mechanical property for rubber matrix would be influenced by the adding of aramid fiber. In this study, the dynamic and quasi-static viscoelasticity were investigated via the dynamic thermal analysis experiment and the mullins experiment with multi-step relaxation, respectively. The frequency and temperature scanning were employed for AFRC with different fiber volume fractions, fiber aspect ratios and fiber orientation distributions. The effects of constituent parameters on the dynamic viscoelasticity were studied as well as the general rule of reinforcing effect of aramid fiber on rubber materials was presented. The stress relaxtion for AFRC were analysed basing on the experimental results. In addition, The ability of the viscoelastic constitutive model to describe the quasi-static viscoelastic behavior of AFRC were explored.

Solution of Integral Equations Arising in Mathematical Problems of Construction Science, using the Bogolyubov – Krylov method

The possibility of a numerical solution of construction problems leading to Fredholm integral equations of the first and second kinds is considered. The following problems of building sciences, for example, lead to such equations: in the theory of elasticity – elastoplastic torsion of bodies, stress concentration in a body during its bending; in the theory of vibrations, in the mechanics of structural failure – in the formation of cracks; when studying non-stationary phenomena in solids – non-stationary heat transfer, quasistatic viscoelasticity, propagation of longitudinal and transverse waves and other processes. According to the formulas of numerical integration, the problem can be reduced to a system of linear algebraic equations with a matrix of coefficients, which can be ill-conditioned, and the task is classically incorrect. This circumstance complicates its decision by traditional methods. To solve ill-posed problems similar to the problem in question, there are classical regularization methods. However, the numerical implementation of regularizing algorithms is not always easy to implement. In this regard, the goal of our study was to develop a numerical method for solving the Fredholm integral equations of the first and second kinds, which allow solving such problems without using regularization methods. The authors propose to solve problems of this type on the basis of the Bogolyubov – Krylov formula for representing certain integrals, in the form of finite-dimensional sums. At the same time, it becomes possible to use a non-uniform grid of nodes for splitting the integration interval. To efficiently define the set of nodes for splitting the integration interval, it is proposed to use a priori information about the properties of the solution of the problem. For example, the number of split points of the integration interval increases where the intended solution undergoes the most rapid changes, and decreases in areas where the intended solution is close to linear. The optimal arrangement of the splitting nodes of the integration interval makes it possible to increase the conditionality of the system of linear equations corresponding to the difference analogue of the problem and, thereby, to prevent the divergence of the iterative process. The article provides an example of calculating the optimal arrangement of the nodes splitting the integration interval when solving a test problem.

Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity

We consider a finite-element-in-space, and quadrature-in-time-discretization of a compressible linear quasistatic viscoelasticity problem. The spatial discretization uses a discontinous Galerkin finite element method based on polynomials of degree r--termed DG(r)--and the time discretization uses a trapezoidal-rectangle rule approximation to the Volterra (history) integral. Both semi- and fully-discrete a priori error estimates are derived without recourse to Gronwall's inequality, and therefore the error bounds do not show exponential growth in time. Moreover, the convergence rates are optimal in both h and r providing that the finite element space contains a globally continuous interpolant to the exact solution (e.g. when using the standard ? k polynomial basis on simplicies, or tensor product polynomials, ? k , on quadrilaterals). When this is not the case (e.g. using ? k on quadri-laterals) the convergence rate is suboptimal in r but remains optimal in h. We also consider a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the DG(r) approximation.

Remarks on the quasistatic problem of viscoelasticity: Existence, uniqueness and homogenization

This article is devoted to the problem (1.1) of quasistatic viscoelasticity. It turns out that (1.1) can be rewritten as an abstract initial value problem of the form (1.2). In Sect. 2 we consider the general abstract initial value problem (1.3). We prove existence and uniqueness of solutions, and stability with respect to the data. In Sects. 3 and 4 we apply our abstract results to the viscoelastic problem (1.1). In Sect. 3 we prove existence and uniqueness of solutions to the n-dimensional problem. In Sect. 4 we develop a stochastic homogenization theory for the 1-dimensional problem. Finally, we close our discussion with some remarks on the homogenization of the n-dimensional problem.

Optimal long-time $L_p}(0,T)$ stability and semidiscrete error estimates for the Volterra formulation of the linear quasistatic viscoelasticity problem

The purpose of this article is to show how the solution of the linear quasistatic (compressible) viscoelasticity problem, written in Volterra form with fading memory, may be sharply bounded in terms of the data if certain physically reasonable assumptions are satisfied. The bounds are derived by making precise assumptions on the memory term which then make it possible to avoid the Gronwall inequality, and use instead a comparison theorem which is more sensitive to the physics of the problem. Once the data-stability estimates are established we apply the technique also to deriving a priori error bounds for semidiscrete finite element approximations. Our bounds are derived for viscoelastic solids and fluids under the small strain assumption in terms of the eigenvalues of a certain matrix derived from the stress relaxation tensor. For isotropic materials we can be explicit about the form of these bounds, while for the general case we give a formula for their computation.

New variational principles in quasi-static viscoelasticity

A “saddle point” (or maximum-minimum) principle is set up for the quasi-static boundary-value problem in linear viscoelasticity. The appropriate class of convolution-type functionals for it is taken in terms of bilinear forms with a weight function involving the Fourier transform. The “minimax” property is shown to hold as a direct consequence of thermodynamic restrictions on the relaxation function. This approach can be extended to further linear evolution problems where initial data are not prescribed.

An existence and uniqueness theorem in quasi-static viscoelasticity

An existence and uniqueness theorem in quasi-static viscoelasticity

Contact Between Viscoelastic Bodies

Similarly to elasticity, it is useful to classify contacts between viscoelastic bodies by comparing the regions of contact in the loaded and load-free states. General properties are obtained for the class of contacts in which the contact region does not exceed the natural (load-free) contact. The displacements, strains, and stresses in such problems are proportional to the level of the loading history, but the scaling of load histories leaves the contact region unaltered. The results are valid within linear viscoelasticity, either quasi-static or dynamic. Additional properties are derived for plane contact problems in quasi-static viscoelasticity when the contact is monotonically receding.

Uniqueness of a solution in nonlinear quasistatic viscoelasticity

Uniqueness of a solution in nonlinear quasistatic viscoelasticity

Existence of solutions to the displacement problem for quasistatic viscoelasticity

Existence of solutions to the displacement problem for quasistatic viscoelasticity

Uniqueness theorems in the linear dynamic theory of anisotropic viscoelastic solids

Abstract : Some uniquenes theorems appropriate to dynamic linear viscoelasticity theory (small deformations) are proved. The first theorem asserts that the mixed problem of dynamic viscoelasticity theory has at most one solution provided the relaxation tensor is initially symmetric and initially positive definite. The proof of this theorem uses techniques which were employed by Volterra to establish a similar theorem in quasistatic viscoelasticity. The second result, the proof of which utilizes the Laplace transform techniques of Breuer and Onat, removes the initial symmetry requirement on the relaxation tensor G. This is done, however, only at the expense of requiring that G be independent of x, and the both G and the solution in question possess a Laplace transform with respect to the time. The third theorem which is the most difficult to prove, says if G is independent of x, initially symmetric, and initially strongly elliptic, then the displacement problem of dynamic viscoelasticity has at most one solution. Notice that if G has the usual symmetries and is initially positive definite, the G is also initially strongly elliptic. In this sense, the third theorem is more general than the first. (Author)