Problems Of Linear Viscoelasticity Research Articles

Numerical methods for treating problems of viscoelastic isotropic solid deformation

Mathematical models for treating problems of linear viscoelasticity involving hereditary constitutive relations for compressible solids are discussed, and their discretization using finite element methods in space together with quadrature rules in time to treat the hereditary integrals is described. The range of applicability of this type of formulation is reviewed in the context of geometric and constitutive linearity/nonlinearity, and the limitations imposed by the availability of physical data are discussed. One of the above models is a Volterra integral equation of the second kind. In this, when the kernel is separable, an established technique due to Goursat (1933) can be exploited to reformulate the problem as a system of ordinary differential equations. This approach will be described. For the special case of a linear viscoelastic, isotropic, homogeneous, synchronous (constant Poisson's ratio) solid this method results in a complete decoupling of the space and time dependence. In this case the problem can be solved at each time level by solving first a problem of linear elasticity and then a system of ordinary differential equations for each point in the spatial mesh at which the viscoelastic displacements are required. The advantages, disadvantages and limitations offered by this, and various other schemes for solving problems of viscoelasticity as outlined below, are discussed.

Inversion and quasi-static problems in linear viscoelasticity

Within the theory of linear viscoelasticity, we seek solutions to the inversion problem of the constitutive equation respectively inL 2 and in the spaceS′ of the tempered distributions. Successively we study the quasi-static problem inS′. Both problems admit one and only one solution if the relaxation function satisfies Graffi's inequality. Finally we show that the inversion problem and the quasi-static one are deeply connected and that every counterexample about the existence or uniqueness of the solutions for the first problem also provides a counterexample for the latter.

Boundary element analysis of linear viscoelastic problems using realistic relaxation functions

This paper presents an efficient method for the stress analysis of realistic viscoelastic solids by the time-domain boundary element method. The fundamental solutions and stress kernels are obtained using the elastic-viscoelastic correspondence principle. Since it is inconvenient to obtain the Laplace transform of the relaxation functions of realistic viscoelastic solids, the method of collocation has been employed and the relaxation function has been expanded in a sum of exponentials. Numerical results of example problems show the effectiveness and applicability of the proposed method.

An energetic formulation for the linear viscoelastic problem. Part I: Theoretical results and first calculations

AbstractAn extremal principle is formulated for the linear viscoelastic problem with general viscous kernel. This is an extension of the classical total potential energy principle of the linear elasticity. Then a discretized formulation in space and time is shown for frame structures, using finite element technique. Several numerical examples, for two different kinds of viscoelastic materials, testify the accuracy and reliability of the proposed method. The matrix conditioning indexes obtained are compared with those achieved by applying the least square method.

To the Construction of Uniformly Valid Forward-Area Asymptotics in Terms of the Ray Method in Dynamic Problems in Linear Viscoelasticity

To the Construction of Uniformly Valid Forward-Area Asymptotics in Terms of the Ray Method in Dynamic Problems in Linear Viscoelasticity

Stress-strain state of multilayered orthotropic shells of revolution made of thermorheologically simple materials

A refined variant of finite-shear theory is used to develop a numerical method for calculating components of the stress-strain state of multilayered orthotropic shells of revolution made of thermorheologically simple materials. The proposed method is based on quadratic approximation of the strains and numerical interaction over time. A boundary-value problem of linear viscoelasticity is solved by the method of discrete orthogonalization. Results are presented from calculations performed for a laminated plate, laminated cylinder, and laminated cyl panel with the use of the proposed method and the method devised by R. A. Schapery.

Method of solving problems of the linear theory of viscoelasticity for anisotropic materials (in the presence of cracks)

1. The solution of quasistatic problems of linear viscoelasticity for anisotropic materials on the basis of the Volterra principle reduces to the problem of deciphering functions of integral Volterra operators. According to this principle the solution of the viscoelastic problem is obtained by solving the corresponding elastic problem and replacing in the latter the elastic characteristics of the material by the corresponding viscoelastic characteristics. The justification of the validity of the Votterra principle in solving viscoelastic problems is discussed in [2, 3], while that for solving quasistatic problems with moving cracks is given in [5]. If the solution of the elastic problem fix) is [7] [(x) =F(Ri)g(x ),

Iteration method for solving linear viscoelasticity problems

Substantiation is given for a new iteration method that makes it possible to solve, with prescribed accuracy, boundary-value problems of quasistatics of a linearly viscoelastic body. A theorem is proved about the convergence of the iteration processes introduced. An approximate correspondence principle, making it possible to construct a solution for viscoelastic problems from known elastic problems, is obtained as a consequence of the theorem. Examples are given of an approximate determination of the connected-creep function, in terms of which numerous analytical solutions to viscoelasticity problems can be expressed.

Fundamental solutions for linear viscoelastic continua

The full set of fundamental solutions for the linear viscoelastic problem is derived from the relevant elastic fundamental solutions using the well-known so-called “correspondence principle”. The fundamental solutions are given for the 3D-continuum, the 2D-plane strain and the 2D-plane stress problem using a wide range of linear viscoelastic constitutive law models.Finally, the possible fields of application of the determined fundamental solutions are given using symmetric BIE formulations in space and time.

Constitutive laws in time and frequency domains for linear viscoelastic materials.

There have been numerous phenomenological approaches to the characterization of linear isothermal viscoelastic material behavior. The particular form of the relaxation function has an impact not only upon how well it can be fit to data from a simple relaxation test but also upon how amenable it is to the integral transform techniques commonly used to solve problems of linear viscoelasticity. A new viscoelastic constitutive law is proposed and compared to the fractional calculus model in the context of a particular problem. The solution procedures for each law demonstrate that the new constitutive law requires less computation. Comparison of the actual results themselves, however, is not meaningful since conclusions drawn would depend upon the particular material being modeled and how well each law can be fit to the particular material. The problem is solved once more using a relaxation modulus defined in terms of a Prony series. Significant computational advantages over the previous two are demonstrated for this constitutive law. The results indicate that the particular constitutive law used for a linear viscoelasticity problem can be chosen based upon a trade-off between material property presentation and computational ease. [Work supported by ONR.]

Ray expansions for the solution of dynamic problems of linear viscoelasticity

Ray expansions for the solution of dynamic problems of linear viscoelasticity

Lagrange identity in linear viscoelasticity

An approach based on the Lagrange identity is developed for the study of the initial boundary value problems of linear viscoelasticity. Estimations are obtained for describing the Liapunov stability and continuous data dependence of solutions. The main differences between the respective estimates consist in the measure of continuity and the constraint sets on which they are valid. The Lagrange identities are also used in order to obtain some reciprocal relations. The Cesàro means and autocorrelations of the velocity field and the strain field are introduced and it is proved that they exist and have a common value.

Boundary Value Problems in Linear Viscoelasticity (J. M. Golden and G. A. C. Graham)

Boundary Value Problems in Linear Viscoelasticity (J. M. Golden and G. A. C. Graham)

Linear viscoelastic analysis in time domain by boundary element method

Isotropic, linear viscoelastic problems are formulated in the time domain by a boundary element method. The viscoelastic fundamental solutions are represented in terms of the constant coefficients of relaxation functions. Starting from the reciprocal work theorem, an alternative form of boundary integral equation is derived by integration by parts. This form requires the regularity of field variables to be one order less than that in the usual formulation. A time marching process is incorporated in the numerical method. The matrices in the resulting algebraic equations are constant over time, except for the matrix for past history. Four test problems in two dimensional space are solved for displacement and stress and checked against exact solutions. It is shown that the formulation is economical and generates stable solutions of good accuracy. The effect of time steps with regard to spatial discretizations is also discussed.

Causality and viscoelastic boundary value problems

The Principle of Causality, for a given system, is related to the analytic structure of the Fourier transform of functions characterizing the system. In the case of linear viscoelastic boundary value problems, these are the Green's functions of the problem. For a certain class of noninertial problems, the causal property of the Green's functions is derived from more elementary constraints. Also, for the tearing mode, semi-infinite, inertial, moving crack problem, it is shown that certain results, already given in the literature, can be derived in a natural and simple way, with the aid of this principle.

A new boundary element technique for solving linear viscoelastic problems and its application.

Viscoelastic analysis is one of the most important subjects in engineering. Several attempts have been so far made for the integral equation approach to viscoelastic problems. To authors' knowlege, however, there is no available literature for the application of a more direct boundary element method to the time dependent problems of this kind using a step-wise time integration scheme. In this paper, we propose a direct boundary integral formulation. This formulation needs only Kelvin's fundamental solution of isotropic elastostatics with material constants prescribed as explicit functions of time. Successive application of this solution scheme can determine the whole behavior of viscoelastic problems. Finally, some sample problems of viscoelastic materials are computed by means of the proposed method of solution to demonstrate its potential usefulness.

Viscoelastic crack analysis by the boundary integral equation method

The linear viscoelastic three-dimensional crack problem is analyzed by combining the correspondence principle and the boundary integral equation method. In a general crack analysis the usual boundary integral equations lead to a nonunique formulation of the problem, because they do not involve information about the loading on the crack surface. Here, the boundary integro-differential equations are applied to the numerical calculation of the crack opening displacement of a penny-shaped crack in an infinite linear viscoelastic body. Moreover, the influence of several parameters of the three-parameter viscoelastic model on the crack opening displacement and the incubation time is shown.

Boundary element method applied to linear viscoelastic analysis

This paper analyses linear viscoelastic problems using the boundary element method together with the numerical inversion of the Laplace transform. This method reduces the amount of manual and computing time as compared with the finite element method. Two numerical examples are included to demonstrate the validity and accuracy of the method.

Approximate Laplace Transform Inversion in Dynamic Viscoelasticity

Laplace transform techniques greatly simplify many problems in linear viscoelasticity. However, if realistic material property representations are used, inversion of the resulting transforms can be difficult. Although approximate transform inversion methods have been widely used in quasi-static viscoelastic problems, the application of these techniques to wave propagation problems has been less successful. Inaccuracy of the transform inversion has been noted previously in the literature. The present work shows that one of the numerical Laplace transform inversion techniques of Bellman can successfully be applied to dynamic viscoelasticity. Comparisons with literature solutions and exact functions indicate accuracies to within ±1 percent can be obtained.

Variant of finite-element method to solve problems of linear viscoelasticity

Variant of finite-element method to solve problems of linear viscoelasticity