Problems Of Linear Viscoelasticity Research Articles

Some problems of scalar and vector-valued optimization in linear viscoelasticity

Optimal and Pareto-optimal stress rate histories are given for some simple scalar and vector-valued optimization problems in one-dimensional linear viscoelasticity.

Variational principle for coupled dynamic problems in linear viscoelasticity

Variational principle for coupled dynamic problems in linear viscoelasticity

Growth estimates for solutions to initial-boundary value problems in viscoelasticity

For the abstract Volterra integro-differential equation u tt − Nu + ∝ −∞ t K( t − τ) u( τ) dτ = 0 in Hilbert space, with prescribed past history u(τ) = U(τ), − ∞ < τ < 0, and associated initial data u(0) = f, u t (0) = g, we establish conditions on K( t), − ∞ < t < + ∞ which yield various growth estimates for solutions u( t), belonging to a certain uniformly bounded class, as well as lower bounds for the rate of decay of solutions. Our results are interpreted in terms of solutions to a class of initial-boundary value problems in isothermal linear viscoelasticity.

An approximate (Wiener-Hopf) kernel for dynamic crack problems in linear elasticity and viscoelasticity

The analysis of dynamic crack problems often leads to results of such complexity that their physical significance cannot readily be apprehended; one must generally have recourse to ad hoc approximations, or to a numerical evaluation of the solutions, before such results can be interpreted. We present in this paper a more radical approximation in the form of an 'approximate kernel’ of the type first considered by Koiter in conjunction with the Wiener-Hopf technique. The approximate solutions to which this leads are compared with the exact solutions for a few basic problems concerning the elastic response to stationary, or moving, point forces applied to the faces of a semi-infinite crack. It is clear that all the essential features of the exact solutions are retained, and of course the approximate solutions are considerably simpler in form. The use of approximate kernels is not tied to the use of the Wiener-Hopf technique: they can also be used in con­junction with a method due to W. E. Williams, which is in fact much better suited to the analysis of crack problems. It is shown that the approximate kernel can also be used to simplify Kostrov’s solution of the arbitrarily moving crack problem and the analysis of crack problems in linear viscoelasticity.

A Priori Bounds for the Secondary Boundary Value Problem in Linear Viscoelasticity

A priori estimates are obtained for the traction problem of isotropic linear viscoelasticity. In these inequalities, energy-like integral measures of the viscoelastic strain are bounded by integrals of the surface tractions and body force. In the case of zero body force, these estimates lead immediately to pointwise bounds for the strain. They also have bearing on the problem of extending Saint-Venant’s principle to viscoelasticity.

Efficient Method for Solution of Viscoelastic Problems

A general method capable of solving a large range of linear viscoelastic problems is presented using a Laplace transform approach. The time-dependent problem is thus reduced to an associated elastic problem in the transform plane. The analysis is accomplished by solving the associated problem in the transform plane by the finite element method and the inversion of the transformed displacements and stresses is achieved by least square collocation. The approach proved to be extremely economical in computation time compared with the step-by-step method and it can be used for a large range of time-dependent problems. The accuracy is checked against some known solutions and a further example from the field of soil mechanics is presented.

Analysis of centrifugal stresses in anisotropic viscoelastic cylinder.

HE stress analysis for anisotropic viscoelastic materials has become the subject of increasing concern in the past few years. The ever increasing use of composites and oriented plastics as structural members has made it necessary to study the effect of anisotropy as well as viscoelasticity on the performance of these components. Problems in linear isotropic viscoelasticity have been analyzed by many authors, such as Lee, 1 Hilton,2 and Bland,3 under various loading conditions. However, similar attempts in the analysis of anisotropic viscoelasticity have been frustrating because of the difficulties faced in the use of Laplace transform, particularly in the inversion of solutions. As a result, even the simple problems which can be solved explicitly, become important in understanding the interaction of anisotropy and the viscoelastic behavior of materials.4 ~ 6 In this paper, some elementary solutions of stress analysis problems for an anisotropic viscoelastic cylinder are given. Specifically, analysis for a long hollow cylinder, rotating about its axis, is considered. In the course of the analysis, the constitutive equations of the theory of infinitesimal viscoelasticity exhibiting a general anisotropic behavior are assumed. These are expressed in terms of different time-dependent modulus functions. A particular type of spacial nonhomogeneity, where the material moduli are assumed to be representable as a production of functions of time and coordinates, is considered. Through the application of integral (Laplace) transform the problem is then solved. Although the analysis is made for a general linear viscoelastic medium, for simplicity, numerical results are presented for a material having simple viscoelastic behavior. Anisotropic Viscoelasticity The basic problem in quasi-static linear viscoelastic stress analysis is to determine a stress field atj(xk, t) and the displacement field wf(xfc, t) as functions of space variables xk and time t. The governing equilibrium equations, which are independent of material constitution, are the displacement-strain relations and the principles of linear momentum and moment of momentum. They are in the usual notation

On the uniqueness of solution and certain approximate methods of solving linear viscoelasticity problems: PMM vol. 37, n≗3, 1973, pp. 505–514

The uniqueness of solution of mixed initial-boundary value problems for a version of linear viscoelasticity is proved, as is also the convergence of some approximate methods of solving these problems. Upon compliance with the conditions μ > 0, m A ⩽ m B and if m A = m B , then λ +2 μ > 0the following cases of the relationship between powers of the polynomials B and C are considered: m B ⩾ m C + 2 ( Sect. 3), m B = m C and m B = m C + 1( Sect. 4). These cases include all the types of relations between the strains and stresses of the type (1.1) used in the theory of viscoelasticity. As a mathematical foundation for such constraints, it can be noted that for m C > m B the Cauchy problem for a body occupying all of space is always incorrect according to G.E. Shilov [1].

Solution of linear-viscoelasticity boundary value problems

The basic elements of the algebra of exponential-fractional operators [1,9] are generalized to include any arbitrary resolvent operator. As a consequence, it is found that solutions of viscoelasticity boundary-value problems within the framework of the Volterra principle can always be realized in time without specifying the starting operators. Analytic functions of resolvent operators are discussed. The results are demonstrated with reference to some common integral operators of viscoelasticity.

Remark on the approximate solution of the problems of linear viscoelasticity: PMM vol. 36, n≗4, 1972, pp. 749–753

Remark on the approximate solution of the problems of linear viscoelasticity: PMM vol. 36, n≗4, 1972, pp. 749–753

The integral representation of fractionally exponential functions and their application to dynamic problems of linear visco-elasticity

Fractionally exponential functions are written in the integral form and distribution functions with an Abelian singularity are obtained for the corresponding relaxation and retardation spectra. A principle is stated, defining the dynamic problems for which weakly singular functions can be used as the kernels of the integral operators. A one-dimensional sound wave traveling in a semiinfinite visco-elastic medium is considered. The generalized exponential functions of fractional order, proposed by Yu. N. Rabotnov [1, 2] as the kernels of Boltzmann-Volterra integral relations, have found wide applications in theory of linear visco-elasticity. This is explained partly by the great mathematical flexibility of the F-operators when applying the Volterra principle to the solution of elastically hereditary problems and partly by the fact that almost all weakly singular kernels possessing an Abelian singularity are connected in some way or other with the F-functions. For example, the resolvent of the elementary weakly singular Abelian kernel is an F-function. The product of an exponential function with an Abelian kernel represents a particular case of the product of two F-functions with different fractional parameters, while the resolvent of such a kernel is the product of an exponential function with an F-function [3, 4]. Since the e-functions are defined by slowly convergent series, their various asymptotic forms [2, 5–8] are commonly used in practical calculations. The theory of F-functions can be developed further in the context of their integral representations, which enables a more exact physical interpretation to be given to their parameters and on occasion simplifies computational operations.

On the approximate solution of problems of linear viscoelasticity: PMM vol. 35, n≗1, 1971, pp. 185–189

On the approximate solution of problems of linear viscoelasticity: PMM vol. 35, n≗1, 1971, pp. 185–189

Integral representations of ε γ-functions and their application to problems in linear viscoelasticity

Techniques for the solution of Boltzmann-Volterra integrals, the kernels of which are expressed by expansions involving fractional indices of the time, i.e. the ε γ-functions proposed by Rabotnov [1,2], are developed. Integral representations of the kernel are obtained by means of Fourier-Mellin and Laplace inversions; they can frequently facilitate computation and clarify the physical significance of the coefficients in the ε γ -function. These advantages are illustrated by examples relating to the propagation of an acoustic wave in a material displaying hereditary viscoelasticity defined by an ε- γ-kernel, and by the damped vibrations of an impulse-excited two-dimensional vibrator.

On the mathematical content of Volterra's principle in the boundary value problem of viscoelasticity: PMM vol. 35, no. 5, 1971, pp. 869–878

A direct proof of Volterra's principle is given by reducing the boundary value problem of homogeneous viscoelasticity to solving the corresponding elastic problem and some operator equations. The conditions of applicability of the symbolic method are formulated as equivalence conditions between the realization of the viscoelastic operator function which arises from the elastic problem and the solutions of the operator equations. We note that the second procedure is more general and can be used in the problems of viscoelasticity whose solutions cannot be constructed by Volterra's principle. Volterra's principle [1 – 5] is widely used for the construction of the solution of boundary value problems in linear homogeneous viscoelasticity. The basis for its applicability is the independence of the operations with respect to the coordinates and time in the complete fundamental system of quasi-static equations of a viscoelastic body. As a result, the problem is divided into solving the corresponding boundary value problem of the elastic body and the determination of the operator functions. The latter are obtained from the elastic solution through the formal replacement of the mechanical moduli by the viscoelastic operators. However, the separation of the space and time operations in the equations of viscoelasticity is, by itself, not a sufficient criterion for the applicability of the operator-symbolic method, if only because the boundary conditions are not taken into account. In connection with this, an investigation of the problem of the rational application of Volterra's principle is required. An attempt for the mathematical foundation of Volterra's principle is contained in [6]. In the case of time-independent viscoelastic properties the identity between the first and second form of the correspondence principle has been established by the methods of operational calculus [7]. The symbolic method is justified by the con struction of an isomorphism between the sets of functions of the viscoelasticity operators and the functions of a complex variable. The conditions of applicability of Volterra's principle are determined by the possibility of performing a Laplace transform in the equations and the boundary conditions of the viscoelasticity problem.

Lateral vibration of a non-linear viscoelastic beam under intial axial tension

The free lateral vibration of a simply-supported metallic beam under axial creep deformation is considered. The constitutive law is formulated with stress power functions in the primary and secondary creep terms; the instantaneous linear elastic deformation is also included. For better physical visualization, a non-linear Maxwell-Kelvin model is used to represent the constitutive law. A reduction to various special models is also obtained. It is assumed that the initial stress is much greater than the increments of stress caused by the oscillation, and a perturbation technique is employed. In the present study, the perturbation essentially results in replacing a non-linear viscoelastic problem by an equivalent linear viscoelastic problem. Analytical solutions for the Maxwell-Kelvin model and for the special models are obtained. Numerical results for a stainless steel and an aluminum alloy are also presented and discussed.

On the Identification Problem in Linear Viscoelasticity

AbstractThis paper discusses a class of identification problems in linear viscoelasticity, in particular a procedure for determining a suitable constitutive functional structure from experimental information is presented.The problem is viewed as a linearization procedure of a general nonlinear stress‐strain functional relationship in the neighborhood of some stress‐strain histories, in the presence of noise. The highly nonlinear optimization problem posed by this approach is solved using ideas of differential approximation and quasilinearinzation. The characteristic viscoelastic are ultimately given as the solutions of suitable differential equations subject to initial conditions, a convenient functional parametrization if a digital computer is to be used in subsequent stress‐analysis. Several examples are presented to illustrate the feasibility and accuracy of the method.

The Transient Response of a Viscoelastic Torsional Pendulum

The pendulum consists of a circular viscoelastic rod fixed at one end and attached to a disk at the other end. The disk, initially at rest, is subjected to a step function of torque. The subsequent motion is analyzed in terms of normal modes and normal coordinates. The normal modes are found by separation of variables. The characteristic numbers associated with these modes depend only on the moments of inertia of the disk and rod. The normal coordinates satisfy integro-differential equations of the Volterra type. The mechanical properties of polyisobutylene at 25°C were represented in differential operator form. The Volterra integral equations were then solved by means of the Laplace transforms. An analysis of the numerical results shows that only the oscillatory part of the fundamental mode should persist after a short time. This result is in qualitative agreement with observations. This method of analysis may be applied to other dynamic problems in linear viscoelasticity provided the characteristic equation can be reduced to a form that does not involve the mechanical properties of the material.