Boundary value problems viscoelastic

In this chapter we analyse a wide class of equilibrium problems with cracks. It is well known that the classical approach to the crack problem is characterized by the equality type boundary conditions considered at the crack faces, in particular, the crack faces are considered to be stress-free (Cherepanov, 1979, 1983 Kachanov, 1974 Morozov, 1984). This means that displacements found as solutions of these boundary value problems do not satisfy nonpenetration conditions. There are practical examples showing that interpenetration of crack faces may occur in these cases. An essential feature of our consideration is that restrictions of Signorini type are considered at the crack faces which do not allow the opposite crack faces to penetrate each other. The restrictions can be written as inequalities for the displacement vector. As a result a complete set of boundary conditions at crack faces is written as a system of equations and inequalities. The presence of inequality type boundary conditions implies the boundary problems to be nonlinear, which requires the investigation of corresponding boundary value problems. In the chapter, plates and shells with cracks are considered. Properties of solutions are established existence of solutions, regularity up to the crack faces, convergence of solutions as parameters of a system are varying and so on. We analyse different constitutive laws elastic, viscoelastic. 

Golden, J.M. and Graham, G.A.C., Boundary Value Problems in Linear Viscoelasticity. Springer-Verlag, Heidelberg, 1988. 

The periodic response of a linear viscoelastic cooling tower to a prescribed recurring sequence of pressure fluctuations and earth accelerations are analyzed. An approximate analysis, based on the bending theory of shells, is presented. The problem is reduced to a double sequence of boundary-value problems of linear ordinary differential equations. 19 refs, cited. 

Bobrov also used this model of a syntactic foam to calculate hydrostatic strengths164). At the same time, he showed that this parameter cannot be obtained theoretically for a syntactic foam using traditional micromechanical, macromechanical, or statistical approaches, as they are unsuitable for these foams. The first approach requires a three-dimensional solution of the viscoelasticity boundary value problem of a multiphase medium, and this is very laborious. The second and third methods assume the material is homogeneous overall, and so produce poor estimates for syntactic materials. 

M. Renardy, A well-posed boundary value problem for supercritical flow of viscoelastic fluids of Maxwell-type, in Nonlinear Evolution Equations That Change Type, B.L. Keyfitz and M. Shearer (eds.), IMA Volumes in Mathematics and its Applications 27, Springer-Verlag, Berlin, 1991, 181-191. 

Various types of coupled non-linear Fickian diffusion processes were numerically simulated using the free-volume approach given by equation [12.8], as well as non-Fickian transport. The non-Fickian transport was modeled as a stress-induced mass flux that typically occurs in the presence of non-uniform stress fields normally present in complex structures. The coupled diffusion and viscoelasticity boundary value problems were solved numerically using the finite element code NOVA-3D. Details of the non-hnear and non-Fickian diffusion model have been described elsewhere [14]. A benchmark verification of the linear Fickian diffusion model defined by equations [12.3]-[12.5] under a complex hygrothermal loading is presented in Section 12.6. 

Morland and Lee (1960) showed how to incorporate tune-temperature shifting into linear viscoelastic boundary-value problems. For the purpose a pseudo-time i(t) is introduced such that the amount of time that passes during an interval d< is given by dt/flj-. Then we have... 

The Phenomenology of the Linear Theory of Viscoelasticity. One of the powers of the linear viscoelasticity theory is that it is predictive. The constitutive law that comes from Boltzmann superposition theory requires simply that the material functions discussed above be known for a given material. Then, for an arbitrary stress or deformation history, the material response can be obtained. In addition, the elastic-viscoelastic correspondence principle can be used so that boundary value problems such as beam bending, for which an elastic solution exists, can be solved for linear viscoelastic materials as well. Both of these subjects are treated in this section. 

Hence, the two sources of nonlinearity in Eq. (54) are moisture concentration c and dilatational strain Consequently, Eqs. (48) and (54) are coupled. The diffusion boundary-value problem must be solved in conjunction with the nonlinear viscoelasticity boundary-value problem by using an iterative procedure. Finite-element formulation of Eq. (54) is standard (see Reddy(47)) and is given by... 

It is possible using transform methods to convert viscoelastic problems into elastic problems in the transformed domain, allowing the wealth of elasticity solutions to be utilized to solve viscoelastic boundary value problems. Although there are restrictions on the applicability of this technique for certain types of boundary conditions (discussed further in Chapter 9), the method is quite powerful and can be introduced here by building on the framework provided by mechanical models. Recall the differential equation for a generalized Maxwell or Kelvin model,... 

Principle. The quantity, E (s), in transform space is analogous to the usual Young s modulus for a Unear elastic materials. Here, the Unear differential relation between stress and strain for a viscoelastic polymer has been transformed into a linear elastic relation between stress and strain in the transform space. It will be shown in the next chapter that the same result can be obtained from integral expressions of viscoelasticity without recourse to mechanical models, so that the result is general and not limited to use of a particular mechanical model. Therefore, the simple transform operation allows the solution of many viscoelastic boundary value problems using results from elementary solid mechanics and from more advanced elasticity approaches to solids such as two and three dimensional problems as well as plates, shells, etc. See Chapters 8 and 9 for more details on solving problems in the transform domain. 

A general issue in working with viscoelastic materials is representing the measured material properties by an appropriate mathematical function. As indicated earlier, a closed mathematical form facilitates solution of boundary value problems, as well as ease of manipulation of data. While viscoelastic properties can be represented by a number of functional forms, the exponential Prony series... 

The fact that Eq. 8.3 can be considered as the equivalent of Hooke s law in the transform domain leads to a general method to solve many practical viscoelastic boundary value problems in a simple manner. This procedure is often attributed to Turner Alfrey and is sometimes referred to as Al-frey s correspondence principle. Simply stated the procedure is as follows ... 

Find a previously solved linear elasticity boundary value problem with the same geometry, loading, and boundary conditions as the linear viscoelastic boundary problem for which a solution is needed. 

Three methods of solving viscoelastic boundary value problems were given early in the Fundamental Concepts section of this chapter. The development of the beam equation serves as a simple method to illustrate these various techniques. Before proceeding with this section, the reader is advised to review the procedure for developing the deflection equation for linear elastic prismatic beams given in elementary texts on solid mechanics. 

Again, the viscoelastic solution for stress is exactly the same as the elastic solution stress. As stated earlier, in general, if the linear elastic solution for stresses for a given boundary value problem does not contain elastic constants, the solution for stresses in a viscoelastic body with equivalent geometry and equivalent loads is identical to that for the elastic body. This means that the stress analysis of most problems considered in elementary solid mechanics such as beams in bending, bars in torsion or axial load, pressure vessels, etc. will have the same solution for stress in a linear viscoelastic material as in a linear elastic material. Further, stress analysis of combined axial, bending, torsion and pressure loads can be handled easily using superposition. 

Describe three analytical approaches for obtaining solutions viscoelastic boundary value problems. 

The various approaches to the solution of viscoelastic boundary value problems discussed in the last chapter for bars and beams carry over to the solution of problems in two and three dimensions. In particular, if the solution to a similar problem for an elastic material already exists, the correspondence principle may be invoked and with the use of Laplace or Fourier transforms a solution can be found. Such solutions can be used with confidence but one must be cognizant of the general equations of elasticity and the methods of solutions for elasticity problems in two and three dimensions as well as any assumptions that might often be applied. To provide all of the necessary information and background for multidimensional elasticity theory is beyond the scope of this text but the procedures needed will be outlined in the following sections. 

This chapter will focus on developing the equations, assumptions and procedures one must use to solve two and three dimensional viscoelastic boundary value problems. The problem of an elastic thick walled cylinder will be used as a vehicle to demonstrate how to obtain the solution of a more difficult reinforced viscoelastic thick walled cylinder. In the process, we first demonstrate how the elasticity solution is developed and then apply the correspondence principle to transform the solution to the viscoelastic domain. Several extensions to this problem will be discussed and additional practice is provided in the homework problems at the end of the chapter. 

Note that with Eqs. 9.10, 9.12 and 9.14, we again have the viscoelastic constitutive law represented in the transform domain in a form equivalent to elasticity. These relationships will then allow us to utilize the correspondence principle as in Chapter 8 to solve 2D and 3D viscoelastic boundary value problems based on elasticity solutions. 

Obviously, the above transformed governing equations for a linear viscoelastic material (Eqs. 9.33- 9.36) are of the same form as the governing equations for a linear elastic material (Eqs. 9.25 - 9.28) except they are in the transform domain. This observation leads to the correspondence principle for three dimensional stress analysis For a given a viscoelastic boundary value problem, replace all time dependent variables in all the governing equations by their Laplace transform and replace all material properties by s times their Laplace transform (recall, e.g., G (s) = sG(s)),... 

Methods for Solving Viscoelastic Problems As mentioned in Chapter 8 on bars and beams, three related methods can be used to solve linear viscoelastic boundary value problems. These are ... 

As a result, at the present time, use of one or more of the assumptions provided earlier in the chapter, together with broadband data for shear or extensional modulus, represent the most fruitful approach to the solution of viscoelastic boundary value problems. 

Name two frequently encountered viscoelastic boundary value problems in solid mechanics that cannot be solved with the standard correspondence prionciple.X... 

Graham, G.A.C., The Correspondence Principle of Linear Viscoelasticity Theory for Mixed Boundary Value Problems Involving Time-... 

This chapter deals with fundamental definitions, constitutive equations of a viscoelastic medium subject to infinitesimal strain, and the nature and properties of the associated viscoelastic functions. General dynamical equations are written down. Also, the boundary value problems that will be discussed in later chapters are stated in general terms. Familiar concepts from the Theory of Linear Elasticity are introduced in a summary manner. For a fuller discussion of these, we refer to standard references (Love (1934), Sokolnikoff (1956), Green and Zerna (1968), Gurtin (1972)). Coleman and Noll (1961) have shown that the theory described here may be considered to be a limit, for infinitesimal deformations, of the general (non-linear) theory of materials with memory. 

We have also allowed the boundary regions to depend on the component being specified. This is required to cover examples such as frictionless contact. These are not the most general boundary conditions that can be conceived. For example the case of frictional contact is not even covered by this scheme. However, we shall see in Chap. 3 that, at least in the plane case, such problems can be handled by methods similar to those used in the frictionless case. The fact is that the methods outlined in later chapters for attacking viscoelastic boundary value problems are all indirect, in the sense that they focus on the boundary quantities, with the aim of determining these quantities everywhere on B. Once this is done, the task of determining any quantity in the interior of the medium is in principle easy. Indeed, it is either a first or second boundary value problem as defined above. 

In this chapter, we discuss methods of solution of viscoelastic boundary value problems in general terms, together with certain relevant theorems. The main emphasis is on non-inertial problems. Also, most of the discussion is confined to the isothermal case. 

General methods of solving viscoelastic boundary value problems are described in this chapter. 

Most methods of solution of viscoelastic boundary value problems developed to date have relied heavily on analogy with Elasticity. The most striking formal difference between (3.1.3) and the corresponding elastic equations is the integral with kernel on the right of (3.1.3c). Unless this can be eliminated, there... 

I. Method of Solution. The method of solution is based on the viscoelastic Kolosov-Muskhelishvili equations, adapted to a half-space. Explicit solutions to the first and second boundary value problems are presented in detail. In these cases no restrictions on material behaviour are necessary. In the case of mixed boundary value problems where surface friction is present, it is necessary to make the proportionality assumption. Limiting frictional contact problems are... 

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