Viscoelastic problems

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

In order to solve viscoelastic problems, we must select the most convenient model for the stress and then proceed to develop the finite element formulation. Doue to the excess in non-linearity and coupling of the viscoelastic momentum equations, three distinct Galerkin formulations are used for the governing equations, i.e., we use different shape functions for the viscoelastic stress, the velocity and the pressure... 

One must note that the balance equations are not dependent on either the type of material or the type of action the material undergoes. In fact, the balance equations are consequences of the laws of conservation of both linear and angular momenta and, eventually, of the first law of thermodynamics. In contrast, the constitutive equations are intrinsic to the material. As will be shown later, the incorporation of memory effects into constitutive equations either through the superposition principle of Boltzmann, in differential form, or by means of viscoelastic models based on the Kelvin-Voigt or Maxwell models, causes solution of viscoelastic problems to be more complex than the solution of problems in the purely elastic case. Nevertheless, in many situations it is possible to convert the viscoelastic problem into an elastic one through the employment of Laplace transforms. This type of strategy is accomplished by means of the correspondence principle. 

Owing to the multiaxial character of the problems addressed in this chapter, the field equations depend not only on time but also on the position defined by their coordinates. Finally, it is necessary to stress that the solution of viscoelastic problems requires, as in the elastic case, specification of adequate boundary conditions. In this chapter, in addition to considering both integral and differential multiaxial stress-strain relationships, some viscoelastic problems of interest in technical applications are solved. 

Among the equations that govern a viscoelastic problem, only the constitutive equations differ formally from those corresponding to elastic relationships. In the context of an infinitesimal theory, we are interested in the formulation of adequate stress-strain relationships from some conveniently formalized experimental facts. These relationships are assumed to be linear, and field equations must be equally linear. The most convenient way to formulate the viscoelastic constitutive equations is to follow the lines of Coleman and Noll (1), who introduced the term memory by stating that the current value of the stress tensor depends upon the past history... 

The balance equations can also be transformed in terms of the operational variable, thus completing the preliminary step in the solutions of a viscoelastic problem. 

The solution of a problem in linear viscoelasticity requires the determination of the stress, strain, and displacement histories as a function of the space coordinates. The uniqueness of the solution was proved originally by Volterra (11). The analysis carried out in this chapter refers exclusively to isotropic materials under isothermal conditions. As a rule, it is not possible to give a closed solution to a viscoelastic problem without previous knowledge of the material functions. The experimental determination of such functions and the relationships among them have been studied in a specific way in separate chapters, and therefore the reader s knowledge of them is assumed. At the same time, the methods of analysis carried out in this chapter and in Chapter 17 will allow us to optimize the calculation of the material functions. 

Boundary conditions allow us to obtain specific results for each three-dimensional viscoelastic problem. If the stresses on the surface of the body are stated (first boundary problem), then the system of 15 basic equations is reduced to one of only six independent differential equations containing the six independent stress components. The strategy to follow implies the formulation of the compatibility equations in terms of the stress (Beltrami-Michell compatibility equations). 

The technique of separation of variables, that is, the possibility of separating the spatial and temporal variables in the stress and strain fields, is particularly useful in the solution of dynamic viscoelastic problems. As a rule, this requires us to assume that the Poisson ratio is constant, a reasonable assumption in many cases. Alternatively, the divergence of the displacement vector must be constant. A particularly important case of application of the variables separation method, where the assumption concerning the constancy of the Poisson ratio is relaxed, occurs in those problems in which the boundary conditions or the forces of volume are... 

The geometric characteristics of some interesting viscoelastic problems have special symmetries. For example, spherical shells have wide applications in pressure vessels, heat exchangers, and nuclear reactors. Loading of these structures can occur not only in accidental conditions but also in normal situations, e.g., during overpressurization. These spherical shells are radially symmetrical. 

Now we are ready to solve the corresponding viscoelastic problem. As usual, a step input angular velocity co = is assumed. According to... 

The solution of the viscoelastic problem is found by assuming G = G (o)). By assuming furthermore that the sample is fixed at z = h and that the torque is applied at z = 0, the solution of the previous differential equation is given by... 

The methods utilized to measure the viscoelastic functions are often close to the stress patterns occurring in certain conditions of use of polymeric materials. Consequently, information of technological importance can be obtained from knowledge of these functions. Even the so-called ultimate properties imply molecular mechanisms that are closely related to those involved in viscoelastic behavior. Chapters 16 and 17 deal with the stress-strain multiaxial problems in viscoelasticity. Application of the boundary problems for engineering apphcations is made on the basis of the integral and differential constitutive stress-strain relationships. Several problems of the classical theory of elasticity are revisited as viscoelastic problems. Two special cases that are of special interest from the experimental point of view are studied viscoelastic beams in flexion and viscoelastic rods in torsion. 

I had come from Australia, where I had been studying the mechanical properties of wool (and where there are about 10 times as many sheep as there are humans) to Utah (where there are about 10 times as many humans as there are sheep) to learn something about the theory of rate processes and its application to viscoelastic problems. Eyring and co-workers had developed the hyberbolic sine law of flow and applied it to many systems—both solid and liquid—and I had expected to find this kind of work going on full pace. 

These relationships are known as Newton s Law of viscous flow a is termed the fluidity and -q the dynamical shear viscosity. Newton s Law is analogous to Hooke s Law, except shear strain has been replaced by shear strain rate and the shear modulus by shear viscosity. As shown later, this analogy is often very important in solving viscoelastic problems. In uniaxial tension, the viscous equivalent to Hooke s Law would be a=7] ds/dt), where q is the uniaxial viscosity. As v=0.5 for many fluids, this equation can be re-written as <7-=3Tj(de/dO using t7=t /[2(1+v)], the latter equation being the equivalent of the interrelationship between three engineering elastic constants, (fi=E/[2il + v)]). 

Viscoelastic problems can also be considered in terms of the compliance of the system and one can define a storage compliance, =efpos8 (T and a loss compliance, "=e(,sin6/a-j. The energy dissipated per cycle At/ can be determined from... 

Solution of the viscoelastic problem is appropriate when a deposited layer has significant coefficients of both elasticity and viscosity which might occur, for example, in the case of a polymeric coating on the crystal surface. 

An approximate method to analyze viscoelastic problems has been outlined by Schap-ery.(30) jn this method, the solution to a viscoelastic problem is approximated by a correspond ing elasticity solution wherein the elastic constants have been replaced by time-dependent creep or relaxation functions. The method may be applied to linear as well as nonlinear problems. Weitsman D used Schapery s quasi-elastic approximation to investigate the effects of nonlinear viscoelasticity on load transfer in a symmetric double lap joint. By introducing a stress-dependent shift factor, he observed that the enhanced creep causes shear stress relief near the edges of the adhesive joint. 

Numerical analysis of viscoelastic problems recherches en mathematiques appliques,... 

These expressions will be used extensively later in Chapter 9 when dealing with viscoelasticity problems in two and three dimensions. 

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,... 

What is now known as the correspondence principle for converting viscoelastic problems in the time domain into elastic problems in the transform domain was first discussed by Turner Alfrey in 1944. As a result, the principle is sometimes referred to as Alfrey s correspondence principle. Later in 1950 and in 1955 the principle was generalized and discussed by W.T. Read and E. H. Lee respectively. (See bibliography for references.)... 

Generally viscoelastic problems can be solved using relations between internal stresses and external loads subject to the geometry of the structure in a similar manner as for elastic materials in the subject areas mentioned above. For both elastic and viscoelastic materials, the state of the material or equations of state must be included. Here elastic and viscoelastic materials are different in that the former does not include memory (or time dependent) effects while the latter does include memory effects. Because of this difference, stress, strain and displacement distributions in polymeric structures are also usually time dependent and may be very different from these quantities in elastic structures under the same conditions. 

Naturally, this assumption also implies that the dilatational strains are always zero. For computer simulations of viscoelastic problems, this assumption can sometimes cause numerical difficulties. Most standard finite... 

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 ... 

Method 1 Solve the viscoelastic problem in the time domain using Eqs. 9.29 9.32. 

Convert the solution of the elastic problem to the solution of the viscoelastic problem in the transform domain by replacing all variables by their Laplace transform and all elastic constants by their equivalent in the transform domain, i.e.,... 

Method 3 allows viscoelastic problems to be solved quite easily providing that the analogous elastic solution exists or can be found. There are im-... 

Since modulus of the outer metallic cylinder is much larger than that of the inner polymer cylinder, it is reasonable to assume the outer shell is rigid. This assumption provides a further simplitication to the problem. In this section, we develop the fully elastic solution based upon the classic Lame solution. Subsequently we will consider the viscoelasticity of the polymer and invoke the correspondence principle to solve the viscoelastic problem. 

Since we have the elasticity solution to the reinforced thick walled cylinder problem, we can now find the solution to the viscoelastic problem by applying the correspondence principle (Method 3 from earlier in this chapter). Replacing the variables in Eqs. 9.50 and 9.51 by the appropriate transforms gives the solution for stresses and displacements of the viscoelastic problem in the transform domain. 

If the material is not incompressible, the solution for stresses in Eq. 9.52 can be used to obtain the stress field with time for the viscoelastic problem given the material properties. In order to examine a particular loading and material, it is convenient to use Eqs. 9.16 to obtain the stress solution in terms of the shear and bulk moduli in order to make reasonable assumptions about the material similar to those outlined earlier in Eqs. 9.20-9.24. The term including Poisson s ratio can be rewritten as... 

Similar to Laplace transforms, Fourier transforms also have special properties under differentiation and integration making them a very effective method for solving differential equations. Due to the form of the constitutive laws in viscoelasticity, Fourier transforms are quite useful in analysis of viscoelastic problems. 

These are formally equivalent to the FT equations governing elastic problems, where the complex moduli fiijki(a>)y the properties of which are discussed in Sect. 1.5, replace the elastic moduli. This immediately suggests that elastic solutions can be modified so that they apply to corresponding viscoelastic problems. We have not mentioned boundary conditions however, and this is the catch. It will emerge that there is a simple correspondence between elastic and viscoelastic solutions only for certain types of boundary conditions. However, under quite general conditions, there is a connection between the two, and the exploration and utilization of this connection forms the subject manner of virtually all later chapters. 

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