Engineering Viscoelasticity

This chapter describes the molecular structure of amorphous polymers, whose mechanical response to loads combines the features of elastic solids and viscous fluids. Materials that respond in such manner are called viscoelastic, and their mechanical properties have an intrinsic dependence on the time and temperature at which the response is measured. To put this into context, the chapter compares the nature of the response of elastic, viscous, and viscoelastic materials to several types of loading programs, examining the physical nature of their mechanical properties, their behavior regarding energy conservation, and the phenomenon of aging. The topics treated in this chapter provide the neophyte and casual reader with a good understanding of what viscoelastic materials are all about.

Materials respond to external load by deforming and straining, and by developing stresses. The internal stresses corresponding to a given set of strains depend on the constitution of the material itself. For this reason, the rules that permit calculation of internal stresses from known strains, or vice versa, are called constitutive laws or constitutive equations. There are two equivalent ways to describe the mathematical relationships between stresses and strains for viscoelastic materials. One form uses integrals to define the constitutive relations, while the other relates stresses and strains by means of differential equations. Starting from Boltzmann’s superposition principle, this chapter develops the integral form of the one-dimensional constitutive equations for linearly viscoelastic materials. This is followed by a discussion of the principle of fading memory, which helps to define the acceptable analytical forms of the material property functions. It is then shown that the closed-cycle condition (i.e., that the steady-state response of a non-aging viscoelastic material to a periodic excitation be periodic) requires that the material property functions depend only on the difference of their arguments. The chapter also examines the relationships between the relaxation modulus and creep compliance functions in the physical time domain as well as in Laplace-transformed space. Various alternative forms of the integral constitutive equations often encountered in practice are discussed as well.

The mechanical response of a viscoelastic material to external loads combines the characteristics of elastic and viscous behavior. On the other hand, as we know from experience, springs and dashpots are mechanical devices which exhibit purely elastic and purely viscous response, respectively. It is then natural to imagine that the equations that relate stresses to strains in a viscoelastic material could be represented with an appropriate combination of equations which relate stresses to strains in springs and dashpots. To develop this idea, Sect. 3.2 examines the response of the linear elastic spring and linear viscous dashpot to externally applied loads. The response equations for these simple mechanical elements are formalized in Sect. 3.3, with the introduction of so-called rheological operators. As it turns out, because combinations of springs and dashpots require the addition and multiplication of constant and first derivative operators, it turns out that the constitutive equation of general arrangements of springs and dashpots, such as are needed to reproduce observed viscoelastic behavior, must be represented by linear ordinary differential equations whose order depends on the number, type, and specific arrangement of the springs and dashpots. The physical significance of the coefficients in the resulting differential equations is examined also, and the proper form of the initial conditions established. As will be seen, the mere presence or absence of some of the coefficients of a differential equation reveals whether the particular arrangement of springs and dashpots it represents will model fluid or solid behavior, and whether it will exhibit instantaneous, elastic response. A general approach to establishing rheological models is presented in Sect. 3.4, and applied in Sects. 3.5 through 3.7 to develop the differential equations, and examine the behavior of simple and general rheological models.

Although the viscoelastic constitutive equations in either integral or differential form apply in general, irrespective of the type of loading, or the point in time at which the response is sought, it is possible to derive from them constitutive equations of a form especially well suited to steady-state oscillations. This chapter uses complex algebra to transform the integral and differential constitutive equations of viscoelasticity, defined in the time domain, into algebraic expressions in the complex plane. The chapter also examines the relationships between the material property functions defined in the time domain, and their complex-variable counterparts, and examines the problem of energy dissipation during steady-state oscillations, important in the design of mounts for vibratory equipment, among others.

This chapter is devoted to structural mechanics, developing the theories of bending, torsion, and buckling of straight bars, and presenting a detailed account of vibration of single-degree-of-freedom viscoelastic systems, including vibration isolation. A balanced treatment is given to stress–strain equations of integral and differential types, and to stress–strain relations in complex-variable form, which are applicable to steady-state response to oscillatory loading. All equations in this chapter are developed from first principles, without presuming previous knowledge of the subject matter being presented. This approach is followed for two reasons: first, because it is necessary for readers without a formal training in mechanics of materials; and secondly, because it provides the reader—even one with formal training in classical engineering—with a method to follow when the use of popular shortcuts, like the integral transform techniques, might be questionable or unclear.

This chapter examines the simultaneous dependence of material property functions on time and on temperature. The time–temperature superposition principle and the concept of time–temperature shifting are introduced first. The dependence of the glass transition temperature both on the time of measurement and on surrounding pressure is examined in detail. The integral and differential constitutive equations are then generalized to include thermal strains and strains due to changes in humidity. Two ways used in practice to represent the material property function that accounts for thermal strains are considered: independent of time and time dependent.

This chapter examines four topics of practical importance. It begins with an introduction to material characterization testing, covering stress relaxation, creep, constant rate, and dynamic tests. The chapter then introduces two types of analytical forms, typically used to describe mechanical constitutive property functions. One type, usually referred to as a Dirichlet-Prony series, is expressed as a finite sum of exponentials; the other form is a power law in time. This treatment is followed by a discussion of methods of inversion of material property functions given in Prony series form; both exact and approximate methods of inversion are presented. The chapter is completed with a discussion of practical ways to establish the numerical coefficients entering the analytical forms used to represent the WLF shift relation, the relaxation modulus, and the creep compliance. The use of a computer application available with the book, which was specifically developed to obtain the exact convolution inverse of function in a Prony series form, is also presented and its use is illustrated by means of some examples.

This chapter generalizes to three dimensions the one-dimensional viscoelastic constitutive equations derived in earlier chapters. The concepts of homogeneity, isotropy, and anisotropy are introduced and the principle of superposition is used to construct three-dimensional constitutive equations for general anisotropic, orthotropic, and isotropic viscoelastic materials. So-called Poisson’s ratios are introduced, and it is shown that uniaxial tensile and shear relaxation and creep tests suffice to characterize orthotropic viscoelastic solids. A rigorous treatment extends applicability of the Laplace and Fourier transforms to three-dimensional conditions, and constitutive equations in both hereditary integral form and differential form for compressible and incompressible isotropic solids are developed and discussed in detailed.

This chapter contains a comprehensive discussion of the types of boundary-value problems encountered in linear viscoelasticity. The chapter presents detailed solution methods for compressible and incompressible solids, including materials with synchronous moduli, whose property functions are assumed to have the same time dependence. The method of separation of variables in the time domain and frequency domains is also described in full, as is the use of the Laplace and Fourier transformations. The elastic–viscoelastic correspondence principle, which allows viscoelastic solutions to be constructed from equivalent elastic ones and as a consequence of the applicability of integral transforms, is also developed and examined in detail.

This chapter examines the propagation of harmonic and shock waves in viscoelastic materials of integral and differential type. For simplicity, the different topics are introduce in one dimension, presenting the balance of linear momentum across the shock front, and the jump equations in stress, strain and velocity without obscuring the subject matter. As must be expected, harmonic waves in viscoelastic media are always damped. Also shown is that shock waves travel at the glassy sonic speed of the viscoelastic material in which they occur; that is, at the speed of sound in an elastic solid with modulus of elasticity equal to the glassy modulus of the viscoelastic material at hand.

This chapter introduces the subject of the variation of a functional and develops variational principles of instantaneous type which are the equivalent of Castigliano’s theorems of elasticity for computing the generalized force associated with a generalized displacement and vice versa, by means of partial derivatives of the potential energy and the complementary potential energy functionals, respectively. A natural consequence of the variational principle of instantaneous type is that the constitutive potentials of viscoelastic materials are not unique. Any dissipative term can be added to them without changing the stress strain law. The viscoelastic versions of the unit load theorem of elasticity, and the theorems of Betti and Maxwell for elastic bodies, are also developed in detail.