Introduction

Composite materials are those that contain two or more distinct constituent materials or phases, on a microscopic or macroscopic size scale larger than the atomic scale, and in which physical properties are significantly altered in comparison with those of a homogeneous material. In this vein, fiberglass and other fibrous materials are viewed as composites, but alloys, such as brass, are not. Semicrystalline polymers, such as polyethylene, have a heterogeneous structure, which can be treated via composite theory. Biological materials also have a heterogeneous structure and are known as natural composites. Composites may contain solid, liquid, and gas phases. For example, composites of gas and liquid include mist and foam; composites of solid and gas include foam and smoke. Composites with a structural role have several solid phases or a connected solid phase with gas or liquid in the interstices (structural foam and honeycomb).

Composite Structures and Properties

Ideal Structures

The properties of composites are greatly dependent upon microstructure. Composites differ from homogeneous materials in that considerable control can be exerted over the larger scale structure; and hence over the desired properties. In particular, the properties of a composite depend upon the shape of the heterogeneities, upon the volume fraction occupied by them, and upon the interface between the constituents. Volume fraction refers to the ratio of the volume of a constituent to the total volume of a composite specimen. The shape of the heterogeneities in a composite is classified as follows. The principal inclusion shape categories (Figure 9.1) are the particle, with no long dimension; the fiber, with one long dimension; and the platelet (flake, lamina), with two long dimensions.

Introduction

Rationale

The purpose of Chapter 7 is to present the viscoelastic behavior of representative real materials so that the reader can gain a sense of orders of magnitude of the effects. Engineers and scientists who deal with elastic behavior of materials are aware of the moduli of various common materials. Similarly, knowledge of the viscoelastic properties of particular materials is essential to rationally apply them. Further examples, with analysis of the physical causes of the viscoelasticity, are provided in Chapter 8. Materials presented here are classified as polymers, metals, ceramics, biological composites and synthetic composites (Chapter 9). In a survey, damping properties of metals, ceramics, and metal matrix composites are compared [1]. Structural metals tend to be low damping, with tan? on the order of 10−3 or less. Experimental results for the same material can differ substantially depending on purity and permanent deformation. Most ceramics also exhibit low damping at ambient temperature, but some exhibit modest damping at elevated temperature.

Overview: Some Common Materials

An overview of modulus and damping of selected classes of materials at small strain is shown in Figure 7.1. At high strain amplitude, some metals exhibit higher damping [2] in such maps.

The loss tangents of some well-known materials at various temperatures and frequencies are presented in Table 7.1. Most of the data are at room temperature, denoted rt in Table 7.1 except as noted, and at audio or subaudio frequencies.

Introduction

For some applications of viscoelastic materials, it is sufficient to understand creep and relaxation properties. For example sometimes structural elements are maintained under steady load or constant extension. In other applications the response to an arbitrary load or strain history is required. To predict this response, constitutive equations that incorporate all possible responses are of use. Various mathematical tools are used in the development of these equations. The viscoelastic functions in the equations are obtained by experimentation.

Prediction of the Response of Linearly Viscoelastic Materials

Prediction of Recovery from Relaxation E(t)

The creep and relaxation properties described above in §1.3 permit one to predict the response of the material to a step function stress or strain. To predict the response of the material to any history of stress or strain (i.e., stress or strain as a function of time), constitutive equations are developed.

The following is restricted to isothermal deformation in one dimension. The symbol E is used to represent an elastic modulus, however, the analysis applies equally to shear deformation corresponding to a shear modulus G or volumetric deformation corresponding to a bulk modulus B, which is sometimes called K.

To develop the constitutive equation for linear materials, we use the Boltzmann superposition principle, which states that the effect of a compound cause is the sum of the effects of the individual causes. This principle is a statement of linearity. First, we consider the strain associated with a relaxation and recovery experiment, with the intention to use the idea of linearity as embodied in the Boltzmann superposition principle to predict the resulting stress history.

Introduction

Rationale

The treatment of viscoelasticity thus far has dealt with phenomena, measurement of material properties, prediction of response to various load histories, and stress analysis. In this chapter we consider the physical causes of viscoelastic response. Study of causal mechanisms is motivated by (1) the desire for scientific understanding; (2) the intention to gain ability to choose or tailor materials with specified viscoelastic properties; and (3) the utility of viscoelastic measurements as a probe of microphysical processes that are causally linked to viscoelasticity. The conceptually simplest of the causal mechanisms are developed in detail here. These processes can occur in crystalline materials including metals, ceramics as well as in polymers. Examples of damping due to fluid–solid interaction in porous materials according to the Biot theory are given in §7.5, biological tissue. Viscoelasticity in tissue also results from stress induced motion at the molecular scale. The treatment is intended to be introductory, not exhaustive.

Survey of Viscoelastic Mechanisms

There are many causal mechanisms [1–3] responsible for viscoelastic response; several of these are as follows. Mechanisms indicated by a * are called fundamental mechanisms because they occur even in an ideal perfect single crystal, and they are not removable, even in principle. Many effects are named for the person who discovered them or who provided a clear analysis of the physical processes responsible for them. The word relaxation is used in these names to refer to viscoelasticity in a general sense, not necessarily a specific stress relaxation experiment.

Introduction and Rationale

The response of viscoelastic materials to sinusoidal load is developed in this section, and this response is referred to as dynamic. Dynamic in this context has no connection with inertial terms or resonance. The dynamic behavior is of interest because viscoelastic materials are used in situations in which the damping of vibration or the absorption of sound is important. The results developed in this section have a bearing on applications of viscoelastic materials, on experimental methods for their characterization, and on the development of physical insight regarding viscoelasticity.

The frequency of the sinusoidal load on an object or structure may be so slow that inertial terms do not appear (the subresonant regime), or high enough that resonance of structures made of the material occurs. At a sufficiently high frequency, dynamic behavior is manifested as wave motion. This distinction between ranges of frequency does not appear in the classical continuum description of a homogeneous material, because the continuum view deals with differential elements of material.

Oscillatory stress and strain histories are represented by sinusoid functions. Suppose we have α(ωt) = α0sinωt, in which t is time, α0 is the amplitude, and ω is the angular frequency. The sine function repeats every 2π radians. So α(ωt + 2π) = α(ωt). The time T required for the sine function to complete one cycle is obtained from ωT = 2π, or T = 2π/ω. T is called the period, the number of seconds required for one cycle.

Introduction and General Requirements

Experimental procedures for viscoelastic materials, as in other experiments in mechanics, make use of a method for applying force or torque, a method for measuring the same, and a method for determining displacement, strain, or angular displacement of a portion of the specimen. If the experimenter intends to explore a wide range of time, and/or frequencies, the requirements for performance of the instrumentation can be severe. Elementary creep procedures are discussed first, because they are the simplest. Methods for measurement and for load application are discussed separately, since many investigators choose to assemble their own equipment from components. Many procedures in viscoelastic characterization of materials have aspects in common with other mechanical testing. Therefore, study of known standard methods [1, 2] for mechanical characterization of materials is useful. As in other forms of mechanical characterization, it is important that the stress distribution in the specimen be well-defined. End conditions in the gripping of specimens are usually not well known. The experimenter often uses elongated specimens for tension, torsion, or bending, to appeal to Saint Venant's principle in using idealized stress distributions for the purpose of analysis. The determination of viscoelastic properties as a function of frequency is at times referred to as mechanical spectroscopy.

Frequency response is an important consideration for instruments used in viscoelasticity; the transducers [3] for force generation and measurement and for measurement of deformation must respond adequately at the frequencies of interest.