The behavior of many materials under an applied load may be approximated by specifying a relationship between the applied load or stress and the resultant deformation or strain. In the case of elastic materials this relationship, identified as Hooke's Law, states that the strain is proportional to the applied stress, with the resultant strain occurring instantaneously. In the case of viscous materials, the relationship states that the stress is proportional to the strain rate, with the resultant displacement dependent on the entire past history of loading. Boltzmann (1874) proposed a general relationship between stress and strain that could be used to characterize elastic as well as viscous material behavior. He proposed a general constitutive law that could be used to describe an infinite number of elastic and linear anelastic material behaviors derivable from various configurations of elastic and viscous elements. His formulation, as later rigorously formulated in terms of an integral equation between stress and strain, characterizes all linear material behavior. The formulation, termed linear viscoelasticity, is used herein as a general framework for the derivation of solutions for various wave-propagation problems valid for elastic as well as for an infinite number of linear anelastic media.

Consideration of material behavior in one dimension in this chapter, as might occur when a tensile force is applied at one end of a rod, will provide an introduction to some of the well-known concepts associated with linear viscoelastic behavior. It will provide a general stress–strain relation from which stored and dissipated energies associated with harmonic behavior can be inferred as well as the response of an infinite number of viscoelastic models.

There are a number of continuum mechanics books in the literature, written with the general tensor formulation. An essential characteristic of general tensors is the use of curvilinear coordinates. We may imagine a Cartesian grid placed inside a continuum. As the continuum deforms, the grid lines deform with it into a mesh of curves. Thus general tensor formulations are convenient to describe continuum mechanics. Although this chapter is not required to follow our presentation of continuum mechanics in terms of Cartesian tensors, it will be of value to extend the students' knowledge through additional reading. In modern computational mechanics, curvilinear grids are often used and the governing equations in general tensor formulation are needed. As an added bonus, students will be able to follow the theory of shells and the general theory of relativity!

In the previous chapter we saw systems of Cartesian coordinates and transformations of tensors between two coordinates that are related linearly. When we have to deal with curvilinear coordinates, the coordinate transformations are in general nonlinear and the coordinates may not form an orthogonal system. Representation of tensors in such a system depends on the directions of local tangents and normals to the coordinate surfaces. It is conventional to use superscripts to denote the coordinate curves. The reason for this will be clarified later. Let us begin with a Cartesian system with labels x1, x2, and x3, transforming into curvilinear system ξ1, ξ2, and ξ3.

Mechanics is the study of the behavior of matter under the action of internal and external forces. In this introductory treatment of continuum mechanics, we accept the concepts of time, space, matter, energy, and force as the Newtonian ideals. Here our objective is the formulation of engineering problems consistent with the fundamental principles of mechanics. To paraphrase Professor Y. C. Fung–there are generally two ways of approaching mechanics: One is the ad hoc method, in which specific problems are considered and specific solution methods are devised that incorporate simplifying assumptions, and the other is the general approach, in which the general features of a theory are explored and specific applications are considered at a later stage. Engineering students are familiar with the former approach from their experience with “Strength of Materials” in the undergraduate curriculum. The latter approach enables them to understand an entire field in a systematic way in a short time. It has been traditional, at least in the United States, to have a course in continuum mechanics at the senior or graduate level to unify the ad hoc concepts students have learned in the undergraduate courses. Having had the knowledge of thermodynamics, fluid dynamics, and strength of materials, at this stage, we look at the entire field in a unified way.

Concept of a Continuum

Although mechanics is a branch of physics in which, according to current developments, space and time may be discrete, in engineering the length and time scales are orders (and orders) of magnitude larger than those in quantum physics and we use space coordinates and time as continuous.

The time-independent, permanent deformation in metals beyond the elastic limit is described by the term plasticity. As shown in Fig. 14.1, the elastic part of the uniaxial stress–strain curve OA is reversible. When we unload from any point beyond point A, corresponding to the zero-stress state, there is a permanent deformation in the specimen. Several theories describing the plastic behavior of metals have been proposed, but none is totally satisfactory. Ideally, the stress σ, separating the elastic part OA from the inelastic part, is called the yield stress σ0 of the material. However, it is difficult to obtain this special point accurately from experiments, and it often depends on the history of the loading the specimen has undergone.

When a specimen is unloaded from a point C to zero stress and reloaded, permanent deformation usually begins at a stress level σC. In other words, the yield stress is higher after the specimen has undergone a certain amount of permanent deformation. This is known as work hardening or strain hardening.

Another feature of metal deformation is that, from a strained state, beyond the yield point, reversal of loading causes the compressive yield stress to be different from the tensile yield stress. It is usually lower in magnitude than the tensile yield stress. This effect is called the Bauschinger effect.