An Introduction to the Theory of Elasticity

Preface

For many years, continuum mechanics has been recognised as a rich and challenging subject for study, and is firmly established in undergraduate courses in many universities and colleges. Over the last two or three decades, intense research activity in this field has inevitably affected the teaching of the subject. The student’s introduction to the once-diverse fields of solid and fluid mechanics has been unified, and as a result, considerable clarity and simplification have been achieved. Our objective here is to place into the hands of second- and third-year undergraduates a text on elasticity, treated from this unified standpoint, which has been prepared while carefully bearing in mind the limited mathematical tools that he is likely to have at his disposal. Much of the material has been given in our own lecture courses over a number of years, and modified in the light of our students’ understanding. We have endeavoured to amplify all those sections which have commonly presented difficulties when they have been met with for the first time. Moreover, we have tried to present the material in a manner which will not only be easily assimilated by the student, but will also serve as a foundation for his reading of the many modern advanced texts, both in the finite and infinitesimal theory, which will become necessary should he wish to embark upon postgraduate or research programmes.

In recent years, many lecturers have found that the finite deformation theory of elasticity is a perfectly suitable subject for undergraduate courses, and it is a most desirable background even if the infinitesimal theory is the ultimate goal. Of course, to keep the volume to a reasonable size, we have had to be selective in our choice of subject matter, but we are hopeful that adequate material will be found here that can be presented to students of average ability in an introductory course. If the material is found to be too much for the time available, the syllabus could easily be weighted in favour of finite deformation or the infinitesimal theory according to taste. Chapters 1 to 3 would provide an introduction to finite deformation while some of Chapter 2 and the whole of Chapter 3 could be omitted to provide a course majoring on the infinitesimal theory.

In order to make the text largely self-contained, we have included in Chapter 1 and the early parts of Chapter 2 a discussion of the general principles of continuum mechanics which are necessary for the foundations of elasticity. Where amplification of this material is required, readers are referred to the companion text by Spencer (1980).

A knowledge of vector analysis and calculus, at a level usually reached in introductory university courses on these topics, is assumed. Some familiarity with elementary complex variable theory, Cartesian tensors, and matrix algebra is also required, but sufficient references are given for the student to make good any deficiency he has in these areas.

There has been a trend in recent years towards presenting continuum theories in coordinate-free notation. While this is a simple, elegant tool in the hands of specialists, we have felt that the use of suffix notation in the major part of the text provides the student with an easier transition from the courses in vector analysis that he is likely to have taken. However, direct notation is employed in parts where it provides a clear alternative to suffixes, and where the student is more likely to recognise standard theorems from matrix algebra. The basic theory is derived using Cartesian coordinates, although in applications a limited use of cylindrical and spherical polars is made.

In our presentation of finite deformation elasticity we have, wherever possible, maintained a close link with experiment. We believe that a student’s appreciation of, and motivation in, the subject is enhanced if he realizes that the novel effects arising from the theory can not only be observed with relatively simple apparatus, but also used in obtaining quantitative information about the strain-energy function for the material. For this reason we have analysed in detail some non-homogeneous deformations, as well as the simple homogeneous ones, and we have described the associated experiments in which this analysis is used.

When studying the infinitesimal theory of elasticity we have found that there is often confusion in the student’s mind concerning the sense in which the deformation is to be regarded as small, and the distinction, if any, which needs to be made between the undeformed and deformed configurations. To clarify these issues we have presented the basic equations in material coordinates so that the displacement and stress are vector and tensor fields defined over the reference configuration. Then, having solved for the displacement field, the deformed configuration can be constructed by superimposing this field on the reference configuration. Some of the worked and unworked examples should clarify potential difficulties in this direction. One price that must be paid in this approach is that the idea of Piola stresses must be introduced. On the whole, we felt that the extra work involved is amply repaid by the gain in understanding of the approximation procedure, and the application of boundary conditions. Moreover, once the equations have been derived, the Piola stresses may be identified with the Cauchy stresses to the degree of approximation used, and so for anyone considering only the applications, a knowledge of the usual Cauchy stress is adequate.

Throughout our presentation we have been greatly influenced by, and have drawn freely from, the articles by Truesdell and Toupin (1960), Truesdell and Noll (1965), and Gurtin (1972). We hope that students will be stimulated to read these major reference books for themselves, as well as the many other volumes listed in our selection of recommendations for further reading (p. 241).

1

Deformation and stress

In our discussion of the macroscopic behaviour of materials we disregard their microscopic structure. We think of the material as being continuously distributed throughout some region of space. At any instant of time, every point in the region is the location of what we refer to as a particle of the material. In this chapter we discuss how the position of each particle may be specified at each instant, and we introduce certain measures of the change of shape and size of infinitesimal elements of the material. These measures are known as strains, and they are used later in the derivation of the equations of elasticity. We also consider the nature of the forces acting on arbitrary portions of the body and this leads us into the concept of stress.

1.1 Motion. Material and spatial coordinates

We wish to discuss the mechanics of bodies composed of various materials. We idealize the concept of a body by supposing that it is composed of a set of particles such that, at each instant of time t, each particle of the set is assigned to a unique point of a closed region e9780486150994_i0002.jpg of three-dimensional Euclidean space, and that each point of e9780486150994_i0003.jpg is occupied by just one particle. We call e9780486150994_i0004.jpg the configuration of the body at time t.

To describe the motion of the body, that is, to specify the position of each particle at each instant, we require some convenient method of labelling the particles. To do this, we select one particular configuration e9780486150994_i0005.jpg and call this the reference configuration. The set of coordinates (X1, X2, X3), or position vector X, referred to fixed Cartesian axes, of a point of e9780486150994_i0006.jpg uniquely determines a particle of the body and may be regarded as a label by which the particle can be identified for all time. We often refer to such a particle as the particle X. In choosing e9780486150994_i0007.jpg we are not restricted to those configurations occupied by the body during its actual motion, although it is often convenient to take e9780486150994_i0008.jpg to be the configuration e9780486150994_i0009.jpg occupied by the body at some instant which is taken as the origin of the time scale t.

The motion of the body may now be described by specifying the position x of the particle X at time t in the form of an equation

(1.1.1)

(see Fig. 1.1) or, in component form,

(1.1.2)

and we assume that the functions X1, X2, and X3 are differentiable with respect to X1, X2, X3, and t as many times as required.

Fig. 1.1

Sometimes we wish to consider only two configurations of the body, an initial configuration and a final configuration. We refer to the mapping from the initial to the final configuration as a deformation of the body. The motion of the body may be regarded as a one-parameter sequence of deformations.

We assume that the Jacobian

(1.1.3)

exists at each point of e9780486150994_i0014.jpg and that

(1.1.4)

The physical significance of these assumptions is that the material of the body cannot penetrate itself, and that material occupying a finite non-zero volume in e9780486150994_i0016.jpg cannot be compressed to a point or expanded to infinite volume during the motion (see Example 1.3.2).

Mathematically (1.1.4) implies that (1.1.1) has the unique inverse

(1.1.5)

Now at the current time t the position of a typical particle P is given by its Cartesian coordinates (x1, x2, x3), but, as mentioned above, P continues to be identified by the coordinates (X1, X2, X3) which denoted its position in e9780486150994_i0018.jpg The coordinates (X1, X2, X3) are known as material (or Lagrangian) coordinates since distinct sets of these coordinates refer to distinct material particles. The coordinates (x1, x2, x3) are known as spatial (or Eulerian) coordinates since distinct sets refer to distinct points of space. The values of x given by equation (1.1.1) for a fixed value of X are those points of space occupied by the particle X during the motion. Conversely, the values of X given by equation (1.1.5) for a fixed value of x identify the particles X passing through the point x during the motion.

From now on, when upper- or lower-case letters are used as suffixes, they are understood generally to range over 1, 2, and 3. Usually upper-case suffixes refer to material coordinates, lower-case to spatial coordinates and repetition of any suffix implies summation over the range. For example, we write xi for (x1, x2, x3), XA for (X1, X2, X3) and xixi denotes e9780486150994_i0019.jpg

When a quantity is defined at each point of the body at each instant of time, we may express this quantity as a function of XA and t or of xi and t. If XA and t are regarded as the independent variables then the function is said to be a material description of the quantity; if xi and t are used then the corresponding function is said to be a spatial description. One description is easily transformed into the other using (1.1.1) or (1.1.5). The material description Ψ(X, t) has a corresponding spatial description ψ(x,t) related by

(1.1.6)

or

(1.1.7)

To avoid the use of a cumbersome notation and the introduction of a large number of symbols, we usually omit explicit mention of the independent variables, and also use a common symbol for a particular quantity and regard it as denoting sometimes a function of XA and t, and sometimes the associated function of xi and t. The following convention for partial differentiation should avoid any confusion.

Let u be the common symbol used to represent a quantity with the material description Ψ and spatial description ψ (these may be scalar-, vector-, or tensor-valued functions) as related by (1.1.6) and (1.1.7). We adopt the following notation for the various partial derivatives:

(1.1.8)

(1.1.9)

Example 1.1.1

Write down equations describing the motion of a rigid body moving with constant velocity V in the 1-direction using the configuration e9780486150994_i0024.jpg of the body at time t = 0 as reference configuration.

If the material description of the temperature u in the body is aX1, where a is a constant, find the spatial description.

Find also Du/Dt and ∂u/∂t, and interpret physically.

Using e9780486150994_i0025.jpg as reference configuration, the motion is given by

x1 = X1 + Vt, x2 = X2, x3 = X3

The material and spatial descriptions of the temperature field are

u = aX1 = a(x1 − Vt)

Hence,

The coordinates XA refer to given particles of the body so the temperature u = aX1, remains fixed at each particle. The time derivative Du/Dt measures the time rate of increase of u at fixed XA, and this is therefore zero. On the other hand, at a fixed point xi, of space the temperature varies as the body passes through and its time rate of increase is −aV.

These two time derivatives are of great importance in continuum mechanics and they are discussed in more detail in the next section.

1.2 The material time derivative

Suppose that a certain quantity is defined over the body, and we wish to know its time rate of change as would be recorded at a given particle X during the motion. This means that we must calculate the partial derivative, with respect to time, of the material description Ψ of the quantity, keeping X fixed. In other words we calculate ∂Ψ(X, t)/∂t. This quantity is known as a material time derivative. We may also calculate the material time derivative from the spatial description Ψ. Using the chain rule of partial differentiation, we see from (1.1.7) that

(1.2.1)

remembering, of course, that repeated suffixes imply summation over 1, 2, and 3.

Consider now a given particle X0. Its position in space at time t is

x = χ(X0, t)

and so its velocity and acceleration are

respectively. We therefore define the velocity field for the particles of the body to be the material time derivative ∂x(X, t)/∂t, and use the common symbol v to denote its material or spatial description:

(1.2.2)

Likewise we define the acceleration field f to be the material time derivative of v:

(1.2.3)

Moreover, in view of (1.2.1) the material time derivative of u has the equivalent forms

(1.2.4)

In particular, the acceleration (1.2.3) may be written as

(1.2.5)

where the operator e9780486150994_i0033.jpg is defined relative to the coordinates xi, that is,

(1.2.6)

In suffix notation, (1.2.5) becomes

(1.2.7)

Example 1.2.1

Find the value of J and the material and spatial descriptions of the velocity field for the motion given by

(1.2.8)

where α and β (≥0) are constants. Calculate ∂v/∂t, (v· e9780486150994_i0037.jpg ) v, and Dv/Dt. Verify the relation (1.2.4).

For this motion,

Using (1.2.2),

(1.2.9)

which is the material description of the velocity field. Equations (1.2.8) may be inverted to give

and so the spatial description of the velocity field may be written

Hence,

and

Also, from (1.2.9),

and so (1.2.4) is verified.

1.3 The deformation-gradient tensor

We have discussed how the motion of a body may be described. In this section we analyse the deformation of infinitesimal elements of the body which results from this motion.

Suppose that e9780486150994_i0045.jpg coincides with the initial configuration e9780486150994_i0046.jpg and that two neighbouring particles P and Q have positions X and X + dX in e9780486150994_i0047.jpg Then at time t their positions in e9780486150994_i0048.jpg are x and x + dx, where

(1.3.1)

and the components of the total differential dx are given in terms of the components of dX and the partial derivatives of χ by

(1.3.2)

The quantities xi,A are known as the deformation gradients. They are the components of a second-order tensor known as the deformation-gradient tensor which we denote by F. (Readers unfamiliar with Cartesian tensors should consult, for example, Spencer (1980) Ch. 3.)

Example 1.3.1

Show that when the coordinate axes are rotated about the origin, and the coordinates xi and XA are transformed into

where lij are the direction cosines of the coordinate transformation, the components FiA of F are transformed into

(This proves that the deformation gradients are the components of a second-order tensor.)

Example 1.3.2

Show that the assumption (1.1.4), J>0, implies that the material of the body cannot penetrate itself, and that material occupying a finite non-zero volume in e9780486150994_i0053.jpg cannot be compressed to a point or expanded to infinite volume during the motion. Deduce also that a volume element dV0 in e9780486150994_i0054.jpg deforms into a volume element dV in e9780486150994_i0055.jpg where dV = J d V0.

Consider three infinitesimal non-coplanar line elements dX(1), dX(2), dX(3) at a point P in e9780486150994_i0056.jpg and suppose that they correspond to line elements dx(1), dx(2), dx(3) in e9780486150994_i0057.jpg . Then

Hence

(1.3.3)

since the determinant of the product of two matrices is equal to the product of their determinants. But

(1.3.4)

Now dX(1). dX(2) × dX(3) is positive or negative according as to whether dX(1), dX(2), dX(3) are ordered in a right-handed or left-handed sense, and this triple scalar product has magnitude equal to the volume of a parallelepiped, three of whose edges are taken to be dX(1), dX(2), dX(3). A similar result holds for dX(1), dX(2), dX(3). From (1.3.3) and (1.3.4) we see therefore that, if