Elasticity

The subject of Elasticity is concerned with the determination of the stresses and displacements in a body as a result of applied mechanical or thermal loads, for those cases in which the body reverts to its original state on the removal of the loads. In this book, we shall further restrict attention to the case of linear infinitesimal elasticity, in which the stresses and displacements are linearly proportional to the applied loads and the displacements are small in comparison with the characteristic length dimensions of the body. These restrictions ensure that linear superposition can be used and enable us to employ a wide range of series and transform techniques which are not available for non-linear problems.

Most engineers first encounter problems of this kind in the context of the subject known as Mechanics of Materials, which is an important constituent of most undergraduate engineering curricula. Mechanics of Materials differs from Elasticity in that various plausible but unsubstantiated assumptions are made about the deformation process in the course of the analysis. A typical example is the assumption that plane sections remain plane in the bending of a slender beam. Elasticity makes no such assumptions, but attempts to develop the solution directly and rigorously from its first principles, which are Newton’s laws of motion, Euclidian geometry and Hooke’s law. Approximations are often introduced towards the end of the solution, but these are mathematical approximations used to obtain solutions of the governing equations rather than physical approximations that impose artificial and strictly unjustifiable constraints on the permissible deformation field.

We can think of an elastic solid as a highly redundant framework — each particle is built-in to its neighbours. For such a framework, we expect to get some equations from considerations of

equilibrium

, but not as many as there are unknowns. The deficit is made up by

compatibility

conditions — statements that the deformed components must fit together. These latter conditions will relate the dimensions and hence the strains of the deformed components and in order to express them in terms of the same unknowns as the stresses (forces) we need to make use of the stress-strain relations as applied to each component separately.

If we were to approximate the continuous elastic body by a system of interconnected elastic bars, this would be an exact description of the solution procedure. The only difference in treating the continuous medium is that the system of algebraic equations is replaced by partial differential equations describing the same physical or geometrical principles.

A problem is two-dimensional if the field quantities such as stress and displacement depend on only two coördinates (x, y) and the boundary conditions are imposed on a line f(x, y)=0 in the xy-plane.

In this sense, there are strictly no two-dimensional problems in elasticity. There are circumstances in which the stresses are independent of the z-coördinate, but all real bodies must have some bounding surfaces which are not represented by a line in the xy-plane. The two-dimensionality of the resulting fields depends upon the boundary conditions on such surfaces being of an appropriate form.

Newton’s law of gravitation states that two heavy bodies attract each other with a force proportional to the inverse square of their distance — thus it is essentially a vector theory, being concerned with forces. However, the idea of a scalar gravitational potential can be introduced by defining the work done in moving a unit mass from infinity to a given point in the field. The principle of conservation of energy requires that this be a unique function of position and it is easy to show that the gravitational force at any point is then proportional to the gradient of this scalar potential. Thus, the original vector problem is reduced to a problem about a scalar potential and its derivatives.

In general, scalars are much easier to deal with than vectors. In particular, they lend themselves very easily to coördinate transformations, whereas vectors (and to an even greater extent tensors) require a set of special transformation rules (e.g. Mohr’s circle).

In certain field theories, the scalar potential has an obvious physical significance. For example, in the conduction of heat, the temperature is a scalar potential in terms of which the vector heat flux can be defined. However, it is not necessary to the method that such a physical interpretation can be given. The gravitational potential can be given a physical interpretation as discussed above, but this interpretation may never feature in the solution of a particular problem, which is simply an excercise in the solution of a certain partial differential equation with appropriate boundary conditions. In the theory of elasticity, we make use of scalar potentials called

stress functions

or

displacement functions

which have no obvious physical meaning other than their use in defining stress or displacement components in terms of derivatives.

The Cartesian coördinate system (x, y) is clearly particularly suited to the problem of determining the stresses in a rectangular body whose boundaries are defined by equations of the form x = a, y = b. A wide range of such problems can be treated using stress functions which are polynomials in x, y. In particular, polynomial solutions can be obtained for ‘Mechanics of Materials’ type beam problems in which a rectangular bar is bent by an end load or by a distributed load on one or both faces.

The solution of §5.2.2 must be deemed approximate insofar as the boundary conditions on the ends x = ±a of the rectangular beam are satisfied only in the weak sense of force resultants, through equations (5.45–5.47). In general, if a rectangular beam is loaded by tractions of finite polynomial form, a finite polynomial solution can be obtained which satisfies the boundary conditions in the strong (i.e. pointwise) sense on two edges and in the weak sense on the other two edges.

The error involved in such an approximation corresponds to the solution of a corrective problem in which the beam is loaded by the difference between the actual tractions applied and those implied by the approximation. These tractions will of course be confined to the edges on which the weak boundary conditions were applied and will be self-equilibrated, since the weak conditions imply that the tractions in the approximate solution have the same force resultants as the actual tractions.

For the particular problem of §5.2.2, we note that the stress field of equations (5.78–5.80) satisfies the boundary conditions on the edges y =±b, but that there is a self-equilibrated normal traction

(6.1)

$$\sigma _{xx} = \frac{p}{{10b^3 }}(3b^2 y - 5y^3 )$$

on the ends x = ±a, which must be removed by superposing a corrective solution if we wish to satisfy the boundary conditions of Figure 5.3 in the strong sense.

A

body force

is defined as one which acts directly on the interior particles of the body, rather than on the boundary. Since the interior of the body is not accessible, it follows necessarily that body forces can only be produced by some kind of physical process which acts ‘at a distance’. The commonest examples are forces due to gravity and magnetic or electrostatic attraction. In addition, we can formulate quasi-static elasticity problems for accelerating bodies in terms of body forces, using D’Alembert’s principle (see §7.2.2 below).

We noted in §4.3.2 that the Airy stress function formulation satisfies the equilibrium equations if and only if the body forces are identically zero, but the method can be extended to the case of non-zero body forces provided the latter can be expressed as the gradient of a scalar potential,

V

.

Polar coördinates (

r

, θ) are particularly suited to problems in which the boundaries can be expressed in terms of equations like

r

=

a

, θ = α. This includes the stresses in a circular disk or around a circular hole, the curved beam with circular boundaries and the wedge, all of which will be discussed in this and subsequent chapters.

So far, we have restricted attention to the calculation of stresses and to problems in which the boundary conditions are stated in terms of tractions or force resultants, but there are many problems in which displacements are also of interest. For example, we may wish to find the deflection of the rectangular beams considered in Chapter 5, or calculate the stress concentration factor due to a rigid circular inclusion in an elastic matrix, for which a displacement boundary condition is implied at the bonded interface.

If the stress components are known, the strains can be written down from the stress-strain relations (1.77) and these in turn can be expressed in terms of displacement gradients through (1.51). The problem is therefore reduced to the integration of these gradients to recover the displacement components.

The method is most easily demonstrated by examples, of which we shall give two — one in rectangular and one in polar coördinates.

If we cut the circular annulus of Figure 8.1 along two radial lines, θ = α, β, we generate a curved beam. The analysis of such beams follows that of Chapter 8, except for a few important differences — notably that (i) the ends of the beam constitute two new boundaries on which boundary conditions (usually weak boundary conditions) are to be applied and (ii) it is no longer necessary to enforce continuity of displacements (see §9.3.1), since a suitable principal value of θ can be defined which is both continuous and single-valued.

We first consider the case in which the curved surfaces of the beam are traction-free and only the ends are loaded. As in §5.2.1, we only need to impose boundary conditions on one end — the Airy stress function formulation will ensure that the tractions on the other end have the correct force resultants to guarantee global equilibrium.

In this chapter, we shall consider a class of problems for the semi-infinite wedge defined by the lines α<θ<β, illustrated in Figure 11.1.

We first consider the case in which the tractions on the boundaries vary with

r

n

, in which case equations (8.10, 8.11) suggest that the required stress function will be of the form

(11.1)

$$\phi = r^{n + 2} f(\theta ).$$

The function f(θ) can be found by substituting (11.1) into the biharmonic equation (8.16), giving the ordinary differential equation

(11.2)

$$\left( {\frac{{d^2 }}{{d\theta ^2 }} + (n + 2)^2 } \right)\left( {\frac{{d^2 }}{{d\theta ^2 }} + n^2 } \right)f = 0.$$

In the previous chapter, we considered problems in which the infinite wedge was loaded on its faces or solely by tractions on the infinite boundary. A related problem of considerable practical importance concerns the wedge with traction-free faces, loaded by a concentrated force

F

at the vertex, as shown in Figure 12.1.

An important characteristic of this problem is that there is no inherent length scale. An enlarged photograph of the problem would look the same as the original. The solution must therefore share this characteristic and hence, for example, contours of the stress function φ must have the same geometric shape at all distances from the vertex. Problems of this type — in which the solution can be mapped into itself after a change of length scale — are described as self-similar.

In this chapter, we shall discuss the applications of two solutions which are singular at an

interior

point of a body, and which can be combined to give the stress field due to a concentrated force (the

Kelvin solution

) and a dislocation. Both solutions involve a singularity in stress with exponent −1 and are therefore inadmissable according to the criterion of §11.2.1. However, like the Flamant solution considered in Chapter 12, they can be used as Green’s functions to describe distributions, resulting in convolution integrals in which the singularity is integrated out. The Kelvin solution is also useful for describing the far field (i.e. the field a long way away from the loaded region) due to a force distributed over a small region.

Most materials tend to expand if their temperature rises and, to a first approximation, the expansion is proportional to the temperature change. If the expansion is unrestrained, all dimensions will expand equally — i.e. there will be a uniform dilatation described by

14.1

$$e_{xx} = e_{yy} = e_{zz} = \alpha T$$

14.2

$$e_{xy} = e_{yz} = e_{zx} = 0$$

where α is the coefficient of linear thermal expansion. Notice that no shear strains are induced in unrestrained thermal expansion, so that a body which is heated to a uniformly higher temperature will get larger, but will retain the same shape.

Thermal strains are additive to the elastic strains due to local stresses, so that Hooke’s law is modified to the form

14.3

$$e_{xx} = \frac{{\sigma _{xx} }}{E} - \frac{{v\sigma _{yy} }}{E} - \frac{{v\sigma _{zz} }}{E} + \alpha T$$

14.4

$$e_{xy} = \frac{{\sigma _{xy} (1 + v)}}{E}.$$

In Chapters 3–14, we have considered two-dimensional states of stress involving the in-plane displacements u

x

, u

y

and stress components σ

xx

, σ

xy

, σ

yy

. Another class of two-dimensional stress states that satisfy the elasticity equations exactly is that in which the in-plane displacements u

x

, u

y

are everywhere zero, whilst the out-of-plane displacement u

z

is independent of z — i.e.

(15.1)

$$u_x = u_y = 0; \quad u_x = f(x,y).$$

Substituting these results into the strain-displacement relations (1.51) yields

(15.2)

$$e_{xx} = e_{yy} = e_{zz} = 0$$

and

(15.3)

$$e_{xy} = 0;e_{yz} = \frac{1}{2}\frac{{\partial f}}{{\partial y}};e_{zx} = \frac{1}{2}\frac{{\partial f}}{{\partial x}}.$$

It then follows from Hooke’s law (1.71) that

(15.4)

$$\sigma _{xx} = \sigma _{yy} = \sigma _{zz} = 0$$

and

(15.5)

$$\sigma _{xy} = 0;\sigma _{yz} = \mu \frac{{\partial f}}{{\partial y}};\sigma _{zx} = \mu \frac{{\partial f}}{{\partial x}}.$$

In other words, the only non-zero stress components are the two shear stresses σ

zx

, σ

zy

and these are functions of

x, y

only. Such a stress state is known as

antiplane shear

or

antiplane strain

.

If a bar is loaded by equal and opposite torques

T

on its ends, we anticipate that the relative rigid-body displacement of initially plane sections will consist of rotation, leading to a twist per unit length β. These sections may also deform out of plane, but this deformation must be the same for all values of z. These kinematic considerations lead to the candidate displacement field

16.1

$$u_x {\rm } = {\rm } - \beta zy;{\rm }u_y {\rm } = {\rm }\beta zx{\rm };{\rm }u_z {\rm } = {\rm }\beta f(x,{\rm }y){\rm },$$

where

f

is an unknown function of x, y describing the out-of-plane deformation. Notice that it is convenient to extract the factor β explicitly in u

z

, since whatever the exact nature of

f

, it is clear that the complete deformation field must be linearly proportional to the applied torque and hence to the twist per unit length.

In this chapter, we shall consider the problem in which a prismatic bar occupying the region z>0 is loaded by transverse forces F

x

, F

y

in the negative x- and y-directions respectively on the end z =0, the sides of the bar being unloaded. Equilibrium considerations then show that there will be shear forces

17.1

$$V_x = \int {\int_\Omega {\sigma _{zx} dxdy} } = F_x ;{\rm }V_y = \int {\int_\Omega {\sigma _{zy} dxdy} } = F_y$$

and bending moments

17.2

$$M_x = \int {\int_\Omega {\sigma _{zz} ydxdy} } = zF_x ;{\rm }M_y \equiv - \int {\int_\Omega {\sigma _{zz} xdxdy} } = - zF_x $$

at any given cross-section Ω of the bar. In other words, the bar transmits constant shear forces, but the bending moments increase linearly with distance from the loaded end.

The position of a point in the plane is defined by the two independent coördinates (x, y) which we here combine to form the complex variable ζ and its conjugate

$$\bar \varsigma$$

, defined as

$$\varsigma = x + iy;{\rm }\bar \varsigma = x - iy.$$

We can recover the Cartesian coördinates by the relations

$$x = \mathcal{R}(\varsigma ),y = \mathfrak{f}(\varsigma )$$

, but a more convenient algebraic relationship between the real and complex formulations is obtained by solving equations (18.1) to give

18.2

$$x = \frac{1}{2}(\varsigma + \,\mathop \varsigma \limits^ - );y = - \frac{i}{2}(\varsigma - \,\mathop \varsigma \limits^ - ).$$

At first sight, this seems a little paradoxical, since if we know ζ, we already know its real and imaginary parts x and y and hence

$$\overline \zeta$$

. However, for the purpose of the complex analysis, we regard ζ as the indissoluble combination of

x+?y

, and hence ζ and

$$\overline \zeta$$

act as two independent variables defining position. In this chapter, we shall always make this explicit by writing f(ζ,

$$\overline \zeta$$

) for a function that has fairly general dependence on position in the plane.

From a mathematical perspective, both two-dimensional vectors and complex numbers can be characterized as ordered pairs of real numbers. It is therefore a natural step to represent the two components of a vector function

V

by the real and imaginary parts of a complex function. In other words,

19.1

$$V \equiv iV_x + jV_y $$

is represented by the complex function

19.2

$$V = V_x + iV_y.$$

In the same way, the vector operator

(19.3)

$$\nabla \equiv i\frac{\partial }{{\partial x}} + j\frac{\partial }{{\partial y}} = \frac{\partial }{{\partial x}} + ii\frac{\partial }{{\partial y}} = 2\frac{\partial }{{\partial \zeta }},$$

from equation (18.6).

In Part II, we chose a representation for stress which satisfied the equilibrium equations identically, in which case the compatibility condition leads to a governing equation for the potential function. In three-dimensional elasticity, it is more usual to use the opposite approach — i.e. to define a potentia l function representation for displacement (which therefore identically satisfies the compatibility condition) and allow the equilibrium condition to define the governing equation.

A major reason for this change of method is simplicity. Stress function formulations of three-dimensional problems are generally more cumbersome than their displacement function counterparts, because of the greater complexity of the three-dimensional compatibility conditions. It is also worth noting that displacement formulations have a natural advantage in both two and three-dimensional problems when displacement boundary conditions are involved — particularly those associated with multiply connected bodies (see §2.2.1).

The Galerkin and Papkovich-Neuber solutions have the advantage of presenting a general solution to the problem of elasticity in a suitably compact notation, but they are not always the most convenient starting point for the solution of particular three-dimensional problems. If the problem has a plane of symmetry or particularly simple boundary conditions, it is often possible to develop a special solution of sufficient generality in one or two harmonic functions, which may or may not be components or linear combinations of components of the Papkovich-Neuber solution. For this reason, it is convenient to record detailed expressions for the displacement and stress components arising from the several terms separately and from certain related displacement potentials.

The first catalogue of solutions of this kind was compiled by Boussinesq

1

and is reproduced by Green and Zerna

2

, whose terminology we use here. Boussinesq identified three categories of harmonic potential, one being the strain potential of §20.1, already introduced by Lamé and another comprising a set of three scalar functions equivalent to the three components of the Papkovich-Neuber vector, Ψ. The third category comprises three solutions particularly suited to torsional deformations about the three axes respectively. In view of the completeness of the Papkovich-Neuber solution, it is clear that these latter functions must be derivable from equation (20.17) and we shall show how this can be done in §21.3 below

3

.

As in the two-dimensional case (Chapter 14), three-dimensional problems of thermoelasticity are conveniently treated by finding a particular solution — i.e. a solution which satisfies the field equations without regard to bound ary conditions — and completing the general solution by superposition of an appropriate representation for the general isothermal problem, such as the Papkovich-Neuber solution.

In this section, we shall show that a particular solution can always be obtained in the form of a strain potential — i.e. by writing

(22.1)

$$2\mu u = \nabla \phi$$

The thermoelastic stress-strain relations (14.3, 14.4) can be solved to give

(22.2)

$$\sigma _{xx} = \lambda e + 2\mu e_{xx} - (3\lambda + 2\mu )\alpha T$$

(22.3)

$$\sigma _{xy} = 2\mu e_{xy} $$

etc.

Singular solutions have a special place in the historical development of potential theory in general and of Elasticity in particular. Many of the important early solutions were obtained by appropriate superposition of singular potentials, noting that any form of singularity is permitted provided that the singular point is not a point of the body

1

.

The generalization of this technique to allow continuous distributions of singularities in space (either at the boundary of the body or in a region of space not occupied by it) is still one of the most widely used methods of treating three-dimensional problems.

In this chapter, we shall consider some elementary forms of singular solution and examine the problems that they solve when used in the various displacement function representations of Chapter 21. The solutions will be developed in a rather ad hoc way, starting with those that are mathematically most straightforward and progressing to more complex forms. However, once appropriate forms have been introduced, a systematic way of developing them from first principles will be discussed in the next chapter.

The singular solutions introduced in the last chapter are particular cases of a class of functions known as spherical harmonics. In this chapter, we shall develop these functions and some related harmonic potential functions in a more formal way. In particular, we shall identify:-

(i) Finite polynomial potentials expressible in the alternate forms P

n

(x, y, z), P

n

(r, z) cos(mθ) or R

n

P

n

(cos β) cos(mθ), where Pn represents a polynomial of degree n.

(ii) Potentials that are singular only at the origin.

(iii) Potentials including the factor ln(R+z) that are singular on the negative z-axis (z < 0, r = 0).

(iv) Potentials including the factor ln{(R+z)/(R?z)} that are singular everywhere on the z-axis.

(v) Potentials including the factor ln(

r

) and/or negative powers of

r

that are singular everywhere on the z-axis.

All of these potentials can be obtained in axisymmetric and non-axisymmetric forms. When used in solutions A,B and E of Tables 21.1, 21.2, the bounded potentials (i) provide a complete set of functions for the sphere, cylinder or cone with prescribed surface tractions or displacements on the curved surfaces. These problems are three-dimensional counterparts of those considered in Chapters 5, 8 and 11. Problems for the hollow cylinder and cone can be solved by supplementing the bounded potentials with potentials (v) and (iv) respectively.

Axisymmetric

problems for the hollow sphere or the infinite body with a spherical hole can be solved using potentials (i,ii), but these potentials do not provide a complete solution to the non-axisymmetric problem1. Functions (ii) and (iii) are useful for problems of the half space, including crack and contact problems.

The bounded harmonic potentials of equations (24.27) in combination with solutions A, B and E provide a complete solution to the problem of a solid circular cylinder loaded by axisymmetric polynomial tractions on its curved surfaces. The corresponding problem for the hollow cylinder can be solved by including also the singular potentials of equation (24.42). The method can be extended to non-axisymmetric problems using the results of §24.7. If strong boundary conditions are imposed on the curved surfaces and weak conditions on the ends, the solutions are most appropriate to problems of ‘long’ cylinders in which L >> a, where L is the length of the cylinder and a is its outer radius. At the other extreme, where L << a, the same harmonic functions can be used to obtain three-dimensional solutions for in-plane loading and bending of circular plates, by imposing strong boundary conditions on the plane surfaces and weak conditions on the curved surfaces. As in Chapter 5, some indication of the order of polynomial required can be obtained from elementary Mechanics of Materials arguments.

The spherical harmonics of Chapter 24 can be used in combination with Tables 21.2, 22.2 to treat problems involving bodies whose boundaries are surfaces of the spherical coördinate system, for example the spherical surface R = a or the conical surface β=β

0

, where a, β

0

are constants. Examples of interest include the perturbation of an otherwise uniform stress field by a spherical hole or inclusion, the stresses in a solid sphere due to rotation about an axis and problems for a conical beam or shaft.

The general solution for a solid sphere with prescribed surface tractions can be obtained using the spherical harmonics of equation (24.24) in solutions A,B and E. The addition of the singular harmonics (24.25) permits a general solution to the axisymmetric problem of the hollow sphere, but the corresponding non-axisymmetric solution cannot be obtained from equations (24.24, 24.25). To understand this, we recall from §20.4 that the elimination of one of the components of the Papkovich-Neuber vector function is only generally possible when all straight lines drawn in a given direction cut the boundary of the body at only two points. This is clearly not the case for a body containing a spherical hole, since lines in any direction can always be chosen that cut the surface of the hole in two points and the external boundaries of the body at two additional points.

We have already remarked in §25.1 that the use of axisymmetric harmonic potential functions in Solution E provides a general solution of the problem of an axisymmetric body loaded in torsion. In this case, the only non-zero stress and displacement components are

27.1

$$\mu u_\theta = - \frac{{\partial \psi }}{{\partial r}};\sigma _{\theta r} = \frac{1}{r}\frac{{\partial \psi }}{{\partial r}} - \frac{{\partial ^2 \psi }}{{\partial r^2 }};\sigma _{\theta z} = - \frac{{\partial ^2 \psi }}{{\partial r\partial z}}.$$

A more convenient formulation of this problem can be obtained by considering the relationship between the stress components and the torque transmitted through the body. We first note that the two non-zero stress components (27.1) both act on the cross-sectional θ-plane, on which they constitute a vector field σ

θ

of magnitude

27.2

$$|\sigma _\theta | = \sqrt {\sigma _{\theta r}^2 + \sigma _{\theta z}^2.} $$

This vector field is illustrated in Figure 27.1. It follows that the stress component σ

θn

normal to any of the ‘flow lines’ in this figure is zero and hence the complementary shear stress σ

nθ

. Thus, the flow lines define traction-free surfaces in

r

, θ,

z

space.

We have seen in Chapter 25 that three-dimensional solutions can be obtained to the problem of the solid or hollow cylindrical bar loaded on its curved surfaces, using the Papkovich-Neuber solution with spherical harmonics and related potentials. Here we shall show that similar solutions can be obtained for bars of more general cross-section, using the complex-variable form of the Papkovich-Neuber solution from §21.6.

We use a coördinate system in which the axis of the bar is aligned with the z-direction, one end being the plane z = 0. The constant cross-section of the bar then comprises a domain Ω in the xy-plane, which may be the interior of a closed curve Г, or that part of the region interior to a closed curve Г

0

that is also

exterior

to one or more closed curves Г

1

, Г

2

, etc. The following derivations and examples will be restricted to the former, simply connected case, but it will be clear from the methods used that the additional complications asssociated with multiply connected cross-sections arise only in the solution of two-dimensional boundary-value problems, for which classical methods exist. As in Chapter 25, we shall apply weak boundary conditions at the ends of the bar, which implies that the solutions are appropriate only for relatively long bars in regions that are not too near the ends.

As we noted in §21.5.1, Green and Zerna’s Solution F is ideally suited to the solution of frictionless contact problems for the half-space, since it identically satisfies the condition that the shear tractions be zero at the surface z=0. In fact the

surface

tractions for this solution take the form

29.1

$$\sigma _{zz} = - \frac{{\partial ^2 \varphi }}{{\partial z^2 }};\sigma _{zx} = \sigma _{zy} = 0;z = 0,$$

whilst the surface displacements are

29.2

$$u_x = \frac{{(1 - 2v)}}{{2\mu }}\frac{{\partial \varphi }}{{\partial x}};u_y = \frac{{(1 - 2v)}}{{2\mu }}\frac{{\partial \varphi }}{{\partial y}};u_z = \frac{{(1 - v)}}{\mu }\frac{{\partial \varphi }}{{\partial z}};z = 0,$$

from Table 21.3.

The simplest frictionless contact problem of the class defined by equations (29.3–29.6) is that in which the contact area

A

is the circle 0 < r < a and the indenter is axisymmetric, in which case we have to determine a harmonic function φ(r, z) to satisfy the mixed boundary conditions

30.1

$$\frac{{\partial \varphi }}{{\partial z}} = - \frac{\mu }{{(1 - v)}}u_0 (r);0 \le r < a,z = 0$$

30.2

$$\frac{{\partial ^2 \varphi }}{{\partial z^2 }} = 0;r > a,z = 0.$$

As in the two-dimensional case, we shall find considerable similarities in the formulation and solution of contact and crack problems. In particular, we shall find that problems for the plane crack can be reduced to boundary-value problems which in the case of axisymmetry can be solved using the method of Green and Collins developed in §30.2.

The problem of a crack at the interface between dissimilar elastic media is of considerable contemporary importance in Elasticity, because of its relevance to the problem of debonding of composite materials and structures. Figure 32.1 shows the case where such a crack occurs at the plane interface between two elastic half-spaces. We shall use the suffices 1,2 to distinguish the stress functions and mechanical properties for the lower and upper half-spaces, z>0, z<0, respectively.

The difference in material properties destroys the symmetry that we exploited in the previous chapter and we therefore anticipate shear stresses as well as normal stresses at the interface, even when the far field loading is a state of uniform tension. However, we shall show that we can still use the same methods to reduce the problem to a mixed boundary-value problem for the half-space.

Energy or variational methods have an important place in Solid Mechanics both as an alternative to the more direct method of solving the governing partial differential equations and as a means of developing convergent approximations to analytically intractable problems. They are particularly useful in situations where only a restricted set of results is required — for example, if we wish to determine the resultant force on a cross-section or the displacement of a particular point, but are not interested in the full stress and displacement fields. Indeed, such results can often be obtained in closed form for problems in which a solution for the complete fields would be intractable.

From an engineering perspective, it is natural to think of these methods as a consequence of the principle of conservation of energy or the first law of thermodynamics. However, conservation of energy is in some sense guaranteed by the use of Hooke’s law and the equilibrium equations. Once these physical premises are accepted, the energy theorems we shall present here are purely mathematical consequences. Indeed, the finite element method, which is one of the more important developments of this kind, can be developed simply by applying arguments from approximation theory (such as a least-squares fit) to the governing equations introduced in previous chapters. For this reason, these techniques are now more often referred to as

Variational Methods

, meaning that instead of seeking to solve the governing partial differential equations directly, we seek to define a scalar function of the physical parameters which is stationary (generally maximum or minimum) in respect to infinitesimal variations about the solution.

In the previous chapter, we exploited the idea that the strain energy

U

stored in an elastic structure must be equal to the work done by the external loads, if these are applied sufficiently slowly for inertia effects to be negligible. Equation (33.1) is based on the premise that the loads are all increased at the same time and in proportion, but the final state of stress cannot depend on the exact history of loading and hence the work done

W

must also be historyindependent. This conclusion enables us to establish an important result for elastic systems known as the reciprocal theorem.