OUTLINE

In this chapter we consider the force equilibrium in a continuous body under the assumption that the underlying deformation is adequately described by the small strain hypothesis. The principle of virtual power occupies a central place in this treatment, since it offers a rational basis for developing equations that apply to a continuum in a state of equilibrium. Furthermore, the concept of stress arises naturally from this analysis as dual to small strain for a solid continuum. Stress equilibrium and traction boundary conditions also appear in the most convenient invariant form. For illustration, virtual power expressions are given for systems obeying different kinematics, such as inviscid fluid flow or beams in simple bending, and the resulting stresslike quantities and their equilibrium equations are readily derived. Cauchy's stress principle and the Cauchy–Poisson theorem are also given.

Once it is established that equlibrium stress states in continuum solids in the absence of body forces are given by divergence-free tensors, the representation of such tensor fields is addressed. Beltrami potential representation of divergence-free tensors is considered, and Donati's theorem is introduced to illustrate a certain duality that exists between stress equilibrium and strain compatibility conditions.

FORCES AND MOMENTA

A body may be subject to a system of exterior forces of the following types:

• Body force f is described by a vector field distributed over the entire volume of body Ω. Denoting body force by the vector f (x, t) represents the fact that the force f (x, t)dv acts on an infinitesimal volume dv at point x at time t. An example of the body force is the force of gravity.

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OUTLINE

In the preceding chapters we have discussed, on the one hand, the kinematics of deformation of continuous media, where the principal unknowns are the displacement vector field u and the strain tensor field ∊. On the other hand, we have introduced the dynamics of deformation, representing the balance of forces in terms of the stress tensor field σ as the principal unknown.

Until now we have made no attempt to relate the strain and stress fields to each other. Before we begin the discussion of the detailed nature of this relationship, we can make the following general remarks:

• Description so far is clearly incomplete, because we have at our disposal only 6 kinematic relations and 3 force balance equations for the determination of the 3 + 6 + 6 unknown functions, that is, the components of displacement u, strain ∊, and stress σ.

• We are so far unable to distinguish between different materials which might assume different deformed configurations under the same external loading. Clearly structures produced out of wood, steel, or ceramic may deform in different ways, so that the complete solutions are different.

The purpose of this chapter is to establish a class of relationships between strains and stresses known as the linear elastic constitutive law and to discuss a series of basic properties of these relationships.

OUTLINE

In this chapter we consider the application of the methods of displacement potential and demonstrate the implementation of these methods in Mathematica.

The fundamental expression for the Papkovich–Neuber representation of the elastic displacement fields is introduced first. Papkovich representations of the simple strain states are next considered, followed by the fundamental singular solutions for the centres of dilatation and rotation and the Kelvin solution for the concentrated force in an infinite solid. From the Kelvin solution the momentless force doublet and the force doublet with moment are derived by differentiation. The combination of three mutually perpendicular momentless force doublets is considered and is shown to be equivalent to the centre of dilatation. This example is used to demonstrate the nonuniqueness of the Papkovich description of elastic solutions. The combination of the centre of dilatation with a force doublet is also shown to correspond to a point eigenstrain solution. The point shear eigenstrain is compared with the combination of two force doublets.

Boussinesq and Cerruti solutions for the concentrated force applied at the boundary of a semi-infinite elastic solid are presented next. Solutions for concentrated forces applied at the vertex of an infinite cone are derived using the same principles from superpositions of known solutions for concentrated forces and lines of centres of rotation and dilatation.

MOTIVATION

The idea for this book arose when the authors discovered, working together on a particular problem in elastic contact mechanics, that they were making extensive and repeated use of Mathematica™ as a powerful, convenient, and versatile tool. Critically, the usefulness of this tool was not limited to its ability to compute and display complex two-and three-dimensional fields, but rather it helped in understanding the relationships between different vector and tensor quantities and the way these quantities transformed with changes of coordinate systems, orientation of surfaces, and representation.

We could still remember our own experiences of learning about classical elasticity and tensor analysis, in which grasping the complex nature of the objects being manipulated was only part of the challenge, the other part being the ability to carry out rather long, laborious, and therefore error-prone algebraic manipulations.

It was then natural to ask the question: Would it be possible to develop a set of algebraic instruments, within Mathematica, that would carry out these laborious manipulations in a way that was transparent, invariant of the coordinate system, and error-free? We started the project by reviewing the existing Mathematica packages, in particular the VectorAnalysis package, to assess what tools had been already developed by others before us, and what additions and modifications would be required to enable the manipulation of second-rank tensor field quantities, which are of central importance in classical elasticity.

OUTLINE

This chapter is devoted to the solution of elastic problems using the stress function approach. The Beltrami potential has already been introduced previously as a convenient form of representation for self-equilibrated stress fields. However, the main emphasis in the chapter is placed on the analysis of the Airy stress function formulation, even though it represents only a particular case of the Beltrami representation. The reason for this is the particular importance of this approach in the context of plane elasticity.

The Airy stress function approach is introduced taking particular care to ensure that conditions of strain compatibility are properly satisfied. The approximate nature of the plane stress formulation is elucidated.

The properties of Airy stress functions in cylindrical polar coordinates are then addressed. Particular care is taken to analyse some important fundamental solutions that serve as nuclei of strain within the elasticity theory, namely the solutions for a disclination, dislocations and dislocation dipoles, and concentrated forces.

Williams eigenfunction analysis of the stress state in an elastic wedge under homogeneous loading is presented next, and the elastic stress fields found around the tip of a sharp crack subjected either to opening or shear mode loading. Finally, two further important problems are treated, namely the Kirsch problem of remote loading of a circular hole in an infinite plate, and the Inglis problem of remote loading of an elliptical hole in an infinite plate.

OUTLINE

This chapter is devoted to the introduction of the fundamental concepts used to describe continuum deformation. This is probably most naturally done using examples from fluid dynamics, by considering the description of particle motion either with reference to the initial particle positions, or with reference to the current (actual) configuration. The relationship between the two approaches is illustrated using examples, and further illustrations are provided in the exercises at the end of the chapter. Some methods of flow visualisation (streamlines and streaklines) are described and are illustrated using simple examples. The concepts are then clarified further using the example of inviscid potential flow.

Placing the focus on the description of deformation, the fundamental concept of deformation gradient is introduced. The polar decomposition theorem is used to separate deformation into rotation and stretch using appropriate tensor forms, with particular attention being devoted to the analysis of the stretch tensor and the principal stretches, using pure shear as an illustrative example. Trigonometric representation of stretch and rotation is discussed briefly.

Discussion is further specialised to the consideration of small strains. Analysis of integrability of strain fields then leads to the identification of the invariant form of compatibility conditions. This subject is important for many applications within elastic theory and is therefore dwelt on in some detail.

So far, we have focused attention on the dynamic behavior of individual slender structural members. There are, of course, many practical situations in which a number of members are connected to form a truss or frame. Often, such systems are analyzed as moment frames designed to resist loads in bending. However, there are occasions in which significant axial loading occurs, and thus we augment the standard methods of structural dynamics to account for this situation. We start with revisiting the partial differential equation description of the dynamics of a slender beam with axial loading. However, we now allow for varying degrees of elastic restraint at the boundaries, because, in a typical framework, the stiffness of a joint depends (in a nonsimple way) on the effects of the members contributing to that joint. In Section 8.2, we developed some general expressions for beams with elastically restrained ends. Initially, we follow the standard approach in which a characteristic equation is developed from the governing partial differential equation. However, this is rarely a practical approach for a frame, and hence the chapter then proceeds to introduce the dynamic stiffness method. This is a systematic, FE technique that provides a powerful, matrix-based approach to solving problems in structural dynamics. We focus on frames consisting of prismatic beam members, and, because of the way in which most frames are designed to resist lateral as well as axial loads, any postbuckling, as such, will be encountered as the growth of large deflections during loading.

Introduction

In the previous two chapters we focused on issues of modeling, equilibrium, and stability largely associated with SDOF systems. As such, the dynamic and stability behavior of a mass was largely characterized, for example, by a single frequency. However, there are many examples of systems with more than a SDOF, in which dynamics and stability issues are more involved. Certain characteristics of a physical system can be lumped at discrete locations, and this leads to sets of coupled ordinary differential equations. Linear algebra plays a key role in their analysis. Of course, most real structures are continuous and have an infinite number of DOFs. Governing equations of motion are typically partial differential equations (depending on both space and time), with boundary, as well as initial, conditions needing to be satisfied for a complete solution. Unlike typical (static) bending problems, which lead to inhomogeneous differential equations, the systems of primary focus in this book concern nontrivial homogeneous differential equations and often, in the analysis process, we will need to formulate and solve an eigenvalue problem: algebraic, for finite DOFs, and differential, for infinite DOFs. This will also typically involve the use of various approximation techniques, discretizations, and computational methods. This chapter serves the purpose of expanding the theoretical basis of dynamics and stability to this wider class of problems.

Multiple-Degree-of-Freedom Systems

Returning to the Lagrangian description (Section 2.5), we again focus attention on conservative systems and develop Lagrange's equation in matrix form.

In the engineered world (and in a good deal of the natural world), stable equilibrium, or some kind of stationary or steady-state behavior, is the order of the day. Systems are designed to operate in a predictable fashion to fulfill their intended functions despite disturbances and changing conditions. Control systems have been spectacularly successful in maintaining a desirable (stable) response given inevitable uncertainty in modeling system physics. However, there are plenty of examples of systems becoming unstable - and often the consequences of instability are severe. This book looks at the interplay between vibrations and stability in elastic structures.

A brief view of an ecological system provides an effective analogy. The competition between certain species can be viewed as a coupled dynamic system in a slowly changing environment. External influences are provided by various factors including the climate, disease, and human influence. The delicate interaction is played out as conditions evolve and populations respond accordingly - usually in a correspondingly slow way also. However, an instability may occur leading to extinction on a relatively short time scale, perhaps when a disease (or massive meteorite) wipes out an entire population. This situation is not that dissimilar to the fluctuations of the stock markets (in which prediction of changes is of concern to individuals and governments).

Introduction

This chapter develops the theoretical basis for the derivation of governing equations of motion. It starts with Newton's second law and then uses Hamilton's principle to derive Lagrange's equations. A number of conservation laws are introduced. The theory is developed initially for a single particle and extended to systems of particles where appropriate. The emphasis is placed on building the theory relevant to the types of physical system of interest in structural dynamics. Other than the usual limitations regarding relativistic and quantum effects, we also restrict ourselves to translational (rather than rotational) systems, which is largely a matter of coordinates. The majority of problems in this book involve systems in which the forces developed during elastic deformation play a crucial role. Certain standard problems in classical mechanics, for example the central force motion leading to the two-body problem or particle scattering, are not relevant here and are not considered. We shall see the important role played by energy methods in studying the dynamics of structures. Classical mechanics has a long history and in-depth treatment of the subject can be found in Goldstein, Whittaker, and Synge and Griffith and, of course, going back to the early developments of Newton, Euler, and Lagrange.

Newton's Second Law

The natural starting point in any text covering an aspect of classical mechanics are Newton's laws of motion.