Many common fluids, including air, water, honey, and liquid metals, are accurately represented by the Newtonian constitutive equation. However, there are also many materials whose behavior is distinctly non-Newtonian: molten polymers, molten glass, semisolid metals, grease, many types of paint, and foods such as mayonnaise, peanut butter, and melted cheese. Non-Newtonian fluids obey the same mass and momentum balance equations as Newtonian fluids, but they have different constitutive equations for stress. In this chapter we show how to develop constitutive equations for non-Newtonian fluids, and we demonstrate their use in the solving of flow problems. We also develop a simplified set of governing equations for viscous flow in narrow gaps, the generalized Hele–Shaw approximation.

The study of the deformation and flow of materials, particularly of non-Newtonian fluids, is the subject called rheology. Rheologists create experimental techniques to measure viscosity and other flow properties, they develop constitutive equations that describe non-Newtonian material behavior, and they relate that material behavior to microscopic structure. A full treatment of rheology is beyond the scope of this book, and readers may wish to consult some of the excellent texts on the subject (Tanner, 1985; Bird et al., 1987; Barnes et al., 1989; Dealy and Wissbrun, 1990; Macosko, 1994).

NON-NEWTONIAN BEHAVIOR

PURELY VISCOUS FLUIDS

There is only one type of Newtonian fluid behavior, and one Newtonian fluid differs from another only by its value of viscosity.

In this chapter we turn our attention to problems involving fluid flow. Such problems are solved by using the mass and momentum balance equations derived in Chapter 2. We start by analyzing Newtonian fluids in simple geometries, such as flow between parallel plates, or flow in a tube. We use scaling to determine whether inertial or viscous effects dominate the problem, and we concentrate initially on viscous-dominated flows. These simple-geometry viscous results are useful building blocks for process modeling, especially in polymer processing where the fluid viscosity is very large. Several examples illustrate how this is done. We show how to formulate problems with free or moving boundaries, a situation that is common in materials processing, and we apply these ideas to some simple models of mold filling. Finally, we consider cases in which the inertial terms are dominant, and we provide basic tools for treating these problems.

NEWTONIAN FLOW IN A THIN CHANNEL

Consider flow in a channel whose thickness is much smaller than its width and length. A great many polymer processing operations involve this type of geometry, for reasons that can be understood by using the results from Chapter 4. Most plastic parts are intended to be inexpensive, which requires that they be produced in a short time, and cooling the polymer melt is the slowest part of the process because polymers are relatively poor conductors of heat.

INTRODUCTION

The balance equations derived in the preceding chapter are the basis for all of the modeling work to come. When combined with constitutive equations and initial or boundary conditions, they provide the governing equations for most models. However, these governing equations are often difficult to solve in their general form, because they are multidimensional and nonlinear.

In this chapter we discuss a method for systematically simplifying the governing equations by determining which terms can be safely neglected in a given problem. We use the values of the physical parameters in the problem to estimate the relative importance of each term in the governing equations, using a particular procedure we call scaling. We then simplify the equations by neglecting terms that are “small” in comparison to terms that are “large.” This produces governing equations that are often simpler than the original forms, but they still reflect all the important phenomena of the problem.

Scaling analysis produces other important results as well. Through scaling we learn the characteristic values of all of the problem variables. We also derive dimensionless parameters that have physical meaning for the particular problem. We can even determine whether the solution is likely to contain a boundary layer, and how large that layer will be. This makes scaling analysis a richer and more useful tool than traditional dimensional analysis. That is why we regard scaling as an essential step in model development.

This appendix summarizes some important mathematical background. We first cover vector and tensor algebra and the relationship between different styles of notation. Other topics include the divergence theorem, the curvature of curves and surfaces, and the Gaussian error function. The material is intended primarily for reference purposes, so no derivations or proofs are provided. Readers desiring more background can consult the bibliography at the end of this appendix.

In this appendix the symbols u, v, and w represent arbitrary vectors, and σ and τ represent arbitrary tensors.

SCALARS, VECTORS, AND TENSORS: DEFINITIONS AND NOTATION

SCALARS AND VECTORS

A scalar is a quantity that has magnitude but no direction. Physical quantities that are scalars include temperature, pressure, and density. Scalar quantities are denoted by lightface italic symbols: T, p, ρ.

A vector is a quantity having both magnitude and direction. We frequently visualize a vector as an arrow in space: the direction of the arrow gives the direction of the vector, and the length of the arrow gives the vector's magnitude. Vector quantities include velocity, surface traction, and heat flux.

This text uses several different notations for vectors. In one of these, called Gibbs notation, the vector is denoted by a boldface letter: v, t, q. The bold letter represents the entire vector, and this notation is not tied to any particular choice of coordinate system. A convenient notation for the blackboard or handwritten work is to underline letters that represent vectors.

Materials processing models often deal with the transport of mass, momentum, and energy. In this chapter we derive governing equations for the conservation of these quantities. We focus on differential forms, which are used in continuum models. We also include some integral forms, which help in the physical interpretation of the equations. The forms we derive are quite general, and you may wonder if the complexity this adds is worthwhile. Remember that we are seeking equations that have a very wide range of validity, so that, when we simplify them for a particular model, we can be sure that no term has been left out.

Although the focus of this chapter is on balance equations, we also introduce a few classical constitutive equations: the Newtonian fluid and Fourier's law of heat conduction. However, we also retain forms of the balance equations that can be used with other constitutive equations.

Insofar as possible, we avoid reference to particular coordinate systems, by using general vector and tensor notation. Readers who are unfamiliar with this style may want to review Appendix A. All of the governing equations derived in this chapter are written out in terms of their components for rectangular, cylindrical, and spherical coordinates in Appendix B. These component forms are the actual starting point for most models.

PRELIMINARIES

MATERIAL AND SPATIAL DESCRIPTIONS

In this chapter we want to develop equations for quantities that associate a value with each point in the material, quantities that include velocity, temperature, pressure, and stress. We also want to deal with materials that are moving and deforming.

WHAT IS A MODEL?

In recent years, modeling has been embraced by the materials processing community as a tool for understanding and improving manufacturing processes. Models are often implemented in computer programs, but there are important differences between a model and the computer code that implements it. A model is a set of equations used to represent a physical process. Finite element or finite difference methods, and the computer programs that implement them, are techniques to solve the equations of the model, but they are not the model itself. Our main emphasis will be on creating models – on reducing a physical process to a set of equations – especially models whose solution accurately describes the behavior of the process. Occasionally we also will explore numerical solution methods, often to point out where unenlightened use can lead you astray.

Whenever we create a model, we make assumptions about what phenomena are important to the behavior of the physical process. This is both good and bad. Assumptions help define the mathematical model and make it amenable to analysis. However, incorrect assumptions and erroneous information become part of the model and may well distort the results. Sometimes assumptions greatly simplify the model and permit an easy solution, but they may also cause important physical phenomena to be misrepresented or overlooked. Limiting the number of assumptions helps to avoid this problem but may make the model overly complex. Then the solution becomes difficult, and important information may be obscured.

In previous chapters we considered homogeneous materials, but there are many significant problems in materials processing in which more than one chemical species is involved. One technologically important case is the solidification of metal alloys. When more than one species is present, we must address two additional considerations: chemical equilibrium between the various phases, and transport of the species by diffusion. This chapter begins by setting up the governing equations for diffusion and shows that species transport is mathematically similar to energy transport. The use of the diffusion equations is demonstrated in several example applications. We then look at solute partitioning in alloy solidification, consider the stability of the solid–liquid interface in solidification, and show how these phenomena control microstructure development.

GOVERNING EQUATIONS FOR DIFFUSION

When two soluble materials are put into contact, they mix through the action of molecular diffusion. If one waits long enough, a final state is achieved in which the local concentration of each species is uniform. As an example, think about pouring milk into hot coffee. Eventually, the coffee reaches a uniform light color, which we associate with complete mixing. We can make “eventually” arrive sooner by stirring the coffee, which transports material by advection, but diffusion operates whether we stir or not.

Our discussion of diffusion will be confined to systems in which the species are completely miscible for any composition that is present in the problem.

After some years of teaching separate courses on metal solidification and polymer processing, we realized that the two subjects shared a substantial base of common material. All the models started with the same basic equations and were built by using the same general procedure. We began to teach a single course on materials processing, and we found that our unified treatment gave students a better overall perspective on modeling. We also discovered that we needed a new book, as existing texts were almost all devoted exclusively to polymers, or to metals, or to ceramics. In this book, we treat metal and polymer processing problems together, building around the transport equations as a unifying theme.

We were also dissatisfied with ad hoc model development, in which terms were arbitrarily dropped from the governing equations, or simplifications were made without a clear explanation. Simplifying the general governing equations is a critical step in modeling, but it is a skill, not an art. In this text we introduce scaling analysis as a systematic way to reduce the governing equations for any particular problem. Scaling provides a way for both novices and experts to simplify a model, while ensuring that all of the important phenomena are included.

fter deriving the governing equations in their general form and introducing scaling analysis, we examine physical phenomena such as heat conduction and fluid flow. We work out many problems that include only a few of these phenomena – problems that can be solved analytically.

In this chapter we discuss applications of the perturbation approach to stability analysis of elastic bodies subjected to large deformations. Various ideas commonly used in the perturbation approach are explained by using simple examples. Two types of bifurcations are distinguished: bifurcations at a non-zero critical mode number and bifurcations at a zero critical mode number. For each type we first explain with the aid of a model problem how stability analysis can be carried out and then explain how the analysis could be extended to problems in Finite Elasticity. Although the present analysis focuses on the perturbation approach, the dynamical systems approach is also discussed briefly and references are made to the literature where more details can be found. In the final section, we carry out a detailed analysis for the necking instability of an incompressible elastic plate under stretching.

Introduction

This chapter is concerned with nonlinear stability analysis of elastic bodies subjected to large elastic deformations. A typical problem we have in mind is the stability of a cylindrical rubber tube that is compressed either by an external pressure or by forces at the two flat ends. In general terms, we consider an elastic body which has an undeformed configuration Br in a three-dimensional Euclidean point space. This elastic body is then subjected to some external forces. It is now customary to refer to such an elastic body as pre-stressed in the Finite Elasticity literature (the pre-stress considered in the present context is not therefore that induced in a manufacturing process).

This chapter is an overview of a theory of a class of nonlinear elastic materials for which the deformation is subject to an internal material constraint described in experiments by James F. Bell on the finite plastic deformation of a variety of annealed metals. Research by Bell and his associates published since about 1979 is reviewed, and Bell's empirically deduced rules and laboratory data are compared with analytical results obtained within the context of nonlinear elasticity theory. First, Bell's empirical characterization of the constrained response of polycrystalline annealed metals in finite plastic strain is sketched. A few kinematical consequences of Bell's constraint, an outline of the constitutive theory developed to characterize the isoteopic, nonlinearly elastic response of Bell materials, and theoretical results that lead to Bell's empirical parabolic laws within the structure of isotropic, elastic and hyperelastic Bell constrained materials are presented. The study concludes with discussion of Bell's empirically based incremental theory of plasticity.

Introduction

It is common in technical writing to begin with a sketch of related research assembled to set the stage for the work ahead. But I'm not going to follow the usual path. There is more to this account than just its technical side - teachers and students, colleagues and associates, family and friends, places and events, life and death - the ingredients of the human side of the story. A reader who feels no interest in this sort of personal, anecdotal retrospection, however, will find immediate relief and surely suffer no loss in skipping ahead to Section 2.3 where Bell's important experiments and his internal material constraint are introduced. We'll return to this shortly.

A deformation or a motion is said to be a universal solution if it satisfies the balance equations with zero body force for all materials in a given class, and is supported in equilibrium by suitable surface tractions alone. On the other hand for a given deformation or motion, a local universal relation is an equation relating the stress components and the position vector which holds at any point of the body and which is the same for any material in a given class. Universal results of various kinds are fundamental aspects of the theory of finite elasticity and they are very useful in directing and warning experimentalists in their exploration of the constitutive properties of real materials. The aim of this chapter is to review universal solutions and local universal relations for isotropic nonlinearly elastic materials.

Introduction

One of the main problems encountered in the applications of the mechanics of continua is the complete and accurate determination of the constitutive relations necessary for the mathematical description of the behavior of real materials.

After the Second World War the enormous work on the foundations of the mechanics of continua, which began with the 1947 paper of Rivlin on the torsion of a rubber cylinder published in the Journal of Applied Physics, has allowed important insights on the above mentioned problem (Truesdell and Noll, 1965).

In this chapter we provide an introductory exposition of singularity theory and its application to nonlinear bifurcation analysis in elasticity. Basic concepts and methods are discussed with simple mathematics. Several examples of bifurcation analysis in nonlinear elasticity are presented in order to demonstrate the solution procedures.

Introduction

Singularity theory is a useful mathematical tool for studying bifurcation solutions. By reducing a singular function to a simple normal form, the properties of multiple solutions of a bifurcation equation can be determined from a finite number of derivatives of the singular function. Some basic ideas of singularity theory were first conjectured by R. Thorn, and were then formally developed and rigorously justified by J. Mather (1968, 1969a, b). The subject was extended further by V. I. Arnold (1976, 1981). In two volumes of monographs, M. Golubitsky and D. G. Schaeffer (1985), and M. Golubitsky, I. Stewart and D. G. Schaeffer (1988) systematized the development of singularity theory, and combined it with group theory in treating bifurcation problems with symmetry. Their work establishes singularity theory as a comprehensive mathematical theory for nonlinear bifurcation analysis.

The purpose of this chapter is to give a brief exposition of singularity theory for researchers in elasticity. The emphasis is on providing a working knowledge of the theory to the reader with minimal mathematical prerequisites. It can also serve as a handy reference source of basic techniques and useful formulae in bifurcation analysis.