Continuum Mechanics as an Effective Theory

Materials exhibit structural features at a number of different scales, all of which can alter their macroscopic response to external stimuli such as mechanical loads or the application of electromagnetic fields. One of the fundamental difficulties faced in the modeling of materials is how to extract those features of the problem that are really necessary, while at the same time attaining some tolerable level of simplification. Traditionally, one of the favored routes for effecting the reduction of problems to manageable proportions has been the use of continuum theories. Such theories smear out features at ‘small’ scales such as the discrete phenomena that are tied to the existence of atoms and replace the kinematic description of materials in terms of atomic positions with field variables. Adoption of these strategies leads to theories that can describe material behavior ranging from deformation of polycrystals to mass transport to the behavior of domain walls in magnetic materials.

The continuum mechanics of solids and fluids serves as the prototypical example of the strategy of turning a blind eye to some subset of the full set of microscopic degrees of freedom. From a continuum perspective, the deformation of the material is captured kinematically through the existence of displacement or velocity fields, while the forces exerted on one part of the continuum by the rest are described via a stress tensor field. For many problems of interest to the mechanical behavior of materials, it suffices to build a description purely in terms of deformation fields and their attendant forces. A review of the key elements of such theories is the subject of this chapter.

Point Defects and Material Response

This chapter is the first in a series that will make the case that many of the important features of real materials are dictated in large measure by the presence of defects. Whether one's interest is the electronic and optical behavior of semiconductors or the creep resistance of alloys at high temperatures, it is largely the nature of the defects that populate the material that will determine both its subsequent temporal evolution and response to external stimuli of all sorts (e.g. stresses, electric fields, etc.). For the most part, we will not undertake an analysis of the widespread electronic implications of such defects. Rather, our primary charter will be to investigate the ways in which point, line and wall defects impact the thermomechanical properties of materials.

Though we will later make the case (see chap. 11) that many properties of materials involve a complex interplay of point, line and wall defects, our initial foray into considering such defects will be carried out using dimensionality as the central organizing principle. In many instances, it is convenient to organize defects on the basis of their underlying dimensionality, with point defects characterized mathematically through their vanishing spatial extent, line defects such as dislocations sharing some of the mathematical features of space curves, and surfaces, grain boundaries and other wall defects being thought of as mathematical surfaces. Indeed, one of the interesting physics questions we will ask about such defects is the extent to which such dimensional classifications jibe with the properties of the defects themselves.

We have made no secret of the pivotal role played by various geometric structures within materials in conspiring to yield observed properties. However, one of the abiding themes of our work thus far has been the observation that this notion of ‘structure’ is complex and scale-dependent. The properties of materials depend upon geometry not only at the microscopic scale, but at larger scales as well. Recall that the hierarchy of geometric structures that populate materials begins with the underlying atomic arrangements. Questions of structure at this level are answered through an appeal to phase diagrams in the way considered in chap. 6. These atomic-level geometries are often disturbed by the various defects that have served as the centerpiece of the previous three chapters, and which introduce a next larger set of length scales in the hierarchy. However, much of the effort in effecting the structure–properties linkage in materials science is carried out at a larger scale yet, namely, that associated with the various microstructures within a material.

The current chapter has as its primary aim a discussion of some of the many types of microstructures that populate materials and how such microstructures and their temporal evolution can be captured from a theoretical perspective. This discussion will also serve as the basis of our later efforts to examine the connection between structure and properties in materials. We begin with an attempt at microstructural taxonomy with the aim being to give an idea of the various types of microstructures that arise in materials.

The Role of the Total Energy in Modeling Materials

A recurring theme in the study of materials is the connection between structure and properties. Whether our description of structure is made at the level of the crystal lattice or the defect arrangements that populate the material or even at the level of continuum deformation fields, a crucial prerequisite which precedes the connection of structure and properties is the ability to describe the total energy of the system of interest. In each case, we seeka function (or functional) such that given a description of the system's geometry, the energy of that system can be obtained on the basis of the kinematic measures that have been used to characterize that geometry. As we have discussed earlier (see chap. 2), the geometry of deformation may be captured in a number of different ways. Microscopic theories are based upon explicitly accounting for each and every atomic coordinate. Alternatively, a continuum description envisions a sort of kinematic slavery in which the motion of a continuum material particle implies the associated evolution of large numbers of microscopic coordinates.

From the standpoint of microscopic theories, the computation of the total energy requires a description in terms of the atomic positions. In this instance, we seeka reliable function Etot({Ri}), where {Ri} refers to the set of all nuclear coordinates and serves to describe the geometry at the microscale. For a more complete analysis, this picture should reflect the electronic degrees of freedom as well.

The purpose of this, our final chapter, is first to revisit some of the key phenomena faced in the task of modeling materials. In particular, we reflect on the complementary objectives of explicating both universality and specificity in materials. Our use of these words is very deliberate and is meant to conjure up two widely different perspectives. Universality refers to those features of material response which are indifferent to the particulars of a given material system. For example, huge classes of materials obey the simple constitutive model of Hooke in the small deformation regime. Similarly, both the low- and high-temperature specific heats of many solid materials obey the same basic laws. And, we have seen that the yield strength scales in a definite way with the grain size, again, in a way that is largely indifferent to material particulars. As a last example, the simple continuum model for diffusion we set forth is relevant for a great variety of materials. By way of contrast, there is a set of questions for which the answer depends entirely upon the details of the material in question. Indeed, our emphasis on material parameters and the numbers that can be found in databooks forms a complementary set of questions about materials. Our hope is to contrast these two conflicting perspectives and to show the way each is important to the overall endeavor of understanding material response.

Once we have finished the business of universality and specificity, the final discussion of the chapter, and indeed of the book as a whole, will serve as a personal reflection on what appear to me to be some of the more intriguing and fertile realms for reflection in…

Permanent Deformation of Materials

Steel conjures up images of an absolute rigidity. Yet, as a visit to the most ordinary of steel mills quickly demonstrates, at high enough temperatures a steel bar can be elongated by many orders of magnitude with respect to its initial length. Indeed, it is staggering to stand near a rolling mill, one's face aglow, as a giant block of steel is flattened and stretched into sheet or wire. Similarly, the tungsten filaments which illuminate our homes have had a history rich in permanent deformation with the length of a given bar undergoing extension by factors of as much as 500 000. The physical mechanisms that make such counterintuitive processes possible pose a puzzle that leads us to the consideration of one of the dominant lattice defects, the dislocation.

Dislocations occupy centerstage in discussions of the permanent deformation of crystalline solids largely because of their role as the primary agents of plastic change. The attempt to model such plasticity can be based upon an entire spectrum of strategies. At the smallest scales, appeal can be made to the atomic structure of dislocation cores which can suggest features such as the fact that in certain materials the flow stress increases with increasing temperature. At the other extreme, continuum models of single crystal plasticity are purely phenomenological and make no reference to dislocations except to the extent that they motivate the choice of slip systems that are taken into account.

The task of this chapter is to introduce the key concepts (both continuum and discrete) used in thinking about dislocations with the aim of explaining a range of observations concerning plastic deformation in crystalline solids.

Background

In the previous chapter we examined the central tenets of continuum mechanics with an eye to how these ideas can be tailored to the modeling of the mechanical properties of materials. During our consideration of continuum mechanics, we found that one of the key features of any continuum theory is its reliance on some phenomenological constitutive model which is the vehicle whereby mechanistic and material specificity enter that theory. As was said before, the equations of continuum dynamics are by themselves a continuous representation of the balance of linear momentum and make no reference to the particulars of the material in question. It is the role of the constitutive model to inform the equations of continuum dynamics whether we are talking about the plastic deformation of metals or the washing of waves across a beach. A key realization rooted in the microscopic perspective is the idea that the constitutive response, used in the continuum settings described in the previous chapter, reflects a collective response on the part of the microscopic degrees of freedom. It is the business of quantum and statistical mechanics to calculate the average behavior that leads to this collective response. In addition to our interest in effective macroscopic behavior, one of our primary aims is to produce plausible microscopic insights into the mechanisms responsible for observed macroscopic behavior. For example, is there a microscopic explanation for the difference in the Young's modulus of lead and silicon? Or, under what conditions might one expect to see creep mediated by grain boundary diffusion as opposed to bulkdif fusion? Or, how does the yield strength depend upon the concentration of some alloying element?

Structures in Solids

A central tenet of materials science is the intimate relation between structure and properties, an idea that has been elevated to what one might call the structure–properties paradigm. However, we must proceed with caution, even on this seemingly innocent point, since materials exhibit geometrical structures on many different scales. As a result, when making reference to the connection between structure and properties, we must askourselv es: structure at what scale? This chapter is the first in a long series that will culminate in a discussion of microstructure, in which we will confront the various geometric structures found in solids. The bottom rung in the ladder of structures that make up this hierarchy is that of the atomic-scale geometry of the perfect crystal. It is geometry at this scale that concerns us in the present chapter.

In crystalline solids, the atomic-scale geometry is tied to the regular arrangements of atoms and is described in terms of the Bravais lattice. At this scale, the modeling of structure refers to our ability to appropriately decipher the phase diagram of a particular material. What is the equilibrium crystal structure? What happens to the stable phase as a function of an applied stress, changes in temperature or changes in composition? And a more subtle question, what are the favored metastable arrangements that can be reached by the system? At the next level of description, there are a host of issues related still to the atomic-level geometry, but now as concerns the various defects that disturb the perfect crystal.

A Material World

Steel glows while being processed, aluminum does not. Red lasers are commonplace, while at the time of this writing, the drive to attain bright blue light is being hotly contested with the advent of a new generation of nitride materials. Whether we consider the metal and concrete structures that fill our cities or the optical fibers that link them, materials form the very backdrop against which our technological world unfolds. What is more, ingenious materials have been a central part of our increasing technological and scientific sophistication from the moment man took up tools in hand, playing a role in historic periods spanning from the Bronze Age to the Information Age.

From the heterostructures that make possible the use of exotic electronic states in optoelectronic devices to the application of shape memory alloys as filters for blood clots, the inception of novel materials is a central part of modern invention. While in the nineteenth century, invention was acknowledged through the celebrity of inventors like Nikola Tesla, it has become such a constant part of everyday life that inventors have been thrust into anonymity and we are faced daily with the temptation to forget to what incredible levels of advancement man's use of materials has been taken. Part of the challenge that attends these novel and sophisticated uses of materials is that of constructing reliable insights into the origins of the properties that make them attractive. The aim of the present chapter is to examine the intellectual constructs that have been put forth to characterize material response, and to take a first look at the types of models that have been advanced to explain this response.

This chapter focuses on the buckling of cylindrical shells with small thickness variations. Two important cases of thickness variation pattern are considered. Asymptotic formulas up to the second order of the thickness variation parameter Є are derived by combining the perturbation and weighted residual methods. The expressions obtained in this study reduce to Koiter's formulas, when only the first-order term of the thickness variation parameter is retained in the analysis. Results from the asymptotic formula are compared with the those obtained through the purely numerical techniques of the finite difference method and the shooting method.

This chapter investigates the buckling mode localization in the periodic multi-span beams and plates. We start our discussion with disorder in two- and three-span elastic plates; then we focus our attention on the multi-span beams and plates with a disorder occurring in an arbitrary single span. The analytical finite difference calculus is employed to derive the transcendental equations from which buckling load is calculated. The underlying treatment is general, and the solution thus obtained is exact within the theory used. Numerical results show that the buckling mode is highly localized in the vicinity of the disordered span of the beam or the plate. In the multi-span, elastic plates are considered with transverse stiffeners, and the discreteness of the stiffeners is accounted for. The torsional rigidity of the stiffener is found to play an important role in the buckling mode pattern. When the torsional rigidity is properly adjusted, the stiffener can be used in passive control; that is, it can serve as an isolator of deformation for the structure at buckling so that the deflection is limited to only a small area.

Localization in Elastic Plates Due to Misplacement in the Stiffener Location

Traditionally, the stability of the stiffened plate has been studied following three different settings.