In Chapter 2, we reviewed the essential concepts of continuum mechanics (which are covered in full detail in the companion book to this one [TME12]) and talked at length about atomistic models in Chapter 5 and static solution methods for these models in Chapter 6. The focus of the current chapter is on ways to couple the two approaches – continuum and atomistic – in search of a “best of both worlds” model that combines their strengths.

The models discussed in this chapter achieve this coupling by using a discretized approximation to continuum mechanics called the finite element method (FEM). We provide a brief review of FEM in the next section (see [TME12] for a more detailed discussion).

Finite elements and the Cauchy–Born rule

An overview of finite elements The problem we wish to solve with FEM is the static boundary value problem embodied in Fig. 12.1(a). A body B0 in the reference configuration has surface ∂B0 with surface normal N. This surface is divided into a portion (∂B0u) over which the displacements are prescribed as ū and the remainder (∂B0t) which is either free or subject to a prescribed traction. Our goal is to determine the resulting stress, strain and displacement fields throughout the body due to the applied loads.

This boundary-value problem is conveniently recast as an energy minimization problem using the principle of minimum potential energy from Section 2.6.

Studying materials can mean studying almost anything, since all of the physical, tangible world is necessarily made of something. Normally, we think of studying materials in the sense of materials science and engineering – an endeavor to understand the properties of natural and man-made materials and to improve or exploit them in some way – but even this includes broad and disparate goals. One can spend a lifetime studying the strength and toughness of steel, for example, and never once concern oneself with its magnetic or electric properties. At the same time, modeling in science can mean many things to many people, ranging from computer simulation to analytical effective theories to abstract mathematics. To combine these two terms “modeling materials” as the title of a single book, then, is surely to invite disaster. How could it be possible to cover all the topics that the product modeling × materials implies? Although this book remains true to its title, it will be necessary to pick and choose our topics so as to have a manageable scope. To start with, then, we have to decide: what models and what materials do we want to discuss?

As far as modeling goes, we must first recognize the fact that materials exhibit phenomena on a broad range of spatial and temporal scales that combine together to dictate the response of a material. These phenomena range from the bonding of individual atoms governed by quantum mechanics to macroscopic deformation processes described by continuum mechanics.

Statistical mechanics provides a bridge between the atomistic world and continuum models. It capitalizes on the fact that continuum variables represent averages over huge numbers of atoms. But why is such a connection necessary? The theory of continuum mechanics is an incredibly successful theory; its application is responsible for most of the engineered world that surrounds us in our daily lives. This fact, combined with the internal consistency of continuum mechanics, has led some of its proponents to adopt the view that there is no need to attempt to connect this theory with more “fundamental” models of nature. (See, for example, the discussion of Truesdell and Toupin's view on this in Section 2.2.1.) However, there are a number of reasons why making such a connection is important.

First, continuum mechanics is not a complete theory since in the end there are more unknowns than the number of equations provided by the basic physical principles. To close the theory it is necessary to import external “constitutive relations” that in engineering applications are obtained by fitting functional forms to experimental measurements of materials. Continuum mechanics places constraints on these functional forms (see Section 2.5) but it cannot be used to derive them. A similar state of affairs exists for failure criteria, such as fracture and plasticity, which are add-ons to the theory. There is a strong emphasis in modern engineering to go beyond the phenomenology of classical continuum mechanics to a theory that can also predict the material constitutive response and failure.

As we explained in the preface, modeling materials is to a large extent an exercise in multiscale modeling. To set the stage for the discussion of the various theories and methods used in the study of materials behavior, it is helpful to start with a brief tour of the structure of materials – and in particular crystalline materials – which are the focus of this book. In a somewhat selective way, we will discuss the phenomena that give rise to the form and properties of crystalline materials like copper, aluminum and steel, with the goal of highlighting the range of time and length scales that our modeling efforts need to address.

Multiple scales in crystalline materials

Orowan's pocket watch

The canonical probe of mechanical properties is the tensile test, whereby a standard specimen is pulled apart in uniaxial tension. The force and displacement are recorded during the test, and usually normalized by the specimen geometry to provide a plot of stress versus strain. In the discussion of an article by a different author on “the significance of tensile and other mechanical test properties of metals,” Egon Orowan states: [Oro44]

In Chapter 12, we discussed a suite of static atomistic–continuum coupling methods, all with the goal of being an approximate, more efficient alternative to molecular statics (MS) (Chapter 6). While there are many such methods, we demonstrated that they can all be described within a common framework, and summarized the various options that one can select within that framework to define a specific variant of the coupling procedure. We deliberately avoided the question of dynamic multiscale methods in that discussion.

Here, we look at methods whose goal is to be an approximate, low-cost alternative to molecular dynamics (MD) (Chapter 9). The general idea is still embodied in Fig. 12.2, as we envision partitioning the body into two regions: one (BA) that will be treated atomistically using full MD and another (BC) that can be approximated using a continuum approach (commonly discretized using finite elements). Instead of seeking an equilibrium configuration for the body, we now have time-dependent boundary conditions (including prescribed displacements, atomic forces and continuous tractions) and a set of initial conditions from which we want to follow the evolution of the system. In the atomistic region, this means the trajectories of all the atoms. In the continuum region, this may include a discretized approximation to the mean displacement, velocity and temperature fields.

Superficially, this sounds like a straightforward extension of static methods, but in fact there are additional difficulties that make the problem much more complex.

Crystalline materials were known from ancient times for their beautiful regular shapes and useful properties. Many materials of important technological value are crystalline, and we now understand that their characteristics are a result of the regular, repeating arrangement of atoms making up the crystal structure. The details of this crystal structure determine, for example, the elastic anisotropy of the material. It helps determine whether the crystal is ductile or brittle (or both depending on the direction of the applied loads). The crystallinity manifests itself in structural phase transformations, where materials change from one crystal structure to another under applied temperature or stress. Defects in crystals (discussed in Chapters 1, 6 and 12) determine the electrical and mechanical response of the material. Indeed, we saw in Chapter 1 that the starting point for understanding any of the properties of crystalline materials is the understanding of the underlying crystal structure itself.

Crystal history: continuum or corpuscular?

The evolution of the modern science of crystallography was a long time in coming. Here, we present a brief overview, partly based on the fascinating detailed history of this science in the article by J. N. Lalena [La106].

Prehistoric man used flint, a microcrystalline form of quartz peppered with impurities, to make tools and weapons. Most likely he never concerned himself with the inner structure of the material he was using. If pressed he would probably have adopted a continuum view of his material since clearly as he formed his tools the chips flying off were always just smaller pieces of flint.

In this chapter, we derive expressions for the free energy, stress and elasticity tensors for crystalline systems under equilibrium conditions. These expressions can be used as constitutive relations in continuum mechanics (see Section 2.5) under the assumption that a continuum system is in a state of “local thermodynamic equilibrium” at each point. The advantage of the “atomistic constitutive relations” derived here is that they inherently possess basic properties of the material such as its symmetries and lattice-invariant shears which are difficult to incorporate into standard continuum models for crystals. It is also hoped that atomistic models are more predictive than macroscopic phenomenological models, but of course this depends on the transferability of the interatomic model as discussed in Section 5.7.2. The use of atomistic constitutive relations within a continuum finite element framework will be our first example of a multiscale method in Chapter 12.

Chapter 8 has already dealt with the derivation of microscopic expressions for stress and elasticity, so why are we revisiting this problem again? The reason is that the statistical mechanics expressions in Chapter 8 place no restrictions on the positions of the atoms aside from overall macroscopic constraints. This is an excellent model for fluids, where atoms move freely through space. However, in solid systems, atoms are arranged in energetically-favorable patterns about which they vibrate with an overall magnitude dictated by the temperature of the system.