An Introduction to Mathematical Modeling: A Course in Mechanics

By J. Tinsley Oden

A modern approach to mathematical modeling, featuring unique applications from the field of mechanics

An Introduction to Mathematical Modeling: A Course in Mechanics is designed to survey the mathematical models that form the foundations of modern science and incorporates examples that illustrate how the most successful models arise from basic principles in modern and classical mathematical physics. Written by a world authority on mathematical theory and computational mechanics, the book presents an account of continuum mechanics, electromagnetic field theory, quantum mechanics, and statistical mechanics for readers with varied backgrounds in engineering, computer science, mathematics, and physics.

The author streamlines a comprehensive understanding of the topic in three clearly organized sections:

• Nonlinear Continuum Mechanics introduces kinematics as well as force and stress in deformable bodies; mass and momentum; balance of linear and angular momentum; conservation of energy; and constitutive equations

• Electromagnetic Field Theory and Quantum Mechanics contains a brief account of electromagnetic wave theory and Maxwell's equations as well as an introductory account of quantum mechanics with related topics including ab initio methods and Spin and Pauli's principles

• Statistical Mechanics presents an introduction to statistical mechanics of systems in thermodynamic equilibrium as well as continuum mechanics, quantum mechanics, and molecular dynamics

Each part of the book concludes with exercise sets that allow readers to test their understanding of the presented material. Key theorems and fundamental equations are highlighted throughout, and an extensive bibliography outlines resources for further study.

Extensively class-tested to ensure an accessible presentation, An Introduction to Mathematical Modeling is an excellent book for courses on introductory mathematical modeling and statistical mechanics at the upper-undergraduate and graduate levels. The book also serves as a valuable reference for professionals working in the areas of modeling and simulation, physics, and computational engineering.

• Language: English
• Publisher: Wiley
• Release date: Feb 23, 2012
• ISBN: 9781118105740

Book preview

PREFACE

This text was written for a course on An Introduction to Mathematical Modeling for students with diverse backgrounds in science, mathematics, and engineering who enter our program in Computational Science, Engineering, and Mathematics. It is not, however, a course on just how to construct mathematical models of physical phenomena. It is a course designed to survey the classical mathematical models of subjects forming the foundations of modern science and engineering at a level accessible to students finishing undergraduate degrees or entering graduate programs in computational science. Along the way, I develop through examples how the most successful models in use today arise from basic principles and modern and classical mathematics. Students are expected to be equipped with some knowledge of linear algebra, matrix theory, vector calculus, and introductory partial differential equations, but those without all these prerequisites should be able to fill in some of the gaps by doing the exercises.

I have chosen to call this a textbook on mechanics, since it covers introductions to continuum mechanics, electrodynamics, quantum mechanics, and statistical mechanics. If mechanics is the branch of physics and mathematical science concerned with describing the motion of bodies, including their deformation and temperature changes, under the action of forces, and if one adds to this the study of the propagation of waves and the transformation of energy in physical systems, then the term mechanics does indeed apply to everything that is covered here.

The course is divided into three parts. Part I is a short course on nonlinear continuum mechanics; Part II contains a brief account of electromagnetic wave theory and Maxwell’s equations, along with an introductory account of quantum mechanics, pitched at an undergraduate level but aimed at students with a bit more mathematical sophistication than many undergraduates in physics or engineering; and Part III is a brief introduction to statistical mechanics of systems, primarily those in thermodynamic equilibrium.

There are many good treatments of the component parts of this work that have contributed to my understanding of these subjects and inspired their treatment here. The books of Gurtin, Ciarlet, and Batra provide excellent accounts of continuum mechanics at an accessible level, and the excellent book of Griffiths on introductory quantum mechanics is a well-crafted text on this subject. The accounts of statistical mechanics laid out in the book of Weiner and the text of McQuarrie, among others, provide good introductions to this subject. I hope that the short excursion into these subjects contained in this book will inspire students to want to learn more about these subjects and will equip them with the tools needed to pursue deeper studies covered in more advanced texts, including some listed in the references.

The evolution of these notes over a period of years benefited from input from several colleagues. I am grateful to Serge Prudhomme, who proofread early versions and made useful suggestions for improvement. I thank Alex Demkov for reading and commenting on Part II. My sincere thanks also go to Albert Romkes, who helped with early drafts, to Ludovic Chamoin, who helped compile and type early drafts of the material on quantum mechanics, and Kris van der Zee, who helped compile a draft of the manuscript and devoted much time to proofreading and helping with exercises. I am also indebted to Pablo Seleson, who made many suggestions that improved presentations in Part II and Part III and who was of invaluable help in putting the final draft together.

J. Tinsley Oden

Austin, Texas

June 2011

Part I

Nonlinear Continuum Mechanics

CHAPTER 1

KINEMATICS OF DEFORMABLE BODIES

Continuum mechanics models the physical universe as a collection of deformable bodies, a concept that is easily accepted from our everyday experiences with observable phenomena. Deformable bodies occupy regions in three-dimensional Euclidean space , and a given body will occupy different regions at different times. The subsets of occupied by a body are called its configurations. It is always convenient to identify one configuration in which the geometry and physical state of the body are known and to use that as the reference configuration; then other configurations of the body can be characterized by comparing them with the reference configuration (in ways we will make precise later).

For a given body, we will assume that the reference configuration is an open, bounded, connected subset Ω0 of ³ with a smooth boundary . The body is made up of physical points called material points. To identify these points, we assign each a vector X and we identify the components of X as the coordinates of the place occupied by the material point when the body is in its reference configuration relative to a fixed Cartesian coordinate system.

It is thus important to understand that the body is a non-denumerable set of material points X. This is the fundamental hypothesis of continuum mechanics: Matter is not discrete; it is continuously distributed in one-to-one correspondence with points in some subset of ³. Bodies are thus continuous media: The components of X with respect to some basis are real numbers. Symbolically, we could write

equation

for some orthonormal basis {e1, e2, e3} and origin 0 chosen in three-dimensional Euclidean space and, thus, identified with ³. Hereafter, repeated indices are summed throughout their ranges; i.e. the summation convention is employed.

Kinematics is the study of the motion of bodies, without regard to the causes of the motion. It is purely a study of geometry and is an exact science within the hypothesis of a continuum (a continuous media).

1.1 Motion

We pick a point 0 in ³ as the origin of a fixed coordinate system (x1, x2, x3) = x defined by orthonormal vectors ei, i = 1, 2, 3. The system (x1, x2, x3) is called the spatial coordinate system. When the physical body occupies its reference configuration Ω0 at, say, time t = 0, the material point X occupies a position (place) corresponding to the vector X = Xiei. The spatial coordinates (X1, X2, X3) of X are labels that identify the material point. The coordinate labels Xi are sometimes called material coordinates (see Fig. 1.1).

Figure 1.1: Motion from the reference configuration Ω0 to the current configuration Ωt.

Remark Notice that if there were a countable set of discrete material points, such as one might use in models of molecular or atomistic dynamics, the particles (discrete masses) could be labeled using natural numbers n , as indicated in Fig. 1.2. But the particles (material points) in a continuum are not countable, so the use of a label of three real numbers for each particle corresponding to the coordinates of their position (at t = 0) in the reference configuration seems to be a very natural way to identify such particles.

Figure 1.2: A discrete set of material particles.

The body moves through over a period of time and occupies a configuration Ωt ³ at time t. Thus, material points X in 0 (the closure of Ω0) are mapped into positions x in t by a smooth vector-valued mapping (see Fig. 1.1)

(1.1) equation

Thus, (X, t) is the spatial position of the material point X at time t. The one-parameter family { (X, t)} of positions is called the trajectory of X. We demand that be differentiable, injective, and orientation preserving. Then is called the motion of the body:

1. Ωt is called the current configuration of the body.

2. is injective (except possibly at the boundary .

3. is orientation preserving (which means that the physical material cannot penetrate itself or reverse the orientation of material coordinates, which means that det .

Hereafter we will not explicitly show the dependence of and other quantities on time t unless needed; this time dependency is taken up later.

The vector field

(1.2) equation

is the displacement of point X. Note that

equation

The tensor

(1.3) equation

is called the deformation gradient. Clearly,

(1.4) equation

where I is the identity tensor and is the displacement gradient.

Some Definitions

A deformation is homogeneous if F = C = constant.

A motion is rigid if it is the sum of a translation a and a rotation Q:

equation

where a , with +³ the set of orthogonal matrices of order 3 with determinant equal to +1.

As noted earlier, the fact that the motion is orientation preserving means that

equation

Recall that

equation

For any matrix A = [Aij] of order n and for each row i and column j, let Aij’ be the matrix of order n - 1 obtained by deleting the ith row and jth column of A. Let dij = (-1)i+j det Aij’. Then the matrix

equation

is the cofactor matrix of A and dij is the (i, j) cofactor of A. Note that

(1.5)

equation

1.2 Strain and Deformation Tensors

A differential material line segment in the reference configuration is

equation

while the same material line in the current configuration is

equation

The tensor

C = FTF = the right Cauchy–Green deformation tensor

is thus a measure of the change in dS0² due to (gradients of) the motion

equation

C is symmetric, positive definite. Another deformation measure is simply

equation

where

(1.6)

equation

Since F = I + μ and C = FTF, we have

(1.7) equation

The tensor

B = FFT = the left Cauchy–Green deformation tensor

is also symmetric and positive definite, and we can likewise define

equation

where

(1.8)

equation

or

(1.9)

equation

where grad μ is the spatial gradient grad μ = (i.e., (grad μ)ij = ; see also Sec. 1.3.

Interpretation of E Take dS0 = dX1 (i.e., dX = (dX1, 0, 0)T). Then

equation

so

equation

We call e1 the extension in the X1 direction at X (which is a dimension-less measure of change in length per unit length)

equation

or

equation

Similar definitions apply to E22 and E33.

Now take dX = (dX1, dX2, 0)T and

equation

The shear (or shear strain) in the X1 - X2 plane is defined by the angle change (see Fig. 1.3),

Figure 1.3: Change of angle through the motion .

equation

Therefore

(1.10) equation

Thus, E12 (and, analogously, E13 and E23) is a measure of the shear in the X1 - X2 (or X1 - X3 and X2 - X3) plane.

Small strains The tensor

(1.11) equation

is called the infinitesimal or small or engineering strain tensor. Clearly

(1.12) equation

Note that if E is small (i.e., |Eij| 1), then we obtain

equation

that is,

equation

and

equation

Thus, small strains can be given the classical textbook interpretation: en is the change in length per unit length and e12 is the change in the right angle between material lines in the X1 and X2 directions. In the case of small strains, the Green–St. Venant strain tensor and the Almansi–Hamel strain tensor are indistinguishable.

1.3 Rates of Motion

If (X, t) is the motion (of X at time t), i.e.,

equation

then

(1.13) equation

is the velocity and

(1.14) equation

is the acceleration. Since is (in general) bijective, we can also describe the velocity as a function of the place x in ³ and time t:

equation

This is called the spatial description of the velocity.

This leads to two different ways to interpret the rates of motion of continua:

The material description (functions are defined on material pointsX in the body in correspondence to points in ³);

The spatial description (functions are defined on (spatial) placesx in ³).

When the equations of continuum mechanics are written in terms of the material description, the collective equations are commonly referred to as the Lagrangian form (formulation) of the equations (see Fig. 1.4). When the spatial description is used, the term Eulerian form (formulation) is used (see Fig. 1.5).

Figure 1.4: Lagrangian (material) description of velocity. The velocity of a material point is the time rate of change of the position of the point as it moves along its path (its trajectory) in ³.

Figure 1.5: Eulerian (spatial) description of velocity. The velocity at a fixed place x in ³ is the speed and direction (at time t) of particles flowing through the place x.

There are differences in the way rates of change appear in the Lagrangian and Eulerian formulations.

In the Lagrangian case: Given a field (the sub script m reminding us that we presume is a function of the material coordinates),

equation

but = 0 because X is simply a label of a material point. Thus,

(1.15) equation

In the Eulerian case: Given a field = (x, t),

equation

but is the velocity at position x and time t. Thus,

(1.16)

equation

Notation We distinguish between the gradient and divergence of fields in the Lagrangian and Eulerian formulation as follows:

equation

In classical literature, some authors write

(1.17) equation

as the material time derivative of a scalar field , giving the rate of change of at a fixed described place x at time t. Thus, in the Eulerian formulation, the acceleration is

equation

v being the velocity.

1.4 Rates of Deformation

The spatial (Eulerian) field

(1.18)

equation

is the velocity gradient. The time rate of change of the deformation gradient F is

equation

or

(1.19) equation

where Lm = L is written in material coordinates, so

(1.20) equation

It is standard practice to write L in terms of its symmetric and skew-symmetric parts:

(1.21) equation

Here

(1.22)

equation

We can easily show that if v is the velocity field,

(1.23) equation

where ω is the vorticity

(1.24) equation

Recall (cf. Exercise 2.6) that

equation

for any invertible tensor A and arbitrary V L(V, V). Also, if f(g(t)) = f g(t) denotes the composition of functions f and g, the chain rule of differentiation leads to

equation

Combining these expressions, we have

equation

(since , where tr L = tr grad v = div v).

Summing up:

(1.25) equation

There is a more constructive way of deriving (1.25) using the definitions of determinant and cofactors of F; see Exercise 4 in Set I.2.

1.5 The Piola Transformation

The situation is this: A subdomain G0 Ω0 of the reference configuration of a body, with boundary and unit exterior vector n0 normal to the surface-area element dA0, is mapped by the motion into a subdomain G = of the current configuration with boundary with unit exterior vector n normal to the deformed surface area dA (see Fig. 1.6).

Figure 1.6: Mapping from reference configuration into current configuration.

Let denote a tensor field defined on G and T(x) n(x) the flux of T across being a unit normal to . Here Ωt is fixed so t is held constant and not displayed. Corresponding to T, a tensor field T0 = T0(X) is defined on G0 that associates the flux T0(X) n0 (X) through being the unit normal to . We seek a relationship between T0(X) and T(x) that will result in the same total flux through the surfaces and , so that

(1.26)

equation

with . This relationship between T0 and T is called the Piola transformation.

Proposition 1.1 (Piola Transformation) The above correspondence holds if

(1.27)

equation

Proof (This development follows that of Ciarlet [2]). We will use the Green’s formulas (divergence theorems)

equation

and

equation

where

equation

We will also need to use the fact that

equation

To show this, we first verify by direct calculation that

equation

where no summation is used. Then a direct computation shows that

equation

Next, set

equation

Noting that

equation

and

equation

we