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Chapter 1: Conceptual and Mathematical Preliminaries 1-1 Chapter 1 Conceptual and Mathematical Preliminaries “The best thing I ever learned in life was that things have to be worked for. A lot of people seem to think there is some sort of magic in making a winning football team. There isn’t, but there’s plenty of work.” Knute Rockne 1.1 Historical Perspectives In the 1950s, most of the mathematical foundations for continuum mechanics were codified, focusing on tensor calculus and non-Euclidean geometry. We owe much of its development to R.S. Rivlin, who wrote broadly and examined finite elasticity, thermodynamics, Non- Newtonian fluids, and electromagnetism [7, 8]. In addition, The Classical Field Theories of Truesdell and Toupin, and Truesdell and Noll’s The Nonlinear Field Theories of Mechanics,” both published originally in The Handbook of Physics (Handb¨ uch der Physik) [9, 10], are im- pressive in scope and exhaustively review the mechanical theories of continua 1 . Subsequent work expanded the range of applications, especially in thermodynamics, plasticity, electro- magnetic continua, and mixture theory. Many of these developments were the result of work 1 For many reasons, the work of Russian mathematicians and physicists was not well-represented in the annals of continuum mechanics until quite recently. Working independently, they made enormous progress, often outpacing their American and European counterparts. In a series of texts, Landau, Lifshitz, and Pitaevskii provide an excellent summary of the development of applied physics in Russia (then the Soviet Union) from the 1950s to the 1980s [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. One is left to wonder what might have been achieved had researchers been able to access each others works more freely.

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Chapter 1: Conceptual and Mathematical Preliminaries 1-1

Chapter 1

Conceptual and MathematicalPreliminaries

“The best thing I ever learned in life was that things have to be worked for.A lot of people seem to think there is some sort of magic in making a winningfootball team. There isn’t, but there’s plenty of work.”

∼ Knute Rockne

1.1 Historical Perspectives

In the 1950s, most of the mathematical foundations for continuum mechanics were codified,focusing on tensor calculus and non-Euclidean geometry. We owe much of its developmentto R.S. Rivlin, who wrote broadly and examined finite elasticity, thermodynamics, Non-Newtonian fluids, and electromagnetism [7, 8]. In addition, The Classical Field Theories ofTruesdell and Toupin, and Truesdell and Noll’s The Nonlinear Field Theories of Mechanics,”both published originally in The Handbook of Physics (Handbuch der Physik) [9, 10], are im-pressive in scope and exhaustively review the mechanical theories of continua1. Subsequentwork expanded the range of applications, especially in thermodynamics, plasticity, electro-magnetic continua, and mixture theory. Many of these developments were the result of work

1For many reasons, the work of Russian mathematicians and physicists was not well-represented in theannals of continuum mechanics until quite recently. Working independently, they made enormous progress,often outpacing their American and European counterparts. In a series of texts, Landau, Lifshitz, andPitaevskii provide an excellent summary of the development of applied physics in Russia (then the SovietUnion) from the 1950s to the 1980s [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. One is left to wonder what mighthave been achieved had researchers been able to access each others works more freely.

1-2 Chapter 1: Conceptual and Mathematical Preliminaries

by Green [21], Naghdi [22, 23, 24, 25], Eringen [26], Tiersten [27], Gurtin [28, 29], Rajagopal[30, 31, 32], and Casey [33, 34, 35] and will be referenced throughout this text. It shouldbe noted that the theory of mixtures is particularly important to the development of biome-chanics given the importance of fluid movement and solute transport to many physiologicalproblems. In particular, porous material formulations coupled with jump discontinuitiesmake it possible to study musculoskeletal interfaces such as the junction between cartilageand bone (Fig. 1.1), the vertebral endplate, and the enthesis. Throughout this text, wewill consider examples taken from many of the aforementioned applications, providing thenecessary anatomical and physiological background along the way.

Figure 1.1: Structure of the femoral head highlighting the bone-cartilage interface, cancellousbone architecture, and cortical bone structure.

The application of continuum mechanics to biological systems was pioneered by Y.C. Fung[4, 3], Richard Skalak [1, 36], and Shu Chien [37, 38]. Their contributions included the de-velopment of nonlinear, history-dependent constitutive laws for soft tissues, the foundationsof cell mechanics, and the quantitative description of the microcirculation. We must alsoconsider the contributions of those who have enjoyed particular insights into biomechanicsof the solid and fluid variety. They include Sir D’Arcy Wentworth Thompson [39], StevenVogel [40, 41, 42], David Burr, Steve Cowin, Sheldon Weinbaum, Rich Hart [43, 44, 45],

Chapter 1: Conceptual and Mathematical Preliminaries 1-3

R. Bruce Martin, Dennis Carter [46, 47], Roger Kamm, and Jay Humphrey among others.As a result of their efforts, the number of applications has grown considerably to includeosteoporotic bone, skin, arteries, liver, the cornea, sclera, lens, optic nerve head, engineeredtissue replacements, drug eluting polymers, and many others. Recently, the continuum me-chanics of the central nervous system have become particularly important, building uponthe impressive efforts of Werner Goldsmith and Ayub Khan Ommaya [48, 49, 50]. Before wecan discuss these topics in detail, however, we must define what we mean by a continuumand review some basic mathematics. After that, we will begin describing the kinematics andkinetics that are important to biomechanics and medical technology, making a concertedeffort along the way to integrate as much mathematics as possible.

Upon completion of this chapter, you should understand,

1. how to define a continuum,

2. the distinction between scalars, vectors, and tensors,

1.2 Definition of a Continuum

The definition of a continuum is a useful starting place. More precisely, our notion of acontinuum actually defines the length scale at which our system of equations might actuallyapply. The continuum approximation allows us to model our tissue or biomaterial as acontinuous region which, when sub-divided, retains the same material properties as the bulkmaterial. As an illustration, a Hersheys Krackel (or a Nestle Crunch) is a very good placeto start because it is a simple mixture of delicious chocolate and crisped rice. If you startat a point located inside a piece of rice and slowly expand your region of interest (ROI)2,you can calculate the volume fraction of the rice, φrice , defined as the volume of rice, V rice

, divided by the volume of your region of interest, V ROI ,

φrice =V rice

V ROI. (1.1)

Initially, that number will be 1, and within that region, you would be correct to say thatwe can model the mechanics of that material using the properties of crisped rice. As thesphere increases in radius, however, you will add some chocolate, so the volume fraction ofrice will decrease. Typically one will find that the volume fraction of the rice approachessome relatively constant value, Fig. (1.2). Here we can also model the mechanics of theKrackel bar, but we will have to define the material as a mixture or composite of chocolateand crisped rice. A similar argument can be made if you start studying the material from a

2Depending on your computational method, your ROI may be shaped like a sphere, cube, or in somecases, a parallelepiped.

1-4 Chapter 1: Conceptual and Mathematical Preliminaries

point located entirely within the chocolate region. At the smallest length scale we consider,the sphere of interest is entirely composed of milk chocolate so our continuum model wouldconsider only the mechanics of the chocolate. As we increase the radius of our sphere, thevolume fraction of chocolate, φchoc, will reach a relatively constant value (in this case itis equal to 1 − φrice). In fact, it does not matter appreciably where you start within theKrackel bar, you should always get a similar curve and similar average volume fraction asyou increase the size of the ROI, provided you do not get too close to the edge of the bar. Itshould be clear that a Nestle Crunch bar or a Hershey bar with almonds will exhibit similarscale effects and the determination of the length scale at which we define a continuum willhave to be performed in the same way3.

φrice

1

0

Size of the Region of Interest (ROI)

µm mm cm

Starting inside the

crisped rice

Starting inside the

delicious chocolate

Figure 1.2: Example volume fraction vs. ROI size diagram (simulated data for a HersheysKrackel bar). Note that the x-axis is logarithmic. Depending on whether you start withinthe rice or the chocolate regions, the calculated value of φrice will be 1 or 0, respectively.Increasing the size of the ROI eventually smoothes the calculation. At length scales wherethe curve is primarily flat, it is possible to define one or more continua. For instance, atsmall length scales, one must model the Krackel bar as rice or chocolate. At large lengthscales, it may be a single continua representing the composite rice-chocolate structure.

For a more traditional biomaterial, cancellous bone is a good starting place. Let’s assumethat we are interested in the stress distribution within the femur due to the application ofmuscle loads and contact between the femoral head and the acetabulum. This is a compli-

3Interestingly, a Snickers bar offers some difficulties with regard to the definition of a continuum lengthscale because it is more of a layered material. For what it is worth, it does provide 18% of your DailyRecommended Allowance of Niacin.

Chapter 1: Conceptual and Mathematical Preliminaries 1-5

cated problem and before we can even approach it, we need to define an appropriate lengthscale for the analysis. Our first consideration is a simple one. We would like our lengthscale to be smaller than the regional variation in the tissue, Fig. (1.3). For the femur shownin Fig. (1.1), it is clear that there is a dense, highly oriented band of cancellous bone thatstarts at the ligamentum teres and moves down towards the lesser trochanter. In contrast,the cancellous bone contained in the greater trochanter appears to be considerably less denseand almost randomly oriented. Consequently, we should probably assume that the upperrange at which we could expect continuum behavior to be approximately 1 cm. But whatabout the lowest level at which we can expect to find a continuum? The answer to thatmay depend on how much effort one is willing to put into the computation. For instance, ifone is willing to model the precise geometry of the cancellous bone throughout the femur,one can use a resolution of 1-10 um [51]. At this length scale it is possible to account forthe precise microstructure of all the bone tissue in the femur. It is also a model that wedo not yet have the computer power to evaluate, at least not for an entire bone. Anotherapproach might be to model the cancellous bone as a continuum. In that case, Harrigan etal. demonstrated that one needs to consider a region that consists of 5-10 unit cells in anygiven direction in order to have a reasonable expectation of continuum-like behavior; i.e.that there are no discontinuities in the strain field. Here we generally use volume fraction asthe fundamental property that determines the range over which a continuum model shouldbe applicable. In truth, it is more of a starting point. There may be other properties that donot vary directly with volume fraction that we may be more interested in for a given prob-lem, but they can be considered in a similar manner. The length scale over which continuumbehavior is applicable is the key to scaling analyses. Once a continuum scale is identified, itis possible to analyze transport, load transfer, tissue damage and interpret medical images,topics important to everything we will consider in this text and beyond. Some of the mostimportant problems require moving between continuum scales, but this can only be accom-plished once we successfully understand continuum phenomena at a single length scale andthen only with great care4.

4This last sentence might be profound or obvious or profoundly obvious. Understanding the translationof tissue level stress to the cells within the tissue has been one of the primary goals of the biomechanics fieldfor the last 50 years. To date, however, moving between length scales has been very difficult. Understandingthe continuum assumption is the first step toward rectifying the situation.

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Figure 1.3: The mechanics and transport characteristics of cancellous bone can be analyzedat two different length scales. If we quantify the volume fraction of bone using its volumefraction, φbone. Beginning with an ROI around 1-10 µm and slowly increasing in size willtypically provides a volume fraction of either 1 or 0 depending on whether the center ofyour ROI is within an individual trabeculae or inside the marrow space. Continuum theorieswork in this region because one can model the bone and marrow as separate continua. Aswe increase the size of the ROI, we observe considerable variability until we reach an ROIapproximately 1 mm in size. At that point the volume fraction levels out again and thecancellous bone and marrow can be treated as a single continuum. If we try to analyze boneat length scales smaller than 1-10 µm, a new difficulty arises because “bone” as it is typicallydescribed does not exist. One must consider the separate constituents, hydroxyapatite (HA)and collagen. The grey boxes illustrate where one might expect to find reasonable continuummodels for HA and collagen. These concepts are critical because continuum behavior is anecessary prerequisite for dimensional analysis, mechanical models, and medical imaging.