In September 1997 an educational programme in Biomedical Engineering, unique in the Netherlands, started at the Eindhoven University of Technology, together with the University of Maastricht, as a logical step after almost two decades of research collaboration between both universities. This development culminated in the foundation of the Department of Biomedical Engineering in April 1999 and the creation of a graduate programme (MSc) in Biomedical Engineering in 2000 and Medical Engineering in 2002.

Already at the start of this educational programme, it was decided that a comprehensive course in biomechanics had to be part of the curriculum and that this course had to start right at the beginning of the Bachelor phase. A search for suitable material for this purpose showed that excellent biomechanics textbooks exist. But many of these books are very specialized to certain aspects of biomechanics. The more general textbooks are addressing mechanical or civil engineers or physicists who wish to specialize in biomechanics, so these books include chapters or sections on biology and physiology. Almost all books that were found are at Masters or post-graduate level, requiring basic to sophisticated knowledge of mechanics and mathematics. At a more fundamental level only books could be found that were written for mechanical and civil engineers.

We decided to write our own course material for the basic training in mechanics appropriate for our candidate biomedical engineers at Bachelor level, starting with the basic concepts of mechanics and ending with numerical solution procedures, based on the Finite Element Method.

In the present and following chapters extensive use will be made of a simple finite element code mlfem_nac. This code, including a manual, can be freely downloaded from the website: www.mate.tue.nl/biomechanicsbook.

The code is written in the program environment MATLAB. To be able to use this environment a licence for MATLAB has to be obtained. For information about MATLAB see: www.mathworks.com.

Introduction

It will be clear from the previous chapters that many problems in biomechanics are described by (sets of) partial differential equations and for most problems it is difficult or impossible to derive closed form (analytical) solutions. However, by means of computers, approximate solutions can be determined for a very large range of complex problems, which is one of the reasons why biomechanics as a discipline has grown so fast in the last three decades. These computer-aided solutions are called numerical solutions, as opposed to analytical or closed form solutions of equations. The present and following chapters are devoted to the numerical solution of partial differential equations, for which several methods exist. The most important ones are the Finite Difference Method and the Finite Element Method. The latter is especially suitable for partial differential equations on domains with complicated geometries, material properties and boundary conditions (which is nearly always the case in biomechanics). That is why the next chapters focus on the Finite Element Method. The basic concepts of the method are explained in the present chapter.

Introduction

Fibres and fibre-like structures play an important role in the mechanical properties of biological tissues. Fibre-like structures may be found in almost all human tissues. A typical example is the fibre reinforcement in a heart valve, Fig. 4.1(a). Another illustration is found in the intervertebral disc as shown in Fig. 4.1(b).

Fibre reinforcement, largely inspired by nature, is frequently used in prosthesis design to optimize mechanical performance. An example is found in the aortic valve prosthesis, see Fig. 4.2.

Fibres are long slender bodies and, essentially, have a tensile load bearing capacity along the fibre direction only. The most simple approximation of the, often complicated, mechanical behaviour of fibres is to assume that they behave elastically. In that case fibres have much in common with springs. The objective of this chapter is to formulate a relation between the force in the fibre and the change in length of a fibre. Such a relation is called a constitutive model.

Elastic fibres in one dimension

Assume, for the time being, that the fibre is represented by a simple spring as sketched in Fig. 4.3. At the left end the spring is attached to the wall while the right end is loaded with a certain force F. If no load is applied to the spring (fibre) the length of the spring equals l0, called the reference or initial length.

Introduction

We experience the effects of force in everyday life and have an intuitive notion of force. For example, we exert a force on our body when we lift or push an object while we continuously (fortunately) feel the effect of gravitational forces, for instance while sitting, walking, etc. All parts of the human body in one way or the other are loaded by forces. Our bones provide rigidity to the body and can sustain high loads. The skin is resistant to force, simply pull on the skin to witness this. The cardiovascular system is continuously loaded dynamically due to the pulsating blood pressure. The bladder is loaded and stretched when it fills up. The intervertebral discs serve as flexible force transmitting media that give the spine its flexibility. Beside force we are using levers all the time in our daily life to increase the ‘ force ’ that we want to apply to some object, for example by opening doors with the latch, opening a bottle with a bottle-opener. We feel the effect of a lever arm when holding a weight close to our body instead of using a stretched arm. These experiences are the result of the moment that can be exerted by a force. Understanding the impact of force and moment on the human body requires us to formalize the intuitive notion of force and moment. That is the objective of this chapter.

Introduction

In the previous chapters the global behaviour of fibres was considered, without much attention to the detailed shape of these structures. Only the length change of the fibre played a role in the analysis. In the present chapter we address in a little bit more detail the deformation of long slender structures. These can be tendons, muscles, but also long bones. The aim is, to generalize the concepts introduced in previous chapters for discrete systems (i.e. springs) to continuous systems. To simplify matters the loading and deformation of a one-dimensional elastic bar is considered.

Equilibrium in a subsection of a slender structure

Consider a straight bar as visualized in Fig. 6.1(a), loaded by an external force F at x = L and fixed in space at x = 0. The figure shows the bar with the x-axis in the longitudinal or axial direction. It is assumed, that each cross section initially perpendicular to the axis of the bar remains perpendicular to the axis after loading. In fact it is assumed that all properties and displacements are a function of the x-coordinate only.

The objective is, to compute the displacement of each cross section of the bar due to the loading. The area of the cross section perpendicular to the central axis of the bar as well as the mechanical properties of the bar may be a function of the x-coordinate. Therefore the displacement may be a non-linear function of the axial coordinate.

Natural Convection Concepts

Introduction

Chapters 4 and 5 focus on forced convection problems in which the fluid motion is driven externally, for example by a fan or a pump. However, even in the limit of no externally driven fluid motion, a solid surrounded by a fluid may not reduce to a conduction problem because the fluid adjacent to a heated or cooled surface will usually not be stagnant. Natural (or free) convection refers to convection problems in which the fluid is not driven mechanically but rather thermally; that is, fluid motion is driven by density gradients that are induced in the fluid as it is heated or cooled. The velocities induced by these density gradients are typically small and therefore the absolute magnitude of natural convection heat transfer coefficients is also typically small.

The flow patterns induced by heating or cooling can be understood intuitively; hot fluid tends to have lower density and therefore rise (flow against gravity) while cold fluid with higher density tends to fall (flow with gravity). The existence of a temperature gradient does not guarantee fluid motion. Figure 6-1(a) illustrates fluid between two plates oriented horizontally (i.e., perpendicular to the gravity vector g ) where the lower plate is heated (to TH ) and the upper plate is cooled (to TC ). The heated fluid will tend to rise and the cooled fluid fall, resulting in the natural convection “cells” that are shown in Figure 6-1(a).

Introduction to EES

This extended section of the book can be found on the website www.cambridge.org/nellisandklein. EES (pronounced ‘ease’) is an acronym for Engineering Equation Solver. The basic function provided by EES is the numerical solution of non-linear algebraic and differential equations. EES is an equation-solver, rather than a programming language, since it does not require the user to enter instructions for iteratively solving non-linear equations. EES provides capability for unit checking of equations, parametric studies, optimization, uncertainty analyses, and high-quality plots. It provides array variables that can be used in finite-difference calculations. In addition, EES provides high-accuracy thermodynamic and transport property functions for many fluids and solid materials that can be integrated with the equations. The combination of these capabilities together with an extensive library of heat transfer functions, discussed throughout this text, makes EES a very powerful tool for solving heat transfer problems. This appendix provides a tutorial that will allow you to become familiar with EES.

Introduction to maple

This extended section of the book can be found on the website www.cambridge.org/nellisandklein. Maple is an application that can be used to solve algebraic and differential equations. Maple has the ability to do mathematics in symbolic form and therefore it can determine the analytical solution to algebraic and differential equations. Maple provides a very convenient mathematical reference; if, for example, you've forgotten that the derivative of sine is cosine, it is easy to use Maple to quickly provide this information.

Chapter 1 discussed the analytical and numerical solution of 1-D, steady-state problems. These are problems where the temperature within the material is independent of time and varies in only one spatial dimension (e.g., x). Examples of such problems are the plane wall studied in Section 1.2, which is truly a 1-D problem, and the constant cross section fin studied in Section 1.6, which is approximately 1-D. The governing differential equation for these problems is an ordinary differential equation and the mathematics required to solve the problem are straightforward.

In this chapter, more complex, 2-D steady-state conduction problems are considered where the temperature varies in multiple spatial dimensions (e.g., x and y). These can be problems where the temperature actually varies in only two coordinates or approximately varies in only two coordinates (e.g., the temperature gradient in the third direction is negligible, as justified by an appropriate Biot number). The governing differential equation is a partial differential equation and therefore the mathematics required to analytically solve these problems are more advanced and the bookkeeping required to solve these problems numerically is more cumbersome. However, many of the concepts that were covered in the context of 1-D problems continue to apply.

Shape factors

There are many 2-D and 3-D conduction problems involving heat transfer between two well-defined surfaces (surface 1 and surface 2) that commonly appear in heat transfer applications and have previously been solved analytically and/or numerically.

Introduction to Heat Exchangers

Introduction

A heat exchanger is a device that is designed to transfer thermal energy from one fluid to another. The term “heat exchanger” like “heat transfer” is inconsistent with the thermodynamic definition of heat; these devices would be more appropriately called thermal energy exchangers. However, the term “heat exchanger” is ubiquitous. Heat exchangers are also ubiquitous; nearly all thermal systems employ at least one and usually several heat exchangers.

The background on conduction and convection, presented in Chapters 1 through 7, is required to analyze and design heat exchangers. This section reviews the applications and types of heat exchangers that are commonly encountered. Subsequent sections provide the theory and tools required to determine the performance of these devices.

Applications of Heat Exchangers

You may be unaware of just how common heat exchangers are in both residential and industrial applications. For example, you live in a residence that is heated to a comfortable temperature in winter and possibly cooled in the summer. Heating is usually accomplished by combusting a fuel (e.g., natural gas, propane, wood, or oil) that provides the desired thermal energy, but also produces combustion gases that can be harmful. Therefore, your furnace includes a heat exchanger that transfers thermal energy from the combustion gases to an air stream that can be safely circulated through the building.

Conduction Heat Transfer

Introduction

Thermodynamics defines heat as a transfer of energy across the boundary of a system as a result of a temperature difference. According to this definition, heat by itself is an energy transfer process and it is therefore redundant to use the expression ‘heat transfer’. Heat has no option but to transfer and the expression ‘heat transfer’ reinforces the incorrect concept that heat is a property of a system that can be ‘transferred’ to another system. This concept was originally proposed in the 1800's as the caloric theory (Keenan, 1958); heat was believed to be an invisible substance (having mass) that transferred from one system to another as a result of a temperature difference. Although the caloric theory has been disproved, it is still common to refer to ‘heat transfer’.

Heat is the transfer of energy due to a temperature gradient. This transfer process can occur by two very different mechanisms, referred to as conduction and radiation. Conduction heat transfer occurs due to the interactions of molecular (or smaller) scale energy carriers within a material. Radiation heat transfer is energy transferred as electromagnetic waves. In a flowing fluid, conduction heat transfer occurs in the presence of energy transfer due to bulk motion (which is not a heat transfer) and this leads to a substantially more complex situation that is referred to as convection.