The elastic behaviour of a material is commonly investigated using a tensile testing system. A sample of the material is clamped into the system and then stretched apart. Both applied force and deformation are measured and can be plotted. Figure 1.2 shows the force-extension behaviour for steel. For many materials, such as steel and glass, the initial behaviour is linear; a doubling of applied force results in a doubling of the extension. In this region the material is elastic in that it will follow the same line on the force-extension graph during loading or unloading. The material is elastic up the point Y, which is called the ‘yield point’ but, after the yield point, the slope of the line decreases. The material is softer in that small changes in force result in large changes in extension. Beyond the yield point the material becomes plastic in that the material does not return to its original shape after removal of the force but is permanently deformed. In Fig. 1.2 further increase in force eventually leads to fracturing of the material at the point U, called the ‘ultimate tensile strength’ (UTS).

The force-deformation behaviour can be understood at the atomic level. The chemical bonds between atoms and molecules are deformable and small deformations from the equilibrium position can be tolerated without change in structure. The equations governing the force-extension behaviour at the atomic level demonstrate linear behaviour and the macroscopic behaviour of a material is the composite of a multitude of interactions at the atomic and molecular level. In the plastic region there are changes in structure at the atomic and molecular level. In many materials this arises through slip processes involving the movement of dislocations or through the creation and propagation of cracks.

Biological materials are generally composite in nature. From a mechanical point of view the most important components are collagen fibres, elastin, reticulin and an amorphous, hydrophilic, material called ‘ground substance’ which contains as much as 90 % water. The elastic behaviour of the biological tissue is determined by the proportion of each component and by their physical arrangement. For example, collagen fibres in the wall of arteries are arranged in a helical pattern. Collagen is especially important in determining mechanical properties of soft biological tissues. Collagen is laid down in an un-stretched state. These unstressed fibres have a wavy, buckled shape, referred to as ‘crimp’. On application of a force, the fibres begin to straighten and the ‘crimp’ disappears and, as a result, the tissue deforms relatively easily. With increasing extension the fibres straighten fully and resist the stretch. This leads to collagen having a non-linear force-extension behaviour, which explains the non-linear force-extension behaviour of most biological soft tissues.

A simple 1D tensile testing system can also be used to demonstrate viscoelasticity. It was stated above that in elastic behaviour the loading and unloading curves are the same. For a viscoelastic material they are different. In elastic behaviour the application of a force results more or less immediately in deformation of the material. Viscoelastic behaviour is associated with a time- lag between the applied force and the resulting deformation. The term ‘viscoelastic’ implies that the material has a mix of elastic and viscous properties. If the tensile testing system stretches the material in a cyclic manner, then as the tissue is loaded and unloaded, the resulting force-deformation curve will be in the shape of an ellipse (Fig. 1.3). During loading the force increases but the extension increases more slowly. During unloading the force decreases but the extension decreases more slowly. If the viscous component is low compared to the elastic component then the loading and unloading curves will be close together. For materials with a higher viscous component the curves are more separated and the width of the ellipse is larger.

In Sect. 1.1.1 the discussion of elastic behaviour was in terms of applied force and deformation. However the quantities stress and strain are more widely used in theory and experiment. The stress, σ, is the force, F, per unit area, A, and has units of pascals. The strain, ε, is the ratio of the extension, δl, divided by the original length, l, and is a dimensionless quantity.

The Young’s modulus, E, is a measure of the elastic behaviour of a material and is a fundamental mechanical property. Young’s modulus is the ratio of stress divided by strain (Eq. 1.3). The units of E are pascals (Pa) or newtons per square metre (N m⁻²).

Young’s modulus is commonly measured using a tensile testing system. The value E is equal to the slope of the line on the stress–strain plot. For a linear elastic material the slope is constant over much of the range of stress/strain and the mechanical properties of the material may be described by a single value of E. For non-linear materials such as rubber or soft biological tissues, the value of E is dependent on the strain. For such materials the ‘incremental elastic modulus’ may be defined as the change in stress over the change in strain over a small section of the stress–strain curve (Eq. 1.4).

Figure 1.4 shows the Young’s modulus of a number of common materials. Note that the scale is logarithmic with a range of 9 orders of magnitude. Hard materials such as ceramics, metals and glasses have very high values of elastic modulus. These are usually quoted in gigapascals (GPa). Wood and wood products have lower values of elastic modulus, but still have a very wide range from very hard woods such as oak, to very soft woods such as balsawood. Rubbers also have a very wide range from the hard vulcanised rubber used in tyres to the soft silicone rubber used in baby’s dummies. The lowest elastic moduli values on the graph are for materials that mimic soft tissue used in phantoms for testing medical imaging systems. These are designed to mimic key properties of soft biological tissues, such as fat and muscle, and have low elastic modulus values in the range 2–500 kPa. Figure 1.5 shows the Young’s modulus of a number of different biological tissues, taken from Sarvazyan et al. (1998). Again, there is a huge range of values. Bone and tooth enamel have the highest values of elastic modulus; liver, muscle and fat the lowest values.

The observant reader might have noted that it has been stated that the constitutive equations for soft biological tissues are complex and that the stress–strain behaviour is non-linear. How then is it justified in reporting Young’s modulus, which generally applies to simple materials with linear stress–strain behaviours? This question will be addressed in Sect. 1.1.8; after more complex constitutive models have been considered.

When a material is stretched in the z direction there is usually compression in the x and y directions, and when a material is compressed there is usually expansion in the other two directions. This is called the Poisson effect. The Poisson ratio ν is given by the fractional change in length in the x direction divided by the fractional change in length in the z direction (Eq. 1.5).

For incompressible materials, that is, materials where the volume does not change when loaded, the Poisson ratio has a value of 0.5. Soft biological tissues contain large amounts of water and have Poisson values close to 0.5. For many materials such as metals, glasses and concrete, the Poisson value is in the range 0.2–0.4.

Elasticity and viscosity can be represented by a spring and a dashpot. The spring responds immediately to being stretched, which represents the purely elastic behaviour of a material. For a dashpot, there is a delay between the stretching force and the extension, which represents the viscous behaviour of a material. A viscoelastic material can be represented as a combination of a spring and a dashpot and there are various configurations, three of which are shown in Fig. 1.6. These are; the Maxwell model where the spring and the dashpot are in series, the Voigt model where the spring and the dashpot are in parallel and a model consisting of two springs and a dashpot. Figure 1.6 also shows the behaviour of each of these models. For the Maxwell model there is a sudden extension which corresponds to the spring stretching immediately followed by a linear increase caused by the dashpot slowly extending. For the Voigt model there is a gradual increase in the deformation. For the 3-parameter model, there is a sudden extension followed by a gradual extension. In practice, for a particular material, the model which best describes the experimentally determined behaviour of the material is chosen.

The Young’s modulus and the Poisson ratio are two examples of parameters which describe the elastic behaviour of materials. In this section other parameters will be defined. In order to simplify things, the approach is taken of assuming that the material is linear elastic with no viscous components. It will also be assumed that the material is isotropic (having the same behaviour in all directions).

The bulk modulus, B, describes the ability of the material to resist a change in volume. A cube of material of volume, V, is subjected to a pressure, P, (Fig. 1.7a). The pressure increases by a small amount, δP. This leads to a reduction in volume by a small amount, δV. The bulk modulus is the change in pressure divided by the change in volume.

The shear modulus, G, describes the ability of a material to withstand a shearing force (Fig. 1.7b). The shear modulus is the shear stress divided by the shear strain.

For the simple elastic material described in this section, G and E are related by Eq. (1.8) which involves both E and the Poisson ratio ν. If the material is also incompressible, then Poisson’s ratio ν is 0.5 and the equation simplifies (Eq. 1.9).

Figure 1.5 shows values of G and B, along with E, for biological tissues. The range of values for bulk modulus is about one order of magnitude, whereas those for shear modulus and Young’s modulus range over 7 orders of magnitude.

For the material described in this section, linear elastic isotropic, there are four parameters which describe elastic behaviour; Young’s modulus E, Poisson ratio ν, bulk modulus B and shear modulus G. However, only 2 of these are independent. In other words knowledge of 2 of these quantities (for example Poisson ratio and bulk modulus) allows estimation of the other 2 (Appendix 2).

The theory described in the previous section will be developed further, but without the constraint of isotropy. So far the theory has been developed by considering mainly stress and deformation in 1D. However in most cases, the behaviour applies in 3D and the solid will deform in all three directions. Figure 1.8 shows the forces in 1 dimension for stretching of a column of material, in 2D for a square of material and in 3D for a cube. For the column of material there is a single extension force. For the square, each face has a force perpendicular to the face and also a shear force. For the cube, each face has a perpendicular force and two shear forces. The types of deformation which result are shown in the lower images. The number of stress and strain components needed to describe behaviour increases from 1D to 3D. For 1D it is just one stress and one strain. For 2D four components are needed and for 3D nine components are needed. Much more data is needed to describe behaviour in 3D than in 2D or 1D.