Elasticity and Hooke's law
Nico A.M. Schellart, Dept. of Med. Physics, AMC

Principle

Hooke's law of elasticity is an approximation which states that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress). Materials for which Hooke's law is a useful approximation are known as linear-elastic or "Hookean" materials.

As a simple example, if a spring is elongated by some distance ΔL, the restoring force exerted by the spring F, is proportional to ΔL by a constant factor k, the spring constant. Basically, the extension produced is proportional to the load. That is,

F = - k ΔL. (1a)

The negative sign indicates that the force exerted by the spring is in direct opposition to the direction of displacement. It is called a "restoring force", as it tends to restore the system to equilibrium. The potential energy stored in a spring is given by:

U = 0.5 k ΔL². (1b)

This comes from adding up the energy it takes to incrementally compress the spring. That is, the integral of force over distance. This potential can be visualized as a parabola on the U- ΔL plane. As the spring is stretched in the positive L-direction, the potential energy increases (the same thing happens as the spring is compressed). The corresponding point on the potential energy curve is higher than that corresponding to the equilibrium position (ΔL =0). The tendency for the spring is to therefore decrease its potential energy by returning to its equilibrium (unstretched) position, just as a ball rolls downhill to decrease its gravitational potential energy. If a mass is attached to the end of such a spring and the system is bumped, it will oscillate with a natural frequency (or resonant angular frequency, see Resonance).

For a Hookean material it also holds that ΔL is reciprocally proportional to the cross sectional area A, so ΔL ~ A ^-1 and ΔL ~ L. When this all holds, we say that the spring is a linear spring. So, Hooke’s law, equation (1a) holds. Generally ΔL is small compared to L.

For many applications, a rod with length L and cross sectional area A, can be treated as a linear spring. Its relative extension (strain) is denoted by ε and the tension in the material per unit area, the tensile stress, by σ. Tensile stress or tension is the stress state leading to expansion; that is, the length of a material tends to increase in the tensile direction.

In formula:

ε = ΔL/L, (2a)

σ = Eε = EΔL/L =F/A, (2b)

where ΔL is the extension and E the modulus of elasticity, also called Young’s modulus. Notice that a large E yields a small ΔL. E is a measure of the stiffness and reciprocal to the mechanical compliance. As (2b) says, it is the ratio σ /ε and so the slope of the stress/strain (σ /ε) curve, see Fig. 1.

Fig. 1 Stress-strain curve. The slope of the linear part is by definition E. 1. ultimate strength, 2 limit of proportional stress. 3 rupture.

The modulus of elasticity E is a measure of the stiffness of a given material. Hooke's law is only valid for the linear part, the elastic range, of the stress strain curve (see Fig. 1). When the deforming force increases more and more, the behavior becomes non-linear, i.e. the stress-strain curve deviates from a straight line. In the non-linear part E is defined as the rate of change of stress and strain, given by the slope of the curve of Fig. 1, obtained by a tensile test. Outside the linear range for many materials like metals stress generally gives permanent deformation.

With further increasing stress the slope diminishes more and more with increasing strain and materials like metals liquidize. Finally it ruptures. See Tensile Strength for more info how materials with different mechanical properties, such as bone and tendon, behave before rupturing.

Materials such as rubber, for which Hooke's law is never valid, are known as "non-hookean". The stiffness of rubber is not only stress dependent, but is also very sensitive to temperature and loading rate.

Applications

Strength calculations of skeleton and cartilage structures, of tendons (especially the archillus and patella tendon, also aging studies) and the elastic behavior of muscles, blood vessels and alveoli. Further aging of bone, thrombolysis (compression modulus, see More Info).There are also applications in the hearing system: calculations of the mechanical behavior of the tympanum, middle ear ossicles, auditory windows and the cochlear membranes.

Failure strain are important to know, especially for biomaterials. For tendons and muscle (unloaded) they go up to 10%, sometimes until 30%. Ligaments have occasionally a failure strain up to 50%. Spider fiber reaches 30% and rubber strands more than 100%.However, these high values all include plastic deformation.

Rubber is a (bio)material with one of the most exotic properties. Its stress-strain behavior exhibits the so called Mullins effect (a kind of memory effect) and Payne effect (a kind of strain dependency) and is often modeled as hyperelastic (strain energy density dependency) . Its applications, also in medicine, are numerous.

More Info

Application of the law as described above can become more complicated.

Linear vs non-linear This has already been discussed. There exist all kind of non-linearity’s, depending on the material (steel-like, aluminum-like, brittle-like as bone, elastic-like as tendons).

Isotropic and anisotropic Most metals and ceramics, along with many other materials, are isotropic: their mechanical properties are the same in all directions. A small group of materials, as carbon fiber and composites have a different E in different directions. They are anisotropic. Many biomaterials, like muscle, bone, tendon, wood, are also anisotropic. Therefore, E is not the same in all three directions. Generally in the length of the structure it is different than in both par-axial directions. Now, σ and ε comprise each 3x3 terms. This gives the tensor expression of Hooke's Law and complicates calculations for biomaterials considerably. Often there is circular symmetry (muscle, tendon) which brings the dimensionality down to 2D (2D elasticity)

Inhomogenity Sometimes, a (biological) material s not homogeneous is some direction, so E changes along some axis. This happens in trabecular bone when it is denser at the surface.

Literature

http://www.vki.ac.be/research/themes/annualsurvey/2002/biological_fluid_ea1603v1.pdf