Elasticity 1: elastic or Young’s modulus
Nico A.M. Schellart, Dept. of Med. Physics, AMC

Principle

There are three primary elastic moduli, each describing a different kind of deformation. These are the elasticity or Young’s modulus E, shear modulus G (see Elasticity 2: Shear Strength) and the bulk modulus K (see Elasticity 3: compressibility and bulk modulus):

Because all elastic moduli can be derived from Young's modulus, it is often referred to simply as the elastic modulus.

The modulus of elasticity E is a measure of the stiffness of a given material or its resistance to be deformed when a force is applied to it. The elastic modulus of an object is defined as the slope of the curve of tensile stress and tensile strain:

E = Δσ /Δε,

where relative extension (strain) is denoted by ε and the tension in the material per unit area, the tensile stress, by σ. Because strain ε is a unitless ratio, σ and E are measured in Pa. For small values of ε and σ, E is constant for some materials (its stress strain curve), but generally the curve is nowhere a straight line.

For the linear behavior (straight line) we have:

Table 1 gives some values of E for trivial and biological material (room temperature):

Table 1 Young’s modulus for some (bio)materials

material	E (GPa)	G (GPa)
rubber	0.001
ZYLON PBO	138-167
steel	200	75.8
titanium		41.4
muscle	10^4
vessel wall	0.0001-0.001
muscle		0.00006
tendon	0.1
bone	0.1-10	4.5 – 8.0*

* human pipe bones

For solids G is about halve K. See for a further description Elasticity and Hooke's law.

Application

In the biomechanics of bone and tendon structures in basic medical science and clinical applications, orthopedic prostheses etc.

Aging makes many biomaterials more brittle with lowering ultimate strength. For bone 2%/decade with a 2% increase of E.

More Info

When an object has an isotropic molecular crystalline structure, E as well as K are larger than G. In other words, the crystal resist stronger against strain and compression than against shear. Strain gives a distance decrease of some particle with the neighboring particles in the direction of the applied force and in the two perpendicular directions, but to a smaller amount. The latter depends on the Laplace ratio (see Elasticity and Hooke’s Law). In the unstressed state this potential energy, the result of solely internal forces, is minimal. Whit strain, external force is added and the potential energy rises.

Compression gives a distance decrease of some particle with all neighboring particles that gives the largest change in potential energy. With shear there is only a distance change in one dimension. Shear and strain makes the crystal anisotropic. This can change various material properties.

Young's modulus is a physical consequence of the Pauli exclusion principle.