Elasticity 3: compressibility and bulk modulus
Nico A.M. Schellart, Dept. of Med. Physics, AMC

 

Principle

 

The bulk modulus (K) describes volumetric elasticity, or the tendency of an object's volume to deform when under pressure. It is defined as volumetric compressive stress over volumetric compression, and is the inverse of compressibility.

Compressive strength is the capacity of a material to withstand axially directed pushing forces. When the limit of compressive strength is reached, materials are crushed (bones, concrete). An object under high pressure may undergo processes that will affect the stress-strain curve (see Tensile Strength), due to chemical reactions.

 

 

Application

 

Material choice for the design of prostheses and strength calculations of biomaterials like bone. When a mammalian or avian pipe bone with thin walls is bended, in the outer curvature there is tensile stress and in the inner curvature compressive strength. The shear stress (see Elasticity 2: Shear Modulus) is small but becomes more important the thicker the wall.

Bone can withstand greater 1D compressive stress than tensile stress. K of bone, ca. 50 GPa, is some 2-4 times that of concrete.

 

 

More info

 

The Poisson ratio μ

Here, for simplicity it is assumed that in the three dimensions material properties are the same (a so called isotropic material).When a beam is stretched by tensil force (see Tensile Strength), its diameter (d) changes. The change in diameter Δd, generally a decrease, can be calculated:

 

Δd/d = - μ ΔL/L.,                    (1a)

 

where L is the length of the beam, and ΔL the length increase by the tensil force F (see also Tensile Strength). The constant of proportionality μ is the Poisson ratio. Since ΔL/L. = F/EA with A is the cross sectional area it holds that:

 

Δd/d = μ F/EA.                       (1b)

 

The range of μ is from 0 to 0.5. Mostly 0.2 < μ < 0.4. For metals μ ≈ 1/3 and for rubber μ ≈ 0.5, which means that rubber hardly changes volume? Some materials behave atypically: increase of d when stretched. This happens with polymer foam.

Since generally ΔL << L, it can easily been proven that the change in volume ΔV (generally increase) is:

 

ΔV = d2ΔL + 2dL Δd.                             (2a)

 

Substitution in (1a) and dividing by V = d2L gives the relative volume change of the beam:

 

ΔV/V = (1-2 μ)ΔL/L = (1-2 μ)F/EA       (2b)

 

With the material constants E and μ the changes in shape of an isotropic elastic body can be described when an object is stretched or compressed along one dimension. Objects can be subjected to a compressive force in all directions. For this case a new material constant, the bulk modulus K, comprising E and μ, is introduced. The bulk modulus can be seen as an extension of Young's modulus E (see Elasticity 1: elastic or Young’s modulus) to three dimensions.

 

Derivation of K

When an elastic body, here as an example a cube (edge length b), is compressed from all sides by a pressure increase of ΔP, then the change of the edge length Δb is:

 

                              (3a)

The volume change is by approximation:

 

ΔV ≈ 3b2 Δb                                           (3b)

 

After substitution of Δb from (3b) in (3a) it follows that:

 

 

The bulk modulus K is defined as:

 

                                      (4a)

 

The value of determines whether K is smaller (brittle materials), about equal (many metals) or larger (highly elastic materials like rubber) than E. Finally, It follows that

 

                                      (4b)

 

This equation says that the higher the compressive strength, the smaller the volume changes.

The material constant K (in Pa) is known for many materials.