Elastic Constants

A material’s elastic constants are the ability of a material to return to its original shape after being deformed, or stretched. Elasticity is the measure of that ability. If a bullet is fired through the wall of an airplane, it expands the hole significantly due to elastic expansion. When released into a vacuum, it will retract back down to its original size due to elastic contraction.  The tensile strength and elongation of rubber bands depend on their elastic constants as well as their diameters.

Definition of Elastic Constants:

Elastic constants are the ratio of the elastic energy (or strain energy) per unit length to the elastic modulus, also referred to as Young’s Modulus. 

Young’s Modulus:

Young’s modulus is a ratio of stress to strain, a measure of the stiffness of a solid material. The units are typically megaPascals (MPa) or giga-Pascals (GPa). Young’s modulus for almost all materials tends to increase with temperature. Young’s modulus is named after the 19th century British scientist Thomas Young. It was first introduced in 1807 by French physicist Augustin Louis Cauchy for describing extension in elastic materials and later named “Young’s Modulus”.

Young modulus = Stress / Strain

Bulk Modulus:

A material’s bulk modulus measures its inherent stiffness. Alternatively, a material’s bulk modulus can be thought of as the ratio of the stress to the strain as applied at any point in the material. A material with a high bulk modulus will absorb a large amount of energy before deforming. 

Bulk modulus = Direct stress / Volumetric strain

Rigidity Modulus:

Rigidity modulus is a material property that represents the ratio of the stress to the strain in a material versus any given direction of deformation. Similar to strength, rigidity modulus can be used to compare substances with different Young’s Modulus. When comparing substances with different rigidity modulus, it is important to remember that “higher” rigidity will not necessarily mean “stronger.” 

Rigidity modulus = Shear stress / Shear strain

Permanent Stiffness:

Permanent stiffness or compressive stiffness is a material property that measures how much energy is required for a given load to deform the material. Unlike elasticity, permanent stiffness does not depend on the amount of strain applied, only on the initial and final load.

Poisson’s Ratio:

Poisson’s ratio is a material property that measures the ellipticity of a material. Poisson’s Ratio shows how much the cross-sectional area of a given shape will produce a particular stress in the material. It is important to realize that for many materials, there are two independent variables used to measure elasticity. These variables are stress and strain. For instance, a rubber band may exhibit elasticity due to its Young’s Modulus or due to its Strain Energy Or vice versa.

Poisson’s ratio = Lateral strain / Longitudinal strain

Relationship Between Elastic Constants:

The relation of these elastic constants is dependent on the material. The following equations represent general descriptions of the relationship between these properties. 

Example of elastic constants in action: 

The relationship between Young’s Modulus, Poisson’s Ratio and Bulk Modulus are demonstrated through the actions of atmospheric pressure in a balloon. Let us assume that our balloon has a diameter of 1 meter and is made out of rubber that has a Young’s Modulus of 5*109 N/m2, a Poisson’s Ratio of 0.28 and a bulk modulus of 2*108 N/m2. Also assume that our balloon is sitting in an atmosphere that has an atmospheric pressure exerted on it.

• The relation between Young’s modulus (E), rigidity modulus (G), and Poisson’s ratio (µ) is,

E = 2G(1+ µ)

• The relation between Young’s modulus (E), bulk modulus (K), and Poisson’s ratio (µ) is,

E = 3K(1-2µ)

• The relation between Young’s modulus (E), bulk modulus (K), and rigidity modulus (G) is,

E = 9KG/(3K+G)

• The relation between rigidity modulus (G), bulk modulus (K), and Poisson’s ratio (µ) is,

µ = (3K-2G)/(6K+2G)

Conclusion:

Elastic constants serve a very important function to our everyday life. As we can see, these constants can be used to measure the elasticity of different objects or materials. In addition, they are useful in the design of many objects that require elasticity to meet a given requirement. However, it is important to understand that elastic constants are dependent on the material, shape and size of an object. This means that there are many other factors that need to be considered when making qualitative and quantitative assessments of a material’s elasticity.