• Sooraj S Nair

Elastic Constants of Materials

In this article, we will look into some relations and constants with respect to structural materials that govern the properties and consequently their use in practical purposes.

Picture representing Elasticity of Material

Table of contents

  1. Introduction to Elastic Constants

  2. Types of Elastic Constants

  3. Young's Modulus of Elasticity (E)

  4. Shear Modulus or Modulus of Rigidity (G)

  5. Bulk Modulus (K)

  6. Poisson's Ratio (µ)

  7. Summary

  8. Short Video

Introduction to Elastic Constants

In the last article on Stress, Strain and Hooke's Law, we looked at some properties of materials that they demonstrate when they are exposed to external forces and how they behave accordingly. Elastic constants are those constants that determine the deformation produced by a given stress system acting on the material.

In short,

  • Elastic constants are used to determine engineering strain theoretically.

  • They are used to obtain a relationship between engineering stress and engineering strain.

  • For a homogeneous and isotropic material, the numbers of elastic constants are 4.

  • For non-isotropic or anisotropic materials have different properties in different directions. They show non- homogeneous behaviour. The number of elastic constants is 21.

Types of Elastic Constants

  • Young’s Modulus or Modulus of Elasticity (E)

  • Shear Modulus or modulus of rigidity (G)

  • Bulk Modulus (K)

  • Poisson’s Ratio (µ)

Young's Modulus Meme

Young's Modulus of Elasticity (E)

Now, let's first discuss Young’s modulus or modulus of Elasticity (E). The Young modulus, or the modulus of elasticity in tension, is a mechanical property that measures the tensile stiffness of a solid material.


Definition of Modulus of Elasticity

As per Hooke’s law, up to the proportional limit, “for small deformation, stress is directly proportional to strain.”

Mathematically, Hooke’s Law expressed as:

Stress α Strain

σ = E ε

In the formula as mentioned above, “E” is the constant of proportionality termed as Modulus of Elasticity.

σ is the Stress, and ε denotes Strain.

We can write the expression for Modulus of Elasticity using the above equation as,

So we can define modulus of Elasticity as the ratio of normal stress to longitudinal strain.

Unit of Modulus of Elasticity

The unit of normal Stress is Pascal, and longitudinal strain has no unit. Because longitudinal strain is the ratio of change in length to the original length. So the unit of Modulus of Elasticity is the same as of Stress, and it is Pascal (Pa). We use most commonly Megapascals (MPa) and Gigapascals (GPa) to measure the modulus of Elasticity.

1 MPa =106 Pa

1 GPa = 109Pa


Experimentally the value of Modulus of Elasticity (E) is equal to the slope of the Stress-strain curve up to Proportionality Limit. If the value of E increases, then longitudinal strain decreases, which means a change in length decreases.

Here are some values of E for the most commonly used materials.

  • Mild Steel E= 200 GPa

  • Cast Iron E= 100 GPa

  • Aluminium E= 200/3 GPa

What are its Applications?

  • It is used in engineering as well as medical science.

  • You can use the elastic modulus to calculate how much a material will stretch and also how much potential energy will be stored.

  • The elastic modulus allows you to determine how a given material will respond to Stress.

  • Elastic modulus is used to characterize biological materials like cartilage and bone as well.

Shear Modulus of Rigidity (G)

Shear modulus or Modulus of rigidity is the measure of the rigidity of the body, given by the ratio of shear stress to shear strain.

Definition of Shear Modulus

Shear Modulus of elasticity is one of the measures of mechanical properties of solids. The shear modulus of the material is the ratio of shear stress to shear strain in a body. It is the elastic constant that we get when a shear force is applied resulting in lateral deformation. It gives us a measure of how rigid a body is. Often denoted by G.

In short,

  • Measured using the SI unit pascal or Pa.

  • The dimensional formula of Shear modulus is [M¹ L⁻¹ T⁻²].

  • It is denoted by G.

Modulus Of Rigidity Formula

Where

  • τxy=FA is shear stress.

  • F is the force acting on the object.

  • A is the area on which the force is acting.

  • γxy=Δxl is the shear strain.

  • Δx is the transverse displacement.

  • l is the initial length.

Unit of Shear Modulus of Rigidity

  • The modulus of rigidity is measured using the SI unit pascal or Pa.

  • Commonly it is expressed in terms of GigaPascal (GPa).

Representation of Bulk Modulus

Bulk Modulus Of Elasticity

It is given by the ratio of pressure applied to the corresponding relative decrease in the volume of the material.

The Bulk elastic properties of material tell us how much a body will compress under a given amount of external pressure.


Definition of Bulk Modulus of Elasticity

The Bulk Modulus of a substance is a measure of how resistant to compression that substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume.

Mathematically, it is represented as follows:

B= (delta P)/(delta V/ V)
Bulk Modulus Formula

Where:

B: Bulk modulus

ΔP: change of the pressure or force applied per unit area on the material

ΔV: change of the volume of the material due to the compression

V: Initial volume of the material in the units in the English system and N/m2 in the metric system.

In short,

  • Measured using the SI unit Pascal (Pa) or Newton per square metre (N/m2).

  • The dimensional formula of Shear modulus is [M¹ L⁻¹ T⁻²].

  • It is denoted by K.

Applications of Bulk Modulus

  • Bulk modulus is used to measure how incompressible a solid is, the more the value of K for a material, the higher is its nature to be incompressible. For example, the value of K for steel is 1.6×10^11 N/m2 and the value of K for glass is 4×10^10N/m2. Here, K for steel is more than three times the value of K for glass. This implies that glass is more compressible than steel.

  • While in solids, the value of K varies in gases, as they are extremely compressible.

  • The concept of Bulk Modulus is also used in liquids.

  • Temperatures of fluid and entrained air content are the two factors highly controlled by the bulk modulus.


Poisson's Ratio

Poisson’s ratio is the negative of the ratio of transverse strain to lateral or axial strain.

Representation of Poisson's Ratio

Definition of Poisson’s Ratio

Poisson’s ratio is the ratio of transverse contraction strain to longitudinal extension strain in the direction of the stretching force.

Poisson's ratio is a measure of the Poisson effect. The Poisson ratio will be the ratio of relative contraction to relative expansion.

Poisson Effect

When a material is stretched in one direction, it tends to compress in the direction perpendicular to that of force application and vice versa. The measure of this phenomenon is given in terms of Poisson’s ratio. For example, a rubber band tends to become thinner when stretched.

Poisson’s ratio values for different material. It is the ratio of transverse contraction strain to longitudinal extension strain, in the direction of the stretching force. There can be a stress and strain relation that is generated with the application of force on a body.

  • For tensile deformation, Poisson’s ratio is positive.

  • For compressive deformation, it is negative.

In certain rare cases, the material will actually shrink in the perpendicular direction when compressed or expand when stretched, which will have a negative value of the Poisson ratio.

The formula for Poisson’s ratio is,

Poisson′s ratio=Transverse strain/Longitudinal strain

Poisson's ratio = ( Lateral Strain)/(Longitudinal Strain)
Poisson's Formula

where,

εt is the Lateral or Transverse Strain

εl is the Longitudinal or Axial Strain

µ is the Poisson’s Ratio

The strain on its own is defined as the change in dimension (length, breadth, area…) divided by the original dimension.

Here, the negative Poisson ratio suggests that the material will exhibit a positive strain in the transverse direction, even though the longitudinal strain is positive as well.

For most materials, the value of Poisson’s ratio lies in the range, 0 to 0.5.

Check out this simple video!

Summary:

  1. Different elastic constants are as follows :

  2. Young’s modulus

  3. Bulk modulus

  4. Rigidity modulus

  5. Poisson’s ratio.

  6. The ratio of applied stress to the strain is constant and is known as Young’s modulus or modulus of elasticity (E).

  7. Within its elastic limits, the ratio of direct stress to the corresponding volumetric strain is found to be constant. This ratio is called Bulk Modulus (K).

  8. The ratio of shear stress to the corresponding shear strain is called Rigidity Modulus or modulus of rigidity.

  9. The ratio of the lateral strain to the longitudinal strain is called Poisson’s ratio(µ)

So, in this article, we have learnt about the different Elastic Constants. In the upcoming articles, we will talk in detail regarding the relationship between these constants and how we can use these practical scenarios. Until then, be safe, wear a mask and always have a smile. Ciao!

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