Learn Relation between Elastic Constants
We know that depending upon the type of stress being applied and the resulting strain, the modulus of elasticity had been classified into the following three types.
Young’s Modulus (E): The ratio of longitudinal stress to longitudinal strain.
Bulk Modulus (K): The ratio of volumetric stress to volume strain.
Shear Modulus (G): The ratio of shear stress to shear strain.
All three elastic constants can be interrelated by deriving a relation between them known as the Elastic constant formula. But young’s modulus (E) and the Poisson ratio(𝝂) are known as the independent elastic constants and they can be obtained by performing the experiments.
The bulk modulus and the shear modulus are dependent constants and they are related to Young’s modulus and the Poisson ratio.
The relation between Young’s modulus and shear modulus is
⇒ E = 2G(1 + v)N/m2
The relation between Young’s modulus and Bulk modulus is:
⇒ E = 3K(1 - 2v) N/m2
Elastic Constant Formula
The relation between different elastic constants is achieved by a small derivation. For the derivation of the relation between elastic constants, we will use the relation between Young’s modulus and the bulk modulus and also the relation between Young’s modulus and the shear modulus.
Derivation of Relation Between Elastic Constants
Consider the relation between Young’s modulus and the shear modulus,
⇒ E = 2G(1 + v)N/m2 ……….(1)
Where,
E - Young’s modulus
G - Shear modulus
v - Poisson ratio
From equation (1) the value of the Poisson ratio is:
⇒ \[v = \frac{E }{2G} - 1 \]……….(2)
We know that the relation between Young’s modulus and the Bulk modulus is
⇒ E = 3K(1 - 2v) N/m2 …………..(3)
Where,
E - Young’s modulus
K - Bulk modulus
v - Poisson ratio
Substituting the value of Poisson ration from equation (2) in (3) and simplify,
⇒\[E = 3K(1 - 2\frac{E} {2G}−1)\]
⇒\[E = 3K(1 -\frac{E}{G}-2) \]
\[E = 3K(3 -\frac{E}{G})\]
Equation (4) is known as the Elastic constant formula and it gives the Relation between elastic constants.
On further simplification,
⇒ \[ E = 9K -\frac{3KE}{G}\]
Taking LCM of G and on cross multiplication,
⇒ EG + 3KE = 9KG
⇒ E(G + 3K) = 9KG
On rearranging the above expression,
⇒ \[ E = \frac{9KG}{G+3K}\] N/m2 ………..(4)
Where,
E - Young’s modulus
G - Shear modulus
K - Bulk modulus
Equation (4) is known as the Elastic constant formula and it gives the Relation between elastic constants.
Did You Know?
The relationship between different elastic constants is also given by the expression,
⇒\[\frac{1}{K} -\frac{3}{G} = \frac{9}{E}\]
Where,
E - Young’s modulus
G - Shear modulus
K - Bulk modulus
These are the different ways of writing the relationship between elastic constants, depending upon the need for the solution we should utilize the formulas.
Elastic Constants:
Elastic constants are the constants that describe the mechanical response of a material when it is elastic. The elastic constant represents the elastic behaviour of objects.
Different Elastic Constants are as Follows:
Young’s modulus
Bulk modulus
Rigidity modulus
Poisson’s ratio
Young’s Modulus
Young's modulus is based on the elastic constant which is defined as the proportionality constant between stress and strain.
Bulk Modulus
Bulk Modulus of Elastic Constants is one of the measures for mechanical properties of solids. It is explained as having the ability of a material to resist deformation in terms of change in volume at the time of subject compression under pressure.
Rigidity Modulus
The modulus of rigidity that is also known as shear modulus is defined as the measure of elastic shear stiffness of a material. This property depends on the material of the member which means the more elastic the member, the higher the modulus of rigidity.
Poisson’s Ratio
Poisson's ratio is defined as the ratio of the change in the width per unit width of a material in order to the change in its length per unit length which will be given as a result of strain.
Relationship between Elastic Constants
Young’s modulus, bulk modulus and Rigidity modulus of an elastic solid together can be explained as Elastic constants. In addition to this when a deforming force is acting on a solid that will result in the change in its original dimension. In such cases, we can use the relation between elastic constants to understand the magnitude of deformation.
FAQs on Relation Between Elastic Constants
1. What are elastic constants and what do they represent?
Elastic constants are physical properties of a material that measure its resistance to being deformed elastically when a force is applied. They define the relationship between stress (force per unit area) and strain (the resulting deformation). In essence, a higher elastic constant means a material is stiffer and deforms less under a given load.
2. What are the four primary elastic constants used in Physics?
The four main elastic constants that describe the elastic behaviour of an isotropic material are:
- Young's Modulus (E): Measures the resistance to linear stretching or compression.
- Shear Modulus (G) or Modulus of Rigidity: Measures the resistance to shearing or twisting deformation.
- Bulk Modulus (K): Measures the resistance to a change in volume when subjected to uniform pressure from all sides.
- Poisson's Ratio (ν): Describes the ratio of transverse strain to axial strain, indicating how much a material narrows when it is stretched.
3. What is the fundamental formula that connects Young's Modulus (E), Bulk Modulus (K), and Shear Modulus (G)?
The key relationship connecting these three important elastic constants is given by the formula:
E = 9KG / (3K + G)
This equation is crucial because it shows that these constants are not independent of each other. If you know the values of any two, you can calculate the third one for a given material.
4. How does Poisson's ratio (ν) relate to Young's Modulus (E) and Shear Modulus (G)?
The relationship between Young's Modulus, Shear Modulus, and Poisson's ratio is expressed by the formula: E = 2G(1 + ν). This equation is particularly useful in engineering and material science as it connects a material's stiffness in tension (E) with its stiffness in shear (G) through its lateral contraction property (ν).
5. Why is it important for engineers to understand the relationships between elastic constants?
Understanding these relationships is critical for designing safe and reliable structures and components. For example, an engineer designing a bridge beam must use Young's Modulus (E) to predict how much it will bend under a load. They also need Shear Modulus (G) to understand how it resists twisting forces. The inter-relations allow them to predict a material's complete behaviour under complex stresses, ensuring the material chosen is suitable and will not fail unexpectedly.
6. Why are Young's Modulus (E) and Poisson's ratio (ν) often called 'independent' elastic constants?
For an isotropic material (one with uniform properties in all directions), only two elastic constants are needed to fully define its elastic behaviour. Young's Modulus (E) and Poisson's Ratio (ν) are typically chosen as the independent pair because they can be most easily and directly measured from a standard tensile test. Once E and ν are known from experiments, other constants like the Bulk Modulus (K) and Shear Modulus (G) can be mathematically calculated using the relationship formulas, making them 'dependent' constants.
7. Can an elastic constant have a negative value?
Generally, Young's Modulus (E), Shear Modulus (G), and Bulk Modulus (K) must be positive, as a positive stress should result in a positive strain in the same direction, indicating resistance to deformation. However, Poisson's ratio (ν) can be negative for certain exotic materials known as auxetic materials. A negative Poisson's ratio means that when the material is stretched, it gets thicker in the perpendicular direction, which is counter-intuitive to how most common materials behave.
