Short Answer:
The relation between the three elastic constants — Young’s Modulus (E), Modulus of Rigidity (G), and Bulk Modulus (K) — helps in understanding the elastic behavior of materials. These constants are interrelated through Poisson’s ratio (μ), which represents the lateral strain to longitudinal strain ratio. The general relation is given by:
This equation allows us to calculate one modulus if the other two and Poisson’s ratio are known. It shows how the deformation characteristics of a material are linked through its elasticity properties.
Detailed Explanation:
Relation between E, G, and K
In the study of elasticity, three important constants are used to describe the mechanical behavior of materials under stress. These are Young’s Modulus (E), Modulus of Rigidity (G), and Bulk Modulus (K). Each of these constants measures a specific type of deformation, but since they all describe the same material, they are related through the material’s Poisson’s ratio (μ).
Let us briefly recall what each constant represents before deriving their relationship:
- Young’s Modulus (E): It is the ratio of normal stress to normal strain within the elastic limit when a material is stretched or compressed. It represents the stiffness of a material in tension or compression.
- Modulus of Rigidity (G): It is the ratio of shear stress to shear strain within the elastic limit. It represents how resistant a material is to shape deformation when subjected to tangential forces.
- Bulk Modulus (K): It is the ratio of volumetric stress to volumetric strain, showing the resistance of a material to uniform compression.
Relation Using Poisson’s Ratio (μ)
Poisson’s ratio (μ) is a property that connects the lateral strain and longitudinal strain in an elastic material when subjected to axial stress. It plays a vital role in linking the three moduli.
From the basic relations of elasticity, we know that:
- Relationship between E, G, and μ:
- Relationship between E, K, and μ:
These two equations can be combined to eliminate μ and derive a direct relation between E, G, and K.
Derivation of the Relation
Starting with:
and
From equation (i):
Substitute this value of μ into equation (ii):
Simplify the equation:
Expanding further:
Rearranging the terms:
Simplify to get the final relationship:
Interpretation of the Relation
This equation shows the interdependence between the three elastic constants. It means that if any two of them are known, the third one can be easily calculated.
- When the bulk modulus (K) is large, it indicates the material is difficult to compress.
- When the modulus of rigidity (G) is large, the material resists deformation by shear.
- Young’s modulus (E) represents overall stiffness.
Thus, their relationship reflects how compressibility, shear resistance, and stiffness are linked together.
Special Case (For Steel Example)
For steel:
Then using , we get:
Using , we get:
Finally, applying the derived formula :
This confirms that the relation holds true for real materials like steel.
Conclusion:
The relation between E, G, and K is a fundamental expression in elasticity that connects the stiffness, rigidity, and compressibility of a material. The equation
shows that these three constants are not independent. Understanding this relation helps engineers to calculate the elastic behavior of materials under different types of loads without needing to measure all constants experimentally.