What is the relation between E, G, and K?

Short Answer:

The relation between E (Young’s Modulus), G (Modulus of Rigidity), and K (Bulk Modulus) shows the interconnection among the three main elastic constants of a material. These constants describe how a material behaves under different types of loading conditions — tension, shear, and compression.

The mathematical relation between them is given by the formula:

This equation helps to find one constant if the other two are known, and it is derived using Poisson’s ratio (μ) as a link among them.

Detailed Explanation:

Relation between E, G, and K

Meaning of Elastic Constants:
Elastic constants describe the behavior of a material under various types of stresses and strains within the elastic limit. The three main elastic constants are:

  • E (Young’s Modulus): Measures stiffness under tensile or compressive stress.
  • G (Modulus of Rigidity or Shear Modulus): Measures resistance to shear stress.
  • K (Bulk Modulus): Measures resistance to uniform compression.

These constants are not independent of each other. They are related through Poisson’s ratio (μ), which connects lateral and longitudinal strains. The relationship helps engineers understand how a material will behave under combined stress conditions.

Step 1: Basic Relationships Involving Poisson’s Ratio

The relationships among the elastic constants and Poisson’s ratio are as follows:

  1. Between E and G:
  1. Between E and K:

Here,

  • E = Young’s modulus
  • G = Modulus of rigidity
  • K = Bulk modulus
  • μ = Poisson’s ratio

These two equations connect the three constants using Poisson’s ratio as a common term.

Step 2: Derivation of Relation between E, G, and K

To find a direct relation between E, G, and K, we eliminate μ (Poisson’s ratio) from the above two equations.

From the first equation,

 

Substitute this value of μ into the second equation:

 

Simplify the equation:

 

Multiply both sides by G:

Simplify further:

Rearrange terms to isolate E:

 

Finally,

This is the relation between Young’s modulus (E), modulus of rigidity (G), and bulk modulus (K).

Step 3: Understanding the Relation

This equation shows how the stiffness of a material under different loading conditions is related.

  • E (Young’s modulus) represents resistance to longitudinal deformation (tension/compression).
  • G (Modulus of rigidity) represents resistance to shear deformation.
  • K (Bulk modulus) represents resistance to volumetric compression.

If any two of these constants are known, the third can be easily calculated using this relationship. The relationship ensures consistency in the behavior of isotropic (uniform) materials where deformation is uniform in all directions.

Step 4: Physical Meaning of the Relation

This relation highlights that the three elastic constants are interdependent and determined by the internal atomic bonding of the material.

  • If K increases (material becomes less compressible), E increases — the material becomes stiffer.
  • If G increases (material resists shear deformation more), E also increases.
    Thus, both G and K contribute to the overall stiffness represented by E.

For isotropic materials, this relationship is always valid. However, for anisotropic materials (like composites or wood), separate directional constants must be used because their properties differ along different directions.

Step 5: Numerical Example

Let’s take an example to understand the practical use of this formula.

Given:

We know,

Substitute the values:

 

Hence, the Young’s modulus of the material is 120 GPa. This shows how the relation helps in material testing and analysis.

Step 6: Importance of the Relation in Engineering

  1. Material Testing and Characterization:
    It helps determine all elastic constants when only two are experimentally measured.
  2. Design and Analysis:
    In engineering design, this relation is used to calculate stiffness and deformation characteristics of structural components like beams, shafts, and columns.
  3. Stress-Strain Analysis:
    The relationship helps predict how a material behaves under different stress conditions — tension, compression, and shear.
  4. Finite Element Modeling (FEM):
    Material properties such as E, G, and K are inputs in simulations to predict stresses and strains in real-world components.
  5. Selection of Materials:
    Helps engineers compare stiffness, shear strength, and compressibility to select appropriate materials for different mechanical applications.

Step 7: Typical Values for Common Materials

Material E (GPa) G (GPa) K (GPa)
Steel 200 80 160
Aluminum 70 25 75
Copper 110 45 140
Rubber 0.01 0.0003 0.001

From these values, it is clear that materials with higher K and G have a higher E, meaning they are more rigid and less compressible.

Conclusion:

The relationship between E (Young’s modulus), G (modulus of rigidity), and K (bulk modulus) is given by the equation

This relation shows that these three elastic constants are interdependent and connected through the material’s mechanical properties. They represent how a material resists different types of deformation — tension, shear, and compression. Understanding this relation is essential in mechanical and structural design, allowing engineers to determine unknown properties, ensure material safety, and optimize performance under various loading conditions.