ELASTIC CONSTANTS

1. What is Elastic Constant ?

Elastic constants are the material properties that define the relationship between stress and strain within the elastic limit of a material. When a material is loaded within this limit, it regains its original shape and size after removal of the load. These constants are extremely important in the design and analysis of machine parts, structures, and mechanical components.

2. Elasticity and Elastic Limit

• Elasticity is the property of a material by which it returns to its original dimensions after the applied load is removed.
• The elastic limit is the maximum stress up to which the material behaves elastically.
• Beyond the elastic limit, permanent deformation (plastic deformation) occurs.

3. Stress and Strain

Before understanding elastic constants, it is necessary to know stress and strain:

• Stress (σ) = Force / Area $$\sigma = \frac{P}{A}$$σ=AP​
• Strain (ε) = Change in dimension / Original dimension $$\varepsilon = \frac{\Delta L}{L}$$ε=LΔL​

4. Hooke’s Law

Hooke’s Law states that:

Within the elastic limit, stress is directly proportional to strain.

$$\sigma \propto \varepsilon \quad \Rightarrow \quad \sigma = E \varepsilon$$σ∝ε⇒σ=Eε

Where E is the modulus of elasticity.

5. Types of Elastic Constants

There are four main elastic constants:

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5.1 Young’s Modulus (E)

Definition:
Young’s modulus is the ratio of longitudinal stress to longitudinal strain within the elastic limit.$$E = \frac{\text{Longitudinal Stress}}{\text{Longitudinal Strain}} = \frac{\sigma}{\varepsilon}$$

Units: N/m² or Pascal (Pa)

Significance:

• Measures the stiffness of a material.
• Higher value of E indicates a stiffer material.

Examples:

• Steel → High Young’s modulus
• Rubber → Low Young’s modulus

5.2 Modulus of Rigidity or Shear Modulus (G)

Definition:
It is the ratio of shear stress to shear strain.$$G = \frac{\text{Shear Stress}}{\text{Shear Strain}}$$G=Shear StrainShear Stress​

Units: N/m²

Significance:

• Important for components subjected to torsion, such as shafts and springs.

5.3 Bulk Modulus (K)

Definition:
Bulk modulus is the ratio of volumetric stress to volumetric strain.$$K = \frac{\text{Volumetric Stress}}{\text{Volumetric Strain}}$$

Significance:

• Indicates resistance to change in volume.
• Mainly used for fluids and materials under hydrostatic pressure.

5.4 Poisson’s Ratio (μ or ν)

Definition:
Poisson’s ratio is the ratio of lateral strain to longitudinal strain.$$\mu = \frac{\text{Lateral Strain}}{\text{Longitudinal Strain}}$$μ=Longitudinal StrainLateral Strain​

Characteristics:

• Dimensionless quantity
• Typical values:
  • Steel: 0.25 – 0.30
  • Rubber: ≈ 0.5
  • Cork: ≈ 0

Significance:

• Important in multi-axial stress analysis.

6. Relationship Between Elastic Constants

Elastic constants are interrelated:

1. Relation between E and G

$$E = 2G(1 + \mu)$$E=2G(1+μ)

2. Relation between E and K

$$E = 3K(1 – 2\mu)$$E=3K(1−2μ)

3. Relation between G and K

$$G = \frac{3K(1 – 2\mu)}{2(1 + \mu)}$$G=2(1+μ)3K(1−2μ)​

These equations are valid for homogeneous and isotropic materials.

7. Stress–Strain Curve (Elastic Region)

8. Engineering Applications

Elastic constants are used in:

• Design of beams, shafts, springs, columns
• Structural analysis of bridges and buildings
• Machine design and vibration analysis
• Pressure vessels and fluid mechanics

9. Comparison of Elastic Constants

Elastic Constant	Symbol	Type of Deformation
Young’s Modulus	E	Axial (tension/compression)
Shear Modulus	G	Shear deformation
Bulk Modulus	K	Volumetric change
Poisson’s Ratio	μ	Lateral deformation