Coefficient Of Elasticity Dimensional Formula Derivation

Elasticity Coefficient Formula:

\[ E = \frac{\text{stress}}{\text{strain}} = \frac{F/A}{\Delta L/L} \]

Force (F):

Area (A):

m²

Change in Length (ΔL):

Original Length (L):

Young's Modulus (E):

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1. What is Coefficient of Elasticity?

The coefficient of elasticity, also known as Young's modulus, is a measure of the stiffness of a solid material. It defines the relationship between stress (force per unit area) and strain (proportional deformation) in a material in the linear elasticity regime of a uniaxial deformation.

2. Dimensional Formula Derivation

The dimensional formula derivation follows from the definition:

\[ E = \frac{\text{stress}}{\text{strain}} = \frac{F/A}{\Delta L/L} \]

Where:

\( E \) — Young's modulus (Coefficient of Elasticity)
\( F \) — Force applied [M L T⁻²]
\( A \) — Cross-sectional area [L²]
\( \Delta L \) — Change in length [L]
\( L \) — Original length [L]

Dimensional Analysis:
Stress = Force/Area = [M L T⁻²]/[L²] = [M L⁻¹ T⁻²]
Strain = ΔL/L = [L]/[L] = Dimensionless
Therefore: E = Stress/Strain = [M L⁻¹ T⁻²]

3. Physical Significance

Details: Young's modulus quantifies a material's resistance to elastic deformation under load. Higher values indicate stiffer materials that deform less under the same stress. It's a fundamental property in materials science and engineering design.

4. Using the Calculator

Tips: Enter force in Newtons, area in square meters, and lengths in meters. All values must be positive. The calculator computes stress, strain, and Young's modulus according to the standard formula.

5. Frequently Asked Questions (FAQ)

Q1: What is the SI unit of Young's modulus?
A: The SI unit is Pascal (Pa), which is equivalent to N/m². Common multiples include MPa and GPa.

Q2: How does Young's modulus differ from other elastic constants?
A: Young's modulus describes tensile/compressive elasticity, while shear modulus describes shear deformation, and bulk modulus describes volumetric compression.

Q3: What are typical Young's modulus values for common materials?
A: Steel: ~200 GPa, Aluminum: ~70 GPa, Concrete: ~30 GPa, Wood: ~10 GPa, Rubber: ~0.01-0.1 GPa.

Q4: When is the linear elastic assumption valid?
A: The linear relationship holds only within the elastic limit (proportional limit) of the material, before plastic deformation occurs.

Q5: Can Young's modulus be negative?
A: No, Young's modulus is always positive for stable materials. Negative values would imply unstable mechanical behavior.