Elastic Modulus Calculator

The modulus of elasticity, or Young's modulus, is the stiffness of a material: how much stress it takes to produce a given strain.

$E = \frac{\sigma}{\varepsilon} = \frac{F/A}{\Delta L / L_0} = \frac{F L_0}{A\, \Delta L}$

• $E$ — Young's modulus, pascals (Pa = N/m²); quoted in GPa because the numbers are large
• $\sigma$ — tensile stress, Pa; force per unit original cross-sectional area
• $\varepsilon$ — strain, dimensionless; extension divided by original length
• $F$ — axial load, newtons (N); $A$ — cross-sectional area, m²
• $L_0$ — original length, m; $\Delta L$ — extension, m

Typical values: steel about 200 GPa, aluminium 70 GPa, concrete 30 GPa, rubber under 0.1 GPa.

When it applies: only in the linear elastic region, below the proportional limit, where Hooke's law holds and the specimen returns to its original length on unloading. Past the yield point $E$ is no longer the slope of anything useful.

The assumption people forget: strain has no units, so $E$ carries the units of stress. Report it in Pa or GPa, never in N or N/m.

On a stress-strain curve, $E$ is the gradient of the straight initial portion:

$E = \frac{\sigma_2 - \sigma_1}{\varepsilon_2 - \varepsilon_1}$

Pick two points that are clearly on the straight part, not near the yield knee. Because strain is dimensionless, the gradient comes out directly in the stress unit — MPa divided by a pure number is MPa.

Young's modulus has two siblings, all defined as stress over strain but for different loadings:

• Shear modulus $G = \tau/\gamma$ — twisting or sliding
• Bulk modulus $K = -\Delta P/(\Delta V/V_0)$ — uniform compression

For an isotropic material they are linked through Poisson's ratio $\nu$:

$G = \frac{E}{2(1+\nu)}, \qquad K = \frac{E}{3(1-2\nu)}$

When it applies: the isotropic relations fail for composites, timber and rolled metals, whose stiffness depends on direction.

• Leaving lengths in millimetres — convert the diameter, area and extension to metres before dividing, or the modulus lands orders of magnitude out.
• Using diameter instead of radius in the area — $A = \pi d^2/4$, or $\pi r^2$ with $r = d/2$. Using $\pi d^2$ makes the area four times too big.
• Giving strain a unit — it is metres per metre, which is a pure number. A strain of 0.075% is $7.5 \times 10^{-4}$.
• Taking the gradient across the yield point — only the initial straight segment gives $E$.
• Confusing stiffness with strength — $E$ describes resistance to elastic deformation, not the stress at which the material breaks.
• Using the deformed area — engineering stress uses the original cross-section throughout.