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Young's Modulus Calculator

Calculate stress, strain, and Young's modulus from applied force, cross-sectional area, original length, and elongation, in pascals. Free.

Frequently Asked Questions

What is Young modulus?

Young modulus is a measure of a material stiffness under tension or compression, equal to stress divided by strain in the elastic region.

Why must I stay in the elastic region?

Young modulus only holds while deformation is recoverable. Past the yield point the material deforms permanently and the linear relationship no longer applies.

What units should I use for force and area?

Enter the force in newtons and the cross-sectional area in square millimeters; the calculator converts the area to square meters internally. With those inputs the stress comes out in megapascals (force divided by area) and Young's modulus in gigapascals (stress divided by strain). Strain is the change in length divided by the original length, so it is dimensionless and the length units cancel as long as both lengths use the same unit. As a sanity check, steel should land near 200 GPa and aluminum near 70 GPa.

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Estimates for informational purposes only.

Important Disclaimer: Estimates for informational purposes only.

This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

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How This Calculator Works

Young modulus measures the stiffness of a solid material under tension or compression. It tells you how much a material stretches for a given load and is a fundamental property used throughout structural and mechanical engineering.

The Formulas

Stress equals the applied force divided by the cross-sectional area, given here in megapascals. Strain equals the change in length divided by the original length and is dimensionless. Young modulus equals stress divided by strain, reported in gigapascals. The area is entered in square millimeters and converted to square meters internally.

Worked Example

A steel rod with a 100 mm² cross-section and 1000 mm original length is pulled with 10,000 N and stretches 0.5 mm.

Stress = F ÷ A = 10,000 N ÷ (100 mm² = 1×10^-4 m²) = 1×10⁸ Pa = 100 MPa.

Strain = ΔL ÷ L = 0.5 ÷ 1000 = 0.0005 (dimensionless).

Young's modulus E = stress ÷ strain = 1×10⁸ ÷ 0.0005 = 2×10¹¹ Pa = 200 GPa, exactly the expected value for steel.

Reading The Result

A higher Young modulus means a stiffer material that deforms less under load. Steel sits near 200 gigapascals, aluminum near 70, and rubber far lower. The relationship holds only in the elastic region, before the material yields permanently.

Tips

Use measurements from within the elastic, recoverable range.
Keep force in newtons and area in square millimeters.
Strain is unitless; stress and modulus share pressure units.

Young's Modulus Across Materials

Young's modulus spans roughly seven orders of magnitude across engineering materials, and understanding where a material sits on this spectrum is central to structural and mechanical design.

Steels (all grades): 190 to 210 GPa. The modulus is nearly the same for mild steel, high-strength steel, stainless steel, and tool steel. Alloying and heat treatment dramatically change yield strength but barely move E. This is why steel structures have predictable deflection regardless of grade.
Aluminum alloys: 69 to 72 GPa, roughly one-third that of steel. Aluminum structures deflect approximately three times more than equivalent steel structures under the same load, which often makes stiffness rather than strength the governing design criterion for aluminum.
Carbon fiber reinforced polymer (CFRP): 70 to 200 GPa depending on fiber orientation and volume fraction. Along the fiber direction, high-modulus carbon fibers can reach 400 GPa. Transverse to the fibers the modulus drops to near that of the polymer matrix (3 to 5 GPa), making CFRP strongly anisotropic - it must be oriented to align fibers with the principal stress direction.
Titanium alloys: 110 to 120 GPa. About 60% of steel's modulus at roughly 57% of steel's density, giving a slightly better specific stiffness than steel and much better corrosion resistance.
Concrete: 25 to 40 GPa in compression (tensile E is similar but concrete is not used in tension without reinforcement). Lower modulus and high density mean concrete structures deflect more than steel for the same geometry.
Cortical bone: 10 to 20 GPa along the bone axis. Bone is anisotropic and viscoelastic, with stiffness increasing at higher loading rates. Implant materials are often chosen to match bone stiffness and avoid stress shielding, where a stiffer implant carries load that should be stimulating the bone.
Rubber and elastomers: 0.001 to 0.1 GPa. These are not linear-elastic, so the "modulus" is a secant or tangent value at a particular strain level rather than a constant. Rubber is essentially incompressible (Poisson's ratio ≈ 0.5).

Stiffness and strength are distinct properties that do not necessarily correlate. Diamond has the highest modulus of any natural material (around 1,000 GPa) but is brittle and fractures easily. Rubber has very low modulus but can stretch 500% before failure. High-strength steels have higher yield strength than mild steel but the same modulus. When selecting materials, both properties must be evaluated for the application's requirements.

Measuring Young's Modulus

The standard method is the tensile test (ASTM E8 for metals, ISO 6892 equivalent). A machined specimen is gripped at both ends, loaded in uniaxial tension, and the force and elongation are recorded simultaneously. An extensometer clipped directly to the gauge length provides accurate strain measurement; relying on crosshead displacement introduces significant error from machine compliance and grip slip.

The elastic modulus is the slope of the linear portion of the stress-strain curve, typically measured between 10% and 40% of the expected yield stress to avoid the nonlinear toe at the start of the curve. For materials with no clear linear region, a secant modulus or chord modulus between two defined strain points is used instead.

Dynamic measurement methods are sometimes preferred because they are non-destructive and more precise. The resonant frequency method (ASTM E1876) excites a specimen at its natural frequency and calculates E from the resonance frequency, mass, and geometry. Ultrasonic pulse-echo methods measure the acoustic wave velocity through the material and compute E from the wave speed and density. These methods give values that may differ slightly from quasi-static tensile test results because viscoelastic and anelastic effects contribute to the quasi-static value.

For thin films and nano-scale structures, nanoindentation (instrumented indentation testing) presses a diamond tip into the surface and extracts modulus and hardness from the load-displacement curve using the Oliver-Pharr method. Reported E values are often ranges rather than single numbers because grain orientation, processing history, temperature, and minor alloying variations all shift the measured value by a few percent.

Engineering Applications

Young's modulus appears in virtually every structural and mechanical calculation that involves elastic deformation. Beam deflection under transverse loads scales with 1/(EI): doubling E halves deflection for the same cross-section. This is why long-span structural members in bridges and aircraft use high-modulus materials even when strength is not the limiting factor.

Natural vibration frequency is proportional to √(E/ρ), where ρ is density. This ratio, called specific stiffness, determines how stiff a structure can be per unit weight. CFRP has a specific stiffness roughly five times that of steel, which is why it dominates in aerospace and high-performance sporting goods where resonant frequency must be maximized with minimal mass.

Thermal stress arises when a material is constrained from expanding or contracting with temperature: σ = E·α·ΔT, where α is the coefficient of thermal expansion. In electronics, mismatched E and α between substrate, solder joint, and package create cycling thermal stresses that eventually cause solder fatigue failure. Choosing materials with matched specific stiffness and thermal expansion is a key objective in precision instrument design.

Spring-back in metal forming occurs because the elastic strain component recovers after the tooling is removed. The amount of spring-back is proportional to yield stress / E. High-strength steels and titanium alloys spring back significantly more than mild steel, requiring the tooling to over-bend the part to achieve the target geometry.

Composite Stiffness

For fiber-reinforced composites, the rule of mixtures gives the longitudinal modulus: E_L = V_fE_f + V_mE_m, where V_f and V_m are the fiber and matrix volume fractions. Transverse modulus follows the inverse rule of mixtures: 1/E_T = V_f/E_f + V_m/E_m. The anisotropy ratio E_L/E_T can be 10:1 or higher for unidirectional carbon/epoxy laminates. Quasi-isotropic layups (equal proportions of fibers at 0, 45, -45, and 90 degrees) give an in-plane modulus of roughly 38% of the unidirectional longitudinal value, but the same in all in-plane directions.