What is Poisson's ratio?

Poisson's ratio (nu) is the negative ratio of lateral strain to axial strain when a material is loaded uniaxially: nu = -epsilon_lateral / epsilon_axial. Most metals have nu of about 0.25 to 0.35. An incompressible material like rubber has nu close to 0.5. Cork is near 0 (no lateral expansion). Values range from -1 to 0.5 for stable materials.

What is the difference between stress and strain?

Stress (sigma) is force per unit area (Pa or psi) and represents the internal resistance of a material. Strain (epsilon) is the dimensionless ratio of deformation to original length (delta L / L). They are related by Hooke's Law: sigma = E * epsilon in the elastic range.

What is von Mises stress?

Von Mises stress is an equivalent stress combining all stress components using the distortion energy theory: sigma_vm = sqrt(0.5*((sigma1-sigma2)^2 + (sigma2-sigma3)^2 + (sigma3-sigma1)^2)). Yielding occurs when sigma_vm >= Sy. It is the most widely used yield criterion for ductile metals.

What is the elastic limit and proportional limit?

The proportional limit is the stress below which Hooke's Law applies (stress proportional to strain). The elastic limit is the stress below which the material returns to its original shape when the load is removed. For most metals these are very close together. Beyond the yield strength, permanent plastic deformation occurs.

Stress, Strain, Young’s Modulus & Poisson’s Ratio Calculator | CalcEngines

Mechanical Engineering Tools · CalcEngines

Stress, Strain, Young’s Modulus & Poisson’s Ratio Calculator

Solve for stress, strain, elastic modulus or deformation using Hooke’s Law. Includes 3D stress state, lateral strain, volumetric strain, von Mises stress, and principal stresses.

σ = Eε 4 solve modes 3D Hooke’s Law von Mises 20 materials SI & Imperial

Solve For

Quick Presets

Material

Material E = 200 GPa · ν = 0.30 · S_y = 250 MPa

Young’s Modulus E

Elastic modulus

Poisson’s Ratio ν Lateral / axial strain ratio

Applied Load Inputs

Stress σ

Axial normal stress

Strain ε Dimensionless (ΔL/L)

Original Length L

For deformation δ = ε×L

Deformation δ

Axial elongation

Yield & Geometry (Optional)

Yield Strength S_y

MPa

For safety factor check

Applied Force F

Optional — σ = F/A

Cross-section Area A

mm²

Optional — σ = F/A

Safety Factor S_f Applied to yield check

Imperial

Key Results

Enter values above and click Calculate.

All Stress & Strain Values

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Lateral Strain & Poisson Effects

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3D Stress State & Principal Stresses

Stress σ_y (transverse)

MPa

y-direction normal stress

Stress σ_z (out-of-plane)

MPa

z-direction normal stress

Shear Stress τ_xy

MPa

In-plane shear stress

Enter σ_x (from above) to compute principal stresses.

Mohr’s Circle (2D)

Mohr’s Circle Diagram

Calculate to see Mohr’s Circle data.

Material Reference

Material	E (GPa)	G (GPa)	ν	S_y (MPa)	S_u (MPa)	ρ (kg/m³)

Formula Reference

Quantity	Formula	Notes
Hooke’s Law	σ = E × ε	Elastic range only
Strain	ε = ΔL / L = σ / E	Dimensionless
Deformation	δ = ε × L = σL / E	Also: δ = FL / (AE)
Lateral strain (Poisson)	ε_lat = −νε_axial	ν = 0.25–0.35 for metals
3D Hooke’s Law (x)	ε_x = (σ_x − ν(σ_y+σ_z)) / E	Full triaxial form
Volumetric strain	ε_v = (1−2ν)(σ_x+σ_y+σ_z) / E	Dilatation
Principal stresses (σ_1,2)	(σ_x+σ_y)/2 ± √((σ_x−σ_y)²/4 + τ_xy²)	2D plane stress
Max shear stress	τ_max = (σ₁−σ₂) / 2	In-plane maximum
Von Mises stress	σ_vm = √(σ₁² − σ₁σ₂ + σ₂²)	Yield if σ_vm ≥ S_y
Shear modulus G	G = E / (2(1+ν))	From E and ν
Bulk modulus K	K = E / (3(1−2ν))	Volumetric stiffness

Common Questions

What is Young’s modulus and how is it measured? +

Young’s modulus (E) is the slope of the linear (elastic) portion of the stress–strain curve: E = σ / ε. It is measured by applying a known tensile or compressive load to a standardised specimen, recording the elongation, and computing σ = F/A and ε = ΔL/L. Steel is ~200 GPa, aluminium ~70 GPa, glass ~65 GPa, and rubber 0.01–0.1 GPa.

How does Poisson’s ratio affect 3D strain? +

When a bar is stretched axially (ε_x > 0), it contracts laterally: ε_y = ε_z = −νε_x. In a full 3D biaxial or triaxial state, each strain component is influenced by all three normal stresses: ε_x = (σ_x − ν(σ_y+σ_z))/E. This means compressive stresses on one axis can cause expansion on another.

What is principal stress and why does it matter? +

Principal stresses (σ₁, σ₂) are the maximum and minimum normal stresses at a point, found by rotating the stress element until shear stress is zero. They occur at the principal planes. Failure criteria (von Mises, Tresca) use principal stresses to predict yielding, which is why structural analyses always transform to principal components.

What is the difference between engineering and true stress/strain? +

Engineering stress uses the original cross-sectional area (F/A₀) and engineering strain uses the original length (ΔL/L₀). True stress and strain account for the actual changing geometry: σ_true = σ_eng(1+ε_eng). For small elastic strains (below ~1%), the difference is negligible. This calculator uses engineering definitions, which is standard for elastic analysis.

How is the safety factor for stress calculated? +

The static safety factor against yielding is SF = S_y / σ_applied for uniaxial loading, or SF = S_y / σ_vm for multiaxial loading using the von Mises criterion. SF > 2 is typical for static structural design; fatigue applications require higher values and use the endurance limit rather than yield strength.