2025-08-21 14 min

Poisson's Ratio Calculator

Calculate Poisson's ratio (v = -e_lateral/e_axial) from lateral and axial strain measurements.

Calculator

Unit System

Preset Values

Lateral Strain (e_lateral) Help

Strain perpendicular to loading (negative for contraction)

Axial Strain (e_axial) Help

Strain in the loading direction (positive for tension)

Decimal Precision

Results update live as you type

Results

Normal

Poisson's Ratio

0.3

Formula

Poisson's ratio equals negative lateral strain divided by axial strain.

v = -e_lateral / e_axial

Variables

Symbol	Name	Unit	Description
v	Poisson's Ratio	dimensionless	Measure of lateral-to-axial strain coupling
e_lateral	Lateral Strain	dimensionless	Strain perpendicular to loading direction
e_axial	Axial Strain	dimensionless	Strain in the loading direction

Explanation

Poisson's ratio describes the tendency of a material to thin when stretched. A higher v means more lateral contraction for the same axial stretch.

Derivation

From the definition: when a bar is loaded in tension with axial strain e_axial, the lateral dimensions decrease. The ratio of lateral to axial strain is defined as -v. Rearranging: v = -e_lateral/e_axial.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Measure the axial strain e_axial in the loading direction.

Result: Axial strain measured

Step 2

Measure the lateral strain e_lateral perpendicular to loading.

Result: Lateral strain measured

Step 3

Apply formula: v = -e_lateral / e_axial.

Result: Poisson's ratio computed

Step 4

Verify the result is physically reasonable (-1 < v < 0.5).

Result: Result validated

Final Answer

0.3

Worked Examples

See how the calculator works with different scenarios.

Scenario

Steel specimen (v≈0.30) under axial strain of 0.001. Calculate lateral strain.

Inputs

Axial Strain:0.001

Poisson's Ratio (known):0.30

Calculation

e_lateral = -v × e_axial = -0.30 × 0.001

Result

e_lateral = -0.0003

Interpretation

The specimen contracts by 0.03% in the lateral direction while stretching 0.1% axially.

Understanding Your Result

Your Result: Poisson's Ratio: 0.3

Typical range: Metals: 0.25-0.35. Ceramics: 0.15-0.25. Polymers: 0.35-0.50. Rubber: 0.4999. Cork: ~0.00. Auxetic: -0.7 to 0.0.

What This Means

Poisson's ratio (v) measures how much a material thins when stretched. A v of 0.30 means that for every 1% stretch in one direction, the material contracts by 0.30% in the perpendicular direction.

Why It Matters

Poisson's ratio connects all three elastic constants (E, G, K). It affects beam deflection, contact mechanics, wave propagation, and multiaxial stress states. Accurate v is essential for FEA.

Professional Interpretation

Compare against literature values. For isotropic materials: v = E/(2G) - 1. For anisotropic materials, multiple Poisson's ratios exist for different loading/direction combinations.

Recommendations

•Use dual extensometers for simultaneous measurement
•For composites, measure v12 and v21 separately
•Verify v satisfies -1 < v < 0.5
•Account for temperature dependence

Educational Content

Introduction

Poisson's ratio is one of the three fundamental elastic constants for isotropic materials. It quantifies the coupling between axial and lateral deformation.

Concept Explanation

v = -e_lateral/e_axial. For isotropic materials: -1 < v < 0.5. Related to other constants: G = E/(2(1+v)), K = E/(3(1-2v)). The sign convention ensures v is positive for most materials.

Applications

▸Multiaxial stress analysis

▸FEA material modeling

▸Contact mechanics

▸Wave propagation

▸Composite laminate analysis

Benefits

•Connects all elastic constants
•Predicts lateral deformation
•Essential for FEA input
•Affects all multiaxial calculations

Limitations

•Assumes isotropy for single v
•Temperature dependent
•May change with strain level
•Anisotropic materials need multiple values

Best Practices

•Measure v at the same conditions as E
•For composites, test both v12 and v21
•Use DIC for full-field v measurement
•Report loading direction for anisotropic materials

Common Misconceptions

Myth: Poisson's ratio is always positive

Truth: Auxetic materials have negative v, expanding laterally when stretched.

Myth: v = 0.5 means perfectly rigid

Truth: v = 0.5 means perfectly incompressible (constant volume), not rigid. Rubber has v≈0.5 but is very flexible.

Practical Tips

•Steel: v = 0.27-0.30. Aluminum: v = 0.33. Rubber: v ≈ 0.5
•v = 0.5 means incompressible (constant volume deformation)
•v = 0 means no lateral deformation (cork)
•v < 0 means auxetic (expands laterally when stretched)
•G = E/(2(1+v)) and K = E/(3(1-2v)) link all elastic constants

Common Mistakes

Ignoring sign convention

v = -e_lateral/e_axial. For tension, e_axial > 0 and e_lateral < 0, giving positive v.

Assuming constant v for composites

Composites are anisotropic. v12 ≠ v21 for most laminate configurations. Measure each independently.

Using v from wrong temperature

v changes with temperature, especially for polymers. Use values at operating temperature.

Assumptions

This calculator makes the following assumptions to provide estimates:

Isotropic Material: Single v value assumes isotropy. Anisotropic materials need multiple values.

Small Strains: Linear elasticity assumption. v may change at large strains.

Homogeneous: Uniform v throughout the material volume.

Limitations

Be aware of these limitations when using this calculator:

Anisotropic Materials: Requires multiple v values for different loading/direction combinations.

Nonlinear Behavior: v may change with stress/strain level in the nonlinear region.

Temperature Dependence: v varies with temperature, particularly for polymers near Tg.

Frequently Asked Questions

Answers to common questions about this calculator.

What is Poisson's ratio?

Poisson's ratio (v) is the ratio of lateral strain to axial strain: v = -e_lateral/e_axial. When you stretch a material, it gets thinner in the transverse direction. v quantifies this effect.

What is the formula for Poisson's ratio?

v = -e_lateral / e_axial, where e_lateral is the transverse (perpendicular) strain and e_axial is the longitudinal (axial) strain. The negative sign ensures v is positive for most materials.

What are typical values of Poisson's ratio?

Steel: 0.27-0.30. Aluminum: 0.33. Copper: 0.33. Rubber: 0.4999 (nearly incompressible). Concrete: 0.15-0.25. Cork: ~0.00. Auxetic materials: negative (-0.1 to -0.7).

Why is Poisson's ratio always less than 0.5?

For isotropic materials, v must be between -1 and 0.5 for thermodynamic stability. v = 0.5 means perfectly incompressible (constant volume). Values above 0.5 violate energy stability.

What is a negative Poisson's ratio?

Auxetic materials expand laterally when stretched (and contract when compressed). Examples include re-entrant foams, certain polymers, and some geological formations. They have v < 0.

What is the relationship between v, E, and G?

G = E/(2(1+v)). For steel (E=200 GPa, v=0.3): G = 200/(2×1.3) = 76.9 GPa. Knowing any two of E, G, v determines the third.

What is the relationship between v and K?

K = E/(3(1-2v)). For incompressible materials (v=0.5), K is infinite. This is why rubber (v≈0.5) has very high K but low E.

What does v = 0.5 mean?

v = 0.5 means the material is perfectly incompressible (constant volume under deformation). Rubber is nearly incompressible (v≈0.4999). Water has v = 0.5 exactly.

What does v = 0 mean?

v = 0 means no lateral strain when axially loaded. Cork has v≈0 because its cellular structure compresses without lateral expansion. This is why cork works well as a bottle stopper.

Does Poisson's ratio change with temperature?

Yes, but the change is usually small. For metals, v typically increases slightly with temperature. For polymers near Tg, v changes significantly. At very low temperatures, v decreases.

How is Poisson's ratio measured experimentally?

Methods include: (1) Two extensometers measuring axial and lateral strain simultaneously, (2) Strain gauges at 0 and 90 degrees, (3) DIC (Digital Image Correlation) full-field measurement.

What is effective Poisson's ratio for composites?

For unidirectional composites, v12 (loading along fibers) is typically 0.25-0.35. v21 (loading transverse to fibers) is much smaller: v21 = v12×E2/E1. Composites are anisotropic.

What is the Poisson effect in beams?

When a beam bends, the top fibers stretch (positive e) and contract laterally (negative e_lat). The bottom fibers compress and expand laterally. This creates anticlastic curvature.

How does Poisson effect affect pressurized vessels?

In thin-walled vessels, hoop stress creates axial strain through Poisson effect: e_L = (s_h - v×s_L)/E = (pD/4tE)(1-2v). This affects seal design and nozzle reinforcement.

What is anisotropic Poisson's ratio?

In anisotropic materials, different Poisson's ratios exist for different loading/direction combinations. For orthotropic materials: v12, v13, v23, v21, v31, v32 (6 independent values).

What is the Poisson ratio of cork?

Cork has v≈ 0.00 because its closed-cell foam structure compresses internally without lateral expansion. This is ideal for bottle stoppers (compresses to fit without pushing out).

What is the Poisson ratio of rubber?

Rubber has v≈ 0.4999 (nearly incompressible). This means it maintains almost constant volume during deformation. The slight deviation from 0.5 accounts for the small compressibility of rubber.

What is the Poisson ratio of auxetic materials?

Auxetic materials have v < 0, meaning they expand laterally when stretched. Examples: re-entrant foam (v = -0.7), certain molecular structures (v = -0.16). They have improved indentation resistance.

How does Poisson ratio affect natural frequency?

Natural frequency depends on E and G, which both depend on v. Higher v increases G for fixed E, raising shear-dominated natural frequencies. It also affects mode shapes in plates and shells.

What is the Poisson ratio of concrete?

Concrete v = 0.15-0.25 depending on mix design and stress level. At higher stress levels, microcracking reduces effective v. Normal-weight concrete: v≈ 0.20. Lightweight concrete: v≈ 0.15.

What is the Poisson ratio of bone?

Cortical bone: v = 0.20-0.40 depending on loading direction. Cortical bone is anisotropic: v12≈0.37, v13≈0.46. These values are important for implant design.

What is the Poisson ratio used for in seismic analysis?

Poisson's ratio affects seismic wave velocities. P-wave velocity depends on both K and G through v. Rock v typically ranges from 0.1 (sandstone) to 0.4 (clay), used to identify subsurface materials.

How does Poisson ratio affect buckling load?

Euler buckling load: P_cr = pi²×E×I/L², independent of v. However, v affects I through cross-section geometry changes and affects lateral-torsional buckling through G = E/(2(1+v)).

What is the Poisson ratio of diamond?

Diamond has v = 0.07 (extremely low). This is due to its very stiff covalent bond network that resists lateral deformation. Diamond also has the highest E (1200 GPa) of any material.

What is the Poisson ratio of gold?

Gold has v = 0.44 (relatively high for a metal). This is because gold is a soft, ductile metal with relatively weak metallic bonds compared to transition metals like steel.

What is the Poisson ratio of lead?

Lead has v = 0.44 (similar to gold). This high value reflects lead's extreme ductility and softness. It also explains lead's effectiveness as a vibration damping material.

How does Poisson ratio affect contact mechanics?

In Hertzian contact, the contact radius and pressure distribution depend on the combined Poisson's ratio of both surfaces. Higher v increases the contact area for a given load.

What is the Poisson ratio of geological materials?

Rocks vary widely: sandstone v = 0.1-0.25, limestone v = 0.25-0.35, granite v = 0.2-0.25, clay v = 0.3-0.45, soil v = 0.3-0.5. These values are critical for geotechnical design.

What is the Poisson ratio of a foam?

Foam v depends on relative density. Open-cell foams: v ≈ 0.2-0.3 (below dense solid value). Closed-cell foams: v ≈ 0.3-0.5 depending on cell gas pressure. Auxetic foams: v < 0.

How does Poisson ratio affect thermal stress?

Thermal stress in constrained structures: s = EαΔT/(1-v) for biaxial constraint. Higher v increases thermal stress. For full 3D constraint: s = EαΔT/((1+v)(1-2v)/(1-v)).

What is the theoretical basis for v = 0.5?

For an incompressible material, volume is conserved: dV/V = ex + ey + ez = 0. For uniaxial loading with ex = e: ey = ez = -v×e. Sum = 0 gives: e(1 - 2v) = 0, so v = 0.5.

What is the Poisson ratio of a fabric or textile?

Fabrics can have effective negative Poisson's ratio depending on weave pattern. Some engineered textiles and knitted fabrics exhibit auxetic behavior due to re-entrant geometry under tension.

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Poisson's Ratio (v): Ratio of lateral strain to axial strain: v = -e_lat/e_ax. Dimensionless.
Auxetic Material: Material with negative Poisson's ratio. Expands laterally when stretched.
Incompressible: Material with v = 0.5. Maintains constant volume during deformation.
Anticlastic Curvature: Curvature perpendicular to beam bending caused by Poisson effect.

Authoritative References

Trusted sources and standards referenced by this calculator.

Hibbeler, Mechanics of Materials Engineering Toolbox - Poisson's Ratio

Applicable Standards

ASTM E132 - Poisson's Ratio at Room TemperatureISO 527 - Plastics tensile properties

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Last Updated:

2025-08-21

Version:

2.0.0

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Free to use for educational and professional purposes.