Reduced (effective) elastic modulus of two contacting bodies

What does this quantity describe?

When two bodies touch, both deform elastically. The reduced elastic modulus E* combines both material stiffnesses into a single effective value: 1/E* = (1−ν1²)/E1 + (1−ν2²)/E2, using elastic moduli E1, E2 and Poisson's ratios ν1, ν2 of both contacting bodies.

Picture two springs in series, each deforming under the same load according to its own compliance: total compliance is the sum of individual compliances, not simply the average of individual stiffnesses. The same addition applies here to the compliance contributions 1/E* of both bodies.

Formula and variables

1/E* = (1−ν1²)/E1 + (1−ν2²)/E2

• 1/E* = (1−ν1²)/E1 + (1−ν2²)/E2
• E1 = (1−ν1²) / [1/E* − (1−ν2²)/E2]

Choose the inputs correctly

Elastic modulus and Poisson's ratio of both contacting bodies set the reduced elastic modulus. For most steels, ν is about 0.3; E varies only slightly between steel grades, around 200 to 210 GPa.

How to use the calculator

Enter E1, ν1, E2 and ν2 to compute E*. For a pure steel-on-steel pair, both value pairs can be set equal; the result is then always about 115 GPa regardless of the exact steel grade.

Worked example

Two steel bodies with E1 = E2 = 210 GPa and ν1 = ν2 = 0.3 give E* = 1 / [2 · (1−0.3²)/210 GPa] ≈ 115.4 GPa.

Understand the result and units

A reduced modulus of 115.4 GPa for steel-on-steel is markedly smaller than a single steel body's elastic modulus (210 GPa), since both bodies contribute to total deformation. For a pairing with a softer partner (e.g. steel on bronze), E* falls further, since the softer body dominates overall compliance.

Elastic moduli are usually given in GPa; Poisson's ratios are dimensionless and fall roughly between 0.25 and 0.35 for most metallic materials.

Why E* isn't simply the average of E1 and E2

The formula 1/E* = (1−ν1²)/E1 + (1−ν2²)/E2 adds COMPLIANCES (reciprocal stiffnesses), not stiffnesses themselves — just like springs in series or parallel. For two identical bodies (E1=E2=E, ν1=ν2=ν), the formula simplifies to E* = E/[2·(1−ν²)], which for steel (E=210 GPa, ν=0.3) gives about 115 GPa — markedly less than E itself. A simple arithmetic average of E1 and E2 would not correctly represent this relationship and would give a falsely high, non-physical result.

Typical applications

The reduced elastic modulus is a necessary intermediate value for any Hertz contact pressure calculation in line or point contact, such as rolling bearings, gear tooth flanks, cam contacts, rollers on rails, and ball or roller contacts generally.

Assumptions, limits and common mistakes

The formula assumes linear-elastic, isotropic material behavior for both bodies. It does not account for anisotropy (for example hardened surface layers with different properties), temperature dependence of E and ν, or plastic or viscoelastic behavior.

Common mistake: A common mistake is simply averaging E1 and E2 instead of applying the formula correctly, which gives a falsely high E*. It is also easy to forget Poisson's ratio entirely or use a blanket value unsuited to the specific material.