Biaxial Stress State⚠ unverified
Mechanical / Stress Analysis · Compute the x-direction strain term for a biaxial stress state
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| sigma_x | σx | Pa | 1.0 | Normal stress along the x-axis |
| sigma_y | σy | Pa | 1.0 | Normal stress along the y-axis |
| nu | ν | — | 1.0 | Dimensionless Poisson's ratio of the material |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | εx | Pa | The combined stress term ``sigma_x - nu * sigma_y`` (divided by unit dimensionless E), in pascals (Pa) |
The science & history
Understanding the Parameters
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Direct stress $\sigma_x$ — stretches the material along x; on its own it would give $\varepsilon_x = \sigma_x/E$ (uniaxial Hooke's law).
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Transverse stress $\sigma_y$ — through the Poisson effect, a stress along y makes the material contract along x by $-\nu\sigma_y/E$. This coupling is what makes a biaxial state different from two independent uniaxial ones.
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Poisson's ratio $\nu$ — the coupling strength, $0$ (cork-like, no lateral coupling) to $0.5$ (rubber-like, incompressible); ~0.3 for steel. The registry default $1.0$ is outside the physical range.
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Elastic modulus $E$ — the material stiffness. It must divide the stress combination to yield strain; the registry's $E = 1$ simplification is why its output is a stress, not a strain.
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Plane stress — this form assumes $\sigma_z = 0$ (a free surface, thin part). The full 3D law adds a $-\nu\sigma_z/E$ term.
Derivation (Approaching a Proof)
Start from 1D Hooke's law, $\varepsilon = \sigma/E$, and add the Poisson contraction from stresses on the other axes. A stress $\sigma_j$ produces a transverse strain $-\nu\sigma_j/E$ on any perpendicular axis. Superposing the contributions of all in-plane stresses onto the x-strain (plane stress, $\sigma_z = 0$):
$$\varepsilon_x = \frac{\sigma_x}{E} - \nu\frac{\sigma_y}{E} = \frac{\sigma_x - \nu\,\sigma_y}{E}.$$
Superposition is valid because linear elasticity is, well, linear — each stress contributes its strain independently. (The companion relations are $\varepsilon_y = (\sigma_y - \nu\sigma_x)/E$ and, out of plane, $\varepsilon_z = -\nu(\sigma_x+\sigma_y)/E$.)
Dimensional check. $\varepsilon_x = \dfrac{\sigma_x - \nu\sigma_y}{E} = \dfrac{\text{Pa}}{\text{Pa}}$ = dimensionless — a strain, as it must be. (With the registry's $E = 1$ the division is dropped, leaving the numerator in Pa — the source of the mislabel.)
History and Development
The generalised Hooke's law — Robert Hooke's 1678 ut tensio, sic vis extended to three dimensions with Poisson's transverse-coupling ratio (Siméon Poisson, 1820s) — is the constitutive foundation of linear elasticity. The biaxial (plane-stress) specialisation is the everyday tool for thin structures and for converting measured strains (from rosettes) back into stresses.
Related Concepts: Strain Energy Density, Principal Stresses, Von Mises Stress, Load and Stress Analysis Fundamentals, Pressure Vessel Design
Notes: Output should be a dimensionless strain — the registry returns $\sigma_x-\nu\sigma_y$ in Pa (takes $E=1$); divide by real $E$. Default $\nu=1.0$ is non-physical (use $\le 0.5$). Plane stress ($\sigma_z=0$); companion $\varepsilon_y=(\sigma_y-\nu\sigma_x)/E$.