Residual Stress Superposition⚠ unverified
Mechanical / Stress Analysis · Compute the total stress by superposing applied and residual stresses
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| sigma_applied | σapplied | Pa | 1.0 | Applied (service) stress |
| sigma_residual | σresidual | Pa | 1.0 | Residual (pre-existing) stress |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | σtotal | Pa | Total stress, in pascals (Pa) |
The science & history
Understanding the Parameters
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Applied stress $\sigma_{applied}$ — the stress from external service loads, computed by ordinary analysis as if the part were stress-free at rest.
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Residual stress $\sigma_{residual}$ — the signed self-equilibrating stress present with no external load. Sign is everything: compressive (negative) residuals subtract from tensile service stress and are beneficial; tensile (positive) residuals add and are harmful.
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Total stress $\sigma_{total}$ — the value that actually drives yielding, fatigue, and fracture. A part with a modest applied stress can still fail if a large tensile residual pushes the total past the limit.
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Self-equilibration — residual stresses must balance over any cross-section (tensile in one region, compressive in another), so a beneficial compressive surface layer (e.g. from shot peening) is always paid for by tension in the interior.
Derivation (Approaching a Proof)
Linear elasticity obeys the principle of superposition: for a linear material, the response to a sum of load systems equals the sum of the responses. Treat the residual stress field as the "response" to the locked-in eigenstrains (from prior plastic deformation, thermal gradients, or phase changes) and the applied stress as the response to external loads. Because both satisfy the same linear equilibrium and stress–strain equations, the fields add pointwise:
$$\sigma_{total}(\mathbf{x}) = \sigma_{applied}(\mathbf{x}) + \sigma_{residual}(\mathbf{x}).$$
The result is exact while the material stays elastic. Once the total stress causes local yielding, the residual field itself redistributes (relaxes), and the simple sum no longer holds — a key limitation for overloaded or thermally cycled parts.
Dimensional check. $\sigma_{total} = \sigma_{applied} + \sigma_{residual} = \text{Pa} + \text{Pa} = \text{Pa}$ — a stress; both terms must share units for the sum to be meaningful.
History and Development
The superposition principle is a cornerstone of linear elasticity (Cauchy, Navier, 19th century). Its application to residual stress underlies major fatigue-life technologies: shot peening, surface rolling, carburising, and induction hardening all deliberately introduce compressive surface residuals to subtract from service tension and delay crack initiation. Conversely, welding residual stresses are a leading cause of fatigue and stress-corrosion cracking, managed by post-weld heat treatment (stress relief). Measuring residuals (X-ray diffraction, hole-drilling) is a field in itself.
Related Concepts: Stress Concentration, Fatigue Notch Factor, Modified Goodman Factor, Principal Stresses, Marin Endurance Limit, Load and Stress Analysis Fundamentals
Notes: Valid in the linear-elastic regime — local yielding relaxes/redistributes residuals. Residual stress is signed: compressive (peening) beneficial, tensile (welding) harmful. Residuals self-equilibrate over the section. Total stress drives yield/fatigue/fracture.