Ramberg Osgood

Ramberg-Osgood Stress-Strain (structures/materials/ramberg-osgood)

Use when the task is the elastic-plastic stress-strain response of a metallic material: total strain from a stress, stress by inversion of the implicit Ramberg-Osgood equation, plastic strain, secant modulus, and tangent modulus for metallic structural analysis beyond the yield point.

Domain quick reference

• Ramberg-Osgood three-parameter model (NACA TN 902): epsilon = sigma / E + 0.002 * (sigma / sigma_0.2) ** n, with stress sigma and elastic modulus E in MPa, sigma_0.2 the 0.2 percent offset yield strength in MPa, and n the strain hardening exponent (typically 3 to 30 for aerospace metals). Strain is dimensionless.
• Elastic strain: epsilon_e = sigma / E.
• Plastic strain: epsilon_p = 0.002 * (sigma / sigma_0.2) ** n; the total strain is the sum epsilon_e + epsilon_p.
• Stress at a given total strain: the equation is implicit in sigma, so solve by bisection on the monotonic residual f(sigma) = sigma / E + 0.002 * (sigma / sigma_0.2) ** n - epsilon, bracketed on [0, E * epsilon] because sigma / E never exceeds the total strain.
• Secant modulus: E_s = sigma / epsilon (chord slope from the origin).
• Tangent modulus: E_t = 1 / (1 / E + 0.002 * n * sigma ** (n - 1) / sigma_0.2 ** n); at sigma = 0 the tangent modulus equals E, and it falls toward the plastic plateau as the stress rises.
• The model is common engineering methodology; the source paper is NACA TN 902 (US government work, public domain).

Workflow

1. Collect the material elastic modulus E, the 0.2 percent offset yield strength sigma_0.2, and the strain hardening exponent n.
2. Compute the total strain at a stress with strain, or the elastic and plastic parts with elastic_strain and plastic_strain.
3. For a required total strain, invert the model with stress_for_strain (bisection; the stress lies between 0 and E * epsilon).
4. Derive the secant modulus with secant_modulus and the tangent modulus with tangent_modulus along the curve.
5. Build the curve as a table of stress, strain, plastic strain, secant modulus, and tangent modulus points for the structural analysis.

Pitfalls

• Using only the elastic term sigma / E above the yield point: the plastic term 0.002 * (sigma / sigma_0.2) ** n dominates once sigma exceeds sigma_0.2, and a purely elastic estimate understates the strain badly.
• Treating the exponent n as a count: it is the strain hardening exponent, not a number of terms; values below 1 break the model monotonicity and the bisection bracket.
• Mixing units: E and sigma_0.2 must share the unit of sigma (MPa); mixing MPa and ksi shifts the plastic term by powers of 10.
• Expecting stress_for_strain to be a closed form: the equation is implicit in sigma and needs the bisection solve; the elastic extrapolation E * epsilon is an upper bound on the stress, not the answer.
• Calling secant_modulus at zero strain: the chord slope is undefined at epsilon = 0; use the tangent modulus E there.
• Ignoring the plastic strain sign: plastic_strain raises on an inconsistent input where sigma / E already exceeds the total strain.

Behavior contract (gate 3)

The Ramberg-Osgood strain, inversion, and modulus logic is exercised by the gate 3 contract test: scripts/test_ramberg_osgood.py against scripts/ramberg_osgood_logic.py (stdlib unittest, offline). Run: python3 scripts/test_ramberg_osgood.py

Compliance

• Standards referenced, not reproduced: NACA TN 902 is US government work (public domain); the Ramberg-Osgood equation and its secant and tangent modulus derivatives are common materials-engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.