Stress–Strain Calculator — Engineering vs True Stress, Hollomon n and K, Live Curve

This interactive calculator converts between engineering and true stress-strain, fits the Hollomon power-law (σ_T = K × ε_Tⁿ) to extract the strain hardening exponent n and strength coefficient K, computes modulus of resilience, toughness, uniform elongation, and the Consière necking strain — all from standard uniaxial tensile test inputs. A live canvas plots both the engineering and true stress-strain curves simultaneously with all key points labelled.

Engineering vs True Stress–Strain: Definitions and Conversions

A tensile test records force F and displacement ΔL at every loading increment. The method of normalising these raw measurements defines whether you obtain engineering or true stress-strain, and the choice has significant consequences for large-deformation analysis.

Engineering (Nominal) Stress and Strain

Engineering (nominal) definitions:

  σ_e = F / A₀          [MPa]   A₀ = original cross-sectional area

  ε_e = ΔL / L₀ = (L - L₀) / L₀    [dimensionless or %]

Advantages:
  • Simple to calculate from load cell + crosshead displacement
  • Conservative (underestimates actual stress at large strains)
  • All standard code properties (UTS, R_p0.2, A%, Z%) are engineering values

Limitations:
  • σ_e decreases after UTS (artefact of fixed A₀ while actual F decreases)
  • ε_e is not additive (two sequential strains ε₁+ε₂ ≠ total true strain)
  • Inaccurate for FEA material models beyond ~5% strain

True (Cauchy) Stress and Logarithmic Strain

True (Cauchy) definitions:

  σ_T = F / A_instantaneous    [MPa]

  ε_T = ∫ dL/L from L₀ to L = ln(L/L₀) = ln(1 + ε_e)

Conversion (assumes isochoric / volume-conserving deformation:
V = A₀·L₀ = A·L  →  A = A₀·L₀/L = A₀/(1+ε_e)):

  σ_T = σ_e × (1 + ε_e)

  ε_T = ln(1 + ε_e)

Valid ONLY in the uniform deformation region (pre-necking).
After necking, the stress state is triaxial and simple conversion
fails — Bridgman necking correction is required for exact values.

True strain at fracture via reduction in area (RA):
  ε_T,f = ln(A₀/A_f) = ln(1 / (1 - RA/100))

Why True Stress-Strain is Required for FEA and Forming Analysis

Finite element material models require the plastic true stress-plastic true strain relationship as input. The engineering curve, with its artificial stress drop after UTS, would cause the FEA solver to predict that the material softens under continued loading — the opposite of the actual hardening behaviour. Sheet metal forming simulations (stamping, deep drawing, hydroforming) use the Hollomon fit to the true stress-strain data to generate forming limit diagrams (FLD), springback predictions, and draw force estimates. See the hardness testing guide for empirical correlations between hardness and UTS that can supplement tensile data.

The Hollomon Power Law: n and K Determination

The Hollomon equation σ_T = K × ε_Tⁿ is the simplest and most widely used flow stress model for metals in the uniform plastic deformation regime. It accurately describes the behaviour of a wide range of metals between the yield point and the onset of necking.

Hollomon Power Law:
  σ_T = K × ε_T^n

Taking natural logarithms:
  ln(σ_T) = ln(K) + n × ln(ε_T)

This is a straight line on a log–log plot:
  slope = n  (strain hardening exponent)
  intercept at ε_T = 1  →  σ_T = K  (strength coefficient)

Practical 2-point determination using yield point and UTS:
  Point 1 (yield):  σ_T1 = σ_y × (1 + ε_y)   ε_T1 = ln(1 + ε_y)
  Point 2 (UTS):    σ_T2 = UTS × (1 + ε_UTS)   ε_T2 = ln(1 + ε_UTS)

  n = [ln(σ_T2) − ln(σ_T1)] / [ln(ε_T2) − ln(ε_T1)]

  K = σ_T2 / ε_T2^n   (or σ_T1 / ε_T1^n — should agree within ~5%)

The Consière Criterion and Uniform Elongation

At the onset of necking, a small local cross-section reduction begins to grow rather than self-stabilise. The mathematical condition for this instability — the Consière criterion — gives a remarkably clean result for Hollomon materials:

Consière necking criterion:
  dF/dε_T = 0  at necking onset
  F = σ_T × A   →   dF = A·dσ_T + σ_T·dA = 0
  Volume conservation: A·dL = −L·dA  →  dA/A = −dε_T
  →  dσ_T / dε_T = σ_T   (necking condition)

For Hollomon material σ_T = K × ε_T^n:
  dσ_T / dε_T = K × n × ε_T^(n-1) = n × σ_T / ε_T

Setting equal to σ_T:   n × σ_T / ε_T = σ_T
  ∴  ε_T,neck = n

Engineering uniform elongation:
  ε_e,uniform = exp(n) − 1

Conclusion: for a Hollomon material, the strain hardening
exponent n is BOTH the log–log slope of the flow curve AND
the true strain at which necking initiates.

Modulus of Resilience and Toughness

Modulus of Resilience (U_r):
  Energy absorbed elastically per unit volume up to yield:
  U_r = (1/2) × σ_y × ε_y = σ_y² / (2E)   [J/m³ = Pa = N/m²]

  Practical: U_r [kJ/m³] = σ_y² / (2 × E × 1000)
  where σ_y in MPa and E in GPa

  Insight: for same σ_y, LOWER E → HIGHER resilience.
  Spring steel (E=200 GPa, σ_y=1400 MPa): U_r = 4900 kJ/m³
  Ti-6Al-4V  (E=114 GPa, σ_y=880 MPa):   U_r = 3395 kJ/m³
  Al 6061-T6 (E=70 GPa,  σ_y=276 MPa):   U_r = 544 kJ/m³

Toughness (approximate, area under eng. curve to fracture):
  Using the Hollomon region + post-UTS approximation:
  U_T ≈ (UTS + σ_y) / 2 × ε_f    [MJ/m³, with ε_f as fraction]

  More accurately, integrate the triangular elastic region +
  Hollomon plastic region + trapezoidal post-UTS region.

Ramberg–Osgood Equation

Where the Hollomon equation applies only in the plastic regime, the Ramberg–Osgood equation provides a continuous single-equation description spanning both elastic and plastic deformation. It is the standard model for cyclic stress-strain curves, fatigue analysis, and the HRR (Hutchinson–Rice–Rosengren) crack-tip field in elastic-plastic fracture mechanics.

Ramberg–Osgood:
  ε_T = σ_T/E + (σ_T/K)^(1/n)
         →     elastic    +   plastic

  Or in the common alternative form:
  ε_T = σ_T/E + α × (σ_T/σ_y)^m

  where α = (σ_y/E) / ε_y^plastic  and m = 1/n (Ramberg exponent)

Cyclic Ramberg–Osgood (for fatigue material models):
  ε_a = σ_a/E + (σ_a/K’)^(1/n’)
  where K’ and n’ are CYCLIC strength coefficient and exponent
  — generally different from monotonic K and n

Typical Hollomon n Values and Strength Coefficients for Engineering Alloys

Industrial Applications of Stress–Strain Analysis

Sheet Metal Forming and Stamping

The Hollomon n value is the primary formability metric for sheet steel grades used in automotive body panels, structural members, and white goods. Steel manufacturers report n as a standard product property alongside yield strength, UTS, and elongation. Press shop engineers use n to estimate the limiting draw ratio (LDR), select blank diameters for deep drawing, and set blank-holding force to prevent wrinkling. The R-value (Lankford coefficient, plastic anisotropy) is an equally important companion: materials with high n AND high R (both approaching 1.8–2.0 for DDQ steel) excel at deep drawing without thinning failure. See the impact testing guide for toughness measurement context.

Structural Steel Design and Safety Factors

Structural codes (Eurocode 3, AISC 360) use yield strength R_eh (or R_p0.2 for alloys without a defined yield point) as the design basis. The ratio UTS/σ_y (strain hardening ratio) is important for seismic design: codes require a minimum ratio of 1.20–1.25 to ensure adequate energy dissipation in plastic hinges during earthquake loading. A material with high UTS/σ_y (equivalently, high n) can redistribute stress across connections and joints rather than concentrating deformation in a single critical section. The martensite formation guide explains how tempering temperature affects the yield-to-UTS ratio in Q&T steels.

FEA Material Model Definition

When setting up a nonlinear FEA simulation (ANSYS, Abaqus, LS-DYNA) for large-deformation forming or crash analysis, the material card requires a true stress vs. true plastic strain table. The workflow is: (1) obtain engineering stress-strain from tensile test; (2) convert to true stress-strain using the calculator formulas above; (3) subtract the elastic strain (ε_T,plastic = ε_T − σ_T/E) to isolate the plastic portion; (4) fit to the Hollomon equation to extrapolate beyond the measured necking strain if needed. For crash simulation, the strain rate dependence (Cowper-Symonds or Johnson-Cook model) must also be characterised from high-rate Split-Hopkinson Bar tests.