What is the Hollomon Equation? Definition and Historical Context
The Hollomon equation (also called the Hollomon power law, the power-law hardening equation, or simply the strain hardening equation) is an empirical relationship that describes how the flow stress of a ductile metal increases with increasing plastic strain during cold deformation. It was proposed by J. H. Hollomon in 1945 in a landmark paper in the Transactions of AIME, and remains the most widely used constitutive model for the strain-hardening region of the stress-strain curve.
In physical terms, plastic deformation multiplies dislocation density in the crystal lattice. As dislocation density \(\rho\) rises, dislocations increasingly obstruct each other's motion, requiring a higher applied stress to continue deforming the material. The Taylor hardening law (\(\sigma \propto \sqrt{\rho}\)) combined with experimental observations of how \(\rho\) evolves with strain led to the power-law relationship that Hollomon formalised. To understand where this equation fits in the broader context of work hardening mechanisms, see our detailed strain hardening guide.
The equation is valid in the plastic strain-hardening region only - between the end of the yield plateau (or the 0.2% proof strain for materials without a plateau) and the onset of necking at the ultimate tensile strength. It is used everywhere from sheet metal forming simulations to quality control of structural steel to prediction of cold-drawing forces for wire manufacture.
Figure 1: True stress–true plastic strain curve showing the Hollomon power law region (shaded), the strength coefficient K, and the strain hardening exponent n for a typical ductile metal.
Key physical insight: The Hollomon equation is not just curve-fitting - it is rooted in dislocation physics. The Taylor hardening law predicts \(\sigma \propto \sqrt{\rho}\). Experiments show that dislocation density \(\rho\) scales approximately as \(\varepsilon^{2n}\) during cold work. Combining these gives \(\sigma \propto \varepsilon^n\) - exactly the Hollomon form. The exponent n therefore encodes information about how rapidly dislocations accumulate and interact in a particular metal's crystal structure.
Hollomon Equation Formula, Variables, and Complete Derivation
Written in full mathematical notation:
\(\varepsilon_p\) = true plastic strain = total true strain minus elastic component = \(\ln(1+\varepsilon_e) - \sigma_T/E\).
\(K\) = strength coefficient (MPa): the true flow stress extrapolated to \(\varepsilon_p = 1.0\); a measure of the overall strength level.
\(n\) = strain hardening exponent: dimensionless slope on a log-log plot of \(\sigma_T\) vs \(\varepsilon_p\). Controls the rate of hardening.
Valid range: from yield point / end of Lüders band (mild steel) or from \(\varepsilon_p = 0.002\) (high-strength steel) up to the onset of necking at UTS.
\(\dfrac{d}{d\varepsilon}(K\varepsilon^n) = nK\varepsilon^{n-1}\).
Setting this equal to \(\sigma = K\varepsilon^n\):
\(nK\varepsilon^{n-1} = K\varepsilon^n \;\Rightarrow\; n = \varepsilon_u\).
Conclusion: for a Hollomon material, the true strain at the onset of necking equals n exactly. This means n is simultaneously the hardening exponent and the uniform elongation index. Higher n → more uniform deformation before necking → better formability.
Critical limitation to remember: The Hollomon equation predicts \(\sigma = 0\) when \(\varepsilon = 0\), which is physically impossible - a metal has a finite yield strength at zero plastic strain. This means the equation is never valid at the yield point itself. It is only fitted to data in the strain-hardening region (typically \(\varepsilon_p > 0.01\) for mild steel and \(\varepsilon_p > 0.002\) for high-strength steel). For a model that is valid from yield onset, use the Ludwik equation: \(\sigma = \sigma_0 + K\varepsilon^n\).
Understanding K and n: Physical Meaning, Ranges, and How They Affect the Curve
Typical range: 200 MPa (very soft annealed aluminium) to 2,500+ MPa (cold-drawn high-carbon steel wire).
Effect on the curve: Increasing K shifts the entire stress-strain curve upward (higher stresses at every strain level) without changing its shape. A material with high K is intrinsically stronger.
Influenced by: Crystal structure, alloying elements (solid solution strengthening), grain size (Hall-Petch), precipitation hardening, and prior cold work. Does not represent the yield strength directly.
Effect on the curve: Higher n = steeper hardening curve = more uniform elongation before necking (Considère: \(\varepsilon_u = n\)). Lower n = rapid saturation of hardening and earlier necking.
Influenced by: Crystal structure (FCC > BCC in general), stacking fault energy (low SFE = high n, e.g. austenitic SS), temperature (higher T = lower n), prior cold work (reduces n dramatically).
Formability index: n is the primary measure of sheet metal formability. Limiting draw ratio (LDR) ≈ e^n. IS 513 CR3 (extra deep drawing) requires n ≥ 0.21.
Figure 2: Effect of K and n on the Hollomon stress–strain curve. Left panel: varying K at constant n shifts the curve vertically. Right panel: varying n at constant K changes the hardening rate and uniform elongation.
| n value | Hardening behaviour | Uniform elongation εu | Typical material | Formability |
|---|---|---|---|---|
| 0.05 | Very low hardening - flow stress barely increases with strain | ≈ 5% | Heavily cold-drawn wire, prestressed concrete wire | Very poor - necks almost immediately after yielding |
| 0.10 | Low hardening | ≈ 10% | High-strength steel (QT), cold-drawn rod | Poor - limited deep drawing capability |
| 0.18–0.22 | Moderate hardening | ≈ 18–22% | Mild steel (IS 2062 E250), structural steel | Moderate - adequate for most stamping |
| 0.26–0.30 | Good hardening | ≈ 26–30% | Annealed low-carbon steel, IF steel | Good - suitable for deep drawing |
| 0.44–0.54 | Very high hardening | ≈ 44–54% | Annealed copper, annealed austenitic SS (304) | Excellent - large elongation before necking |
True vs Engineering Stress-Strain: Why Hollomon Requires True Values
This is the most common source of error when applying the Hollomon equation. The equation uses true (Cauchy) stress and true (logarithmic) strain - not the engineering (nominal) values that a tensile testing machine records directly. You must always convert before fitting K and n.
| Eng. Strain εe | True Strain εT | Eng Stress σe (MPa) | True Stress σT (MPa) | Error if engineering used |
|---|---|---|---|---|
| 0.01 (1%) | 0.00995 | 280 | 282.8 | +1% - negligible |
| 0.05 (5%) | 0.04879 | 340 | 357.0 | +5% - noticeable |
| 0.10 (10%) | 0.09531 | 375 | 412.5 | +10% - significant |
| 0.15 (15%) | 0.13976 | 395 | 454.3 | +15% - major error |
| 0.20 (20%) - near UTS | 0.18232 | 410 | 492.0 | +20% - completely wrong |
Never plug engineering stress and engineering strain directly into the Hollomon equation. At 20% engineering strain (typical mild steel UTS), the true stress is 20% higher than the engineering stress. Using engineering values gives a K that is 20% too low and an n that is distorted. All K and n values in published tables and in this article refer to true stress and true plastic strain.
K and n Values Table: 20+ Metals and Alloys
The following table gives Hollomon parameters K and n for the most common engineering metals in their annealed or as-received condition (unless noted). All values correspond to true stress and true plastic strain. Use these for quick estimates - for precise work, always fit K and n from your own tensile test data using the log-log method.
Figure 3: Strain hardening exponent n for common engineering metals (annealed condition). Austenitic stainless steel and copper have the highest n values; heavily cold-drawn wire and high-strength steels have the lowest.
| Material | Condition | K (MPa) | n | σy (MPa) | UTS (MPa) | Elong. (%) | Notes |
|---|---|---|---|---|---|---|---|
| Annealed Copper (OFHC) | Annealed | 315–530 | 0.44–0.54 | 70 | 220–250 | 50–65 | Very high n due to FCC + low SFE; excellent formability; used for deep-drawn pressure vessels |
| Annealed Brass (70Cu-30Zn) | Annealed | 600–900 | 0.35–0.50 | 100–200 | 380–450 | 60–70 | High n FCC; twinning-induced plasticity contributes; excellent for cartridge cases |
| Annealed Low-Carbon Steel (IS 2062 E250) | Hot-rolled / normalised | 530–800 | 0.20–0.26 | 250 | 410–500 | 23–28 | Most common structural steel; moderate n; IS 2062 Grade E250 |
| Annealed Medium-Carbon Steel (0.4%C) | Annealed | 750–1000 | 0.15–0.20 | 350 | 600–700 | 18–22 | Higher C reduces n; used for machine parts, axles |
| IS 1786 Fe 415 TMT Rebar | Hot-rolled TMT | 820–900 | 0.18–0.22 | 415 | 485–600 | 14.5–18 | Tempered martensite rim; adequate n for seismic ductility (IS 13920) |
| IS 1786 Fe 500D TMT Rebar | Hot-rolled TMT | 900–990 | 0.18–0.20 | 500 | 565–650 | 16–20 | Enhanced ductility grade; UTS/yield ≥ 1.15 per IS 1786 |
| ASTM A36 Structural Steel | As-rolled | 600–830 | 0.20–0.26 | 250 | 400–550 | 20–25 | US equivalent to IS 2062 E250; similar K and n |
| SS 304 Austenitic Stainless (annealed) | Annealed | 1200–1400 | 0.45–0.55 | 210 | 515–600 | 50–70 | Very high n due to FCC + very low SFE + TRIP effect; extensive cold work possible |
| SS 316L Austenitic Stainless (annealed) | Annealed | 1100–1300 | 0.44–0.52 | 170 | 485–550 | 40–55 | Slightly lower K than 304; higher Mo content; marine and pharmaceutical applications |
| Annealed Aluminium (1100-O) | Annealed | 140–180 | 0.22–0.26 | 35 | 90–110 | 35–45 | Pure Al; low K but moderate n; used for packaging foil and conductor wire |
| Al 6061-T4 (solution treated) | T4 (nat. aged) | 200–240 | 0.12–0.16 | 145 | 240–280 | 22–25 | Precipitation-hardenable; lower n than pure Al due to prior strengthening |
| Al 5052-H32 (work hardened) | H32 (quarter hard) | 300–380 | 0.10–0.14 | 195 | 230–270 | 12–16 | Marine Al alloy; lower n due to prior cold work |
| Annealed Nickel (200) | Annealed | 700–900 | 0.35–0.45 | 150 | 380–450 | 40–55 | FCC; high n; used for chemical plant and battery electrodes |
| Annealed Titanium (Grade 2) | Annealed | 700–950 | 0.14–0.20 | 275 | 345–450 | 20–28 | HCP structure; lower n than FCC; significant strain rate sensitivity |
| IS 1785 PC Wire (cold-drawn) | Heavy CW (60–70%) | 2000–2200 | 0.04–0.07 | 1570 | 1770–1860 | 3–5 | Near-exhausted hardening capacity; very low n; fractures with little post-yield deformation |
| Cold-Drawn High-Strength Wire (1080 Steel) | Heavy CW | 1700–2000 | 0.05–0.10 | 1200 | 1400–1600 | 5–8 | Spring wire, fasteners; low n from prior cold work |
| Annealed Magnesium AZ31B | Annealed | 200–300 | 0.10–0.16 | 150 | 250–290 | 10–15 | HCP; limited slip systems; low n; warm forming required for better ductility |
| TRIP 800 AHSS | CR + annealed | 900–1150 | 0.20–0.28 | 420–480 | 750–850 | 25–35 | Transformation-induced plasticity raises effective n during straining |
| Dual Phase DP 600 | HR + QT | 900–1050 | 0.16–0.20 | 330–380 | 590–650 | 22–27 | Ferrite + martensite; good n for automotive structural use |
| Annealed Zinc (pure) | Annealed | 180–250 | 0.06–0.12 | 30 | 120–150 | 30–40 | HCP at RT; partial recovery at room temperature limits n |
How to read this table: K and n values given as ranges reflect variability between different sources, heats, and test conditions. For design calculations, use the mid-range value and verify against your own material certificates. Values are for monotonic tensile loading at room temperature and quasi-static strain rates (0.0001–0.001 s⁻¹). Cold-worked conditions will have higher K and lower n than the annealed values shown.
Worked Examples: Annealed Copper and Structural Steel
Example 1 - Annealed Copper: Flow Stress, UTS Prediction, and Cold-Drawing Force
Given: Annealed copper bar. Hollomon parameters from tensile test: K = 450 MPa, n = 0.50. Young's modulus E = 120,000 MPa. Original diameter d₀ = 10 mm (A₀ = 78.54 mm²).
Questions: (a) Find the true flow stress at εp = 0.30 and at εp = 0.50. (b) Predict the true and engineering UTS. (c) A wire-drawing die reduces diameter from 10 mm to 8 mm in a single pass - find the drawing force and the yield strength of the drawn wire.
Flow stress at εp = 0.30:
\(\sigma_T = K\varepsilon_p^n = 450 \times (0.30)^{0.50} = 450 \times 0.5477 = \mathbf{246.5 \text{ MPa}}\)
Flow stress at εp = 0.50 (= n):
\(\sigma_T = 450 \times (0.50)^{0.50} = 450 \times 0.7071 = \mathbf{318.2 \text{ MPa}}\)
True UTS (Considère: necking at εu = n = 0.50):
\(\sigma_{UTS,true} = K \cdot n^n = 450 \times (0.50)^{0.50} = 450 \times 0.7071 = \mathbf{318.2 \text{ MPa}}\)
Engineering UTS:
At necking, engineering strain \(\varepsilon_e = e^n - 1 = e^{0.50} - 1 = 1.6487 - 1 = 0.6487\).
Engineering UTS \(= \sigma_{UTS,true} / (1 + \varepsilon_e) = 318.2 / 1.6487 = \mathbf{193.0 \text{ MPa}}\).
Check: published UTS for annealed OFHC copper ≈ 210–250 MPa. Our estimate of 193 MPa is in the correct range - the actual K for OFHC copper is slightly higher (≈530 MPa).
Wire drawing - true strain imparted:
New area: \(A_f = \tfrac{\pi}{4} \times 8^2 = 50.27 \text{ mm}^2\).
True strain \(= \varepsilon_{CW} = \ln(A_0/A_f) = \ln(78.54/50.27) = \ln(1.5625) = \mathbf{0.446}\)
Flow stress during drawing (exit):
Exit flow stress \(= 450 \times (0.446)^{0.50} = 450 \times 0.668 = 300.5 \text{ MPa}\)
Drawing force \(F = \sigma_{flow} \times A_f = 300.5 \times 50.27 = \mathbf{15{,}106 \text{ N} \approx 15.1 \text{ kN}}\)
Yield strength of the cold-drawn wire:
After drawing to \(\varepsilon_{CW} = 0.446\), the wire's new yield strength = flow stress at that strain = 300.5 MPa (up from the annealed yield of ≈70 MPa - a 4.3× increase).
Remaining uniform elongation \(= n - \varepsilon_{CW} = 0.50 - 0.446 = \mathbf{0.054}\) (5.4% true strain) - a significant reduction from the annealed 50% elongation.
Example 2 - Structural Steel IS 2062 E250: K–n Fitting from Tensile Test Data
Given: Tensile test data for IS 2062 E250 (10 mm diameter bar, A₀ = 78.54 mm², E = 200,000 MPa). Data points in the strain-hardening region (after Lüders band, before necking):
| Pt. | εe | σe (MPa) | σT (MPa) | εT | εp | ln εp | ln σT |
|---|---|---|---|---|---|---|---|
| 1 | 0.025 | 305 | 312.6 | 0.02469 | 0.02313 | −3.766 | 5.746 |
| 2 | 0.050 | 325 | 341.3 | 0.04879 | 0.04708 | −3.057 | 5.833 |
| 3 | 0.080 | 355 | 383.4 | 0.07696 | 0.07504 | −2.590 | 5.949 |
| 4 | 0.110 | 375 | 416.3 | 0.10436 | 0.10228 | −2.280 | 6.031 |
| 5 | 0.140 | 390 | 444.6 | 0.13103 | 0.12881 | −2.050 | 6.098 |
| 6 | 0.170 | 400 | 468.0 | 0.15700 | 0.15466 | −1.865 | 6.149 |
| 7 (UTS) | 0.195 | 410 | 489.9 | 0.17805 | 0.17561 | −1.740 | 6.194 |
Linear regression of ln σT on ln εp:
Mean ln εp = −17.348 / 7 = −2.478 Mean ln σT = 42.000 / 7 = 6.000
Slope (= n):
\(n = \dfrac{\displaystyle\sum(\ln\varepsilon_p - \overline{\ln\varepsilon_p})(\ln\sigma_T - \overline{\ln\sigma_T})}{\displaystyle\sum(\ln\varepsilon_p - \overline{\ln\varepsilon_p})^2} = \mathbf{0.213}\)
Intercept (= ln K):
\(\ln K = \overline{\ln\sigma_T} - n \cdot \overline{\ln\varepsilon_p} = 6.000 - 0.213 \times (-2.478) = 6.528\)
\(K = e^{6.528} = \mathbf{685 \text{ MPa}}\)
Hollomon equation for this IS 2062 E250 steel:
\(\sigma_T = 685 \; \varepsilon_p^{0.213}\)
Considère check: predicted necking strain = n = 0.213. Actual UTS at point 7: \(\varepsilon_p = 0.176\). Discrepancy ≈ 17% - acceptable for Hollomon (slight Voce saturation near UTS). \(R^2 \approx 0.998\) - excellent fit.
Predicted true UTS:
\(\sigma_{UTS,true} = 685 \times (0.213)^{0.213} = 685 \times 0.724 = 496 \text{ MPa}\)
Engineering UTS \(= 496 / e^{0.213} = 496 / 1.237 = \mathbf{401 \text{ MPa}}\). Actual recorded = 410 MPa - within 2.2%. ✓
The Log-Log Plot Method: How to Determine K and n from a Tensile Test
Figure 4: Log-log plot of true stress (ln σT) vs true plastic strain (ln εp) for annealed copper (slope n ≈ 0.50) and mild steel (slope n ≈ 0.21). The y-intercept = ln K for each material.
The log-log linearisation of the Hollomon equation provides a straightforward graphical and numerical procedure to extract K and n from any standard tensile test. The procedure is fully described in IS 1608 / ASTM E8M and is used routinely in material qualification and forming process design.
Step-by-Step Procedure to Determine K and n (IS 1608 / ASTM E8M)
Conduct the tensile test at constant crosshead speed giving strain rate 0.00025–0.0025 s⁻¹ (IS 1608 Clause 10). Record load F (N) and extension Δl (mm) continuously. At least 15–20 data points in the plastic region are needed for reliable regression.
Select data range: Start after the yield plateau (engineering strain > 2% for mild steel; > 0.2% for high-strength steel). Stop before necking onset (engineering strain < εUTS). Discard data near the plateau where Lüders bands distort measurements.
Compute true stress and true plastic strain using the conversion formulas for each data point: \(\sigma_T = \sigma_e(1+\varepsilon_e)\), \(\varepsilon_T = \ln(1+\varepsilon_e)\), \(\varepsilon_p = \varepsilon_T - \sigma_T/E\).
Take natural logarithms of both σT and εp for each data point. Plot ln σT (y-axis) against ln εp (x-axis). A straight line confirms Hollomon behaviour.
Linear regression: Slope = n. Y-intercept at ln εp = 0 gives ln K, so K = eintercept. In Excel: use =LINEST(ln_sigma, ln_eps) or =SLOPE() and =INTERCEPT(). In Python: numpy.polyfit(np.log(eps_p), np.log(sigma_T), 1).
Quality check: R² > 0.99 → excellent Hollomon fit. R² = 0.97–0.99 → acceptable. R² < 0.97 → consider Ludwik or Voce equation instead, or check for data errors. Also verify: necking strain ≈ n (Considère check); predicted UTS ≈ actual UTS within 5%.
Quick estimate without full test data: If you only have yield strength σy and UTS from a material certificate (e.g. IS 2062 mill test report), you can estimate K and n numerically using:
\(UTS_{eng} = K \cdot n^n / e^n\) and \(\sigma_y \approx K \cdot \varepsilon_{yield}^n\) where \(\varepsilon_{yield} \approx \sigma_y / E\).
Solving with Excel Solver or our calculator below gives approximate K and n. Accuracy is lower than full regression but useful for preliminary forming process design.
Limitations of the Hollomon Equation and Comparison with Ludwik, Voce, and Swift Models
The Hollomon equation is powerful but not universal. Understanding its limitations helps you decide when to use it and when to choose an alternative model. For a comprehensive side-by-side comparison of all major constitutive models including Swift and Johnson-Cook, refer to our strain hardening overview article.
Hollomon (1945)
Pros: Simplest form; only 2 parameters; analytically tractable; gives closed-form necking strain (= n); most widely tabulated.
Cons: σ = 0 at ε = 0 (invalid at yield onset); overestimates stress at large strains (no saturation); only valid from yield plateau end to necking.
Best for: Mild steel and copper in moderate strain range (2–20%); forming process quick estimates; Considère instability analysis.
Ludwik (1909)
Pros: Physically correct at ε = 0 (gives σ = σ₀ = yield strength); valid from yield onset; standard form for FE material cards (ABAQUS combined hardening).
Cons: 3 parameters to fit; still overestimates at very large strains; K and n have different numerical values than Hollomon for same material.
Best for: High-strength steel and stainless steel (no distinct yield plateau); FE simulations; when elastic-to-plastic transition must be captured.
Voce (1948)
Pros: Correctly predicts saturation of flow stress at large strains (Stage III hardening); physically motivated by Kocks-Mecking recovery model; best accuracy for austenitic SS, Cu, Al at large strains.
Cons: 3 parameters; no closed-form necking criterion; may underestimate hardening at low strains for some steels.
Best for: Large-strain forming (stamping, deep drawing); stainless steel and aluminium; where Hollomon overestimates at ε > 0.30.
| Limitation | Hollomon | Ludwik | Voce | Impact on Engineering Use |
|---|---|---|---|---|
| Valid at ε = 0? | No (σ=0) | Yes (σ=σ₀) | Yes (σ=σ₀) | Hollomon cannot model the elastic-plastic transition; never use it below ε ≈ 0.01 |
| Predicts saturation? | No | No | Yes | For large-strain forming (ε > 0.3), Hollomon and Ludwik overestimate force by 5–20% |
| Closed-form necking strain? | Yes (εu = n) | Approximately | No | Hollomon uniquely useful for quick formability estimation |
| Number of parameters | 2 (K, n) | 3 (σ₀, K, n) | 3 (σ₀, σ∞, εr) | Hollomon is simplest; others require more test data points for reliable fitting |
| Valid range | εy to εu | 0 to εu | 0 to very large ε | Hollomon is least general but adequate for most cold-forming process design |
| Rate/temperature effects | Not included | Not included | Partially (via εr) | For high-rate or elevated-temperature problems, use Johnson-Cook instead |
Engineering Applications of the Hollomon Equation
The Hollomon equation underpins quantitative analysis in nearly every metal-forming and structural application that involves plastic deformation. The following are the most important uses in practice.
Cold-Rolling & Sheet Metal Forming
Rolling force F = σ_flow × contact area. Since σ_flow = Kεn and the strain increases with reduction, the Hollomon equation predicts how rolling load increases with each pass. IS 1079 cold-rolled sheet grades are specified partly by their Hollomon n value (minimum n for drawing quality grades).
Wire Drawing Force Prediction
For a drawing die reducing area from A₀ to A_f: true strain ε = ln(A₀/A_f). Average draw stress ≈ K·εn × (1 + die friction factor). Total drawing force = draw stress × A_f. Multi-pass schedules are planned so each pass stays within the wire's tensile strength (prevents wire breakage).
Deep Drawing - Limiting Draw Ratio
For a Hollomon material, the limiting draw ratio LDR ≈ e^n (blank diameter / punch diameter at failure). Higher n → larger LDR → deeper cups possible. IS 513 CR3 (extra deep drawing) specifies n ≥ 0.21 to achieve LDR ≥ 1.23, sufficient for automotive door panels and pressure vessel heads.
Forming Limit Diagram (FLD)
FLD curves separate safe from failed strain states in sheet metal. The maximum major strain ε₁ on the FLD at ε₂ = 0 (plane strain) ≈ n for a Hollomon material (Keeler-Goodwin equation). FLDs are the primary tool for stamping die design and die-try-out in automotive manufacturing.
Cold Forging & Heading
Bolt and fastener blanks are cold-headed from wire rod. The forging pressure = K·εn × geometric constraint factor. The Hollomon equation predicts how strength increases after each heading blow, enabling fatigue-life estimates for IS 1367 bolts without additional testing.
Strain Hardening in Structural Rebar
IS 1786 requires TMT rebar to have UTS/yield ≥ 1.15. This ratio is controlled by n: higher n gives a higher ratio. Fe 500D's mandatory elongation ≥ 16% is directly linked to a minimum effective n ≈ 0.08–0.10 per IS 13920 seismic design requirements.
Residual Stress After Bending
When a bar is bent beyond yield and springback occurs, the Hollomon equation predicts the through-thickness stress distribution in the plastic zone. Higher n → more uniform through-thickness hardening → less springback - important for precision press-brake forming of structural sections.
FE Simulation Material Cards
ABAQUS, Ansys, and LS-DYNA accept tabular true stress vs true plastic strain data derived directly from the Hollomon equation. For a given K and n, generate σ-ε pairs at ε = 0.01, 0.02, … 0.30 and input as piecewise linear plasticity. The Hollomon curve gives physically consistent data even at strains not directly measured in the test.
Hollomon Equation Calculator
Hollomon Power Law Calculator - \(\sigma = K\varepsilon^n\)
Enter K, n, and a true plastic strain to compute flow stress. Also calculates UTS (true and engineering), Considère necking strain, and post-cold-work yield strength. Select a preset material or enter custom values.
Frequently Asked Questions - Hollomon Equation
1. What is the Hollomon equation and when was it proposed?
The Hollomon equation is σ_T = K·ε_p^n, proposed by J. H. Hollomon in 1945 in the Transactions of AIME (American Institute of Mining and Metallurgical Engineers). It is an empirical power law relating true flow stress (σ_T, in MPa) to true plastic strain (ε_p) in the strain-hardening region of the stress-strain curve. K is the strength coefficient (MPa) - the flow stress extrapolated to ε_p = 1.0 - and n is the strain hardening exponent (dimensionless), which controls the rate of hardening. The equation is also called the power-law hardening equation or simply the strain hardening equation. It is the most widely used constitutive model for the plastic region of ductile metals in forming process analysis, FE simulations, and material qualification.
2. What is the strain hardening exponent n and what values are typical?
The strain hardening exponent n is the slope of the log-log plot of true stress vs true plastic strain (ln σ vs ln ε). It quantifies the rate at which a metal hardens as it deforms plastically: n = 0 means perfectly plastic (no hardening - constant stress); n = 1 means linearly strain hardening. Real metals fall between these extremes. Typical values: heavily cold-drawn steel wire n ≈ 0.04–0.07 (nearly exhausted hardening capacity); mild structural steel (IS 2062 E250) n ≈ 0.20–0.26; annealed copper n ≈ 0.44–0.54; annealed austenitic stainless steel (SS 304) n ≈ 0.45–0.55. A critical result from the Considère criterion is that for a Hollomon material, n equals the true uniform strain at the onset of necking - so higher n also means greater uniform elongation and better formability.
3. What is the strength coefficient K in the Hollomon equation?
K is the strength coefficient (in MPa), defined as the true flow stress the material would have at a true plastic strain of ε_p = 1.0. In physical terms, K represents the extrapolated stress level at very large strain - it sets the overall height of the hardening curve. On the log-log plot, K is found from the y-intercept: K = e^(intercept). K does not equal the yield strength. For mild steel, K ≈ 530–800 MPa while yield strength is only 250 MPa. For annealed copper, K ≈ 315–530 MPa while yield is only 70 MPa. Higher K means the material reaches higher stresses at any given strain, indicating a stronger material with greater inherent resistance to plastic flow. K is influenced by alloying, grain size, and prior deformation history.
4. Why must true stress and true strain be used - not engineering values?
The Hollomon equation is derived for and must be applied with true (Cauchy) stress and true (logarithmic) strain because these quantities correctly account for the changing geometry of the specimen during deformation. Engineering stress σ_e = F/A₀ uses the original area and underestimates actual stress as the specimen necks. Engineering strain ε_e = Δl/l₀ is not additive for successive deformations and overestimates the total true strain. Conversion formulas (valid up to necking): true stress σ_T = σ_e(1+ε_e); true strain ε_T = ln(1+ε_e). At a typical mild steel UTS with ε_e ≈ 0.20, the true stress is 20% higher than engineering stress. Plugging engineering values directly into Hollomon gives a K that is too low and an n that is distorted, making all subsequent calculations wrong.
5. What is the Considère criterion and what does it tell us about n?
The Considère criterion (1885) defines the condition for the onset of necking (plastic instability) in a tensile specimen: necking begins when the rate of strain hardening dσ/dε equals the current flow stress σ. Substituting the Hollomon equation σ = Kε^n: dσ/dε = nKε^(n-1). Setting this equal to σ = Kε^n gives n·Kε^(n-1) = Kε^n, which simplifies to n = ε_u. This remarkable result means that for a Hollomon material, the true strain at the onset of necking (the maximum load point / UTS) equals the strain hardening exponent n exactly. Practical uses: (1) n is directly the uniform elongation index - materials with higher n sustain more uniform deformation before necking; (2) IS 513 deep-drawing steel grades specify minimum n values to ensure adequate Limiting Draw Ratio; (3) predicted UTS strain ≈ n provides a cross-check for the quality of Hollomon fitting.
6. What is the difference between the Hollomon and Ludwik equations?
The Hollomon equation σ = Kε^n has two parameters (K, n) and predicts zero stress at zero strain - making it physically unrealistic at the yield point but mathematically simple. The Ludwik equation σ = σ₀ + Kε^n adds the initial yield strength σ₀ as a third parameter, making it physically correct (σ = σ₀ at ε = 0). Both Hollomon and Ludwik overestimate flow stress at large strains because neither accounts for the saturation of strain hardening (Stage III recovery). The Voce equation correctly models this saturation and is more accurate for austenitic stainless steel, copper, and aluminium at strains above 0.30. For most structural steel applications and cold-forming processes at moderate strains (ε < 0.25), the Hollomon equation is accurate enough and preferred for its simplicity.
7. How do I determine K and n from a tensile test?
Extract K and n using the log-log linearisation of Hollomon: ln σ_T = ln K + n·ln ε_p. Steps: (1) Conduct tensile test per IS 1608 / ASTM E8M. (2) Convert engineering values to true stress and true plastic strain for each data point in the hardening region (after yield plateau, before necking). (3) Plot ln σ_T vs ln ε_p - a straight line confirms Hollomon behaviour. (4) Linear regression: slope = n; y-intercept = ln K so K = e^intercept. In Excel use SLOPE() and INTERCEPT() functions, or LINEST(). (5) Quality checks: R² > 0.99 for good fit; predicted necking strain ≈ n; predicted engineering UTS = K·n^n/e^n ≈ actual UTS within 5%.
8. How does prior cold work affect K and n?
Cold work (plastic pre-deformation) consumes some of the metal's hardening capacity. After a metal has been cold-worked to true strain ε_CW, it starts subsequent loading from a higher point on the Hollomon curve. The effective remaining hardening exponent for further deformation is approximately n_remaining = n − ε_CW. If ε_CW ≥ n, the material has essentially no remaining uniform elongation and will neck immediately upon reloading. Practically: cold-drawn copper wire at 60% area reduction (ε_CW ≈ 0.92, much greater than n ≈ 0.50) has exhausted its hardening capacity - it behaves like a nearly perfectly plastic material with very low elongation. This is why PC wire (IS 1785) has n ≈ 0.04–0.07 and elongation only 3–5%, compared to annealed copper with n ≈ 0.50 and elongation 50–65%.
9. What is the relationship between n and the limiting draw ratio (LDR) in deep drawing?
For a Hollomon material, the Limiting Draw Ratio in deep drawing is approximately LDR = e^n, where e is Euler's number ≈ 2.718. This comes from the analysis of the force balance between the punch load (which tends to thin and fracture the sheet at the punch nose) and the resistance in the flange (which thickens and work-hardens during drawing). Materials with higher n distribute the strain more uniformly, allowing deeper cups before tearing. Examples: mild steel n = 0.22 → LDR ≈ e^0.22 = 1.25; IF steel n = 0.28 → LDR ≈ 1.32; austenitic SS n = 0.50 → LDR ≈ 1.65. IS 513 specifies minimum n values for CR2 (ordinary drawing) ≥ 0.18 and CR3 (extra deep drawing) ≥ 0.21 to ensure adequate LDR for their intended applications.
10. Can the Hollomon equation be used for elevated temperature or high strain rate conditions?
The Hollomon equation as written applies only to quasi-static (low strain rate ≈ 0.001 s⁻¹) monotonic loading at room temperature. For elevated temperature, both K and n change significantly: higher temperature decreases K (metal is softer) and usually decreases n (dynamic recovery is faster, limiting hardening). For high strain rate conditions (seismic loading ≈ 0.01–0.1 s⁻¹; impact/blast ≈ 100–1000 s⁻¹), the Johnson-Cook model extends Hollomon: σ = (A + Bε^n)(1 + C·ln ε̇*)(1 − T*^m), which multiplies the Hollomon-like hardening term by strain-rate and temperature factors. For hot forming above the recrystallisation temperature (above ≈720°C for steel), strain hardening is completely absent because new strain-free grains continuously form - the Hollomon equation does not apply at all in hot working conditions.
Key References
Hollomon, J.H. (1945). Tensile deformation. Transactions of the American Institute of Mining and Metallurgical Engineers, 162, 268–290. [Original paper proposing σ = Kε^n.]
Ludwik, P. (1909). Elemente der Technologischen Mechanik. Springer, Berlin. [Proposed σ = σ₀ + Kε^n as an improvement on Hollomon.]
Considère, A. (1885). Mémoire sur l'emploi du fer et de l'acier dans les constructions. Annales des Ponts et Chaussées, 9, 574–775. [Derived the necking criterion ε_u = n for power-law materials.]
Taylor, G.I. (1934). The mechanism of plastic deformation of crystals. Proceedings of the Royal Society A, 145(855), 362–387. [Dislocation-based derivation of the physical basis for σ ∝ √ρ and hence the power law.]
Dieter, G.E. (1986). Mechanical Metallurgy, 3rd ed. McGraw-Hill. [Chapter 8: comprehensive treatment of strain hardening, Hollomon fitting, and formability.]
Callister, W.D. and Rethwisch, D.G. (2018). Materials Science and Engineering: An Introduction, 10th ed. Wiley. [Tables of K and n for common metals; Chapter 7 on dislocations and strengthening mechanisms.]
BIS (2005). IS 1608: Metallic Materials - Tensile Testing at Ambient Temperature. Bureau of Indian Standards. [Standard test method; specimen geometry; properties measured including yield, UTS, elongation used to fit Hollomon parameters.]
BIS (2008). IS 1786: High Strength Deformed Steel Bars and Wires for Concrete Reinforcement. Bureau of Indian Standards. [Ductility requirements (UTS/yield ≥ 1.15, elongation ≥ 16% for Fe 500D) linked to minimum n values.]
Kocks, U.F. and Mecking, H. (2003). Physics and phenomenology of strain hardening: the FCC case. Progress in Materials Science, 48(3), 171–273. [Physical derivation of dislocation-density evolution and its connection to Hollomon behaviour.]
Related article on this site: The Hollomon equation is one of several constitutive models covered in depth in our full guide: Strain Hardening in Steel and Metals - Complete Guide. That article covers the Bauschinger effect, cold-working processes, and the role of strain hardening in IS 456 / IS 800 structural design.
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