Work Hardening Calculator
The Hollomon power law sigma = K * epsilon^n is the standard engineering model for describing the flow stress of metals in the plastic regime. K is the strength coefficient (related to the stress at unit strain) and n is the strain hardening exponent (the slope of the log-log stress-strain curve). Given K = 965 MPa and n = 0.14 (annealed 1045 steel), at a true plastic strain of 0.10: sigma = 965 * 0.10^0.14 = 965 * 0.712 = 687 MPa. This calculator also identifies the critical strain at which necking begins (epsilon = n by the Considere criterion).
Hollomon work hardening formula
sigma = K * epsilon^n
Necking strain (Considere): epsilon_neck = n
Where sigma is true flow stress (MPa), K is the strength coefficient (MPa), epsilon is true plastic strain (dimensionless), and n is the strain hardening exponent. The Considere criterion states that necking (localized thinning) begins when the true strain equals n, providing a useful upper bound on uniform elongation in forming operations.
Metal forming applications
The Hollomon equation is used in metal forming simulations (finite element and analytical) to compute forming forces, springback, and required press capacity. For deep drawing, stamping, and sheet metal forming, high n values (such as in austenitic stainless steels) enable greater stretching before necking and improve formability. Cold-worked materials (low n) are used where dimensional stability and strength are more important than ductility.
Work hardening: frequently asked questions
What is work hardening?
Work hardening (also called strain hardening) is the increase in yield strength and hardness of a metal when it is plastically deformed at room temperature. Dislocation density increases during plastic deformation, making further deformation progressively more difficult. Cold rolling, drawing, and forging all exploit work hardening.
What is the Hollomon equation?
The Hollomon equation (or power law) relates true stress to true plastic strain in the strain-hardening regime: sigma = K * epsilon^n. K is the strength coefficient (MPa) and n is the strain hardening exponent (0 to 1). It is the most widely used empirical model for metal forming analysis.
What is the strain hardening exponent n?
The strain hardening exponent n characterizes how quickly a metal strengthens with plastic strain. n = 0 means perfectly plastic (no hardening). n = 1 means linear hardening. Most structural metals have n between 0.1 and 0.5. High n also delays necking (Considere criterion: necking begins at true strain = n).
What are typical K and n values for common metals?
Annealed AISI 304 stainless: K = 1,400 MPa, n = 0.45. Annealed 1045 steel: K = 965 MPa, n = 0.14. Annealed 6061 aluminum: K = 205 MPa, n = 0.20. Commercially pure copper: K = 530 MPa, n = 0.34. Values vary with heat treatment and prior deformation history.
How is true strain different from engineering strain?
True (logarithmic) strain epsilon_t = ln(L/L0) = ln(1 + e), where e is engineering strain. True strain accounts for the continuously changing gauge length during deformation. For small strains they are nearly equal; for large strains such as in cold drawing, true strain can be significantly larger.
Official sources
- ASTM E646, "Standard Test Method for Tensile Strain-Hardening Exponents (n-Values) of Metallic Sheet Materials": astm.org.
- ASM International, "ASM Handbook Vol. 14: Forming and Forging": asminternational.org.
- NIST, Materials Measurement Science: nist.gov/mml.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.