Johnson-Cook Constitutive Model
Johnson-Cook Constitutive Model: Theoretical Foundations
What is the Johnson-Cook Constitutive Law?
Professor, what is the Johnson-Cook constitutive law?
The Johnson-Cook (JC) model (1983) is a strain rate and temperature dependent elastoplastic + ductile damage model. It is the most widely used model for metal deformation and fracture under impact and collision.
Constitutive Equation
Flow stress:
- $A$ โ Yield Stress
- $B, n$ โ Strain hardening coefficient and exponent
- $C$ โ Strain rate sensitivity
- $m$ โ Thermal softening exponent
- $\dot{\varepsilon}^* = \dot{\varepsilon}/\dot{\varepsilon}_0$ โ Dimensionless strain rate
- $T^* = (T-T_{room})/(T_{melt}-T_{room})$ โ Homologous temperature (dimensionless)
It's the multiplication of three factors (hardening ร rate ร temperature)!
Simple yet practical. It can describe high-speed metal deformation over a wide range using just five parameters ($A, B, n, C, m$). JC parameters for many metals are reported in the literature.
JC Failure Criterion
Equivalent plastic strain at ductile fracture:
$\eta = \sigma_m / \sigma_{vm}$ is the stress triaxiality. $D_1 \sim D_5$ are the fracture parameters.
The fracture strain changes with stress triaxiality $\eta$. It behaves more brittle under tension ($\eta > 0$) and more ductile under shear ($\eta \approx 0$).
Summary
Key Points:
- $\sigma = (A+B\varepsilon^n)(1+C\ln\dot{\varepsilon}^)(1-T^{m})$ โ Hardening ร Rate ร Temperature
- Five material constants โ Literature values available for many metals
- JC failure criterion โ Stress triaxiality dependent ductile fracture
- Standard model for impact/collision analysis โ LS-DYNA MAT_15, Abaqus PLASTIC+DAMAGE
Year of JC Model Proposal
This model, proposed by Gordon Johnson and William Cook in 1983, expresses stress as a multiplicative form of plastic strain, strain rate, and temperature. Originally developed to organize U.S. Army ballistic penetration test data, its adoption as a standard material model for high-speed deformation analysis spread within two years of the paper's publication.
Computational Methods for the Johnson-Cook Constitutive Model
LS-DYNA
```
*MAT_JOHNSON_COOK
$ A, B, n, C, m, Tmelt, Troom, eps0
350., 275., 0.36, 0.022, 1.0, 1793., 293., 1.0
```
Abaqus
```
*PLASTIC, HARDENING=JOHNSON COOK
A, B, n, m, Tmelt, Troom
*RATE DEPENDENT, TYPE=JOHNSON COOK
C, eps0
*DAMAGE INITIATION, CRITERION=JOHNSON COOK
D1, D2, D3, D4, D5, Tmelt, Troom
*DAMAGE EVOLUTION, TYPE=DISPLACEMENT
u_f
```
So in Abaqus, plasticity + rate dependence + damage are set with three separate definitions.
LS-DYNA uses a single *MAT card for everything. Abaqus defines them separately, offering flexibility but requiring more settings.
Summary
Identification Experiments for the 5 Parameters
The five Johnson-Cook constants (A, B, n, C, m) are identified stepwise. First, A, B, n are determined from quasi-static tests, then C from Split Hopkinson Bar tests (strain rates 10ยฒโ10โด/s), and m from heating tests. Representative values for Al6061-T6, widely cited, are A=276MPa, B=406MPa, n=0.51, C=0.00519, m=1.0.
Johnson-Cook Constitutive Model in Practice
JC in Practice
Used for ballistic impact (armor plate penetration), high-speed metal cutting, and metal fracture in crash safety.
Representative JC Parameter Values
| Material | A (MPa) | B (MPa) | n | C | m |
|---|---|---|---|---|---|
| Mild Steel (AISI 1018) | 220 | 750 | 0.40 | 0.022 | 1.0 |
| Al 6061-T6 | 324 | 114 | 0.42 | 0.002 | 1.34 |
| Ti-6Al-4V | 1098 | 1092 | 0.93 | 0.014 | 1.1 |
Practical Checklist
Application to Bird Strike Analysis
For bird strike analysis on aircraft engine fan blades, Johnson-Cook parameters for Ti-6Al-4V are essential. In FEA verification preceding FAR 33.76 certification tests, LS-DYNA and Abaqus Explicit are primarily used, with multiple cases reported in aerospace journals predicting blade tip deformation at 200 m/s impact velocity with ยฑ5 mm accuracy.
Johnson-Cook Constitutive Model: Software & Solver Comparison
JC Tools
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