The Johnson-Cook (J-C) model is a structural material model. It is widely used to simulate the behavior of metals under high strain rates, large plastic deformations, and high temperatures. In this paper, we will fully explain this model, which is implemented in Abaqus/Explicit software. Based on our experience in finite element simulation, this model is particularly effective for impact analysis, ballistics, and machining simulations.
Johnson-Cook Theoretical Foundation
The J-C model, introduced by Johnson and Cook in 1983 , focuses on the characterization of the yield stress and failure strain of materials during deformation. This model takes into consideration three key factors: temperature, strain, and strain rate . It primarily aims to describe the influence of these factors on the material’s behavior, allowing for a better understanding of its mechanical response under varying conditions. By integrating these factors, the J-C model provides a comprehensive understanding of the material’s behavior, allowing for a visualization of the impact of different factors on strain. This enables researchers and practitioners to gain insights into the material’s response to various process conditions and optimize the grinding accordingly.
The J-C model expresses flow stress (σ) as a product of three components:
1. Strain Hardening Component
Defines plastic deformation at reference conditions:
- A: Initial yield stress (MPa)
- B: Strain hardening coefficient (MPa)
- n: Strain hardening exponent
2. Strain Rate Hardening Component
- C: Strain rate sensitivity coefficient
- ε̇0: Reference strain rate (s-1) – Typically 1 s-1 for metals
3. Thermal Softening Component
- m: Thermal softening exponent
- Troom: Reference temperature (20-25°C)
- Tmelt: Melting temperature (material’s liquidus temperature)
- Ttransition: Optional transition temperature threshold
4. Damage and Failure Criterion
The Johnson-Cook damage model predicts material failure through damage accumulation:
- d1-d5: Damage parameters (dimensionless)
- η: Stress triaxiality (p/q)
- εf: Equivalent plastic strain at failure
- D: Damage variable (failure when D ≥ 1)
Material Parameters
| Parameter | Description | Typical Units |
|---|---|---|
| A | Yield stress at zero strain | MPa |
| B | Hardening modulus | MPa |
| n | Hardening exponent | Dimensionless |
| C | Strain rate coefficient | Dimensionless |
| m | Thermal softening exponent | Dimensionless |
| d1-d5 | Damage model coefficients | Dimensionless |
| Tmelt | Melting temperature | °C |
| ε̇0 | Reference strain rate | s-1 |
Do you want to know our suggested ideas for simulation in Abaqus?
Our experience can help you solve complex finite element simulation problems..
Defining Johnson-Cook Damage Parameters (d₁-d₅)
Parameter Significance
The damage parameters d₁-d₅ control material failure prediction in the Johnson-Cook model. These dimensionless coefficients determine how plastic strain accumulates toward failure under different conditions:
Determination Methods
| Parameter | Physical Significance | Calibration Method | Typical Range |
|---|---|---|---|
| d₁ | Base failure strain at η=0 | Uniaxial tension tests | 0.01-0.5 |
| d₂ | Stress triaxiality sensitivity | Notched specimen tests | 0.1-5.0 |
| d₃ | Triaxiality exponential factor | Multi-axial loading tests | -3.0-0.0 |
| d₄ | Strain rate sensitivity | High-rate testing (SHPB) | 0.001-0.1 |
| d₅ | Temperature sensitivity | Heated specimen tests | 0.0-2.0 |
Calibration Process
- Experimental Testing:
- Uniaxial tension tests (for d₁)
- Notched specimen tests (for d₂, d₃)
- Split-Hopkinson Pressure Bar (SHPB) tests (for d₄)
- High-temperature tests (for d₅)
- Numerical Optimization:
Minimize: Σ(εexpf – εmodelf)²
Use optimization algorithms to match experimental failure strains
Johnson-Cook Parameters for AISI 4340 Steel
Abaqus Implementation
*MATERIAL, NAME=STEEL_JC *JOHNSON COOK DAMAGE 0.05, 3.44, -2.12, 0.002, 0.61 *DAMAGE EVOLUTION, TYPE=DISPLACEMENT 0.1, 0.0001
Where the damage evolution parameters control post-initiation softening
Important Notes
- Parameters are highly material-specific
- Require validation with multiple test configurations
- Negative d₃ values are typical for metals
- d₅ > 1 indicates thermal softening accelerates damage
Implementation in Abaqus
Basic Configuration
- Define material plasticity using *PLASTIC option
- Specify strain rate parameters using *RATE DEPENDENT
- Include thermal expansion with *EXPANSION
- Define temperature-dependent properties using *DEPVAR
Advanced Implementation Johnson-Cook suboptions
Johnson-Cook suboptions configuration in Abaqus/Explicit:
*JOHNSON COOK DAMAGE
*DAMAGE EVOLUTION, TYPE=DISPLACEMENT
- Plasticity Suboption: Defines strain hardening parameters (A, B, n)
- Rate Dependency Suboption: Specifies C and ε̇0
- Damage Initiation: Sets d1-d5 parameters
- Damage Evolution: Controls post-damage behavior using:
- Linear or exponential softening laws
- Element deletion criteria (D ≥ 1)
- *DAMAGE EVOLUTION parameters for displacement at failure
Model Limitations
- Assumes multiplicative decomposition of effects
- Limited accuracy for extreme strain rates (>104 s-1)
- Damage parameters require extensive calibration
- Assumes proportional loading for damage accumulation
- Limited to isotropic damage representation
- Temperature calculation assumes adiabatic conditions
Conclusion
The Johnson-Cook model provides an efficient framework for modeling rate-dependent plasticity in Abaqus. Based on our experience, this model is particularly useful for dynamic events in the simulation of the aforementioned cases. Although it has some limitations, its computational efficiency was very interesting and helpful for industrial applications that require the analysis of large deformations. Also, according to the results of various papers, we found that the extension of the damage evolution allows for the prediction of material failure, making it suitable for impact and crash simulation.
