Johnson-Cook構成則
Theory and Physics
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:
- $A$ — Yield Stress
- $B, n$ — Strain hardening coefficient and exponent
- $C$ — Strain rate sensitivity
- $m$ — Thermal softening exponent
- $\dot{\varepsilon}^* = \dot{\varepsilon}/\dot{\varepsilon}_0$ — Dimensionless strain rate
- $T^* = (T-T_{room})/(T_{melt}-T_{room})$ — Homologous temperature (dimensionless)
It's the multiplication of three factors (hardening × rate × 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:
- $\sigma = (A+B\varepsilon^n)(1+C\ln\dot{\varepsilon}^)(1-T^{m})$ — Hardening × Rate × Temperature
- Five material constants — Literature values available for many metals
- JC failure criterion — Stress triaxiality dependent ductile fracture
- Standard model for impact/collision analysis — 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.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Haven't you experienced your body being thrown forward during sudden braking? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly enough that acceleration is negligible". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber band. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong". Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (e.g., pressure, contact forces). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A typical pitfall here: getting the load direction wrong. Intending "tension" but actually applying "compression"—it sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades away. That's because the vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would keep shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic inhomogeneities
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions)
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity and creep, constitutive law extensions are needed
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation) |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
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 10²–10⁴/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.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but lower stress accuracy. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with Midside Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2–3 times. Recommended when stress evaluation is critical.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Meshing
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates the tangent stiffness matrix every iteration. Achieves quadratic convergence within the convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Incrementation Method
Applies the full load not all at once, but in small increments. The arc-length method (Riks method) can trace beyond limit points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but accuracy improves with each iteration. It's the same principle as looking up a word in a dictionary: opening to an estimated page and adjusting forward/backward (iterative method) is more efficient than searching sequentially from the first page (direct method).
Relationship Between Mesh Order and Accuracy
First-order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. Second-order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judgment should be based on total cost-effectiveness.
Practical Guide
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 ±5 mm accuracy.
Analogy for Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy the ingredients (prepare the CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work". If mesh quality is poor, the results will be a mess no matter how excellent the solver is.
Common Pitfalls for Beginners
Are you checking mesh convergence? Do you think "the calculation ran = the result is correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer will be far from reality. Confirm that results stabilize across at least three mesh density levels—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct".
Thinking About Boundary Conditions
Setting boundary conditions is the same as "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface really fully fixed?" "Is this load really uniformly distributed?"—Correctly modeling real-world constraint conditions is often the most critical step in the entire analysis.
Software Comparison
JC Tools
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