Strain Rate Dependent Plasticity
Theory and Physics
Strain Rate Dependent Plasticity
Professor, does the yield stress change with strain rate?
When metals deform at high speeds, the yield stress increases. In impacts ($\dot{\varepsilon} = 10 \sim 1000$ /s), it becomes 1.2 to 1.5 times the static yield stress.
Representative Models
| Model | Equation | Application |
|---|---|---|
| Cowper-Symonds | $\sigma_Y(1+(\dot{\varepsilon}/C)^{1/p})$ | LS-DYNA MAT_24. Most common |
| Johnson-Cook | $\sigma(1+C\ln\dot{\varepsilon}^*)$ | High-speed deformation + temperature |
| Perzyna | $\dot{\varepsilon}^p = \gamma(f/\sigma_0)^n$ | Overstress model |
Is Cowper-Symonds the most widely used?
Cowper-Symonds is standard in LS-DYNA MAT_24 for automotive crash. Representative parameters for steel (mild steel): $C = 40$ /s, $p = 5$.
Summary
Fundamentals of Viscoplasticity: Perzyna Rule
Piotr Perzyna formulated a viscoplastic model in 1963 at the Polish Academy of Sciences, where the plastic strain rate becomes a function of the overstress (difference between current stress and static yield stress). The relation $\dot{\varepsilon}_p = \gamma \langle f(\sigma,\kappa)/k \rangle^n$ allows unified treatment of creep and dynamic plasticity. This "Perzyna viscoplasticity" became the theoretical starting point for high-speed deformation analysis.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced 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 is "left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly enough that acceleration is negligible". It cannot be omitted in 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. 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"—different concepts.
- External force term (load term): Body forces $f_b$ (gravity, etc.) and surface forces $f_s$ (pressure, contact forces, etc.). 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 it becomes "compression"—sounds like a joke, but it actually happens when coordinate systems rotate 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. That's because vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—deliberately absorbing vibration energy for a smoother ride. What if damping were zero? Buildings would keep swaying 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 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 equilibrium 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
FEM Settings
LS-DYNA MAT_24:
```
*MAT_PIECEWISE_LINEAR_PLASTICITY
$ ..., C, p
, , , , , , 40., 5.
```
```
*RATE DEPENDENT, TYPE=POWER LAW
D, p $ Cowper-Symonds
```
Or:
```
*RATE DEPENDENT, TYPE=JOHNSON COOK
C, eps0
```
Summary
Identification of the Cowper-Symonds Equation
The Cowper-Symonds equation σy=σ₀[1+(ε̇/D)^(1/q)] is a representative approximation for strain rate dependent yield stress. The widely cited Jones (1989) values for mild steel are D=40.4 s⁻¹, q=5. The standard procedure involves obtaining data at 10²〜10⁴/s via Split Hopkinson Bar (SHPB) tests and determining the slope 1/q and D^(1/q) through least squares regression on a log-log plot.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Mid-side Nodes)
Capable of representing 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 Mesh
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 tangent stiffness matrix every iteration. Achieves quadratic convergence within convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using 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 $\epsilon = 10^{-3}$〜$10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies total load not all at once, but in small increments. The arc-length method (Riks method) can track 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 improves accuracy 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) is more efficient than searching sequentially from the first page (direct).
Relationship Between Mesh Order and Accuracy
1st order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd order elements are like "flexible curves"—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
Practical Checklist
Application to Explosive Protection Design
In IED (Improvised Explosive Device) protection analysis, the strain rate dependence of armor steel RHA (Rolled Homogeneous Armour) directly affects design quality. Set Cowper-Symonds coefficients in LS-DYNA MAT_024 to predict deformation behavior within 1ms after explosion. Practical accuracy has been verified since the 2010s through protection performance simulations for different NATO STANAG 4569 levels.
Analogy of Analysis Flow
The analysis flow is actually very similar to cooking. First, you shop for ingredients (prepare 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.
Pitfalls Beginners Often Fall Into
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. Verify that results stabilize across at least three levels of mesh density—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 like "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 truly fully fixed?" "Is this load truly uniformly distributed?"—Correctly modeling real-world constraint conditions is actually the most critical step in the entire analysis.
Software Comparison
Tools
LS-DYNA Strain Rate Options
LS-DYNA offers diverse SR (strain rate effect) options in material cards. For MAT_024, VP (viscoplastic flag)=0 uses equivalent plastic strain rate, VP=1 uses volumetric strain rate. MAT_019 (STRAIN_RATE_DEPENDENT_PLASTICITY) directly implements the Perzyna rule and is used for high-precision impact analysis. From R14 (2023), a diagnostic display function for effects was added.
Three Most Important Questions for Selection
- "What are you solving?": Does it support the required physical models/element types for strain rate dependent plasticity? For example, in fluids, presence of LES support; in structures, contact/large deformation capability makes a difference.
- "Who will use it?": For beginner teams, tools with rich GUI are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between automatic transmission (GUI) and manual transmission (script) in cars.
- "How far will it expand?": Selection considering future analysis scale expansion (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technology
Advanced
Crystal Plasticity and Strain Rate
In crystal plasticity theory (Crystal Plasticity), slip system
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