動的破壊力学
Theory and Physics
Dynamic Fracture Mechanics
Professor, how is dynamic fracture different from static fracture?
Dynamic fracture is a phenomenon where cracks propagate at high speeds. The crack tip velocity approaches the Rayleigh wave speed (the surface wave speed of the material). Brittle fracture due to impact loading and rapid crack propagation in pipelines are typical examples.
Dynamic SIF
$k(\dot{a})$ is a dynamic correction factor dependent on crack velocity $\dot{a}$. As $\dot{a} \to c_R$ (Rayleigh wave speed), $k \to 0$.
Dynamic Fracture in FEM
Summary
Why the Charpy Test Was Created
The Charpy impact test was devised in 1901 by French engineer Georges Charpy to easily evaluate the brittle-ductile transition of steel. At the time, winter brittle fracture of steel bridges and pressure vessels had become a social issue. The requirement of 27J or more at -40°C (ISO standard) is still used today. It remains famous as a pioneering test in dynamic fracture mechanics.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, meaning "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried forward" 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, which assumes "forces are applied slowly enough that acceleration can be ignored". 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 is 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" is incorrect. 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 common mistake here: getting the load direction wrong. Intending "tension" but ending up with "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 continue 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 or 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 deformations). |
| 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) | Unified as N in mm system, N in m system. |
Numerical Methods and Implementation
FEM for Dynamic Fracture
```
*MAT_ADD_EROSION
$ Element deletion based on maximum principal strain
```
Abaqus/Explicit:
```
DAMAGE INITIATION + DAMAGE EVOLUTION + STATUS
```
Elements are deleted (STATUS) when damage reaches 1.0.
Summary
Calculation Method for Dynamic Stress Intensity Factor Kd
In dynamic fracture, the stress field changes over time due to stress wave propagation, so the quasi-static KIc cannot be used directly. The dynamic stress intensity factor KId is best determined from observations of crack tip stress fringes (coating method, isochromatic fringes) using high-speed cameras, but the virtual node force method in FEM can also achieve accuracy within ±10%. When propagation speed exceeds 0.3 times the Rayleigh wave speed, the KI/KId ratio changes drastically.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Computational cost is low, but stress accuracy is low. 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 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 Increment Method
Applies the full load not all at once, but in small increments. The arc-length method (Riks method) can traverse 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: it's more efficient to open it at an estimated location and adjust forward/backward (iterative) than to search sequentially from the first page (direct).
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 "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
Dynamic Fracture in Practice
Rapid crack propagation in pipelines (DF: Dynamic Fracture). Accidents where cracks propagate at sonic speeds in gas pipelines.
Practical Checklist
Ductile Fracture in Automotive Crash Analysis
In automotive crash safety analysis, determining whether steel sheets will fracture is crucial. Impact fracture of high-strength steel (590 MPa class) occurs at displacement speeds of 5-10 m/s, with fracture energy 20-30% lower than quasi-static fracture. Toyota used LS-DYNA's dynamic fracture model (Johnson-Cook criterion) to improve prediction accuracy for occupant protection structure fracture in side impacts to within ±15%.
Analogy of the 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 (post-processing visualization). 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 results are 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 can be far from reality. Confirm 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 often the most critical step in the entire analysis.
Software Comparison
Tools
Capability of LS-DYNA Dynamic Fracture Analysis
LS-DYNA implements multiple dynamic fracture analysis methods including SPH, EFG, and peridynamics. GM used LS-DYNA's XFEM feature (added in 2016) to analyze bumper crack propagation, achieving over 95% agreement with tests at time intervals of 5-10 ms. The combination with Material Type 24 (Johnson-Cook) in impact analysis is the standard in practice.
The Three Most Important Questions for Selection
- "What are you solving?": Does it support the physical models and element types needed for dynamic fracture mechanics? For example, for fluids, the presence of LES support; for structures, the capability for contact and large deformation makes the difference.
- "Who will use it?": For beginner teams, tools with rich GUIs are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between automatic (GUI) and manual (script) transmission cars.
- "How far will it be extended?": Selection considering future expansion of analysis scale (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technologies
Advanced Topics in Dynamic Fracture
Material Data for Strain Rate Dependent Fracture Toughness
Dynamic fracture toughness KId strongly depends on strain rate dε/dt. For carbon steel A508, it decreases from quasi-static KIc=120 MPa√m to impact KId=60-80 MPa√m (at strain rate 1000 /s). This difference is critical in thermal shock analysis of reactor pressure vessels. ASME Code Appendix G specifies a procedure to determine correction factors from estimated strain rates.
Troubleshooting
Dynamic Fracture Troubles
Common Mistakes in Dynamic Fracture Analysis
A frequent mistake is using static material properties for dynamic fracture. For example, the fracture toughness of steel can drop by 30-50% under impact. Another is neglecting stress wave reflections from boundaries, which can cause unrealistic stress concentrations. In pipeline rapid crack propagation analysis, using the wrong gas decompression model leads to significant errors in arrest prediction.
Troubleshooting Mindset
When an analysis fails, don't just randomly change parameters. First, simplify the problem. For example, if a complex impact analysis diverges, try a simple cantilever beam impact first. If that works, gradually add complexity back. This "divide and conquer" approach is the fastest way to identify the root cause.
Checklist When Not Converging
- Material nonlinearity: Are you using an appropriate constitutive model? Plasticity models require proper hardening rules.
- Contact settings: Is the contact stiffness appropriate? Too high causes convergence issues, too low causes penetration.
- Load increments: Are the increments too large? For path-dependent problems, smaller steps are often necessary.
- Mesh distortion: In large deformation, elements may become too distorted. Consider adaptive remeshing or ALE methods.
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