衝撃解析（落下・衝突）
Theory and Physics
Fundamentals of Impact Analysis
Professor, how is impact analysis different from regular dynamic analysis?
Impact is a phenomenon where a large force acts over an extremely short time ($\mu s \sim ms$). The time scale is orders of magnitude shorter than that of typical vibration analysis.
Classification of Impact
| Type | Time Scale | Example | Analysis Method |
|---|---|---|---|
| Low-Velocity Impact | 1–100 ms | Drop, Vehicle Collision | Explicit FEM |
| High-Velocity Impact | 0.1–1 ms | Ballistic Impact, Tool Impact | Explicit FEM + SPH |
| Hyper-Velocity Impact | < 0.1 ms | Space Debris, Explosion | SPH, ALE |
| Shock Wave | $\mu s$ | Blast, Underwater Explosion | ALE, Eulerian Method |
So the analysis method changes depending on the time scale.
Low-velocity impact is sufficiently handled by standard explicit FEM. For high-velocity impact, elements undergo large distortion, requiring methods like SPH (Smoothed Particle Hydrodynamics) or ALE.
Mechanics of Impact
Basic impact parameters:
- Impact Velocity $v$ — Kinetic energy $E_k = mv^2/2$
- Impact Duration $\Delta t$ — Time from contact to separation
- Peak Force $F_{max}$ — Maximum value of impact force
- Impulse $I = \int F dt \approx m \Delta v$ — Change in momentum
Can we roughly estimate the impact result using energy conservation?
Assuming all $E_k = mv^2/2$ is converted into deformation energy:
Comparing FEM results with this rough estimate serves as a sanity check.
Impact Analysis in FEM
In explicit FEM:
1. Model the impactor (rigid or deformable body)
2. Model the target (shell/solid + Material Nonlinearity)
3. Define contact (Penalty Method)
4. Set initial velocity
5. Execute time-history analysis
6. Evaluate force-time, deformation-time, energy-time
Summary
Key Points:
- Impact involves large forces over short times — $\mu s \sim ms$ scale
- Explicit FEM is standard — LS-DYNA, Abaqus/Explicit
- SPH/ALE for high-velocity impact — Avoids large element distortion
- Rough estimate check with energy conservation — $E_k = mv^2/2$
- Force-time curve and deformation pattern are primary results
The Essence of Impact as Wave Propagation
Impact in solids propagates as an elastic longitudinal wave (P-wave) at the speed of sound c₀=√(E/ρ). For steel, c₀≈5000 m/s, meaning a stress wave takes only 20 μs to traverse a 100 mm component. The one-dimensional wave propagation theory organized by Kolsky in the 1950s remains the analytical foundation for Hopkinson bar tests today, serving as an essential method for evaluating material properties at strain rates of 10³–10⁴/s.
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 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, 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 pull a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. So, 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 force $f_b$ (e.g., gravity) and surface force $f_s$ (e.g., pressure, contact force). 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 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 applying "compression"—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. 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 to improve ride comfort. 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 heterogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and stress-strain relationship is linear.
- Isotropic material (unless specified otherwise): 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: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behaviors like plasticity and creep require constitutive law extensions.
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. Note unit system inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note 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
Impact Analysis Settings
Please tell me the specific FEM settings for impact analysis.
LS-DYNA
```
*KEYWORD
*CONTROL_TERMINATION
0.010 $ 10 ms
*CONTROL_TIMESTEP
0.0, 0.9 $ dt safety factor 0.9
*INITIAL_VELOCITY_SET
1, 0., 0., -5000. $ 5 m/s downward (in mm/s units)
*CONTACT_AUTOMATIC_SURFACE_TO_SURFACE
1, 2
```
Abaqus/Explicit
```
*STEP, NAME=impact
*DYNAMIC, EXPLICIT
, 0.010 $ 10 ms
*INITIAL CONDITIONS, TYPE=VELOCITY
impactor, 1, 0.
impactor, 2, 0.
impactor, 3, -5.0 $ 5 m/s
*CONTACT
...
*END STEP
```
So you set the initial velocity, and then the solver tracks the contact and deformation.
The explicit method is a "set the situation and let physics play out" approach. The user only defines initial conditions (velocity, position) and contact. The results emerge automatically according to physical laws.
Mesh Size Guideline
Mesh size for impact analysis:
| Target | Element Size |
|---|---|
| Contact Surface (Impact Area) | 1–5 mm |
| Remote Areas | 5–20 mm |
| Impactor (Rigid Body) | Coarse is OK |
Fine at the contact surface, coarse in remote areas.
Contact resolution directly affects the results. The area around the impact point needs 1–2 mm elements. However, finer meshes lead to smaller $\Delta t$, so balance with computational cost.
Summary
Designs Ignoring Strain Rate Dependence Are Dangerous
The yield stress of steel materials increases by 1.3–2 times at a strain rate of 10³/s compared to quasi-static (10⁻³/s) (Cowper-Symonds law). In LS-DYNA's MAT_003, velocity dependence is expressed by D·n parameters, with D=40.4, n=5.0 being widely used standard values for mild steel. Multiple studies report experimental data showing that ignoring this velocity effect in automotive bumper crash analysis leads to 20–40% overestimation of deformation.
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)
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 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 each 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 the full load in small increments rather than all at once. The arc-length method (Riks method) can track beyond extremum 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
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 judge based on total cost-effectiveness.
Practical Guide
Impact Analysis Practice
Main applications of impact analysis:
| Application | Standard | Condition |
|---|---|---|
| Smartphone Drop | Internal Company Standard | 1.5 m drop, concrete surface |
| Electronic Equipment Drop |