層間剥離解析
Theory and Physics
What is Delamination?
Professor, is "delamination" the most dangerous failure mode for composites?
Exactly. Delamination is failure where the layers of a laminate separate, and it is the most common and dangerous damage mode for composites. It is an "internal damage" that is invisible from the surface, and this difficulty in detection increases its danger.
Invisible from the surface! That's scary.
With low-velocity impact (falling objects, dropped tools, etc.), there may be almost no trace on the surface, yet extensive delamination has spread internally. This is called BVID (Barely Visible Impact Damage) and is the most important design condition for composite aircraft structures.
Mechanism of Delamination
Delamination occurs when the interlaminar shear stresses ($\tau_{xz}, \tau_{yz}$) and peel stress ($\sigma_z$) exceed the interface strength.
From a fracture mechanics perspective, there are three modes:
| Mode | Stress | Deformation | Standard Test |
|---|---|---|---|
| Mode I (Opening) | $\sigma_z$ (Tension) | Layers open | DCB |
| Mode II (In-plane shear) | $\tau_{xz}$ | Layers slide | ENF / 4ENF |
| Mode III (Out-of-plane shear) | $\tau_{yz}$ | Layers twist | ECT |
Is Mode I the most dangerous?
The critical energy release rate for Mode I, $G_{Ic}$, is the lowest ($\approx 0.1 \sim 0.3$ kJ/m²). Mode II's $G_{IIc}$ is 2 to 4 times higher. Therefore, Mode I opening often initiates first.
Energy Release Rate
The condition for delamination propagation is described by the Energy Release Rate (ERR):
$$ G \geq G_c $$
The condition for delamination propagation is described by the Energy Release Rate (ERR):
$G$ is the current energy release rate, $G_c$ is the critical value (material property).
For mixed-mode (Mode I + Mode II acting simultaneously), a mixed-mode criterion is used:
$\alpha = \beta = 1$ is a special case of the Benzeggagh-Kenane (BK) criterion.
Modeling Delamination in FEM
How do you model delamination in FEM?
Two main methods:
1. VCCT (Virtual Crack Closure Technique)
Calculates the energy release rate from nodal forces and opening displacements at the crack tip. A method for tracking the propagation of an existing crack.
2. CZM (Cohesive Zone Model)
Places cohesive elements at the interface and represents delamination using a constitutive law (traction-separation law) relating stress to opening displacement. Can handle both crack nucleation and propagation.
Is CZM more versatile?
CZM does not require the crack location to be assumed in advance (cohesive elements can be placed on all interfaces). VCCT is only for propagation of existing cracks. CZM is currently the mainstream method.
Summary
Let me organize the theory of delamination.
Key points:
- Delamination is the most dangerous damage in composites — invisible from the surface (BVID)
- Three fracture modes — Mode I (opening), II (shear), III (twisting)
- Propagation occurs when Energy Release Rate $G \geq G_c$ — mixed-mode criterion
- CZM (Cohesive Zone Model) is mainstream — can handle both nucleation and propagation
- VCCT is only for existing crack propagation — more limited than CZM but computationally lighter
BVID (Barely Visible Impact Damage) dictates aircraft design... It's a design battle against an "invisible enemy".
Exactly. Aircraft composite design is performed under the premise that "the worst BVID exists." That's why CAI (Compression After Impact) strength becomes the design allowable value.
Discovery of Delamination and its Impact on Aerospace
The CFRP interlaminar delamination problem became a serious challenge during the introduction of CFRP in aircraft in the 1970s. Delamination discovered in the CFRP horizontal stabilizer of the F-14 fighter caused a 40% reduction in design strength, prompting the US Navy to conduct NDT inspections on all aircraft in 1975. This incident was the catalyst for treating "delamination" as a primary failure mode in CFRP design.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "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 $F=kx$, and it's the essence of the stiffness term. So here's a question — an iron rod and a rubber band, which stretches more under the same force? 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 $f_b$ (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), 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 it's actually "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 deliberately 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 heterogeneity.
- 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: Large deformation/large rotation problems require geometric nonlinearity. Plasticity, creep, and other nonlinear material behaviors require constitutive law extensions.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify load/elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit 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
CZM (Cohesive Zone Model) Implementation
Please teach me the specific implementation of CZM.
CZM describes interface behavior using a traction-separation law.
Bilinear traction-separation law:
1. Linear elastic region — Stress increases with initial stiffness $K$.
2. Damage initiation — Stress reaches interface strength $t^0$.
3. Softening region — Stress decreases while opening displacement increases.
4. Complete separation — Fracture occurs when the energy released reaches $G_c$.
What parameters are needed?
Abaqus Settings
```
*COHESIVE SECTION, RESPONSE=TRACTION SEPARATION
1.0,
*SURFACE INTERACTION, NAME=cohesive_prop
*COHESIVE BEHAVIOR
1e6, 1e6, 1e6
*DAMAGE INITIATION, CRITERION=QUADS
60., 90., 90.
*DAMAGE EVOLUTION, TYPE=ENERGY, MIXED MODE BEHAVIOR=BK, POWER=1.5
0.28, 0.79, 0.79
```
The initial stiffness of $10^6$ is quite large.
The initial stiffness is a penalty parameter expressing "no deformation in the bonded state". Too large and the condition number worsens, causing convergence difficulties. Too small and deformation occurs before separation. A guideline is $K \approx E / t_{ply}$ (ply elastic modulus / ply thickness) times 10 to 100.
VCCT vs. CZM
| Characteristic | VCCT | CZM |
|---|---|---|
| Crack Nucleation | × | ○ |
| Existing Crack Propagation | ○ | ○ |
| Mesh Dependency | Present | Low (regularized by $G_c$) |
| Parameters | Only $G_{Ic}, G_{IIc}$ | Strength + $G_c$ + Stiffness |
| Computational Cost | Low | High |
| Stability | Somewhat unstable | More stable |
So CZM is more versatile and stable, but has more parameters and higher cost.
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