VCCT（仮想亀裂閉合法）
Theory and Physics
What is VCCT?
Professor, what is VCCT?
VCCT (Virtual Crack Closure Technique) is a method that directly calculates the energy release rate $G$ from nodal forces and opening displacements at the crack tip. It is a crack propagation technique alongside CZM.
$F_y$ is the nodal force at the crack tip, $\delta_y$ is the opening displacement behind the crack tip, and $\Delta a$ is the element size.
VCCT vs. CZM
Characteristic VCCT CZM Crack Nucleation × ○ Propagation of Existing Crack ○ ○ Parameters Only $G_c$ Strength + $G_c$ + Stiffness Mesh Dependency Present Low (regularized by $G_c$) Computational Cost Low High
VCCT is only for propagation of existing cracks. CZM can also handle nucleation.
VCCT is optimal for delamination propagation (problems where cracks propagate from known crack tips). CZM is more versatile.
Summary
| Characteristic | VCCT | CZM |
|---|---|---|
| Crack Nucleation | × | ○ |
| Propagation of Existing Crack | ○ | ○ |
| Parameters | Only $G_c$ | Strength + $G_c$ + Stiffness |
| Mesh Dependency | Present | Low (regularized by $G_c$) |
| Computational Cost | Low | High |
VCCT is only for propagation of existing cracks. CZM can also handle nucleation.
VCCT is optimal for delamination propagation (problems where cracks propagate from known crack tips). CZM is more versatile.
Rybicki and Energy Release Rate
VCCT (Virtual Crack Closure Technique) was proposed in 1977 by Rybicki and Kanninen (US invention). It is a method to calculate the energy release rate when a crack propagates by a small amount Δa, using the forces at the crack tip nodes and the displacement differences when the crack is virtually closed. Its efficiency in calculating GI, GII, and GIII simultaneously from a single FEM calculation has made it the standard method for interface delamination analysis in composite materials.
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 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 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'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 $f_b$ (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 ending up with "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 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 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 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. Nonlinear material behavior like plasticity or creep requires 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. 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) | Unify to N in mm system, N in m system. |
Numerical Methods and Implementation
FEM for VCCT
```
*DEBOND, SLAVE=crack_surface, MASTER=intact_surface
*FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVIOR=BK
G_Ic, G_IIc, G_IIIc, eta
```
Define the upper and lower surfaces of the crack as slave/master. Nodes are automatically released when $G \geq G_c$.
Summary
VCCT Mesh Requirements and Error Evaluation
VCCT depends on the element size Δa at the crack tip. Smaller Δa improves accuracy but increases computational cost. Practically, Δa = t/10 to t/20 relative to plate thickness t is recommended; coarser meshes can lead to errors exceeding 10%. Also, because it's a simple calculation of the product of nodal forces and displacements rather than an integral path, its calculation speed is 2-3 times faster than the J-integral.
Linear Elements (1st-order Elements)
Linear interpolation between nodes. Low computational cost but lower stress accuracy. Beware of shear locking (mitigated by 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 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 each iteration. Provides quadratic convergence within the convergence radius but has high computational cost.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the initial value or every few iterations. Cost per iteration is low, but convergence 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 total load in small increments rather than all at once. 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 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 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
VCCT in Practice
Used for delamination in composite materials (simulation of DCB, ENF tests), and adhesive joint debonding.
Practical Checklist
Example of VCCT Application for CFRP Wing Skin Delamination
VCCT is used to evaluate delamination growth at the joint between CFRP wing skin and ribs on the Airbus A320. It combines the mixed-mode fracture criterion (GI/GIc+GII/GIIc=1) with VCCT calculated values to predict delamination front propagation. Boeing's internal standard BGS-33 "Composite Delamination Evaluation Procedure" has adopted VCCT as a standard method since the 1990s and is also used for CFRP component certification on the 737MAX.
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), put it on the 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 the mesh quality is poor, the results will be a mess no matter how good 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 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
VCCT Tools
ANSYS Mechanical VCCT Implementation
ANSYS Mechanic
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