Cohesive Zone Model (CZM)
Theory and Physics
What is CZM?
Professor, CZM was also mentioned on the interlaminar delamination page, but can it be used for general fracture as well?
CZM (Cohesive Zone Model) is a versatile fracture model that describes interface failure using a traction-separation law. It is applicable not only to interlaminar delamination but also to adhesive joint debonding, ductile cracking in metals, and concrete cracking.
Traction-Separation Law
Bilinear type:
1. Linear Elasticity — Stress increases with stiffness $K$
2. Damage Initiation — Reaches strength $t^0$
3. Softening — Stress decreases while opening increases
4. Complete Separation — Fracture after consuming energy $G_c$
Advantages of CZM
Summary
The Competing Works of the Dugdale-Barenblat Model
The cohesive zone model was independently published in 1960 by Dugdale (UK) and Barenblatt (USSR). Dugdale approximated the plastic zone in steel plates as a strip, while Barenblatt formulated a more general attraction relationship. Due to the Cold War, there was no information exchange for several years, and they only learned of each other's papers in the 1970s. Today's CZM was born from the competing works of these two individuals.
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 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, which is the assumption that "acceleration can be ignored because forces are applied slowly". It absolutely cannot be omitted in impact loading 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 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. 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 $f_b$ (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 common mistake here: getting the load direction wrong. Intending "tension" but it becomes "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 heterogeneity
- 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 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 loads and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistencies 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
CZM in FEM
Two implementations:
1. Cohesive Elements — Place thin cohesive elements at the interface (COH3D8, etc.)
2. Surface-based CZM — Set CZM on contact surfaces (no additional elements)
```
*COHESIVE SECTION, RESPONSE=TRACTION SEPARATION
*COHESIVE BEHAVIOR
K_n, K_s, K_t
*DAMAGE INITIATION, CRITERION=QUADS
t_n, t_s, t_t
*DAMAGE EVOLUTION, TYPE=ENERGY, MIXED MODE BEHAVIOR=BK
G_Ic, G_IIc, G_IIIc
```
Mesh Requirements
3 to 5 elements within the process zone. $l_{cz} \approx EG_c/(t^0)^2$.
Summary
Relationship Between TSL Curve Shape and Fracture Toughness
The traction-separation law (TSL) in cohesive zone models has several shapes such as triangular, trapezoidal, and exponential. Different shapes change crack propagation behavior even for the same fracture energy Gc. Trapezoidal shapes often better represent experiments for high-strength adhesives, while exponential shapes fit rubber-based adhesives. The peak strength σmax and area Gc are the minimum required two parameters; using only one of them worsens FEM convergence.
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 important.
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 the convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial values 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
Instead of applying the full load at once, apply it in small increments. The arc-length method (Riks method) can trace beyond extremum points in the load-displacement relationship.
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 judgment should be based on total cost-effectiveness.
Practical Guide
CZM in Practice
Interlaminar delamination in composites, strength evaluation of adhesive joints, concrete cracking, fracture in welded joints.
Practical Checklist
Delamination Simulation of Aircraft Panels
For the adhesive interface between carbon fiber reinforced plastic (CFRP) panels and metal frames on the Airbus A380, delamination simulation using cohesive zone models has become a design standard. Setting TSL parameters of Gc=800 J/m² and σmax=50 MPa independently for Mode I and Mode II achieved prediction accuracy of ±15% for damage area after drop impact (from Airbus demonstration tests in the 2010s).
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 (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 can be far from reality. Confirm that results stabilize with 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 the same as "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 really fully fixed?" "Is this load really uniformly distributed?" — Correctly modeling real-world constraint conditions is often the most important step in the entire analysis.
Software Comparison
CZM Tools
Abaqus CZM Implementation and Contact Elements
In Abaqus/Standard, CZM can be implemented in two ways: surface-based cohesive behavior and dedicated cohesive elements (COH2D4, etc.). Boeing used Abaqus CZM to perform certification analysis for adhesive joints in the B787 wing spar. Analysis of a 300mm×100mm test specimen based on FAA regulations kept prediction error for delamination load within ±10%.
The Three Most Important Questions for Selection
- "What are you solving?": Does the tool support the physical models and element types required for the Cohesive Zone Model (CZM)? For example, in fluids, the presence of LES support; in structures, the ability to handle contact and large deformations makes a 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 in cars.
- "How far will it expand?": 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 CZM
Application of CZM to Biomaterials
Since the 2010s, cohesive zone models have also been applied to fracture healing simulations. Converting cortical bone fracture toughness Kc=2.2 MPa√m to CZM parameters can reproduce debonding at bone-implant interfaces. In 2018, a group at KIT predicted bone fixation strength of hip implants using CZM and obtained results matching actual measurements 3 months post-surgery within 10% error.
Troubleshooting
CZM Troubles
Related Topics
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