🔬 applied physics

Experimental characterization of cohesive laws for mode-II interlaminar fracture in geometrically scaled composites using through-thickness deformation analysis

This study proposes and validates an experimental framework using through-thickness deformation analysis and digital image correlation to characterize a single cohesive law for mode-II interlaminar fracture in geometrically scaled composites, revealing that while a unified law can capture local fracture behaviors, it yields lower fracture energy and smaller process zone sizes compared to traditional size effect analysis.

Original authors: Han-Gyu Kim, Ryan Howe, Richard Wiebe, S. Michael Spottswood, Patrick J. O'Hara, Marco Salviato

Published 2026-02-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Han-Gyu Kim, Ryan Howe, Richard Wiebe, S. Michael Spottswood, Patrick J. O'Hara, Marco Salviato

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand how a stack of sticky notes tears apart when you try to slide one layer over another. In the world of high-speed aircraft, engineers use carbon-fiber composites (which are like incredibly strong, layered sticky notes) to build planes that are light but tough. However, when these planes fly fast, the air gets hot and pushes hard, creating stress that can cause the layers to slide apart and fail. This is called Mode-II interlaminar fracture.

The big question this paper asks is: "How do we predict exactly how and when these layers will fail, regardless of how big the piece of material is?"

Here is the story of their experiment, explained simply:

1. The Problem: The "Goldilocks" Dilemma

Engineers usually test materials by breaking small pieces to predict how big structures will behave. But with these composite materials, size matters a lot.

The Old Way: If you test a tiny piece, it seems to break easily. If you test a huge piece, it seems stronger.
The Confusion: It's like trying to guess how heavy a whole watermelon is by weighing a single seed. The old math said the "strength" changed depending on the size of the test piece, which doesn't make sense. Strength should be a property of the material, not the size of the sample.

2. The Experiment: The "Microscope vs. The Telescope"

To solve this, the researchers built three sets of test pieces: Small, Medium, and Large. They used a special bending test (like bending a ruler) to try to slide the layers apart.

To see what was happening inside the material, they used two different "eyes":

The Microscope (Small & Medium pieces): They used a high-powered microscope to watch the crack start. It was like looking at the tear in a piece of paper with a magnifying glass, seeing every tiny fiber move.
The Telescope (Large piece): The big piece was too huge for the microscope, so they used a standard 3D camera system. It was like looking at the whole paper from across the room.

3. The Discovery: The "Partial Tear"

When they analyzed the data, they found something fascinating.

The Small Pieces: When they broke, the layers only started to separate a tiny bit before snapping. The "tear zone" (where the material is slowly giving way) was very short. It was like the paper started to rip but stopped before the tear got long.
The Large Pieces: The layers separated much more before breaking. The "tear zone" was long and wide.

The Analogy: Imagine pulling apart a piece of taffy.

If you pull a tiny piece of taffy, it snaps almost instantly.
If you pull a giant block of taffy, it stretches out for a long time before finally breaking.
The researchers realized that the "rules" for how the material breaks (called the Cohesive Law) looked different depending on the size of the taffy block.

4. The Solution: Finding the "Master Rule"

The researchers wanted to find one single rule that could explain the behavior of the tiny, medium, and giant pieces all at once. This is the "Holy Grail" of material science.

Step 1: They tried to make a rule for each size separately. It worked perfectly for that specific size, but failed miserably for the others.
Step 2: They tried to find a "Master Rule" (a single cohesive law). They built a computer model based on the idea that the material has a specific "stretch limit" and "breaking strength."

The Result:
The "Master Rule" worked surprisingly well!

It predicted the overall strength of the medium-sized piece almost perfectly.
It predicted the breaking point of the small and large pieces reasonably well.
Most importantly, it captured the local behavior: It showed that in the small pieces, the material only partially developed its "tear zone" before breaking, while the large pieces got to stretch out fully.

5. Why This Matters

Think of this like a video game. Before this paper, the game had different physics engines for small objects and big objects. If you tried to use the "small object" physics on a "big object," the game would crash.

This paper successfully wrote one physics engine that works for all sizes.

For Engineers: This means they can now design safer, lighter aircraft. They don't need to build a massive, expensive prototype to test if a tiny part will fail. They can use this new "Master Rule" to simulate failures on a computer with high confidence.
For the Future: It helps us understand that when materials break, they don't just snap; they have a "process zone" where they struggle and stretch. Understanding this struggle helps us build planes that can survive the extreme heat and pressure of supersonic flight.

Summary

The researchers took a complex puzzle (how size affects breaking strength), used microscopes and cameras to watch the layers slide, and discovered that while small pieces break "prematurely," there is one underlying rule that governs all of them. They found the "secret sauce" that lets us predict how composite materials will fail, no matter how big or small the piece is.

1. Problem Statement

Modern aircraft structures increasingly rely on carbon-fiber composites, which are susceptible to mode-II interlaminar fracture (delamination) under extreme aerothermodynamic environments. A critical challenge in modeling this failure is characterizing the cohesive law (traction-separation relationship) for quasibrittle materials.

The Size Effect Paradox: Traditional Linear Elastic Fracture Mechanics (LEFM) methods, such as the compliance calibration method used in ASTM standards, assume fracture energy ( $G_f$ ) is a unique material property. However, experimental evidence shows that for geometrically scaled composites, LEFM-based methods yield different $G_f$ values depending on specimen size (larger specimens appear to have higher fracture energy).
Cohesive Zone Modeling (CZM) Limitations: While CZMs are widely used to simulate damage, determining a single, unique cohesive law for a material across different scales is difficult. Previous studies often used single-size specimens, ignoring size effects, or struggled to capture the complex, unstable crack propagation and small separation values typical of mode-II failure in composites.
Measurement Challenges: Accurately measuring the separation ( $\Delta u$ ) and the Fracture Process Zone (FPZ) size at the crack tip is difficult due to the small scale of damage and the "snapback" instability often seen in mode-II tests.

2. Methodology

The authors proposed a comprehensive experimental framework combining geometric scaling, multi-scale Digital Image Correlation (DIC), and numerical simulation.

A. Experimental Setup

Specimens: End-Notched Flexure (ENF) specimens made of Toray T700G/2510 carbon/epoxy prepreg.
Scaling: Three geometrically scaled sizes (Size-1, Size-2, Size-3) were created with scaling factors of 1/4, 1/2, and 1 (baseline). Dimensions (thickness, gauge length, crack length) were scaled, while width remained constant.
Loading: Three-point bending tests under displacement control.
Imaging Systems:
- Microscopic DIC (2D): Used for Size-1 and Size-2 specimens to capture high-resolution details of the crack tip and FPZ.
- Macroscopic DIC (3D): Used for Size-3 specimens to capture global deformation, though with lower resolution.

B. Data Processing Framework

The DIC data underwent a novel three-step post-processing procedure to characterize local fracture behavior:

Coordinate Transformation: Raw displacement data ( $u_x, u_z$ ) measured in geometric axes were rotated into principal material axes ( $u_1, u_3$ ) to account for beam bending rotation.
Curve Fitting: Polynomial fitting was applied to smooth noise and interpolate data points along the crack path.
Through-Thickness Deformation Analysis: The authors analyzed the variation of in-plane displacement ( $u_1$ ) across the thickness ( $z$ -axis). By extrapolating the displacement curves and accounting for "superficial traction" (paint adhesion preventing immediate separation), they estimated the true separation values ( $\Delta u_{max}$ and $\Delta u_{min}$ ) and the FPZ size.

C. Modeling and Simulation

Finite Element Analysis (FEA): Models were built in Abaqus using 2D plane-strain elements.
Cohesive Laws: Three types of traction-separation laws were tested: Linear (CZM-L), Bilinear (CZM-biL), and Plateau-Linear (CZM-pL).
Validation Strategy:
1. Individual Calibration: Parameters were tuned to match each specific specimen size.
2. Single Law Validation: A single cohesive law was developed to see if it could simultaneously predict the behavior of all three scaled sizes, addressing the material property uniqueness requirement.

3. Key Contributions

Novel Experimental Framework: Introduced a specific workflow (Coordinate Transformation $\to$ Curve Fitting $\to$ Through-Thickness Analysis) to extract cohesive parameters from DIC data, specifically addressing the difficulty of measuring small separations in mode-II failure.
Quantification of Size Effects: Demonstrated that fracture energy and FPZ size are not constant across scales when using standard LEFM methods, confirming the quasibrittle nature of the composite.
Identification of Partial CZM Development: Showed that smaller specimens exhibit partial development of the cohesive zone at peak load, meaning the full separation ( $\Delta u_f$ ) is not reached before failure, which explains why smaller specimens appear to have lower fracture energy in standard analyses.
Single Law Development: Successfully proposed a single bilinear cohesive law that captures the global fracture behavior (load-displacement) and local separation trends across all scales, despite discrepancies in FPZ size predictions.

4. Key Results

Global Behavior (Size Effect):
- LEFM-based compliance calibration yielded increasing fracture energy with specimen size (violating material uniqueness).
- Applying Bažant's Type-II Size Effect Law to the experimental data yielded a single, unique fracture energy value: $G_f = 1.347$ N/mm and an effective FPZ size of $c_f = 7.126$ mm.
Local Behavior (DIC Analysis):
- Separation values ( $\Delta u$ ) and FPZ sizes increased with specimen size.
- Size-1: $\Delta u_{max} \approx 6.92$ $\mu$ m, FPZ $\approx 0.7–2.0$ mm.
- Size-2: $\Delta u_{max} \approx 8.75$ $\mu$ m, FPZ $\approx 1.5–3.0$ mm.
- Size-3: $\Delta u_{max} \approx 25.47$ $\mu$ m, FPZ $\approx 22.0$ mm.
- This confirmed that smaller specimens fail with only partial cohesive zone development.
Simulation Outcomes:
- Individual Calibration: Models calibrated to specific sizes matched experimental load-displacement curves with <1% error.
- Single Cohesive Law: A single bilinear law ( $\tau_f = 350$ $τ_{f} = 350$ MPa, $G_f = 0.72$ $G_{f} = 0.72$ N/mm) was developed.
  - It successfully captured the post-peak response (snapdown) for all sizes.
  - It matched the local separation values for Size-1 and Size-2 well.
  - Discrepancies: The single law underestimated the peak load for Size-1 and overestimated it for Size-3. The simulated FPZ sizes were significantly smaller than experimental measurements, suggesting that damage initiation in the model occurred at higher traction levels than in reality.

5. Significance and Conclusion

This work provides a robust methodology for characterizing cohesive laws in quasibrittle composites, moving beyond the limitations of LEFM.

Practical Implication: The study proves that a single cohesive law can be defined for a material, but it requires accounting for partial cohesive zone development in smaller specimens.
Methodological Advance: The "through-thickness deformation analysis" allows researchers to extract separation data from DIC images even when the crack faces are not fully separated, overcoming a major hurdle in mode-II testing.
Future Directions: The authors note that discrepancies in the single law (specifically regarding FPZ size and peak load) may be due to crack-parallel stresses and unmodeled plastic dissipation. Future work will focus on refining the single law to incorporate the size-effect-derived fracture energy and investigating the impact of crack-parallel stress on mode-II failure.

In summary, the paper bridges the gap between experimental observation of size effects and the theoretical requirement for a unique material cohesive law, offering a validated framework for high-fidelity damage modeling in aerospace composites.