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A new approach for the determination of through-thickness and free-edge stresses in composite laminates based on structural elements

This paper proposes a novel method using structural cohesive elements and an accurate penalty stiffness derived from resin-rich layers to recover out-of-plane stress components in Kirchhoff-Love-based thin shell elements, thereby enabling accurate prediction of damage initiation in composite laminates.

Original authors: Xiaopeng Ai, Christos Kassapoglou, Boyang Chen

Published 2026-07-31

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Xiaopeng Ai, Christos Kassapoglou, Boyang Chen

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 building a skyscraper out of thousands of tiny, colorful Lego bricks. In the real world, engineers don't use plastic; they use "composite laminates," which are like super-strong sandwiches made of layers of fabric soaked in glue (resin). These materials are the secret sauce behind lightweight airplanes and fast cars because they are incredibly strong but weigh very little. However, there's a tricky problem: when you bend or stretch these sandwiches, the layers want to slide past each other or peel apart, especially near the edges. This is called "delamination," and it's a silent killer of structures.

To predict if a plane wing will hold up, engineers use computer simulations. For thin structures, they usually use a shortcut called "shell elements." Think of a shell element like a flat sheet of paper in a computer program. It's great at calculating how the paper stretches or bends across its surface, but it's terrible at figuring out what's happening inside the paper's thickness. It's like trying to understand the filling of a sandwich by only looking at the top slice of bread; you miss the cheese and meat entirely. Because these standard computer models can't see the "inside" stresses (the forces pushing the layers apart), they often miss the warning signs before a structure fails. This paper steps in to fix that blind spot, allowing engineers to see the hidden forces inside the sandwich without having to build a massive, slow, and complicated 3D model of every single grain of glue.

The Paper's Big Idea: Seeing the Invisible Layers

The authors, Xiaopeng Ai, Christos Kassapoglou, and Boyang Chen, have cooked up a new way to peek inside these composite sandwiches using a clever trick. They are building on a method that treats each layer of the laminate as a "structural element" (a fancy, high-tech version of our paper sheet) and connects them with "cohesive elements." You can think of these cohesive elements as invisible, stretchy glue strips between the layers.

In the past, these glue strips were a bit of a mystery. Engineers had to guess how stiff they should be. If they guessed wrong, the computer would either think the layers were glued together with super-strong epoxy (too stiff) or weak tape (too squishy), leading to wrong answers about how much stress the structure could handle. The paper argues that the old ways of guessing this stiffness were flawed because they treated the glue as if it had zero thickness, which isn't true in the real world where there's always a tiny bit of resin-rich space between layers.

The "Resin-Rich" Secret Sauce

The team's main discovery is a new formula for calculating exactly how stiff these "glue strips" should be. Instead of guessing, they realized that the stiffness depends on the actual properties of the resin-rich layer—that tiny, glue-heavy zone between the fabric layers.

Imagine you are stretching a stack of sticky notes. The force you feel depends on how thick the sticky part is and how strong the glue is. The authors derived a mathematical rule that says: "The stiffness of the glue strip is simply the strength of the glue divided by the thickness of that tiny resin layer."

They tested this idea by running simulations on different types of composite sandwiches:

Cross-ply laminates: Layers stacked like a checkerboard (0° and 90°).
Angle-ply laminates: Layers stacked at angles like a zigzag (+45° and -45°).
Quasi-isotropic laminates: Complex stacks that act the same in all directions.

In these simulations, they applied a uniform stretch to the ends of the laminate, just like pulling a rubber band. They then compared their new method against two things:

Analytical solutions: These are perfect mathematical answers derived by other scientists (like Pagano and Mittelstedt) that serve as the "gold standard."
Solid-element models: These are the heavy-duty, slow computer models that break the sandwich down into millions of tiny 3D blocks to see every detail.

What They Found

The results were impressive. When the authors used their new "resin-rich" stiffness formula, their simple shell models predicted the internal stresses almost perfectly.

Accuracy: Their results matched the "gold standard" mathematical solutions and the heavy-duty 3D block models. They could accurately predict the interlaminar normal stress (the force trying to peel the layers apart, denoted as $\sigma_z$ ) and the shear stress (the force trying to slide the layers past each other, denoted as $\tau_{xz}$ and $\tau_{yz}$ ).
Free-Edge Effects: One of the hardest things to predict is what happens at the very edge of the material, where the layers are exposed to the air. This is where "stress singularities" (extreme spikes in force) happen. The new method captured these spikes accurately, whereas older methods often missed them or required incredibly fine, slow meshes to see them.
Robustness: The team also checked if their method was sensitive to small changes. They asked: "What if the resin layer is slightly thicker or thinner? What if the glue is slightly stronger or weaker?" They found that their method was very stable. Even if the resin layer thickness varied between 0.6 × 10⁻³ inch and 0.9 × 10⁻³ inch, or if the glue's stiffness varied between 3 GPa and 5 GPa, the results didn't change much. This means engineers don't need to know the exact microscopic details of the glue to get a good answer.

The Speed Boost

Perhaps the most exciting part is the speed. The heavy-duty 3D block models (the "solid-element" models) are accurate but slow. To get a good answer, they needed to slice the laminate into eight tiny blocks for every single layer. The new method, however, only needed one shell element per layer.

In their tests, the new method cut the computer processing time by about 45%. For a specific test case, the old method took 9.52 seconds per calculation step, while the new method took only 5.21 seconds. That's a huge saving when you are designing a whole airplane wing with thousands of parts.

What This Means for the Future

The paper doesn't claim to have solved every problem in the universe. They admit that for very complex stress patterns inside a single layer, their method uses a "linear interpolation" (a straight line) to guess the values between layers, which might miss tiny, wiggly details. However, for the big picture of how layers peel and slide, it works beautifully.

By proving that you can get accurate 3D stress data using simple, fast shell elements, this work opens the door for engineers to simulate damage and delamination much more efficiently. It's like upgrading from a slow, blurry telescope to a sharp, fast one, allowing us to see the hidden dangers in our lightweight structures before they ever take flight. The authors suggest this framework could be the foundation for future tools that simulate both in-plane damage (cracks in the fabric) and delamination (peeling layers) all at once, making our aircraft and cars safer and lighter.

Technical Summary: A New Approach for Through-Thickness and Free-Edge Stresses in Composite Laminates

Problem Statement
Fiber-reinforced composites are critical for lightweight aerospace structures, yet predicting their strength and damage tolerance remains challenging. While thin-shell elements based on the Kirchhoff–Love hypothesis are widely used for their computational efficiency, they inherently account for only three in-plane stress components ( $\sigma_x$ , $\sigma_y$ , and $\tau_{xy}$ ). Consequently, the out-of-plane stress components required by conventional three-dimensional (3D) damage criteria are unavailable. This limitation prevents the accurate prediction of damage initiation, particularly interlaminar delamination and free-edge effects, which are driven by out-of-plane normal and shear stresses.

Existing methods to recover these stresses often rely on Mindlin–Reissner shell theories or post-processing techniques that either neglect cohesive elements (limiting delamination simulation) or face mesh-density bottlenecks imposed by cohesive zone models. Furthermore, previous formulations for penalty stiffness in cohesive elements often assume zero thickness or rely on kinematic assumptions inconsistent with the Kirchhoff–Love hypothesis, leading to overestimated stress levels.

Methodology
The authors propose a unified framework based on structural elements that captures free-edge effects in multilayer composite laminates with arbitrary lay-ups. The methodology consists of three core components:

Structural Element Formulation: The laminate is modeled using cubic shell elements for each ply, connected by structural cohesive elements. This approach retains the Kirchhoff–Love kinematic assumptions (where normals remain straight and perpendicular to the mid-plane) while eliminating the cohesive-zone mesh-density limitation found in traditional cohesive zone modeling.
Stress Recovery Mechanism: Out-of-plane stress components ( $\sigma_z$ , $\tau_{xz}$ , $\tau_{yz}$ ) are recovered directly from the tractions transmitted by the structural cohesive elements. By enforcing equilibrium at the interface, the cohesive tractions are equated to the out-of-plane stresses in the adjacent plies. These interfacial stresses are then interpolated through the thickness to reconstruct the full 3D stress field.
Physically Representative Penalty Stiffness: A critical innovation is the derivation of a new penalty stiffness formulation for the cohesive elements. Unlike previous approaches that assume zero thickness or simple shear deformation, this work defines the penalty stiffness as a function of the mechanical properties and thickness of the resin-rich layer between adjacent plies.
- The normal penalty stiffness ( $K_n$ ) is derived as $E_3 / h_{rr}$ , where $E_3$ is the out-of-plane elastic modulus and $h_{rr}$ is the resin-rich layer thickness.
- The shear penalty stiffness ( $K_s, K_t$ ) is derived as $G / h_{rr}$ , where $G$ is the shear modulus of the resin-rich layer.
- This formulation acknowledges that the effective thickness of the shell-based cohesive element is half the sum of the adjacent shell thicknesses, aligning with the kinematic assumptions of the Kirchhoff–Love theory.

Key Results
The proposed method was validated against analytical solutions (e.g., Pagano, Mittelstedt) and high-fidelity 3D solid-element models (C3D8 in Abaqus) across various laminate configurations:

Cross-ply laminates ([0°/90°]s): The method accurately predicted interlaminar normal ( $\sigma_z$ ) and shear ( $\tau_{yz}$ ) stresses at the interface and mid-plane, showing excellent agreement with analytical solutions.
Angle-ply laminates ([+θ/−θ]s): The method successfully captured the significant interlaminar shear stress ( $\tau_{xz}$ ) near free edges, matching both analytical and solid-element results.
Quasi-isotropic and General Lay-ups: The approach was extended to complex stacking sequences (e.g., [45/0/−45/90]s and arbitrary lay-ups) where no analytical solutions exist. In these cases, the structural element results demonstrated close agreement with solid-element models, accurately capturing stress singularities and peak values.
Sensitivity Analysis: The results showed that the predicted stress distributions are relatively insensitive to variations in the resin-rich layer's elastic modulus and Poisson's ratio within realistic ranges. The method remains robust even with approximate resin-rich layer properties.
Computational Efficiency: By using a single shell element per ply rather than multiple solid elements through the thickness, the proposed method reduced CPU time per increment by approximately 45% while maintaining comparable accuracy in stress prediction.

Significance and Claims
The paper claims to provide a dedicated approach for recovering the complete 3D stress field within Kirchhoff–Love shell elements, a capability previously unavailable for high-order shell formulations. By introducing a penalty stiffness formulation based on the resin-rich layer, the authors address the discrepancy between theoretical zero-thickness assumptions and the finite effective thickness of structural cohesive elements.

The significance of this work lies in its ability to:

Enable accurate prediction of through-thickness and free-edge stresses without resorting to computationally expensive 3D solid models.
Facilitate the simulation of combined in-plane damage and delamination on coarse meshes by removing the mesh-density limitations of traditional cohesive zone models.
Provide a physically based foundation for future developments in structural cohesive element models for composite laminates.

The authors conclude that this method offers a computationally efficient and accurate alternative for analyzing complex composite laminates, particularly for applications requiring the assessment of delamination initiation and free-edge effects.

The Paper's Big Idea: Seeing the Invisible Layers

The "Resin-Rich" Secret Sauce

What They Found

The Speed Boost

What This Means for the Future

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