古典積層理論（CLT）
Theory and Physics
What is CLT?
Professor, what is Classical Lamination Theory (CLT)?
CLT is the fundamental theory describing the mechanics of fiber-reinforced composite (FRP) laminates. It is an extension of Kirchhoff plate theory to anisotropic laminates, calculating the overall stiffness of the laminate from the material properties and fiber angles of each layer.
How is it different from isotropic plate theory?
For isotropic plates, everything is determined by a single parameter, the bending stiffness $D = Et^3/12(1-\nu^2)$. In CLT, a 6×6 matrix called the ABD matrix is required. It includes membrane stiffness, bending stiffness, and membrane-bending coupling.
ABD Matrix
The central concept of CLT, the ABD matrix:
Where:
- $\{N\}$ — Membrane force resultants ($N_x, N_y, N_{xy}$)
- $\{M\}$ — Bending moment resultants ($M_x, M_y, M_{xy}$)
- $\{\varepsilon^0\}$ — Midplane strains
- $\{\kappa\}$ — Curvatures
What do $[A]$, $[B]$, and $[D]$ represent?
$[B]$ is interesting. For isotropic plates, $[B] = 0$, right?
Exactly. For isotropic plates or symmetric laminates (e.g., $[0/90]_s$), $[B] = 0$. For unsymmetric laminates, $[B] \neq 0$, causing peculiar behavior like warping when tension is applied.
ABD Matrix Calculation
Calculation formulas for each matrix:
Where $\bar{Q}_{ij}^{(k)}$ is the transformed reduced stiffness matrix of the $k$-th layer (considering fiber angle), and $z_k$ is the layer position.
So the overall ABD matrix of the laminate can be calculated from the position and fiber angle of each layer.
Yes. CLT is the theory that derives the "macroscopic stiffness of the laminate" from "material properties + fiber angle + stacking sequence of each layer". When composite materials are handled by shell elements in FEM, CLT is used internally.
Laminate Notation
How do you read notations like "$[0/90/\pm 45]_s$"?
Fiber angles are listed from the bottom up:
- $[0/90/\pm 45]_s$ = $[0/90/+45/-45/-45/+45/90/0]$
- $s$ means symmetric. It is mirrored about the midplane.
- For symmetric laminates, $[B] = 0$ (no membrane-bending coupling).
Is using symmetric laminates a basic design principle?
Basically, yes. Unsymmetric laminates cause warpage (curing warpage) after molding, which is also problematic for manufacturing. Aircraft structures are, in principle, symmetric laminates.
Summary
Let me organize the CLT theory.
Key points:
- The ABD matrix describes laminate stiffness — $[A]$: membrane, $[B]$: coupling, $[D]$: bending
- Calculate the ABD matrix from each layer's $\bar{Q}$ and position $z_k$ — Layer material properties and fiber angle are the inputs.
- If $[B] = 0$, there is no membrane-bending coupling — Achieved with symmetric laminates.
- Symmetric laminates are the design standard — Prevents warpage.
- FEM composite shell elements use CLT internally — CLT is the foundational theory for FEM.
So, without understanding CLT, you can't properly set up or interpret the results of FEM composite analysis.
Exactly. CLT is the "alphabet" of composite design. You cannot discuss FEM analysis of composites without it.
Origin of CLT and Contributions of Reissner & Mindlin
Classical Lamination Theory (CLT) was established by Whitney, Leissa, and others in the 1960s-70s. CLT assumes each lamina is in a state of plane stress and integrates based on Kirchhoff plate theory (out-of-plane deformation keeps the normal rigid) to obtain the overall stiffness matrix [ABD]. Reissner's shear deformation theory (1945) and Mindlin's improvement (1951) evolved into the First-order Shear Deformation Theory (FSDT), which compensates for the "no shear stiffness" limitation of CLT.
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 is negligible". 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 $F=kx$, the essence of the stiffness term. 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" is incorrect. Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External force term (load term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (e.g., pressure, contact forces). 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 applying "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 vibrational 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 vibrational energy to improve ride comfort. What if damping were zero? Buildings would continue 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 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: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity or creep, constitutive law extensions are needed.
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 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) | In mm system: N, in m system: N (consistent). |
Numerical Methods and Implementation
CLT Implementation in FEM
How is CLT implemented in FEM shell elements?
For composite shell elements, the ABD matrix is constructed from each layer's $\bar{Q}$ matrix and laminate information (fiber angle, thickness, position), and reflected in the element stiffness matrix. Integration points through the thickness are placed in each layer.
Nastran
```
PCOMP, 1, , , , SYM
, 1, 0.125, 0., YES,
, 1, 0.125, 90., YES,
, 1, 0.125, 45., YES,
, 1, 0.125, -45., YES
```
The PCOMP card specifies material ID, thickness, and fiber angle for each layer. SYM indicates a symmetric laminate.
Abaqus
```
*SHELL SECTION, COMPOSITE, ELSET=panel
0.125, 3, CFRP, 0.
0.125, 3, CFRP, 90.
0.125, 3, CFRP, 45.
0.125, 3, CFRP, -45.
```
Each line defines one layer: thickness, number of integration points, material name, fiber angle.
Ansys
In Workbench, laminates are defined using "ACP (Ansys Composite PrePost)". GUI operations allow intuitive setting of draping and fiber angles.
Why is Nastran's PCOMP widely used in aerospace?
PCOMP has a history dating back to the 1980s and has extensive verification records for aerospace certification (type certification). It can directly output strain/stress for each layer and has robust integration with failure criteria (Tsai-Wu, Hashin, etc.).
Through-Thickness Integration Points
How many integration points are needed per layer?
A minimum of 3 points per layer (Simpson integration) is recommended. For elastoplastic analysis, 5 or more points are needed.
Total: $n$ layers × 3 points = $3n$ points. For a 20-layer laminate, that's 60 points. More through-thickness integration points increase computational cost but are crucial for accuracy.
Material Coordinate System
Relative to which coordinate system is the fiber angle defined?
The material coordinate system is defined for each layer. Typically, the fiber angle is specified relative to the in-plane direction (axis 1 direction) of the shell element. Since the fiber angle reference changes with element orientation, verifying material direction is essential.
To accurately represent draping (variation of fiber angle on curved surfaces), the material coordinate system must be set individually for each element. ACP (Ansys) and Fibersim (Siemens) automate this task.
Summary
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