Analysis of Sandwich Panels
Theory and Physics
What is a Sandwich Structure?
Professor, a sandwich panel is a structure where a core is sandwiched between two face sheets, right?
Correct. It's a combination of thin, high-stiffness face sheets and a lightweight core that resists shear. It's analogous to an I-beam: the flanges correspond to the face sheets, and the web corresponds to the core. It achieves high bending stiffness with low weight.
Where are they used?
Mechanics of Sandwich Structures
Bending stiffness of a sandwich panel:
The first term is dominant, right? The farther the face sheets are from the neutral axis, the higher the bending stiffness.
It's the same principle as an I-beam. The distance $d$ between the face sheets and the core determines the bending stiffness. Doubling the core thickness makes the bending stiffness four times greater.
Shear in Core Materials
The most important characteristic of sandwich structures is shear deformation of the core. Because core materials are orders of magnitude softer than face sheets, shear deformation can account for the majority of the total deflection.
Ratio of shear deflection to bending deflection:
$E_f/G_c$ can be 100 or more... so shear deflection can be several times greater than bending deflection.
That's why Kirchhoff plate theory cannot be used for sandwich panels. It is essential to use Mindlin plate theory (including shear deformation) or higher-order theories. Solving a sandwich beam with Euler-Bernoulli beam theory is also incorrect.
Failure Modes of Sandwich Structures
Sandwich panels have inherent failure modes:
| Failure Mode | Cause | Severity |
|---|---|---|
| Face sheet yield/failure | Excessive bending stress | High |
| Core shear failure | Exceeding core shear strength | High |
| Face sheet buckling (dimpling) | Local buckling of face between cell walls | Medium |
| Face sheet wrinkling | Short-wavelength buckling of entire face | High |
| Core crushing | Core collapses under concentrated load | Medium |
| Face-core debonding | Adhesive failure, impact damage | High (BVID) |
There are that many failure modes?
Sandwich structures are lightweight but have complex failure modes. Design must consider all modes.
Summary
Let me organize the theory of sandwich panels.
Key points:
- Combination of face sheets + core — Lightweight with high bending stiffness
- Shear deformation of the core is dominant — Kirchhoff plate theory cannot be used. Mindlin or higher is mandatory.
- Six inherent failure modes — Face failure, core shear, buckling, debonding
- Impact damage (BVID) is the most dangerous — Debonding at the face-core interface
- $D \propto d^2$ — Doubling core thickness quadruples bending stiffness
So, sandwich structures are "complex failure modes as the price for being lightweight"?
It's a trade-off between performance and complexity. Sandwich design requires comprehensive checking of all failure modes, which is difficult without the aid of FEM.
The "Engineering Analogy" of Sandwich Structures
Sandwich structures are likened to a jumbo (large sandwich). The outer skin is the bread, the core (honeycomb, etc.) is the filling. With good design, a dramatic increase in bending stiffness can be achieved with only a slight increase in total weight. Doubling the skin-core distance d increases bending stiffness eightfold (Ei×I ∝ d²). When used in aircraft floor structures, structures that are lighter than solid aluminum plates and 3 to 10 times stiffer can be realized.
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 pulled" 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 assumes "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's Hooke's law $F=kx$, the essence of the stiffness term. So, 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". 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 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"—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. 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 inhomogeneity.
- 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 equilibrium 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 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) | In mm system: N, in m system: N (unified). |
Numerical Methods and Implementation
Modeling Sandwich Structures in FEM
How do you model a sandwich panel in FEM?
Three approaches:
| Method | Model | Accuracy | Cost |
|---|---|---|---|
| Equivalent Shell | A single shell element. Stiffness expressed via ABD matrix. | Medium (global behavior) | Low |
| Layered Shell | Shell element + layup definition (face + core + face). | Medium ~ High | Medium |
| 3D Solid | Model faces and core separately: faces as shells, core as solids. | High | High |
The equivalent shell is the simplest.
Simple, but cannot evaluate core shear failure or local buckling. Use only for estimating overall deflection or buckling load.
The recommended practical approach is layered shell. Define face sheets and core as separate layers and set each layer's material properties correctly. Core shear stiffness is automatically considered.
Nastran
```
PCOMP, 1, , , , ,
, 1, 0.5, 0., YES, $ Face 1 (CFRP)
, 2, 20., 0., YES, $ Core (Honeycomb)
, 1, 0.5, 0., YES $ Face 2 (CFRP)
```
Abaqus
```
*SHELL SECTION, COMPOSITE
0.5, 3, CFRP, 0.
20., 3, CORE, 0.
0.5, 3, CFRP, 0.
```
What material properties are needed for the core material?
Key properties for core materials (honeycomb, foam):
| Property | Honeycomb (Nomex) | PVC Foam |
|---|---|---|
| $E_c$ (out-of-plane compression) | 130~300 MPa | 50~150 MPa |
| $G_{xz}$ (out-of-plane shear) | 30~80 MPa | 20~50 MPa |
| $G_{yz}$ (out-of-plane shear) | 15~40 MPa | 20~50 MPa |
| Crushing strength | 1~5 MPa | 0.5~3 MPa |
Honeycomb has different shear stiffness depending on direction. $G_{xz} \neq G_{yz}$.
Honeycomb has different shear properties in the L-direction (ribbon direction) and W-direction (expansion direction). It must be set as orthotropic. Foam cores are generally isotropic.
Detailed Modeling of the Core
When is a 3D solid model used?
The standard approach is to model the core with solid elements and the faces with shell elements, then connect the interface (TIE constraint or CZM).
Summary
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