Sandwich structures — face sheets, core

What Is a Sandwich Structure?

The faces carry bending/axial loads; the core resists shear and maintains face separation distance.

WHY this configuration? In beam bending, stress $\sigma = \frac{My}{I}$ is maximum at outer fibers. By placing strong material at $y_{\text{max}}$ (far from neutral axis) and using lightweight core to maintain that distance, we get: $\text{Bending stiffness} \propto \frac{Et^3}{12} +2 E_f A_f \left(\frac{h}{2}\right)^2$ where the second term (face contribution) dominates for $h \gg t_f$.

Why Sandwich Structures in Spacecraft?

Three critical advantages:

1. High specific stiffness ($E/\rho$): Launch costs scale with mass. A sandwich panel can be 5-10× stiffer per unit mass than a solid plate of the same material.

2. High specific strength ($\sigma_y/\rho$): Faces placed at maximum moment arm $h/2$ multiply their contribution to moment capacity by $(h/2)^2$.

3. Multifunctionality: Core can provide thermal insulation, acoustic damping, or house wiring/piping (spacecraft bus panels, satellite solar arrays).

WHAT makes it work? The section modulus $S = I/c$ benefits enormously from increased depth. For a sandwich with thin faces of thickness $t_f$ and total depth $h$: $I \approx 2 \cdot \frac{b t_f h^2}{4} = \frac{b t_f h^2}{2}$ compared to a solid plate of same total mass with thickness $t_s$: $I_{\text{solid}} = \frac{b t_s^3}{12}$

If we equate masses: $2\rho_f b t_f + \rho_c b h \approx \rho_s b t_s$ and $\rho_c \ll \rho_f \approx \rho_s$, then $t_s \approx 2t_f$ and: $\frac{I_{\text{sandwich}}}{I_{\text{solid}}} \approx \frac{b t_f h^2 / 2}{b(2t_f)^3/12} = \frac{6 t_f h^2}{8 t_f^3} = \frac{3h^2}{4t_f^2} \gg 1$

Derivation: Bending Stiffness of Sandwich Beam

Goal: Find effective bending stiffness $(EI)_{\text{eff}}$ for a sandwich beam with:

• Face sheets: thickness $t_f$, Young's modulus $E_f$
• Core: thickness $c$, shear modulus $G_c$ (assume $E_c \ll E_f$)
• Total depth: $h =2t_f + c$

Step 1: Parallel Axis Theorem

Each face is a rectangle whose centroid is at distance $d = \frac{c}{2} + \frac{t_f}{2} = \frac{h}{2} - \frac{t_f}{2}$ from the neutral axis. For thin faces ($t_f \ll h$), we approximate $d \approx h/2$. This approximation is only valid when $t_f \ll h$ and should always be stated explicitly.

$I_{\text{face}} = \frac{b t_f^3}{12} + b t_f \, d^2 \approx \frac{b t_f^3}{12} + b t_f \left(\frac{h}{2}\right)^2$

Why neglect the first term? For thin faces, $t_f \ll h$, so $t_f^3 \ll t_f h^2$. The parallel-axis term dominates.

$I_{\text{faces}} \approx 2 b t_f \left(\frac{h}{2}\right)^2 = \frac{b t_f h^2}{2}$

Step 2: Core Contribution

Core bending stiffness: $I_{\text{core}} = \frac{bc^3}{12}$. If $c \sim h$ but $E_c \ll E_f$, its contribution is minor: $E_c I_{\text{core}} \ll E_f I_{\text{faces}}$

Typical: $E_c/E_f \sim 10^{-3}$ (aluminum honeycomb vs. aluminum face), so we neglect core bending.

Step 3: Total Bending Stiffness

$(EI)_{\text{sandwich}} \approx E_f \cdot \frac{b t_f h^2}{2} = \frac{E_f b t_f h^2}{2}$

HOW to use this? For deflection $\delta = \frac{FL^3}{3(EI)}$ of a cantilever, sandwich deflects less by factor $\sim h^2/t_f^2$ compared to solid plate of same mass.

Core Types and Selection

WHY honeycomb dominates spacecraft?

1. Highest $G_c/\rho$ ratio — resists shear buckling of faces
2. No continuous path for heat conduction (thermal insulation)
3. Vacuum compatibility — cells vent, no trapped volatiles

Selection criterion (strength-based). Core shear failure is governed by comparing the actual core shear stress to the core shear strength $\tau_{c,\text{allow}}$, not by the shear modulus. For a panel of width $b$ carrying transverse shear $V$: $\tau_c \approx \frac{V}{b\,h} \leq \tau_{c,\text{allow}}$ Separately, the shear modulus $G_c$ (a stiffness property) controls the extra deflection from shear deformation. For long panels ($L/h > 20$), the shear-deflection term $\frac{VL}{G_c b h}$ becomes significant and must be added to the bending deflection.

Example 1: Solar Array Panel Design

Given: Design a solar array panel, $L = 1.5 \text{ m} \times b = 0.8 \text{ m}$, modeled as a simply-supported beam of span $L$ and width $b$, carrying a uniform pressure during $5g$ launch acceleration. Target: deflection $< 5 \text{ mm}$.

Solution:

Step 1: Choose materials

• Face sheets: CFRP (carbon fiber), $E_f = 150 \text{ GPa}$, $\rho_f = 1600 \text{ kg/m}^3$
• Core: Aluminum honeycomb, $\rho_c = 50 \text{ kg/m}^3$, $G_c = 300 \text{ MPa}$

Step 2: Estimate load Uniform pressure $p \approx 200 \text{ N/m}^2$ from panel + cells under $5g$. Line load per unit length along the span: $w = p \, b = 200 \times 0.8 = 160 \text{ N/m}$

Why line load? We are treating the panel as a simply-supported beam of width $b$ under a distributed line load $w$, so the beam deflection formula applies consistently.

Step 3: Solve for required $(EI)$ Max deflection of a simply-supported beam under uniform load: $\delta = \frac{5 w L^4}{384 (EI)} \;\Rightarrow\; (EI)_{\text{req}} = \frac{5 w L^4}{384 \, \delta}$ $(EI)_{\text{req}} = \frac{5 \times 160 \times 1.5^4}{384 \times 0.005} = \frac{5 \times 160 \times 5.0625}{1.92} \approx 2637 \text{ Nm}^2$

Why this value? This is the minimum $(EI)$ needed to keep deflection under 5 mm.

Step 4: Choose geometry Try $h = 20 \text{ mm}$, $t_f = 0.5 \text{ mm}$: $(EI) = \frac{E_f b t_f h^2}{2} = \frac{150 \times 10^9 \times 0.8 \times 0.0005 \times 0.02^2}{2}$ $= \frac{150\times10^9 \times 0.8 \times 0.0005 \times 4\times10^{-4}}{2} = 1.2 \times 10^4 \text{ Nm}^2$

Why oversized? Safety factor $1.2\times10^4 / 2637 \approx 4.5$ covers bonding imperfections and local buckling. We could reduce $t_f$ or $h$ to save mass while keeping SF $\gtrsim 2$.

Step 5: Check mass (be careful with units!) $m = 2 \rho_f b L t_f + \rho_c b L h$ $= 2 \times 1600 \times 0.8 \times 1.5 \times 0.0005 + 50 \times 0.8 \times 1.5 \times 0.02$ $= 1.92 + 1.2 = 3.12 \text{ kg}$

Steel-man the earlier blunder: A naive computation writing $t_f = 0.005$ m (i.e., 5 mm instead of 0.5 mm) gives face mass $2\times1600\times0.8\times1.5\times0.005 = 19.2$ kg — a 10× error from a decimal slip. Always double-check the unit of $t_f$ in millimetres vs. metres. With the correct $t_f = 0.5\text{ mm} = 0.0005\text{ m}$, face mass is $1.92$ kg and total $\approx 3.12$ kg.

Example 2: Face Wrinkling vs. Core Shear

Given: Sandwich panel, width $b = 0.8 \text{ m}$ (same panel as Example 1), $E_f = 70 \text{ GPa}$ (Al faces), $t_f = 0.8 \text{ mm}$, $h = 25 \text{ mm}$, honeycomb $E_c = 0.1 \text{ GPa}$, $G_c = 300 \text{ MPa}$. Under compression $\sigma_x = 50 \text{ MPa}$ in faces, what fails first: face wrinkling or core shear?

Solution:

Step 1: Face wrinkling stress. The face acts as a plate on an elastic foundation (the core). Energy minimization over the wrinkle wavelength gives the general result: $\boxed{\sigma_{\text{wr}} = C \left(E_f \, E_c \, G_c\right)^{1/3}}$ with $C \approx 0.5$–$0.82$ depending on boundary conditions. When the core is isotropic so that $G_c \approx E_c/[2(1+\nu_c)] \sim E_c/3$, substituting $G_c \sim E_c/3$ gives the commonly quoted simplified form $\sigma_{\text{wr}} \approx C'(E_f E_c^2)^{1/3}$. These are the same formula — one keeps $G_c$ explicit, the other folds $G_c$ into $E_c$ for an isotropic core. We use the explicit form: $\sigma_{\text{wr}} = 0.5 \left(70\times10^9 \times 0.1\times10^9 \times 0.3\times10^9\right)^{1/3}$ $= 0.5 \left(2.1\times10^{27}\right)^{1/3} = 0.5 \times 1.28\times10^{9} \approx 0.64 \text{ GPa}$

Step 2: Compare to applied stress $\sigma_{\text{applied}} = 50 \text{ MPa} \ll 640 \text{ MPa}$

Verdict: Face wrinkling has a comfortable margin (~13×). BUT this assumes a perfect bond; delamination could occur at much lower stress if the adhesive is weak.

Step 3: Core shear check (strength, not modulus) For the panel loaded in bending as a cantilever with tip load $F = 1000 \text{ N}$, the core carries transverse shear $V = F$: $\tau_c = \frac{V}{b\,h} = \frac{1000}{0.8 \times 0.025} = 5.0\times10^4 \text{ Pa} = 50 \text{ kPa}$

Honeycomb shear strength $\tau_{c,\text{allow}} \sim 1$–$2 \text{ MPa}$ $\gg 50$ kPa → safe by a large margin.

Why face wrinkling rarely governs in spacecraft? Panels are usually large and thin, so overall panel buckling between supports occurs before local wrinkling. Wrinkling matters mostly for thick cores with soft foams.

Common Failure Modes

Why It Feels Right: Faces carry the load, core just spaces them.

The Fix: Core must transfer shear between faces. If $G_c$ too low, faces slip relative to each other → no composite action. Also, peel stresses at free edges can delaminate faces even with strong adhesive.

Criterion: Design so the bond shear strength exceeds the core shear strength, $\tau_{\text{bond}} > 1.5\,\tau_{c,\text{allow}}$, so the core fails before the bond (easier to detect and predict).

Why It Feels Right: Metal honeycomb looks robust.

The Fix: Honeycomb has high in-plane stiffness but very low through-thickness compression strength ($\sigma_c \sim 1-5 \text{ MPa}$). Point loads (bolts, inserts) require potting (filling cells with epoxy) or inserts (metal bushings) to distribute load.

Design rule: For attachment, pot diameter $\geq 4 \times$ bolt diameter.

Why It Feels Right: Faces cover the core.

The Fix: Core cells are open at edges (cut during machining). On Earth, moisture enters and in vacuum (space), water vaporizes → pressure buildup → face blowout.

Solution: Seal edges with edge closeout (epoxy fillet or metal channel). For Nomex (aramid) honeycomb, this is critical — Nomex absorbs moisture.

Optimization: The 80/20 Rule

80% of sandwich performance comes from:

1. Core depth $h$ — stiffness scales as $h^2$
2. Face material $E_f$ — use composites (CFRP) for maximum $E/\rho$

Diminishing returns from:

• Over-thickening faces (mass penalty for small stiffness gain)
• Exotic core materials (aluminum honeycomb is already near-optimal)

Quick design heuristic: $t_f \approx \frac{h}{50} \quad \text{(thin face assumption valid)}$ $\rho_c \approx \frac{\rho_f}{20} \quad \text{(core mass negligible)}$

If $t_f > h/30$, a solid plate may be more mass-efficient.

The smart trick: Take two thin sheets of fiberglass (the faces) and glue them to the top and bottom of the foam (the core). Now when you stand on it, the top sheet gets squished and the bottom sheet gets stretched — but the foam keeps them apart so they have to work really hard. It's like a tug-of-war where the rope is longer, so each side pulls with more force!

Spacecraft use this same idea with aluminum honeycomb (looks like a bee's home) instead of foam, because it's even lighter and doesn't get crushed. The panels in satellites are like super-tech surfboards — crazy light, crazy strong.

• Stiffness scales as $h^2$
• Point loads need Potting
• Aluminum honeycomb → Aerospace standard
• Crushing strength low through-thickness
• Expensive to repair → design conservatively

Connections

• Beam Bending Theory — sandwich extends simple beam to distributed cross-section
• Composite Materials — CFRP faces, anisotropic properties
• Buckling and Instability — face wrinkling is a local buckling mode
• Thermal Protection Systems — sandwich with ceramic face sheets for reentry
• Vibration and Modal Analysis — sandwich panels have high fundamental frequency
• Adhesive Bonding — film adhesives (epoxy, phenolic) for face-core bond
• Finite Element Analysis — modeling sandwich requires shell elements + volumetric core

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Concept Map

Hinglish (regional understanding)