Sandwich structures — face sheets, core — Hinglish notes

3.6.17 · Physics › Spacecraft Structures & Systems Engineering

What Is a Sandwich Structure?

Faces bending/axial loads carry karti hain; core shear resist karta hai aur face separation distance maintain karta hai.

YE configuration KYU? Beam bending mein, stress $σ = \frac{M y}{I}$ outer fibers par maximum hoti hai. Strong material ko $y_{max}$ par rakhke (neutral axis se door) aur lightweight core se wo distance maintain karke, hume milta hai: $Bending stiffness \propto \frac{E t ^{3}}{12} + 2 E_{f} A_{f} (\frac{h}{2})^{2}$ jahan doosra term (face ka contribution) $h ≫ t_{f}$ ke liye dominant hota hai.

Why Sandwich Structures in Spacecraft?

Teen critical advantages:

High specific stiffness ( $E / ρ$ ): Launch costs mass ke saath scale karte hain. Ek sandwich panel solid plate se 5-10× zyada stiff ho sakta hai same material ke liye per unit mass mein.
High specific strength ( $σ_{y} / ρ$ ): Faces maximum moment arm $h /2$ par rakhe jaate hain jo moment capacity mein unka contribution $(h /2)^{2}$ se multiply kar deta hai.
Multifunctionality: Core thermal insulation, acoustic damping provide kar sakta hai, ya wiring/piping house kar sakta hai (spacecraft bus panels, satellite solar arrays).

YE kaise kaam karta hai? Section modulus $S = I / c$ increased depth se bahut zyada benefit karta hai. $t_{f}$ thickness ki thin faces aur $h$ total depth wale sandwich ke liye: $I \approx 2 \cdot \frac{b t _{f} h ^{2}}{4} = \frac{b t _{f} h ^{2}}{2}$ same total mass ke solid plate se comparison, jiska thickness $t_{s}$ ho: $I_{solid} = \frac{b t _{s}^{3}}{12}$

Agar hum masses equal karein: $2 ρ_{f} b t_{f} + ρ_{c} bh \approx ρ_{s} b t_{s}$ aur $ρ_{c} ≪ ρ_{f} \approx ρ_{s}$ , to $t_{s} \approx 2 t_{f}$ aur: $\frac{I _{sandwich}}{I _{solid}} \approx \frac{b t _{f} h ^{2} /2}{b ( 2 t _{f} ) ^{3} /12} = \frac{6 t _{f} h ^{2}}{8 t _{f}^{3}} = \frac{3 h ^{2}}{4 t _{f}^{2}} ≫ 1$

Derivation: Bending Stiffness of Sandwich Beam

Goal: Sandwich beam ke liye effective bending stiffness $(E I)_{eff}$ nikalna, jisme:

Face sheets: thickness $t_{f}$ , Young's modulus $E_{f}$
Core: thickness $c$ , shear modulus $G_{c}$ (assume $E_{c} ≪ E_{f}$ )
Total depth: $h = 2 t_{f} + c$

Step 1: Parallel Axis Theorem

Har face ek rectangle hai jiska centroid neutral axis se $d = \frac{c}{2} + \frac{t _{f}}{2} = \frac{h}{2} - \frac{t _{f}}{2}$ door hai. Thin faces ke liye ( $t_{f} ≪ h$ ), hum approximate karte hain $d \approx h /2$ . Ye approximation tabhi valid hai jab $t_{f} ≪ h$ ho aur ise hamesha explicitly state karna chahiye.

$I_{face} = \frac{b t _{f}^{3}}{12} + b t_{f} d^{2} \approx \frac{b t _{f}^{3}}{12} + b t_{f} (\frac{h}{2})^{2}$

Pehla term kyun neglect karein? Thin faces ke liye, $t_{f} ≪ h$ , isliye $t_{f}^{3} ≪ t_{f} h^{2}$ . Parallel-axis term dominate karta hai.

$I_{faces} \approx 2 b t_{f} (\frac{h}{2})^{2} = \frac{b t _{f} h ^{2}}{2}$

Step 2: Core Contribution

Core bending stiffness: $I_{core} = \frac{b c ^{3}}{12}$ . Agar $c \sim h$ hai lekin $E_{c} ≪ E_{f}$ , to uska contribution minor hai: $E_{c} I_{core} ≪ E_{f} I_{faces}$

Typical: $E_{c} / E_{f} \sim 1 0^{- 3}$ (aluminum honeycomb vs. aluminum face), isliye hum core bending neglect karte hain.

Step 3: Total Bending Stiffness

$(E I)_{sandwich} \approx E_{f} \cdot \frac{b t _{f} h ^{2}}{2} = \frac{E _{f} b t _{f} h ^{2}}{2}$

Iska use KAISE karein? Cantilever ke deflection $δ = \frac{F L ^{3}}{3 ( E I )}$ ke liye, sandwich $\sim h^{2} / t_{f}^{2}$ factor se kam deflect karta hai same mass ke solid plate ke comparison mein.

Core Types and Selection

Core Type	Density (kg/m³)	$G_{c}$ (MPa)	Applications	Trade-offs
Aluminum honeycomb	30-80	200-500	Satellite panels, launch fairings	Best stiffness, point loads se crush ho sakta hai
Nomex honeycomb	30-100	50-150	Interior non-structural panels	Lower cost, moisture-sensitive
Foam (polyurethane)	30-200	10-80	Contoured surfaces, radomes	Machine karna aasaan, lower shear strength
Corrugated	50-150	100-300	Low-cost prototypes	Anisotropic, direction-dependent

Spacecraft mein honeycomb KYU dominate karta hai?

Highest $G_{c} / ρ$ ratio — faces ki shear buckling resist karta hai
Heat conduction ke liye koi continuous path nahi (thermal insulation)
Vacuum compatibility — cells vent ho jaate hain, koi trapped volatiles nahi

Selection criterion (strength-based). Core shear failure tab hoti hai jab actual core shear stress ko core shear strength $τ_{c, allow}$ se compare karein, shear modulus se nahi. Width $b$ ke panel pe transverse shear $V$ ke liye: $τ_{c} \approx \frac{V}{b h} \leq τ_{c, allow}$ Alag se, shear modulus $G_{c}$ (ek stiffness property) shear deformation se extra deflection control karta hai. Long panels ke liye ( $L / h > 20$ ), shear-deflection term $\frac{V L}{G _{c} bh}$ significant ho jaata hai aur use bending deflection mein add karna padta hai.

Example 1: Solar Array Panel Design

Given: Design karo ek solar array panel, $L = 1.5 m \times b = 0.8 m$ , simply-supported beam of span $L$ aur width $b$ ki tarah model kiya gaya, jo $5 g$ launch acceleration ke dauran uniform pressure carry karta hai. Target: deflection $< 5 mm$ .

Solution:

Step 1: Materials chunna

Face sheets: CFRP (carbon fiber), $E_{f} = 150 GPa$ , $ρ_{f} = 1600 kg/m^{3}$
Core: Aluminum honeycomb, $ρ_{c} = 50 kg/m^{3}$ , $G_{c} = 300 MPa$

Step 2: Load estimate karo Uniform pressure $p \approx 200 N/m^{2}$ panel + cells se $5 g$ ke under. Span ke saath per unit length line load: $w = p b = 200 \times 0.8 = 160 N/m$

Line load kyun? Hum panel ko width $b$ ke simply-supported beam ki tarah treat kar rahe hain distributed line load $w$ ke saath, isliye beam deflection formula consistently apply hota hai.

Step 3: Required $(E I)$ solve karo Simply-supported beam ka uniform load mein max deflection: $δ = \frac{5 w L ^{4}}{384 ( E I )} \Rightarrow (E I)_{req} = \frac{5 w L ^{4}}{384 δ}$ $(E I)_{req} = \frac{5 \times 160 \times 1. 5 ^{4}}{384 \times 0.005} = \frac{5 \times 160 \times 5.0625}{1.92} \approx 2637 Nm^{2}$

Ye value kyun? Ye minimum $(E I)$ hai jo deflection ko 5 mm se neeche rakhne ke liye chahiye.

Step 4: Geometry chunno $h = 20 mm$ , $t_{f} = 0.5 mm$ try karo: $(E I) = \frac{E _{f} b t _{f} h ^{2}}{2} = \frac{150 \times 1 0 ^{9} \times 0.8 \times 0.0005 \times 0.0 2 ^{2}}{2}$ $= \frac{150 \times 1 0 ^{9} \times 0.8 \times 0.0005 \times 4 \times 1 0 ^{- 4}}{2} = 1.2 \times 1 0^{4} Nm^{2}$

Oversized kyun? Safety factor $1.2 \times 1 0^{4} /2637 \approx 4.5$ bonding imperfections aur local buckling cover karta hai. Mass bachane ke liye hum $t_{f}$ ya $h$ reduce kar sakte hain jabki SF $≳ 2$ rakhen.

Step 5: Mass check karo (units par dhyan do!) $m = 2 ρ_{f} b L t_{f} + ρ_{c} b L h$ $= 2 \times 1600 \times 0.8 \times 1.5 \times 0.0005 + 50 \times 0.8 \times 1.5 \times 0.02$ $= 1.92 + 1.2 = 3.12 kg$

Pehli galti ka steel-man: Ek naive computation jo $t_{f} = 0.005$ m (yaani 0.5 mm ki jagah 5 mm) likhta hai, face mass deta hai $2 \times 1600 \times 0.8 \times 1.5 \times 0.005 = 19.2$ kg — decimal slip se 10× error. Hamesha $t_{f}$ ki unit millimetres vs. metres mein double-check karo. Sahi $t_{f} = 0.5 mm = 0.0005 m$ se, face mass $1.92$ kg aur total $\approx 3.12$ kg hai.

Example 2: Face Wrinkling vs. Core Shear

Given: Sandwich panel, width $b = 0.8 m$ (Example 1 jaisa panel), $E_{f} = 70 GPa$ (Al faces), $t_{f} = 0.8 mm$ , $h = 25 mm$ , honeycomb $E_{c} = 0.1 GPa$ , $G_{c} = 300 MPa$ . Faces mein compression $σ_{x} = 50 MPa$ ke under, pehle kya fail hoga: face wrinkling ya core shear?

Solution:

Step 1: Face wrinkling stress. Face ek plate ki tarah kaam karta hai elastic foundation (core) par. Wrinkle wavelength par energy minimization ye general result deta hai: $σ_{wr} = C (E_{f} E_{c} G_{c})^{1/3}$ jahan $C \approx 0.5$ – $0.82$ boundary conditions par depend karta hai. Jab core isotropic ho to $G_{c} \approx E_{c} / [2 (1 + ν_{c})] \sim E_{c} /3$ , $G_{c} \sim E_{c} /3$ substitute karne par commonly quoted simplified form milta hai $σ_{wr} \approx C^{'} (E_{f} E_{c}^{2})^{1/3}$ . Ye same formula hain — ek $G_{c}$ explicit rakhta hai, doosra isotropic core ke liye $G_{c}$ ko $E_{c}$ mein fold kar deta hai. Hum explicit form use karte hain: $σ_{wr} = 0.5 (70 \times 1 0^{9} \times 0.1 \times 1 0^{9} \times 0.3 \times 1 0^{9})^{1/3}$ $= 0.5 (2.1 \times 1 0^{27})^{1/3} = 0.5 \times 1.28 \times 1 0^{9} \approx 0.64 GPa$

Step 2: Applied stress se compare karo $σ_{applied} = 50 MPa ≪ 640 MPa$

Verdict: Face wrinkling mein comfortable margin hai (~13×). LEKIN ye perfect bond assume karta hai; delamination bahut kam stress par ho sakti hai agar adhesive weak ho.

Step 3: Core shear check (strength, modulus nahi) Panel jo cantilever ki tarah tip load $F = 1000 N$ mein loaded hai, core transverse shear $V = F$ carry karta hai: $τ_{c} = \frac{V}{b h} = \frac{1000}{0.8 \times 0.025} = 5.0 \times 1 0^{4} Pa = 50 kPa$

Honeycomb shear strength $τ_{c, allow} \sim 1$ – $2 MPa$ $≫ 50$ kPa → large margin se safe hai.

Spacecraft mein face wrinkling rarely kyun govern karta hai? Panels usually bade aur patle hote hain, isliye supports ke beech overall panel buckling local wrinkling se pehle hoti hai. Wrinkling mostly thick cores with soft foams ke liye matter karta hai.

Common Failure Modes

Ye Sahi Kyun Lagta Hai: Faces load carry karti hain, core bas unhe space karta hai.

Fix: Core ko faces ke beech shear transfer karna hota hai. Agar $G_{c}$ bahut low hai, faces ek doosre ke relative slip ho jaati hain → koi composite action nahi. Saath hi, free edges par peel stresses strong adhesive ke saath bhi faces ko delaminate kar sakti hain.

Criterion: Design karo taki bond shear strength core shear strength se zyada ho, $τ_{bond} > 1.5 τ_{c, allow}$ , taaki bond se pehle core fail ho (detect aur predict karna aasaan).

Ye Sahi Kyun Lagta Hai: Metal honeycomb robust dikhta hai.

Fix: Honeycomb ki high in-plane stiffness hoti hai lekin through-thickness compression strength bahut low hoti hai ( $σ_{c} \sim 1 - 5 MPa$ ). Point loads (bolts, inserts) ke liye potting (cells ko epoxy se bharna) ya inserts (metal bushings) chahiye taaki load distribute ho sake.

Design rule: Attachment ke liye, pot diameter $\geq 4 \times$ bolt diameter.

Ye Sahi Kyun Lagta Hai: Faces core ko cover karti hain.

Fix: Core cells edges par open hoti hain (machining ke dauran cut hoti hain). Zameen par moisture andar jaati hai aur vacuum (space) mein, paani vapourize ho jaata hai → pressure buildup → face blowout.

Solution: Edges ko edge closeout se seal karo (epoxy fillet ya metal channel). Nomex (aramid) honeycomb ke liye ye critical hai — Nomex moisture absorb karta hai.

Optimization: The 80/20 Rule

Sandwich performance ka 80% in se aata hai:

Core depth $h$ — stiffness $h^{2}$ se scale karti hai
Face material $E_{f}$ — maximum $E / ρ$ ke liye composites (CFRP) use karo

Diminishing returns in se:

Faces ko over-thicken karna (small stiffness gain ke liye mass penalty)
Exotic core materials (aluminum honeycomb already near-optimal hai)

Quick design heuristic: $t_{f} \approx \frac{h}{50} (thin face assumption valid)$ $ρ_{c} \approx \frac{ρ _{f}}{20} (core mass negligible)$

Agar $t_{f} > h /30$ , to solid plate zyada mass-efficient ho sakti hai.

Recall 12-Saal-Ke-Bachche Ko Samjhao

Socho tumhe ek super-light lekin strong surfboard banana hai. Agar tum use solid foam se banao, to wo light hai lekin aasaani se toot jaata hai. Agar tum use solid fiberglass se banao, to wo strong hai lekin carry karne ke liye bahut bhaari hai.

Smart trick: Do patli fiberglass sheets lo (faces) aur unhe foam ke upar aur neeche glue karo (core). Ab jab tum uske upar khade ho, upar wali sheet squish hoti hai aur neeche wali stretch hoti hai — lekin foam unhe apart rakhta hai taaki unhein bahut mehnat karni pade. Ye ek tug-of-war jaisa hai jahan rope lambi hai, isliye har side zyada force se kheenchti hai!

Spacecraft isi idea ko aluminum honeycomb (madhumakhi ke ghar jaisa dikhta hai) ke saath use karte hain foam ki jagah, kyunki ye aur bhi halka hai aur crush nahi hota. Satellites ke panels super-tech surfboards ki tarah hain — pagalon ki tarah halke, pagalon ki tarah strong.

Stiffness $h^{2}$ se scale karti hai
Point loads ko Potting chahiye
Aluminum honeycomb → Aerospace standard
Crushing strength through-thickness mein low hai
Expensive to repair → conservatively design karo

Connections

Beam Bending Theory — sandwich simple beam ko distributed cross-section tak extend karta hai
Composite Materials — CFRP faces, anisotropic properties
Buckling and Instability — face wrinkling ek local buckling mode hai
Thermal Protection Systems — reentry ke liye ceramic face sheets wala sandwich
Vibration and Modal Analysis — sandwich panels ki high fundamental frequency hoti hai
Adhesive Bonding — face-core bond ke liye film adhesives (epoxy, phenolic)
Finite Element Analysis — sandwich modeling ke liye shell elements + volumetric core chahiye

#flashcards/physics

Sandwich structure ke teen components kya hain?

Do face sheets (patli, high-strength), lightweight core (honeycomb/foam), aur unke beech adhesive bond.

Solid plate use karne ki bajaye face sheets ko door-door kyun rakhte hain?

Bending stiffness

(E I) \propto h^{2}

. Lightweight core se faces ke beech

h

distance maximize karke, stiffness quadratically badhti hai jabki mass linearly badhta hai.

Sandwich panel mein face sheets kaunse loads carry karti hain?

Bending stresses (axial tension aur compression) aur in-plane loads. Ye I-beam ke flanges ki tarah kaam karti hain.

Sandwich structure mein core ka primary role kya hai?

Shear stresses resist karna (faces ko slide hone se rokna) aur faces ke beech separation distance

h

maintain karna. Buckling support bhi provide karta hai.

Thin faces wale sandwich beam ki bending stiffness derive karo.

Har face parallel axis theorem se

I_{face} \approx b t_{f} (h /2)^{2}

contribute karta hai (valid jab

t_{f} ≪ h

). Total:

(E I)_{sandwich} = E_{f} \cdot 2 b t_{f} (h /2)^{2} = E_{f} b t_{f} h^{2} /2

Spacecraft structures mein foam ki jagah aluminum honeycomb kyun prefer kiya jaata hai?

Zyada shear modulus

G_{c}

(buckling resist karta hai), vacuum mein outgassing nahi, better thermal insulation (conduction ke liye koi continuous path nahi), aur superior strength-to-weight ratio.

Face wrinkling kya hai aur kab hoti hai?

Core se elastic foundation ki tarah support ki gayi face sheet ki local buckling. Tab hoti hai jab compressive stress

σ_{wr} = C (E_{f} E_{c} G_{c})^{1/3}

se zyada ho jaaye.

Core shear failure shear modulus se govern hoti hai ya shear strength se?

Shear FAILURE shear STRENGTH se govern hoti hai (

τ_{c} \leq τ_{c, allow}

). Shear MODULUS

G_{c}

shear DEFLECTION (extra stiffness loss) govern karta hai, failure nahi.

Honeycomb edges kyun seal karni chahiye?

Core cells cut edges par open hoti hain. Moisture zameen par andar ja sakti hai aur space vacuum mein vapourize ho jaati hai, pressure buildup hota hai jo face sheets blow off kar deta hai.

Potting kya hai aur bolted joints ke liye kyun zaroori hai?

Insert location ke around honeycomb cells ko epoxy se bharna. Zaroori hai kyunki honeycomb ki through-thickness compression strength bahut low hoti hai (~1-5 MPa) aur bolt loads se crush ho jaata.

Same stiffness ke solid plate se sandwich panel mass mein kaise compare karta hai?

Same bending stiffness ke liye, sandwich typically 5-10× halka hota hai, kyunki ratio

I_{sandwich} / I_{solid} \approx 3 h^{2} / (4 t_{f}^{2}) ≫ 1

Kaunsa failure mode faces ke ek doosre ke relative slip hone se involve karta hai?

Core shear failure. Tab hoti hai jab core shear strength exceed ho jaaye ya bond fail ho jaaye, faces ke beech composite action rok deta hai.

Sandwich structures mein stiffness

h^{2}

se kyun scale karti hai?

h /2

distance par thin faces ke liye second moment of area

I

(h /2)^{2} = h^{2} /4

se scale karta hai. Har face

I

mein

b t_{f} (h /2)^{2}

contribute karta hai.

Typical face-thickness-to-total-depth ratio kya hai?

t_{f} / h \approx 1/50

to $1

Sandwich structures — face sheets, core