Composite materials — fiber-matrix, ply properties, laminate theory: Question bank

3.6.15 · D5 · Physics › Spacecraft Structures & Systems Engineering › Composite materials — fiber-matrix, ply properties, laminate

Traps se pehle, ek chhota sa glossary taaki neeche use hone wala har symbol pehle se samajh aa jaye.

Definition Is page par use hone wale har symbol ka matlab

==Fiber volume fraction $V_{f}$ ==: composite ke volume ka woh hissa jo fiber hai (jaise $V_{f} = 0.6$ matlab 60% fiber). ==Matrix volume fraction $V_{m}$ == baaki hissa hai, toh $V_{f} + V_{m} = 1$ .
== $E_{f}$ , $E_{m}$ ==: pure fiber aur pure matrix ki apni stiffness (Young's modulus) — carbon fiber $E_{f} \sim 230$ GPa, epoxy $E_{m} \sim 3$ GPa.
== $E_{1}$ , $E_{2}$ ==: composite ki stiffness fibers ke saath (axis 1) aur fibers ke aar-paar (axis 2).
==Strain $ϵ$ ==: koi piece kitna khichta hai, apni length ke fraction ke roop mein; $ϵ_{f}$ aur $ϵ_{m}$ fiber part aur matrix part ke stretches hain.
==Shear modulus $G_{12}$ ==: khichne ki bajaye sliding (angle banana) ke khilaaf stiffness, 1–2 plane mein.
==Poisson ratio $ν_{12}$ (major)==: axis 1 ke saath khiincho aur material axis 2 mein squeeze hota hai — $ν_{12}$ batata hai kitna. == $ν_{21}$ (minor)== wahi coupling hai jo axis 2 ke saath kheenchne se referenced hai, aur $ν_{21} = ν_{12} E_{2} / E_{1}$ .

Recall Woh do directions jinhe har answer mein refer kiya gaya hai

Material axes "1" aur "2" ka kya matlab hai? ::: Axis 1 fibers ke saath chalta hai (stiff, strong); axis 2 fibers ke aar-paar chalta hai, weak matrix se hokar — ply ka poora point yahi hai ki ye do directions bilkul alag behave karti hain.

Do pictures hain jo neeche milne wale do rules aur chaar matrices ko anchor karti hain.

Figure — Composite materials — fiber-matrix, ply properties, laminate theory

Sahi ya galat — justify karo

TF1.

E_{1} = 140

GPa wala composite "utna hi acha hai" jitna 140 GPa wala isotropic metal.

Galat — woh 140 GPa sirf fibers ke saath exist karta hai; unke aar-paar

E_{2}

~10 GPa ho sakta hai, toh material transversely lagbhag 14× zyada soft hai aur ise isotropic aluminium ki tarah treat nahi kiya ja sakta.

TF2. Longitudinal Rule of Mixtures

E_{1} = E_{f} V_{f} + E_{m} V_{m}

assume karta hai ki fiber aur matrix ko same stress feel hota hai.

Galat — yeh same strain assume karta hai (woh bonded hain aur parallel mein saath stretch karte hain, jaise parallel springs); phir har phase apni stiffness ke proportion mein stress carry karta hai.

TF3. Transverse modulus

E_{2}

E_{f}

aur

E_{m}

ke average ke kareeb hota hai.

Galat —

E_{2}

inverse (series) rule use karta hai aur sabse kamzor link (matrix) se dominate hota hai, toh

E_{2}

average ke paas nahi balki

E_{m}

ke paas hota hai.

TF4. Kyunki

ν_{21} = ν_{12} E_{2} / E_{1}

hai, minor Poisson ratio

ν_{21}

major

ν_{12}

se chhota hota hai.

Sahi — kyunki

E_{2} < E_{1}

, factor

E_{2} / E_{1} < 1

, toh

ν_{21} < ν_{12}

; dono wahi physical coupling describe karte hain, sirf alag loading directions se referenced hain.

TF5.

Q_{11}

exactly

E_{1}

ke barabar hota hai.

Galat —

Q_{11} = E_{1} / (1 - ν_{12} ν_{21})

E_{1}

se thoda sa bada hota hai kyunki Poisson coupling constrained response ko stiffen karta hai, haalaanki fark aam taur par 1% se kam hota hai.

TF6. Zyada fiber daalna (

V_{f} \to 1

) hamesha better composite banata hai.

Galat — ~0.65 se aage fibers ko wet aur support karne ke liye kaafi matrix nahi bachta, toh voids ban jaate hain, load transfer fail ho jaata hai, aur strength actually gir jaati hai.

TF7. Off-axis load karne par ek single unidirectional ply pure tension mein bhi shear develop karta hai.

Sahi — jab fibers angle

θ

par baithe hon, tensile pull fibers ke saath aur aar-paar dono components mein resolve ho jaata hai, aur transformed stiffness

[\overset{ˉ}{Q}]

mein extension–shear coupling terms aa jaate hain.

TF8. Fatigue resistance ek selling point hai kyunki composites, metals ke unlike, cracks ko usi tarah propagate nahi karte.

Sahi — koi single crystal-lattice crack ander se march nahi karta; damage distributed fiber/matrix micro-events ke roop mein failta hai, jo bahut flatter fatigue curves deta hai (launch load cycling se relevant).

Galti dhundho

SE1. "Series (transverse) model mein maine

ϵ_{f} = ϵ_{m}

set kiya kyunki dono phases same stretch karte hain."

Galat — series loading mein equal stress hota hai, equal strain nahi; softer matrix zyada strain karta hai (

ϵ_{m} > ϵ_{f}

), aur total strain dono ka volume-weighted sum hai.

SE2. "

1/ E_{2} = V_{f} / E_{f} + V_{m} / E_{m}

, toh

V_{f} = 0.6

ke saath fiber

E_{2}

ko dominate karta hai."

Galat — inverse sum mein sabse bada denominator sabse chhota term contribute karta hai, toh tiny

V_{m} / E_{m}

term (chhota

E_{m}

)

1/ E_{2}

ko dominate karta hai; transverse stiffness ko fiber nahi, matrix control karta hai.

SE3. "Stiffness matrix

[Q]

mein ek nonzero

Q_{16}

term hai jo normal stress ko shear se couple karta hai."

Galat — material axes mein ply orthotropic hai, toh

Q_{16} = Q_{26} = 0

; woh coupling terms sirf transformed

[\overset{ˉ}{Q}]

mein appear karte hain jab angled layup ke liye rotate kiya jaaye.

SE4. "Compliance

[S]

se stiffness paane ke liye main bas har entry ka reciprocal le leta hoon."

Galat —

[Q] = [S]^{- 1}

ek full matrix inverse hai, entry-by-entry nahi; yahi reason hai ki shared determinant

1 - ν_{12} ν_{21}

har

Q

term mein appear karta hai.

SE5. "

ν_{12}

aur

ν_{21}

independent material constants hain jinhein maine alag se measure karna hai."

Galat — woh symmetry relation

ν_{21} E_{1} = ν_{12} E_{2}

se linked hain, toh sirf ek independent hai; dono measure karna ek redundancy check hai, do alag data points nahi.

SE6. "Ek quasi-isotropic laminate isotropic hai, toh main har jagah isotropic hand-book formulas use kar sakta hoon."

Galat — "quasi-isotropic" ka matlab sirf yeh hai ki in-plane extensional stiffness direction-independent hai; bending, interlaminar shear, aur through-thickness behaviour strongly anisotropic rehte hain.

SE7. "Maine shear modulus

G_{12}

ke liye parallel Rule of Mixtures use kiya."

Galat — in-plane shear (

G_{12}

, sliding ke against resistance) load matrix se hokar pass karta hai jaise transverse case mein, toh yeh inverse (series) rule follow karta hai,

E_{1}

ke form se nahi balki

E_{2}

ke form se milta hai.

Kyun wale questions

WHY1. Matrix weak hone ke bawajood matter kyun karta hai?

Kyunki woh fibers ke beech load transfer karta hai, unki spacing/alignment pakad ke rakhta hai, aur crack tips ko blunt karta hai — uske bina ek bare fiber bundle compression mein buckle ho jaata aur kisi bhi off-axis load mein bikh jaata.

WHY2. Hum compliance ko invert karke stiffness kyun lete hain balki seedha compliance ke saath kaam kyun nahi karte?

Kyunki laminate assembly aur FEA (dekho 5.3-finite-element-analysis) stacked plies ki stiffnesses jodte hain laminate response build karne ke liye; tum un layers ki compliances simply add nahi kar sakte jo same strain share karti hain.

WHY3. Low thermal expansion composites ko optics aur antennas ke liye kyun prized banata hai?

Kyunki near-zero (even negative) fiber expansion ko tune kiya ja sakta hai taaki mirror ya antenna geometry orbital hot/cold swings mein stable rahe — yeh 4.2-thermal-design mein handle hota hai — jahan ek metal defocus ho jaata.

WHY4. Spacecraft ke liye material selection mein raw stiffness nahi balki specific stiffness (

E / ρ

) kyun drive karta hai?

Kyunki har kilogram launch mass cost karta hai (2.4-mass-budgets); aluminium ki stiffness ka 60% lekin density bhi 60% wala composite stiffness-per-kilogram mein phir bhi jeetta hai, jo structural frequency aur mass set karta hai.

WHY5. Ply count nahi balki ply orientation track kyun karni chahiye?

Kyunki har ply sirf apne fibers ke saath load resist karta hai; chaar 0° plies huge axial stiffness dete hain lekin transversely almost kuch nahi, jabki balanced 0/±45/90 stack capability wahan failaata hai jahan loads actually point karte hain.

WHY6. Term

1 - ν_{12} ν_{21}

kisi real material ke liye hamesha positive aur ek se kam kyun hota hai?

Kyunki thermodynamic stability ke liye strain energy positive honi zaroori hai, jo force karta hai ki

ν_{12} ν_{21} < 1

;

\geq 1

ki value imply karegi ki material load hone par energy release karta hai — physically impossible.

WHY7. Trade studies mein composite ko steel ke bajaye aluminium se compare kyun karte hain?

Kyunki aluminium incumbent aerospace baseline hai; material selection ke dauran meaningful engineering question yeh hai ki "kya composite us metal ko beat karta hai jo hum otherwise fly karte?"

WHY8. Sandwich structures thin composite faces ko light core ke saath kyun pair karte hain?

Kyunki stiff faces ko alag karna (dekho 3.6.16-sandwich-structures) almost bina mass ke bending stiffness multiply karta hai, ek strong-but-thin laminate ko thick beam ki tarah act karne deta hai.

Edge cases

EC1. Limit

V_{f} \to 0

(sab matrix) mein

E_{2}

kya hoga?

Inverse rule collapse hokar

1/ E_{2} = 1/ E_{m}

ban jaata hai, toh

E_{2} \to E_{m}

— bina fibers ke composite sirf matrix hai, exactly jaisa hona chahiye.

EC2.

V_{f} \to 1

(theoretical sab-fiber) hone par

E_{1}

ka kya hoga?

E_{1} \to E_{f}

; parallel rule smoothly pure-fiber modulus recover kar leta hai, formula ke end behaviour ko confirm karta hai chahe yeh

V_{f}

unmanufacturable kyun na ho.

EC3.

θ = 0°

par, transformation matrix

[T]

kis cheez mein reduce ho jaata hai?

Identity matrix mein, toh

[\overset{ˉ}{Q}] = [Q]

— loading axis ke saath aligned ply ko kisi transformation ki zaroorat nahi, yahi sanity check hai jo har rotation formula ko pass karna chahiye.

EC4.

θ = 90°

par,

E_{1}

aur

E_{2}

ke roles kaise swap ho jaate hain?

Fibers ab load ke aar-paar chalte hain, toh effective axial stiffness

E_{2}

(matrix-dominated) ban jaata hai — "90° ply" ko pull karna actually use transversely load karna hai.

EC5. Agar laminate ko exactly ek fiber direction ke saath load kiya jaaye lekin ±45° plies bhi hon, toh kya woh angled plies zero load carry karti hain?

Nahi — kyunki sab plies same in-plane strain share karti hain, ±45° plies phir bhi stretch hoti hain aur stress (aur shear) carry karti hain, sirf aligned plies se kam efficiently.

EC6. Agar

E_{f} = E_{m}

ho (fiber aur matrix equally stiff)?

Dono rules dete hain

E_{1} = E_{2} = E_{f}

; bina stiffness contrast ke material isotropic ban jaata hai aur poori anisotropy story gayab ho jaati hai — contrast hi composite hai.

EC7. Kya koi ऐसा orientation hai jahan off-axis ply koi extension–shear coupling feel nahi karta?

Haan — sirf

θ = 0°

aur

θ = 90°

par; un angles par

sin θ cos θ = 0

hota hai, toh

[\overset{ˉ}{Q}]

ke coupling terms vanish ho jaate hain aur pure tension se pure extension milta hai.

EC8. Jab

E_{2} \to 0

(imaginary infinitely soft matrix) ho toh

ν_{21}

kiske paas approach karta hai?

ν_{21} = ν_{12} E_{2} / E_{1} \to 0

; vanishingly stiff transverse direction stiff direction ko andar nahi kheench sakti, toh minor Poisson coupling disappear ho jaata hai.