3.6.15 · D1Composite materials — fiber-matrix, ply properties, laminate
Composite do materials ko ek mein glue karke banata hai: patle stiff fibers jo apni length ke along load carry karte hain, ek soft matrix ke andar band hote hain jo unhe bind karta hai aur unke beech load pass karta hai. Is topic mein sab kuch ek hi sawaal ka bookkeeping hai — agar main is layered material ko kisi direction mein kheenchu, dhakelu, ya twist karun, toh kitna stretch hoga aur andar kitna stress build up hoga?
Parent page par koi bhi stiffness matrix padhne se pehle, aapko uske har letter ka ownership lena hoga. Yeh page har symbol ko us order mein introduce karta hai jahan koi cheez use hone se pehle build ki jaati hai. Upar se neeche ek baar padho aur parent note plain English ban jaata hai.
Sab kuch material ke ek block ko kheenchne se shuru hota hai. Do sawaal uthte hain: main surface ke har unit par kitni jor se kheench raha hun? aur block apni original length ke comparison mein kitna stretch hua?
Figure 1 neeche teeno ideas side by side dikhata hai: ek block ko stretch kiya ja raha hai (normal stress), aur ek block ko skew kiya ja raha hai (shear stress). Cyan outline (original shape) ko amber outline (deformed shape) se compare karo — poora page usi difference ko dekhne par tikaa hai.
σ τ Stress force ko us area se divide karna hai jis par woh act karta hai: σ = F / A A F Pa = N/m 2 1 GPa = 1 0 9 Pa
Normal stress σ : force face ke perpendicular act karta hai (alag kheenchna ya ek saath push karna).Shear stress τ : force face ke along act karta hai, ek layer ko doosri par sliding karaata hai — jaise cards ki deck ka top sideways push karna.
Definition Sign convention — tension positive, compression negativeStress aur strain ka ek sign hota hai, aur sabko uss par agree karna hota hai nahi toh matrices galat answers deti hain:
Tension (alag kheenchna, block lamba hota hai) → σ > 0 ϵ > 0 Figure 1 mein amber block cyan se lamba hai, isliye dono positive hain.Compression (ek saath push karna, block chhota hota hai) → σ < 0 ϵ < 0 σ = F / A negative force plug in karte ho kyunki woh andar ki taraf point karta hai.Direction matters : har stress ek named face aur axis par rehta hai. σ 1 σ 2 kaunsa face, kaunsi taraf, aur kya sign .
ϵ γ Strain fractional stretch hai: ϵ = Δ L / L ϵ = 0.01 ϵ apne aap ka fraction .
Engineering shear strain γ : ek corner ke right angle mein total change , radians mein. Precisely: ek corner lo jo exactly 9 0 ∘ 9 0 ∘ − γ γ decrease hai, koi "lean" nahi — angle ka poora shrinkage, donon edges ka tilt count karke. Figure 1 mein yeh amber corner ka square se departure hai.Yeh engineering definition hai (jo parent ki matrices use karti hain); ek related "tensorial" version γ /2 γ γ ≈ tan γ
Intuition Normal aur shear mein alag kyun karte hain?Material ke flat piece ko deform karne ke kisi bhi tarike mein sirf teen moves ka mixture hai: left-right stretch, up-down stretch, aur skew (shear). Isliye parent page ki har matrix mein exactly teen rows hain. Yeh teen master karo aur aapne in-plane loading ke sab cases cover kar liye.
Zyada jor se kheeencho, zyada stretch hoga. Unka ratio material ki property hai, block ke size ki nahi. Woh ratio stiffness hai.
Definition Young's modulusE Young's modulus stress divided by strain hai: E = σ / ϵ E E
Socho do springs: ek stiff wala haath ke neeche barely hilta hai, ek floppy wala bahut hilta hai. E
Units: stress jaisi, GPa. Carbon fiber E f ≈ 230 E m ≈ 3
G G 12 Shear modulus G G = τ / γ E
Isotropic material ke liye ek number G Directional ply ke liye hum batana padega kaunsa plane shear ho raha hai, isliye subscripts aate hain: ==G 12 G 12 G
E G E G
Rubber band stretch karo aur woh patla ho jaata hai. Ek block ko ek taraf kheecho aur woh doosri taraf shrink ho jaata hai. Uss coupling ka apna symbol hai.
Figure 2 yeh directly dikhata hai: cyan block (pehle) amber block (baad mein) ban jaata hai — lamba bhi aur paatla bhi . Donon cyan inward arrows woh sideways shrink hain jo Poisson's ratio measure karta hai.
Definition Poisson's ratioν Poisson's ratio ν ν = − lengthwise strain sideways strain ν 0.2 0.35
Socho block ek saath lamba aur paatla ho raha hai (Figure 2 ). ν
ν = 0 ν = 0.5
Kyunki composite har direction mein same nahi hoti, isko do Poisson's ratios milte hain:
Definition Major aur minor Poisson's ratiosν 12 ν 21 ν 12 ν 21 equal nahi hain, lekin parent use karne wali symmetry rule se tied hain:
ν 21 = ν 12 E 1 E 2
Intuition DO Poisson ratios koν 21 = ν 12 E 2 / E 1 kisi material ko stretch karne mein jo kaam karte ho woh loads apply karne ke order par depend nahi karta. Yahan teen visible steps mein argument hai.
Step 1 — do cross-couplings likho. Parent ki compliance relations se, sirf direction 1 mein stress σ 1 ϵ 2 = − E 1 ν 12 σ 1 σ 2 ϵ 1 = − E 2 ν 21 σ 2 E 1 ν 12 E 2 ν 21
Step 2 — energy ko order ki parwah nahi. Block ko final state ( σ 1 , σ 2 ) σ 1 σ 2 σ 2 σ 1 same number hai dono taraf se (yeh final squeezed state ki property hai, jaise pahad par height — path-independent). Dono paths ke liye kaam likhne par, donon totals tab hi match ho sakte hain har load ke liye jab do cross-terms equal hon:
E 1 ν 12 = E 2 ν 21 symmetric hona chahiye (apne diagonal ke across mirror-image). Isliye parent Q 12
Step 3 — rearrange karo. Dono sides ko E 2 ν 21 = ν 12 E 1 E 2 E 2 ≪ E 1 ν 21 ν 12 0.3 → 0.0214
Aluminium jaisi ek single material chahe jis taraf bhi kheeencho same behave karti hai — woh isotropic hai (Greek se: "sabhi directions mein equal"). Composite aisa nahi karta: fibers ek taraf run karte hain, isliye woh taraf stiff hai aur across soft hai. Yeh anisotropic hai. Track rakhne ke liye, hum do axes name karte hain jo fibers se chipke hain.
Figure 3 ek tilted ply dikhata hai: amber lines fibers hain, 1 labelled white arrow unke along run karta hai, cyan arrow 2 labelled across run karta hai, aur amber arc structure ke x θ
Definition Material axes: direction 1 aur direction 2
Direction 1 : fibers ke along . Yeh strong, stiff direction hai — modulus E 1 Direction 2 : fibers ke across (perpendicular, ply ke plane mein). Yeh weak hai — modulus E 2 ≪ E 1 E 1 , E 2 , G 12 , ν 12 G 12
Definition Fiber orientation angleθ nahi karte. ==θ x 0 ∘ x 9 0 ∘ 4 5 ∘ x Figure 3 mein).
Intuition Do coordinate systems, ek materialEk material system hai (1, 2 — fibers se chipka) aur ek laminate system (x , y θ x y [ Q ˉ ] x y
Composite stiffness predict karne ke liye recipe chahiye: kitna fraction material fiber hai versus matrix.
Definition Volume fractionsV f V m V f total volume ka woh fraction hai jo fiber hai; V m V f + V m = 1 V f = 0.6 V m = 0.4 V f ≈ 0.55 0.65
Subscript f m
Mnemonic Subscripts padhnaf f iber, m m atrix, numbers 1/2 x / y kaunsa piece aur kaunsi direction quantity ke baare mein sochne se pehle.
Parent teen stresses aur teen strains ko columns mein stack karta hai aur unhe numbers ki grid se link karta hai. Ek grid jo numbers ki ek list ko doosri mein turn kare woh matrix hai.
Definition Matrix yahan kya karta haiEk matrix numbers ka ek rectangular table hai jo quantities ka ek column doosre mein map karta hai. Strains ( ϵ 1 , ϵ 2 , γ 12 ) ( σ 1 , σ 2 , τ 12 ) is strain ka us stress mein kitna contribution hai."
[ Q ] stiffness matrix: strains in → stresses out. Bade numbers = stiff.[ S ] compliance matrix: stresses in → strains out. Yeh exact reverse hai, [ S ] = [ Q ] − 1 [ T ] transformation matrix: numbers rebuild karta hai jab aap axes 1–2 se axes x y θ cos θ sin θ
Q 66 = G 12 1/ E strains push karte hain aur stresses chahte hain, isliye unhe inverse [ Q ] Q 66 = G 12 symmetry bhi kaam aati hai: yeh [ Q ] Q 12
Neeche ka diagram ek flowchart render karta hai — neeche se upar padho: poora parent topic yahi building blocks stack karke bana hai.
Force over area = stress sigma tau
Fractional stretch = strain eps gamma
Poisson ratio nu couples directions
Angle theta and transform T
Transformed stiffness Qbar in x y
Poora parent topic (Laminate Theory ) yahi building blocks stack karke bana hai. Yeh foundation aage material selection se connect karta hai, finite-element models ko feed karta hai, aur iske stiffness numbers ultimately launch-load aur mass-budget trades drive karte hain. Anisotropy thermal design ke liye bhi matter karti hai aur layers stack hokar sandwich structures banaate hain.
Khud ko test karo — aap parent page ke liye tabhi ready hain jab har jawab instantly aaye.
Stress σ Force per unit area, F / A
Compressive stress ka sign kya hota hai, aur kyun? Negative — force andar ki taraf point karta hai, isliye σ = F / A σ < 0 ϵ < 0
Strain ϵ Strain fractional stretch Δ L / L
Engineering shear strain γ Ek originally right angle ki decrease (radians mein) jab block shear hota hai: corner 9 0 ∘ 9 0 ∘ − γ
Young's modulus E E = σ / ϵ E
Shear modulus G G 12 Skewing/sliding — right angle mein change (shear strain γ
Ply mein axis 1 versus axis 2 kaunsi direction hai? 1 = fibers ke along (stiff, E 1 E 2
Angle θ Structure ke x
V f V m Yeh sirf do ingredients hain; saath mein poora volume fill karte hain.
[ Q ] [ S ] [ Q ] [ S ] [ S ] = [ Q ] − 1
ν 21 = ν 12 E 2 / E 1 Elastic energy path-independent hai, jo compliance table ko symmetric hone par force karta hai (ν 12 / E 1 = ν 21 / E 2
