Hooke's law in 3D — generalized stress-strain (tensor): Visual walkthrough

3.6.4 · D2 · Physics › Spacecraft Structures & Systems Engineering › Hooke's law in 3D — generalized stress-strain (tensor)

Step 1 — "Little cube of material" hota kya hai?

KYA. Kisi bhi solid ke andar koi bhi point lo — maano ek fuel tank ki aluminium wall. Tab tak zoom karo jab tak tum imagine kar sako ki ek tiny cube of material wahan baitha hai. Jo kuch bhi hum discuss karte hain woh is ek cube ke chhe faces pe hota hai.

CUBE kyun? Kyunki ek cube humein teen saaf directions deta hai baat karne ke liye — left/right, front/back, up/down. Hum unhe $x$ , $y$ , $z$ naam dete hain. Material pe koi bhi push ya slide describe ki ja sakti hai jo in chhe flat faces pe hota hai. Ek sphere ke paas push karne ke liye koi flat face nahi hoga; cube sabse simple shape hai jo humein free mein teen perpendicular directions deta hai.

PICTURE. Neeche, cube apne teen axes ke saath. Opposite faces ke har pair ka ek outward normal hota hai (woh chhoti arrow jo face se seedha bahar nikal rahi hai). Normal hi hum face ko naam dete hain: " $x$ -face" woh face hai jiska normal $x$ ke along point karta hai.

Figure — Hooke's law in 3D — generalized stress-strain (tensor)

Step 2 — Stress: do arrows chahiye, isliye do indices chahiye

KYA. Ek face pe push ya pull karo. Area pe spread hui force ko stress kehte hain. Lekin yahan subtle baat hai: stress describe karne ke liye tumhe do cheezein kehni padti hain — kaun sa face hai jis pe act kar rahe ho, aur kaun si taraf force us face pe point karti hai.

DO cheezein kyun? $x$ -face dekho. Main use seedha bahar khich sakta hoon (force along $x$ ) — yeh cube ko stretch karta hai. Ya main use sideways kheench sakta hoon (force along $y$ ) — yeh cube ko shear karta hai, jaise cards ki deck ka top slide karna. Wohi face, bilkul alag effect. Ek arrow (face) dono ko capture nahi kar sakta. Isliye stress ko do labels chahiye: $σ_{ij}$ jahan $i$ = face ki normal direction aur $j$ = force ki direction.

PICTURE. $x$ -face jo ek pull carry kar raha hai ( $σ_{xx}$ , orange, seedha bahar) aur ek drag ( $σ_{x y}$ , teal, sideways). Pehla subscript face hai; doosra arrow hai.

3 faces $\times$ 3 arrow-directions $= 9$ numbers hain. Yeh ek $3 \times 3$ table mein rehte hain — dekho Stress tensor and Cauchy's relation jisme bataya hai kyun $σ_{ij} = σ_{j i}$ (table symmetric hai, isliye sirf 9 mein se 6 independent hain).

Step 3 — Strain: measure karna ki cube ka shape actually kitna badla

KYA. Stress woh hai jo hum cube ke saath karte hain. Strain woh hai jaise cube react karta hai — kitna stretch hua ya shear hua, ek fraction ke roop mein. Agar $10 mm$ ka edge $10.01 mm$ ho jaata hai, toh strain hai $0.01/10 = 0.001$ . Yeh ek pure ratio hai, koi units nahi.

FRACTION kyun, length kyun nahi? Kyunki ek bade cube aur ek chhote cube ka same material same pull ke neeche same fraction se stretch hoga. Fractions material property ko star banate hain, sample size ko nahi. Hum ek point ke tiny displacement $u$ ko track karte hain (woh kitna move hua) aur usse uske neighbour se compare karte hain.

PICTURE. Left: normal strain $ε_{xx}$ — cube $x$ mein longer ho jaata hai, fractional stretch. Right: shear strain — do edges ke beech ka right angle thoda tip ho jaata hai; woh tilt hi shear hai. Dono milke shape ki har possible chhoti change describe karte hain.

Cloze check:

Displacement gradient ko symmetrize kyun karte hain?

Antisymmetric part pure rotation hai, jo koi elastic energy store nahi karta aur strain count nahi hona chahiye.

Step 4 — Naive guess aur kyun ek number $E$ fail hota hai

KYA. 1D mein, Hooke's law ek line hai: $σ = E ε$ — "pull stretch ke proportional hai," stiffness $E$ constant ke roop mein. Kya hum sirf ek $E$ cube ke liye reuse kar sakte hain?

KYUN fail hota hai. Cube ko $x$ mein pull karo. Woh $x$ mein longer ho jaata hai — theek hai, $ε_{xx}$ badhta hai. Lekin woh $y$ aur $z$ mein thinner bhi ho jaata hai! Woh sideways shrinking Poisson effect hai. Ek akela number $E$ jo " $x$ mein pull" ko " $x$ mein stretch" se relate karta hai, uske paas $y$ aur $z$ shrinking predict karne ka koi tarika nahi. Isliye humein ek rule chahiye jo allow kare ki ek direction mein stress kaafi directions mein strain cause kare.

PICTURE. Cube ko $x$ along pull karo (orange arrows). Dashed original outline dekho: yeh lamba stretch hota hai lekin sides andar squeeze hoti hain (plum arrows). Ek input, teen outputs.

Step 5 — Tables ke beech map: 4-index tensor aata hai

KYA. Ek rule jo 9 strains mein se har ek ko 9 stresses mein se har ek mein convert kare. "Strain $ε_{k l}$ ka stress $σ_{ij}$ pe effect" point karne ke liye humein chaar directions name karne padenge: $i, j$ (kaun sa stress) aur $k, l$ (kaun sa strain). Woh chaar-labelled object stiffness tensor $C_{ij k l}$ hai.

CHAAR indices kyun? Do us stress ke liye jo hum produce karte hain, do us strain ke liye jo use produce karta hai. Hum har strain ka contribution add karte hain — woh summation letters $k$ aur $l$ repeat karke likhi jaati hai (Einstein ka shorthand: repeated letter matlab "sum over $1, 2, 3$ ").

PICTURE. Ek wiring diagram: left pe 6 strains, right pe 6 stresses, aur $C_{ij k l}$ wires ka bundle jo har strain ko har stress se connect karta hai. Har wire ka ek gain hota hai — woh gain $C$ ke andar ek number hai.

Step 6 — Symmetry ek shredder ki tarah: 81 → 21 → 2

KYA. Ab hum redundant numbers delete karte hain teen facts use karke jo already page pe hain.

HAR CUT KYUN KAAM KARTA HAI:

Stress table symmetric hai ( $σ_{ij} = σ_{j i}$ , Step 2) $\Rightarrow$ $i, j$ swap karna kuch nahi badal sakta: $C_{ij k l} = C_{j ik l}$ .
Strain table symmetric hai ( $ε_{k l} = ε_{l k}$ , Step 3) $\Rightarrow$ $C_{ij k l} = C_{ij l k}$ .
Stored elastic energy ek single scalar hai, aur uske mixed second derivatives commute karte hain, isliye pura front pair back pair se swap ho jaata hai: $C_{ij k l} = C_{k l ij}$ (dekho Elastic strain energy density).

Yeh teen cuts $81 \to 36 \to 21$ karte hain. 21 independent numbers wala material fully anisotropic hai (har direction mein alag) — socho carbon-fibre laminate, dekho Isotropic vs anisotropic materials (composites).

PICTURE. Ek funnel: 81 grains of sand upar se girte hain, symmetry sieves sirf 21 ko through jaane dete hain, aur final "isotropy" sieve (har direction mein same) sirf 2 ko through jaane deta hai — do Lamé numbers $λ, μ$ .

Step 7 — Isotropic law assemble karna aur har term padhna

KYA. Sirf do knobs $λ$ aur $μ$ ke saath, stiffness forced hai ki woh ho $C_{ij k l} = λ δ_{ij} δ_{k l} + μ (δ_{ik} δ_{j l} + δ_{i l} δ_{j k}) .$ Ise $σ_{ij} = C_{ij k l} ε_{k l}$ mein feed karo aur yeh workhorse formula mein collapse ho jaata hai.

KYUN yeh do clean pieces mein split hota hai. $λ δ_{ij} δ_{k l}$ block, jab $ε_{k l}$ ke against sum kiya jaata hai, trace $ε_{k k} = ε_{11} + ε_{22} + ε_{33}$ — fractional volume change — pull out karta hai aur use $δ_{ij}$ ke zariye sirf diagonal pe stamp karta hai. $μ$ block har component ko directly hit karta hai, shape change (shear) govern karta hai. Volume vs shape: do effects, do knobs.

PICTURE. Cube ka response do mein split: ek uniform "breathing" (volume, $λ$ -term) plus ek "twisting/squaring" (shape, $2 μ$ -term). Koi bhi deformation in donon ka sum hai.

Step 8 — Edge & degenerate cases (reader ko kabhi stranded mat chhodna)

KYA. Formula ko uski limits tak push karo aur check karo ki woh phir bhi sense banata hai.

KYUN. Ek law jis par tum sirf beech mein trust karte ho woh koi law nahi. Hum corners test karte hain: koi shear nahi, all-around pressure, aur incompressible limit.

PICTURE. Teen mini-cubes: (a) pure uniaxial pull $σ = E ε$ recover karta hai; (b) sab taraf equal pressure — pure volume change, koi shear nahi; (c) $ν \to \frac{1}{2}$ , cube volume change karne se mana kar deta hai.

Ek-picture summary

Yeh single figure puri walkthrough chain karta hai: cube → two-index stress → two-index strain → one-number guess fails (Poisson) → 4-index map $C$ → symmetry shredder (81→21→2) → isotropic law split into volume ( $λ$ ) aur shape ( $2 μ$ ) parts.

Aage, yeh law Thin-walled pressure vessels (tank walls), Von Mises yield criterion (kab toot jaata hai?), aur Finite Element Method — stiffness matrices ke element blocks ko feed karta hai.

Recall Feynman: puri walkthrough plain words mein

Ek tiny sugar-cube of metal socho. Describe karne ke liye ki tum use kaise push karte ho, ek arrow kaafi nahi — tumhe kehna padega kaun sa face aur kaun si taraf, isliye pushes ko do labels milte hain (yahi stress hai). Describe karne ke liye ki woh kaise squish hoti hai, tum compare karte ho ki har point kaise move hua, phir se do labels ke saath (yahi strain hai). Ab tricky part: usse upar se dabao aur woh sirf chhota nahi hoti, woh sides se bahar bulge karta hai — isliye ek stiffness number kaam nahi kar sakta; ek taraf push karna shape har taraf change karta hai. Honest bookkeeping ek giant table hai chaar labels ke saath, $C$ , jo tumhe batata hai ki har squish se har push kaise hota hai. Woh table 81 numbers se shuru hoti hai, lekin "stress table symmetric hai," "strain table symmetric hai," aur "energy sirf ek number hai" redundancies cross out karte hain jab tak 21 nahi bache. Agar material har direction mein same hai, sirf do bachte hain: ek ke liye ki woh volume change se kaise ladhta hai, ek ke liye ki woh shape change se kaise ladhta hai. Un do knobs ko $λ$ aur $2 μ$ likho, aur poora law pop out ho jaata hai — wohi $E$ aur $ν$ jo tum pehle se jaante the, bas 3D mein rehne ke liye kaafi grown up.

Flashcards

Stress ko do indices kyun chahiye?

Ek index face ko naam deta hai (uska normal), doosra us direction ko naam deta hai jis taraf force us face pe point karti hai.

Ek single modulus

E

3D response kyun describe nahi kar sakta?

Kyunki ek direction mein push doosri directions mein bhi shape change karta hai (Poisson effect), isliye humein pure strain table se pure stress table tak ek map chahiye.

Stiffness tensor mein kitne indices hone chahiye aur kyun?

Chaar — do produced stress ko naam dene ke liye, do use produce karne wale strain ko naam dene ke liye.

Symmetry chain stiffness constants ko kaise reduce karta hai?

81 \to 36

(minor symmetries)

\to 21

(major/energy symmetry)

\to 2

(isotropy).

σ_{ij} = λ ε_{k k} δ_{ij} + 2 μ ε_{ij}

mein, kaun sa term volume vs shape control karta hai?

λ ε_{k k} δ_{ij}

volume control karta hai (diagonal pe trace);

2 μ ε_{ij}

shape control karta hai (har component).