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

3.6.4 · D1 · Physics › Spacecraft Structures & Systems Engineering › Hooke's law in 3D — generalized stress-strain (tensor)

Yeh page assume karta hai ki tumne kuch nahi dekha. Hum har letter, index, aur picture ko build karenge jo parent note pe depend karta hai, ek aisi order mein jahan har idea sirf usse pehle waale idea pe rest karta hai.

0 · Directions aur axes — woh stage jahan sab kuch rehta hai

Kisi bhi physics se pehle, humein ek tarika chahiye yeh kehne ka ki "yeh direction, woh nahi".

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

Letters ki jagah numbers kyun? Kyunki jaldi hi humein aise quantities milenge jo do directions ek saath carry karte hain (ek face + ek force). $σ_{12}$ likhna kaafi cleaner hai "the-thing-on-the-x-face-pointing-in-y" se. Upar ki picture puri coordinate stage hai — is topic mein har arrow, angle aur tensor slot inhi teen axes mein se kisi ek ki taraf point karta hai.

1 · Subscripts (indices) — labels jo ek slot choose karte hain

$M_{ij}$ ko zor se padho "M-row-i-column-j". Yeh aadat akele poore topic ko unlock kar deti hai.

2 · Vectors — ek arrow, teen numbers

Bold $u$ matlab hai "poora arrow", jabki $u_{i}$ matlab hai "uska ek chosen piece". Same object, do zoom levels.

3 · Tensor — numbers ki ek table jisme directions hain

Topic ko rank-2 tensor kyun chahiye? Kyunki dono stress aur strain deformation states describe karte hain jinhe pin down karne ke liye do directions chahiye. Rank-4 kyun? Kyunki ek $3 \times 3$ grid ko doosre $3 \times 3$ grid se, slot by slot, relate karne ke liye ek aisa rule chahiye jisme chaar address slots hon — yahi stiffness tensor $C_{ij k l}$ hai.

4 · Stress $σ_{ij}$ — force per area, direction bookkeeping ke saath

$σ_{11}$ : $x$ ke along force, $x$ face karne wale face pe → face ke seedha andar pull/push (normal stress).
$σ_{12}$ : $y$ ke along force, $x$ face karne wale face pe → face ke along sideways drag (shear stress, aksar $τ_{12}$ likha jaata hai).

Greek letter hai sigma $σ$ — yahan iska matlab hamesha stress hai. Symbol $τ$ (tau) sirf off-diagonal shear entries ka nickname hai.

Definition Sign convention (ise ek baar yaad kar lo)

Normal stress: tension positive hai, compression negative. Toh ek rod jo pull ho rahi hai uska $σ_{11} > 0$ ; squeeze ho rahi rod ka $σ_{11} < 0$ . Inward push karne wala pressure $p$ deta hai $σ_{11} = σ_{22} = σ_{33} = - p$ .
Normal strain: stretch positive hai, shrink negative. Rod ko pull karo aur $ε_{11} > 0$ ; Poisson sideways shrink deta hai $ε_{22}, ε_{33} < 0$ .
Shear sign: $σ_{ij}$ (jab $i \neq = j$ ) positive hota hai jab, us face pe jiska outward normal $+ i$ ke along point karta hai, force $+ j$ ke along point kare. Face ya force direction dono mein se kisi ek ko flip karo aur sign flip ho jaata hai. Shear strain $ε_{ij}$ positive hoti hai jab originally-perpendicular $+ i$ aur $+ j$ material lines ke beech ka angle kam hota hai.

Parent note mein har worked example exactly isi "tension +, compression −" convention use karta hai.

5 · Strain $ε_{ij}$ — fractional deformation

Fractional kyun, absolute nahi? 2 mm wire ke liye 1 mm stretch bahut zyada hai aur 2 m beam ke liye kuch nahi. Original length se divide karne se strain deformation ki ek property ban jaati hai, object ke size se independent — exactly wahi jo ek material law ko chahiye. Greek letter hai epsilon $ε$ ; iska matlab hamesha strain hai.

6 · Symmetry $σ_{ij} = σ_{j i}$ — grid ek mirror hai

Is mirror ki wajah se, 9 slots mein sirf 6 independent numbers hain: 3 diagonal, plus 3 off-diagonal pairs. Yahi "6" hai isiliye engineers ek tensor ko ek 6-slot list mein pack kar sakte hain (Voigt notation).

7 · $σ_{ij}$ actually kahan se aata hai — kisi bhi plane pe traction

Nau numbers $σ_{ij}$ teen coordinate faces pe defined hain. Lekin ek real crack ya bolt material ko kisi bhi tilted plane ke along cut kar sakta hai. Woh arbitrary plane kaisi force feel karta hai?

Yahan yeh kyun matter karta hai: yeh batata hai kyun stress ko do indices chahiye. Ek index ( $j$ , summed) plane ki orientation read karta hai; free index ( $i$ ) resulting force direction deliver karta hai. Toh $σ_{ij}$ sirf "teen faces pe nau numbers" nahi hai — yeh us point ke through har imaginable slice pe force ka complete rule hai.

8 · Kronecker delta $δ_{ij}$ — identity switch

Topic ko yeh kyun chahiye: yeh ek ek direction-blind building block hai jo available hai. Jab ek material har direction mein same behave karta hai (isotropic), toh aap sirf $δ$ 's se bani tensors se uska law build kar sakte ho — isi tarah parent note 81 constants se 2 tak pahunchta hai.

9 · Summation (Einstein) convention — hidden totals

10 · Partial derivative $\partial_{j} \equiv \partial / \partial x_{j}$ — local rate of change

Yeh tool kyun, plain fraction nahi? Ek plain ratio $Δ u /Δ x$ depend karta hai ki tum kitna bada step lete ho. Derivative woh limit hai jab step zero tak shrink ho jaata hai, ek single point pe local stretch rate deta hai — exactly wahi jo strain ko chahiye, kyunki strain ek bent structure ke andar point to point vary kar sakti hai. Nau slopes $\partial_{j} u_{i}$ displacement-gradient grid form karte hain; uska symmetric half strain tensor hai (dekho Strain tensor and displacement gradient).

11 · Hooke's law index form mein — poora topic ek line mein

Ab jab stress, strain, indices, aur summation sab build ho gaye, hum parent note ka central rule uski apni definition ke roop mein state kar sakte hain.

12 · Material constants — do dials

Yeh topic mein kaise feed hote hain

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle answer do.

u_{i}

mein subscript

i

kya pick out karta hai?

Vector

u

i

-wa direction-piece (

u_{1} = x

-part, etc.).

Ek symmetric

3 \times 3

tensor mein kitne independent numbers hote hain, aur kyun?

6 — teen diagonal plus teen off-diagonal pairs, kyunki grid mirror karta hai (

σ_{ij} = σ_{j i}

σ_{12}

mein do indices ka kya matlab hai?

Direction

2

(

y

) mein point karta force, us face pe act karta hai jiska normal direction

1

(

x

) mein point karta hai — ek shear stress.

Normal stress aur strain ka sign convention kya hai?

Tension/stretch positive, compression/shrink negative; ek pressure

p

deta hai

σ_{ii} = - p

Traction relation

t_{i} = σ_{ij} n_{j}

tumhe kya batata hai?

Stress tensor plane ke normal

n

ko force-per-area

t

mein turn karta hai jo woh plane feel karta hai (Cauchy's theorem).

δ_{ij}

i = j

hone pe aur

i \neq = j

hone pe kya equal hai?

1

jab

i = j

0

jab

i \neq = j

(identity switch).

ε_{k k}

ko poora likh ke batao ki physically iska kya matlab hai.

ε_{11} + ε_{22} + ε_{33}

— volume mein fractional change (volumetric strain).

C_{ij k l} ε_{k l}

mein kaun sa index summed hai aur kaun sa free?

k

aur

l

summed hain (har ek repeated hai);

i

aur

j

free hain (equation label karte hain).

Strain displacement-gradient

\partial_{j} u_{i}

ka sirf symmetric half kyun rakhti hai?

Antisymmetric half pure rotation hai, jo koi shape change nahi karta aur koi elastic energy store nahi karta.

\partial_{j}

kiska shorthand hai?

\partial / \partial x_{j}

ka shorthand, axis

j

ke along step karte waqt change ki rate.

Small-strain formula

ε_{ij} = \frac{1}{2} (\partial_{j} u_{i} + \partial_{i} u_{j})

kab break down karta hai?

Large deformations ke liye — yeh quadratic gradient terms drop kar deta hai, toh tab finite-strain measures zaroori hain.

Fully anisotropic solid ke liye kitne independent constants hain, aur isotropy ke baad?

21 (81 se minor + major symmetries ke through), isotropy ke liye 2 tak gir jaata hai.

Engineering shear

γ_{12}

aur tensor shear

ε_{12}

ke beech kya relation hai?

γ_{12} = 2 ε_{12}

Strain ke liye hum ratio

Δ u /Δ x

ki jagah partial derivative kyun use karte hain?

Derivative ek point pe local, step-size-independent stretch rate hai; ek ratio depend karta hai ki tum kitna bada step lete ho.

Ek isotropic solid ko kitne material constants chahiye?

Do (jaise

E

aur

ν

, ya

λ

aur

μ