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

3.6.4 · Physics › Spacecraft Structures & Systems Engineering

Tensor ki zaroorat KYUN hai?

KYA relate kiya ja raha hai? Do quantities jinhe direction information chahiye:

Stress $σ_{ij}$ : force per area. Pehla index = face normal, doosra index = force direction. 9 components (6 independent, kyunki $σ_{ij} = σ_{j i}$ moment balance se).
Strain $ε_{ij}$ : fractional deformation. Yeh bhi symmetric hai, 6 independent.

KYUN sirf ek matrix nahi? Kyunki har point par loading ki state khud ek $3 \times 3$ object hoti hai. Ek $3 \times 3$ tensor ko linearly doosre se relate karne ke liye ek aisa object chahiye jisme 4 indices hon:

σ_{ij} = C_{ij k l} ε_{k l} (i, j, k, l \in {1, 2, 3})

Repeated $k, l$ par summation (Einstein convention). Yeh Hooke's law ka sabse general linear-elastic form hai.

Displacement se strain nikalna (first principles)

Yeh form kyun? Do paas-paas ke points lo jo $d x$ se alag hain. Deformation ke baad separation ban jaata hai $d x_{i} + \frac{\partial u _{i}}{\partial x _{j}} d x_{j}$ . Squared length mein change, first order tak, $\partial u_{i} / \partial x_{j}$ ke symmetric part ko involve karta hai. Antisymmetric part pure rotation hai (koi stretch nahi), isliye hum use discard kar dete hain — yahi reason hai symmetrize karne ka. Rotation elastic energy store nahi karta, isliye woh $ε$ mein appear nahi hona chahiye.

Constants ginana: 81 → 21 → 2

Energy se major symmetry KYUN milti hai? Ek conservative elastic material mein stress, stored-energy density ka gradient hota hai: $σ_{ij} = \partial W / \partial ε_{ij}$ . Phir $C_{ij k l} = \partial^{2} W / \partial ε_{ij} \partial ε_{k l}$ , aur mixed partials commute karte hain — isliye $ij$ aur $k l$ pairs ko swap karne se $C$ nahi badalta.

Isotropic law (sabse zyada kaam aane wala)

Yahan $ε_{k k} = ε_{11} + ε_{22} + ε_{33}$ volumetric strain hai (trace = fractional volume change). $λ$ aur $μ$ Lamé parameters hain; $μ \equiv G$ shear modulus hai.

Sirf do building blocks kyun? Isotropy ka matlab hai ki $C$ ko kisi bhi rotation ke baad same dikhna chahiye — ek isotropic tensor. Sirf isotropic 4th-rank tensors hi $δ_{ij} δ_{k l}$ , $δ_{ik} δ_{j l}$ , $δ_{i l} δ_{j k}$ ke combinations hote hain. Minor symmetry force karta hai ki last do saath appear karein, jisse do coefficients bachte hain.

Strain-from-stress nikalna (engineering form)

Upar wale ko $ε$ ke liye solve karne par woh form milta hai jo students yaad karte hain:

ε_{ij} = \frac{1 + ν}{E} σ_{ij} - \frac{ν}{E} σ_{k k} δ_{ij}

Component-wise (normal strains):

ε_{xx} = \frac{1}{E} [σ_{xx} - ν (σ_{y y} + σ_{z z})]

$- ν (\dots)$ term hi Poisson effect hai: $y, z$ mein stress se $x$ sikhad jaata hai.

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

Voigt (matrix) notation — engineers aslmein kaise compute karte hain

Kyunki dono tensors symmetric hain, unke 6 independent components ko vectors mein pack karo:

{σ} = [σ_{xx}, σ_{y y}, σ_{z z}, τ_{y z}, τ_{x z}, τ_{x y}]^{T}, {ε} = [ε_{xx}, ε_{y y}, ε_{z z}, γ_{y z}, γ_{x z}, γ_{x y}]^{T}

jisme engineering shear $γ_{x y} = 2 ε_{x y}$ hai. Phir ${σ} = [C] {ε}$ ek $6 \times 6$ matrix ke saath. Isotropic compliance $[S] = [C]^{- 1}$ :

[S] = \frac{1}{E} 1 - ν - ν 000 - ν 1 - ν 000 - ν - ν 1000 000 2 (1 + ν) 00 0000 2 (1 + ν) 0 00000 2 (1 + ν)

Worked examples

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho ek foam ka block hai. Agar tum usse upar se dabao, toh woh sirf chhota nahi hota — woh sides se baahar bhi nikal aata hai. Toh "kitna squish hota hai" yeh depend karta hai ki tum kaunsi sides ko push kar rahe ho aur kitna, sab ek saath. Ek number yeh describe nahi kar sakta. Isliye hum numbers ki ek badi table (ek tensor) use karte hain jo kehti hai: "agar tum is taraf push karo, toh exactly yahan batao ki woh har direction mein kaise badlega." Zyaadatar simple materials ke liye poori badi table bas do numbers tak aa jaati hai: yeh kitna stiff hai ( $E$ ) aur yeh sideways kitna bulge karta hai ( $ν$ ). Yahi poora trick hai.

Flashcards

General 3D Hooke's law index form mein

σ_{ij} = C_{ij k l} ε_{k l}

k, l

par sum karo.

Ek fully anisotropic stiffness tensor mein kitne independent components hote hain, aur kyun?

21;

3^{4} = 81

se do minor symmetries (

σ, ε

symmetric) se 36 ho jaata hai, phir major symmetry (energy) se 21.

21 constants ko 2 tak kya reduce karta hai?

Isotropy — sab rotations ke under invariance, sirf Lamé

λ

aur

μ

bachte hain.

Isotropic stress–strain law (Lamé form)

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

Small-strain tensor ki definition

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

; displacement gradient ka symmetric part (rotation remove kiya hua).

Displacement gradient ko symmetrize kyun karte hain?

Antisymmetric part rigid rotation hai, jo koi elastic energy store nahi karta.

Major symmetry

C_{ij k l} = C_{k l ij}

kahan se aati hai?

σ_{ij} = \partial W / \partial ε_{ij}

, isliye

C = \partial^{2} W / \partial ε \partial ε

aur mixed partials commute karte hain.

Normal component ke liye engineering strain-from-stress

ε_{xx} = \frac{1}{E} [σ_{xx} - ν (σ_{y y} + σ_{z z})]

E, ν

ke terms mein shear modulus

G = μ = E / [2 (1 + ν)]

Bulk modulus aur physical limit

K = E / [3 (1 - 2 ν)]

; positive

K

ke liye

ν < 1/2

;

ν \to 1/2

= incompressible.

Factor-of-2 shear trap

Engineering shear

γ_{x y} = 2 ε_{x y}

; compliance

γ

use karta hai isliye shear entry

= 1/ G = 2 (1 + ν) / E

Connections

Stress tensor and Cauchy's relation
Strain tensor and displacement gradient
Elastic strain energy density
Poisson's ratio and material limits
Isotropic vs anisotropic materials (composites)
Thin-walled pressure vessels (spacecraft tanks)
Von Mises yield criterion
Finite Element Method — stiffness matrices

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

Tensor ki zaroorat KYUN hai?

Displacement se strain nikalna (first principles)

Constants ginana: 81 → 21 → 2

Isotropic law (sabse zyada kaam aane wala)

Strain-from-stress nikalna (engineering form)

Voigt (matrix) notation — engineers aslmein kaise compute karte hain

Worked examples

Flashcards

Connections

Concept Map

Deep Dive