# ho2 - MIT 16.20 Fall 2002 16.20 HANDOUT#2 Fall 2002 Review...

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MIT - 16.20 Fall, 2002 16.20 HANDOUT #2 Fall, 2002 Review of General Elasticity NOTATION REVIEW ( e.g., for strain ) Engineering Contracted Engineering “Tensor” Tensor ε x = ε 1 = ε xx = ε 11 ε y = ε 2 = ε yy = ε 22 ε z = ε 3 = ε zz = ε 33 γ yz = ε 4 = 2 ε yz = 2 ε 23 γ xz = ε 5 = 2 ε xz = 2 ε 13 γ xy = ε 6 = 2 ε xy = 2 ε 12 EQUATIONS OF ELASTICITY y 3 , z y 2 , y y 1 , x Right-handed rectangular Cartesian 15 equations/15 unknowns coordinate system 1. Equilibrium (3) ∂σ 11 + ∂σ 21 + ∂σ 31 + f 1 = 0 y 1 y 2 y 3 ∂σ 12 + ∂σ 22 + ∂σ 32 + f 2 = 0 ∂σ mn + f n = 0 y 1 y 2 y 3 y m ∂σ 13 + ∂σ 23 + ∂σ 33 + f 3 = 0 y 1 y 2 y 3 Paul A. Lagace © 2002 Handout 2-1

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2212 13 MIT - 16.20 Fall, 2002 2. Strain-Displacement (6) ε 11 = u 1 ε 21 = ε 12 = 1 u 1 + u 2 y 1 2 y 2 y 1 ε 22 = u 2 ε 31 = ε 13 = 1 u 1 + u 3 1 u m + u n y 2 2 y 3 y 1 ε mn = 2 y n y m ε 33 = u 3 y 3 ε 32 = ε 23 = 1 u 2 + u 3 2 y 3 y 2 3. Stress-Strain (6) Generalized Hooke’s Law: σ mn = E mnpq ε pq • Anisotropic: σ 11 E 1111 E 1122 E 1133 2E 1123 1113 1112 ε 11 E 1122 E 2222 E 2233 2223 2213 2212 ε 22 σ 22 σ 33 E 1133 E 2233 E 3333 3323 3313 3312 ε 33 = σ 23 E 1123 E 2223 E 3323 2323 1323 1223 ε 23 σ 13 E 1113 E 2213 E 3313 1323 1313 1213 ε 13 σ 12 E 1112 E 2212 E 3312 1223 1213 1212 ε 12 • Orthotropic: σ 11 E 1111 E 1122 E 1133 0 0 0 ε 11 σ 22 E 1122 E 2222 E 2233 0 0 0 ε 22 σ 33 E 1133 E 2233 E 3333 0 0 0 ε 33 = σ 23 0 0 0 2323 0 0 ε 23 σ 13 0 0 0 0 1313 0 ε 13 σ 12 0 0 0 0 0 1212 ε 12 Compliance Form: ε mn = S mnpq σ pq where: E -1 = S ~ ~ Paul A. Lagace © 2002 Handout 2-2
MIT - 16.20 Fall, 2002 DEFINITION OF ENGINEERING CONSTANTS 1. Longitudinal (Young’s) (Extensional) Moduli: σ mm E mm = ε due to σ mm applied only (no summation on m) mm 2. Poisson’s Ratios: ε ν nm = mm due to σ nn applied only (for n m) ε nn Reciprocity: ν nm E m = ν mn E n (no sum) (m n) 3. Shear Moduli: σ mn

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ho2 - MIT 16.20 Fall 2002 16.20 HANDOUT#2 Fall 2002 Review...

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