# unit5 - MIT 16.20 Fall 2002 Unit 5 Engineering Constants...

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MIT - 16.20 Fall, 2002 Unit 5 Engineering Constants Readings : Rivello 3.1 - 3.5, 3.9, 3.11 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace © 2001

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MIT - 16.20 Fall, 2002 We do not characterize materials by their E mnpq . The E mnpq are useful in doing transformations, manipulations, etc. We characterize materials by their “ENGINEERING CONSTANTS” ( or , Elastic Constants) (what we can physically measure) There are 5 types 1. Longitudina l (Young’s) (Extensional) Modulus : relates extensional strain in the direction of loading to stress in the direction of loading. (3 of these) 2. Poisson ’s Ratio : relates extensional strain in the loading direction to extensional strain in another direction. (6 of these…only 3 are independent) Paul A. Lagace © 2001 Unit 5 - p. 2
MIT - 16.20 Fall, 2002 3. Shear Modulus : relates shear strain in the plane of shear loading to that shear stress. (3 of these) 4. Coefficient of Mutual Influence : relates shear strain due to shear stress in that plane to extensional strain or , relates extensional strain due to extensional stress to shear strain. (up to 18 of these) 5. Chentsov Coefficient : relates shear strain due to shear stress in that plane to shear strain in another plane. (6 of these) Let’s be more specific: 1. Longitudina l Modulus 1) E 11 or E xx or E 1 or E x : contribution of ε 11 to σ 11 2) E 22 or E yy or E 2 or E y : contribution of ε 22 to σ 22 3) E 33 or E zz or E 3 or E z : contribution of ε 33 to σ 33 Paul A. Lagace © 2001 Unit 5 - p. 3

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MIT - 16.20 Fall, 2002 In general : E mm = σ mm due to σ mm applied only ε mm (no summation on m) 2. Poisson ’s Ratios ( negative ratios) 1) ν 12 or ν xy : (negative of) ratio of ε 22 to ε 11 due to σ 11 2) ν 13 or ν xz : (negative of) ratio of ε 33 to ε 11 due to σ 11 3) ν 23 or ν yz : (negative of) ratio of ε 33 to ε 22 due to σ 22 4) ν 21 or ν yx : (negative of) ratio of ε 11 to ε 22 due to σ 22 5) ν 31 or ν zx : (negative of) ratio of ε 11 to ε 33 due to σ 33 6) ν 32 or ν zy : (negative of) ratio of ε 22 to ε 33 due to σ 33 In general : ν nm = ε mm due to σ nn applied only ε nn (for n m) Important : ν nm ≠ ν mn Paul A. Lagace © 2001 Unit 5 - p. 4
MIT - 16.20 Fall, 2002 However, these are not all independent. There are relations known as “reciprocity relations” (3 of them) ν 21 E 11 = ν 12 E 22 ν 31 E 11 = ν 13 E 33 ν 32 E 22 = ν 23 E 33 3. Shear Moduli 1) G 12 or G xy or G 6 : contribution of (2) ε 12 to σ 12 2) G 13 or G xz or G 5 : contribution of (2) ε 13 to σ 13 3) G 23 or G yz or G 4 : contribution of (2) ε 23 to σ 23 In general : G mn = σ mn due to σ mn applied only 2 ε mn factor of 2 here since it relates physical quantities shear stress τ mn G mn = shear deformation (angular charge) γ mn Paul A. Lagace © 2001 Unit 5 - p. 5

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MIT - 16.20 Fall, 2002 4. Coefficients of Mutual Influence ( negative ratios) (also known as “coupling coefficients”) Note : need to use contracted notation here: 1) η 16 : (negative of) ratio of (2) ε 12 to ε 11 due to σ 11 2) η 61 : (negative of) ratio of ε 11 to (2) ε 12 due to σ 12 3) η 26 (5) η 36 (7) η 14 (9) η 24 4) η 62 (6) η 63 (8) η 41 (10) η 42 11) η 34 (13) η 15 (15) η 25 (17) η 35 12) η 43 (14) η 51 (16) η 52 (18) η 53 5. Chentsov Coefficients
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