MT-1620 al.2002 Unit 5 Engineering Constants Readings Rivello 31-35,39,3.11 Paul A Lagace, Ph. D Professor of aeronautics Astronautics and Engineering Systems Paul A Lagace @2001
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
MT-1620 Fall 2002 We do not characterize materials by their enno the emon are mpg 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. Longitudinal young s)(Extensional)Modulus: relates extensional strain in the direction of loading to stress in the direction of loading (3 of these) 2. Poissons 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 it 5-p. 2
MIT - 16.20 Fall, 2002 We do not characterize materials by their Emnpq. The Emnpq 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. Longitudinal (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
MT-1620 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. Chentsoy 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 Longitudinal Modulus 1)E or Exx or E, or Ex: contribution of E,1 to O11 2 or Ew or E2 or E: contribution of E22 to O22 3)E33or Ez or E3 or E: contribution of e33 to O33 Paul A Lagace @2001 lt 5-p. 3
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. Longitudinal Modulus 1) E11 or Exx or E1 or Ex: contribution of ε11 to σ11 2) E22 or Eyy or E2 or Ey: contribution of ε22 to σ22 3) E33 or Ezz or E3 or Ez: contribution of ε33 to σ33 Paul A. Lagace © 2001 Unit 5 - p. 3
MT-1620 Fall 2002 In general mm mm due to Omm applied only mm ( no summation on m) 2. Poisson's Ratios (negative ratios) 1)v12 or vx:(negative of) ratio of E22 to e,1 due to O, 2)v13 or vxz:(negative of) ratio of E33 to e11 due to 3)V23 or Vyz: (negative of) ratio of E33 to E22 due to O22 4)V21 or vox:(negative of) ratio of E, to E22 due to O22 5)V31 or Vzx:(negative of) ratio of E,1 to e33 due to 33 6)V32 or vxy:(negative of) ratio of E22 to E33 due to O33 In general: Vom Emm due to onn applied only (forn≠m Important:vm≠Vm Paul A Lagace @2001 lt 5-p. 4
MIT - 16.20 Fall, 2002 In general: Emm = σmm due to σmm applied only εmm (no summation on m) 2. Poisson’s Ratios (negative ratios) 1) ν12 or νxy: (negative of) ratio of ε22 to ε11 due to σ11 2) ν13 or νxz: (negative of) ratio of ε33 to ε11 due to σ11 3) ν23 or νyz: (negative of) ratio of ε33 to ε22 due to σ22 4) ν21 or νyx: (negative of) ratio of ε11 to ε22 due to σ22 5) ν31 or νzx: (negative of) ratio of ε11 to ε33 due to σ33 6) ν32 or νzy: (negative of) ratio of ε22 to ε33 due to σ33 In general: νnm = − εmm due to σnn applied only εnn (for n ≠ m) Important: νnm ≠ νmn Paul A. Lagace © 2001 Unit 5 - p. 4
MT-1620 al.2002 However, these are not all independent. There are relations known as "reciprocity relations"(3 of them) 21-11 12-22 31-11 13-33 v E 2-22 23-33 3. Shear modul 1)G12 or Gxy or Gg: contribution of (2)E12 to O12 2)G13or Gx or G: contribution of (2)E13 to O13 3)G23 or Gvz or G4: contribution of (2)E23 to O23 general:Gmm=mm due to Omn applied only factor of 2 here since it relates physical quantities shear stress shear deformation(angular charge Paul A Lagace @2001 lt 5-p. 5
MIT - 16.20 Fall, 2002 However, these are not all independent. There are relations known as “reciprocity relations” (3 of them) ν21 E11 = ν12 E22 ν31 E11 = ν13 E33 ν32 E22 = ν23 E33 3. Shear Moduli 1) G12 or Gxy or G6: contribution of (2)ε12 to σ12 2) G13 or Gxz or G5: contribution of (2)ε13 to σ13 3) G23 or Gyz or G4: contribution of (2)ε23 to σ23 In general: Gmn = σmn due to σmn applied only 2εmn factor of 2 here since it relates physical quantities shear stress ⇒ G τmn mn = shear deformation (angular charge) γ mn Paul A. Lagace © 2001 Unit 5 - p. 5