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530.352 Materials Selection

530.352 Materials Selection. Lecture #6 : Elastic properties Wednesday September 21 st , 2005. Anisotropic elasticity :.  i = C ij  j.  i = S ij  j. For Cubic crystals :. C 11 = C 22 = C 33 , C 44 = C 55 = C 66 , C 12 = C 21 = C 13 = C 31 =C 23 = C 32 , all others = 0.

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530.352 Materials Selection

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  1. 530.352 Materials Selection Lecture #6 : Elastic propertiesWednesday September 21st, 2005

  2. Anisotropic elasticity : i = Cijj i = Sijj For Cubic crystals : C11= C22 = C33, C44= C55 = C66 , C12= C21 = C13 = C31 =C23 = C32 , all others = 0

  3. E[100], E[110], and E[111] E[100] = 1 s11 E[110] = 2 [s11 + s12 + s44/2] E[111] = 3 [s11 + 2s12 + s44]

  4. Relations : c11 = s11 + s12 (s11 - s12) (s11 + 2s12) c12 = -s12 (s11 - s12) (s11 + 2s12) c44 = 1 s44 s11 = c11 + c12 (c11 - c12) (c11 + 2c12) s12 = -c12 (c11 - c12) (c11 + 2c12) s44 = 1 c44

  5. Isotropic Elasticity All directions the same polycrystalline What are grains ?? • grains are individual crystals that have nucleated with a random orientation.

  6. Lamé Coefficient (l) xx yy Isotropic Elasticity  =E =G p = -K =yyxx yyxx All directions the same 5 constants only 2 independent polycrystalline

  7. 1-D example :  = 5,000 N = 1,250 MPa 4 x 10-6 m2  = .2 mm = 0.02 = 2% 10 mm E =  = 1.25 GPa = 62.5 GPa  0.02 Ao = 4 mm2 Lo = 10 mm L = 200 microns = 0.2 mm W W = 5,000 kg

  8. What about 3-D ?? zz = 0.01 Does  = Estill hold? Glue

  9. zz = 0.01 z y x What about 3-D ?? Note: glue cannot contract in xy plane : zz = 0.01 xx = yy =  xy = yz = zx = 

  10. 3-D isotropic elasticity : Axial Draws in Draws in from from side other side z y x

  11. 3-D isotropic elasticity : - or -

  12. 3D Hooke’s Law:

  13. zz = 0.01 z y x Example of 3D Hooke’s Law xx = 0 = 1/E [xx-(yy+zz)] xx = yy = T (1-)T - zz = 0 zz= (1-)T zz = 0.01 = 1/E [zz-(2T)] zz = 0.01 = 1/E [(1-)T -2T]  = 1/3 ; E = 2 GPa T = xx = 0.01E /(2 - ) = 15 MPa zz = ( / 1) xx = 30 MPa

  14. Comparing 3D and 1D : Note : 3D: zz = 30 MPa is greater than 1D: zz = E = 2,000 MPa x 0.01 = 20 MPa

  15. Measuring Elastic Moduli • Static loading • Dynamic measurements • Theoretical calculations

  16. Static Loading W L E =  = W Lo AoL Lo

  17. Dynamic Measurements 1-D bar Eq. of motion:  d2u/dt2 = d/dx = d(E/dx =Ed(du/dx)/dx d2u/dt2 - (E/ d2u/dx2 = 0 (Wave Eq.) VL = ( E /  ) 0.5

  18. Longitudinal Transverse VL = ( ( + 2G) /  ) 0.5 VT = ( G /  ) 0.5 Dynamic Measurements: 3-D

  19. Theoretical Calculations : • First-principles • Solve Schrodinger Eq. • Atomistics • curve fit: • ao, E, phase, Tm, SFE Physicists Calculate from: U r

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