Transcription of Strain E and Displacement u(x)
1 Physics for Solid State Applications Lecture 4: Vibrations in Solids February 11, 2004. Outline 1-D Elastic Continuum 1-D Lattice Waves 3-D Elastic Continuum 3-D Lattice Waves Strain E and Displacement u(x). dx dx'. u(x) u(x+dx) u(L). (dx) = dx' dx how much does a differential length change (dx) = u(x+dx) - u(x) difference in displacements Strain : 1. Uniform Strain & Linear Displacement u(x). dx dx'. u(x) u(x+dx) u(L). Linear Displacement : u(x) = E0 x Constant Strain : More Types of Strain dx Non-uniform dx' E = E(x). u(x) u(x+dx) u(L). Zero Strain : u(x) is constant Just a Translation We will ignore this 2. 1-D Elastic Continuum stress and Strain uniaxial loading Lo L. stress : Elongation: Normal Strain : If ux is uniform there is no Strain , just rigid body motion.
2 1-D Elastic Continuum METALS : Young's Modulus Tungsten (W) 406. Chromium (Cr) 289. Berylium (Be) 200 - 289. Nickel (Ni) 214. Iron (Fe) 196. Low Alloy Steels 200 - 207. Stainless Steels 190 - 200. Young's Modulus For Various Materials (GPa). Cast Irons 170 - 190. from Christina Ortiz Copper (Cu) 124. CERAMICS GLASSES AND SEMICONDUCTORS Titanium (Ti) 116. Diamond (C) 1000 Brasses and Bronzes 103 - 124. Tungsten Carbide (WC) 450 -650 Aluminum (Al) 69. Silicon Carbide (SiC) 450. Aluminum Oxide (Al2O3) 390 PINE WOOD (along grain): 10. Berylium Oxide (BeO) 380. Magnesium Oxide (MgO) 250 POLYMERS : Zirconium Oxide (ZrO) 160 - 241 Polyimides 3-5. Mullite (Al6Si2O13) 145 Polyesters 1-5. Silicon (Si) 107 Nylon 2-4. Silica glass (SiO2) 94 Polystryene 3 - Soda-lime glass (Na2O - SiO2) 69 Polyethylene Rubbers / Biological Tissues 3.
3 Dynamics of 1- 1-D Continuum 1-D Wave Equation Net force on incremental volume element: Dynamics of 1- 1-D Continuum 1-D Wave Equation Velocity of sound, c, is proportional to stiffness and inverse prop. to inertia 4. Dynamics of 1- 1-D Continuum 1-D Wave Equation Solutions Clamped Bar: Standing Waves Dynamics of 1- 1-D Continuum 1-D Wave Equation Solutions Periodic Boundary Conditions: Traveling Waves 5. 3-D Elastic Continuum Volume Dilatation Lo apply load L. Volume change is sum of all three normal strains 3-D Elastic Continuum Poisson's Ratio is Poisson's Ratio ratio of lateral Strain to axial Strain Poisson's ratio can not exceed , typically 6. 3-D Elastic Continuum Poisson's Ratio Example Aluminum: EY= GPa, = 5kN. 5kN 75mm 20mm 3-D Elastic Continuum Poisson's Ratio Example Aluminum: EY= GPa, = 5kN.
4 5kN 75mm 20mm 7. 3-D Elastic Continuum Poisson's Ratio Example Aluminum: EY= GPa, = 5kN. 5kN 75mm 20mm 3-D Elastic Continuum Shear Strain Shear loading Shear plus rotation Pure shear . 2 .. Pure shear Strain Shear stress G is shear modulus 8. 3-D Elastic Continuum stress and Strain Tensors For most general isotropic medium, Initially we had three elastic constants: EY, G, e Now reduced to only two: , . 3-D Elastic Continuum stress and Strain Tensors If we look at just the diagonal elements Inversion of stress / Strain relation: 9. 3-D Elastic Continuum Example of Uniaxial stress Lo L. Dynamics of 3- 3-D Continuum 3-D Wave Equation Net force on incremental volume element: Total force is the sum of the forces on all the surfaces 10. Dynamics of 3- 3-D Continuum 3-D Wave Equation Net force in the x-direction: Dynamics of 3- 3-D Continuum 3-D Wave Equation Finally, 3-D wave equation.
5 11. Dynamics of 3- 3-D Continuum Fourier Transform of 3- 3-D Wave Equation Anticipating plane wave solutions, we Fourier Transform the equation . Three coupled equations for Ux, Uy, and Uz . Dynamics of 3- 3-D Continuum Dynamical Matrix Express the system of equations as a matrix . Turns the problem into an eigenvalue problem for the polarizations of the modes (eigenvectors) and wavevectors q (eigenvalues) . 12. Dynamics of 3- 3-D Continuum Solutions to 3- 3-D Wave Equation Transverse polarization waves: Longitudinal polarization waves: Dynamics of 3- 3-D Continuum Summary 1. Dynamical Equation can be solved by inspection 2. There are 2 transverse and 1 longitudinal polarizations for each q 3. The dispersion relations are linear 4. The longitudinal sound velocity is always greater than the transverse sound velocity 13.
