ESM 5344
Wave Propagation in Solids

Through out this semester students are encouraged to contact Dr. R.D. Kriz via email at rkriz@vt.edu

Interactive Web based simulations of wave propagation topics disccused in class:
(NOTE: Simulations require a Java enabled Web browser with VRML1-2 plugins)

• Isotropic 1-D wave progation: Exact solution vs. Finite Element Model (FEM): Access Simulation
• Anisotropic 2-D wave propagation demonstration of energy flux deviation, dispersion, and dissipation.:
  • Small FEM mesh (30x60): Access Simulation: 10 elements modeled over one wave length
  • Larger FEM mesh (45X180): Access Simulation: 20 elements modeled over one wave length
• Anisotropic 3D wavesurfaces corresponding to C_ijkl up to triclinic symmetry: Access Simulation
  NOTE: Generates only VRML-1 files. Newer (3 years old) web browsers plugins may not work.

Topics discussed in class with information provide in html format:

• Simulated Wave Guides: Simulation-visualization of quide modes propagating in a graphite fiber composite where waves originating from a scanning acoustic microscope at different wave lengths (frequencies) are used to measure the gradients in elastic properties in the fiber/matrix interphase region:
  

• Simulated Energy Flux Deviation of Planar Waves: Simulation-visualization of plane waves propagating in a unidirectional graphite/epoxy composite as a function of fiber orientation / simulation results archived for discussion in class:
  

• Simulated Wave Propagation Dispersion: Simulation-visualization of dispersion of plane wave propagation in unidirectional graphite/epoxy for 4, 8, 12, 16, 20 and 24 elements modeled over one wave length. Simulation results archived for dicussion in class:
  

• Solutions to Christoffel's Equations for different crystal class symmetries: Wave surface representations ("Glyphs") of fourth order stiffness tensors: numerical solutions to Christoffel's equations. Web notes on introduction to elastic aniosotropy which includes the derivation of Christoffel's equation.

  3D Cijkl Wavesurfaces: Wave surface geometry defined as it represents the fourth order stiffenss tensor, C_ijkl. VRML files viewable with Cosmo Plugin ( download for Windows Web browsers) or with the Cortona Plugin ( ParallelGraphics Inc.: for Windows, Mac and MacOS-X). Students can also access details on how-to-create these wave surfaces. NOTE: links below access older VRML-1 and newer VRML-2 files. Future development of Cijkl glyphs using the X3D ISO standard file format will take about six months to develop. Let's hope the X3D ?standard? lasts longer than five years. Comments on these every changing ?standards? .

  • Isotropic (polycrystalline): Stainless Steel
  • Cubic Symmetry: Aluminum
  • Hexagonal Symmetry: Zinc
  • Transversly Isotropic: Unidirectional Graphite/Epoxy (Fiber Volume Fraction = 66%)
  • Tetragonal Symmetry: Tellurium Dioxide: Discussion on wave surface "cross-overs".
  • Orthorhombic Symmetry:
    • Type-1: Benzene (Most Common: Inner and Intermediate Wave Surfaces Connected)
      
    • Type-1: Barium-Yitrium-CopperOxide (Wave Surfaces Connected due to superconducting phase transformation)
      
    • Type-2: Calcium Formate (Rarest: All Wave Surfaces Connected Into a Single Surface)
      
    • Type-3: Hypothetical (Outer, Intermediate, and Inner Wave Surfaces Not Connected)
      
    • Type-4: Hypothetical (Outer and Intermediate Wave Surfaces Connected)
      
    Special thanks goes to Mr. Sanjiv Parhik and Mr. Robert Hunter
    who helped develop the underlying code used to generate these results.

• Fourth Order Stiffness Tensor Glyphs: Envisioning Fourth-Order Stiffness Tensors C_ijkl in a Continuum
  

End of Information for ESM5344

Engineering Science and Mechanics: ESM5344
Spring 2008

Any comments and suggestions about the material presented above,
please contact Ron Kriz at rkriz@vt.edu

bio
Virginia Tech
College of Engineering

http://www.sv.vt.edu/classes/ESM5344/ESM5344_kriz_NoteBook/ESM5344_kriz.html