ESM 5344
Wave Propagation in Solids

Final Exam: 1:05-3:05PM, Wednesday May 11, 2011

Homework Assignments Posted

Class information on weighting, tests, homework, project, lecture outline, references

Interactive Web-based simulations on topics discussed in class:

Simulations require a Java enabled browser with VRML-1 / VRML-2 plugins.
Information on plugins and your system configuraiton is produced by NIST.
(NOTE: RED links: simulations setup and submitted to the SV-server which may be down.
            GREEN links: Simulations setup on ESM-server and submitted to SV-server.

• Isotropic 1D wave propagation: Exact solution vs. Finite Element Model (FEM).
  
  • Very small FEM mesh (4x40): Simulation: SV-submit / ESM-setup, submit to SV
    

• Anisotropic 2D wave propagation: energy flux deviation, dispersion, and dissipation:
  • Small FEM mesh (30x60): Simulation: SV-submit / ESM-setup, submit to SV
    : 10 elements modeled over one wave length
  • Larger FEM mesh (45x180): Simulation: SV-submit / ESM-setup, submit to SV
    : 20 elements modeled over one wave length

• Anisotropic 3D wavesurfaces used to envision C_ijkl up to triclinic symmetry.
  Christoffel's equation: Numerical*: SV-submit / ESM-setup only, no submit
  * eigenvalues (wave veloctiy / eigenvector (vibration direction color)
  NOTE: Only VRML-1 files are created of these colored wave surface shapes.
  Work in progress: Same wave surfaces created as VRML-2 and X3D files.

Topics discussed in class based on results using simulations above:

• Simulated Energy Flux Deviation of Planar Waves: Simulation-visualization
  summary of plane waves propagating in a unidirectional graphite/epoxy composite
  as a function of fiber orientation.
  
  Simulation results for small mesh (30x60) archived in the directory:
  
  README file, which will be discussed in class, where waves are launched
  as:  1) Pure L or T,  or  2) Preferential (Pref.) QT or QL modes over a range
  of fiber orientations, from 0 to 90 degrees (0->90) with respect to the plane
  wave normal. For comparison results are combined and presented as:
  • images (0->90): 1) Pure L & T,   2) Pref. QT & QL, and their
  • movies (0->90): 1) Pure L & T,   2) Pref. QT & QL
    
  Suggestion: download high resolution movies and adjust to fit your screen.
  
  Simulation results for large mesh (45x180) archived in the directory:
  
  README file, will be discussed in class. Waves are launched either as pure
  or preferential (Pref.) modes:  1) Pure L,   2) Pure T,   3) Pref. QT, or 4) Pref. QL
  over a range of fiber orientations, from 0 to 90 degrees (0->90), with respect to
  the plane wave normal. For comparison results are combined and presented as:
  • images (0->90): 1) Pure L,   2) Pure T,   3) Pref. QL,   4) Pref. QT,
  • movies (0->90): 1) Pure L,   2) Pure T,   3) Pref. QL,   4) Pref. QT
  Suggestion: download high resolution movies and adjust to fit your screen.
  

• Simulated Wave Propagation Disperson: Simulation-visualization of dispersion in
  graphite/epoxy modeled using a (45x180) FEM mesh where 4, 8, 12, 16, 20 and 24
  elements are simulated over one wave length equivalent to the transducer diameter.
  
  Simulation results archived in the directory:
  README file, description on how to setup dispersion simulation
  which will be discussed in class. Dispersion is measured as a decrease
  in wave speed caused by the coarseness of the FEM mesh and simulated
  by modeling various elements per wavelength for a pure transverse wave
  orientated: at:   1) 0 degrees   and   2) 90 degrees   fiber orientations with
  respect to the plane wave normal. For comparison results are combined and
  presented as:
  • images at various wave lengths:   1) 0 degrees,   2) 90 degrees),
  • movies at various wave lengths:   1) 0 degrees,   2) 90 degrees.
  Suggestion: download high resolution movies and adjust to fit your screen.
  

• Solutions to Christoffel's Equations for different crystal class symmetries:
  

  Envisioning fourth order stiffness tensors, C_ijkl, as complex 3D wave surface glyphs can be understood in context of the derivation of Christoffel's equation, see web notes: Introduction on Elastic Anisotropy. The VRML-1 amd VRML-2 file formats used here enables the viewer to interactive with and interpret complex 3D wave surface geometries as a fourth order tensor glyphs. The FreeWRL application is open-source with web-plugins that supports VRML-2 and X3D file formats and works on current OS: Linux, Mac OSX, and Windows. Older versions of FreeWRL before 2009 support VRML-1. Future development of X3D glyphs will require a six month commitment of time.

  To foster future content development, as 3D file formats change, the reader can access code either at the SV-server or the ESM-server together with a description on how-to-create 3D wave surface C_ijkl-glyphs for different crystal class symmetries.

  Envisioning Crystal Classes as 3D C_ijkl-glyphs: Geometry defined for C_ijkl-glyph.

  Isotropic (polycrystalline): Stainless Steel
  Cubic Symmetry: Aluminum
  Hexagonal Symmetry: Zinc
  Transversly Isotropic: Unidirectional Graphite/Epoxy (Fiber Volume Fraction 66%)
  Tetragonal Symmetry: Tellurium Dioxide: Surface "Cross-Overs" defined.
  Orthorhombic Symmetry:

  • Type-1: Benzene (Most Common: Inner and Intermediate Surfaces Connected)
    
  • Type-1: Barium-Yitrium-CopperOxide (Phase Transformation Connect Surfaces)
    
  • Type-2: Calcium Formate (Rarest: All Surfaces Connected Into a Single Surface)
    
  • Type-3: Hypothetical (Outer, Intermediate, and Inner Surfaces Not Connected)
    
  • Type-4: Hypothetical (Outer and Intermediate Surfaces Connected)
    
  Special thanks goes to Mr. Sanjiv Parhik and Mr. Robert Hunter
  who developed the first generation code used to create these glyphs.

  Below are web links related to envisioning tensors and their equations:
        Envisioning Zeroth Order Tensors: 3D Scalar Gradients and Function Extraction.
        Envisioning Second and Fourth Order Tensor Glyphs: Eigen-value/vector Glyphs.
        Envisioning tensors and their invariant equations: General discussion.

• Stress induced anisotropy modelled using sixth order tensors: Content first
  -- envisioning sixth order tensors as C_ijklmn-Glyphs pending.

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

Engineering Science and Mechanics: ESM5344

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

Virginia Tech
College of Engineering

http://www.esm.vt.edu/~rkriz/classes/ESM5344/ESM5344_NoteBook/ESM5344_kriz.html
Original: http://www.sv.vt.edu/classes/ESM5344/ESM5344_kriz_NoteBook/ESM5344_kriz.html

Revised January 2011