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Modeling Waves in Linear and Nonlinear Solids by First-Order Hyperbolic Differential Equations

Yang, Lixiang
http://rave.ohiolink.edu/etdc/view?acc_num=osu1303846979

Abstract Details

2011, Doctor of Philosophy, Ohio State University, Mechanical Engineering.
In this dissertation, a new theoretical and numerical approach to model linear and nonlinear stress waves in complex solids has been developed. The model equations are derived based on the conservation laws in the Eulerian frame in conjunction with the constitutive relations of several types of media, including anisotropic elastic solids, piezoelectric crystals, hypoelastic solids, nonlinear elastic solids, soft tissues modeled by viscoelasticity, and plastic deformation in metals. For waves in a thin rod, detailed formulations are provided, including a two-equation model, in which Hookean elasticity is assumed, and a three-equation model with formal hypo-elasticity relationship. For piezoelectric solids, the model equations include the equations of motion, a part of the Maxwell equations, and the constitutive relations for anisotropic and piezoelectric solids. For waves in soft tissues, the governing equations include the equation of motion, the viscoelastic constitutive relations, and the equations for internal variables. At the end of the dissertation, a hypo-plasticity relationship coupled with conservation of mass and momentum was developed to model wave propagation in plastic medium. For all these media, the model equations are composed of a set of first-order, linear or nonlinear, coupled hyperbolic partial differential equations (PDEs) with velocity and stress components as the unknowns. To understand the governing equations and to facilitate numerical solution, various forms of each model have been derived and reported in the dissertation, including the conservative form, the non-conservative form, the diagonal characteristic form if it can be derived. The eigen systems of the model equations are then analyzed by deriving the eigenvalues and eigenvectors of the Jacobian matrices of the first order PDEs. The Conservation Element and Solution Element (CESE) method is then used to solve these model equations for time-accurate solutions of propagating waves. For nonlinear elastic waves, I conducted simulations of wave propagating in a thin rod, including a sudden expansion wave, a compression wave, resonant waves, etc. In the linear elastic regime, numerical solution is directly compared to the classical analytical solution. For nonlinear waves, numerical solution shows salient features of nonlinearity, including the appearance of super harmonics of imposed oscillations, etc.
Sheng-Tao (John) Yu, Ph.D. (Advisor)
Stephen Bechtel, Ph.D. (Committee Member)
Marcelo Dapino, Ph.D. (Committee Member)
Daniel Mendelsohn, Ph.D. (Committee Member)
350 p.
Acoustics; Biomechanics; Biomedical Engineering; Energy; Engineering; Geophysical; Systems Science; Theoretical Mathematics; Theoretical Physics
Wave Propagation; Numerical Simulation, Hyperbolic System; hypoelastic, viscoelastic, hypoplastic, anisotropic

Recommended Citations

Citations

  • Yang, L. (2011). Modeling Waves in Linear and Nonlinear Solids by First-Order Hyperbolic Differential Equations [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1303846979

    APA Style (7th edition)

  • Yang, Lixiang. Modeling Waves in Linear and Nonlinear Solids by First-Order Hyperbolic Differential Equations. 2011. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1303846979.

    MLA Style (8th edition)

  • Yang, Lixiang. "Modeling Waves in Linear and Nonlinear Solids by First-Order Hyperbolic Differential Equations." Doctoral dissertation, Ohio State University, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=osu1303846979

    Chicago Manual of Style (17th edition)

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This open access ETD is published by The Ohio State University and OhioLINK.