Lixiang Yang

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 (i) anisotropic elastic solids, (ii) piezoelectric crystals, (iii) hypoelastic solids, (iv) nonlinear elastic solids, (v) soft tissues modeled by viscoelasticity, and (vi) plastic deformation in metals. For waves in a thin rod, detailed formulations are provided, including a twoequation 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

Gent's model is explained by using non-Gaussian behavior of an individual freely-joint chain. The physical meanings of two parameters in Gent's model are shown to be a function of temperature and the number of bonds. Under certain assumptions, the Gent's model and Arruda–Boyce model will become identical.

This paper reports a theoretical framework to analyze wave propagation in elastic solids of hexagonal symmetry. The governing equations include the equations of motions and partial differentiation of elastic constitutive relations with respect to time. The result is a set of nine, first-order, fully-coupled, hyperbolic partial differential equations with velocities and stress components as the unknowns. The equation set is

This paper reports the application of the space-time conservation element and solution element (CESE) method to the numerical solution of nonlinear waves in elastic solids. The governing equations consist of a pair of coupled first-order nonlinear hyperbolic partial differential equations, formulated in the Eulerian frame. We report their derivations and present conservative, nonconservative, and diagonal forms. The conservative form is solved numerically by the CESE method; the other forms are used to study the eigenstructure of the hyperbolic system (which reveals the underlying wave physics) and deduce the Riemann invariants. The proposed theoretical/numerical approach is demonstrated by directly solving two benchmark elastic wave problems: one involving linear propagating extensional waves, the other involving nonlinear resonant standing waves. For the extensional wave problem, the CESE method accurately captures the sharp propagating wavefront without excessive numerical diffus...

This paper reports an extension of the space-time conservation element and solution element (CESE) method to simulate stress waves in elastic solids of hexagonal symmetry. The governing equations include the equation of motion and the constitutive equation of elasticity. With velocity and stress components as the unknowns, the governing equations are a set of 9, first-order, hyperbolic partial differential equations. To assess numerical accuracy of the results, the characteristic form of the equations is derived. Moreover, without using the assumed plane wave solution, the one-dimensional equations are shown to be equivalent to the Christoffel equations. The CESE method is employed to solve an integral form of the governing equations. Space-time flux conservation over conservation elements (CEs) is imposed. The integration is aided by the prescribed discretization of the unknowns in each solution element (SE), which in general does not coincide with a CE. To demonstrate this approac...

ABSTRACT This paper reports the construction of viscoelasticity models for simulation of wave propagation in soft tissues. Aided by the Carson transform, Fung’s model and Iatridis’s model are transformed into the frequency domain with the imaginary part of the transformed relaxation function shown to be related to wave absorption effect. Based on measured wave absorption coefficients, the viscoelasticity models in the time domain are determined. Results show that Fung’s model does not support the frequency-dependent damping effect, but Iatridis’ model does. To validate the approach, numerical simulation of propagating wavelets in a skeletal pig muscle is performed. The constructed relaxation function is discretized by parallelly connected Standard Linear Solid (SLS) models. The hereditary integration is then transformed into a set of partial differential equations by introducing the internal variables. Together, the governing equations include the equation of motion, the constitutive equation, and the equations of internal variables. Velocity, stress, and internal variables are the primary unknowns. The model equations are solved by using the space–time CESE method. The calculated wave absorption effect compares well with the measured data.

This paper reports the eigenstructure of a set of first-order hyperbolic partial differential equations for modeling waves in solids with a trigonal 32 symmetry. The governing equations include the equation of motion and partial differentiation of the elastic constitutive relation with respect to time. The result is a set of nine, first-order, fully coupled, hyperbolic partial differential equations with velocity and stress components as the unknowns. Shown in the vector form, the model equations have three 9×9 coefficient matrices. The wave physics are fully described by the eigenvalues and eigenvectors of these matrices; i.e., the nontrivial eigenvalues are the wave speeds, and a part of the corresponding left eigenvectors represents wave polarization. For a wave moving in a certain direction, three wave speeds can be identified by calculating the eigenvalues of the coefficient matrix in a rotated coordinate system. In this process, without using the plane-wave solution, we recover the Christoffel matrix and thus validate the formulation. To demonstrate this approach, two- and three-dimensional slowness profiles of quartz are calculated. Wave polarization vectors for wave propagation in several compression directions as well as noncompression directions are discussed.