Acoustoelastic Wave Research Articles

3D acoustoelastic FD modeling of elastic wave propagation in prestressed solid media

AbstractSeismic exploration of deep oil/gas reservoirs involves the propagation of seismic waves in high-pressure media. Traditional elastic wave equations are not suitable for describing such media. The theory of acoustoelasticity establishes the dynamic equation of wave propagating in prestressed media through constitutive relation using third-order elastic constants. Many studies have been carried out on numerical simulations for acoustoelastic waves, but are mainly limited to 2D cases. A standard staggered-grid (SSG) finite-difference (FD) approach and the perfectly matched layer (PML) absorbing boundary are combined to solve 3D first-order velocity-stress equations of acoustoelasticity to simulate wave propagating in 3D prestressed solid medium. Our numerical results are partially validated by plane-wave analytical solution through the comparison of calculated and theoretical P-/S-wave velocities as a function of confining prestress. We perform numerical simulations of acoustoelastic waves under confining, uniaxial and pure-shear prestressed conditions. The results show the stress-induced velocity anisotropy in acoustoelastic media, which is closely related to the direction of prestresses. Comparisons to seismic simulations based on the theory of elasticity illustrate the limitation of conventional elastic simulations for prestressed media. Numerical simulations prove the significant effect of prestressed conditions on seismic responses.

Reconfigurable coupled-resonator acoustoelastic waveguides in fluid-filled phononic metaplates

We study numerically and experimentally acoustoelastic wave propagation in a two-dimensional phononic metaplate consisting of a periodic array of cups sitting on a thin epoxy plate that is perforated with cross holes. When all cups are filled with water, the metaplate possesses a complete band gap. Reconfigurable coupled-resonator acoustoelastic waveguides (CRAEWs) are created by locally emptying certain cups, thus introducing local resonances that are evanescently coupled. Straight and 90° bent periodic waveguides are considered, together with an aperiodic chain of 11 coupled resonators. The aperiodic chain has no definite spatial periodicity but supports collective resonances resulting from the coupling of nearest resonators. Lamb waves are experimentally excited by a piezoelectric patch and received by a scanning optical vibrometer. Experimental results for acoustoelastic wave propagation along both periodic and aperiodic CRAEWs are compared to a three-dimensional finite element model taking fluid–structure interaction into account. The propagation of confined acoustoelastic waves in the 90° bent waveguides and the collective resonances of the aperiodic chain of defected resonators are observed experimentally. Reconfigurability are realized based on the coupling of acoustoelastic waves in a phononic metaplate. Our results show plenty of potential possibilities for the practical design of reconfigurable and programmable elastic wave devices.

Acoustoelastic wave propagation in initially stressed solids using a pure stress formulation

Wave propagation in homogeneous elastic solids is governed by Cauchy's first law of motion, which relates the acceleration of material points to the divergence of stress. For Hookean solids, the stress is typically written in terms of displacements to form an equation of motion with displacement as the dependent variable, i.e., Navier's equation of elastodynamics. Previously, the authors considered wave propagation in anisotropic elastic solids by formulating the equation of motion using only stress as the dependent variable. Eigenvalue solutions provided the (stress) wave phase velocity in terms of elastic compliance constants. In this presentation, the stress formalism is extended and applied to the nonlinear problem of stress wave propagation in an initially stressed material (also known as acoustoelasticity). The stress formulation is viewed as a more natural fit over historical derivations, which were based on wave displacements superimposed on an initially strained material state. The stress formulation of elastodynamics is firstly reviewed. Then, the extension to stress wave propagation in an initially stressed material is given. The phase velocity of the stress wave is related to the initial stress through second- and third-order compliance constants (instead of elastic stiffnesses in traditional acoustoelasticity).

Reconfigurable waveguides defined by selective fluid filling in two-dimensional phononic metaplates

We investigate two-dimensional phononic metaplates consisting of a periodic array of cups on a thin epoxy plate that is perforated with periodic cross holes. The cups are individually filled with water or remain empty, in view of creating reconfigurable phononic waveguides. Phononic band gaps exist for empty or filled epoxy cups, leading to waveguides defined with either positive or negative contrast. Straight and 90° bent waveguides are considered experimentally. Lamb waves are excited by a piezoelectric patch glued onto the metaplate and are imaged using a scanning laser vibrometer. Experimental results are compared to a three-dimensional finite element model of fluid–structure interaction. Passing and forbidden frequency ranges are identified for positive and negative contrast, and confined propagation is observed along the waveguides. Significantly, the propagation of acoustoelastic waves in the 90° bent waveguides is observed experimentally. Reconfigurability and reusability are thus realized based on the coupling of elastic waves in the solid and acoustic waves in the fluid. The results show plenty of potentiality for the practical design of multiplexed and programmable acoustic devices implemented with reconfigurable waveguides printed on demand in a phononic metaplate.

Dynamic Fracture of Expanding Cavities in Nonlinear Soft Solids

Abstract Recent experimental observation [Milner, M. P., and Hutchens, S. B., 2021, “Multi-Crack Formation in Soft Solids During High Rate Cavity Expansion,” Mech. Mater., 154, p. 103741] suggests that crack formation during rapid cavity expansion in low modulus, highly deformable solids depends on the ratio of the rate of expansion and the acoustoelastic wave speed, similar to observations in rock and metal [Grady, D., and Kipp, M., 1987, “Dynamic Rock Fragmentation,” Fracture Mechanics of Rock, Elsevier, p. 429475]. Here, we explore the effect of material nonlinearity on predictions of the number of cracks formed at the cavity surface. We find that nonlinearity influences crack formation only when the cavity size normalized elasto-fracture length is greater than one and the cavity’s rate of expansion is greater than the acoustoelastic wave speed. The sensitivity of these predictions for two idealized fracture geometries, either a spherical damaged zone or discrete cracks, suggests a direction for further experimentation that may illuminate crack formation mechanisms in soft solids under dynamic loading.

Investigation of thermo-acoustoelastic guided waves by semi-analytical finite element method

Guided waves are sensitive to variations in propagation environments. Many recent studies have focused on the uniform thermal effect on Lamb waves. However, there is little research on the thermal effect in a more complex situation, such as a nonuniform thermal effect and wave propagation in an arbitrary cross-section. In this study, a thermo-acoustoelastic theory combined with the semi-analytical finite element (TAE-SAFE) method is proposed to investigate both uniform and nonuniform thermal effects on acoustoelastic guided wave propagation. In the TAE-SAFE method, effective thermo-acoustoelastic constants including third-order elastic constants are employed. Then, an acoustoelastic wave equation of the thermal effect is formulated by Hamilton’s principle and computed by the semi-analytical finite element (SAFE) method. The phase velocity, group velocity, velocity thermal sensitivity, and displacement mode shape shift can be extracted by the proposed method. To validate this method, numerical results of Lamb waves in an aluminum plate subjected to a uniform thermal effect are compared with the results of a previous theoretical analysis. The results show computational veracity and validity. Two typical cases are investigated: (1) an isotropic aluminum plate under a linear temperature gradient condition; (2) a uniform temperature case in a rail track with a constant irregular cross-section. This study provides an effective numerical method to analyze thermo-acoustoelastic guided wave propagation.

Plane residual stress measurement using ultrasonic considering weak anisotropy of metallic material

Ultrasonic velocity variation (UVV) along different directions induced by material weak anisotropy will lead to residual stress measurement error using ultrasonic inevitably. In this research, the influence of weak anisotropy on residual stress measurement is analyzed theoretically and experimentally. Firstly, the in-plane acoustoelastic wave equation is modified by considering material anisotropy, which can be characterized by two in-plane anisotropic parameters. Further, an analysis model is also established to express the linear superposition of weak anisotropy and acoustoelastic effect on UVV. Based on this, a plane residual stress measurement method has been developed, which was verified using the rolling aluminum alloy plate after friction stir welding. From the comparison with hole-drilling method, it was adequate and capable to measure the plane residual stress by the relative UVV along three particular directions.

Stiffness matrices for fluid and anisotropic soil layers with applications in soil dynamics

In this study, we introduce and discuss features and improvements of the well-established stiffness matrix method that is used in simulation of wave propagation in layered media. More specifically, we present stiffness matrices for an acoustic layer and a vertically transverse isotropic (VTI) viscoelastic soil layer. Combining these stiffness matrices enables a straightforward technique for modeling of acousto-elastic wave propagation in layered infinite media. In addition, we propose a technique to simulate discontinuity seismic sources, which was not used earlier in the context of the stiffness matrix method. Finally, we propose a framework to derive a key parameter of the absorbing boundary domain technique Perfectly Matched Layer (PML). Numerical examples are presented in order to help understanding the features and improvements discussed in the study from the fields of geophysics and soil dynamics. It is believed that the features and improvements discussed herein will make the application of the stiffness matrix method even wider and more flexible.

Analysis and experimental verification of multiple scattering of acoustoelastic waves in thin plates for enhanced energy harvesting

Acoustoelastic wave energy harvesting in thin plates and other structures has recently gained attention from the energy harvesting research community. Metamaterial-inspired concepts for enhancing wave power generation have been investigated, including metamaterial funnels, mirrors, and defect-based resonators. In support of such concepts, this paper presents an electromechanically coupled, multiple scattering formulation for accurately modeling, exploring, and optimizing metamaterial-based harvesting systems incorporating scatterers (e.g., cylindrical inclusions and voids). Following development, the formulation is applied to determining optimal arrangements of scatterers, nominally in a semi-elliptical path, which maximize electrical power harvested, This is done, in part, by diminishing side lobes resulting from ellipse truncation. Optimization results exhibit minimized side lobes and harvester power nearly ten times that of the non-optimized case. Finally, an experimental study is presented which confirms many of the model predictions.

A discontinuous Galerkin method with a modified penalty flux for the propagation and scattering of acousto-elastic waves

We develop an approach for simulating acousto-elastic wave phenomena, including scattering from fluid-solid boundaries, where the solid is allowed to be anisotropic, with the Discontinuous Galerkin method. We use a coupled first-order elastic strain-velocity, acoustic velocity-pressure formulation, and append penalty terms based on interior boundary continuity conditions to the numerical (central) flux so that the consistency condition holds for the discretized Discontinuous Galerkin weak formulation. We incorporate the fluid-solid boundaries through these penalty terms and obtain a stable algorithm. Our approach avoids the diagonalization into polarized wave constituents such as in the approach based on solving elementwise Riemann problems.

In situ estimation of applied biaxial loads with Lamb waves

Spatially distributed arrays of piezoelectric disks are being applied to monitor structural integrity using Lamb waves. Applied loads directly affect waves propagating between array elements because of dimensional changes and the acoustoelastic effect. Resulting changes in phase velocity depend upon the propagation direction as well as the Lamb wave mode and frequency. This paper shows from numerical solutions of the acoustoelastic wave equation for an isotropic plate that it is possible to decouple the effects of a homogeneous biaxial stress into its two principal components. As a consequence of both this decoupling and material isotropy, the acoustoelastic response of a specific mode and frequency is described by only two constants, which can be determined from a uniaxial loading experiment. Using this formulation, a method is developed and verified via simulations to estimate an arbitrary biaxial load from phase velocity changes measured along multiple directions of propagation. Results from uniaxial loading experiments on two different plates further demonstrate the efficacy of the method. It is also shown that opening fatigue cracks may significantly degrade results by interfering with Lamb wave direct arrivals, but that this degradation can be mitigated by using a reduced set of data from unaffected paths of propagation.

Formulation and selected applications of a regularized elastodynamics integral equation approach to large scale scattering problems involving inhomogeneous objects

Elements are presented of the formulation and selected applications of the combined surface and volume integral-equation approach for finding pressure, displacement, and traction fields in composite objects consisting of piece-wise homogeneous regions characterized by different Lame's material parameters, as well as of highly inhomogeneous regions. The proposed elastodynamics surface/volume-based integral equations, in which the unknowns are volumetric pressure or displacement variables and surface displacement and traction fields on the interfaces, involve only weakly singular integral-equation kernels with at most 1/r singularity. The proposed method is found to provide well-convergent and accurate solutions for a large spectrum of problems, in particular those involving high-contrast material interfaces. The proposed method's accuracy and its capability (achieved through the (non-lossy FFT-based) Adaptive Integral Method matrix compression) of handling large-scale problems involving several millions of unknowns, is demonstrated through comparison with analytically constructed solution for a layered material sphere and in selected numerical simulations of propagation of acousto-elastic waves in human head and inner-ear region through air- and bone-conduction pathways. The proposed approach can be used in assessing effectiveness of noise-protection devices. This work is supported by a grant from the Air Force Office of Scientific Research.

Time-harmonic solution for acousto-elastic interaction with controllability and spectral elements

The classical way of solving the time-harmonic linear acousto-elastic wave problem is to discretize the equations with finite elements or finite differences. This approach leads to large-scale indefinite complex-valued linear systems. For these kinds of systems, it is difficult to construct efficient iterative solution methods. That is why we use an alternative approach and solve the time-harmonic problem by controlling the solution of the corresponding time dependent wave equation. In this paper, we use an unsymmetric formulation, where fluid-structure interaction is modeled as a coupling between pressure and displacement. The coupled problem is discretized in space domain with spectral elements and in time domain with central finite differences. After discretization, exact controllability problem is reformulated as a least-squares problem, which is solved by the conjugate gradient method.