Bojan B. Guzina

We investigate the boundary layers in periodic media, termed surface Bloch (SB) waves, affiliated with rational surface cuts and homogeneous Neumann data. We adopt a 2D scalar wave equation as a testbed and develop a unit cell-of-periodicity-based, reduced order model of the SB waves that is capable of describing their dispersion, waveforms, and “skin depth”. The centerpiece of our analysis is a quadratic eigenvalue problem for the effective unit cell of periodicity that seeks the complex wavenumber quantifying the evanescence away from the cut plane given (i) the excitation frequency, and (ii) wavenumber in the direction of the cut plane.

Much like their counterparts in homogeneous elastic solids, waves in periodic media can be broadly classified into Floquet–Bloch body waves and evanescent surface waves. Our goal is to elucidate the latter boundary layers, termed surface Bloch (SB) waves, affiliated with rational surface cuts and homogeneous Neumann data. To this end, we adopt a two-dimensional (2D) scalar-wave equation with periodic coefficients (describing antiplane shear waves in phononic crystals) as a test bed and develop a unit cell-of-periodicity-based, reduced-order model (ROM) of the SB waves that is capable of describing their dispersion, waveforms and ‘skin depth’. The centrepiece of our analysis is a quadratic eigenvalue problem (QEP) for the effective unit cell of periodicity—deriving from a geometric interplay between the mother Bravais lattice and orientation of the surface cut—that seeks the complex wavenumber quantifying the evanescence away from the cut plane given (i) the excitation frequency and (ii) wavenumber in the direction of the cut plane. In this way, the sought boundary layer is obtained by superposition of the evanescent QEP eigenstates, whose relative amplitudes are obtained by imposing the homogeneous boundary condition. With the QEP eigenspectrum at hand, evaluation of an SB wave—in terms of both dispersion characteristics and evanescent waveforms—entails only a low-dimensional eigenvalue problem. This feature caters for rapid exploration of the effect of (periodic) surface undulations, and so enables manipulation of SB waves via optimal design of the surface cut. Our analysis also includes an account for the power flow and skin depth of SB waves, both of which are critical for the energetic relevance of boundary layers.

In this work, we explore the possibility of reconstructing microseismic events in a solid body, in terms of both event locations and respective seismic moment tensors, using full-waveform observations of acoustic pressure fluctuations generated in the surrounding fluid. Our developments are motivated by the prospect of locating and characterizing, via hydrophone measurements, reaction-driven cracking in (coastal or seabed) geologic formations underpinning mineralization-driven, permanent carbon storage in the subsurface. On representing the microseismic events as dipole and double-couple point sources in the solid, the prototype (two-dimensional) analysis is reduced to an inverse source problem that deploys (i) time-harmonic forward map for the solid–fluid system; (ii) combinatorial search over a prescribed grid spanning the source region; (iii) global and local grid refinements catering for the stability and veracity of source reconstruction, and (iv) multi-frequency components of the acoustic pressure traces observed in the surrounding fluid. Numerical simulations demonstrate the feasibility of reconstructing multiple (contemporaneous) microseismic events, in terms of both their locations and respective seismic moment tensors, using a modest number of acoustic pressure observations in the fluid. The simulations highlight the importance of observation aperture on the quality of source reconstruction, an item that features two modalities: (a) spatial—manifested via the number and location of pointwise sensors (hydrophones), and (b) temporal—synthesized via the number and linear independence of multi-frequency observations. The two aperture modalities are found via numerical simulations to work in unison in that the scarcity of one can, to a degree, be compensated by the abundance of another. In a physical setting, (a) is controlled by the sensing setup, while (b) hinges on the spectral bandwidth of microseismic events.

Mineral carbon storage in rock formations has gained significant interest in recent years. In principle, changes in mechanical rock properties driven by carbon mineralization could be quantified using seismic methods, opening the door toward field monitoring of (the progress of) carbon storage. However, these changes may vary spatially within a rock mass when reactive transport occurs. In this vein, full-field ultrasonic characterization of reacted specimens can help shed light on the process. We use a 3D Scanning Laser Doppler Vibrometer to perform full-field monitoring of one-dimensional (1D) ultrasonic waves in rod-shaped sandstone specimens exposed to NaCl-rich fluid. Our initial experiments were conducted on intact sandstone specimens with high aspect ratio (length/diameter $\simeq 15$) to cater for 1D axial wave propagation. To investigate the evolution of the Young's modulus and attenuation of rock due to reactive transport, we exposed the specimens to an under-saturated NaCl solution, achieving supersaturation -- and so mineralization -- through evaporation. The upward movement of the fluid, supplied at the bottom of each specimen, was achieved through capillary action. We deploy an elastography-type approach to back-analysis, known as modified-error-in-constitutive-relation (MECR) approach, to expose the spatially-heterogeneous evolution of mechanical rock properties due to reactive transport. Our results consistently demonstrate (i) degradation of the Young's modulus, and (ii) increase in attenuation due to mineralization. To better understand the root causes of these changes, we made use of the X-ray micro-computed tomography ($μ$CT) and scanning electron microscopy (SEM) of selected cross-sections. The grain-scale information suggests that microcracking (resp. pore filling) is predominantly (resp. partly) responsible for the observed macroscopic changes.

Motivated by the recent experimental and analytical developments enabling high-fidelity material characterization of (heterogeneous) sheet-like solid specimens, we seek to elucidate the constitutive behavior of linear viscoelastic solids under the plane stress condition. More specifically, our goal is to expose the relationship between the plane-stress viscoelastic constitutive parameters and their (native) “bulk” counterparts. To facilitate the sought reduction of the three-dimensional (3D) constitutive behavior, we deploy the concept of projection operators and focus on the frequency-domain behavior by resorting to the Fourier transform and the mathematical framework of tempered distributions, which extends the Fourier analysis to functions (common in linear viscoelasticity) for which Fourier integrals are not convergent. In the analysis, our primary focus is the on class of linear viscoelastic solids whose 3D rheological behavior is described by a set of constant-coefficient ordinary differential equations, each corresponding to a generic arrangement of “springs” and “dashpots”. On reducing the general formulation to the isotropic case, we proceed with an in-depth investigation of viscoelastic solids whose bulk and shear modulus each derive from a suite of classical “spring and dashpot” configurations. To enable faithful reconstruction of the 3D constitutive parameters of natural and engineered solids via (i) thin-sheet testing and (ii) applications of the error-in-constitutive-relation approach to the inversion of (kinematic) sensory data, we also examine the reduction of thermodynamic potentials describing linear viscoelasticity under the plane stress condition. The analysis is complemented by a set of analytical and numerical examples, illustrating the effect on the plane stress condition on the behavior of isotropic and anisotropic viscoelastic solids.

Motivated by the recent experimental and analytical developments enabling high-fidelity material characterization of (heterogeneous) sheet-like solid specimens, we seek to elucidate the constitutive behavior of linear viscoelastic solids under the plane stress condition. More specifically, our goal is to expose the relationship between the plane-stress viscoelastic constitutive parameters and their (native) ``bulk'' counterparts. To facilitate the sought reduction of the three-dimensional (3D) constitutive behavior, we deploy the concept of projection operators and focus on the frequency-domain behavior by resorting to the Fourier transform and the mathematical framework of tempered distributions, which extends the Fourier analysis to functions (common in linear viscoelasticity) for which Fourier integrals are not convergent. In the analysis, our primary focus is the on class of linear viscoelastic solids whose 3D rheological behavior is described by a set of constant-coefficient ordinary differential equations, each corresponding to a generic arrangement of ``springs'' and ``dashpots''. On reducing the general formulation to the isotropic case, we proceed with an in-depth investigation of viscoelastic solids whose bulk and shear modulus each derive from a suite of classical ``spring and dashpot'' configurations. To enable faithful reconstruction of the 3D constitutive parameters of natural and engineered solids via (i) thin-sheet testing and (ii) applications of the error-in-constitutive-relation approach to the inversion of (kinematic) sensory data, we also examine the reduction of thermodynamic potentials describing linear viscoelasticity under the plane stress condition. The analysis is complemented by a set of analytical and numerical examples, illustrating the effect on the plane stress condition on the behavior of isotropic and anisotropic viscoelastic solids.