Nico F. Declercq

Higher-order material constants govern nonlinear elasticity, bias-field electromechanics, acoustoelasticity, and a wide range of modern applications in nonlinear acoustics, piezoelectric transduction, and engineered crystalline media. Yet even for well-studied crystals, the symmetry reduction of higher-rank tensors remains scattered across the literature, derived incompletely, or presented in incompatible conventions. This book provides a unified, invariance-based framework for deriving symmetry relations of material tensors in crystalline media. Starting from exact tensor transformation laws and point-group generators, all admissible tensor components and their linear relations are obtained systematically—without heuristic assumptions or undocumented shortcuts. The method applies uniformly across tensor ranks and physical couplings, yielding canonical reduced forms that are transparent, reproducible, and internally consistent. A fully worked trigonal benchmark, grounded in the author’s earlier Annalen der Physik study of biased piezoelectric crystals, validates the approach. The treatment then extends exhaustively across all crystallographic point groups and culminates in consolidated summary tables for rapid reference. This volume provides a rigorous, generator-based invariance procedure applicable at any tensor rank, explicit symmetry reductions for elastic, piezoelectric, dielectric, electrostrictive, and higher-order coupled tensors, complete coverage of all crystallographic point groups and the isotropic limit, carefully curated summary tables of independent constants by symmetry class, and a consistent Voigt and generalized-Voigt convention designed to eliminate sign and indexing ambiguities. Intended as both a reference and a methodological guide, this book offers a dependable foundation for researchers and engineers working with higher-order constitutive models in crystalline and multiphysics media.

Higher-order material constants govern nonlinear elasticity, bias-field electromechanics, acoustoelasticity, and a wide range of modern applications in nonlinear acoustics, piezoelectric transduction, and engineered crystalline media. Yet even for well-studied crystals, the symmetry reduction of higher-rank tensors remains scattered across the literature, derived incompletely, or presented in incompatible conventions. This book provides a unified, invariance-based framework for deriving symmetry relations of material tensors in crystalline media. Starting from exact tensor transformation laws and point-group generators, all admissible tensor components and their linear relations are obtained systematically—without heuristic assumptions or undocumented shortcuts. The method applies uniformly across tensor ranks and physical couplings, yielding canonical reduced forms that are transparent, reproducible, and internally consistent. A fully worked trigonal benchmark, grounded in the author’s earlier Annalen der Physik study of biased piezoelectric crystals, validates the approach. The treatment then extends exhaustively across all crystallographic point groups and culminates in consolidated summary tables for rapid reference. This volume provides a rigorous, generator-based invariance procedure applicable at any tensor rank, explicit symmetry reductions for elastic, piezoelectric, dielectric, electrostrictive, and higher-order coupled tensors, complete coverage of all crystallographic point groups and the isotropic limit, carefully curated summary tables of independent constants by symmetry class, and a consistent Voigt and generalized-Voigt convention designed to eliminate sign and indexing ambiguities. Intended as both a reference and a methodological guide, this book offers a dependable foundation for researchers and engineers working with higher-order constitutive models in crystalline and multiphysics media.

Conventional nuclear models, from liquid-drop parametrizations to shell and density-functional theories, describe many global trends in binding and decay but remain probabilistic and heavily parameterized. They do not explain, from first principles, persistent anomalies such as deep sub-barrier fusion hindrance, odd-even staggering in fission yields, or the long half-life of 14C. This work extends Trembling Spacetime Relativity Theory (TSRT) into the nuclear domain as a deterministic geometric framework for structure, stability, and transformation. In TSRT, each nucleus is a localized trembling-curvature eigenmode whose stability results from sustained suppression of intrinsic spacetime curvature. Decay is a causal relaxation of this suppression and is modeled as curvature reconfiguration rather than a stochastic transition. Binding, fission, fusion, and decay thus emerge from the same geometric dynamics. Using a single global normalization fixed once on 60Co, the TSRT formulation reproduces experimental half-lives across beta decay, alpha decay, and spontaneous fission over more than twenty orders of magnitude, with a mean logarithmic deviation below one part in a million. No shell closures, pairing corrections, empirical preformation factors, or per-nucleus tuning are invoked. The 14C anomaly follows from its geometric beta-path curvature without adjustable hindrance. All Q-values, barrier actions, and emission rates follow deterministically from the spacetime metric and its curvature energy, and TSRT provides a causal mechanism for mass–energy conversion as curvature redistribution. TSRT also reproduces the emergence of neutron magic numbers as geometric minima of the curvature–stiffness map, matching the conventional magic sequence without quantum postulates. Deep sub-barrier fusion hindrance arises from geometric suppression of trembling-mode overlap; odd-even staggering in fission yields originates from phase-locked neck eigenmodes at scission; and electron-screening shifts reflect renormalization of near-field electromagnetic curvature. These effects are consistent with the trembling-spacetime geometry that also underlies atomic structure, photon emission, and gravitational redshift. Quantitatively, TSRT reproduces absolute nuclear half-lives from milliseconds up to about ten quintillion years, with mean logarithmic deviation below one millionth, using only three global constants fixed once per mode. This accuracy surpasses conventional microscopic and empirical models—which typically yield mean logarithmic deviations between one tenth and one hundredth—by roughly ten thousand times, showing that nuclear decay is a deterministic geometric phenomenon rather than a stochastic process. All physical quantities are expressed in absolute SI and nuclear units, with TSRT calibration constants tied to fundamental curvature and lifetime anchors without empirical scaling. TSRT also reproduces absolute fission energetics, yielding a total energy release of 170.0 MeV for thermal-neutron-induced U-235 (n,f) without any per-observable tuning. By deriving nuclear binding, decay, reaction timescales, and mass–energy conversion from a single geometric principle, TSRT establishes a unified, predictive, and reproducible description of nuclear stability and transformation. It bridges microscopic nuclear structure with macroscopic relativistic consistency, showing that mass–energy balance and decay kinetics are natural manifestations of curvature redistribution in trembling spacetime.

This paper demonstrates that all known fundamental particles arise as geometric eigenmodes of trembling spacetime: localized, causally stable oscillations of the spacetime metric itself. Within the framework of Trembling Spacetime Relativity Theory (TSRT), particle identity, mass, spin, and charge are not introduced as independent assumptions but emerge deterministically as intrinsic properties of symmetry-bound deviations in the underlying geometry. Building on earlier foundational work, TSRT identifies exactly four causally admissible symmetry classes—U(1), SU(2), SU(3), and gravitational curvature—each corresponding uniquely to one of the known fundamental interactions. This classification is both highly restrictive and, within the constraints analyzed, empirically complete: it recovers the Standard Model gauge groups and predicts, to the extent that no additional causally admissible trembling symmetry classes have been identified, the absence of further fundamental forces or generations. In contrast to quantum field theory, which permits arbitrary field extensions, TSRT excludes superpartners, extra families, or exotic gauge sectors unless they can be realized as causally coherent metric deviations—an outcome shown to be incompatible with the theory’s variational structure under the current formulation. Charge quantization, spin-statistics correspondence, confinement of color charge, and the discreteness of particle families all emerge from the causal coherence and topological closure of trembling geodesics, without recourse to field quantization or probabilistic interpretation. Beyond classification, TSRT provides a geometric foundation for principles traditionally regarded as axiomatic. The derivation of rest energy and inertial mass follows directly from proper-time action accumulation, yielding the Einstein relation between energy and mass as a deterministic outcome of geodesic resistance to curvature. Interactions are formulated as curvature-mediated coupling among trembling modes, replacing virtual particle exchanges with deterministic geometric correlations. The Higgs boson is reinterpreted not as a quantized scalar field excitation but as a metastable curvature-bound resonance decaying through geometric fragmentation. Finally, TSRT yields empirical predictions distinguishing it from the Standard Model, including geometric constraints on fermion generations, minimal action scales linked to Planck’s constant, and testable signatures of curvature-induced trembling. By deriving the complete set of known particles and interactions from first principles—while identifying precise conditions for falsification—this work advances the geometric unification program and shows that matter originates from a single underlying principle: causal coherence in trembling spacetime.

Trembling Spacetime Relativity Theory (TSRT) provides a deterministic and fully geometric foundation for atomic structure, replacing the probabilistic framework of quantum mechanics with causally governed geodesic motion in a dynamically fluctuating spacetime. Previous work established TSRT as a unifying theory of all four fundamental forces, deriving interference without wave-particle duality, Planck’s radiation law without quantization postulates, and entanglement from geometric correlations. In this work, TSRT advances from theoretical coherence to high-precision empirical predictions, confronting experimental spectroscopic data across hydrogen, Deuterium, and Muonic Hydrogen. With only a single calibration line, the full spectrum of bound-state energy levels is derived from first principles without free parameters. TSRT reproduces excited states and fine-structure splittings to within a few millielectronvolts of experiment. The energy difference between the two fine-structure components of the second principal level is recovered with sub-percent accuracy, without invoking spin, operator algebra, or field-theoretic corrections. In Muonic Hydrogen, where curvature effects are amplified, refinements incorporating proton size, nuclear mass, and realistic density profiles yield agreement with experiment at the per-thousand level. These corrections arise naturally from the variational geometric structure of TSRT, requiring no renormalization or perturbative methods. Photon emission is described as a deterministic transition between complete geodesic ensembles, and the energy of the emitted photon is subsequently redshifted as it escapes the curved trembling geometry surrounding the atom. Further redshift may occur if the atom resides in a larger gravitational field, such as in a stellar environment. Atomic shell structure emerges as the equilibrium configuration of curvature-induced confinement, and an effective atomic coupling constant is derived from gravitational scaling, linking electromagnetism and gravity through geometry. Gravitational contributions, often neglected in atomic physics, are shown to be essential for spectral precision within TSRT. Even in weak fields, spacetime curvature determines energy levels with high accuracy, while in strong gravity environments, TSRT predicts additional geometric shifts beyond the standard redshift. These results demonstrate that trembling spacetime is not merely an alternative conceptual framework, but a viable and predictive physical foundation for atomic structure, unifying quantum and gravitational phenomena within a single geometric continuum.

The quantization of radiation energy, first introduced by Max Planck to resolve the ultraviolet catastrophe, marked the historical origin of quantum theory. However, Planck's postulate has long remained a phenomenological insertion without a first-principles derivation. In this work, we derive both Planck’s postulate and Planck’s radiation law from the geometric foundations of Trembling Spacetime Relativity Theory (TSRT). Here, spacetime is not smooth but exhibits bounded microscopic fluctuations, constrained by the requirement that proper time remains real and forward-directed. Photons are modeled not as wave–particle dual entities but as corpuscular excitations traveling along null geodesics perturbed by this geometric trembling. The quantization of energy emerges as a direct consequence of coherence conditions in geodesic deviation, without invoking operator formalism, wave superposition, or probabilistic postulates. A statistical treatment of these geodesic configurations leads to the Planck spectrum, even at zero temperature, where residual trembling persists. Furthermore, we derive consistent, geometric definitions of entropy and temperature in this framework, showing that TSRT not only reproduces known thermodynamic limits but also predicts a maximal temperature associated with the breakdown of causal geodesic structure. Despite the strict corpuscular nature of photons in TSRT, the theory exactly reproduces the relativistic Doppler shift, demonstrating that spectral and thermodynamic observables transform covariantly across inertial frames. This offers a deterministic and falsifiable foundation for black body radiation and the thermodynamics of light, grounded entirely in the causal geometry of spacetime. We further derive a local, curvature-suppressed entropy formula from trembling geodesics, which reduces to the Bekenstein–Hawking area law in strongly curved regions, providing a first-principles geometric foundation for gravitational thermodynamics without invoking quantum field theory.

Abstract This study presents a novel experimental methodology designed to assess damage in woven glass fibers reinforced polyamide 6,6/6 composites, specifically subjected to low-velocity impact and cyclic tensile loading. Conventional ultrasonic testing techniques often fail to detect subtle material degradation, particularly when dealing with barely visible impact damage (BVID), which can go unnoticed but still significantly compromise structural integrity. In contrast, the proposed approach utilizes multi-directional ultrasonic Lamb wave analysis, a more advanced technique that offers greater sensitivity and precision in identifying damage at various stages of the composite’s lifespan. In this work, a damage indicator is defined based on the velocity profile of Lamb waves, which are sensitive to changes in material properties such as stiffness degradation. The Lamb wave-based methodology is rigorously validated through detailed comparisons with X-ray tomography. These comparisons reveal strong correlations between the two techniques, highlighting the effectiveness of the proposed ultrasonic approach in detecting BVID. Moreover, the study demonstrates that this methodology is not only highly sensitive but also scalable, making it suitable for industrial applications where automated inspection of composite components is essential. The proposed method offers a significant advancement in non-destructive testing (NDT) techniques based on Lamb wave diagnostic tools in composite material testing.

The double-slit experiment has long been interpreted as evidence of wave-particle duality and quantum superposition. Here we propose a deterministic alternative rooted in Trembling Spacetime Relativity Theory (TSRT), where particles always follow localized proper-time geodesics within a causally structured but metrically fluctuating spacetime. Apparent interference fringes arise not from wavefunction superposition, but from the deterministic redirection of particle geodesics due to curvature perturbations shaped by the slit geometry. The observed distribution of arrival positions is governed by two geometric mechanisms: (i) the slit separation, which sets the angular deflection range of redirected geodesics, and (ii) a minimal action threshold that enforces a resolution limit based on causal distinguishability. Planck’s constant emerges naturally from this causal constraint as the smallest resolvable geodesic action difference, without any quantum postulates. TSRT provides analytical predictions for fringe spacing and intensity distribution across a wide range of particles—photons, electrons, and neutrons—by applying geometric deflection laws to causally permitted geodesics. These predictions align closely with established experimental results. In addition, the TSRT framework enables simulation of arbitrary configurations by modeling ensembles of geodesics subject to metric trembling, providing a fully geometric alternative to quantum superposition.

Diffraction gratings, with their periodically ordered structures, have been critical components in acoustics, optics, and spectroscopy for over a century. The classical grating equation describes the emergence of diffraction phenomena by gratings, considering the groove periodicity and the characteristics of the incident wave. These gratings find extensive applications in communication, spectroscopy, architectural acoustics, and underwater research, and they are foundational to pioneering investigations in phononic crystals and meta-materials. While much attention has been given to understanding the diffraction behavior of linear acoustics concerning gratings, the literature lacks research regarding the influence of high-amplitude ultrasonic waves, which introduce observable nonlinear effects. This experimental enquiry presents a pioneering methodology for isolating higher harmonics from these nonlinear phenomena. We have developed a spatial filtering apparatus with a single-frequency transducer and a specially designed grating profile, enabling precise frequency selection or rejection.

Biofilm formation in process industries, water treatment devices, and drinking water pipe networks pose a significant risk to public health and cause a variety of operational issues. One of the major challenges is the inspection of biofilms in heat-exchanging devices such as plate heat exchangers. A plate heat exchanger (PHE) is integral to food industries, water treatment plants, and others. Conventional techniques for cleaning biofilm formation in these plates (foulant) include chemical cleaning, steam, and hydro-blasting. However, they are inefficient and labor-intensive because of limitations such as increased handling risk, over-cleaning, and corrosion. Thus, developing novel techniques for real-time monitoring of biofilm growth on these devices is critical for efficient working. Previous studies were restricted to the development of ultrasound-assisted heat exchangers to reduce the deposits in these plates. However, only a few studies have investigated ultrasound as a probable monitoring tool. Thus, this research aims to explore the nonlinear ultrasonic parameters using the second harmonic generation technique as a real-time tool for monitoring biofilms in PHE. The proposed research will help design more effective ultrasonic-assisted plate heat exchangers to achieve maximum heat transfer efficiency.

In light of industrial sectors' rapid and exponential growth, the deleterious specter of water pollution looms ominously over our environment. One of the significant challenges is obtaining a sustainable and energy-efficient water purification/treatment system. Since the 1990s, several research studies have been proposed to demonstrate the usefulness of ultrasound as a water purification tool. As a result, newly designed devices such as ultrasound-assisted electrochemical treatment and ultrasound-assisted heat exchanging devices are becoming more common. However, such devices' high voltage ultrasonic emission is a significant problem because the nonlinear acoustic effects are not well understood and are, therefore, not adequately integrated into the design of the devices. Furthermore, the presence of biofilms in these devices creates more complexity due to the interaction of high-amplitude ultrasonic waves with the biofilm network. To better overcome the inefficient functioning of such devices and adverse operational issues, the current study aims to investigate and explain the nonlinear ultrasonic effects in ultrasonic-assisted purification devices, with and without biofilm deposits. The obtained insight will help develop an effective design strategy for high efficiency.

Solar photovoltaic modules are versatile power sources in diverse materials and configurations, including compact and flexible variants for portable electronic devices. Ensuring the reliability of these modules is crucial for sustaining the functionality of the devices they power. Manufacturing or handling-induced defects, such as cracks or scratches, pose a threat to the performance of solar modules. Hence, non-destructive inspection becomes essential in the quality control process. Ultrasonic C-scan has been an established inspection technique within various industries; however, its application on solar modules remains uncommon. On the other hand, Scanning Acoustic Microscopy (SAM) has been implemented for observing defects in solar cells, yet employing SAM for comprehensive module scanning is inefficient. This study aims to assess the capability of ultrasonic c-scan in detecting micro-cracks within semi-flexible solar panels and to evaluate the effects of frequency selection on the results. This work investigates the specimen with an Ultrasonic C-Scanner at different frequencies. Subsequently, the outcomes are validated by comparing them with the results from the SAM. The potential of using a widely known ultrasonic technique for this purpose, while understanding its limitations, such as the c-scan, will enable a more straightforward integration of the technique into the solar module quality control process.

The acoustic signature is essential in the characterization of liquids and the non-destructive evaluation of microstructural damages in materials.It closely relates to the sample's molecular structure, providing insight into its material properties.Studies have reported the measurement of B/A for the characterization of industrial materials ranging from geophysics to biological fluids such as silicon oil and gelatin.However, none of the studies has focused on investigating B/A for quality monitoring of viscous fluids such as syrups.As viscosity is a critical factor for these fluids' shelf-life, processing, and packaging, it is vital to monitor their quality during production.However, most current techniques rely only on the laboratory-based rheological method for quality monitoring and assessment.Thus, this paper aims to provide a non-destructive, real-time monitoring tool to monitor the viscosity of honey using a nonlinear acoustical approach based on the Second harmonic generation method (SHGM).A simultaneous study is performed to obtain the linear acoustical parameter, such as speed of sound and attenuation, for a similar setup.Both results are compared to see the viscosity correlation with each parameter.This research will help the food industry provide a fast and efficient way of inspecting consumable food products.

Metal plate structures, crucial components in various industrial sectors, demand meticulous inspection methods for the maintenance of their structural integrity. This review article not only serves as a contemporary introduction to this research field but also underlines the vital role of this field in ensuring the safety and reliability of these structures. The study delves into Lamb wave generation and detection techniques, highlighting the challenges and advancements in transducer technologies. Two detailed case studies are presented to contextualize and illustrate the practical applications of these techniques. The first case study demonstrates the detection of weld joints and stiffeners in steel plates, particularly relevant to the shipbuilding industry. Through a combination of numerical simulations and experimental validations designed for this narrative, this study highlights the capability of the A0 Lamb wave mode in identifying these features. The second case study, equally supported by new experiments, focuses on detecting thickness reductions in aluminum plates using high-order Lamb modes in a multimodal excitation setup. This scenario simulates conditions such as corrosion or wear that induce material thinning. By creating blind holes of varying depths on one side of the plate and conducting inspections from the opposite side, the study demonstrates the method’s precision in identifying hidden defects. The case studies involving aluminum and steel specimens exemplify the efficacy of Lamb waves in the nondestructive evaluation of metal plates. They provide critical insights into the method’s ability to deliver precise and efficient detection of structural anomalies despite inherent challenges in signal interpretation and analysis.

This paper presents a Graph-based Simultaneous Localization And Mapping (GraphSLAM) approach for a robotic system relying on the reflections of ultrasonic guided waves to enable long-range inspection tasks on plate-based metal structures. A measurement model that can leverage multi-order acoustic echoes is introduced for accurate localization, and beamforming is used for mapping the boundaries of individual metal panels. These two elements are subsequently integrated within a nonlinear least squares optimizer to solve the full offline SLAM problem. We experimentally evaluate the potential of this approach in a laboratory environment. We observe the improved localization accuracy of the multi-order echo model compared to a second model, from previous works, that relies solely on first-order echoes. We also show that the proposed approach can yield accurate SLAM results, hence showcasing the standalone capability of ultrasonic-based GraphSLAM for envisioned long-range inspection applications.

Abstract Background. Donated blood is routinely preserved for about six weeks. After that, a considerable amount of unused blood is discarded for safety. We carried out sequential measurements of the ultrasonic parameters (Velocity of propagation of ultrasound, its attenuation, and relative nonlinearity coefficient B/A) for red blood cells (RBCs) bags in their physiological preserving conditions in the blood bank, in a given experimental setup, to investigate the gradual deteriorations in the biomechanical properties of RBCs. Materials and Methods . We discuss our primary findings, which indicate the applicability of ultrasound techniques as a quantitative quick, non-invasive routine check for the validity of sealed blood bags. The technique can be applied during and beyond the regular preservation period, thus enabling deciding for each bag to either further preserve or withdraw. Results and Discussion . Considerable increases in the velocity of propagation ( Δ V = 966 m s −1 ) and ultrasound attenuation ( Δα = 0.81 dB C −1 m −1 ) were detected to take place during the preservation time. Likewise, the relative nonlinearity coefficient showed a generally rising trend during the preservation period ( Δ (B/A) = 0.0129). At the same time, a distinctive feature characteristic of a specific blood group type is realized in all cases. Due to the complex stress-strain relations and their reflection on the hydrodynamics and flow rate of non-Newtonian fluids, the increased viscosity of long-preserved blood may justify the known post-transfusion flow complications.

Current evolutions in transducer design, such as phased arrays, but more importantly, metamaterials-based acoustic lenses, potentially enable generating specific beam shapes earlier unconsidered. It is known that the Schoch effect, when a bounded incident beam on a submerged solid reflects at the Rayleigh angle, depends on the beam width and the frequency. This work numerically explores the consequence of the shape of such beams on the Schoch effect and invites further experimental work. The study investigates square shapes and beams with exponential flanks compared to Gaussian reference profiles and incorporates diffraction upon sound propagation to resemble reality better. It is shown that stunning differences occur depending on the beam shape, particularly for square beams.

Compared to standard plain glass panes, laminated glass has many advantages, such as improved glass panes’ safety, structural strength, and aesthetics. An essential quality criterium of laminated glass, tunable during production, is the adhesion strength of the interlayer. Traditional quality control is destructive and involves peeling, tensioning, or impacting the specimens, while ultrasound may, if properly applied, reveal the quality parameters nondestructively and reduce production costs. In facilities equipped with ultrasonic phased array transducers, bounded beam effects may be exploited to investigate the considered glass panels. In this work, we provide a proof of concept of a bounded beam effect, known as the Schoch displacement, caused by ultrasonic guided waves, to map variations in adhesion nondestructively. Even though in industrial production lines, one may opt for phased array techniques to measure the effects, in this work, we provide images obtained by acousto-optic Schlieren photography for demonstrative purposes.

Wiring, of different degrees of complexity, is a dominant part of mechanical support in constructions, electromagnetic and telecommunication signal transmission cables, among other applications. Single and manifold twisted wires are prominent examples of such utilities and are susceptible to mechanical irritations and deterioration. They require ultrasonic non-destructive testing and health monitoring. The objective is to develop an ultrasound-based technique to automatically measure the number of twists per meter in winded wire strands implementable in the industry, to be used during an ultrasonic scan and provide the number of twists per meter during cable production, for instance, to verify that calibration is still in place. Fourier transformation is applied as an expedited non-destructive testing method of twisted wires. Digital signal processing to obtain spatial and time spectral representation recognition due to amplitude variance, induced by the varying distance between the transducer and wire, is developed depending on the number of twists. Two different spatial spectral analyses satisfactorily quantify the number of twists by providing the distance between each twist. The method is robust and applicable when the distance between the transducer and strand is not constant, as the industry requires.

Metamaterials are intensely explored for their capabilities to modify sound beams. In addition to frequency filtering, acoustic lenses offer intriguing possibilities for shaping sound beams. For the time being, the versatility of metamaterials remains limitless. In beam-shape adaptation, however, their complexity suggests that manufacturers of transducers could benefit from combining metamaterials with more conventional materials. This paper investigates the transmission of a circumscribed beam through a stratum of anisotropic material to examine the change in beam shape after transmission. The incident sound is presumed to originate from a conventional transducer, possibly coated with a metamaterial to modify the sound field, before being transmitted through the anisotropic layer. Different incident beam shapes, such as conical-like, Gaussian, and pillar beams, are investigated. While the results are not exhaustive, they demonstrate the beam shape’s adaptability.