Reciprocity Theorem Research Articles

A new analytical model for predicting the welding deformation of large welded plates with equivalent moments

The efficient prediction of welding deformation is important for the deviation control of large welded structures. In this paper, an analytical model is proposed to predict the welding deformation of large aluminium alloy plates. The welding-induced equivalent moments are calculated based on the inherent strain of a small-scale welded plate. The large welded plate is regarded as a corner-supported plate subjected to the equivalent moments. The welding deformation of the large welded plate is analytically derived by using the reciprocal theorem of work, and is validated by the experimental data. The effect of heterogeneous weld properties and plate dimensions on the welding deformation is quantitatively discussed. The proposed model can facilitate the deformation prediction of large welded structures.

The motion of a self-propelling two-sphere swimmer in a weakly viscoelastic fluid

We study analytically the propulsion of a force- and torque-free swimmer composed of two counterrotating spheres of differing radii through a viscoelastic fluid described by the Giesekus constitutive model. Our analysis includes both swimmers composed of directly touching spheres and those composed of spheres separated by some finite distance. The propulsion speed of the swimmer is calculated by first expanding the equations of motion and the Giesekus constitutive model as a power series in the Weissenberg number, and then using the Lorentz reciprocal theorem to determine the first-order propulsion speed using the known flow fields for rotating and translating two-sphere geometries at zeroth order. We calculate the relative rotation speeds of the two spheres necessary to maintain the torque-free condition, including an approximate correction for fluid elasticity. The impact of the separation distance between the two spheres, the ratio of their radii, and the value of the Giesekus mobility parameter α on the propulsion speed are all examined; we find that the propulsion speed of the swimmer is maximized for two touching spheres with a radius ratio of approximately 0.7, with a Giesekus mobility parameter of α=0, corresponding to an Oldroyd-B fluid. We also quantify how the increased shear-thinning in the fluid, represented by increasing values of α, results in a significant decrease in the swimmer speed. Finally, through calculations of the fluid stresses around the two-sphere swimmer, we demonstrate the development of enhanced hoop stresses around the smaller sphere, which drive the expulsion of stretched fluid behind the smaller sphere and induce motion of the swimmer in the direction of the larger sphere.

Numerical and experimental crack-tip cohesive zone laws with physics-informed neural networks

The cohesive zone law represents the constitutive traction versus separation response along the crack-tip process zone of a material, which bridges the microscopic fracture process to the macroscopic failure behavior. Elucidating the exact functional form of the cohesive zone law is a challenging inverse problem since it can only be inferred indirectly from the far-field in experiments. Here, we construct the full functional form of the cohesive traction and separation relationship along the fracture process zone from far-field stresses and displacements using a physics-informed neural network (PINN), which is constrained to satisfy the Maxwell-Betti's reciprocal theorem with a reciprocity gap to account for the plastically deforming background material. Our numerical studies simulating crack growth under small-scale yielding, mode I loading, show that the PINN is robust in inversely extracting the cohesive traction and separation distributions across a wide range of simulated cohesive zone shapes, even for those with sharp transitions in the traction-separation relationships. Using the far-field elastic strain and residual elastic strain measurements associated with a fatigue crack for a ZK60 magnesium alloy specimen from synchrotron X-ray diffraction experiments, we reconstruct the cohesive traction-separation relationship and observe distinct regimes corresponding to transitions in the micromechanical damage mechanisms.

Experimental and numerical estimation of complex stress intensity factor for the completely debonded anti-crack embedded into a weak matrix using domain integral method

The problem of interface debonding is of fundamental importance in understanding the stress transfer mechanism and behavior of weak interfaces present due to rigid stiffeners. A domain integral method based on Betti’s reciprocal theorem was adopted to compute the complex stress intensity factor (SIF) for the completely debonded anti-crack (discontinuity) with mixed boundary conditions. The domain integral approach is based on two admissible mechanical states, which comprise of actual and auxiliary elastic fields. The singular auxiliary fields in terms of angular variations for the completely debonded anticrack were obtained using the full-field solution. The obtained angular variations were compared with the Williams eigenvalue problem for completeness. The displacement and strains were obtained using digital image correlation (DIC) experiments and numerical simulation for parallel and perpendicular orientations of the discontinuity. The obtained DIC strain contours revealed the presence of anti-symmetric and symmetric variation on the bonded side of the anti-crack. The complex SIF is estimated experimentally and numerically based on the domain integral method to seek a better comparison with the analytical estimate. The SIF estimated for the perpendicular orientation is higher than the parallel orientation.

Some basic theorems on the modified Green–Lindsay model of linear generalized thermoelasticity

The present paper is concerned with a theory of thermoelasticity based upon a newly developed temperature-rate-dependent thermoelastic model that was proposed by Yu et al. [Meccanica. 2018;53:2543–2554], which has eliminated the discontinuity observed in the displacement field during transient motion. The basic governing equations for a homogeneous and isotropic medium are formulated to deduce the significant theorems. Firstly, we prove the uniqueness theorem concerning a mixed initial and boundary value problem within the framework of linear thermoelasticity. Furthermore, we establish a reciprocity theorem utilizing the Laplace transform, which provides insight into the mutual influence between different aspects of the system under consideration. Consequently, we proved the Somigliana and Green theorems. This generalized theorem extends the applicability and versatility of the classical theorems and also assists the issue of thermal and mechanical fields as well as is beneficial for geophysics and mining.

A Simple Path to the Small Perturbation Method for Scattering from Slightly Rough Dielectric Surfaces

We propose a perturbative method to compute electromagnetic scattering from slightly rough dielectric surfaces, which leads to the same result as the usual Small Perturbation Method (SPM) in a surprisingly simple way. The proposed method is based on three pillars: the volumetric perturbative approach, the reciprocity theorem, and a proper approximation of the electric field within the perturbation volume, that we name Internal Field Approximation (IFA). The proposed new mathematical derivation of the SPM turns out to be much simpler and more concise than the classical one. In addition, being based on a volumetric perturbation approach, it has the potential of dealing in future with surface and volume scattering within a unitary framework, which is useful in modelling scattering from, e.g., vegetated soil, snow-covered terrain, and inhomogeneous soils. Therefore, although the presented result is mainly theoretical, it can have important applications in remote sensing.

Assessing Onsager’s reciprocity difference in signal transduction cascade model: a new metric for evaluating cellular response dynamics

The cell signaling system translates extracellular changes into biochemical reactions within the cell, ultimately affecting gene expression. This system is crucial for responding to environmental stressors through a series of chemical interactions among intracellular molecules. Although Onsager’s reciprocity theorem is foundational in analyzing non-equilibrium systems, its limitations are evident in the nonlinear responses characteristic of the chemical reactions essential for signal transduction. In this study, we demonstrate that the reciprocity coefficients between signal molecule Xj, Lj:j+1 in the transmission from the jth step to the j + 1th step and molecule Xj+1, Lj+1:j from the j + 1th step to the jth step are not necessarily equal due to nonlinear interactions. This disparity, quantified as Jj:j+1 = −Lj:j+1 + Lj+1:j, reflects the amount of signal transduced between these steps, coupled with the phosphorylation rate of signaling molecules. Through kinetic model simulations, we analyzed the cellular response to external radiation exposure, monitoring how signal transduction progresses via phosphorylation reactions over time. The simulation showed a transition from an unstable to a stable, unimodal signaling pattern at varying radiation doses, correlating well with actual cellular responses. In conclusion, our findings suggest that the discrepancy in Onsager’s reciprocity coefficients correlates with signal transduction velocity, providing novel insights into the thermodynamic underpinnings of cellular signaling mechanisms.

Clustering induces switching between phoretic and osmotic propulsion in active colloidal rafts

Active particles driven by chemical reactions are the subject of intense research to date due to their rich physics, being intrinsically far from equilibrium, and their multiple technological applications. Recent attention in this field is now shifting towards exploring the fascinating dynamics of active and passive mixtures. Here we realize active colloidal rafts, composed of a single catalytic particle encircled by several shells of passive microspheres, and assembled via light-activated chemophoresis. We show that the cluster propulsion mechanism transits from diffusiophoretic to diffusioosmotic as the number of colloidal shells increases. Using the Lorentz reciprocal theorem, we demonstrate that in large clusters self-propulsion emerges by considering the hydrodynamic flow via the diffusioosmotic response of the substrate. The dynamics in our active colloidal rafts are governed by the interplay between phoretic and osmotic effects. Thus, our work highlights their importance in understanding the rich physics of active catalytic systems.

Waves in space-dependent and time-dependent materials: A systematic comparison

Waves in space-dependent and in time-dependent materials obey similar wave equations, with interchanged time- and space-coordinates. However, since the causality conditions are the same in both types of material (i.e., without interchangement of time- and space-coordinates), the solutions are dissimilar.We present a systematic treatment of wave propagation and scattering in 1D space-dependent and in 1D time-dependent materials. After formulating unified equations, we discuss Green’s functions and simple wave field representations for both types of material. Next we discuss propagation invariants, i.e., quantities that are independent of the space coordinate in a space-dependent material (such as the net power-flux density) or of the time coordinate in a time-dependent material (such as the net field-momentum density). A discussion of general reciprocity theorems leads to the well-known source–receiver reciprocity relation for the Green’s function of a space-dependent material and a new source–receiver reciprocity relation for the Green’s function of a time-dependent material. A discussion of general wave field representations leads to the well-known expression for Green’s function retrieval from the correlation of passive measurements in a space-dependent material and a new expression for Green’s function retrieval in a time-dependent material.After an introduction of a matrix–vector wave equation, we discuss propagator matrices for both types of material. Since the initial condition for a propagator matrix in a time-dependent material follows from the boundary condition for a propagator matrix in a space-dependent material by interchanging the time- and space-coordinates, the propagator matrices for both types of material are interrelated in the same way. This also applies to representations and reciprocity theorems involving propagator matrices, and to Marchenko-type focusing functions.

Momentum and angular momentum reciprocity theorems for acoustic fields.

As a universal theorem applicable to linear physical systems, reciprocity theorems describe the reciprocity between the input and output of linear systems. As for the electromagnetic field, there have been detailed studies on the reciprocity theorems of energy, momentum, and angular momentum, which reveal the correlation between two electromagnetic field sources from different aspects. Based on the fundamentals of linear acoustic theory, the momentum-based and angular momentum-based reciprocity theorems for acoustic fields are derived. They indicate the correlation between two acoustic systems from the perspective of momentum and angular momentum, respectively. On this basis, a systematic summary and analogical study is also performed on various reciprocity theorems for acoustic and electromagnetic fields, which deepens our understanding of the reciprocity theorem as a universal law for linear physical fields. The results obtained in this work can further improve the reciprocity theory of linear acoustic systems, as well as provide theoretical guidance for practical applications, such as audio signal processing and acoustic measurement.

Examining basic theorems and plane waves in the context of thermoelastic diffusion using a multi-phase-lag model with temperature dependence

This study employs a novel mathematical formulation of temperature-dependent thermoelastic diffusion with multi-phase delays (TDMT). The TDMT model, which is assumed, is obtained by converting the fundamental rules of Fourier’s and Fick’s laws into higher order time derivatives of the heat flow vector, the temperature gradient, diffusing mass flux, and the chemical potential gradient. Fundamental theorems such as energy, uniqueness, reciprocity theorems, and variational criterion are established using the fundamental equations for the assumed model TDMT. The reciprocity theorem is applied to a particular scenario by using instantaneous concentrated body forces, heat sources, chemical potential sources, and moving heat sources as examples. It is noted that both the variational criterion and these theorems depend on the field variables’ susceptibility as well as the fluctuations in the higher order temporal derivatives’ parameters. Additionally, the assumed model’s two-dimensional (2D) plane wave propagation is described. Three longitudinal waves are identified: the longitudinal (P) wave, the thermal (T) wave, the chemical potential (CP) wave, and the linked transverse (SV) wave. Wave properties (phase velocity and attenuation coefficient) are calculated numerically and shown visually. A few special examples are also investigated and compared with the established findings. The framework for additional research into basic issues in thermoelastic continua under various physical field variables is provided by this work. Numerous applications in material science, geomechanics, soil dynamics, and the electronic sector can be made of the current findings.

A reciprocity theorem for the Cohen–Ramanujan sums and its application to Cohen–Ramanujan expansions

For an arithmetical function [Formula: see text], its Ramanujan expansion is a series expansion in the form [Formula: see text] where [Formula: see text] are complex numbers and [Formula: see text] is the Ramanujan sum. Here, we prove a reciprocity result on Cohen–Ramanujan sums [Formula: see text] to change the position of [Formula: see text] and [Formula: see text] in a twisted function and use it to prove that for certain arithmetical functions [Formula: see text], Cohen–Ramanujan series expansions in the form [Formula: see text] exist if and only if expansions in the form [Formula: see text] exist.

Broadband nonreciprocal gyromagnetic metasurface via magnetic Kerker-type dimers

Optical nonreciprocity, stemming from the deviation of the Lorentz reciprocity theorem, holds significant interest in the realm of optics and electromagnetics. Here, we propose and experimentally demonstrate broadband nonreciprocal transmission via a low-biased magnetic Kerker-type dimer metasurface. The designed magneto-optical metasurface comprises three layers of metal sandwiched between two gyromagnetic near-zero thickness slabs. The Kerker-type dimers broaden the isolation bandwidth utilizing multiple resonances where the double-stacked metallic disks act as Kerker-type dipoles, enhancing the transmissibility of the metasurface. The multipole decomposition reveals that the magnetic dipole contribution arising from magnetization is the primary cause of the metasurface's nonreciprocal response. Microwave measurement demonstrates that the bandwidth for an isolation ratio exceeding 10 dB is over 3 GHz. The broadband nonreciprocal performance remains relatively stable, exhibiting strong robustness against the bias disturbance. Our findings provide an alternative avenue for enhancing broadband nonreciprocity transmission under a low-biased magnetic field.

Some qualitative results in hyperbolic two-temperature generalized thermoelasticity

This study aims to establish the convolutional-type variational and reciprocity theorems within the framework of the hyperbolic two-temperature generalized thermoelasticity theory for an isotropic thermoelastic material, with the help of alternate formulation of the mixed boundary initial value problem, in which initial conditions are combined with field equations (using the Laplace transform). The convolutional-type variational principle adapts readily to numerical solutions based on the Ritz method and is useful in the finite element method. The reciprocity theorem is helpful in the theoretical development of boundary and finite element methods. The current effort can be valuable for the problem of coupling effects of thermal and mechanical fields, especially in geophysics and mining.

SH-wave based defect imaging method for CFRP plates with variable thickness

ABSTRACT Classical guided wave imaging methods find applicability in uniform-thickness plates or quasi-isotropic plates. To be compared with, the method presented in this paper advances damage imaging accuracy for both uniform-thickness and variable-thickness anisotropic plates by incorporating new compensation terms grounded in the theoretical excitation directivity of the quasi-SH0 mode in anisotropic material. Specifically, the nondispersive nature of the wave speed and skew angle of the quasi-SH0 mode in the low- fd states (frequency-thickness-product lower than the cut-off value of SH1 mode) is established through dispersion curves and wave structure calculations. Additionally, employing the elastic dynamics reciprocity theorem reveals that the directivity of this mode is also nondispersive in the low- fd states. Therefore, the quasi-SH0 mode is theoretically insensitive to thickness variation. Experiment validated these theoretical calculations. On this basis, this paper proposed an integrated imaging method that utilises transmitted and reflected wave information by incorporating matching pursuit algorithm, cross-correlation analysis, probabilistic reconstruction and delay-and-sum method that incorporates new compensation terms based on the quasi-SH0 mode theory. Experimental validation of uniform-thickness carbon fibre-reinforced polymer (CFRP) plates and variable-thickness CFRP plates shows the insensitivity of this method to structural thickness variation, enabling effective imaging of damage in such anisotropic structures.

On uniqueness and Somigliana theorem for a thermoelastic theory

The main concern of the present article is based on unified generalized thermoelasticity in an isotropic and homogeneous medium. The uniqueness theorem and Betti type reciprocal principle are derived. Based on the reciprocal theorem, the generalization of the Somigliana and Green theorem is also established in the present context.

A permittivity imaging method in the polar coordinates: Electrical capacitance tomography based on the electromagnetic momentum

Electrical capacitance tomography (ECT) is widely used in petroleum, chemical, and other industrial detection. The ECT equations are nonlinear, and their corresponding inverse problems are ill-posed, so the imaging resolution is low. The ECT technique detects scalar data. The permittivity imaging method based on the electromagnetic momentum reciprocity theorem detects vector data, which carries more information than scalar data. To improve the ability of the imaging system to recognise permittivity boundaries, we propose a permittivity imaging method in the polar coordinate system, which is centred on electrical capacitance tomography based on the electromagnetic momentum (ECT-EMM). ECT-EMM principle with the electrodes-moving scheme and the object-moving scheme is proposed. We reconstruct the permittivity gradient from the capacitance gradient. We adopt the object-moving scheme and use the Tikhonov regularisation algorithm to reconstruct the permittivity through the permittivity gradient. The image reconstruction results and image evaluation metrics show that the proposed method clearly recognises the object boundaries compared to ECT. The proposed method performs better when using threshold filtering.

Study the Cross Section of 67Ga(n,p)67Zn Reaction by using the Revers Reaction in the Ground State

In this study the excitation function for 67Ga(n,p)67Zn nuclear reaction in the ground state has been calculated using the reciprocity theorem. The decay energy of proton particles equal to (-1.78369) MeV according to the available International Atomic Energy Agency (IAEA) for 67Zn (p,n) 67Ga nuclear reaction with threshold energy (1.81054 MeV) and this data replicated in fine proceedings for specified energy range. The cross-section calculations in a present work shows same behavior of the corresponding experimental data and with previous theoretical work using statistical model.

Quaternion form of the electromagnetic reciprocity theorem

In this paper, the general forms of quaternion electromagnetic field mutual energy-momentum equations (QEMF-MEME) and quaternion electromagnetic field energy-momentum reciprocal equations (QEMF-EMRE) in frequency domain are derived based on the quaternion-electromagnetic-field equations. It has been discovered that QEMF- EMRE is a reflection of the electromagnetic interaction between lagging and leading waves. The real and imaginary parts of scalar, the real and imaginary parts of vector of QEMF-EMRE correspond to the Feld-Tai reciprocity theorem, Lorentz reciprocity theorem, and the two sets of momentum reciprocity theorems, respectively. While, QEMF-MEME reflects the electromagnetic interaction of two sets of lagging waves, and its real scalar, imaginary scalar, real-vector, and imaginary-vector parts are the Feld-Tai reciprocal energy theorem, the reciprocal energy theorem, the two sets of reciprocal momentum theorem.

From port-based teleportation to Frobenius reciprocity theorem: partially reduced irreducible representations and their applications

In this paper, we present the connection of two concepts as induced representation and partially reduced irreducible representations (PRIR) appear in the context of port-based teleportation protocols. Namely, for a given finite group G with arbitrary subgroup H, we consider a particular case of matrix irreducible representations, whose restriction to the subgroup H, as a matrix representation of H, is completely reduced to diagonal block form with an irreducible representation of H in the blocks. The basic properties of such representations are given. Then as an application of this concept, we show that the spectrum of the port-based teleportation operator acting on n systems is connected in a very simple way with the spectrum of the corresponding Jucys–Murphy operator for the symmetric group S(n-1)⊂S(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S(n-1)\\subset S(n)$$\\end{document}. This shows on the technical level relation between teleporation and one of the basic objects from the point of view of the representation theory of the symmetric group. This shows a deep connection between the central object describing properties of deterministic PBT schemes and objects appearing naturally in the abstract representation theory of the symmetric group. In particular, we present a new expression for the eigenvalues of the Jucys–Murphy operators based on the irreducible characters of the symmetric group. As an additional but not trivial result, we give also purely matrix proof of the Frobenius reciprocity theorem for characters with explicit construction of the unitary matrix that realizes the reduction in the natural basis of induced representation to the reduced one.