Enhanced avionic sensing based on Wigner’s cusp anomalies

Typical sensors detect small perturbations by measuring their effects on a physical observable, using a linear response principle (LRP). It turns out that once LRP is abandoned, new opportunities emerge. A prominent example is resonant systems operating near Nth-order exceptional point degeneracies (EPDs) where a small perturbation ε ≪ 1 activates an inherent sublinear response ∼εN≫ε in resonant splitting. Here, we propose an alternative sublinear optomechanical sensing scheme that is rooted in Wigner’s cusp anomalies (WCAs), first discussed in the framework of nuclear reactions: a frequency-dependent square-root singularity of the differential scattering cross section around the energy threshold of a newly opened channel, which we use to amplify small perturbations. WCA hypersensitivity can be applied in a variety of sensing applications, besides optomechanical accelerometry discussed in this paper. Our WCA platforms are compact, do not require a judicious arrangement of active elements (unlike EPD platforms), and, if chosen, can be cavity free.

Unconventional singularities and energy balance in frictional rupture

A widespread framework for understanding frictional rupture, such as earthquakes along geological faults, invokes an analogy to ordinary cracks. A distinct feature of ordinary cracks is that their near edge fields are characterized by a square root singularity, which is intimately related to the existence of strict dissipation-related lengthscale separation and edge-localized energy balance. Yet, the interrelations between the singularity order, lengthscale separation and edge-localized energy balance in frictional rupture are not fully understood, even in physical situations in which the conventional square root singularity remains approximately valid. Here we develop a macroscopic theory that shows that the generic rate-dependent nature of friction leads to deviations from the conventional singularity, and that even if this deviation is small, significant non-edge-localized rupture-related dissipation emerges. The physical origin of the latter, which is predicted to vanish identically in the crack analogy, is the breakdown of scale separation that leads an accumulated spatially-extended dissipation, involving macroscopic scales. The non-edge-localized rupture-related dissipation is also predicted to be position dependent. The theoretical predictions are quantitatively supported by available numerical results, and their possible implications for earthquake physics are discussed.

Boundary Element Analysis of Anisotropic Thermomagnetoelectroelastic Solids with 3D Shell-Like Inclusions

The paper presents novel boundary element technique for analysis of anisotropic thermomagnetoelectroelastic solids containing cracks and thin shell-like soft inclusions. Dual boundary integral equations of heat conduction and thermomagnetoelectroelasticity are derived, which do not contain volume integrals in the absence of distributed body heat and extended body forces. Models of 3D soft thermomagnetoelectroelastic thin inclusions are adopted. The issues on the boundary element solution of obtained equations are discussed. The efficient techniques for numerical evaluation of kernels and singular and hypersingular integrals are discussed. Nonlin-ear polynomial mappings are adopted for smoothing the integrand at the inclusion’s front, which is advantageous for accurate evaluation of field intensity factors. Special shape functions are introduced, which account for a square-root singularity of extended stress and heat flux at the inclusion’s front. Numerical example is presented.

Interaction Between Coexisting Orbits in Impact Oscillators

Impact oscillators exhibit an abrupt onset of chaos close to grazing due to the square-root singularity in their discrete time maps. In practical applications, this large-amplitude chaotic vibration needs to be avoided. It has been shown that this can be achieved if the ratio of the natural frequency of the oscillator ω0 and the forcing frequency is an even integer. But, in practice, it is difficult to set a parameter at such a precise value. We show that in systems with square-root singularity (prestressed impacting surface), there exists a range of ω0 around the theoretical value over which the chaotic orbit does not occur, and that this is due to an interplay between the main attractor and coexisting orbits. We show that this range of forcing frequency has exponential dependence on the amount of prestress as well as on the stiffness ratio of the springs.

Exact solution for the Poisson field in a semi-infinite strip

The Poisson equation is associated with many physical processes. Yet exact analytic solutions for the two-dimensional Poisson field are scarce. Here we derive an analytic solution for the Poisson equation with constant forcing in a semi-infinite strip. We provide a method that can be used to solve the field in other intricate geometries. We show that the Poisson flux reveals an inverse square-root singularity at a tip of a slit, and identify a characteristic length scale in which a small perturbation, in a form of a new slit, is screened by the field. We suggest that this length scale expresses itself as a characteristic spacing between tips in real Poisson networks that grow in response to fluxes at tips.

Extended series solutions and bifurcations of the Dean equations

Steady, incompressible flow down a slowly curving circular pipe is considered. Both real and complex solutions of the Dean equations are found by analytic continuation of a series expansion in the Dean number, $K$. Higher-order Hermite–Padé approximants are used and the results compared with direct computations using a spectral method. The two techniques agree for large, real $K$, indicating that previously reported asymptotic behaviour of the series solution is incorrect, and thus resolving a long-standing paradox. It is further found that a second solution branch, known to exist at high Dean number, does not appear to merge with the main branch at any finite $K$, but appears rather to bifurcate from infinity. The convergence of the series is limited by a square-root singularity on the imaginary $K$-axis. Four complex solutions merge at this point. One corresponds to an extension of the real solution, while the other three are previously unreported. This bifurcation is found to coincide with the breaking of a symmetry property of the flow. On one of the new branches the velocity is unbounded as $K\rightarrow 0$. It follows that the zero-Dean-number flow is formally non-unique, in that there is a second complex solution as $K\rightarrow 0$ for any non-zero $\vert K\vert $. This behaviour is manifested in other flows at zero Reynolds number. The other two complex solutions bear some resemblance to the two solution branches for large real $K$.

Exact Solution for an Anti-Plane Interface Crack between Two Dissimilar Magneto-Electro-Elastic Half-Spaces

This paper investigated the fracture behaviour of a piezo-electro-magneto-elastic medium subjected to electro-magneto-mechanical loads. The bimaterial medium contains a crack which lies at interface and is parallel to their poling direction. Fourier transform technique is used to reduce the problem to three pairs of dual integral equations. These equations are solved exactly. The semipermeable crack-face magneto-electric boundary conditions are utilized. Field intensity factors of stress, electric displacement, magnetic induction, cracks displacement, electric and magnetic potentials, and the energy release rate are determined. The electric displacement and magnetic induction of crack interior are discussed. Obtained results indicate that the stress field and electric and magnetic fields near the crack tips exhibit square-root singularity.

J-Integral Analysis of Surface Cracks in Round Bars under Tension Loadings

This paper presents a non-linear numerical investigation of surface cracks in round bars under tension stresses by using ANSYS finite element analysis (FEA). Due to the symmetrical analysis, only quarter finite element (FE) model was constructed and special attention was given at the crack tip of the cracks. The surface cracks were characterized by the dimensionless crack aspect ratio, a/b = 0.6, 0.8, 1.0 and 1.2, while the dimensionless relative crack depth, a/D = 0.1, 0.2 and 0.3. The square-root singularity of stresses and strains were modeled by shifting the mid-point nodes to the quarter-point locations in the region around the crack front. The proposed model was validated with the existing model before any further analysis. The elastic-plastic analysis under tension loading was assumed to follow the Ramberg-Osgood relation with n = 5 and 10. J values were determined for all positions along the crack front and then, the limit load was predicted using the J values obtained from FEA through the reference stress method.