Infinite Strip Research Articles

INTERACTION OF BENDING WAVES PROPAGATING IN AN INHOMOGENEOUS PLATE WITH AN OBSTACLE, WHICH IS A ROD RESTING ON A VISCOELASTIC FOUNDATION

A single interaction of a wave propagating in an inhomogeneous plate in the form of an infinite strip with a one-dimensional distributed mechanical object is considered. A distributed object is understood as a rod lying on a viscoelastic foundation that performs bending and torsional vibrations. It is considered that the plate has different parameters to the left and to the right of the rod. Physically and mathematically correct conditions at the interface between a plate and a rod are obtained as a consequence of the formulation of contact problems of the dynamics of two-dimensional elastic systems with one-dimensional loads, based on the variational principle of Hamilton – Ostrogradsky. The frequencies and wave numbers of the secondary (reflected and transmitted) waves, as well as the critical frequency, below which the wave does not propagate in the plate, are determined. Based on the solution of the kinematics problem from a system of linear algebraic equations obtained from the boundary conditions, the coefficients of reflection and transmission of bending waves are found. These coefficients resonantly depend on the frequency of the incident wave. Calculated graphs of the transmission coefficient versus the frequency of the incident wave are given for various parameters of the rod. The conditions for self-isolation and reflectionless passage of a wave through an obstacle are determined. It has been established that the frequency of the maximum vibration isolation is located above the frequencies at which the waves completely pass through the obstacle. An expression is obtained for the force due to the pressure of bending waves on a one-dimensional object. Its constant component is calculated, which vanishes (for a homogeneous plate) in the absence of waves reflected from the obstacle. It is shown that in degenerate cases the results obtained coincide with previous studies by other authors.

New Exact Solutions with a Linear Velocity Field for the Gas Dynamics Equations for Two Types of State Equations

In this paper, exact solutions with a linear velocity field are sought for the gas dynamics equations in the case of the special state equation and the state equation of a monatomic gas. These state equations extend the transformation group admitted by the system to 12 and 14 parameters, respectively. Invariant submodels of rank one are constructed from two three-dimensional subalgebras of the corresponding Lie algebras, and exact solutions with a linear velocity field with inhomogeneous deformation are obtained. On the one hand of the special state equation, the submodel describes an isochoric vortex motion of particles, isobaric along each world line and restricted by a moving plane. The motions of particles occur along parabolas and along rays in parallel planes. The spherical volume of particles turns into an ellipsoid at finite moments of time, and as time tends to infinity, the particles end up on an infinite strip of finite width. On the other hand of the state equation of a monatomic gas, the submodel describes vortex compaction to the origin and the subsequent expansion of gas particles in half-spaces. The motion of any allocated volume of gas retains a spherical shape. It is shown that for any positive moment of time, it is possible to choose the radius of a spherical volume such that the characteristic conoid beginning from its center never reaches particles outside this volume. As a result of the generalization of the solutions with a linear velocity field, exact solutions of a wider class are obtained without conditions of invariance of density and pressure with respect to the selected three-dimensional subalgebras.

Differential-symbol method of solving the Dirichlet problem in an infinite strip for a system of differential equations with shifts

Differential-symbol method of solving the Dirichlet problem in an infinite strip for a system of differential equations with shifts

Light scattering by an infinite strip grating

Light scattering by an infinite strip grating

Scattering of SH guided wave by a circular cavity and a semicircular salient in an infinite piezoelectric strip

In this article, the problem of a salient with complex boundary conditions in a piezoelectric strip is studied by Fourier series expansion method, and SH guided wave is considered as an external load acting on piezoelectric strip. First, on the basis of repeated image superposition, the analytical expression of scattering wave is conducted, which satisfies the stress free and electric insulation conditions on the upper and lower horizontal boundaries of the strip. Next, with the aid of boundary condition, integral equations are setup and solved by applying orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor and electric field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Dynamic anti-plane analysis for an elliptic inclusion in an infinite piezoelectric strip

Dynamic incident anti-plane shearing (SH guided wave) is considered to calculate dynamic stress and electric intensity in an infinite piezoelectric strip that contains an elliptic inclusion. By the means of the conformal mapping method, the field of an ellipse can be transformed into a unit circle. With the aid of series expansion method, the expression of SH guided wave is established, which satisfies the stress-free and electric insulation conditions on the upper and lower horizontal boundaries of the strip. The expression of scattering waves is conducted based on repeated image superposition. Boundary conditions are solved by applying the orthogonal function expansion technique. Results clarified the influence on the dynamic stress concentration factor and electric field intensity concentration factor under proper conditions.

Well-posedness of the Stokes-transport system in bounded domains and in the infinite strip

We consider the Stokes-transport system, a model for the evolution of an incompressible viscous fluid with inhomogeneous density. This equation was already known to be globally well-posed for any L1∩L∞ initial density with finite first moment in R3. We show that similar results hold on different domain types. We prove that the system is globally well-posed for L∞ initial data in bounded domains of R2 and R3 as well as in the infinite strip R×(0,1). These results contrast with the ill-posedness of a similar problem, the incompressible porous medium equation, for which uniqueness is known to fail for such a density regularity.

Transverse size of interacting directed lattice animals studied by Yang–Lee approach

Distributions of zeros of the grand canonical partition function in the complex fugacity plane are examined numerically for a model of self interacting directed lattice animals on infinite strips of the square lattice. It is shown that patterns of these zeros have a simple circular-like forms for large values of the interaction strength, with a small fraction of points scattered over the plane. As the interaction weakens, the zeros make up a more complicated structure which consists of a large number of branches. For weak interaction strength, the zeros lying closest to the real axis approach it by following the power-law behavior as the function of strip width. This allowed us to get quite good estimate of transverse size critical exponent for animals in the swollen state as well as at the point of their collapse transition. For stronger interactions, we found that these zeros move much faster toward the real axis—following an exponential law, which leads us to conjecture that animal transverse size diverges in a logarithmic way in the thermodynamic limit. An additional analysis, based on finite-size scaling analysis of the animal longitudinal correlation length, supports this conjecture.

Scattering of SH-guided wave by an elliptic inclusion in an infinite strip region

Dynamic incident anti-plane shearing (SH-guided wave) is considered to calculate dynamic stress in an infinite strip region with an elliptic inclusion. By the means of the conformal mapping method, the field of an ellipse can be transformed into a unit circle. With the aid of series expansion method, the expression of SH-guided wave is established, which satisfies the stress free condition on the upper and lower horizontal boundaries of the strip. The expression of scattering wave is conducted based on repeated image superpositions. Boundary conditions are solved by applying the orthogonal function expansion technique. Results clarified the influence on the dynamic stress concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Seismic Rotational Stability Analysis of Gravity Retaining Wall under Heavy Rainfall

A new methodology for the rotational stability analysis of a gravity retaining wall supporting inclined backfill under earthquake and heavy rainfall conditions has been presented. According to the movement mode of retaining wall and the characteristics of backfill sliding and rainwater infiltration, a sliding model of the infinite soil strip and rainwater infiltration model were established respectively. By calculating the internal energy dissipation rate and external loads power of the wall-soil system mechanism, a formula to calculate seismic yield acceleration coefficient under coupling conditions of earthquakes and rainfall was deduced. The results revealed a large effect size of infiltration depth of rainwater and the backfill inclination on the seismic yield acceleration coefficient. When the rainwater reaches 1/5 the height of the retaining wall and the backfill inclination exceeds 15°, the seismic yield acceleration coefficient will decrease rapidly. Moreover, the results obtained in this paper showed good consistency with those obtained by numerical simulation.

On the existence of the global conformal gauge in string theory

The global conformal gauge is playing the crucial role in string theory providing the basis for quantization. Its existence for two-dimensional Lorentzian metric is known locally for a long time. We prove that if a Lorentzian metric is given on a plain then the conformal gauge exists globally on the whole {{mathbb {R}}}^2. Moreover, we prove the existence of the conformal gauge globally on the whole worldsheets represented by infinite strips with straight boundaries for open and closed bosonic strings. The global existence of the conformal gauge on the whole plane is also proved for the positive definite Riemannian metric.

An exact solution for a partially clamped rectangle with a crack

The article deals with a boundary value problem for a rectangle whose horizontal sides are rigidly clamped, and the ends are free. In the centre of the rectangle, a vertical cut is made on which a discontinuity of the longitudinal displacements is given. An exact solution to the problem is constructed in the form of series in Papkovich–Fadle eigenfunctions. First, the corresponding boundary value problem for an infinite clamped strip is solved, then the solution for a rectangle is superimposed on this solution, with the help of which the boundary conditions at its ends are satisfied. Examples are given in which discontinuities of three types are considered which differ in the smoothness of the discontinuity contour near its ends.

Verification of mixing properties in two-dimensional shifts of finite type

This work introduces constructive and systematic methods for verifying the topological mixing and strong specification (or strong irreducibility) of two-dimensional shifts of finite type. First, we define transition matrices on infinite strips of width n for all n ≥ 2. To determine the primitivity of the transition matrices, we introduce the connecting operators that reduce the high-order transition matrices to lower-order transition matrices. Then, two sufficient conditions for primitivity are provided; they are invariant diagonal cycles and primitive commutative cycles of connecting operators. Then, the primitivity, corner-extendability, and crisscross-extendability are used to demonstrate the topological mixing. Finally, we show that the hole-filling condition yields the strong specification property. The application of all the above-mentioned conditions can be verified in a finite number of steps.

Dynamic anti-plane analysis for a semi-circular salient in an infinite elastic strip region

Dynamic incident anti-plane shearing (SH guided wave) is considered to calculate dynamic stress in an infinite strip region with a semi-salient. With the aid of series expansion method, the expression of SH guided wave is established, which satisfies the stress free condition on the upper and lower horizontal boundaries of the strip. The expression of standing wave is constructed by the means of Fourier series expansion. The expression of scattering wave is conducted based on repeated image superposition. Boundary conditions are solved by applying orthogonal function expansion technique. Results clarified the influence on the dynamic stress concentration factor under proper conditions.

Configuration spaces of disks in an infinite strip

We study the topology of the configuration spaces \(\mathcal {C}(n,w)\) of n hard disks of unit diameter in an infinite strip of width w. We describe ranges of parameter or “regimes”, where homology \(H_j [\mathcal {C}(n,w)]\) behaves in qualitatively different ways. We show that if \(w \ge j+2\), then the homology \(H_j[\mathcal {C}(n, w)]\) is isomorphic to the homology of the configuration space of points in the plane, \(H_j[\mathcal {C}(n, \mathbb {R}^2)]\). The Betti numbers of \(\mathcal {C}(n, \mathbb {R}^2) \) were computed by Arnold (The cohomology ring of the colored braid group. Springer Berlin, pp 183–186, 2014), and so as a corollary of the isomorphism, \(\beta _j[\mathcal {C}(n,w)]\) is a polynomial in n of degree 2j. On the other hand, we show that if \(2 \le w \le j+1\), then \(\beta _j [ \mathcal {C}(n,w) ]\) grows exponentially with n. Most of our work is in carefully estimating \(\beta _j [ \mathcal {C}(n,w) ]\) in this regime. We also illustrate, for every n, the homological “phase portrait” in the (w, j)-plane—the parameter values where homology \(H_j [ \mathcal {C}(n,w)]\) is trivial, nontrivial, and isomorphic with \(H_j [ \mathcal {C}(n, \mathbb {R}^2)]\). Motivated by the notion of phase transitions for hard-spheres systems, we discuss these as the “homological solid, liquid, and gas” regimes.

Singular integral equations in plane wave scattering by infinite graphene strip grating with brake of periodicity

Abstract In this paper, the solution of the H-polarized wave scattering problem by infinite graphene strip grating is obtained. The structure is periodic except two neighboring strips. The distance between these two strips is arbitrary. In particular, such a problem allows to quantify the mutual interaction of graphene strips in the array. The total field is represented as a superposition of the field of currents on the ideally-periodic grating and correction currents induced by the shift of the strips. The analysis is based on the convergent method of singular integral equations. It enables us to study the influence of the correction currents in a wide range from 10 GHz to 6 THz. It is shown that the interaction between graphene strips is strong near plasmon resonances and near the Rayleigh anomaly.

Broad band gaps for flexural wave manipulation in plates with embedded periodic strip acoustic black holes

Acoustic black holes (ABHs) achieved by thinning structural thickness following a power-law profile have shown fantastic applications in vibration and noise suppression, energy harvesting, and wave manipulation. For the latter, many passband properties exemplified by self-collimation, focusing, and bi-refraction, have been reported on metamaterial plates with embedded circular ABHs in periodic arrangement. To date, however, band gaps (BGs), being the main feature for metamaterials, have not yet been observed in ABH plates. In this paper, we propose a new class of phononic crystal consisting in two crossed strip ABHs to open broad and complete BGs in plates, based on the dual benefit of local resonance and Bragg scattering effect. The dispersion curves of infinite strip ABHs are recovered by means of the Gaussian expansion method, then validated by a finite element model. The influences of ABH radius, ABH order and residual thickness, on the GBs have been carried out by parametric studies. The results show that the BGs can be modulated to very low frequencies via adjusting the ABH parameters, while maintaining the broadband characteristics. Finally, the wave propagation properties including isolation, guiding, and shielding, in plates with finite periodic arrays have been characterized, indicating that a small number of ABHs are very efficient to quarantine flexural waves. Experimental samples have been fabricated and tested, showing very close results compared to the BGs predicted by the GEM. Even though the proposed strip ABHs seems fragile in strength, peripheral frame can be adopted to partially alleviate this problem. The proposed PCs possess a great potential to broaden ABH applications in vibration control and wave manipulation in plates based on the highlighted BG feature.

Global conformal gauge in string theory

It was supposed for fifty years that the conformal gauge in string theory exists globally on the whole string world sheet. In fact, almost all results were obtained under this assumption. However, this statement was proved only locally in some neighbourhood of an arbitrary point on the worldsheet, and its extension is far from being obvious. In the present paper we prove that the conformal gauge does exists globally on the string worldsheet represented by an infinite strip with straight parallel boundaries.

On null‐controllability of the heat equation on infinite strips and control cost estimate

AbstractWe consider an infinite strip , , , and study the control problem of the heat equation on with Dirichlet or Neumann boundary conditions, and control set . We provide a sufficient and necessary condition for null‐controllability in any positive time , which is a geometric condition on the control set ω. This is referred to as “thickness with respect to ” and implies that the set ω cannot be concentrated in a particular region of . We compare the thickness condition with a previously known necessity condition for null‐controllability and give a control cost estimate which only shows dependence on the geometric parameters of ω and the time T.

An abstract Logvinenko-Sereda type theorem for spectral subspaces

We provide an abstract framework for a Logvinenko-Sereda type theorem, where the classical compactness assumption on the support of the Fourier transform is replaced by the assumption that the functions under consideration belong to a spectral subspace associated with a finite energy interval for some lower semibounded self-adjoint operator on a Euclidean L2-space. Our result then provides a bound for the L2-norm of such functions in terms of their L2-norm on a thick subset with a constant explicit in the geometric and spectral parameters. This recovers previous results for functions on the whole space, hyperrectangles, and infinite strips with compact Fourier support and for finite linear combinations of Hermite functions and allows to extend them to other domains. The proof follows the approach by Kovrijkine and is based on Bernstein-type inequalities for the respective functions, complemented with a suitable covering of the underlying domain.