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This paper investigates the bending fracture problem of a micro/nanoscale cantilever thin plate with surface energy, where the clamped boundary is partially debonded along the thickness direction. Some fundamental mechanical equations for the bending problem of micro/nanoscale plates are given by the Kirchhoff theory of thin plates, incorporating the Gurtin-Murdoch surface elasticity theory. For two typical cases of constant bending moment and uniform shear force in the debonded segment, the associated problems are reduced to two mixed boundary value problems. By solving the resulting mixed boundary value problems using the Fourier integral transform, a new type of singular integral equation with two Cauchy kernels is obtained for each case, and the exact solutions in terms of the fundamental functions are determined using the Poincare-Bertrand formula. Asymptotic elastic fields near the debonded tips including the bending moment, effective shear force, and bulk stress components exhibit the oscillatory singularity. The dependence relations among the singular fields, the material constants, and the plate's thickness are analyzed for partially debonded cantilever micro-plates. If surface energy is neglected, these results reduce the bending fracture of a macroscale partially debonded cantilever plate, which has not been previously reported. [ABSTRACT FROM AUTHOR]

In this paper, we study variation-inequality initial-boundary value problems with fouth order p(x)-Kirchhoff operators. First, an operator is constructed based on the Leray Schauder principle, and the existence of solutions is obtained. Secondly, the stability and uniqueness of the solution are analyzed after the conditions are appropriately relaxed on the Kirchhoff operators. [ABSTRACT FROM AUTHOR]

From an initial boundary value problem for the system of nonlinear differential equations, which describes the large deflection of a rectangular shell, we obtain a nonlinear integrodifferential equation for the transverse displacement. This equation is analogous by its structure to the Kirchhoff equation for a string, the Woinowsky-Krieger equation for a beam and the Berger equation for a plate which are united under the common name Kirchhoff type equations. [ABSTRACT FROM AUTHOR]

Highlights • A unified size-dependent plate model for buckling analysis of nanoplates is developed. • The governing equations are derived based on nonlocal strain gradient theory incorporating surface effects. • Various higher-order plate theories as well as Kirchhoff and Mindlin plate theories are included. • Analytical solutions for critical buckling load of rectangular nanoplates under various boundary conditions are obtained. • The effects of small scale parameters, geometric parameters, boundary condition, shear deformation and surface energy are studied. Abstract Based on the nonlocal strain gradient theory and surface elasticity theory, a unified size-dependent plate model is developed for buckling analysis of rectangular nanoplates. The developed model is capable of capturing nonlocal effect, strain gradient effect as well as surface energy effects simultaneously. Moreover, by selecting appropriate shape function, the present model can be reduced to not only Kirchhoff and Mindlin plate models but also various higher-order shear deformation plate models. The non-classical governing equations and associated boundary conditions are established by using the principle of minimum potential energy. Analytical solutions for critical buckling load of rectangular nanoplates under various boundary conditions are obtained. Verification of the proposed model is carried out by comparing the degenerated results with those reported in open literature. The effects of nonlocal parameter, material length scale parameter, geometric parameters, shear deformation and surface energy on the buckling behavior of rectangular nanoplates under different boundary conditions are discussed in detail. The numerical results show that the critical buckling load evaluated by nonlocal strain gradient theory is lower than that predicted by classical continuum theory when the nonlocal parameter is larger than the material length scale parameter, and is higher than that evaluated by classical continuum theory when the nonlocal parameter is smaller than the material length scale parameter. However, when taking surface effects into account, the critical buckling load is mainly affected by surface effects at large length-to-thickness ratio, and depends on the combined effects of nonlocality, strain gradient and surface energy at small length-to-thickness ratio. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of nontrivial solutions for critical Hardy–Schrödinger–Kirchhoff systems driven by the fractional p-Laplacian operator. Existence is derived as an application of the mountain pass theorem and the Ekeland variational principle. The main features and novelty of the paper are the presence of the Hardy terms as well as critical nonlinearities. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following critical nonlocal problem: { M ⁢ (∬ℝ2N |u(x) − u(y)|2/|x − y|N+2s⁢ 𝑑x 𝑑y) ⁢(− Δ)su = λ/uγ + u2s*−1 in ⁢ Ω, u > 0 in Ω, u = 0 in ⁢ ℝN∖Ω, where Ω is an open bounded subset of ℝ N with continuous boundary, dimension N > 2s with parameter s ∈ (0 , 1), 2*s = 2N /(N − 2s) is the fractional critical Sobolev exponent, λ > 0 is a real parameter, γ ∈ (0 , 1) and M models a Kirchhoff-type coefficient, while (−Δ)s is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is, when the Kirchhoff function M is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and uniqueness of global solutions and uniform stabilization of the energy of initial‐boundary value problems for quasilinear equations of the Kirchhoff type. The asymptotic stabilization of the solutions of these problems are obtained without any dissipative mechanism acting on the displacement variable. [ABSTRACT FROM AUTHOR]

A simple skew-symmetric Nitsche’s formulation is introduced into the framework of isogeometric analysis (IGA) to deal with various problems in small strain elasticity: essential boundary conditions, symmetry conditions for Kirchhoff plates, patch coupling in statics and in modal analysis as well as Signorini contact conditions. For linear boundary or interface conditions, the skew-symmetric formulation is parameter-free. For contact conditions, it remains stable and accurate for a wide range of the stabilization parameter. Several numerical tests are performed to illustrate its accuracy, stability and convergence performance. We investigate particularly the effects introduced by Nitsche’s coupling, including the convergence performance and condition numbers in statics as well as the extra “outlier” frequencies and corresponding eigenmodes in structural dynamics. We present the Hertz test, the block test, and a 3D self-contact example showing that the skew-symmetric Nitsche’s formulation is a suitable approach to simulate contact problems in IGA. [ABSTRACT FROM AUTHOR]

We establish the existence of at least three distinct weak solutions for a perturbed nonlo- cal fourth-order Kirchhoff-type problem with Navier boundary conditions under appropriate hypotheses on nonlinear terms. Our main tools are based on variational methods and some critical points theorems. We give some examples to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the existence of solutions for nonlinear nonmonotone evolution equations in Banach spaces involving anti-periodic boundary conditions. Our approach in this study relies on the theory of monotone and maximal monotone operators combined with the Schaefer fixed-point theorem and the monotonicity method. We apply our abstract results in order to solve a diffusion equation of Kirchhoff type involving the Dirichlet p-Laplace operator. [ABSTRACT FROM AUTHOR]

This article studies a partially clamped flexible ultrathin elastic nanoplate with surface effect. An emphasis is to analyze the singular elastic behavior near the tips of partially clamped boundary. A mixed boundary value problem associated with a partially clamped nanoplate having parabolic deflection is presented and solved. Using the Kirchhoff plate theory incorporating the Gurtin–Murdoch surface elasticity, some basic equations for nanoplates with surface elasticity are established. With the aid of the Fourier transform and the superposition principle, two coupled singular integral equations with Cauchy kernel are derived during bending of the nanoplate and then converted to a singular integral equation with two Cauchy kernels. The exact solution to the singular integral equation with two Cauchy kernels is determined and the oscillatory singular elastic fields near the tips of the clamped boundary are given in elementary functions. The bending moments and the normal stress components in the vicinity of the tips simultaneously have the oscillator singular behavior like r − 3 / 2 − i ɛ , r being the distance from the tip and ɛ relying on the material properties. Besides, the effective shear force possesses an r − 5 / 2 − i ɛ singular behavior. The influences of the geometrical parameters and material properties of nanoplates on the oscillatory singular behavior are presented graphically. When surface effect is neglected, the solution is applicable to classical thin plates with partially clamped boundary. • Disturbed field for bending of a partially clamped nanoplate with surface effect is studied. • Oscillatory singularity occurs near the tips of clamped edge. • Bending moments and normal stresses exhibit the oscillatory behavior of r̂{-3/2-i ɛ }. • Effective shear force exhibits the oscillatory behavior of r̂{-5/2-i ɛ }. [ABSTRACT FROM AUTHOR]

The object of this work is to study the existence of solutions for a class of p(x)-Kirchhoff type problem under no-flux boundary conditions with dependence on the gradient. We establish our results by using the degree theory for operators of (S+) type in the framework of variable exponent Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, for investigating a boundary value problem for a nonlinear biharmonic equation of Kirchhoff type, we propose an efficient approach by the reduction of the problem to an operator equation for the nonlinear part of the equation. The result is that we have established the existence and uniqueness of a solution under some easily verified conditions. Also, we propose an iterative method for finding the solution. Some examples demonstrate the applicability of the theoretical results and the efficiency of the iterative method. [ABSTRACT FROM AUTHOR]

We recast the displacement-traction problem of the Kirchhoff-Love theory of linearly elastic plates as a boundary value problem with the bending moments and stress resultants inside the middle section of the plate as the sole unknowns, instead of the displacement field in the classical formulation. To this end, we show in particular how to recast the Dirichlet boundary conditions satisfied by the displacement field of the middle surface of a plate as boundary conditions satisfied by the bending moments and stress resultants. [ABSTRACT FROM AUTHOR]

The boundary-value problem for a block structure consisting of an elastic half-space that occupies the lower part of a Cartesian coordinate system and interacts with two block elements in the form of unbounded wedges with right angles at the vertices is considered. The horizontal boundaries of the wedges are rigidly connected with the half-space boundary, and vertical ones are parallel planes. We consider the antiplane problem for cases where the distance between the vertical boundaries of block elements is present or absent. By the example of a three-dimensional problem degenerating into a two-dimensional one, it is shown that an initial earthquake is possible, like in the case of using Kirchhoff plates instead of three-dimensional block elements. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence of multiple nontrivial solutions for the following fractional p-Laplacian Kirchhoff problems { (∬R2N |u(x)− u(y) |p/|x−y|N+ps dxdy)θ−1 (−Δ)psu = λ|u|θp−2u +|u|ps∗−2u in Ω, u = 0 in RN∖Ω, where parameter λ > 0 belongs to some left neighbourhood of the eigenvalue of the nonlocal operator (∬R2N|u(x)−u(y)|p/|x−y|N+ps dxdy)θ−1(−Δ)ps. The main feature and difficulty of our problems is the fact that the problem is degenerate. [ABSTRACT FROM AUTHOR]

In the present paper, we consider nonlocal linear elastic Kirchhoff plates, where we restrict to thin, isotropic, and homogeneous plates under the action of static transverse forces, utilizing the differential equation form of the Eringen nonlocal continuum theory. For the case of simply supported plates of polygonal planform, we derive analogies between the solutions of the nonlocal Kirchhoff theory and its local counterpart. First, we extend the Marcus decomposition method for local Kirchhoff plates, where we show that, analogous to the local case considered by Marcus, the moment sum and the nonlocal deflection are both governed by Poisson boundary value problems, which correspond to auxiliary (local) membrane problems. In the present context, it eventually follows that there is a nonvanishing term responsible for a correction of the deflection due to the nonlocal effect, while the moments and shear forces of the local and the nonlocal plate coincide. The nonlocal correction of the deflection turns out to be governed by a membrane-type boundary value problem again. From this fact, it follows immediately that the correction for the deflection due to the nonlocal effect can be derived alternatively from both the local deflection and the local moment sum, which represents a substantial simplification of the nonlocal computations. For the sake of demonstration, we consider examples with closed form solutions. We first consider (as limiting case) infinite plates under the action of single forces, where the singular behavior of Green's function of the nonlocal Kirchhoff theory can be clarified. Then we discuss the bending of nonlocal plate strips for comparison sake, as well as equilateral triangular plates. To our best knowledge, no results for bending of nonlocal triangular plates have been presented in the literature so far. The present paper has been strongly influenced by a former contribution on analogies between simply supported polygonal Kirchhoff plates rigid in shear and shear-deformable plate solutions. This paper was co-authored together with our teacher, the late Professor Franz Ziegler, to whom we dedicate the present paper. In light of this latter contribution, and extending an idea promoted by Professor Franz Ziegler and his co-workers in the framework of various other fields, we finally show in the Appendix that the deflections of simply supported nonlocal Kirchhoff plates can be considered as deflections of a local "background" Kirchhoff plate under the action of additional fictitious eigenstrain loadings, such as thermal moments. [ABSTRACT FROM AUTHOR]

We consider the existence of ground state solutions for the Kirchhoff type problem − ( a + b ∫ R N | ∇ u | 2 d x ) △ u + V ( x ) u = | u | p − 2 u , x ∈ R N , u ∈ H 1 ( R N ) , where a , b > 0 , N = 1 , 2 , 3 and 2 < p < 2 ∗ . Here we are interested in the case that 2 < p ≤ 4 since the existence of ground state for 4 < p ≤ 2 ∗ is easily obtained by a standard variational argument. Our method is based on a Poho z ̆ aev type identity. [ABSTRACT FROM AUTHOR]

This paper is devoted to the question of global and local asymptotic stability for nonlinear damped Kirchhoff systems, with homogeneous Dirichlet boundary conditions, under fairly natural assumptions on the external force f = f ( t , x , u ) , the distributed damping Q = Q ( t , x , u , u t ) , the perturbation term μ | u | p − 2 u and the dissipative term ϱ ( t ) M ( [ u ] s p ) | u t | p − 2 u t , with ϱ ≥ 0 and in L loc 1 ( R 0 + ) , when the initial data are in a special region. Here u = ( u 1 , … , u N ) = u ( t , x ) represents the vectorial displacement, with N ≥ 1 . Particular attention is devoted to the asymptotic behavior of the solutions in the linear case specified in Section 5 . Finally, the results are extended to problems where the fractional p -Laplacian is replaced by a more general elliptic nonlocal integro-differential operator. The paper extends in several directions recent theorems and covers also the so-called degenerate case, that is the case in which M is zero at zero. [ABSTRACT FROM AUTHOR]

The initial boundary value problem of a class of Kirchhoff equations with strong damping and nonlinear dissipation is considered. By modifying Vitillaro's argument, we prove a blow-up result for solutions with positive and negative initial energy respectively. [ABSTRACT FROM AUTHOR]

In this paper, we present a modified Kirchhoff theory for analyzing thermo-mechanical static and buckling responses of functionally graded material (FGM) isotropic and sandwich plates. In comparison to the classical Kirchhoff theory, the proposed method is taken account into the shear deformation effects, then it can be applied for both moderately thick and thin plates. On the other hand, comparing to the first order shear deformation theory (FSDT) or Reissner-Mindlin plate theory, the number of independent unknowns reduces one variable and retains four degrees of freedom (Dofs) per node. The shear locking phenomenon is also negligible due to extending from the classical Kirchhoff theory. Furthermore, a new normalization of a quartic form is used to build the shape functions in the radial point interpolation method (RPIM). The novel imposing essential boundary conditions related to slopes of deflections has been proposed, then the enforcement of these gradient deflections becomes easily and directly as the same in finite element analysis. The obtained numerical results for the thermo-mechanical static and buckling problems from the proposed method are stable and well accurate prediction as comparing to the exact solutions and other published analyses. [ABSTRACT FROM AUTHOR]

We consider a weak viscoelastic Kirchhoff plate model with time-varying delay in the boundary. By using a suitable energy and Lyapunov function, we obtain a decay rate for the energy, which depends on the behavior of g and α. [ABSTRACT FROM AUTHOR]

We investigate the existence of multiple solutions for perturbed nonlocal fourth-order equations of Kirchhoff type under Navier boundary conditions. We give some new criteria for guaranteeing that the perturbed fourth-order equations of Kirchhoff type have at least three weak solutions by using a variational method and some critical point theorems due to Ricceri. We extend and improve some recent results. Finally, by presenting two examples, we ensure the applicability of our results. [ABSTRACT FROM AUTHOR]

A hybrid method is proposed to predict the dynamic behavior of functionally graded (FG) plate subjected to a moving mass. The governing equations of motion of FG plate are derived using the Kirchhoff plate theory and Lagrange equation. Improved Rayleigh–Ritz solution is used to treat the spatial partial derivatives. Penalty method is employed to deal with the constraints, and the energy terms due to boundary conditions are included in Lagrange, hence it is not necessary to particularly consider the constraints in the modeling process. And the combination of simple polynomials and trigonometric functions is selected as the admissible functions. The advantage of this improvement in Rayleigh–Ritz method is that it is not needed to find satisfied admissible functions for different boundary conditions while the convergence of the solution is improved. Meanwhile, the method can be used to handle the versatile boundary conditions. Differential quadrature method (DQM) as a step-by-step time integration scheme is employed for discretization of temporal derivatives. The validated results show that the presented method is very reliable and efficient, and its convergence and accuracy are also better compared to finite element method for solving the dynamic problems of FG plate with moving loads (force and mass). Moreover, the influences of material properties and boundary conditions on maximum dynamic deflections are investigated, as well as moving speeds and inertial effects of loads (mass and force). Although only four edge boundary conditions are addressed in the present work, the proposed procedure is applicable for any arbitrary edge boundary conditions. [ABSTRACT FROM AUTHOR]

The non-symmetric variant of Nitsche's method was recently applied successfully for variationally enforcing boundary and interface conditions in non-boundary-fitted discretizations. In contrast to its symmetric variant, it does not require stabilization terms and therefore does not depend on the appropriate estimation of stabilization parameters. In this paper, we further consolidate the non-symmetric Nitsche approach by establishing its application in isogeometric thin shell analysis, where variational coupling techniques are of particular interest for enforcing interface conditions along trimming curves. To this end, we extend its variational formulation within Kirchhoff-Love shell theory, combine it with the finite cell method, and apply the resulting framework to a range of representative shell problems based on trimmed NURBS surfaces. We demonstrate that the non-symmetric variant applied in this context is stable and can lead to the same accuracy in terms of displacements and stresses as its symmetric counterpart. Based on our numerical evidence, the non-symmetric Nitsche method is a viable parameter-free alternative to the symmetric variant in elastostatic shell analysis. [ABSTRACT FROM AUTHOR]

In this paper we consider the existence and general energy decay rate of global solution to the mixed problem for the Kirchhoff-type equation with memory boundary and acoustic boundary conditions. In order to prove the existence of solutions, we employ the Galerkin method and compactness arguments. Besides, we establish an explicit and general decay rate result using the perturbed modified energy method and some properties of the convex functions. Our result is obtained without imposing any restrictive assumptions on the behavior of the relaxation function at infinity. These general decay estimates extend and improve some earlier results, i.e., exponential or polynomial decay rates. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following p-Kirchhoff equation: with Dirichlet boundary conditions, where Ω is a bounded domain in $\mathbb{R}^{N}$ . Under proper assumptions on M and f, we obtain three existence theorems of infinitely many solutions for problem (P) by the fountain theorem. Moreover, for a special nonlinearity $f(x,u)=\lambda |u|^{q-2}u+|u|^{r-2}u$ ( $1< q< p< r< p^{*}$ ), we prove that problem (P) has at least two nonnegative solutions via the Nehari manifold method and a sequence of solutions with negative energy by the dual fountain theorem. [ABSTRACT FROM AUTHOR]

On the basis of the modified strain gradient theory, the present paper deals with the theoretical analysis of the free vibration of coupled double-FGM viscoelastic nanoplates by Kelvin-Voigt visco-Pasternak medium. To establish static equilibrium of atoms on the each nanoplate surface, the effects of the surface layers are considered. The properties of material in the thickness direction vary according to the power low distribution. Kirchhoff plate assumption and Hamilton's variational principle are employed to achieve the partial differential equations for three different cases of vibration (out-of-phase, in-phase, and one nanoplate of the system being stationary) and corresponding boundary conditions. Navier's approach which satisfies the simply supported boundary conditions applied to analytically investigate the size effect on the natural frequencies of double-FGM viscoelastic nanoplate systems. Numerical studies are carried out to illustrate the influence of viscoelastic damping structural of the nanoplates, damping coefficient of the visco-Pasternak medium, independent length scale parameter, aspect ratio, surface properties, and other factors on the frequency behavior system. Some numerical results of this research illustrate that the frequencies may increase or decrease with respect to the sign of the surface properties of FGMs. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of positive solutions to a class of nonlocal boundary value problem of the p-Kirchhoff type [ABSTRACT FROM AUTHOR]

Sixth-order boundary value problems for gradient-elastic Kirchhoff plate bending models are formulated in a variational form within an H 3 Sobolev space setting. Existence and uniqueness of the weak solutions are then established by proving the continuity and coercivity of the associated symmetric bilinear forms. Complete sets of boundary conditions, including both the essential and the natural conditions, are derived accordingly. In particular, the gradient-elastic Kirchhoff plate models feature two different types of clamped and simply supported boundary conditions in contrast to the classical Kirchhoff plate model. These new types of boundary conditions are given additional attributes singly and doubly ; referring to free and prescribed normal curvature, respectively. The formulations and results of the analyzed strain gradient plate models are compared to two other generalized Kirchhoff plate models which follow a modified strain gradient elasticity theory and a modified couple stress theory. It is clarified that the results are extendable to these model variants as well. Finally, the relationship of the natural boundary conditions to external loadings is analyzed. [ABSTRACT FROM AUTHOR]

The Kirchhoff approximation (KA) is used to model backscatter of sound from a partially exposed, rigid sphere at a flat free interface of two homogenous media. Scattered wavefields are calculated through numerical integration on the sphere of the Kirchhoff integral, requiring detailed knowledge of the illuminated region for each scattering path. This approach avoids amplitude discontinuities resulting from geometric transitions in the number of reflected rays. Reflections from the interface are modeled through use of an image source, positioned symmetrically relative to the real source. Results are compared to experimentally obtained backscattering records from elastic spheres at an air-water interface, as well as to an exact partial wave series for a half exposed sphere. These comparisons highlight the omission of Franz-type reflections from consideration within the KA, and the consequences of this omission are discussed. The results can be extended to boundary conditions beyond the ideal free surface limit, and are applicable to the problem of scattering by underwater objects partially buried in sand. [ABSTRACT FROM AUTHOR]

A comprehensive method is proposed to predict the dynamic behaviors of flat plate of arbitrary boundary conditions subjected to moving loads, based on Kirchhoff plate theory. The governing equations of motion are derived using the Lagrange equation. Rayleigh–Ritz method is employed and extended to treat the spatial partial derivatives. Different with conventional Rayleigh–Ritz solutions, the admissible functions adopted here integrate the advantages of both polynomials and trigonometric functions, which just satisfy a totally unconstrained condition, and Courant's penalty method is used to handle constraints. Differential quadrature method is used for discretization of temporal derivatives. The results show that the presented method is very reliable and efficient, and its convergence and accuracy are also better compared to finite element method. Moreover, the method is good for dealing with the boundary conditions due to employing suitable admissible functions. To illustrate this, the method presented evaluates the dynamic response of a plate example, whose three edges are usual constrains, and the fourth edge connects to a real spring with arbitrary length and stiffness value. [ABSTRACT FROM AUTHOR]

Kirchhoff's 1882 theory of optical diffraction forms the centerpiece in the long-term development of wave optics, one that commenced in the 1820s when Fresnel produced an empirically successful theory based on a reinterpretation of Huygens' principle, but without working from a wave equation. Then, in 1856, Stokes demonstrated that the principle was derivable from such an equation albeit without consideration of boundary conditions. Kirchhoff's work a quarter century later marked a crucial, and widely influential, point for he produced Fresnel's results by means of Green's theorem and function under specific boundary conditions. In the late 1880s, Poincaré uncovered an inconsistency between Kirchhoff's conditions and his solution, one that seemed to imply that waves should not exist at all. Researchers nevertheless continued to use Kirchhoff's theory-even though Rayleigh, and much later Sommerfeld, developed a different and mathematically consistent formulation that, however, did not match experimental data better than Kirchhoff's theory. After all, Kirchhoff's formula worked quite well in a specific approximation regime. Finally, in 1964, Marchand and Wolf employed the transformation of Kirchhoff's surface integral that had been developed by Maggi and Rubinowicz for other purposes. The result yielded a consistent boundary condition that, while introducing a species of discontinuity, nevertheless rescued the essential structure of Kirchhoff's original formulation from Poincaré's paradox. [ABSTRACT FROM AUTHOR]

In this article, using Mountain Pass Theorem, we investigate the existence of a nontrivial weak solution for nonlocal equations driven by p(x)-Laplacian, under Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

In the previous works, the boundary conditions of Kirchhoff micro-plate model are derived by only considering the cases of four-edge simply supported and/or four-edge clamped micro-plate, while it is invalid for the plates which have at least one free edge. The general boundary conditions as well as the governing equation of Kirchhoff micro-plate are re-derived based on a strain gradient elasticity theory. The general boundary conditions can degenerate into the corresponding ones of the classical continuum model. With the general boundary conditions, the deflection of some typical cases of plate are studied and discussed. Moreover, the effect of material length scale parameters on the deflection and the size effect of these plates are also discussed. The present work extends the former works to be a generalized one since the generalized boundary conditions may deal with different reasonable boundary conditions. [ABSTRACT FROM AUTHOR]

The aim of this note is to investigate the existence of signed and sign-changing solutions to the Kirchhoff type problem [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the boundary value problem ... where Ω is a bounded smooth domain of ℝN (N = 1; 2 or 3), a > 0, b ≥ 0, and f :?&#10799 ℝℝℝ; is a continuous function which is globally 3-superlinear. By using some variants of the mountain pass theorem established in this paper, we show that this problem has at least three solutions: one positive, one negative, and one which changes its sign. Furthermore, in case f is odd with respect to u we obtain an unbounded sequence of sign-changing solutions. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the existence of weak solutions for a Kirchhoff type problem driven by a non-local integro-differential operator of elliptic type with homogeneous Dirichlet boundary conditions as follows: { M ( ∫ R 2 N | u ( x ) − u ( y ) | p K ( x − y ) d x d y ) L K p u = f ( x , u ) in Ω , u = 0 in R N ∖ Ω , where L K p is a non-local operator with singular kernel K , Ω is an open bounded subset of R N with Lipshcitz boundary ∂ Ω , M is a continuous function and f is a Carathéodory function satisfying the Ambrosetti–Rabinowitz type condition. We discuss the above-mentioned problem in two cases: when f satisfies sublinear growth condition, the existence of nontrivial weak solutions is obtained by applying the direct method in variational methods; when f satisfies suplinear growth condition, the existence of two nontrivial weak solutions is obtained by using the Mountain Pass Theorem. [ABSTRACT FROM AUTHOR]

Expressions for the cross correlation of omnidirectional, broadband continuous-wave (CW) acoustic signals scattered by a pressure-release random surface are derived for both vertically and horizontally spaced hydrophones using the Kirchhoff approximation for the boundary values and the Fresnel phase approximation. The resulting expressions for the spatial coherence are simple in form and require no approximations in the description of the random surface. As the surface becomes perfectly smooth, the expressions reduce to the image solution, thus ensuring the proper limiting behavior. Two narrowband models for vertical coherence were compared to experimental data collected during a sea test at the frequencies of 14, 16, 18, and 20 kHz with a maximum hydrophone spacing of 1.4 m. One model used the measured sound-speed profile (SSP) to estimate input parameters, and the other model assumed isospeed conditions. The agreement between the experimental data and the model based on the measured SSP is very good, and the agreement with the isospeed model is almost as good. [ABSTRACT FROM PUBLISHER]

Simultaneously, considering the viscous effect of material, damping of medium, and rotational inertia, we study a kind of more general Kirchhoff-type extensible beam equation utt-uxxtt+uxxxx-σ(∫0l‍(ux)2dx)uxx-ϕ(∫0l‍(ux)2dx)uxxt=q(x), in  [0,L]×R+ with the structural damping and the rotational inertia term. Little attention is paid to the longtime behavior of the beam equation under nonlinear boundary conditions. In this paper, under nonlinear boundary conditions, we prove not only the existence and uniqueness of global solutions by prior estimates combined with some inequality skills, but also the existence of a global attractor by the existence of an absorbing set and asymptotic compactness of corresponding solution semigroup. In addition, the same results also can be proved under the other nonlinear boundary conditions. [ABSTRACT FROM AUTHOR]

Aeroelastic properties of two-dimensional panels neglecting and considering the shear deformation in supersonic airflow are analytically investigated. The two types of panel are known as the Kirchhoff and Mindlin panels. A method for estimating the flutter bound of the aeroelastic structural system is proposed. The classical plate theory and the first-order shear deformation theory are used in the structural modeling. The unsteady aerodynamic pressure is evaluated by the supersonic piston theory. The governing equation of motion of the structural system is formulated applying Hamilton's principle. The exact solution for the partial differential equation of motion of the aeroelastic structural system is calculated, and the aeroelastic modes (AEMs) of the panels are obtained and compared with the vacuo natural modes of the structures. The correctness of the present methodology used in the aeroelastic analysis is verified. Aeroelastic characteristics of panels with different boundary conditions are investigated, and the time response history of the panel is calculated using the AEMs obtained in this study. Moreover, the accuracy of the assumed mode method in the aeroelastic analysis is also researched. Some interesting and useful results are obtained and analyzed. [ABSTRACT FROM AUTHOR]

We obtain a numerical-analytic solution of the thermal problem of friction for two semibounded bodies with regard for the heat sensitivity of materials and changes in the relative sliding velocity with time (braking with constant deceleration). The linearization of the corresponding boundary-value problem of heat conduction is performed by using the Kirchhoff substitution and the method of linearizing parameters. We study the changes in the temperatures of the working surfaces of the pad and the disk in the process of braking for two frictional couples with and without taking into account the temperature dependences of their thermal properties. [ABSTRACT FROM AUTHOR]

The article presents a study that determines the structure of the spectrum of a net of quantum waveguides and bounded solutions. It describes the method of the study that uses a rigorous asymptotic analysis of the problem and illustrates that asymptotically justified are not the Kirchhoff transmission conditions. It outlines the findings of the study which indicates that the boundary layer problems in the infinite domains have unique stable bounded solution in Neumann boundary conditions.

In this paper, by the Kirchhoff transformation, a Dirichlet-Neumann (D-N) alternating algorithm which is a non-overlapping domain decomposition method based on natural boundary reduction is discussed for solving exterior anisotropic quasilinear problems with circular artificial boundary. By the principle of the natural boundary reduction, we obtain natural integral equation for the anisotropic quasilinear problems on circular artificial boundaries and construct the algorithm and analyze its convergence. Moreover, the convergence rate is obtained in detail for a typical domain. Finally, some numerical examples are presented to illustrate the feasibility of the method. [ABSTRACT FROM AUTHOR]

In this paper we consider the boundary value problem of some nonlinear Kirchhoff-type equation with dissipation. We also estimate the total energy of the system over any time interval [0, T] with a tolerance level γ. The amplitude of such vibrations is bounded subject to some restrictions on the uncertain disturbing force f. After constructing suitable Lyapunov functional, uniform decay of solutions is established by means of an exponential energy decay estimate when the uncertain disturbances are insignificant. [ABSTRACT FROM AUTHOR]

We present the small-amplitude vibrations of a circular elastic ring with periodic and clamped boundary conditions. We model the rod as an inextensible, isotropic, naturally straight Kirchhoff elastic rod and obtain the vibrational modes of the ring analytically for periodic boundary conditions and numerically for clamped boundary conditions. Of particular interest are the dependence of the vibrational modes on the torsional stress in the ring and the influence of the rotational inertia of the rod on the mode frequencies and amplitudes. In rescaling the Kirchhoff equations, we introduce a parameter inversely proportional to the aspect ratio of the rod. This parameter makes it possible to capture the influence of the rotational inertia of the rod. We find that the rotational inertia has a minor influence on the vibrational modes with the exception of a specific category of modes corresponding to high-frequency twisting deformations in the ring. Moreover, some of the vibrational modes over or undertwist the elastic rod depending on the imposed torsional stress in the ring. [ABSTRACT FROM AUTHOR]

Abstract: We develop a new numerical scheme for a fourth order elliptic partial differential equation based on Kirchhoffʼs thin plate theory. In particular we extend the ultra weak variational formulation (UWVF) to thin plate problems with clamped plate boundary conditions. The UWVF uses a finite element mesh and non-polynomial basis functions. After deriving the new method we then prove norm convergence on the boundary. Finally we investigate numerically the feasibility of the UWVF for both homogeneous and inhomogeneous problems and show examples of p- and h-convergence. [Copyright &y& Elsevier]