Oblique wave scattering by a semi-infinite elastic plate with finite draft floating on a step topography

On the basis of a potential theory and Euler-Bernoulli beam theory, an analytical solution for oblique wave scattering by a semi-infinite elastic plate with finite draft floating on a step topography is developed using matched eigenfunction expansions. Different from previous studies, the effects of a wave incident angle, a plate draft, three different plate edge conditions （free, simply supported and built-in） and a sea-bottom topography are all taken into account. Moreover, the plate edge conditions are directly incorporated into linear algebraic equations for determining unknown expansion coefficients in velocity potentials, which leads to a simple and efficient solving procedure. Numerical results show that the convergence of the present solution is good, and an energy conservation relation is well satisfied. Also, the present predictions are in good agreement with known results for special cases. The effects of the wave incident angle, the plate draft, the plate edge conditions and the sea-bottom topography on various hydrodynamic quantities are analyzed. Some useful results are presented for engineering designs.

Draft effect on wave action with a semi-infinite elastic plate

A method for analyzing reflection and transmission of ocean waves from a semi-infinlte elastic plate with a draft is developed. The relation of energy conservation for plates with three different edge conditions （ free, simply supported and built-in） is also derived. It is found that the present method satisfies the energy relation very well. The effects of draft on wave reflection and transmission coefficients as well as on the vertical vibration of the plates are examined through numerical tests. It is demonstrated that the zero draft assumption works well for low wave frequencies, but the effect of plate draft becomes significant for high wave frequencies.

Data-driven prediction of dimensionless quantities for semi-infinite target penetration by integrating machine-learning and feature selection methods

This study employs a data-driven methodology that embeds the principle of dimensional invariance into an artificial neural network to automatically identify dominant dimensionless quantities in the penetration of rod projectiles into semi-infinite metal targets from experimental measurements.The derived mathematical expressions of dimensionless quantities are simplified by the examination of the exponent matrix and coupling relationships between feature variables.As a physics-based dimension reduction methodology,this way reduces high-dimensional parameter spaces to descriptions involving only a few physically interpretable dimensionless quantities in penetrating cases.Then the relative importance of various dimensionless feature variables on the penetration efficiencies for four impacting conditions is evaluated through feature selection engineering.The results indicate that the selected critical dimensionless feature variables by this synergistic method,without referring to the complex theoretical equations and aiding in the detailed knowledge of penetration mechanics,are in accordance with those reported in the reference.Lastly,the determined dimensionless quantities can be efficiently applied to conduct semi-empirical analysis for the specific penetrating case,and the reliability of regression functions is validated.

Analytical solutions of steady vibration of free rectangular plate on semi-infinite elastic foundation

The method of double Fourier transform Was employed in the analysis of the semi-infinite elastic foundation with vertical load. And an integral representations for the displacements of the semi-infinite elastic foundation was presented. The analytical solution of steady vibration of an elastic rectangle plate with four free edges on the semi-infinite elastic foundation was also given by combining the analytical solution of the elastic rectangle plate with the integral representation for displacements of the semiinfinite elastic foundation. Some computational results and the analysis on the influence of parameters were presented.

Conceptual Analysis of a Time-Dependent Double-Diffusive Flow over a Semi-Infinite Vertical Plate

This study is devoted to the analysis of a one-dimensional time-dependent double-diffusive flow over a semi-infinite vertical plate, under a convective surface boundary condition. Using similarity variable, the governing nonlinear partial differential equations have been transformed into a set of coupled nonlinear ordinary differential equations, which are solved numerically by using shooting method alongside with Runge-Kutta integration scheme as embedded in Maple software programme. The numerical results of the skin-friction coefficient, the Nusselt and Sherwood numbers are discussed and depicted graphically.

Scattering of Surface Water Waves Involving Semi-infinite Floating Elastic Plates on Water of Finite Depth

Two problems of scattering of surface water waves involving a semi-infinite elastic plate and a pair of semi-infinite elastic plates,separated by a gap of finite width,floating horizontally on water of finite depth,are investigated in the present work for a two-dimensional time-harmonic case.Within the frame of linear water wave theory,the solutions of the two boundary value problems under consideration have been represented in the forms of eigenfunction expansions.Approximate values of the reflection and transmission coefficients are obtained by solving an over-determined system of linear algebraic equations in each problem.In both the problems,the method of least squares as well as the singular value decomposition have been employed and tables of numerical values of the reflection and transmission coefficients are presented for specific choices of the parameters for modelling the elastic plates.Our main aim is to check the energy balance relation in each problem which plays a very important role in the present approach of solutions of mixed boundary value problems involving Laplace equations.The main advantage of the present approach of solutions is that the results for the values of reflection and transmission coefficients obtained by using both the methods are found to satisfy the energy-balance relations associated with the respective scattering problems under consideration.The absolute values of the reflection and transmission coefficients are presented graphically against different values of the wave numbers.

无界域上具有记忆的非自治Plate方程随机吸引子的存在性

研究无界域上一类具有衰退记忆和加性噪声的非自治Plate方程解的长时间行为.利用一致估计验证了解的拉回渐近紧性,获得了其随机吸引子的存在性.

Interaction between many parallel screw dislocations and a semi-infinite crack in a magnetoelectroelastic solid

Based on the fundamental equations of magnetoelectroelastic material and the analytic theory, and using the Muskhelishvili-introduced well-known elastic techniques combined with the superposition principle, the closed form solution of the generalized stress field of the interaction between many parallel screw dislocations and a semi-infinite crack in an infinite magnetoelectroelastic solid is obtained, on the assumption that the surface of the crack is impermeable electrically and magnetically. Besides, the Peach-Koehler formula of n parallel screw dislocations is given. Numerical examples show that the generalized stress varies with the position of point z and is related to the material constants. The results indicate that the stress concentration occurs at the dislocation core and the tip of the crack. The result of interaction makes the system stay in a lower energy state.

Investigation on the penetration of jacketed rods with striking velocities of 0.9-3.3 km/s into semi-infinite targets

In this study, a combined experimental, numerical and theoretical investigation is conducted on the penetration of semi-infinite 4340 steel targets by a homogeneous 93 W rod and two types of jacketed rods with striking velocities of 0.9-3.3 km/s. The results show that the jacketed rods produced typical“co-erosion” damage at all test velocities, except for the 93 W/1060 Al jacketed rod, which switched from an early “bi-erosion” damage to later “co-erosion” damage at a striking velocity of 936 m/s. However, the homogeneous 93 W rod always forms a large mushroom head during the penetration process. The damage mechanisms of these two types of jacketed rods differ for striking velocities of 0.9-2.0 km/s, but this difference gradually decreases with increased striking velocity. For velocities of 2.0-3.3 km/s, all three types of projectiles exhibit typical hydrodynamic penetration characteristics, and the damage mechanisms of the two types of jacketed rods are almost identical. For the same initial kinetic energy, the penetration performance of the jacketed rods is distinctly superior to that of the homogeneous 93 W rods.Compared with jacket density, jacket strength shows a more significant influence on the damage mechanism and penetration performance of the jacketed rod. Finally, an existing theoretical prediction model of the penetration depth of jacketed rods on semi-infinite targets in the co-erosion mode is modified. It transpires that-in terms of penetration depth-the modified theoretical model is in good agreement with the experimental and numerical observations for 93 W/TC4 and 93 W/1060 Al jacketed rods penetrating semi-infinite 4340 steel targets.

ASYMPTOTIC APPROXIMATION METHOD AND ITS CONVERGENCE ON SEMI-INFINITE PROGRAMMING

The aim of this article is to discuss an asymptotic approximation model and its convergence for the minimax semi-infinite programming problem. An asymptotic surrogate constraints method for the minimax semi-infinite programming problem is presented by making use of two general discrete approximation methods. Simultaneously, the consistence and the epi-convergence of the asymptotic approximation problem are discussed.

Interaction between infinitely many dislocations and a semi-infinite crack in one-dimensional hexagonal quasicrystal

By means of analytic function theory, the problems of interaction between infinitely many parallel dislocations and a semi-infinite crack in one-dimensional hexagonal quasicrystal are studied. The analytic solutions of stress fields of the interaction between infinitely many parallel dislocations and a semi-infinite crack in one-dimensional hexagonal quasicrystal are obtained. They indicate that the stress concentration occurs at the dislocation source and the tip of the crack, and the value of the stress increases with the number of the dislocations increasing. These results are the development of interaction among the finitely many defects of quasicrystals, which possesses an important reference value for studying the interaction problems of infinitely many defects in fracture mechanics of quasicrystal.

Exact solutions of two semi-infinite collinear cracks in piezoelectric strip

Using the complex variable function method and the conformal mapping technique, the fracture problem of two semi-infinite collinear cracks in a piezoelectric strip is studied under the anti-plane shear stress and the in-plane electric load on the partial crack surface. Analytic solutions of the field intensity factors and the mechanical strain energy release rate are derived under the assumption that the surfaces of the crack are electrically impermeable. The results can be reduced to the well-known solutions for a purely elastic material in the absence of an electric load. Moreover, when the distance between the two crack tips tends to infinity, analytic solutions of a semi-infinite crack in a piezoelectric strip can be obtained. Numerical examples are given to show the influence of the loaded crack length, the height of the strip, the distance between the two crack tips, and the applied mechanical/electric loads on the mechanical strain energy release rate. It is shown that the material is easier to fail when the distance between two crack tips becomes shorter, and the mechanical/electric loads have greater influence on the propagation of the left crack than those of the right one.

Analytical models for the penetration of semi-infinite targets by rigid,deformable and erosive long rods

A theoretical study is presented herein on the pen- etration of a semi-infinite target by a spherical-headed long rod for Yp 〉 S, where Yp is the penetrator strength and S is the static target resistance. For Yp 〉 S, depending upon initial impact velocity, there exist three types of penetration, namely, penetration by a rigid long rod, penetration by a deforming non-erosive long rod and penetration by an erosive long rod. If the impact velocity of the penetrator is higher than the hydrodynamic velocity （VH）, it will penetrate the target in an erosive mode; if the impact velocity lies between the hydrodynamic velocity （VH） and the rigid body velocity （VR）, it will penetrate the target in a deformable mode; if the impact velocity is less than the rigid body velocity （VR）, it will penetrate the target in a rigid mode. The critical conditions for the transition among these three penetration modes are proposed. It is demonstrated that the present model predictions correlate well with the experimental observations in terms of depth of penetration （DOP） and the critical transition conditions.

Anti-plane analysis of semi-infinite crack in piezoelectric strip

Using the complex variable function method and the technique of the conformal mapping, the fracture problem of a semi-infinite crack in a piezoelectric strip is studied under the anti-plane shear stress and the in-plane electric load. The analytic solutions of the field intensity factors and the mechanical strain energy release rate are presented under the assumption that the surface of the crack is electrically impermeable. When the height of the strip tends to infinity, the analytic solutions of an infinitely large piezoelectric solid with a semi-infinite crack are obtained. Moreover, the present results can be reduced to the well-known solutions for a purely elastic material in the absence of the electric loading. In addition, numerical examples are given to show the influences of the loaded crack length, the height of the strip, and the applied mechanical/electric loads on the mechanical strain energy release rate.

A Problem of a Semi-Infinite Medium Subjected to Exponential Heating Using a Dual-Phase-Lag Thermoelastic Model

The problem of a semi-infinite medium subjected to thermal shock on its plane boundary is solved in the context of the dual-phase-lag thermoelastic model. The expressions for temperature, displacement and stress are presented. The governing equations are expressed in Laplace transform domain and solved in that domain. The solution of the problem in the physical domain is obtained by using a numerical method for the inversion of the Laplace transforms based on Fourier series expansions. The numerical estimates of the displacement, temperature, stress and strain are obtained for a hypothetical material. The results obtained are presented graphically to show the effect phase-lag of the heat flux and a phase-lag of temperature gradient on displacement, temperature, stress.

Necessary Optimality Conditions for Multi-Objective Semi-Infinite Variational Problem

In this paper, necessary optimality conditions for a class of Semi-infinite Variational Problems are established which are further generalized to a class of Multi-objective Semi-Infinite Variational Problems. These conditions are responsible for the development of duality theory which is an extremely important feature for any class of problems, but the literature available so far lacks these necessary optimality conditions for the stated problem. A lemma is also proved to find the topological dual of as it is required to prove the desired result.

Deriving Vibration Modes of Semi-infinite Chain Model by “Invariant Eigen-operator”Method

For the first time, we introduce a fully quantum mechanical Hamiltonian for a semi-infinite chain model of atoms. We then derive the vibration modes of this model by virtue of the ＂invariant eigen-operator＂ method in two different cases, which is concise and revealing.

Quantum diffusion in semi-infinite periodic and quasiperiodic systems

This paper studies quantum diffusion in semi-infinite one-dimensional periodic lattice and quasiperiodic Fibonacci lattice. It finds that the quantum diffusion in the semi-infinite periodic lattice shows the same properties as that for the infinite periodic lattice. Different behaviour is found for the semi-infinite Fibonacci lattice. In this case, there are still C（t） - t^-δ and d（t） - t^β. However, it finds that 0 〈δ 〈 1 for smaller time, and δ = 0 for larger time due to the influence of surface localized states. Moreover, β for the semi-infinite Fibonacci lattice is much smaller than that for the infinite Fibonacci lattice. Effects of disorder on the quantum diffusion are also discussed.

FINITE LAYER ANALYSIS FOR SEMI-INFINITE SOILS (Ⅱ)——SPECIAL CASES AND NUMERICAL EXAMPLES

In the present paper reductions of the finite layer mathod once studied in detail by the authors for the elastodvnamics of transverse isotropic bodies are given to several special cases. Two-dimensional problems, axisymmetric problems and static problems are discussed, respectively, and this finite layer method is also generalized to the problems in which materials possess viscous properties. Two numerical examples have been presented for the axisymmetric case. From these two examples it can be concluded that the finite layer method can be used to analyse semi-infinite layered soils and to deal with the problem of the interaction between soils and structures.

FINITE LAYER ANALYSIS FOR SEMI-INFINITE SOILS (Ⅰ)——GENERAL THEORY

In the present paper a finite layer method is studied for the flastodynarnics of transverse isotropic bodies. With this method, semi-infinite soils can be considered as an transverse isotropic half-space, its material functions varying with depth. Dividing the half-space into a scries of layers in the direction of depth, the material junctions in each layer are simulated by exponential functions Consequently, the fundamental equations to be solved can be simplified if the Fourier transform with repsect to coordinates is used. We have obtained the relationship between the 'layer forces' and 'layer displacements'. This finite layer method, in fact, can also be called a semi-analytical method. It possesses those advantages as the usual semi-analytical methods do, and can be used to analyse the problem of the interaction between soils and structures.