Transient bending waves in plates studied by hologram interferometry

Propagating bending waves are studied in plates made of aluminum and wood. The waves are generated by the impact of a ballistic pendulum. Hologram interferometry, with a double pulsed ruby laser as the light source, is used to record the out of plane motion of the waves. Elliptic-like fringes visualize differences in wave speed for different directions in the anisotropic plate and circular ones are obtained for the isotropic plate. The experimental data for the isotropic plate compare favorably with analytical results derived from the Kirchhoff-plate equation with a point impact of finite duration. A similarity variable is found when starting conditions are modeled as a Dirac pulse in space and time, that brings new understanding to the importance of specific parameters for wave propagation in plates. A formal solution is obtained for a point force with an arbitrary time dependence. For times much larger than the contact time, the plate deflection is shown to be identical to that from a Dirac pulse applied at the mean contact time. A method for determining material parameters, and the mean contact time, from the interferograms is hence developed.     

Coupled hydrodynamic and structural analysis of compressible jet impact onto elastic panels

This paper is concerned with the initial stage of a compressible liquid jet impact onto an elastic plate. The fluid flow is governed by the linear wave equation, while the response of the plate is governed by the classical linear dynamical plate equation. The coupling between the fluid flow and the plate deflection is taken into account through the dynamic and kinematic conditions imposed on the wetted part of the plate. The problem is solved numerically by the normal mode method. The principal coordinates of the hydrodynamic pressure and plate deflections satisfy a system of ordinary differential and integral equations. A time stepping method based on the Runge–Kutta scheme is used for the numerical integration of the system. Calculations are performed for two-dimensional, axisymmetric and three-dimensional jet impacts onto an elastic plate. The effects of the impact conditions and the elastic properties of the plate on the magnitudes of the elastic deflections and bending stresses are analysed.     

Effect of permanent indentation on the delamination threshold for small mass impact on plates

Small mass impacts on composite structures are common cases caused by hailstones and runway debris. Small mass impactors usually result in a wave controlled local response, which is independent of boundary conditions. This response occurs before the reflection of waves from the boundaries and cannot be modeled by large mass drop weight tests. An elasto-plastic contact law, which accounts for permanent indentation and damage effects, was used here to study small mass impact on laminated composite plates. By comparing with results from the Hertzian contact law, it was found that damage can change the dynamic response of the structure significantly with increasing impact velocity. Due to smaller contact force generated for the case of using elasto-plastic contact, the central displacement of the plate is also less than the one using Hertzian contact law. The linearized version of the contact law was then used to derive the closed-form approximations of the contact force, indentation and plate central displacement for the impact loading of composite laminates. The threshold velocity for delamination onset under small mass impact was predicted analytically based on the obtained peak contact forces by combining with an existing quasi-static delamination threshold load criterion. A good agreement was found between the predicted threshold values and published experimental results.     

Generation of ramp waves using variable areal density flyers

Ramp loading using graded density impactors as flyers in gas-gun-driven plate impact experiments can yield new and useful information about the equation of state and the strength properties of the loaded material. Selective Laser Melting, an additive manufacturing technique, was used to manufacture a graded density flyer, termed the “bed-of-nails” (BON). A 2.5-mm-thick \(\times \) 99.4-mm-diameter solid disc of stainless steel formed a base for an array of tapered spikes of length 5.5 mm and spaced 1 mm apart. The two experiments to test the concept were performed at impact velocities of 900 and 1100 m/s using the 100-mm gas gun at the Institute of Shock Physics at Imperial College London. In each experiment, a BON flyer was impacted onto a copper buffer plate which helped to smooth out perturbations in the wave profile. The ramp delivered to the copper buffer was in turn transmitted to three tantalum targets of thicknesses 3, 5 and 7 mm, which were mounted in contact with the back face of the copper. Heterodyne velocimetry (Het-V) was used to measure the velocity–time history, at the back faces of the tantalum discs. The wave profiles display a smooth increase in velocity over a period of \(\sim \!\!2.5 \, {\upmu } \hbox {s}\), with no indication of a shock jump. The measured profiles have been analysed to generate a stress vs. volume curve for tantalum. The results have been compared with the predictions of the Sandia National Laboratories hydrocode, CTH.     

Numerical modeling of PZT-induced Lamb wave-based crack detection in plate-like structures

This paper presents the numerical modeling and simulations of PZT-induced Lamb wave propagation in plate-like structures by using the spectral finite element method. A novel spectral plate finite element, which can efficiently model the three-dimensional (3D) behavior of Lamb waves, is proposed. In the formulation, linear displacement distributions in the thickness direction are assumed for both the PZT layer and the base plate. A way to avoid the thickness locking is proposed and used in the formulations. Two examples, one for the validation of the proposed two-dimensional (2D) spectral finite element and the other for the demonstration of crack detection in plates, are presented and discussed. The contact between the two faces of crack is considered. Numerical results show that (1) only the anti-symmetric mode is prone to thickness locking thus remedy should be made only on this part, (2) the proposed 2D spectral finite element can adequately model the Lamb wave propagation in plate-like structures and the complex scattering for the crack, and (3) crack location can be well determined by a PZT-induced Lamb wave-based diagnosis algorithm.     

Plane problem of asymmetrical wave impact on an elastic plate

The problem of wave impact on the edge of an elastic horizontal plate is studied within the framework of the Wagner approach using the normal-modes method. The plate is governed by the Euler beam equation with simply supported ends. The liquid is assumed to be ideal and incompressible. The problem is coupled: the elastic and hydrodynamic characteristics of the impact process and the dimension of the contact region should be found simulatenously. An algorithm that permits a detailed study of the impact on an elastic plate is proposed. The phenomenon of unlimited increase of hydrodynamic loads owing to the plate flexibility (blockage) is revealed for fairly long plates. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirisk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 148–158, September–October, 1998.     

Problem of the impact interaction of an elastic rod with a Uflyand-Mindlin plate

Conclusions A problem on the impact of an elastic rod of circular cross-section onto an Uflyand-Mindlin isotropic elastic plate is considered. The radial method, which allows construction of an approximate solution behind fronts of discontinuity surfaces up to the boundary of the contact region by means of segments of power series with variable coefficients, is used as a method of solution. It is, however, established that segments of radial series are not uniformly suitable throughout the region of existence of wave motion. A method of multiple scales is used for their regularization. The method, based on asymtotics of the radial series, permits construction of a uniformly suitable expansion containing a smaller number of terms than segments of radial series, but equal to them in accuracy.Voronezh Structural Engineering Institute, Russia. Translated from Prikladnaya Mekhanika, Vol. 29, No. 2, pp. 39–46, February, 1993.     

Linear interactions affecting the propogation of waves in a linear elastic fluid

Small linear interactions affecting the propogation of waves in a linear elastic fluid are investigated. These linear interactions may occur as a result of impurities on the surface of a linear elastic fluid. These interactions are imposed on the linear wave equations which were investigated in Momoniat (Propogation of waves in a linear elastic fluid, submitted for publication) using the non-classical contact symmetry method. The occurrence of a small parameter in the wave equations under consideration in this paper makes the problem ideal for analysis using an approximate non-classical contact symmetry method. Approximate contact symmetries and approximate solutions are determined and discussed for the problems under consideration. Comparisons are made with the case of no interaction.     

Transient wave propagation of isotropic plates using a higher-order plate theory

Transient wave propagation of isotropic thin plates using a higher-order plate theory is presented in this paper. The aim of this investigation is to assess the applicability of the higher-order plate theory in describing wave behavior of isotropic plates at higher frequencies. Both extensional and flexural waves are considered. A complete discussion of dispersion of isotropic plates is first investigated. All the wave modes and wave behavior for each mode in the low and high-frequency ranges are provided in detail. Using the dispersion relation and integral transforms, exact integral solutions for an isotropic plate subjected to pure impulse load and a number of wave excitations based on the higher-order theory are obtained and asymptotic solutions which highlight the physics of waves are also presented. The axisymmetric three-dimensional analytical solutions of linear wave equations are also presented for comparison. Results show that the higher-order theory can predict the wave behavior closely with exact linear wave solutions at higher frequencies.     

A sensing-plate method for measuring force and duration of impact in elastic-plastic impact of bodies

A new method for measuring an impulsive force generated at the impact end in collision and the duration of impact is developed. The measurement method is based on a simple structure, here called the special sensing plate, which produces the same effect as strain gages embedded in a solid. The sensing-plate method is confirmed by calibration tests to be effective for measuring not only the impulsive force generated at the impact end of a body but also the duration of the force. The measurement is almost entirely free from disturbance caused by interference from reflected waves. Plastic impact experiments are also performed for pure aluminum bars with several different lengths by impacting them on a sensing circular plate. Impact specimens are shot out from a high-pressure air gun. The time variation of stress at the impact end of the specimen bars colliding with the sensing plate and the duration of impact are measured at various impact velocities. The strain-rate dependent theory of plastic wave propagation is applied to obtain a theoretical prediction of those results. The theoretical predictions have shown to agree well with the observed results. Paper was presented at the 1988 SEM Spring Conference on Experimental Mechanics held in Portland, OR on June 5–10.     

Linear Contact Between Plates and Unilateral Elastic Supports*

Abstract

The problem of frictionless linear contact between elastic line supports (beam or Winkler-type) and an elastic plate that is loaded laterally is considered. Contact is called linear if there is no initial gap or no initial pre-force on the potential contact surface. The shear deformation of the plate is taken into account. The contact problem is discretized at points on the potential contact surface. Numerical results are calculated with an optimization algorithm that is based on the force method. The effects of the thickness and Poisson's ratio of the plate and the elastic constants of the line supports are considered. Two loading cases are dealt with: a point load in the middle of the plate and a uniform load over the plate. The plate is rectangular, with side ratios 1 and 2.     

Rayleigh-type waves in a water-saturated porous half-space with an elastic plate on its surface

The paper deals with the plane problem of steady-state time harmonic vibrations of an infinite elastic plate resting on a water-saturated porous solid. The displacements of the plate are described by means of the linear theory of small elastic oscillations. The motion of the two-phase medium is studied within the framework of Biot's linear theory of consolidation. The main interest is focused on the investigation of properties of the Rayleigh-type waves propagating alongside of the contact surface between the plate and the porous half-space. In particular, the dependence of the phase velocity and attenuation of the waves on the plate stiffness, mass coupling coefficient, and degree of saturation of the medium is studied. Besides, for the limiting case of an infinitely thin plate, the comparison of the wave characteristics is carried out with those of the pure Rayleigh waves.     

SPH modeling of fluid–solid interaction for dynamic failure analysis of fluid-filled thin shells

This work concerns the prediction of failure of a fluid-filled tank under impact loading, including the resulting fluid leakage. A water-filled steel cylinder associated with a piston is impacted by a mass falling at a prescribed velocity. The cylinder is closed at its base by an aluminum plate whose characteristics are allowed to vary. The impact on the piston creates a pressure wave in the fluid which is responsible for the deformation of the plate and, possibly, the propagation of cracks. The structural part of the problem is modeled using Mindlin–Reissner finite elements (FE) and Smoothed Particle Hydrodynamics (SPH) shells. The modeling of the fluid is also based on an SPH formulation. The problem involves significant fluid–structure interactions (FSI) which are handled through a master–slave-based method and the pinballs method. Numerical results are compared to experimental data.     

Contact problem of rubber rings with large deformation

A contact problem with friction between a rubber ring with large deformation and a linear elastic thin plate is solved by means of the substructuring technique in this paper. A study of the influence of fractional contact, of the influence of plate thickness on rubber ring’ deformation is presented.     

Elastic–plastic contact force history and response characteristics of circular plate subjected to impact by a projectile

A new elastic–plastic impact–contact model is proposed in this paper. By adopting the principle of minimum acceleration for elastic–plastic continue at finite deformation, and with the aid of finite difference method, the proposed model is applied in the problem of dynamic response of a clamped thin circular plate subjected to a projectile impact centrally. The impact force history and response characteristics of the target plate is studied in detail. The theoretical predictions of the impact force and plate deflection are in good agreements with those of LDA experimental data. Linear expressions of the maximum impact force/transverse deflection versus impact velocity are given on the basis of the theoretical results. The project supported by the National Natural Science Foundation of China (10532020).     

Interaction of a rigid punch and a circular plate with a fixed side and a stress-free face

The axisymmetric contact problem of a rigid punch indentation into an elastic circular plate with a fixed side and a stress-free face is considered. The problem is solved by a method developed for finite bodies which is based on the properties of a biorthogonal system of vector functions. The problem is reduced to a Volterra integral equation (IE) of the first kind for the contract pressure function and to a system of two Volterra IE of the first kind for functions describing the derivative of the displacement of the plate upper surface outside the punch and the normal (or tangential) stress on the plate lower fixed surface. The last two functions are sought as the sum of a trigonometric series and a power-law function with a root singularity. The obtained ill-conditioned systems of linear algebraic equations are regularized by introducing small parameters and have a stable solution. A method for solving the Volterra IE is given. The contact pressure functions, the normal and tangential stresses on the plate fixed surface, and the dimensionless indentation force are found. Several examples of a plane punch computation are given.     

Wave scattering by a floating porous elastic plate of arbitrary shape: A semi-analytical study

In this paper, a semi-analytical model based on linear potential flow theory and an eigenfunction expansion method is developed to study wave scattering by a porous elastic plate with arbitrary shape floating in water of finite depth. The water domain is divided into the interior and exterior regions, corresponding to the domain beneath the plate and the rest extending towards infinite distance horizontally, respectively. The unknown coefficients in the potential expressions are determined by satisfying the continuity conditions for pressure and velocity at the interface of the two regions, together with the conditions for the motion/force at the edge of the plate, where the Fourier series expansion method is employed to deal with the terms associated with the radius function. A plate with three shapes – circular, cosine and elliptical – and three edge conditions are considered. We find that the largest deflection of the plate with a simply supported edge and a clamped edge is more sensitive to the change in porosity when the porosity is small. The power dissipated by an elliptical plate with its major axis perpendicular to the incident wave direction is the largest among the case studies for the majority of the porosity values tested.     

孤立波与淹没平板相互作用的三维波面和水动力实验研究

在波浪水池中进行了孤立波作用下有限长度和有限宽度淹没平板的三维模型水池实验. 首次应用多目视觉立体重构技术测量局部三维自由表面变形, 该系统的有效测量水平范围为1.7 m$\times $1.6 m. 用4个三分力测力传感器组成水下测力系统, 在不影响波面的情况下测量孤立波对平板的作用力和力矩. 针对波浪不破碎的情况, 选择0.4 m水深和0.16 m波高的来波条件, 平板淹没深度为0.1 m. 实验结果表明, 孤立波经过淹没平板时自由面有明显的三维变形, 导致孤立波波幅的时空变化. 波幅在平板尾缘中心线处达到最大值, 并沿展向逐渐减小. 利用多目视觉立体重构系统得到的波面变化过程与浪高仪给出定点波面时间序列相互印证, 表明建立标识码波面测量方法是有效的. 孤立波对淹没平板作用的水动力载荷变化分为6个典型阶段, 并与利用波面三维重构得到的波面测量标识码并讨论. 基于多目视觉立体重构技术得到了垂向力和俯仰力矩极值点出现时的三维波面形态. 建立的多目视觉立体重构系统将为海洋工程结构物的水池物理模型实验提供新的波面测量手段.      

TiNi合金圆薄板的横向冲击特性实验

利用改装的霍普金森压杆装置对周边固支伪弹性TiNi合金圆薄板进行了冲击实验,初步得到了该结构在时空2个尺度上的动态力学响应的演变发展现象和规律,包括板中弯曲波的传播、相变区的演化和全场的离面位移等,并和A3钢做了对比。结果表明,由于圆板的二维扩散效应,冲击过程中仅在TiNi板中心很小区域(约5 mm)内形成相变区和相变铰,卸载后相变铰消失,钢试件则留下明显的残余变形。TiNi合金圆板的冲击特性受热弹性马氏体相变和逆相变的支配,不同于传统的弹塑性机制。     

Triangular grid method for stress-wave propagation in 2-D orthotropic materials

A triangular grid method is presented to calculate propagation problems of elastic stress waves in 2-D orthotropic materials. This method is based on the dynamic equilibrium equations of the computational cells formed among the auxiliary triangular grids. The solution is obtained by calculating alternately the nodal displacements and the central point stresses of the spatial grids. The numerical results are compared with the corresponding solutions of the finite element method. Comparisons show that the triangular grid method yields a higher calculational speed than the finite element method. The stress concentrations are investigated from wave-field analyses when the stress wave propagates within an orthotropic plate with a hole. Finally, the presented numerical method is used to study the features of wave propagation and diffraction in a square orthotropic plate with a hole when an impact load is applied to the top of the plate.This work was supported by National Natural Science Foundation of China (Nos. 10025212 and 10232040) and Natural Science Foundation of Liaoning province (No. 20021070).