Coupled axisymmetric vibration of nonlocal fluid-filled closed spherical membrane shell

In this paper, the axisymmetric vibration of a fluid-filled spherical membrane shell is studied based on nonlocal elasticity theory. The membrane shell is considered elastic, homogeneous and isotropic. The shell model is reformulated using the nonlocal differential constitutive relations of Eringen. The membrane shell is completely filled with an inviscid fluid. The motion of the fluid is governed by the wave equation. Nonlocal governing equations of motion for the fluid-filled spherical membrane shell are derived. Along the contact surface between the membrane and the fluid, the compatibility requirement is applied and Legendre polynomials, associated Legendre polynomials and spherical Bessel functions are used to obtain the natural frequencies of the fluid-filled spherical membrane shells. The frequencies for both empty and fluid-filled spherical membrane shell are evaluated, and their comparisons are performed to confirm the validity and accuracy of the proposed method. An excellent agreement is found between the present and previous ones available in the literature. The variations of the natural frequencies with the small-scale parameter, density ratio, wave speed ratio and Poisson’s ratio are also examined. It is observed that the frequencies are affected when the size effect is taken into consideration.     

Axisymmetric wave propagation in a solid viscoelastic sphere

Two problems of linear wave propagation in a viscoelastic solid sphere are solved. The waves are generated by two types of impact on the surface of the sphere. The deformation has symmetry with respect to an axis through the center of the sphere. The solution is based on a superposition principle which reduces the general solution to a static elastic solution, an elastic solution of an eigenvalue problem and an integral equation of the Volterra type involving time only. The solutions are given in double infinite series involving spherical Bessel functions, Legendre polynomials and Legendre functions of the first kind and order one.     

Non-elastic models of interaction of an impactor and an Uflyand–Mindlin Plate

This paper is devoted to the mathematical modeling of the low velocity impact of a solid body upon a thin elastic isotropic plate whose dynamic behavior is described by wave equations. As a method of analysis the ray method, the Laplace transform method for the viscoelastic impact and the method of spliced asymptotic expansions received for small times in the contact area and outside it are used. The local deformations in the contact area are taken into account in terms of two elastoplastic models (Aleksandrov–Kadomtsev and Kilchevskii models) and the viscoelastic Maxwell model. The stiffness of the Maxwell model’s element is expressed in terms of the integral operator with the exponential kernel of relaxation and thus the relationship for the contact force takes the integral form with a function of relaxation. The results of the present investigation can be used for the projecting and analysis of the planar constructions and their elements to impact loading in civil engineering and automotive industry.     

Transverse impact of a ball on a sphere with allowance for waves in the target

The transverse impact of a solid projectile on an elastic spherical shell with a pivoting contour support has been studied. Inside the contact zone, the projectile-target interaction is described by a solution of the standard system of equations. Outside the contact zone, the points of the shell are displaced and the shell is deformed due to propagation of a nonstationary wave front. A solution in this region is constructed using ray series with variable coefficients representing jumps of the time derivatives of the unknown functions on the wave surface of strong discontinuity. These coefficients are determined to within arbitrary constants using momentless equations of motion of the shell points. The constants are determined by matching two solutions at the contact zone boundary. Using the obtained analytical expressions and plotted dependences for the contact force and dynamic inflection, it is possible to judge on the influence of the shell structure design on the dynamic characteristics of impact interaction.     

低速冲击下含损伤层合中厚浅球壳的非线性动力响应分析

基于中厚壳几何非线性理论和损伤理论 , 采用应变描述的失效准则 , 导出了层合浅球壳含基体损伤和纤维2基体剪切损伤的损伤本构关系 , 建立了低速冲击下轴对称正交对称铺设层合中厚浅球壳的非线性运动控制方程。对未知函数在空间域采用正交配置法离散 , 时间域采用 Newmark2 β方法离散 , 对整个问题进行迭代求解。数值结果表明 , 损伤、 冲击物的初速度以及结构的几何参数都在不同程度上影响着低速冲击下结构所受的冲击载荷和结构的非线性动力响应。      

Guided waves in functionally graded viscoelastic plates

In this paper, a dynamic solution for the propagating viscoelastic waves in functionally graded material (FGM) plates subjected to stress-free conditions is presented in the context of the Kelvin–Voigt viscoelastic theory. The FGM plate is composed of two orthotropic materials. The material properties are assumed to vary in the thickness direction according to a known variation law. The three obtained wave equations are divided into two groups, which control viscoelastic Lamb-like wave and viscoelastic SH wave, respectively. They are solved respectively by the Legendre orthogonal polynomial series approach. The validity of the method is confirmed through a comparison with the Lamb wave solution of a pure elastic FGM plate and a comparison with the SH wave solution of a viscoelastic homogeneous plate. The dispersion curves and attenuation curves for the graded and homogeneous viscoelastic plates are calculated to highlight their differences. The viscous effect on dispersion curves is shown. The influences of gradient variations are illustrated.     

Fractional-derivative viscoelastic model of the shock interaction of a rigid body with a plate

The impact of a rigid body upon an elastic isotropic plate is investigated for the case when the equations of motion take rotary inertia and shear deformation into account. The impactor is considered as a mass point, and the contact between it and the plate is established through a buffer involving a linear-spring–fractional-derivative dashpot combination, i.e., the viscoelastic features of the buffer are described by the fractional-derivative Maxwell model. It is assumed that a transient wave of transverse shear is generated in the plate, and that the reflected wave has insufficient time to return to the location of the spring’s contact with the plate before the impact process is completed. To determine the desired values behind the transverse-shear wave front, one-term ray expansions are used, as well as the equations of motion of the impactor and the contact region. As a result, we are led to a set of two linear differential equations for the displacements of the spring’s upper and lower points. The solution of these equations is found analytically by the Laplace-transform method, and the time-dependence of the contact force is obtained. Numerical analysis shows that the maximum of the contact force increases, tending to the maximal contact force when the fractional parameter is equal to unity.     

Generalized theory of resonance excitation by sound scattering from an elastic spherical shell in a nonviscous fluid

This work presents the general theory of resonance scattering (GTRS) by an elastic spherical shell immersed in a nonviscous fluid and placed arbitrarily in an acoustic beam. The GTRS formulation is valid for a spherical shell of any size and material regardless of its location relative to the incident beam. It is shown here that the scattering coefficients derived for a spherical shell immersed in water and placed in an arbitrary beam equal those obtained for plane wave incidence. Numerical examples for an elastic shell placed in the field of acoustical Bessel beams of different types, namely, a zero-order Bessel beam and first-order Bessel vortex and trigonometric (nonvortex) beams are provided. The scattered pressure is expressed using a generalized partial-wave series expansion involving the beam-shape coefficients (BSCs), the scattering coefficients of the spherical shell, and the half-cone angle of the beam. The BSCs are evaluated using the numerical discrete spherical harmonics transform (DSHT). The far-field acoustic resonance scattering directivity diagrams are calculated for an albuminoidal shell immersed in water and filled with perfluoropropane gas, by subtracting an appropriate background from the total far-field form function. The properties related to the arbitrary scattering are analyzed and discussed. The results are of particular importance in acoustical scattering applications involving imaging and beam-forming for transducer design. Moreover, the GTRS method can be applied to investigate the scattering of any beam of arbitrary shape that satisfies the source-free Helmholtz equation, and the method can be readily adapted to viscoelastic spherical shells or spheres.     

Viscoelastic damped response of cross-ply laminated shallow spherical shells subjected to various impulsive loads

In this paper, the viscoelastic damped response of cross-ply laminated shallow spherical shells is investigated numerically in a transformed Laplace space. In the proposed approach, the governing differential equations of cross-ply laminated shallow spherical shell are derived using the dynamic version of the principle of virtual displacements. Following this, the Laplace transform is employed in the transient analysis of viscoelastic laminated shell problem. Also, damping can be incorporated with ease in the transformed domain. The transformed time-independent equations in spatial coordinate are solved numerically by Gauss elimination. Numerical inverse transformation of the results into the real domain are operated by the modified Durbin transform method. Verification of the presented method is carried out by comparing the results with those obtained by the Newmark method and ANSYS finite element software. Furthermore, the developed solution approach is applied to problems with several impulsive loads. The novelty of the present study lies in the fact that a combination of the Navier method and Laplace transform is employed in the analysis of cross-ply laminated shallow spherical viscoelastic shells. The numerical sample results have proved that the presented method constitutes a highly accurate and efficient solution, which can be easily applied to the laminated viscoelastic shell problems.     

A sensing block method for measuring impact force generated at a contact part

A new method has been devised for measuring the impact force generated at the contact part in a collision. This new method, here called the sensing block method, is based on a simple structure which is made of a steel cylindrical block with a small cylindrical projection. The top of this projection serves as the contact plane in a collision. The impact force generated at the contact part can be measured using small strain gages mounted on the side surface of the projection. Through the calibration tests, the new method was confirmed to be more effective for direct measurement of impact force with relatively high sensitivity for both normal and oblique impacts. The obtained measurements are almost entirely free from disturbances caused by reflected wave interference. Using this method, an impact experiment on commercially available pure aluminum bars with different lengths and spherical ends was performed. From the experimental results, it was found that the generated maximum force gradually increased in proportion to the square root of potential energy, and that the generated mean stress on the contact part was constant and independent of the radius of curvature of the spherical end and impact velocity.     

求解一类非线性粘弹性问题的弹性回复对应原理

本文提出了一个全新的粘弹性理论体系,它把传统的线粘弹性理论作为特殊情况包括在内。其主要结果是两个用于求解一类物理非线性粘弹性问题的弹性回复对应原理。利用它,只要知道相应的非线性弹性问题的解,就可以求出非线性粘弹性问题的解答。对应原理不是基于本构关系的相似性,而是基于从粘弹性现时响应到瞬时弹性响应的可回复性[1]。首先找到非线性粘弹性与非线性弹性本构关系之间的对应,然后导出了两个弹性回复对应原理。通过对改性聚丙烯材料的实验验证了该对应原理的正确性和对此类材料的实用性。     

The elastic impact response of glass/epoxy laminated ogival shells

This study examines the elastic response of glass/epoxy laminated composite ogival shells subjected to low velocity impact at an arbitrary location by a solid striker. An analytic solution for prediction of transient displacement and induced stress and strain is presented. This solution, which has not been obtained previously, considers both contact and transverse shear deformation. Based on this solution, the contact force and transient strains induced in a shell by impact are studied. Impact experiments are also conducted on a simply-supported glass/epoxy ogival shell and the resulting strain responses examined. The results show that the contact force and strain response predicted using the present solution agree well with experimental measurements.     

Transformation of sliding motion to rolling during spheres collision

In this research, we have investigated the three-dimensional elastic collision of two balls, based on friction in the tangential plane. Our aim is to offer analytical closed form relations for post collision parameters such as linear and angular velocities, collision time and tangential and normal impulse in three dimensions. To simplify the problem, stick regime is not considered. In other words, balls have a low tangential coefficient of restitution. Sliding, sliding then rolling, and rolling at the beginning of contact are three cases that can occur during impact which have been considered in our research. The normal interaction force is described by the Hertz contact force and dimensionless analysis is used for investigating normal interaction force; furthermore, Coulomb friction is considered during sliding. Experimental data for collisions show when sliding exists through the impact, tangential impulses can be taken as frictional impulses using the Coulomb law if the dynamic regime is not stick regime. To identify transformation of sliding motion to rolling or sticking during the impact process, linear and trigonometric functions are considered as an approximation for the normal interaction force. Afterwards, we have obtained the condition for the possibility of this transformation; moreover, we can estimate the duration of sliding and rolling or sticking. We have obtained an analytical solution for maximum force and deformation, collision time, impulses and post-collision linear and angular velocities in three dimensions.     

A linear model of elasto-plastic and adhesive contact deformation

Rigorous non-linear models of elasto-plastic contact deformation are time-consuming in numerical calculations for the distinct element method (DEM) and quite often unnecessary to represent the actual contact deformation of common particulate systems. In this work a simple linear elasto-plastic and adhesive contact model for spherical particles is proposed. Plastic deformation of contacts during loading and elastic unloading, accompanied by adhesion are considered, for which the pull-off force increases with plastic deformation. Considering the collision of a spherical cohesive body with a rigid flat target, the critical sticking velocity and coefficient of restitution in the proposed model are found to be very similar to those of Thornton and Ning’s model. Sensitivity analyses of the model parameters such as plastic, elastic, plastic-adhesive stiffnesses and pull-off force on work of compaction are carried out. It is found that by increasing the ratio of elastic to plastic stiffness, the plastic component of the total work increases and the elastic component decreases. By increasing the interface energy, the plastic work increases, but the elastic work does not change. The model can be used to efficiently represent the force-displacement of a wide range of particles, thus enabling fast numerical simulations of particle assemblies by the DEM.     

Torsion of viscoelastic spheres in contact

The theory of elastic contact between two spherical bodies is used as a basis for an extension to include the contribution of the viscous effects to the total stress for viscoelastic spheres subjected to twisting moments. Expressions relating twisting moment to penetration of slip and penetration of slip to twist angle are derived. Two term power series truncations of the relations are then used to derive approximate expressions for torsional compliance of the bodies. Validation experiments for the extended model were performed by use of a rheometer device. Applications for the model in post-harvest agriculture include extraction of material properties for use in Discrete Element Modelling of mechanical interactions of fruits and other regular shaped produce during machine handling. A specific application is performed involving the use of a rheometer to measure the coefficient of friction for fruit-fruit contact.     

Time dependent response of low velocity impact induced composite conical shells under multiple delamination

This paper presents the time dependent response of multiple delaminated angle-ply composite pretwisted conical shells subjected to low velocity normal impact. The finite element formulation is based on Mindlin’s theory incorporating rotary inertia and effects of transverse shear deformation. An eight-noded isoparametric plate bending element is employed to satisfy the compatibility of deformation and equilibrium of resultant forces and moments at the delamination crack front. A multipoint constraint algorithm is incorporated which leads to asymmetric stiffness matrices. The modified Hertzian contact law which accounts for permanent indentation is utilized to compute the contact force, and the time dependent equations are solved by Newmark’s time integration algorithm. Parametric studies are conducted with respect to triggering parameters like laminate configuration, location of delamination, angle of twist, velocity of impactor, and impactor’s displacement for centrally impacted shells.     

颗粒弹塑性碰撞理论模型

以Hertz弹性接触力学为基础,假设材料为弹塑性强化材料,并考虑接触压力作用下颗粒球体接触面上材料进入塑性阶段后的应力调整与释放,提出了一种新的颗粒弹塑性接触理论。以此为基础,在准静态假设的基础上,研究了颗粒间的弹塑性碰撞问题,推导了相应的计算公式,并与其它弹塑性接触模型进行了比较。结果表明:按照该文模型预测的荷载变位曲线介于Hertz弹性解和Thornton解之间;强化系数越高,荷载变位曲线越接近Hertz弹性解,强化系数越低,越接近Thornton解,Thornton模型为该文模型的一个特例。在颗粒弹塑性碰撞过程中,本文理论预测的冲击力远远小于Hertz弹性解的预测结果,更与实际情况相符。     

A contact algorithm for 3D discrete and finite element contact problems based on penalty function method

A contact algorithm in the context of the combined discrete element (DE) and finite element (FE) method is proposed. The algorithm, which is based on the node-to-surface method used in finite element method, treats each spherical discrete element as a slave node and the surfaces of the finite element domain as the master surfaces. The contact force on the contact interface is processed by using a penalty function method. Afterward, a modification of the combined DE/FE method is proposed. Following that, the corresponding numerical code is implemented into the in-house developed code. To test the accuracy of the proposed algorithm, the impact between two identical bars and the vibration process of a laminated glass plate under impact of elastic sphere are simulated in elastic range. By comparing the results with the analytical solution and/or that calculated by using LS-DYNA, it is found that they agree with each other very well. The accuracy of the algorithm proposed in this paper is proved.     

Free vibration of deep spherical sandwich shells

Summary This paper deals with axisymmetric vibrations of deep sandwich spherical shells. The shell consists of a thick core and two face sheets of the same isotropic material with equal thickness. The appropriate differential equations have been obtained in terms of two new arbitrary functions which replace the meridional displacement components. The solution is in terms of Legendre functions from which numerical results are computed by using an iterative technique. Effects of geometric and elastic properties of the core and the face sheets are presented in the form of graphs.