Dynamic analysis of an axially moving sandwich beam with viscoelastic core

A development of the beam model of the axially moving sandwich continua with elastic faces and the core characterized by viscoelastic properties is presented in this paper. Two-parameter Kelvin–Voigt rheological model is used to describe material properties of the core. The Galerkin method is used to solve the governing partial differential equation. Dynamic analysis of the composite with two aluminum facings and a polyurethane core is carried out. The effect of the transport speed, the core thickness and the internal damping of the core material on the dynamic behavior of the system is investigated in undercrtitical and supercritical range of transport speed.     

A numerical study of free and forced vibration of composite sandwich beam with viscoelastic core

In this article, higher order theory for sandwich beam with composite faces and viscoelastic core is achieved by considering independent transverse displacements on two faces and linear variations through the depth of the beam core. In addition, the effects of Young modulus, rotational inertia and core kinetic energy are considered to modify the “Mead & Markus” theory that is used frequently for sandwich beam. These assumptions have not been considered together in previous articles. A finite element code is developed for structural response analysis of the free and forced vibration. The obtained results are compared with the corresponding results of previous researches. The effects of impressive parameters including fiber angle, thickness of faces and core thickness on the loss factors and natural frequencies of the beam are examined. Frequency response of the beam for two cases, constant and frequency dependent core shear modulus are obtained. Finally, time response of the beam is presented based on the Newmark method. Obtained results show that, when the core is soft or hard, “Mead & Markus” theory cannot accurately predict the frequency responses of the system in comparison with the presented theory in this paper; whereas for moderately hard core, both methods lead to the same results. In addition, when the beam is unsymmetrical about its neutral axis, i.e. one face sheet is weaker than the other face sheet, the inaccuracy of the “Mead & Markus” theory increases, even at low frequencies.     

两端固支泡沫铝夹芯梁在冲击荷载作用下的动力响应

提出两端固支泡沫铝夹芯梁在跨中受到冲击荷载作用下动力响应的简化理论计算方法。运用该方法及有限元软件LS-DYNA分别计算了泡沫铝夹芯梁在冲击荷载作用下的动力响应,着重考查了面板材料及芯材厚度对泡沫铝夹芯梁跨中位移的影响情况。并通过试验测量结果对理论计算结果及数值模拟结果进行了验证。研究显示,在不同冲量作用下,泡沫铝夹芯梁跨中位移理论值与实验结果两者符合程度较好,最大误差仅为14%;HRB335级钢面板泡沫铝夹芯梁较304#不锈钢面板泡沫铝夹芯梁在相同冲量作用下具有更小的跨中位移;芯材厚度的增加对提高泡沫铝夹芯梁抵抗冲击荷载的性能也有一定的贡献,夹芯梁芯材厚度由10mm增加至20mm,其跨中位移减小了33%左右。     

Low-velocity heavy-mass impact response of slender metal foam core sandwich beam

The objective of this work is to investigate the dynamic large deflection response of fully clamped metal foam core sandwich beam struck by a low-velocity heavy mass. Analytical solution and ‘bounds’ of dynamic solutions are derived, respectively. Also, finite element analysis is carried out to obtain the numerical solution of the problem. Comparisons of the dynamic, the quasi-static and numerical solutions for the non-dimensional maximum deflection of the sandwich beam with non-dimensional initial kinetic energy of the striker are presented for different cases of mass ratio, impact velocity and location. It is seen that the dynamic solution approaches the quasi-static one as the mass ratio of the striker to the beam is large enough, the quasi-static solution is in good agreement with the numerical results and both solutions lie in the ‘bounds’ of dynamic solutions. The quasi-static and numerical results for the impact force against the maximum deflection of the sandwich beam are obtained. It shows that the quasi-static solution can offer adequate accuracy to predict the low-velocity heavy-mass impact response of fully clamped sandwich beam.     

Vibration modeling of large repetitive sandwich structures with viscoelastic core

A double scale asymptotic method (DSAM) is proposed for vibration modeling of large repetitive sandwich structures with a viscoelastic core. The method decomposes the initial nonlinear vibration problem into two small linear ones. The first one is defined on few basic cells while the second is a differential global amplitude equation with complex coefficients. Their numerical computations permit determination of the damping properties as well as pass and stop bands avoiding the direct computation on the whole structure. Viscoelastic frequency dependent core with fractional and anelastic displacement field models are considered. The resulting nonclassical problems are solved by asymptotic numerical method coupled with automatic differentiation. Based on the presented method, a large reduction of the needed computational time and memory is obtained. The accuracy and efficiency of the proposed method are validated with comparisons to the direct simulations by discretization of the whole structure using asymptotic numerical method coupled with automatic differentiation.     

Multiscale design of a rectangular sandwich plate with viscoelastic core and supported at extents by viscoelastic materials

This work presents a multiscale model of viscoelastic constrained layer damping treatments for vibrating plates/beams. The approach integrates a finite element (FE) model of macroscale vibrations and a micromechanical model to include effects of microscale structure and properties. The FE model captures the shear deformation of the viscoelastic core, rotary inertial effects of all layers, and viscoelastic boundaries of the plate. Comparison with analytical and FE results validates the proposed FE model. A self-consistent (SC) model makes the micro to macro scale transition to approximate the effective behavior a heterogeneous core. Modal damping resulting from the presence of voids and negative stiffness regions in the core material is modeled. Results show that negative stiffness regions in the viscoelastic core material, even at low volume fractions, yield superior macroscopic damping behavior. The coupled SC and FE models provide a powerful multiscale predictive design tool for sandwich beams and plates.     

On displacement-based theories of sandwich plates with soft core

The paper is concerned with a family of refined models of elastic sandwich plates with soft core. Construction of this family is based upon the kinematic assumptions of Dundrová, Kovaik and Slapák [4]. The model energy-consistent with this hypothesis turns out to be nonelliptic. However, this model makes it possible to find a generalization of Hoff's [7] model in which transverse normal deformations of the core are partly considered. A proof is given that both this and Hoff's model is correctly stated irrespective of the choice of fields that describe the angles of rotation of the plate cross-sections. On the other hand, in the model of Reissner [18] this flexibility is tost and only one choice of fields standing for rotations is admissible-namely that in which the assumptions of the Lax-Milgram lemma are fulfilled.     

Transient response of a sandwich beam with functionally graded porous core traversed by a non-uniformly distributed moving mass

International Journal of Mechanics and Materials in Design - The aim of the present work is to develop a high-order shear deformable beam model and investigate the transient response of a sandwich...     

粘弹性复合材料夹芯板的稳态响应分析

芯材采用Kelvin粘弹性本构模型,推导了复合材料夹芯板的动力学方程,运用模态正交原理,以Navier完备解形式求解了四边简支正交对称铺层层合板的稳态响应,并给出了固有频率和结构损耗因子的解析解。通过固有频率的有限元解对比验证了数值计算的可靠性。分析了芯材剪切模量和芯材厚度对结构固有频率和损耗因子的影响。探讨了稳态响应的收敛性,并得到结构稳态响应振幅与频率的关系,分析了芯材损耗因子对结构稳态响应的影响。结果表明：芯材剪切模量存在最佳设计值;结构首阶模态特性主导结构的稳态动态响应。     

Impact response of sandwich plates with a pyramidal lattice core

The ballistic performance edge clamped 304 stainless-steel sandwich panels has been measured by impacting the plates at mid-span with a spherical steel projectile whose impact velocity ranged from 250 to 1300 m s⁻¹. The sandwich plates comprised two identical face sheets and a pyramidal truss core: the diameter of the impacting spherical projectile was approximately half the 25 mm truss core cell size. The ballistic behavior has been compared with monolithic 304 stainless-steel plates of approximately equal areal mass and with high-strength aluminum alloy (6061-T6) sandwich panels of identical geometry. The ballistic performance is quantified in terms of the entry and exit projectile velocities while high-speed photography is used to investigate the dynamic deformation and failure mechanisms. The stainless-steel sandwich panels were found to have a much higher ballistic resistance than the 6061-T6 aluminum alloy panels on a per volume basis but the ballistic energy absorption of the aluminum structures was slightly higher on a per unit mass basis. The ballistic performance of the monolithic and sandwich panels is almost identical though the failure mechanics of these two types of structures are rather different. At high impact velocities, the monolithic plates fail by ductile hole enlargement. By contrast, only the proximal face sheet of the sandwich plate undergoes this type of failure. The distal face sheet fails by a petalling mode over the entire velocity range investigated here. Given the substantially higher blast resistance of sandwich plates compared to monolithic plates of equal mass, we conclude that sandwich plates display a potential to outperform monolithic plates in multi-functional applications that combine blast resistance and ballistic performance.     

Dynamic response of clamped sandwich beam with aluminium alloy foam core subjected to impact loading

The dynamic response of clamped sandwich beam with aluminium alloy open-cell foam core subjected to impact loading is investigated in the paper. The face sheet and the core of the sandwich beam have the different thickness. And the sandwich beam is impacted by a steel projectile in the mid-span. The impact force is recorded by using accelerometer. The results show that tensile crack and core shear are the dominant failure modes. And the impact velocity and the thickness of the face sheet and the foam core have a significant influence on the failure modes and the impact forces. Combining with the inertia effect and experimental results, the failure mechanisms of the sandwich beams are discussed. The thickness of the foam core plays an important role in the failure mechanism of the sandwich beam. In present paper, the failure of the sandwich beam with a thin core is dominated by the bending moment, while the sandwich beam with a thick core fails by bending deformation in the front face sheet and the bottom face sheet in opposite direction due to the plastic hinges in the front face sheet.     

A new FE model based on higher order zigzag theory for the analysis of laminated sandwich beam with soft core

A new finite element (FE) model has been developed based on higher order zigzag theory (HOZT) for the static analysis of laminated sandwich beam with soft core. In this theory, the in-plane displacement variation is considered to be cubic for both the face sheets and the core. The transverse displacement is assumed to vary quadratically within the core while it remains constant in the faces beyond the core. The proposed model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the beam. The nodal field variables are chosen in an efficient manner to overcome the problem of continuity requirement of the derivatives of transverse displacements. A C₀ quadratic beam finite element is implemented to model the HOZT for the present analysis. Numerical examples covering different features of laminated composite and sandwich beams are presented to illustrate the accuracy of the present model. Many new results are also presented which should be useful for future research.     

Bending response of inhomogeneous fiber-reinforced viscoelastic sandwich plates

The static response of an inhomogeneous fiber-reinforced viscoelastic sandwich plate is investigated by using the first-order shear deformation theory. Several types of sandwich plates are considered taking into account the symmetry of the plate and the thickness of each layer. In addition, two cases are considered depending on the viscoelastic material which are included in the core or the faces of the sandwich plates. The method of effective moduli and Illyushin’s approximation method are used to solve the equations governing the bending of simply supported inhomogeneous fiber-reinforced viscoelastic sandwich plates. Numerical computations were carried out to study the effect of the time parameter on deflections and stresses at different values of the aspect ratio, side-to-thickness ratio and constitutive parameter.     

Forced harmonic response of viscoelastic sandwich beams by a reduction method

The aim of this article is to develop a reduction method to determine the forced harmonic response of viscoelastic sandwich structures at a reasonable computational cost. The numerical resolution is based on the asymptotic numerical method. This type of problem is complex, and its number of degrees of freedom is double the number of the undamped structure, leading to a high computational time. To address the problem, three reduction methods are evaluated, which differ from the projection operator. Numerical tests have been performed in the case of cantilever sandwich beams. The comparison of the results obtained by the reduction order resolution with those given by the full order resolution shows both a good agreement and a significant reduction in computational cost.     

Linear stress relations for a metal matrix composite sandwich beam with any core material

In this paper, it is shown that shear stresses are developed in the interface between the facing material and the core of a sandwich beam. The sandwich beam is composed of a core of any suitable material sandwiched between an upper unreinforced metal facing and a bottom facing made from metal matrix composite (MMC) material. The shear stress is shown to be a consequence of the differences in the core and facing elastic moduli. The magnitude of the shear stress increases as the core stiffness is made to diminish. The shear stress can exceed the bond strength between facing and core, resulting in delamination. Consequently, structural materials using this type of construction and particularly flexural experiments should contain a relatively stiff core. The magnitude of the facing stresses is shown to be relatively insensitive to the assumption or neglect of these shear stresses. In the worst case considered, neglecting the interfacial shear stresses results in an overestimation of the compressive and tensile stresses by less than 5%.     

Vibration analysis of composite sandwich beams with viscoelastic core by using differential transform method

This paper deals with the vibration analysis of a three layered composite beam with a viscoelastic core. First, the equations of motion that govern the free vibrations of the sandwich beam are derived by applying Hamilton’s principle. Then, these equations are solved by using differential transform method (DTM) in the frequency domain. The variation of modal loss factor with system parameters is evaluated and presented graphically. Also, the results obtained with DTM are checked against the findings of previous studies and a good agreement is observed. It is the first time that DTM is used for the eigenvalue analysis of a sandwich structure.     

The dynamic response of end-clamped sandwich beams with a Y-frame or corrugated core

The dynamic response of end-clamped monolithic beams and sandwich beams has been measured by loading the beams at mid-span using metal foam projectiles. The AISI 304 stainless-steel sandwich beams comprise two identical face sheets and either prismatic Y-frame or corrugated cores. The resistance to shock loading is quantified by the permanent transverse deflection at mid-span of the beams as a function of projectile momentum. The prismatic cores are aligned either longitudinally along the beam length or transversely. It is found that the sandwich beams with a longitudinal core orientation have a higher shock resistance than the monolithic beams of equal mass. In contrast, the performance of the sandwich beams with a transverse core orientation is very similar to that of the monolithic beams. Three-dimensional finite element (FE) simulations are in good agreement with the measured responses. The FE calculations indicate that strain concentrations in the sandwich beams occur at joints within the cores and between the core and face sheets; the level of maximum strain is similar for the Y-frame and corrugated core beams for a given value of projectile momentum. The experimental and FE results taken together reveal that Y-frame and corrugated core sandwich beams of equal mass have similar dynamic performances in terms of rear-face deflection, degree of core compression and level of strain within the beam.     

轴向运动软夹层梁横向振动分析

针对传统夹层梁沿厚度方向不可压缩缺点,以上下约束层与夹心层中面横向位移为独立变量,提出全新的夹层梁理论。将夹层内任意点横向位移假设沿厚度方向变化的二次待定多项式,利用界面位移协调条件,获得以夹心层中面、上下约束层中面横向位移表示的夹心层横向位移模式,由此获得厚度方向正应变及相应剪应变。基于Hamilton原理,建立轴向运动软夹层梁横向振动控制方程组,用Galerkin法求解控制方程。研究表明,软夹层梁一阶模态为上下约束层与夹层一起作横向运动,两层之间无相对变形,与传统夹层梁理论一致;软夹层梁二阶模态为上下约束层向两相反方向运动,软夹层中面相对上下约束层不动,夹层处于上下拉伸或压缩状态;软夹层梁三阶模态为上下约束层向同一方向运动,夹心层中面向相反方向运动,夹心层上下处于不同变形状态（拉或压）。通过对振型、模态函数、自由振动响应、轴向运动速度对频率影响等因素分析表明,传统夹层梁模型为夹层梁模型的特殊形式。     

Free vibration response of composite sandwich cylindrical shell with flexible core

This study deals with the free vibration analysis of composite sandwich cylindrical shell with a flexible core using a higher order sandwich panel theory. The formulation uses the classical shell theory for the face sheets and an elasticity theory for the core and includes derivation of the governing equations along with the appropriate boundary conditions. The model consists of a systematic approach for the analysis of sandwich shells with a flexible core, having high-order effects caused by the nonlinearity of the in-plane and the vertical displacements of the core. The behavior is presented in terms of internal resultants and displacements in the faces, peeling and shear stresses in the face–core interface and stress and displacement field in the core. The accuracy of the solution is examined by comparing the results obtained with the analytical and numerical results published in the literatures. The parametric study is also included to investigate the effect of geometrical properties such as radius of curvature, length and sector angle of the shell.