First applications of a novel unified model for global and local buckling of sandwich columns

Due to the intrinsic heterogeneity of sandwich structures, phenomena at various scales can co-exist in these layered-like assembly of thick-soft and thin-stiff materials. Especially under in-plane compression loadings, geometrical instabilities can occur at both global (structure) and local (skins) scales. Therefore, the in-plane compressive response of sandwich structures is of major concern in designing structural applications. In the present paper, the first applications of a novel unified model for sandwiches are presented, with closed-form solutions for both global and local buckling. For the perfect structure, analytical critical loads are extracted for a simply supported beam, through the calculation of two eigenvalues leading to three buckling modes: it appears that the eigenvalue associated with the antisymmetrical mode can correspond to the occurrence of either global or local (wrinkling) buckling. These global and local loads from the present unified model are shown to compare very well with the predictions given by the most complete specific models from the literature. Moreover, it is shown that conversely to the classical models, our approach yields critical loads that depend only on rigorous well-founded mechanical hypotheses. The simple but general analytical expressions from the unified model permit to select quickly configurations against local and global buckling. In this simplified framework, conclusions can be drawn from this unified model capable of properly predicting the phenomena at both scales. This simplified study is essential in getting an insight in the role played by each geometrical and material parameter, the combination of which is of importance for subsequent non-linear interactive post-buckling analyses (Léotoing et al., 2001).     

热荷载作用下Timoshenko功能梯度夹层梁的静态响应

在精确考虑轴线伸长和一阶横向剪切变形的基础上建立了Timoshenko功能梯度夹层梁在热载荷作用下的几何非线性控制方程.采用打靶法数值求解所得强非线性边值问题,获得了两端固支功能梯度夹层梁在横向非均匀升温作用下的静态热过屈曲和热弯曲变形数值解.分析了功能梯度材料参数变化、不同表层厚度和升温参数对夹层梁弯曲变形、拉-弯耦...     

Coupled bending-torsion vibration of a homogeneous beam with a single delamination subjected to axial loads and static end moments

Delaminations in structures may significantly reduce the stiffness and strength of the structures and may affect their vibration characteristics. As structural components, beams have been used for various purposes, in many of which beams are often subjected to axial loads and static end moments. In the present study, an analytical solution is developed to study the coupled bending-torsion vibration of a homogeneous beam with a single delamination subjected to axial loads and static end moments. Euler–Bernoulli beam theory and the "free mode" assumption in delamination vibration are adopted. This is the first study of the influences of static end moments upon the effects of delaminations on natural frequencies, critical buckling loads and critical moments for lateral instability. The results show that the effects of delamination on reducing natural frequencies, critical buckling load and critical moment for lateral instability are aggravated by the presence of static end moment. In turn, the effects of static end moments on vibration and instability characteristics are affected by the presence of delamination. The analytical results of this study can serve as a benchmark for finite element method and other numerical solutions.     

Predicting buckling loads from vibration data

There are analytical methods for predicting the buckling loads of columns with the boundaries ideally fixed, i.e., simply supported or built-in, or partially fixed. Vibration-test results may furnish a practical method of measuring the fixity. In this investigation a beam, that may or may not be loaded as a column, is assumed to have a torsional spring at each end such that a zero torsional stiffness corresponds to a simply supported end and an infinite torsional stiffness corresponds to a built-in end. From a Rayleigh-Ritz analysis, the buckling load and the fundamental frequency of the beam are each computed as a function of the torsional stiffness. This procedure leads to a one-to-one nondimensional relationship between the buckling load and the natural frequency. From these calculations, it is seen that regardless of the degree of clamping of one end relative to the other end, all that is needed to predict the buckling load within a 15-percent range is a knowledge of the theoretical buckling load of the simply supported column; the theoretical fundamental frequency of the simply supported beam; and the experimental fundamental frequency. Experimental results are presented to support the theory.     

Solitary waves in longitudinally wrinkled and creased helicoids

Elastic ribbons subjected to twist and stretch handle multiple morphological instabilities, amongst others, the longitudinally wrinkled and creased helicoids are investigated in the present paper as promising periodic nonlinear waveguides. Modeling the ribbon by isogeometric Kirchhoff–Love shells, the first longitudinal buckling mode is recovered numerically and used into the Bloch–Floquet method to obtain dispersion curves. After analyzing the effects of the buckling pattern on the different wavemodes, it is shown that classical linear axial waves interact with bending ones and become dispersive. Additionally, as buckling involves geometrical nonlinearities, the structure is expected to host stable nonlinear waves. Indeed, clear supersonic rarefaction trains are observed experimentally and their characteristics are found in agreement with the weakly nonlinear Boussinesq model.     

S型功能梯度扁球壳非线性屈曲的渐近分析

本文利用渐近迭代法获得了边界弹性支撑的功能梯度扁球壳的非线性屈曲问题的理论解．假设材料组分体积分数沿壳体厚度方向呈sigmoid幂函数变化,边界上考虑一般的弹性支撑约束．基于经典的薄壳理论和几何非线性关系,导出了S型功能梯度扁球壳的非线性屈曲问题的控制方程．采用渐近迭代法通过两次迭代得到了无量纲挠度和均布荷载之间的非线性特征关系．通过与已有有限元方法和其他数值方法的结果对比,验证了本文解的有效性和高精度．同时,通过计算阐述了缺陷因子、材料参数、边界约束系数及特征几何参数对壳体临界屈曲荷载的影响．     

Use of magnetorheological elastomer in an adaptive sandwich beam with conductive skins. Part II: Dynamic properties

The second part of this research is to present the field-dependant dynamic property of a sandwich beam with conductive skins and a soft core composed of a magnetorheological elastomer (MRE) part and two non-MRE parts. The MRE part of the core is configured to operate in shear mode and hence the dynamic properties of the sandwich beam can be controlled by magnetic fields due to the field-dependant shear modulus of MRE material. According to the analytical solution for the magnetoelastic loads applied to conductive deformable bodies presented in the first part, the model of the proposed sandwich beam is developed via Hamilton principle. A simply supported MRE-based sandwich beam excited by a vertical force distributed uniformly in a narrow region around the center of the beam is simulated. The anti-resonant frequencies and the resonant frequencies are found to change with the applied magnetic field up to 30%. The procedure to seek the optimal length of MRE part is also presented. Although MRE is a soft material with shear modulus about 0.4 Mpa, this research indicates that the sandwich configuration can well utilize the controllable property of MRE to design applicable smart devices with controllable stiffness.     

THERMAL POST-BUCKLING OF FUNCTIONALLY GRADED MATERIAL TIMOSHENKO BEAMS

Analysis of thermal post-buckling of FGM (Functionally Graded Material) Timoshenko beams subjected to transversely non-uniform temperature rise is presented. By accurately considering the axial extension and transverse shear deformation in the sense of theory of Timoshenko beam, geometrical nonlinear governing equations including seven basic unknown functions for functionally graded beams subjected to mechanical and thermal loads were formulated. In the analysis, it was assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate. By using a shooting method, the obtained nonlinear boundary value problem was numerically solved and thermal buckling and post-buckling response of transversely non-uniformly heated FGM Timoshenko beams with fixed-fixed edges were obtained. Characteristic curves of the buckling deformation of the beam varying with thermal load and the power law index are plotted. The effects of material gradient property on the buckling deformation and critical temperature of beam were discussed in details. The results show that there exists the tension-bend coupling deformation in the uniformly heated beam because of the transversely non-uniform characteristic of materials.     

Nonlinear delamination buckling and expansion of functionally graded laminated piezoelectric composite shells

In this paper, an analytical method is presented to investigate the nonlinear buckling and expansion behaviors of local delaminations near the surface of functionally graded laminated piezoelectric composite shells subjected to the thermal, electrical and mechanical loads, where the mid-plane nonlinear geometrical relation of delaminations is considered. In examples, the effects of thermal loading, electric field strength, the stacking patterns of functionally graded laminated piezoelectric composite shells and the patterns of delaminations on the critical axial loading of locally delaminated buckling are described and discussed. Finally, the possible growth directions of local buckling for delaminated sub-shells are described by calculating the expanding forces along the length and short axis of the delaminated sub-shells.     

Deteriorated Geometrical Stiffness for Higher Order Finite Elements with Application to Panel Flutter

Higher order elements were first design for linear problems where, in certain situations, they present advantages over the lower order elements. A method to efficiently extend their use to geometrical nonlinear problems as panel flutter and postbuckling behavior is presented. The chaotic and limit-cycle oscillations of an isotropic plate are obtained based on direct integration of the discretized equation of motion. The plate is modeled using the von Karman theory and the geometrical nonlinearities are separated in a nonlinear term of the first kind which manifests especially in the prebuckling and buckling regimes, and a nonlinear term of the second kind which is responsible for the postbuckling behavior. A fifth order, fully compatible element has been used to model thin plates while the inplane loads where introduced through a membrane element. The aerodynamics was modeled using the first order 'piston theory. The method introduces the concept of a deteriorated form of the second geometric matrix which is equivalent to neglecting higher order terms in the strain energy of the plate. This allows for a drastic reduction in the computational effort with no observable loss of accuracy. Well established results in the literature are used to validate the method.     

复合材料层合梁的屈曲

本文在铁摩辛柯梁理论基础上,利用迭合刚度方法及Hamilton原理建立了层合梁屈曲问题控制方程,并用此控制方程求解了在具体边界条件下层合梁的屈曲问题,得出了无论在什么边界条件下层合梁的最小屈曲载荷不会大于等效剪切刚度系数C的结论.     

Buckling driven debonding in sandwich columns

A compression loaded sandwich column that contains a debond is analyzed using a geometrically non-linear finite element model. The model includes a cohesive zone along one face sheet/core interface whereby the debond can extend by interface crack growth. Two geometrical imperfections are introduced; a global imperfection of the sandwich column axis and a local imperfection of the debonded face sheet axis. The model predicts the sandwich column to be very sensitive to the initial debond length and the local face sheet imperfection. The study shows that the sensitivity to the face sheet imperfection results from two mechanisms: (a) interaction of local debond buckling and global buckling and (b) the development of a damaged zone at the debond crack tip. Based on the pronounced imperfection sensitivity, the author predicts that an experimental measurement of the strength of sandwich structures may exhibit a large scatter caused by geometrical variations between test specimens.     

Comparative studies of localized buckling in sandwich struts with different core bending models

Analytical models with geometric non-linearities accounting for interactions between local and global instability modes leading to localized buckling in sandwich struts are formulated. For the core material response, two increasingly sophisticated bending models are compared against each other: Timoshenko beam theory (TBT) and Reddy-Bickford beam theory (RBT). Numerical solutions of the analytical models are validated with the commercial finite element code ABAQUS. It is found that there is a small but significant difference in the critical load between the two models and that the previously obtained solution slightly underestimates the linear buckling strength. More importantly, it is found that the RBT model predicts the onset of interactive buckling before the TBT model and, according to the results from the finite element study, matches the actual behaviour of a strut in both its initial and advanced post-buckling states with excellent correlation.     

Thermal buckling analysis of truss-core sandwich plates

Truss-core sandwich plates have received much attention in virtue of the high values of strength-to-weight and stifness-to-weight as well as the great ability of impulseresistance recently. It is necessary to study the stability of sandwich panels under the influence of the thermal load. However, the sandwich plates are such complex threedimensional(3D) systems that direct analytical solutions do not exist, and the finite element method(FEM) cannot represent the relationship between structural parameters and mechanical properties well. In this paper, an equivalent homogeneous continuous plate is idealized by obtaining the efective bending and transverse shear stifness based on the characteristics of periodically distributed unit cells. The first order shear deformation theory for plates is used to derive the stability equation. The buckling temperature of a simply supported sandwich plate is given and verified by the FEM. The efect of related parameters on mechanical properties is investigated. The geometric parameters of the unit cell are optimized to attain the maximum buckling temperature. It is shown that the optimized sandwich plate can improve the resistance to thermal buckling significantly.     

Free vibration analyses of axially loaded laminated composite beams based on higher-order shear deformation theory

The dynamic stiffness matrix method is introduced to solve exactly the free vibration and buckling problems of axially loaded laminated composite beams with arbitrary lay-ups. The Poisson effect, axial force, extensional deformation, shear deformation and rotary inertia are included in the mathematical formulation. The exact dynamic stiffness matrix is derived from the analytical solutions of the governing differential equations of the composite beams based on third-order shear deformation beam theory. The application of the present method is illustrated by two numerical examples, in which the effects of axial force and boundary condition on the natural frequencies, mode shapes and buckling loads are examined. Comparison of the current results to the existing solutions in the literature demonstrates the accuracy and effectiveness of the present method.     

Features of nonlinear deformation and critical states of shell systems with geometrical imperfections

Results from theoretical and experimental investigations into the nonlinear deformation (geometrical nonlinearity, plastic deformation, creep) and critical states (limit loads, buckling) of shell-frame systems with geometric imperfections are analyzed. The presence of prestresses is allowed. Diverse effects of various geometrical imperfections and plastic deformation for different load histories are studied. New qualitative effects of the mutual influence of these factors are established. Relevant experimental results are outlined __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 12, pp. 3–47, December 2006.     

Three-point bending of honeycomb sandwich beams with facesheet perforations

A novel square honeycomb-cored sandwich beam with perforated bottom facesheet is investigated under threepoint bending,both analytically and numerically.Perforated square holes in the bottom facesheet are characterized by the area ratio of the hole to intact facesheet(perforation ratio).While for large-scale engineering applications like the decks of cargo vehicles and transportation ships,the perforations are needed to facilitate the fabrication process(e.g.,laser welding)as well as service maintenance,it is demonstrated that these perforations,when properly designed,can also enhance the resistance of the sandwich to bending.For illustration,fair comparisons among competing sandwich designs having different perforation ratios but equal mass is achieved by systematically thickening the core webs.Further,the perforated sandwich beam is designed with a relatively thick facesheet to avoid local indention failure so that it mainly fails in two competing modes:(1)bending failure,i.e.,yielding of beam cross-section and buckling of top facesheet caused by bending moment;(2)shear failure,i.e.,yielding and buckling of core webs due to shear forcing.The sensitivity of the failure loads to the ratio of core height to beam span is also discussed for varying perforation ratios.As the perfo-ration ratio is increased,the load of shear failure increases due to thickening core webs,while that of bending failure decreases due to the weakening bottom facesheet.Design of a sandwich beam with optimal perforation ratio is realized when the two failure loads are equal,leading to significantly enhanced failure load(up to 60%increase)relative to that of a non-perforated sandwich beam with equal mass.     

Low-velocity impact response of sandwich beams with functionally graded core

The problem of low-speed impact of a one-dimensional sandwich panel by a rigid cylindrical projectile is considered. The core of the sandwich panel is functionally graded such that the density, and hence its stiffness, vary through the thickness. The problem is a combination of static contact problem and dynamic response of the sandwich panel obtained via a simple nonlinear spring-mass model (quasi-static approximation). The variation of core Young’s modulus is represented by a polynomial in the thickness coordinate, but the Poisson’s ratio is kept constant. The two-dimensional elasticity equations for the plane sandwich structure are solved using a combination of Fourier series and Galerkin method. The contact problem is solved using the assumed contact stress distribution method. For the impact problem we used a simple dynamic model based on quasi-static behavior of the panel—the sandwich beam was modeled as a combination of two springs, a linear spring to account for the global deflection and a nonlinear spring to represent the local indentation effects. Results indicate that the contact stiffness of the beam with graded core increases causing the contact stresses and other stress components in the vicinity of contact to increase. However, the values of maximum strains corresponding to the maximum impact load are reduced considerably due to grading of the core properties. For a better comparison, the thickness of the functionally graded cores was chosen such that the flexural stiffness was equal to that of a beam with homogeneous core. The results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites.     

轻质金属点阵夹层板热屈曲临界温度分析

本文针对均匀温度场下四边简支和四边固支金属点阵夹层板的临界热屈曲温度进行了求解和参数影响分析。将点阵夹芯等效为均匀连续体,并且将夹层板的剪切刚度近似为点阵夹芯的抗剪切刚度,忽略夹芯的抗弯刚度且认为夹层板主要由面板来提供抗弯刚度。对于无法获得解析解的四边固支条件,通过对未知变量进行双傅里叶展开的方法求解了Ressiner夹层板模型的临界屈曲温度,理论分析结果与有限元计算结果吻合良好。进一步分析了不同边界条件、点阵胞元构型、点阵材料相对密度、面板厚度等对临界屈曲温度的影响规律。     

两端固支复合材料浅拱的动力屈曲分析

本文研究两端固支层合复合材料浅拱在阶跃载荷作用下的动力稳定性问题。通过对浅拱动力响应的数值计算结果,然后利用B-R动力屈曲准则,着重分析了集中阶跃载荷作用下几种铺层顺序及铺层数对浅拱动力临界载荷的影响,并给出了能够产生‘跳跃失稳’的最小的结构参数γ0。此外,在利用伽辽金法求解浅拱动力学控制方程时,通过取梁的自由振动模态和柱的静力屈曲模态作为浅拱的动力屈曲模态,分别进行计算并比较了二者的结果,进而讨论了二级数解的收敛性。