Mechanical Properties of two novel planar lattice structures

Two novel statically indeterminate planar lattice materials are designed: a new Kagome cell (N-Kagome) and a statically indeterminate square cell (SI-square). Their in-plane mechanical properties, such as stiffness, yielding, buckling and collapse mechanisms are investigated by analytical methods. The analytical stiffness is also verified by means of finite element (FE) simulations. In the case of uniaxial loading, effective modulus, yield strength, buckling strength and critical relative density are compared for various lattice structures. At a critical relative density, the collapse mode will change from buckling to yielding. Elastic buckling under macroscopic shear loading is found to have significant influence on failure of lattice structures, especially at low relative densities. Comparison of the analytical bulk and shear moduli with the Hashin–Shtrikman bounds indicates that the mechanical properties of the SI-square honeycomb are relatively close to being optimal. It is found that compared with the other existing stretching-dominated 2D lattice structures, the N-Kagome cell possesses the largest continuous cavities for fixed relative densities and wall thicknesses, which is convenient for oil storage, disposal of heat exchanger, battery deploying and for other functions. And the initial yield strength of the N-Kagome cell is slightly lower than that of the Kagome cell. The SI-square cell has similar high stiffness and strength as the mixed cell while its buckling resistance is about twice than that of the mixed cell.     

Buckling of stochastically heterogeneous beams,using a functional perturbation method

The buckling load and its probabilistic nature (average and variance) of Bernoulli beams with stochastic material (bending stiffness) properties is derived analytically by a new functional perturbation method (FPM). A buckling shape function is assumed, based on the homogeneous solution and additional terms to account for the morphology effects. The buckling load in the transcendental equation is treated as a functional of the bending modulus (stiffness or compliance) field. Applying a functional perturbation to the above equation, the buckling load is found analytically to any desired degree of accuracy, as a function of material morphology. The FPM is executed using both stiffness and compliance statistical data. The impact of each of the two data sources on the solution accuracy is examined, showing that compliance based solutions are accurate for small correlation lengths. Statically indeterminate problems can be treated with no additional effort. An example of a simply supported beam is solved in detail. Comparison with previous studies, where stochastic finite element and Monte Carlo simulation were used, showed the relative accuracy and insight capabilities of the method. The clamped-free case is also studied to demonstrate that symmetry conditions, used for homogeneous beams to find the buckling load on the basis of a simply supported case, are not valid for heterogeneous beams.     

Mechanical properties of lattice grid composites

An equivalent continuum method only considering the stretching deformation of struts was used to study the in-plane stiffness and strength of planar lattice grid com- posite materials. The initial yield equations of lattices were deduced. Initial yield surfaces were depicted separately in different 3D and 2D stress spaces. The failure envelope is a polyhedron in 3D spaces and a polygon in 2D spaces. Each plane or line of the failure envelope is corresponding to the yield or buckling of a typical bar row. For lattices with more than three bar rows, subsequent yield of the other bar row after initial yield made the lattice achieve greater limit strength. The importance of the buckling strength of the grids was strengthened while the grids were relative sparse. The integration model of the method was used to study the nonlinear mechanical properties of strain hardening grids. It was shown that the integration equation could accurately model the complete stress-strain curves of the grids within small deformations.     

Interaction effects of multiple damage mechanisms in composite sandwich beams subject to time dependent loading

Theoretical models are formulated to explain evolution and interaction of the damage mechanisms for multiple delamination of the face-sheet and core crushing in composite sandwich beams subjected to dynamically applied out-of-plane loading and continuously supported by rigid planes. The models are based on simplified one-dimensional formulations and describe the impacted face of the sandwich as a set of Timoshenko beams joined by cohesive interfaces and resting on a nonlinear Winkler foundation, which approximates the response of the core; the dimensionless formulation highlights the material/structure groups that control the mechanical response. The characteristic features of the problem and transitions in damage progression are explored on varying geometrical parameters and material properties and magnitude and duration of the applied load. For quasi-static loading and low velocity impact, core/face-sheet interactions generate energy barriers to the propagation of delaminations; the efficacy of the barriers in controlling damage in the face-sheets depends on the relative stiffnesses of face-sheet and core and on the foundation yielding strength. For dynamic loading conditions, significant dynamic effects arise in certain regimes and cause substantial changes in behavior: shielding of the crack tip stress fields provided by the foundation is reduced, especially after the load is removed when important delamination openings occur; core plasticity generally opposes this behavior and limits damage in the face-sheet.     

Kagome plate structures for actuation

A class of planar, pin-jointed truss structures based on the ancient Kagome basket weave pattern with exceptional characteristics for actuation has been identified. Its in-plane stiffness is isotropic and has optimal weight among planar trusses for specified stiffness or strength. The version with welded joints resists plastic yielding and buckling, while storing minimal energy upon truss bending during actuation. Two plate structures are considered which employ the planar Kagome truss as the actuation plane. It is shown that these plates can be actuated with minimal internal resistance to achieve a wide range of shapes, while also sustaining large loads through their isotropic bending/stretching stiffness, and their excellent resistance to yielding/buckling.     

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。     

Imperfection sensitivity of axially compressed stringer reinforced cylindrical sandwich panels

This paper presents some numerical results of the effects of several nondimensional parameters on the buckling and initial post buckling behaviors of shallow sandwich panels under axial compression. Results are presented that show these effects due to transverse shearing resistance of the core material, different face-sheet thicknesses, and different core thicknesses. Further effects on the buckling and initial postbuckling behaviors of sandwich panels are presented due to the torsional resistance of longitudinal edge stiffeners.The results show that the range of flatness parameter, δ/d, for which sandwich panels remain imperfection-insensitive increases with increases in transverse shearing resistance of the core material and with larger core thicknesses. These results also indicate that this range of δ/d is smallest when the face-sheet thicknesses are equal. Finally, as in the case of homogeneous panels, torsional resistance of the longitudinal edge stiffeners has the effect of making the sandwich panel less imperfection-sensitive.     

Nonlinear in-plane thermal buckling of rotationally restrained functionally graded carbon nanotube reinforced composite shallow arches under uniform radial loading

The nonlinear in-plane instability of functionally graded carbon nanotube reinforced composite (FG-CNTRC) shallow circular arches with rotational constraints subject to a uniform radial load in a thermal environment is investigated. Assuming arches with thickness-graded material properties, four different distribution patterns of carbon nanotubes (CNTs) are considered. The classical arch theory and Donnell’s shallow shell theory assumptions are used to evaluate the arch displacement field, and the analytical solutions of buckling equilibrium equations and buckling loads are obtained by using the principle of virtual work. The critical geometric parameters are introduced to determine the criteria for buckling mode switching. Parametric studies are carried out to demonstrate the effects of temperature variations, material parameters, geometric parameters, and elastic constraints on the stability of the arch. It is found that increasing the volume fraction of CNTs and distributing CNTs away from the neutral axis significantly enhance the bending stiffness of the arch. In addition, the pretension and initial displacement caused by the temperature field have significant effects on the buckling behavior.     

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.     

基于态型近场动力学的准静态数值仿真方法

近场动力学PD (Peridynamics)是一种通过空间积分方程描述力学行为的方法,对不连续问题的求解有重要的应用前景。本文构建了一种基于态型PD理论的准静态数值仿真方法,推导了节点刚度矩阵和结构刚度矩阵的表达关系,改进了Kilic在键型PD理论中运用的动力松弛法,将其引入态型PD理论,综合绝对和相对收敛准则得到了一种修正的迭代收敛准则。用该方法对薄板轴压稳定性进行了数值仿真,采用了方板晶格的离散方法,以此减小计算规模提高计算效率,成功捕获了三次屈曲现象,对比经验公式和试验结果等过往研究结论,一致性良好。     

Analysis of microbuckling for monomolecular layers adhering to a substrate

A two-dimensional linear spring model is established to study the microbuckling of a plane monomolecular layer adhering to a substrate. The model is for the layer subjected to a compressive load having an arbitrary angle with the chemical bond of the layer. The effects of the load angle, the strength of adhesion and the bending stiffness and shearing stiffness (the capability of resisting transverse bending and in-plane shearing) of the layer on the minimal buckling force and the critical buckling mode are discussed. It is found that the minimal buckling force increases with increasing load angle and, for a given bending stiffness, increases with increasing strength of adhesion and decreasing shearing stiffness. Furthermore, a critical condition under which the buckling of the layer can just occur is obtained, which is helpful to avoid buckling in an engineering application. The project supported by the National Distinguished Young Scientist Fund, Cheung Kong Scholars Programme, the National Natural Science Foundation of China (10272082, 10172068) and Shanghai Postdoctoral Science Foundation     

Postbuckling and imperfection sensitivity of fixed-end and free-end struts on elastic foundation

Summary A study of the postbuckling and imperfection sensitivity of fixed-end and free-end struts on a Winkler elastic foundation is carried out. The configuration and stability of the postbuckling paths bifurcating from the critical points are analysed. For the most part of foundation stiffness, the corresponding postbuckling paths are shown to be falling with respect to load and be unstable. This indicates that, for almost all values of foundation stiffness, the buckling loads of the struts will be sensitive to imperfections. We also obtain imperfection sensitivity of the struts with respect to geometric imperfections having the shape of buckling modes. Received 30 October 1998, accepted for publication 30 March 1999     

热载荷作用下梯度多孔材料梁弯曲及过屈曲力学行为的研究

基于Bernoulli-Euler梁理论,引入物理中面解耦了复合材料结构的面内变形与横向弯曲特性,研究了梯度多孔材料矩形截面梁在热载荷作用下的弯曲及过屈曲力学行为．假设沿梁厚度方向材料的性质是连续变化的,利用能量法推导了矩形截面梁的控制微分方程和边界条件,并用打靶法对无量纲化的控制方程进行数值求解．利用计算得到的结果分析了材料的性质、热载荷、边界条件对矩形截面梁非线性力学行为的影响．结果表明,对称材料模型下,固支梁与简支梁均显示出了典型的分支屈曲行为特征,而其临界屈曲热载荷值均会随着孔隙率系数的增加而单调增加．非对称材料模型下,固支梁仍显示出分支屈曲行为特征,但其临界屈曲热载荷不再随着孔隙率系数的变化而单调变化;而对于两端简支梁,发生了弯曲变形,弯曲挠度随载荷的增大而增大．     

Stiffness threshold of randomly distributed carbon nanotube networks

For carbon nanotube (CNT) networks, with increasing network density, there may be sudden changes in the properties, such as the sudden change in electrical conductivity at the electrical percolation threshold. In this paper, the change in stiffness of the CNT networks is studied and especially the existence of stiffness threshold is revealed. Two critical network densities are found to divide the stiffness behavior into three stages: zero stiffness, bending dominated and stretching dominated stages. The first critical network density is a criterion to judge whether or not the network is capable of carrying load, defined as the stiffness threshold. The second critical network density is a criterion to measure whether or not most of the CNTs in network are utilized effectively to carry load, defined as bending–stretching transitional threshold. Based on the geometric probability analysis, a theoretical methodology is set up to predict the two thresholds and explain their underlying mechanisms. The stiffness threshold is revealed to be determined by the statical determinacy of CNTs in the network, and can be estimated quantitatively by the stabilization fraction of network, a newly proposed parameter in this paper. The other threshold, bending–stretching transitional threshold, which signs the conversion of dominant deformation mode, is verified to be well evaluated by the proposed defect fraction of network. According to the theoretical analysis as well as the numerical simulation, the average intersection number on each CNT is revealed as the only dominant factor for the electrical percolation and the stiffness thresholds, it is approximately 3.7 for electrical percolation threshold, and 5.2 for the stiffness threshold of 2D networks. For 3D networks, they are 1.4 and 4.4. And it also affects the bending–stretching transitional threshold, together with the CNT aspect ratio. The average intersection number divided by the fourth root of CNT aspect ratio is found to be an invariant at the bending–stretching transitional threshold, which is 6.7 and 6.3 for 2D and 3D networks, respectively. Based on this study, a simple piecewise expression is summarized to describe the relative stiffness of CNT networks, in which the relative stiffness of networks depends on the relative network density as well as the CNT aspect ratio. This formula provides a solid theoretical foundation for the design optimization and property prediction of CNT networks.     

Buckling behavior of perfect and defective DWCNTs under axial, bending and torsional loadings via a structural mechanics approach

The buckling behavior of perfect and defective double-walled carbon nanotubes (DWCNTs) under axial compressive, torsional and bending loadings is investigated using a structural mechanics model. The effects of van der Waals (vdW) forces are further modeled using a nonlinear spring element. Critical buckling loads, critical buckling moments and the effects of vacancy defects were studied for armchair nanotubes with various aspect ratios. The results show that vacancy defects greatly reduce the critical buckling load of DWCNTs. The density of defects plays an important role in buckling of DWCNTs. The results of this numerical model are in good agreement with their comparable existing works.     

Asymptotic analysis of the non-linear behavior of long anisotropic tubes

An asymptotically correct beam model is obtained for a long, thin-walled, circular tube with circumferentially uniform stiffness (CUS) and made of generally anisotropic materials. By virtue of its special geometry certain small parameters cause unusual non-linear phenomena, such as the Brazier effect, to be exhibited. The model is constructed without ad hoc approximations from 3D elasticity by deriving its strain energy functional in terms of generalized 1D strains corresponding to extension, bending, and torsion. Large displacement and rotation are allowed but strain is assumed to be small. Closed-form expressions are provided for the 3D non-linear warping and stress fields, the 1D non-linear stiffness matrix and the bending moment–curvature relationship. In bending, failure could be caused by limit-moment instability, local buckling or material failure of a ply. A procedure to determine the failure load is provided based on the non-linear response, neglecting micro-mechanical failure modes, post-failure behavior, and hygrothermal effects. Asymptotic considerations lead to the neglect of local shell interlaminar and transverse shear stresses for the thin-walled configuration. Results of the theory are illustrated for a few symmetric, antisymmetric angle-ply and unsymmetric layups and show that some previously published theories are not asymptotically correct.     

Stability of an elastic cytoskeletal tensegrity model

An elastic cytoskeletal tensegrity structure composed by six inextensible elastic struts and 24 elastic cables is considered. The model is studied, adopting delay convention for stability. Critical conditions for simple and compound instabilities are defined. Post-critical behavior is also described. Equilibrium states with buckling of the struts are also considered. It is revealed that critical Euler buckling load of the struts is a necessary but not a sufficient condition for the existence of bifurcated equilibrium states, caused by buckling of the struts.     

几种变截面点阵承受面外压缩载荷的力学行为

增材制造工艺的出现使得轴向变截面点阵结构设计成为可能,变截面点阵的刚度、强度和稳定性等力学行为研究具有重要的工程意义。假定杆件横截面沿轴向呈双曲线或椭圆型变化,积分推导变截面对于杆件刚度产生的影响,在平衡方程中进行参数变换,推导出变截面对于杆件失稳特征值的解析解,并最终将椭圆型截面杆件应用于实际的金字塔点阵设计。最后引入了特征值算法和弧长法,前者证实了解析解的正确性,后者用于计算含几何缺陷的金字塔胞元后屈曲行为。解析解与数值方法均表明,对于相对密度较低的金字塔点阵,椭圆型轴向变截面杆件设计可以使得点阵刚度在基本保持不变的情况下,有效提高点阵等效压缩强度。研究结果可以为高性能变截面点阵的设计提供理论基础。     

Effects of high order deformations on the strength of planar lattice materials

Lattice materials have been attractive over the last decade for use as load-carrying structures, energy absorbing elements and heat exchanging structures because of their excellent mechanical properties and multifunctional characters. However, the quantitative analysis accounting for high order deformations upon the collapse of lattice materials, which is important for their applications, has not been reported. An analytical investigation of yield surfaces with respect to the high order deformations was carried out for two typical planar lattice materials： triangular and Kagome lattices separately. The analytical results were validated by the finite element method （FEM） simulations. It was found that the effect of high order deformation on the yield strength increases with the relative density. The bending effect of the Kagome lattice is more obvious than that of the triangular one with the same relative density and stress state. The yield strength of the Kagome lattice calculated by neglecting the bending effect overestimates the result by more than 10% when the relative density is higher than about 11.1%, which may not be ignored in engineering applications. The yielding surfaces of the two lattice materials demonstrated in the paper also confirm the analytical results.     

The predicted compressive strength of a pyramidal lattice made from case hardened steel tubes

A sandwich panel with a core made from solid pyramidal struts is a promising candidate for multifunctional application such as combined structural and heat-exchange function. This study explores the performance enhancement by making use of hollow struts, and examines the elevation in the plastic buckling strength by either strain hardening or case hardening. Finite element simulations are performed to quantify these enhancements. Also, the sensitivity of competing collapse modes to tube geometry and to the depth of case hardening is determined. A comparison with other lattice materials reveals that the pyramidal lattice made from case hardened steel tubes outperforms lattices made from solid struts of aluminium or titanium and has a comparable strength to a core made from carbon fibre reinforced polymers.