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Deformation and failure mechanisms of lattice cylindrical shells under axial loading
Yihui Zhang, Zhenyu Xue, Liming Chen, Daining Fang *
Department of Engineering Mechanics, FML, Tsinghua University, Beijing 100084, PR China
1. Introduction
The interest of lattice structures with various core topologies has grown rapidly over the last decade for their superior properties of high speci?c stiffness and strength, effective energy absorption, shock mitigation and heat insulation [1–4]. These studies have shed light on that well-designed lattice structures are able to outperform the solid plate and shell components in many applications. In general, the structural topology, as the primary concern in the design, plays a signi?cant role in dominating the overall mechanical response of the structures. Understanding the deformation mechanisms of various topologies undoubtedly aids to attain the best design.
Most of the previous studies were focused on the twodimensional planar lattices. Figs. 1(a)–(d) exhibit four types of the planar lattice con?gurations, namely diagonal square, hexagonal, Kagome and triangular, respectively. Each of them has the periodic patterning formed from a two-dimensional geometric shape with an in?nite out-of-plane thickness. The overall effective in-plane stiffness and strength of the diagonal square, hexagonal, Kagome and triangular lattices have been analyzed recently, and they show a rich diversity in deformation [5–8]. For the diagonal square and hexagonal lattice plates, each truss member undergoes bending deformation under most in-plane loading conditions, except for the diagonal square lattice plate uniaxially loaded along the axial directions of its truss member. For the triangular and Kagome lattice plates, the deformations of their truss members are always dominated by their axial stretching or compressing, resulting in higher stiffness and load capacity than the former two. The hexagonal lattice structure can be processed easily using standard sheet metal fabrication method. The elastic modulus, plastic yield as well as buckling behavior of the hexagonal honeycomb have been extensively explored [1,9,10]. A new kind of fabrication method named powder processing technology has been developed recently [11], thus activating more varieties of complicated con?gurations to be fabricated by this approach. Wang and McDowell [8,12] systematically analyzed the stiffness, strength and yield surfaces of several types of planar lattice patterns. Fleck and Qiu [13] estimated the fracture toughness of elastic-brittle planar lattices using ?nite element method for three topologies: the hexagonal, triangular and Kagome lattices. Zhang et al. [14,15] proposed two novel statically indeterminate planar lattice structures and furthermore formulated their initial yield surfaces and utmost yielding surfaces.
As an ultra-light-weight material, lattice material is an ideal candidate of traditional material in aerospace engineering. For example, utilizing the winding technology, one can manufacture lattice cylindrical shells, which, as depicted in Figs. 1(e)–(h), are the key components of aerospace craft and airplane. The three dominating geometrical parameters of the representative unit cell are demonstrated in Fig. 2 by exemplifying the triangular lattice cylindrical shell, where is the arc
Fig. 1. Con?gurations of four 2D lattice plates and the corresponding cylindrical shells: (a) diagonal square lattice plate; (b) hexagonal lattice plate; (c) Kagome lattice plate; (d) triangular lattice plate; (e) diagonal square lattice cylindrical shell; (f) hexagonal lattice cylindrical shell; (g) Kagome lattice cylindrical shell; (h) triangular lattice
cylindrical shell.
length of each beam,and denote the thickness of the beam in the radial direction of the cylinder and the thickness of the beam in the shell face, respectively. The hexagonal lattice sandwich cylindrical shell has been popularly utilized in practical applications as fuselage section of aircrafts and load-barring tubes of satellites for several decades [16–19]. Under axial compression, this lattice sandwich cylindrical shell possesses better mechanical performance than the traditional axial stiffened cylindrical shells. Although much attention has been paid on the mechanical behavior of hexagonal lattice, the previous investigations were mainly focused on the simple planar hexagonal lattice structures such as beams and plates, and the delicate investigation on the mechanical behavior of hexagonal lattice cylindrical shell has been scarce. Therefore, the lattice cylindrical shell of hexagonal topology is one focus in this paper. The lattice cylindrical shells made from
Fig. 2. The sketch of triangular lattice cylindrical shell with one unit cell in the axial direction (a) and the three-dimensional ?gure of the representative unit cell with illustration of its geometric parameters (b).
Fig. 3. Deformation modes and the non-dimensional axial elastic moduli of hexagonal lattice cylindrical shells with the beam of rectangular cross section and the geometric parameters,; ;;
stretching- dominated topologies, such as the triangular and Kagome lattices, due to their better in-plane mechanical property than that of the hexagonal one [1,8], are likely to be better candidates to the axial stiffened cylindrical shell than the hexagonal lattice cylindrical shell. Under operating, the cylindrical shell must be able to bear relatively large axial load and resist buckling.
Sophisticated analyses of the axial elastic modulus, the axial yield strength and the axial bulking of the lattice cylindrical shell is crucial when assessing its performance under axial loading. Noting that the studies are still comparatively scarce for the mechanical behavior of these lattice cylindrical shells though there are some limited experiments results reported [20], the study on deformation mechanisms and failure analysis of lattice cylindrical shells should be valuable and bene?cial for practical applications of lattice structures in engineering.
The outline of the paper is as follows. Section 2 focuses on the type of cylindrical shells made from the bending-dominated planar lattices. The emphasis is placed on the in?uence of the geometric dimensions on the overall effective stiffness and elastic buckling behavior. In Section 3, we ?rst present simple models capable of quantitatively predicting the effective elastic modulus and yield strength of Kagome and triangular lattice cylindrical shells. The models are veri?ed by the corresponding ?nite element calculations. Furthermore, we explore the failure modes of the Kagome lattice cylindrical shell and construct a failure mechanism map to identify them. Finally, a comprehensive comparison of their load capacities versus their weights is made among three types of lattice cylindrical shells, indicating that Kagome and triangular lattice cylindrical shells have similar load capacities and both outperform the hexagonal one.
2. Axial mechanical properties of cylindrical shells made from the bending-dom-Inated planar lattices
In this section, the deformation mechanisms of cylindrical shells made from bending-dominated planar lattices are studied for the case of an axial stress uniformly distributed through the whole cylindrical shell. Two topologies of lattices, diagonal square and hexagonal, are considered as sketched in Figs. 1(e) and (f). The beams of the lattice structures are all ideally welded with each other. A static analysis based on ?nite element method has been carried out to identify the effective elastic modulus as a function of the geometric and material parameters. An analytical solution for the critical elastic buckling load is also provided utilizing the homogenization method. Because of the similarity of the deformation mechanism between the diagonal square and hexagonal lattices, only the result of the hexagonal topology is selected for demonstration.
Finite element analyses using the commercial software ABAQUS are carried out for the hexagonal lattice cylindrical shells. Both circular and rectangular shapes are considered as the beam sections. The aspect ratio of the rectangular section, ,is de?ned as . The parent material of the lattice structure is isotropic, with the material parameters ?xed as and , where and denote Young’s modulus and Poisson ratio of the material, respectively. Beam elements (B32 in ABAQUS denotation) and re?ned meshes (30 elements for each beam) are adopted to ensure accuracy. The numerical results for the case of rectangular beam cross-section are summarized in Fig. 3. The axial elastic modulus is de?ned according to the average axial strain of the cylindrical shell under a given uniform axial stress. In FE calculations, a uniform axial stress is applied and the measured average axial displacement of the free end is used to estimate the effective axial strain. The effective axial modulus of lattice cylindrical shell is just calculated using the uniform axial stress and effective axial strain. As to the axial yield strength discussed in Section 3, it is measured based on the maximum stress within the lattice members. The yield of the cylindrical shell is assumed to be indexed by the yield of the lattice member. It is observed that the cross-section of the cylinders does not remain circular if their beam members have non-square cross-section in the elastic deformation range. While for a hexagonal cylindrical shell with beam members of a circular cross-section, its cylindrical crosssection keeps circular as the structure is deforming. The numeric results on the deformation mechanism can be summarized as that the cross-section of axially loaded hexagonal cylindrical shells will not remain circular if the inertia principal direction of the beam section is not arbitrary. It should be stated that this phenomenon may not be inconsistent with its application since this lattice cylindrical shell is commonly utilized as the core of sandwich structure and the rigidity of the two panels constrains the local bending to a great extent.
Based on the ?nite element calculations, the normalized effective axial elastic modulus of the hexagonal cylindrical shells with a rectangular cross-section of beam member is also plotted in Fig. 3 as a function of the aspect ratio a. The relative density of the lattice cylindrical shell, , is fixed as . The number of unit cells in the axial and circumferential directions of the lattice cylindrical shells, and , are also kept as,,and only the thickness in the radial direction of the shell, b, is changed. We de?ne the relative thickness of the cylindrical shell, , as the ratio of the thickness of the cylindrical shell to the radius, i.e. . Simple analysis gives that the effective modulus of the bending-dominated hexagonal planar lattice structure, can be expressed as [1]. then is employed as the reference to normalize the calculated effective elastic modulus of the lattice cylindrical shells. It is found that the normalized elastic moduli of the lattice cylindrical shells is linearly dependent on the relative thickness approximately expressed by the common equation, , where denotes the axial effective elastic modulus of the lattice cylindrical shell. When the beam of cylindrical shell has a square cross-section, its axial modulus approximately equals to that of the planar counterpart, otherwise the axial modulus monotonously increases as the aspect ratio of the beam section increases. The qualitative explanation on their underlying deformation mechanisms is given as follows: the cross-section of the hexagonal cylindrical shell with the beams having square cross-section remains circular under axial loading. In this case, the axial deformation of the cylindrical shell is mainly contributed by the bending of the curved beams in the shell face, which is approximately the same as the bending of the beams in its planar counterpart under the same loading condition. Therefore, the effective axial elastic modulus of the hexagonal cylindrical shell should be approximately equal to its planar counterpart correspondingly. When increasing the aspect ratio, , and ?xing the product, bt, the bending deformation out of the shell face of the curved beam decreases, resulting in that the variation of the shell radius as well as the circumferential deformation decrease. Furthermore, the axial deformation decreases due to the Poisson effect. Hence, the axial effective modulus monotonously increases as the aspect ratio of the beam section increases.
Elastic buckling of the hexagonal cylindrical shells is investigated both numerically and analytically. In the ?nite element calculations, the attention is ?rst restricted to the role of aspect ratio, . The geometric parameters are ,,and in FE calculations, where the product of the thickness and is ?xed in order to ?x the weight of the lattice shell. Fig. 4 shows that the critical buckling load will be varied dramatically with the change of the aspect ratio in the range roughly between 0 and 4. Its value attains the maximum when. Akin to the fact that the initial curvature of a straight beam can signi?cantly reduce its critical buckling load, the nonuniform deformation of the lattice cylindrical shell that we described previously introducing an imperfection into its deformed con?guration results in the reduction of the critical buckling load for the cases of
Fig. 4. The buckling mode and the relationship of the non-dimensional overall elastic buckling load with the aspect ratio of rectangular beam section for the hexagon cylindrical shell with the geometric parameters and the Poisson ratio of the parent material, ,,,. The product of the thickness and is ?xed in order to ?x the weight of the lattice shell.
To derive the solution for the critical buckling load when , we employ a homogenization approach such that the discrete lattice cylindrical shell is smeared out as a homogeneous solid shell of the same dimensions whose effective properties mimic those of the lattice one. For the homogeneous solid cylindrical shell, the critical buckling load can be written as
where and are the Young’s modulus and Poisson ratio of the effective solid material, is the thickness of the shell and is a knockdown factor that multiplies the classical critical load for a perfect cylindrical shell. The overall buckling of a cylindrical shell is generally imperfection-sensitive and takes this into account. The empirical choice for as a function of is suggested by NASA [21,22] such that
where
This dependence of on is based on a lower bound to reams of critical load data on cylindrical shell buckling. The microstructure of the lattice cylindrical shell causes the diversity of the buckling modes. The geometry of the lattice microstructure determines the eigenvalues for different buckling modes, and therefore can change the coincidence of eigenmodes. So it is responsible for the extreme imperfection sensitivity of lattice cylindrical shells. In this paper, the NASA critical load knockdown factor is adopted as an approximation to consider the in?uence of imperfection sensitivity. Both the analytical results with and without this knockdown factor, , are calculated for comparison. The effective elastic modulus and Poisson ratio of the hexagonal lattice cylindrical shell are approximately the same as that of its unwound counterpart. By assuming that the overall structural deformation is dominated by bending of each beam member and ignoring the contribution of its axial deformation, previous studies [23] provided the expression of the effective elastic modulus, Eplanar, and Poisson ratio, nplanar, of the planar hexagonal lattice as
,
However, the above equation is not suitable for evaluating the critical buckling load since it combining with Eq. (1) predicts an in?nite value, which is meaningless. The more rigorous derivation of the effective elastic modulus, Eplanar, and Poisson ratio, nplanar,of the planar hexagonal lattice is performed by considering both the contribution of the bending and axial deformation of each beam. The detailed derivation can be found in the monograph of Gibson and Ashby [1], and here only the results are given as
,
By comparison, Eq. (4) overestimates both the effective elastic modulus and Poisson ratio. The analytical solution of critical load for the hexagonal lattice cylindrical shell, , can be available by substituting Eq. (5) into (1), i.e.
Fig. 5 illustrates the relationships of the non-dimensional critical buckling load with the number of unit cells in the circumferential direction at different weight levels. The non-dimensional critical buckling load and weight index of the lattice cylindrical
Fig. 5. The non-dimensional overall elastic buckling loads of the hexagon cylindrical shell versus the circumferential numbers of unit cells at different weight levels with the beam of square cross section and the parameters, ,
Fig. 6. The comparison of the analytical predictions and FE results for the nondimensional overall elastic buckling loads of the hexagon cylindrical shell at different relative densities with the beam of square cross-section.
shell are de?ned as and , , respectively, where denotes the density of the material, is the height of the cylindrical shell and is the weight of the cylindrical shell, i.e. . The ratio of the height of the cylindrical shell to its diameter, , is kept constant as . Both analytical and numerical predictions are plotted. It is shown that at the same weight level, the analytical critical load incorporating the knockdown factor decreases gradually to a steady value with the increase of the circumferential number of unit cells. The prediction based on the analytical results without knockdown factor overestimates the buckling load by about 50%. Discrepancy between the numerical results and the analytical solutions with knockdown factor displays when the circumferential number of unit cells, , is small, e.g. . While the results from two methods are more consistent when n are larger than about 8. This is mainly due to the difference of loading conditions of the two methods such that the loads are discretely applied at limited points in FEM, while uniformly applied in the analytical homogenization method. The circumferential numbers of unit cells of the lattice cylindrical shells are usually larger than 8 in practical applications, so the proposed analytical model (Eq. (6)) provides a good approximation of the critical elastic buckling load for the lattice cylindrical shells. Based on the FE results, there exists a maximum value in the range of n between 10 and 15 for each of the three different weight levels. Additional calculations are performed for the hexagonal cylindrical shells of equaling 10 and 15, respectively. For such two speci?ed cases, the analytical predictions of the critical buckling loads varying with the relative density are compared with the corresponding ?nite element results in Fig. 6. Good agreement is found between the FE results and the analytical solution with the knockdown factor. Since the present analytical solution with the knockdown factor can predict the in?uence of both two geometric parameters with high accuracy, as shown in Figs. 5 and 6, the present solution (Eq. (6)) is suggested in practical applications.
It should be noted that no initial geometrical imperfection is introduced in the ?nite element model of the lattice cylindrical shell. But as is different from the solid shell, the lattice shell is made of discrete curved beam, resulting in the high discreteness of the shell structure. This kind of discreteness of the cylindrical shell can be deemed as a kind of imperfection. The fact that the analytical model incorporating the knockdown factor