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Deformation and failure mechanisms of lattice cylindrical shells under axial loadingYihui Zhang, Zhenyu Xue, Liming Chen, Daining Fang *Department of Engineering Mechanics, FML, Tsinghua University, Beijing 100084, PR China1. IntroductionThe interest of lattice structures with various core topologies has grown rapidly over the last decade for their superior properties of high specic stiffness and strength, effective energy absorption, shock mitigation and heat insulation 14. 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 signicant 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 congurations, namely diagonal square, hexagonal, Kagome and triangular, respectively. Each of them has the periodic patterning formed from a two-dimensional geometric shape with an innite 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 58. 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 congurations 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. Congurations 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 latticecylindrical 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 1619. 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 benecial 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 inuence 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 veried 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 latticesIn 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 dened as . The parent material of the lattice structure is isotropic, with the material parameters xed as and , where and denote Youngs modulus and Poisson ratio of the material, respectively. Beam elements (B32 in ABAQUS denotation) and rened 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 dened 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 dene 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 signicantly reduce its critical buckling load, the nonuniform deformation of the lattice cylindrical shell that we described previously introducing an imperfection into its deformed conguration 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 aswhere and are the Youngs 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 inuence 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 innite 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 dened 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 specied 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 inuence 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 is capable of capturing the dependences of both the two parameters and on the buckling load with a high delity indicates that the NASA knockdown factor also estimates well the inuence of imperfection induced by the discreteness of the lattice cylindrical shell.Then we make a brief discussion of the local Euler buckling of the beam within the lattice shell. With the lattice cylindrical shell under an axially compressed load , each beam along the axial direction supports a compressed load of . Assuming the end constraint to be simple support, the Euler buckling of the axial beam then requires that where denotes the buckling load of the hexagonal lattice cylindrical corresponding to local Euler buckling of the beam, and is the moment of inertia along the direction where the buckling of the beam occurs. In the case of square beam section, substituting into Eq. (7) gives The ratio of the local Euler buckling load, , to the shell buckling load (Eq. (6), can be given as Noting that is always less than 1 and the value of is commonly larger than 10, the ratio, , is commonly larger than 2.7. The shell buckle half-wavelength is also derived by the homogenization approach, and it can be simplified as for the case of square beam section. This indicates that the shell buckle half-wavelength is commonly in the range of 1.5l, 2.5l for . Therefore, the local Euler buckling of beam usually does not occur before the shell buckling for the hexagonal lattice shell.3. Failure analyses of cylindrical shells made from the stretching-dominated planar latticesTwo types of cylindrical shells, referenced as Kagome and triangular as shown in Figs.1 (g) and (f), respectively, are considered here. The deformation of their planar counterparts is dominated by stretching. FE calculations show that the cross-section of this type of lattice cylindrical shells always remains circular under axial loading, which is different from that of the hexagonal lattice cylindrical shells. An analytical model of predicting their effective axial elastic modulus and yield strength are provided. The Kagome lattice cylindrical shell is exemplified to demonstrate the derivation of failure loads as well as the analyses of failure mechanisms.Table1 The results of fitting parameters in the model Eq.(10) for the Kagome and triangular lattice cylindrical shells.3.1. Axial mechanical properties For the planar Kagome and triangular lattice plates, the effective elastic modulusand yield strengthare given by Wang and McDowell 8 as and . The yield strength in the perpendicular direction of Figs.1 (c) or (d), where , is the relative density of the planar lattice, and are theYoungs modulus and yield strength of the parent material, respectively. For the two cylindrical shells, each curved beam member has a rectangular cross-section. Denote as the number of unit cells in the circumferential direction and as the normalized thickness. A dimensional analysis implies that the axial elastic modulus and yield strength, and can be expressed asandrespectively. Finite element calculations were performed to identify the expressions of and. With the best choices ofand, the non-dimensional elastic modulus and yield strength of the Kagome and triangular cylindrical shells can be formulated as ,Where the parameters, can be specified by fitting the numerical results, whose values are listed in Table1. Figs. 7(a)(c)verify that Eq. (10) is indeed capable of capturing the dependences of the dominant parameters and on the effective elastic modulus and yield strength of these two cylindrical shells with a high fidelity. The detailed geometric parameters in FE calculations are given in corresponding labels of the figures. The curves asymptotically tend to flat when the relative thickness and circumferential number of unit cells are relatively large. The asymptotical values coincide with those of the corresponding unwounded planar structures. This indicates that bending of the beam members has a non-negligible influence on the axial response of the lattice cylindrical shells for the cases where the relative thickness and circumferential number of unit cells ares mall, which must be taken into account in modeling. By comparing the performance of these two lattice cylindrical shells under the same situation, the present numerical results uncover that the Kagome one has slightly higher elastic modulus and yield strength than the triangular one of same geometric parameters. This is an improvement of the previous conclusions made by Wang and McDowell 8,12 that the Kagome and triangular lattice plates have the same elastic moduli and yield strengths.3.2. Failure mechanism map Here, we first concentrate on a Kagome lattice cylindrical shell of radius and height . The axially uniformly distributed compressive load is applied to its one Fig. 7. The FE results and predicted results based on the present model for the normalized elastic modulus and yield strength of the Kagome and triangular cylindrical shells versus (a) the circumferential number of unit cells ; (b) the relative thickness ; (c) the relative density .and the other end is simply supported. The cross-section of the curved beam is rectangular. As an example, the yield strain, elastic modulus and Poisson ratio of the material have the following values:and respectively. Four possible failure modes are considered for this axially compressed lattice cylindrical shell: (1) plastic yielding; (2) overall elastic buckling of the shell; (3) local elastic buckling of beam out of the shell face; (4) local elastic buckling of beam in the shell face. The buckling modes may not really be independent in the presence of small geometrical imperfections and they may be susceptible to a strong non-linear mode interaction 24,25. The actual strength may be degraded to some extent due to this mode interaction. It should be stated that this mode interaction is not taken into account in this paper due to the length of the paper and the analytical complexity. It will be considered in our future investigation. According to Eq.(10), the yielding load, , can be obtained easily, i.e. (11)where is the yielding load of the Kagome lattice cylindrical shell and the superscript denotes the Kagome lattice. Analogous to buckling analysis of the hexagonal lattice cylindrical shell described in Section 2, the homogenization method is adopted here for evaluating the critical buckling load of axially loaded Kagome lattice cylindrical shell. Its effective Poisson ratio is approximately taken to be that of the planar counterpart, that is, the overall critical buckling load of the shell, can be given as (12)where the knockdown facto is defined in Eq.(2). In order to identify the local elastic buckling load of the curved beam within the Kagome lattice cylindrical shell analytically, the resultant forces and bending moments acting on the two ends of each beam should be determined first. However, it is infeasible for such a high-order statically indeterminate structure. Alternatively, we make such an approximation that the axial forces of each curved beam are equal to that of the corresponding straight beam in the unwound lattice plate. Consequently, the critical load of the curve beam, can be approximated by the Euler buckling load of the straight beam, i.e. (13)where is the moment of inertia along the direction where the buckling of the beam occurs, and is the end constraint factor that depends on the degree of constraint to rotation at the end nodes. For the local buckling of beam in the shell face, we assume that the buckling mode of the periodic lattice is near to that of the fixedfixed boundary conditions . The approximate buckling loads by taking correspond to the upper bounds to the buckling strengths. The moment of inertia is .For the local buckling out of the shell face, as sketched in Fig. 8, since the end has no out of shell face constraint, two beams only construct one sinusoidal half-wave-length mode. Therefore, the end constraint can be approximated by simple support and the local buckling load of beam out of shell face can be written as , where the moment of inertia is . Furthermore, the two local buckling loads of the cylindrical shell for buckling of beam in and out of shell face, and , can be expressed in terms of non-dimensional geometrical parameters, i.e. (14) A failure mechanism map is constructed to illustrate the dominant regions of the failure modes introduced above. The active failure mode is the one associated with the lowest failure load. The failure map for Kagome lattice cylindrical shell with relativeFig. 8. The sketch of local buckling mode out of the shell face for Kagome cylindrical shell.Fig. 9. Failure mechanism map for Kagome lattice cylindrical shell with the relative density.density and the yield strain of the parent materialas a function of two non-dimensional parameters and is shown in Fig. 9. We define the normalized load index and weight index asand , respectively. The contours of the load index are also marked in the map. It is notable that plastic yielding is the dominating failure mode for the cases of either or , while local buckling of beam in the shell face does not display in the whole range considered here. Finite element calculations are also carried out to verify the failure map for three different geometries. For each case, both the yielding load and the elastic buckling load are computed and the less one is specified as the corresponding failure load. The results are marked in Fig. 9, all of which are located in the region predicted by the analytical method. Similar analyses can be conducted for the triangular lattice cylindrical shell.is taken for evaluating its in-the-shell face bulking load. For the local buckling out of the shell face, each beam within the triangular lattice cylindrical shell possesses one sinusoidal half-wave-length. Therefore, the local buckling load of beam out of the shell face can be, where the superscript denotes the triangular lattice. By simplification, the two local buckling loads of the cylindrical shell for buckling of beam in and out of shell face,and, for triangular lattice cylindrical shell are expressed as (15)where the value of for the Kagome lattice cylindrical shell is the same as that of the triangular one, while the value of for the Kagome lattice cylindrical shell is four times that of the triangular one.4. Minimum weight design of lattice cylindrical shellsIn this section, an effort to identify optimal configurations for the lattice cylindrical shells subject to an axial compression will be made. For comparison, three topologies: hexagonal, Kagome and triangular, are considered here. The specific stiffness index will be introduced first. The results are based on the analyses presented in previous sections. The relationships of the weight index with the load index will be illustrated to evaluate the load capacities of the three lattice cylindrical shells.4.1. Specific stiffness indexThe specific stiffness index of the lattice cylindrical shell is defined as , where is the weight index defined before. As introduced in Section 2, the axial modulus is for the hexagonal lattice cylindrical shell with square shape of beam section. Hence its specific stiffness index can be simply given as (16)where the superscript denotes the hexagonal lattice. For the Kagome and triangular lattice cylindrical shells with rectangular shape of beam section, the specific stiffness indices can be obtained from Eq.(10), i.e. and Both of them are independent on the relative density, but increase monotonously with the circumferential number of unit cells. By taking the partial derivative of with respect to to be zero, one obtains that the specific stiffness indices will attain the extreme values at and (18)for the Kagome and triangular lattice cylindrical shells, respectively and the corresponding extreme values are and (19) The relationships of the maximum stiffness index versus the circumferential number of unit cells are demonstrated in Fig. 10, with the relative density of the hexagonal one equal to 0.1 and the relative thickness in the range of. The maximum specific stiffness of the Kagome and triangular cylindrical shells are linearly dependent on when and , respectively. It is clear that the Kagome and triangular cylindrical shells possess much higher specific stiffness than the hexagonal one especially at high value of .4.2. Minimum weight designFor an axially compressed lattice cylindrical shell, the optimization solution is the one that minimizes the weight index for a prescribed value of load index . Here, we make a comparison among the load-bearing performances of hexagonal, Kagome and triangular cylindrical shells. The shapes of beam cross-sections are square for hexagonal one and rectangular for Kagome and triangular ones. The material is taken to be elastic-perfectly plastic. The four failure mechanisms: (1) plastic yielding; (2) overall elastic shell buckling; (3) local elastic buckling of beam out of the shell face; (4) local elastic buckling of beam in the shell face, are taken into account for optimization of the Kagome and triangular lattice cylindrical shells. The formulas of the four failure load have been provided in Section 3.2. For the hexagonal lattice cylindrical shell, two failure mechanisms, yielding and overall elastic shell buckling, are considered, since the local buckling of beam hardly occurs. The yielding load of the hexagonal lattice cylindrical shell is approximated by that of the planar hexagonal lattice plate 1,8, i.e., which can be deemed as the upper bounds to the yielding load.Fig. 10. The maximum specific stiffness indices for hexagonal, Kagome and triangular lattice cylindrical shells.Fig. 11. The load capacities of hexagonal, Kagome and triangular lattice cylindrical shells with the geometric parameters. The predictions of the minimum weight of the three lattice cylindrical shells are plotted in Fig. 11 as a function of the load index, with the yield strain of the parent material,. The ranges of the non-dimensional geometric parameters are in the optimization, which are accordance with the practical applications. It can be concluded that the Kagome and triangular lattice cylindrical shells have equivalent load-bearing capacity, both of which are much more efficient than the hexagonal one for all load indices.5. Concluding remarksThe deformation and failure mechanisms of lattice cylindrical shell are systematically studied in this paper. The results indicate that the deformation mechanisms of the lattice cylindrical shells are distinct from that of the lattice plates or lattice beams. This investigation reveals a number of remarkable features of lattice cylindrical shells, as summarized below: (i) For the cylindrical shells made from bending-dominated planar lattices, the cross-section of the shells will not remain circular if the inertia principal direction of the beam section is not arbitrary. If the beam of cylindrical shell has a square or circular cross-section, its axial modulus equals to that of the planar counterpart, otherwise the axial modulus normalized by that of the planar counterpart monotonously increases as the aspect ratio of the beam section increases. Our study also indicates that the lattice cylindrical shell is more difficult to buckle under axial compression if the cross-section of its beam is square. In addition, a homogenization method was proposed to provide good analytical solutions for the overall elastic buckling load in the case of optimum square beam sections. (ii) For the cylindrical shells made from stretching-dominated planar lattices, we presented a model of predicting the overall effective axial elastic modulus and strength. By fitting numerical results to identify several coefficients, the model provides reasonably accurate approximation of these effective mechanical properties. Compared to the planar lattice plate, the effective elastic modulus and yield strength of the cylindrical shell are weakened due to bending of its curved beams. Failure map is constructed for the Kagome lattice cylindrical shells made from an elastic ideally-plastic material. The map incorporates various failure mechanisms, including yielding, global buckling and local buckling, thus providing a good guidance for identifying the possible failure modes of the structures under axial compression. (iii) Comparison has been made among the optimal cylindrical shells of three topologies based on the design stratagem that is to minimize the weight for a given stiffness or load- carrying capacity. It is revealed that the maximum specific stiffness of the Kagome lattice cylindrical shell is slightly higher than that of the triangular one, and the axially loaded Kagome and triangular lattice cylindrical shells have equivalently high load-carrying efficiency which is much higher than that of the hexagonal one.AcknowledgementsThe authors are grateful for the support by National Natural Science Foundation of China under Grant #10632062. 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Journal of the Mechanics and Physics of Solids 1972;20:1019.晶格圆柱壳受轴向负荷时的变形破坏机制导言在过去十年中,因为各种核心拓扑的网络结构具有很高的强度和硬度,能有效的吸收能量,缓震隔热,所以其影响很大14.研究表明,在一些应用中设计良好的网格结构相对于实心板具有更好的性能.一般来说,拓扑结构是设计中首要考虑的问题,因为其在结构的整个力学响应中发挥了重要作用.只有理解了各种拓扑结构的变形机制,才能实现最佳的设计.以往的研究大多数研究的是二维平面格.图1(a)(d)分别为四种平面网格配置,即正方形,六角形, Kagome,三角形.每种结构都是由二维几何平面图形组成,最近对正方形,六角形, Kagome,三角形网格的强度和刚度进行了分析,它们表现出丰富的多样性变形,如图5-8.对于正方形和六角形网格板来说,其每一个网格都承受的起由大多数载荷受的起受的起由大多数载荷引起的弯曲变形,但是正方形网格在承受轴向载荷时的承受能力会相对小一些.对于三角形和可果美网格板,网格的变形主要是轴向拉伸或压缩,所以两者具有更高的刚度和承载能力.用标准的金属板材加工方法可以很容易的加工出六角形网格结构.六角形结构的弹性模量,塑料制品产量和屈曲行为被广泛讨论 1,9,10 。最近已经有了一种新的被命名为粉加工技术的加工方法11.王和麦克道尔8,12系统的分析了几种平面网格模式的刚度,强度和屈服强度. 弗莱克和秋 13 用有限元方法估计了三种弹性脆性平面格的断裂韧性: 六边形，三角形和可果美.张等人14,15提出了两个新的超静定平面网络结构,而且制定了其表面屈服强度的范围.因为晶格材料是一种超轻质材料,所以这种传统材料在航空航天工程中被广泛采用.例如,利用绕组技术,人们可以制造晶格圆柱壳,如图1(e)-(h), 晶格圆柱壳是航空航天器和飞机的重要组成部分.三角晶格圆柱壳晶胞的主要几何参数代表了这三种晶格的主要几何参数,(l)表示晶格晶胞边线的长度,(b)(t)分别表示径向方向和晶格壳表面晶胞边线的密度.六角形晶格夹层圆柱壳在过去几十年中被广泛应用于飞机的机身和卫星的电子管16-19.受轴向压缩时,夹层圆柱壳具有比传统的轴向加强圆柱壳更好的力学性能.虽然人们对六边形晶格的力学性能进行了深入研究,但以前的研究主要集中在简单的六角形晶格平面结构上,如晶胞边线和由晶胞组成的平面,对于六角形晶格圆柱壳的力学行为没有进行过深入调查.因此,六角形拓扑晶格圆柱壳是本文的讲述重点.由主要受拉伸作用的拓扑结构组成的晶格圆柱壳,如三角形晶格和可过没晶格,其平面力学性能好于六角形晶格圆柱壳1,8,所以轴向加强圆柱壳更适合采用受拉伸作用的拓扑结构.实际应用中,圆柱壳必须能够承受相当大的轴向载荷,能够抵制屈曲.轴向弹性模量的复杂分析,受轴向载荷时,晶格圆图1 四种二维点阵板和相应圆柱壳的结构:(a) 正方形晶格板;(b)六边形晶格板;(c) Kagome晶格板;(d)三角形晶格板;(e)正方形晶格圆柱壳;(f)六边形晶格圆柱壳;(g) Kagome晶格圆柱壳;(h)三角形晶格圆柱壳柱壳的性能取决于它的轴向屈服强度和轴向膨胀系数.尽管有一些有限的实验报告,但对这些晶格圆柱壳的力学行为的研究还是比较少的,晶格圆柱壳的变形机制壳的变形机制和失效分析的研究网格结构的工程应用中是非常有价值的.全文的纲要如下,第二节的重点是由主要承受弯曲载荷的平面格组成的圆柱壳的类型.主要是分析几何尺寸对全面有效刚度和弹性屈曲行为的影响.第三节中,我们首先介绍定量预测有效弹性模量的简单模型和Kagome与三角形晶格圆柱壳的屈服强度.通过相应的有限元计算验证模型.此外,我们通过研究Kagome晶格圆柱壳的失效模型建立其失效机理图.最后,通过全面比较三种晶格圆柱壳的负载能力和重量,得出Kagome和三角形晶格圆柱壳的负载能力相当,且都强于六角形晶格圆柱壳的负载能力.图2 (a)三角形晶格圆柱壳的一个晶胞沿轴线方向的示意图(b)晶胞三维图及其几何参数图3 六角形晶格圆柱壳的矩形截面梁的变形模式和无量纲轴向弹性模量及其几何参数; ;1 圆柱壳受挤压时的轴向力学性能在这一节中,研究的是圆柱壳受均匀轴向挤压载荷时的变形机制.图1中的e图和f图分别为正方形和六角形的拓扑结构.晶格结构的梁与梁都很好的结合在一起.通过基于有限元方法的静态分析可以确定有效弹性模量,有效弹性模量可以作为几何和材料参数.通过均匀化方法来解析弹性屈服载荷.因为正方形晶格和六角形晶格变形机制相似,所以这里只对六边形晶格进行分析.利用商业软件ABAQUS对六角形晶格圆柱壳进行有限元分析.圆形和矩形都是梁.矩形截面的宽高比等于,晶格结构的材料是各向同性的, 和,其中和分别表示材料的杨氏模量和泊松比.通过梁和完善网格以确保准确性.矩形梁截面图及数值如图3.轴向弹性模量是指圆柱壳在受一轴向应力时的平均轴向应变.用有限元计算法,通过受轴向应力时自由端的平均轴向位移量估计出有效轴向应变.晶格圆柱壳的有效轴向模量被用于计算的前提是晶格圆柱壳受同一轴向应力和有效轴向应变.轴向屈服强度在第3节讨论,它是通过晶格所承受的应力中的最大应力确定的.圆柱壳的屈服可以认为是晶格单元的屈服.在弹性变形范围内,如果圆柱壳晶格为非方形,那么圆柱壳的截面将不能保持圆形.对于六角形晶格圆柱壳来说,当它的结构发生变形时,若它的晶格是圆形时则圆柱壳的截面为圆形.如果晶格梁的惯量主要方向不是任意的,则可以通过变形机制的数值结果判断出轴向负载的六角形晶格圆柱壳的截面不会是圆形.应当指出,这一现象和晶格圆柱壳的应用是不矛盾的,因为晶格圆柱壳通常被用做夹层结构的核心,较硬的上下夹层在很大程度上制约了局部变形.图3是通过有限元计算得出的晶格的截面为矩形的六角形晶格圆柱壳的有效轴向弹性模量，a为截面宽高比。晶格圆柱壳的相对密度，。和分别为晶格圆柱壳轴向和圆周方向的晶胞数目，,，仅仅是圆柱壳的厚度改变。圆柱壳的相对厚度为厚度与半径比，。简单分析得出以弯曲变形为主的六角形平面晶格结构的有效模量用表示，。用来标准化晶格圆柱壳的有效弹性模量。晶格圆柱壳的正常弹性模量和相对厚度近似成正比，其中表示晶格圆柱壳的轴向有效弹性模量。当圆柱壳的晶格为正方形时，圆柱壳的轴向模量与此时晶格的弹性模量相当，否则轴向模量将随着晶格宽高比的增加而增加。基本变形机制的定性解释如下：承受轴向载荷时，晶格为正方形的六角形晶格圆柱壳的截面为圆形。在这种情况下，圆柱壳的轴向变形是由圆柱壳表面梁的弯曲引起，相同载荷下圆柱壳的弯曲变形和单个梁的弯曲变形大致相当。因此，六角形晶格圆柱壳和六角形晶格板的弹性模量大致相等。宽高比越大，由得出圆柱壳的弯曲变形减小，导致圆柱壳的半径和圆周变形减小。此外，由于泊松效应轴向变形也减小。因此，圆柱壳的轴向有效模量随着晶格宽高比的增加而增加。我们通过数值分析六角形圆柱壳的弹性屈曲。用有限元计算法，首先注意的是宽高比，。,中的几何参数和是不变的，可以确定网格的重量。图4表明临界屈曲载荷随着宽高比的改变而在0和4之间改变。当其值达到最大。类似这样的事实，未变形时梁的最初曲率会显著降低梁的临界屈曲载荷，我们前面介绍的非理想状态下的晶格圆柱壳的非均匀变形导致临界屈曲载荷减少，应用。图4 屈曲模式,具有一定几何参数的六角形晶格圆柱壳的无量纲总体弹性屈曲载荷与材料的泊松比的关系, ,.厚度和是固定的,用来确定网壳的重量. 当我们采用均匀化的方法,把晶格圆柱壳分成若干个尺寸相同的立体壳.对于均匀的固体圆柱壳,其临界弯曲载荷可用下面公式表示: 其中和分别表示有效的固体物质的杨氏模量和泊松比,表示壳的厚度.圆柱壳的整体弯曲是非理想的,主要是因为的存在.的值通过经验确定.的功能是由美国航天局建议的, where 晶格圆柱壳的微观模型导致弯曲模型的多样性.晶格微观结构的几何形状决定了弯曲模型的不同特征值.因此,它导致了晶格圆柱壳的极不理想的敏感度.本文中,美国航天局的临界载荷作为一个近似值被采纳.分析有临界载荷和没有临界载荷的结果,计算出进行对比.六角形晶格圆柱壳的有效弹性模量和泊松比与其在非理想条件下的弹性模量和泊松比大致相同.假设总结构的变形只是梁的弯曲变形,忽略轴向变形,则从以往的研究23得出 ,但是,上述方程不适用于评测临界屈服载荷的,因为预测出的结果是无限值.在对平面六角形晶格的有效弹性模量的推导是应该同时考虑梁的弯曲变形和轴向变形.详细的推倒过程可以从吉布森和阿什比1的著作中找到,这里只给出结果公式 ,相比之下, Eq. (4)高估了有效弹性模量和泊松比.通过分析,把公式5代入公式1得: 图5 晶格为方形的晶胞重量不同的六角形晶格圆柱壳对应的非三维整体弹性弯曲载荷及其参数, , 图5说明了在不同重量水平无量纲临界弯曲载荷与圆周方向晶胞个数的关系.晶格圆柱壳的无量纲临界弯曲载荷和重量指数的关系,其中表示物质的密度,表示圆柱壳的高度, 表示圆柱壳的重量,即.圆柱壳的高度与直径的比值是不变的.结果标在图上,结果表明,在同一重量级别,圆周方向晶胞数量的增加将使临界载荷减小.不考虑的knockdown factor结果高于实际弯曲载荷的50%.在考虑时计算结果与预测值之间的差异大小由圆周方向晶胞数目多少决定, ,当的取值大于8时,两种方法得来的结果更一致.两种方法结果的差异主要是由于装载条件不同引起的,一种载荷作用于物的有限的离散点,一种载荷是均匀的作用于物体.实践应用中,晶格圆柱壳的圆周方向的晶胞数量大于8,图6 用有限元和分析预测两种方法得出的不同相对密度不同晶格截面的六角形晶格圆柱壳的无量纲总体弹性弯曲载荷值.所以模型的临界弹性弯曲载荷与晶格圆柱壳的实际临界弹性弯曲载荷相当.基于有限元计算的结果中,当在10到15之间变动时,三个不同重量级别的六角形晶格圆柱壳的无量纲总体弹性弯曲载荷都存在一个最大值.分别计算出取10和15时六角形晶格圆柱壳的无量纲总体弹性弯曲载荷.对于这两种特定的情况,在图6中比较通过分析预测和有限元计算两种方法得出的临界弯曲载荷的值.根据两种结果取一个适当值.由于目前的考虑knockdown factor分析方法可以准确预测两个集合几何的影响,所以这个方法在实践应用中被采用.应当指出的是,晶格圆柱壳的有限元模型适用于理想状态.若网壳由离散的弯曲梁组成,导致壳的结构高度离散,这种离散的圆柱壳是非理想的.我们对网壳内梁的局部欧拉弯曲作一个简短