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Effective elastic properties and initial yield surfaces of two 3D lattice structures
Accepted Manuscript
Effective elastic properties and initial yield surfaces of two 3D lattice structures Mingyang Zhang , Zhenyu Yang , Zixing Lu , Baohua Liao , Xiaofan He PII: DOI: Reference:
S0020-7403(17)32904-1 10.1016/j.ijmecsci.2018.02.008 MS 4168
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
15 October 2017 1 February 2018 3 February 2018
Please cite this article as: Mingyang Zhang , Zhenyu Yang , Zixing Lu , Baohua Liao , Xiaofan He , Effective elastic properties and initial yield surfaces of two 3D lattice structures, International Journal of Mechanical Sciences (2018), doi: 10.1016/j.ijmecsci.2018.02.008
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Highlights
Effective elastic properties of two lattice structures are investigated theoretically.
The coupling effect of axial force and bending moment in a yielding beam is considered.
Yield surfaces under complex stress states are depicted and verified by FE
The evolution of yield surfaces is discussed systematically.
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calculation.
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Effective elastic properties and initial yield surfaces of two 3D lattice structures Mingyang Zhanga, Zhenyu Yanga*, Zixing Lua, Baohua Liaob, Xiaofan Hea Institute of Solid Mechanics, Beihang University (BUAA), Beijing 100083, P. R. China b
AVIC Chengdu Aircraft Design & Research Institute, Chengdu 610091, P. R. China
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a
Abstract
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The effective elastic properties of two bending-dominated lattice structures subjected to uniaxial loading are investigated with using the classical beam theory. In particular, considering the coupling effect of axial force and bending moment, a theoretical approach for predicting the yield surface of lattice structures under complex
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stress state is proposed. The analytical solutions are verified by comparing the predictions with finite element method (FEM) simulations. It is shown that the
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deformation mode of the struts is strongly dependent on the loading directions, and the lattice can be either bending-dominated or stretching-dominated in different stress states,
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resulting in narrow yield surfaces. The comparisons reveal that both the theoretical analysis and numerical simulations present a considerably good description of the initial
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yield surfaces, which provides new insight into understanding the mechanical behaviors
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of the lattice structures and applications in lightweight structures. Keywords: Elastic properties; Initial yield surface; Lattice structure; Beam theory;
FEM simulation
*
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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1. Introduction Lightweight structures such as honeycombs, foams, and lattice materials are widely used in aerospace because of their superior specific stiffness and specific strength [1-5]. High porosity and unique structure provide lightweight structures with many other excellent properties, such as outstanding designability, great potential for wave
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propagation control, and high energy absorption capacity [6-10]. The mechanical behavior of a lattice material is mainly dependent on its internal architecture, which have been widely studied in the previous literatures with using analytical models and numerical simulations [11-13]. Based on homogenization approach and classical beam theory, Mohr [14] developed a general micromechanics-based finite-strain constitutive
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model for truss lattice materials. Liu et al. [15] proposed a constitutive model and carried out optimization design for truss-cored sandwiches. Gibson and Ashby [16] deduced that the effective mechanical properties of a foam material are functions of the relative density of the material. Deshpande et al. [17] analyzed the topography of foams
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and proposed a criterion for lattice materials to determine the minimum node connectivity of a special class of lattice structured materials. Deshpande et al. [18]
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discussed the effective properties of an octet-truss lattice material and depicted its 2D yield surface. Fan and Wei [19] analyzed some stretching dominated lattice materials
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and calculated 3D yield surfaces for the equivalent continuum. However, traditional 3D lattice (lattice materials made by traditional processing technology) is very limited in
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topology type due to technical limitations [20]. The cell size of traditional 3D lattice material is usually of
structures
[21,
22]. Although
some of them
can be
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stretching-dominated
the order of centimeter and most of them are
bending-dominated in topology, the lattice materials are often used as core materials and are fixed on the laminate, and the additional constrains make them stretching-dominated [23-25].
With the development of the technology of additive manufacturing (AM) like selective laser melting (SLM) [26-29], many 3D lattice materials with complex configurations and smaller cell size can be produced for applications [6, 8, 30]. With the cell
size
reaching
to millimeter
scale,
many
new
configurations
including
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stretching-dominated and bending-dominated structures are designed and produced. In turn, plenty of researches were focused on the mechanical properties of 3D printing lattice materials. Gümrük and Mines [31] observed the structure of lattice materials produced by SLM and found imperfections on the surface of struts, which was then considered to
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improve the calculation precision of relative density. They also predicted the compressive strength of body-centered cubic (BBC) lattice block by finite element simulations. Gümrük et al. [32] carried out a series of experiments on the static mechanical behaviors of lattice structures under different loading conditions including uniaxial compression and shear, with the initial collapse envelope obtained
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experimentally. In order to consider the variation of diameter of the struts, Ravari et al. [33] and Tancogne-Dejean et al. [8] proposed a solid finite element model with variable cross sections, and a more accurate solution was obtained. To consider the compound effect of viscoelasticity and stochasticity, Mukhopadhyay et al. [34] analyzed irregular
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lattices and developed a practically relevant stochastic modelling approach. Tancogne-Dejean et al. [35] analyzed elasticity and initial yield surfaces of some other
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truss lattice materials theoretically, numerically and experimentally, and found elastically-isotropic truss lattices are still plastically anisotropic. Moreover, 3D printing
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technology can bring not only geometric imperfections but also internal defects in lattice material such as tiny voids and imperfect welding, which is attributed to the effects of
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complex thermodynamic process on the metal lattice structure. These all have an impact on the mechanical properties of the lattice structures. Instead of efforts of modeling the
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complex imperfections and unique internal structure, some researchers have carried out mechanical tests to get the stress-strain curves of the defected single strut for further analysis [30, 31, 36]. In this paper, the effective moduli of two 3D lattice structures in the principal directions are obtained as functions of the aspect ratio of the stuts and the initial yield surfaces are constructed under in-plane and limited 3D stress states. Issues related to the initial yield surface of the two 3D lattice structures are discussed in details, including the
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effects of the aspect ratio of struts and coupling stress states. Systematic FEM simulations are also carried out to validate the theoretical prediction. 2 . Geometries of two 3D lattice structures Fig. 1 shows the geometries of the two lattice structures analyzed in this study. The first lattice structure (Fig. 1 (a)) is named as BCCZ (body-centered cubic with a strut in
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Z direction) lattice, which is presented in the previous literatures [6, 37, 38]. The second lattice structure (Fig. 1 (d)) is a rhombic dodecahedron (RD) lattice. It is worth noting that the two kinds of representative unit cell of lattice BCCZ showed in Fig. 1(c) are equivalent. Fig. 1(f) illustrates the unit cell for the RD lattice structure. The cross-section
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unit cells in Fig. 1(c) and (f) are both L.
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of each strut is assumed to be circular, with a constant diameter d and the cell length of
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Fig. 1. The geometries of the two lattice structures: (a) specimen of lattice BCCZ produced by SLM, (b) geometric model of lattice BCCZ in isometric view, (c) two
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equivalent unit cells of lattice BCCZ, (d) specimen of lattice RD make by SLM, (e) geometric model of lattice RD in isometric view, (f) unit cell of lattice RD. The struts colored with different colors in (b) and (e) show the positions of unit cells in lattice structures. Through the derivation of geometric relations, the relative densities for a unit cell of the lattice BCCZ and RD with very low values of the ratio d/L can be determined as
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2 1 4 3 d BCCZ 4 L 2 d RD 2 3 L
(1)
These formulas are limited to the lattice materials with very low density. For the lattice materials with relative high density, the overlap of the volumes at the joints cannot be
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neglected. The precise expression for the relative density can be determined by subtracting the volumes double counted in Eq. (1), and can be expressed as:
(2)
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2 3 1 4 3 d k1 d BCCZ 4 6 L L 2 3 d k2 d 2 3 RD 6 L L
Then, the parameters k1 and k2 can be obtained by fitting Eq. (2) with the results measured by the commercial software. Finally, we can get k1=11 and k2=29 for
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lattice BCCZ and RD, respectively. It is worth noting that k1 and k2 are obtained by
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fitting, so it can only be used for lattice BCCZ and RD.
Fig. 2. Comparison of the relative densities obtained from analytical considerations and numerical calculations with software of CATIA.
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Fig. 2 shows the comparison of the predictions for relative density by theoretical results (Eqs. (1) and (2)) and the results measured from geometric models in the CATIA software. As can be seen from Fig. 2, Eq. (2) matches the geometric models in the CATIA software much better especially when the aspect ratio L/d is less than 10. When L/d is larger than 10, the difference between Eq. (1) and Eq. (2) can be reasonably 3 d L .
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neglected because the overlap of the volumes at the joints is at the order of
Consequently, Eq. (1) can be applied appropriately for small relative density lattice materials. In addition, for the same aspect ratio L/d, the RD lattice structure shows
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higher relative density.
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3. Elastic properties
Fig. 3. Representative unit cells of the two lattice structures: (a) 3D unit cell model of the lattice BCCZ, (b) Simplified geometry model of the unit cell of the lattice BCCZ,
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with each strut numbered, (c) 3D unit cell model of the lattice RD, (d) theoretical model of unit-cell of the lattice RD and struts numbering. For the convenience of mechanical analysis, appropriate representative unit cells (Fig. 1(c) and (f)) are taken from the two lattice structures to establish theoretical analysis models, as shown in Fig. 3. For the lattice BCCZ structure, struts ②-⑨ have
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the same length (Fig. 3(b)) and strut ① is parallel to the Z axis. Considering of the symmetry of the unit cell of the lattice RD, only 1/8 of the unit cell (Fig. 3(c)) of lattice RD is needed to be taken into account, as shown in Fig. 3(d), in which the length of the struts ①-④ are equal and they are connected at the geometrical center of the cubic
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model.
Conventional beam theory has been widely used in the determination of initial stiffness of lattice materials [39-43]. Beam theory is also adopted in this paper with the struts simplified as beams.
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3.1. Young’s moduli and shear moduli of lattice BCCZ
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Fig. 4. Sketch of mechanical systems for calculation of the effective moduli.
The lattice BCCZ can be disassembled to a BCC lattice and a separate strut and Ez
of lattice BCCZ is the combination of Ea and Eb, as illustrated in Fig. 4. Ea was deduced by Ushijima [37] and Eb is easy to deduce, finally we get 3 Ez Ea Eb 2
d d E E 4L L 4
2
(3)
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Usually, d/L is small and we ignore higher order terms, thus the effective Young’s modulus in Z-direction is
E z Eb
d
2
E 4L
(4)
For the BCCZ lattice structure under loading of an axial stress in X or Y direction,
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the internal forces of each strut in the unit cell (Fig. 3(b)) should be the same, so we can get Ex = Ey for lattice BCCZ. The internal force and deformation are determined by the
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following analysis as illustrated in Fig. 5.
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Fig. 5 Schematic diagram of the deformation due to an uniaxial loading σy: (a) the internal forces Fy on struts, (b) the deformation profile of the struts from top view of (a),
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(c) deflection of the strut.
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Considering the unit cell under uniaxial stress σy, the equivalent internal forces Fy
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in struts can be determined as
Fy
y L2 2
(5)
Fy can be decomposed into Fy1 and Fy2 :
Fy 2
2 Fy 2
(6)
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According to the theory of material mechanics, displacement caused by Fy1 is much smaller than Fy2 so that we can ignore it. The displacement δ caused by Fy2 can be solved equivalently through Fig. 5 (c) and can be expressed as 3 3 Fy 2 L 64 EI
From the relationship in the Fig. 5 (c), we have
y
2 d4 ,I 2 64
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Finally, we can get:
(7)
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Ey
y 2 3 d4 4E y 3 L
(8)
(9)
As for shear moduli of lattice BCCZ, by simply analyzing, the vertical strut in Fig.
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6(a) only has rigid displacement when lattice BCCZ is under the loading of shear in
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plane. So the internal forces of each strut in the unit cell (Fig. 3(b)) should be the same, and we can conclude Gxy= Gyz= Gxz. The internal force and deformation are determined
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by the following analysis as illustrated in Fig. 6.
Fig. 6 Schematic diagram of the internal forces and deformation of the unit cell subjected to a shear loading in x-y plane (a) the unit cell under a shear stress σxy and
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internal forces Fτ on struts, (b) the deformation profile of the unit cell from top view of (a), (c) deformation of the slanted struts.
Considering the unit cell under in-plane shear stress, the equivalent internal forces
2 xy L2
F
2
The displacement δ caused by Fτ is 3 3F L 2 E d 2
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Fτ in struts can be determined as
(10)
(11)
The shear deformation can be deduced as:
M
xy 2
2
(12)
L
xy 3 d2 Gxy E xy 9 L2
(13)
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Combining the Eqs. (10) to (12) gives
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3.2. Young’s moduli and shear moduli of lattice RD Similar to the analysis method for lattice BCCZ, when the RD lattice structure is
loaded by an axial stress in X, Y or Z direction, the internal forces of each strut in the unit cell (Fig. 3(d)) should be the same, so it is easy to get Ex = Ey = Ez for lattice RD. It is worth noting that the 1/8 part of lattice RD is just 1/2 of the BCC structure presented in Ushijima [37], so we can say
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4
1 d Ez Ebcc 4 3 E 2 L
(14)
For Gxy ,Gyz and Gxz of lattice RD, still due to symmetry, we can get Gxy= Gyz= Gxz. When lattice RD is loaded by shear force in x-y plane as illustrated in Fig. 7(a), the
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internal force and deformation are determined by the following force analysis.
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Fig. 7. (a) The unit cell of lattice RD subjected to shear loading in x-y plane, (b) equivalent internal forces Fτ on struts, (c) the deformation profile of the unit cell from
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top view of (a), (d) the relationship between the displacement δ and deflection Δ.
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Considering the unit cell under shear stress in x-y plane, the equivalent internal forces Fτ in struts (Fig. 7(b)) can be determined as
L 2 xy 2 F 2
2
The defection Δ of the strut in Fig. 7(d) can be expressed as
(15)
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3 3 F L 3 8 2 3EI
3
(16)
The displacement caused by Fτ (Fig. 7(c)) can be solved by the relationship between δ and the deflection Δ,
3 3
(17)
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Considering the deformation profile shown in Fig. 7, the effective shear deformation can be written as 4 2 L
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xy
(18)
Finally, the shear modulus can be obtained as
4
M
d Gxy 4 3 E L
Effective elastic properties of lattice BCCZ and RD under uniaxial loading. Items
BCCZ
Ez
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Lattice type
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Table 1
(19)
Ex= Ey
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BCCZ
Solutions
√
√
BCCZ
Gxy=Gyz=Gxz
RD
Ex = Ey= Ez
√
RD
Gxy=Gyz=Gxz
√
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Table 1 lists the Young’s moduli and shear moduli of lattice BCCZ and RD under uniaxial loading. It is worth to focus on the exponential difference between Ez and Ex of lattice BCCZ. For example, when d/L=0.1, Ez of lattice BCCZ is about 20 times bigger than Ex and Ey . That means bending-dominated structures perform differently under different load directions. Moreover, the exponential difference between Gxy of lattice BCCZ and RD indicates that the topology type can also affect the mechanical properties.
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Based on the above analysis, the effective moduli were shown in the principal system of the lattice. According to the theoretical analysis [44, 45], the elastic properties of these lattice materials on a general direction can also be determined by the stiffness tensor.
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4. Initial yield surfaces 4.1. Ultimate load of circular homogeneous beam
Through effective elastic properties analysis in section 3, a strut in the unit cell can be bending-dominated or stretching-dominated in different stress state (strut ② in
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lattice BCCZ under the stress of σy and σxy). Consequently, there is a transition from bending-dominated mechanism to stretching-dominated mechanism in the structure,
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which means the coupling of axial load and bending moment should be considered in the calculation of initial yield surfaces. The material is assumed to be perfectly
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elastic-plastic with a yield strength σs and the cross section is assumed to be circular
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with a diameter d=2r.
Fig. 8. The stress distribution on the cross section: (a) liner elastic stage, (b) complete plastic stage, (c) axial force on the section, (d) bending moment on the section, (e) schematic diagram of solving method for (d) , (f) schematic diagram of solving method for (d).
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According to the plastic theory [46], when plastic hinges are formed, the stress distribution on the cross section is shown in Fig. 8, in which point F is the centroid of semi-circle CDG and point E is the centroid of section ABCD. When a beam is loaded under the combination of axial force and bending moment, the neutral axis shifts, and we assume that the offset is h, as shown in Fig. 8. Fig. 8. (b)
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shows the stress distribution when the plastic hinge is formed, and it can be decomposed into two parts: the axial force F as shown in Fig. 8(c) and the bending moment M as shown in Fig. 8(d). The axial force F as shown in Fig. 8(c) is easy to calculate [46]
F 2 SABCD s
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(20)
The area of the cross section ABCD can be expressed as SABCD r 2 arcsin
h h r 2 h2 r
(21)
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As it is explained in the Fig. 8(d), (e) and (f), the bending moment can be written as (22)
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M (2 SCDG yF 2 SABCD yE ) s
For convenience, the axial force F and bending moment M can be
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non-dimensionalized with d 2 s 4 and d 3 s 6 , respectively, as following
n
F
d 2 s 4
2 h h h arcsin 1 r r r
2
(23)
3
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r 2 h2 2 m 3 d s 6 r2 M
(24)
Finally, we get the relationship between non-dimensional axial force and bending
moment when the circular section yields n
2 arcsin( 1 m 2/3 ) m1/3 1 m 2/3
(25)
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If the lattice structures are subjected to compressive stress in the axial direction, there is a competition between the elastic buckling and plastic yielding . As for strut buckling, according to the geometric parameters and boundary conditions, the max flexibility [47] of struts in lattice BCCZ and RD can be expressed as : (26)
I A
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max L /
I is the moment of inertia of the circular cross section, A is the area, and μ is a length factor. For a clamped-clamped beam, we take μ=0.5 and max 2L d . The critical
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flexibility for Euler buckling is :
p
E
p
(27)
E is the elastic modulus for a material and σp is its proportional limit. For commonly
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used materials, λp is usually greater than 50. In this study, the ratio of L/d is less than 25, which means λmax is less than λp, so elastic buckling can be avoided in the analysis of the
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initial yield surfaces.
4.2. Initial yield surfaces of unit-cell under complex stress state
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Through simple mechanical analysis, we can get the axial force and max bending moment of each strut under uniaxial stress state, with the results listed in Table 2~Table
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5. The axial force is positive if it is subjected to tensile stress and vice versa. Linear superposition can be used when the unit-cell is under complex stress state.
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Table 2 The axial force of each strut in lattice BCCZ under uniaxial stress state Strut number
Axial force under different stress states
④
√
⑤
√
⑥
√
⑦
√
⑧
√
⑨
√
σyz
σ
σ
σ
0
0
0
σ
σ
σ
σ
σ
σ
σ
σ
√
√
√
√
√
√
√
σ
0
√
σ
0
√
σ
0
√
σ
0
√
σ
0
σ
0
σ
√
σ
σ
σ
σ
√
√
√
0
√
0
√
σ
σ
σ
σ
σ
√
σ
σ
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√
σxz
√
√
√
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③
σxy
M
√
σz
√
√
√
√
√
σ
√
σ
√
σ
√
σ
√
σ
√
σ
√
σ
σ
σ
σ
σ
σ
σ
σ
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②
σy
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①
σx
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Table 3 The max bending moment of each strut in lattice BCCZ under uniaxial stress state Strut number
Max bending moment under different stress states
①
②~⑨
√
σx ,σy
σz,σxy,σxz,σyz
0
0
|σ σ |
0
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① F
√
σy √
σ
① MX ① MY
1
① MZ
1
② F
√
1
σ
② MX ② MY
1
② MZ
1
√ σ 1
√ σ 1
σ σ
√
σ
1
σ
1
σ
√ σ 1 1 1
0
σ
1
0
σ
σ
1
σ
1
0
σyz
σ
√
σ
σxz
1
σ 0
σ
σxy
√
σ
1
0
σz
σ σ 0
√ σ 1 1
σ
1
σ
1
0
σ
0
σ 1
0
1
σ
√ σ 1 1
1
1
σ
√ σ 1 0
σ
1
σ
0
σ
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σx
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Strut number
1
σ
σ σ
M
Table 5 The axial force and max bending moment of strut ③ and ④ in lattice RD under uniaxial stress state
σx
③MX
σ 0
1
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③MZ ④F
√
1
√
④MX
σ
σ 0
④MY
1
④MZ
1
1
σ
1
√ 1
1
σz
σxy
σxz
σyz
σ
√ σ 1
√ σ 1
√ σ 1
√ 1 1
σ σ
σ
σ σ
1 1
0
σ
0
σ σ
σ
0
σ
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③MY
√
σy
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③F
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Axial force or max bending moment under different stress states
Strut number
√ 1 1
σ σ σ 0
σ
√ σ 1
1
σ
1
σ
√ σ 1 1
1 1
σ
σ
0
1 1
σ
√ σ 1
σ
σ 0
0
σ 0
σ 0
1
1
1
σ σ
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The max bending moment of each strut in lattice BCCZ is only depended on x and y . Every strut under other stress states ( z , xy , yz , xz ) is stretched only and the bending moment can be ignored. In addition, for lattice BCCZ under z , strut ① is stretched only while struts ②~⑨ are bending-dominated. Consequently, the axial force
applied on strut ①.
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in struts ②~⑨ is tiny enough to be ignored, and we can assume that z is wholly .
From Table 2~Table 5, we can calculate the absolute value of axial force | | and bending moment | | of every strut by linear superposition when the unit-cell is under
them into the formula f n
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any stress state, then we nondimensionalize them through Eq. (23) and Eq. (24) and put
2 arcsin( 1 m 2/3 ) m1/3 1 m 2/3
(28)
M
With the value of f, the initial yield surface of a single strut is composed of the set of stress components which makes f equal to 0, and the initial yield surface of an unit-cell
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is the minimum envelop surface created from those surfaces of all the struts that the unit-cell contains [48]. In mathematics, that can be expressed as
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f unit-cell
max( fstrut-1, fstrut-2
, fstrut-n )
0
(29)
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In addition, if we neglect the coupling effect of axial force and bending moment, Eq.
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(28) is then changed to
5. Results and discussion 5.1. Finite element modelling
f max n 1, m 1
(30)
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In order to validate the theoretical approach, nonlinear finite element calculation were carried out using the commercial software ANSYS15.0 with both beam element and solid element. The material is set to be isotropic, elastic perfectly-plastic, and follows the Mises yield criterion with a yield strength of s . Fig. 9 illustrates the finite element model of lattice BCCZ and RD. Here we use BEAM189, a 3D and 3-node
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element, for beam model and SOLID 185, a 3D and 8-node element, for solid model. Periodic boundary conditions [49] are applied to the unit cell models. In order to verify the theoretical results, we must conduct a great many of calculations of different stress states. Therefore, a script is programmed to achieve automatic calculation and data
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M
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extraction.
Fig. 9. Finite element models of lattice BCCZ and RD: (a) beam model of lattice BCCZ, (b) beam model of lattice RD, (c) solid model of lattice BCCZ, (d) solid model of lattice RD.
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5.2. Effective moduli in principal directions
Fig. 10. Comparisons of relative effective moduli under uniaxial load: (a) Ez and Gxy of
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lattice BCCZ, (b) Ex of lattice BCCZ and Ex, Gxy of lattice RD. The red dash line overlaps with the dash dot line, because Ex equals Gxy for the lattice RD.
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In Fig. 10, the comparisons of relative effective moduli under uniaxial load obtained by theoretical results and finite element models are illustrated. For lattice BCCZ, theoretical
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prediction agrees well with FEM simulations when the aspect ratio L/d is larger than 10, however, it is 14 for lattice RD. It is reasonable because for the same unit length and
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aspect ratio, the shortest strut in lattice RD is shorter than the one in lattice BCCZ, which means using convention beam theory bring more error. As for solid model, it is
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always bigger than beam model because it considers the effect of the mass associated with the joints. The difference between these three results becomes larger with
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decreasing of the ratio L/d. Such a difference arises from the neglect of shear force for Euler beam and the inadequate modeling at the joint of the strut. That is, as pointed out by Luxner et al. [50], the possible constraints in the vicinity of the joint cannot be taken into account. However, for large L/d, these three solutions agree well with each other, similar trends can be found in many other researches [6, 18, 31].
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In addition, for a same L/d, effective moduli in Fig. 10(a) is much bigger than that in Fig. 10(b). It is reasonable because effective moduli in Fig. 10(a) is proportional to 2 4 d L while it is proportional to the d L for the effective moduli in Fig. 10(b).
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5.3. Ultimate load function of circular homogeneous beam
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Fig. 11. Comparison of ultimate load between theoretical results and numerical
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solutions.
Fig. 11 illustrates the ultimate load predicted by theoretical model and finite element
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simulations. Theoretical solutions and numerical simulations agree very well with each other, which indicate that the equation is adequate to predict ultimate load for
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homogeneous circular beam. From Fig. 11 we can see that without the coupling effect, the beam yields when max(m,n)=1, but if we consider the effect, the beam yields at the curve n=f(m) from Eq. (25). It is obvious that axial force has an effect on the ultimate bending moment, the coupling effect of axial force and bending moment should be considered. By comparing the envelope area of m=1 and n=1 with n=f(m), we can conclude that the coupling effect makes the yield surface shrink, that means, the beam becomes easier to yield.
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5.4. Initial yield surfaces under complex stress states In order to compare the theoretical analysis with the numerical simulations, lattice models with L/d of 10 are selected for following comparisons. Considering the computational efficiency, only the simulations on the FEM model with beam elements
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better displaying the initial yield surfaces.
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are carried out. In addition, the axis of z s is rescaled by multiplying a fraction for
Fig. 12. Initial yield surfaces of lattice BCCZ, L/d=10: (a) 3D yield surface under the subspace of
,
and
, (b) 2D yield surface when
=0, (d) 2D yield surface when
=0.
=0, (c) 2D yield surface when
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Fig. 12~Fig. 15 show the theoretical initial yield surfaces of lattice BCCZ and RD, all of them are verified by FEM. Fig. 12(a)~Fig. 15(a) illustrate the 3D yield surfaces, other 2D yield surfaces are degenerated from the 3D yield surfaces. From Fig. 12~Fig.
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15, we can conclude that all the FEM results agree well with theoretical surfaces.
Fig. 13. Yield surfaces of lattice RD, L/d=10: (a) 3D yield surface under the subspace of ,
and
, (b) 2D yield surface when
2D yield surface when
=0.
=0, (c) 2D yield surface when
=0, (d)
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of
,
and
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Fig. 14.Yield surfaces of lattice BCCZ, L/d=10: (a) 3D yield surface under the subspace , (b) 2D yield surface when
=0,
=0.
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(d) 2D yield surface when
=0, (c) 2D yield surface when
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Some yield surfaces in Fig. 12~Fig. 15 are the same, that is consistent with the symmetry of the unit cell. There is a distinct difference between the yield surfaces
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showed in Fig. 12~Fig. 15, and other researches [48, 51]. Surfaces showed in Fig. 12~Fig. 15, are much narrower and slender due to anisotropy (exclude the influence of sacle,
as
mentioned
at
the
beginning
of
section
5.4).
For
example,
a
stretching-dominated lattice is expected to be about ten times as stiff and about three times as strong as a bending-dominated lattice for a relative density 0.1 [17]. So when the struts in a bending-dominated structures are stretched only in some specific stress state, the yield strength will become several times bigger, that makes its yield surface narrow and slender in visual, similar results can also been seen in Demiray [52].
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Fig. 16 shows the different deformation form and von Mises stress distribution of the same unit-cell (lattice RD) in two typical different stress state. It is clearly that struts in Fig. 16(a) deform as bending-dominated struts, however, struts in Fig. 16(b) are stretched only. Both the deformation and max von Mises stress in Fig. 16(a) are much
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larger than those in Fig. 16(b).
Fig. 15. Yield surfaces of lattice RD, L/d=10: (a) 3D yield surface under the subspace of ,
and
, (b) 2D yield surface when
2D yield surface when
=0.
=0, (c) 2D yield surface when
=0, (d)
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Fig. 16. The different deformation form and von Mises stress distribution of the same unit-cell in different stress state : (a) Uniaxial tensile stress state ( x, (b) Hydrostatic stress state ( x,
xz)=(0.25,0,0,0,0,0),
5.5. Evolution of the yield surfaces 5.5.1 Influence of coupling effect
z,
xy,
z,
yz,
xy,
yz,
xz)=
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(0.25,0.25,0.25,0,0,0). (MPa)
y,
y,
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Fig. 17 shows the 3D yield surfaces of considering the coupling effect of axial force
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and bending moment (red opaque surface, from Eq. (28)) or not (black translucent surface, from Eq. (30)). In Fig. 17 we can see that the red opaque surfaces are smaller
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than those black translucent surfaces. In seceion 5.3, we conclude that the coupling effect makes the yield surface of a beam shrink, so do the yield surfaces of lattice BCCZ
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and RD. FEM results agree well with the result from Eq. (28) indicates we should consider the coupling effect. However, in Fig. 17(d), those two surfaces are nearly the
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same because, for lattice RD, the axial force is always tiny enough to ignore in this stress space ( x,
y,
xy).
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Fig. 17. 3D yield surfaces of considering the coupling effect or not, L/d=10: (a) yield surfaces of lattice BCCZ under the subspace of
and
, (b) yield surfaces of
,
under the subspace of
, (d) yield surfaces of lattice RD under the
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lattice BCCZ under the subspace of
,
and
and
, (c) yield surfaces of lattice RD
.
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subspace of
,
and
,
5.5.2.
Influence of geometric parameters
Structural geometric parameters do have an effect on the mechanical properties. To give a clear view, we take the 2D yield surface in Fig. 12(b) as an example. Fig. 18 shows the influence of geometric parameters on the yield surface. In general, the bigger L/d is, the smaller the yield surface becomes. Form Fig. 2 we know that bigger L/d
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means lower relative density, which leads to weaker mechanical properties, that is consistent with the result in Deshpande [17]. It is worth noting that point B and C are not on the line of OA, which indicates that the surfaces in Fig. 18 are not proportional in geometry. It is caused by the different failure modes in the yield surface. Take the surface when L/d equals to 10 for
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example(depicted in blue solid line): the failure mode of line segment CD and EF is the collapse of strut ① caused by axial compression or stretching, while curve CF and DE represent the collapse of the other struts caused by strut bending. As it is pointed out by Deshpande et al. [17] and Gümrük and Mines [31], the strength of struts stretching is 2 3 d L while it is proportional to d L for struts bending.
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proportional to
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Therefore, for different values of L/d, the surfaces are not proportional in geometry.
Fig. 18. The influence of geometric parameters on the yield surface of lattice BCCZ, the stress state is ( x, 5.5.3.
y,
0, 0, 0, 0), L/d=10, 12 or 15.
Influence of the change of single stress component
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Fig. 19. The evolution of the yield surface in Fig. 17(b) under the change of
z:
(a)
illustration of the yield surface and cutting plane in 3D view, (b) the evolution of the 2D z.
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yield surface under the change of
Taking the yield surface in Fig. 17(b) as an example, Fig. 19 illustrates its evolution
yield surface under
z. z.
Fig. 19(a) gives an intuitive explanation for the evolution of the
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under the change of
When
z
is given, the 3D yield surface degenerate into 2D curve
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[52], which is composed by the intersection curve between the 3D yield surface and cutting plane. Fig. 19(b) gives a detail comparison and explanation. With the increase of the yield surface using Eq. (28) in Fig. 19(b) not only shift but also change its shape,
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z,
similar conclusions can also be found in Demiray [52] and Deshpande [18]. It is
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reasonable that the surface become smaller: with the increase of
z,
deserve larger load and become easier to yield, so the envelope in
struts in the unit-cell x and
y
will shrink.
FEM simulations agree well with Eq. (28) indicates that the coupling effect of axial force and bending moment should be taken into account. All the evolutionary trends can be seen intuitively through the 3D yield surface using an imaginary translational cutting plane, which means that the initial yield surfaces in 3D view is more intuitive and can offer more information than 2D.
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6. Conclusions A theoretical study on the elastic properties and initial yield surfaces of two lattice structures is conducted. Based on the classical beam theory, the effective Young’s moduli and shear moduli are obtained as functions of aspect ratio, theoretical results agrees well with FEM when the aspect ratio L/d is larger than 10. The larger is the value
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of L/d, the smaller the deviation between theoretical results and numerical solutions will be.
In addition, the explicit expression of ultimate load for circular section is derived. The yield surfaces of lattice BCCZ and RD under complex stress states are calculated
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both theoretically and numerically, with good agreement obtained. The results reveal that the struts in a bending-dominated structure deform differently under different stress state; they can even be stretched only in some specific stress state. For example, when lattice RD is under the stress state, x y z , all the struts in lattice RD are stretched,
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which makes the yield surfaces slender and narrow. The coupling effect of axial force and bending moment makes yield surfaces shrink, especially when non-dimensional
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axial force and bending moment are close to each other. It is necessary to consider the effect when a bending-dominated lattice is under the stress state that just makes its axial
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force not minor. Geometric parameters and the change of single stress component do have an effect on the evolution of the yield surfaces. Due to the diversities in failure
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modes, the yield surfaces with different geometric parameters are not proportional in
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geometry.
As a conclusion, this paper proposed a method for predicting the elastic properties
and initial yield surfaces of two lattice materials, and this analysis method is also adequate for other lattice materials, which can be used for strength prediction under complex stress state. However, this method can be improved by considering strut buckling or plastic strengthening in further study. With more and more lattice materials are produced by additive manufacturing, it is also worth to pay attention to the influence of technology on the mechanical properties. Based on the results, some performance
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characteristics of lattice materials are summarized, these can be useful for material design and structural optimization.
Acknowledgements
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The authors thank the support from the National Natural Science Foundation of China (11672014, 11672013 and 11772027), Defense Industrial Technology Development Program (No. JCKY2016601B001 and JCKY2016205C001), Aviation Science Foundation of China (28163701002) and the Fundamental Research Funds for the
CAS is also gratefully acknowledged. References
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Central Universities. The support from Super Computing Center ScGrid/CNGrid of
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Graphical abstract
Effective elastic properties and initial yield surfaces of two 3D lattice structures Mingyang Zhang , Zhenyu Yang , Zixing Lu , Baohua Liao , Xiaofan He PII: DOI: Reference:
S0020-7403(17)32904-1 10.1016/j.ijmecsci.2018.02.008 MS 4168
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
15 October 2017 1 February 2018 3 February 2018
Please cite this article as: Mingyang Zhang , Zhenyu Yang , Zixing Lu , Baohua Liao , Xiaofan He , Effective elastic properties and initial yield surfaces of two 3D lattice structures, International Journal of Mechanical Sciences (2018), doi: 10.1016/j.ijmecsci.2018.02.008
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights
Effective elastic properties of two lattice structures are investigated theoretically.
The coupling effect of axial force and bending moment in a yielding beam is considered.
Yield surfaces under complex stress states are depicted and verified by FE
The evolution of yield surfaces is discussed systematically.
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calculation.
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Effective elastic properties and initial yield surfaces of two 3D lattice structures Mingyang Zhanga, Zhenyu Yanga*, Zixing Lua, Baohua Liaob, Xiaofan Hea Institute of Solid Mechanics, Beihang University (BUAA), Beijing 100083, P. R. China b
AVIC Chengdu Aircraft Design & Research Institute, Chengdu 610091, P. R. China
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a
Abstract
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The effective elastic properties of two bending-dominated lattice structures subjected to uniaxial loading are investigated with using the classical beam theory. In particular, considering the coupling effect of axial force and bending moment, a theoretical approach for predicting the yield surface of lattice structures under complex
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stress state is proposed. The analytical solutions are verified by comparing the predictions with finite element method (FEM) simulations. It is shown that the
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deformation mode of the struts is strongly dependent on the loading directions, and the lattice can be either bending-dominated or stretching-dominated in different stress states,
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resulting in narrow yield surfaces. The comparisons reveal that both the theoretical analysis and numerical simulations present a considerably good description of the initial
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yield surfaces, which provides new insight into understanding the mechanical behaviors
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of the lattice structures and applications in lightweight structures. Keywords: Elastic properties; Initial yield surface; Lattice structure; Beam theory;
FEM simulation
*
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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1. Introduction Lightweight structures such as honeycombs, foams, and lattice materials are widely used in aerospace because of their superior specific stiffness and specific strength [1-5]. High porosity and unique structure provide lightweight structures with many other excellent properties, such as outstanding designability, great potential for wave
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propagation control, and high energy absorption capacity [6-10]. The mechanical behavior of a lattice material is mainly dependent on its internal architecture, which have been widely studied in the previous literatures with using analytical models and numerical simulations [11-13]. Based on homogenization approach and classical beam theory, Mohr [14] developed a general micromechanics-based finite-strain constitutive
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model for truss lattice materials. Liu et al. [15] proposed a constitutive model and carried out optimization design for truss-cored sandwiches. Gibson and Ashby [16] deduced that the effective mechanical properties of a foam material are functions of the relative density of the material. Deshpande et al. [17] analyzed the topography of foams
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and proposed a criterion for lattice materials to determine the minimum node connectivity of a special class of lattice structured materials. Deshpande et al. [18]
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discussed the effective properties of an octet-truss lattice material and depicted its 2D yield surface. Fan and Wei [19] analyzed some stretching dominated lattice materials
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and calculated 3D yield surfaces for the equivalent continuum. However, traditional 3D lattice (lattice materials made by traditional processing technology) is very limited in
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topology type due to technical limitations [20]. The cell size of traditional 3D lattice material is usually of
structures
[21,
22]. Although
some of them
can be
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stretching-dominated
the order of centimeter and most of them are
bending-dominated in topology, the lattice materials are often used as core materials and are fixed on the laminate, and the additional constrains make them stretching-dominated [23-25].
With the development of the technology of additive manufacturing (AM) like selective laser melting (SLM) [26-29], many 3D lattice materials with complex configurations and smaller cell size can be produced for applications [6, 8, 30]. With the cell
size
reaching
to millimeter
scale,
many
new
configurations
including
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stretching-dominated and bending-dominated structures are designed and produced. In turn, plenty of researches were focused on the mechanical properties of 3D printing lattice materials. Gümrük and Mines [31] observed the structure of lattice materials produced by SLM and found imperfections on the surface of struts, which was then considered to
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improve the calculation precision of relative density. They also predicted the compressive strength of body-centered cubic (BBC) lattice block by finite element simulations. Gümrük et al. [32] carried out a series of experiments on the static mechanical behaviors of lattice structures under different loading conditions including uniaxial compression and shear, with the initial collapse envelope obtained
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experimentally. In order to consider the variation of diameter of the struts, Ravari et al. [33] and Tancogne-Dejean et al. [8] proposed a solid finite element model with variable cross sections, and a more accurate solution was obtained. To consider the compound effect of viscoelasticity and stochasticity, Mukhopadhyay et al. [34] analyzed irregular
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lattices and developed a practically relevant stochastic modelling approach. Tancogne-Dejean et al. [35] analyzed elasticity and initial yield surfaces of some other
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truss lattice materials theoretically, numerically and experimentally, and found elastically-isotropic truss lattices are still plastically anisotropic. Moreover, 3D printing
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technology can bring not only geometric imperfections but also internal defects in lattice material such as tiny voids and imperfect welding, which is attributed to the effects of
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complex thermodynamic process on the metal lattice structure. These all have an impact on the mechanical properties of the lattice structures. Instead of efforts of modeling the
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complex imperfections and unique internal structure, some researchers have carried out mechanical tests to get the stress-strain curves of the defected single strut for further analysis [30, 31, 36]. In this paper, the effective moduli of two 3D lattice structures in the principal directions are obtained as functions of the aspect ratio of the stuts and the initial yield surfaces are constructed under in-plane and limited 3D stress states. Issues related to the initial yield surface of the two 3D lattice structures are discussed in details, including the
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effects of the aspect ratio of struts and coupling stress states. Systematic FEM simulations are also carried out to validate the theoretical prediction. 2 . Geometries of two 3D lattice structures Fig. 1 shows the geometries of the two lattice structures analyzed in this study. The first lattice structure (Fig. 1 (a)) is named as BCCZ (body-centered cubic with a strut in
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Z direction) lattice, which is presented in the previous literatures [6, 37, 38]. The second lattice structure (Fig. 1 (d)) is a rhombic dodecahedron (RD) lattice. It is worth noting that the two kinds of representative unit cell of lattice BCCZ showed in Fig. 1(c) are equivalent. Fig. 1(f) illustrates the unit cell for the RD lattice structure. The cross-section
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unit cells in Fig. 1(c) and (f) are both L.
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of each strut is assumed to be circular, with a constant diameter d and the cell length of
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Fig. 1. The geometries of the two lattice structures: (a) specimen of lattice BCCZ produced by SLM, (b) geometric model of lattice BCCZ in isometric view, (c) two
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equivalent unit cells of lattice BCCZ, (d) specimen of lattice RD make by SLM, (e) geometric model of lattice RD in isometric view, (f) unit cell of lattice RD. The struts colored with different colors in (b) and (e) show the positions of unit cells in lattice structures. Through the derivation of geometric relations, the relative densities for a unit cell of the lattice BCCZ and RD with very low values of the ratio d/L can be determined as
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2 1 4 3 d BCCZ 4 L 2 d RD 2 3 L
(1)
These formulas are limited to the lattice materials with very low density. For the lattice materials with relative high density, the overlap of the volumes at the joints cannot be
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neglected. The precise expression for the relative density can be determined by subtracting the volumes double counted in Eq. (1), and can be expressed as:
(2)
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2 3 1 4 3 d k1 d BCCZ 4 6 L L 2 3 d k2 d 2 3 RD 6 L L
Then, the parameters k1 and k2 can be obtained by fitting Eq. (2) with the results measured by the commercial software. Finally, we can get k1=11 and k2=29 for
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lattice BCCZ and RD, respectively. It is worth noting that k1 and k2 are obtained by
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fitting, so it can only be used for lattice BCCZ and RD.
Fig. 2. Comparison of the relative densities obtained from analytical considerations and numerical calculations with software of CATIA.
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Fig. 2 shows the comparison of the predictions for relative density by theoretical results (Eqs. (1) and (2)) and the results measured from geometric models in the CATIA software. As can be seen from Fig. 2, Eq. (2) matches the geometric models in the CATIA software much better especially when the aspect ratio L/d is less than 10. When L/d is larger than 10, the difference between Eq. (1) and Eq. (2) can be reasonably 3 d L .
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neglected because the overlap of the volumes at the joints is at the order of
Consequently, Eq. (1) can be applied appropriately for small relative density lattice materials. In addition, for the same aspect ratio L/d, the RD lattice structure shows
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higher relative density.
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3. Elastic properties
Fig. 3. Representative unit cells of the two lattice structures: (a) 3D unit cell model of the lattice BCCZ, (b) Simplified geometry model of the unit cell of the lattice BCCZ,
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with each strut numbered, (c) 3D unit cell model of the lattice RD, (d) theoretical model of unit-cell of the lattice RD and struts numbering. For the convenience of mechanical analysis, appropriate representative unit cells (Fig. 1(c) and (f)) are taken from the two lattice structures to establish theoretical analysis models, as shown in Fig. 3. For the lattice BCCZ structure, struts ②-⑨ have
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the same length (Fig. 3(b)) and strut ① is parallel to the Z axis. Considering of the symmetry of the unit cell of the lattice RD, only 1/8 of the unit cell (Fig. 3(c)) of lattice RD is needed to be taken into account, as shown in Fig. 3(d), in which the length of the struts ①-④ are equal and they are connected at the geometrical center of the cubic
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model.
Conventional beam theory has been widely used in the determination of initial stiffness of lattice materials [39-43]. Beam theory is also adopted in this paper with the struts simplified as beams.
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3.1. Young’s moduli and shear moduli of lattice BCCZ
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Fig. 4. Sketch of mechanical systems for calculation of the effective moduli.
The lattice BCCZ can be disassembled to a BCC lattice and a separate strut and Ez
of lattice BCCZ is the combination of Ea and Eb, as illustrated in Fig. 4. Ea was deduced by Ushijima [37] and Eb is easy to deduce, finally we get 3 Ez Ea Eb 2
d d E E 4L L 4
2
(3)
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Usually, d/L is small and we ignore higher order terms, thus the effective Young’s modulus in Z-direction is
E z Eb
d
2
E 4L
(4)
For the BCCZ lattice structure under loading of an axial stress in X or Y direction,
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the internal forces of each strut in the unit cell (Fig. 3(b)) should be the same, so we can get Ex = Ey for lattice BCCZ. The internal force and deformation are determined by the
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following analysis as illustrated in Fig. 5.
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Fig. 5 Schematic diagram of the deformation due to an uniaxial loading σy: (a) the internal forces Fy on struts, (b) the deformation profile of the struts from top view of (a),
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(c) deflection of the strut.
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Considering the unit cell under uniaxial stress σy, the equivalent internal forces Fy
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in struts can be determined as
Fy
y L2 2
(5)
Fy can be decomposed into Fy1 and Fy2 :
Fy 2
2 Fy 2
(6)
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According to the theory of material mechanics, displacement caused by Fy1 is much smaller than Fy2 so that we can ignore it. The displacement δ caused by Fy2 can be solved equivalently through Fig. 5 (c) and can be expressed as 3 3 Fy 2 L 64 EI
From the relationship in the Fig. 5 (c), we have
y
2 d4 ,I 2 64
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Finally, we can get:
(7)
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Ey
y 2 3 d4 4E y 3 L
(8)
(9)
As for shear moduli of lattice BCCZ, by simply analyzing, the vertical strut in Fig.
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6(a) only has rigid displacement when lattice BCCZ is under the loading of shear in
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plane. So the internal forces of each strut in the unit cell (Fig. 3(b)) should be the same, and we can conclude Gxy= Gyz= Gxz. The internal force and deformation are determined
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by the following analysis as illustrated in Fig. 6.
Fig. 6 Schematic diagram of the internal forces and deformation of the unit cell subjected to a shear loading in x-y plane (a) the unit cell under a shear stress σxy and
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internal forces Fτ on struts, (b) the deformation profile of the unit cell from top view of (a), (c) deformation of the slanted struts.
Considering the unit cell under in-plane shear stress, the equivalent internal forces
2 xy L2
F
2
The displacement δ caused by Fτ is 3 3F L 2 E d 2
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Fτ in struts can be determined as
(10)
(11)
The shear deformation can be deduced as:
M
xy 2
2
(12)
L
xy 3 d2 Gxy E xy 9 L2
(13)
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Combining the Eqs. (10) to (12) gives
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3.2. Young’s moduli and shear moduli of lattice RD Similar to the analysis method for lattice BCCZ, when the RD lattice structure is
loaded by an axial stress in X, Y or Z direction, the internal forces of each strut in the unit cell (Fig. 3(d)) should be the same, so it is easy to get Ex = Ey = Ez for lattice RD. It is worth noting that the 1/8 part of lattice RD is just 1/2 of the BCC structure presented in Ushijima [37], so we can say
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4
1 d Ez Ebcc 4 3 E 2 L
(14)
For Gxy ,Gyz and Gxz of lattice RD, still due to symmetry, we can get Gxy= Gyz= Gxz. When lattice RD is loaded by shear force in x-y plane as illustrated in Fig. 7(a), the
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internal force and deformation are determined by the following force analysis.
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Fig. 7. (a) The unit cell of lattice RD subjected to shear loading in x-y plane, (b) equivalent internal forces Fτ on struts, (c) the deformation profile of the unit cell from
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top view of (a), (d) the relationship between the displacement δ and deflection Δ.
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Considering the unit cell under shear stress in x-y plane, the equivalent internal forces Fτ in struts (Fig. 7(b)) can be determined as
L 2 xy 2 F 2
2
The defection Δ of the strut in Fig. 7(d) can be expressed as
(15)
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3 3 F L 3 8 2 3EI
3
(16)
The displacement caused by Fτ (Fig. 7(c)) can be solved by the relationship between δ and the deflection Δ,
3 3
(17)
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Considering the deformation profile shown in Fig. 7, the effective shear deformation can be written as 4 2 L
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xy
(18)
Finally, the shear modulus can be obtained as
4
M
d Gxy 4 3 E L
Effective elastic properties of lattice BCCZ and RD under uniaxial loading. Items
BCCZ
Ez
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Lattice type
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Table 1
(19)
Ex= Ey
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BCCZ
Solutions
√
√
BCCZ
Gxy=Gyz=Gxz
RD
Ex = Ey= Ez
√
RD
Gxy=Gyz=Gxz
√
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Table 1 lists the Young’s moduli and shear moduli of lattice BCCZ and RD under uniaxial loading. It is worth to focus on the exponential difference between Ez and Ex of lattice BCCZ. For example, when d/L=0.1, Ez of lattice BCCZ is about 20 times bigger than Ex and Ey . That means bending-dominated structures perform differently under different load directions. Moreover, the exponential difference between Gxy of lattice BCCZ and RD indicates that the topology type can also affect the mechanical properties.
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Based on the above analysis, the effective moduli were shown in the principal system of the lattice. According to the theoretical analysis [44, 45], the elastic properties of these lattice materials on a general direction can also be determined by the stiffness tensor.
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4. Initial yield surfaces 4.1. Ultimate load of circular homogeneous beam
Through effective elastic properties analysis in section 3, a strut in the unit cell can be bending-dominated or stretching-dominated in different stress state (strut ② in
M
lattice BCCZ under the stress of σy and σxy). Consequently, there is a transition from bending-dominated mechanism to stretching-dominated mechanism in the structure,
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which means the coupling of axial load and bending moment should be considered in the calculation of initial yield surfaces. The material is assumed to be perfectly
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elastic-plastic with a yield strength σs and the cross section is assumed to be circular
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with a diameter d=2r.
Fig. 8. The stress distribution on the cross section: (a) liner elastic stage, (b) complete plastic stage, (c) axial force on the section, (d) bending moment on the section, (e) schematic diagram of solving method for (d) , (f) schematic diagram of solving method for (d).
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According to the plastic theory [46], when plastic hinges are formed, the stress distribution on the cross section is shown in Fig. 8, in which point F is the centroid of semi-circle CDG and point E is the centroid of section ABCD. When a beam is loaded under the combination of axial force and bending moment, the neutral axis shifts, and we assume that the offset is h, as shown in Fig. 8. Fig. 8. (b)
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shows the stress distribution when the plastic hinge is formed, and it can be decomposed into two parts: the axial force F as shown in Fig. 8(c) and the bending moment M as shown in Fig. 8(d). The axial force F as shown in Fig. 8(c) is easy to calculate [46]
F 2 SABCD s
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(20)
The area of the cross section ABCD can be expressed as SABCD r 2 arcsin
h h r 2 h2 r
(21)
M
As it is explained in the Fig. 8(d), (e) and (f), the bending moment can be written as (22)
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M (2 SCDG yF 2 SABCD yE ) s
For convenience, the axial force F and bending moment M can be
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non-dimensionalized with d 2 s 4 and d 3 s 6 , respectively, as following
n
F
d 2 s 4
2 h h h arcsin 1 r r r
2
(23)
3
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r 2 h2 2 m 3 d s 6 r2 M
(24)
Finally, we get the relationship between non-dimensional axial force and bending
moment when the circular section yields n
2 arcsin( 1 m 2/3 ) m1/3 1 m 2/3
(25)
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If the lattice structures are subjected to compressive stress in the axial direction, there is a competition between the elastic buckling and plastic yielding . As for strut buckling, according to the geometric parameters and boundary conditions, the max flexibility [47] of struts in lattice BCCZ and RD can be expressed as : (26)
I A
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max L /
I is the moment of inertia of the circular cross section, A is the area, and μ is a length factor. For a clamped-clamped beam, we take μ=0.5 and max 2L d . The critical
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flexibility for Euler buckling is :
p
E
p
(27)
E is the elastic modulus for a material and σp is its proportional limit. For commonly
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used materials, λp is usually greater than 50. In this study, the ratio of L/d is less than 25, which means λmax is less than λp, so elastic buckling can be avoided in the analysis of the
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initial yield surfaces.
4.2. Initial yield surfaces of unit-cell under complex stress state
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Through simple mechanical analysis, we can get the axial force and max bending moment of each strut under uniaxial stress state, with the results listed in Table 2~Table
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5. The axial force is positive if it is subjected to tensile stress and vice versa. Linear superposition can be used when the unit-cell is under complex stress state.
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Table 2 The axial force of each strut in lattice BCCZ under uniaxial stress state Strut number
Axial force under different stress states
④
√
⑤
√
⑥
√
⑦
√
⑧
√
⑨
√
σyz
σ
σ
σ
0
0
0
σ
σ
σ
σ
σ
σ
σ
σ
√
√
√
√
√
√
√
σ
0
√
σ
0
√
σ
0
√
σ
0
√
σ
0
σ
0
σ
√
σ
σ
σ
σ
√
√
√
0
√
0
√
σ
σ
σ
σ
σ
√
σ
σ
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√
σxz
√
√
√
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③
σxy
M
√
σz
√
√
√
√
√
σ
√
σ
√
σ
√
σ
√
σ
√
σ
√
σ
σ
σ
σ
σ
σ
σ
σ
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②
σy
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①
σx
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Table 3 The max bending moment of each strut in lattice BCCZ under uniaxial stress state Strut number
Max bending moment under different stress states
①
②~⑨
√
σx ,σy
σz,σxy,σxz,σyz
0
0
|σ σ |
0
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① F
√
σy √
σ
① MX ① MY
1
① MZ
1
② F
√
1
σ
② MX ② MY
1
② MZ
1
√ σ 1
√ σ 1
σ σ
√
σ
1
σ
1
σ
√ σ 1 1 1
0
σ
1
0
σ
σ
1
σ
1
0
σyz
σ
√
σ
σxz
1
σ 0
σ
σxy
√
σ
1
0
σz
σ σ 0
√ σ 1 1
σ
1
σ
1
0
σ
0
σ 1
0
1
σ
√ σ 1 1
1
1
σ
√ σ 1 0
σ
1
σ
0
σ
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σx
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Strut number
1
σ
σ σ
M
Table 5 The axial force and max bending moment of strut ③ and ④ in lattice RD under uniaxial stress state
σx
③MX
σ 0
1
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③MZ ④F
√
1
√
④MX
σ
σ 0
④MY
1
④MZ
1
1
σ
1
√ 1
1
σz
σxy
σxz
σyz
σ
√ σ 1
√ σ 1
√ σ 1
√ 1 1
σ σ
σ
σ σ
1 1
0
σ
0
σ σ
σ
0
σ
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③MY
√
σy
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③F
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Axial force or max bending moment under different stress states
Strut number
√ 1 1
σ σ σ 0
σ
√ σ 1
1
σ
1
σ
√ σ 1 1
1 1
σ
σ
0
1 1
σ
√ σ 1
σ
σ 0
0
σ 0
σ 0
1
1
1
σ σ
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The max bending moment of each strut in lattice BCCZ is only depended on x and y . Every strut under other stress states ( z , xy , yz , xz ) is stretched only and the bending moment can be ignored. In addition, for lattice BCCZ under z , strut ① is stretched only while struts ②~⑨ are bending-dominated. Consequently, the axial force
applied on strut ①.
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in struts ②~⑨ is tiny enough to be ignored, and we can assume that z is wholly .
From Table 2~Table 5, we can calculate the absolute value of axial force | | and bending moment | | of every strut by linear superposition when the unit-cell is under
them into the formula f n
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any stress state, then we nondimensionalize them through Eq. (23) and Eq. (24) and put
2 arcsin( 1 m 2/3 ) m1/3 1 m 2/3
(28)
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With the value of f, the initial yield surface of a single strut is composed of the set of stress components which makes f equal to 0, and the initial yield surface of an unit-cell
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is the minimum envelop surface created from those surfaces of all the struts that the unit-cell contains [48]. In mathematics, that can be expressed as
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f unit-cell
max( fstrut-1, fstrut-2
, fstrut-n )
0
(29)
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In addition, if we neglect the coupling effect of axial force and bending moment, Eq.
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(28) is then changed to
5. Results and discussion 5.1. Finite element modelling
f max n 1, m 1
(30)
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In order to validate the theoretical approach, nonlinear finite element calculation were carried out using the commercial software ANSYS15.0 with both beam element and solid element. The material is set to be isotropic, elastic perfectly-plastic, and follows the Mises yield criterion with a yield strength of s . Fig. 9 illustrates the finite element model of lattice BCCZ and RD. Here we use BEAM189, a 3D and 3-node
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element, for beam model and SOLID 185, a 3D and 8-node element, for solid model. Periodic boundary conditions [49] are applied to the unit cell models. In order to verify the theoretical results, we must conduct a great many of calculations of different stress states. Therefore, a script is programmed to achieve automatic calculation and data
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PT
ED
M
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extraction.
Fig. 9. Finite element models of lattice BCCZ and RD: (a) beam model of lattice BCCZ, (b) beam model of lattice RD, (c) solid model of lattice BCCZ, (d) solid model of lattice RD.
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5.2. Effective moduli in principal directions
Fig. 10. Comparisons of relative effective moduli under uniaxial load: (a) Ez and Gxy of
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lattice BCCZ, (b) Ex of lattice BCCZ and Ex, Gxy of lattice RD. The red dash line overlaps with the dash dot line, because Ex equals Gxy for the lattice RD.
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In Fig. 10, the comparisons of relative effective moduli under uniaxial load obtained by theoretical results and finite element models are illustrated. For lattice BCCZ, theoretical
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prediction agrees well with FEM simulations when the aspect ratio L/d is larger than 10, however, it is 14 for lattice RD. It is reasonable because for the same unit length and
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aspect ratio, the shortest strut in lattice RD is shorter than the one in lattice BCCZ, which means using convention beam theory bring more error. As for solid model, it is
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always bigger than beam model because it considers the effect of the mass associated with the joints. The difference between these three results becomes larger with
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decreasing of the ratio L/d. Such a difference arises from the neglect of shear force for Euler beam and the inadequate modeling at the joint of the strut. That is, as pointed out by Luxner et al. [50], the possible constraints in the vicinity of the joint cannot be taken into account. However, for large L/d, these three solutions agree well with each other, similar trends can be found in many other researches [6, 18, 31].
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In addition, for a same L/d, effective moduli in Fig. 10(a) is much bigger than that in Fig. 10(b). It is reasonable because effective moduli in Fig. 10(a) is proportional to 2 4 d L while it is proportional to the d L for the effective moduli in Fig. 10(b).
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5.3. Ultimate load function of circular homogeneous beam
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Fig. 11. Comparison of ultimate load between theoretical results and numerical
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solutions.
Fig. 11 illustrates the ultimate load predicted by theoretical model and finite element
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simulations. Theoretical solutions and numerical simulations agree very well with each other, which indicate that the equation is adequate to predict ultimate load for
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homogeneous circular beam. From Fig. 11 we can see that without the coupling effect, the beam yields when max(m,n)=1, but if we consider the effect, the beam yields at the curve n=f(m) from Eq. (25). It is obvious that axial force has an effect on the ultimate bending moment, the coupling effect of axial force and bending moment should be considered. By comparing the envelope area of m=1 and n=1 with n=f(m), we can conclude that the coupling effect makes the yield surface shrink, that means, the beam becomes easier to yield.
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5.4. Initial yield surfaces under complex stress states In order to compare the theoretical analysis with the numerical simulations, lattice models with L/d of 10 are selected for following comparisons. Considering the computational efficiency, only the simulations on the FEM model with beam elements
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better displaying the initial yield surfaces.
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are carried out. In addition, the axis of z s is rescaled by multiplying a fraction for
Fig. 12. Initial yield surfaces of lattice BCCZ, L/d=10: (a) 3D yield surface under the subspace of
,
and
, (b) 2D yield surface when
=0, (d) 2D yield surface when
=0.
=0, (c) 2D yield surface when
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Fig. 12~Fig. 15 show the theoretical initial yield surfaces of lattice BCCZ and RD, all of them are verified by FEM. Fig. 12(a)~Fig. 15(a) illustrate the 3D yield surfaces, other 2D yield surfaces are degenerated from the 3D yield surfaces. From Fig. 12~Fig.
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15, we can conclude that all the FEM results agree well with theoretical surfaces.
Fig. 13. Yield surfaces of lattice RD, L/d=10: (a) 3D yield surface under the subspace of ,
and
, (b) 2D yield surface when
2D yield surface when
=0.
=0, (c) 2D yield surface when
=0, (d)
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of
,
and
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Fig. 14.Yield surfaces of lattice BCCZ, L/d=10: (a) 3D yield surface under the subspace , (b) 2D yield surface when
=0,
=0.
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(d) 2D yield surface when
=0, (c) 2D yield surface when
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Some yield surfaces in Fig. 12~Fig. 15 are the same, that is consistent with the symmetry of the unit cell. There is a distinct difference between the yield surfaces
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showed in Fig. 12~Fig. 15, and other researches [48, 51]. Surfaces showed in Fig. 12~Fig. 15, are much narrower and slender due to anisotropy (exclude the influence of sacle,
as
mentioned
at
the
beginning
of
section
5.4).
For
example,
a
stretching-dominated lattice is expected to be about ten times as stiff and about three times as strong as a bending-dominated lattice for a relative density 0.1 [17]. So when the struts in a bending-dominated structures are stretched only in some specific stress state, the yield strength will become several times bigger, that makes its yield surface narrow and slender in visual, similar results can also been seen in Demiray [52].
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Fig. 16 shows the different deformation form and von Mises stress distribution of the same unit-cell (lattice RD) in two typical different stress state. It is clearly that struts in Fig. 16(a) deform as bending-dominated struts, however, struts in Fig. 16(b) are stretched only. Both the deformation and max von Mises stress in Fig. 16(a) are much
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larger than those in Fig. 16(b).
Fig. 15. Yield surfaces of lattice RD, L/d=10: (a) 3D yield surface under the subspace of ,
and
, (b) 2D yield surface when
2D yield surface when
=0.
=0, (c) 2D yield surface when
=0, (d)
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Fig. 16. The different deformation form and von Mises stress distribution of the same unit-cell in different stress state : (a) Uniaxial tensile stress state ( x, (b) Hydrostatic stress state ( x,
xz)=(0.25,0,0,0,0,0),
5.5. Evolution of the yield surfaces 5.5.1 Influence of coupling effect
z,
xy,
z,
yz,
xy,
yz,
xz)=
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(0.25,0.25,0.25,0,0,0). (MPa)
y,
y,
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Fig. 17 shows the 3D yield surfaces of considering the coupling effect of axial force
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and bending moment (red opaque surface, from Eq. (28)) or not (black translucent surface, from Eq. (30)). In Fig. 17 we can see that the red opaque surfaces are smaller
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than those black translucent surfaces. In seceion 5.3, we conclude that the coupling effect makes the yield surface of a beam shrink, so do the yield surfaces of lattice BCCZ
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and RD. FEM results agree well with the result from Eq. (28) indicates we should consider the coupling effect. However, in Fig. 17(d), those two surfaces are nearly the
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same because, for lattice RD, the axial force is always tiny enough to ignore in this stress space ( x,
y,
xy).
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ED
Fig. 17. 3D yield surfaces of considering the coupling effect or not, L/d=10: (a) yield surfaces of lattice BCCZ under the subspace of
and
, (b) yield surfaces of
,
under the subspace of
, (d) yield surfaces of lattice RD under the
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lattice BCCZ under the subspace of
,
and
and
, (c) yield surfaces of lattice RD
.
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subspace of
,
and
,
5.5.2.
Influence of geometric parameters
Structural geometric parameters do have an effect on the mechanical properties. To give a clear view, we take the 2D yield surface in Fig. 12(b) as an example. Fig. 18 shows the influence of geometric parameters on the yield surface. In general, the bigger L/d is, the smaller the yield surface becomes. Form Fig. 2 we know that bigger L/d
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means lower relative density, which leads to weaker mechanical properties, that is consistent with the result in Deshpande [17]. It is worth noting that point B and C are not on the line of OA, which indicates that the surfaces in Fig. 18 are not proportional in geometry. It is caused by the different failure modes in the yield surface. Take the surface when L/d equals to 10 for
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example(depicted in blue solid line): the failure mode of line segment CD and EF is the collapse of strut ① caused by axial compression or stretching, while curve CF and DE represent the collapse of the other struts caused by strut bending. As it is pointed out by Deshpande et al. [17] and Gümrük and Mines [31], the strength of struts stretching is 2 3 d L while it is proportional to d L for struts bending.
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proportional to
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ED
M
Therefore, for different values of L/d, the surfaces are not proportional in geometry.
Fig. 18. The influence of geometric parameters on the yield surface of lattice BCCZ, the stress state is ( x, 5.5.3.
y,
0, 0, 0, 0), L/d=10, 12 or 15.
Influence of the change of single stress component
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Fig. 19. The evolution of the yield surface in Fig. 17(b) under the change of
z:
(a)
illustration of the yield surface and cutting plane in 3D view, (b) the evolution of the 2D z.
M
yield surface under the change of
Taking the yield surface in Fig. 17(b) as an example, Fig. 19 illustrates its evolution
yield surface under
z. z.
Fig. 19(a) gives an intuitive explanation for the evolution of the
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under the change of
When
z
is given, the 3D yield surface degenerate into 2D curve
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[52], which is composed by the intersection curve between the 3D yield surface and cutting plane. Fig. 19(b) gives a detail comparison and explanation. With the increase of the yield surface using Eq. (28) in Fig. 19(b) not only shift but also change its shape,
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z,
similar conclusions can also be found in Demiray [52] and Deshpande [18]. It is
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reasonable that the surface become smaller: with the increase of
z,
deserve larger load and become easier to yield, so the envelope in
struts in the unit-cell x and
y
will shrink.
FEM simulations agree well with Eq. (28) indicates that the coupling effect of axial force and bending moment should be taken into account. All the evolutionary trends can be seen intuitively through the 3D yield surface using an imaginary translational cutting plane, which means that the initial yield surfaces in 3D view is more intuitive and can offer more information than 2D.
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6. Conclusions A theoretical study on the elastic properties and initial yield surfaces of two lattice structures is conducted. Based on the classical beam theory, the effective Young’s moduli and shear moduli are obtained as functions of aspect ratio, theoretical results agrees well with FEM when the aspect ratio L/d is larger than 10. The larger is the value
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of L/d, the smaller the deviation between theoretical results and numerical solutions will be.
In addition, the explicit expression of ultimate load for circular section is derived. The yield surfaces of lattice BCCZ and RD under complex stress states are calculated
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both theoretically and numerically, with good agreement obtained. The results reveal that the struts in a bending-dominated structure deform differently under different stress state; they can even be stretched only in some specific stress state. For example, when lattice RD is under the stress state, x y z , all the struts in lattice RD are stretched,
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which makes the yield surfaces slender and narrow. The coupling effect of axial force and bending moment makes yield surfaces shrink, especially when non-dimensional
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axial force and bending moment are close to each other. It is necessary to consider the effect when a bending-dominated lattice is under the stress state that just makes its axial
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force not minor. Geometric parameters and the change of single stress component do have an effect on the evolution of the yield surfaces. Due to the diversities in failure
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modes, the yield surfaces with different geometric parameters are not proportional in
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geometry.
As a conclusion, this paper proposed a method for predicting the elastic properties
and initial yield surfaces of two lattice materials, and this analysis method is also adequate for other lattice materials, which can be used for strength prediction under complex stress state. However, this method can be improved by considering strut buckling or plastic strengthening in further study. With more and more lattice materials are produced by additive manufacturing, it is also worth to pay attention to the influence of technology on the mechanical properties. Based on the results, some performance
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characteristics of lattice materials are summarized, these can be useful for material design and structural optimization.
Acknowledgements
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The authors thank the support from the National Natural Science Foundation of China (11672014, 11672013 and 11772027), Defense Industrial Technology Development Program (No. JCKY2016601B001 and JCKY2016205C001), Aviation Science Foundation of China (28163701002) and the Fundamental Research Funds for the
CAS is also gratefully acknowledged. References
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Central Universities. The support from Super Computing Center ScGrid/CNGrid of
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Graphical abstract