Mechanical and energy absorption characteristics of additively manufactured functionally graded sheet lattice structures with minimal surfaces

International Journal of Mechanical Sciences 167 (2020) 105262

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Mechanical and energy absorption characteristics of additively manufactured functionally graded sheet lattice structures with minimal surfaces Miao Zhao a, David Z. Zhang a,b,∗, Fei Liu a, Zhonghua Li c, Zhibo Ma a, Zhihao Ren a a

State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing, 400044, China College of Engineering, Mathematics and Physical Sciences, University of Exeter, North Park Road, EX4 4QF, UK c School of Mechanical Engineering, North University of China, Taiyuan, 030051, China b

a r t i c l e

i n f o

Keywords: Functionally graded lattice structure Triply periodic minimal surface Mechanical properties Energy absorbing Finite element analysis Selective laser melting

a b s t r a c t Functionally graded (FG) lattice structures have attracted attention for their potential application in lightweighting and energy absorption. In this study, functionally graded sheet (FGS) structures with primitive (P) and gyroid (G) minimal surfaces were fabricated by selective laser melting (SLM) using Ti-6Al-4V powder. The mechanical properties, deformation behavior, and energy absorption performance of uniform sheet (US) and FGS lattice structures were systematically investigated using compression tests and the finite element method (FEM). The FGS structures were found to eliminate abrupt shear failure and to exhibit the predictable layer-by-layer deformation accompanied by sub-layer collapse. The cumulative energy absorption per unit volume of FGS samples increased as a power of strain function throughout the compression process, while the US samples exhibited a linear relationship, thus resulting in an excellent energy absorption capability for FGS structures. The energy absorption of FGS samples was higher than for US samples by approximately 60%. In addition, the results of FEM with the Johnson-Cook models demonstrated a high capability for predicting the deformation behaviors as well as mechanical properties (especially in terms of yield strength, compressive strength and energy absorption) of lattice structures. which can be used to provide guidance on selecting a lattice structure to meet multifunctional demands.

1. Introduction Lattice structures are seen as multifunctional materials, providing benefits in applications including lightweight design [1], energy absorption [2], heat exchangers [3], and biomedical scaffolds [1,4]. Conventional methods of fabricating metallic porous structures include melt gas injection [5], investment casting [6], physical vapor deposition [7], and sheet metal technology [8]. However, lattice structures with controllable volume fraction and well-defined internal architectures are still unachievable through these conventional fabrication methods. Thanks to additive manufacturing (AM) technology, such as selective laser melting (SLM) [9] and electron beam melting (EBM) [10], metallic lattice structures in the micro scale range with complex geometries and well-defined internal configurations are now able to be fabricated. The study of Gibson and Ashby [11] revealed that the mechanical properties of a lattice structure are closely related to its volume fraction. Thus, lattice structures with the graded change in volume fraction, namely functionally graded (FG) lattice structures, are attracting atten-

∗

tion among researchers due to their distinctive mechanical properties, energy absorption performance, and biomedical properties. Previous research has investigated several FG structures, such as cubic [12], diamond [13], and body-centred cubic (BCC) [14]. For instance, Maskery et al. [14] manufactured graded BCC lattice structures with Al-Si-10Mg by SLM technology and found that the FG lattice structures absorbed more energy than the uniform ones. Choy et al. [12] investigated the FG cubic and honeycomb lattice structures with the strut diameters continuously changed along the load direction, and the compression tests demonstrated that the energy absorption behavior of the graded structures was improved due to a distinctive layer-by-layer deformation. In addition, the cell proliferation experiments demonstrated that FG octet truss and tetrahedron structures were more suitable for bone tissue implantation because of their high cell proliferation rate, in comparison to uniform structures [15]. However, most of their FG models were designed with strut diameters or pore size changes at every layer separately, and these strut-like FG lattice structures were generated by CADbased methods, which are more time-consuming for the designer and resource-consuming for the computer. Moreover, these lattice structures

Corresponding author. E-mail address: [email protected] (D.Z. Zhang).

https://doi.org/10.1016/j.ijmecsci.2019.105262 Received 10 September 2019; Received in revised form 11 October 2019; Accepted 18 October 2019 Available online 18 October 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.

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International Journal of Mechanical Sciences 167 (2020) 105262

tend to experience more stress concentrations near the nodes during loading, which results in low mechanical properties [16,17]. Currently, an increasing level of interest has focused on a novel mathematical approach to create FG lattice structures with triply periodic minimal surfaces (TPMS). The boundary between the void and solid material is represented by TPMS, and specific volume fractions of lattice structures are able to be achieved by modifying the offset parameter in TPMS formulas [16]. Therefore, FG lattice structures were directly generated by adding a linear z-value offset term to the TPMS formula [18-20]. Additionally, it has already been shown that the smooth convergence of struts with TPMS can reduce stress concentrations [16], and a high surface to volume ratio can help cells to adhere and stimulate cell ingrowth in biomedical applications [21]. Apart from these skeleton TPMS-based lattice, another form of the TPMS-based lattice, known as the sheet structure, was created by thickening the surfaces [22]. The sheet structures were recently identified as having a higher stiffness, strength, and energy absorption capability compared to the skeleton ones at the same volume fraction, making it particularly more suitable for use in engineering applications [22,23]. It is found that most investigations in mechanical properties and energy absorption behavior of FG lattice structures are limited to experimental work, which can be time-consuming and expensive for the high costs of metal powder, machines, operators, processing, and experiment. Recently, using the finite element method (FEM) to predict the properties of FG lattice structures has attracted interest. For example, the Von Mises stress distribution and elastic modulus of FG lattice structures in the elastic region have been studied using FEM [20,24–26]. Further, the compressive behaviors of FG lattice structures with brittle metallic material were experimentally and numerically investigated, and periodic hardening and softening behaviors were observed in the fluctuating stage due to the brittle fracture mechanisms [27]. However, as an important factor to influence the energy absorption performance, this behavior was not being considered in the finite element model. In order to predict the deformation and energy absorption behavior of FG structures, FEM for their plastic and fluctuating responses should be developed. While advantages of TPMS-based uniform sheet (US) lattice structures were previously reported in literature, limited literature has been published to design and investigate novel TPMS-based functionally graded sheet (FGS) lattice structures. Therefore, the aim of this study was to experimentally and numerically investigate the mechanical properties, deformation behavior, and energy absorption performance of TPMS-based FGS lattice structures fabricated by SLM with Ti-6Al-4V. For this purpose, FGS lattice structures based on two popular minimal surfaces, namely primitive (P) and gyroid (G), with volume fractions continuously changing from 30% to 10% were generated using the TPMS formula and fabricated by means of SLM. The corresponding uniform sheet (US) lattice structures with 20% volume fraction were also fabricated for comparison. The morphological characteristics, mechanical properties, and energy absorption capability of the fabricated lattice structures were investigated. In addition, finite element analysis with the Johnson-Cook plasticity model and Johnson-Cook damage model were performed to simulate the mechanical and energy absorption characteristics of the FGS and US structures under compressive loading. 2. Materials and methods 2.1. Design of sheet lattice structures The boundary between solid and void material lattice structures can be mathematically defined by TPMS equations using implicit methods. The primitive (P) and gyroid (G) lattice boundaries are expressed as: ( ) ( ) ( ) 𝜑𝑃 (𝑥, 𝑦, 𝑧) = cos 𝑘𝑥 𝑥 + cos 𝑘𝑦 𝑦 + cos 𝑘𝑧 𝑧 = 𝑐 (1) ( ) ( ) ( ) ( ) ( ) ( ) 𝜑𝐺 (𝑥, 𝑦, 𝑧) = cos 𝑘𝑥 𝑥 sin 𝑘𝑦 𝑦 + cos 𝑘𝑦 𝑦 sin 𝑘𝑧 𝑧 + cos 𝑘𝑧 𝑧 sin 𝑘𝑥 𝑥 =𝑐

(2)

Fig. 1. CAD representation of (a) sheet P and (b) sheet G unit cell.

Fig. 2. Relationship between parameter c and volume fraction (𝜌∗ ) for sheet P and sheet G lattice structures.

in which, 𝑛 𝑘𝑖 = 2𝜋 𝑖 (𝑖 = 𝑥, 𝑦, 𝑧) 𝐿𝑖

(3)

where, c is an offset parameter used to control the position of the solidvoid boundary, ki (i = x, y, z) are the TPMS function periodicities to control the unit size in x, y, and z directions, ni is the number of lattice units in Li length in x, y, and z directions. For the sheet TPMS lattice structures, the solid-void boundaries are described by two TPMS equations. Therefore, the solid regions shown in Fig. 1 are defined by the inequality of − c ≤ 𝜙(x, y, z) ≤ c, and volume fraction (𝜌∗ ) of sheet lattice structures are calculated by the triply integral method as follows: 𝜌∗ =

∭ (−𝑐 ≤ 𝜑(𝑥, 𝑦, 𝑧)≤𝑐 ) ∩ (|𝑥|≤𝐿∕2) ∩ (|𝑦|≤𝑊 ∕2) ∩ (|𝑧|≤𝐻∕2)𝑑 𝑥𝑑 𝑦𝑑 𝑧 𝐿⋅𝑊 ⋅𝐻 (4)

where, L, W, and H are the length, width, and height of the sheet lattice structures, respectively. The thickness of the sheet lattice structures is varied by parameter c, which controls the volume fraction (𝜌∗ ) of lattice structures. With this in mind, sheet P and G lattice structures with predefined volume fractions (𝜌∗ ) were generated in order to develop the relationship between the volume fraction (𝜌∗ ) and parameter c, shown in Fig. 2. Thus, the parameter c for a uniform sheet P and sheet G lattice structure with a predefined volume fraction can be generated by using the linear fitted equations c = 0.01753𝜌∗ and c = 0.01552𝜌∗ , respectively. In this study, the design samples had an overall dimension of 20 mm × 20 mm × 20 mm corresponding to 5, 5, and 5 lattice units along each direction, which was achieved by setting the parameters kx = ky = kz = 0.5𝜋 and x, y, z ∈ [0, 20]. Both lattice structures were designed with an average volume fraction of 20%. For US lattice structures, the offset parameter c of sheet P and sheet G structure was 0.3506 and 0.3104, respectively. For the FGS lattice structures, the offset parameter c = 0.5259 − 0.01753z and c = 0.4656 − 0.01552z corresponded to sheet P and sheet G structures with decreased volume fractions from the bottom (30%) to the top (10%) in the z direction. The CAD models were performed by a MATLAB code to generate the STL files and processed in Magics software, shown in Fig. 3. Then, a plate with 0.5 mm thickness was added to the bottom of each lattice

M. Zhao, D.Z. Zhang and F. Liu et al.

Fig. 3. Three-dimensional models of (a) US P, (b) US G, (b) FGS P, and (b) FGS G lattice structures.

International Journal of Mechanical Sciences 167 (2020) 105262

morphology of lattice samples before compression tests were characterized by a scanning electron microscope (SEM) (VEGA3 LMH, Tescan Inc., Brno, The Czech Republic). In order to evaluate the manufacturability of the sheet lattice structures, a digital vernier caliper (smallest scale division of 0.01 mm) and a digital scale (smallest scale increments of 0.0001 g) were used to measure the dimensions and masses of the lattice samples, respectively. The designed mass of the lattice structures were calculated from m = V𝜌s , where V is designed volume of the lattice structures, and 𝜌s is the density of the bulk Ti-6Al-4V (4.42 g/mm3 ). Uniaxial compression tests were conducted with a universal mechanical testing machine (CMT5105, Shenzhen Wance Testing Machine Co., Ltd., Shenzhen, China) equipped with a 100KN load cell. Three duplications of each sample were subjected to the uniaxial compress test with a strain rate of 2 mm/min. In order to simultaneously record the deformation procedure of the samples, two Canon 60D cameras were placed at the front and left side view of the samples. The compressive force and displacement were recorded by a computer. The stress (𝜎) was obtained by dividing the compressive force into the apparent cross-sectional area of the samples and the strain value (𝜀) was calculated by dividing displacement into the initial height of the samples. To this end, the slope of the linear fitted stress–strain curve in the elastic deformation region was regarded as the elastic modulus (EL ). The yield strength (𝜎 L ) was defined as the stress at 0.2% plastic deformation. The compression strength (𝜎 c ) was regarded as the stress at the first peak on the stress-strain curves. The plateau stress (𝜎 pl ) is defined as the mean of stresses at the strain interval between 0.2 and 0.4 [28], and the densification stress (𝜀D ) was determined using the energy efficiency method [14]. 2.4. Finite element modelling

Fig. 4. SEM micrographs of Ti6Al4V powder.

sample to reduce the removed material of lattice samples by the wire electrical discharge (EDM) process. 2.2. Additive manufacturing process The scanning electron microscope (SEM) micrographs of the commercial Ti-6A-l4V powder (Ti64-53/20, Tekna Advanced Materials, Canada) used in this study are demonstrated in Fig. 4. The powder showed a spherical shape and smooth surface with an excellent flowability. The particle size exhibited a narrow distribution between 25.0 𝜇m (D10) and 52.3 𝜇m (D90) with an average diameter of 35.4 𝜇m. Three duplications of each lattice structure were manufactured by SLM using an EOSINT-M290 machine (EOS GmbH, Krailling, Germany). The processing occurred in an Argon atmosphere with an O2 content less than 0.1%, and the processing parameters adopted in this study were as follows: laser power of 175 W, laser beam diameter of 0.1 mm, hatch spacing of 0.1 mm, layer thickness of 30 𝜇m and scanning speed of 1250 mm/s. In addition, scanning direction was rotated 67○ between adjacent layers. Subsequently, the fabricated samples were removed from the base plate by wire electrical discharge machining. No heat treatment was applied to these samples. Using these manufacturing processes, the density of bulk Ti-6Al-4V was measured as 4.42 g/mm3 through the Archimedes method.

In order to predict the deformation behavior and mechanical properties of the lattice structures, the finite element analysis was carried out using ABAQUS/Standard 2016 software with a dynamic explicit procedure. The TPMS equations (− c ≤ 𝜙(x,y, z) ≤ c) were used to select the distributions of finite elements, and a Matlab code was developed to generate the 8-node hexahedral finite elements (C3D8R) for each lattice structure. Additionally, convergence studies with a range of mesh numbers were conducted in order to obtain a desirable mesh size, as shown in Fig. 5, and approximately 600,000 hexahedral elements were produced for each model. This was in good agreement with the works of Afshar [19] and Kadkhodapour [29], who found that mesh sizes of around 4300 elements per unit cell were sufficient. Then, these finite element models demonstrated in Fig. 6 were imported into ABAQUS software and constrained between two rigid shell plates (meshed with the 4-node 3D bilinear rigid quadrilateral element, R3D4) simulating boundary conditions in the compression process. The bottom plate remained stationary, and the upper plate moved downward 16 mm. The total simulation time was defined as 2 ms, and the kinetic energy was negligible as compared to the internal energy to ensure that the finite

2.3. Measurements An optical microscope (OM) (AM6000, Seepack Inc., Shenzhen, China) was used to illustrate the geometrical characteristics of the fabricated samples. The morphology of the Ti-6Al-4V powder and surface

Fig. 5. Convergence of stress-strain curves corresponding to the three mesh numbers for the US P structure.

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International Journal of Mechanical Sciences 167 (2020) 105262

3. Results and discussion 3.1. Geometric and structural characteristics

Fig. 6. Representation of hexahedral mesh models generated for finite element analysis of (a) US P, (b) US G, (c) FGS P, and (d) FGS G.

element model fulfilled quasi-static conditions [30]. The central point of the upper plate was used to obtain the displacement and force of the numerical compression process. The elastic properties were considered with a Poisson’s ratio of 0.3 and an elastic modulus of 107 GPa [31]. The plastic deformation of materials used in FEM is described by the Johnson–Cook plasticity model [32], which accounts for strain hardening, strain-rate, and thermal softening, and it is commonly used in the dynamic response analysis. According to the model, the yield stress can be represented by the equation as follows: [ ( 𝑝 )] [ ( ) ] [ ( )𝑁 ] 𝑇 − 𝑇𝑟𝑜𝑜𝑚 𝑀 𝜀̇ 𝜎𝑠 = 𝐴 + 𝐵 𝜀 𝑒 ⋅ 1− ⋅ 1 + 𝐶 ln 𝑇𝑚 − 𝑇𝑟𝑜𝑜𝑚 𝜀̇ 0

(5)

where, the constant values of A, B, N, C, and M are material-related parameters based on the flow stress data obtained from mechanical tests, 𝜀e is the equivalent plastic strain, 𝜀̇ 𝑝 is the equivalent plastic strain rate, 𝜀̇ 0 is the reference equivalent plastic strain rate, Troom is the room temperature, and Tm is the melting temperature. In order to evaluate the fracture behavior of lattice structures, the Johnson–Cook damage model was employed to remove the failure elements once they exceeded the limits [32]. The fracture strain is described as: [ ( 𝑝 )] [ ( )] [ ( )] 𝑇 − 𝑇𝑟𝑜𝑜𝑚 𝜀̇ 𝜀𝑓 = 𝐷1 + 𝐷2 𝑒𝑥𝑝 𝐷3 𝜎 ∗ ⋅ 1 + 𝐷4 ln ⋅ 1 + 𝐷5 𝑇𝑚 − 𝑇𝑟𝑜𝑜𝑚 𝜀̇ 0 (6) where, D1 , D2 , and D3 , are damage constant parameters related to the relationships between failure strain and stress triaxiality, D4 and D5 are constants depending on strain rate and temperature, respectively, and 𝜎 ∗ is stress triaxiality defined as the ratio of hydrostatic stress and the equivalent stress. In this study, all compression tests were performed at a constant strain rate and room temperature, and hence, C, M, D4 and D5 were ignored. The input constants of Johnson-Cook models with Ti6Al-4V were based on previous studies [31,33], as shown in Table 1. All simulations were performed on an Intel Core i7-8750H CPU with 10 cores and 16 GB RAM, and the computer time taken to solve each finite element model was approximately 12 h.

Table 1 Constitutive parameters of Johnson-Cook plasticity and damage model used in FEM. A(MPa)

B(MPa)

N

D1

D2

D3

1567

952

0.4

0.005

0.43

−0.48

As demonstrated in Fig. 7, unconnected solid regions were observed when the volume fraction was lowered below a particular threshold for skeleton P and skeleton G lattice structures. The connectivity problem is attributed to the thinnest features degrade entirely, resulting in unconnected solid sections [24]. These unconnected lattices exhibited poor manufacturability and could not to bear the load, which limited the design space. In contrast, the sheet P and sheet G lattice structures with low volume fractions still had the smooth and continuous connection between adjacent lattice cells shown in Fig. 7. Therefore, the volume fraction of sheet P and sheet G lattices were mainly restricted to the processing constraints in AM technology such as minimum wall size and minimum feature size. The SLM-processed samples are shown in Fig. 8, and the FGS structures were found to have a continuous graded thickness that conformed to the design. As seen in Table 2, the dimensions along x and y directions in SLM-manufactured samples were higher than in the designed samples, which are consistent with other previous experimental studies [34,35]. However, the phenomenon of loss of height was only found in some SLM-manufactured samples. This was because a plate with 0.5 mm thickness was added to the bottom of each lattice sample to reduce the removed material of lattice samples by the wire electrical discharge (EDM) process. It is clear from Table 2 that measured mass of lattice samples were slightly higher than the designed mass. It was mainly attributed to the partially-melted powder particles on the lattice surfaces, found in the high magnification SEM micrograph in Fig. 9, which has been discussed in previous studies [9,16]. Moreover, it should be noted that the difference between designed mass and real mass was also affected by the amount of removed material by EDM process [34], the gradient in the lattice [34], the accuracy of STL models [36], the porosities of fabricated samples [17], and the processing conditions [17], but these are beyond the remit of this paper. 3.2. Deformation behavior and mechanical properties The stress-strain curves of US and FGS lattice structures obtained from compression tests are illustrated in Fig. 10. Three distinct stages existed: the elastic–plastic stage, fluctuation stage, and densification stage. For the US samples, following the initial stress peak, a significant drop in stress values was observed in Fig. 10(a), resulting in the US P and US G samples losing around 64% and 54% of their strength with poor recovery, respectively. This softening behavior was attributed to the abrupt shear band failure that two neighboring halves separating and sliding with weak load-bearing capability at approximately 0.14 and 0.1 strain for the US P and US G samples, respectively, as shown by the white lines in Fig. 11(a) and (b). Similar failure behavior was observed in other uniform structures, such as the BCC lattice structure [16], honeycomb lattice structure [12], and diamond lattice structure [23], whose resolved shear stress was at a maximum at 45° to the load direction [17]. In the fluctuating stage, a long plateau-like region can be seen in stress-strain curves for US G structures, whereas the stress values of US P fluctuated over a large interval. One may assign it to the fact that internal material distribution is an important factor to influence lattice mechanical characteristics [11]. Therefore, a geometrical analysis was applied to the US P and US G unit cells, and the variation of the lattice relative load-bearing area in the xy plane along the load direction is illustrated in Fig. 12. The relative load-bearing area of the sheet P unit varied in the range of 0.154 to 0.293, and this variation occurred 4 times over the unit height. However, sheet G unit varied more frequently but in a small range of 0.167 to 0.258. For this reason, the fractured part of sheet G structures could be rapidly supported by the rest of the remaining part after collapse, which led to a relatively uniform changes in stress values in the fluctuating stage.

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International Journal of Mechanical Sciences 167 (2020) 105262

Fig. 7. Geometric characteristics of (a) skeleton P and sheet P lattice structures and (b) skeleton G and sheet G lattice structures with different volume fractions.

Fig. 8. Images of the fabricated samples of (a) US P, (b) US G, (c) FGS P, and (d) FGS G. Table 2 Measured characteristics of lattice samples. Samples

Measured dimensions x × y × z (mm3 )

Measured mass (g)

Designed mass (g)

US P

20.04 × 20.02 × 19.94 20.02 × 20 × 19.9 20.08 × 20.04 × 19.82 20.02 × 20 × 20 20.08 × 20 × 20.02 20 × 20.02 × 19.9 20.01 × 20.02 × 20.08 20.02 × 20.03 × 20 20.04 × 20.04 × 20.06 20.02 × 20.08 × 20 20.04 × 20.06 × 19.92 20.02 × 20.06 × 20.02

7.1071 7.0812 7.0744 7.0897 7.1571 7.0872 7.1324 7.1091 7.1443 7.1174 7.1067 7.1528

7.072

US G FGS P FGS G

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International Journal of Mechanical Sciences 167 (2020) 105262

Table 3 Mechanical properties of lattice samples.

Fig. 9. SEM micrograph of the surface morphology for the FGS G sample.

In contrast to the US samples with diagonal or V-shaped shear band failures, the FGS samples illustrated completely different deformation behaviors, as seen in Fig. 13. Beginning with the fracture in the top layer with the lowest volume fraction, lattice samples sequentially collapsed in a predictable layer-by-layer manner. Although the initial peak in stress-strain curves of all FGS samples was smaller seen in Fig. 10(b), the subsequent peaks were increasingly higher, and the maximum peak of FGS P samples and FGS G samples was approximately 4 times and 3.5 times higher than the initial peak, respectively, which indicated that their load-bearing capability was enhanced to resist the further damage. However, a linear stress rise and a stable plateau in the stress-strain curves for each collapse of graded lattice structures were observed by Maskery [37]. This is because the material for that lattice structure was polymeric material with good ductility, resulting in a stable plateau region after failure. In addition to the type of material used, due to a more uniform internal material distribution in the sheet G unit shown in Fig. 12, the FGS G structure provided more stable strength recoveries in the fluctuating stage. It should be noted that the number of stress peaks in stress-strain curves was not consistent with the collapsed layers, as shown in Fig. 10(b). These differences might be attributed to the layer-by-layer deformation associated with sub-layer collapses for the FGS samples. This was in contrast to the observation of FG lattice structures with volume fractions that varied in a step-wise manner, where the number of stress peaks in stress-strain curves was consistent with the collapsed layers [14,37], which will be further discussed in Section 3.4. Mechanical properties of US and FGS lattice samples are listed in Table 3. The elastic modulus, yield strength, and compressive strength of all the US lattice samples were higher than the respective values of the FGS ones, which is understandable as the FGS samples initially collapsed in the top layer where the volume fraction was lower than that of uniform ones. However, the G structures showed higher mechanical properties than P structures in both US and FGS designs, which indicates that the mechanical properties of lattice structures were determined not

Properties

US P

US G

FGS P

FGS G

EL (MPa) 𝜎 L (MPa) 𝜎 c (MPa) 𝜎 pl (MPa) 𝜀D

1477 ± 29 54.7 ± 3.5 72.1 ± 0.66 51.91 ± 0.66 0.61 ± 0.021

1699 ± 8 62.1 ± 1.2 77.1 ± 0.9 51.42 ± 1.3 0.643 ± 0.002

1188 ± 30 29.8 ± 0.6 33.1 ± 0.7 51.83 ± 0.22 0.727 ± 0.011

1495 ± 62 36.4 ± 1.2 37 ± 1.3 52.02 ± 1.1 0.762 ± 0.008

only by the change of volume fraction, but also by the design of the lattice unit. The higher property values were caused by the existence of vertical area in the sheet G structure as shown in Fig. 1, and these will be further discussed in Section 3.4. It was of interest to compare the deformation and mechanical properties in this study with the most significant theoretical work of Gibson and Ashby using the following equations [11]: ( )2 𝜎𝑝𝑙 ( )1.5 𝐸𝐿 = 𝐶 1 𝜌∗ = 𝐶 5 𝜌∗ 𝜀𝐷 = 1 − 𝛼𝜌∗ 𝐸𝑆 𝜎𝑆

(9)

where, ES and 𝜎 S are the elastic modulus and yield stress of the bulk material set as 107 GPa and 997 MPa, respectively [31]. Therefore, in setting the volume fraction to 20%, the Gibson-Ashby coefficients C1 , C5 and 𝛼 of the lattice samples were obtained through Eqs. (7)–(9), and the results are tableted in Table 4. The determined values of C1 for the US and FGS samples fell within the range of 0.1 to 4.0 given by Gibson and Ashby. However, the calculated C5 values for all samples were out of the given Gibson-Ashby range, being relatively higher than those found by Gibson and Ashby. It should be noted that the calculated C5 values are determined not only by the type of material used, but also by the architectural design of lattice units. For example, Li et al. [38] reported that the calculated C5 values for polymeric US G lattice structures was 0.179. Maskery et al. [37] found that the C5 values of BCC and BCCz lattice structures with polymeric material were 0.202 and 0.285, respectively. For the coefficient 𝛼, only the values of US samples were in the range of 1.4 to 2.0 given by Gibson and Ashby, while the values of all FGS samples were slightly lower than the range, which means that the densification strains for FGS samples were higher than US samples. This was in contrast to what happened in some results of studies where the densification strains of FG structures with BCC [14] and F2BCC [27] unit designs were lower than their respective uniform structures. However, it is worth highlighting that the difference is possibly attributed to the types of lattice units. For sheet G [38] and skeleton G [20] lattice structures, the densification strains of FG structures were also experimentally found to be lower than their respective uniform ones. 3.3. Energy absorption One of the major uses of lattice structures is in energy absorption applications, such as packaging and protective devices. The investigation Fig. 10. Stress–strain curves of (a) US and (b) FGS samples during compression tests.

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International Journal of Mechanical Sciences 167 (2020) 105262

Fig. 11. Experimental and simulation deformation and failure mechanisms for (a) US P and (b) US G lattice structures at different compressive strains. Table 4 Gibson-Ashby coefficients of lattice samples. Coefficients

Given range of Gibson-Ashby coefficients

US P

US G

FGS P

FGS G

C1 C5 𝛼

0.1–0.4 0.25–0.35 1.4–2

0.345 ± 0.007 0.582 ± 0.039 1.95 ± 0.104

0.397 ± 0.002 0.576 ± 0.015 1.783 ± 0.011

0.277 ± 0.009 0.581 ± 0.003 1.365 ± 0.051

0.349 ± 0.015 0.583 ± 0.012 1.19 ± 0.042

Table 5 Energy absorption property of lattice samples. Properties 3

Wvt (MJ/m )

US P

US G

FGS P

FGS G

31.05 ± 0.85

33.82 ± 0.29

49.72 ± 0.52

54.44 ± 0.85

of lattice structures in packaging and protective devices, and it can be calculated using numerical integration as follows: 𝜀𝐷

𝑊𝑣𝑡 = ∫ 𝜎(𝜀)𝑑𝜀

(10)

0

Fig. 12. Relative load-bearing area of sheet P and sheet G unit cells with 20% volume fraction along the load direction.

of energy absorption capability of lattice structures provides a quantitative measure of comparing and selecting suitable candidate lattice structures in energy absorbing applications. The energy absorbed per unit volume up to densification point (Wvt ), namely the energy absorption capability, is an important factor that determines the applications

where 𝜎(𝜀) is the stress (𝜎) related to strain (𝜀) during the compression tests. The results are outlined in Table 5. It can be observed that the FGS samples absorbed a higher amount of energy as compared to US samples, and the difference in values showed an approximate increase of 60% in the value of energy absorption with both FGS P and FGS G designs. The higher values of energy absorption were found to agree with the results from other literature that studied the FG structures. For example, in the case of FG design, the increase of energy absorption was 23% for F2BCC design [27], 80% for BCC design [37], and 114% for BCCz design [37]. The excellent energy absorption capability of FGS samples was basically attributed to their distinctive deformation behav-

M. Zhao, D.Z. Zhang and F. Liu et al.

International Journal of Mechanical Sciences 167 (2020) 105262

Fig.13. Experimental and simulation deformation and failure mechanisms for (a) FGS P and (b) FGS G lattice structures at different compressive strains.

Fig. 14. Cumulative energy absorption per unit volume (Wv ) versus strain (𝜀) curves for US and FGS lattice samples under compression tests.

ior during compression tests. Failure of the FGS structure started from the layer with the lowest mechanical properties, and then it was directly supported by the remaining part, which eliminated the high strength reductions and gradually increased the load-bearing capability due to the higher volume fractions. In contrast, the diagonal and V-shaped shear band failure occurred for US structures when their stress reached the peak point of stress, leading to a large drop in stress and a low capability to against the applied load after the initial collapse. In order to further investigate the energy absorption behavior of lattice structures, the changes in cumulative energy absorption per unit volume (Wv ) versus strain curves under compression tests are illustrated in Fig. 14 and then fitted by the power law. The fitted exponent values represent an increasing rate of cumulative energy absorption of lattice structures throughout the compression process. It can be seen that the exponent values of US samples were close to 1, and the linear relation-

ship was also found to agree with the findings of other uniform lattice structures such as BCC lattice structures [16], honeycomb lattice structures, and cubic lattice structures [12]. In comparison, the exponent values of FGS samples were higher than the respective US ones, being 1.89 for the FGS P structure and 1.68 for the FGS G structure, demonstrating that the energy absorption capability of FGS lattice structures was gradually improved throughout the compression process. This distinctive energy absorption behavior of FGS lattice structures is basically attributed to the increasing volume fractions in FGS samples throughout the compression process. The high exponent values for FG lattice structures were also experimentally observed in other previous literatures. For example, Choy [12] found the exponent values of graded cubic and honeycomb lattice structures with Ti-6Al-4V were in the range of 1.28 to 2.68. Similarly, the exponent value of graded F2BCC made of AlSi-12 was around 1.48 observed by Dheyaa [27]. However, it should be noted that the exponent values in different architectures of the lattice unit were significantly different in FGS samples but comparably less so in US samples. Therefore, it allows the designers to choose one FGS lattice structure most suited to a particular application in energy absorption by adjusting the architecture of the lattice unit. 3.4. Simulation analysis The computational plastic deformation patterns of US and FGS lattice structures at different strains are illustrated in Figs. 11 and 13, respectively. For the US lattice structures, plastic strain was more concentrated in the diagonal area throughout the whole lattice structure when the strain was 0.14 for US P and 0.1 for US G. When structures partly compacted in the 0.5 strain, simulation results with V-shaped bands of local densification were in good agreement with the experimental results. For the FGS lattice structures, the collapse started with the high plastic strain in the top layer with the lowest volume fraction and was followed by layer-by-layer crushing, due to the graded volume fraction

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International Journal of Mechanical Sciences 167 (2020) 105262

Fig. 15. Plastic strain distribution in the top layer of (a) FGS P and (b) FGS G lattice structure at initial fracture.

Fig. 16. Comparing experimental stress-strain curves with simulation results of (a) US P, (b) US G, (c) FGS P, and (d) FGS G lattice structures.

in the loading direction, which corresponded to the experimental deformation behaviors of all FGS samples. In order to further reveal the deformation behaviors of FGS lattice structures, the plastic strain of FGS structures in the top layer at the initial failure strain are illustrated in Fig. 15. The plastic strain was distributed mainly in the top area of the units, which confirmed that the top layer failure consisted of some sub-layer collapses. As demonstrated by the analysis of lattice units in Fig. 12, it can be found that the lattice unit can be further subdivided into some sub-layers along the load direction due to the periodic internal material distribution. Therefore, for the FGS samples with continuous volume fraction changes, layer failure started from the upper sub-layer with the lowest volume fraction to the sub-layers with higher volume fractions because of the gradually changing volume fraction over the sub-layers. For this reason, several stress peaks were observed in the experimental stress-strain curves between the collapses of adjacent layers, as shown in Fig. 10(b). However, for the FG lattices with volume fractions that changed in every layer of the lattice unit, layer failure occurred across the whole unit of a single layer [14,37], owing to the same volume fraction of each sub-layer, and even-

tually resulted in the same number of collapse layers and stress peaks in stress-strain curves. The results show that the deformation behavior of FG lattice structures is possible to be more precisely controlled by optimizing the internal volume fraction distribution in the range of the lattice unit. In order to validate the reliability of the FEM in predicting the mechanical properties and energy absorption of US and FGS lattice structures, the predicted and experimental stress-strain curves are illustrated in Fig. 16, and the relative errors of the predicted and experimental mechanical properties are summarized in Table 6. In general, it can be seen by comparing the experimental data and simulation results that the finite element model could properly predict the trend of stress-strain curves for US and FGS lattice structures, and the yield strength and compressive strength of finite element models were in good agreement with the experimental data, with relative errors of less than 31.9%. Although the elastic modulus and stress values in the elastic region for all finite element models were higher than the experimental results, the order of the elastic modulus was correctly simulated by the finite element models. The differences in the elastic region might be assigned to the fact

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International Journal of Mechanical Sciences 167 (2020) 105262

Table 6 Comparison of mechanical properties between experimental and computational data for lattice samples. Sample US P US G FGS P FGS G

EL (MPa) Exp FEM

Error

𝜎 L (MPa) Exp FEM

Error

𝜎 c (MPa) Exp FEM

Error

Wvt (MJ/mm3 ) Exp FEM

Error

1477 1699 1188 1495

13.3% 44.8% 26.5% 38.9%

54.7 62.1 29.8 36.4

29.8% 8.9% 30.2% 19.8%

72.1 77.1 33.1 37

31.9% 18.4% 25.7% 13.2%

31.05 33.82 49.72 54.44

8.3% 15% 30% 25.4%

1673 2461 1503 2076

38.4 56.6 20.8 29.2

49.1 62.9 24.6 32.1

28.48 28.75 34.82 40.6

FEM with the Johnson-Cook plasticity model and Johnson-Cook damage model can be used to evaluate the deformation behavior of lattice structures and provide guidance in designing multifunctional lattice structures which can exhibit a combination of high performance features, such as a relatively low density accompanied by a high strength and energy absorption capability. 4. Conclusions In this paper, FGS lattice structures were generated by TPMS formulas and successfully fabricated by SLM technology with Ti-6Al-4V powder. The mechanical properties, deformation behaviors, and energy absorption characteristics of the FGS structures were experimentally and numerically investigated and compared to those of the US ones. The following conclusions can be drawn:

Fig. 17. The Von Mises Stress distribution of (a) US P and (b) US G structures in two cross-sectional areas in the yz plane at 𝜀 = 0.01.

that the porosities and surface morphologies in fabricated samples were not considered in the voxel-based model used in this numerical study [39]. Moreover, at the beginning of the elastic stage, a non-linear and concave upward region was observed in experimental results, owing to the effect of the distortion of lattice samples when they were cut from the base plate [9], and these effects were ignored in the finite element model. It should be noted that all simulation and experimental results showed higher mechanical properties for G structures than P structures in both US and FGS designs, which indicates that the mechanical properties of lattice structures are mainly determined by the design of lattice unit with the same design of volume fraction. From the Von Mises stress distributions of lattice cross-sectional area in Fig. 17, for the P design, stress was mainly concentrated in the inclined region, while for the G design, stress was the lowest at the horizontal region but highest at the vertical and inclined regions. The observation indicates that the G structure experienced a combination of buckling at the vertical region and bending at the inclined region upon the compressive load, while the P structure underwent more bending at the angled region during loading. The more stretch-dominated behavior for G structures was in good agreement with the experimental study by Al-Ketan [23], and it was accountable for the higher mechanical properties of both US and FGS structures with the G design. Following the predicted curves that reached their first peak, simulation results with the Johnson-Cook damage model showed a good ability to predict the decline of strength. The softened behavior was a consequence of the failure elements which were removed by the JohnsonCook damage model in the finite element model. In the fluctuating stage, the increased peaks of FGS structures were properly predicted, as seen in Fig. 16(c) and (d). Moreover, the small fluctuations of sheet G structure in the fluctuating stage, seen in Fig. 16(b) and (d), were also predicted because of the more uniform distribution of internal material. It is worth noting that the numerical simulation and experimental results were highly consistent in the fluctuating stage, which enabled the energy absorption of the lattice to be predicted with relative errors in the range of 8.3%–30%. Therefore, it can be concluded that the

1 The sheet P and G lattice structures could eliminate the unconnected regions of the skeleton lattice structures, which led to the geometric continuity and connectivity of lattice structures with low volume fractions. 2 The internal material distribution of sheet G lattice unit was more uniform than that of sheet P, leading to a stable change in strength for both US and FGS G lattice structures in the fluctuating stage. 3 The deformation behaviors of all FGS lattice structures included layer-by-layer failure accompanied by sub-layer collapses, and their load-bearing capability gradually increased throughout the compression process. 4 The cumulative energy absorption per unit volume of FGS lattice structures increased as a power of strain function, while the US ones exhibited a linear relationship. The distinctive energy absorption behavior rendered the FGS structures superior to US ones, and the value of the total cumulative energy absorbed approximately increased by 60%. 5 Deformation behaviors of finite element models with the JohnsonCook plasticity and damage model were in good agreement with experimental results. The stress-strain curves of simulation results showed the capability for predicting the mechanical properties of lattice structures (especially in yield strength, compressive strength and energy absorption). The FGS lattice structures demonstrated significant advantages in terms of deformation behavior and energy absorption capability, which were competitive with US ones for protective and energy-absorbing applications, such as helmets, packaging materials, and energy absorbers. For helmets and packaging materials, the lower initial compressive stress of FGS structures can impart small forces to avoid harming a person or a package. In addition, when impacted, the lowest-density layer of FGS structures collapses and serves to absorb the energy, while the highest-density layer is still in tact to protect a person or a package. For energy absorbers, FGS structures are possible to replace the thin-walled tubular structures as energy absorbers in automobile, high-speed railway and aerospace industries [40,41] or replace the honeycomb blocks as shock absorbers in the landing systems of space vehicles [42], owing to their high energy absorption capability. However, the FGS structures in this study showed only linear changes in volume fractions along the load direction. Further investigation will focus on optimizing the distri-

M. Zhao, D.Z. Zhang and F. Liu et al.

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