*Article* **Comparative Study on the Uniaxial Behaviour of Topology-Optimised and Crystal-Inspired Lattice Materials**

#### **Chengxing Yang, Kai Xu and Suchao Xie \***

Key Laboratory of Traffic Safety on Track, Ministry of Education, School of Traffic & Transportation Engineering, Central South University, Changsha 410075, China; Chengxing\_Yang\_Hn@163.com (C.Y.); csu\_xukai@csu.edu.cn (K.X.)

**\*** Correspondence: xsc0407@csu.edu.cn; Tel.: +86-1370-7481-946

Received: 10 March 2020; Accepted: 7 April 2020; Published: 8 April 2020

**Abstract:** This work comparatively studies the uniaxial compressive performances of three types of lattice materials, namely face-centre cube (FCC), edge-centre cube (ECC), and vertex cube (VC), which are separately generated by topology optimisation and crystal inspiration. High similarities are observed between the materials designed by these two methods. The effects of design method, cell topology, and relative density on deformation mode, mechanical properties, and energy absorption are numerically investigated and also fitted by the power law. The results illustrate that both topology-optimised and crystal-inspired lattices are mainly dominated by bending deformation mode. In terms of collapse strength and elastic modulus, VC lattice is stronger than FCC and ECC lattices because its struts are arranged along the loading direction. In addition, the collapse strength and elastic modulus of the topology-optimised FCC and ECC are close to those generated by crystal inspiration at lower relative density, but the topology-optimised FCC and ECC are obviously superior at a higher relative density. Overall, all topology-generated lattices outperform the corresponding crystal-guided lattice materials with regard to the toughness and energy absorption per unit volume.

**Keywords:** lattice material; topology optimisation; crystal inspiration; mechanical properties; energy absorption

#### **1. Introduction**

As a mimic of nature cellular material, lattice material is designed with its unit cells arranged periodically along tessellation directions. Metallic lattice materials can provide excellent mechanical properties, e.g., ultra light weight, better durability, high specific strength and stiffness, and a superior energy absorption capability [1–4]. Methods for lattice material design can be generally classified into two categories, i.e., manual generation and mathematical generation [5].

Manual generation means designing a lattice material by using beams and trusses with joints modified to create seamless transitions between unit cell elements [5]. There are numerous manually designed lattice materials and some of them are inspired by crystal structures. Typical examples are simple cubic [6], diamond [7], face-centred cubic (FCC) [8], FCCZ (i.e., FCC with enhanced vertical struts) [9], body-centred cubic (BCC) [10], Kagome [11], F2BCC [12]; Gurtner-Durand [13], Octet-truss [14], Octahedron [13], Octahedron-cross [15], Octahedral [16], tetrakaidecahedron [13], rhombic dodecahedron [17], etc. Additive manufacturing (AM) technology can be employed to fabricate lattice materials [18]. Lozanovski et al. [19,20] numerically investigated the strut defects of lattice materials fabricated by the AM method, and a Monte Carlo simulation-based approach was proposed to predict the stiffness of a lattice material with defects.

The mathematical generation method utilises algorithms and constraints to create a lattice structure [5]. Examples of mathematically generated lattice materials are triply periodic minimal surfaces (TPMS) (e.g., skeletal-TPMS and sheet-TPMS lattices) [17,21,22], all face-centred cubic (AFCC) [23], FCC [1], edge-centre cube (ECC) [1], vertex cube (VC) [1], cuttlebone-like lattice (CLL) [24], etc.) The topology optimisation algorithm can also be used to determine the optimal material distribution for lightweight structure design.

Two common solutions for topology optimisation are the optimality criteria (OC) [25] and the Method of Moving Asymptotes (MMA) [26], and details are referred to [27]. Its applications on engineering structures are reviewed by Sigmund and Maute [28] and Kentli [29]. Topology optimisation methods based on discrete elements are the ground structure approach (GSA) [30], solid isotropic material with penalization (SIMP) method [31–34], homogenization method (HM) [35], evolutionary structural optimization (ESO) [36], level-set method (LSM) [37,38], and the hybrid cellular automata (HCA) algorithm [39]. In terms of lattice material design by topology optimisation, Hu et al. [24] proposed a CLL material that exhibited high compression-resistant capability. However, it can only suffer uniaxial loading rather than multiple loadings from triaxial directions. Xiao et al. [1] proposed three topology-optimised lattice materials, i.e., FCC, VC, and ECC, under three different loading modes, which were similar to manually designed lattices. Yang et al. [40] addressed four different lattice cells under various extreme loading modes, which expanded the library of structural metamaterials. However, the differences between topology-optimised and manually generated lattices have not been assessed.

In the present work, our objective is to systematically explore the similarities and differences of three types of lattice materials guided by two methods, i.e., topology optimisation and crystal inspiration. Section 2 presents the detailed design process and topological geometries, as well as the relationship between relative density and aspect ratio for crystal-inspired lattices. Finite element models are built in Section 3, and the results are presented in Section 4, including deformation modes, mechanical properties, and energy absorption. Finally, some meaningful conclusions are reported in Section 5.

#### **2. Cell Architecture Design**

#### *2.1. Topology Optimisation Problem*

In this study, two kinds of design methods, i.e., topology optimisation and crystal inspiration, are adopted to generate lattice cells. Topology optimisation is carried out in LS-TaSCTM by integrating with Ls-dyna® (LSTC, Livermore, CA, USA). The implicit algorithm is employed. The optimisation problem can be mathematically stated as follows:

$$\begin{cases} \min \quad F(\mathbf{x}) = c(\mathbf{x}) \\ \text{s.t.} \quad \mathbf{K}(\mathbf{x})\mathbf{u} = \mathbf{f}, \\ \qquad M(\mathbf{x})/M\_0 \le M\_{f\_i} \\ \quad 0 < \mathbf{x}\_{\min} \le \mathbf{x}\_i \le 1, \ i = 1 \ldots n \end{cases} \tag{1}$$

The goal of the topology optimisation in Equation (1) is to minimise the compliance *c*(*x*). The design variable *x*, with 0 < *xmin* ≤ *x* ≤ 1 is known as relative density. *n* denotes the total number of design variables. **f** and **u** are, respectively, the vectors of nodal force and displacement, while **K**(*x*) is the structure stiffness matrix. *M*<sup>0</sup> and *M*(*x*) represent the initial mass and the current mass of the design domain. A constraint is imposed on the mass fraction *Mf* . The base material is a high-strength steel referred to Yang et al. [41], with the key parameters presented in Table 1. Three different loading conditions, i.e., face-centred loadings, edge-centred loadings and vertexes loadings, represented by arrows, are illustrated in Table 2. The magnitude of loadings (1 N) is well selected to make sure that the design domain only suffers elastic deformation and the loading points are fixed during the optimisation process. Increasing the value of loading may lead to different optimised cells and it may also cause a

problem in designing structures with a high level of porosity [1]. The topology optimisation method is a hybrid cellular automata (HCA) algorithm [42], and the solid isotropic material with penalization (SIMP) model is adopted. Two termination conditions are used to stop the optimisation process: (i) the number of iterations exceeds the maximum number of iterations, or (ii) the change in the topology is smaller than the tolerance [43]. Corresponding optimal cells are also presented in Table 2.

**Table 1.** Material properties of the base material, data from [41].


**Table 2.** Lattice materials with various relative densities generated from topology optimisation and crystal inspiration.


#### *2.2. Analogy with Crystal Structures*

Crystal structures are favourable sources of inspiration in manual designs with struts and joints [44]. As shown in Figure 1, a simple cube (7.5 <sup>×</sup> 7.5 <sup>×</sup> 7.5 mm3) is composed of six faces, twelve edges, and eight vertexes. Therefore, three lattice materials can be generated by, respectively, connecting the face-centre points, edge-centre points, and vertexes. The obtained lattice materials are referred to as the face-centred cube (FCC), edge-centred cube (ECC), and vertex cube (VC), which is consistent with Xiao et al. [1].

**Figure 1.** Schematic illustration of manually designed lattice materials: (**a**) FCC (face-centre cube); (**b**) edge-centre cube (ECC); (**c**) vertex cube (VC).

To mathematically analyse a lattice material, the individual struts shown in Figure 2 are considered. The materials overlapping in joints are removed to calculate the actual volume occupied by the lattice material. It is assumed that the lattice is composed of several struts and each strut contains one cylinder and two cones. Thus, the relative density (ρ) of three cells can be obtained via Equations (3) and (4), respectively.

$$
\overline{\rho} = \rho\_L / \rho\_0 \tag{2}
$$

$$\text{FCC and ECC}: \ \overline{\rho} = 3\pi \left(\frac{d}{l\_1}\right)^2 \left(\frac{\sqrt{2}}{2} - \frac{5d}{6l\_1}\right) \tag{3}$$

$$\text{VC}: \ \overline{\rho} = \frac{3\pi}{2} \left(\frac{d}{l\_1}\right)^2 \left(\frac{1}{2} - \frac{d}{3l\_1}\right) \tag{4}$$

where ρ<sup>0</sup> and ρ*<sup>L</sup>* are the density of the base material and of the lattice material, respectively, *d* represents strut diameter, *l*<sup>1</sup> is cube length, and *d*/*l*<sup>1</sup> denotes the aspect ratio.

It should be mentioned that FCC and ECC share the same expression of relative density (Equation (3)). Figure 3 plots the curves showing the change in the relative density versus aspect ratio *d*/*l*<sup>1</sup> given by Equations (3) and (4). Computer-aided design (CAD) predictions are also given in the same figure to properly validate the responses of both equations. As can be seen from Figure 3, good consistency is observed between theoretical and CAD predictions.

#### *2.3. Numerical Results*

Lattice materials derived from topology optimisation (labelled as -TO) and crystal inspiration (labelled as -CI) are presented in Table 2. The relative density of the generated cells varies in the interval [0.1, 0.3], which is comparable to that of aerogel, alumina nanolattices, and other ultralight materials [45]. The boundary surface of the final optimised topology of the lattice is fitted smoothly by using FreeCAD software (open source); thus, the geometry model of the lattice cell is obtained. High consistency is observed in geometry between topology-optimised and crystal-inspired materials, but the solutions provided by topology optimisation are generally non-uniform in terms of strut thickness and joint shape, unlike the regular topology of crystal-inspired cells. By increasing the

relative density, walls are likely to be formed between neighbouring struts (see lattice materials with ρ = 0.30). It should be mentioned that the FCC in this work is similar to the octahedral lattice in [5,46]. The ECC in this work is also named as octahedron in Gross et al. [13], and the VC in this work has the same cell topology as the cubic lattice in Mei et al. [2].

**Figure 2.** Individual strut geometries of: (**a**) FCC and ECC; (**b**) VC. p

**Figure 3.** Comparison of the relative density predicted by theory and the CAD model.

#### **3. Finite Element Modelling**

Finite element models are built in Ls-dyna® to predict the uniaxial compressive behaviour of the lattice materials. The geometry models of lattice materials are meshed into tetrahedral elements, which are constant stress solid elements. Instead of using periodic boundary conditions on a single unit cell, a 3 <sup>×</sup> <sup>3</sup> <sup>×</sup> 3 cell model with dimension of 22.5 <sup>×</sup> 22.5 <sup>×</sup> 22.5 mm<sup>3</sup> is built. The influence of the number of cells on the elastic modulus of lattice material was numerically studied by Maskery et al. [47], and they found that the converged modulus of the 3 × 3 × 3 cell diamond lattice was just 1% below the upper bound of the theoretical elastic modulus. The FE model of FCC-TO is shown in Figure 4. The lattice materials are placed between two parallel plates, which are modelled as rigid bodies by \*Mat.020\_Mat\_Rigid [48]. The bottom plate is fixed while the top plate impacts the specimen at a constant speed (10 m/s). "\*Mat.024\_Mat\_Piecewise\_Linear\_Plasticity" [48] is selected to bilinearly approximate the stress–strain curve of elastic-plastic material in Table 1. The impact force is captured by using the \*Automatic\_Surface\_To\_Surface contact algorithm [48] applied among specimens and boundaries. In this algorithm, the stiffness of contact elements is penalised by a scale factor [48]. Mesh

size is determined, as a result of a sensitivity analysis, as a compromise between accuracy and low computing time. Figure 5 plots the stress–strain curves of the FCC-TO sample calculated by various element sizes, i.e., 0.10~0.15 mm, 0.125~0.200 mm, 0.15~0.25 mm, 0.20~0.30 mm, and 0.25~0.35 mm. In Figure 5, convergence can be obtained when element size is equal to 0.125~0.200 mm. Therefore, this element size is employed for computation throughout this study, which means the main parts of the lattice have an element size of 0.200 mm, while other parts (e.g., joints) are characterised by a smaller element size, i.e., 0.125 mm. The finite element model has been validated by means of experimental results obtained for a body-centred-cubic (BCC) lattice made of 316L stainless steel, taken from [49]. As shown in Figure 5b, a good agreement is observed between experimental and numerical data. It should be noted that the simulation curve in Figure 5b is filtered by using Butterworth Filter with a frequency of 20,000 Hz. The frequency is carefully selected to filter the numerical perturbation while maintaining the characteristics of the stress–strain curve.

**Figure 4.** Finite element model of the FCC-TO with its unit cell enlarged.

**Figure 5.** (**a**) Effects of element size on the stress-strain curve of the FCC-TO lattice material; (**b**) validation of the modelling method by using experimental data reproduced from [49], copyright permission: Elsevier, 2020.

#### **4. Results and Discussion**

#### *4.1. Deformation Modes*

Table 3 reports the Von Mises stress distributions in lattice materials with a relative density of 0.10 up to 50% overall deformation. Generally, each layer of lattice materials collapses almost synchronously, although the middle layer deforms relatively faster than the top and bottom layers, which was also observed in sheet-IWP latice by Al-Ketan et al. [50]. There are strong stress concentrations occurring on some supporting struts and connecting joints for each lattice material. However, the level of equivalent stress is the lowest at the horizontal rods of VC-CI because these struts are orthogonal to the loading direction. The gap between the stresses of vertical and horizontal rods results in unstable deformation in VC-CI at a low relative density, which exhibits a failure band at ε = 0.35. Liu et al. [51] obtained microscopy images of local stress concentrations in VC-CI where slip bands and grain boundaries were found. Comparing the topology-guided and manually generated lattices, FCC-TO and ECC-TO exhibit highly similar deformation modes to the FCC-CI and ECC-CI. However, an obvious difference is found between VC-TO and VC-CI lattices (marked in Table 3) and no shear band is observed in the VC-TO lattice. Thus, the topology-optimised VC lattice undergoes a more stable deforming process.


**Table 3.** Deformation features of lattice materials with ρ = 0.10 at different compressive strains.

It should be noted that the stress level related to the colour fringe of each figure in Tables 3 and 4 varies to have a better visualisation. In general, the blue colour represents the lowest stress level, while the red colour denotes the highest stress level in each figure.


**Table 4.** Deformation features of lattice materials with ρ = 0.20 at different compressive strains.

To study the effect of relative density, the deformation features of lattice materials with a relative density ranging from 0.10 to 0.30 are carefully examined. The stress distributions are similar for a relative density higher than 0.20, although the higher the relative density, the higher the Von Mises stress due to the higher stiffness. Thus, the Von Mises stress distributions in the lattice materials with relative density of 0.20 are presented in Table 4. It can be found that lattice materials deform more uniformly and collectively at a higher relative density. In addition, the shear band disappears in the VC-CI lattice when its relative density is higher than 0.20.

#### *4.2. Stress-Strain Curves*

The stress-strain (σ-ε) curves extracted from the impact simulation on the designed lattice materials with various relative densities are shown in Figure 6. Additionally, the initial compression stage with ε < 5‰ is enlarged. The stress and strain are calculated in accordance with the initial cross-sectional area and length of each specimen, respectively.

*Metals* **2020**, *10*, 491

**Figure 6.** Stress-strain curves of lattice materials generated from two methods with various relative densities: (**a**) FCC; (**b**) ECC; (**c**) VC.

The curves are similar to those of common porous or cellular materials [52], which exhibit three ideal regimes under uniaxial compression, i.e., a pre-collapse regime (including the linear elastic stage), followed by a plateau regime with approximately constant stress and a final densification regime with steeply increasing stress. The linear elastic regime is characterized by elastic modulus (*EL*), which is driven by the bending or stretching for the inclined or vertical cell struts/walls, respectively [53]. The plateau regime due to the plastic hinges at sections or joints can be measured by plateau stress (σ*pl*). The densification regime starts from a densification strain (ε*cd*), where the individual cell strut/wall comes into contact with each other, and exhibits dramatically increasing strength [53].

The modulus *EL* is defined as the slope of the initial linear elastic region; the initial highest peak stress is defined as the strength σ*b*; and the plateau stress σ*pl* is calculated as the arithmetical mean of the stress at a strain interval between 20% and 40% according to ISO 13314: 2011 [54].

$$
\sigma\_{pl} = \frac{1}{\varepsilon\_2 - \varepsilon\_1} \int\_{\varepsilon\_1}^{\varepsilon\_2} \sigma(\varepsilon) d\varepsilon,\tag{5}
$$

where ε<sup>1</sup> and ε<sup>2</sup> equal 0.2 and 0.4, respectively.

The ε*cd* is identified by using the energy absorption efficiency (η) method.

$$\eta(\varepsilon) = \frac{1}{\sigma(\varepsilon)} \int\_0^\varepsilon \sigma(\varepsilon) d\varepsilon. \tag{6}$$

The onset of densification is given by using the following equation:

$$\left. \frac{d\eta(\varepsilon)}{d\varepsilon} \right|\_{\varepsilon = \varepsilon\_{cd}} = 0,\tag{7}$$

for which the η(ε) reaches a maximum on the η(ε) − ε curve. For example, the η(ε) − ε curves of the FCC-TO lattices are depicted in Figure 7.

**Figure 7.** The η(ε) − ε curves of FCC-TO lattices with various relative densities.

Figure 6 shows that the slopes and initial peak stresses of the σ-ε curves go up as the relative density increases and all the curves reach the maximum strength value at about 1‰~5‰ overall deformation. The plateau regime is steady at lower relative density. However, a nearly linear hardening phenomenon is observed in lattice materials with higher relative density due to an increasing stiffness and hardening effect of base material. With an increasing relative density, the onset of densification also occurs earlier, resulting in a decreasing ε*cd* (Figure 7). Comparing the two lattice generation methods, topology-guided lattices generally produce higher σ-ε curves than manually generated structures. Specifically, the gap between TO- and CI- lattices is larger at a higher relative density, although the difference is not obvious at a lower relative density. This is because of the cell walls formed in topology-optimised lattices at high mass fraction, as shown in Table 2. In general, σ*pl* of optimised structures is higher than that of manually designed structures, especially for FCC (see Table 5). However, three cases (i.e., ECC with ρ ≈ 0.15, VC with ρ ≈ 0.15 and 0.20) are excepted, which may be attributed to the slightly lower relative density of topology-optimised lattices comparing with the corresponding crystal-inspired lattices. For example, the ρ of VC-TO is 0.147, while that of corresponding VC-CI is 0.150 (see Figure 6).


**Table 5.** Plateau stress, σ*pl*/[MPa], of the as-designed lattice materials with various relative densities.

#### *4.3. Mechanical Properties and Energy Absorption*

In general, the compressive strength and elastic modulus of lattice materials would increase if the relative density increases because there is a larger amount of material withstanding the impact force. This relationship could be fitted by a power law proposed by Gibson and Ashby [55]. The elastic Modulus, *EL*, and collapse strength, σ*b*, scale with relative density,ρ, according to the relationships:

$$\frac{\sigma\_b}{\sigma\_y} = \mathbb{C}\_1(\overline{\rho})^{n\_1},\tag{8}$$

$$\frac{E\_L}{E\_0} = \mathcal{C}\_2(\overline{\rho})^{\mathbb{N}\_2},\tag{9}$$

where *n*<sup>1</sup> and *n*<sup>2</sup> represent the structural bending/stretching dominated mode. For bending-dominated structures (e.g., body-centred lattice), *n*<sup>1</sup> = 1.5, and *n*<sup>2</sup> = 2; for stretching-dominated structures (e.g., Octet-truss lattice), both *n*<sup>1</sup> and *n*<sup>2</sup> are equal to 1. *C*<sup>1</sup> and *C*<sup>2</sup> are constants related to the lattice's architecture as well as the base material properties.

The power law and simulation data are plotted in Figure 8, reflecting that the results of this study are fitted very well with the formulae. The coefficients of constant *C* (*C*<sup>1</sup> and *C*2) and exponent *n* (*n*<sup>1</sup> and *n*2) are tabulated in Table 6. It is noticed that the exponents, *n*<sup>1</sup> and *n*2, of the as-designed lattice materials are respectively close to 1.50 and 1.70, which indicates a bending-dominated behaviour mixed with a light stretching mode. Montemayor and Greer [46] pointed out that the FCC lattice behaves as a bending-dominated structure due to the rotation between and within unit cells, although the unit cell is a stretching-dominated structure. This phenomenon can be observed in Table 3. Gross et al. [13] made it clear that the ECC lattice is also a bending-dominated structure. Compression tests were carried out on the VC-CI lattices by Mei et al. [2], where the power law with *n*<sup>1</sup> = 1.5 and *n*<sup>2</sup> = 2 was used to fit the experimental data.


**Table 6.** Values of the parameters of the power laws used in fitting mechanical properties and energy absorption.

Comparing the six bending-dominated structures in Figure 8a,b, it is predicted that VC lattices are stronger than ECC lattices, while ECC lattices are superior to FCC lattices. Additionally, the collapse strength and elastic modulus of FCC-TO and ECC-TO are close to those of FCC-CI and ECC-CI at lower relative density. However, the topology-optimised FCC and ECC obviously outperform the corresponding crystal-inspired structures at a higher relative density. Interestingly, the inverse phenomenon is found in VC lattices where the collapse strength and elastic modulus of VC-CI are higher than those of VC-TO.

**Figure 8.** The relationship between the mechanical properties/energy absorption and relative density of lattice materials: (**a**) normalised collapse strength; (**b**) normalised elastic modulus; (**c**) toughness; (**d**) energy absorption per unit volume.

The energy absorption ability, namely the energy absorbed per unit volume of cellular materials, is defined by the area under the stress-strain curve up to the densification strain.

$$\mathcal{W}\_V = \bigcap\_{0}^{\varepsilon\_{cl}} \sigma(\varepsilon) d\varepsilon \tag{10}$$

Among which, the toughness (*UT*) is defined as the amount of energy per unit volume up to the strain of 0.25 [50].

$$
\mathcal{U}\_T = \int\_0^{\varepsilon\_d} \sigma(\varepsilon) d\varepsilon, \quad \varepsilon\_d = 0.25 \tag{11}
$$

The *UT* and *WV* are also be fitted by the power law.

$$\mathcal{U}\_T = \mathbb{C}\_3(\overline{\rho})^{n\_3} \tag{12}$$

$$\mathcal{W}\_V = \mathbb{C}\_4(\overline{\rho})^{n\_4} \tag{13}$$

As shown in Figure 8c,d, the toughness and energy absorption of all materials rise with an increasing relative density. The relationship among three kinds of lattice materials keeps as VC > FCC > ECC, and the differences become more distinct at a higher relative density. In addition, topology-optimised lattices are characterised by performances better than those of crystal-inspired lattices and the differences also become more distinct with an increasing relative density.

Engineering application generally requires that an energy absorber ought to absorb impact energy as much as possible while maintaining a low maximum stress [55]. Cellular material with high porosity produces low plateau stress; however, the quantity of absorbed energy may also be low. In contrast, a dense material is able to absorb a large amount of energy, but a high plateau stress may exceed the stress limitation [55]. The plot of energy absorption in Figure 8d misses the information of the maximum allowable stresses. Therefore, Figure 9 shows the diagrams of FCC-TO, ECC-TO, and VC-TO, in which the *WV* is plotted with respect to the stress (σ) to simplify the relationships of different compressive stages. The maximum allowable stress (σ*max*) under a certain energy absorption ability can also be obtained. This curve helps to find a cellular material that bears the required σ*max* by maximising the energy absorption capability [56]. An energy-efficient structure gives a high envelope. In Figure 9, VC-TO lattice is able to absorb more energy than others with the same allowable stress. For instance, if the σ*max* is 300 MPa, the values of *WV* for FCC-TO, ECC-TO, and VC-TO with ρ = 0.196 are 64 MJ/m3, 47 MJ/m3, and 80 MJ/m3, respectively.

**Figure 9.** The energy absorption diagrams of topology-optimised lattice materials.

#### **5. Conclusions**

Three types of lattice materials, i.e., FCC, ECC, and VC, have been separately designed based on two different methods: topology optimisation and crystal inspiration. Numerical compression tests have been conducted to comparatively characterise their deformation modes, mechanical properties, and energy absorption capability. The main conclusions and contributions are summarised as follows:


However, differences are found between VC-TO and VC-CI lattices. Shear band is observed in VC-CI structures at a low relative density while the VC-TO lattice deforms stably.

d) In terms of collapse strength and elastic modulus, the VC lattice is stronger than the FCC and ECC lattices because its struts are arranged along the loading direction. On the other hand, topology-generated lattices outperform the corresponding crystal-guided lattices in aspects of toughness and energy absorption per unit volume.

**Author Contributions:** Conceptualization, C.Y., K.X. and S.X.; methodology, C.Y. and S.X.; software, C.Y. and K.X.; validation, C.Y. and K.X.; formal analysis, C.Y. and S.X; investigation, C.Y., K.X. and S.X.; data curation, C.Y. and K.X.; writing-original draft preparation, C.Y.; writing-review and editing, K.X. and S.X; project administration, S.X.; funding acquisition, S.C.X. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research was funded by the National Natural Science Foundation of China (No. 51775558) and the Nature Science Foundation for Excellent Youth Scholars of Hunan Province (No. 2019JJ30034).

**Acknowledgments:** The authors are grateful to Ping Xu and Shuguang Yao at CSU for insightful discussion.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Elasto-Plastic Behaviour of Transversely Isotropic Cellular Materials with Inner Gas Pressure**

**Zhimin Xu 1, Kangpei Meng 2, Chengxing Yang 2, Weixu Zhang 1,\*, Xueling Fan <sup>1</sup> and Yongle Sun 2,\***


Received: 7 July 2019; Accepted: 14 August 2019; Published: 16 August 2019

**Abstract:** The fabrication process of cellular materials, such as foaming, usually leads to cells elongated in one direction, but equiaxed in a plane normal to that direction. This study is aimed at understanding the elasto-plastic behaviour of transversely isotropic cellular materials with inner gas pressure. An idealised ellipsoidal-cell face-centred-cubic foam that is filled with gas was generated and modelled to obtain the uniaxial stress–strain relationship, Poisson's ratio and multiaxial yield surface. The effects of the elongation ratio and gas pressure on the elasto-plastic properties for a relative density of 0.5 were investigated. It was found that an increase in the elongation ratio caused increases in both the elastic modulus and yield stress for uniaxial loading along the cell elongation direction, and led to a tilted multiaxial yield surface in the mean stress and Mises equivalent stress plane. Compared to isotropic spheroidal-cell foams, the size of the yield surface of the ellipsoidal-cell foam is smaller for high-stress triaxiality, but larger for low-stress triaxiality, and the yield surface rotates counter-clockwise with the Lode angle increasing. The gas pressure caused asymmetry of the uniaxial stress–strain curve (e.g., reduced tensile yield stress), and it increased the nominal plastic Poisson's ratio for compression, but had the opposite effect for tension. Furthermore, the gas pressure shifted the yield surface towards the negative mean stress axis with a distance equal to the gas pressure. The combined effects of the elongation ratio and gas pressure are complicated, particularly for the elasto-plastic properties in the plane in which the cells are equiaxed.

**Keywords:** foam; enclosed gas; anisotropy; elasticity; plasticity; multiaxial yielding

#### **1. Introduction**

Cellular materials, either natural or manmade, are unique with regard to mechanical, thermal, acoustic and electromagnetic properties, benefitting from their high porosity, which reduces the overall density, enhances the energy absorption capacity and enables the integration of multiple functions. Foams, made of metals, polymers, ceramics, etc., are typical cellular materials, and they are widely used in transport, aerospace, defence, building and biomedical industries [1,2].

In the fabrication of foams, the cell structure is determined by the foaming process, which is sensitive to the foaming agent used and the state of the base material during the foaming [3]. It is common that the cell structure is anisotropic after fabrication, particularly when gravity plays an important role in the foaming process. Foams usually consist of cells which are elongated in one direction, but equiaxed in the plane perpendicular to the elongation direction, exhibiting transversely isotropic structural characteristics [4]. However, most previous studies focused on the elasto-plastic behaviour of presumably isotropic foams [5–8], and there is still a paucity of experimental and

modelling data for better understanding the anisotropic elasto-plastic behaviour of foams. Therefore, more studies on anisotropic foams are needed for both uniaxial and multiaxial loadings, which are pertinent to applications of foams as energy absorbers and cores of sandwich structures.

Closed-cell foams contain inner trapped gas in the cells after fabrication and the inner gas pressure can considerably affect the macroscopic elasto-plastic properties [1]. It is also of fundamental significance and scientific interest to study the effect of gas pressure on the elasto-plastic behaviour of closed-cell foams. For static loading, Ozgur et al. [9] analysed the effect of gas pressure on the elastic properties of cellular materials using 2D finite element (FE) models, in which hexagonal, rectangular and circular cells were considered. Öchsner and Mishuris [10] developed a 3D simple cubic cell model and numerically analysed the gas effects on the stress–strain relationship, Poisson's ratio, Young's modulus and yield surface. Zhang et al. [11] established a theoretical model using second-order moment of stress with consideration of inner gas pressure, and they overcame the limitation of an earlier theoretic model [12] and clarified the gas effect. Based on the Gurson yield function [13], Guo et al. [14–17] developed a thick-walled spherical unit cell model and systematically studied the yield behaviour of metal foams with inner gas pressure. For dynamic loading, Sun and Li [18] investigated the effects of gas pressure on the dynamic strength and deformation of cellular materials, and their findings have been applied to explain experimental observations on dynamic compressive behaviour of closed-cell foams [2]. However, these previous studies all focused on isotropic cellular materials. There is still a lack of modelling studies on the elasto-plastic behaviour of anisotropic cellular materials with inner gas pressure.

In this study, an idealised transversely isotropic cellular material with inner gas pressure, i.e., gas-filled ellipsoidal-cell face-centred-cubic (FCC) foam, is studied numerically. The ellipsoidal cell is elongated in one direction, but equiaxed in the plane normal to the elongation direction. Different elongation ratios are considered to investigate the effects of anisotropy on static elasto-plastic properties (e.g., uniaxial stress–strain relationship, Poisson's ratio and multiaxial yield surface). The effects of gas pressure on the anisotropic elasto-plastic properties are also investigated.

#### **2. Material and Methods**

The typical cell structure of a transversely isotropic closed-cell foam is shown in Figure 1a, along with the inner gas pressure. To facilitate modelling without losing key physics, one representative volume (RV) unit is considered, as shown in Figure 1b. The RV simplification of geometry implies that the cells are periodically arranged in 3D space, as well as cell deformation, in contrast to the random distribution of irregular cells and cell deformation in an actual closed-cell foam that is normally seen. Nevertheless, provided that the closed-cell foam possesses transversely isotropic macro-properties, a mechanical model based on RV geometry is deemed reliable to capture qualitative behaviour and gain general insights. The enhanced computational efficiency obtained by the geometric simplification enables the analysis of sufficient loading cases in modelling to investigate both the uniaxial and multiaxial elasto-plastic behaviour of the closed-cell foam. A similar RV approach has been widely used for the analysis of the elasto-plastic behaviour of cellular materials [10,19,20].

The RV unit is idealised to be an FCC unit with ellipsoidal cells, as shown in Figure 1b. The ellipsoid is elongated in the y direction (Figure 1c) and mathematically described as

$$\frac{(x^2+z^2)}{a\_1^2} + \frac{y^2}{a\_2^2} \le 1\tag{1}$$

where *a*<sup>1</sup> and *a*<sup>2</sup> are the axial radii in the isotropic plane (i.e., x-z plane) and along the elongation direction (i.e., y direction), respectively. The elongation ratio is thus defined as *R* = *a*2/*a*1.

The relative density of a cellular material is

$$f = \rho / \rho\_s \tag{2}$$

where ρ and ρ*<sup>s</sup>* are the densities of the cellular material and the base material, respectively. For the closed-cell FCC foam, the relative density can also be determined from geometric parameters, viz.

$$f = 1 - \frac{16\pi a\_1^2 a\_2}{3L^3} \tag{3}$$

Here, *L* is the size of the cubic unit, which is kept constant. We adopted *f* = 0.5 in this study. It should be noted that, for given *R*, *L*, and *f*, the values of *a*<sup>1</sup> and *a*<sup>2</sup> can be uniquely determined.

**Figure 1.** (**a**) Typical cell structure of transversely isotropic closed-cell foam with inner gas pressure; (**b**) representative volume (RV) unit of ellipsoidal-cell face-centred-cubic (FCC) foam; (**c**) orientation of the ellipsoidal cell, which is equiaxed in the x-z plane, but elongated in the y direction; (**d**) definition of directions parallel and perpendicular to the elongation direction (i.e., y-direction).

The multiaxial plastic behaviour of cellular materials is complicated. It has been demonstrated that the yield surface of an isotropic foam is dependent on both the first and second stress invariants [2,6]. For transversely isotropic cellular materials, the yield surface may be more complicated and the locus of yield points may not be unique if only the first and second stress invariants are used in characterisation. Therefore, a full description of the stress state should be provided. In general, three parameters are needed to determine each point in the principal stress space, and here, the mean stress, Mises equivalent stress and Lode angle are adopted. It should be noted that these stress parameters are solely associated with the type of loading and are independent of the constitutive behaviour of the material which is either isotropic or not isotropic. The mean stress is expressed as

$$
\sigma\_{\rm ll} = \frac{\sigma\_{\rm kk}}{3} \tag{4}
$$

with Einstein's summation convention applied. The familiar Mises equivalent stress is given by

$$
\sigma\_{\mathfrak{e}} = \sqrt{3l\_2} = \sqrt{\frac{3}{2} \mathbf{s}\_{ij} \mathbf{s}\_{ij}} \tag{5}
$$

where *J*<sup>2</sup> is the second deviatoric stress invariant, and *sij* is the deviatoric stress, i.e., *sij* = σ*ij* − σ*m*δ*ij*, (*i*, *j* = 1, 2, 3). The stress triaxiality is thus obtained via

$$X\_{\Sigma} = \frac{\sigma\_m}{\sigma\_{\mathfrak{C}}} \tag{6}$$

The Lode angle is defined as follows:

$$L = -\cos(3\theta) = -\left(3\frac{\sqrt[3]{\frac{1}{2}f\_3}}{\sigma\_\varepsilon}\right)^3 = -\frac{27}{2}\frac{f\_3}{\sigma\_\varepsilon^3} = -\frac{27}{2}\frac{\text{Det}(s\_{ij})}{\sigma\_\varepsilon^3} \tag{7}$$

where *L* is the Lode parameter, θ is the Lode angle and *J*<sup>3</sup> is the third deviatoric stress invariant.

It is assumed that the principal stress direction coincides with the axial direction of the ellipsoidal cell, i.e., σ<sup>1</sup> = σ11, σ<sup>2</sup> = σ<sup>22</sup> and σ<sup>3</sup> = σ33. Accordingly, the relationship between the principal stress, mean stress, Mises equivalent stress and Lode angle is given as follows:

$$\frac{3}{2\sigma\_{\ell}} \{\sigma\_1, \sigma\_2, \sigma\_3\} = \left\{-\cos\left(\theta + \frac{\pi}{3}\right), -\cos\left(\theta - \frac{\pi}{3}\right), \cos\theta\right\} + \frac{3}{2\varepsilon} \{1, 1, 1\} \tag{8}$$

In the ellipsoidal-cell FCC foam model, only one eighth of the RV unit is considered (Figure 2a) owing to symmetry. Multiaxial loading is applied according to Equation (8), as shown in Figure 2b, where *T*1, *T*<sup>2</sup> and *T*<sup>3</sup> are axial loads in three principal directions. Any state of loading stress can be obtained by varying the ratio of the three axial loads.

**Figure 2.** (**a**) Schematic of ellipsoidal-cell face-centred-cubic (FCC) foam model (only 1/8 volume is considered owing to symmetry); (**b**) application of multiaxial loading.

Details of the loading process are described below:

1) Inner gas pressure is applied and meanwhile, an additional load is applied to balance the inner gas pressure. Note that the gas pressure refers to the gauge pressure after the subtraction of ambient air pressure. Öchsner and Mishuris [10] concluded that the inner gas pressure only causes slight deformation and thus the gas pressure is assumed to be constant throughout the loading process. Taking a uniaxial loading case as an example, the area *A*<sup>1</sup> of the solid in Figure 2a is on the loading boundary, where a balancing load to the inner gas pressure *P* needs to be applied, in addition to the main load. The equilibrium equation is given as

$$\int\_{A\_2} P \mathrm{d}A\_2 = -\int\_{A\_1} F' \mathrm{d}A\_1 \tag{9}$$

where *A*<sup>1</sup> and *A*<sup>2</sup> are the areas of the solid and gas, respectively, as shown in Figure 2a. *F* denotes the balancing load to the inner gas pressure *P*. The method for balancing gas pressure is the same when applying loads in other directions;

2) The relationship between triaxial loads and invariant stress parameters has been given in Equation (8). Accordingly, the loads to be applied in the three axial directions are calculated and determined for different stress states. This provides guidance on the application of multiaxial loads to maximise the attainable stress states in the principal stress space, and the multiaxial loading is applied in a proportional manner;

3) The macroscopic stress and strain is calculated. Taking uniaxial loading as an example, the macroscopic stress can be calculated using the following formula:

$$\overline{\sigma} = \frac{1}{A\_1 + A\_2} \left( \int\_{A\_1} F \text{d}A\_1 + \int\_{A\_2} P \text{d}A\_2 + \int\_{A\_1} F' \text{d}A\_1 \right) \tag{10}$$

where *F* is the main load applied to the FCC foam. Taking into account Equation (9), the macroscopic stress can be expressed as

$$\overline{\sigma} = \frac{1}{A\_1 + A\_2} \int\_{A\_1} F \mathbf{d} A\_1 \tag{11}$$

The corresponding macroscopic strain ε can be expressed as

$$
\overline{\varepsilon} = \overline{\varepsilon}\_F - \overline{\varepsilon}\_P - \overline{\varepsilon}\_F \tag{12}
$$

where ε*F*, ε*<sup>P</sup>* and ε*F* are the strains corresponding to the main load *F*, inner gas pressure *P* and the balancing load *F* to the gas pressure, respectively. Similarly, macroscopic stresses and strains in other directions can be obtained. Finally, the macroscopic Mises equivalent stress and equivalent strain can be calculated via

$$
\overline{\sigma}\_{\mathfrak{c}} = \sqrt{\frac{3}{2} \overline{S}\_{ij} \overline{S}\_{ij}} \tag{13}
$$

$$
\overline{\varepsilon}\_{\varepsilon} = \sqrt{\frac{2}{3} \overline{\varepsilon}\_{ij} \overline{\varepsilon}\_{ij}} \tag{14}
$$

where *Sij* and *eij* are the macroscopic deviatoric stress and strain tensors, respectively.

General purpose FE software ANSYS® was employed in numerical modelling. Linear solid elements (ANSYS designation SOLID185) were used and the FE mesh consisted of 61,091 elements, as shown in Figure 3. A mesh sensitivity analysis was performed, which showed that the numerical results of stress and strain were unchanged when further refining the mesh.

It is assumed that the base material conforms to Hooke's elasticity law and von Mises plasticity theory, i.e.,

$$
\sigma = \begin{cases}
 E\_s \varepsilon\_r & \text{for } \varepsilon \le \varepsilon\_0 \\
 \sigma\_s + E\_p(\varepsilon - \varepsilon\_0), & \text{for } \varepsilon > \varepsilon\_0
\end{cases}
\tag{15}
$$

where *Es* and *Ep* are the elastic modulus and plastic hardening modulus, respectively; σ*<sup>s</sup>* and ε<sup>0</sup> are the initial yield stress and yield strain, respectively. For a qualitative study on cellular materials in general, the following parameters are assumed: *Es* = 3.5 GPa, *Ep* = 0.1 GPa, σ*<sup>s</sup>* = 80 MPa and Poisson's ratio = 0.33.

The initial yield stress of the ellipsoidal-cell FCC foam is defined as the intersection of the extrapolated linear elastic portion and plastic portion in the macroscopic stress–strain curve, as proposed by Deshpande and Fleck [21]. For multiaxial loading, the Mises equivalent stress–strain curve and mean stress–strain curve are used to determine the initial yield surface. The Poisson's ratio of the ellipsoidal-cell FCC foam is defined as the absolute value of the ratio of macroscopic strains perpendicular and parallel to the loading direction, respectively.

**Figure 3.** Finite element (FE) mesh of ellipsoidal-cell face-centred-cubic (FCC) foam model (only 1/8 volume is considered owing to symmetry).

#### **3. Results and Discussion**

#### *3.1. Uniaxial Stress-Strain Relationship*

Figure 4 shows the uniaxial stress–strain curves of the FCC foam loaded along the ellipsoidal-cell elongation direction, when different values of gas pressure and elongation ratio are considered. For a typical elongation ratio (*R* = 2), increasing the gas pressure does not change the linear elastic portion of the stress–strain curve, but the plastic portion is significantly affected, as shown in Figure 4a. Without inner gas pressure, the stress–strain curve is antisymmetric between tension and compression, but such anti-symmetry is violated by the presence of gas pressure. When the gas pressure increases, the tensile yield stress markedly decreases, while the compressive yield stress only slightly decreases (in magnitude). For spheroidal-cell foam, Xu et al. [20] found a similar effect of gas pressure on tensile yield stress, but a slight increase in compressive yield stress due to gas pressure, in contrast to the observation in Figure 4a. It is surmised that the gas pressure can exacerbate the deformation concentration at the end of the long axis when compressive load is applied along the long axis, thereby promoting macroscopic yielding at a lower stress level for the ellipsoidal-cell foam, compared to the compressive behaviour of spheroidal-cell foam [20]. Nevertheless, for both spheroidal-cell and ellipsoidal-cell foams, the effect of gas pressure on the uniaxial stress–strain curve is more pronounced in tension than in compression.

Figure 4b shows the uniaxial stress–strain curves of the gas-filled ellipsoidal-cell FCC foam with different elongation ratios. For a given value of gas pressure (*P* = 15 MPa), increasing the elongation ratio leads to an increase in yield stress for both tension and compression, to a lesser extent for the latter. The elastic modulus (i.e., slope of the linear elastic portion) also increases. The above observation demonstrates a strong effect of the elongation ratio on the uniaxial elasto-plastic behaviour of the gas-filled FCC foam loaded along the cell elongation direction. The increases in both yield stress and elastic modulus can be attributed to the fact that a higher elongation ratio implies more material is distributed to bear the load applied in the cell elongation direction.

**Figure 4.** Uniaxial stress–strain curves of the ellipsoidal-cell face-centred-cubic (FCC) foam loaded along the cell elongation direction for different values of inner gas pressure (**a**) and elongation ratio (**b**).

Figure 5 shows the uniaxial stress–strain curves of the FCC foam when the loading direction is perpendicular to the cell elongation direction. Figure 5a shows the gas effect when the elongation ratio is two. Again, the gas pressure hardly affects the elastic modulus and the effect of gas pressure on the tensile stress–strain curve is similar to that in the parallel loading case (Figure 4a). However, the gas pressure enhances the magnitude of compressive yield stress, unlike the decreasing trend found in the parallel loading case (Figure 4a). It is expected that the gas pressure can contribute to an enhanced resistance of the FCC foam to compressive load, since the gas pressure can be directly additive to the compressive load-bearing capacity, except that undesirable deformation potentially induced by gas pressure could cause an adverse effect (e.g., slight decrease in compressive yield stress, as observed in Figure 4a). Figure 5b shows the effect of the elongation ratio on the uniaxial stress–strain curve of the gas-filled ellipsoidal-cell FCC foam. It appears that increasing the elongation ratio does not affect the elastic modulus, but causes a reduction in tensile yield stress, in contrast to the increasing trend found in the parallel coating case (Figure 4b). The effect of the elongation ratio on the compressive yield stress is similar between the perpendicular and parallel loading cases.

**Figure 5.** Uniaxial stress–strain curves of the ellipsoidal-cell face-centred-cubic (FCC) foam loaded perpendicularly to the cell elongation direction for different values of inner gas pressure (**a**) and elongation ratio (**b**).

Comparing the results shown in Figures 4 and 5, one can conclude that (1) given an elongation ratio greater than one, the elastic modulus and tensile yield stress are larger in the long axial direction than in the short axial direction, whether inner gas pressure is present or not; however, the compressive yield stress is affected by gas pressure in opposite ways between the parallel loading and perpendicular loading, although when gas pressure is absent, the compressive yield stress is higher in the long axial direction than in the short axial direction; (2) given the gas pressure, the elongation ratio plays a more significant role in the uniaxial elasto-plastic behaviour under parallel loading than under perpendicular loading; (3) in general, the gas pressure hardly affects the elastic modulus, and it has less effect on the uniaxial plastic behaviour under parallel loading than under perpendicular loading.

#### *3.2. Poisson's Ratio*

Figure 6a,b shows the Poisson's ratio of the FCC foam uniaxially loaded parallelly and perpendicularly to, respectively, the cell elongation direction. It should be noted that the Poisson's ratio of isotropic spheroidal-cell foam (*R* = 1) is independent of the loading direction. By contrast, for ellipsoidal-cell foam, the Poisson's ratio does not vary in the equiaxed-cell plane (x-z plane) for the parallel loading (Figure 6a), but it differs between the long and short axes for the perpendicular loading (Figure 6b). The prominent feature is that when the gas pressure is absent, the Poisson's ratio is symmetric between tension and compression and such symmetry is independent of the elongation ratio and loading direction. When the elongation ratio increases, the Poisson's ratio becomes larger in the parallel loading case (Figure 6a), but smaller in the perpendicular loading case (Figure 6b). The gas pressure does not affect the Poisson's ratio in the elastic stage, but its effect is significant in the plastic stage, wherein the gas pressure decreases the Poisson's ratio for tension, but increases the Poisson's ratio for compression. As a result, the Poisson's ratio becomes asymmetrical between tension and compression. The effect of gas pressure on Poisson's ratio can be explained as follows. For uniaxial tension, the gas pressure and tensile loading are aligned, which favours the deformation along the loading direction and thus reduces the Poisson's ratio. Conversely, the gas pressure counteracts the compressive load and thereby increases the Poisson's ratio under uniaxial compression. Such trends hold for different elongation ratios. A similar gas effect on Poisson's ratio was also reported by Xu et al. [20] and Öchsner and Mishuris [10] for isotropic cellular materials.

**Figure 6.** Poisson's ratio of the ellipsoidal-cell face-centred-cubic (FCC) foam loaded parallelly (**a**) and perpendicularly to (**b**) the cell elongation direction. Note that for perpendicular loading, the Poisson's ratio differs between the short and long axes.

#### *3.3. Multiaxial Yield Surface*

Figure 7 shows the initial yield surfaces of the FCC foam under multiaxial loading at different Lode angles. For a Lode angle of 0◦, when the elongation ratio is one (i.e., spheroidal-cell foam) and the gas pressure is zero, the yield surface is an ellipse, being symmetrical about the σ<sup>e</sup> axis in the σ<sup>e</sup> − σ<sup>m</sup> plane, but such symmetry is violated when the elongation ratio is two, leading to a yield surface manifested as a tilted ellipse in the σ<sup>e</sup> − σ<sup>m</sup> plane. It is clearly seen that the yield surface of the transversely isotropic ellipsoidal-cell FCC foam is distinct from the yield surface of the isotropic spheroidal-cell FCC foam. Particularly, compared to the spheroidal-cell foam, the ellipsoidal-cell foam is more susceptible to yielding when approaching a hydrostatic stress state (i.e., yield stress is lower when *X*<sup>Σ</sup> → ∞), but it is more resistant to yielding when approaching a shear stress state (i.e., yield stress is higher when *X*<sup>Σ</sup> → 0). The Lode angle does not affect the yield surface of the spheroidal-cell

foam (*R* = 1), but it does affect the orientation of the tilted yield surface of the ellipsoidal-cell foam (*R* = 2), i.e., the yield surface rotates counter-clockwise when the Lode angle is increasing.

**Figure 7.** Initial multiaxial yield surfaces of the ellipsoidal-cell face-centred-cubic (FCC) foam for Lode angles of 0◦ (**a**), 60◦ (**b**) and 90◦ (**c**).

The effect of gas pressure on the yield surface is straightforward, i.e., the gas pressure shifts the yield surface towards the negative mean stress axis with a distance equal to the gas pressure in the σ<sup>e</sup> − σ<sup>m</sup> plane. Such an effect is independent of the Lode angle and elongation ratio, which is consistent with the previous studies by Xu et al. [20] and Zhang et al. [11].

#### **4. Conclusions**

A qualitative study on the elasto-plastic properties of transversely isotropic cellular materials with inner gas pressure is presented, which is focused on a gas-filled ellipsoidal-cell FCC foam with a relative density of 0.5. The findings are summarised as follows:

(1) The elasto-plastic behaviour of the gas-filled ellipsoidal-cell FCC foam is dependent on the loading direction. The effect of the elongation ratio is most pronounced for uniaxial loading along the cell elongation direction. Increasing the elongation ratio leads to increases in the elastic modulus, yield stress and Poisson's ratio, due to more load-bearing material being distributed in the elongation direction. For perpendicular loading, increasing the elongation ratio does not affect the elastic modulus, but reduces the tensile yield stress and Poisson's ratio, and increases the compressive yield stress. In general, the elastic modulus, tensile yield stress and Poisson's ratio are higher for parallel loading than for perpendicular loading. For multiaxial loading, the initial yield surface of the ellipsoidal-cell FCC foam is a tilted ellipse in the σ<sup>e</sup> − σ<sup>m</sup> plane and it rotates counter-clockwise when the Lode angle is increasing;

(2) The inner gas pressure causes asymmetry of the uniaxial stress–strain curve of the FCC foam. It reduces the tensile yield stress, but it has the opposite effects on the compressive yield stress for

loadings parallel and perpendicular to the cell elongation direction, i.e., slight reduction for the former, while a considerable increase for the latter. The effect of gas pressure on the uniaxial stress–strain curve is more pronounced in tension than in compression. Poisson's ratio is independent of gas pressure in the elastic regime, but it increases in the plastic compression regime and decreases in the plastic tension regime, due to the effect of gas pressure. Furthermore, the gas pressure shifts the tilted multiaxial yield surface of the ellipsoidal-cell foam towards the negative mean stress axis with a stress value identical to the gas pressure value.

**Author Contributions:** Conceptualization, W.Z. and Y.S.; methodology, Z.X. and X.F.; software, Z.X. and X.F.; validation, K.M. and C.Y.; formal analysis, Z.X. and Y.S.; investigation, Z.X., K.M. and C.Y.; resources, W.Z., X.F. and Y.S.; data curation, K.M. and C.Y.; writing—original draft preparation, Z.X. and Y.S.; writing—review and editing, K.M., C.Y., X.F. and W.Z.; visualization, Z.X.; supervision, W.Z.; project administration, X.F.; funding acquisition, W.Z. and Y.S.

**Funding:** This research was funded by the China State Key Laboratory for Strength and Vibration of Mechanical Structures through Open Project (grant number SV2018-KF-37) and the National Natural Science Foundation of China (grant numbers 11772246, 11472203, 11172227, U1530259).

**Acknowledgments:** The authors are grateful to Fulin Shang at XJTU for insightful discussion.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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