Large deformations of soft metamaterials fabricated by 3D printing

Materials & Design 131 (2017) 81–91

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Large deformations of soft metamaterials fabricated by 3D printing M. Bodaghi, A.R. Damanpack, G.F. Hu, W.H. Liao

MARK

⁎

Smart Materials and Structures Laboratory, Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, People's Republic of China

A R T I C L E I N F O

A B S T R A C T

Keywords: Metamaterial Soft poly-lactic acid 3D printing Large deformation Experimental validation FEM

The aim of this paper is to explore large-deformation responses of hyper-elastic porous metamaterials made by three-dimensional (3D) printing technology. They are designed as a repeating arrangement of unit-cells in parallelogram and hexagonal shapes. Fused deposition modeling is implemented to fabricate metamaterial structures from soft poly-lactic acid. 3D printed metamaterials are tested under in-plane tension-compression in axial and transverse directions. Experiments reveal that unit-cell shape, direction, type and magnitude of mechanical loading have significant effects on metamaterial anisotropic response and its instability characteristics. To replicate experimental observations, a finite element solution is developed adopting the hyper-elastic Mooney-Rivlin constitutive equations and non-linear Green-Lagrange strains. Iterative Newton-Raphson approach is implemented to solve governing equations with material-geometric non-linearities. The accuracy of the computational tool is verified by capturing main features observed in the experiments. It is found that modeling of hyper-elasticity and large strain is essential to accurately predict non-linear responses of the 3D printed soft metamaterials. Due to the absence of similar results in the specialized literature, this paper is likely to advance the state of the art of metamaterial printing, and provide pertinent results that are instrumental in the design of hyper-elastic metamaterial structures and infill patterns for printing purpose.

1. Introduction Metamaterials are appearing at the frontier of structural engineering owing to their unique mechanical features and extraordinarily properties originated from their structures rather than their constituents [1]. Metamaterials comprise a repeating arrangement of unit-cells with tunable behaviors. Significant examples of metamaterials are pentamodes with vanishing shear modulus [2], mono/multi-stable materials [3,4] and auxetic with negative Poisson's ratio [5]. Recently, three-dimensional (3D) printing technology [6,7] has gained significant attention for processing metamaterials. It has provided flexibility in creating metamaterials with complex structures and shapes, fast and with low cost. 3D printing procedure relies on CAD data and comprises depositing material in consecutive layers. It is worthy to mention that most of 3D printers fill the interior of an object by some infill patterns that are similar to porous metamaterials [8,9]. Although there is a variety of different infills, unit-cells with parallelogram, rectangular and hexagonal shapes are mostly used. Some researchers have been conducted to explore material properties of metamaterials experimentally and theoretically. For instance, Scarpa et al. [5] calculated in-plane Poisson's ratio and Young's moduli of re-entrant cell auxetic honeycombs for different geometric layout

⁎

Corresponding author. E-mail address: [email protected] (W.H. Liao).

http://dx.doi.org/10.1016/j.matdes.2017.06.002 Received 4 April 2017; Received in revised form 13 May 2017; Accepted 1 June 2017 Available online 01 June 2017 0264-1275/ © 2017 Elsevier Ltd. All rights reserved.

combinations by means of finite element (FE) simulations and experiments. Florijn et al. [10] created metamaterials whose response to uniaxial compression can be tuned by lateral confinement, allowing monotonic, non-monotonic, and hysteretic feature. Using PolyJet 3D printing system, Wang et al. [11] fabricated auxetic metamaterials with elastic joints and stiff beams/walls without buckling issue. They investigated the influence of material selections and stiff material fraction on Young's Modulus, Poisson's ratio, and volume reduction by experiments and FE simulations. Mousanezhad et al. [12] investigated small and large deformation in-plane elastic response of auxetic spiderweb honeycombs fabricated by PolyJet 3D printing under compression through analytical modeling, numerical simulation, and experiments. They also examined the behavior of auxetic metamaterials with structural hierarchy by experimental and computational studies [13]. The results showed that hierarchy-dependent buckling introduced at early steps of deformation decreased the Poisson's ratio as the structure was uniaxially compressed leading to auxeticity in next deformation steps. Ren et al. [14] experimentally and numerically investigated the loss of auxetic feature of buckling-induced metallic metamaterials under compression to understand the mechanism behind this phenomenon. They found that the auxetic performance can be tuned by the microstructure geometry while strength and stiffness can be tuned through

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Fig. 1. Metamaterials: (a) schematic sketch; (b) geometries and dimensions of unit-cells; (c) periodicity.

3D auxetic metamaterial on the basis of 2D re-entrant honeycombs and tested it numerically and experimentally. The results showed that the 3D structure can gain a negative Poisson's ratio in two principal orthogonal directions under compression in other principal axis. Recently, Naddeo et al. [19] presented an algorithmic procedure to replace continuous mass of convex solids with a cancellous bone-inspired lattice structure showing curved beams oriented according to the external forces, sharing with its border and boundary conditions. They selected a cubic representative volume element (RVE) and performed FE sensitivity analysis by implementing periodic boundary conditions to investigate structural mechanical behaviors. The computational results were compared with experimental data of compression tests on polymeric 3D printed cubic specimens that confirmed validity of the beam element-based lattice structure. The literature survey reveals that, most of research works have been focused on the behaviors of polymeric metamaterials under either

plasticity of the main material. Hedayati et al. [15] derived analytical relationships for elastic modulus, yield stress, buckling stress and Poisson's ratio of 3D printed octagonal honeycombs using linear beam theories of Euler-Bernoulli and Timoshenko. They compared the results from analytical solutions with experiments and those computed by ANSYS FE code. They found that there is a good agreement between the results from Timoshenko beam theory, FE and experiments. In another study [16], they investigated influence of assuming exact apparent density rather than approximate density on the elastic modulus, yield stress and Poisson's ratio. They validated the accuracy of the proposed model by comparing predicted mechanical properties with experimental data. Tang and Yin [17] explored the designs for obtaining extreme stretchability and/or compressibility in auxetic metamaterials by combining line cuts, cut-outs, and hierarchical structures. They verified their results through experiments, geometrical modeling, and FE simulation. Using 3D printing technology, Fu et al. [18] fabricated a 82

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lower height for compressive testing as shown in Fig. 2b and d. The counterpart of samples for metamaterials with hexagonal unit-cells are also illustrated in Fig. 2e–h. Dimensions of metamaterials depicted in Fig. 2a–h in terms of height × width × thickness are 132 × 86.5 × 3, 66 × 86.5 × 10, 142.2 × 82.5 × 3, 56.8 × 82.5 × 10, 228.6 × 132 × 3, 85.7 × 132 × 10, 231 × 143.6 × 3 and 82.5 × 143.6 × 10 mm, respectively. Finally, to explore mechanical behaviors of soft PLA material used for printing, tensile test samples are printed based on the geometry and dimension described by the ASTM D638 standard in a dog-bone shape (Type IV, 3 mm thickness) [23]. The overall length, width and gauge length and width of the narrow section are 115, 19, 25 and 6 mm, respectively. The dog-bone shape is filled with a perimeter raster in which the print raster is parallel to the length of the gauge section (0° raster angle in the gauge section). Three specimens for each sample including metamaterial and dog-bone shape samples are printed and tested. Experiments are performed on a Tinius Olsen® H5kS (Tinius Olsen, Horsham, PA, USA) tensile testing machine equipped with a 5 kN load cell. This is a displacement-controlled machine that can measure positive forces. A home-made fixture is also designed and employed to adopt the machine for compressive tests. All tests are conducted at a constant speed of 1 mm min− 1. Furthermore, three repetition tests are performed on each specimen to get a stable response. All stress and strain values are computed using initial cross-sectional area and length of the test specimens in the calculations. For each series of tests, arithmetic mean of all values is reported as an average value. In this respect, the testing results of all the specimens are considered due to close agreement. The uniaxial tensile behaviors of the 3D printed dog-boneshape sample in terms of engineering stress-strain, F/A0 − ΔL/L, are illustrated in Fig. 3. F indicates force while A0 and L indicate initial cross-sectional area and length, respectively. As it can be seen, the 3D printed soft PLA has a non-linear elastic response. Although 30% engineering strain is moderately large, the material behaves elastically in a non-linear manner experiencing no residual strain. In fact, it acts like incompressible rubber materials [24]. Therefore, the 3D printed soft PLA can be categorized as incompressible hyper-elastic materials [24]. It should be noted that, the layer-by-layer FDM printing process may fabricate transversely isotopic material in which the plane of isotropy would be normal to the print raster. For the current 3D printed objects, the print raster is along the length of the gauge section of the dog-boneshape sample and also of beam members of metamaterials. Therefore, the uniaxial response presented in Fig. 3 can be considered as uniaxial stress-strain behavior for a material point of beam members in the metamaterials.

compression or tension in the small strain regime. In real applications, metamaterials may experience large deformation during loadings [20,21]. In this case, they may be fabricated by largely deformable rubber-like materials called hyper-elastics. For instance, using 3D printing technology, Jakus et al. [21] developed metamaterial bones made of hyper-elastics, which may undergo recoverable strains of 25%. The present paper aims at exploring mechanical behaviors of metamaterials made of hyper-elastics under both tension and compression in the large-strain range, experimentally and numerically. Fused deposition modeling (FDM), as the most common consumer 3D printing technology, is employed to fabricate metamaterials with parallelogram and hexagonal unit-cells from soft poly-lactic acid (PLA). FDM is of particular interest due to its association to desktop 3D printers. Experiments are conducted to demonstrate in-plane mechanical responses of metamaterials in both tension and compression modes revealing buckling instability. An FE solution is developed to replicate experimental results with instability characteristics. The equilibrium equations are derived based on the hyper-elastic Mooney-Rivlin constitutive equations and non-linear Green-Lagrange strains. The governing equations with both material and geometric non-linearities are then solved by implementing iterative incremental Newton-Raphson technique to trace non-linear equilibrium path in the large-strain regime. It is shown that the structural-material model and established FE solution are able to predict key non-linear characteristics observed in the experiments. It is also noticed that modeling of hyper-elasticity and large strain is essential for an accurate prediction of non-linear responses of the 3D printed soft metamaterials. Numerical and experimental results reveal that unit-cell shape, direction type and magnitude of loading may have significant influence on metamaterial anisotropic response in large deformations. In this respect, performing multivariate analysis and parametric sensitivity analysis could identify the parameters that strongly influence the structural responses (for example, see Ref. [22]). These analyses would be considered in future studies. Due to lack of similar work in the specialized literature, the results of this research are expected to contribute to understanding of the hyper-elastic metamaterial behaviors and to be instrumental towards a reliable design of soft metamaterial structures and infill patterns for printing purpose. 2. Materials and methods 2.1. Materials and fabrication Metamaterials are composed of repeatedly arranged unit-cells. In the present work, two metamaterials designed by parallelogram and hexagonal unit-cells are investigated as shown in Fig. 1a. Geometries and dimensions of the unit-cells are also denoted in Fig. 1b. Both unitcells are composed of six beam-like members with l length and thick1 ness of 2 t in parallelogram and hexagonal shapes. Based on the periodicity concept [13,19], parallelogram and hexagonal unit-cells are arranged in the x-z plane to form a periodic arrangement, see Fig. 1c. Filament-based FDM as one of well-known 3D printing techniques is implemented to fabricate metamaterial structures. In this respect, a FlashForge New Creator Pro 3D printer is used. Commercial soft PLA materials are also selected since they may behave hyper-elastically. Printing parameters like layer height and temperatures of nozzle extrusion, build platform and chamber are set as 0.2 mm and 185, 70 and 25 °C while printing head speed is 40 mms− 1. Metamaterials with geometric parameters of t = 1 mm and l = 16 mm with parallelogram and hexagonal unit-cells are printed, see Fig. 2. Metamaterials are filled with a perimeter raster in which the tool paths are along the beam-like members, i.e., the raster angle is 0°. To facilitate the mechanical testing, two handles have been considered at top and bottom of the metamaterials, as shown in Fig. 2. Two samples as demonstrated in Fig. 2a and c are used for tensile tests of metamaterials with parallelogram unit-cells in axial, x, and transverse, z, directions. They are also printed with a

2.2. Theoretical modeling In this section, a theoretical model is developed to simulate mechanical behaviors of hyper-elastic soft PLA and 3D printed metamaterials. Metamaterial structures are comprised of plane array of unit-cells in (x, z), see Fig. 1a. Every unit-cell is made of some straight beams oriented in different directions. First of all, a generic beam is discretized to some elements. A Ritz-based finite element formulation is developed to derive equilibrium governing equations for the base element in its local coordinate system. Then, they are transformed to the global coordinate system through transformation rules. Global finite element governing equations of the metamaterial are established by assembling elements. Finally, mechanical behaviors of the metamaterial are obtained by solving the governing equations. Consider a generic beam element with length a, width b and thickness h, as depicted in Fig. 4. A local 2-D Cartesian coordinate system (X, Z) is mounted on the beam element aligned at an angle β with respect to the global x-direction. It is considered that a material point on the mid-plane of the beam (Z = 0) moves along X and Z coordinates equal to u and w. Furthermore, it is assumed that the crosssection may undergo a large rotation θ as shown in Fig. 4. The 83

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(a)

(e)

(b)

(c)

(f)

(d)

(g)

(h)

Fig. 2. 3D printed metamaterials for mechanical tests in axial (a, b, e, f) and transverse (c, d, g, h) directions under tension (a, c, e, g) and compression (b, d, f, h).

Fig. 4. Beam element deformation. Fig. 3. Uniaxial tensile response of the 3D printed dog-bone-shape sample.

EXX = U , X + displacement field can be expressed as:

1 ((U , X )2 + (W , X )2) 2

1 1 (U , Z + W , X ) + (U , X U , Z + W , X W , Z ) 2 2

(2a)

U (X , Z ) = u (X ) + Z sin θ (X )

(1a)

EXZ =

W (X , Z ) = w (X ) + Z (cos θ (X ) − 1)

(1b)

where EXX and EXZ indicates normal and transverse shear strains. Also, a comma followed by X or Z denotes partial differentiation with respect to the coordinates. By substituting Eqs. (1a) and (1b) into (2a) and (2b) and neglecting quadratic terms in Z, normal and shear strains can be derived as:

It is worthy to mention that the above displacement field is reduced to the Timoshenko beam theory when the rotation angle is small and sin θ and cos θ can be approximated by θ and 1. The deformation of the generic beam element is described in terms of the non-linear Green-Lagrange strain-displacement expressions. They are valid for finite strains and large rotations. It is formulated as:

EXX = u, X + 84

1 ((u, X )2 + (w , X )2) + 2ZEXZ , θ θ, X 2

(2b)

(3a)

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(4b)

SXZ = 4RT where 1

R (EXX ) = c1 + c2 (1 + 2EXX )− 2 T (EXX , EXZ ) = (1 + 2EXX )−1EXZ

(5)

Eqs. (4a) and (4b) reveal that stress-strain relations are highly nonlinear for hyper-elastic materials. Normal and shear stresses are also coupled in terms of strain components with polynomial non-linearity. Parameters c1 and c2 are considered as material constants that can be calibrated to match uniaxial tension test. The static version of the principle of virtual work known as the principle of minimum total potential energy is implemented to derive governing equations of equilibrium of the beam element and associated boundary conditions. It is formulated as: a

Fig. 5. Comparison between model predictions and experimental data for uniaxial test on the 3D printed dog-bone-shape sample.

h 2

n

∫ ∫ δEXX SXX + 2δEXZ SXZ dZdX = 1b ∑ δui Pi + δui Vi + δθi Mi i=1

0 −h 2

EXZ

1 = ((1 + u, X ) sin θ + w , X cos θ) 2

where P and V are concentrated axial and transverse shear forces while M indicates concentrated bending moment acting on the ith local point of the beam element. The external variables u , w and θ are approximated using the following interpolations:

(3b)

As it can be found, the coefficient of thickness coordinate in Eq. (3a) shows change in rotation of cross-sections (curvature) that is related to derivative of shear strain with respect of X. Adopting Mooney-Rivlin model for isotropic incompressible hyperelastic materials, stress-strain constitutive equations for the beam element can be derived as:

(

3

)

(

1

SXX = 2 1 − (1 + 2EXX )− 2 R − 4T 2 2c1 + 3c2 (1 + 2EXX )− 2

)

(6)

∑ ui ϕi (X ) i=1

m

m

m

u (X ) =

, w (X ) =

∑ wi ϕi (X ) , i=1

θ (X ) =

∑ θi ϕi (X ) i=1

(7)

where ϕi′s denote interpolation functions. In this investigation, quadratic Lagrange interpolation functions are chosen (m = 3) as:

(4a)

Fig. 6. Experimental results of the metamaterial with parallelogram unit-cells under axial and transverse loadings in tension (a) and compression (b).

(a)

(b)

85

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Fig. 7. Experimental results of the metamaterial with hexagonal unit-cells under axial and transverse loadings in tension (a) and compression (b).

(a)

(b)

(

ϕ2 =

4 a

ϕ3 =

1 a

displacement vector of u owing to the geometric non-linearity of the Green-Lagrange strain-displacement. The variation of the strain field can also be expressed as:

)(1 − X ) X (1 − X ) X ( X − 1) 2

1 a

ϕ1 = 1 − a X

1 a

2 a

δEXX = r (X , Z , u) δ u δEXZ = s (X , Z , u) δ u

(8)

The displacement components can be expressed in the discretized form as:

By substituting the discretized displacement and strain fields (9), (12) and (13) into the principle of the minimum total potential energy (6), the governing equations of equilibrium can be derived in the local coordinate system as:

u = {ϕ1 ϕ2 ϕ3 0 0 0 0 0 0} u w = { 0 0 0 ϕ1 ϕ2 ϕ3 0 0 0 } u θ = { 0 0 0 0 0 0 ϕ1 ϕ2 ϕ3 } u

(9)

K (u) = f

where u is called the element nodal displacement vector and defined as:

u = {u1 u2 u3 w1 w2 w3 θ1 θ2 θ3 }

T

(13)

(14)

where a

(10)

K= In the next step, the discretized displacement components (9) need to be substituted in the strain field ((3a) and (3b)). In this respect, trigonometric functions in Eqs. (3a) and (3b) are first expanded by the following Maclaurin expansions.

h 2

∫ ∫ rTSXX + 2sT SXZ dZdX 0 −h 2

f=

1 { P1 P2 P3 V1 V2 V3 M1 M2 M3 }T b

(15a) (15b)

Afterwards, by substituting Eq. (9) into Eqs. (3a) and (3b), the strain components can be rewritten in terms of nodal variable vector, u, as:

in which K denotes the non-linear elastic stiffness vector while f is force vector. Eq. (14) is a highly non-linear algebraic equation in terms of nodal variables. Next, the elemental governing equations (14) should be expressed in the global coordinate system (x, z). Using the transformation rule, dis and placement and force vectors in global coordinate system, called u f ,̂ can be related to their counterparts in the local coordinate system as:

EXX = r (X , Z , u) u EXZ = s (X , Z , u) u

 u = Qu f = Qf ̂

p

sin(θ) = ∑ i=0 p

cos(θ) = ∑ i=0

(−1)i θ (2i + 1) factorial (2i + 1) (−1)i θ 2i factorial (2i)

(11)

(12)

It should be mentioned that r and s have dependency on the nodal

where 86

(16)

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Fig. 8. Model predictions compared to experimental data for the metamaterial with parallelogram unit-cells under axial (a, c) and transverse (b, d) loadings in tension (a, b) and compression (b, c).

(a)

(b)

(c)

(d)

⎡ cos(β ) I sin(β ) I 0 ⎤ ⎡1 0 0⎤ ⎡0 0 0⎤ Q = ⎢− sin(β ) I cos(β ) I 0 ⎥ , I = ⎢ 0 1 0 ⎥ , 0 = ⎢ 0 0 0 ⎥ ⎢ ⎥ ⎣0 0 1⎦ ⎣0 0 0⎦ 0 0 I⎦ ⎣

Newton-Raphson [25] is implemented tracing non-linear equilibrium paths. It is worthy to mention that, due to periodic nature of the current metamaterials, the present problem could be modeled by considering an appropriate sized RVE and applying periodic boundary conditions in the FE analysis (for example, see Ref. [19]). However, it is beyond the focus of the present work and possible REV-based FE analysis could be considered in future development efforts.

(17)

The matrix Q is known as transformation matrix. By substituting Eq. (16) into Eq. (14), the governing equations of equilibrium are obtained in global coordinate system as:

 (u ) = f ̂ K

(18)

in which

 = QT K K

3. Results and discussions

(19)

is called transformed stiffness vector. Finally, Eq. (18) is employed to construct global finite element equilibrium equations of the metamaterial composed of all beams oriented in various directions by assembling scheme to result in: ∼∼ K (u) = f͠ (20)

3.1. Material behavior First, in order to verify the accuracy of the Mooney-Rivlin constitutive model (4) implemented for analyzing soft PLA materials, uniaxial response of the 3D printed dog-bone-shape sample presented in Fig. 3 is replicated. Experimental data and model prediction for a tension-compression test are illustrated in Fig. 5. In addition, predictions based on the linear elastic model (SXX = 3GEXX) with small and large

In order to solve the algebraic equation with both geometric and material non-linearities, an iterative incremental method such as 87

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Fig. 9. Model predictions compared to experimental data for the metamaterial with hexagonal unit-cells under axial (a, c) and transverse (b, d) loadings in tension (a, b) and compression (b, c).

(a)

(b)

(c)

(d)

elastic metamaterials are studied experimentally and numerically. First, metamaterials with parallelogram and hexagonal unit-cells are experimentally tested in axial and transverse directions, i.e. x and z directions, under tension and compression. They are fixed through two handles printed at top and bottom of the lattices. For simulation, boundary conditions at top and bottom of the metamaterial are assumed to be fully clamped in both x and z directions whereas left and right sides are considered to be free to move horizontally and vertically. Mechanical responses of the metamaterial lattice with parallelogram unit-cells in axial and transverse directions are presented in Fig. 6a and b respectively for tension and compression tests. The counterpart of this figure is illustrated in Fig. 7 for the metamaterial lattice with hexagonal unitcells. These experimental results are also employed to validate the accuracy of the present structural-material model and developed FEM solution. Comparison between model predictions and experimental data is displayed in Figs. 8 and 9 for the metamaterials with parallelogram and hexagonal unit-cells, respectively. Finally, configurations of the metamaterials at various strain-step associated with Figs. 8 and 9 are illustrated in Figs. 10 and 11, respectively. The primary conclusion from Figs. 6 and 7 is the fact that the

strain assumptions by considering non-linear Green-Lagrange strain (2a) are also included in Fig. 5. Material parameters of c1 and c2 in hyper-elastic model are calibrated as −1 and 6 MPa to fit experimental data. Furthermore, shear modulus is obtained as G = 2(c1 + c2) = 10 MPa, which is the constant slope of the stress-strain curve in the small strain regime. As it can be found from Fig. 5, both linear and hyper-elastic theories result in the same linear response in small strain range until 3%, which is in a good correlation with the experiment. Beyond this strain, the present hyper-elastic model replicates softeningtype mechanical response of the 3D printed soft PLA very well. However, it is found that, while small-strain linear elastic model predicts an overestimated constant material stiffness, assuming non-linear GreenLagrange strain leads to a hardening behavior that is in contrast with softening response of the soft PLA. This emphasizes that modeling of hyper-elasticity is essential to accurately predict mechanical responses of the 3D printed soft PLAs. 3.2. Metamaterial behavior In this section, the mechanical responses of the 3D printed hyper88

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Fig. 10. Configurations of metamaterial with parallelogram unit-cells under axial (a, c) and transverse (b, d) loadings in tension (a, b) and compression (b, c) at different strain-steps. (black, blue, green, purple and red colors mean 0, 3, 9, 18 and 30% strains). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(a)

(b)

(c)

(d)

Next, experimental and numerical results presented in Figs. 6–11 are discussed in details. The results in Figs. 6 and 7 show that each metamaterial has similar responses in both x and z directions within the small strain regime. For instance, the metamaterial with parallelogram unit-cells has a similar response in both axial and transverse directions under tension and compression till strains of 2.5 and 0.6%, respectively. These strains become larger, 3.5 and 1.5%, for metamaterial with hexagonal unit-cells. Beyond these strains, metamaterial response becomes different in axial and transverse directions revealing anisotropy. In this respect, differences between axial and transverse responses of the metamaterial with hexagonal unit-cells are more obvious than those of parallelogram one. It means anisotropy of the parallelogram unit-cell is more/less than hexagonal one in small/large strain ranges due to geometric properties of the unit-cell. As it can be seen in Fig. 6a, the transverse stress-strain response is separated from the axial response at 2.5% strain and its slope is decreased. It can be related to the buckling that occurs in horizontal beams due to contraction in the axial x-

mechanical behaviors of metamaterials with parallelogram and hexagonal unit-cells are very different. Furthermore, it can be found that the mechanical responses are totally different not only in axial and transverse directions but also under tension and compression. Metamaterial response trend is also changed during mechanical loading due to material and geometrical non-linearities. All these observations imply that the response of metamaterial is anisotropic and has a strong dependency on the unit-cell shape, direction, type and magnitude of mechanical loadings. Most of 3D printers fill the interior of an object by infill patterns in parallelogram and hexagonal shapes like metamaterial structures under investigation. Therefore, the present results can provide useful insights into design of metamaterial-like infill patterns for reinforcing 3D printed soft objects and providing more stiffness (c.f. Ref. [19]). Regarding comparative studies, the results presented in Figs. 8 and 9 reveal that the structural-material model and solution methodology are capable of well replicating the experiments, and can serve as an accurate computational procedure for digital design. 89

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Fig. 11. Configurations of metamaterial with hexagonal unit-cells under axial (a, c) and transverse (b, d) loadings in tension (a, b) and compression (b, c) at different strain-steps. (black, blue, green, purple and red colors mean 0, 3, 9, 18 and 30% strains). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(a)

(b)

(c)

(d)

cell shape has a significant effect on the local and global buckling behaviors of the metamaterial. As it can be seen, the metamaterial experiences a linear stress-strain response in the pre-buckling state while a non-linear equilibrium path exists in the post-buckling regime. It is found that the metamaterial has a higher critical buckling load in transverse direction than through x axis. This is due to the fact that local buckling of straight beams in the x-direction decreases the axial buckling strength of the metamaterial, see Fig. 10c and d. After global buckling, the metamaterial has different behaviors in axial and transverse directions. The metamaterial experiences a snapping global response in the transverse direction while the response is somehow hardening in the axial direction. This can be associated with local postbuckling behavior of beam members in the unit-cell through axial and transverse directions. In this respect, structural-material model overestimates axial hardening response while it well replicates snapping phenomenon in the axial tensile loading, see Fig. 8c and d. Fig. 6b also shows that, in the deep post-buckling regime, both axial and transverse

direction, see Fig. 10b. It implies that metamaterial with parallelogram unit-cells experience local buckling during tension. In the moderate strain regime, 2.5–16.5%, as it can be seen in Fig. 6a, metamaterial with parallelogram unit-cells is stronger in axial direction than transverse one. It is because a parallelogram unit-cell has two straight beams, two 60°-beams and one thick 60°-beam in axial loading direction while it has two 30°-beams and one thick 30°-beam in transvers loading direction. The more equivalent straight members in the loading direction, the more the stiffness in that direction. By further loading, the buckled beams are deformed more and two 30°-beams and one thick 30°-beam in transvers loading direction get more freedom to become straight along the loading direction, see Fig. 10b. As it can be seen in Fig. 6a, beyond 16.5% strain, the structure shows more transverse strength compared to axial one. It means that, in the large strain domain, the buckled unit-cell becomes stiffer and the metamaterial shows a larger transverse stiffness. In conjunction with the compression test, Fig. 6b reveals that unit90

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and infill patterns for printing purpose.

stress-strain responses approach to each other and tend to coincide. This behavior is because all beam members have been deformed deeply and the deformed metamaterial tends to behave isotropically in axial and transverse directions, see Fig. 10c and d. Finally, it should be mentioned that by compressing the metamaterial more, beam members meet each other. That is why compressive response has been investigated till 18% strain. Regarding mechanical responses of the metamaterial with hexagonal unit-cells, Fig. 7a reveals that, after an isotropic response in the small strain regime, the tensile strength of the structure in axial direction is higher than its transverse one within moderate and large strain ranges. It should be mentioned that a hexagonal unit-cell has four 30°-beams in axial loading direction while it has two straight beams and four 60°-beams in transvers loading direction. As it can be seen, in the moderate strain, 3.5–20%, the metamaterial experiences a hardeningtype response in both directions. However, beyond 20% strain, the axial response softens while the transverse one keeps hardening. This trend change can be associated to metamaterial configuration under deep axial deformation. As it can be seen in Fig. 11a and b, almost all oblique beams become straight in large strain regime and metamaterial gets a uniform distribution of square-shape unit-cells and behaves like a hyper-elastic material point, see Fig. 3. Focusing on results in Figs. 6b and 7b reveals that metamaterial with hexagonal unit-cells has a very low compressive strength, around one-tenth, compared with metamaterial with parallelogram unit-cells. It is due to lower structural density of the hexagonal unit-cell. It is also found that metamaterial with parallelogram unit-cells in both directions and metamaterial with hexagonal unit-cells in transverse direction have linear and non-linear compressive responses in pre- and postbuckling regimes. However, metamaterial with hexagonal unit-cells results in a non-linear axial response in all compressive load steps. This behavior can be associated to configuration of hexagonal unit-cells during compressive axial loading as it doesn't undergo any obvious local buckling, see Fig. 11c. Comparing results in Fig. 7b, it is found that the structure has a higher compressive transverse strength than the axial one till 11% strain. This is directly related to straight beams in transverse loading direction that can undergo larger compressive loads. However, beyond 9% strain, as it can be seen in Fig. 11d, straight beams rotate and the metamaterial loses its transvers strength experiencing a plateau softening that is lower than the axial stress-strain response.

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4. Conclusion The in-plane mechanical behaviors of 3D printed hyper-elastic metamaterials under both tension and compression in the large-deformation regime were studied in this paper. Metamaterials comprising a repeating arrangement of parallelogram and hexagonal unit-cells were designed and fabricated by FDM 3D printing technique from soft hyperelastic PLAs. They were tested under tension and compression in axial and transverse directions. To simulate experimental results, an FE formulation on the basis of the Mooney-Rivlin constitutive equations and non-linear Green-Lagrange strains was derived. The governing equations with material and geometric non-linearities were then solved using iterative incremental Newton-Raphson method to track nonlinear equilibrium path. The accuracy of the developed formulation and solution methodology was confirmed by replicating experimental results. It was also shown that hyper-elastic behavior and finite strain needed to be considered for an accurate prediction of soft metamaterial responses in the large deformations. Experimental and numerical results revealed that soft metamaterials have a strong anisotropy and unit-cell shape, direction, type and magnitude of loading influence their responses and instability characteristics. They can serve as benchmark for future work dealing with the design of hyper-elastic metamaterials

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