Adaptive metamaterials by functionally graded 4D printing

Materials and Design 135 (2017) 26–36

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Adaptive metamaterials by functionally graded 4D printing M. Bodaghi, A.R. Damanpack, W.H. Liao ⁎ Smart Materials and Structures Laboratory, Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China

H I G H L I G H T S

G R A P H I C A L

A B S T R A C T

• Fabricating adaptive metamaterials by functionally graded 4D printing technology • Building performance-driven functionality directly into SMPs by FDM • Designing adaptive metamaterials by self-folding/coiling capabilities • Characterizing self-folding in terms of printing-speed and liquefier-temperature • Developing FEM simulations with material and geometric non-linearities

a r t i c l e

i n f o

Article history: Received 20 July 2017 Received in revised form 26 August 2017 Accepted 30 August 2017 Available online 1 September 2017 Keywords: Material programming 4D printing SMPs Self-folding/coiling Experimental demonstration FE analysis

a b s t r a c t This paper shows how fused decomposition modeling (FDM) as a three-dimensional (3D) printing technology can engineer adaptive metamaterials with performance-driven functionality built directly into materials. The tactic is based on an understanding of thermo-mechanics of shape memory polymers (SMP) and fabrication concept behind FDM as well as experiments to explore how FDM can program self-foldable metamaterials. Self-folding mechanism is investigated in terms of fabrication parameters like printing-speed and liquefier-temperature that affect layer-by-layer programming process and shape-change. It can be called a functionally graded 4D printing so that the structure is fabricated additively and programmed functionally. A finite element (FE) formulation based on the non-linear Green-Lagrange kinematic relations coupled with a robust SMP constitutive model is established to describe material tailoring in fabrication stage and deformation. Governing equations with material-geometric non-linearities are solved by implementing iterative Newton-Raphson method to trace large-deformation non-linear equilibrium path. FDM and FE solution are then applied to digitally design and fabricate straight/curved beams as structural primitives for adaptive metamaterials that show 1D/2D-to-2D/3D shapeshifting by self-folding or/and self-coiling. Finally, it is experimentally shown that the 4D printed metamaterials have great potential in mechanical/biomedical applications like structural/dynamical switches, self-conforming substrates, self-tightening surgical sutures, self-conforming splints and self-coiling/deploying stents. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Additive manufacturing (AM), also known as three-dimensional (3D) printing, has been identified as the second industrial revolution ⁎ Corresponding author. E-mail address: [email protected] (W.H. Liao).

http://dx.doi.org/10.1016/j.matdes.2017.08.069 0264-1275/© 2017 Elsevier Ltd. All rights reserved.

of the epoch [1]. It has drawn considerable attention in scientific/public community and even allowed anyone to be a designer. 3D printers can fabricate complex and intricate components with significant reduction in manufacturing time/cost and material wastage. AM techniques available commercially encompass PolyJet or stereolithography (STL) of a photo-polymer liquid, material extrusion according to ISO/ASTM 52900 [2] or Stratasys fused deposition modeling (FDM) of polymeric

M. Bodaghi et al. / Materials and Design 135 (2017) 26–36

polymers by inducing a compressive strain during photo-polymerization in PolyJet process. Although it would be straightforward to program 3D printed objects/metamaterials for simple shape change and/or function, it might be a significant challenge for complicated changes/functions. Furthermore, it is difficult to perform manipulations needed to program materials manually in certain situations on large or small scales or in remote locations such as in space or inside human-body. This paper aims at demonstrating an approach to design and manufacture adaptive metamaterials enabled by functionally graded (FG) 4D printing technology without application of any programming process and external manipulation. The approach is based on an understanding of thermo-mechanics of SMPs and FDM printing technique. Five parameters as material/surface of platform, its temperature, printing speed, liquefier temperature and delay time for printing each layer are assumed as effective parameters that may influence material tailoring. Experiments are conducted to explore self-folding features in terms of relevant fabrication parameters as well as theoretical framework to describe phenomena/mechanisms. It is experimentally shown that FDM printing can program SMPs during fabrication in a graded manner for desire functions. An FE formulation on the basis of the non-linear Green-Lagrange strains coupled with a robust SMP constitutive model is developed and solved implementing iterative incremental NewtonRaphson technique. It is then applied to explore mechanisms behind FG 4D printing and structural shape-change during activation. The printing method and FE solution are then employed to digitally design and fabricate self-foldable straight and curved beams as structural primitives for adaptive metamaterials. The feasibility of 1D/2D-to-2D/3D shape-shifting by self-folding or/and self-coiling features is shown, experimentally and numerically. Finally, some experiments are conducted to demonstrate the potential applications of adaptive metamaterials in mechanical and biomedical engineering devices like structural/dynamical switches, conformal substrates, surgical sutures, stents and splints. 2. Concepts and models The functionally graded 4D printing concept is developed here based on an understanding of the FDM printing technology and shape memory effects. The approach also includes experiments to show its feasibility as well as a mathematical modeling to describe the mechanism behind it. 2.1. FG 4D printing concept The main idea is combination of FDM printing method with SMPs to create potentials in the design and development of adaptive metamaterials. SMPs have capacity of retaining a temporary shape and recovering to original shape when subjected to environmental stimulus like heat.

III IV

Strain

filaments and selective laser sintering (SLS) from metallic or polymeric powders. Metamaterials as a class of multi-scale structures exhibit thermomechanical properties that are not found in the nature. Their unusual characteristics arise from their structure and geometry rather than the matter of which they are composed [3]. In recent years, 3D printing technology has demonstrated great potential in fabricating metamaterials. For instance, employing Object350 Stratasys 3D printer based on the PolyJet technology, Wang et al. [4] introduced dual-material auxetic metamaterials composed of elastic joints and stiff walls without instability during deformation. Computational results from COMSOL finite element (FE) software and mechanical testing indicated that the dualmaterial auxetic metamaterial has distinctly different auxeticity and mechanical properties form traditional single-material auxetic metamaterials. In another research work [5], they fabricated metamaterials with micro-structured reinforcement embedded in soft polymeric matrix. Simulations conducted by ANSYS FE code and uniaxial tension tests revealed that dual-material designs exhibit strain-stiffening features. Using SLS technology for printing Nylon-based rubber-like materials, Rafsanjani et al. [6] proposed 2D metamaterials that exhibit multistability in response to tension. The results from ABAQUS FE and mechanical tests revealed that multi-stable responses were triggered by snap-through buckling. Mousanezhad et al. [7] investigated 2D auxetic metamaterials with structural hierarchy fabricated by Eden260V Stratasys 3D printer based on PolyJet technique. Simulations from ABAQUS FE code and experiments showed that Poisson's ratio was deceased by hierarchy-dependent elastic buckling introduced at early steps of compression. Che et al. [8] proposed methods of small geometric variations to tune the deformation sequence of 2D metamaterial with multiple stable states fabricated by PolyJet Object260 3D printer. There was a good agreement between experimental data and those from analytical solutions and ABAQUS FE code. Naddeo et al. [9] presented experimental validation of an algorithmic procedure, running in ANSYS FE code, able to replace continuous mass of a 3D convex solid with a cancellous bone-inspired space frame organized for having fibers oriented according to boundary conditions. A cubic unit cell was designed and sized to analyze mechanical performances of the new porous material. The numerical results obtained from a unit-cell-based analysis were compared with experimental tensile tests on metallic 3D printed cubic specimens confirming validity of the beam elementbased metamaterial. Recently, Bodaghi et al. [10] studied hyper-elastic behaviors of soft metamaterials fabricated by FDM 3D printing technology in the large-strain regime experimentally and numerically. Results revealed that unit-cell shape, direction, type and magnitude of loading had significant influences on anisotropic response of metamaterial and its instability. The 3D printed metamaterials reviewed above all are static and inanimate. Recently, some researchers have employed 3D printing combined with smart materials to offer an innovative method for crafting active structures [11,12] and metamaterials [12–15]. It is called 4D printing where a capacity for movement is latent within the 3D printed structure. 4D printed structures are able to actively transform configurations and evolve over time in response to surrounding stimuli. Among active materials, shape memory polymers (SMPs) [16–19] and hydrophilic polymers have been used because of their capability to produce shape change subjected to an external stimulus such as heat and humidity fields. For instance, Mao et al. [13] introduced a 2D metamaterial lattice with shape-changing feature via printing SMPs and hydrogels by PolyJet Objet260. The 4D printed structure expanded by immersing in water with low and high temperatures, cooling down, drying and then shrinked upon putting in high-temperature water. Bodaghi et al. [14] proposed a self-expanding/shrinking 2D metamaterial lattice by printing fibers of SMPs into flexible matrix using PolyJet Object500 3D printer. The metamaterials needed to be programmed through a heatingloading-cooling-unloading process after fabrication. Recently, Ding et al. [15] printed self-foldable metamaterials consisting of stiff and soft

27

II V

I

Temperature Fig. 1. Schematic of the shape memory effect in the strain-temperature plane.

28

M. Bodaghi et al. / Materials and Design 135 (2017) 26–36

Fig. 1 displays a standard step-by-step thermo-mechanical protocol of programming for SMPs in the framework of a strain-temperature phase diagram. It starts at a strain/stress-free state at a low temperature less than transition temperature, T = Tl b Tg, where SMP is stable in a glassy phase. It is then heated above the transition temperature range, T = Th N Tg (step I). Next, SMP stable at rubbery phase is loaded mechanically to achieve a maximum strain, εm (step II). The strain is kept fixed and the material is cooled down to the low temperature, Tl, (step III). In this step, the rubbery phase gradually converts to the glassy phase memorizing inelastic strains. SMP is finally unloaded while a pre-strain remains in the material, εp (step IV). For the sake of activation, SMP is heated to the high temperature, Th, that relaxes strain and recovers the permanent shape (step V). This is known as a free-strain recovery. Now the scope is to implement the above programming protocol during printing process. Among various standard 3D printing technologies, FDM technology applies a similar thermo-mechanical procedure on the material during fabrication. Fig. 2 displays a schematic of the base of FDM procedure. The printer performs hot extrusion of a polymeric filament through a circular nozzle. The material is heated inside liquefier in temperature Tln that is above its Tg and then deposited onto the platform by the printer head moving at speed Sp. In fact, the material with high temperature is stretched that is similar to the heating-loading process described above (steps I and II). After deposition, the printed layer cools and solidifies and bonds with the platform that is again similar to the step III in Fig. 1. Once a layer is completed, the build platform advances downward and printing head proceeds to deposit the next layer. Finally, the thermomechanical programming protocol is completed by mechanical unloading through removing the printed object from the platform, like step IV. Therefore, it can be found that FDM has potential to fabricate 3D SMP architectures and impose inelastic pre-strain that may drive shape memory effects in a prescribed way. In this respect, some printing parameters are supposed to be effective on SMP programming process. For instance, printing speed may have a key role in determining prestrain value. It would reasonable that more speed provides more mechanical loading that may induce more pre-strain, see Fig. 1, step II. While all layers are printed with the same printing speed, their surface experiences different thermal/connection conditions. For example, while the lower layer, as the first printing layer, is deposited on the stiff polymeric printer bed, other layers above it are laid on the already printed SMP layers, see Fig. 2. Thus, platform material/surface conditions may be important and affect bonding and fixed strain conditions in the first layer during cooling and solidifying. Regarding thermal surface conditions, temperature of build platform may also affect bonding conditions and pre-strain value in the first layer deposited on it. Furthermore, the first layer and layers above it, except the end layer, are

Liquefier Printing direction

Extrusion nozzle Upper layer (end layer)

heated by the hot extruded material when the machine is depositing the next layer above them, see Fig. 2. This extra heat definitely affects the pre-strain value and reduces it. Therefore, the nozzle temperature takes account into a significant parameter affecting the pre-strain. Moreover, the temperature of the layer that is being heated is another parameter. This temperature is directly related to the delay time between printing each layer. On the other hand, the upper layer as the end layer never gets any heat since the nozzle leaves it at the end of printing stage so that its pre-strain might be the highest one. Finally, it can be concluded that printed layers may be programmed in different manners and get various values of pre-strain. Also, five parameters as material/surface of platform, its temperature, printing speed, nozzle temperature and delay time for printing each layer can be considered as effective parameters that may influence pre-strain values in printed layers. From literature terminology, this printing process can be called functionally graded 4D printing. The 4D printing process could program SMPs during fabrication inducing pre-strain. On the other hand, this programming could be done in functionally graded manner. While inelastic material property like pre-stain may change layer-by-layer gradually, there are five parameters to control and program the object for desire functions. In this research, FDM printing technology is implemented to fabricate adaptive metamaterials with performance-driven functionality built directly into the materials. 2.2. Materials and printing In this study, polyurethane-based SMP filaments with a diameter of 1.75 mm and glass transition temperature of 60°C is employed (SMP Technologies Inc., Tokyo, Japan). All SMP samples are manufactured by a New Creator Pro desktop 3D printer developed by FlashForge. New Creator Pro is a low cost desktop printer that extrudes filaments with a 0.4 mm nozzle size. It can be controlled with any open-source software. In this work, Craft-Ware software is utilized to produce G-code files from STL files and to command and control the procedure parameters of liquefier temperature and printing speed. As default, liquefier temperature is set as 230°C while temperatures of build platform and chamber remain as room temperature, 24°C. Also, the printing speed and layer height are set as 20 mm/s and 0.2 mm, respectively. Thermo-mechanical properties of the printed SMPs are identified by a dynamic-mechanical analyzer (DMA, NETZSCH, Model 242). To this end, a beam sample is printed with the dimension of 15 mm (length) × 1.6 mm (width) × 1 mm (thickness). DMA test is carried out in an axial tensile way with frequency of force oscillation 1 Hz and heating rate 5°C/min ranging from −20 to 90°C. Applied dynamic stress to static stress is around 1.5. The temperature-dependent results in terms of storage modulus, ES, and tan (δ) are presented in Fig. 3. As it can be found, the storage modulus in glassy phase, e.g. in Tl =20°C, is around ESg =1656MPa while it reduces drastically to 3.18MPa at rubbery phase, e.g. in Th =90°C. A high ratio of storage modulus downfall equal to 520 is observed. Using the formula qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for Young's modulus defined as E ¼ ES 1 þ ð tanðδÞÞ2 , Eg = 1657 MPa

Molten material Re-heated zone

Middle layers

Solidified material

Lower layer (first layer)

Build platform Fig. 2. FDM printing process.

Fig. 3. DMA measurement results for the 3D printed SMP in terms of tan (δ) and storage modulus.

M. Bodaghi et al. / Materials and Design 135 (2017) 26–36

and Er =3.26 MPa can be calculated from the DMA test. The prominent peak in the graph of tan(δ) associated with the glass transition temperature is read as Tg =60°C. Henceforward, sub-scripts ‘g’ and ‘r’ signify the glassy and rubbery phases, respectively. Next, Young's modulus (E), Poisson's ratio (ν) and thermal expansion coefficient (α) of the 3D printed SMPs in glassy and rubbery phases are specified. To this end, a sample is printed based on the geometry and dimensions characterized by the ASTM D638 standard [20]. For tensile tests at low and high temperatures, a Tinius Olsen® H5kS machine (Tinius Olsen, Horsham, PA, USA) with a 5 kN load-cell and a homemade thermal chamber is utilized. Material properties extracted from thermo-mechanical tests are presented in Table 1. The results show that there is an excellent agreement between measured Young's modulus from DMA and tensile tests. Also, it is observed that, while Poisson's ratio has an increasing trend during phase transformation from glassy to rubbery, thermal expansion coefficient does not change so that it is the same at both phases. 2.3. Feasibility demonstration In this section, feasibility of fabricating adaptive SMP metamaterials by FG 4D printing is demonstrated experimentally. Metamaterials are designer matter with exotic thermo-mechanical characteristics generally adjusted by their architecture rather than their chemical compositions. They comprise an arrangement of mechanical elements with tunable behaviors. SMP straight and curved beams as structural primitives for metamaterials are fabricated by FDM printing method. Geometry and dimensions of a typical printed curved beam with L mean length, b width, h thickness, R mean radius and opening angle of θ are shown in Fig. 4. A Cartesian coordinate system (xi, i = 1, 2, 3) is located on the mid-plane of the curved beam. Among five parameters that may affect pre-strain regime in the 4D printed objects, the effects of the printing speed, Sp, and nozzle temperature are examined here. It should be noted that, the nozzle temperature is unknown while temperature of liquefier, Tln, is known and controllable by the Craft-Ware software. Therefore, the effect of nozzle temperature is investigated through temperature of liquefier. Four straight 1D beams with L = 15, b = 1.6, h= 1 mm , θ ≅ 0 are fabricated by printing parameters of Sp = 30 , 40 mm/s and Tln = 210 , 230°C as shown in Fig. 5a. The print raster is assumed to be along the length direction, x2 direction. In order to activate the printed objects, they are heated by immersing into hot water with a prescribed temperature of 90°C that is 30°C more than Tg = 60°C. All the following samples are also activated by the same method and with the same temperature. The configurations of the elements after hating-cooling process are presented in Fig. 5b. Their geometric parameters, L′ , θ′, measured in the x2 − x3 plane are also included in Fig. 5b. The preliminary conclusion drawn from Fig. 5 is this primitive is able to change its configuration over time in the presence of hot water as a heating source. It means they are already programmed and pre-stained during fabrication and are able to react to changes in the environment to evoke the shape-memory effect. When the primitive printed in the x1 − x2 plane is exposed to high temperature, it shrinks and bends to form a curved beam in the x2 − x3 plane and remains folded permanently, see Fig. 5b. It can be considered as a self-folding 1D-to-2D process that may enable pre-designed planar templates to transform into 3D structures. This self-bending observed is due to an unbalanced prestrain induced during fabrication and deposited along x2 direction. In

Fig. 4. Illustration of the 4D printed primitive.

fact, unbalancing in pre-strain distributed through the thickness leads a mismatch in free-strain recovery inducing curvatures. It can also be concluded that pre-strain may have an increasing trend through the thickness upward from the lower to the upper layer that may enable the overall shape to be changed toward the upper layer. Fig. 5 reveals that printing speed and liquefier temperature have significant effects on pre-stain regime through the thickness direction. It is found that any increase in the liquefier temperature has a negative effect on the pre-strain. This is in accordance with what has been explained in Section 2.2. The hotter the liquefier, the less the pre-strain. It is also seen that enhancing printing speed increases the pre-stain value and consequently bending deformation. It is reasonable since more speed provides more mechanical loading that induces more pre-strain, c.f. Fig. 1. Therefore, it can be called a functionally graded 4D printing so that the structure is fabricated additively and programmed functionally. Next, it is needed to explore the pre-strain regime in the 4D printed primitives for enhancing design capabilities. 2.4. Theoretical modeling Adaptive structures like metamaterials fabricated by FG 4D printing method can be designed based on the theoretical understanding of the programming mechanism and thermo-mechanical behaviors of the printed elements. This section is dedicated to develop constitutive modeling and mathematical formulation to describe thermo-mechanical mechanisms behind programming in fabrication stage and deformation during activation. 2.4.1. SMP phase transformation A robust phenomenological constitutive model basically presented in Refs [21,22]. is reformulated here to estimate SMP phase transformations during thermo-mechanical loadings in a straightforward manner. SMP materials show a combination of glassy and rubbery features. The volume fractions of the glassy and rubbery phases are defined as: ηg ¼

Vg Vr ;η ¼ V r V

Parameter

Glassy phase

Rubbery phase

E (MPa) ν α(10−4K−1)

1660 0.35 1

3.3 0.4 1

ð1Þ

where V represents the total volume of the SMP while Vg and Vr are volumes of the glassy and rubbery phases, respectively. Since the summation of these volume fractions should be unity, i.e. ηg + ηr = 1, the rubbery volume fraction can be expressed in terms of glassy one as: ηr ¼ 1−ηg

Table 1 Thermo-mechanical material properties of the 3D printed SMP.

29

ð2Þ

that means there is just one independent volume fraction. As a generally well-known assumption, ηg and Vg are considered to be dependent only on temperature variable, T: ηg ¼ ηg ðT Þ; V g ¼ V g ðT Þ

ð3Þ

30

M. Bodaghi et al. / Materials and Design 135 (2017) 26–36

Fig. 5. Exprimental illustration on the concept of the functionally graded 4D printing: (a) straight beam printed with different printing speed and liquefier temperature, (b) configuration of the printed beam after heating-cooling process.

The volume fraction of the glassy phase can be determined according to the nature of SMP phase transformation fitting experimental DMA data. Considering the results presented in Fig. 3, an explicit function in terms of trigonometric functions is introduced to interpolate storage modulus from DMA results as: ηg ¼ −

    tanh a1 T g −a2 T − tanh a1 T g −a2 T h     tanh a1 T g −a2 T h − tanh a1 T g −a2 T l

ð4Þ

ε_ in ¼ η_ g εr

in which ai (i = 1, 2) are selected to match DMA curve. SMP phase transformation occurs by nucleation and growth of platelet inclusions, mainly directed in the stress direction [21]. Thus, it is considered that glassy and rubbery phases are connected to each other in series. By considering the fact that the 4D printed objects may experience small strains and moderately large rotations as observed in Fig. 5, a general assumption of additivity of strains is adopted in the form   ε ¼ ηg εg þ 1−ηg εr þ εin þ εth

ð5Þ

where ε denotes total strain vector signifying total deformation of the material; εg and εr represent elastic strains in the glassy and rubbery phases, respectively; εin is associated to inelastic strain due to phase transformations; and εth is called thermal strain and defined as: εth ¼

Z T    α r þ α g −α r ηg ðT Þ dT T0

in which α is thermal expansion vector and the expression inside the integral follows the rule of mixture. T0 is also reference temperature. Next, εin that is an unknown quantity is described based on glassyrubbery phase transformation mechanisms. During cooling, the material is frozen and the rubbery phase changes gradually to the glassy phase. In this respect, the strain is assumed to be stocked in the SMP in proportion to the transformed rubbery phase strain. It is formulated as:

ð6Þ

ð7Þ

where a superposed dot denotes rating function. Regarding the heating, the strain stored is assumed to be released gradually in proportion to the volume fraction of the glassy phase with respect to the previous glassy phase and stored strain. It is stated as: ε_ in ¼

η_ g ηg

εin

ð8Þ

and implies that the strain release is independent of the stress/strain state during heating. In order to derive stress state, the second law of thermo-dynamics in the sense of Clausius-Duhem inequality should be fulfilled. By considering ε and T as external control variables, and εg ,εr , εin and ηg as internal variables and adopting Helmholtz free energy density functions and following standard arguments, stress quantity, σ, can be derived as: σ ¼ σg ¼ σr

ð9Þ

M. Bodaghi et al. / Materials and Design 135 (2017) 26–36

31

Heating:

in which σg ¼ Cg εg ; σr ¼ Cr εr

ð10Þ

εin ¼

ηg ηkg

εkin

ð17Þ

and C represents elasticity stiffness matrix defined as: 2

1−ν 6 ν 6 6 ν 6 6 0 E 6 C¼ ð1 þ νÞð1−2ν Þ 6 6 6 0 6 4 0

ν 1−ν ν

ν ν 1−ν

0

0

0

0

0

0

0 0 0 ð1−2ν Þ 2

0 0 0

0 0 0

0

0

0

ð1−2νÞ 2

0

0

0

3 7 7 7 7 7 7 7 7 7 7 5

ð1−2νÞ 2 ð11Þ

Eq. (9) is consistent with the fact that stress evolutions of each phase are equal in the series model. Next, in order to derive constitutive stress-strain relationship for SMP materials consists of glassy and rubbery phases, strains from Eq. (10) are substituted into Eq. (5) that yields: σ ¼ Ce ðε−εin −εth Þ

ð12Þ

in which   −1 Ce ¼ Sr þ ηg Sg −Sr

ð13Þ

is called an equivalent stiffness matrix and looks like Reuss formula in the general theory of homogenization. Also, S denotes the compliance matrix defined as C−1. From a computational point of view, the non-linear SMP behavior during heating/cooling can be treated as an explicit time-discrete stress/strain-temperature-driven problem. The time interval [0, t] is partitioned into sub-increments and the discretized problem is solved over a general interval [tk, tk + 1]. For notation simplicity, the variables with superscript k are associated to the preceding time-step whereas the ones without superscript are referred to the current step k + 1. It is assumed that control variables ε (or σ), T and ηg at time t as well as the solution at time tk are known. The evolution Eqs. (7) and (8) for the inelastic strain during cooling/heating can be discretized by implicit backward-Euler integration scheme as: εin ¼ εkin þ Δηg εr εin ¼ εkin þ

Δηg ηg

ð14aÞ

εin

ð14bÞ

Δηg ¼ ηg −ηkg

ð15Þ

By using Eqs. (10) and (12) to substitute εr and σ, and preforming some mathematical simplifications, the inelastic strain (Eqs. (14a) and (14b)) can be updated explicitly for cooling and heating processes in stress and strain control manners as: Cooling: εin ¼

  σ ¼ Cu ε−β εkin −εth

þ Δηg Sr σ

for stress control mode

 −1   εin ¼ I þ Δηg Sr Ce εkin þ Δηg Sr Ce ðε−εth Þ

ð16aÞ

for strain control mode

ð16bÞ

ð18Þ

where Cu and β called unified parameters need to be set for heating and cooling as: 8  −1 > > Ce ; < Cu ¼ I þ Δηg Sr Ce ηg > > : Cu ¼ Ce ; β ¼ ηk g

_ β ¼ 1 Tb0

ð19Þ

_ TN0

Eq. (18) reveals that the developed constitutive model for SMPs looks like a linear thermo-elastic model with non-linear material properties and having inelastic strains known from the preceding time step. 2.4.2. Equilibrium equations The deformation of the 4D printed objects are expressed in terms of the Green-Lagrange strain vector accounted for small strains and moderately large rotations. The non-linear strain-displacement kinematic relations can be expressed in a matrix operator form as: ε¼

  1 DL þ DN u 2

ð20Þ

where u indicates generalized displacement vector defined as: 8 9 < u1 = u ¼ u2 : ; u3

ð21Þ

and ui (i = 1, 2, 3) represents displacement components of a generic point within the printed object domain along xi (i = 1, 2, 3) axes, respectively. Denoting partial differentiation with respect to the spatial coordinates xi (i = 1,2,3) as: ∂ ¼ ∂1 ; ∂x1

∂ ¼ ∂2 ; ∂x2

∂ ¼ ∂3 ∂x3

ð22Þ

DL and DN known as linear and non-linear operator matrices are defined by: 2

where

εkin

Finally, by considering updated values for inelastic strain, the stressstrain relationship (Eq. (12)) can be updated and unified for heating and cooling processes as:

∂1 60 6 60 DL ¼ 6 60 6 4 ∂3 ∂2

0 ∂2 0 ∂3 0 ∂1

2 3 0 ∂1 u1 ∂1 6 07 ∂2 u1 ∂2 6 7 6 ∂3 u1 ∂3 ∂3 7 7; DN ¼ 6 6 ∂2 u1 ∂3 þ ∂3 u1 ∂2 ∂2 7 6 7 4 ∂1 u1 ∂3 þ ∂3 u1 ∂1 ∂1 5 0 ∂1 u1 ∂2 þ ∂2 u1 ∂1

∂1 u2 ∂1 ∂2 u2 ∂2 ∂3 u2 ∂3 ∂2 u2 ∂3 þ ∂3 u2 ∂2 ∂1 u2 ∂3 þ ∂3 u2 ∂1 ∂1 u2 ∂2 þ ∂2 u2 ∂1

3 ∂1 u3 ∂1 7 ∂2 u3 ∂2 7 7 ∂3 u3 ∂3 7 ∂2 u3 ∂3 þ ∂3 u3 ∂2 7 7 ∂1 u3 ∂3 þ ∂3 u3 ∂1 5 ∂1 u3 ∂2 þ ∂2 u3 ∂1

ð23Þ It can be shown that variation of strain field is derived as: δε ¼ ðDL þ DN Þδu

ð24Þ

Equations of equilibrium as well as the corresponding boundary conditions are derived in a straightforward manner from the principle of the minimum total potential energy as: δU−δW ¼ 0

ð25Þ

32

M. Bodaghi et al. / Materials and Design 135 (2017) 26–36

in which variations δU of the strain energy and δW of the work done by the external loads need to be substituted with their definitions in terms of stresses, forces, virtual strains and virtual displacements. Starting from the variation of the strain energy δU ¼ ∭V δεT σ dV

ð26Þ

and using stress definition (Eq. (18)), it can be rewritten as:   δU ¼ ∭ V δεT Cu ε−β εkin −εth dV

ð27Þ

where V signifies the volume of the printed structure. The potential energy of external loads can be also expressed as: δW ¼ δuT Q

ð28Þ

where 9 8 < Q1 = Q ¼ Q2 ; : Q3

ð29Þ

denotes point forces in the xi (i = 1,2,3) directions. 2.4.3. FE methodology In this division, a Ritz-based FE solution is developed to predict functionally graded 4D printing process and thermo-mechanical behaviors of 4D printed metamaterials. A 20-noded quadratic serendipity hexahedron element with 3 degrees of freedom (i.e. ui (i =1,2, 3)) per node is proposed for the present problem. Considering quadrilateral shape functions ψi(x1, x2, x3) (i = 1…20) [23], the displacement vector u can be approximated in terms of nodal-mechanical variables as: u ¼ ΨU

ð30Þ

in which Ψ and U are elemental interpolating matrix and generalized nodal displacement vector defined as: 2

3 ψ 0 0 4 Ψ ¼ 0 ψ 05 0 0 ψ ψ ¼ fψ1 ⋯ψ20 g 0 ¼ f8 0⋯09 g 9 8 < u1 = < ui−1 = ; ⋮ U ¼ u2 ; ui ¼ ; : ; : ui−20 u3

ð31Þ i ¼ 1; 2; 3

where ui−j (i = 1…3;j = 1…20) are mechanical nodal variables. By substituting the discretized displacement field (Eq. (30)) into the strain field (Eq. (20)) and its variation (Eq. (24)) and subsequent results into the virtual energies (Eqs. (27) and (28)) and the principle of minimum total potential energy (Eq. (25)), FE governing equations for a generic element can be derived as: ðK0 þ KL ðUÞ þ KN ðUÞÞU ¼ Fm þ Fkin þ Fth

They all are defined as: K0 ¼ ∭ V  ðDL ΨÞT Cu ðDL ΨÞT dV  1 KL ¼ ∭V ðDL ΨÞT Cu ðDN ΨÞT þ ðDN ΨÞT Cu ðDL ΨÞT dV 2 1 KN ¼ ∭V ðDN ΨÞT Cu ðDN ΨÞT dV 9 8 29 8 < Q1 = < Q i−1 = ; i ¼ 1; 2; 3 ⋮ Fm ¼ Q 2 ; Q i ¼ ; ; : : ui−20 Q 2

ð33Þ

Fkin ¼ β∭ V ðDL ΨÞT þ ðDN ΨÞT Cu εkin dV   F th ¼ ∭V ðDL ΨÞT þ ðDN ΨÞT Cu εth dV

It should be mentioned that, εkin and εth are known variables through the SMP element that are held into the volumetric integrals. In this respect, T is assumed to be constant through the element volume while the Gauss-Legendre numerical integration method is implemented to calculate the integral including εkin along xi (i = 1, 2, 3) coordinates. A 3 × 3 ×3 discrete grid Gauss points having 3 points along each xi axis is considered in this study. Finally, in order to construct global FE governing equations, Eq. (32) is assembled for all elements that results in: 

    k K0 þ KL U þ KN U U ¼ Fm þ Fin þ Fth

ð34Þ

Eq. (34) represents a non-linear set of equations due to geometrical non-linearity. Consequently, an iterative solution procedure such as Newton-Raphson technique [23] is implemented to solve the problem in a step-by-step mode. By knowing temperature T, mechanical loading Fm and inelastic strain field εkin from the previous load-step, new value of U is calculated iteratively. Afterwards, the strain and stress state can be computed using Eqs. (30), (20) and (18), respectively. Finally updated inelastic strains at all Gauss points are calculated through either Eqs. (16a) and (16b) or (17) for cooling or heating process. It should be noted that, in the printer nozzle, the polymer melt is under stress, with part of the deformation energy being stored elastically. As the polymer melt leaves the nozzle, stresses are relaxed and the elastically stored energy is released, leading to radial expansion of the melt referred to as die swelling. The phenomenon may affect pre-strain regime induced by 4D printing process. For predicting die swelling and investigating its effects on pre-strain regime, viscoelastic-based SMP constitutive model should be developed, even though it is beyond the objective of the present work. For more details on modeling of viscoelasticity in SMPs and die swelling, one may refer to [24–26]. 3. Adaptive metamaterials: design-fabrication-simulation In this part, the developed computational tool is first implemented to describe thermo-mechanical mechanisms behind SMP material tailoring during fabrication stage. It is then applied to digitally design primitives fabricated by FG 4D printing for adaptive metamaterials

ð32Þ

where K0 represents the constant stiffness matrix while KL and KN are linear and quadratic stiffness matrices which are dependent on the nodal displacements. Also, Fm , Fkin and Fth are load vectors induced by mechanical loading, SMP phase transformation and thermal loading due to temperature change from a reference temperature, respectively.

Fig. 6. Model prediction for Young's modulus and comparison with experimental DMA data.

M. Bodaghi et al. / Materials and Design 135 (2017) 26–36 Table 2 Predictions for pre-strain (%) regime of the SMP beam printed in various printing speeds and liquefier temperatures. Location

Upper layer Middle layers Lower layer

Sp = 30 mm/s

Sp = 40 mm/s

Tln = 210°C

Tln = 230°C

Tln = 210°C

Tln = 230°C

30.5 25.3 20

15 10.1 5.6

37 29 21

31.3 24 17

that can reveal self-assembly functions. Finally, several 3D adaptive metamaterials with shape-shifting features that illustrate broad potential applications are presented. 3.1. Pre-strain regime prediction The first step is to predict pre-strain value induced in all printed layers during fabrication. In order to achieve the investigation, phasetransformation parameters including a1 , a2 , Tl and Th presented in Eq. (4) need to be calibrated. Using DMA results already displayed in Fig. 3, they are set as 0.15, 0.145 and 20, 100°C, respectively. Young's modulus calculated from Eq. (13) is illustrated in Fig. 6 and compared with experimental data from DMA. The results reveal that the trigonometric function is able to successfully simulate SMA phase transformation in a smooth and gradual manner as observed in experiments. The initial and final status of the straight beams as displayed in Fig. 5 are given as inputs and applied to FE tool. Implementing a try and error scheme, pre-strain regime in terms of its value at lower, middle and upper layers is computed and presented in Table 2 for various printing speeds and liquefier temperatures. As it can be seen, prestrain has an almost linearly increasing trend through the thickness upward inducing self-folding feature. During heating, the pre-strain is released non-uniformly. This un-balancing produces bending deformation and leads to self-folding. In this respect, any increase in printing speed or liquefier temperature has positive and negative effects on the induced pre-strain value and consequently structural deformation as observed in experiments, see Fig. 5. The results presented in Fig. 5

33

and Table 2 also reveal that the 4D printing process with parameters of Sp = 40 mm/s , Tln = 230°C and Sp = 30 mm/s , Tln = 210°C induces similar pre-strain into the beam structure and results in nearly similar configuration after activation. Finally, it can be concluded that control of printing speed and liquefier temperature may allow the realization of self-folding 3D structures. The pre-strain regime and its values reported in Table 2 may serve as benchmark data to researchers dealing with design, simulation and fabrication of FG 4D printed object. From a technical viewpoint, objects printed with high temperatures may have a strong interface bonding. Therefore, the following results are printed with fabrication parameters of Sp = 40mm/s and Tln = 230°C. 3.2. Primitives for adaptive metamaterials Five 2D primitives with different shapes are proposed for designing adaptive metamaterials with self-folding feature. First, one straight and two curved beams with the same length, width and thickness (L = 34, b = 1.6,h = 1 mm) and different angels of θ = 0 , 60 and 90° are printed as illustrated in Fig. 7a. The configurations of the primitives after hating-cooling process are displayed in Fig. 7b. The FE model is also implemented to simulate structural deformations using pre-strain regime reported in Table 2 for Sp = 40 mm/s and Tln = 230°C. Simulations are also included in Fig. 7c. The color bar in Fig. 7c displays transverse displacement (u1), along X1-direction, to maximum transverse displacement (u1m). In this aspect, u1m is reported in Fig. 7c. The results show that, while straight beam has a shape-shifting via self-bending mechanism, curved beams experience both bending and cross-sectional rotation at the same time. In fact, once activated by hot water, the freestrain recovery stands the curved beam up while natural curvature leads the structure to rotate along the length direction. It results in forming a conical ring with rectangular cross-section that is oblique with respect to the ring axial direction. From a topology point of view, in fact, inner and outer edges of the curved beam with small and large radii make the apex and base of truncated conical shell-like structures. Furthermore, it is interesting to note that all beams with the same length either in curve shape or straight transform to rings with similar mean diameters (≅ 7 mm) so that both end edges meet each other.

(a) (b) (c)

u1m=0 mm

u1m=17.4 mm

u1m=22.4 mm

Fig. 7. Straight and curved 4D printed primitives: (a) initial configuration after printing, (b) final configuration after heating-cooling, (c) simulation.

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M. Bodaghi et al. / Materials and Design 135 (2017) 26–36

Fig. 8. Wave-shaped (a–c) and extended wave-shaped (d–f) 4D printed primitives: (a, d) initial configuration after printing, (b, e) final configuration after heating-cooling, (c, f) simulation.

Their difference is cross-section angle with respect to the ring axial direction is measured as 22 and 31° for initial primitives printed with angels of 60 and 90°. Regarding simulation, it is noted that the FE model-solution is able to correctly replicate main characteristics observed in the experiments. In this aspect, maximum error of 2% is achieved for mean diameters that is acceptable. It should be mentioned that the simulations have been performed using pre-strain information obtained from the previous test (Table 2). The good qualitative and quantitative agreement obtained with the same pre-strain for both tests confirms accuracy and reliability of the predicted pre-strains in the previous section. These self-foldable primitives can serve as elements of adaptive metamaterials for complicated shape-changes or complex functions. It is worth mentioning that, while the mass of the printed elements remains constant during phase transformation, the geometry and stiffness are changed. It means dynamic properties are changed as the 4D printed object is heated. They could be a good candidate to be employed as adaptive structural/dynamic switches providing different temperature-dependent dynamic characteristics. For instance, they may be promising in designing self-foldable metamaterials for affecting resonance behaviors [27]. Next, another primitive with geometrical parameters of the curved beam described above with L = 34 , b = 1.6 , h = 1 mm and θ = 60° is printed in a symmetric wave-shaped form. In fact, it is composed of two arcs with θ = 30°. The configurations of the sample after printing and heating-cooling are demonstrated in Fig. 8a and b. Simulations are also embedded in the third part of the figure. As it can be seen, the 2D wave-shaped primitive gets into a ring shape but has a derivation along the circumferential direction. A qualitatively and quantitatively good agreement between experiment and simulation is also observed. The final shape looks like a helix of helical spring. It brings the idea that the planar wave-shaped component may be employed for designing adaptive metamaterials in spring shape. The wave shape template is then extended in both directions to have four segments of arcs with θ=30°. The counterpart of Fig. 8a–c for the current example is depicted in Fig. 8d–f. The results reveal that the extended wave-shape template transfers to a helical spring with two coils after heating-cooling process. It can be a demonstration of 2D to 3D shape-shifting by self-coiling mechanism. Thus, it can be called a self-coiling spring. Beyond self-coiling feature, a static spring could be printed directly by FDM 3D printing. However, it

will be required to print some supportive material that may affect surface quality and mechanical properties/strengths of the spring. It is interesting to note that the helical is permanent shape of the selfcoiling structure. It means it is possible to program it into any other shape as a temporary shape and then achieve free shape recovery upon heating. For instance, the spring as depicted in Fig. 9a is heated by immersing in hot water with temperature of 90°C and then straightened. Next, it is kept fixed and cooled down to the room temperature followed by mechanical unloading, see Fig. 9b. It looks like a straight beam with 54 mm length. Heating the straightened 1D beam will result in fully shape recovery as shown in Fig. 9.

Fig. 9. Thermo-mechanical programming on the self-coiling spring: (a) initial configuration after printing-heating-cooling process, (b) final configuration after heating-straightening-cooling-unloading (the straightened beam is recovered back to the spring upon heating).

M. Bodaghi et al. / Materials and Design 135 (2017) 26–36

35

Fig. 10. 4D printed self-conforming device: (a) straightened beam and a hex key, (b) conforming to radius of 6.8 mm, (c) configuration after full shape recovery; 4D printed self-tightening fiber: (d) straightened beam and soft polymeric substrate, (e) forming a loose knot, (f) shrinkage of the fiber by heating.

3.3. Potential applications This section is dedicated to demonstrate potential applications of self-folding primitives and self-coiling springs in mechanical and biomedical engineering devices. For example, the self-coiling spring demonstrated in Fig. 9 can serve as spiral stents. In the straightened shape, the 1D beam can be placed inside a catheter and then delivered to the required location. Once the constraint is removed, the 1D beam may be heated by blood temperature and mechanically self-coil. It may act like a supporting function for opening the artery keeping blood flowing normally or encompasses any object like a blood clot for treating stroke. In the next example, the 1D straightened beam presented in Fig. 9b is used as a 1D element that softens and conforms to 3D environments upon heating. Fig. 10a displays the 1D component and a hex key with a diagonal of 6.8 mm. The experimental results presented in Fig. 10b and c reveal that, after warming, the 1D smart component is able to softly conform to the 3D hexagonal supporting environment. Even after cooling, the device retains its conformed geometry and shape fixing, see Fig. 10c. It can be found that this adaptive element has potential to be used as flexible and conformal components for future mechanical/ biomedical devices fabricated by 4D printing technology. Motivated by the research work of on shape memory suture [28], the feasibility of the current 1D-to-3D self-coiling metamaterial for a similar purpose is checked experimentally. A two-part soft polymeric substrate that looks like tissue, (30 mm × 15 mm × 1 mm) × 2, as shown in Fig. 10d, is

loosely connected by the 1D straightened beam at room temperature, see Fig. 10e. When the temperature is increased above the critical temperature, Tg, the adaptive metamaterial is activated triggering the shape recovery and removes the gap and tighten the tissue-like substrate, see Fig. 10f. In fact, it acts like a self-tightening fiber. This may pave the way for 4D printing patient-design self-tightening fibers for surgical sutures. As a final example, straight beam element presented in Fig. 7 is used to design and fabricate planar metamaterial (34 mm × 17.6mm × 1 mm) by self-folding ability. It is composed of a periodic arrangement of the six straight primitives (34 mm × 1.6 mm × 1 mm) programmed along their length direction during printing stage as depicted in Fig. 11a. Upon heating, the 2D lattice transforms to shape a 3D tubular metamaterial as shown in Fig. 11b. The inner dimeter is also similar to that of self-foldable primitive. It confirms reliability of SMP programming during FG printing process so that all beam-like elements are programed in a similar way. This 4D printed planar metamaterial with self-folding capability may has potentials to be used as adaptive custom-designed structures that may conform to cylindrical environment and perform supporting function. It can be used in biomedical devices for different supporting usages like broken bones or tracheal splint. Such a 4D metamaterial may exhibit another great potential to serve as a self-deployable stent, see Fig. 11c and d. To this end, it needs to be heated and rolled to get like a smaller cylinder as shown in Fig. 11d. It is able to retain the temporary shape by fixing-cooling-unloading process. The small tube can be stored inside a catheter and then delivered

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M. Bodaghi et al. / Materials and Design 135 (2017) 26–36

(a)

(b)

(c)

(d)

heating

programming

Fig. 11. 4D printed metamaterial lattice with self-folding capability: (a) after printing, (b) after heating-cooling process; 4D printed self-deploying stent: (c) another view of part b, (d) configuration after heating-rolling-cooling-unloading (the rolled cylindrical lattice deploys upon heating).

to the required position. Once the catheter is removed, it is heated to self-deploy and recover its original shape automatically, see Fig. 11c. 4. Conclusion The aim of this paper was to describe functionally graded 4D printing to fabricate planar adaptive metamaterials enabled to transform to 2D/3D configurations by self-folding and self-coiling features. FDM printing technology was used to program SMP materials during deposition stage in a functionally graded manner. Experiments were conducted to explore self-folding features characterized in terms of relevant fabrication parameters like printing speed and liquefier temperature. Boundary value problems were solved to describe thermo-mechanical mechanisms behind programming during fabrication stage and shape-change during activation. In this respect, FE governing equations were derived based on the non-linear Green-Lagrange strains and coupled with constitutive model developed for SMP phase transformations during heating/cooling processes. They were then solved by iterative incremental Newton-Raphson scheme to trace non-linear equilibrium path in the large-deformation regime. The understanding derived from modeling and experiments was employed to design and fabricate straight and curved beams as structural primitives for adaptive metamaterials that revealed 1D-to-2D and 1D/ 2D-to-3D shape transformation through self-folding or/and self-coiling. Finally, it was experimentally shown that the adaptive metamaterials had the potential to be used in mechanical and biomedical applications like structural/dynamical switches, self-conforming substrates, self-tightening surgical sutures, self-conforming splints and self-coiling/deploying stents. This research is likely to advance the state of the art 4D printing and unlock potentials in the design and development of functional structures with self-folding/coiling/conforming/deploying features in a controllable manner. Acknowledgments The work described in this paper was supported by the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK 14202016) and the Chinese University of Hong Kong (Project ID: 3132823). References [1] Y.W. Zhai, D.A. Lados, J.L. Lagoy, Additive manufacturing: making imagination the major limitation, JOM 66 (2014) 808–816. [2] ISO/ASTM 52900, Additive Manufacturing-General Principles-Terminology, 2015. [3] A.A. Zadpoor, Mechanical meta-materials, Mater. Horiz. 3 (2016) 371–381. [4] K. Wang, Y.H. Chang, Y.W. Chen, C. Zhang, B. Wang, Designable dual-material auxetic metamaterials using three-dimensional printing, Mater. Des. 67 (2015) 159–164.

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