Comprehensive Study on Shape Shifting Behaviors of Thermally Activated Hinges in FDM-based 4D Printing

4D printing of shape shifting structures , aka “hinges”, ha s raised a new standard in many fields. By using these hinges in certain parts of a 3D printed structures, a pre designed complex 3D shape with potential multifunctional application can be achieved from flat structure. This paper proposes a comprehensive semi-empirical model to predict the final shape shifting behavior and magnitude of the hinges printed by FDM process. First, all FDM main parameters are selected and reduced by design of experiment to printing speed, lamina thickness, nozzle temperature as well as printing pattern. In order to develop the model, a time-dependent constitutive model with these four process parameters were extracted for strain of an SMP homogeneous single layer structure using a fractional Zener model accompanied with Multiple Linear Regression (MLR) technique. Thereafter, the mathematical relations for shape shifting behavior of bilayer 4D printed structures were developed for beam bending and twisting by modifying Timoshenko’s constitutive equations . A comprehensive shape-shifting model was established including 3D printing parameters, angles, thickness ratios, activation time and temperature which was compared to the experimental data and results predicted both shape shifting behavior and magnitude of the hinges with good agreement. In addition, a novel flowchart was suggested to design and achieve the desired shape shifting behaviors. The proposed model and flowchart are novel tools to design 4D structures through desired shape-shifting of the hinges.


INTRODUCTION
The term "4D printing" was introduced by Tibbits in 2013 [1] and refers to the 3D printing processes in which the resulting object could changes its shape with time (i.e., the 4 th dimension) [1][2][3]. In most cases, the working basis of 4D printing is shape-shifting in synthetic materials, which is achieved either by a physical effect such as residual stresses or through the energy released from an active or smart material [4]. Most of the 3D printing techniques, such as material jetting [5][6][7][8], DW [9], SLA [10]& [11], and customized 3D printers [12,13] are suitable for use in 4D printing processes, but not all can be used equally well as they usually require specialized equipment that put them out of the reach of most research labs; Moreover, the materials used in these processes tend to be expensive and proprietary. Recently, material extrusion technique (FDM), which is one of the most commonly available and inexpensive 3D printing processes that uses affordable off-the-shelf materials, is receiving increasing attention [14][15][16][17][18][19][20][21][22] as a good replacement for the abovementioned processes.
There are two approaches to 4D printing using an FDM 3D printer. The first approach is an approach that requires post-processing after 3D printing, i.e., manually deforming the printed object into a temporary shape after 3D printing, which complicates the manufacturing process. Unlike the first approach, the second approach uses only a 3D printer, without the need for post-processing to manually change the shape of the printed structure [23].
It also eliminates the need to print supports and reduces the printing time compared with direct printing of the targeted shapes [24]. In this approach a flat bilayer structure composed of two layers with different strains while exposed to an external stimulus is printed. In recent years, 4D printing of these shape-shifting flat structures aka active hinges has raised a new standard in many fields such as bio-technology, electronics, aerospace, etc. [25].
These hinges can be used in certain parts of an structure, enabling the user to bend or twist the structure where needed and produce a complex shape out of a simple structure [26]. Making the flat structure, aside from the simplicity, will give the user access to the surface of the structure, which can be used for many purposes including drug delivery devices [27]. Applying active hinges in certain parts of a complex structures enables the structure to be more compact and portable. On the other hand, controlled and precise self-shape-shifting of these hinges eliminates the need for human intervention. Thus, there are a lot of potential and actual applications for them specially for remote and space-limited environments like spaceships. Figure 1 shows one of these potential applications. A device that carries two planar antennas or two solar panels which uses active hinges. Each side of the device have one antenna/panel. As it can be seen in Figure 1, the device is flat, compact and portable before activation. Right antenna/panel is attached to two hinges that are expected to only bend 60 degrees and left antenna/panel is attached to a hinge that is expected to bend 60 degrees and twist 35 degrees. We used our model to set the process parameters so that the desired conditions are satisfied. Printed specimen is also shown in Figure   1. printing have focused on predicting the mechanical and thermo-mechanical behavior of these structures through physical and material parameters. There are only handful of studies concerning the shape-shifting predictions through 3D printing process parameters, and they only model the bending behaviors.
here in this study, we provide a semi-empirical constitutive model to predict the time dependent shape-shifting magnitude and behavior of the hinges including bending, twisting and a combination of both, through printing parameters namely printing speed, lamina thickness and nozzle temperature which were selected among all important process parameters using S/N ratio analysis. In addition, the model is compatible with angle laminae. Thus this proposed model will grant the most comprehensive control over 4D printed structures through active hinges.

Working principle
4D printed structures require an interaction mechanism in order to properly respond to the applied stimulus.
Different interaction mechanisms have been proposed in the literature [25], of which a specific one called "constrained-thermo-mechanics" happens in FDM process. Figure 2 shows a schematic of FDM process where SMP filament is extruded through the hot end nozzle. First, the SMP is heated to that is above its glass transition temperature ( ) through the hot end nozzle and is extruded on the bed at this temperature which causes the orientation of the polymer chains in nozzle movement direction (print pattern). while nozzle moving forward, the printed SMP is cooled to that is below its . When cooled to , the orientation imposed onto the SMP will be maintained; this is simply due to the freezing of the polymer chains that locks in the oriented chain conformation, enabling the storage of entropic energy which produces a stored strain in the SMP. when it is heated above the again in the absence of external stress, the SMP will release the strain and recover to its permanent shape. heating activates the molecular mobility, which releases the entropic energy, thus driving the molecular chains back into its highest entropic state corresponding to the permanent shape [35]. Now to extract the out of plane shape shifting behaviors such as bending and twisting from this shape memory strains, the concept of bilayer structures is being used. Figure 3 shows the concept of a bilayer strip (hinge) in FDM 3D printing process, with each layer having a different printing pattern and/or process parameters.
As it can be seen in Figure 3 (right) each layer consists of an arbitrary number of laminae and each lamina consists of an arbitrary number of rasters. This difference in printing pattern and/or process parameters leads to different stored strain in each layer which according to Timoshenko's bilayer beam theory [36] will cause out of plane shape shifting behaviors (Figure 3, left). By changing the strain difference of two layers (longitudinal, transverse or shear strains), the shape-shifting behavior and its magnitude can be controlled. Figure 4 shows six apparently similar printed samples with different shape-shifting behaviors indicating the importance of programming these strains through printing parameters and printing patterns.

Shape recovery ratio test
Four specimens (40102 mm) were printed according to the printing parameters given in Figure 5. A tensile strain of 10 percent were applied to Each specimen using SANTAM STM-20 tensile tester following the thermomechanical mechanism: the specimens were first heated to their Tg (65 °C) in a hot water bath installed on the tensile tester, then were loaded to the predetermined strain while the temperature was held at 65 °C. the specimens then were cooled down to room temperature without removing the external stress so as to maintain the strain. For the final step, each specimen were reheated to a different temperature (65-95°C). A Dino-Light AM3713BT digital microscope were used to capture the shape recovery in time.

Design of experiments
In this study, all of the main printing parameters that may affect the amount of strain which is the cause of shapeshifting behavior in the 4D printing process, were selected. Using the Taguchi method and signal to noise (S/N) ratio analysis the parameters with significant effect on the strain were determined. Table 1 shows the parameters and their levels (only two levels are required for screening). According to the number of parameters and their levels, L8 Taguchi Orthogonal Array design [37] were chosen to implement S/N ratio analysis. Table 2 shows the L8 design for printing parameters. S/N ratio analysis indicates that the optimal condition is where the signal-to-noise ratio is maximum. The parameter of this analysis, , is shown as follows and the more it changes for different levels of the input factor, the greater the effect of that factor on the output [37]: Where is the output value for kth experiment and s is the number of iterations. Table 3 shows the values of and the output (strain), which was measured three times using DIC method. All samples were printed with a printing pattern of [0]10 at room temperature. Figure 6 shows versus different printing parameters, based on which the parameters of layer thickness, nozzle temperature and print speed have a significant effect on the output and were chosen to further study in this paper.

Model description
Here we propose a simple but straightforward model to predict shape-shifting of a hinge; a model that enables user to change the magnitude of FDM process parameters as desired and get the resulting final shape-shifting behavior. Thus by controlling the shape-shifting of the hinges, this model is a mean to design 4D printed structures such as active origami.

Free recovery of a single layer
To obtain the free recovery strain of a homogeneous thin beam of an SMP, constitutive equations were extracted using the Zener fraction model. Figure 7 shows a schematic of the Zener fraction model, which includes an equilibrium and a nonequilibrium branch. In this model, the dash-pot is replaced by an element termed a 'springpot' by Koeller [38]. This simple two branch fractional model has proven to be as accurate as a 12 branch generalized maxwell model or a five branch modified generalized maxwell model [39,40].

Figure 7. Schematic illustration of Zener fractional model
According to the mechanical model shown in Figure 7, The constitutive equation for the fractional Zener model can be written as: where ( ), ( ), , and are the recovery stress, recovery strain, time, modulus of elasticity and relaxation time, respectively. is known as memory parameter or the fractional derivative parameter, and indicates fractional differentiation. Finally, 1 = √1 + 1 0 ⁄ .
There are nine parameters in Eqs. (4) & (5) to be identified for prediction of free recovery of a single layer, which they are listed in Table 5. First five parameters namely 0 , 1 , , 0 , 0 where determined using [42]. To determine 1 and 2 , DMA tests were performed in different frequencies of 0.5, 1, 10, 20, and 50 Hz and based on the TTSP, a master curve at reference temperature were constructed with shifting factors. finally to determine a , tan delta curves for different frequencies were used (Figure 8.a). The g at different frequencies is determined from the peak of the tan delta curves and a is calculated from the slope of the plot of ln(f) versus 1 g ⁄ (Figure 8.b) using the following equation [43]: Where is frequency (Hz). After identifying all parameters, now we can use Eqs. (4) & (5) to predict free recovery strain of each layer through time and temperature. Figure 9 compares the model predictions and experimental data for strain recovery ratio. As it can be seen, the results of the model has a good agreement with experiments.

Relation between stored strain and printing parameters
Since the strain will be stored in printed layer during the printing process, It is obvious that printing parameters namely printing speed, layer thickness and nozzle temperature will directly impact the strain. Considering that the

Curvature modelling of hinges
According to Timoshenko [36], a beam made of two layers of metal with different coefficients of thermal expansion (CTE), will bend if heated due to the strain difference created in the layers. Figure 11 shows a bilayer beam which undergoes bending over activation. Considering the fact that here each of the two layers forming the hinge is of the same material and differs only in printing conditions, the difference in CTE of the layers is neglectable; moreover, Timoshenko's theory is for solids. For these reasons, it cannot predict the curvature of the self-bending hinges that have viscoelastic properties. However, the prediction of the curvature is expected via modification of the relationship for SMPs. The Curvature could be in x or y direction: The modulus of elasticity must be obtained for each layer to calculate the value of . Assuming the printed rasters as fibers in composites, elastic modulus for different printing patterns has been modeled by [30] using CLT.
It can be shown that when the thickness of the two layers are the same, i.e. = 1, n does not produce any substantial effect on curvature [36], thus, Eq (17) can be rewritten as:

Torsion modelling of hinges
Timoshenko's constitutive equations for beam torsion are used to derive the torsion of a two-layer structure. Figure 12 shows a bilayer beam which undergoes twisting over activation. All the forces acting over the section of the first layer can be represented by a shear force 1 and a torsional moment 1 . For the second layer, All forces acting on the cross-section can be represented by a shear force 2 and a torsional moment 2 . Since no external force is applied to the beam, all forces acting on any cross section of the beam must be in equilibrium.

Figure 12. twisting of a bilayer hinge over activation (uniformly heated)
At the interface of the two layers, shear strain can be written as below: Thus: (20) 2 + Like curvature, It can be shown that when the thickness of the two layers are the same, ́ does not produce any substantial effect on torsion, thus, Eq (21) can be rewritten as:

stored strain and printing pattern
To verify Equation (12), samples with different printing patterns (angles) were printed and their longitudinal, transverse and shear strains were measured carefully after activation. Figure 13 shows good agreement between the experimental data and predictions of the model. The printing conditions are given in Figure 5.

modification of curvature and torsion of bilayer beams
The model predictions for curvature and torsion are expected to differ from the experimental results by an extent, necessitating the modification of the model. There are two major reasons behind this difference: first, the calculated free recovery strain of a single layer will be different from that of the same layer while in a bilayer structure as it is constrained by the next layer, and second, the other dimensions of each layer such as thickness and width, will also undergo a different amount of strain through the activation process which consequently alters the value of ́, and the total thickness, ℎ, in addition. For the sake of model simplicity, we won't apply these conditions to the model and instead will add a correction factor using experimental data. Process parameters are intentionally set in a way that the resulting strains increase incrementally. Figure 14.a compares the experimental data of the final curvature, which is at ( ) ≈ 1, with its predicted value by the proposed model for different values of Δεi (longitudinal and transverse strain differences between two layers). The slope of the graph as shown in Figure 14.b is the correction factor ́=0.67 for curvature. As for torsion, Figure 15.a compares the experimental data of the final torsion, , with its predicted value by the proposed model for different values of (shear strain differences between two layers). Here slope of the graph as shown in Figure 15.b would be the correction factor ́= 1.4 for torsion.

4D printed bilayer shape shifting model
By combining Eqs (6), (11), (13), (17) and (21), the curvature and torsion relations of bilayer 4D printed structures can be written in a matrix form: If the thickness of the two layers is equal which in most cases they are, Eq (25) can be rewritten as:

Model verification
According to the application of these bilayer structures, is not considered and to verify the accuracy of the model, only the predicted values of and were compared with xperimental data. As shown in Figure 17, three samples were designed and printed in such a way that the sample (a) goes under pure bending, sample (b) pure torsion and sample (c) shows Both bending and twisting shape shifting behaviors. The printing conditions of the three samples are shown in Table 6. Activation time and temperature for all samples (40 × 10 × 2 mm) were set to 90 ° C and 30 seconds, respectively.
As it can be seen in Figure 17, model results show good agreement with experimental data, thus model is capable of predicting the final shape shifting behavior and it's magnitude.

Discussion
It was shown that a single layer 4D printed SMP beam would shrink to an extend after being activated in higher temperatures than its Tg. Printing a bilayer beam with each layer having a different strain, would cause out of plane shape shifting behaviors like bending and twisting. In order to have two layers with different strains, 3D printing process parameters should be changed for each layer. We have experimentally determined the parameters with significant effect on strains. Then a model were developed based on that parameters to predict the shape shifting behaviors and their magnitude. According to the proposed model, printing parameters affect the magnitude of the strains, whereas the printing pattern affect the direction of the stains, also difference in longitudinal and transverse strains will cause bending behavior in x and y directions respectively, and difference in shear strains will cause twisting behavior in bilayer structure. To achieve both behaviors simultaneously, both longitudinal/transverse strains and shear strains of the two layers should be different. In this regard, a shapeshifting flowchart as shown in Figure 18 were designed so that one can achieve desired shape-shifting behavior by following this flowchart.

Conclusions
In this paper, we developed a semi-empirical model to control and predict the shape-shifting behavior of the bilayer structures aka hinges which printed using FDM 3D printing process. Our model allows the user to desirably change the process parameters including printing parameters and patterns and see the resulting shape shifting behavior and its magnitude skipping the trial and error experiments. The results of the model shows a good agreement with experiment data and is accurate enough to be used in practice. The model also shows the effective 3D printing parameters and their impact on shape shifting magnitude where increasing the print speed increases it and decreasing the nozzle temperature and layer heigh decrease it. The next step would be to predict the shapeshifting behavior of the hinges through the activation time. In addition, model can predict the shape-shifting behavior of the hinges through activation time and temperature. By predicting the behavior through time, one can design multiple and/or multi-step desired shapes, not to mention the further control over the shape-shifting magnitude.

Funding
No funding was received to assist with the preparation of this manuscript.