Shape morphing cycle
In contrast to shape morphing structures where individual elements or bars have to be actuated separately to achieve a predefined target shape, we introduce a concept where uniform heating activates the structure and enables shape morphing. The concept is based on exploiting the different mechanical properties at different temperatures of materials that are 3D printed with the Objet500 Connex3 3D printer (Stratasys, Eden Prairie, MN, USA). The 3D printer uses the material jetting technology and allows for printing multiple materials at the same time. Figure 1a shows the Young’s modulus at different temperatures of the two materials “VeroWhite+” (VW) and “High Temperature” (HT). The values are obtained by tensile tests according to the respective ASTM norm [31], conducted on an Instron ElectroPuls E3000 mechanical testing machine and an Instron 3119 environmental chamber.
The markers indicate the mean value of five samples, and vertical error bars indicate the standard deviation. At room temperature and up to about 50 °C, both materials VW and HT have approximately the same Young’s modulus and decrease almost linearly. At higher temperatures, the stiffness of VW drops, while the stiffness of HT further decreases, almost linearly. Even though both materials become relatively compliant, the stiffness of HT remains higher than the stiffness of VW at temperatures up to 80 °C. This behavior is caused by the two different glass transition temperatures of about 55–60 °C (VW) and about 70 °C (HT).
Instead of achieving a target deformation by recovering a pre-programmed state, as often seen in literature, we use this difference in the thermo-mechanical properties of the materials to introduce local stiffness gradients in the structures at higher temperatures, which cause the target deformations under a single actuation input displacement. A similar concept was introduced by Weeger et al. for semi-compliant, voxelized beam structures [30]. Figure 1b schematically shows a 2D lattice structure where each bar either consists of VW (gray lines) or HT (black lines). The solid black line and the dashed black line show two different possible distributions of HT material. At room temperature, all bars have the same Young’s modulus of about 2100 MPa and the lattice has a homogeneous stiffness distribution. At 80 °C, the stiffness of the HT bars (black) is reduced to about 305 MPa and the stiffness of the VW bars (gray) is reduced to about 8 MPa. Since the HT bars are about 38 times stiffer than the VW bars, local stiffness gradients are introduced to the structure and the overall mechanical response is now mostly governed by the distribution of the stiffer HT material. Hence, by controlling the material distribution in the structure, the mechanical properties and the deformation under external displacements can be controlled.
In the first case, HT material is deposited as indicated by the solid black lines and forms a re-entrant cell, which is generally associated with auxetic behavior [33]. Hence, at high temperatures a horizontal expansion of the structure will result in a vertical expansion (Fig. 1c top), the overall Poisson’s ratio of the structure is negative \(\nu < 0\). In the second case, HT material is deposited as indicated by the dashed black lines and forms a diamond cell, which will contract vertically under a horizontal expansion with \(\nu > 0\) (Fig. 1c bottom). This simple example shows that it is possible to control the overall deformation of a structure at high temperatures by assigning the two materials VW and HT to the individual bars. Figure 1d shows schematically how this concept can be implemented in a mechanical structure. The gray region represents a general lattice structure, which is fixed at the left side and which can be mechanically deformed by a local input displacement at the right side, for example via a pneumatic actuator. The dash-dotted lines indicate the global deformations of the structure at high temperatures, which are determined by the exact material distribution in the base lattice.
By controlling the input deformation and the temperature of the structure, this setup enables reversible shape morphing behavior. Figure 2 schematically shows a 2D lattice structure and the respective stiffness distribution during all states of the shape morphing cycle. In step one, the structure is at room temperature (23 °C) and all bars have the same stiffness, independent of the material distribution. In step two, the structure is heated up to 80 °C. The bars that consist of VW, indicated by the gray lines, become more compliant than the bars that consist of HT, indicated by the black lines.
Next, an input displacement is applied locally to the right side of the structure. Due to the heterogeneous stiffness distribution, the structure deforms into the desired shape, which is the sine shape at the top and at the bottom here. To fix the deformations, the structure is cooled down to room temperature in step 4 and the stiffness in all bars is again the same. Since the overall deformations are based on deformations in the members, the overall topology of the structure is preserved. Due to the shape memory properties of the materials, the applied and fixed strains in the bars are reversible, and the original shape can be recovered when the structure is heated once again to 80 °C.
Since different material distributions yield different deformations for given input displacements, it can be difficult to find a material distribution for which the structure deforms to meet the target shape. For small lattices, intuitive configurations such as the re-entrant or diamond configurations in Fig. 1b might yield qualitatively good solutions. However, a computational design framework is required to systematically design larger scale lattices with more complex deformations while taking into account that the optimization problem is nonlinear and non-convex. In this work, we develop a two-step computational design optimization framework to solve the discrete material distribution problem. To make use of gradient-based optimization techniques for efficiently solving large-scale problems, the discrete problem is relaxed and solved with an interior point line search filter method (IPOPT) as a first step using a random starting point. As the relaxed problem can yield solutions that are not fully discrete, i.e. some bars with intermediate material properties remain, in a second postprocessing step, a Genetic Algorithm (GA) is used to find fully discrete solutions based on the almost discrete solutions from the first step. The optimization model and methods are described in detail in “Methodology”.
Optimized Structures
To show the applicability to complex 3D problems and the generality of the method, Fig. 3 introduces the three example structures “2D Sine”, “3D Sine”, and “Airfoil”. The example “2D Sine” in Fig. 3a consists of 6 × 4 square unit cells with 154 bars, where each unit cell has the side length \(L = 17.0\;{\text{mm}}\) and a central node in the middle. Hence, the length of the horizontal and vertical bars is \(l = L\) diagonal bars is \(l = (\sqrt 2 /2) \times L\). The \(n_{{\text{T}}}\) target displacements at the top and at the bottom, indicated by the dashed blue lines, are defined on in total 10 individual nodes and describe mirrored sine curves with an amplitude of a = 3.0 mm. The input actuation is a displacement of all nodes at the right side of the structure, indicated by the black arrows. The structure “3D Sine” consists of 4 × 4 × 2 octet-truss unit cells with 896 individual bars. All bars are of equal length \(l = 14.1\;{\text{mm}}\). In contrast to the 2D example, this structure is deformed by compressive displacements at the top. The target displacements on the left side and on the right side describe mirrored sine shapes with an amplitude of a = 3.0 mm and are defined on in total 46 individual nodes. The structure “Airfoil” has the shape of the back half of a NACA 6420 airfoil [34]. This shape is extruded in z-direction and discretized with 18 × 2 × 8 body-centered cubic unit cells, which results in 3516 bars in total. Due to the mapping of the unit cells to the airfoil geometry, the bar lengths \(l\) vary between 12.4 mm at the root and 0.7 mm at the tip. The target shape describes a deflection of the tip of the airfoil downwards by a = 10.0 mm, as it could be used for example for shape-morphing wings. The target shape is defined by, in total, 105 individual nodes at the top and at the bottom of the airfoil. A bar radius of r = 0.75 mm is chosen for all structures.
Each example structure is optimized 10 times, due to the random starting point, and the arithmetic mean and the standard deviation for both optimization steps are given in Table 1. For all three example structures, the best solution after both steps is selected out of the 10 optimization runs, fabricated and experimentally tested. It can be observed that the solution of the relaxed problem after the first optimization step achieves lower objective function values for all three examples than after the postprocessing step, as the restriction on purely discrete values in the postprocessing step further constrains the design space. On the left side of Fig. 4, the best optimized solution for each example is shown in the deformed state. The deformations are computed by a linear-elastic, truss FE analysis with material properties at 80 °C. The colors of the bars indicate the material distribution, where light gray represents VW material and dark gray indicates HT material. The transparent blue dashed lines and the transparent blue planes indicate the predefined target displacements, respectively. In general, the target shapes are matched by the optimized structures well. However, the absolute values of the objective function of the three examples after the postprocessing step are significantly different, as summarized in Table 1.
TABLE 1 Statistical results of ten consecutive optimization runs for each example. This stems from the definition of the objective function \(f\), i.e. the sum of the squared error over all target nodes. Larger structures with more target nodes such as the “Airfoil” more likely have larger objective function values than smaller structures with fewer target nodes. To make these values comparable, we compute the “mean error” of the optimization at every node via \(e_{{\text{opt,mean}}} = \sqrt {f/n_{{\text{T}}} }\), with the best objective function value after the postprocessing step \(f\) and the number of target displacements \(n_{{\text{T}}}\). This metric can be interpreted as the mean error at every target node, i.e. the mean distance between the deformed shape predicted by the simulation and the target shape, evaluated at all target nodes of the optimized structure. The corresponding values are found in Table 1 and show that the optimization mean error at all nodes is between \(0.12\) and \(0.39\,{\text{mm}}\), which is the same order of magnitude for all examples.
To experimentally validate the feasibility of our approach, the selected structures are 3D printed on a Stratasys Objet500 Connex3 printer with the default options of the digital material mode. The structures “2D Sine” and “3D Sine” are heated up to 80 °C in an Instron 3119 environmental chamber. The actuation displacement is applied to the structure “2D Sine” by an Instron ElectroPuls E3000 mechanical testing machine. Solid material is added to the left and to the right of the structure, where the tensile grippers of the testing machine are attached. The structure “3D Sine” is deformed in the same testing machine using compression plates. Since the structure “Airfoil” is too big to fit in the environmental chamber, the structure is mounted to a rig, immersed in a 400 mm × 400 mm × 400 mm glass container filled with 80 °C hot water, and actuated manually with a metal rod. On the right side of Fig. 4, the fabricated structures are shown in the deformed state at 80 °C. Both fabricated materials VW and HT have the same color and cannot be distinguished in the 3D-printed structures. The target shapes are indicated by transparent blue dashed lines, respectively. In Fig. 4a, the full deformed structure as computed by the FE analysis is additionally shown on top of the fabricated structure with transparent blue lines. For better comparability, the fabricated structures in Fig. 4b and c are shown in the front view. In Fig. 4c, the thin solid green line indicates the initial shape of the “Airfoil” structure.
The deviations between the experimentally deformed structures and the numerically predicted shapes at the target nodes are determined with the image processing software ImageJ [35]. For the 3D structures, the experimental error is measured at all visible nodes in the front view. The mean error at the target nodes \(e_{{{\text{exp}},{\text{mean}}}}\) is reported in Fig. 4 on the right side below the images and in Tab. 1. Even though the simulated structures match the target shapes with almost no visible deviations, small deviations on the millimeter scale are visible between the target shapes and the fabricated structures. The mean error is smaller than 0.6 mm for the examples “3D Sine” and “Airfoil”, but it is significantly larger for the structure “2D Sine” with 1.28 mm. Further it can be observed that individual beams buckle, which does not occur in the linear-elastic truss FE simulation.
The material distribution in the structure “2D Sine” is almost symmetric with respect to the horizontal symmetry line. The more rigid HT material, which dominates the structural behavior, forms a re-entrant shaped cell in the left part of the structure and causes the structure to expand vertically due to its auxetic behavior. In the right part of the structure, HT material is found at the top and bottom, which are connected to the actuated nodes on the right side by diagonal lines of HT material. Like this, these regions of the structure are “pulled” towards the center and the structure contracts. Between the left side and the right side of the structure, a vertical line of HT material prevents large vertical displacements of the top and bottom. The material distributions in Fig. 4b and c are more complex and geometric patterns that are related to the deformation behavior of the structures as in Fig. 4a are not directly visible anymore.
Figure 5 shows the convergence behavior of the three optimization runs during both optimization steps of the structures in Fig. 4 and the corresponding computation times. All computations are carried out on a commercially available Dell XPS 15 9500 notebook with an Intel Core i7-10750H processor. The computation times are in the order of magnitude of seconds to minutes, and the computation time increases with the complexity of the problem. Oscillations at low objective function values, as seen for example in the black “Airfoil” line, occur as the algorithm tries to find mechanically valid configurations while simultaneously reducing the objective function. The objective function values increase between the end of the first and the beginning of the second step as the remaining intermediate values are discretized. This effect is most noticeable in the “Airfoil” example as its objective function is generally higher than for the other examples.
To experimentally show the reversibility of the programmed deformations, the structure “3D Sine” is cooled down for about 1 minute after its deformation at 80 °C by removing the environmental chamber. Next, the upper compression plate, which was used to apply the input displacements, is moved back up to the initial position. The structure is now in a deformed, stable state where both materials have again the same Young’s modulus of about 2100 MPa. To recover the initial shape, the environmental chamber is reinstalled, heated up to 80 °C, and the recovery of the structure is visually recorded. Figure 6 shows the recovery of the “3D sine” structure at 80 °C. The structure is initially fixed in the deformed shape and stable at room temperature. Upon heating up to 80 °C, the structure slowly recovers its initial shape, which is reached after about 6 min.
To investigate the mechanical behavior of structures in the deformed state, the structure “3D Sine” is mechanically tested at room temperature in both the initial state and in the deformed state, respectively. One structure in each state is compressed vertically until failure in the Instron ElectroPuls E3000 testing machine with a testing speed of 2.5 mm/min. The load and displacement data are recorded. Figure 7 shows the load carrying capacities of the “3D Sine” structure in the initial state (solid black line) and in the deformed state (dashed black line) at room temperature. In its initial state, the structure can carry a maximum compressive load of about 1250 N, while in the deformed state the maximum load is reduced by about 62% to about 470 N. The maximum displacement before failure however increases from about 2.6 mm to about 4.4 mm. While the load-displacement curve is theoretically linear at the beginning, some slack is observed here as not all parts at the top and bottom of the lattice are immediately touching the compression plates of the testing machine due to small fabrication inaccuracies. The blue solid and blue dashed lines indicate the stiffness of the structure in the initial and in the deformed state as predicted by the linear-elastic, truss FE simulation. It can be observed that the stiffness is almost the same, which indicates that the reduction in stiffness observed in the experiments does not stem from the shape change but rather from the buckling of the individual struts.