# Development of a Material Design Space for 4D-Printed Bio-Inspired Hygroscopically Actuated Bilayer Structures with Unequal Effective Layer Widths

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Development of Mechanical Models

_{a}, t

_{p}and Young’s moduli E

_{a}, E

_{p}of the active and passive layer, respectively, and the total thickness h = t

_{a}+ t

_{p}of the beam, Timoshenko’s equation provides an analytical solution for the curvature radius r and the curvature κ caused by a swelling strain ε

_{swell}in the longitudinal direction in the active layer.

_{a}, b

_{p}into the equation. The effective layer widths b

_{a}and b

_{p}were obtained by subtracting the width of gaps g between adjacent printing strands in the longitudinal direction from the total width of the respective layer and thus only considering the materialized part of the total width (Figure 1a). The analytical solution for the curvature then reads:

#### 2.2. Measurements of Established System’s Parameters

#### 2.2.1. Sample Production

#### 2.2.2. Measurements of Young’s Modulus and Swelling Strains

_{a}and E

_{p}and the swelling strain ε

_{swell}have to be determined for the curved humid and dry states. Additionally, for the deflection due to self-weight, Young’s moduli in wet state are required.

#### 2.2.3. Bilayer Deflection and Curvature Measurements

## 3. Results

#### 3.1. Material Parameters of the Established Bilayer System

_{swell}= 2.6% from wet to humid and 3.5% from wet to dry state for batch 1 (2.8% and 3.7% for batch 2, respectively). For PLA no shrinkage could be observed (Figure 3).

#### 3.2. Comparison of Measured Bilayer Deflections and Curvatures with Computed Values

## 4. Discussion

#### 4.1. Evaluation of Computed Deflections and Curvatures

_{p}the analytical model predicts an opposite trend for the curvature. Additionally, all values show slight variations between measured and calculated results. To examine the potential of the modified model in Equation (1) to predict the experimentally measured curvature, we fit the swelling strain ε

_{swell}and the ratio of Young’s moduli n to the measured curvature values of the experiments by least squares approximation. Thus, we were able to separate effects from strongly scattering material parameters from our validation of the structural model. Here, we used again the hypothetical, median-based cantilever geometries together with the medians of measured curvatures (Table 2).

_{p}(Figure 5).

_{swell}from the optimization approach (Table 2) are slightly larger than those measured in single-material experiments. These differences can be explained by inaccuracies of length measurements in wet state. The extremely soft material might have been compressed during the manual measurements, resulting in an underestimation of wet sample length. The resulting ratios of Young’s moduli n from the optimization approach differ strongly from those measured in all single-material tests, where we found ratios of about n = 40. This could be due to a violation of the assumption of linear material behavior, which is a major element of Timoshenko’s model [23]. Tensile tests showed the end of the elastic range at approximately 2% for Laywood in humid and in dry state (Supplementary Materials File S1). Additionally, in single-material deflection tests of humid and dry state, a maximum elastic strain of only approximately 0.1% could be calculated (Supplementary Materials File S2), while for curved bilayers a maximum elastic strain of about 5% resulted from calculations using the modified model, which means an exceedance of the elastic limit (Supplementary Materials Files S3 and S4).

#### 4.2. Development of a Design Space

_{a}= 1.43 mm in dry and t

_{a}= 1.45 mm in wet state (Figure 7). The density of the passive material is chosen as measured with ρ

_{p}= 1.43 mg/mm

^{3}and the passive layer’s Young’s moduli E

_{p}= 1785 MPa (lower surface in Figure 7b) and E

_{p}= 1000 MPa (upper surface in Figure 7b) are considered.

_{dry}= 30. Below this ratio, small changes of n

_{dry}lead to drastic changes in the resulting curvature. The swelling strain, however, has only a linear influence on the curvature. When analyzing deflections, the active layer’s density ρ

_{a}and n

_{wet}have an almost linear influence on the deflection. It can be lowered by decreasing either ρ

_{a}or n

_{wet}. Another option would be the simultaneous increase of E

_{p}and E

_{a}.

#### 4.3. Usage and Limitations of Material Design Space for Material Selection or Development

_{dry}and swelling strains. Here, attention has to be drawn to existing kinks in the solution surfaces, as can be seen in the exemplary solution in Figure 7a. The values should not be chosen too close-by to achieve a stable prediction, if the material parameters used later for the actual bilayer system show typical scattering due to production errors. If commercially available or already developed materials are used, the corresponding value n

_{wet}, which will differ from n

_{dry}, if material’s stiffnesses are moisture dependent, can be calculated. If not, we recommend an educated guess to determine the range of n

_{wet}. With the set value of n

_{wet}, absolute values for E

_{a}and E

_{p}can be chosen from the analytical solution for deflections, leading to a resulting density of the active material. With these values at hand, it is possible to reach the target values for curvature and deflection. Additionally, further fine-tuning or variation of curvature and deflection can be realized by reiterating on the geometric parameters, such as layer thicknesses and spacing. This can also be useful to program the timing of the curvature process [18], which is so far not included in the presented mechanical model and material design space.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{a}, b

_{a}) and passive layer (t

_{p}, b

_{p}) and the total thickness of the strip h = t

_{a}+ t

_{p}. The material parameters are the Young’s moduli of both layers (E

_{a}, E

_{p}) and the longitudinal swelling strain of the active layer ε

_{swell}.

_{swell}in the active layer, prevention of the passive layer from pure expansion leads to an axial compressive force P

_{a}in the active layer. As we assume no additional external force loads, static equilibrium states a tensile force P

_{p}in the passive layer, equal in amount: ${P}_{\mathrm{a}}={P}_{\mathrm{p}}=P$.

## Appendix B

Active Layer Laywood Meta 5 | Passive Layer PLA | |
---|---|---|

Layer height | 0.2 mm | 0.2 mm |

E-value | 0.040 | 0.033 |

Nozzle temperature | 210 °C | 210 °C |

Bed temperature | 55 °C | 55 °C |

Printing speed (feed rate) | 1200 mm/min | 1200 mm/min |

## Appendix C

**Table A2.**Hypothetical, median-based cantilever geometry and material input data for deflection calculations of bilayer experiments A, B and C in wet state.

Experiment | t_{a}mm | t_{p}mm | b_{a}mm | b_{p}mm | L mm | ρ_{a}mg/mm ^{3} | ρ_{p}mg/mm ^{3} | E_{a}MPa | E_{p}MPa | |
---|---|---|---|---|---|---|---|---|---|---|

wet state | A | free | 0.15 | 20.61 | 8.45 | 72.94 | 1.03 | 1.43 | 6.3 | 1785.4 |

B | 1.45 | free | 20.69 | 9.10 | 73.11 | 1.02 | 1.03 | 9.7 | 1785.4 | |

C | 1.46 | 0.22 | 20.69 | free | 72.64 | 1.02 | 0.99 | 9.7 | 1785.4 |

**Table A3.**Hypothetical, median-based cantilever geometry and material input data for curvature calculations of bilayer experiments A, B and C, in humid and in dry state.

Experiment | t_{a} in mm | t_{p} in mm | b_{a} in mm | b_{p} in mm | E_{a} in MPa | E_{p} in MPa | ε_{swell} | |
---|---|---|---|---|---|---|---|---|

humid state | A | free | 0.15 | 20.42 | 8.45 | 37.6 | 2188.7 | 0.026 |

B | 1.38 | free | 20.43 | 9.10 | 61.9 | 2188.7 | 0.028 | |

C | 1.37 | 0.22 | 20.48 | free | 61.9 | 2188.7 | 0.028 | |

dry state | A | free | 0.15 | 20.30 | 8.45 | 99.0 | 2467.9 | 0.035 |

B | 1.36 | free | 20.36 | 9.10 | 110.4 | 2467.9 | 0.037 | |

C | 1.37 | 0.22 | 20.37 | free | 110.4 | 2467.9 | 0.037 |

## Appendix D

^{0}-continuous. Additionally, transverse shear effects are considered layer-wise. We performed simulations using the commercial finite element software ANSYS (Release 20 R1, ANSYS Inc. Canonsburg, PA, USA). We modelled the bilayer cantilever beams using two stacked layers of eight-node structural solid shell finite elements (SOLSH190) that are capable of handling large rotations and large strains and we used geometrically non-linear static analysis with a load controlled path following algorithm. Convergence studies showed that a mesh of 100 elements in the longitudinal direction and 30 elements in the direction of the cantilever width are needed per layer. To account for the porous mesostructure of the 3D-printed layers, Young’s modulus of the passive layer was scaled, using the ratio of effective layer widths. The cantilever beam was clamped at one end and we applied a temperature load case, while defining the thermal expansion coefficient of the active layer in the longitudinal direction as the swelling strain ε

_{swell}. For post-processing, we defined three nodes in the longitudinal direction, at both ends and in the middle of the cantilever, all located on the mid-surface. From the displacements of these nodes, we calculated the curvature radius and the curvature of the resulting arc. For the material, we choose a linear elastic definition on the one hand and a hyperelastic Neo-Hookean material law on the other hand, using the measured Young’s moduli and a Poisson’s ratio of ν = 0.0.

_{a}= 1.38 mm), the differences in curvature results between the modified Timoshenko model, which corresponds to CLT, and the layer-wise modeling and simulation approach using a linear elastic material law are small. Consideration of a more realistic hyperelastic material law shows differing results for the highly curved samples that experience compressive elastic strains (Supplementary Materials File S3).

**Figure A1.**Computed curvatures for Laywood/PLA bilayers of experiment B in humid state using different modeling approaches. Open circles: Modeling and simulation based on layer-wise theory using ANSYS assuming linear elastic material. Open triangles: Modeling and simulation based on layer-wise theory using ANSYS assuming hyperelastic material. Open squares: Modeling based on classical laminate theory using the modification of Timoshenko’s model in Equation (1). For highly curved samples with compressive strains, the more realistic hyperelastic model shows an increasing difference.

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**Figure 1.**Geometry of bilayer samples. (

**a**) Schematic of bilayer sample consisting of active (blue) and passive (red) layer. l: Length of bilayer. t

_{p}: Thickness of passive layer. t

_{a}: Thickness of active layer. w: Width of printed strands (extrusion width). g: Gap between adjacent strands. s: Spacing between adjacent printing paths. Effective width of active layer b

_{a}= b

_{tot}. Effective width of passive layer was calculated as b

_{p}= b

_{tot}− n

_{gap}∙ g, with number of gaps n

_{gap}. Filament deposition along the x-axis was defined as transversal, along the y-axis as longitudinal. Single-material layers of the same geometry were used as samples for measurements of Young’s moduli. (

**b**) Photograph of bilayer sample in dry and therefore curved state. Grey strands of passive material are visible.

**Figure 2.**Evaluation of deflection and curvature. (

**a**) Deflected sample in wet state. Measurement of deflection as the vertical line between sample base and tip; (

**b**) curved bilayer sample in dry state. Evaluation of curvature by selecting three points on the sample, getting the defined triangle and calculating the radius of its circumcircle.

**Figure 3.**Length development of repeatedly tested samples of Laywood meta 5 and PLA. Blue: wet state. Gray: humid state (80% RH). Red: dry state (30% RH). Roman numerals refer to rinsing cycles. Laywood 1 and 2 refer to different batches of material. No differences can be seen between the second and third rinsing cycles in Laywood. Only Laywood samples show a major length change between humidity levels. N = 8.

**Figure 4.**Measured and computed values for deflection and curvature of Laywood/PLA bilayers. (

**a**–

**c**) Blue: Deflection in wet state; (

**d**–

**f**) Grey: Curvature values in humid state; (

**g**–

**i**) Red: Curvature values in dry state. Dark squares: measured values. Light circles: computed values using the analytical models and geometric input values as measured for every individual sample. Lines represent the solution of the analytical models for a hypothetical bilayer cantilever with median values of measured geometries. All computed deflections are in good accordance with the measured values. For increasing thicknesses of active and passive layers, the computed curvatures show similar trends to the measured values with slight differences in absolute numbers. For increasing passive layer effective width, the computed and measured curvatures show opposing trends.

**Figure 5.**Analytical curves with optimized ratios of Young’s moduli and swelling strains. Squares: Median values from experiments. Solid line: Using optimized parameters for the modified model. Dashed line: Using optimized parameters for Timoshenko’s original model. For Experiment A with different active layer thickness t

_{a}, lines from optimizations are congruent. Both models are able to depict the measured trends of experiments A and B, while only the modified model is able to reproduce a decreasing trend of curvature for increasing passive layer effective width b

_{p}in experiment C.

**Figure 6.**Length development of Laywood meta 5 monolayers from deconstructed bilayers. Blue: wet state; black: humid state; red: dry state; A1 to A5: increasing active layer thickness; B1 to B5: increasing passive layer thickness; C1 to C5: decreasing passive layer width. Laywood 1 and 2 refer to different batches of used Laywood meta 5 filament. Dashed lines mark the median length of pure Laywood meta 5 monolayers. With increasing passive layer thickness, a distinct increase in active layer length is visible. N = 7.

**Figure 7.**Exemplary design space for the choice of parameters of an active material for a given geometry and different assumptions on the passive material. (

**a**) Solution surface of curvature for various swelling strains ε

_{swell}and ratios of Young’s moduli n. Colors indicate magnitude of curvature. (

**b**) Solution surfaces of deflection for various densities ρ

_{a}and ratios of Young’s moduli n. The lower surface corresponds to an absolute value of the passive layer’s Young’s modulus E

_{p}= 1785 MPa, the upper surface corresponds to E

_{p}= 1000 MPa. For both surfaces the density of the passive material is chosen as ρ

_{p}= 1.43 mg/mm

^{3}. Colors indicate magnitude of deflection. Black dots mark the solutions for the given Laywood/PLA bilayers of samples from experiment A with t

_{a}= 1.43 mm in dry and t

_{a}= 1.45 mm in wet state.

**Table 1.**Young’s moduli in all states. Measurements are given as median (IQR). Values were recalculated from deflections under self-weight. N = 8.

Young’s Modulus in MPa | |||
---|---|---|---|

Wet | Humid | Dry | |

Laywood, batch 1 | 6.3 (0.8) | 37.6 (11.4) | 99.0 (12.8) |

Laywood, batch 2 | 9.7 (0.5) | 61.9 (10.5) | 110 (36.9) |

PLA | 1785 (558) | 2189 (303) | 2467 (569) |

**Table 2.**Results for least squares fit of material parameters n and

**ε**

**to measured curvatures in humid and in dry state for original and modified Timoshenko model.**

_{swell}Modified Model (Equation (1)) | Original Model (Timoshenko) | ||||
---|---|---|---|---|---|

Experiment | n | ε_{swell} | n | ε_{swell} | |

Humid state | A | 691.2 | 0.034 | 286.0 | 0.034 |

B,C | 610.4 | 0.043 | 149.2 | 0.037 | |

Dry state | A | 561.9 | 0.045 | 233.9 | 0.045 |

B,C | 299.7 | 0.051 | 111.6 | 0.048 |

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**MDPI and ACS Style**

Krüger, F.; Thierer, R.; Tahouni, Y.; Sachse, R.; Wood, D.; Menges, A.; Bischoff, M.; Rühe, J. Development of a Material Design Space for 4D-Printed Bio-Inspired Hygroscopically Actuated Bilayer Structures with Unequal Effective Layer Widths. *Biomimetics* **2021**, *6*, 58.
https://doi.org/10.3390/biomimetics6040058

**AMA Style**

Krüger F, Thierer R, Tahouni Y, Sachse R, Wood D, Menges A, Bischoff M, Rühe J. Development of a Material Design Space for 4D-Printed Bio-Inspired Hygroscopically Actuated Bilayer Structures with Unequal Effective Layer Widths. *Biomimetics*. 2021; 6(4):58.
https://doi.org/10.3390/biomimetics6040058

**Chicago/Turabian Style**

Krüger, Friederike, Rebecca Thierer, Yasaman Tahouni, Renate Sachse, Dylan Wood, Achim Menges, Manfred Bischoff, and Jürgen Rühe. 2021. "Development of a Material Design Space for 4D-Printed Bio-Inspired Hygroscopically Actuated Bilayer Structures with Unequal Effective Layer Widths" *Biomimetics* 6, no. 4: 58.
https://doi.org/10.3390/biomimetics6040058