# A Review on Tailoring Stiffness in Compliant Systems, via Removing Material: Cellular Materials and Topology Optimization

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## Abstract

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## 1. Introduction

## 2. Tailoring Stiffness via Cellular Materials

#### Cellular Materials in Compliant Systems as Flexure Elements

## 3. Tailoring Stiffness via Topology Optimization

#### Topology Optimization in Compliant Systems as Flexure Elements

## 4. Remarks on Tailoring Stiffness via Cellular Materials and Topology Optimization

## 5. Conclusions

- Due to their advantages in tailoring the mechanical properties, exploring the use of closed-cell structures, 3D arrays, auxetic, and zero-Poisson architectures in tailoring the stiffness of flexure elements could lead to benefits compliant systems.
- As flexures base their functioning on the elastic response of their elements, other material properties, i.e., Poisson’s ratio, could complicate their use to build up other kinematic pairs, e.g., cylindrical. However, a correct selection of cellular pattern could be designed so that it restricts the Poisson’s ratio to 0. This opens up a new range of possibilities in compliant systems applications.
- Combining both Cellular Materials and Topology Optimization approaches (TOAUC, MTOCM, and TOFCM) to expand the possibilities in tailoring stiffness of flexure elements. This combination leads to obtain the best out of each. An example of the use of these two could be by implementing Topology Optimization at the conceptual design stage for the proposal of a flexure element. Then, employ a Cellular Material to adjust the stiffness by modifying the parameters that define the structure.
- Analytical parameterization of the Cellular Material employed can result in an accurate and precise tracking of the stiffness. This allows a controlled tailoring of stiffness in compliant systems.
- Another aspect to keep in mind is that lattice honeycombs might have in-plane anisotropy. Therefore, when used in flexures, the proper selection of orientation becomes crucial. Failure in considering this could lead to deformations in non-desired directions, i.e., parasitic motions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MEMS | Microelectromechanical Systems |

3D | Three-dimensional |

DOF | Degrees of Freedom |

2D | Two-dimensional |

SIMP | Solid Isotropic Material with Penalization |

TORCP | Topology Optimization Resulting in Cellular Patterns |

TOAUC | Topology Optimization Applied to Unit Cells |

MTOCM | Multiscale Topology Optimization with Cellular Materials |

TOFCM | Topology Optimization of Functionally-graded Cellular Materials |

SARFJ | Single Axis Rotational Flexure Joints |

FEA | Finite Element Analysis |

QVFH | Quasi V-shaped Flexure Hinge |

MCFH | Multi-Cavity Flexure Hinge |

QLPFH | Quasi-Leaf Porous Flexure Hinge |

SATFJ | Single-Axis Translational Flexure Joints |

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**Figure 2.**Process to tailor stiffness of flexure elements via (

**a**) Cellular Materials and (

**b**) Topology Optimization (Figure based on that in [52]).

**Figure 3.**Typical cellular structures: (

**a**) 2D cellular structure known as “Honeycomb”, (

**b**) 3D open-cell structure (lattice structure), and (

**c**) 3D closed-cell structure. The structures in subfigures (

**b**,

**c**) are 3D structures termed as “Foams”.

**Figure 5.**(

**a**) Complete schematic overview of airfoil morphing and main categories of 2D cellular structures, and (

**b**) unit-cell of the zero-Poisson’s ratio cellular material termed as“Fish cell” (figure adapted from in [125]).

**Figure 6.**Unit cells of some of the most commonly used cellular structures for tailoring stiffness. Unit cells for increase stiffness are categorized in: 2D lattice structures (square [92], triangular [93], and kagome [100]), 3D-lattice structures (octet truss [94], and body centered cubic [81]), and closed-cell structures (body centered cubic [96]). Unit cells for reduce stiffness are only in the category of 2D lattice structures, and they are separated according to their Poisson’s ratio behavior in: positive Poisson’s ratio (hexagonal [119]), auxetic (hexagonal chiral [116], and re-entrant [117]), and zero Poisson’s ration (chevron [122], and accordion [121]). Some of the unit cells were generated using nTopology

^{®.}

**Figure 9.**Process to tailor stiffness using Topology Optimization applied to the unit cells of Cellular Materials.

**Figure 10.**Process to tailor stiffness using the multiscale process that combines Topology Optimization with Cellular Materials.

**Figure 11.**Process to tailor stiffness using Topology Optimization applied to cellular design domains that leads to functionally graded Cellular Materials.

**Figure 12.**Schematic description of the Topology Optimization process to design the optimal compliant limbs of 2D compliant parallel mechanisms: (

**a**) 3-DoF ($x,y,\theta $) compliant parallel mechanism proposed by Lum et al. [200] (Adapted from [200] 2015, Elsevier); subfigures (

**b**,

**c**) are the 2-DoF ($x,y$) and the 3-DoF ($x,y,\theta $) compliant parallel mechanisms, respectively, proposed by Jin et al. found in (i) [201] (Adapted from [201] 2016, Elsevier); and (ii) [202] (Adapted from [202] 2018, ASME).

**Figure 13.**Schematic description of the Topology Optimization process to design the optimal configuration of SARFJ using a 2D rectangular design domain. Optimal designs were obtained in (and adapted from) (

**a**) [207]; (

**b**) [208]; (

**c**) [209]; (

**d**) [210] i-design based on linear FEA and ii-design based on nonlinear FEA; (

**e**) [211]; (

**f**) [212] i-design with less stress allowed for a mesh of 50 × 50 and ii-design with less stress allowed for a mesh of 100 × 100; (

**g**) [213]; (

**h**) [214] i-design for general distributed flexure hinges, ii-design for distributed flexure hinges with prescribed compliances, and iii-design for distributed flexure hinges with minimal parasitic motion; and (

**i**) [56] i-optimal topology, ii-simplified design named “Snowflake”, and iii-simplified design named “‘Half-Snowflake”.

**Figure 14.**Schematic description of the Topology Optimization process to design the optimal configuration of SARFJ using a 2D design domain based on the geometry of a right circular flexure hinge. Optimal designs obtained: (

**a**) i-optimal topology and ii-final shape (Adapted with permission from ref. [215]. 2021, Elsevier) and (

**b**) optimal topology obtained in [209] (Adapted from ref. [209]. 2019, Springer).

**Figure 15.**Schematic description of the Topology Optimization process to design the optimal configuration of SARFJ using a 3D cubic design domain. Optimal designs obtained in (

**a**) i-optimal topology and ii-final shape (Adapted with permission from ref. [220]. 2021, Elsevier), (

**b**) i-optimal topology and ii-final shape (Adapted from ref. [221]. 2020, Springer).

**Figure 17.**Schematic description of the Topology Optimization process to design the optimal configuration of SATFJ using a 2D design domain based on the geometry of blade flexures. (

**a**) Optimal design obtained in [52] i-optimal topology and ii-final shape (reprinted with permission from [52]. 2019, Elsevier).

**Figure 18.**Summary diagram of the Cellular Materials and Topology Optimization approaches and their combinations.

**Table 1.**Stiffness reduction via Cellular Materials: Airfoil-core morphing. Principal works found in literature by author and cellular structure used (P: positive Poisson, Z: zero-Poisson, A: auxetic). Young’s modulus ratios ${E}_{c}/E$ between the obtained by the application of Cellular Material, ${E}_{c}$, and that of the parent material E for in-plane ${E}_{cx}/E$ and ${E}_{cy}/E$ stiffness, and percentage of weight reduction. “−” indicates that there is no sufficient information reported to obtain that parameter. The percentage of weight reduction was obtained based on the reported relative density $\overline{\rho}$ of the cellular structure.

Author | Cellular | Young’s Modulus Ratios | Weight Reduction | |
---|---|---|---|---|

Structure | $\left({\mathit{E}}_{\mathbf{cx}}/\mathit{E}\right)\times {10}^{5}$ | $\left({\mathit{E}}_{\mathbf{cy}}/\mathit{E}\right)\times {10}^{5}$ | $\left(\%\right)$ | |

Bornengo et al. [116] | A—Hex. Chiral | 1.18–0.60 | 1.18–0.60 | 90−95 |

Dong et al. [117] | A—Re-entrant | 2.51 | 1.35 | − |

Zhang et al. [118] | P—Cross-shaped | 0.25 | 0.13 | − |

Heo et al. [119] | A—Hex. Chiral | 11.58 | 11.58 | 90 |

P—Hexagonal | 69.32 | 69.32 | 94 | |

A—Re-entrant | 427.19 | 427.19 | 81 |

**Table 2.**Stiffness reduction via Cellular Materials: Airfoil-skin morphing. Principal works found in literature by author and cellular structure used (P: positive Poisson, Z: zero-Poisson, A: auxetic). Main direction Young’s modulus ratio ${E}_{cx}/E$, and shear stiffness ratio ${G}_{c}/E$. “−” indicates that there is no sufficient information reported to obtain that parameter.

Author | Cellular | Young’s Modulus Ratios | |
---|---|---|---|

Structure | ${\mathit{E}}_{\mathbf{cx}}/\mathit{E}$ | ${\mathit{G}}_{\mathit{c}}/\mathit{E}$ | |

Olympio et al. [121] | Z—hybrid | 0.096–0.0047 | − |

Z—Accordion | 0.0047–0.10 | − | |

Olympio et al. [120] | P—Hexagonal | 0.39–2.88 × ${10}^{-5}$ | 0.022–1.07 × ${10}^{-5}$ |

Chen et al. [122] | Z—Chevron | 0.033–2.92 × ${10}^{-5}$ | 5.2 × ${10}^{-5}$–6.74 × ${10}^{-4}$ |

Vigliotti et al. [123] | Z—Chevron | 3.29 × ${10}^{-6}$ | 2.05 × ${10}^{-2}$ |

Chang et al. [124] | P—Hexagonal | 0.015 | 0.0025 |

Z—Accordion | 0.023 | 0.0026 | |

Zadeh et al. [125,126] | Z—Fish cell | 1.84 × ${10}^{-6}$–5.75 × ${10}^{-7}$ | 1.30 × ${10}^{-7}$ |

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**MDPI and ACS Style**

Arredondo-Soto, M.; Cuan-Urquizo, E.; Gómez-Espinosa, A.
A Review on Tailoring Stiffness in Compliant Systems, via Removing Material: Cellular Materials and Topology Optimization. *Appl. Sci.* **2021**, *11*, 3538.
https://doi.org/10.3390/app11083538

**AMA Style**

Arredondo-Soto M, Cuan-Urquizo E, Gómez-Espinosa A.
A Review on Tailoring Stiffness in Compliant Systems, via Removing Material: Cellular Materials and Topology Optimization. *Applied Sciences*. 2021; 11(8):3538.
https://doi.org/10.3390/app11083538

**Chicago/Turabian Style**

Arredondo-Soto, Mauricio, Enrique Cuan-Urquizo, and Alfonso Gómez-Espinosa.
2021. "A Review on Tailoring Stiffness in Compliant Systems, via Removing Material: Cellular Materials and Topology Optimization" *Applied Sciences* 11, no. 8: 3538.
https://doi.org/10.3390/app11083538