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Sun X, Yue L, Yu L, Forte CT, Armstrong CD, Zhou K, Demoly F, Zhao RR, Qi HJ. Machine learning-enabled forward prediction and inverse design of 4D-printed active plates. Nat Commun 2024; 15:5509. [PMID: 38951533 PMCID: PMC11217466 DOI: 10.1038/s41467-024-49775-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/01/2023] [Accepted: 06/13/2024] [Indexed: 07/03/2024] Open
Abstract
Shape transformations of active composites (ACs) depend on the spatial distribution of constituent materials. Voxel-level complex material distributions can be encoded by 3D printing, offering enormous freedom for possible shape-change 4D-printed ACs. However, efficiently designing the material distribution to achieve desired 3D shape changes is significantly challenging yet greatly needed. Here, we present an approach that combines machine learning (ML) with both gradient-descent (GD) and evolutionary algorithm (EA) to design AC plates with 3D shape changes. A residual network ML model is developed for the forward shape prediction. A global-subdomain design strategy with ML-GD and ML-EA is then used for the inverse material-distribution design. For a variety of numerically generated target shapes, both ML-GD and ML-EA demonstrate high efficiency. By further combining ML-EA with a normal distance-based loss function, optimized designs are achieved for multiple irregular target shapes. Our approach thus provides a highly efficient tool for the design of 4D-printed active composites.
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Affiliation(s)
- Xiaohao Sun
- The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA
| | - Liang Yue
- The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA
| | - Luxia Yu
- The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA
| | - Connor T Forte
- The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA
| | - Connor D Armstrong
- The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA
| | - Kun Zhou
- Singapore Centre for 3D Printing, School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore, Singapore
| | - Frédéric Demoly
- ICB UMR 6303 CNRS, Belfort-Montbeliard University of Technology, UTBM, Belfort, France
- Institut universitaire de France (IUF), Paris, France
| | - Ruike Renee Zhao
- Department of Mechanical Engineering, Stanford University, Stanford, CA, USA
| | - H Jerry Qi
- The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA.
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2
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Czajkowski M, Rocklin DZ. Duality and Sheared Analytic Response in Mechanism-Based Metamaterials. PHYSICAL REVIEW LETTERS 2024; 132:068201. [PMID: 38394578 DOI: 10.1103/physrevlett.132.068201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2022] [Revised: 11/17/2023] [Accepted: 11/28/2023] [Indexed: 02/25/2024]
Abstract
Mechanical metamaterials designed around a zero-energy pathway of deformation known as a mechanism, challenge the conventional picture of elasticity and generate complex spatial response that remains largely uncharted. Here, we present a unified theoretical framework to showing that the presence of a unimode in a 2D structure generates a space of anomalous zero-energy sheared analytic modes. The spatial profiles of these stress-free strain patterns is dual to equilibrium stress configurations. We show a transition at an exceptional point between bulk modes in structures with conventional Poisson ratios (anauxetic) and evanescent surface modes for negative Poisson ratios (auxetic). We suggest a first application of these unusual response properties as a switchable signal amplifier and filter for use in mechanical circuitry and computation.
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Affiliation(s)
- Michael Czajkowski
- School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
| | - D Zeb Rocklin
- School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
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3
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Castro D, Aharoni H. Shape Morphing of Planar Liquid Crystal Elastomers. PHYSICAL REVIEW LETTERS 2023; 130:178101. [PMID: 37172258 DOI: 10.1103/physrevlett.130.178101] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/15/2022] [Accepted: 03/17/2023] [Indexed: 05/14/2023]
Abstract
We consider planar liquid crystal elastomers: two-dimensional objects made of anisotropic responsive materials that remain flat when stimulated, however change their planar shape. We derive a closed form, analytical solution based on the implicit linearity featured by this subclass of deformations. Our solution provides the nematic director field on an arbitrary domain starting with two initial director curves. We discuss the different gauge choices for this problem and the inclusion of disclinations in the nematic order. Finally, we propose several applications and useful design principles based on this theoretical framework.
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Affiliation(s)
- Daniel Castro
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
| | - Hillel Aharoni
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
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4
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Hebner TS, Bowman RGA, Duffy D, Mostajeran C, Griniasty I, Cohen I, Warner M, Bowman CN, White TJ. Discontinuous Metric Programming in Liquid Crystalline Elastomers. ACS APPLIED MATERIALS & INTERFACES 2023; 15:11092-11098. [PMID: 36791283 DOI: 10.1021/acsami.2c21984] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
Liquid crystalline elastomers (LCEs) are shape-changing materials that exhibit large deformations in response to applied stimuli. Local control of the orientation of LCEs spatially directs the deformation of these materials to realize a spontaneous shape change in response to stimuli. Prior approaches to shape programming in LCEs utilize patterning techniques that involve the detailed inscription of spatially varying nematic fields to produce sheets. These patterned sheets deform into elaborate geometries with complex Gaussian curvatures. Here, we present an alternative approach to realize shape-morphing in LCEs where spatial patterning of the crosslink density locally regulates the material deformation magnitude on either side of a prescribed interface curve. We also present a simple mathematical model describing the behavior of these materials. Further experiments coupled with the mathematical model demonstrate the control of the sign of Gaussian curvature, which is used in combination with heat transfer effects to design LCEs that self-clean as a result of temperature-dependent actuation properties.
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Affiliation(s)
- Tayler S Hebner
- Department of Chemical and Biological Engineering, University of Colorado Boulder, 596 UCB, Boulder, Colorado 80309, United States
| | - Riley G A Bowman
- Department of Chemical and Biological Engineering, University of Colorado Boulder, 596 UCB, Boulder, Colorado 80309, United States
| | - Daniel Duffy
- Department of Engineering, University of Cambridge, Cambridge, England CB2 1PZ, U.K
| | - Cyrus Mostajeran
- Department of Engineering, University of Cambridge, Cambridge, England CB2 1PZ, U.K
- School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
| | - Itay Griniasty
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853-2501, United States
| | - Itai Cohen
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853-2501, United States
| | - Mark Warner
- Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom
| | - Christopher N Bowman
- Department of Chemical and Biological Engineering, University of Colorado Boulder, 596 UCB, Boulder, Colorado 80309, United States
- Materials Science and Engineering Program, University of Colorado Boulder, 596 UCB, Boulder, Colorado 80309, United States
| | - Timothy J White
- Department of Chemical and Biological Engineering, University of Colorado Boulder, 596 UCB, Boulder, Colorado 80309, United States
- Materials Science and Engineering Program, University of Colorado Boulder, 596 UCB, Boulder, Colorado 80309, United States
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5
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Feng F, Duffy D, Warner M, Biggins JS. Interfacial metric mechanics: stitching patterns of shape change in active sheets. Proc Math Phys Eng Sci 2022; 478:20220230. [PMID: 35814332 PMCID: PMC9240917 DOI: 10.1098/rspa.2022.0230] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/01/2022] [Accepted: 06/09/2022] [Indexed: 11/22/2022] Open
Abstract
A flat sheet programmed with a planar pattern of spontaneous shape change will morph into a curved surface. Such metric mechanics is seen in growing biological sheets, and may be engineered in actuating soft matter sheets such as phase-changing liquid crystal elastomers (LCEs), swelling gels and inflating baromorphs. Here, we show how to combine multiple patterns in a sheet by stitching regions of different shape changes together piecewise along interfaces. This approach allows simple patterns to be used as building blocks, and enables the design of multi-material or active/passive sheets. We give a general condition for an interface to be geometrically compatible, and explore its consequences for LCE/LCE, gel/gel and active/passive interfaces. In contraction/elongation systems such as LCEs, we find an infinite set of compatible interfaces between any pair of patterns along which the metric is discontinuous, and a finite number across which the metric is continuous. As an example, we find all possible interfaces between pairs of LCE logarithmic spiral patterns. By contrast, in isotropic systems such as swelling gels, only a finite number of continuous interfaces are available, greatly limiting the potential of stitching. In both continuous and discontinuous cases, we find the stitched interfaces generically carry singular Gaussian curvature, leading to intrinsically curved folds in the actuated surface. We give a general expression for the distribution of this curvature, and a more specialized form for interfaces in LCE patterns. The interfaces thus also have rich geometric and mechanical properties in their own right.
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Affiliation(s)
- Fan Feng
- Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK
| | - Daniel Duffy
- Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK
| | - Mark Warner
- Department of Physics, University of Cambridge, Cambridge CB3 0HE, UK
| | - John S. Biggins
- Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK
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6
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Kam D, Levin I, Kutner Y, Lanciano O, Sharon E, Shoseyov O, Magdassi S. Wood Warping Composite by 3D Printing. Polymers (Basel) 2022; 14:polym14040733. [PMID: 35215644 PMCID: PMC8877370 DOI: 10.3390/polym14040733] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/03/2022] [Revised: 02/08/2022] [Accepted: 02/12/2022] [Indexed: 11/16/2022] Open
Abstract
Wood warping is a phenomenon known as a deformation in wood that occurs when changes in moisture content cause an unevenly volumetric change due to fiber orientation. Here we present an investigation of wood warped objects that were fabricated by 3D printing. Similar to natural wood warping, water evaporation causes volume decrease of the printed object, but in contrast, the printing pathway pattern and flow rate dictate the direction of the alignment and its intensity, all of which can be predesigned and affect the resulting structure after drying. The fabrication of the objects was performed by an extrusion-based 3D printing technique that enables the deposition of water-based inks into 3D objects. The printing ink was composed of 100% wood-based materials, wood flour, and plant-extracted natural binders cellulose nanocrystals, and xyloglucan, without the need for any additional synthetic resins. Two archetypal structures were printed: cylindrical structure and helices. In the former, we identified a new length scale that gauges the effect of gravity on the shape. In the latter, the structure exhibited a shape transition analogous to the opening of a seedpod, quantitatively reproducing theoretical predictions. Together, by carefully tuning the flow rate and printing pathway, the morphology of the fully dried wooden objects can be controlled. Hence, it is possible to design the printing of wet objects that will form different final 3D structures.
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Affiliation(s)
- Doron Kam
- Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel; (D.K.); (Y.K.); (O.L.)
- Plant Sciences and Genetics in Agriculture, the Hebrew University of Jerusalem, Rehovot 7610001, Israel
| | - Ido Levin
- Racah Institute of Physics, the Hebrew University, Jerusalem 9190401, Israel;
| | - Yinnon Kutner
- Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel; (D.K.); (Y.K.); (O.L.)
- Alpha Program, Future Scientist Center, Jerusalem 9190401, Israel
| | - Omri Lanciano
- Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel; (D.K.); (Y.K.); (O.L.)
| | - Eran Sharon
- Racah Institute of Physics, the Hebrew University, Jerusalem 9190401, Israel;
- Correspondence: (E.S.); (O.S.); (S.M.)
| | - Oded Shoseyov
- Plant Sciences and Genetics in Agriculture, the Hebrew University of Jerusalem, Rehovot 7610001, Israel
- Correspondence: (E.S.); (O.S.); (S.M.)
| | - Shlomo Magdassi
- Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel; (D.K.); (Y.K.); (O.L.)
- Correspondence: (E.S.); (O.S.); (S.M.)
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7
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Meiri S, Efrati E. Cumulative geometric frustration in physical assemblies. Phys Rev E 2021; 104:054601. [PMID: 34942847 DOI: 10.1103/physreve.104.054601] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2021] [Accepted: 10/13/2021] [Indexed: 11/07/2022]
Abstract
Geometric frustration arises whenever the constituents of a physical assembly locally favor an arrangement that cannot be realized globally. Recently, such frustrated assemblies were shown to exhibit filamentation, size limitation, large morphological variations and other exotic response properties. While these unique characteristics can be shown to be a direct outcome of the geometric frustration, some geometrically frustrated systems do not exhibit any of the above phenomena. In this work we exploit the intrinsic approach to provide a framework for directly addressing the frustration in physical assemblies. The framework highlights the role of the compatibility conditions associated with the intrinsic fields describing the physical assembly. We show that the structure of the compatibility conditions determines the behavior of small assemblies and in particular predicts their superextensive energy growth exponent. We illustrate the use of this framework to several well-known frustrated assemblies.
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Affiliation(s)
- Snir Meiri
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
| | - Efi Efrati
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
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8
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Duffy D, Javed M, Abdelrahman MK, Ware TH, Warner M, Biggins JS. Metric mechanics with nontrivial topology: Actuating irises, cylinders, and evertors. Phys Rev E 2021; 104:065004. [PMID: 35030939 DOI: 10.1103/physreve.104.065004] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/20/2021] [Accepted: 11/22/2021] [Indexed: 11/07/2022]
Abstract
Liquid crystal elastomers contract along their director on heating and recover on cooling, offering great potential as actuators and artificial muscles. If a flat sheet is programed with a spatially varying director pattern, then it will actuate into a curved surface, allowing the material to act as a strong machine such as a grabber or lifter. Here we study the actuation of programed annular sheets which, owing to their central hole, can sidestep constraints on area and orientation. We systematically catalog the set of developable surfaces encodable via axisymmetric director patterns and uncover several qualitatively new modes of actuation, including cylinders, irises, and everted surfaces in which the inner boundary becomes the outer boundary after actuation. We confirm our designs with a combination of experiments and numerics. Many of our actuators can reattain their initial inner or outer radius upon completing actuation, making them particularly promising, as they can avoid potentially problematic stresses in their activated state even when fixed onto a frame or pipe.
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Affiliation(s)
- D Duffy
- Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, United Kingdom
| | - M Javed
- Department of Bioengineering, University of Texas at Dallas, Richardson, Texas 75080, USA.,Department of Biomedical Engineering, Texas A&M University, College Station, Texas 77843, USA
| | - M K Abdelrahman
- Department of Bioengineering, University of Texas at Dallas, Richardson, Texas 75080, USA.,Department of Materials Science and Engineering, Texas A&M University, College Station, Texas 77843, USA
| | - T H Ware
- Department of Bioengineering, University of Texas at Dallas, Richardson, Texas 75080, USA.,Department of Biomedical Engineering, Texas A&M University, College Station, Texas 77843, USA.,Department of Materials Science and Engineering, Texas A&M University, College Station, Texas 77843, USA
| | - M Warner
- Department of Physics, University of Cambridge, 19 JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - J S Biggins
- Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, United Kingdom
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9
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Griniasty I, Mostajeran C, Cohen I. Multivalued Inverse Design: Multiple Surface Geometries from One Flat Sheet. PHYSICAL REVIEW LETTERS 2021; 127:128001. [PMID: 34597088 DOI: 10.1103/physrevlett.127.128001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/09/2021] [Revised: 05/19/2021] [Accepted: 08/04/2021] [Indexed: 06/13/2023]
Abstract
Designing flat sheets that can be made to deform into three-dimensional shapes is an area of intense research with applications in micromachines, soft robotics, and medical implants. Thus far, such sheets were designed to adopt a single target shape. Here, we show that through anisotropic deformation applied inhomogeneously throughout a sheet, it is possible to design a single sheet that can deform into multiple surface geometries upon different actuations. The key to our approach is development of an analytical method for solving this multivalued inverse problem. Such sheets open the door to fabricating machines that can perform complex tasks through cyclic transitions between multiple shapes. As a proof of concept, we design a simple swimmer capable of moving through a fluid at low Reynolds numbers.
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Affiliation(s)
- Itay Griniasty
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853-2501, USA
| | - Cyrus Mostajeran
- Department of Engineering, University of Cambridge, Cambridge, England CB2 1PZ, United Kingdom
| | - Itai Cohen
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853-2501, USA
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10
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Al-Izzi SC, Morris RG. Active flows and deformable surfaces in development. Semin Cell Dev Biol 2021; 120:44-52. [PMID: 34266757 DOI: 10.1016/j.semcdb.2021.07.001] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2021] [Revised: 06/30/2021] [Accepted: 07/01/2021] [Indexed: 12/15/2022]
Abstract
We review progress in active hydrodynamic descriptions of flowing media on curved and deformable manifolds: the state-of-the-art in continuum descriptions of single-layers of epithelial and/or other tissues during development. First, after a brief overview of activity, flows and hydrodynamic descriptions, we highlight the generic challenge of identifying the dependence on dynamical variables of so-called active kinetic coefficients- active counterparts to dissipative Onsager coefficients. We go on to describe some of the subtleties concerning how curvature and active flows interact, and the issues that arise when surfaces are deformable. We finish with a broad discussion around the utility of such theories in developmental biology. This includes limitations to analytical techniques, challenges associated with numerical integration, fitting-to-data and inference, and potential tools for the future, such as discrete differential geometry.
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Affiliation(s)
- Sami C Al-Izzi
- School of Physics and EMBL Australia Node in Single Molecule Science, School of Medical Sciences, University of New South Wales - Sydney, 2052, Australia
| | - Richard G Morris
- School of Physics and EMBL Australia Node in Single Molecule Science, School of Medical Sciences, University of New South Wales - Sydney, 2052, Australia.
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11
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Pedrini A, Virga EG. Ridge energy for thin nematic polymer networks. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2021; 44:7. [PMID: 33616761 PMCID: PMC7900098 DOI: 10.1140/epje/s10189-021-00012-1] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/11/2020] [Accepted: 12/23/2020] [Indexed: 06/10/2023]
Abstract
Minimizing the elastic free energy of a thin sheet of nematic polymer network among smooth isometric immersions is the strategy purported by the mainstream theory. In this paper, we broaden the class of admissible spontaneous deformations: we consider ridged isometric immersions, which can cause a sharp ridge in the immersed surfaces. We propose a model to compute the extra energy distributed along such ridges. This energy comes from bending; it is shown under what circumstances it scales quadratically with the sheet's thickness, falling just in between stretching and bending energies. We put our theory to the test by studying the spontaneous deformation of a disk on which a radial hedgehog was imprinted at the time of crosslinking. We predict the number of folds that develop in terms of the degree of order induced in the material by external agents (such as heat and illumination).
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Affiliation(s)
- Andrea Pedrini
- Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, 27100 Pavia, Italy
| | - Epifanio G. Virga
- Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, 27100 Pavia, Italy
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12
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Duffy D, Biggins JS. Defective nematogenesis: Gauss curvature in programmable shape-responsive sheets with topological defects. SOFT MATTER 2020; 16:10935-10945. [PMID: 33140798 DOI: 10.1039/d0sm01192d] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Flat sheets encoded with patterns of contraction/elongation morph into curved surfaces. If the surfaces bear Gauss curvature, the resulting actuation can be strong and powerful. We deploy the Gauss-Bonnet theorem to deduce the Gauss curvature encoded in a pattern of uniform-magnitude contraction/elongation with spatially varying direction, as is commonly implemented in patterned liquid crystal elastomers. This approach reveals two fundamentally distinct contributions: a structural curvature which depends on the precise form of the pattern, and a topological curvature generated by defects in the contractile direction. These curvatures grow as different functions of the contraction/elongation magnitude, explaining the apparent contradiction between previous calculations for simple +1 defects, and smooth defect-free patterns. We verify these structural and topological contributions by conducting numerical shell calculations on sheets encoded with simple higher-order contractile defects to reveal their activated morphology. Finally we calculate the Gauss curvature generated by patterns with spatially varying magnitude and direction, which leads to additional magnitude gradient contributions to the structural term. We anticipate this form will be useful whenever magnitude and direction are natural variables, including in describing the contraction of a muscle along its patterned fiber direction, or a tissue growing by elongating its cells.
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Affiliation(s)
- Daniel Duffy
- Engineering Dept., University of Cambridge, Trumpington St., Cambridge, CB2 1PZ, UK.
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13
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Gao T, Siéfert E, DeSimone A, Roman B. Shape Programming by Modulating Actuation over Hierarchical Length Scales. ADVANCED MATERIALS (DEERFIELD BEACH, FLA.) 2020; 32:e2004515. [PMID: 33073431 DOI: 10.1002/adma.202004515] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2020] [Revised: 08/19/2020] [Indexed: 06/11/2023]
Abstract
Many active materials used in shape-morphing respond to an external stimulus by stretching or contracting along a director field. The programming of such actuators remains complex because of the single degree of freedom (the orientation) in local actuation. Here, texturing this field in zigzag patterns is shown to provide an extended family of biaxial active stretches out of an otherwise single uniaxial active deformation, opening a larger parameter space. By further modulating the zigzag patterns at the larger scale of the structure, its deployed shape can be controlled. This notion of texturing over hierarchical length scales follows geometrical principles, and is robust against changes in size and materials. The robustness of the approach is demonstrated by considering three different responsive materials: inextensible flat fabrics, channel-bearing elastomer (respectively, contracting and expanding perpendicularly to the director field when actuated pneumatically), and 3D-printed thermoplastic (composed of extruded filaments that contract when heated). It is shown that large-scale shape-morphing structures can be generated and that their geometry can be controlled with high accuracy.
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Affiliation(s)
- Tian Gao
- PMMH, CNRS, ESPCI Paris, Université PSL, Sorbonne Université, Université de Paris, F-75005, Paris, France
| | - Emmanuel Siéfert
- PMMH, CNRS, ESPCI Paris, Université PSL, Sorbonne Université, Université de Paris, F-75005, Paris, France
| | - Antonio DeSimone
- MathLab, SISSA-International School for Advanced Studies, 34136, Trieste, Italy
- The BioRobotics Institute and Department of Excellence in Robotics and AI, Scuola Superiore Sant'Anna, 56127, Pisa, Italy
| | - Benoît Roman
- PMMH, CNRS, ESPCI Paris, Université PSL, Sorbonne Université, Université de Paris, F-75005, Paris, France
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14
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Ozenda O, Sonnet AM, Virga EG. A blend of stretching and bending in nematic polymer networks. SOFT MATTER 2020; 16:8877-8892. [PMID: 33026035 DOI: 10.1039/d0sm00642d] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Nematic polymer networks are (heat and light) activable materials, which combine the features of rubber and nematic liquid crystals. When only the stretching energy of a thin sheet of nematic polymer network is minimized, the intrinsic (Gaussian) curvature of the shape it takes upon (thermal or optical) actuation is determined. This, unfortunately, produces a multitude of possible shapes, for which we need a selection criterion, which may only be provided by a correcting bending energy depending on the extrinsic curvatures of the deformed shape. The literature has so far offered approximate corrections depending on the mean curvature. In this paper, we derive the appropriate bending energy for a sheet of nematic polymer network from the celebrated neo-classical energy of nematic elastomers in three space dimensions. This task is performed via a dimension reduction based on a modified Kirchhoff-Love hypothesis, which withstands the criticism of more sophisticated analytical tools. The result is a surface elastic free-energy density where stretching and bending are blended together; they may or may not be length-separated, and should be minimized together. The extrinsic curvatures of the deformed shape not only feature in the bending energy through the mean curvature, but also through the relative orientation of the nematic director in the frame of the directions of principal curvatures.
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Affiliation(s)
- Olivier Ozenda
- Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, 27100 Pavia, Italy.
| | - André M Sonnet
- Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK.
| | - Epifanio G Virga
- Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, 27100 Pavia, Italy.
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15
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Siéfert E, Warner M. Inflationary routes to Gaussian curved topography. Proc Math Phys Eng Sci 2020; 476:20200047. [PMID: 32922150 DOI: 10.1098/rspa.2020.0047] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/25/2020] [Accepted: 03/24/2020] [Indexed: 11/12/2022] Open
Abstract
Gaussian-curved shapes are obtained by inflating initially flat systems made of two superimposed strong and light thermoplastic impregnated fabric sheets heat-sealed together along a specific network of lines. The resulting inflated structures are light and very strong because they (largely) resist deformation by the intercession of stretch. Programmed patterns of channels vary either discretely through boundaries or continuously. The former give rise to faceted structures that are in effect non-isometric origami and that cannot unfold as in conventional folded structures since they present the localized angle deficit or surplus. Continuous variation of the channel direction in the form of spirals is examined, giving rise to curved shells. We solve the inverse problem consisting in finding a network of seam lines leading to a target axisymmetric shape on inflation. They too have strength from the metric changes that have been pneumatically driven, resistance to change being met with stretch and hence high forces like typical shells.
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Affiliation(s)
- Emmanuel Siéfert
- Laboratoire de Physique et Mécanique des Milieux Hétérogènes, CNRS UMR7636, Ecole Supérieure de Physique et Chimie Industrielles de Paris (ESPCI), Sorbonne Université, Université de Paris, 75005 Paris, France
| | - Mark Warner
- Cavendish Laboratory, University of Cambridge, 19 JJ Thomson Avenue, Cambridge CB3 0HE, UK
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Siéfert E, Reyssat E, Bico J, Roman B. Programming stiff inflatable shells from planar patterned fabrics. SOFT MATTER 2020; 16:7898-7903. [PMID: 32729603 DOI: 10.1039/d0sm01041c] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Lack of stiffness often limits thin shape-shifting structures to small scales. The large in-plane transformations required to distort the metrics are indeed commonly achieved by using soft hydrogels or elastomers. We introduce here a versatile single-step method to shape-program stiff inflated structures, opening the door for numerous large scale applications, ranging from space deployable structures to emergency shelters. This technique relies on channel patterns obtained by heat-sealing superimposed flat quasi-inextensible fabric sheets. Inflating channels induces an anisotropic in-plane contraction and thus a possible change of Gaussian curvature. Seam lines, which act as a director field for the in-plane deformation, encode the shape of the deployed structure. We present three patterning methods to quantitatively and analytically program shells with non-Euclidean metrics. In addition to shapes, we describe with scaling laws the mechanical properties of the inflated structures. Large deployed structures can resist their weight, substantially broadening the palette of applications.
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Affiliation(s)
- Emmanuel Siéfert
- PMMH, CNRS, ESPCI Paris, Université PSL, Sorbonne Université, Université de Paris, F-75005, Paris, France.
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Liu M, Domino L, Vella D. Tapered elasticæ as a route for axisymmetric morphing structures. SOFT MATTER 2020; 16:7739-7750. [PMID: 32743628 DOI: 10.1039/d0sm00714e] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Transforming flat two-dimensional (2D) sheets into three-dimensional (3D) structures by combining carefully made cuts with applied edge-loads has emerged as an exciting manufacturing paradigm in a range of applications from mechanical metamaterials to flexible electronics. In Kirigami, patterns of cuts are introduced that allow solid faces to rotate about each other, deforming in three dimensions whilst remaining planar. In other scenarios, however, the solid elements bend in one direction. In this paper, we model such bending deformations using the formulation of an elastic strip whose thickness and width are tapered (the 'tapered elastica'). We show how this framework can be exploited to design the tapering patterns required to create planar sheets that morph into desired axisymmetric 3D shapes under a combination of horizontal and vertical edge-loads. We exhibit this technique by recreating miniature structures with positive, negative, and variable apparent Gaussian curvatures. With sheets of constant thickness, the resulting morphed shapes may leave gaps between the deformed elements. However, by tapering the thickness of the sheet too, these gaps can be closed, creating tessellated three-dimensional structures. Our theoretical approaches are verified by both numerical simulations and physical experiments.
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Affiliation(s)
- Mingchao Liu
- Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK.
| | - Lucie Domino
- Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK.
| | - Dominic Vella
- Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK.
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18
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Feng F, Biggins JS, Warner M. Evolving, complex topography from combining centers of Gaussian curvature. Phys Rev E 2020; 102:013003. [PMID: 32795049 DOI: 10.1103/physreve.102.013003] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/03/2020] [Accepted: 07/06/2020] [Indexed: 11/07/2022]
Abstract
Liquid crystal elastomers and glasses can have significant shape change determined by their director patterns. Cones deformed from circular director patterns have nontrivial Gaussian curvature localized at tips, curved interfaces, and intersections of interfaces. We employ a generalized metric compatibility condition to characterize two families of interfaces between circular director patterns, hyperbolic and elliptical interfaces, and find that the deformed interfaces are geometrically compatible. We focus on hyperbolic interfaces to design complex topographies and nonisometric origami, including n-fold intersections, symmetric and irregular tilings. The large design space of threefold and fourfold tiling is utilized to quantitatively inverse design an array of pixels to display target images. Taken together, our findings provide comprehensive design principles for the design of actuators, displays, and soft robotics in liquid crystal elastomers and glasses.
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Affiliation(s)
- Fan Feng
- Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom
| | - John S Biggins
- Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom
| | - Mark Warner
- Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom
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