{"title":"从杆件和铰链弹性推导平行四边形折纸的有效板理论","authors":"","doi":"10.1016/j.jmps.2024.105832","DOIUrl":null,"url":null,"abstract":"<div><p>Periodic origami patterns made with repeating unit cells of creases and panels bend and twist in complex ways. In principle, such soft modes of deformation admit a simplified asymptotic description in the limit of a large number of cells. Starting from a bar and hinge model for the elastic energy of a generic four parallelogram panel origami pattern, we derive a complete set of geometric compatibility conditions identifying the pattern’s soft modes in this limit. The compatibility equations form a system of partial differential equations constraining the actuation of the origami’s creases (a scalar angle field) and the relative rotations of its unit cells (a pair of skew tensor fields). We show that every solution of the compatibility equations is the limit of a sequence of soft modes — origami deformations with finite bending energy and negligible stretching. Using these sequences, we derive a plate-like theory for parallelogram origami patterns with an explicit coarse-grained quadratic energy depending on the gradient of the crease-actuation and the relative rotations of the cells. Finally, we illustrate our theory in the context of two well-known origami designs: the Miura and Eggbox patterns. Though these patterns are distinguished in their anticlastic and synclastic bending responses, they show a universal twisting response. General soft modes captured by our theory involve a rich nonlinear interplay between actuation, bending and twisting, determined by the underlying crease geometry.</p></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":null,"pages":null},"PeriodicalIF":5.0000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Derivation of an effective plate theory for parallelogram origami from bar and hinge elasticity\",\"authors\":\"\",\"doi\":\"10.1016/j.jmps.2024.105832\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Periodic origami patterns made with repeating unit cells of creases and panels bend and twist in complex ways. In principle, such soft modes of deformation admit a simplified asymptotic description in the limit of a large number of cells. Starting from a bar and hinge model for the elastic energy of a generic four parallelogram panel origami pattern, we derive a complete set of geometric compatibility conditions identifying the pattern’s soft modes in this limit. The compatibility equations form a system of partial differential equations constraining the actuation of the origami’s creases (a scalar angle field) and the relative rotations of its unit cells (a pair of skew tensor fields). We show that every solution of the compatibility equations is the limit of a sequence of soft modes — origami deformations with finite bending energy and negligible stretching. Using these sequences, we derive a plate-like theory for parallelogram origami patterns with an explicit coarse-grained quadratic energy depending on the gradient of the crease-actuation and the relative rotations of the cells. Finally, we illustrate our theory in the context of two well-known origami designs: the Miura and Eggbox patterns. Though these patterns are distinguished in their anticlastic and synclastic bending responses, they show a universal twisting response. General soft modes captured by our theory involve a rich nonlinear interplay between actuation, bending and twisting, determined by the underlying crease geometry.</p></div>\",\"PeriodicalId\":17331,\"journal\":{\"name\":\"Journal of The Mechanics and Physics of Solids\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":5.0000,\"publicationDate\":\"2024-08-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of The Mechanics and Physics of Solids\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022509624002989\",\"RegionNum\":2,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of The Mechanics and Physics of Solids","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022509624002989","RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, MULTIDISCIPLINARY","Score":null,"Total":0}
Derivation of an effective plate theory for parallelogram origami from bar and hinge elasticity
Periodic origami patterns made with repeating unit cells of creases and panels bend and twist in complex ways. In principle, such soft modes of deformation admit a simplified asymptotic description in the limit of a large number of cells. Starting from a bar and hinge model for the elastic energy of a generic four parallelogram panel origami pattern, we derive a complete set of geometric compatibility conditions identifying the pattern’s soft modes in this limit. The compatibility equations form a system of partial differential equations constraining the actuation of the origami’s creases (a scalar angle field) and the relative rotations of its unit cells (a pair of skew tensor fields). We show that every solution of the compatibility equations is the limit of a sequence of soft modes — origami deformations with finite bending energy and negligible stretching. Using these sequences, we derive a plate-like theory for parallelogram origami patterns with an explicit coarse-grained quadratic energy depending on the gradient of the crease-actuation and the relative rotations of the cells. Finally, we illustrate our theory in the context of two well-known origami designs: the Miura and Eggbox patterns. Though these patterns are distinguished in their anticlastic and synclastic bending responses, they show a universal twisting response. General soft modes captured by our theory involve a rich nonlinear interplay between actuation, bending and twisting, determined by the underlying crease geometry.
期刊介绍:
The aim of Journal of The Mechanics and Physics of Solids is to publish research of the highest quality and of lasting significance on the mechanics of solids. The scope is broad, from fundamental concepts in mechanics to the analysis of novel phenomena and applications. Solids are interpreted broadly to include both hard and soft materials as well as natural and synthetic structures. The approach can be theoretical, experimental or computational.This research activity sits within engineering science and the allied areas of applied mathematics, materials science, bio-mechanics, applied physics, and geophysics.
The Journal was founded in 1952 by Rodney Hill, who was its Editor-in-Chief until 1968. The topics of interest to the Journal evolve with developments in the subject but its basic ethos remains the same: to publish research of the highest quality relating to the mechanics of solids. Thus, emphasis is placed on the development of fundamental concepts of mechanics and novel applications of these concepts based on theoretical, experimental or computational approaches, drawing upon the various branches of engineering science and the allied areas within applied mathematics, materials science, structural engineering, applied physics, and geophysics.
The main purpose of the Journal is to foster scientific understanding of the processes of deformation and mechanical failure of all solid materials, both technological and natural, and the connections between these processes and their underlying physical mechanisms. In this sense, the content of the Journal should reflect the current state of the discipline in analysis, experimental observation, and numerical simulation. In the interest of achieving this goal, authors are encouraged to consider the significance of their contributions for the field of mechanics and the implications of their results, in addition to describing the details of their work.