Large elastic deformations of thin rubber membranes
Abstract
Numerical solutions are given for the problem of the inflation of a circular membrane of uniform thickness clamped around its circumference. The membrane is assumed to be composed of an incompressible highly elastic material whose elasto-mechanical properties can be represented by the exponential-hyperbolic elasticity parameters introduced by Hart-Smith [1]. These parameters apply to rubber and to the long-chain polymer materials. In turn, it is shown how this experiment is eminently suitable for determining the three finite elasticity constants appearing in the two elasticity parameters. It proves useful in the process to solve an associated problem, namely the determination of the thickness distribution of an initially flat sheet inflated to a spherical form; closed form analytical solutions are given for this problem.
Résumé
Les auteurs donnent des solutions numériques du problème de la déformation d'une membrane circulaire, d'égale épaisseur, encastrée sur sa circonférence. Il est admis que la membrane est constituée par un matériau incompressible, hautement élastique, dont les propriétés élasto-mécaniques peuvent être exprimées par les paramètres d'élasticité exponentiels-hyperboliques introduits par Hart-Smith [1], Ces paramètres peuvent s'appliquer aux matériaux en caoutchouc et aux polymères à longue chaine. Il est montré, en outre, comment cette expérience se prête particulièrement bien à la détermination des trois constantes élastiques, finies, qui apparaissent dans les deux paramètres d'élasticité. Elle est utile, également, pour résoudre un problème associé, en l'espèce, la détermination de la distribution de l'épaisseur d'une plaque, initialement plane, et déformée en forme de sphère. Les auteurs donnent des solutions analytiques précises de ce problème.
Zusammenfassung
Für das Problem einer aufgeblasenen, gleichmässig dicken Rundmembrane, die an ihrer gesamten Peripherie festgeklemmt ist, werden zahlenmässige Lösungen angegeben. Dabei wird angenommen, dass die Membrane aus einem inkompressiblen, hochelastischen Stoff besteht, dessen elasto-mechanische Eigenschaften sich durch die von Hart-Smith [1] eingeführten exponential-hyperbolischen Elastizität-sparameter beschreiben lassen. Diese Parameter sind für Gummi und die aus langen Molekülketten bestehenden Polymerstoffe gültig. Es wird gezeigt, dass sich dieses Experiment ausgezeichnet zur Bestimmung der drei, in den zwei Elastizitätsparametern erscheinenden endlichen Elastizitätskonstanten eignet und dass dieses auch zur Lösung eines verknüpften Problems nützlich ist, wo die Dickenverteilug eines anfänglich flachen und später zu einer Kugel aufgeblasenen Blattes bestimmt werden soll. Für dieses Problem werden analytische Lösungen in geschlossener Form angegeben.
Sumàrio
Si danno le soluzioni numeriche del problema dell'inflazione di una membrana circolare di spessore uniforme bloccata sulla circonferenza. Si presume ehe la membrana sia composta da un materiale altamente elastico e incomprimibile le cui caratteristiche elastomeccaniche possano venire rapp esentate con i parametri di elasticità esponenziale-iperbolica introdotti da Hart-Smith [1]. Questi parametri valgono nei confronti della gomma e dei materiali polimerici a catena lunga. Si dimostra inoltre come questo esperimento sia ottimamente indicate per determinare le tre costanti finite di elasticità che compaiono nei due parametri di elasticità. Esso ha dimostrato la sua utilità nella soluzione di un problema associate, cioè la determinazione della distribuzione dello spessore di un foglio infinitamente piatto gonfiato in forma sferica. Per questo problema si danno soluzioni analitiche di forma chiusa.
Реферат
Дaюoтcя чиcлeнныe peшeния пpoблeмы нaдyвaния кpyглoй мeмбpaны paвнoмepнoй тoлщины, зapeплeннoй пo пepифepии. Пoлaгaeтcя, чтo мeмбpaнa cocтгoит из нecжимaeмoгo oчeнь yпpyгoгo мaтpиaлa, yпpyгo-мeчaничecкиe cвoйcтвa кoтopoгo мoжнo пpeдcтaвить чapaмeтpaмими экcпoнeциaльнo-гипepбoличecкoй yпpyгocти, ввдeнными Чapт-Cмитoм [1]. Эти лapaмeтpы пpимeнютcя и к peзин, и к пoлимepaм c qdлнннoй цeпьы. Пoкaзывaeтcя, кa пoдчoдит этoт экcпepимeнт дл. oпpeдлeнитpeч финитныч пocтoянныч yпpyгocти, кoтopыe фигypиpyют в пapaмeтpaч yпpyгocти. Oн явлтc. пoлeныμ пpи пpoцecce peщeни. cвяэaннoй пpoблeмы, a имeннo, oпpeлeнии pacпpeлeния тoлщины пepвoнaчaльнo плocкoгo плacтa, paздyтoгo дo cфepичecoй фopмы. Дaютcя qzaмкнчтыe aнaлитичecкиы peщeния для этoй пpoблeмы.
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