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Conservation laws in elasticity of the J-integral type

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Abstract

Conservation laws which are expressible as functionals linear in the strain energy and its derivatives are laws of the same type as theJ-integral. For finite elastic deformations of homogeneous bodies, relations between the conservation laws are shown through the use of inverse deformation results. Completeness of the laws are established for homogeneous materials and for materials whose strain energies satisfy objectivity, isotropy, or are homogeneous functions. Laws for a class of membranes inflated by pressure are derived and applied to a cylindrical membrane. For infinitesimal deformations of linear elastic bodies, new laws which relate two independent equilibrium states are presented and applied to the problem of a line crack in a plate under mixed-mode loading conditions. A relation is shown to exist between theJ-integral and the reciprocal work theroem of Betti.

Zusammenfassung

Erhaltungssätze, welche sich durch lineare Funktionale in der Verformungsenergie und ihren Ableitungen ausdrücken lassen, sind von gleicher Art wie dasJ-Integral. Für endliche elastische Deformationen homogener Körper werden mit Hilfe inverser Deformationsergebnisse Beziehungen zwischen den Erhaltungssätzen aufgezeigt. Vollständigkeit der Gesetze wird für homogene Materialien und für solche nachgewiesen, deren Verformungsenergie den Bedingungen der Objektivität und Isotropie genügen oder homogene Funktionen sind. Es werden für eine Klasse durch Druck aufgeblasener Membranen Beziehungen hergeleitet und auf eine zylindrische Membran angewandt. Für infinitesimale Verformungen linear-elastischer Körper werden neue Gesetze, welche zwei unabhängige Gleichgewichtszustände verknüpfen, aufgestellt und auf das Problem des Haarrisses in einer Platte unter gemischten Lastbedingungen angewandt. Schliesslich wird gezeigt, dass zwischen demJ-Integral und dem Bettischem Satz der reziproken Arbeit ein Zusammenhang besteht.

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Authors and Affiliations

  1. Dept of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

    Francis H. K. Chen & Richard T. Shield

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  1. Francis H. K. Chen
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  2. Richard T. Shield
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Chen, F.H.K., Shield, R.T. Conservation laws in elasticity of the J-integral type. Journal of Applied Mathematics and Physics (ZAMP) 28, 1–22 (1977). https://doi.org/10.1007/BF01590704

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