Summary
Within the framework of the linearized theory of couple-stresses for perfectly elastic, non-homogeneous, anisotropic materials, a general reciprocal theorem is derived and applied in the proof of other general theorems in both elastostatics and elastodynamics. In particular, a general work theorem is derived; for small periodic vibrations it is shown that the normal mode functions are orthogonal; and these results are combined to prove that if the strain energy function is positive definite, then the normal mode frequencies are real and non-vanishing. In addition,Castigliano's theorem and several other theorems that are common to structural analysis are shown to follow in a rigorous and natural way from the theory of couple-stresses.
Zusammenfassung
Im Rahmen der linearisierten Theorie der Momentenspannungen in vollkommen elastischen, inhomogenen und anisotropen Körpern wird ein allgemeines Reziprozitätstheorem hergeleitet und für den Beweis anderer allgemeiner Theoreme der Elastostatik und Elastodynamik verwendet. Insbesondere wird ein allgemeiner Arbeitssatz hergeleitet, und es wird gezeigt, daß die Eigenschwingungsformen für kleine periodische Schwingungen orthogonal sind. Diese Ergebnisse werden kombiniert, um zu beweisen, daß die Eigenfrequenzen reell sind und nicht verschwinden, wenn die Verzerrungsenergiefunktion positiv definit ist. Ferner wird gezeigt, daß der Satz vonCastigliano und verschiedene andere Sätze der Festigkeitslehre in strenger und natürlicher Weise aus der Theorie der Momentenspannungen folgen.
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Beatty, M.F. A reciprocal theorem in the linearized theory of couple-stresses. Acta Mechanica 3, 154–166 (1967). https://doi.org/10.1007/BF01453712
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DOI: https://doi.org/10.1007/BF01453712