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Second-order effects in problems for a class of elastic materials

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Abstract

For a special class of elastic materials, the inverse deformation is always possible in the same material. This result is used to simplify the determination of second-order effects in problems for this special class. The second-order solution for indentation of a half-space by a smooth rigid sphere is determined. For indentation by a flat-ended circular punch it is found that the second-order solution for compressible materials is physically inadmissible, while the solution for incompressible materials involves a concentrated line load at the edge of the punch. A rigid spherical inclusion embedded in an elastic matrix with uniform stresses at infinity is considered, and the interface stresses are found to second order.

Zusammenfassung

Für seine Sonderklasse von elastischen Stoffen ist die inverse Verformung für das gleiche Material immer möglich. Dieses Resultat wird angewendet, um bei Problemen mit dieser Sonderklasse von Materialien die Ermittlung von Effekten zweiter Ordnung zu vereinfachen. Die Lösung zweiter Ordnung für das Kontaktproblem zwischen einem Halbraum einer glatten starren Kugel wird hergeleitet. Bei dem Kontaktproblem mit dem kreiszylindrischen starren Stempel findet man, daß die Lösung zweiter Ordnung mit kompressiblem Material physikalisch unzulässig ist, während für das inkompressible Material am Rand des Stempels eine konzentrierte Linienlast entsteht. Schließlich wird eine starre kugelförmige Inklusion betrachtet, welche in einer elastischen Matrix mit gleichförmig verteilten Spannungen im Unendlichen eingebettet ist. Die Spannungen am Rand der Inklusion werden bis zur zweiten Ordnung hergeleitet.

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Authors and Affiliations

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

    I. Choi & R. T. Shield

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  1. I. Choi
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  2. R. T. Shield
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Choi, I., Shield, R.T. Second-order effects in problems for a class of elastic materials. Z. angew. Math. Phys. 32, 361–381 (1981). https://doi.org/10.1007/BF00955616

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  • DOI: https://doi.org/10.1007/BF00955616

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