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Radially Symmetric Motions of Nonlinearly Viscoelastic Bodies Under Live Loads

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Abstract

This paper treats radially symmetric motions of nonlinearly viscoelastic circular-cylindrical and spherical shells subjected to the live loads of centrifugal force and (time-dependent) hydrostatic pressures. The governing equations are exact versions of those for 3-dimensional continuum mechanics (so shell does not connote an approximate via some shell theory). These motions are governed by quasilinear third-order parabolic-hyperbolic equations having but one independent spatial variable. The principal part of such a partial differential equation is determined by a general family of nonlinear constitutive equations. The presence of strains in two orthogonal directions requires a careful treatment of constitutive restrictions that are physically natural and support the analysis. The interaction of geometrically exact formulations, the compatible use of general constitutive equations for material response, and the presence of live loads show how these factors play crucial roles in the behavior of solutions. In particular, for different kinds of live loads there are thresholds separating materials that produce qualitatively different dynamical behavior. The analysis (using classical methods) covers infinite-time blowup for cylindrical shells subject to centrifugal forces, infinite-time blowup for cylindrical shells subject to steady and time-dependent hydrostatic pressures, finite-time blowup for spherical shells subject to steady and time-dependent hydrostatic pressures, and the preclusion of total compression. This paper concludes with a sketch (using some modern methods) of the existence of regular solutions until the time of blowup.

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Authors and Affiliations

  1. Quantitative Research Division, Intercontinental Exchange, 5660 New Northside Dr, Atlanta, GA, 30328, USA

    Alexey B. Stepanov

  2. Department of Mathematics, Institute for Physical Science and Technology, Institute for Systems Research, University of Maryland, College Park, MD, 20742, USA

    Stuart S. Antman

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  1. Alexey B. Stepanov
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  2. Stuart S. Antman
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Correspondence to Stuart S. Antman.

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Communicated by C. Dafermos

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Stepanov, A.B., Antman, S.S. Radially Symmetric Motions of Nonlinearly Viscoelastic Bodies Under Live Loads. Arch Rational Mech Anal 226, 1209–1247 (2017). https://doi.org/10.1007/s00205-017-1153-9

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