Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Folding of a layered viscoelastic medium derived from an exact stability theory of a continuum under initial stress


Author: M. A. Biot
Journal: Quart. Appl. Math. 17 (1959), 185-204
MSC: Primary 73.00
DOI: https://doi.org/10.1090/qam/106609
MathSciNet review: 106609
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Abstract: A layer of viscoelastic material embedded in an infinite medium also viscoelastic tends to fold through instability when the system is compressed in a direction parallel with the layer. This phenomenon which was treated previously by using an approximate plate theory, [1, 2], is analyzed here by using an exact theory for the deformation of a continuum under prestress [3, 4, 5, 6], The effect of the compressive prestress in the embedding medium is taken into account, and it is found that although it is not always negligible, it tends to be very small under conditions where the instability of the layer is strong. The same conclusion holds for the error involved in the use of a plate theory for the layer instead of the exact equations for a prestressed continuum. In the course of the analysis we have also treated the problem of a semi-infinite viscoelastic half space subject to a uniform internal compression parallel with the boundary and a surface load normal to this boundary. The compressive load produces an increase of the surface deflection under the normal load. This effect appears through an amplification factor which is evaluated numerically for the particular example of an elastic body. It is shown that under certain conditions the free surface of the compressed semi-infinite medium may become unstable and will tend to wrinkle. This is suggested as a probable explanation for the wrinkles which appear on the surface of a body subjected to a plastic compression.


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  • [1] M. A. Biot, Folding instability of a layered viscoelastic medium under compression, Proc. Roy. Soc. London. Ser. A. 242 (1957), 444–454. MR 0092454, https://doi.org/10.1098/rspa.1957.0187
  • [2] M. A. Biot, On the instability and folding deformation of a layered viscoelastic medium in compression, J. Appl. Mech. 26 (1959), 393–400. MR 0110315
  • [3] M. A. Biot, Theory of elasticity with large displacements and rotations, Proc. Fifth Intern. Congr. Appl. Mechanics, 1938
  • [4] M. A. Biot, Non-linear theory of elasticity and the linearized case for a body under initial stress, Phil. Mag., Sec. 7, XXVII, 468-489 (April 1939)
  • [5] Maurice A. Biot, Elastizitätstheorie zweiter Ordnung mit Anwendungen, Z. Angew. Math. Mech. 20 (1940), 89–99 (German). MR 0002311, https://doi.org/10.1002/zamm.19400200202
  • [6] Maurice A. Biot, The influence of initial stress on elastic waves, J. Appl. Phys. 11 (1940), 522–530. MR 0002310
  • [7] M. A. Biot, Theory of stress-strain relations in anisotropic viscoelasticity and relaxation phenomena, J. Appl. Phys. 25, No. 11, 1385-1391 (November 1954)

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DOI: https://doi.org/10.1090/qam/106609
Article copyright: © Copyright 1959 American Mathematical Society


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