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A Comparison of Stability and Bifurcation Criteria for a Compressible Elastic Cube

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Abstract

A version of Rivlin’s cube problem is considered for compressible materials. The cube is stretched along one axis by a fixed amount and then subjected to equal tensile loads along the other two axes. A number of general results are found. Because of the homogeneous trivial and non-trivial deformations exact bifurcation results can be found and an exact stability analysis through the second variation of the energy can be performed. This problem is then used to compare results obtained using more general methods. Firstly, results are obtained for a more general bifurcation analysis. Secondly, the exact stability results are compared with stability results obtained via a new method that is applicable to inhomogeneous problems. This new stability method allows a full nonlinear stability analysis of inhomogeneous deformations of arbitrary, compressible or incompressible, hyperelastic materials. The second variation condition expressed as an integral involving two arbitrary perturbations is replaced with an equivalent nonlinear third order system of ordinary differential equations. The positive definiteness condition is thereby reduced to the simple numerical evaluation of zeros of a well behaved function.

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References

  1. R.S. Rivlin (1948) ArticleTitleLarge elastic deformations of isotropic materials. II. Some uniqueness theorems for pure homogeneous deformation Phil. Trans. R Soc. London A 240 491–508

    Google Scholar 

  2. R.S. Rivlin (1974) ArticleTitleStability of pure homogeneous deformations of an elastic cube under dead loading Q. Appl. Math. 32 265–271

    Google Scholar 

  3. C.H. Wu O.E Widera (1969) ArticleTitleStability of a thick rubber solid subject to pressure loads Int. J. Solids Struct. 5 1107–1117 Occurrence Handle10.1016/0020-7683(69)90006-7

    Article  Google Scholar 

  4. K.N. Sawyers R.S Rivlin (1974) ArticleTitleBifurcation conditions for a thick elastic plate under thrust Int. J. Solids Struct. 10 483–501 Occurrence Handle10.1016/0020-7683(74)90054-7

    Article  Google Scholar 

  5. R.W. Ogden (1997) Non-Linear Elastic Deformations Dover Publications New York 532

    Google Scholar 

  6. R.W. Ogden (1985) ArticleTitleLocal and global bifurcation phenomena in plane strain finite elasticity Int. J. Solids Struct. 21 121–132 Occurrence Handle10.1016/0020-7683(85)90029-0

    Article  Google Scholar 

  7. C.E. Pearson (1956) ArticleTitleGeneral theory of elastic stability Q. Appl. Math. 14 133–144

    Google Scholar 

  8. R. Hill (1957) ArticleTitleOn uniqueness and stability in the theory of finite elastic strain J. Mech. Phys. Solids 5 229–241 Occurrence Handle10.1016/0022-5096(57)90016-9

    Article  Google Scholar 

  9. K.N. Sawyers R.S. Rivlin (1982) ArticleTitleStability of a thick elastic plate under thrust J. Elasticity 12 101–125 Occurrence Handle10.1007/BF00043707

    Article  Google Scholar 

  10. R.W. Ogden (1984) ArticleTitleOn non-uniqueness in the traction boundary-value problem for a compressible elastic solid Q. Appl. Math. 42 337–344

    Google Scholar 

  11. R.S. Rivlin M.F. Beatty (2003) ArticleTitleDead loading of a unit cube of compressible isotropic elastic material Z.A.M.P. 54 954–963

    Google Scholar 

  12. Y.C. Chen D.M. Haughton (2003) ArticleTitleStability and bifurcation of inflation of elastic cylinders Proc. R Soc. London A. 459 137–156

    Google Scholar 

  13. D.M. Haughton (2004) ArticleTitleOn non-linear stability in unconstrained non-linear elasticity Int. J. Non-Linear Mech. 39 1181–1192 Occurrence Handle10.1016/j.ijnonlinmec.2003.07.002

    Article  Google Scholar 

  14. D.M. Haughton E. Kirkinis (2002) ArticleTitleA comparison of stability and bifurcation criteria for inflated spherical elastic shells Math. Mech. Solids 8 561–572 Occurrence Handle10.1177/10812865030085008

    Article  Google Scholar 

  15. C.A. Truesdell W. Noll (1965) NoChapterTitle S. Flügge (Eds) The non-linear field theories of mechanics. Handbuch der Physik, Vol. III/3 Springer Heidelbrg 1–1602

    Google Scholar 

  16. M.F. Beatty (1967) ArticleTitleA theory of elastic stability for incompressible hyperelastic bodies Int. J. Solids Struct. 3 23–37 Occurrence Handle10.1016/0020-7683(67)90042-X

    Article  Google Scholar 

  17. M.F. Beatty (1968) ArticleTitleStability of the undistorted states of an isotropic elastic body Int. J. Nonlinear Mech. 3 337–349 Occurrence Handle10.1016/0020-7462(68)90006-1

    Article  Google Scholar 

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  1. Department of Mathematics, University of Glasgow, University Gardens, Glasgow, G12 8QW, UK

    D. M. Haughton

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  1. D. M. Haughton
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Haughton, D.M. A Comparison of Stability and Bifurcation Criteria for a Compressible Elastic Cube. J Eng Math 53, 79–98 (2005). https://doi.org/10.1007/s10665-005-4752-7

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  • DOI: https://doi.org/10.1007/s10665-005-4752-7

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