The Unexpected Kinematics of Simple Extension and Contraction of Incompressible Materials

  • C. O. Horgan
  • J. G. MurphyEmail author


Simple extension and contraction of incompressible non-linearly elastic materials is considered here, with cuboid specimens stretched or contracted in one of the principal directions. Although the average stretch of infinitesimal line elements in such deformations is always extensile, the local response is much more nuanced. We examine the variation of the squared stretch with applied stretch and line element orientation in the undeformed configuration. The response of interest here is the percentage of directions at a point for which the material is in compression which we call the amount of material in compression. It is shown that the maximum amount of compression attainable is approximately \(61\%\) which occurs for infinitesimal contractions. Further contraction results in a decrease in the amount of material in compression, with the material being extensile almost everywhere in the limit of zero axial stretch. On the other hand, for simple extension, the amount of material in compression is a monotonically decreasing function of axial stretch, with \(39\%\) in compression for infinitesimal strains. The amount of material in compression therefore exhibits a discontinuity in the reference configuration. The amount of material in compression is still significant for moderate extensions, with, for example, \(25\%\) of the material still in compression for \(100\%\) stretch. Suspecting that this somewhat surprising response is a result of assuming perfect incompressibility, the effect of compressibility on the amount of material in compression is examined within the context of the linear theory for isotropic elastic materials.


Incompressible elastic materials Isochoric simple extension and contraction 

Mathematics Subject Classification

74B20 74G55 



  1. 1.
    Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics. Springer, Berlin (2004) CrossRefzbMATHGoogle Scholar
  2. 2.
    Atkin, R.J., Fox, N.: An Introduction to the Theory of Elasticity. Longman, London (1980) zbMATHGoogle Scholar
  3. 3.
    Ogden, R.W.: Non-linear Elastic Deformations. Dover, New York (1984) zbMATHGoogle Scholar
  4. 4.
    Treloar, L.R.G.: The Physics of Rubber Elasticity. Oxford University Press, London (1975) Google Scholar
  5. 5.
    Horgan, C.O., Murphy, J.G.: Magic angles for fibrous incompressible elastic materials. Proc. R. Soc. A 474, 20170728 (2018) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Murphy, J.G.: Transversely isotropic biological soft tissue must be modeled using both anisotropic invariants. Eur. J. Mech. A, Solids 42, 90–96 (2013) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Horgan, C.O., Murphy, J.G.: The counterintuitive out-of-plane strength of incompressible orthotropic hyperelastic materials. Int. J. Solids Struct. 115, 170–179 (2017) CrossRefGoogle Scholar
  8. 8.
    De Rosa, E., Latorre, M., Montans, F.J.: Capturing anisotropic constitutive models with WYPiWYG hyperelasticity; and on consistency with the infinitesimal theory at all deformation levels. Int. J. Non-Linear Mech. 96, 75–92 (2017) CrossRefGoogle Scholar
  9. 9.
    Horgan, C.O., Murphy, J.G.: Magic angles and fibre stretch in arterial tissue: insights from the linear theory. J. Mech. Behav. Biomed. Mater. 88, 470–477 (2018) CrossRefGoogle Scholar
  10. 10.
    Goriely, A., Tabor, M.: Rotation, inversion and perversion in anisotropic elastic cylindrical tubes and membranes. Proc. R. Soc. A 469, 20130011 (2013) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Demirkoparan, H., Pence, T.J.: Magic angles for fiber reinforcement in rubber-elastic tubes subject to pressure and swelling. Int. J. Non-Linear Mech. 68, 87–95 (2015) ADSCrossRefGoogle Scholar
  12. 12.
    Goriely, A.: The Mathematics and Mechanics of Biological Growth. Springer, New York (2017) CrossRefzbMATHGoogle Scholar
  13. 13.
    Horgan, C.O., Murphy, J.G.: Magic angles in the mechanics of fibrous soft materials. Mech. Soft Mater. 1, 1–6 (2019) ADSCrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Engineering and Applied ScienceUniversity of VirginiaCharlottesvilleUSA
  2. 2.Department of Mechanical EngineeringDublin City UniversityDublinIreland
  3. 3.School of Mathematics, Statistics, and Applied MathematicsNational University of Ireland GalwayGalwayIreland

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