The occurrence of surface instabilities and shear bands in plane-strain deformation of an elastic half-space.

*(English)*Zbl 0531.73029The quasi-static bifurcation of an incompressible isotropic hyperelastic half-space \(x_ 2\geq 0\) under plane strain with stretches \(\lambda_ 1=\lambda\), \(\lambda_ 2=1/\lambda\), \(\lambda_ 3=1\) is considered. The incremental equations needed for the bifurcation analysis are derived. Two modes of bifurcation are investigated: a diffuse mode corresponding to surface instability, and a localized deformation, also known as a shear band. Criteria for bifurcation into either a diffuse or localized mode are established. It is shown that, under very mild restrictions on the strain-energy function, the diffuse mode will precede the localized mode.

Up to this point, all results hold for arbitrary forms of strain-energy functions. For a class of strain-energy functions proposed by R. W. Ogden [Proc. R. Soc. London, Ser. A 326, 565-584 (1972; Zbl 0257.73034)] it is illustrated that, if bifurcation occurs at all, it will do so in the diffuse mode. A necessary condition for bifurcation under tension into either mode is given. Further specific implications are discussed.

Up to this point, all results hold for arbitrary forms of strain-energy functions. For a class of strain-energy functions proposed by R. W. Ogden [Proc. R. Soc. London, Ser. A 326, 565-584 (1972; Zbl 0257.73034)] it is illustrated that, if bifurcation occurs at all, it will do so in the diffuse mode. A necessary condition for bifurcation under tension into either mode is given. Further specific implications are discussed.

Reviewer: M.Biermann

##### MSC:

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74B20 | Nonlinear elasticity |

74G60 | Bifurcation and buckling |