An asymptotic finite deformation analysis for an isotropic incompressible hyperelastic half-space subjected to a tensile point load (Q2783703)

The paper is devoted to the study of Boussinesq classical problem of a point load acting on a nonlinear incompressible hyperelastic half-space. Only the case of tensile point load is considered. The authors, using the well-known conservation laws of isotropic nonlinear elasticity first derived by \textit{J. K. Knowles} and \textit{E. Sternberg} [Arch. Ration. Mech. Anal. 63, 321-336 (1977; Zbl 0351.73061)], provide some asymptotic tests that give necessary conditions for a given material (i.e. an explicit functional form of strain-energy density function) to support a finite deflection under a point load. The problem is very complex and truly challenging, and therefore also formal results, as the one obtained in this paper, are hard to obtain. To understand clearly the role of the conditions here derived, the reader has to remark that in page 123 the authors establish that the generalized power-law material \(W=\mu /(2b)\{ [ 1+(b/k)(I_{1}-3) ] ^{k}-1\} \) (see formula (131) where \(\mu \) has been settled by the authors equal to \(1\)) can support a finite deflection under a point load provided \(k\) satisfies \(k>3/2\). On the other hand, if we let \(k\to \infty \) in this strain-energy density function, we recover the strain-energy density function \(W=\mu /(2b))( e^{b( I_{1}-3) }-1) \) which is up to a trivial change of notation exactly the material (167) in page 126 of the present paper. About this second strain-energy density function the authors claim that the material (167) cannot support a point load. Therefore it is clear that the conditions derived in this paper seems to be only a subset of necessary conditions needed to determine whether the tensile point load may be supported. We point out that a paper related to the problem here considered has recently appeared in [\textit{Y. C. Gao}, J. Elasticity 64, 111-130 (2001; Zbl 1034.74009)].