Discussion on “A. Aguiar and R. Fosdick, Self-intersection in elasticity”: [International Journal of Solids and Structures 38 (28) (2001) 4797–4825]

henomenon in linear elasticity. Specifically, it is observed that there are many problems in linear elasticity theory whose solutions guarantee the injectivity of the deformation field not everywhere in the domain. In such cases, the Jacobian of elastic deformation transformation becomes negative, i.e. IðxÞ ¼ det dij þ oui oxj < 0; ð1Þ where dij is Kronecker’s delta, uiðxÞ is the elastic displacement vector, and x denotes the triplet of Cartesian coordinates ðx1; x2; x3Þ. Such a situation is, of course, related to inter-penetration of material particles. These writers then illustrate the occurrence of this phenomenon in Abramov’s bonded punch problem. However, these results are not new. They are obviously unaware of the fact that Savin and Rvachev had first discovered this phenomenon in 1963 (Savin and Rvachev, 1963a, 1963b, 1964). Their results are also summarized as a separate chapter in the book by Rvachev and Protsenko (1977). In particular, Savin and Rvachev showed that the Jacobian in Abramov’s problem is given by Iðx1; 0Þ ¼ 1 Pm pl ffiffiffi v p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 x21 p cos b log l þ x1 l x1 ; ð2Þ where b ¼ log v=2p, v ¼ 3 4m (m is the Poisson ratio of the material of the half-space). As can be seen from Eq. (2), the inequality Iðx1; 0Þ > 0 is violated infinite number of times as the punch corner is approached. Away from the punch corner, there is no such anomalous behavior. My own interest (1990) in their results was aroused by the problem of a punch moving across an elastic half-space. It is well known that there are problems when the punch moves at a transonic speed, which cannot be resolved within the scopes of classical linear elasticity theory. Since the governing integral equation for this problem resembles that corresponding to Abramov’s bonded punch problem, I anticipated that the anomalous behavior of the contact stresses under a punch moving at a transonic speed might be related to Savin and Rvachev’s observation. Several years later, in view of the fact that their observation