Abstract :
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
