By L. A. Galin
L.A. Galin’s publication on touch difficulties is a awesome paintings. truly there are books: the 1st, released in 1953 bargains with touch difficulties within the classical conception of elasticity; this can be the one who used to be translated into English in 1961. the second one e-book, released in 1980, integrated the 1st, after which had new sections on touch difficulties for viscoelastic fabrics, and tough touch difficulties; this part has no longer formerly been translated into English. during this new translation, the unique textual content and the mathematical research were thoroughly revised, new fabric has been extra, and the fabric showing within the 1980 Russian translation has been thoroughly rewritten. furthermore there are 3 essays by means of scholars of Galin, bringing the research modern.
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Extra info for Contact Problems: The legacy of L.A. Galin (Solid Mechanics and Its Applications)
When such a rigid body presses into an elastic body, the surface of the elastic body is displaced relative to it, and slides along it. , stick) in the contact region. 1) Here w1 (z) → iP /(2πz) as z → ∞. The potential w1 (z) may have singularities of the form (z − c)α , where c = ±a, ±b, and 0 < α < 1. From symmetry, the singularities at x = +a and −a must be the same, as for those at x = +b and −b. 4), namely ω(x) = arctan[d(x)/c(x)] where c(x), d(x) characterise the boundary condition 3. Plane Static Isotropic Contact Problems 59 Fig.
We consider this complex combination of stresses for the half-plane under the assumption that the stresses tend to zero at inﬁnity. , for large values of |z|: (z) = γ1 1 +o , z z (z) = − (z) = γ2 1 +o z z γ1 1 +o 2 2 z z We recall the elements of the theory of Cauchy integrals, see Gladwell (1980) for a fuller version. We start with the deﬁnition of a holomorphic function of a complex variable z. The function f (z) is said to be holomorphic (sometimes the term regular is used) in a ﬁnite region Sof the complex plane if it is single-valued in S, and its complex derivative f (z) exists at every point in S.
At those points where the stress is less than a critical value the two bodies stick together; and where the stress exceeds the critical value, one body slips relative to the other. Thus the contact region can be partitioned into two separate parts: stick zones and slip zones. These problems present some severe mathematical difﬁculties. The boundary problems for frictional contact problems are as follows: on parts of the elastic body in contact with the punch (or punches) we are given the displacement normal to the surface.