ISSN 1028-3358, Doklady Physics, 2006, Vol. 51, No. 5, pp. 268–271. © Pleiades Publishing, Inc., 2006. Original Russian Text © I.I. Argatov, 2006, published in Doklady Akademii Nauk, 2006, Vol. 408, No. 2, pp. 188–191. 268 The asymptotic solution to the problem is con- structed under the assumption that the contact pressure under the punch base is slightly varied during the time of travel of the Rayleigh wave along the distance equal to the diameter of the contact area. It is assumed that, during the motion, the contact area is fixed and there is no friction under the punch base. The construction of terms of the asymptotic expansion for the contact pres- sure is reduced to the solution of the integral equation for a quasistatic contact problem. STATEMENT OF THE PROBLEM Let 0 < ε be a small parameter. The contact area ω ε is obtained from domain ω 1 through ε –1 -fold contrac- tion, i.e., ω ε = {(x 1 , x 2 ): ε –1 (x 1 , x 2 ) ∈ ω 1 }. The idea of extended coordinates x = ε –1 (x 1 , x 2 ) is widely exploited in the method of matched asymptotic expan- sions [1]. We require that the diameter of domain ω 1 be comparable to value Tc 3 , where T is the character- istic time of motion (the period in the case of a peri- odic motion). Denote the given vertical displacement of the punch by δ 0 (t) and the angles of rotation of the punch about the horizontal axes by β 1 (t) and β 2 (t). The shape of the punch base is specified in the extended coordinates by the equation x 3 = –Φ(ε –1 x 1 , ε –1 x 2 ). According to the solution to Lamb’s problem (see, e.g., [2, 3]), the contact pressure density p(t, x 1 , x 2 ) sat- isfies the following integral equation [3, 4]: (1) D ε p ( ) tx 1 x 2 , , ( ) δ 0 t () β 2 t () x 1 – = + β 1 t () x 2 Φ x 1 x 2 , ( ) , x 1 x 2 , ( ) ω ε . ∈ – Here, D ε is the linear integral operator acting by the for- mula (2) By G 3 (t, r), we denote the vertical displacement of the points of the half-space surface under the action of a unit concentrated force that is suddenly applied at the origin at time instant t = 0 and retains its value from then on. For the Poisson ratio ν < 0.263, the following repre- sentation is valid [2]: (3) Here, H(t) is the Heaviside function; c 1 , c 2 , and c 3 are the velocities of the tensile wave, shear wave, and the Ray- leigh wave, respectively; T 3 (r) = ϑr –1 is the displace- ment in the Boussinesq problem; ϑ = (πE) –1 (1 – ν 2 ), E is the Young modulus; ν is the Poisson ratio; and func- tions f 12 (x) and f 23 (x) are given. Setting G 3 (t, r) = T 3 (r) + g 3 (t, r) and integrating (2) by parts, we obtain (4) D ε p ( ) tx 1 x 2 , , ( ) = ∂ ∂ t τ – ( ) ----------------- G 3 t τ – x y – , ( ) p τ y , ( ) y τ . d d ∫ ω ε ∫ 0 t ∫ G 3 tr , ( ) T 3 r () 1 2 -- Ht r c 1 ---- – ⎝ ⎠ ⎛ ⎞ 1 2 -- Ht r c 2 ---- – ⎝ ⎠ ⎛ ⎞ + = + f 12 c 2 t r ------ ⎝ ⎠ ⎛ ⎞ Ht r c 1 ---- – ⎝ ⎠ ⎛ ⎞ H r c 2 ---- t – ⎝ ⎠ ⎛ ⎞ – f 23 c 2 t r ------ ⎝ ⎠ ⎛ ⎞ Ht r c 2 ---- – ⎝ ⎠ ⎛ ⎞ H r c 3 ---- t – ⎝ ⎠ ⎛ ⎞ . D ε p ( ) tx 1 x 2 , , ( ) g 3 t x y – , ( ) p 0 y , ( ) y d ∫ ω ε ∫ = + T 3 x y – ( ) pt y , ( ) y d ∫ ω ε ∫ + g 3 t x y – , ( ) ∂ p ∂ t ----- t y , ( ) y . d ∫ ω ε ∫ Slow Vertical Motions of a Punch on an Elastic Half-Space I. I. Argatov Presented by Academician N.F. Morozov October 29, 2005 Received November 14, 2005 PACS numbers: 46.40.–f, 62.30.+d DOI: 10.1134/S1028335806050090 Admiral Makarov State Maritime Academy, Kosaya liniya 15A, St. Petersburg, 199026 Russia e-mail: argatov@home.ru MECHANICS