<p>Boundary-contact problems (BCPs) are studied within the frames of</p><p>classical mathematical theory of elasticity and plasticity</p><p>elaborated by Landau, Kupradze, Timoshenko, Goodier, Fichera and</p><p>many others on the basis of analysis of two- and three-dimensional</p><p>boundary value problems for linear partial differential equations.</p><p>A great attention is traditionally paid both to theoretical</p><p>investigations using variational methods and boundary singular</p><p>integral equations (Muskhelishvili) and construction of solutions</p><p>in the form that admit efficient numerical evaluation (Kupradze).</p><p>A special family of BCPs considered by Shtaerman, Vorovich,</p><p>Alblas, Nowell, and others arises within the frames of the models</p><p>of squeezing thin multilayer elastic sheets. We show that</p><p>mathematical models based on the analysis of BCPs can be also</p><p>applied to modeling of the clich\'{e}-surface printing contacts</p><p>and paper surface compressibility in flexographic printing.</p><p>The main result of this work is formulation and complete</p><p>investigation of BCPs in layered structures, which includes both</p><p>the theoretical (statement of the problems, solvability and</p><p>uniqueness) and applied parts (approximate and numerical</p><p>solutions, codes, simulation).</p><p>We elaborate a mathematical model of squeezing a thin elastic</p><p>sheet placed on a stiff base without friction by weak loads</p><p>through several openings on one of its boundary surfaces. We</p><p>formulate and consider the corresponding BCPs in two- and</p><p>three-dimensional bands, prove the existence and uniqueness of</p><p>solutions, and investigate their smoothness including the behavior</p><p>at infinity and in the vicinity of critical points. The BCP in a</p><p>two-dimensional band is reduced to a Fredholm integral equation</p><p>(IE) with a logarithmic singularity of the kernel. The theory of</p><p>logarithmic IEs developed in the study includes the analysis of</p><p>solvability and development of solution techniques when the set of</p><p>integration consists of several intervals. The IE associated with</p><p>the BCP is solved by three methods based on the use of</p><p>Fourier-Chebyshev series, matrix-algebraic determination of the</p><p>entries in the resulting infinite system matrix, and</p><p>semi-inversion. An asymptotic theory for the BCP is developed and</p><p>the solutions are obtained as asymptotic series in powers of the</p><p>characteristic small parameter.</p><p>We propose and justify a technique for the solution of BCPs and</p><p>boundary value problems with boundary conditions of mixed type</p><p>called the approximate decomposition method (ADM). The main idea</p><p>of ADM is simplifying general BCPs and reducing them to a chain</p><p>of auxiliary problems for 'shifted' Laplacian in long rectangles</p><p>or parallelepipeds and then to a sequence of iterative problems</p><p>such that each of them can be solved (explicitly) by the Fourier</p><p>method. The solution to the initial BCP is then obtained as a</p><p>limit using a contraction operator, which constitutes in</p><p>particular an independent proof of the BCP unique solvability.</p><p>We elaborate a numerical method and algorithms based on the</p><p>approximate decomposition and the computer codes and perform</p><p>comprehensive numerical analysis of the BCPs including the</p><p>simulation for problems of practical interest. A variety of</p><p>computational results are presented and discussed which form the</p><p>basis for further applications for the modeling and simulation of</p><p>printing-plate contact systems and other structures of</p><p>flexographic printing. A comparison with finite-element solution</p><p>is performed.</p>
| Identifer | oai:union.ndltd.org:UPSALLA/oai:DiVA.org:kau-773 |
| Date | January 2007 |
| Creators | Kotik, Nikolai |
| Publisher | Karlstad University, Faculty of Technology and Science, Fakulteten för teknik- och naturvetenskap |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Doctoral thesis, monograph, text |
| Relation | Karlstad University Studies, 1403-8099 ; 2007:8 |
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