# Solution to boundary-contact problems of elasticity in mathematical models of the printing-plate contact system for flexographic printing

University dissertation from Fakulteten för teknik- och naturvetenskap

Abstract: Boundary-contact problems (BCPs) are studied within the frames ofclassical mathematical theory of elasticity and plasticityelaborated by Landau, Kupradze, Timoshenko, Goodier, Fichera andmany others on the basis of analysis of two- and three-dimensionalboundary value problems for linear partial differential equations.A great attention is traditionally paid both to theoreticalinvestigations using variational methods and boundary singularintegral equations (Muskhelishvili) and construction of solutionsin the form that admit efficient numerical evaluation (Kupradze).A special family of BCPs considered by Shtaerman, Vorovich,Alblas, Nowell, and others arises within the frames of the modelsof squeezing thin multilayer elastic sheets. We show thatmathematical models based on the analysis of BCPs can be alsoapplied to modeling of the clich\'{e}-surface printing contactsand paper surface compressibility in flexographic printing.The main result of this work is formulation and completeinvestigation of BCPs in layered structures, which includes boththe theoretical (statement of the problems, solvability anduniqueness) and applied parts (approximate and numericalsolutions, codes, simulation).We elaborate a mathematical model of squeezing a thin elasticsheet placed on a stiff base without friction by weak loadsthrough several openings on one of its boundary surfaces. Weformulate and consider the corresponding BCPs in two- andthree-dimensional bands, prove the existence and uniqueness ofsolutions, and investigate their smoothness including the behaviorat infinity and in the vicinity of critical points. The BCP in atwo-dimensional band is reduced to a Fredholm integral equation(IE) with a logarithmic singularity of the kernel. The theory oflogarithmic IEs developed in the study includes the analysis ofsolvability and development of solution techniques when the set ofintegration consists of several intervals. The IE associated withthe BCP is solved by three methods based on the use ofFourier-Chebyshev series, matrix-algebraic determination of theentries in the resulting infinite system matrix, andsemi-inversion. An asymptotic theory for the BCP is developed andthe solutions are obtained as asymptotic series in powers of thecharacteristic small parameter.We propose and justify a technique for the solution of BCPs andboundary value problems with boundary conditions of mixed typecalled the approximate decomposition method (ADM). The main ideaof ADM is simplifying general BCPs and reducing them to a chainof auxiliary problems for 'shifted' Laplacian in long rectanglesor parallelepipeds and then to a sequence of iterative problemssuch that each of them can be solved (explicitly) by the Fouriermethod. The solution to the initial BCP is then obtained as alimit using a contraction operator, which constitutes inparticular an independent proof of the BCP unique solvability.We elaborate a numerical method and algorithms based on theapproximate decomposition and the computer codes and performcomprehensive numerical analysis of the BCPs including thesimulation for problems of practical interest. A variety ofcomputational results are presented and discussed which form thebasis for further applications for the modeling and simulation ofprinting-plate contact systems and other structures offlexographic printing. A comparison with finite-element solutionis performed.