Acta Mech 221, 147–174 (2011) DOI 10.1007/s00707-011-0473-3 Dimosthenis I. Bardzokas · S. M. Mkhitaryan · Georgios I. Sfyris The methods of complex potentials, singular integral equations and integral transformations in a series of problems for the reinforcement of cracked plates Received: 23 March 2010 / Published online: 5 May 2011 © Springer-Verlag 2011 Abstract We study the initiation and propagation of a vertical crack in an elastic semi-infinite plate, reinforced on its boundary by an infinite discontinuous stringer within the limits of the theory of brittle failure. The plate is subjected to uniform distributed tensile forces at infinity, as well as to contact stresses due to application of forces to the stringer. We find the appropriate loading of the coherent stringer, and consequently we consider a problem where the stringer is cracked and a vertical crack has developed within the plate. We deduce the exact analytical solution for the principal singular integral equation for this case; hence the stringer is perfectly rigid and we calculate characteristic parameters of the problem. The results show that the crack tip has a logarithmic singularity, and the tangential contact stresses under the stringer at that end point are finite and generally differ from zero. 1 Introduction Contact problems of thin-walled elements (in the form of straight line stringers) with massive deformable bodies of different geometric forms were considered by many authors, in the fields of composite materials, metrical techniques and applied mechanics in general. Melan [1] presents the contact problem between an infinite stringer and an elastic semi-infinite plate. The stringer is considered to be a one-dimensional con- tinuum with negligible bending stiffness, and the normal contact stresses are equal to zero. In the case of two-dimensional elasticity, Bullfer [2], Muki and Steinberg [3] substantiated the above model. The funda- mental achievements in the field of generalizing and developing Melan’s work are reflected in the works of Sternberg [4], Erdogan [5], Grigoluk and Tolkachev [7], Cherepanov [8], Alexandrov and Mkhitaryan [9] and in the book “The development of the theory of contact problems in USSR” [6] edited by Galine. The problems of contact interaction of a discontinuous stringer with elastic massive media are of great interest from the fracture mechanics point of view and are closely connected with the denoted circle of problems. Theoretical investigations on the questions of how the cracks in composites are formed and propagated are also of great interest. In this field we should mention the works of Muki and Sternberg [10], Greif and Sanders [11], Delale, Erdogan [12], Theocaris and Bardzokas [13–15], Bardzokas and Exadaktylos and Anastaselos [16], Antipov et al. [17]. In ideal and methodological aspects the aforementioned problems are highly close to problems of D. I. Bardzokas: Deceased. D. I. Bardzokas · G. I. Sfyris (B ) National Technical University of Athens, Athens, Greece E-mail: bardim@central.ntua.gr S. M. Mkhitaryan Institute of Mechanics, National Academy of Sciences of Republic of Armenia, 24b Marshal Baghramianave., Erevan 0019, Republic of Armenia E-mail: Lilit_Dashtoyan@mechins.sci.am