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We studied the stress concentration in the neighborhood of the vertices of the internal crack located on the bisector of an infinite elastic wedge. Normal forces are applied to the edges of the crack. The edges of the wedge are supported by a thin flexible covering, free from stresses from the outside. The effect of the coating on the stress-strain state of the wedge is modeled by a special boundary condition. The correctness of which is confirmed by numerical simulations. The integral Mellin transform made it possible to reduce the problem to the solution of a singular integral equation of the first kind with a Cauchy kernel with respect to the derivative of the crack opening function. Solutions of the integral equation are constructed by the collocation method for various combinations of geometric and physical parameters of the problem.

Ðàññìàòðèâàåòñÿ ñòàòè÷åñêàÿ çàäà÷à òåîðèè óïðóãîñòè î êîíöåíòðàöèè íàïðÿaeåíèé â îêðåñòíîñòè âåðøèí âíóòðåííåé òðåùèíû êîíå÷íîé äëèíû â ïîëîñå, óñèëåííîé òîíêèì ãèáêèì ïîêðûòèåì. Òðåùèíà ðàñïîëîaeåíà ïàðàëëåëüíî ãðàíèöàì ïîëîñû, áåðåãà åå íå âçàèìîäåéñòâóþò. Çàäà÷à ñèììåòðè÷íà îòíîñèòåëüíî ëèíèè òðåùèíû. Èññëåäîâàíèå îñíîâàíî íà ìåòîäå èíòåãðàëüíûõ ïðåîáðàçîâàíèé, êîòîðûé ïîçâîëèë ñâåñòè çàäà÷ó ê ðåøåíèþ ñèíãóëÿðíîãî èíòåãðàëüíîãî óðàâíåíèÿ ïåðâîãî ðîäà ñ ÿäðîì Êîøè.  êà÷åñòâå ìîäåëè ïîêðûòèÿ èñïîëüçîâàíû ñïåöèàëüíûå ãðàíè÷íûå óñëîâèÿ, ñôîðìóëèðîâàííûå íà îñíîâå àñèìïòîòè÷åñêîãî àíàëèçà ðåøåíèÿ çàäà÷è äëÿ òîíêîé óïðóãîé ïîëîñû, èçãèáíîé aeåñòêîñòüþ êîòîðîé ìîaeíî ïðåíåáðå÷ü. Ïðîâåäåíî èññëåäîâàíèå ðåãóëÿðíîé ÷àñòè ÿäðà â çàâèñèìîñòè îò ñîîòíîøåíèé ôèçè÷åñêèõ õàðàêòåðèñòèê ìàòåðèàëîâ ïîëîñû è ïîêðûòèÿ, à òàêaeå òàêèõ ãåîìåòðè÷åñêèõ ïàðàìåòðîâ, êàê ðàçìåð òðåùèíû è òîëùèíû ïîëîñû è ïîêðûòèÿ. Ðåøåíèå èíòåãðàëüíîãî óðàâíåíèÿ ïîñòðîåíî ìåòîäîì êîëëîêàöèé â âèäå ðàçëîaeåíèÿ ïî ïîëèíîìàì ×åáûøåâà ñ çàðàíåå âûäåëåííîé îñîáåííîñòüþ. Ïðîâåäåí àíàëèç ñõîäèìîñòè ìåòîäà â çàâèñèìîñòè îò ñîîòíîøåíèÿ çíà÷åíèé ïàðàìåòðîâ çàäà÷è. Ïîëó÷åíû çíà÷åíèÿ ôàêòîðà âëèÿíèÿ, ïðèâåäåííîãî êîýôôèöèåíòà èíòåíñèâíîñòè íîðìàëüíûõ íàïðÿaeåíèé â îêðåñòíîñòè âåðøèí òðåùèíû äëÿ ðàçëè÷íûõ êîìáèíàöèé ãåîìåòðè÷åñêèõ è ôèçè÷åñêèõ ïàðàìåòðîâ çàäà÷è.  ÷àñòíîñòè, óñòàíîâëåíî, ÷òî óâåëè÷åíèå òîëùèíû è aeåñòêîñòè ïîêðûòèÿ âåäåò ê ñíèaeåíèþ âåëè÷èíû ôàêòîðà âëèÿíèÿ. Óâåëè÷åíèå äëèíû òðåùèíû èëè óìåíüøåíèå øèðèíû ïîëîñû ïðèâîäèò ê óâåëè÷åíèþ âåëè÷èíû ôàêòîðà âëèÿíèÿ. Ðàññìîòðåíû èçâåñòíûå ÷àñòíûå ñëó÷àè óêàçàííîé çàäà÷è.  ÷àñòíîñòè, â ñëó÷àå îòñóòñòâèÿ ïîêðûòèÿ ðåçóëüòàòû ñîïîñòàâëåíû ñ èìåþùèìèñÿ â ëèòåðàòóðå äàííûìè. Êëþ÷åâûå ñëîâà: òðåùèíà, òîíêîå ãèáêîå ïîêðûòèå, êîýôôèöèåíò èíòåíñèâíîñòè íàïðÿaeåíèé, ôàêòîð âëèÿíèÿ, ìåòîä èíòåãðàëüíûõ ïðåîáðàçîâàíèé, ìåòîä êîëëîêàöèé, ñèíãóëÿðíîå èíòåãðàëüíîå óðàâíåíèå, ÿäðî Êîøè. Ââåäåíèå Ýêñïëóàòàöèÿ äåòàëåé ìàøèí è êîíñòðóêöèé ïðîèñõîäèò â óñëîâèÿõ âîçíèêíîâåíèÿ êîððîçèè, áîëüøèõ íàãðóçîê, ïîâûøåííîãî èçíàøèâàíèÿ, êîãäà êîíöåíòðà

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The first part of this work is devoted to the location of defects in a coated bar and the identification of their geometrical parameters. Using the methods of finite element modeling, ultrasonic non-destructive testing and machine learning technologies (artificial neural networks), the inverse problem of mechanics has been solved. A finite element model of ultrasonic wave propagation in a bar with a coating and an internal defect is constructed. Compared with previous works, the model used PML (Perfectly Matched Layer) structures, which suppress multiple reflections of the probe ultrasound pulse inside the bar and prevent signal noise. Based on the conducted numerical calculations of the finite element model, a data set was constructed. It contains the geometrical parameters of the defect and the corresponding amplitude-time characteristic of the ultrasonic signal. The architecture of a direct propagation neural network has been developed. The neural network was trained on the basis of previously processed data. As a result, on the basis of ultrasound data obtained from the outer surface of the bar, it is possible to restore the values of such defect parameters as depth, length and thickness. At the second stage, analytical-numerical technology for studying the stress intensity factor (SIF) at the crack tip is described using the example of the problem of a longitudinal internal crack of finite length located in an elastic strip reinforced with a thin flexible coating. The solution to this problem is based on the method of integral transformations, which made it possible to reduce it to a singular integral equation of the first kind with a Cauchy kernel, which is solved by the collocation method in the form of expansion in Chebyshev polynomials with a factor that explicitly takes into account a feature in the vicinity of the crack vertices. The latter allows you to directly find the SIF and evaluate the effect on it of various combinations of geometric and physical parameters of the problem.