For two types of large‐scale bodies, a half‐plane and a strip, each of which is weakened by a straight transverse crack, the static problem of elasticity theory is considered. The upper boundary of each body is reinforced by a thin flexible coating. The coating is modeled by special boundary conditions on the upper faces of considered bodies. Three different cases of boundary conditions on the lower face of the strip were studied. By application of generalized integral transforms to the equilibrium equations in displacements the problems were reduced to the solutions of singular integral equations of first kind with Cauchy kernel to the respect of derivative of the crack opening function. In all considered cases the integral equations consists of a singular term, corresponding to crack behavior in an infinite plate, and a regular term, reflecting the influence of various geometric and physical parameters. For various sets of model parameters the solutions of the integral equations were built by small parameter and collocation methods; their structure was analyzed. The values of stress intensity factor in the vicinity of the tips of the crack were obtained and analyzed for different coating materials and geometric parameters of the crack. From the analysis of the numerical results of the problem, it can be concluded that thin flexible coatings significantly reduce stress intensity at a crack tip and therewith significantly increase a reliability of considered elastic bodies.
Static elasticity problems for a half-plane and a strip weakened by a rectilinear transverse crack are studied. In each case, the upper boundary of the body is reinforced by a flexible patch. Various versions of conditions on the lower boundary are considered in the case of the strip. The crack is maintained in the open state by distributed normal forces. The method of generalized integral transforms reduces solving the problem for the equations of equilibrium to solving a singular integral equation of the first kind with the Cauchy kernel with respect to the derivative of the crack opening function. The solutions of the integral equation are constructed by the small parameter and collocation methods for various combinations of the geometric and physical parameters of the problem, and the structure of the solutions is analyzed. The values of the stress intensity factor (SIF) near the crack vertex are obtained.
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
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.