ON AXIAL MOTION OF THE PANEL UNDER MECHANICALAND TEMPERATURE ACTION
О ПРОДОЛЬНОМ ДВИЖЕНИИ ПАНЕЛИ ПРИ МЕХАНИЧЕСКИХ И ТЕМПЕРАТУРНЫХ ВОЗДЕЙСТВИЯХ

N. V. Banichuk, S. Yu. Ivanova
2016 Problems of Strength and Plasticity  
2 Ìîñêîâñêèé ôèçèêî-òåõíè÷åñêèé èíñòèòóò, Ìîñêâà, Ðîññèéñêàÿ Ôåäåðàöèÿ banichuk@ipmnet.ru Ïîñòóïèëà â ðåäàêöèþ 28.03.2016 Íà îñíîâå êëàññè÷åñêèõ ìåòîäîâ ìàòåìàòè÷åñêîé ôèçèêè è ìåõàíèêè èññëåäóþòñÿ ïðîáëåìû óñòîé÷èâîñòè äâèaeóùèõñÿ â îñåâîì íàïðàâëåíèè ìàòåðèàëîâ è ÷óâñòâèòåëüíîñòè ê âíåøíèì âîçäåéñòâèÿì ïîâåäåíèÿ óïðóãîãî ïîëîòíà. Èçó÷àåòñÿ ïðîöåññ ïðÿìîëèíåéíîãî äâèaeåíèÿ óïðóãîãî ïîëîòíà, îïåðòîãî íà ñèñòåìó øàðíèðíûõ ïîäêðåïëåíèé è ìîäåëèðóåìîãî óïðóãîé ïàíåëüþ, êîòîðàÿ äâèaeåòñÿ ñ
more » ... é ñêîðîñòüþ è ïîäâåðaeåíà íàãðåâó è âíåøíèì ìåõàíè÷åñêèì âîçäåéñòâèÿì. Ìàëûå ïîïåðå÷íûå óïðóãèå ïåðåìåùåíèÿ ïàíåëè îïèñûâàþòñÿ äèôôåðåíöèàëüíûì óðàâíåíèåì ÷åòâåðòîãî ïîðÿäêà, ñîäåðaeàùèì ÷ëåíû, îáóñëîâëåííûå äåéñòâèåì âíóòðèïëîñêîñòíîãî íàòÿaeåíèÿ è öåíòðîñòðåìèòåëüíûõ ñèë, à òàêaeå ÷ëåíû, ó÷èòûâàþùèå òåðìîìåõàíè-÷åñêèå âîçäåéñòâèÿ. Îòäåëüíî ðàññìàòðèâàþòñÿ äâà âàaeíûõ ñëó÷àÿ: ñëó÷àé äèâåðãåíöèè ïàíåëè ïðè îäíîðîäíîì âíóòðèïëîñêîñòíîì òåðìîìåõàíè÷åñêîì ðàñòÿaeåíèè, êîãäà èññëåäóåòñÿ ñòàòè÷åñêàÿ íåóñòîé÷èâîñòü (äèâåðãåíöèÿ) äâèaeóùåéñÿ èçîòðîïíîé ïàíåëè, è ñëó÷àé äåôîðìàöèè ïàíåëè ïðè îäíîâðåìåííîì òåðìîìåõàíè÷åñêîì èçãèáå è ðàñòÿaeåíèè. îáîèõ ñëó÷àÿõ çàäà÷è ðåøàþòñÿ àíàëèòè÷åñêè è ðåøåíèÿ ïîëó÷åíû â ÿâíîì âèäå. Êëþ÷åâûå ñëîâà: ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå, ïðîäîëüíî äâèaeóùàÿñÿ ïàíåëü, äèâåðãåíöèÿ, òåðìîóïðóãàÿ íåóñòîé÷èâîñòü Ââåäåíèå Èçó÷åíèå ìåõàíè÷åñêîãî ïîâåäåíèÿ ïðîäîëüíî äâèaeóùèõñÿ óïðóãèõ ñèñòåì áûëî íà÷àòî â ðàáîòå [1] è ïðîäîëaeåíî çíà÷èòåëüíî ïîçäíåå â [2−14]. ÷èñëå ñîâðåìåííûõ ðàáîò â ýòîé îáëàñòè ñëåäóåò îòìåòèòü ñòàòüè [15−18], à òàêaeå ìîíîãðàôèè [19, 20], â êîòîðûõ ïðèâåäåí øèðîêèé îáçîð íàó÷íîé ëèòåðàòóðû ïî ðàññìàòðèâàåìîé ïðîáëåìàòèêå. Öåëüþ íàñòîÿùåãî èññëåäîâàíèÿ ÿâëÿåòñÿ ðàçðàáîòêà ìàòåìàòè÷åñêîé ìîäåëè äëÿ äâèaeóùåãîñÿ ìàòåðèàëà è óñòàíîâëåíèå êðèòåðèåâ óñòîé÷èâîñòè ïðè òåðìîìå-õàíè÷åñêèõ âîçäåéñòâèÿõ, ïðèâîäÿùèõ ê äåôîðìàöèÿì â ïîïåðå÷íîì íàïðàâëåíèè è ê ñòàòè÷åñêèì ôîðìàì ïîòåðè óñòîé÷èâîñòè (äèâåðãåíöèè). Îòäåëüíî ðàññìàòðè-ÏÐÎÁËÅÌÛ ÏÐÎ×ÍÎÑÒÈ È ÏËÀÑÒÈ×ÍÎÑÒÈ, ò. 78, ¹ 2, 2016 ã. * Âûïîëíåíî ïðè ôèíàíñîâîé ïîääåðaeêå ÐÔÔÈ (ãðàíò ¹14-08-00016à). The problems of the stability of axially moving materials and the elastic web behavior sensitivity with respect to an external loading are investigated on the basis of the classic mathematical physics and mechanics methods. It is considered the process of rectilinear motion of a simply supported elastic web modelled as a panel which moves with a constant velocity and is on the external thermomechanical actions. Small transverse elastic displacements of the panel are described by a fourth-order differential equation that includes the terms expressing the action of in-plane tension and out-of-plane centrifugal forces and the terms taking into account a thermomechanical action. Two important cases are investigated separately: the divergence of the panel loaded by in-plane thermomechanical tension and the deformation of the beam subjected to combined thermomechanical bending and tension. In both cases the considered problems have been studied analytically and the solutions have been found in an explicit form.
doi:10.32326/1814-9146-2016-78-2-123-130 fatcat:kf67m7nk5nbn3nytpycyaexmoe