Abstract
Introduction . Loss of stability and buckling of a round plate may be observed if the plate is loaded on the lateral surface. The solution to this problem is based on a bifurcation approach. In this case, a plate is considered as a nonlinear elastic body. It is important to choose the relation between stresses and deformations in sustainability problems of nonlinear elasticity. Simple laws of state (constitutive equations) were considered in early works devoted to this problem, for example, material of the “harmonic type” (Sensenig). Materials and Methods . Equations of neutral equilibrium for round plates made of Murnaghan and Blatz-Ko materials are obtained. Assuming a uniform initial deformation on the plate, the stability problem is considered. Strict threedimensional neutral equilibrium equations provide exploring related forms of equilibrium taking into account physical and geometric nonlinearity. Derivation of these equations is based on the application of the theory of superposition of small deformation on the final one. Results . Progress in solution to the corresponding secular equation (with non-linear parameter entry) for practically important laws of elasticity of Murnaghan and Blatz-Ko is possible using the numerical methods only. The developed method for calculating bifurcation values of loading parameters makes it possible to analyze the effect of nonlinearity. Discussion and Conclusions. The influence of physical and geometric nonlinearity on the upper critical value of the initial deformation parameter is explored. The results obtained can be used under the assessment of reliability of elastic third-order moduli for various physical materials. Data on these moduli is still scarce. The numerical research has shown that the constants given in some sources should be treated with caution. The use of elasticity moduli in the law of state of Blatz-Ko is also discussed.
Highlights
Loss of stability and buckling of a round plate may be observed if the plate is loaded on the lateral surface
A plate is considered as a nonlinear elastic body
It is important to choose the relation between stresses and deformations in sustainability problems of nonlinear elasticity
Summary
ФГБОУ ВО «Донской государственный технический университет» (г. Ростов-на-Дону, Российская Федерация). Если круглая пластина нагружена по боковой поверхности, возможны потеря ее устойчивости и выпучивание. Для круглой пластины из материалов Мурнагана и Блейтца и Ко получены уравнения нейтрального равновесия. В предположении однородной начальной деформации рассмотрена задача устойчивости этой пластины. Строгие трехмерные уравнения нейтрального равновесия позволяют исследовать смежные формы равновесия с учетом физической и геометрической нелинейности. Продвижение в решении соответствующего векового уравнения (с нелинейным вхождением параметра) для практически важных законов упругости Мурнагана и Блейтца и Ко возможно лишь с использованием численных методов. Исследовано влияние физической и геометрической нелинейности на величину верхнего критического значения параметра начальной деформации. Полученные результаты могут быть использованы при оценке достоверности модулей упругости третьего порядка для разных физических материалов. Обсуждается также использование модулей упругости в законе состояния Блейтца и Ко. Ключевые слова: конечная деформация, напряжения, выпучивание, пластина, устойчивость. И. Устойчивость нелинейно-упругой пластины при боковом сжатии / Г.
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