Tensile bifurcations in a truncated hemispherical thin elastic shell
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Tensile bifurcations in a truncated hemispherical thin elastic shell Ciprian D. Coman
Abstract. The work described in this paper is concerned with providing a rational asymptotic analysis of the wrinkling bifurcation experienced by a thin elastic hemispherical segment subjected to vertical tensile forces on its upper rim. This is achieved by considering the interplay between two boundary layers and matching the corresponding solutions associated with each separate region. Our key result is a four-term asymptotic formula for the critical load in terms of a small parameter proportional to the ratio between the thickness and the radius of the shell. Comparisons of this formula with direct numerical simulations provide further insight into the range of validity of the results derived herein. Mathematics Subject Classification. 74K25, 74G60, 34D15, 34B09. Keywords. Wrinkling, Boundary layers, Shallow shell equations, Matched asymptotics.
1. Introduction The mechanical failure of structural elements under tensile loading is arguably far less common than in the case of compressive forces. It is therefore unsurprising that despite a vast literature on elastic buckling, the role played by tensile forces in triggering elastic instabilities is much less well documented. Perhaps, one of the possible explanations for the scarcity of this latter topic has to do with the fact that compressive stresses are not always triggered in structures subjected to tensile loads. When they do, such stresses are usually part of inhomogeneous stress fields that lead to bifurcation equations with variable coefficients, and whose analytical solutions tend to lie beyond the scope of most engineering curricula. In recent years, there has been a resurgence of interest in the onset of buckling-type instabilities in thin elastic plates under tension (e.g. [1–6]). Such interest has been mainly driven by various practical applications (e.g. [7–10]) and has generated many new insights into such an old and well-ploughed field as elastic buckling. On the other hand, examples of buckling in thin shells subjected to tensile forces have attracted comparatively less attention, although there are a number of common configurations known to be prone to such instabilities. An example in this direction is that of a truncated hemispherical shell subjected to vertical pulling forces uniformly distributed on its upper rim, and clamped along the equator. Yao [11] was the first to take up a brief numerical and experimental exploration of this interesting problem. He noted that the buckled shape was similar to that of a cylinder buckled under radial pressure, consisting of one axial half-wave and a short-wavelength rippling pattern around the shell’s lateral surface (i.e. in the azimuthal direction). Although his numerical work was based on a rather crude Galerkin method, he managed to obtain reasonably accurate values for the critical tension responsible for the instability. The starting point of this numerical work w
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