Bristol myers squibb bms

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Thin films are among the ubiquitous examples of flexible structures that buckle under compressive loads. More interestingly, these buckling instabilities usually develop into wrinkled patterns that provide a dramatic display of the applied stress field (1, 2). Wrinkles align perpendicularly to the bristol myers squibb bms direction, depicting the principal lines of stress and a geometric tool for mechanical characterization.

Traditional buckling theory is regularly used to understand these patterns in the near-threshold (NT) regime, in which the deformations are small perturbations of the initial flat state. However, it has been known since Wagner (3, 4) that, when the exerted loads are well in excess of those necessary to initiate buckling, the asymptotic state of the plate bristol myers squibb bms very different from the one observed under NT conditions.

In this far-from-threshold (FFT) regime, the stress nearly vanishes in the compression direction and bristol myers squibb bms mark the region where the compressive посмотреть больше has collapsed. Two complementary approaches have provided some insight into wrinkled sheets under FFT loading conditions.

In a 1961 paper (5), Stein and Hedgepeth computed the asymptotic stress field in infinitely thin sheets under compression by assuming a vanishing component of bristol myers squibb bms stress по этому адресу along the compression direction. They further showed how such an asymptotic stress field yields the extent of wrinkles in several basic examples.

A similar formalism that builds on the same assumption was advanced later by Pipkin (6). A second approach, which does address the wavelength of wrinkled sheets in the FFT regime, has been introduced recently by Cerda et al. Using the simplicity of the stress tensor in this geometry (where the dominant stress component is approximately T everywhere) and assuming a balance of bending, compressive, and tensile forces, these authors derived an asymptotic scaling law for the FFT wavelength and amplitude of wrinkles.

Although this idea bristol myers squibb bms been very successful in characterizing the asymptotic wrinkling pattern of tensed rectangular sheets, its implementation in situations characterized by a spatially varying stress distribution, with a wrinkled state spanning a finite region, remains obscure.

The lack of a theoretical setup that enables a quantitative distinction between wrinkling patterns in the NT and FFT regimes has led to confusion in interpreting experimental observations. For instance, the length and number of wrinkles in nanofilms have been measured in ref. In another experiment (10), the onset of wrinkling was identified by slowly increasing the exerted loads or modifying the setup geometry.

These and other experiments have shown various scaling laws for the length and number of wrinkles. In this paper, we present an FFT theory of wrinkling in very thin sheets that connects the tension field theory (5, 6) to the study of the wrinkle wavelength (7, 8).

We show that the extent of the wrinkled region (5) comes from the leading order of that expansion, whereas the wavelength and amplitude of wrinkles (7, 8) result from bristol myers squibb bms subleading order.

Furthermore, through a quantitative analysis of the FvK equations, our approach enables a clear identification of the NT and FFT regimes of wrinkling patterns and exposes the subtleties in interpretation of bristol myers squibb bms observations.

In order to bristol myers squibb bms the basic principles bristol myers squibb bms the theory, we focus on a model problem of fundamental interest: bristol myers squibb bms very thin annular sheet under planar axisymmetric loading (Fig.

Our main findings are summarized in Fig. The NT analysis is valid below the blue dashed line (see text). After a cross-over region (purple), the sheet is under FFT conditions (red). Curves a and b show the stress profile as predicted by Eq. However, curve c, which is well within the FFT region, shows that the hoop stress has collapsed in a manner compatible bristol myers squibb bms Eq.

To emphasize the collapse of compressive stress, the red dashed line in the inset shows the hoop stress given by Eq. We show how the resulting stress field bristol myers squibb bms to scaling laws for the extent and number of wrinkles, which are markedly different from the NT behavior. We emphasize insights provided by our model, explain experimental observations, and conclude with open questions and future по этому адресу. We study the essential differences between the NT and FFT ссылка by focusing on the configuration shown in Bristol myers squibb bms. Similar geometries have been used to study wrinkling under different types of central loads, such as the impact of fast projectiles (12), the de-adhesion and wrinkling of a thin sheet loaded at a point (13), and the wrinkling and folding of floating membranes (9, 14).

We included here the shear components, although they will not be required bristol myers squibb bms our analysis. Whereas the energy and wrinkled extent in узнать больше здесь NT regime are determined by Eqs. The forces in Bristol myers squibb bms. These relations allow us to estimate by inspection the different terms in Eq.

Motivated by experiments (9, 10) and following the formalism developed bristol myers squibb bms refs. Thus, wrinkling appears as a mechanism for releasing elastic energy in the film. For Rin r L, Eq. For a given wrinkle extent L, the FFT stresses are now fully characterized by Eqs.

The bristol myers squibb bms extent will be determined by minimizing the energy over L. Before turning to energy calculations, let us highlight some important aspects of the FFT solution. An analysis of the right-hand side Eq.



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