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Lipid membranes constitute very particular materials: on the one hand, theybreak very easily under microscopical stretching; on the other hand, they areextremely flexible, presenting deformations even at small scales. Consequently,a piece of membrane has an area excess relative to its optically resolvablearea, also called projected area. From a mechanical point of view, we can thusidentify three tensions associated to lipid membranes: the mechanical effectivetension $\tau$, associated to an increase in the projected area and to theflattening of the fluctuations; the tension $\sigma$, associated to themicroscopical area of the membrane and thus non measurable, but commonly usedin theoretical predictions; and its macroscopical counterpart measured throughthe fluctuation spectrum, $r$. Up to now, the equality between these quantitieswas taken for granted when analyzing experimental data. In this dissertation,we have studied, using the projected stress tensor, whether and under whichconditions it is justified to assume $\tau = \sigma$. We studied threegeometries (planar, spherical and cylindrical) and obtained the relation $\tau\approx \sigma - \sigma_0$, where $\sigma_0$ is a constant depending only onthe membrane's high frequency cutoff and on the temperature. Accordingly, weconclude that neglecting the difference between $\tau$ and $\sigma$ isjustifiable only to membranes under large tensions: in the case of smalltensions, corrections must be taken into account. We have studied theimplications of this result to the interpretation of experiments involvingmembrane nanotubes. Regarding $r$, we have questioned a former demonstrationconcerning its equality with $\tau$. Finally, the force fluctuation for planarmembranes and membrane nanotubes was quantified for the first time.

Forces and fluctuations in planar, spherical and tubular membranes