| Резюме: | <p>Wrinkling is a universal instability occurring in a wide variety of engineering and biological materials. Despite extensive study across different systems, a full description is still lacking, particularly in the case of growing materials. Most studies consist of some combination of a substantially simplified elastic model, an analysis that does not go beyond first order, and numerical simulations in commercial software packages that do not necessarily correspond to the mathematical system studied. This thesis addresses all three of these shortcomings by providing a systematic analysis of a fully hyperelastic bilayer past the linear stability threshold into the weakly nonlinear regime, along with a carefully discretised numerical bifurcation analysis of the system. </p>
<p>For comparison, we assume that wrinkling is generated either by the isotropic growth of the film or by the lateral compression of the entire film-substrate system (both in plane strain). We adopt a stream-function-based formulation and perform an exhaustive linear analysis of the wrinkling problem for all stiffness ratios and under a variety of additional boundary and material effects. Namely, we consider the effect of added pressure, surface tension, an upper substrate, and fibres. We obtain analytical estimates of the instability in the two asymptotic regimes of long and short wavelengths.</p>
<p>We then carry out a weakly-nonlinear analysis to derive an amplitude equation that describes the evolution of the wrinkling amplitude beyond the bifurcation point, followed by a comprehensive numerical bifurcation analysis of the problem using the finite element method. We demonstrate excellent agreement between the weakly-nonlinear analysis and the numerical experiments and are also able to directly solve for the bifurcation point in our discretised system and characterise the effect of implementation details such as the aspect ratio of the computational domain on the observed bifurcation point. </p>
<p>We then explore solutions of the amplitude equation in the case that the wrinkling amplitude is allowed to vary over long spatial and/or temporal scales. Finally, we demonstrate that our numerical experiments are able to identify secondary bifurcations in the system similar to those observed in experiments for which analytical methods have thus far been unable to provide a complete explanation.</p>
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