Shell buckling: from morphogenesis of soft matter to prospective applications

Geometric mapping from rectilinear material orthotropy to isotropy: Insights into plates and shells

Orthotropic shell structures are ubiquitous in biology and engineering, from bacterial cell walls to reinforced domes. We present a rescaling transformation that maps an orthotropic shallow shell to an isotropic one with a different local geometry. The mapping is applicable to any shell section for which the material orthotropy directions match the principal curvature directions, assuming the commonly used Huber form for the orthotropic shear modulus. Using the rescaling transformation, we derive exact expressions for the buckling pressure as well as the linear indentation response of orthotropic cylinders and general ellipsoids of revolution, which we verify against numerical simulations. Our analysis disentangles the separate contributions of geometric and material anisotropy to shell rigidity. In particular, we identify the geometric mean of orthotropic elastic constants as the key quantifier of material stiffness, playing a role akin to the Gaussian curvature which captures the geometric stiffness contribution. Besides providing insights into the mechanical response of orthotropic shells, our work rigorously establishes the validity of isotropic approximations to orthotropic shells and also identifies situations in which these approximations might fail.

Buckling of Thin-Walled Cylinders from Three Dimensional Nonlinear Elasticity

The famous bifurcation analysis performed by Flügge on compressed thin-walled cylinders is based on a series of simplifying assumptions, which allow to obtain the bifurcation landscape, together with explicit expressions for limit behaviours: surface instability, wrinkling, and Euler rod buckling. The most severe assumption introduced by Flügge is the use of an incremental constitutive equation, which does not follow from any nonlinear hyperelastic constitutive law. This is a strong limitation for the applicability of the theory, which becomes questionable when is utilized for a material characterized by a different constitutive equation, such as for instance a Mooney-Rivlin material. We re-derive the entire Flügge's formulation, thus obtaining a framework where any constitutive equation fits. The use of two different nonlinear hyperelastic constitutive equations, referred to compressible materials, leads to incremental equations, which reduce to those derived by Flügge under suitable simplifications. His results are confirmed, together with all the limit equations, now rigorously obtained, and his theory is extended. This extension of the theory of buckling of thin shells allows for computationally efficient determination of bifurcation landscapes for nonlinear constitutive laws, which may for instance be used to model biomechanics of arteries, or soft pneumatic robot arms.

Characterising the diffusion of biological nanoparticles on fluid and cross-linked membranes

Tracing the motion of macromolecules, viruses, and nanoparticles adsorbed onto cell membranes is currently the most direct way of probing the complex dynamic interactions behind vital biological processes, including cell signalling, trafficking, and viral infection. The resulting trajectories are usually consistent with some type of anomalous diffusion, but the molecular origins behind the observed anomalous behaviour are usually not obvious. Here we use coarse-grained molecular dynamics simulations to help identify the physical mechanisms that can give rise to experimentally observed trajectories of nanoscopic objects moving on biological membranes. We find that diffusion on membranes of high fluidities typically results in normal diffusion of the adsorbed nanoparticle, irrespective of the concentration of receptors, receptor clustering, or multivalent interactions between the particle and membrane receptors. Gel-like membranes on the other hand result in anomalous diffusion of the particle, which becomes more pronounced at higher receptor concentrations. This anomalous diffusion is characterised by local particle trapping in the regions of high receptor concentrations and fast hopping between such regions. The normal diffusion is recovered in the limit where the gel membrane is saturated with receptors. We conclude that hindered receptor diffusivity can be a common reason behind the observed anomalous diffusion of viruses, vesicles, and nanoparticles adsorbed on cell and model membranes. Our results enable direct comparison with experiments and offer a new route for interpreting motility experiments on cell membranes.