Spreading speeds in reducible multitype branching random walk

This paper gives conditions for the rightmost particle in the $n$th generation of a multitype branching random walk to have a speed, in the sense that its location divided by n converges to a constant as n goes to infinity. Furthermore, a formula for the speed is obtained in terms of the reproduction laws. The case where the collection of types is irreducible was treated long ago. In addition, the asymptotic behavior of the number in the nth generation to the right of na is obtained. The initial motive for considering the reducible case was results for a deterministic spatial population model with several types of individual discussed by Weinberger, Lewis and Li [J. Math. Biol. 55 (2007) 207-222]: the speed identified here for the branching random walk corresponds to an upper bound for the speed identified there for the deterministic model.Comment: Published in at http://dx.doi.org/10.1214/11-AAP813 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Characterization of soft stripe-domain deformations in Sm-C and Sm-C* liquid-crystal elastomers

The neoclassical model of Sm-C (and Sm-C*) elastomers developed by Warner and Adams predicts a class of “soft” (zero energy) deformations. We find and describe the full set of stripe domains—laminate structures in which the laminates alternate between two different deformations—that can form between pairs of these soft deformations. All the stripe domains fall into two classes, one in which the smectic layers are not bent at the interfaces, but for which—in the Sm-C* case—the interfaces are charged, and one in which the smectic layers are bent but the interfaces are never charged. Striped deformations significantly enhance the softness of the macroscopic elastic response

Exactly isochoric deformations of soft solids

Many materials of contemporary interest, such as gels, biological tissues and elastomers, are easily deformed but essentially incompressible. Traditional linear theory of elasticity implements incompressibility only to first order and thus permits some volume changes, which become problematically large even at very small strains. Using a mixed coordinate transformation originally due to Gauss, we enforce the constraint of isochoric deformations exactly to develop a linear theory with perfect volume conservation that remains valid until strains become geometrically large. We demonstrate the utility of this approach by calculating the response of an infinite soft isochoric solid to a point force that leads to a nonlinear generalization of the Kelvin solution. Our approach naturally generalizes to a range of problems involving deformations of soft solids and interfaces in 2 dimensional and axisymmetric geometries, which we exemplify by determining the solution to a distributed load that mimics muscular contraction within the bulk of a soft solid

Elasticity of Polydomain Liquid Crystal Elastomers

We model polydomain liquid-crystal elastomers by extending the neo-classical soft and semi-soft free energies used successfully to describe monodomain samples. We show that there is a significant difference between polydomains cross-linked in homogeneous high symmetry states then cooled to low symmetry polydomain states and those cross-linked directly in the low symmetry polydomain state. For example, elastomers cross-linked in the isotropic state then cooled to a nematic polydomain will, in the ideal limit, be perfectly soft, and with the introduction of non-ideality, will deform at very low stress until they are macroscopically aligned. The director patterns observed in them will be disordered, characteristic of combinations of random deformations, and not disclination patterns. We expect these samples to exhibit elasticity significantly softer than monodomain samples. Polydomains cross-linked in the nematic polydomain state will be mechanically harder and contain characteristic schlieren director patterns. The models we use for polydomain elastomers are spatially heterogeneous, so rather than solving them exactly we elucidate this behavior by bounding the energies using Taylor-like (compatible test strain fields) and Sachs (constant stress) limits extended to non-linear elasticity. Good agreement is found with experiments that reveal the supersoft response of some polydomains. We also analyze smectic polydomain elastomers and propose that polydomain SmC* elastomers cross-linked in the SmA monodomain state are promising candidates for low field electrical actuation.Comment: 13 pages, 11 figure

Supersoft elasticity in polydomain nematic elastomers

We consider the equilibrium stress-strain behavior of polydomain liquid crystal elastomers (PLCEs). We show that there is a fundamental difference between PLCEs cross-linked in the high temperature isotropic and low temperature aligned states. PLCEs cross-linked in the isotropic state then cooled to an aligned state will exhibit extremely soft elasticity (confirmed by recent experiments) and ordered director patterns characteristic of textured deformations. PLCEs cross-linked in the aligned state will be mechanically much harder and characterized by disclination textures