Transitions through Critical Temperatures in Nematic Liquid Crystals

We obtain ‘dynamic’ estimates for critical nematic liquid crystal (LC) temperatures with a slowly varying temperature-dependent control variable. We focus on two critical temperatures : the supercooling temperature below which the isotropic phase loses stability and the superheating temperature above which the ordered nematic states do not exist. In contrast to the static problem, the isotropic phase exhibits a memory effect below the supercooling temperature. This delayed loss of stability is independent of the rate of change of temperature and depends purely on the initial value of the temperature

Hemihelical local minimizers in prestrained elastic bi-strips

We consider a double layered prestrained elastic rod in the limit of vanishing cross section. For the resulting limit Kirchoff-rod model with intrinsic curvature we prove a supercritical bifurcation result, rigorously showing the emergence of a branch of hemihelical local minimizers from the straight configuration, at a critical force and under clamping at both ends. As a consequence we obtain the existence of nontrivial local minimizers of the $3$-d system.Comment: 16 pages, 2 figure

The role of the patch test in 2D atomistic-to-continuum coupling methods

For a general class of atomistic-to-continuum coupling methods, coupling multi-body interatomic potentials with a P1-finite element discretisation of Cauchy--Born nonlinear elasticity, this paper adresses the question whether patch test consistency (or, absence of ghost forces) implies a first-order error estimate. In two dimensions it is shown that this is indeed true under the following additional technical assumptions: (i) an energy consistency condition, (ii) locality of the interface correction, (iii) volumetric scaling of the interface correction, and (iv) connectedness of the atomistic region. The extent to which these assumptions are necessary is discussed in detail.Comment: Version 2: correction of some minor mistakes, added discussion of multiple connected atomistic region, minor improvements of styl

On approaches to modelling lattice dislocations

By proposing a sinusoidal relationship between slip discontinuity and the associated mismatch force, Peierls and Nabarro famously developed a dislocation model that eliminates the stress singularity from the Volterra dislocation model. Recently, Lubarda and Markenscoff (Appl. Phys. Lett. 89:151923, 2006) developed a model in which the Burgers vector of the dislocation is applied over some finite distance, , described as the ‘core radius’. They found that the shear stress on the glide-plane predicted in the Lubarda-Markenscoff model is identical to that predicted by the Peierls-Nabarro model. In this paper, we investigate generalisations of both the Lubarda-Markenscoff and Peierls-Nabarro models, demonstrating that different distributions of infinitesimal dislocations in a generalised Lubarda-Markenscoff model can be associated with different expressions for the misalignment force in a generalised Peierls-Nabarro model. Our results indicate that the generalised Lubarda-Markenscoff framework is a versatile and useful method for modelling the core of a dislocation that neatly complements the well established Peierls-Nabarro framework

The mechanics of a chain or ring of spherical magnets

Strong magnets, such as neodymium-iron-boron magnets, are increasingly being manufactured as spheres. Because of their dipolar characters, these spheres can easily be arranged into long chains that exhibit mechanical properties reminiscent of elastic strings or rods. While simple formulations exist for the energy of a deformed elastic rod, it is not clear whether or not they are also appropriate for a chain of spherical magnets. In this paper, we use discrete-to-continuum asymptotic analysis to derive a continuum model for the energy of a deformed chain of magnets based on the magnetostatic interactions between individual spheres. We find that the mechanical properties of a chain of magnets differ significantly from those of an elastic rod: while both magnetic chains and elastic rods support bending by change of local curvature, nonlocal interaction terms also appear in the energy formulation for a magnetic chain. This continuum model for the energy of a chain of magnets is used to analyse small deformations of a circular ring of magnets and hence obtain theoretical predictions for the vibrational modes of a circular ring of magnets. Surprisingly, despite the contribution of nonlocal energy terms, we find that the vibrations of a circular ring of magnets are governed by the same equation that governs the vibrations of a circular elastic ring

Analysis of a moving mask approximation for martensitic transformations

In this work we introduce a moving mask approximation to describe the dynamics of austenite to martensite phase transitions at a continuum level. In this framework, we prove a new type of Hadamard jump condition, from which we deduce that the deformation gradient must be of the form $\mathsf{1} +\mathbf{a}\otimes \mathbf{n}$ a.e. in the martensite phase. This is useful to better understand the complex microstructures and the formation of curved interfaces between phases in new ultra-low hysteresis alloys such as Zn45Au30Cu25, and provides a selection mechanism for physically-relevant energy-minimising microstructures. In particular, we use the new type of Hadamard jump condition to deduce a rigidity theorem for the two well problem. The latter provides more insight on the cofactor conditions, particular conditions of supercompatibility between phases believed to influence reversibility of martensitic transformations