Two types of interface defects

The solution of a plane problem in the theory of elasticity for a two-component body with an interface, a finite part of which is either weakly distorted or is a weakly curved crack is constructed using the perturbation method. In the first case, it is assumed that the discontinuities in the forces and displacements at the interface are known, and, in the second case, the non-equilibrium nature of the load in the crack is taken into account. General quadrature formulae are derived for the complex potentials, which enable any approximation to be obtained in terms of elementary functions in many important practical cases. An algorithm is indicated for calculating each approximation. Families of defects are studied, the form of which is determined by power functions. The effect of the amplitude of the distortion and the shape of the interface crack on the Cherepanov–Rice integral as well as the shape of the distorted part of the interface on the stress concentration is investigated in the first approximation. An analysis of the applicability of the oscillating solution for a distorted interface crack is carried out. The results of the calculations are shown in the form of graphical relations.     

A transient electromechanical process in a liquid crystal dipole

On the basis of the equations of the electromagnetic field of an orthotropic medium and the Lagrange equations for rotational motion, we obtain the equations that describe the electromechanical state of a liquid crystal dipole in an external electric field. The results of computer modeling of the transient process when an electric field is abruptly imposed are given. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 160–162.     

Existence of minimal energy configurations of nematic liquid crystals with variable degree of orientation

Summary We prove existence of minimizers of the functional recently suggested by Ericksen [8] for the statics of nematic liquid crystals. A set of necessary conditions for the minimizers and a monotonicity formula are also found.     

Mathematical problems of liquid crystal flows

This article deals with flow phenomena in liquid crystal materials. The model that I analyse is that due to Ericksen for liquid crystals with variable degree of orientation. One of the goals is understanding the role of the order parameter in the model. I examine the multiplicity of solutions, phases, that occur in shear flow regimes and analyse their stability under physically realistic conditions. In particular, I show that ellipticity of the linearized system of governing equations is a consequence of the Clausius–Duhem inequality.     

Balance laws for liquid crystal mixtures

Equations of balance of mass, linear momentum, angular momentum and energy are obtained for each of the species in a chemically reacting mixture of liquid crystals. The forms of these equations for the mixture as a whole are then arrived at.     

Concentration of electroelastic fields in a composite piezoceramic plate with defects crossing the interface

The characteristics of fracture and strength of a composite piezoceramic plate with defects in the form of cracks and holes situated in both of the plate components are investigated. The corresponding boundary-value problems of electroelasticity are reduced to systems of singular integral equations by constructing integral representations of the complex potentials. The results of numerical realization of the constructed algorithms are reported.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 6, pp. 777–786, November–December, 1998.     

Reproductivity for a nematic liquid crystal model

In this paper we prove existence of weak solution with the reproductivity in time property, for a penalized PDE’s system related to a nematic liquid crystal model. This problem is relatively explict when time-independent Dirichlet boundary conditions are imposed for the orientation of crystal molecules. Nevertheless, for the time-dependent case, the treatment of the problem is completely different. The verification of a maximum principle for weak reproductive solutions is fundamental in the argument.     

Reproductivity for a nematic liquid crystal model

In this paper we prove existence of weak solution with the reproductivity in time property, for a penalized PDE’s system related to a nematic liquid crystal model. This problem is relatively explict when time-independent Dirichlet boundary conditions are imposed for the orientation of crystal molecules. Nevertheless, for the time-dependent case, the treatment of the problem is completely different. The verification of a maximum principle for weak reproductive solutions is fundamental in the argument. Finally, the relation between reproductive and periodic in time (regular) solutions will be pointed out, differenting the 2D and 3D cases. Basically, in two-dimensional domains every reproductive solution is regular and time periodic, whereas the problem remains open for three-dimensional domains.     

Smectic Liquid Crystals: Materials with One-dimensional, Periodic Order

Smectic liquid crystals are materials formed by stacking deformable, fluid layers. Although smectics prefer to have flat, uniformly-spaced layers, boundary conditions can impose curvature on the layers. Since the layer spacing and curvature are intertwined, the problem of finding minimal configurations for the layers becomes nontrivial. We discuss various topological and geometrical aspects of these materials and present recent progress on finding some exact layer configurations. We also exhibit connections to the study of certain embedded minimal surfaces and briefly summarize some important open problems.     

Regularity of solutions to the liquid crystal flows with rough initial data

In this paper, we are concerned with the regularity of solutions to the liquid crystal flows with rough initial data in Rⁿ$R^n$. We prove that the solution constructed by Wang (2011) in [23] has higher regularity. Moreover we obtain a decay estimate in time for any space derivative.     

A blowup criterion for the compressible nematic liquid crystal flows in dimension two

In this paper, we establish a blowup criterion for the two-dimensional compressible nematic liquid crystal flows. The criterion is given in terms of the density and the gradient of direction field, where the later satisfies the Serrin-type blowup criterion. For this result, we do not need the initial density to be positive.     

Scattering by small defects in the neighbourhood of a fluid-solid interface

The scattering of incident plane elastic waves by a varietyof different defects that lie upon a fluid-solid interface isconsidered here using matched asymptotic expansions. The expansionscheme is developed in terms of a parameter , the ratio of typicaldefect length scale to a typical wavelength of the incidentfield, taken to be small. Three different canonical situations occur and these are illustratedvia three specific examples treated here: a rigid strut, anedge crack, and a rigid strip. In each case the leading-ordermatching is performed to identify the leading-order contributionof the defect to the acoustic field in the far field. In particular,each defect is identified with a source of dipole response ininterfacial stress of displacement. It is shown in the limit as s<<s1 that in the inner problemsthe fluid and solid pieces uncouple in a particularly convenientmanner allowing analytical solutions to be deduced. These arethen matched with appropriate outer solutions.     

Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties

We study the asymptotic behavior of global classical solutions to hydrodynamical systems modeling the nematic liquid crystal flows under kinematic transports for molecules of different shapes. The coupling system consists of Navier–Stokes equations and kinematic transport equations for the molecular orientations. We prove the convergence of global solutions to single steady states as time tends to infinity as well as estimates on the convergence rate both in 2D for arbitrary regular initial data and in 3D for certain particular cases.     

Regularity of the solutions to the liquid crystal equations with small rough data

In [30], the author has proved the existence of Koch–Tataru [12] type solution to the incompressible nematic liquid crystal flow with initial data (_u0,_d0)$(u_0,d_0)$ in BMO⁻¹×BMO${\mathit{BMO}}^{-1}\times \mathit{BMO}$. In this paper, we shall present that the solution in [30] has arbitrary spatial regularity. Furthermore, we shall give the decay estimates for any space derivative.     

On tactical configurations with no four-cycles

An improved lower bound is given for the band sizes of tactical configurations of rank exceeding two having no 4-cycles. This bound is applied to find an optimal configuration with certain specified parameters. A formula is given for the maximum number of cycle types one must examine to demonstrate that a rank r configuration has no g-cycle. This result, which has appeared in our earlier work as the number of types of closed walks, answers a question of Longyear.