Surface waves in nematic liquid crystals

The problem of small-amplitude harmonic wave propagation along the surface of an incompressible nematic liquid crystal is considered when the evolution of the orientation vector is specified by viscous stresses and the orientational elasticity can be ignored. An analytic solution and a dispersion relation are obtained in the case of the planar and homeotropic initial orientation of this vector.     

Sharp-Interface Nematic–Isotropic Phase Transitions without Flow

We derive a supplemental evolution equation for an interface between the nematic and isotropic phases of a liquid crystal when flow is neglected. Our approach is based on the notion of configurational force. As an application, we study the behavior of a spherical isotropic drop surrounded by a radially oriented nematic phase: our supplemental evolution equation then reduces to a simple ordinary differential equation admitting a closed-form solution. In addition to describing many features of isotropic-to-nematic phase transitions, this simplified model yields insight concerning the occurrence and stability of isotropic cores for hedgehog defects in liquid crystals.     

各向异性介质中弹性波的数值模拟

提出了一种非均匀各向异性介质中弹性波传播的数值模拟算法。该方法可以灵活地运用于具有任意地表形状、内部孔洞、固液边界和不规则内部交界面的介质情况,另外,该方法自然满足复杂几何边界的自由表面条件。这种基于三角形和四边形离散网格的算法使用的是围绕每个节点的积分平衡方程,而不是其它有限差分法中使用的各个节点满足的弹性动力学的微分方程。该文工作是非均匀各向同性介质中弹性波传播格子法研究的继续。除了研究各向异性介质中波的传播以外,还给出了一种能够省时的四边形网格的格子法。     

Sharp-Interface Nematic-Isotropic Phase Transformations With Flow

We develop a sharp-interface theory for phase transformations between the isotropic and uniaxial nematic phases of a flowing liquid crystal. Aside from conventional evolution equations for the bulk phases and corresponding interface conditions, the theory includes a supplemental interface condition expressing the balance of configurational momentum. As an idealized illustrative application of the theory, we consider the problem of an evolving spherical droplet of the isotropic phase surrounded by the nematic phase in a radially-oriented state. For this problem, the bulk and interfacial equations collapse to a single nonlinear second-order ordinary differential equation for the radius of the droplet—an equation which, in essence, expresses the balance of configurational momentum on the interface. This droplet evolution equation, which closely resembles a previously derived and extensively studied equation for the expansion of contraction of a spherical gas bubble in an incompressible viscous liquid, includes terms accounting for the curvature elasticity and viscosity of the nematic phase, interfacial energy, interfacial viscosity, and the ordering kinetics of the phase transformation. We determine the equilibria of this equation and study their stability. Additionally, we find that motion of the interface generates a backflow, without director reorientation, in the nematic phase. Our analysis indicates that a backflow measurement has the potential to provide an independent means to determine the density difference between the isotropic and uniaxial nematic phases.     

Adhesion of oriented nematic droplets

We propose and analyze a two-dimensional model for the equilibrium of oriented droplets of nematic liquid crystals that may adhere to a rigid substrate, while surrounded by an isotropic environment. We obtain the contact condition at the edge where the liquid crystal, the substrate, and the environment come together. We further develop a fairly general method to arrive at the equilibrium shapes of a drop, which is then applied to the case where the surface tension at the liquid crystal interface is given by Rapini and Papoular's expression. In this case, we also predict the existence of concave equilibrium shapes. Here is indeed the main difference between this method and Wulff's construction, which always yields convex equilibrium shapes for a drop free from adhesion. Received February 22, 2000     

Elastic wave propagation in anisotropic spherical curved plates

Spherical plate-like structures are used in pressure vessels, spherical domes of power plants, and in many other industrial applications. For non-destructive evaluation of such spherical structures, the mechanics of elastic wave propagation in spherical curved plates must be understood. The current literature shows some valuable studies on Rayleigh surface wave propagation in isotropic solids with spherical boundaries. However, the guided wave propagation problem in an anisotropic spherical curved plate, which has not been studied before, is solved for the first time in this paper.The wave propagation, in both isotropic and anisotropic spherical curved plates, is investigated. The differential equations of motion and the stress-free boundary conditions on the inner and outer surfaces of a hollow sphere are approximately solved by a general solution technique. This solution technique was successfully utilized by the authors for solving the wave propagation problem in cylindrical plates, in their earlier works. Dispersion curves for spherical plates made of isotropic aluminum, steel, and anisotropic composite material are presented as well.     

Finite amplitude elastic waves due to non-uniform traction at the surface of a cylindrical cavity

This paper is concerned with two spatial dimension, finite amplitude wave propagation emanating from the surface of an initially circular cylindrical cavity in an unbounded isotropic compressible isotropic hyperelastic solid. The solid is initially in the natural reference configuration and the wave propagation is due to an azimuthally non-uniform, sudden application of compressive nominal traction at the surface of the cavity. Governing equations for the problem are obtained in Lagrangian form in terms of cylindrical polar coordinates and two different classes of strain energy functions are considered. Numerical solutions, for a particular application of traction, are obtained from a fully explicit finite difference scheme. It was found that the responses to the particular application of traction differed negligibly for the various strain energy functions considered.     

Shear response of monodomains of flow-aligning nematic liquid crystals: TIF model comparisons and effect of pre-tilt

This work is the continuation of a previous study [Van Horn BL, Winter HH (2000) Dynamics of shear aligning nematic liquid crystal monodomains. Rheol Acta 39:294–300] on the shear dynamics of monodomains of shear aligning nematic liquid crystals [NLC]. The strain dependence of director orientation has been experimentally investigated for monodomains of a NLC with various initial orientations. Comparison of experimental results to predictions using Ericksen's transversly isotropic fluid [TIF] model supports the validity of the TIF model for systems of low molecular weight NLCs. The TIF model has been used to examine the effect of pre-tilt on the dynamics of flow-aligning NLC monodomains. It is shown that small deviations from planar alignment (no pre-tilt) have a large effect on orientation dynamics.     

Dynamic interfacial crack propagation in elastic–piezoelectric bi-materials subjected to uniformly distributed loading

In transversely isotropic elastic solids, there is no surface wave for anti-plane deformation. However, for certain orientations of piezoelectric materials, a surface wave propagating along the free surface (interface) will occur and is called the Bleustein–Gulyaev (Maerfeld–Tournois) wave. The existence of the surface wave strongly influences the crack propagation event. The nature of anti-plane dynamic fracture in piezoelectric materials is fundamentally different from that in purely elastic solids. Piezoelectric surface wave phenomena are clearly seen to be critical to the behavior of the moving crack. In this paper, the problem of dynamic interfacial crack propagation in elastic–piezoelectric bi-materials subjected to uniformly distributed dynamic anti-plane loadings on crack faces is studied. Four situations for different combination of shear wave velocity and the existence of MT surface wave are discussed to completely analyze this problem. The mixed boundary value problem is solved by transform methods together with the Wiener–Hopf and Cagniard–de Hoop techniques. The analytical results of the transient full-field solutions and the dynamic stress intensity factor for the interfacial crack propagation problem are obtained in explicit forms. The numerical results based on analytical solutions are evaluated and are discussed in detail.     

Rayleigh Surface Waves on a Kelvin-Voigt Viscoelastic Half-Space

In this paper we consider the propagation of Rayleigh surface waves in an exponentially graded half-space made of an isotropic Kelvin-Voigt viscoelastic material. Here we take into account the effect of the viscoelastic dissipation energy upon the corresponding wave solutions. As a consequence we introduce the damped in time wave solutions and then we treat the Rayleigh surface wave problem in terms of such solutions. The explicit form of the secular equation is obtained in terms of the wave speed and the viscoelastic inhomogeneous profile. Furthermore, we use numerical methods and computations to solve the secular equation for some special homogeneous materials. The results sustain the idea, existent in literature on the argument, that there is possible to have more than one surface wave for the Rayleigh wave problem.     

Orientational instability of a lyotropic nematic liquid crystal layer

The problem of periodic domain initiation in a thin lyotropic nematic liquid crystal layer is studied. This layer has a planar director initial orientation, but the anchoring energy is minimized by the homeotropic one. The periodic structures whose wave vector is perpendicular to the director exist during the director reorientation process from the planar orientation to the homeotropic one when the reorientation wave front appears. It is shown that the divergent terms of the Prank orientation elasticity energy plays an important role in this effect. The saddle-splay Prank constant and the anisotropic anchoring energy coefficient are estimated.     

Thermoelastic surface waves on an exponentially graded half-space

In this paper, we consider the propagation of Rayleigh surface waves in a functionally graded isotropic thermoelastic half-space, in which all thermoelastic characteristic parameters exponentially change along the depth direction. The propagation condition is established in the form of a bicubic equation whose coefficients are complex numbers while the analytical solutions (eigensolutions) of the thermoelastodynamic system are explicitly obtained in terms of the characteristic solutions. The concerned solution of the Rayleigh surface wave problem is subsequently expressed as a linear combination of the three eigensolutions while the secular equation is established in an implicit form. The explicit secular equation is written when an isotropic and homogeneous thermoelastic half-space is considered and some numerical simulations are given for a specific material.     

Shear cell rupture of nematic liquid crystal droplets in viscous fluids

We model the hydrodynamics of a shear cell experiment with an immiscible nematic liquid crystal droplet in a viscous fluid using an energetic variational approach and phase-field methods [86]. The model includes the coupled system for the flow field for each phase, a phase-field function for the diffuse interface and the orientational director field of the liquid crystal phase. An efficient numerical scheme is implemented for the two-dimensional evolution of the shear cell experiment for this initial data. The same model reduces to an immiscible viscous droplet in a viscous fluid, which we simulate first to compare with other numerical and experimental behavior. Then we simulate drop deformation by varying capillary number (independent of liquid crystal physics), liquid crystal interfacial anchoring energy and Oseen–Frank distortional elastic energy. We show the number of eventual droplets (one to several) and “beads on a string” behavior are tunable with these three physical parameters. All stable droplets possess signature quadrupolar shear and normal stress distributions. The liquid crystal droplets always possess a global surface defect structure, called a boojum, when tangential surface anchoring is imposed. Boojums [79], [32] consist of degree +1/2 and ?1/2 surface defects within a bipolar global orientational structure.     

Skin friction measurements on reflective surfaces using nematic liquid crystal

 A new liquid crystal technique for full surface skin friction measurements is introduced. With the new technique, the transmission of light through nematic liquid crystal viewed in reflection provides a quantitative measurement of the skin friction. The measurement technique is discussed with the aid of a model which describes the rotation dynamics of the liquid crystal molecules. Calibration experiments performed in a laminar flow duct demonstrate that the model captures the essential physics of the new technique. The measurement of skin friction downstream of a three dimensional roughness element in an incompressible laminar boundary layer is then presented as a demonstration of the utility of the technique. Received: 5 June 1998/Accepted: 13 November 1998     

Rayleigh Waves on an Exponentially Graded Poroelastic Half Space

In this paper we consider the propagation of seismic waves in isotropic poroelastic half spaces with continuously varying elastic properties, namely with an exponentially decaying depth profile. The present paper shows that the problem leads naturally to a bicubic equation. We obtain explicit inhomogeneous plane wave solutions in an exponential evanescent form with respect to the depth of half space. Further, these solutions are used to solve the boundary value problem of a Rayleigh surface wave and the secular equation is established. The results obtained theoretically are exemplified for numerical data and represented graphically for a representative poroelastic material.     

Wave propagation in transversely isotropic cylinders

Various approaches have been used for model1ing problems dealing with interaction of acoustic/elastic waves with transversely isotropic cylinders. The authors developed the first mathematical model for the scattering of acoustic waves from transversely isotropic cylinders [Honarvar, F., Sinclair, A.N., 1996. Acoustic wave scattering from transversely isotropic cylinders. Journal of the Acoustical Society of America 100, 57–63.]. In the current paper, this model is used for derivation of the frequency equations of longitudinal and flexural wave propagation in free transversely isotropic cylinders. Consistency of this model with the physics of the problem is demonstrated and a systematic solution to the corresponding equations is developed. Numerical results obtained for a number of transversely isotopic cylinders are used for verification of the mathematical model.     

The role of divergent terms in the Frank energy of nematic liquid crystals

The effect of divergent terms in the Frank orientation energy of nematic liquid crystals on the equilibrium state of the director field is studied. Such terms have no effect on the equations of motion or on the equilibrium of the medium under consideration; however, they should be taken into account in the derivation of boundary conditions. It is shown that, in the case of boundary perturbations or in the case of polar orientation angle perturbations, the divergent terms can be considered as a surface energy for the azimuth angle (this energy is similar to the Rapini-Papoular energy). In addition, these terms may cause a deviation of the director in the plane parallel to the boundary. The equilibrium problem for a nematic liquid crystal is considered as an example in the case of small periodic boundary perturbations.     

Modeling Phononic Crystals via the Weighted Relaxed Micromorphic Model with Free and Gradient Micro-Inertia

In this paper the relaxed micromorphic continuum model with weighted free and gradient micro-inertia is used to describe the dynamical behavior of a real two-dimensional phononic crystal for a wide range of wavelengths. In particular, a periodic structure with specific micro-structural topology and mechanical properties, capable of opening a phononic band-gap, is chosen with the criterion of showing a low degree of anisotropy (the band-gap is almost independent of the direction of propagation of the traveling wave). A Bloch wave analysis is performed to obtain the dispersion curves and the corresponding vibrational modes of the periodic structure. A linear-elastic, isotropic, relaxed micromorphic model including both a free micro-inertia (related to free vibrations of the microstructures) and a gradient micro-inertia (related to the motions of the microstructure which are coupled to the macro-deformation of the unit cell) is introduced and particularized to the case of plane wave propagation. The parameters of the relaxed model, which are independent of frequency, are then calibrated on the dispersion curves of the phononic crystal showing an excellent agreement in terms of both dispersion curves and vibrational modes. Almost all the homogenized elastic parameters of the relaxed micromorphic model result to be determined. This opens the way to the design of morphologically complex meta-structures which make use of the chosen phononic material as the basic building block and which preserve its ability of “stopping” elastic wave propagation at the scale of the structure.     

Breather-like director reorientations in a nematic liquid crystal with nonlocal nonlinearity

The nature of nonlinear molecular deformations in a homeotropically aligned nematic liquid crystal (NLC) is presented. We start from the basic dynamical equation for the director axis of a NLC with elastic deformations and adopt space curve mapping procedure to analyze the dynamics. The NLC is governed by an integro-differential perturbed nonlocal nonlinear Schrödinger equation and we solve the same using Jacobi elliptic function method aided with symbolic computation and construct an exact solitary wave solution. In order to better understand the effect of nonlocality on the director reorientations of nematic liquid crystal, we have constructed the component forms of director axis using Darboux vector transformation. This intriguing property as a result of the relation between the coherence of the breather-like solitary deformation and the nonlocality reveals a strong need for a deeper understanding in the theory of self-localization in NLC systems.     

Transient torsional wave propagation in a transversely isotropic tube

Summary By use of the separation of variables method and the Laplace transformation, the two-dimensional transient torsional wave propagation problem in a transversely isotropic tube is studied when a torque is suddenly applied to its end surface. The results show that, for the discontinuous distribution of the impact shear stress, the region of the 2D stress distribution in a transversely isotropic tube becomes large with the increase of the anisotropy of the material. Received 13 June 1997; accepted for publication 17 June 1998