Thick smectic shells

The known ground state of ultrathin smectic films confined to the surface of a sphere is described by four +1/2 defects assembled on a great circle and a director which follows geodesic lines. Using a simple perturbative approach we show that for thick smectic films on a sphere with planar anchoring this solution breaks down, distorting the smectic layers. The instability happens when the bend elastic constant exceeds the anchoring strength times the radius of the inner sphere. Above this threshold, the formation of a periodic chevron-like structure, observed experimentally as well, relieves geometric frustration. We quantify the effect of thickness and curvature of smectic shells and provide insight into the wavelength of the observed texture.     

Landau theory for biaxial nematic liquid crystals with two order parameter tensors

We present a phenomenological theory for the homogeneous phases of nematic liquid crystals constituted by biaxial molecules. We propose a general polynomial potential in two macroscopic order parameter tensors that reproduces the mean-field phase diagram confirmed by Monte Carlo simulations [De Matteis et al. in Phys Rev E 72:041706 (2005)] and recently recognized to be universal [Bisi et al. in Phys Rev E 73:051709 (2006)] for dispersion force molecular pair-potentials enjoying the D _2h symmetry. The requirement that the phenomenological theory comply uniquely with this phase diagram reduces considerably the admissible phenomenological coefficients, both in their number and in the ranges where they can vary.      

Soft elasticity and microstructure in smectic C elastomers

Smectic C elastomers are layered materials exhibiting a solid-like elastic response normal to the layers and a rubbery response in the plane. The set of strains K _C minimizing the elastic energy contains a one-parameter family of simple stretches associated with an internal degree of freedom, coming from the in-plane component of the director. We investigate soft elasticity and the corresponding microstructure by determining the quasiconvex hull of the set K _C , and use this to propose experimental tests that should make the predicted soft response observable.      

The rheology of layered liquids: lamellar block copolymers and smectic liquid crystals

The frequency-dependence of the viscoelastic shear modulus at low frequencies in a lamellar polystyrene-polyisoprene block copolymer is qualitatively identical to that measured in small-molecule smectics, namely, the rod-like 4-cyano-4-octylbiphenyl and the flexible n-nonyl-1-O--D-glucopyranoside. All three materials were studied after quenching from the isotropic state, and during and after alignment by large-amplitude oscillatory shearing. The kinetics of aligning, as measured by changes in moduli during shearing, are similar, despite great differences in molecular characteristics. These moduli and the aligning process are evidently controlled by smectic fluctuations and defects, the dynamics of which have universal features.     

A rheological theory for liquid crystal thin films

A macroscopic rheological theory for compressible isothermal nematic liquid crystal films is developed and used to characterize the interfacial elastic, viscous, and viscoelastic material properties. The derived expression for the film stress tensor includes elastic and viscous components. The asymmetric film viscous stress tensor takes into account the nematic ordering and is given in terms of the film rate of deformation and the surface Jaumann derivative. The material function that describes the anisotropic viscoelasticity is the dynamic film tension, which includes the film tension and dilational viscosities. Viscous dissipation due to film compressibility is described by the anisotropic dilational viscosity. Three characteristic film shear viscosities are defined according to whether the nematic orientation is along the velocity direction, the velocity gradient, or the unit normal. In addition the dependence of the rheological functions on curvature and film thickness has been identified. The rheological theory provides a theoretical framework to future studies of thin liquid crystal film stability and hydrodynamics, and liquid crystal foam rheology. Received: 9 October 2000 Accepted: 6 April 2001     

Theory of linear viscoelasticity of chiral liquid crystals

The governing equations of monodomain isothermal cholesteric liquid crystals subjected to small amplitude oscillatory rectilinear shear have been derived for three representative helix orientations. The imposition of oscillatory flow excites splay-bend-twist deformations when the helix is aligned along the flow direction, splay-bend deformations when the helix is along the vorticity gradient, and twist deformations when aligned along the velocity axis. The different nature of the excited elastic modes as well as the anisotropic viscosities are reflected in the anisotropy of the linear viscoelastic material functions for small amplitude rectilinear oscillatory shear. When the helix is aligned along the flow direction, cholesteric viscoelasticity is strongest, and exists in a relatively narrow band of intermediate frequencies. When the helix is aligned along the vorticity direction cholesteric viscoelasticity is significant in a relatively broad range of intermediate frequencies. Finally, when the helix is aligned along the velocity gradient direction, cholesteric viscoelasticity is relatively insignificant and only exists in a narrow band of frequencies. The cholesteric pitch controls the location of viscoelastic region on the frequency spectrum, but only when the helix is not oriented along the vorticity axis.     

A unified constitutive theory for polymeric liquids I. Basic considerations and a simplified model

By combining a continuum mechanical approach with considerations of network theory and thermodynamics of irreversible processes, a set of differentialtype constitutive equations for polymeric liquids are obtained which provide expressions for the stress tensor, evolution equations of the effective Finger strain and Cauchy strain for the network deformation, and a first order differential equation governing the rigidity modulus. Unlike Giesekus' recent unified approach that starts from the bead-spring model, the theory lends itself more readily to a better understanding of most of the current theories based on continuum mechanics and molecular network concepts. Different recent models such as those due to Leonov, Dashner—Van Arsdale, Phan Thien—Tanner, and Acierno et al. (or Marrucci) can be unambiguously interpreted as resulting from specific approximations or additional assumptions.     

Orienting properties of discotic nematic liquid crystals in Jeffrey-Hamel flows

The orienting properties of incompressible discotic nematic liquid crystals for creeping flows between converging and diverging planar walls (Jeffrey-Hamel) are analyzed using the Leslie-Ericksen theory. The dependence of director orientation on the reactive parameter and the flow kinematics is presented. Closed form stationary solutions for the director orientation are found when elastic effects are neglected. Stationary numerical solutions for the velocity and director fields using the full Leslie-Ericksen theory are presented. The director field in converging flow is characterized by azimuthal (radial) centerline orientation, by being asymmetric with respect to the azimuthal (radial) direction, and by having an allowed orientation range that spans two half-quadrants (full quadrants). In the limiting case of perfectly flat disk ( –) the flow-induced director orientation in converging flow is the azimuthal direction, while in diverging flow the director rotates by a full n radians. By reducing the vertex angle between the walls to vanishingly small values, converging flow solutions properly reduce to those of flow between parallel plates, but diverging flows are expected to lead to a new instability.     

Replacing ordinary derivatives by gauge derivatives in the continuum theory of dislocations to compensate the action of the symmetry group

When the symmetry group of a body is continuous it plays a fundamental role on the nonlinear continuum theory of dislocations: it induces a non-uniqueness to the field that describes the defects – the uniform reference – and affects also other fundamental ingredients of the theory. The purpose of the present paper is to examine how certain important quantities of the dislocation theory are affected from symmetry's group action. Apart from the uniform reference we study how the deformation gradient, the first and second Piola–Kirchhoff stress tensors, the elasticities of the material and the momentum equation are affected from the action of the symmetry group. This action is inhomogeneous, namely, differs from point to point. A similar inhomogeneous action of a group may be found in gauge theories. Prompt by the gauge approach, we propose the use of the gauge covariant exterior derivative to compensate for the action of the symmetry group on the uniform reference. The main advantage of using this derivative is that the momentum equation for the static case retains its divergence form. It remains an open question how the Yang–Mills potentials may be determined for the present theory.     

A coupled zigzag theory for the dynamics of piezoelectric hybrid cross-ply plates

A recently developed coupled third-order zigzag theory for the statics of piezoelectric hybrid cross-ply plates is extended to dynamics. The theory combines a third-order zigzag approximation for the in-plane displacements and a sub-layerwise linear approximation for the electric potential, considering all components of the electric field. The nonuniform variation of the transverse displacement due to the piezoelectric field is accounted for. The conditions for the absence of shear traction at the top and bottom surfaces and continuity of transverse shear stresses in the presence of electromechanical loading are satisfied exactly, thereby reducing the number of displacement variables to five, which is the same as in a first- or third-order equivalent single-layer theory. The governing equations of motion are derived from the extended Hamilton's principle. The theory is assessed by comparing the Navier solutions for the free and forced harmonic vibration response of simply supported plates with the exact three-dimensional piezoelasticity solutions. Comparisons for hybrid test, composite and sandwich plates establish that the present theory is quite accurate for the dynamic response of moderately thick plates.     

A coupled solid/fluids mixture theory that suffices for diffusion problems

In this paper, I begin with the general formulation of mixture theory by Bowen and present the derivation of a minimal set of field equations, constitutive relations, and material parameters suitable for the solutions of meaningful diffusion problems. The specific results are for a single solid and two fluids, and they may be extended to any number of fluids. I allude to the results of three problems, viz. (1) the injection of a fluid into a geological formation saturated with another fluid, (2) the drainage of two dissimilar fluids from a geological formation due to in-situ fluid pore pressures, and (3) the process of squeezing a sponge dry, in order to illustrate the general applicability of the derived theory.     

A new non-linear higher-order shear deformation theory for large-amplitude vibrations of laminated doubly curved shells

A consistent higher-order shear deformation non-linear theory is developed for shells of generic shape, taking geometric imperfections into account. The geometrically non-linear strain-displacement relationships are derived retaining full non-linear terms in the in-plane displacements; they are presented in curvilinear coordinates in a formulation ready to be implemented. Then, large-amplitude forced vibrations of a simply supported, laminated circular cylindrical shell are studied (i) by using the developed theory, and (ii) keeping only non-linear terms of the von Kármán type. Results show that inaccurate results are obtained by keeping only non-linear terms of the von Kármán type for vibration amplitudes of about two times the shell thickness for the studied case.     

A closure approximation for nematic liquid crystals based on the canonical distribution subspace theory

 A closure approximation for nematic polymers is presented. It approximates the fourth rank order tensor in terms of lower rank tensors, and is derived in the framework of the canonical distribution subspace theory. This approach requires a choice of the class of distributions: Here the set of Bingham distributions is chosen, as already introduced by Chaubal and Leal (1998). The closure is written in a generic frame of reference, and in an explicit form, so that it can be easily implemented. Such formulation also permits studying the closure approximation with continuation tools, which rather completely describe the system dynamics. The predictions can then be compared with those obtained with the exact model. The shear flow is considered as a test, since this flow condition appears to be the most demanding for closure approximations for nematic polymers. Received: 30 November 1999/Accepted: 30 November 1999     

Invariant recording of elasticity theory equations

An invariant (with respect to rotations) formalization of equations of linear and nonlinear elasticity theory is proposed. An equation of state (in the form of a convex generating potential) for various crystallographic systems is written. An algebraic approach is used, which does not require any geometric constructions related to the analysis of symmetry in crystals. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 127–142, September–October, 2008.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CON-TINUUM THEORIES (Ⅱ)—MICROMORPHIC CONTINUUM THEORY AND COUPLE STRESS THEORY

The purpose is to reestablish the balance laws of momentum, angular momentum and energy and to derive the corresponding local and nonlocal balance equations for micromorphic continuum mechanics and couple stress theory. The desired results for micromorphic continuum mechanics and couple stress theory are naturally obtained via direct transitions and reductions from the coupled conservation law of energy for micropolar continuum theory, respectively. The basic balance laws and equations for micromorphic continuum mechanics and couple stress theory are constituted by combining these results derived here and the traditional conservation laws and equations of mass and microinertia and the entropy inequality. The incomplete degrees of the former related continuum theories are clarified. Finally, some special cases are conveniently derived. Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931≈)     

Poiseuille flow of Leslie–Ericksen discotic liquid crystals: solution multiplicity, multistability, and non-Newtonian rheology

Computational modeling of the steady capillary Poiseuille flow of flow-aligning discotic nematic liquid crystals (DNLCs) using the Leslie–Ericksen (LE) equations predicts solution multiplicity and multistability. The phenomena are independent of boundary conditions. The steady state solutions are classified into: (a) primary, (b) secondary, and (c) hybrid. Primary solutions exist for all orientation boundary conditions and all flow rates, and are characterized by a flow-alignment angle that is closest to the anchoring angle at the bounding surface. Secondary solutions exist for all orientation boundary conditions and flow rates above a certain critical value. The secondary solutions are characterized by a flow-alignment angle which can be either the nearest neighbor below the primary solution or any multiple of π above. Hybrid solutions interpolate between the primary and the nearest secondary solutions, and hence exhibit two alignment angles. All solutions are stable to planar finite amplitude perturbations. Hybrid solutions are unstable to front propagation and lead to primary or secondary solutions. The non-Newtonian rheology of the primary and secondary solutions is characterized by non-classical shear thinning and thickening apparent viscosity behavior. Well-aligned monodomains can lead to shear thickening, thinning, or a sequence of both. The degree of rheological uncertainty is present for planar and homeotropic anchoring conditions. The non-Newtonian rheology of non-aligned samples leads to shear thinning and lack the uncertainty of well-aligned samples, since the apparent viscosity becomes insensitive to orientation.     

A unified constitutive theory for polymeric liquids II. Applications to basic problems

A unified constitutive theory for polymeric liquids has been recently proposed. Its derivation is based on a combination of continuum mechanical approach, transient-network concept and thermodynamics of irreversible processes. In the resulting model, many modes may be present for each of which there are two time scales, associated with the loss rate and the nonaffine motion of transient network junctions, respectively. A single effective relaxation time, constructed from the two time scales, governs the behavior in the linear regime of deformation. Two new parameters for each mode, in comparison with other models, are introduced: (i) the ratio r of the two time scales, and (ii) the index a distinguishing the rates of loss and creation of junctions. Both are important only for the nonlinear regime of deformation. In this paper, the theory is applied to predict the following cases: (i) stress growth at constant shear strain rate, (ii) steady shear-rate-dependent viscosity and first normal-stress difference and (iii) transient elongational viscosity at constant elongational strain rate. Determination of the model parameters based on usual characterization experiments is described. Comparison of calculated and observed behavior of low-density polyethylene at 150 °C available in the literature are presented. In general, the agreement of the predictions with experiment appear gratifying even with the simplest version of the new model.