Theory of interfacial dynamics of nematic polymers

 The interfacial momentum and torque balance equations for deforming interfaces between nematic polymers and isotropic viscous fluids are derived and analyzed with respect to shape selection and interfacial nematic ordering. It is found that the interfacial momentum balance equation for nematic interfaces involves bending forces that act normal to the interface, and that interfacial pressure jumps may exist even for planar surfaces. In addition tangential forces on nematic interfaces arise in the presence of surface gradients of the tensor order parameter. The torque balance equation shows that couple stress jumps are balanced by the surface molecular field. The interfacial balance equations are shown to be coupled such that nematic ordering depends on shape and vice versa. The governing dimensionless numbers for deforming nematic polymer interfaces are identified and the limiting regimes are discussed in reference to related experimental data. It is found that the ratio of Frank elasticity to surface anchoring controls whether the surface tensor order parameter deviates from its preferred equilibrium value. Whether the shape is affected, depends on the relative magnitudes of the isotropic surface tension, Frank bulk elasticity, and anchoring energy, and capillary number. Received: 16 April 1999/Accepted: 19 August 1999     

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     

Waves on the interface between a nematic liquid crystal and an ideal isotropic fluid

The problem of surface wave propagation over the interface between a nematic liquid crystal and an ideal isotropic fluid is considered. For the nematic liquid crystal the Frank-Oseen model with an isotropic viscous stress tensor is used. Anisotropic surface tension is described by the Rapini model. In this formulation, for the problem of harmonic small-amplitude surface wave propagation, in the case of infinite depths of both phases, an analytical solution is obtained. The dispersion relation is derived and its properties are investigated.     

Kinetic theories and mesoscopic models for solutions of nonhomogeneous liquid crystal polymers

We present a systematic derivation of hydrodynamic theories for nonhomogeneous nematic liquid crystal polymers (LCPs) by approximating the molecules as rigid ellipsoids, which can be either uniaxial molecules (spheroids) or biaxial ones. The short range interaction is assumed to be dominated by the excluded volume effect. Additional molecular properties with ellipsoidal molecules, e.g., a dipole–dipole interaction in extended nematics and chiral molecular structure in cholesterics, are accounted for through additional intermolecular potentials. Long-range molecular interaction is implemented through an averaged mean-field potential characterized by interaction functions. The extra elastic stress tensor is calculated using an extended virtual work principle consistent with conservation of angular momentum on the material volume, whereas the extra viscous stress is obtained by Batchelor’s volume averaging method. In the isothermal case, the theories are shown to satisfy the second law of thermodynamics, i.e., they admit positive production of entropy or energy dissipation. In the case of cholesterics, the kinetic theory reduces to the Leslie–Ericksen theory in the limit of weak translational diffusion, weak long range interaction, and weak flow.     

Dissipative soft modes in viscous nematodynamics

The paper analyses a possible occurrence of soft and semi-soft viscous modes in slow (low Reynolds number) flows of uniaxially anisotropic nematic liquids as described by the five parametric Leslie-Ericksen-Parodi (LEP) constitutive equations (CEs). As in the similar elastic case, the soft viscous modes theoretically cause no resistance to flow, nullifying the corresponding components of the viscous part of the total stress tensor, and do not contribute to the dissipation. That is why these modes can also be called dissipative soft modes. In some flows, these dissipative soft modes may cause the effect of nematic superfluidity. As in the theories of nematic elastic solids, this effect is caused by a marginal thermodynamic stability. The analysis is simplified in a specific local, rotating orthogonal coordinate system whose one axis is directed along the director. We demonstrate that depending on closeness of material parameters to the marginal stability conditions, LEP CEs describe the entire variety of soft, semi-soft and harder behaviors of nematic viscous liquids. When the only shearing dissipative modes are soft, the viscous part of stress tensor is symmetric, and LEP CE for stress, scaled with isotropic viscosity is reduced to a one-parametric, stress-strain rate anisotropic relation. When additionally the elongation dissipative mode is also soft, this scaled relation has no additional parameters and shows that the dissipation is always less than that in isotropic phase. Simple shearing and simple elongation flows illustrate these possible effects.     

Flow in a Porous Matrix with Anisotropic Structure

The flow of a viscous fluid through a porous matrix undergoing only infinitesimal deformation is described in terms of intrinsic variables, namely, the density, velocity and stress occurring in coherent elements of each material. This formulation arises naturally when macroscopic interfaces are conceptually partitioned into area fractions of fluid–fluid, fluid–solid, and solid–solid contact. Such theory has been shown to yield consistent jump conditions of mass, momentum and energy across discontinuities, either internal or an external boundary, unlike the standard mixture theory jump conditions. In the previous formulation, the matrix structure has been considered isotropic; that is, the area fractions are independent of the interface orientation. Here, that is not assumed, so in particular, the cross-section area of a continuous fluid tube depends on its orientation, which influences the directional fluxes, and in turn the directional permeability, anisotropy of the structure. The simplifications for slow viscous flow are examined, and particularly for an isotropic linearly elastic matrix in which area partitioning induces anisotropic elastic response of the mixture. A final specialization to an incompressible fluid and stationary matrix leads to potential flow, and a simple plane flow solution is presented to illustrate the effects of anisotropic permeability.     

Capillary instabilities in a thin nematic liquid crystalline fiber embedded in a viscous matrix

$m$ to take into account non-axisymmetric modes. Capillary instabilities in nematic fibers reflect the anisotropic nature of liquid crystals, such as the orientation contribution to the surface elasticity and surface bending stresses. Surface gradients of bending stresses provide additional anisotropic contributions to the capillary pressure that may renormalize the classical displacement and curvature forces that exist in any fluid fiber. The exact nature (stabilizing and destabilizing) and magnitude of the renormalization of the displacement and curvature forces depend on the nematic orientation and the anisotropic contribution to the surface energy, and accordingly capillary instabilities may be axisymmetric or non-axisymmetric, with finite or unbounded wavelengths. Thus, the classical fiber-to-droplet transformation is one of several possible instability pathways while others include surface fibrillation. The contribution of the viscosity ratio to the capillary instabilities of a thin nematic fiber in a viscous matrix is analyzed by two parameters, the fiber and matrix Ohnesorge numbers, which represent the ratio between viscous and surface forces in each phase. The capillary instabilities of a thin nematic fiber in a viscous matrix are suppressed by increasing either fiber or matrix Ohnesorge number, but estimated droplet sizes after fiber breakup in axisymmetric instabilities decrease with increasing matrix Ohnesorge number. Received November 26, 2001 / Published online May 21, 2002 Communicated by Epifanio Virga, Pavia     

Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions

In this paper we propose a formulation of polyconvex anisotropic hyperelasticity at finite strains. The main goal is the representation of the governing constitutive equations within the framework of the invariant theory which automatically fulfill the polyconvexity condition in the sense of Ball in order to guarantee the existence of minimizers. Based on the introduction of additional argument tensors, the so-called structural tensors, the free energies and the anisotropic stress response functions are represented by scalar-valued and tensor-valued isotropic tensor functions, respectively. In order to obtain various free energies to model specific problems which permit the matching of data stemming from experiments, we assume an additive structure. A variety of isotropic and anisotropic functions for transversely isotropic material behaviour are derived, where each individual term fulfills a priori the polyconvexity condition. The tensor generators for the stresses and moduli are evaluated in detail and some representative numerical examples are presented. Furthermore, we propose an extension to orthotropic symmetry.     

Non-symmetric viscoelasticity of anisotropic polymer liquids

The liquid crystalline (LC) polymers are considered as anisotropic viscoelastic liquids with nonsymmetric stresses. A simple constitutive equation for nematic polymers describing the coupled relaxation of symmetric and antisymmetric parts of the stress tensor is formulated. For illustration of non-symmetric anisotropic viscoelasticity, the simplest viscometric flows of polymeric nematics in the magnetic field are considered. The frequency and shear rate dependencies of extended set of Miesowicz viscosities are predicted. Received: 23 March 1999/Accepted: 13 December 1999     

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.     

A Natural Generalization of Hypoelasticity and Eulerian Rate Type Formulation of Hyperelasticity

According to the classical hypoelasticity theory, the hypoelasticity tensor, i.e. the fourth order Eulerian constitutive tensor, characterizing the linear relationship between the stretching and an objective stress rate, is dependent on the current stress and must be isotropic. Although the classical hypoelasticity in this sense includes as a particular case the isotropic elasticity, it fails to incorporate any given type of anisotropic elasticity. This implies that one can formulate the isotropic elasticity as an integrable-exactly classical hypoelastic relation, whereas one can in no way do the same for any given type of anisotropic elasticity. A generalization of classical theory is available, which assumes that the material time derivative of the rotated stress is dependent on the rotated Cauchy stress, the rotated stretching and a Lagrangean spin, linear and of the first degree in the latter two. As compared with the original idea of classical hypoelasticity, perhaps the just-mentioned generalization might be somewhat drastic. In this article, we show that, merely replacing the isotropy property of the aforementioned stress-dependent hypoelasticity tensor with the invariance property of the latter under an R-rotating material symmetry group R⋆ G ₀, one may establish a natural generalization of classical theory, which includes all of elasticity. Here R is the rotation tensor in the polar decomposition of the deformation gradient and G ₀ any given initial material symmetry group. In particular, the classical case is recovered whenever the material symmetry is assumed to be isotropic. With the new generalization it is demonstrated that any two non-integrable hypoelastic relations based on any two objective stress rates predict quite different path-dependent responses in nature and hence can in no sense be equivalent. Thus, the non-integrable hypoelastic relations based on any given objective stress rate constitute an independent constitutive class in its own right which is disjoint with and hence distinguishes itself from any class based on another objective stress rate. Only for elasticity, equivalent hypoelastic formulations based on different stress rates may be established. Moreover, universal integrability conditions are derived for all kinds of objective corotational stress rates and for all types of material symmetry. Explicit, simple, integrable-exactly hypoelastic relations based on the newly discovered logarithmic stress rate are presented to characterize hyperelasticity with any given type of material symmetry. It is shown that, to achieve the latter goal, the logarithmic stress rate is the only choice among all infinitely many objective corotational stress rates. This revised version was published online in August 2006 with corrections to the Cover Date.     

From coherent to incoherent mismatched interfaces: A generalized continuum formulation of surface stresses

The equilibrium of coherent and incoherent mismatched interfaces is reformulated in the context of continuum mechanics based on the Gibbs dividing surface concept. Two surface stresses are introduced: a coherent surface stress and an incoherent surface stress, as well as a transverse excess strain. The coherent surface stress and the transverse excess strain represent the thermodynamic driving forces of stretching the interface while the incoherent surface stress represents the driving force of stretching one crystal while holding the other fixed and thereby altering the structure of the interface. These three quantities fully characterize the elastic behavior of coherent and incoherent interfaces as a function of the in-plane strain, the transverse stress and the mismatch strain. The isotropic case is developed in detail and particular attention is paid to the case of interfacial thermo-elasticity. This exercise provides an insight on the physical significance of the interfacial elastic constants introduced in the formulation and illustrates the obvious coupling between the interface structure and its associated thermodynamics quantities. Finally, an example based on atomistic simulations of Cu/Cu₂O interfaces is given to demonstrate the relevance of the generalized interfacial formulation and to emphasize the dependence of the interfacial thermodynamic quantities on the incoherency strain with an actual material system.     

Blends of two immiscible and rheologically different fluids

The Doi and Ohta theory of the rheology of immiscible blends is extended by considering the field of the velocity with which the interface moves as an independent state variable. This type of generalization is needed in order to be able to take into account the difference in the rheological properties of the fluids involved. Expression for the extra stress tensor as well as equations governing the time evolution of the interfacial area, orientation of the interface and its velocity are derived.     

考虑温度效应的弹性损伤一般理论

现有的各种损伤理论基本上都是关于等温问题的 ,且在不同程度上依赖于某些经验假设。本文在严格的不可逆热力学理论基础之上 ,建立了考虑温度效应的弹性损伤一般理论。推导出热弹性各向同性与各向异性损伤材料全部本构方程的一般形式 ,其中包括应力 应变关系、熵密度方程、损伤对偶张量表达式、热 固 损伤耦合的热传导方程和损伤演化方程。它们的特殊形式包含了等温各向同性与各向异性弹性损伤的本构方程     

非饱和颗粒材料的多孔连续体有效压力与有效广义Biot应力

多孔连续体理论框架下的非饱和多孔介质广义有效压力定义和Bishop参数的定量表达式长期以来存在争议,这也影响了对与其直接相关联的非饱和多孔介质广义Biot有效应力的正确预测.基于随时间演变的离散固体颗粒-双联液桥-液膜体系描述的Voronoi胞元模型,利用由模型获得的非饱和颗粒材料表征元中水力-力学介观结构和响应信息,文章定义了低饱和度多孔介质局部材料点的有效内状态变量:非饱和多孔连续体的广义Biot有效应力和有效压力,导出了其表达式.所导出的有效压力公式表明,非饱和多孔连续体的有效压力张量为各向异性,它不仅对非饱和多孔连续体广义Biot有效应力张量的静水应力分量的影响呈各向异性,同时也对其剪切应力分量有影响.文章表明,非饱和多孔连续体中提出的广义Biot理论和双变量理论的基本缺陷在于它们均假定反映非混和两相孔隙流体对固相骨架水力-力学效应的有效压力张量为各向同性.此外,为定义各向同性有效压力张量和作为加权系数而引入的Bishop参数并不包含对非饱和多孔连续体中局部材料点水力-力学响应具有十分重要效应的基质吸力.所导出的非饱和多孔介质广义Biot有效应力和有效压力公式(包括反映有效压力...     

Constitutive modeling of complex interfaces based on a differential interfacial energy function

An important theory on the dynamics of complex interfaces is the Doi and Ohta theory where the interfacial contribution to the Cauchy stress tensor is determined from an interfacial conformation tensor. For a uniform deformation field in the Eulerian framework, Doi and Ohta adopted a decoupling approximation to reduce a fourth-order tensor into two second-order tensors and derived a differential equation governing the evolution of the interfacial conformation tensor. In this paper, a different formulation is presented for establishing the Cauchy stress tensor based on a path-independent interfacial energy function. By differentiating this interfacial energy function against a Lagrangian strain tensor, a nearly closed-form solution for the stress tensor was determined, involving only elementary algebraic and matrix operations. From this process, the stress-conformation relation proposed by Doi and Ohta is also confirmed from a thermodynamic perspective. The testing cases with uniaxial elongation and simple shear further showed improved fitting to the analytical or exact solutions.     

A new modified couple stress theory for anisotropic elasticity and microscale laminated Kirchhoff plate model

A new modified couple stress theory for anisotropic elasticity is proposed. This theory contains three material length scale parameters. Differing from the modified couple stress theory, the couple stress constitutive relationships are introduced for anisotropic elasticity, in which the curvature (rotation gradient) tensor is asymmetric and the couple stress moment tensor is symmetric. However, under isotropic case, this theory can be identical to modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The differences and relations of standard, modified and new modified couple stress theories are given herein. A detailed variational formulation is provided for this theory by using the principle of minimum total potential energy. Based on the new modified couple stress theory, composite laminated Kirchhoff plate models are developed in which new anisotropic constitutive relationships are defined. The First model contains two material length scale parameters, one related to fiber and the other related to matrix. The curvature tensor in this model is asymmetric; however, the couple stress moment tensor is symmetric. Under isotropic case, this theory can be identical to the modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The present model can be viewed as a simplified couple stress theory in engineering mechanics. Moreover, a more simplified model of couple stress theory including only one material length scale parameter for modeling the cross-ply laminated Kirchhoff plate is suggested. Numerical results show that the proposed laminated Kirchhoff plate model can capture the scale effects of microstructures.     

Perturbation Formula for Phase Velocity of Rayleigh Waves in Prestressed Anisotropic Media

Herein we consider Rayleigh waves propagating along the free surface of a macroscopically homogeneous, anisotropic, prestressed half-space. We adopt the formulation of linear elasticity with initial stress and assume that the deviation of the prestressed anisotropic medium from a comparative ‘unperturbed’, unstressed and isotropic state, as formally caused by the initial stress and by the anisotropic part of the incremental elasticity tensor, be small. No assumption, however, is made on the material anisotropy of the incremental elasticity tensor. With the help of the Stroh formalism, we derive a first-order perturbation formula for the shift of phase velocity of Rayleigh waves from its comparative isotropic value. Our perturbation formula does not agree totally with that which was derived some years ago by Delsanto and Clark, and we provide another argument as further support for our version of the formula. According to our first-order formula, the anisotropy-induced velocity shifts of Rayleigh waves, taken in totality of all propagation directions on the free surface, carry information only on 13 elastic constants of the anisotropic part of the incremental elasticity tensor. The remaining eight elastic constants are those which would become zero if were monoclinic with the two-fold symmetry axis normal to the free surface of the material half-space in question.