A continuum theory of dense suspensions

A continuum theory is introduced for viscous fluids carrying dense suspensions (such as blood) or emulsions of arbitrary shape and inertia. Suspended particles possess microinertia that make the mixture an anisotropic fluid whose viscosity changes with motion and orientation of suspensions. The microinertia balance law coupled with the equations of motion of an anisotropic fluid govern the ultimate outcome. By means of the second law of thermodynamics, constitutive equations are obtained in terms of the frame-independent tensors. In a special case, a theory of bar-like suspensions is obtained. The field equations, boundary and initial conditions are given for both the arbitrarily-shaped suspensions and the bar-like suspensions. The theory is demonstrated with the solution of the channel flow problem. The mean viscosity of the fluid with suspensions is determined. The motions of suspensions down flow are demonstrated.     

A continuum theory of dense suspensions

A continuum theory is introduced for viscous fluids carrying dense suspensions (such as blood) or emulsions of arbitrary shape and inertia. Suspended particles possess microinertia that make the mixture an anisotropic fluid whose viscosity changes with motion and orientation of suspensions. The microinertia balance law coupled with the equations of motion of an anisotropic fluid govern the ultimate outcome. By means of the second law of thermodynamics, constitutive equations are obtained in terms of the frame-independent tensors. In a special case, a theory of bar-like suspensions is obtained. The field equations, boundary and initial conditions are given for both the arbitrarily-shaped suspensions and the bar-like suspensions. The theory is demonstrated with the solution of the channel flow problem. The mean viscosity of the fluid with suspensions is determined. The motions of suspensions down flow are demonstrated.     

Modeling of induced anisotropy at large deformations for polymers

In this work we develop a model to describe the induced plasticity of polymers at large deformations. Polymers such as stretch films exhibit a pronounced strength in the loading direction. The undeformed state of the films is isotropic, whereas after the uni-axial loading the material becomes anisotropic. In order to consider this induced anisotropy during the stretch process, a spectral decomposition of the inelastic right CAUCHY-GREEN tensor is done. Therefore, the yield function can be formulated as a function of the anisotropic tensor, where again the anisotropic tensor is a function of the maximum eigenvalue. A backward EULER scheme is used for updating the evolution equations, and the algorithmic tangent operator is derived. The numerical implementation of the resulting set of constitutive equations is used in a finite element program and for parameter identification. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the effective viscosity of a periodic suspension – analysis of primal and dual formulations for Newtonian and non‐Newtonian solvents

Computer simulations of the injection molding process of fiber‐reinforced plastics critically depend on the accuracy of the constitutive models. Of prime importance for the process simulation is the precise knowledge of the viscosity. Industrial applications generally feature both high shear rates and high fiber volume fractions. Thus, both the shear‐thinning behavior of the melt and the strong anisotropic effects induced by the fibers play a dominant role. Unfortunately, the viscosity cannot be determined experimentally in its full anisotropy, and analytical models cease to be accurate for the high fiber volume fractions in question. Computing the effective viscosity by a simplified homogenization approach serves as a possible remedy. This paper is devoted to the analysis of a cell problem determining the effective viscosity. We provide primal as well as dual formulations and prove corresponding existence and uniqueness theorems for Newtonian and Carreau fluids in suitable Sobolev spaces. In the Newtonian regime, the primal formulation leads to a saddle point problem, whereas a dual formulation can be obtained in terms of a coercive and symmetric bilinear form. This observation has deep implications for numerical formulations. As a by‐product, we obtain the invertibility of the effective viscosity, considered as a function, mapping the macroscopic shear rate to the macroscopic shear stress. Copyright © 2016 John Wiley & Sons, Ltd.     

An anisotropic flow law for incompressible polycrystalline materials

New and explicit anisotropic constitutive equations between the stretching and deviatoric stress tensors for the two- and three-dimensional cases of incompressible polycrystalline materials are presented. The anisotropy is assumed to be driven by an Orientation Distribution Function (ODF). The polycrystal is composed of transversally isotropic crystallites, the lattice orientation of which can be characterized by a single unit vector. The proposed constitutive equations are valid for any frame of reference and for every state of deformation. The basic assumption of this method is that the principle directions of the stretching and of the stress deviator are the same in the isotropic as well as in the anisotropic case. This means that the proposed constitutive laws are able to model the effects of anisotropy only via a change of the fluidity due to a change of the ODF. Such an assumption is justified to guarantee that, besides knowledge of the parameters involved in the isotropic constitutive equation, the anisotropic material response is completely characterized by only one additional parameter, a type of enhancement factor. Explicit comparisons with experimental data are conducted for Ih–ice.     

Features of the Formation of Composite Materials in Anisotropic Liquid Systems with Elongated Suspended Particles

The equations of rotational motion of nondeformable spherical and axisymmetric elongated particles and a rheologic equation for stresses in arbitrary gradient flows of dilute suspensions of such particles in an anisotropic carrying fluid are obtained within the framework of a structural-phenomenological approach. As a rheologic model of the suspension-carrying fluid and a hydrodynamic model of the suspended particles, we use the Ericksen simple anisotropic fluid and a symmetric triaxial dumbbell, respectively. The constitutive equations obtained are used to study the effect of anisotropy of the carrying fluid on the dynamics of suspended particles and on the rheologic properties of suspensions in a simple shear flow. A stationary orientation of the elongated suspended particles under the action of hydrodynamic forces is discovered. The possibility of applying this phenomenon to the formation of composite materials is discussed.     

An anisotropic flow law for incompressible polycrystalline materials

New and explicit anisotropic constitutive equations between the stretching and deviatoric stress tensors for the two- and three-dimensional cases of incompressible polycrystalline materials are presented. The anisotropy is assumed to be driven by an Orientation Distribution Function (ODF). The polycrystal is composed of transversally isotropic crystallites, the lattice orientation of which can be characterized by a single unit vector. The proposed constitutive equations are valid for any frame of reference and for every state of deformation. The basic assumption of this method is that the principle directions of the stretching and of the stress deviator are the same in the isotropic as well as in the anisotropic case. This means that the proposed constitutive laws are able to model the effects of anisotropy only via a change of the fluidity due to a change of the ODF. Such an assumption is justified to guarantee that, besides knowledge of the parameters involved in the isotropic constitutive equation, the anisotropic material response is completely characterized by only one additional parameter, a type of enhancement factor. Explicit comparisons with experimental data are conducted for Ih–ice. Dedicated to Prof. L.W. Morland on the occasion of his 70^th birthday Received: July 6, 2004; revised: November 8, 2004     

A macroscopic consitutive model on induced anisotropy for polymers with weighting functions

The alignment of polymer chains is a well known microstructural evolution effect due to straining of polymers. This has a drastic influence on the macroscopic properties of the initially isotropic material, such as a pronounced strength in the loading direction of stretched films. Experiments on strain induced anisotropy at room temperature are analyzed by optical measurements. For modeling the effect of strain induced anisotropy a macroscopic constitutive model is presented. As a key idea, weighting functions are introduced to represent a strain-softening/hardening-effect to account for induced anisotropy. These functions represent the ratio between the total strain rate and a structural tensor. In this way, material parameters are used as a sum of weighted direction related quantities. In the finite element examples we simulate the cold-forming of amorphous thermoplastic films below the glass transition temperature subjected to different re-loading directions. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-linear laws of fluid flow through anisotropic porous media

Non-linear laws of fluid flow through anisotropic porous media are written out in invariant tensor form for all crystallographic point symmetry groups. The equations, as is customary in seepage theory [1, 2], are represented by expressions containing the seepage velocity up to and including the third degree. Expressions defining non-linear flow resistances are given and it is shown that, when one transfers from linear to non-linear seepage laws, the symmetry group of the flow properties may change. For example, the isotropic flow properties manifested in Darcy's law may become essentially anisotropic in a non-linear law and display asymmetry, that is, they may be different along one straight line in the positive and negative directions. It is shown that, compared with linear seepage laws for anisotropic media, when flow properties may be defined by just four essentially different types of equation, in non-linear laws the appearance of anisotropy is highly diversified and the number of distinct types of equation increases considerably.     

Verification of an anisotropic Coulomb adhesion-friction law for contact surfaces with periodic structure

Sliding friction forces and so-called adhesion forces are the main mechanical characteristics to describe contact interaction. Both together are representing 2D surface constitutive laws in analogy to e.g. elasto-plasticity for 3D continua. The classical model to generalize the Coulomb friction law into anisotropic domains is to introduce an anisotropic friction tensor. Michalowski and Mroz in [1] proposed the structure of the friction tensor considering the sliding of a rigid block on an inclined surface. Zmitrowicz in [2] developed the theoretical basis for the structure of the friction tensor on symmetry groups for the tensor. The current contribution is aimed at verification of this modeling process based on a homogenization procedure for a very fine discretization representing the exact structure of the surface. The validation issue with realistic experiments given in [4] is discussed as well. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulating of a pressing process of a viscoelastic polymer melt

A well known and often used method to obtain anisotropic polymer films is the so-called pressing process. Here, films are squeezed under high temperatures, pressure and deformation rates. To simulate such a process, the polymeric matrix is treated as a non-Newtonian, viscoelastic melt. The modeling of such melts is done with the anisotropic molecule movement tensor generalization of the Maxwell Model for high deformation rates. The viscoelastic flow simulations are done with DEVSS stabilization techniques and an ALE based dynamic mesh Method. In this work we present simulations in order to show the difference between classical approaches using a generalized Newtonian viscosity to model the melt and the used viscoelastic models. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Falling film flow of a viscoelastic fluid along a wall

In this paper, we study the heat transfer in the fully developed flow of a viscoelastic fluid, a slag layer, down a vertical wall. A new constitutive relation for the stress tensor of this fluid is proposed, where the viscosity depends on the volume fraction, temperature, and shear rate. For the heat flux vector, we assume the Fourier's law of conduction with a constant thermal conductivity. The model is also capable of exhibiting normal stress effects. The governing equations are non‐dimensionalized and numerically solved to study the effects of various dimensionless parameters on the velocity, temperature, and volume fraction. The effect of the exponent in the Reynolds viscosity model is also discussed. The different cases of shear‐thinning and shear‐thickening, cooling and heating, are compared and discussed. The results indicate that the viscous dissipation and radiation (at the free surface) cause the temperature to be higher inside the flow domain. Copyright © 2013 John Wiley & Sons, Ltd.     

准各向同性复合材料弹性和强度的各向异性效应(Ⅰ)──模型*

实验观测表明,准各向同性材料,如N轴纤维增强复合材料层合板和编织材料,其面内刚度和强度具有不同程度的方向性,且强度的各向异性程度往往明显高于弹性性质的各向异性程度。本文根据张量函数表示理论所提出的本构方程和强度准则的一般模型,结合有关实验数据,分析了材料弹性性能和强度的非各向同性效应。具体给出了几类本构模型和强度准则的特殊形式并讨论了本文所得到结果的若干力学性质。本文第Ⅱ部分具体讨论了含单个椭圆孔或裂纹的无限大板的有关强度的各向异性效应,并用细观力学方法检验了本文的模型。     

关于损伤张量的阶次

本文首先讨论了较为广泛的连续介质材料的应力变形本构关系,得到了通常以泛函表示的应力变形本构关系的张量表达式.以此为基础,研究了各向异性材料各向异性损伤时,无论从连续介质力学模型出发还是从缺陷模型出发,描述损伤的张量都存在最高阶次的限制;指出了在什么条件下,损伤变量可用低于最高阶次的张量来描述.     

Modeling residual stresses in arterial walls based on anisotropic growth

With the aim of obtaining a general local formulation for anisotropic growth in soft biological tissues, a model based on the multiplicative decomposition of the growth tensor is formulated. The two parts of the growth tensor are associated with the main anisotropy directions. Together with an anisotropic driving force, the model enables an effective stress reduction by including growth-induced residual stresses, which is demonstrated in a numerical example of an idealized arterial segment. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Anisotropic stress-softening model for compressible solids

A general constitutive theory for anisotropic stress softening in compressible solids is presented. The constitutive equation describes anisotropic strain induced behaviour of an initially “isotropic” virgin material. Parameters which characterise damage are proposed together with a concept of damage function. In order to develop an anisotropic stress-softening theory for compressible materials in close parallel to a recent incompressible anisotropic theory, the right stretch tensor is decomposed into its isochoric and dilatational parts. The ’free’ energy is expressed as a function of the dilatation, modified principal stretches, a volume change parameter and invariants of the dyadic products of the principal directions of the right stretch tensor and two structural tensors. A class of free energy functions is discussed and a special form of this class which satisfies the Clausius–Duhem inequality is proposed. Results of the theory applied to uniaxial tension, bulk compression and simple shear deformations are given. A sequence of deformations involving shear, hydrostatic-compression and hydrostatic-tension deformations is also investigated. In the case of hydrostatic-tension deformation, the stress softening is due to cavitation damage. The theoretical results obtained are consistent with expected behaviour and compare well with experimental data.     

N–Term Approximation in Anisotropic Function Spaces

In L₂(0, 1)²) infinitely many different biorthogonal wavelet bases may be introduced by taking tensor products of one–dimensional biorthogonal wavelet bases on the interval (0, 1). Most well–known are the standard tensor product bases and the hyperbolic bases. In [23, 24] further biorthogonal wavelet bases are introduced, which provide wavelet characterizations for functions in anisotropic Besov spaces. Here we address the following question: Which of those biorthogonal tensor product wavelet bases is the most appropriate one for approximating nonlinearly functions from anisotropic Besov spaces? It turns out, that the hyperbolic bases lead to nonlinear algorithms which converge as fast as the corresponding schemes with respect to specific anisotropy adapted bases.     

Effective Flow Potentials for Anisotropic Polycrystals

In this paper a micromechanically based flow potential for anisotropic fcc polycrystals is derived that takes into account the crystallite orientation distribution function (codf) in terms of tensorial texture coefficients. The effective flow potential is based on a representation theorem for anisotropic scalar functions depending on a 2nd-order tensor. A priori unknown functions in the representation are determined by defining and solving explicitly a minimization problem over SO(3). Important analytical properties of the coefficient matrix of the minimization problem are discussed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)