Stress and strain-rate formulations for fabric evolution in polar ice

Re-orientation of individual crystal glide planes, as isotropic surface ice is deformed during its passage to depth in an ice sheet, creates a fabric and associated anisotropy. We re-examine an orthotropic viscous law which was developed to reflect the induced anisotropy arising from the mean rotation of crystal axes during deformation. This expresses the deviatoric stress, the stress formulation, in terms of the strain-rate, strain, and three structure tensors based on the principal stretch axes, and involves two fabric response coefficient functions which determine the strength of the anisotropy. A validity condition implicitly relates the two response functions, so the model law has only one independent fabric response function. A modified formulation is now presented in which the two fabric response coefficients are expressed as functions of different invariant arguments, and the validity condition becomes an explicit algebraic relation between the two functions. The response can therefore be described explicitly in terms of a single fabric response function. An analogous orthotropic viscous law for the strain-rate, the strain-rate formulation, akin to the conventional “flow law” for isotropic ice, expressed in terms of the deviatoric stresss, strain and the three structure tensors, is also constructed. Correlations with complete (idealised) uni-axial compression and shearing responses are made for the stress formulation, to determine the fabric response function which would yield these responses. Received January 30, 2002 / Published online October 15, 2002 RID="*" ID="*" On leave from the Institute of Hydroengineering, Polish Academy of Sciences, ul. Waryńskiego 17, 71-310 Szczecin, Poland Communicated by Kolumban Hutter, Darmstadt     

Stress-induced crystal preferred orientation in the poromechanics of in-pore crystallization

A non-hydrostatic stress field affects the orientation of crystals growing in the pore network of an elastic porous medium. The hypothesis of a hydrostatic state of stress within the crystal has been implicitly made in the recent extension of poromechanics to in-pore crystalization (Coussy, 2006). This underlying hypothesis is revisited on a small-scale conceptual model based on Eshelby's problem and shows that chemo-mechanical equilibrium requires that the crystal adapts its shape and orientation to the far-field stress, therefore resulting at equilibrium in a hydrostatic state of stress within the crystal. The optimum crystal shape as a function of the far-field stress is consistently investigated, highlighting limiting cases. The small scale model allows to understand the macroscopic effects associated with deviatoric stresses in the poromechanics of in-pore crystallization. Moreover, it provides the building block for an up-scaling of the macroscopic tangent poroelastic properties, which depend on both the current crystal saturation and the state of stress. A dilute micromechanical scheme illustrates the variation of the macroscopic Biot's coefficient tensor as a function of deviatoric stresses. A simple configuration akin to a potential laboratory experiment finally illustrates the strong induced anisotropy of the crystallization induced macroscopic strain when deviatoric stresses are applied to the material prior to crystallization.     

PAMPS-PAAM互穿网络凝胶的摩擦性能研究

选用2-丙烯酰胺-2-甲基丙磺酸(AMPS)凝胶和丙烯酰胺(AAM)溶液制备PAMPS-PAAM互穿网络凝胶,在自制销-盘式摩擦试验机上进行PAMPS-PAAM互穿网络凝胶的摩擦性能试验,研究接触压力、滑行速率和润滑条件对该凝胶的摩擦性能的影响.结果表明:PAMPS-PAAM凝胶受压变形时的性质与天然软骨相似.在较高的压力下(0.7 MPa),凝胶的软骨泵机理发生作用,有助于润滑膜的形成;滑行速率的提高和润滑剂黏度的增加也有利于润滑状态的改善,降低摩擦系数.PAMPS-PAAM凝胶有良好的自恢复功能.     

Role of strain-rate sensitivity in the crystal plasticity of hexagonal structures

In the first part of this paper the stress and strain-rate response of hexagonal crystal structures are examined when slip is viscoplastic according to a power law. The stress and strain-rate equi-potential surfaces are constructed and discussed as a function of the strain-rate sensitivity index m. The second part of this paper deals with the case of linear viscous slip; i.e., for the case when m is equal to one. A simple analytic solution is presented to obtain the deviatoric stress state for a given strain-rate. It is shown that the plastic spin is not zero for m = 1 in hexagonal crystal structures, contrary to the cubic case where the plastic spin vanishes. In addition, the rate of texture evolution in simple shear of a magnesium polycrystal is examined as a function of m.     

Deformations of an elastic,internally constrained material. Part 1: Homogeneous deformations

The nonlinear elastic response of a class of materials for which the deformation is subject to an internal material constraint described in experiments by James F. Bell on the finite deformation of a variety of metals is investigated. The purely kinematical consequences of the Bell constraint are discussed, and restrictions on the full range of compatible deformations are presented in geometrical terms. Then various forms of the constitutive equation relating the stress and stretch tensors for an isotropic elastic Bell material are presented. Inequalities on the mechanical response functions are introduced. The importance of these in applications is demonstrated in several examples throughout the paper.This paper focuses on homogeneous deformations. In a simple illustration of the theory, a generalized form of Bell's empirical rule for uniaxial loading is derived, and some peculiarities in the response under all-around compressive loading are discussed. General formulae for universal relations possible in an isotropic elastic, Bell constrained material are presented. A simple method for the determination of the left stretch tensor for essentially plane problems is illustrated in the solution of the problem of pure shear of a materially uniform rectangular block. A general formula which includes the empirical rule found in pure shear experiments by Bell is derived as a special case. The whole apparatus is then applied in the solution of the general problem of a homogeneous simple shear superimposed on a uniform triaxial stretch; and the great variety of results possible in an isotropic, elastic Bell material is illustrated. The problem of the finite torsion and extension of a thin-walled cylindrical tube is investigated. The results are shown to be consistent with Bell's data for which the rigid body rotation is found to be quite small compared with the gross deformation of the tube. Several universal formulas relating various kinds of stress components to the deformation independently of the material response functions are derived, including a universal rule relating the axial force to the torque.Constitutive equations for hyperelastic Bell materials are derived. The empirical work function studied by Bell is introduced; and a new constitutive equation is derived, which we name Bell's law. On the basis of this law, we then derive exactly Bell's parabolic laws for uniaxial loading and for pure shear. Also, form Bell's law, a simple constitutive equation relating Bell's deviatoric stress tensor to his finite deviatoric strain tensor is obtained. We thereby derive Bell's invariant parabolic law relating the deviatoric stress intensity to the corresponding strain intensity; and, finally, Bell's fundamental law for the work function expressed in these terms is recovered. This rule is the foundation for all of Bell's own theoretical study of the isotropic materials cataloged in his finite strain experiments on metals, all consistent with the internal material constraint studied here.     

Elongational and shear rheology of carbon nanotube suspensions

Rheological behavior of concentrated suspensions of chemical vapor deposition carbon nanotubes in uniaxial elongation and simple shear is studied experimentally and theoretically. Nanotubes are suspended in viscous host liquids—castor oil or its blends with n-decane. The elongational measurements are performed by analyzing self-thinning (due to surface tension effect) liquid threads of nanotube suspensions. A quasi-one-dimensional model is used to describe the self-thinning process, whereas corrections accounting for thread nonuniformity and necking are introduced a posteriori. The effects of nanotube concentration and aspect ratio, viscosity of the suspending liquid, and initial diameter of the self-thinning thread in uniaxial elongation are elucidated. The results for uniaxial elongation are compared with those for simple shear. The correspondence in the results of the shear and elongational measurements is addressed and interpreted. The results conform to the Herschel–Bulkley rheological constitutive equation (i.e., power law fluids with yield stress). However, the yield stress in elongation is about 40% higher than in simple shear flow, which suggests that the original Herschel–Bulkley model need modification with the yield stress being a function of the second invariant of the deviatoric stress tensor. The present effort is the first to study capillary self-thinning of Herschel–Bulkley liquids, which are exemplified here by suspensions of carbon nanotubes.     

Viscosity change in aqueous hectorite suspension activated by DC electric field

The change in the viscosity of deionized aqueous hectorite suspensions by applying an electric field was investigated. The deionized suspensions had low viscosity, which was almost the same as that of its solvent, water. Upon applying a DC electric field of the order of a few volt per millimeter to the deionized suspension, the viscosity started to increase gradually and reached a constant value that was high enough relative to the original value. Regarding the mechanism of the electrically induced stress increase, it is highly plausible that a three-dimensional network structure formed under the electric field due to a deformation of the electrical double layer.     

The relevance of entry flow measurements for the estimation of extensional viscosity of polymer melts

The extensional viscosity of some flexible chain polymers and a thermotropic liquid crystalline polymer was measured in uniaxial extensional flow at constant extension rate. Power law functions were found for the dependence of the extensional viscosity at constant accumulated strain on strain rate. The stress growth curves were compared with measurements in axisymmetric entry flow, where both elongation and shear occur. The comparison showed that the values of the extensional viscosity calculated from the measurements in the entry flow correspond to the ones calculated from the viscosity growth measured in uniaxial elongation and averaged over extensional strain equal to what is accumulated on the fluid as it flows from the barrel into the capillary.     

可压缩理想塑性体中逐步扩展裂纹顶端的弹塑性场

本文利用理想塑性固体平面应变问题的基本方程,分析了可压缩理想塑性体中逐步扩展裂纹顶端的弹塑性场,得到了关于应力的渐近场,分析了弹性卸载区的演变过程和修正的中心扇形区的发展过程,预示了出现二次塑性区的可能性,弹性可压缩性的影响明显表现在经典的中心扇形区必需加以修正,垂直于板面方向的应力偏量不再为零,而且随着新裂纹面的形成,裂纹前方的均匀应力场和紧连着的修正的中心扇形区的应力偏量将发生变化,这种变化是由于垂直于板面方向的应力偏量发生变化造成的。     

Strain path theory of plastic constitutive relation

Based on Illiushin's two hypotheses, a new plastic constitutive equation of integral type is proposed. The deviatoric stress induced by a strain path consists of two parts which vary according to different laws. Comparison of theoretical calculation with recent experiments is satisfactory. The present theory is not an endochronic theory.     

Controllable states of stress for incompressible elastic solids

It is shown that the equilibrium states of Cauchy stress which can exist, in the absence of body force, in every incompressible, homogeneous, isotropic, elastic solid whose deviatoric stress range allows them, must have uniform deviatoric stress invariants. There is at least one such non-uniform stress state. The related problem for incompressible non-Newtonian fluids is also discussed.     

The influence of flow-induced non-Newtonian fluid properties on turbulent drag reduction

Friction factors and velocity profiles in turbulent drag reduction can be compared to Newtonian fluid turbulence when the shear viscosity at the wall shear rate is used for the Reynolds number and the local shear viscosity is used for the non-dimensional wall distance. On this basis, an apparent maximum drag reduction asymptote is found which is independent of Reynolds number and type of drag reducing additive. However, no shear viscosity is able to account for the difference between the measured Reynolds stress and the Reynolds stress calculated from the mean velocity profile (the Reynolds stress deficit). If the appropriate local viscosity to use with the velocity fluctuation correlations includes an elongational component, the problem can be resolved. Taking the maximum drag reduction asymptote as a non-Newtonian flow, with this effective viscosity, leads to agreement with the concept of an asymptote only when the solvent viscosity is used in the non-dimensional wall distance.     

Numerical simulation study on particle breakage behavior of granular materials in confined compression tests

In order to study the fragmentation law, the confined compression experiment of granular assemblies has been conducted to explore the particle breakage characteristic by DEM approach in this work. It is shown that contact and contact force during the loading process gradually transform from anisotropy to isotropy. Meanwhile, two particle failure modes caused by different contact force states are analyzed, which are single-through-crack failure and multi-short-crack failure. Considering the vertical distribution of the number of cracks and the four characteristic stress distributions (the stress related to the maximum contact force, the major principal stress, the deviatoric stress and the mean stress), it is pointed out that the stress based on the maximum contact force and the major principal stress can reflect the distribution of cracks accurately. In addition, the size effect of particle crushing indicates that small size particles are prone to break. The lateral pressure coefficient of four size particles during the loading process is analyzed to explain the reason for the size effect of particle breakage.     

Effect of deviatoric nonassociativity on the failure prediction for elastic–plastic materials

This paper presents a theoretical study of the effect of nonassociativity of the plastic flow rule on the critical plastic modulus for discontinuous bifurcation in an elastic–plastic material. Nonassociativity in both the spherical and the deviatoric spaces are considered, with an emphasis on the effect of nonassociativity in the deviatoric space. A particular form of nonassociativity in the deviatoric space is introduced, where the projections of the plastic flow direction and the normal to the yield surface are assumed to have the same length but the projection of plastic flow direction is allowed to lag that of the normal by an angle. It is shown that even for the simple yield surface of von Mises, nonassociativity in the deviatoric space can lead to a bifurcation for a load parameter significantly lower than the value predicted with an associated flow rule.     

True triaxial cyclic behavior of concrete and rock in compression

Results from true triaxial cyclic tests on concrete are presented and compared with corresponding ones for dense rocks (mainly marbles) obtained under the same laboratory conditions. Common basic characteristics are confirmed regarding the essential effect of intermediate stress, the cataclastic mode of deformation under low confinement, the coupling between volumetric strain and deviatoric stress, the dependence of the equivalent modulus E_q and the hysteresis loops. Common characteristics are also revealed in meridian and deviatoric sections of peak strength surfaces, moreover a stress invariant and its work conjugate plastic strain increment are approximately equal. Concrete behavior is different on the postpeak plastic contraction under high confinement, on the creep rate, on the validity of the condition of convexity and on the considerable degree of violation of the condition of normality. A simple empirical equation is proposed and experimentally verified for predicting deviatoric sections of strength surfaces.     

Time-dependent plane Poiseuille flow of a Johnson–Segalman fluid

We numerically solve the time-dependent planar Poiseuille flow of a Johnson–Segalman fluid with added Newtonian viscosity. We consider the case where the shear stress/shear rate curve exhibits a maximum and a minimum at steady state. Beyond a critical volumetric flow rate, there exist infinite piecewise smooth solutions, in addition to the standard smooth one for the velocity. The corresponding stress components are characterized by jump discontinuities, the number of which may be more than one. Beyond a second critical volumetric flow rate, no smooth solutions exist. In agreement with linear stability analysis, the numerical calculations show that the steady-state solutions are unstable only if a part of the velocity profile corresponds to the negative-slope regime of the standard steady-state shear stress/shear rate curve. The time-dependent solutions are always bounded and converge to different stable steady states, depending on the initial perturbation. The asymptotic steady-state velocity solution obtained in start-up flow is smooth for volumetric flow rates less than the second critical value and piecewise smooth with only one kink otherwise. No selection mechanism was observed either for the final shear stress at the wall or for the location of the kink. No periodic solutions have been found for values of the dimensionless solvent viscosity as low as 0.01.     

Negative wake in the uniform flow past a cylinder

The upstream/downstream streamline shift and the associated negative wake generation (streamwise velocity overshoot in the wake) in a viscoelastic flow past a cylinder are studied in this paper, for the Oldroyd-B, UCM, PTT, and FENE-CR fluids, using the Discrete Elastic Viscous Split Stress Vorticity (DEVSS-ω) scheme (Dou HS, Phan-Thien N (1999). The flow of an Oldroyd-B fluid past a cylinder in a channel: adaptive viscosity vorticity (DAVSS-ω) formulation. J Non-Newtonian Fluid Mech 87:47–73). The numerical algorithm is a parallelized unstructured Finite Volume Method (FVM), running under a distributed computing environment through the Parallel Virtual Machine (PVM) library. It is demonstrated that both the normal stress and its gradient are responsible for the negative wake generation and streamline shifting. Fluid extensional rheology plays an important role in the generation of the negative wake. The negative wake can occur in flows where the fluid extensional viscosity does not increase rapidly with strain rate. The formation of the negative wake does not depend on whether the streamlines undergo an upstream or a downstream shift. Shear-thinning viscosity weakens the velocity overshoot and while shear-thinning first normal stress coefficient enhances the velocity overshoot. Wall proximity is not necessary for the velocity overshoot; however, it enhances the strength of the negative wake. For the Oldroyd-B fluid, the ratio of the solvent viscosity to the zero-shear viscosity plays an important role in the streamline shift. In addition, mesh dependent behaviour of normal stresses along the centreline at high De in most cylinder/sphere simulations is due to the convection of normal stress from the cylinder to the wake, which results in the maximum of the normal stress being located off the centreline by a short distance at high De.