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     

On inner constraints in plane circular arches

Summary A one-dimensional model of plane circular arches with rigid sections is introduced. Suitable strain measures are defined as deviations from rigid displacements. If the arch is thin, constitutive arguments make the shearing strain negligible. Hence, the shearing indeformability will be assumed as inner constraint. By means of a formal power series expansion of the exact measures of deformation it is shown that the shearing indeformability implies some constraints on the axial strain. In particular, the first-order axial strain must vanish in the case of infinitesimal displacements. The same procedure is applied to pure flexible arches, in order to compare the two sets of results. It is shown that the hypothesis of finite pure flexibility is not compatible with small deformations of the arch. An example is provided to evaluate the effects of the two constraints at the first non-linear step of the perturbation expansions. This work has been supported by the “Progetto giovani ricercatori” grant of the University of Rome “La Sapienza” for the year 2002.     

A-priori testing of an eddy viscosity model for the density-weighted subgrid scale stress tensor in turbulent premixed flames

In this study, we report on the direct measurement of the density-weighted subgrid scale (SGS) stress tensor in turbulent premixed flames. In large-eddy simulations (LES), this unresolved tensor is typically modelled using eddy viscosity approaches. Additionally to the direct measurement, we provide a pure experimentally based a-priori test of the commonly used eddy viscosity model suggested by Smagorinsky. For two turbulent premixed V-shaped methane–air flames, a statistical analysis is presented where the correlation between the directly measured SGS stress tensor and the eddy viscosity model following Smagorinsky is tested. The measurement strategy is based on the application of a dual-plane stereo-PIV technique which enables the measurement of the 3D flow field in two parallel planes. This allows the determination of velocities as well as velocity gradients in all three directions. Here, a vector resolution of 118 μm was achieved. For a priori testing, the data are subjected to a spatial filtering procedure that reproduces the application of the filter function in LES. The calculation of velocity gradients is performed after the application of this spatial averaging. Additionally to the velocity field, the flame front position is deduced from the clearly observable step in the tracer particle number density between burnt and unburnt regions of the flame. This facilitates the direct single-shot-based evaluation of all components of the density-weighted SGS stress tensor. Additionally, the model expressions related to these terms can be determined, which is done in this first study for the static Smagorinsky model. With that, the instantaneous local comparison between directly measured stress terms and modelled terms is possible, based on the instantaneous local evaluation procedure. The measurement procedure is described, and first results are presented and discussed. They show a rather poor performance of the static form of the Smagorinsky model (with fixed Smagorinsky constant). Our future aims are to use the directly measured SGS data for the a-priori comparison with more advanced models.     

Slope and curvature measurement by a double-frequency-grating shearing interferometer

This paper describes a shearing interferometer capable of the direct measurement of the slopes and curvatures of reflecting plates. The wavefront shear is produced by a double-frequency grating which is simply a grating containing two closely spaced frequencies. The first-order waves associated with each frequency form the interferogram, the contours of which are related to the model's slope or curvature depending upon system arrangement and recording procedure. Four arrangements are described: two for slope and two for curvature. In one, the slope contours are obtained directly and in real time. In the second slope arrangement, an extra spatial-filtering step is necessary to obtain the slope contours. However, this arrangement, as opposed to the first, measures the slope only due to the loading, compensating for initial model slope and optical-system aberrations. The two curvature techniques can be described as double-shearing interferometers having a primary and secondary shear. The primary shear for both arrangements is provided by the double-frequency grating. In one technique, the secondary shear is provided by a translation of the recording film between two exposures during the recording step. A subsequent spatialfiltering step displays the curvature fringes. The second technique requires only a single exposure and places a beam splitter at the location of one of the first-order diffraction spectra during the filtering step. The field equations defining the fringe values are derived for all four arrangements with experimental results also being given.     

Determining polymer molecular weight distributions from rheological properties using the dual-constraint model

Although multiple models now exist for predicting the linear viscoelasticity of a polydisperse linear polymer from its molecular weight distribution (MWD) and for inverting this process by predicting the MWD from the linear rheology, such inverse predictions do not yet exist for long-chain branched polymers. Here, we develop and test a method of inverting the dual-constraint model (Pattamaprom et al., Rheol Acta 39:517–531, 2000; Pattamaprom and Larson, Macromolecules 34:5229–5237, 2001), a model that is able to predict the linear rheology of polydisperse linear and star-branched polymers. As a first step, we apply this method only to polydisperse linear polymers, by comparing the inverse predictions of the dual-constraint model to experimental GPC traces. We show that these predictions are usually at least as good, or better than, the inverse predictions obtained from the Doi–Edwards double-reptation model (Tsenoglou, ACS Polym Prepr 28:185–186, 1987; des Cloizeaux, J Europhys Lett 5:437–442, 1988; Mead, J Rheol 38:1797–1827, 1994), which we take as a “benchmark”—an acceptable invertible model for polydisperse linear polymers. By changing the predefined type of molecular weight distribution from log normal, which has two fitting parameters, to GEX, which has three parameters, the predictions of the dual-constraint model are slightly improved. These results suggest that models that are complex enough to predict branched polymer rheology can be inverted, at least for linear polymers, to obtain molecular weight distribution. Further work will be required to invert such models to allow prediction of the molecular weight distribution of branched polymers.     

A Novel Workflow to Model Permeability Impairment through Particle Movement and Deposition in Porous Media

This article presents a practical transfer function type solution to a complex problem in which variations in a number of parameters can be taken into account. A new mathematical model, which is based on mass balance transfer function of particles movement/retention in porous media, has been derived. It is used to predict permeability reduction as a function of time. The linear forms as well as the radial forms of the model are described. Although the differential equations derived are similar to the general form of diffusion–convection equations, the marked difference is the suitability of the model, for being applied for variation of parameters, such as particle concentration in the fluid, injection rate, density of solid particles, against the depth and time of invasion. This transfer function has been solved, and the results of the simulation run agree reasonably well with the experimental damage data obtained in laboratory. Owing to its simplicity, this model is more practical to describe permeability reduction for the flow of suspended particles in porous media.     

Time-dependent Viscoelastic Behavior of an LDPE Melt

Two differential constitutive equations, i.e. Giesekus model and Johnson–Segalman model were employed here to predict the time-dependent viscoelastic behavior of an LDPE melt in thixotropy-loop experiments and step shear rate experiment. Multiple relaxation modes were adopted, and the parameters used to describe the nonlinear viscoelasticity in the two models were obtained by fitting the shear-thinning viscosity. The predictions on those transient shear characteristics by the two models are found in qualitative agreement with our previous experiments. Johnson– Segalman model predicts oscillation behavior in the thixotropy-loop and step shear rate experiments, whereas Giesekus model does not. Both models predict higher shear stresses than the experimental data in the case of long time shearing, implying that both models are not able to completely characterize the time-dependent shear stress of the melt at high shear rate.The project was supported by the National Natural Science Foundation of China (10402024, 50335010).The English text was polished by Yunming Chen.     

Multi-scale modeling of grouted sand behavior

The mechanical properties of sand: stiffness, cohesion and, to a less extent, friction angle can be increased through the process of grouting. A constitutive model adapted for cohesive-frictional materials from a homogenization technique which allowed us to integrate constitutive relations at the grain level has been developed to obtain constitutive equations for the equivalent continuous granular medium. A representative volume was obtained by mobilizing particle contacts in all orientations. Thus, the stress–strain relationship could be derived as an average of the behavior of these local contact planes. The local behavior was assumed to obey a stress-dependent elastic law and Mohr–Coulomb’s plastic law. The influence of the cement grout was modeled by means of adhesive forces between grains in contact, which were added to the contact forces created by an external load. The intensity of these adhesive forces is a function of nature and amount of grout present inside the material and can be reduced due to a damage mechanism at the grain contact during loading. In this paper, we present several examples of simulation which show that the model can reproduce with sufficient accuracy the mechanical improvement induced by grouting as well as the damage of the grain cementation during loading.     

基于构型力内变量的界面损伤问题研究

本文基于材料构型力的基本理论和损伤力学中含有内变量的热力学框架,提出了新的损伤变量定义方式,为研究界面损伤问题提供了一种新思路。首先,基于双相弹性体的能量分析,给出界面材料构型力表达式,通过构型力的离散化方法,实现了其在有限元中的数值计算。其次,定义构型力为界面损伤内变量,进而提出一种新的损伤演化模型,并采用刚度劣化的方法,对该界面损伤模型进行数值实现。最后,通过对复合材料界面损伤问题（有裂纹或无裂纹）进行数值模拟,分析了其界面损伤发展趋势,探讨了此模型的合理性和优越性。基于构型力内变量的界面损伤模型,可为研究复合材料的界面损伤失效问题提供一种普适性的方法。     

The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity

A three-dimensional photoelastic body can be represented by an optically equivalent model, which consists of a linear retarder, δ, at a certain angle, θ, and a pure rotator, χ. These have been described as the characteristic retardation, δ, and the primary and secondary characteristic directions, θ and θ+χ. Until now these characteristic parameters have only been determined using manual, point-by-point collection methods which are involved and time consuming. Therefore an automated phase-stepping method has been developed to enable the determination of the three characteristic parameters for three-dimensional or integrated photoelasticity. Expressions have been derived to obtain δ, θ and θ+χ from six phase-stepped images. These images are collected using a CCD camera and the full-field data is processed using a standard personal computer. This novel method allows accurate, full-field maps of all three characteristic parameters to be obtained in a relatively short time, which makes full-field tomographic reconstruction of photoelastic data a real possibility. Both are SEM Members.     

The order of a damage tensor

In this paper, the stress deformation constitutive relations for continua are discussed and a stress deformation constitutive relation expressed by functional tensorial expression is found. When we study the anisotropic damage of anisotropic materials either from a macroscopic continuum mechanics model or from a microdefect model, there exists a limit to the order of a damage tensor, and the condition under which the damage variable may be described by a tensor lower than those of the highest order is found.     

Similarity solutions of three-dimensional boundary layer equations of non-Newtonian fluids

An analysis of possibility of finding similaarity solutions to the three-dimensional, steady, incompressible, boundary layer equations in rectangular co-ordinates for a power law fluid has been discussed in the literature. In the present paper a similarity analysis is made of the steady, three-dimensional, incompressible, Iaminar, boundary layer flow of all time independent non-Newtonian fluids. The important conclusion drawn from this analysis in that for a non-Newtonian fiuid of any model, a similarity solution exists for the fluid for which shearing stress and rate of strain can be related by an arbitrary continuous function.     

Limiting Chain Extensibility Constitutive Models of Valanis–Landel Type

A new general constitutive model in terms of the principal stretches is proposed to reflect limiting chain extensibility resulting in severe strain-stiffening for incompressible, isotropic, homogeneous elastic materials. The strain-energy density involves the logarithm function and has the general Valanis–Landel form. For specific functions in the Valanis–Landel representation, we obtain particular strain-energies, some of which have been proposed in the recent literature. The stress–stretch response in some basic homogeneous deformations is described for these particular strain-energy densities. It is shown that the stress response in these deformations is similar to that predicted by the Gent model involving the first invariant of the Cauchy–Green tensor. The models discussed here depend on both the first and second invariants.      

Deformation and long-term damage of fibrous materials with the stress-rupture microstrength of the matrix described by a fractional-power function

The theory of long-term damage is generalized to unidirectional fibrous composites. The damage of the matrix is modeled by randomly dispersed micropores. The damage criterion for a microvolume is characterized by its stress-rupture strength. It is determined by the dependence of the time to brittle failure on the difference between the equivalent stress and its limit, which is the ultimate strength, according to the Huber–Mises criterion, and assumed to be a random function of coordinates. An equation of damage (porosity) balance in the matrix at an arbitrary time is formulated. Algorithms of calculating the time dependence of microdamage and macrostresses or macrostrains are developed and corresponding curves are plotted in the case of stress-rupture microstrength described by a fractional power function     

Coupled processes of deformation and long-term damage of fibrous materials with the microdurability of the matrix described by an exponential power function

The theory of long-term damage is generalized to fibrous composites. The damage of the matrix is modeled by randomly dispersed micropores. The damage criterion for a microvolume is characterized by its stress-rupture strength. It is determined by the dependence of the time to brittle failure on the difference between the equivalent stress and its limit, which is the ultimate strength, according to the Huber–von Mises criterion, and assumed to be a random function of coordinates. An equation of damage (porosity) balance in the matrix at an arbitrary time is formulated. Algorithms of calculating the time dependence of microdamage and macrostresses or macrostrains are developed and corresponding curves are plotted in the case of stress-rupture microstrength described by an exponential power function     

Finite amplitude, free vibrations of an axisymmetric load supported by a highly elastic tubular shear spring

The finite amplitude, free vibrational characteristics of a simple mechanical system consisting of an axisymmetric rigid body supported by a highly elastic tubular shear spring subjected to axial, rotational, and coupled shearing motions are studied. Two classes of elastic tube materials are considered: a compressible material whose shear response is constant, and an incompressible material whose shear response is a quadratic function of the total amount of shear. The class of materials with constant shear response includes the incompressible Mooney-Rivlin material and certain compressible Blatz-Ko, Hadamard, and other general kinds of models. For each material class, the quasi-static elasticity problem is solved to determine the telescopic and gyratory shearing deformation functions needed to evaluate the elastic tube restoring force and torque exerted on the body. For all materials with constant shear response, the differential equations of motion are uncoupled equations typical of simple harmonic oscillators. Hence, exact solutions for the forced vibration of the system can be readily obtained; and for this class, engineering design formulae for the load-deflection relations are discussed and compared with experimental results of others'. For the quadratic material, however, the general motion of the body is characterized by a formidable, coupled system of nonlinear equations. The free, coupled shearing motion for which either the axial or the azimuthal shear deformation may be small is governed by a pair of equations of the Duffing and Hill types. On the other hand, the finite amplitude, pure axial and pure rotational motions of the load are described by the classical, nonlinear Duffing equation alone. A variety of problems are solved exactly for these separate free vibrational modes, and a number of physical results are presented throughout.     

Dynamic shearing resistance of molten metal films under high pressures and extremely high shearing rates

In the present study plate-impact pressureshear experiments have been conducted to study the dynamic shearing resistance of molten metal films at shearing rates of approximately 10⁷ s⁻¹. These molten films are generated by pressure-shear impact of relatively low melt-point metals such as 7075-T6 Al alloy with high hardness and high flow-strength tool-steel plates. By employing high impact speeds and relatively smooth impacting surfaces, normal interfacial pressures ranging from 1–3 GPa and slip speeds of over 100 m/s are generated during the pressure-shear loading. The resulting friction stress (∼100 to 400 MPa) combined with the high slip speeds generate conditions conductive to interfacial temperatures approaching the fully melt temperature regime of the lower melt-point metal (7075-T6 aluminum alloy) comprising the tribo-pair. During pressure-shear loading, laser interferometry is employed to measure normal and transverse motion at the rear surface of the target plate. The normal component of the particle velocity provides the interfacial normal traction while the transverse component provides the shearing resistance of the interface as it passes through melt. In order to extract the critical interfacial parameters, such as the interfacial slip-speed and interfacial temperatures, a Lagrangian finiteelement code is developed. The computational procedure accounts for dynamic effects, heat conduction, contact with friction, and full thermo-mechanical coupling. At temperatures below melt the flyer and target materials are described as an isotropic thermally softening elastic-viscoplastic solid. For material elements with temperatures in excess of the melt point, a purely Newtonian fluid constitutive model is employed. The results of this hybrid experimental-computational study provide insights into the dynamic shearing resistance of molten metal films at high pressures and extremely high shearing rates.