The existence of real distinct slownesses in photoelastic media

The continuum mathematical models of photoelasticity rely upon the ability of an optical medium, rendered anisotropic by the presence of a stress field, to sustain two transmitted light waves, with distinct slownesses and polarisations. The purpose of this paper is to examine the mathematical models with a view to determining whether they validly support birefringence over a wide range of material properties and impressed stress fields, and to determine conditions under which birefringence is guaranteed by the mathematical models. The classical models of Maxwell and Neumann and the recent finite deformation model of Smith and Rivlin are treated. In each case the secular equation is given and conditions are derived in the form of inequalities for the existence of distinct real slowness solutions. Methods of direct inspection and function extremisation are employed to verify these inequalities, with the aid of a geometric analogy. It is found that for magnetically isotropic media the distinctness of the three eigenvalues of the dielectric tensor is a necessary and sufficient condition for birefringence for all directions of propagation. For magnetically anisotropic media, birefringence is guaranteed by the distinctness of at least two of the eigenvalues of the matrix product of the dielectric tensor and the specific magnetic reluctance tensor.     

Tresca-type plastic materials in the theory of hypoelasticity II. Optical constitutive equations and birefringence in simple shear

Behavior of a Tresca type plastic dielectric is investigated theoretically from a continuum mechanical point of view. The optical constitutive equations are defined as special cases of a hypo-elastic dielectric of grade two. The singularity condition of the constitutive equations satisfies the Tresca yield criterion. The index deviator tensor is proportional to the stress deviator tensor and, then, the birefringence and the extinction angle are expressed by the stress deviator. Their numerical variations with the angle of shear in simple shear deformation are shown.     

A microstructure theory of photoelastic behavior in amorphous solids

Available theories to explain the phenomenon of photomechanical behavior in solid materials below the transition temperature are qualitative in nature, and several distinct mechanisms are capable of producing a deformation birefringence. A new mechanism is proposed on the atomic level to account quantitatively for deformation birefringence in an ideal amorphous elastic solid. The material is an isotropic, statistically homogeneous, elastic medium consisting of a random spatial arrangement of heavy mass points (atom nuclei) with positive electric charge in static equilibrium with a corresponding number of continuous, spherical, negatively charged regions (electron clouds). The medium has a nonzero random initial polarization; changes in the components of the dielectric tensor at optical frequencies are computed for infinitesimal uniaxial strain using the Lorentz field approximation for isotropic media. The stress-optical constant is then computed from the associated change in refractive index, and is shown to be in good agreement with experimental values for ordinary photoelastic materials.     

Characteristic relations of flow birefringence

This is the first of two closely related papers on the flow-birefringence response to the velocity vector field of a particular liquid representing a certain class of birefringent bodies.^* The flowing material under study was the aqueous solution of the compound known under the name NGS 1828 and commonly known as “milling yellow” or “acid yellow”. This solution appears to exhibit all three major mechanisms of birefringence. The physical parameters characterizing this material depend strongly on temperature, concentration and age and, therefore, it can be considered as representing a typical class of liquids used in flow-model experiments. The paper presents the experimental evidence that the flow birefringence cannot be explained and described by the simple mathematical model of birefringence in solid continuum which relates the changes of the components of dielectric tensor to the components of stress and strain tensors, or their derivatives, and which neglects the influence of the spectral frequency (wavelength of radiation). Results are presented for transmission birefringence (and for scattered-light birefringence in the second paper) in the visible and the infrared bands of radiation. It is shown that:

- the amount of birefringence depends strongly and non-monotonically on wavelength of radiation;

- the linear range of optical response to shear-strain rate depends on wavelengths of radiation;

- the directions of optic axes strongly depend both on the shear-strain rate and on the wavelength of radiation, even in the linear range of mechanical response.

It is further shown that there exists a relation between the absorption bands, the maximum transmittance, the dispersion of birefringence, the spectral dependence of optic-axes direction, and the linear range of optical response. Within the maximum transmittance band and the linear range of mechanical response the linear range of birefringence is maximum and the dispersion of birefringence is minimum with respect to the shear-strain rate; the corresponding dispersion of optic axis is also minimum. Samples of typical recordings are given in the visible and the infrared radiation for typical flow patterns. One of the practical conclusions is that to optimize the flow-birefringence studies of engineering problems it is advisable to choose the radiation in the near-infrared range. The evidence presented shows that the common trend in engineering research toward simplification of the model of the flow-birefringence response is not necessary.     

A study of the interacting FENE dumbbell model for semi-dilute polymer solutions in extensional flows

The effect of polymer concentration on the conformation of semidilute polymer solutions in extensional flows is studied via the interacting elastic dumbbell model proposed by Hess (1984), here modified to include a nonlinear Warner spring (FENE dumbbell) instead of the linear Hookean spring of the original model. The length of flow-induced conformation changes for the polymer is predicted to be a decreasing function of concentration. In particular, increasing concentration tends to inhibit large extension of the polymer due to polymer-polymer interaction. The specific birefringence is thus proportional to c ^–1 for semi-dilute solutions, in contrast to dilute solutions where it is known to be independent of concentration. However, the correlation between birefringence and the principle eigenvalue of the velocity gradient tensor, also found originally for dilute solutions, is predicted to occur in the semi-dilute regime. All of these predictions agree qualitatively with experimental observations.Some recent exceptions to the neglect of segmental stretch can be found in Marrucci and Grizzuti (1988), Pearson et al. (1991), Mead et al. (1992).     

Birefringence measurements on polymer melts in an axisymmetric flow cell

 The stress-optical rule relates birefringence to stress. Consequently, measurement of flow birefringence provides a non-intrusive technique of measuring stresses in complex flows. In this investigation we explore the use of an axisymmetric geometry to create a uniaxial elongational flow in polymer melts. In axisymmetric flows both birefringence and orientation angle change continuously along the path of the propagating light. The cumulative influence of the material's optical properties along the light's integrated path makes determination of local birefringence in the melt impossible. One can nevertheless use birefringence measurements to compare with predictions from computer simulations as a means of evaluating the constitutive equations for the stress. More specifically, in this investigation we compare the light intensity transmitted through the experimental set-up vs entry position, with the theoretically calculated transmitted intensity distribution as a means of comparing experiment and simulation. The main complication in our experiments is the use of a flow cell that necessarily consists of materials of different refractive indices. This introduces refraction and reflection effects that must be modeled before experimental results can be correctly interpreted. We describe how these effects are taken into account and test the accuracy of predictions against experiments. In addition, the high temperatures required to investigate polymer melts mean that a further complication is introduced by thermal stresses present in the flow cell glass. We describe how these thermal-stresses are also incorporated in the simulations. Finally, we present some preliminary results and evaluate the success of the overall method. Received: 2 April 2001 Accepted: 27 August 2001     

Tricolor photoviscoelastic technique and its application to moving contact

A new technique for simultaneous determination of both fringe order and principal direction of birefringence in practical photoviscoelastic analysis using white incident light with a set of the primary colors, called tricolor photoviscoelasticity, is described. This method can determine both the fringe order and principal direction of birefringence from a single-color photoviscoelastic image under plane polarization. Then, the authors evaluate time dependent stresses and strains around a contact region in a viscoelastic strip plate under nonproportional loading condition. The variations of the principal stresses and strains are easily obtained over a wide time range by use of the optical constitutive equations of photoviscoelasticity and the characteristic material property functions.     

Prediction of birefringence,stresses and kinematics in fibre spinning for solutions of finitely extensible macromolecules

A study of fibre spinning using an equation of state which accounts for the finite extensibility of the macromolecule is described. The results in this study show the birefringence to increase in a less than linear fashion with spinning stress, which is in agreement with experimental data. Previous studies use equations of state which allow the macromolecule to extend infinitely and therefore predict birefringence to increase linearly with the spinning stress. Accounting for the finite extensibility allows the simulation to be performed at higher throughputs which are more representative of those encoutered in industry. Draw ratios are predicted which are greater than the limit imposed by the infinitely extensible macromolecule model.     

Flowing colloidal suspensions in non-Newtonian suspending fluids: Decoupling the composite birefringence

The rheology of dilute, colloidal suspensions in polymeric suspending fluids can be studied with simultaneous dichroism and birefringence measurements. The dichroism provides a direct measure of the particle dynamics, but the birefringence is a composite property with independent contributions from the suspended particles and the polymer molecules. For suspensions where the contribution from the particles is significant, the composite birefringence must be decoupled in order to analyze the dynamics of the polymeric suspending fluid. A method to perform the decoupling is derived and then demonstrated through transient shear flow experiments with dilute suspensions ofFeOOH particles in semi-dilute, xanthan gum suspending fluids. The birefringence of the xanthan gum suspending fluid is calculated from experimental measurements of the composite birefringence and the dichroism of the suspension. To gather information on particle/polymer interactions, the calculated birefringence is compared to the birefringence of xanthan gum solutions containing no suspended particles and the dirchoism is compared to that of a suspension in a Newtonian fluid.     

The solid phase stress tensor in porous media mechanics and the Hill-Mandel condition

An assessment of the stress tensors used currently for the modeling of partially saturated porous media is made which includes concepts like total stress, solid phase stress, and solid pressure. Thermodynamically constrained averaging theory is used to derive the solid phase stress tensor. It is shown that in the upscaling procedure the Hill conditions are satisfied, which is not trivial. The stress tensor is then compared to traditional stress measures. The physical meaning of two forms of solid pressure and of the Biot coefficient is clarified. Finally, a Bishop-Skempton like form of the stress tensor is obtained and a form of the total stress tensor that does not make use of the effective stress concept.     

Birefringence and stress growth in uniaxial extension of polymer solutions

Simultaneous measurements of extensional stresses and birefringence are rare, especially for polymer solutions. This paper reports such measurements using the filament stretch rheometer and a phase modulated birefringence system. Both the extensional viscosity and the birefringence increase monotonically with strain and reach a plateau. Estimates of this saturation value for birefringence, using Peterlin’s formula for birefringence of a fully extended polymer chain are in agreement with the experimental results. However, estimates of the saturation value of the extensional viscosity using Batchelor’s formula for suspensions of elongated fibres are much higher than observed. Reasons for the inability of the flow field to fully unravel the polymer chain are examined using published Brownian dynamics simulations. It is tentatively concluded that the polymer chain forms a folded structure. Such folded chains can exhibit saturation in birefringence even though the stress is less than that expected for a fully extended molecule.Simultaneous measurements of stress and birefringence during relaxation indicate that the birefringence decays much more slowly than the stress. The stress-birefringence data show a pronounced hysteresis as predicted by bead-rod models. The failure of the stress optic coefficient in strong flows is noted.Experiments were also performed wherein the strain was increased linearly with time, then held constant for a short period before being increased again. The response of the stress and birefringence in such experiments is dramatically different and can be traced to the different configurations obtained during stretching and relaxation. The results cast doubt on the appropriateness of pre-averaging the non-linear terms in constitutive equations.     

n the fourth order tensor valued function of the stress in return map algorithm

针对各向同性材料,基于一组相互正交的基张量,建立了一套有 效的相关运算方法. 基张量中的两个分别是归一化的二阶单位张量和偏应力张量,另一个则 使用应力的各向同性二阶张量值函数经过归一化构造所得,三者共主轴. 根据张量函数表示 定理,本构方程和返回映射算法中所涉及到的应力的二阶、四阶张量值函数及其逆都由这组 基所表示. 推演结果表明：这些张量之间的运算,表现为对应系数矩阵之间的简单 关系. 其中,四阶张量求逆归结为对应的3\times3系数矩阵求逆,它对二阶张量的变换 则表现为该矩阵对3times 1列阵的变换. 最后,对这些变换关系应用于返回映 射算法的迭代格式进行了相关讨论.     

Development of a double-beam rheo-optical analyzer for full tensor measurement of optical anisotropy in complex fluid flow

The design and testing of the Double-Beam Rheo-Optical Analyzers, DB-ROA is described. The DB-ROA contains the two optically modulated laser beams that are transmitted through a sheared sample in different angles. The full components of the stress tensor can be evaluated simultaneously by a single measurement of the birefringence and orientation angle of each beam. The mechanical properties are also measured simultaneously with the optical properties. This system can be applied to both steady and transient shear flows. Three types of DB-ROAs are designed. They are distinguished by the optical modulation system, that is (1) a single modulator system, (2) a synchronization of two modulators, and (3) a two individual modulator system. The performance tests were carried out and its validity was demonstrated by comparison between the optical and the mechanical measurements of the first normal stress difference. The signal-to-noise ratio was strongly affected by the choice of the oblique angle of the second beam. The design features for the different optical modulation systems are discussed.     

Experimental analysis of rolling contact stresses in a viscoelastic strip

An experimental approach to two-dimensional, viscoelastic, steadily moving rolling contact is described. The photoviscoelastic technique is employed for the analysis of rolling contact stresses between a viscoelastic plate and a rigid rolling cylinder in which the principal axes of stress, strain and birefringence are not coincident with each other. Using an elliptically polarized white light, the distribution of isochromatic fringe order and the principal axes of birefringence at an instant are determined from a single photoviscoelastic image. The time variations of the differences of the principal stresses and strains, as well as their directions, are obtained by use of the optical constitutive equations of photoviscoelasticity. The experimental results involving the time variation of the stresses around the contact surface and their distributions are analyzed.     

Use of the parametrix method for estimating effective elastic moduli or randomly nonhomogeneous elastic bodies

The magnitude of the elasticity tensor of a comparison body remains unclarified if we use a singular approximation [1] to estimate the effective values of the elasticity tensor. Below we will use a parametrix method [2] to determine the first approximation of the random component of the deformation tensor and the effective values of the elasticity tensor, and will also compare the exact solution for one particular heterogeneous and a previously used approximation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 171–175, January–February, 1976.     

A study of conformation-dependent friction in a dumbbell model for dilute solutions

We consider predictions of rheological behavior in a variety of shear and extensional flows for an elastic dumbbell model with a nonlinear spring, and conformation-dependent hydrodynamic properties. The latter include a conformation-dependent anisotropic bead friction coefficient, and a related conformation-dependent degree of inefficiency for rotation in straining flows. With these features, the dumbbell exhibits hydrodynamic behavior consistent with a particle of finite axis ratio over the complete set of possible polymer conformations, from random-coil to a fully extended thread-like configuration. The predicted rheological behavior in shear flow is improved, relative to data, by the inclusion of anisotropy and strain-inefficiency in the frictional properties of the model, while other desirable features such as the sudden onset of fully extended states at a critical value of the velocity gradient, the presence of a hysteresis-loop in end-to-end dimension as a function of the velocity gradient, and the correlation of end-to-end distance (or birefringence) with the eigenvalue of the velocity gradient tensor for a wide variety of two-dimensional flows, are maintained.     

Rheo-optical measurements of the first and third normal stresses of homopolymer poly(vinyl methyl ether) melt

Normal stresses play a key role in polymer processing, yet accurate measurements are still challenging. Simultaneous rheo-optical measurements are conducted on a poly(vinyl methyl ether) homopolymer melt over a wide range of temperatures and oscillatory shear frequencies, in an effort to measure the normal stresses, by using quantitative flow birefringence measurements. The stress optical rule holds well for this polymer as expected, with the value of the stress optic coefficient of (6.38±0.19)×10⁻¹¹ cm²/dyn at 30°C. The first and third normal stress difference coefficients, calculated using a single memory constitutive equation applied to the stress and birefringence data, are in excellent agreement. The ratio of the measured third and first normal stress difference coefficients, (1−β)=0.71±0.05, agrees well with the result of the Doi–Edwards model with independent alignment approximation (β=0.28). The measurement of normal stress difference coefficients with such small deviations proves the robust nature of the improved rheo-optical instrument and its ability to measure complete stress tensor.     

Highlights of the optical anisotropy of unstressed transparent polymeric plates

The photoelastic effect is cumulative, therefore the presence of material birefringence in unstressed transparent polymers might lead to erroneous photoelastic analyses. This presence is more influential in the scattered-light photoelasticity. Direct-transmission polariscopes are not suitable for detecting all material birefringence in normally illuminated plates because the birefringence in question coincides with the wave normal of the propagating light. The present paper describes phenomenologically the presence of the material birefringence in an arbitrarily selected group of unstressed cross-linked polymers and presents means of their determination. The material-birefringence information obtained can be used as a means of taking proper precautions for conducting faultless scattered-light photoelastic analyses. Octagonally shaped plates were machined to permit data acquisition using four different light-propagation directions. The orientation and diffusion of molecular chains appear to be the major source of the material birefringence observed.     

Photoelastic analysis of frozen stressed specimens using spectral-contents analysis

The complete birefringence, or isochromatic fringe order, in stress-frozen photoelastic models has been found by measuring the spectral contents of idividual points. A calibration procedure is porposed. The effects of nonuniform birefringence and dispersion of birefringence are considered. The results are presented from the analysis of models of a plate with a central hole and a disk subject to three radial loads.