A stress-optic law for photoelastic analysis of orthotropic composites

The feasibility for utilizing transparent filament-resin composites for photoelastic stress analysis was investigated. Satisfactory photoelastic stress patterns were demonstrated in simple models with undirectional and bidirectional fiber orientations. A stress-optic law was formulated, based on the concept that the birefringence components contributed by each component of plane stress are combined according to a Mohr circle of birefringence. Applying this concept, the difference of the physical and optical principal directions was accounted for, and a general method of photoelastic solution for the plane-stress problem in orthotropic sheets was developed. The method of analysis is little more complex than the well-known procedures for isotropic materials, but at least three experimental measurements are required to characterize the optical response of the material to plane stress. Partial confirmation of the proposed stress-optic law was obtained by comparison of the theory to limited experimental data obtained in uniaxial-stress samples. It remains to establish a more positive verification by experiments in a variety of biaxial-stress conditions.     

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.     

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.     

聚合物注射成型流动残余应力的数值分析

建立了可压缩黏弹性聚合物熔体在薄壁型腔中充模/保压过程中非等温、非稳态流动 的数学模型,用数值方法实现了注射成型过程中流动应力和取向建立及松弛过程的模拟,研 究了熔体温度、模具温度和注射速率等工艺条件对分子冻结取向的影响,取得了与实验相符 的结果.     

Half-fringe photoelasticity: A new approach to whole-field stress analysis

This paper presents a new method for whole-field stress analysis based on a symbiosis of two techniques—classical photoelasticity and modern digital image analysis. The resulting method is called ‘half-fringe photoelasticity (HFP)’. Classical photoelasticity demands materials with high birefringence, which leads to extensive use of plastics as model materials. Since the behavior of these materials is often different from that of the prototype materials, their use distorts the similitude relationships. In many contemporary problems this distortion is untenable. HFP offers a way out of this dilemma. It permits materials and loads to be chosen so that no more than one half of a fringe order appears in the area of interest. Thus, for example, glass, which behaves linearly up to high stress levels and over a wide range of temperatures, could be used as model material. Alternatively, models from polymeric materials could be used under very low load in order to stay within the linear part of the stress-strain diagram and to prevent large deformations. The half-fringe-photoelasticity system, which is described here, utilizes the resulting low levels of birefringence for effective stress analysis. This paper describes the system. It outlines a calibration routine and illustrates its application to two simple problems using glass models.     

On the use of stereolithography for the manufacture of photoelastic models

Three-dimensional photoelasticity using the stress-freezing technique is dependent on the production of resin models that do not possess any residual stresses from the manufacturing process. The traditional methods of production involve casting to shape or machining from solid blocks using thermo-setting resins. These methods are expensive and time-consuming, with models typically requiring days for preparation. The rapid-prototyping technique of stereolithography employs similar resins and allows complex components to be built in a matter of hours. However, the residual birefringence associated with the stereolithographic process has so far inhibited its routine use in photoelasticity. A four-centre study has been conducted in an attempt to understand the mechanisms generating this birefringence and to investigate methodologies for producing models free of stress and birfringence. The mechanical behavior of stereolithographic and thermo-setting resins have been compared at room temperature and under stress-freezing conditions.     

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.     

Photovisco-elastoplastic behavior of polycarbonate material under creep and tension tests

Polymers are widely used as photomechanical models of a prototype material (often a metal). Photoplasticity is one of the methods used in order to show the behavior of plastic materials stressed beyond the linear elastic limit. To illustrate this process we have analyzed the photovisco-elastoplastic behavior of polycarbonate as a photoplastic material. In this paper a technique for local and simultaneous measurement of birefringence and principal strains is presented. The mechanical and optical properties, at room temperature, have been evaluated by means of uniaxial tension tests. A series of creep tests has been carried out in order to study the photovisco-elastoplastic behavior of polycarbonate. In two different experiments we analyzed nonlinear birefringence and the amplitude of the corresponding strains. We could thus evaluate the distribution of strains and the distribution of uniaxial stress for each birefringence state and vice versa.     

Flow birefringence of disordered block copolymer melts

The birefringence relaxation after a step strain S(t), and the oscillatory flow birefringence S ^* are calculated for disordered block copolymer melts, on the basis of four models for the chain dynamics: the Rouse model, the Doi-Edwards reptation model, the reptation model with constraint release, and the reptation model with orientational coupling. All the calculations are performed in the mechanically-uniform limit, i.e., the average subchain length and friction coefficient are independent of block. The net birefringence is assumed to contain no form contribution, and the approach of Kuhn and Gruen is employed in the computation of the intrinsic birefringence. The most important feature of the results is that the stress-optic relation does not apply in general for block copolymers; therefore, unique information about chain relaxation mechanisms can be obtained from measurements of flow birefringence. It is shown that the phase angle of S ^* can be particularly sensitive to the lengths and chain locations of the various blocks, with the most striking effects occurring when two (or more) blocks have optical anisotropies of opposite sign. In contrast, in the mechanically-uniform limit the viscoelastic properties are independent of block length and location.Dedicated to Professor Arthur S. Lodge on the occasion of his 70th birthday and his retirement from the University of Wisconsin.     

A rheo-optical study of shear-thickening and structure formation in polymer solutions. Part II: Light scattering analysis

Light scattering calculations based on Anomalous Diffraction Theory (AD), Rayleigh spheroids, and flexible macromolecules are used to propose a phenomenological explanation for the relationship between shear-thickening and structure formation in polymer solutions. Quantitative comparisons are made to experimental data for the rheo-optical behavior of fractionated polystyrene solutions presented in part I of this paper. Results from the ADA calculations suggest that the viscosity and dichroism behavior can be attributed to the production and growth of micron-size, optically isotropic structures during flow. The saturation dichroism behavior exhibited by the solutions which shear thin can be attributed to the formation of entanglement regions which achieve a fixed size and act as Rayleigh spheroids in their scattering behavior. The magnitude and shear rate dependence of the observed birefringence can be accounted for on the basis of the non-linear, flexible macromolecule model, implying that birefringence is governed by the polymer chains remaining in solution which do not take part in the structure formation. The latter result is consistent with the experimental observation that the birefringence dependence on shear rate is the same whether the solution exhibits shear thickening or shear thinning in its viscosity behavior.     

Dynamic streaming birefringence

Summary A compact formulation is given of the phenomenon of birefringence and some results in oscillatory deformation are derived in terms of a birefringence tensor. The birefringence tensor then enables invariant forms of rheo-optic equations to be formulated involving a birefringence relaxation spectrum. An important practical aspect of the work is that the use of streaming birefringence in engineering studies of inhomogeneous or unsteady fluid flow fields is rather more complicated than the use of photoelastic effects in the study of deformed solids.     

Photoviscoelastic analysis of thermal stress in a quenched epoxy beam

This paper reports on a procedure for photoviscoelastic analysis where the axes of principal stress, principal strain and polarization of light coincide in the presence of a large temperature change. More specially, the transient-thermal stress and strain due to stress in an epoxy beam subjected to quenching from both the upper and lower surfaces, are determined using the time-temperature-equivalent law for stress, strain and birefringence. The transient-thermal stress and strain in the beam were determined experimentally using hereditary integrations from the measurement of the transient temperature and birefringence due to the quenching of the beam. The transient thermal stress and strain were also calculated theoretically using the linear-viscoelastic theory. The experimentally determined thermal stress agrees closely with the theoretical results. The experimentally determined strain agrees qualitatively with the theoretical values. Thus, it is concluded that the photoviscoelastic technique is useful in analyzing the proposed problem.Paper was presented at 1982 SESA/JSME Spring Meeting held in Maui and Oahu, HI on May 23–28, 1982.     

Application of the GIMP software in the analysis of birefringence images obtained in a multipass rheometer

Flow-induced birefringence is a technique with numerous potential applications in polymers, because the bright and dark fringe patterns of the birefringence images can be related to principal stress difference (PSD) profile along the flow centerline. In this work, we developed an easy-to-use methodology for birefringence image processing using the GIMP open-source software. In order to illustrate its application, images obtained in previous work during the flow of two grades of general purpose polystyrene through the slit-die geometry in the multipass rheometer MPR4, located at the University of ?Cambridge, UK, were reprocessed. The results showed that the PSD module profile along the flow centerline obtained with the proposed methodology is more accurate than the manual technique and is obtained with shorter processing time.     

The gamma-ray-irradiation method applied to three-dimensional thermal photoelasticity

It has recently been shown that a gamma-ray irradiation fixes the birefringence in photoelastic models of Araldite B. The method has hitherto only been used to fix the birefringence in specimens subjected to constant forces. This work applies the irradiation method to determine the thermal stresses in a thick-walled cylinder with a temperature gradient along the radius. The model behaved in the following way during the experiment. The material relaxed when the temperature gradient was applied, and the model was irradiated when it was stress free. Real stresses, which were equal to the thermal stresses but with opposite sign, appeared when the temperature gradient was removed. As the problem is mathematically two-dimensional, it is possible to determine the birefringence by means of the slicing technique. Good agreement was obtained between the stress determined experimentally and those calculated analytically. The irradiation sensitivity of a number of materials has been determined, and some new materials have been used that are superior in two important ways to that used earlier in that they require a much lower irradiation dose and their mechanical properties are considerably better.     

Photoviscoelastic stress analysis of a plate with a central hole

An epoxy resin containing excessive plasticizer was developed and characterized. The material, which deforms viscously at room temperature, has optical properties that depend on stress and strain. A tensile specimen was prepared from the epoxy resin so that the mechanical and optical properties of the epoxy resin could be characterized. The elastic and plastic behavior was determined at 37°C using tensile stresses between 4 and 26 MPa. The birefringence was also recorded as a function of time and stress. From these results, a photoviscoelastic constitutive equation was constructed to describe the dependence of the birefringence on stress and strain. The constitutive equation was then applied to study the deformation of a tensile specimen containing a central circular hole. By using the isochromatic fringes in combination with the isoclinic, the time-dependent variation of the stress field in the specimen was solved.     

Structure and rheology of wormlike micelles

In a semi-dilute aqueous solution under certain conditions, surfactant molecules will self assemble to form wormlike micelles. The micelles are dynamic in structure since they can break and reform, providing an additional mode of relaxation. The viscoelastic properties of the wormlike micelles can be predicted using simple theological models. For many surfactant solutions the mechanical data can be related to the optical data by the stress-optical rule. From the viscoelastic data it is possible to estimate the breaking time of the micelle. The techniques of birefringence and small angle light scattering are used to study the microstructure of a surfactant solution under simple shear and extensional flow. The sample under investigation is a solution of cetyltrimethylammonium bromide and sodium salicylate in water, with a salt to surfactant ratio of 7.7. Below a critical shear rate, the birefringence increases linearly with shear rate and the stress-optical rule is valid. The SALS patterns reveal distinctive butterfly patterns indicating that scattering is a result of concentration fluctuations that moderately couple to the flow. However, above a critical shear rate the birefringence plateaus and the stress-optical rule is no longer valid. SALS patterns show both a bright streak and a butterfly pattern. The bright streak is caused by elongated structures aligned in the direction of the flow. The oriented structures occur when the characteristic time of flow is faster than the breaking time of the micelles.Dedicated to Prof. Dr. J. Meissner on the occasion of his retirement from the chair of Polymer Physics at the Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland     

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.     

Comparison of experimental data with the numerical simulation of planar entry flow: Role of the constitutive equation

The goal of this research was to determine whether there is any interaction between the type of constitutive equation used and the degree of mesh refinement, as well as how the type of constitutive equation might affect the convergence and quality of the solution, for a planar 4:1 contraction in the finite eiement method. Five constitutive equations were used in this work: the Phan-Thien–Tanner (PTT), Johnson–Segalman (JS), White–Metzner (WM), Leonov-like and upper convected Maxwell (UCM) models. A penalty Galerkin finite element technique was used to solve the system of non-linear differential equations. The constitutive equations were fitted to the steady shear viscosity and normal stress data for a polystyrene melt. In general it was found that the convergence limit based on the Deborah number De and the Weissenberg number We varied from model to model and from mesh to mesh. From a practical point of view it was observed that the wall shear stress in the downstream region should also be indicated at the point where convergence is lost, since this parameter reflects the throughput conditions. Because of the dependence of convergence on the combination of mesh size and constitutive equation, predictions of the computations were compared with birefringence data obtained for the same polystyrene melt flowing through a 4:1 planar contraction. Refinement in the mesh led to better agreement between the predictions using the PTT model and flow birefringence, but the oscillations became worse in the corner region as the mesh was further refined, eventually leading to the loss of convergence of the numerical algorithm. In comparing results using different models at the same wall shear stress conditions and on the same mesh, it was found that the PTT model gave less overshoot of the stresses at the re-entrant corner. Away from the corner there were very small differences between the quality of the solutions obtained using different models. All the models predicted solutions with oscillations. However, the values of the solutions oscillated around the experimental birefringence data, even when the numerical algorithm would not converge. Whereas the stresses are predicted to oscillate, the streamlines and velocity field remained smooth. Predictions for the existence of vortices as well as for the entrance pressure loss (ΔP_ent) varied from model to model. The UCM and WM models predicted negative values for ΔP_ent.     

Acoustical birefringence and the use of ultrasonic waves for experimental stress analysis

Stress-induced optical birefringence in transparent materials has long been a common technique of stress analysis. Although stress-induced acoustic birefringence was discovered more than 20 years ago, its development and actual applications are still limited. This paper will look at the similarities and differences between the propagation of light waves in photoelastic materials and the propagation of ultrasonic waves in deformed solids. Critical comparisons of the experimental methods employed in photoelasticity with those available in modern ultrasonic measuring technique show why previous studies on ultrasonic measurement of stresses were not very successful. A new experimental technique is devised for using ultrasonic waves for stress analysis. The technique employs a single rotatable 10-MHz shear transducer as the transmitter and receiver of ultrasonic pulses. The enlarged display of the 10-MHz modulated-pulse pattern of reflected echoes provides a convenient way to determine the directions of principal axis of the stress within ±3 deg. The pulse-echo-overlap method is used to measure the absolute velocities of the two principal shear waves. The difference in principal stresses is then calculated from the velocity measurements. Test results of common structural-aluminum and steel specimens under uniaxial compression show a linear relation between the velocity changes and the applied stress. Ultrasonic measurements of stress distribution in a 6.35-cm diameter, 1.9-cm-thick aluminum disk under diametric compression are also reported. Paper was presented at Third SESA International Congress on Experimental Mechanics held in Los Angeles, CA on May 13–18, 1973.     

Experimental analysis of plates using the method of model curing

The method of model curing is applicable to the analysis of plates. When the polymerization process is complete and model is unloaded, the deflection and the birefringence remain permanent. The stereometric method is used for evaluation of bending and torsion moments, and photoelastic method is used as a control method. The example given of the method of curing is a square plate supported at three points and loaded by a concentrated load.