Contribution to the analysis of errors in photoelasticity

Photoelasticity data are expressed in terms of the direction and the differences of the principal stresses. To determine the principal stresses, separation is necessary. The author deals with some of the numerical methods of separating principal stresses, permitting evaluation of the stresses along straight sections. In the first part of this paper the author presents methods of separation and in the second part the errors involved.     

A numerical method for separation of stresses in photo-orthotropic elasticity

A numerical method is suggested for separation of stresses in photo-orthotropic elasticity using the numerical solution of compatibility equation for orthotropic case. The compatibility equation is written in terms of a stress parameter S analogous to the sum of principal stresses in two-dimensional isotropic case. The solution of this equation provides a relation between the normal stresses. The photoelastic data give the shear stress and another relation between the two normal stresses. The accuracy of the numerical method and its application to practical problems are illustrated with examples.     

Phase Shifting Full-Field Interferometric Methods for Determination of In-Plane Tensorial Stress

A new method that combines phase shifting photoelasticity and transmission Coherent Gradient Sensing (CGS) is developed to determine the tensorial stress field in thin plates of photoelastic materials. A six step phase shifting photoelasticity method determines principal stress directions and the difference of principal stresses. The transmission CGS method utilizes a standard four step phase shifting method to measure the x and y first derivatives of the sum of principal stresses. These stress derivatives are numerically integrated using a weighted preconditioned conjugate gradient (PCG) algorithm, which is also used for the phase unwrapping of the photoelastic and CGS phases. With full-field measurement of the sum and difference of principal stresses, the principal stresses may be separated, followed by the Cartesian and polar coordinate stresses using the principal stress directions. The method is demonstrated for a compressed polycarbonate plate with a side V-shaped notch. The experimental stress fields compare well with theoretical stress fields derived from Williams solution for a thin plate with an angular corner.     

Graphical methods for determining the resulting photoelastic effect of compound states of stress

The aim of this work is to give a simplified representation of an interesting subject, namely, the calculation of the photoelastic effect behind a system in which the principal stresses or secondary principal stresses rotate about the direction of the propagation of light. The methods are described in a popular form for understanding the essential features and the necessary graphical operations for solving these complicated problems. Further, the relations between the different methods are shown. Thej-circle technique is improved in order to simplify the operations and a new possibility in applying Wulff's grid is introduced in photoelasticity. The graphical tools can then be applied as so-called “rotation rules.” Some examples related to recent papers which present theoretical results or new methods are given in order to study the function of these “rotation rules” and to recognize the power of these methods.     

On the determination of the sum of the principal stresses in two-dimensional problems

The well-known principle of the determination of the sum of principal stresses in plates by measuring thickness changes is applied in this paper to “frozen” plates and to models made with low-modulus materials, such as rubbers, deformed inside portable frames. It is shown that a sufficiently precise measurement of thickness changes is possible with a machine-shop comparator, rather than with the more delicate laboratory-type instruments. The necessary corrections to be introduced to the comparator readings when rubber models are used are described in the Appendix. Two other methods based on moiré fringes, using the same two kinds of models, are also presented. The use of moiré on “frozen” specimens yields patterns of large response. Gratings on rubber models also yield precise moiré patterns that can be combined with isochromatics to separate the principal stresses. Two ways of conducting the moiré analyses are presented. The advantages and limitations of the several alternative methods are pointed out, and applications are given.     

Experiments in mechanical and optical coincidence in photoplasticity

In the extension of the shear-difference method to the separation of interior principal stresses in the elasto-plastic state¹, a basic question arises whether the optical isoclinics give the directions of the principal stresses, i.e., whether optical coincidence exists. Experiments are described aiming to answer this question, and preliminary results are given for cellulose nitrate as a model material. Experiments are also described dealing with mechanical coincidence.     

边界元法与光弹性实验相结合的应力分析法

本文介绍了边界无法与光弹性实验相结合进行应力分析的方法。叙述了用边界无法求解主应力的基本原理。结合光弹性实验等差线条纹图,可分离出主应力。     

Stress Separation of Interferometrically Measured Isopachics in a Perforated Plate

A method for the stress separation of interferometrically measured isopachics using an Airy stress function is proposed in this study. A Poisson equation that represents the relationship between the sum of principal stresses and an Airy stress function is solved using a finite element method. The Dirichlet boundary condition for solving the Poisson equation is determined by the approximation of an assumed Airy stress function along the boundary of the model. Therefore, the distribution of the Airy stress function is obtained from the measured isopachic contours. Then, the stresses are obtained from the computed Airy stress function. The effectiveness of the proposed method is validated by applying the proposed method to the isopachic contours in a perforated plate obtained by Mach-Zehnder interferometry. Results indicate that stress components around a hole in a plate can be obtained from isopachics by the proposed method.     

Separation of principal stresses in dynamic photoelasticity

A new and effective method used to separate the transient principal stresses for dynamic photoelasticity is proposed. This is a hybrid method combining the optical method of dynamic caustics and the boundary element numerical method. Firstly, a modified Cranz-Schardin spark camera is used to record simultaneously the isochromatic fringe patterns of photoelasticity and the shadow spot patterns in the dynamic process. By means of the isochromatic fringe patterns, the difference between transient principal stresses in the whole domain and the principal stresses along the free boundary can be solved. In addition, the method of caustics is a very powerful technique for measuring the concentrative load. Then, the sum of the principal stresses is calculated by the boundary integral equation obtained from the Laplace integral transform of the wave equation. So, the transient principal stresses can be determined from the experimental and numerical results. As an example, the transient principal stresses in a polycarbonate disk under an impact load are resolved. Concurrently published in the Chinese Edition of Acta Mechanica Sinica, Vol. 26, No. 1, 1994     

Photoelastic stress analysis of machines under dynamic load

Stress processes within the elastic limit are investigated by the method of single shots in repeating the impact. Model laws and methods of interpreting results of experiments in different types of processes have been developed. Their application in practical problems and separation of principal stresses are explained.     

Full-field separation of principal stresses by combined thermo- and photoelasticity

The combined use of thermoelastic stress analysis and full-field reflection photoelasticity based on the phase-stepping technique has been developed for twodimensional problems. The first method determines the sum of the principal stresses, the latter evaluates the difference of the principal stresses. Thus the principal stresses were separated at each point in the field of view without reference to neighboring points. An evaluation of this approach has been performed using a tensile plate with a central circular hole. The results show that the analysis carried out combining thermo- and photoelasticity incurred errors no larger than those of each system working independently.     

A new holographic interferometer for stress analysis

In holographic interferometry with a photoelastic model, two families of fringes are generated simultaneously when the model is stressed. One family represents the isochromatic-fringe pattern normally associated with photoelasticity which yields the difference between the principal stresses. The other family represents the isopachic-fringe pattern associated with interferometry which yields the sum of the principal stresses. From these complementary patterns, the magnitudes of the principal stresses can readily be determined throughout the field of observation. Unfortunately, these fringe patterns are not completely independent but interact in such a way as to make interpretation difficult in critical regions of the model. A new system has been developed which readily permits simultaneous acquisition of these fringe patterns without their undesirable mutual interaction, as well as providing increased sensitivity. This new interferometer uses a double-pass object beam and an optical rotator to eliminate the isochromatic-fringe pattern and its effect from the isopachic interferogram. Such a system has considerable value in experimental mechanics for applications to both static and dynamic model studies and to materials investigation.     

Analysis of scattered-light methods in photoelasticity

The use of scattered light for nondestructive analysis of general photoelastic models is becoming quite common and, during the last few years, several methods have appeared in the literature to determine the directions of the secondary principal stresses and their differences at any general point. Among the methods suggested, some are mentioned as “exact” and some “approximate”. Even the exact methods have limitations in their applicability for a generally stressed model. The present discussion attempts to bring out the inaccuracies involved in the various exact methods. Also, a few modifications to improve the accuracy and a new method called the Mini-max method are proposed. References 1, 2 and 3 give the general methods and the derivations of equations used in the present analysis.     

动态光弹性方法的主应力分离的研究

本文提出了动态光弹性、动态焦散线实验方法同边界元法结合的混合法，并用这种方法解决了动态光弹性主应力分离的问题，首先对现有的多火花高速摄影系统进行了改造，在动态实验过程中，成功地得到了不同瞬时的清晰的动态光弹性的等差线条纹和动态焦散线系列图像，这样，便可提取不同瞬时的边界应力值、全场主应力差值及边界上的外载。继而提出用Ｌａｐｌａｃｅ变换域上的边界元法来求解在冲击载荷作用下二维弹性模型全场的主应力和。最后，以受冲击载荷作用的圆盘为例，进行实验及边界元法计算，得到了分离的主应力场。     

Automated Stress Separation Along Stress Trajectories

A procedure for the separation of principal stresses in automated photoelasticity is presented. It is based on the integration of indefinite equations of equilibrium along stress trajectories, also known as Lamè–Maxwell equations. A new algorithm for precise and reliable stress trajectory calculation, which is an essential feature of the procedure, has also been developed. Automated stress separation is carried out along stress trajectories starting from free boundaries. Experimental tests were performed on a disc in diametral compression and on a ring with internally applied pressure. Full-field principal stress values were obtained and results were compared with those from the theory of elasticity and with those obtained from the classical shear difference method. It was shown that the proposed method is more accurate and less affected by the presence of residual stresses or experimental errors at the boundaries than the shear difference method. In addition, the method requires little human interaction and is therefore well-suited for automated photoelasticity.     

混合光载波法分离动静态主应力

本文将等差载波位相调制原理与夹层全息术结合形成的混合光载波法,可同时获得等差载波条纹图与混合载波条纹图.对混合载波纹图进行光学付里叶变换,利用空间滤波技术可将等和载波条纹从中分离出来.利用图象处理技术实现了对两种载波条纹图从预处理到应力计算的自动化处理.将混合光载波法应用于动态全息光弹性,分离了动态主应力. 文中还提出了一个经济实用的大尺寸载波片制作技术.     

Shear stresses in elastic beams: an intrinsic approach

The theory of non-uniform flexure and torsion of Saint-Venant's beam with arbitrary multiply connected cross section is revisited in a coordinate-free form to provide a computationally convenient context. Numerical implementations, by Matlab, are performed to evaluate the maximum elastic shear stresses in beams with rectangular cross sections for different Poisson's ratios. The deviations between the maximum and mean stresses are then diagrammed to adjust the results provided by Jourawski's method.     

Separation of principal stresses in the freezing method

In order to separate principal stresses in the photoelastic freezing method, the stress sum is determined by measuring the dilatation or variation of specific weight of small particles of frozen-in models. A floating teat has proved to give very accurate results.     

An Integrated Approach to the Separation of Principal Surface Stresses Using Combined Thermo-Photo-Elasticity

A new instrument capable of the full-field separation of principal stresses on the surface of a component is presented. The instrument combines the techniques of thermoelastic stress analysis and reflection photoelasticity in a single optical head, permitting the simultaneous capture of both data from the same point of view. A single strain witness coating is employed for the acquisition of both the thermoelastic and photoelastic data, which is both birefringent under applied stress conditions and opaque at the infrared wavelengths to which the thermoelastic analysis system is sensitive. This enables the combined technique to be performed continuously from the same surface during loading. The performance of the new instrument is validated in the analysis of a classical laboratory specimen of known geometry. Separated stress data from the experiment is compared to simulated data, demonstrating that the accuracy of the stress separation technique is comparable to that of the individual thermoelastic and photoelastic techniques, and it is concluded that combined thermo-photo-elasticity is a powerful tool for the experimental separation of principal surface stresses.