Linear and Nonlinear Algorithms for Stress Separation in Photoelasticity

An experimental-numerical hybrid method for the stress separation in photoelasticity is proposed in this study. In the proposed method, boundary conditions for a local finite element model, that is, tractions along boundaries are inversely determined from photoelastic fringes. Two algorithms are proposed for determining the boundary condition. One is a linear algorithm in which the tractions are obtained by the method of linear least-squares from both principal stress difference and principal direction. Another is the nonlinear algorithm in which the tractions are determined only from the principal stress difference. After determining the boundary conditions for the local finite element model, the stresses can be obtained by finite element direct analysis. The effectiveness is demonstrated by applying the proposed method to a perforated plate under tension and contact problems. Results show that the boundary conditions of the local finite element model can be determined from the photoelastic fringes and then the individual stresses can be obtained by the proposed method. Furthermore, the stresses can be evaluated even if the boundary condition is complicated such as at the contact surface. It is expected that the proposed method can be powerful tool for stress analysis.     

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.     

Stress separation in the photoelastic study of centrifugal stresses

This technical note refers to the problem of stress separation in the photoelastic analysis of plane models under centrifugal stresses. Two methods are described in order to determine the sum of principal stresses. These methods, which are based on the compatibility equation, reduce the determination of the sum of principal stresses to the solution of a Laplace's or Poisson's equations. As an example of application, the separation of stresses in a rotating disk with two eccentric holes is shown and comparison with the stresses obtained by using the shear-difference method is made.     

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.     

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.     

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     

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.     

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。     

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.     

2D and 3D separation of stresses using automated photoelasticity

A procedure for the separation of full-field photoelastic images for use with an automated polariscope is described. Regions of background in the image are identified thus producing the boundary of the model. The shear difference method is used to calculate the components of stress along all raster lines in the image using photoelastic parameters at the boundary points to calculate the initial values of stress. Algorithms were also used to evaluate the stress components along raster lines which did not contain boundary points. A plastic template was used to evaluate the efficiency of the boundary routine. It was found that it was able to identify edges to within approximately one pixel on screen. The complete procedure for stress separation was evaluated using a stress frozen disc in compression and a turbine slot. The values of stress found using the automated polariscope with the stress-separation procedure were found to agree well with theory and with results determined using the method of Tardy compensation and manual analysis. The automated polariscope was also used to analyze three-dimensional stress components along arbitrary lines of a 3D model. A two-model slicing regime was used to analyze a strut subjected to a vertical load. This work was compared to results obtained by Frocht and Guernsey on an identical model machined from Fosterite and subjected to a higher load. Good agreement was found between the results for points away from the region of loading. Significant differences were found near to the load point, however. A finite element analysis of the same problem suggested that this was due to the effects of plasticity.     

Measurement of Applied Stresses and Residual Stresses on a Residual Stress Model by Applying Two Different Loads

Applied stresses on a residual stress model have previously been obtained by measuring the residual stresses and the resultant stresses generated by applying a load. The present paper reports that the applied stresses and the residual stresses on the residual stress model can be obtained by measuring two resultant stresses generated by applying loads of two different magnitudes. In the proposed method, the residual stresses need not be obtained from the residual stress model before applying a load. The residual stress model used to test the proposed method is a circular disk with frozen stresses that is subjected to a diametral compressive load at a certain angle. The applied stresses and the residual stresses on a residual stress model were experimentally and precisely obtained by digital photoelasticity using linearly polarized light.     

A photoelastic study of contact between a cylinder and a half-space

Three-dimensional photoelasticity was employed to study a cylinder in contact with a half-space. Both bodies were modeled in epoxy resin. Three loading cases were examined, namely, the cylinder lying on its side subject to a load normal to the plane, the cylinder on its side subject to both normal and tangential loads and the cylinder standing on its end and subject to a normal compressive load, i.e., as a circular punch. The cylinders and the half-space, which was represented by a large block, were stress frozen with a known coefficient of friction and using relatively small loads so that the strain levels were low. After slicing the cylinders, which resulted in lower fringe orders than could be readily analyzed manually, an automated system based on phase stepping was used to record and process the data. Distributions of maximum shear stress and Cartesian shear stress were obtained for a large area of the slice. Stress separation was performed, using the shear difference method, to obtain the Cartesian stress components in the plane of symmetry of the half-space. These results provide confirmation, by experiment, of the theoretical and numerical models of this type of contact obtained by other investigators.     

A rational theory of oblique incidence and its extension to stress-separation in birefringent composites

A rational theory of oblique incidence for a birefringent medium is developed. The theory rules out the concept of secondary principal stresses as an arbitrary one and provides more adequate interpretation of isochromatics and isoclinics in oblique incidence. The method is shown to be applicable for stress-separation in an anisotropic-birefringent composite, and the stress-optical equations deduced require an additional stress-fringe constant to those involved in normal incidence. The stress-birefringence relations for isotropic photoelasticity are obtained as a particular case of the general theory.     

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.     

Dynamic Measurement of Two Dimensional Stress Components in Birefringent Materials

We adopt classical methods of photoelasticity and Mach–Zehnder interferometry in a combined arrangement in order to determine both principal stresses and their orientations simultaneously. The method is equally applicable to static and dynamic problems. In dynamic problems the measurement may be made with a high-speed photodetector at very high temporal resolution at a single point or a small array of points depending on the detector array and recording device; this eliminates the need for a high-speed photographic system, but more importantly the method described here provides complete, time-resolved evolution of all stress components. Examples of application of the method are demonstrated.     

由应力集中处附近的应变值推求最大应力的一种方法

本文提出了一种用电阻片测定构件应力集中处最大应力的方案,以及相应的数据处理公式。典型实测与算例表明,本方法具有精度高的特点。并可以用于有限元、光弹性、云纹等方法。     

Stress-strain state and stress intensity factors of an inclined elliptic defect in a plate under biaxial loading

We consider a mathematical model of the stress-strain state of a plate with an inclined elliptic defect. We obtain approximate formulas for the stress tensor, the displacements, and the principal stresses near the defect vertex. The obtained formulas are compared with the results obtained by the holographic photoelasticity method.     

Fast evaluation of friction forces in traction-drive contacts including thermal effects

Summary  In automotive traction drives, power is transmitted by friction forces. The friction forces result from the shear stresses developed in lubricated and highly loaded contacts between rolling bodies. Due to the kinematics of a traction drive, shear velocities occur in both the rolling direction and perpendicular to it. Due to these shear velocities and by normal pressure, the lubricant is forced to build up shear stresses. The increase of the shear stresses may be modelled by a nonlinear viscous element. The describing differential equations are coupled by the equivalent shear stress, which defines the nonlinear behaviour of the element. A fast method is described to evaluate the coupled differential equations. By using a known analytical approximation for the equivalent shear stress, the differential equations are decoupled and can be solved analytically. In an iterative procedure the equivalent shear stress is updated, and the complete solution is found. The iterative method is extended to account for thermal effects in the contact. Received 17 June 1999; accepted for publication 26 October 1999     

Observation of Stress Field Around an Oscillating Crack Tip in a Quenched Thin Glass Plate

The variation of stress field around an oscillating crack tip in a quenched thin glass plate is observed using instantaneous phase-stepping photoelasticity. The successive images around the propagating crack are recorded by a CCD camera that is equipped with a pixelated micro-retarder array. Then, the phase maps of the principal stress difference and the principal direction are easily obtained even though the photoelastic fringes cannot be visualized. The path of the crack growth as well as the stress intensity factors and the crack tip constraint are obtained from these phase distributions. Results show that the mode I stress intensity factor and the crack tip constraint vary remarkably with the crack growth. In addition, the results show that the mode-II stress intensity factor exists even though the crack propagates smoothly.     

Stress Generated During Drying of Saturated Porous Media

The article is a contribution for the modelling of heat and mass transfers coupled to strain–stress equations during drying of deformable two-phase media. Both unidirectional and bidirectional configurations are examined. In order to compare the results, one assumes the material of a convectively dried clay slab in two configurations. Numerical calculations of the temperature, drying curves variations and the spatio-temporal distributions of moisture, temperature and drying induced stresses are evaluated. A significant difference was observed between the results obtained for both configurations particularly in intensity of the shear stress that caused cracking.