Obtaining stress maps from photoelastic data using a digital computer

A general procedure for the analysis of photoelastic isochromatic and isoclinic photographs by digital computer to produce plots of coordinate normal and shear stresses is presented. The solution for normal stresses, involving a finite-difference iterative harmonization of the Laplace equation, is discussed, including treatment of the boundaries and selection of an appropriate network grid. The results of a classical problem, solved by this technique, are included.     

Solution of the plane stress problems of strain-hardening materials described by power-law using the complex pseudo-stress function

In the present paper,the compatibility equation for the plane stress problems of power-law materials is transformed into a biharmonic equation by introducing the so-calledcomplex pseudo-stress function,which makes it possible to solve the elastic-plastic planestress problems of strain hardening materials described by power-law using the complexvariable function method like that in the linear elasticity theory.By using this generalmethod,the close-formed analytical solutions for the stress,strain and displacementcomponents of the plane stress problems’of power-law materials is deduced in the paper,which can also be used to solve the elasto-plastic plane stress problems of strain-hardeningmaterials other than that described by power-law.As an example,the problem of a power-law material infinite plate containing a circular hole under uniaxial tension is solved byusing this method,the results of which are compared with those of a known asymptoticanalytical solution obtained by the perturbation method.     

Studies of contact problems using photoelastic isodynes

Experimental determination of contact stresses is difficult and not often satisfactorily accurate, especially when shear stresses exist in contact areas. Several examples presented in the paper show that, by using isodyne techniques, it is relatively easy and inexpensive to determine not only all stress components in contact regions but also the boundaries of contact areas as well.     

On the solution of plane boundary-value problems of elasticity in a rectangle

An algorithm for solving plane boundary-value problems of elasticity for a rectangular domain is expounded. The algorithm is based on a complex-valued representation of the general solution to the differential equations of the plane problem and on the use of Lagrange polynomials to satisfy the boundary conditions. The algorithm can quite easily be implemented in a computer program. This is probably the simplest way of solving boundary-value problems of this class __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 97–102, January 2006.     

Combined photoelastic and electrical-analog method for solution of plane-stress plasticity problems

A new application of the electrical-analog method is introduced for the evaluation of the\(\varepsilon _z - principal\) strain, normal to the surface of a thin sheet loaded under conditions of plane stress in plasticity. The analogy relating the two physical phenomena of\(\varepsilon _z - strain\) distribution in a plane-stress plastic field and the potential φ of a plane electrostatic field is based on the assumption that the Cartesian components of strain parallel to the surface of the body vary along the thickness of the strip. They are expressed as a sum of a term independent ofz and a second term, which is a second degree function of thez-coordinate normal to the surface of the body. The boundary conditions of the\(\varepsilon _z - strain\) distribution may be easily determined by a photoelastic method using birefringent coatings cemented on the surface of the metallic specimens. Then, the electrical analogy can be applied for the evaluation of the\(\varepsilon _z - strain\) distribution all over the field. The graphited paper was used as conducting surface in the application of the analogy. The values of\(\varepsilon _z - strain\), together with data obtained by the birefringent coating and concerning the two other Cartesian components of strain, yield an explicit analytic solution of the elastic-plastic plane-stress problem. The method is applied to a plane-stress restricted plasticity problem of a thin slab, with two semicircular grooves in pure tension. The results were compared with those obtained by a photoelastic pointwise solution using normal and oblique incidence. The agreement between these results shows the accuracy of the method.     

具有孔洞的双周期热弹性平面问题的复势

首先论述了针对调和变温场的热弹性平面问题的复变函数方法,给出了受调和变温时有限 多连通域中复势的一般表达式. 然后基于已知实部的双周期解析函数及其原函数的表示和多 值性分析,导出了双周期调和变温下具有双周期分布孔洞物体平面弹性问题的两个复势函数 的一般公式,由此论证了两种特殊情形下物体的温度应力为零.     

On complex potentials for finite plane deformation of a transversely isotropic material

We consider the finite deformation of plane equilibrium problem for a transversely isotropic layer, using the complex variable approach. We give the general expression for the pertinent complex potentials and state the corresponding fundamental problems. We discuss in detail the boundary value problem for fundamental problem-one. As an application of the espoused method, an analytical solution of “Lame's problem” for an infinite layer is obtained. The nonlinear effect of this is highlighted in the obtained figure.     

Hybrid stress analysis by digitized photoelastic data and numerical methods

A method is presented that determines photoelastic isochromatic values at the nodal points of a grid mesh which in turn is generated by a computer program that accepts digitized input. Values of σ₁ - σ₂ are computed from the digitized fringe orders. The Laplace equation is solved to separate the principal stresses at each nodal point. The method is extended to digitize isoclinics. Subsequently, σ _x - σ _y and τ _xy are calculated to be used as starting values for the solution of the pertaining partial differential equations to enhance convergence. For further accelerating the rate of convergence, superfluous boundary conditions are added from the digitized data; significant improvement is demonstrated. Estimated values of σ _x - σ _y from the digitized data are further used in conjunction with the solution of the Laplace equation to determine the state of stress without solving the boundary value problems. Paper was presented at the 1988 SEM Spring Conference on Experimental Mechanics held in Portland, OR on June 5–10.     

Numerical-analytical solution of plane problems in thermoelasticity

The scaled-boundary method is used as a numerical-analytical method to solve problems of thermoelasticity. As an example, the stress intensity factor for a heated thin circular orthotropic disk with an internal crack is evaluated __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 122–126, December 2007.     

Construction of complex photoelastic models using thin molded-epoxy sheets

The construction and analysis of complex thin-shell photoelastic models are discussed. Examples of both two- and three-dimensional applications are presented. The first example concerns an investigation of the structural supports of a new and unique blastfurnace design. The second example is directly related to the design and analysis of complex thin-shell pressure vessels and manifolds. Existing laboratory techniques were used successfully to construct the photoelastic models. The construction method is basically an extension of the techniques previously associated with the contour-sheet method of preparing photoelastic coatings. As such, it does not involve new or untried concepts, but rather it extends the existing in-house capabilities of many laboratories.     

On the extraction of stress-intensity factors from near-tip photoelastic data

Existing algorithms and a new algorithm called ‘quadratic’, for extracting stress-intensity factors (SIF) from frozen-stress photoelastic data are evaluated for application to near-tip data in three-dimensional problems. It is shown that for this class of problems, modifications to some of the existing algorithms may be necessary, in some cases, in order to ensure reliable SIF values. But that for all the algorithms, the two-parameter form of the governing equations of linear-elastic fracture mechanics (LEFM) is sufficient for use with near-tip data, even for complex three-dimensional geometries. Paper was presented at the 1983 SEM Spring Conference on Experimental Mechanics held in Cleveland, OH on May 15–20.     

On the displacement potential solution of plane problems of structural mechanics with mixed boundary conditions

The present paper describes the advancement of displacement potential approach in relation to solution of plane problems of structural mechanics with mixed mode of boundary conditions. Both the conditions of the plane stress and the plane strain are considered for analyzing the displacement and stress fields of the structural problem. Using the finite difference technique based on the present displacement potential approach for the case of the plane stress and the plane strain conditions, firstly an elastic cantilever beam subjected to a pure shear at its tip is solved and these two solutions (plane stress and plane strain) are compared with Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951); secondly the above-mentioned displacement potential approach for the case of the plane stress and the plane strain conditions are applied to solve a one-end fixed square plate subjected to a combined loading at its tip. Effects of plane stress and plane strain on the elastic field of the plate are discussed in a comparative fashion. Limitations of Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951) over the displacement potential approach for the case of the plane stress and the plane strain conditions are not only discussed but also the superiority of the present displacement potential approach for the case of the plane stress and the plane strain conditions are reflected in the present research work.     

THE CALCULATION OF THE MULTIPLY－CONNECTED ELASTIC PLANE PROBLEMS BY MEANS OF STRESS FUNCTIONS OF MULTIPLE COMPLEX VARIABLES

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Determination of stress-intensity factors from photoelastic data with applications to surface-flaw problems

ATaylor-series correction to the maximum inplane shear stress was studied as a means of extending the data zone in photoelastic determination of stress-intensity factors beyond the singular region of a two-degree-of-freedom analysis. Convergence properties were obtained by comparing with several complete two-dimensional solutions. Experiments were performed on two kinds of three-dimensional problems, plates containing surface flows in both bending and extension. Results were analyzed by both a two-degree-of-freedom and aTaylor-series correction method (TSCM). Results were compared to theories of F. W. Smith and A. S. Kobayashi and R. C. Shah. It was concluded that:

1. The TSCM program converges rapidly to accurateK _I values and will accommodate the scatter inherent in experimental data if the series is properly truncated.
2. The TSCM program is essentially equivalent to the two-parameter representation when only the crack-surface effects dominate.
3. When effects other than crack surfaces are important, TSCM requires more terms but still predictsK _I with reasonable accuracy.

     

Superposition of photoelastic data

The fringe order and twice the isoclinic angle define a vector which allows to superimpose photcelastic data originating from identical locations by vectorial addition. The method holds true also for slices of the stress-freezing method. In a check of this method, the curves of pure bending result.     

Concepts of the photoelastic stress gage

With the photoelastic stress gage birefringence readings are made with light that traverses a path parallel to the surface of the workpiece. Individual stresses are determined in the elastic range of deformation, rather than stress or strain differences. The theory of a circular and linear stress gage is developed, including the influence of Poisson's ratio, and stress gradients. Stresses in the surface of the workpiece are expressed in terms of measured birefringence. Instrumentation is extremely simple. High sensitivity is derived from the relatively long optical-path length through the transducer. Applications should include stress analysis, load analysis and transducer design.     

Filtering of the experimental data in plane stress and strain fields

The paper presents a filtering algorithm which corrects the results of measurements of strain and stress fields in order to satisfy the fundamental equations of the continuum. It is proved that the algorithm is very efficient and requires small computational time. The developed filter can be used to correct the measuring data from any experiments provided some additional information about the measured system is available.