often ill-posed. Therefore, various additional techniques should be introduced to the inverse analysis for obtaining stable and accurate results. In the present study, an alternative and simple hybrid method for stress separation in photoelasticity is proposed. Boundary conditions for a local finite element model, that is, tractions along the boundaries are determined by inverse analysis from photoelastic fringes. Two algorithms are presented. One is linear algorithm that the tractions are determined from the principal stress difference and the principal direction using the method of linear least-squares. In another algorithm, on the other hand, the tractions are determined only from the principal stress difference using nonlinear least-squares. After determining the tractions, the stress components are obtained by finite element direct analysis. The effectiveness of the proposed method is validated by analyzing the stresses around a hole in a plate under tension. Results show that the boundary conditions of the local finite element model can be determined from the photoelastic fringes and then the stresses can be obtained by the proposed method. INVERSION OF BOUNDARY CONDITIONS Basic Principle Figure 1 shows a typical optical setup for photoelasticity, that is, a circular polariscope. A birefringent specimen is placed in the polariscope, and then, photoelastic fringes are appeared when the specimen is loaded. The angle of the principal axis of the specimen is interpreted as the principal direction, i.e., the isoclinic parameter. Similarly, the retardation of the specimen, that is, the isochromatic parameter is related to the principal stress difference as [21] 1 2 fs 2h , (1) where fs is the material fringe value, h is the thickness of a specimen, and 1 and 2 express the principal stresses, respectively. Various techniques such as a phase-stepping method can be used for obtaining the isochromatic and isoclinic parameters [22-25]. Therefore, the principal stress difference and the principal direction are obtained in the region of interest or the whole field of the specimen by introducing one of the data acquisition and processing techniques. In a finite element method, on the other hand, it can be considered that the reasonably accurate stress distributions are obtained when the appropriate boundary conditions are given, provided that appropriate finite element model is used and material properties are known. In the proposed method, therefore, the boundary conditions of the region for analysis, that is, the tractions along the boundaries, are inversely determined from photoelastic fringes. Then, the stresses are determined by finite element direct analysis by applying the computed boundary conditions. Figure 2 schematically shows a two-dimensional finite element model of the region for analysis. The displacements of some nodes are fixed so that the rigid body motion is not allowed. Then, a unit force along one of the direction of the coordinate system is applied to a node at the boundary of the model. That is, the finite element analysis is performed under the boundary condition of the unit force on the boundary. The analysis is repeated by changing the direction of the unit force and the node at which the unit force is applied. The stress components at a point (xi, yi) for the applied unit force Pj = 1 (j = 1~N) are represented as (x')ij, (y')ij, and (xy')ij. Here, i (= 1~M) is the data index, j is the index of the applied force, M is the number of the data points, and N is the number of the forces to be determined at the nodes along the boundary of the model. The stress components (x)i, (y)i, and (xy)i at the point (xi, yi) under the actual applied forces Fj (j = 1~N) can be expressed using the principle of superposition as (x )i (x )ij Fj (y )i (y )ij Fj (xy )i (xy )ij Fj (i 1~ M, j 1~ N), (2) where the summation convention is used. That is, for example, 110
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