PHOTOELASTIC STUDY OF A DOUBLE EDGE NOTCHED PLATE FOR DETERMINATION OF THE WILLIAMS SERIES EXPANSION 1

In this work, digital photoelasticity method is applied for assessment of the crack tip linear fracture mechanics parameters for a plate with double edge notches and diﬀerent other crack conﬁgurations. The overarching objective of the study is to obtain the coeﬃcients of the Williams series expansion for the stress and displacement ﬁelds in the vicinity of the crack tip by the digital photoelasticity technique for the double edge notched plate. The digital image processing tool for experimental data obtained from the photoelasticity experiments is developed and utilized. The digital image processing tool is based on the Ramesh approach but allows us to scan the image in any direction and to analyse the image after any number of logical operations. In the digital image processing isochromatic fringe analysis, the optical data contained in the transmission photoelastic isochromatics were converted into text ﬁle and then the points of isochromatic fringes with minimum light intensity were used for evaluating fracture mechanics parameters. The multi-parameter stress ﬁeld approximation is used. The mixed mode fracture parameters, especially stress intensity factors (SIF) are estimated for specimen conﬁgurations like double edge notches and inclined center crack using the proposed algorithm based on the classical over-deterministic method. The eﬀects of higher-order terms in the Williams expansion were analysed for diﬀerent cracked specimens. It is shown that the higher order terms are needed for accurate characterization of the stress ﬁeld in the vicinity of the crack tip. The experimental SIF values estimated using the proposed method are compared with analytical/ﬁnite element analysis (FEA) results, and are found to be in good agreement.


Introduction
Experimental characterization of the crack-tip stress field in isotropic linear elastic materials has been an area of active research for many decades and the problem continues to actual and important at the present 1 The study is supported by the Russian Foundation for Basic Research, project 19-01-00631.
time [1][2][3][4][5][6][7][8][9]. The stress distribution in the immediate vicinity of a crack tip can be determined experimentally using optical methods such as caustic, holography, moire, photoelasticity or digital image correlation method. Amid them photoelasticity has always played an important role in the experimental fracture mechanics. The overarching objective of this study is to obtain the stress field in the vicinity of the crack tip in an isotropic linear elastic material experimentally by the digital photoelasticity method based on the multi-parameter Williams series expansion including the higher-order terms. The use of the multi-parametric representation of the stress field is not just for academic curiosity but a necessity in many cases of engineering interest [3][4][5][6][7][8][9][10][11]. The effects of higher-order terms in the Williams expansion were analysed for different cracked specimens by different authors [3; 6]. Nowadays the multi-point over-deterministic technique for evaluating the multi-parameter stress field is used [10]. The over-deterministic approach can be based on the experimental evaluation of the stress or displacement fields, for instance, on the interference-optical methods of measurements [5], or on the finite element analysis [12]. However, many questions as in digital image processing methods and in the technique of the multi-point over-deterministic method are still open. Thus, the aim of the contribution is to obtain the stress fields near the crack tip reconstructed based on the stress data obtained experimentally via optical measurements and to compare the stress field approximations with the stress field derived from finite element analysis. Experimental data obtained from the photoelasticity method are taken as inputs. Digital photoelasticity is an experimental technique used by many engineering applications to evaluate the stress fields in bodies under mechanical loads. The photoelasticity method is currently undergoing a Renaissance [13][14][15]. After being developed and then largely abandoned in 2000-2010, the method is now in active use. The rebirth of modern photoelasticity led to review of the whole procedure of processing and interpretation of experimental data. Interest in using the digital photoelasticity is now being fueled by possibility of digital processing of the entire set of experimental information. Owing to the increase in computing resources the digital photoelasticity is now one of the powerful tools for investigating the stress field in solids. The appearance and distribution of computers coupled with developments in digital image processing has had a great influence in developments of modern photoelasticity [2; 13].
At present the technique of digital photoelasticity is being developed in many directions. The most important direction is automation of experimental data collection in interference-optical methods of mechanics [2]. Automation is necessary for rapid processing of experimental information. Thus, the vital step in the digital photoelasticity is the extraction of isochromatic and isoclinic data from the fringe pattern seen on the model under stress [14]. In general, the interference fringes appear as broad bands rather than as thin lines. To identify the actual fringe from the broad band and to extract data for further processing, various algorithms have been proposed invoking techniques from the area of digital image processing (DIP) [14]. In DIP, the image is identified as an assembly of picture elements (pixels). The intensity of light transmitted or reflected by each pixel is assigned a number, say between 0 and 255, and the image is transferred into a matrix of numbers. Subsequent manipulations of the matrix using a digital computer can be effectively employed to extract various features of the image. These algorithms are, in general, time consuming and complex [10]. Further, these algorithms fail in zones of high stress concentration. In the fringe band, the minimum intensity positions actually form the fringe contour. Thus, the digital image processing in the photoelasticity method is clearly still subject to ongoing studies.
The second direction of development of the digital photoelasticity method is diverse applications of the technique photoelasticity [2; 14], such that hydraulic fracture propagation in rock materials [11], dentistry and other applications in biomechanics [16] and integrated use of experimental, manufacturing and numerical methods, for instance, rapid 3D prototyping and photoelasticity [15]. Thus, in [2] the succinct review proposes readers to extrapolate the photoelasticity technique to tackle newer problems in the uncharted territory and domains. Finally, the third area of research in the field of photoelasticity is aimed at building the multi-parameter Williams series expansion for the stress field in the vicinity of the crack tip in a linear elastic isotropic material [5; 8; 17].
The higher order terms of stress field influence significantly the stress distribution in the vicinity of the notch tip and crack tip and consequently can play an important role in brittle fracture. Therefore, in the brittle fracture assessment of interface notches, it is important that to take into account not only the singular stresses but also the higher order terms. However, it is imperative to develop and improve algorithms of accurate photoelastic data extraction from interference fringe patterns obtained in experiments. Some questions remain a challenge even today. In this paper for a better understanding of the multi -parameter approximation of the stress field in the vicinity of the tip of cracks and notches in linear elastic materials the experimental technique of photoelasticity has been utilized for calculating the coefficients of higher order terms of the multi-parameter stress field. In the present study a new algorithm is presented which utilizes the minimum intensity criterion to identify the fringe skeletons. Then, the experimental photoelasticity results were compared with the corresponding values obtained from finite element analysis and a good correlation was observed.

The Williams series expansion of the stress and displacement fields in neighborhood of the crack tip
The present study is aimed at determination of the higher order coefficients in Williams' series expansion in the classical specimen for linear elastic fracture mechanics -a plate with double edge notches using digital photoelasticity method. Williams proposed an infinite series expansions of the elastic stress fields around the crack tip, namely the Williams series expansion based on stress analysis foe a plane crack problem in the form [18]: where r and θ are the local polar coordinate system, f m,ij (θ) are the angular functions; a m k are the coefficients of the Williams series expansion corresponding to the mode I and mode II cracks, respectively, the constant m corresponds to loading mode.The angular functions related to the polar coordinate θare described as The displacement field near the crack tip can be defined as an infinite series. For Mode I, Mode II and Mixed Mode loading the displacement fields are expressed as shown where G is the shear modulus, g m,i (θ) are the functions describing the circumferential behavior of the displacements where κ is the constant of the complex variable theory used in linear plane elastostatics κ = (3 − ν)/(1 + ν) for plane stress conditions, ν is the Poisson's coefficient. The coefficients a 1 k and a 2 k are the unknown mode I and mode II parameters. The stress intensity factors can be determined as Williams demonstrated that the stress field around the crack tip in an isotropic elastic material can be expressed as an infinite series. They were called Williams' series expansion and they are now considered as the most favoured analytical tool for the description of mechanical fields near crack-tips in planar domains [19]. For practical use, these series are generally truncated taking into account a number of terms. Whereas there is a common belief to consider that more terms provide more accurate results, this fact has not been totally demonstrated. These coefficients are widely available for the first two terms. The first term corresponds to the stress intensity factors whereas the second term is related to the T-stress value. However, little information is found in the literature about higher-order terms. Thus, the objective of this work is to obtain the higher-order coefficients in the Williams series expansion for the plate with two double notches.
One can observe the stress distribution given by the theoretical analytical solutions based of the complex variable theory in plane elasticity and the stress distribution obtained by the Williams series expansion (1.1). It should be noted that the coefficients of the Williams series expansion for an infinite plate with the central crack are known [18]. The coefficients of the Williams series expansion are found by the classical approach based on the complex variable theory of plane elasticity: for Mode I a 2 k = 0 One can compare the circumferential distribution of the stress tensor components given by the theoretical solution and the multi-point series expansion. The exact solution is shown by blue points in fig. 1.1-1.3. The colored lines show the angular distributions of the stress component σ 11 (r, θ) given by the various number of the terms in the Williams series expansion at different distances from the crack tip. One can see that the greater the distance from the tip of the crack, the more terms of the Williams series expansion we need to hold in the asymptotic expansion. When we need to describe the stress field in the vicinity of the crack tip with use the isochromatic fringe pattern we don't know apriori how many terms it is necessary to keep. Thus, it is imperative to analyse the distance from the crack tip at which the isochromatic fringe is located. Our considerations show that for the isochromatic fringes of fourth and fifth orders it is indispensable to retain fifteen terms in the Williams series expansion. In view of this exploration we will keep the first fifteen terms in the asymptotic solution and we will find them from the photoelastic experiments and from the finite element analysis and then compare the obtain results.

Photoelastic experiments: experimental setup and procedure
The experimental setup used is shown in fig. 2.1. All the specimens in this work were made by casting of polycarbonate. Experimental specimens were machined from the sheet to get the test specimens. Material properties of the photoelastic material are Young's modulus E = 3GP a, Poisson's ration ν = 0.3 and the material fringe constant is found to be f σ = 10.41P am/f ringe.   [2]. The skeleton of the fringe is identified first to accurately collect the experimental data from the fringes. The programme is specially developed for the interpretation and processing of experimental data from the photoelasticity measurement experiments. The developed and approved tool allows us to find points that belong to isochromatic fringes with minimal light intensity. The analysis of the experimental data uses a Java application programmed for the advanced determination of the fracture mechanics characteristics: coefficients of the Williams series expansion (WE) for the stress field in the vicinity of the crack tip. The programming tool allows us to collect experimental points from the photoelasticity tests on the cracked specimens. The skeleton of isochromatic fringes is shown in figs. 1.3, 2.1. The skeleton is shown by green points. Finally, the chosen points are shown in fig. 2.2. It should be noted that, as it is revealed in the process of the experimental data processing, in order to produce theoretically the isochromatic fringe shown in fig. 2.2 it is necessary to keep twenty terms in the Williams series expansion (1.1).

Evaluation of experimental isochromatic data.
Over-deterministic method The stress optic law establishes that the isochromatic fringe order N is proportional to the differences between the principal stresses at a given point [20; 21] where f σ is the material stress fringe value and h is the model thickness. Note that, the fringe order can be determined at any point from photoelastic data. For methodological purposes we will expound the full procedure used for the determination of coefficients of the Williams series expansion. For plane stress and strain problems the principal stresses are Substituting (3.2) into (3.1) a function g m can be introduced for the mth data point as follows [20]: Initial estimates are made for these unknown parameters a m k and possibly the error will not be zero since the estimates are not accurate. The estimates are corrected using an iterative process based on Taylor series expansion of g m . Then, in accordance with the now-classic procedure the system of nonlinear equations (3.3) with respect to the coefficients a m k the function g m is represented by Taylor's series expansion.Thus, one can obtain the error function can be expressed for ith iteration as where the index i refers to the ith iteration step and ∆a m k denote the corrections to the previous estimates of a m j [20]. Corrections are obtained from the requirements (g m ) i+1 = 0 and eq. Then, this iteration scheme results in an over determined set of linear equations in terms of the unknown corrections ∆a 1 1 , ∆a 1 2 , ... ∆a 1 K and ∆a 2 1 , ∆a 2 2 , ... ∆a 2 L . [20] which can be presented in the matrix form One can arrive at the solution of the incremental change by solving a simple matrix problem. The results are given in Table 1

Finite element solution
This part of the paper aims at determining the stress intensity factor, T-stress and coefficients of higher order terms in the Williams series expansion for a double edge notched specimen with finite element analysis. The computer simulation of the double edge notched plate was performed. In this work, 2D finite element analysis (FEA) of cracked specimens is carried out using SIMULIA Abaqus software to estimate stress intensity factor, T-stress and higher order coefficients for Mode I mode loading and Mixed Mode loading problems. The finite element analysis is done with 8-noded plane strain elements. The quarter point element is used to capture square root singularity at the crack tip. The double edge notched crack model is of dimension 100 mm ×50 mm having two edge notches of 8 mm length. The mesh pattern around the crack tip is kept very fine to capture the high-stress gradient. The mesh convergence is achieved with 72 elements along circumferential and 50 along the radial direction. In total, there are 10742 elements corresponding to 14346 degrees of freedom. The results of the computational analysis are shown in figs. 4.1, 4.2.
During the computational experiment, the concentric circles covering the crack tip were selected. For each contour, seventy-three values of each component of the stress tensor in the plane problem are known. So each contour gives 219 values. The total number of contours varied from four to ten. Thus, the total number of equations in the over-deterministic system varied from 876 to 2190. In the Williams decomposition, the first fifteen terms were retained. Then the stress components are introduced in the procedure of the over-deterministic method as the matrix σ p . Thus, the Williams series expansion can be represented in the matrix form where C is a rectangular matrix of order 3mP × 2n, P is the number of contours surrounding the crack tip, m is the number of experimental points, n is the number of the terms retained in the Williams serries expansion, A is the vector consisting of unknown mode I and mode II fracture parameters. The values of A are estimated by minimizing the objective function ( [6]) The objective function J is of quadratic form for stress expressions in terms of unknown parameters. The closed form solution exists for the parameters A.
Having obtained the numerical solution it is possible to compare the numerical results with the experimental solution. As a comparison, the first fifteen coefficients of the Williams series expansion were likened. The comparison shows the full coincidence the first seven coefficients whereas the larger the coefficient sequence number, the more the coefficients begin to differ from each other, up to ten percent. Considering all experimental and computational results it is reasonably concluded that the chosen methodology is efficient and feasible. A reasonable agreement between the over-deterministic approach that has been used for the experimental and finite element results was reached. The closed form solution for the unknown vector of parameters A where the onjective function has a global minimum is as follows [6]:

Conclusions
The paper delves our knowledge and our considerations of the digital photoelasticity technique and its applications to fracture mechanics, namely, to the determination of higher order terms in the Williams series expansion. In this study higher order coefficients of the multi-parameter Williams series expansion for the stress and displacement fields in the vicinity of the tip of the edge notch in the rectangular plate are obtained by the use of the digital photoelasticity method and finite element analysis. The higher-order terms in the Williams asymptotic expansion are kept. It allows us to have more accurate estimation of stress, strain and displacement fields and to extend the domain of validity for the Williams series expansion. The program is specially developed for the interpretation and processing of experimental data from the photoelasticity measurement experiments. The developed tool allows us to find points that belong to isochromatic fringes with minimal light intensity. The analysis of the experimental data uses a Java application programmed for the advanced determination of the fracture characteristics: coefficients of the Williams series expansion (WE) for the stress field in the vicinity of the crack tip. The tool allows us to collect experimental points from the photoelasticity tests on the cracked specimens. An automatic routine implemented as a Java application permits to determine the values of coefficients of higher order terms of the WE that describe crack-tip fields. These values are calculated using the over-deterministic method which is also applied to the results of the finite element analysis of some mode I and mixed mode test geometry. Hence, the Java application provides an analytical reconstruction of the crack-tip stress field through the truncated WE and enables detailed analysis of the crack-tip stress field approximation. The developed procedures simplify the analysis of the description of mechanical fields at a greater distance from the crack tip considerably. The digital image processing with the aid of the developed tool is performed. The points determined with the adopted tool are used further for the calculations of stress intensity factor, T-stresses and coefficients of higher-order terms in the Williams series expansion.