# Photoelastic study of a double edge notched plate for determination of the Williams series expansion

**Authors:**Stepanova L.V.^{1}, Aldebeneva K.N.^{1}-
**Affiliations:**- Samara National Research University

**Issue:**Vol 26, No 4 (2020)**Pages:**56-67**Section:**Articles**URL:**https://journals.ssau.ru/est/article/view/9199**DOI:**https://doi.org/10.18287/2541-7525-2020-26-4-56-67**ID:**9199

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## Full Text

## Abstract

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.

## Full Text

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

1The study is supported by the Russian Foundation for Basic Research, project 19-01-00631.

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time [1–9]. 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 [3–11]. 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 [13–15]. 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.

Stepanova L.V., Aldebeneva K.N. Photoelastic study of a double edge notched plate for determination...

58Степанова Л.В., Альдебенева К.Н. Исследование пластины с двумя боковыми надрезами с помощью...

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]:

2

σij (r, θ) = ∑

∞

∑ amf (k) (θ)rk/2−1, (1.1)

k

m=1 k=−∞

m,ij

where r and θ are the local polar coordinate system, f (k) (θ) are the angular functions; am are the coeﬃcients

m,ij k

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

f (k)

1,11(θ) = k [(2 + k/2 + (−1)k ) cos(k/2 − 1)θ − (k/2 − 1) cos(k/2 − 3)θ] /2,

f (k)

1,22(θ) = k [(2 − k/2 − (−1)k ) cos(k/2 − 1)θ + (k/2 − 1) cos(k/2 − 3)θ] /2,

f (k)

1,12(θ) = k [− (k/2 + (−1)k ) sin(k/2 − 1)θ + (k/2 − 1) sin(k/2 − 3)θ] /2,

f (k)

2,11(θ) = −k [(2 + k/2 − (−1)k ) sin(k/2 − 1)θ − (k/2 − 1) sin(k/2 − 3)θ] /2,

f (k)

2,22(θ) = −k [(2 − k/2 + (−1)k ) sin(k/2 − 1)θ + (k/2 − 1) sin(k/2 − 3)θ] /2,

f (k)

2,12(θ) = k [− (k/2 − (−1)k ) cos(k/2 − 1)θ + (k/2 − 1) cos(k/2 − 3)θ] /2.

(1.2)

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

m=2 k=∞ 1

∑

ui(r, θ) = ∑

amrk/2g(k) (θ), (1.3)

G k

m=1 k=1

m,i

m,i

where G is the shear modulus, g(k) (θ) are the functions describing the circumferential behavior of the

displacements

g(k)

g(k)

1,1 (θ) = (κ + k/2 + (−1)k )cos(θ/2) − (k/2)cos(k/2 − 2)θ,

g(k)

g(k)

1,2 (θ) = (κ − k/2 − (−1)k )sin(θ/2) + (k/2)sin(k/2 − 2)θ, 2,1 (θ) = −(κ + k/2 − (−1)k )sin(θ/2) + (k/2)sin(k/2 − 2)θ, 2,2 (θ) = (κ − k/2 + (−1)k )cos(θ/2) + (k/2)cos(k/2 − 2)θ,

(1.4)

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 a1

and a2

__a__re the unknown mo__d__e Ik k √ √

1

1

and mode II parameters. The stress intensity factors can be determined as KI = a1

2π and KII = −a2

2π,

T-stress is expressed as

2

T = −4a2.

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

a1 (−1)

n+1

σ

22

(2n)! ∞

a1

−

2n+1 = 23n+1/2(n!)2(2n

1) an−1/2 , n ) 0,

(1.5)

2 = σ∞

k

(α − 1)/4, a1 = 0,

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for Mode II

n

a2 (−1)

σ

12

(2n)! ∞

2n+1 = 23n+1/2(n!)2(2n

a2

k = 0

−

1) an−1/2 , n ) 0, (1.6)

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 σ11(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.

/

o σ

11 22

0.8

r^=r/a=0.75

/

o σ

11 22

0.2

r^=r/a=0.75

N=9,11,13,15,

0.6

0.4

0.2

0

N=1

N=3

N=1

N=7

0

-0.2

exact solution

50,75,100

N=7

-0.2

-0.4

N=5 N=5

exact solution

N=3

N=9,11,13,15,

50,75,100

-0.4

-0.6

-0.6

-0.8

N=2 N=2

-0.8

-1 -3

-2 -1 0 1

2 3 θ

-1 -3

-2 -1 0 1

2 3 θ

Fig. 1.1. Circumferential distributions of σ11(r, θ) at diﬀerent distances from the crack tip

o /σ

σ

r^=r/a=1.25 σ /

11 22

0.6

N=1

1

1 22

0.2

r^=r/a=1.25

N=20

0.4

0.2

0

N=7

N=5

N=3

N=3

N=1 N=5

N=7

N=9

N=11 N=13

0

-0.2

exact solution

N=50,75,100

-0.2

exact solution

-0.4

-0.4

-0.6

-0.6

-0.8

N=2

N=2

-0.8

-1 -3 -2 -1

0 1 2 3 θ

-1 -3

-2 -1 0 1

2 3 θ

Fig. 1.2. Circumferential distributions of σ11(r, θ) at diﬀerent distances from the crack tip

/

o σ

11 22

r^=r/a=1.75

N=9

/

o σ

11 22

r^=r/a=1.75

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

N=5

N=1

exact solution

N=3

N=5

N=2

N=7

N=50

N=11 N=13 N=15

0.2

0

-0.2

-0.4

-0.6

-0.8

N=75

N=100

exact solution

N=50

-1 -3 -2 -1

0 1 2 3 θ

-1 -3 -2 -1 0

1 2 3 θ

Fig. 1.3. Circumferential distributions of σ11(r, θ) at diﬀerent distances from the crack tip

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60Степанова Л.В., Альдебенева К.Н. Исследование пластины с двумя боковыми надрезами с помощью...

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 = 3GPa, Poisson’s ration ν = 0.3 and the material fringe constant is found to be fσ = 10.41Pam/fringe.

Fig. 2.1. Photograph of the experimental apparatus used to visualize and capturing the fringe patterns

Fig. 2.2. Photoelastic fringe patterns for the plate with double edge notches at 70 kg, 75 kg and 100 kg

Isochromatic phase maps obtained for the plate with double edge crack notches under diﬀerent loads are shown in ﬁgs. 2.2, 2.3. One can see from ﬁg. 1.2 that it is not possible to provide an accurate description of the stress ﬁeld in the vicinity of the crack tip using the one-term asymptotic series expansion. Fringe tracking and fringe order assignment have become the central topic of current research in digital photoelasticity [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

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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).

Fig. 2.3. Photoelastic fringe patterns for the plate with double edge notches at 125 kg and 150 kg The results of the digital image processing are shown in ﬁgs. 2.4 and 2.5. The points with the minimum

light intensity are shown in green.

Fig. 2.4. Digital images obtained from photoelastic measurement experiments: isochromatic images in the double edge cracked specimen subject to 75 kg and 100 kg,

Fig. 2.5. Digital images obtained from photoelastic measurement experiments: isochromatic images in the double edge cracked specimen subject to 125 kg and 150 kg

The developed tool allows us to collect the points with the minimum light intensity. The points chosen for the analysis are shown in ﬁg. 2.6.

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62Степанова Л.В., Альдебенева К.Н. Исследование пластины с двумя боковыми надрезами с помощью...

Fig. 2.6. Digital images obtained from photoelastic measurement experiments: isochromatic images in the double edge notched plate

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]

Nfσ/h = σ1 − σ2, (3.1)

where fσ 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

σ1 + σ2 1 √ 2 2

σ1, σ2 =

2 ± 2

(σ11 − σ22)

+ 4σ12. (3.2)

Substituting (3.2) into (3.1) a function gm can be introduced for the mth data point as follows [20]:

2

gm = (σ11 − σ22)

12

+ 4σ2

2

— (Nmfσ/h)

(3.3)

k

Initial estimates are made for these unknown parameters am

and possibly the error will not be zero since

k

the estimates are not accurate. The estimates are corrected using an iterative process based on Taylor series expansion of gm. Then, in accordance with the now-classic procedure the system of nonlinear equations (3.3) with respect to the coeﬃcients am the function gm is represented by Taylor’s series expansion.Thus, one can obtain the error function can be expressed for ith iteration as

gm

a1 +

1

∂gm

a1 + ...

2

∂gm

a1

K

1

∂gm

2 K

∆a2 + ∂gm ∆a2 + ... + ∂gm ∆a

∂a2

1

1

∂a2

2

2

∂a2

M

∂

(gm)i+1 = (gm)i + ∂a1 ∆

+

∂a1 ∆

+ ∆

∂a1

+ (3.4)

M

1 , (3.5)

k

where the index i refers to the ith iteration step and ∆am denote the corrections to the previous estimates

j

of am

[20]. Corrections are obtained from the requirements (gm)i+1 = 0 and eq. (3.5) gives

∂gm

1 ∂gm 1

∂gm 1

−(gm)i =

∂a1 ∆a1 + ∂a1 ∆a2 + ... + ∂a1 ∆aK + (3.6)

1

∂gm

1

+ ∂a2

1

∆a2 +

2

∂gm

2

∂a2

2

∆a2 + ... +

K

∂gm

M

∂a2

L

∆a1 , (3.7)

Then, this iteration scheme results in an over determined set of linear equations in terms of the unknown

corrections ∆a1, ∆a1, ... ∆a1

and ∆a2, ∆a2, ... ∆a2 .[20] which can be presented in the matrix form

1 2 K

1 2 L

Gi = −Bi(∆A)i, (3.8)

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where

g1

g2

Gi = g3

...

gm

, (3.9)

∂g1

1

∂a1

∂g1

2

∂a1

...

∂g1

K

∂a1

∂g1

1

∂a2

∂g1

2

∂a2

...

∂g1

L

∂a2

Bi =

∂g2

1

∂a1

∂g2

1

∂a2

∂g2

... 1

∂aK

∂g2

2

∂a1

∂g2

2

∂a2

∂g2

... 2

∂aL

, (3.10)

.

..

...

...

...

...

.

..

∂g1

1

∂a1

∂gM

2

∂a1

∂gM

...

K

∂a1

∂gM

1

∂a2

M

∂gM

2

∂a2

∂g

...

L

∂a2

∆a1

1

∆a1

2

...

...

K

(∆A)i = ∆a1

∆a2

. (3.11)

1

∆a2

2

...

L

∆a2

One can arrive at the solution of the incremental change by solving a simple matrix problem. The results are given in Table 1. Having obtained the coeﬃcients of the Williams series expansion for the stress and displacement ﬁelds experimentally one can compare the results with the numerical ones. For comparison a series of ﬁnite element calculations for the same type of the cracked specimen has been performed. The results of ﬁnite element simulations are shown in ﬁg. 2.3. The veriﬁcation has proved the experimental results. It is well-known that the multi-purpose program Simulia Abaqus allows us to ﬁnd SIFs and T-stress directly. The experimental and numerical results coincide.

a1 1/2

a1

1 = 45.498985Pam

2 = −8.494524Pa

a1 −1/2

3 = 8.675513Pam

a1 −1

4 = −3.546581Pam

a1 −3/2

5 = 1.185281Pam

a1 −2

6 = 1.660511Pam

a1

−5/2

7 = −1.432483Pam

a1 −3

8 = −0.628455Pam

a1 −7/2

9 = 1.051328Pam

a1 −4

10 = −0.017221Pam

a1 −9/2

11 = −0.639445Pam

a1 −5

12 = −0.015332Pam

a1 −11/2

13 = 0.423673Pam

a1 −6

14 = −0.007816Pam

a1 −13/2

15 = −0.413682Pam

Table 1. Coeﬃcients of the Williams series expansion

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

64Степанова Л.В., Альдебенева К.Н. Исследование пластины с двумя боковыми надрезами с помощью...

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 σp. Thus, the Williams series expansion can be represented in the matrix form

σp = CA, (4.1)

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])

J (A) =

T

[σp − CA]

[σp − CA] . (4.2)

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.

Fig. 4.1. Finite element solution for the plate with double edge notches

Fig. 4.2. Finite element solution for the plate with double edge notches

Вестник Самарского университета. Естественнонаучная серия. 2020. Том 26, № 4. С. 56–67

Vestnik of Samara University. Natural Science Series. 2020, vol. 26, no. 4, pp. 56–67 65

The closed form solution for the unknown vector of parameters A where the onjective function has a global minimum is as follows [6]:

p

A = (CT C)−1 CT σ , (4.3)

where = (CT C)−1 CT is the pseudo-inverse of C.

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.

## About the authors

### L. V. Stepanova

Samara National Research University
**Author for correspondence.**

Email: stepanovalv@samsu.ru

ORCID iD: 0000-0002-6693-3132

Doctor of Physical and Mathematical Sciences, professor, Department of Mathematical Modelling in Mechanics

Russian Federation### K. N. Aldebeneva

Samara National Research University
Email: aldebeneva.kn@ssau.ru

ORCID iD: 0000-0002-0169-3561

postgraduate student of the Department of Mathematical Modeling in Mechanics

Russian Federation## References

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