evaluation of stress field parameters in fracture mechanics by photoelasticity—revisited

$: Evaluation of stress field parameters in fracture mechanics by photoelasticity—Revisited$
Pergamon Engineering Fracture Mechanics Vol. 56, No. 1, pp. 25-45, 1997 Copyright © 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved P l h S0013-7944(96)00098-7 o013-7944/97 $17.oo + o.0o

E V A L U A T I O N O F S T R E S S F I E L D P A R A M E T E R S IN

F R A C T U R E M E C H A N I C S BY P H O T O E L A S T I C I T Y - -

R E V I S I T E D

K. RAMESH,t S. GUPTA and A. A. KELKAR

Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, India 208 016

Abstract--Multi-parameter stress field equations have been reported in the past two decades to model the elastic stress field in the neighbourhood of a crack lying in finite bodies. Notably among them are the generalised Westergaard equations proposed by Sanford, Williams' eigen function expansion and the recent one of Atluri and Kobayashi. The equivalence among these equations is brought out. Using the equations of Atluri and Kobayashi an over-deterministic least squares technique is proposed to evaluate the mixed-mode stress field parameters by the technique of photoelasticity. It is shown that the use of Atluri and Kobayashi's equations leads to a very simple software where one can independently increase the number of terms in Mode I or Mode II series depending on the specific problem requirement until the experimental fringes are correctly modelled. The software is validated by solving three example problems. The example problems emphasise that the use of multi-parameter stress field equations is not just an academic curiosity but a practical necessity to apply concepts of Fracture Mechanics to solve real life problems. Further, it is shown that the multi-parameter solution allows the collection of data from a larger zone which helps to simplify the data collection from experiments. Copyright (~ 1996 Elsevier Science Ltd

1. INTRODUCTION

THE CRACKS in most of the practical structures of engineering importance may be located or developed in-service in zones of stress concentration and the cracks may be large enough so that the crack-tip may be closer to a boundary. The basic stress field equations of fracture mechanics are developed for cracks lying in a body of infinite dimensions (i.e. the crack-tips are far away from the boundaries) subjected to a uniform stress field. When only the first term (singular solution) of these equations is attempted to be applied for solving real life problems, one gets erro- neous results. The recognition of this fact has lead the researchers to account for this effect and multi-parameter stress field equations have been developed in the past two decades. The role of photoelasticity in the development of multi-parameter stress field equations is brought out in ref.[1].

Sanford[2] in 1979 brought out the inadequacy of modified Westergaard equations and proposed an additional stress function Y(z) in a series form and he called the new set of equations the Generalised Westergaard equations. Williams' [3] eigen function solution in a series form with upto six terms in Mode-I was used by Kalthoff [4] while investigating the higher order effects on the Caustic shadow. Recently, Atluri and Kobayashi [5] have reported another set of stress field equations in a series form. The original reference does not state the methodology of obtaining Atluri and Kobayashi's equations. In this paper, the equivalence between these multi- parameter stress field equations is brought out.

Among the several methods for evaluating the stress field parameters from the isochromatic fringes, the over-deterministic method of Sanford and Dally[6] has come to stay as the most useful technique. They reported a three-parameter solution involving KI, KII and aox for the mixed- mode case. Sanford[7] extended the methodology to evaluate the stress field parameters for long cracks in Mode-I loading. He used generalised Westergaard equations to analyse the stress field and reported that upto six terms in the series solution may be required. With the rapid developments in applying Digital Image Processing (DIP) to process data from the fringe patterns, the drudgery of data collection is removed[8-10]. Further these techniques have also helped to fully utilise the whole field data effectively from an experimental technique such as photoelasticity.

tAuthor to whom all correspondence should be addressed.

25

26 K. RAMESH et al.

In this paper, following the approach of Sanford and Dally, using the stress field equations of Atluri and Kobayashi, an over-deterministic least squares approach is proposed for evaluating the mixed-mode stress field parameters in fracture mechanics. A priori one does not know exactly how many terms in the series solution are necessary to model the stress field. It is shown in this paper that the use of Atluri and Kobayashi's equations leads to a very simple software where one can independently increase the number of terms in Mode I or Mode II series depending on the specific problem requirement until the experimental fringes are correctly modelled. The software is validated by solving three example problems of which two deal with Mode I situation and one deals with the combination of Mode I and Mode II. The required data for all these problems is collected using DIP techniques. The fringe patterns are reconstructed to show that in practical situations by taking a sufficient number of terms, the experimental fringe patterns are better modelled and also that the multi-parameter solution allows the collection of data from a larger zone which helps to simplify the data collection from experiments. For the sake of completeness, the developments in stress field equations starting from the singular solution and the developments in SIF evaluation from the two parameter solution are briefly sum- marised.

2. DEVELOPMENTS IN THE DESCRIPTION OF THE STRESS FIELD EQUATIONS IN THE N E I G H B O U R H O O D OF A CRACK-TIP

2.1. Mode-I stress field equations

In the early developments of fracture mechanics, more emphasis was given to Mode-I loading and various stress field equations have been reported. It is instructive to survey this development to understand the importance of including non-singular stress terms for practical applications.

2.1.1. Single parameter stress field equation. The Westergaard[11] complex stress function technique for solving opening mode crack problems has played an important role in the development of linear elastic fracture mechanics. Westergaard in his paper gave the stress functions for the case of a central crack and a series of equally spaced straight cracks of length 2a in an infinite plate with a biaxial field of tension a. Westergaard originally reported that his stress function models the stress field for the case of a crack/cracks in a uniaxial stress field. However, it has been shown by later investigators that Westergaard's solution is valid only for a biaxial field. Irwin[12] in 1957, following the semi-inverse procedure suggested by Westergaard, added three additional examples. They are

(1) Single crack along the x-axis extending from - a to a with a wedge action applied to produce a pair of "splitting forces" of magnitude P located at x = b.

(2) The situation of the above example with an additional pair of forces of magnitudes P at x = - b .

(3) Example (1) repeated along the x-axis at intervals l and with the wedge action centred so that b is zero.

Irwin[12] showed that in all the above five cases the stress distribution near the end of the crack can be expressed independently of the type of loading if the region of interest is very close to the crack-tip [i.e. r/a and r / ( a - b) may be neglected in comparison to unity] and introduced the concept of Stress Intensity Factor (SIF). The expressions thus obtained are known as classical Westergaard equations and for a Mode I loading situation they are given as,

0 . 30 1 - sin ~ sm-~

/ . x / , 0 03o ay m = C O S ~ 1 +sin~sin-~- (1) rxy ~/2zrr

0 3O sin ~cos-~-

Evaluation of stress field parameters 27

2.1.2. Two parameter stress field equation. Irwin [12] observed that when an extending crack moves across a plate of finite width, the crack may attain sufficient length so that the stress field equations obtained from the consideration of a crack in an infinite body may not be accurate. He reported that a correction needs to be applied for a finite body and introduced a constant stress term aox to the trx term. Thus, the modified Westergaard equations become,

0 . 30 1 - sin ~ sin -~-

0 3 0 try = = c o s ~ l + s i n ~ s i n ~ - + 0 . (2) rxy ~/2zrr 0

0 3O s in~cos-~

In a discussion of the paper by Wells and Post[13], Irwin[14] in 1958 reported a method to evaluate KI as well as aox stress term from the isochromatic fringe patterns. Using these values he theoretically plotted the contours of maximum shear stress (isochromatic fringes) and showed that the prominent tilt seen in the experiment is well represented.

On the other hand if one uses classical Westergaard's equations one gets the fringes symmetric with respect to both the x and y axis. This situation occurs in the experiment in a region very close to the crack-tip. In a sense the use of the second term has helped in enlarging the zone of data collection. It is usually reported in the literature that Irwin introduced the aox stress term to explain the tilt of the isochromatics. However, it is to be understood that Irwin realised the need for a correction term to the ax stress term much before and showed a practical method of evaluating KI as well as the ~ox stress term while discussing the work of Wells and Post.

Though Irwin introduced the aox term intuitively, analytical justifications for this additional term in the near-field equations of fracture mechanics came much later. Sih [15] in 1966 starting from the Goursa t -Kolosov complex representation of the plane problem showed that the symmetry condition zxy = 0 on y = 0 could be satisfied by a less restrictive assumption than employed by Westergaard. He obtained near field stress equations which included a constant term to the ~x term. Later Eftis and Liebowitz [16] in 1972 showed that Westergaard's stress function can be related to the Goursat -Kolosov complex representation. They confirmed that Sih's equations are the same as Irwin's modification to the Westergaard equations.

Eftis et al. [17] in 1977 showed that Williams' [3] eigen function approach includes a constant stress term to the trx stress term if two terms are considered in its series. Following the approach of Paris and Sih [18] they introduced complex SIF and using Muskhelishvili's stress function, they could show that the aox stress term can be expressed in terms of the biaxial stress at infinity, i.e.

trox = (1 - k)tr, (3)

where k is the biaxiality ratio. This brought out the fact that the Westergaard stress function is valid only for the case of k = 1. For other cases a constant term is necessary. The importance of Eftis et al.'s work is that they brought out the influence of the aox stress term in evaluating the displacement, local strain energy and local strain energy rate.

2.1.3. Multi-parameter stress field equation. Sanford in 197912] observed for the first time that when the crack length increases in a Modified Compact Tension (MCT) specimen, several isochromatics cross the crack-axis. Sanford reported that this phenomenon is due to the finite dimensions of the specimen. In order to account for finite bodies, Tada et al.[19] introduced a series form of the Westergaard stress function. Sanford observed that the modified Wester- gaard functions predict only a constant fringe order along the crack axis. However, for cracks approaching a boundary, several fringe orders cross the crack axis. This counter example raises questions about the validity of the modified Westergaard equations.

Starting from this standpoint, Sanford reported that an additional stress function Y(z) should be added to the series form of the Westergaard stress function Z(z) where z is x + iy and i is v/L-1. He showed that the condition Zxy = 0 on y = 0 can be achieved by a less restrictive assumption than used by Westergaard and selected the additional stress func-

28 K. RAMESH et al.

tion Y(z) such that it satisfies the condition Im[Y(z)] = 0 on y = 0. He expressed the stress function Y(z) in a series form. With this additional stress function, Sanford [7] showed that crack-tip stress fields can be better characterised. In all the papers reported by Sanford and his co-workers the stress field is represented only in terms of complex stress functions and they are

ay = rxy

ReZ - yImZ' - y l m Y ' + 2ReY I ReZ + yImZ' + yIm Y' J - y R e Z ' - yRe Y' - Im Y

(4)

where Z' and Y' are the first derivatives of Z and Y, respectively, with respect to z. For a single ended crack, with the origin of coordinates at the crack tip and the negative x-

axis coinciding with the crack faces, the functions Z(z) and Y(z) can be expressed as,

Z(z) = y~czjzJ-½ (5) j=0

J Y(z) = ZC2j+1zJ,

j=0 (6)

where C2j and C2j + 1 are real coefficients. One can obtain stress field equations in terms of r, 0 by substituting eqs (5,. 6) into eq. (4)

and expressing z in terms of r and 0. However, the resulting equations are lengthy and can be found in ref. [20].

Equation (4) is known as the Generalised Westergaard Equation. The classical Westergaard equations are obtained by making Y(z) zero and Irwin's modified equations are obtained if the function Y(z) is taken to be a real constant.

2.2. Mixed mode (combination of Mode-I and Mode-II) stress field equations

The stress field equation of Williams (in polar co-ordinates) upto six terms each in Mode-I and Mode-II is given by[20],

{or] cr 0 = FrO

r-~Cll cos ~ - - ~ cos + 4Cl2(COS20)+ r23c13 cos ~ + ~ cos

+r2cl4(COS 0 -k- 3 cos 30) + r~cl5 COS ~- + -~- cos ~ -k- r2c16(12 Cos 40)

r-:~Cll COS ~ + ~ COS +4c12 sin20 + r:-~-cl3 ~cos ~ - - ~ cos

+6rcl4(cos 0 -- cos 30) + -~-r~cl5 ~cos -~- -- ff cos + 12r2cl6(COS 20 -- cos 40)

r-~ ~Cll sin ~ + ~ sin -- (2c12 sin 20) + r½ ~c13 ~ sin ~ - - ~ sin -~-

+2rc14(sin 0 -- 3 sin 30) + r ~ cl5 sin ~- -- ~ sin + 3r2c16(2 sin 20 -- 4 sin 40)


+

-r-½c21 sin ~ - ~ sin + r~3c23 ~ sin ~ + ~ sin -~-

+r2c24 (sin 0 + sin 30) + r~c25 sin ~- + ~- sin + r2c26(6 sin 40)

[ 13 (sin 0 3---~) ~-15c23(sin 0 " 50 -r-~ ~ czl ~ + sin + r2 -~- ~- -sm )

1 30) 35 ~ ( s i n ~ - 70 1 sin40) } +6rcz4(sinO-~ sin + ~-r~c25 sin -~- )+ 12r2c26(sin20--~

{ (1 0 3 30) 3 ( 1 0 5 ~ ) r-½~c21 ~ COS ~ + ~ Cos -~- - - r 1 2 c23 5 COS 5 - - 5 COS

-2rc24(cos O -- cos 30) 35 (~ 3 0 7 ~ ) } - r ~ C25 COS -~- -- ~ COS -- 3r2c26(2 Cos 20 -- 2 cos 40)

(7)

Atluri and Kobayashi[1] also reported the stress field equations for the mixed mode case in a general form. They are described as follows

O'y = A In r-U 72xy =

n n 1 n 1)0 3)0 n n 1 n_ )cos( 3)0

-1(-1)~ + 2 } s i n ( 2 - 1)0 + ( 2 - 1 ) s i n ( 2 - 3)0

oo n

--,~:1= ~ Allnr~;Z~

{ 2 - ( - 1 ) n + 2 } s i n ( 2 - 1 ) O - ( 2 - 1 ) s i n ( 2 - 3 ) O

{2 + (-1)n - 2 } s i n ( 2 - 1)0 + ( 2 - 1 ) s i n ( 2 - 3)0

-{(-1)n - 2 } c o s ( 2 - 1 ) 0 - ( 2 - 1 ) c o s ( 2 - 3) 0

(8)

where All = Kl/,,/2~ and AII1 = -KIl/~r-~. (Note: in the original reference it is reported that All = Kl , , /~ and AI2 = - K I I ~ which is a typographical error. Further, there is a typographical error in the ay term of the Mode II series. The first term is a "sin" term and the second term is denoted as a "cos" term. It was found out that the second term must also be a "sin" term. This was made possible with the help of computer graphics software which simulated fringes observed in experiments. It was also counter checked by comparing the first term of the series solution to the singular solution reported in literature.) The aox stress term has played a very important role in fracture mechanics and the second term in the Mode-I series is related to tTox as 4At2 = - 6ox.

2.3. Equivalence between the multi-parameter stress field equations In view of the compactness and generality of Atluri and Kobayashi's equations, they are

taken as the basis for comparison. It is found that when eq. (4) is expressed in terms of r and 0, and compared with Mode-I

terms of eq. (8) a relationship exists between the coefficients of the two series expressions and is

3O

given by

K. RAMESH et al.

Ci (9) A I ( i + I ) - i + 1 '

When the Williams stress field equations are expressed in terms of rectangular stress com- ponents using the standard transformation law and compared with eq. (8) one gets

Aii = eli and (10)

A l l / : --c2i. (11)

3. DEVELOPMENTS IN SIF EVALUATION METHODOLOGY

The Irwin method for SIF extraction from photoelastic patterns was the accepted method for analysis for many years. In this method the positional coordinates and fringe order for a specific point that satisfies the criteria

Ormo0 . . . . ;O:Om = 0 (12)

is used for evaluating KI and aox. Kobayashi and Bradley[21] modified the formulation, but fun- damentally, the approach was unaltered. Since then several investigators reported various methods, and Etheridge and Dally [22] reported a review on the relative accuracy of the various two-parameter methods. The review brought out the fact that the zone of data collection has to be very small, of the order of rm/a < 0.03 and 73 ° < Om < 139 °.

Though photoelasticity gives whole field information, the previous reluctance to use information from the entire field stemmed primarily from practical limitations on data reduction. The positional co-ordinates of each data point used had to be measured manually and the results are calculated with the aid of a hand held calculator or with a main frame computer. This approach generally required manual data entry. With the introduction of PC based image processing systems, the drudgery of data reduction is removed[8-10].

Improvements in data collection methodology have led to the development of several techniques to process the data from the field rather than restricting the analysis to a point (as in Irwin's method) or a few points along a specified line. Among the several methods [23] the over- deterministic approach of Sanford and Dally [6] has become the most useful technique for evaluating the SIFs and aox.

The approaches described above use the near field equations of fracture mechanics to extract the SIF from the full field fringe patterns. In principle, the accuracy of these methods

Yt j.~--------.~Mixed .field Inelasticreg~ ~ region

Singular ~ J region ~ j Farfield region

Fig. 1. Figure showing qualitatively the various zones near the crack-tip.


should improve as the region of data acquisition is reduced to small regions around the crack- tip. However, it has been independently verified by several researchers [24, 25] that the plane stress assumption ceases to be valid in a small region at the crack-tip. In this region the stress field is three-dimensional and experimental observations are influenced by stress gradients through the thickness. In addition, the stress field is altered by localised crack-tip blunting. On the other hand, if this region is excluded from the analysis, the region of data acquisition may not lie fully within the domain in which the inverse square root singularity dominates the fringe pattern (Fig. 1). The size and shape of the singularity dominated zone are important factors to be determined and such studies as refs [26, 27] emphasised the fact that the singularity dominated zone is very small and indicated the use of a multi-parameter stress field equation for data reduction. The zone of data collection in such a case can be larger and one can collect data from the mixed field zone also (Fig. 1).

4. EVALUATION OF MIXED-MODE STRESS FIELD PARAMETERS USING LEAST SQUARE T ECH N IQ U E

4.1. Formulation of the equations The stress optic law relates the fringe order N and the principal stress as

NFo - - 0 - 1 - - 0 " 2 , (13)

t

where F~ is the material fringe value and t is the model thickness. For a plane stress problem the principal stresses are

0"x + 0"y /(0"x -- O'y) 2 (rx0 2

0-1 ,0"2 = 2 + W 4 ~- - (14)

Substituting eq. (14) into eq. (13), a function g is defined for the mth data point as follows

0-X - - 0-y gm ----- ~ +(rxy)2m- (15)

m

If eq. (8) is substituted into eq. (15), then eq. (15) is non-linear in terms of the unknown An, AI2 ..... A~k, Ain, AIIz ..... Ant, where k is the number of Mode-I parameters and l is the number of Mode-II parameters considered. If initial estimates are made for An, AI2 ..... Aik, Ain, AII2 ..... Am and substituted into eq. (15) it is possible that gm is not equal to zero since the estimates may not be accurate. To correct the estimates, a series of iterative equations based on a Taylor series expansion of gm is written as

Ogm Ogm . Ogm (gm)i+l = (gm)i + ~ ( AAII)i + ~ ( A A I 2 ) i + - - - -t- ~ ( A A I k ) i

+ 0 ~ ( A A I I 1 ) i + Ogre (AAII2)i + . +~--(AAlll) i , (16) II1 0AII2 "" I1/

where the subscript 'T ' refers to the ith iteration step and AAn, AAI2 ..... AAIk, AAIII, AAII 2 ..... AAII l denote the corrections to the previous estimates of An, A,2 ..... Axk, AII1, AII2 ..... AtII.

Corrections are determined such that (gm)i + 1 = 0 and thus eq. (16) gives

Ogrn 00A~2 (AAI2)i + . . . --(gm)i = 0---~i1 (AAII)i + + O0~lk(AAlk)i

(17) "[- ~ ( A A I I 2 ) i + . . .

Applying this iteration scheme to M data points results in an over determined set of linear equations in terms of the unknown corrections AAn, AA~2 ..... AAlk, AAin, AAn2 ..... AAIu given

32 K. RAMESH et al.

in matrix form by

where

{g}i : -[b]i{AA}i, (18)

gl g2

{g}i = " ; gm

gM i

and

~]i =

8gl 8gl Ogl 0gl Ogl Ogl

OAI1 OAt2 OAIk ÙAn1 8AII2 8AIIl

Og2 Og2 Og2 Og2 Og2 Og2 OAII OAI2 OAlk 0AIII 0AII2 OAtII

OgM OgM 8gM 8gM 8gM agm

- OAI1 OAI2 OAIk AIII 0AII2 OhIII -

(19)

A A I I AAI2

AAIk {AA}i : AAm

AAII2

• AAIII i

In forming eq. (18) one needs to calculate (Ogl/OAn), (Ogl/OAI2).•. etc. and at first it appears that it may be difficult to do it in closed-form• However, one can determine them very easily as follows• A typical derivative for the mth data point with respect to Axn is obtained by taking the derivative of eq. (15).

Ogml ( O 0 " x q O O ' y ) (OZ'xY) = ~ (Crx - ~ry)m + 2 rxy (20)

OAIn OAIn OAIn m OAIn/m"

Since, .din is a linear coefficient, the terms (Oax/OAin) etc. in eq. (19) are simply the terms shown in the Mode-I column vector of eq. (8) multiplied by (n/2 r (n -2)/2). Thus, (OgMOAin) can be determined in a closed form way.

Now {AA}i can be determined in a least square sense from eq. (18) as

where

{AA}i = -[c]T i {d }i, (21)

[C]i ~--- [bff[b]i; {d}i = [blTtg}i •

The solution to eq. (21) gives {AA}i which is used to find {A}i + 1 for the next iteration as

{A}i+l : {A}i -k- {AA} i. (22)


4.2. Convergence criteria

In order to make the solution such that it is not influenced significantly by the choice of data points, from the total collected data points, at a time M data points are randomly selected to perform the iteration. Six such data sets are usually found to be sufficient and the average of the parameter values obtained from these is taken as the solution. In the present work the total number of data points selected ranged from 75 to 100 and the data points per set is kept as 35.

The convergence can be decided by either of the two following criteria, namely (1) the parameter error minimisation and (2) the fringe order error minimisation. In the parameter error minimisation the iterations are stopped when the values of {AA}i become reasonably small (say of the order of 10-6). In the fringe order error minimisation, using the newly calculated values of {A}i + 1, the fringe orders corresponding to the selected data points are calculated theoretically during every iteration step and are compared with the experimentally obtained fringe orders. The convergence criteria is satisfied if

Z INtheory -- Nexp [ < convergence error.

total no. of data points - (23)

It has been found that the fringe order minimisation error gives better results and in the subsequent discussions only fringe order minimisation is used as a criteria for stopping the iterations. The best solution is obtained when the convergence error is of the order of 0.05 to 0.1.

4.3. Implementation

A software has been implemented on the PC-386 environment in Turbo Pascal for the least square analysis and the theoretical reconstruction, using up to 13 terms for each of Mode-I and Mode-II of the Atluri and Kobayashi[1] form of the stress field equations. The software requires a data file containing the X, Y coordinates and the fringe order as the input. The software can accept the input data from manual data entry or directly from the DIP data collection software. The least square procedure is invoked and the results of the parameters are written to another data file. Using these parameters the program recalculates ( a l - a2) values, hence the fringe order at every point in the data field and the fringes are then plotted using the graphics utilities of the Pascal environment.

This software has two loops, one for converging a data set and the other for converging a required number of sets. Usually, the maximum number of iterations for a data set to con- verge is of the order of 10. Once a data set is converged, the converged values are taken as the starting values for the iteration of the next data set. This leads to a small reduction in the time taken for convergence of subsequent data sets and reduces the total time taken consider- ably.

The software uses eqs (18, 19) for solving the least square problem. The [b] matrix is to be calculated using eqs (19, 20). Differentiating eq. (8) with respect to AIn gives

OtXx O Aln Oay O Aln Orxy OAIn

n n-2 ~ r-'2-

+ n n - 1 " n

.. (24)

In this it has to be noted that the &rx/OA~n vector is independent of A~n, i.e. they do not change in each iteration. It remains constant for a given data point. However, Ogm/OAIn varies in

34 K. RAMESH et al.

each iteration since ax, ay and rxy are also present in eq. (20). Hence the [b] matrix also varies in each iteration. Therefore, the [b] matrix has to be calculated, but the Oax/OA~n vectors need not be calculated in each iteration. This has been implemented in the following way. First the &rx/OA~n vectors are calculated for each AI and An terms at every data point. These vectors are stored in a file. The least square routine reads a set of data points randomly from the given data file. While this is done, corresponding Oax/OAin vectors are also read. ax, ay and Zxy are calculated using previous values of A1 and AII terms. These crx, ay and rxy values and aa~/OA~n vectors are used to find the [b] matrix. This reduces the time taken for each iteration.

All the intermediate storage and input/output operations in the software are done with files. The input file from DIP software containing co-ordinates in pixel dimensions is accepted with extension ".dat". The software uses " .dat" file to produce another data file in real world co-ordinates with extension ".out". While having manual data entry, the software expects the data file with extension ".out". The Oax/OA~ vectors for all data points are stored in a file with extension ".mas". When a set of data is read randomly a new file is created with ".sco" extension containing Oax/OA~ vectors only of the data points selected randomly for the set. This file is used in calculating the [b] matrix. The ".mas" and ".sco" files are automatically erased when leaving the software.

A priori one does not know how many terms are required to model the stress field. It is always recommended to start from a minimum number of parameters and to progressively increment them by one as the problem demands. If the problem demands say six parameters, but the solution process is started with two parameters, then convergence will not be achieved unless a large convergence error of the order of 0.3 is specified in eq. (23). The converged solution at this stage does not model the stress field correctly. However, it does provide a good starting guess for the two parameters for the subsequent iterations. The number of parameters is then incremented by one and the iterations are initiated with a smaller convergence error. The process is to be repeated until one could achieve a convergence error of 0.1 or less.

The software is written such that it will do this automatically. What one needs to do is to specify the convergence error required at every stage of the iteration. It stores the parameter values after each set is converged into a file with extension ".par". This file has the following information

(1) Number of mode I parameters. (2) Converged values of mode I parameters. (3) Number of mode II parameters. (4) Converged values of mode II parameters.

When the software is run by increasing the number of terms, it reads the starting values from this file for the available terms and assumes the starting values as zero for remaining terms. This procedure has successfully worked for a large number of problems.

The reconstruction routine of fringe patterns is written such that the crack line can be rotated to suit the experimental situation so as to give a realistic representation of the fringe field for easy comparison. The entire screen is scanned row by row and pixel by pixel. While reconstructing only Mode-I or Mode-II fringes, advantage is taken of the symmetry and only one half of the screen is scanned. At the end of the reconstruction, the data points are echoed back. A cursor routine inbuilt in the software, which can be moved by arrow keys, is used to label the fringes.

5. E X P E R I M E N T A L VALIDATION

5.1. System configuration The system consists of a CCD camera (TM 560 PULNIX) that has a pixel resolution of

512 × 512 pixels and digitises the image at video rates. The camera is connected to a PC based


2 Parameters

I I/ I f

I I I I

-

I" ) 3 Parameters

4 Parameters 5 Parameters

Jio ~.e 14r . &8 IR Y i J

6 Parameters

1"1.9 £.9 .9,9

.O 3

7 Parameters

8 Parameters

Fig. 4. Theoretically reconstructed fringe patterns with data points echoed for various parameters for the fringe patterns shown in Fig. 3.

EFM 56/1--B

36 K. RAMESH et aL

3 0 Z ._c

~ 2 0

C

0 .80

r l I"1

I I I I ~ I

2 4 6 8

Number of Parameters

0 n

0 . 6 0

0 .40 /

r l

0 ' 0 . 2 0 ~ I , i , I , 0 I0 0 2 4 6 8 I0

N u m b e r o f P a r a m e t e r s

Fig. 5. Graph showing the total percentage error in N as a function of the number of Mode-I parameters.

Fig. 6. Graph showing the Mode-I SIF as a function of the number of parameters.

(a) (b) Fig. 7. Isochromatic fringe patterns in the neighbourhood of a crack in a SEN specimen obtained using

the finite element method. (a) a/w =0.3. (b) a/w = 0.7.


image processing system equipped with an image processing card (PIP, 1024B Matrox corporation) connected to a high-resolution video monitor. The system has, in addition to PC memory, four frame buffers, each of 512 x 512 pixels x8 bits to store the image being processed and the intermediate results. Figure 2 shows the video monitor with the fringe patterns observed and the PC monitor with the fringe patterns reconstructed. A special macro-lens is available which can provide an on-line magnification of 8 to 16 times.

5.2. Mode-I loading

The problem chosen is that of analysing the stress field in the neighbourhood of radial cracks in an internally pressurised thick cylinder. The model is cut from an epoxy sheet made of CY230 resin and HY951 hardener mixed in the proportion 100:9 by weight. An aluminium disk slightly larger than the internal diameter of the epoxy disk was inserted to simulate the internal pressure. Using a specialised cutter, thin radial slits with thickness of the order of 0.1 mm were made to simulate the cracks.

Figure 3 shows the dark field fringe patterns observed in the case of a radial crack emanating from the outer boundary of an internally pressurised thick ring [28]. The figure shows fringe skeletons (white lines) superimposed on the original fringe patterns. The algorithm of Ramesh and Pramod[8] is used for fringe thinning and Fig. 3 shows the superiority of the algorithm in comparison to other fringe thinning methodologies[10]. It is interesting to note that, ahead of the crack-tip, there is a sink point in the fringe field [29]. After fringe thinning, 75 data points were collected from the fringes interactively. Figure 4 shows the reconstructed fringe patterns with the data points echoed starting from two parameters upto eight parameters in eq. (8). The figure clearly shows that only a multi-parameter solution beyond six parameters actually models the stress field better. Figure 5 shows the percentage error in N as a function of the number of parameters and Fig. 6 shows KI as a function of the number of parameters. It is interesting to note that in the present example, ahead of the crack-tip, several fringes cross the crack axis. This can be attributed to the fact that the crack is located in a non-uniform stress field and the crack-tip is closer to the boundary.

The fact that several isochromatic fringes cross the crack-axis when the crack-tip is closer to a boundary has been observed for various specimen geometries as well. Chona et al. [26] observed this for a Rectangular Double Cantilever Beam (RDCB) specimen at large crack lengths (a/w = 0.9). A finite element analysis [30] of the Single Edge Notched (SEN) specimen using quarterpoint singular elements for modelling the crack-tip shows that for larger crack lengths, several fringes cross the crack-axis (Fig. 7). High quality fringe patterns were possible with the development of a novel software for fringe plotting from FEM results[31]. The result brings out the fact that the experimental observation is not just due to crack-tip blunting or other such experimental difficulties. The FEM results also brought out the fact that workers involved in the numerical studies did not bother about the higher order solution for a long time. The reason is that in the numerical studies, one can go close to the crack-tip and extract data for SIF evaluation. On the otherhand, due to practical difficulties, experimentalists have to collect data from a zone away from the near tip zone in which a higher order solution is necessary to model the stress field.

Next, the SIF is calculated for the situation shown in Fig. 8. Close scrutiny of the fringe patterns reveals that closer to the crack-tip, the fringes are forward tilted and at distances away, the fringes are backward tilted. In principle, by collecting data from only the forward tilted loops, it is possible to evaluate KI and aox using a two-parameter over-deterministic solution. The resulting reconstructed fringe patterns are shown in Fig. 9(a). The reconstructed fringe patterns clearly bring out the fact that the solution models only a small field closer to the crack-tip. The convergence error achievable was 0.15 and r/a lies in the range 0.116 < r~ a < 0.412. By collecting data from a larger zone, with an eight-parameter solution it was possible to model the stress field better and the resulting reconstructed fringe patterns are shown in Fig. 9(b). The convergence error achievable was 0.11. The value of r/a lies in the range 0.116 < r/a < 1.349.

38 K. RAMESH e t al.

k1=9.457 I~dm I~.e.011 If~m , si~x=-t.~ I~a

2 Mode I and 2 Mode II Parameters

.

1 ~ ~. Ol l l l l l l~

/ I ' \ ' . , - - k1=9,,~19 I~l'm 1~8,1m I~a,l'm , s|Mix~9,4~ I~a



k1=9,515 I~ed,, k2.=9.915


. . . .1. N

k1=6.469 I~l'm Id:O,Me If~la . si~,t:-l,.~i I~a


Fig. 11. Theoretically reconstructed fringe patterns with data points echoed for various parameters for the fringe patterns shown in Fig. 10.


Fig. 2. Figure showing the PC based image processing system with the video monitor showing the experimental isochromatics and the PC monitor showing the theoretically reconstructed patterns.

Fig. 3. Dark field isochromatics observed for a radial crack emanating from the outer boundary of an internally pressurised thick ring. The figure also shows the comparative performance evaluation of various fringe thinning algorithms. The fringe skeletons are made white and superimposed on the experimental fringe patterns for easy comparison. (Authors' processing time on a PC-Xt based system) (a)

Yatagai et al., 1188 s. (b) Umezaki et al., 1530 s. (c) Chen and Taylor, 3579 s. (d) Ramesh, 400 s.

40 K. RAMESH e t al.

Fig. 8. Dark field isochromatics observed for a radial crack emanating from the inner boundary of an internally pressurised thick cylinder. Note that the direction of the fringe tilt changes as one moves

towards the crack-tip.

Two term solution kt~O,G3H ifa~n k~8,888 ~ a sig8~-5.145

Eight term solution

Fig. 9. Theoretically reconstructed fringe patterns. (a) Two parameter solution. (b) Eight parameter solution. Note that the zone of data collection is substantially increased and the fringes are better mod-

elled in this case.


Fig. 10. Dark field isochromatics observed for a crack emanat ing from the tensile-root fillet of a spur gear.


0.35

Z 0.25 C

t -

I~ 0.15

0.05

A

A A

I | I I | I I 0 2 3 4 5 6 7 8 9

Number of Perometers

Fig. 12. Graph showing the error in N as a function of the number of parameters.

0.60

0.55

0.50

0 . 4 5

L~ 0.40 0 o. ~E 0.35

0.30

0.25

O. 20

0.15 ! I I I I I I

2 3 4 5 6 7 8 9

Number of Poromelers

Fig. 13. Graph showing the Mode-I SIF as a function of the number of parameters.

0.100

0 (1.

0.050

0.000

- 0,050

-0 . I00 I I I I I I I

2 3 4 5 6 7 8 9

Humber of Parameters

Fig. 14. Graph showing the Mode-II SIF as a function of the number of parameters.

44 K. RAMESH et al.

5.3. Mixed mode loading

Figure 10 shows the fr inge pa t te rns in the n e i g h b o u r h o o d o f a c rack- t ip loca ted in the tensile r oo t fillet o f a spur gear. The stress field in the n e i g h b o u r h o o d o f the crack is also influenced by the con tac t stress field due to the t oo th pai rs in contact . The fringe pa t te rns are qui te complex and a s imple three p a r a m e t e r so lu t ion involving only Kb Kn and aox is no t sufficient to mode l the stress field. F igure 11 shows tha t a mu l t i - pa r a me te r so lu t ion involving six M o d e - I pa rame te r s and six M o d e - I I pa rame te r s was necessary to mode l the stress field. F igure 12 shows the convergence e r ror as a funct ion o f the number o f parameters , F igures 13 a n d l 4 show the var ia t ion o f KI and Kn as a funct ion o f the n u m b e r o f parameters .

6. C O N C L U S I O N S

The equivalence o f the m u l t i - p a r a m e t e r equa t ions p r o p o s e d by At lu r i and K o b a y a s h i with tha t o f the general ised W e s t e r g a a r d equa t ions o f Sanfo rd and Wi l l i ams eigen funct ion expans ion is es tabl ished. The elegance o f using the equa t ions p r o p o s e d by At lur i and K o b a y a s h i in eva lua t ing these pa r ame te r s in a least squares sense f rom the exper imenta l iso- ch romat ics is shown. The example p rob l ems revealed the advan tage o f using a mul t i -pa r - ameter so lu t ion in terms o f col lect ing the da t a f rom a larger zone. Fur the r , the example p rob l ems also es tabl i shed the fact tha t use o f a mu l t i - pa r a me te r stress field equa t ion is not jus t an academic cur ios i ty bu t a prac t ica l necessity to evaluate f rac ture mechanics pa rame te r s in p rob l ems o f prac t ica l impor tance .

Acknowledgements---The research work reported in this paper is in part supported by the Aeronautics Research and Development Board through Project No. 786 and the Department of Mechanical Engineering IIT Kanpur. The authors thank Mr Amit Gupta and V. Ganapathy for making the software user friendly, and Mr Sanjeev Deshmukh for Fig. 9. The authors also thank Prof. B. Dattaguru, Prof. K. Rajaiah and Prof. N. K. Gupta for their interest in the work.

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Communications numer. Meth. Engng, 1995, 11,839 847.

(Received 22 December 1995)

evaluation of stress field parameters in fracture mechanics by photoelasticity—revisited

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