Determining Fracture Parameters with Full.field Optical Methods

by Robert J. Sanford

ABSTRACT--In order to increase the accuracy of stress- intensity-factor measurements and to obtain data on additional parameters which may influence fracture behavior (such as crack branching and crack curvature) a technique for full- field fringe-pattern analysis (referred to as the local collocation method) has been developed. This method removes the restric- tion of limiting the data analysis region to the near-field region by including additional nonsingular terms in the algorithm. In this paper the theory of the method is developed and sample results using photoelastic, holographic and moir6 full-field patterns are provided.

Introduction Over the last three decades full-field optical stress-

analysis methods have played an important role in exper- imental fracture mechanics. These methods are particularly well suited to determining the geometric stress-intensity factor for particular geometry/loading configurations in situations where analytical or numerical methods fail to provide answers. Alternatively, techniques of experimental stress analysis can be used to verify solutions obtained by other methods. In the area of dynamic fracture mechanics optical techniques (particularly photoelasticity) have been the primary tools used to study crack-arrest behavior. In fact, one of the earliest applications of photoelasticity to the study of fracture behavior was the recording of the isochromatic patterns for a running crack by Wells and Post.

The Wells and Post paper focuses on the qualitative aspects of the fringe pattern around the crack tip. More significantly, it demonstrates the utility of the Cranz- Schardin camera system s in dynamic fracture applications. This approach has been used extensively by other researchers in the field. 3 Irwin, ' in a discussion of the Wells and Post paper, added the quantitative analysis of the fringe pattern needed to determine the stress-intensity factor, He observed that the isochromatic fringes formed

Robert J. Sanford (SEM Fellow) is Professor, Mechanical Engineering Department, University of Maryland, College Park, MD 20742. Paper was presented at the 1988 SEM Spring Conference on Experimental Mechanics held in Portland, OR on June 5-10. Original manuscript submitted: July 26, 1988. Final manuscript received." January 20, 1989.

closed loops in the near field of the crack tip and used this observation to develop the equations necessary to compute the stress-intensity factor from the apogee point on a fringe (see Fig. 1). Irwin also observed that the tilt of these fringe loops was the consequence of the constant stress, Oox, in a direction parallel to the crack line and demonstrated that this term must be included in the near- field formulation of the stress equations if a correct inter- pretation of the isochromatic fringe distribution is to be obtained. Analytical justifications for this additional term in the near-field equations of fracture mechanics were provided subsequently by various researchers.~-8

The Irwin method, also called the apogee method, for stress-intensity-factor extraction from photoelastic patterns was the accepted method for analysis for many years. Kobayashi and Bradley 9 modified the formulation, but, fundamentally, the approach was unaltered. The primary application of this approach has been dynamic fracture behavior of transparent, birefringent materials. References 10 and 11 are typical of the results obtained with this approach.

Stress fields in the neighborhood of cracks in three- dimensional bodies have been studied extensively by Smith et aL using the stress-freezing method (e.g., Refs. 12, 13 and 14). The approach used to extract the stress-intensity factor in these studies differs from the apogee method and, instead, uses fringe-order information taken along a line perpendicular to the crack tip. By restricting data acquisition to a radial line, the angular dependence of the near-field equations is eliminated and the stress field can be approximated by a function of position only. A similar approach was used by Smith et al. ~ to determine the stress-intensity factor from displacement information obtained with moir~ interferometry.

Sanford and Dally ~6 developed yet another procedure for determining stress-intensity-factor information from photoelastic patterns. In their approach, points on fringes in the region around the crack tip were selected at random and the set of points matched in the least squares sense to an expression for the maximum shearing stress based on the modified near-field equations. This procedure, called the over-deterministic, least-squares method, was used to determine K,, K , and Oox. An example of the application of this method to failure analysis of an air- craft component is presented in Ref. 17.

Experimental Mechanics �9 241

Nearly every full-field optical stress analysis method has at one time or another been used to study stress fields in the neighborhood of a crack tip. In addition to those already cited, the work of Dudderar and Gorman 18 should be mentioned. They examined the isopachic field near a crack tip using holographic interferometry. Using analysis procedures similar to those of Smith et al., they examined the fringe order along the crack plane and obtained an estimate of the stress-intensity factor from the inverse- square-root dependence of the isopachic fringe order along the line.

All of the approaches described above share a common feature, namely that the analysis procedure used to extract the stress-intensity factor from the full-field fringe pattern relied on the modified near-field equations of fracture mechanics. In principle, the accuracy of these methods should improve as the region of data acquisition is reduced to smaller regions around the crack tip. However, it is generally recognized and has been independently verified by several researchers 19'=~ 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 localized 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-r singularity dominates the fringe pattern. It has been demonstrated that, in specimens of finite dimensions, the singularity dominated region is very small. 2' To overcome these problems the author and various collaborators 22-2' have developed procedures for analyzing fringe patterns from a variety of optical methods in which data acquisition is not restricted to the near- field region.

These procedures, which are an extension of the over- deterministic, least-squares method of Sanford and Dally, retain higher order terms as variables in the analytical expression of the field description for the type of fringe pattern being considered. In this procedure the influence of the singularity is separated from the higher order terms as a consequence of the analysis. Furthermore, by pur- posely eliminating the very near-field, three-dimensional region from the analysis, the problem can be fully defined

(/1 arm/aO =0

~*~ "Cm = constant

Or.

Fig. I--Illustration of a typical fringe loop near a crack tip showing the coordinates of the apogee point used to determine K from Irwin's method

in terms of a truncated series representation of the field equations involving a limited number of terms. Because of the mathematical similarities between this approach and the purely numerical method of boundary collocation, the method is called the local collocation method. In the following sections of this paper the mathematical formula- tion of the local collocation method for several popular full-field optical fringe patterns are developed and an example of each is presented.

Formulation of the General Equations Methods for determining the stress-intensity factor

from full-field optical patterns which are based on the modified near-field equations for the stresses or displace- ments in the neighborhood of the crack tip fall to take advantage of the additional information contained in the full-field pattern. This information can he used to increase the accuracy of the analysis if the influence of the non- singular terms in the general analytical solution for the problem is included in the feature extraction algorithm. In order to use this additional information, the governing equation which describes the optical pattern is represented by a truncated form of the general-series solution for the problem with unknown coefficients. These coefficients, which inchide the parameters of interest (K, Oox, etc.), are to be determined from the analysis of the fringe pattern. To implement this method, a set of equations is written which relate the measured fringe orders at a number of points in an expanded region around the crack tip to the underlying general expression involving the unknown coefficients. These equations are then solved in the least- squares sense to obtain estimates of the coefficients. The number of terms retained in the series depends on the size of the region. Convergence of the truncated series as the number of retained terms is increased can be used as a solution criterion. This approach is analogous to the boundary collocation method except that the number of terms necessary to obtain an accurate estimate of the stress-intensity factor is dramatically reduced since the stress state in the region around the crack is dominated by the lower order coefficients.

Although the general series solution for the problem being considered can be formulated in a variety of ways, the author has found that the generalized Westergaard equations in complex coordinates provide an algebraically convenient formulation. For purposes of illustration of the local collocation method this formulation is used as the basis for the development of the governing equations in the following sections.

It has previously been shown that the stress state for opening-mode crack problems can be completely described in Westergaard notation by an Airy stress function of the form 8

F = Re Z ( z ) + y I m 2 - ( z ) + y I m ~ ( z ) (1)

where the notations

Z-(z) = d Z ( z ) ,

and

d - - Z'(z) d z ( z ) Z(z) = ~ Z ( z ) , =

(2)

d - d Y(z) ~(z) = -~td ~(z), Y(z) = --~tY(z), Y ' ( z ) = - d t

(3) have been employed.

242 �9 September 1989

The functions Z ( z ) and Y ( z ) are suitable Wester- gaard-type stress functions for the geometry/loading being considered and are subject to the constraints that R e Z ( z ) = 0 on the stress-free portions of the crack faces and I m Y(Z) = 0 along the crack line. From eq (1), the stresses and plane-stress displacements, u and v, can be written as

ax = R e Z - y l m Z ' - y I m Y ' -e 2 R e Y ' (4)

ay = R e Z + y l m Z ' + y l m Y ' (5)

r~, = - y R e Z ' - y R e Y ' - I m Y (6)

E u = ( l - v ) R e Z - ( l + v ) y l m Z

- ( l + v ) y l m Y + 2 R e Y

E v = 2 I m Z - - ( l + v ) y R e Z

- ( l + v ) y R e Y + ( l - v ) I m Y

where E is the elastic modulus and v is Poisson's ratio. For a single-ended, stress-free crack with the origin of

the coordinates at the crack tip the functions Z ( z ) and Y ( z ) can be represented as

Z(Z) = ~ A , Z "-1/2 n = 0

and

Y ( z ) = ~ B . z ~ (10) m=O

where Ao and Bo are real constants and K = Ao~/27r and Oo~ = 2B0. It is important to note that the functions given by eqs (9) and (10) are the Westergaard equivalents of the symmetric part of the Williams function ~5 and can be used to solve exactly the same types of problems for which the Williams solution is suitable. For other types of opening-mode crack problems, e.g., the internal crack problem or single-ended cracks with point or distributed loads, alternative forms of the functions Z ( z ) and Y ( z ) must be used. Some guidelines for selecting suitable stress functions have recently been suggested by Sanford and Berger. ~6

Expressions similar to eqs 0)-(8) for the forward shear mode crack problem in Westergaard notation can easily be developed and are presented in Refs. 27 and 28. In addition, these references a l so contain forward shear mode forms of the stress functions Z ( z ) and Y ( z ) for a single-ended, stress-free crack with the origin of the coordinates at the crack tip. These latter stress functions are analogous to the antisymmetric part of the Williams formulation in real variables 2~ and the earlier comments on the equivalence of the Westergaard and Williams representations and the restrictions on their use still apply. The general solution for the mixed-mode plane problem can be obtained by superposition.

Linear Algorithms

Isopachic Patterns

Although interferometric methods have not been widely used for fracture studies, the implementation of the local

collocation method for this type of fringe pattern is particularly simple and results in a set of linear equations for the fracture parameters. The solution scheme for both classical two beam interferometers and holographic interferometry is identical. In either case, the fringes are contour lines of constant sum of the in-plane normal stresses, i.e., isopachics. A typical isopachic fringe pattern in the region of a crack tip in Plexiglas recorded with double-exposure holographic interferometry is shown in Fig. 2. From eqs (4) and (5), the governing optical equa- tion is, ideally, of the form

= a=+ae = R e Z + R e Y (11) 2 t 2

where N is the fringe order, fp is an optical constant and t (7) is the model thickness. Equation (11) assumes that the

wavefronts for each of the two beams in the interferometer are in perfect phase over the field of view. In practice this condition may be difficult to achieve and additional terms

(8) are necessary to account for any initial pattern which may be present. Since the success of the local collocation method depends on matching the observed fringe pattern to an analytical expression containing unknown coefficients which describes the fringe pattern, any additional factors which influence the fringe pattern must be included in the formulation. In the simplest case, a uniform phase difference between the reference and object wave fronts may be present which biases all of the fringes by a con-

(9) stant fractional fringe order. Accordingly, eq (11) must be modified to account for this unknown fringe shift and the governing optical equation should be written as

1 Nfv2t = R e Z + R e r + ~ No (12)

where No is the unknown initial fringe order. For the stress functions given by eqs (9) and (10), the modified optical equation can be written in real, polar coordinates as

N f p = ~ A , r "-1/2 c o s ( n - �89

2t n=0

+ ~ Bmrmcos mO + 1

m=0 T No (13)

Fig. 2--Holographic interference pattern at a crack tip in Plexiglas

Experimental Mechanics �9 243

At any point in the isopachic pattern the coordinates, r and 0, and the fringe order, N, can be substituted into eq (13). The result is a linear equation in the unknown coefficients A , , Bm and No. For a large number of such points, an over-determined set of equations is obtained which can only be solved in the least-squares sense. In matrix notation these equations are of the form

[NI = [S] I t ] (14)

where [N] is the row vector of measured fringe orders, i.e., the left side of eq (13), [S] is the matrix of coefficients of the unknowns, and [C] is the row vector of unknown coefficients in the modified optical equation, eq (13). The solution of this set of equations in the least-squares sense is

[ e l = i s ' s ] -1 i s ] T IN ] (15)

It should be noted that the unknowns Bo(= 00=/2) and No have the same functional form and can be merged into one unknown constant. As a consequence, the local collocation algorithm cannot distinguish between them and Oox cannot be uniquely determined from isopachic patterns.

Moir~ Patterns The displacement patterns obtained with moir~ or

moird interferometry can also be analyzed by the linear algorithm. 2' For grating lines parallel to the Cartesian axes the idealized optical equations are

N,, = u / fa (16)

o r

N v = v / f a (17)

where fd is the fringe sensitivity. Following the same procedure as previously described, the optical equations in real, polar coordinates can be obtained by substituting the appropriate Westergaard stress functions into eqs (7) or (8). For those problems for which eqs (9) and (10) are suitable stress functions the optical equations are the following. For u-field patterns:

Co . . + 1 / 2 "

NuEfa = A,, ~ [(1 - u) cos(n + �89 n

- ( 1 + v) (n + �89 sin 0 sin ( n - �89

+ ~ B rm+----~x m=0 " m + 1 [2 cos(m + 1)0)

- (1 + v) (m + 1) sin 0 sin(m0)] (18)

and for wfield patterns: r rn+ll2

N, Efd = n=O ~" A , ~ [2 sin(n + �89

- (1 + v) (n + �89 sin 0 cos(n - �89 + ~ r m+t

m=0 B m ~ [(1 - v) sin(m + 1)0

- (1 + v) (m + 1) sin 0 cos(toO)] (19)

As was the case for isopachic patterns, these equations must be modified to include additional effects which result from the practical aspects of performing the experiment. In particular, the displacement equations assume that the loads are applied in such a way as to produce complete symmetry about the crack line and the original stays fixed. In practice the fixed point is usually at one of the load points and the crack tip undergoes both a rigid-body translation and rotation. This nonstrain related fringe field can be expressed as

M = Pr cos O + Qr sin O + R (20)

where P , Q, and R correspond to a constant fringe field of P fringes per unit length in the x direction, Q fringes per unit length in the y direction and a constant shift of magnitude R. These additional terms must be appended to the right-hand side of eqs (18) and (19) to obtain modified optical equations which properly describe the observed fringe pattern.

As a practical matter, it is not necessary to determine the absolute order of the fringes since the constant, R, can be interpreted as a fringe-order shift factor which includes the effect of rigid-body motion and an unknown origin. It is necessary only that the relative order of the fringes be maintained for the algorithm to work. Functionally, the resulting set of modified optical equa- tions for the displacement at each measured point is the same as eq (14), the solution of which is given by eq 05).

An example of the application of this method for deter- mining the stress-intensity factor and additional series coefficients for a crack-line loaded, compact-tension geometry from displacement data is shown in Fig. 3. This figure shows computer-generated reconstructions of the fringe patterns using the constants obtained with a six- term (Ao, AI, Bo, B1, As, B2) approximation to the u and v displacement fields plus the three rigid-body motion terms. The white dots shown in the figure are the data points from moir&interferometry experiments used in the analysis. The excellent agreement between the experimental points and the computer-generated reconstructions demonstrates that the analysis faithfully accounted for all of the salient features of the experiments. The use of computer reconstructions, such as those shown in Fig. 3, is suggested as one method for verifying the solution.

Nonlinear Algorithms (Photoelastic Patterns) The most widely used optical technique for studying

fracture behavior is the photoelastic method. The local collocation method can also be applied to the analysis of photoelastic fringe patterns. The solution scheme results in a set of nonlinear equations to be solved. The approach is somewhat different from that previously described; however, the simplicity of a photoelastic experiment compensates for the additional complexity in the analysis.

As in the previous cases, the starting point for the formulation is the governing optical equation which, for photoelasticity, can be expressed as:

t " = (Tmax)2 (21)

where fo is the photoelastic constant for the specimen material. In terms of Westergaard stress functions, the optical equation becomes

244 �9 September 1989

(•---•fr = ( - - q ~ + r~y ~ + t " )2 = D ~ T 2 (22)

where

D = ~y - a. 2 = y l m Z ' + y l m Y ' - R e Y (23)

and

T = z~ = - y R e Z ' - y R e Y ' - I m Y (24)

Unlike the other experimental methods, the optical equa- tion for photoelasticity normally does not need to be modified to include additional experimental effects in order to completely describe the fringe pattern.

For the stress functions given by eqs (9) and (10), eqs (23) and (24), in real variables, become

D = ~ ( n - �89 A . r "-1/2 sin 0 s i n ( n - 3/2)0 n=O

+ ~ B~r~[ms inOs in(mO)+cos(mO)] (25) m=O

T = - ~ ( n - � 8 9 A . r ~- l /2 s inOcos (n -3 /2 )O n=O

- ~ B~r ~ [m sin 0 cos(m0) + sin(m0)] (26) m=O

Substitution of eqs (25) and (26) into eq (22) provides the desired optical equation to be solved in the least- squares sense. Clearly, eq (22) is nonlinear in the unknown constants A . and Bin, and the solution scheme of the previous section does not apply.

There are a variety of methods available to solve non- linear, algebraic systems of equations in the least-squares sense. An iterative procedure based on the Newton- Raphson method can be used to solve the optical equa- tions if eq (22) is cast in a modified form. Let

( N k f u ~2 = 0 (27) gk = D2k+ T~k-- . 2t "

where the subscript k denotes the value of the function evaluated at a point in the fringe field (rk, Ok) at which

the fringe value is Nx. Accordingly

Ok = Ok(An,B. ,)

Tk = T k ( A . , B . ) (28)

Taking a Taylor 's series expansion of eq (27) and retain- ing only the linear terms yields

ag~ ag~ gk ,+, = + + + . . .

Og~ Og~ + ' ~ o ABo + ~ A B ~ + . . . (29)

where superscript i denotes the iterative step and AAo, AA, . . . . . ABo, AB, . . . . are corrections to the previous estimates of the unknowns Ao, AI . . . . Bo, B, . . . . . respectively.

Note from eq (27) that the desired result " i+1 IS gk = 0

yields a system of linear equations (one for each point k) in the correction terms AA. , AB,. of the form

Ogk Ogk �9 A --gk = "-~o AAo + - - ~ za.'t, + . . .

Ogk d3gk + ~ ABo + ~ AB . . . . . (30)

For simplicity, the iteration counter i has been eliminated. Equation (30) can be expressed in matrix form as

[g] = [c] [z~] (31)

where

- g, AAo

[ g ] = A = AAN (32)

ABo - - g L

Z~BM

and

Fig. 3--Computer reconstruction of the u and v moir~ patterns containing rigid-body motion. The white dots denote the locations of data points from the experimental pattern

Experimental Mechanics �9 245

[c] =

Og,

OAo

OgL

OAo

Og, Og, Og, q OAn OBo " " " OBu

] OgL Og~ OgL OA~ OBo OBM

(33)

where L is the total number of data points used in the least-squares algorithm, L > M + N + 2, and N and M are the upper limits of the truncated series approxima- tions to Z(z) and Y(z).

At first glance it might appear that obtaining the elements of matrix [c] involves evaluating a large number of partial derivatives; however, this is not the case�9 The functional form of each column of [c] is identical, only the coordinates used to evaluate the row elements are changed. Furthermore, the partial derivatives in each of the column elements in [c] can be rewritten as

Ogk = 2Dk ODk OTk OA, ~ + 2Tk--OA,

or (34)

[Jgk = 2Dk ODk 0 Tk OB---~ ~ + 2 T OBm

Since the functions D k and Tk are linear functions of the differentiation variables (A , , Bin), the partial derivatives in eq (34) can be obtained from eqs (25) or (26) by inspection�9 Finally, the expressions Dk and Tk, which need to be evaluated in eq (34), are the same as those in eq (27) from which the row elements of matrix [g] are also obtained.

From the above discussion, it is clear that obtaining the numerical values of the matrix elements necessary to form eq (31) is a straightforward algebraic exercise which can be implemented recursively in a variety of computer languages. Similarly, since the functional form of eq (31) is identical to eq (14), the solution of eq (31) in the least- squares sense is given as

[zX] = [c~c1-1 [c] r [g] (35)

Unlike the linear algorithms in which the solution of the matrix equation gives the desired series coefficients directly, the solution of eq (35) results in correction terms for prior estimates of the coefficients A , and B=.

Fig. 4--Typical photoelastic fringe pattern for a three- point bend specimen at a l W = 0.6. Circle denotes the region of data acquisitiOn used in the local collocation analysis

Accordingly, an iterative procedure must be used to obtain the best-fit set of coefficients. The steps in the procedure are the following.

(a) From the experimental fringe pattern, select a sufficiently large number of data points with co- ordinates (rk, Ok, Nk), selected so as to characterize the features of the fringe pattern�9

(b) Assume initial values for the unknown coefficients Ao, AI . . . . . An, Bo . . . . . BM (usually estimates for only the first few leading terms are required, the re- mainder are initially set equal to zero).

(c) Compute the elements of [ g] and [c]. (d) Solve eq (35). (e) Revise the estimates of the unknowns, i.e.,

A~ *1 = A ~ + A A o

A~ +' : A~+ AA~ Bi*l = Bio + ABo o

i+1 i ~ M = B M -t" A B M

(f) Repeat steps (c) through (e) until the elements of [A] become acceptably small.

An example of the application of this approach applied to the determination of the stress-intensity factor for a three-point bend specimen is shown in Fig. 4 (only a por- tion of the fringe pattern in the region of interest is shown). Note that the load was applied through an auto- calibration disk to eliminate errors due to imprecise mea- surement of the applied load and the photoelastic fringe constant. Figure 5 shows the portion of the experimental fringe pattern used for data acquisition, the data set extracted from this region and the computer-generated reconstruction of the fringe pattern using a six-coefficient approximation to the maximum shear stress field�9 Al- though the equations are nonlinear and an iterative procedure was used to solve for the coefficients, the solution converged rapidly. A comparison of the results from the local collocation method for determining the stress-intensity factor with the results from the boundary- collocation method for same problem is shown in Fig. 6. Several additional examples of the application of this algorithm for the analysis of dynamic isochromatic fringe patterns from running cracks are presented in Ref. 27. This reference also describes some practical matters related to the implementation of the method including: selection of points, number of terms to be retained, convergence criteria, etc.

S u m m a r y

The local collocation method provides a general procedure for analyzing full-field optical fringe patterns�9 The method does not depend on unique features of the fringe pattern, as in the apogee method, nor is it confined to the near-field region around the crack tip, as previous methods were. The method is also general in the sense that it can be used with a variety of stress functions and does not depend on the single-ended crack-tip expansion for the stress-field equations�9 By extension, the methods described can be applied to mixed-mode loading by adding to the optical equations the additional mode II terms for

246 �9 September 1989

the stresses. The method has proven to provide accurate answers with a small number of unknown coefficients. In part this increase in accuracy is the result of the high degree of data redundancy used in the method (typically six to ten data points are used for each unknown retained in the analysis).

Although the methods described in this paper are somewhat more difficult to implement than previous methods, the use of modern desk-top computers and digitizing equipment makes the methods practical. The increase in accuracy and the ability to infer additional information about the fracture parameters from the non- singular terms more than compensates for the additional time it takes to implement the algorithms. It is not difficult to envision how new image-processing equipment and feature-extraction algorithms could be incorporated into the experimental analysis in order to fully automate the procedure and produce results while the experiment is in progress.

Acknowledgments The development of the local collocation method

described in this paper has evolved over a period of ten years under the sponsorship from a variety of sources, including U.S. Nuclear Regulatory Commission, Oak Ridge National Laboratory, Sandia Laboratories, Office of Naval Research and the National Science Foundation. Additional financial support was also received from the Computer Science Center of the University of Maryland.

Fig. 5--Sequence of events for local collocation of the fringe pattern of Fig. 4

16

1 4

12

IC

6

4

2

C

i i i i

PHOTOELASTIC RESULTS

BOUNDARY COLLOCATION (Srowley, i976)

I [ I I I O.I 0!2 0.3 0!4 O,S 0.6 0.7 01.8

o/w

Fig. 6--Comparison of the dimensionless shape function for the three-point-bend geometry from boundary and local collocation

0.9

The author wishes to express his appreciation to Dr. G.R. Irwin for his advice and encouragement. Finally, the contributions of Dr. R. Chona over these last ten years are gratefully acknowledged.

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