Complex stress intensity factors evaluated by pseudocaustics

0013-7944/92 S5.00 + 0.00 0 1992 Pergamon Press pk.

Engineering Fracture Mechanics Vol. 41, No. 5, pp. 707-720, 1992 Printed in Great Britain.

COMPLEX

STRESS INTENSITY FACTORS BY PSEUDOCAUSTICS

EVALUATED

P. S. THEOCARIS National Academy of Athens, P.O. Box 77230, Athens 175 10, Greece Ahatract-A pseudocaustic is an illuminated pair of lines created by the same procedure as the caustics, but from reflections along any line on the boundaries or inside a stress field. The pair of pseudocaustics formed by a straight line, either traced and indexed at the vicinity of the tip of a crack or projected on the surface of the specimen by a coarse line grating, yields immediately by a simple and accurate measurement of the distance between the central points of these curves the values of the components of the complex stress intensity factor. The generatrix lines may be traced by a very fine means, the respective pseudocaustics may be as fine as desired and the distance between their central points accurately measured by simple methods of distance measuring. Then, the evaluation of the respective SIFs can be highly accurate, much more so than their respective evaluation through caustics. Furthermore, the method is very simple and versatile since the generatrix line may be displaced to any distance from the crack tip. It may be further used for the evaluation of the plastic stress intensity factor in an elastic-plastic field and allows the study of the progressive transition of K from the purely plastic into the purely elastic domain of the deformed body. Experimental evidence and comparison with theoretical values of SIFs indicates the validity and the accuracy of the method.

1. INTRODUCTION of reflected caustics is an optical method using the reflections of light rays from a curve around and in the vicinity of a singularity in a stress field, which form a strongly illuminated curve, the caustic, and which, among other things, yields in a simple and accurate manner the stress intensity factor at the respective singularity[l]. If, instead of this particular generatrix curve around an eventual singularity, a generic line inside or at the boundaries of the stress field is used, which should be conveniently marked and indexed, the illuminated curve constitutes a pseudocaustic. The method of pseudocaustics was initially applied to define the real length of the contact of two elastic bodies and determine the distribution of normal and tangential forces along the contact zone[2-6]. Thus, in ref. [2] the method was used to determine the stresses resulting from a concentrated load applied at the boundary of a half-plane or a wedge under plane stress, whereas in ref. [3] the case of a uniformly distributed normal or tangential load was anticipated. More important is the inverse problem where an arbitrarily distributed load on a straight boundary was evaluated by using data from the caustics created by the end load-singularities of the distribution[4] and the pseudocaustics generated by the deformed boundary of the half plane[5]. This was achieved by using a hybrid method of solving an ordinary integral equation where data from the pseudocaustics along the contact zone were used in order to define the kernel of the equation. Furthermore, the method of caustics and pseudocaustics was also used to define the intensity, slope and curvature discontinuities in the distributions of loads along any straight contact of two bodies[6], whereas the method of pseudocaustics was successfully used for the experimental solution of the general elastic plane-stress contact problem of two bodies of different materials and arbitrary shapes, one of which was finite and the other infinite[7,8]. Another interesting application of the method of pseudocaustics is to problems of rolling processes. The pseudocaustics yield the exact contact length between two elastic discs[9] and allow the evaluation of the coefficient of friction in the contact zone between an elastic disc and a semi-infinite body deformed plastically[lO]. Using data from pseudocaustics a practical application of the method was to define the roll pressure distribution and the coefficient of friction in hot-rolling where the assumption of plane-strain conditions in the rolls and stock is valid[l 11. As an extension of this method the three-dimensional deformation of rolls and stock for ingot hot-rolling was successfully defined in ref. [12]. If the nature of the deformation of the contacting boundaries is THE METHOD generatrix

707

708

P. S. THEOCARIS

large and plasticity is introduced along the contact, secondary pseudocaustics may be used defined by lines scribed at a distance from the contact boundaries which when combined with a representation through cubic-spline polynomials for the stress distribution may again allow the accurate evaluation of the friction coefficient[l3]. It was shown in an exhaustive study[l4] of the contact zone and the friction coefficient in hot-rolling that the method can be applied in an elegant and accurate way to problems of flat strip cold- and hot-rolling (plane-strain cases), as well as to ingot rolling (three-dimensional cases), yielding the real picture of variation of these quantities in a rather difficult set of problems. Another important application of the method of pseudocaustics concerns the general solution of any elasticity problem, especially those where two phases of a body are interacting. According to these methods, data from the pseudocaustics are used to evaluate the complex potentials Q(z) and Y(z) of Muskhelishvili describing completely the state of stress in either phase. The pseudo~austic provides discrete vaiues for the first derivatives @;,z(t) along the whole boundary (L,,) of either phase, which in turn allow the evaluation of the functions @Jz) inside the whole field of the problem. Furthermore, from the boundary conditions along both phases the other pair of complex functions Y,,Z(t) along the boundaries can be determined, which, in turn, yield the complex functions Y,,*(z) inside the stress field and in this way they complete the solution of the problem. This method was applied to the contact of fibers with their matrix in composite materials with successful results[ 15-l 71. While the method of caustics is a convenient method for determining, among other things, the stress intensity factors at singularities of the stress field, the method of pseudocaustics proved to be an efficient method for evaluating the order of singularities (SO), which is a critical quantity for the de~nition of SIF. The simplifying assumption that the order of singularity in an infinite elastic field of a homogeneous and isotropic material is always equal to f- l/2) is not true for a great number of practical problems. The method of pseudocaustics was used for determining the exact value of the order of singularity by tracing or projecting small circles at the vicinity of the singularity and measuring the lengths of the semi-axes of the ellipses and the orientation to which the circles were transformed in the projections. The method proved to be an efficient means for evaluating the SOS in various types of stress fields[l&201. In this paper the domain of applications of this important theory of pseudocaustics is completed by proving that simple pseudocaustics created by the tracing or projection of straight line segments at the vicinity of the singularity yield directly and by a simple length measurement the values of the components of the stress intensity factor holding at the singularity. The results of the method were tested over simpfe typical cases of stress singularities at crack tips of homogeneous elastic materials and were found to yield accurate and sensitive results.

2, THE PSEUDOCAUSTIC OF A STRAIGHT LINE INSIDE A MODE I CRACK TIP STRESS FIELD Consider the simple case of an elastic infinite plate containing an edge crack whose axis of symmetry is normal to the applied normal stress 6, at infinity (see Fig. 1). If a straight line segment AB is scribed or projected from a projecting screen on the surface of the specimen at a distance a from the crack tip and if this line segment is normal to the axis of the crack, the reflected image of this segment at a reference screen SC according to the theory of far-field caustics is expressed parametrically by the relations[l]: X = &,[Re(z) + 4C Re(@‘(z))] Y = &[Im(z) - 4C Im(@‘(z))]

(1)

where a(z) is the complex-potential function expressing the stress field of the plate, primes mean derivatives with respect to the complex variable z = (x +- iy) and X and Y are the coordinates of the points of the reflected image of the line on the screen SC placed at a distance z0 from the surface of the specimen.

Stress intensity factors evaluate

by pseudocaustics

709

Fig. I. ‘I%e front and rear caustics around the tips of edge cracks in plates under plane-stress conditions subjected to simple tension and the respective pseudocaustics formed by straight lines subtending an angle (a/2-2/?) with tbe crack axes j3 = 0” (a) and @ = IS” (b).

Moreover, A,,,denotes the optical magni~~tion

factor expressed by:

Zi+ ZO L, = ~ zi

where ri is the distance between the focus of the illuminating light source of the specimen. For a parallel light beam &, = 1.O since Zi-+O. Finally, the global constant C takes the following values for the three distinct cases of the fo~ation of caustics: (a) for caustics formed from light rays reflected from the front face of the specimen, C, = z,dq/i,,,

where cf = v/E;

(3)

(b) for caustics formed from light rays reflected from the rear face of the specimen, c, = -2z,dcJ&; (c) for caustics formed from light rays traversing the transparent C, = - q de, /A,,,;

(4) specimen, (9

where d is the thickness of the specimen and c, and c, are the stress optical coefficients of the material for the light rays either reflected from the rear or those traversing the specimen. For a crack subjected to a mode I deformation, as indicated by Fig. 1, the complex stress function Q(z) is expressed by its first-term approximation valid only at the vicinity of the crack tip and given by: @(z)= X2@+‘) (6) where p is the order of singularity which for the crack in a homogeneous and elastic body is equal to -3/2. The derivative of e’(z) with respect to z is then given by W(z) = KO, + I).?‘.

(7)

Along the straight line AB for which it is valid that: 2 = (a -I”iy)ly/ < b

00

where a is the intercept of the line AB, relation (7) becomes: V(z) = KO, + l)(a + iy)P.

(9)

710

P. S. THEOCARIS

Setting 2 = reie, then by means of eq. (8) one has for the generic point L on the line AB: rL = (a* + y*)‘/*

and

eL = tan-’ b/a).

(10)

Thus, eq. (7) becomes: Q’(z) = K(p -t- l)r{e’Q.

(11)

Then, the parametric equations of the pseudocaustic created by the linear segment by: X = &[a + 4CK(p + I)rf cospfl,] Y = &[a

-

AB are given

4CK(p + I)$ sin@J

(121

with rL and eL given by eqs (IO). In order to examine the form and position of the various types of pseudocaustics studied in this paper the following typical example is used throughout this paper. An infinite plate made of polymethylmethacrylate (PMMA = Plexiglas) was considered containing either a transverse edge crack, whose axis is normal to the direction of the applied external tensile load CJ, (Fig. la), or oblique, whose axis subtends an angle fl = 15” with the direction of the normal stresses 6, (Fig. 1b). The characteristic values for the material used are as follows: elastic modulus, E = 3.2 x lo4 kpfcm’; Poisson’s ratio, v = 0.33; stress optical coefficient for reflections from the rear, c, = 1.03 x 1O-5cm2/kp; stress optical coefficient for reflections from the front face, cf = - 3.22 x lo-’ cm2/kp. For the optical arrangement producing the caustics, a divergent light beam was used with a magnification ratio il, = 6.0 and a distance between the specimen and the reference screen 2, = 250 cm. Figures 2a, b and 3a, b present the pseudocaustics created by straight lines normal to the crack axis placed at distances a = 9 and 3 mm and a = - 16 and - 9 mm from the crack tip respectively. These distances a are measured on the reference screen in order to compare the pseudocaustics with the respective caustics. The positive distances are related to the rear face caustics while the negative distances correspond to the front face caustics. The radii of the initial curves of these caustics, measured on the reference plane SC, are given by: rp =

6.981 mm

r: =

14.532 mm.

(13)

It is clear from these figures that as the normal to the crack-axis straight line approaches the dimple created by the deformations of the lateral faces at the vicinity of the crack tip, its front and rear

admm rpt6980~ la)

lb)

r,‘“:14.522mm

Fig. 2. Caustics and p~ud~~tics formad by straight lines normal to the crack axis lying at a distance D from the crack tip. {a) a = 9 mm. (b) a = 3 mm.

Stress intensity facton evaluated by pseudocawtics

711

Fig. 3. Caustics and pseudocaustics formed by straight lines normal to the crack axis lying at a distance a from the crack tip. (a) o = - 16 mm. (b) D = -9 mm.

face pseudocaustics start to split with the pseudocaustic corres~n~ng to the rear face reflections always lying outside the respective caustic, whereas the front face ~ud~~tic is allowed to enter the domain of the external caustic, but again always remains outside the domain of the internal caustic. It is obvious that the external pseudocaustics approach and touch the rear face caustic much earlier (that is for larger distances a from the crack tip) than the internal pseudocaustics approach and touch the internal caustic. This is because the radius of the external initial curve r: is much larger than the radius rfin of the internal initial curve and therefore the extent of the influence of the rear face dimple is much larger in creating the rear face caustic. Figure 4 presents the family of pseudocaustics created by a bundle of parallel equidistant lines normal to the crack axis, as these are found from reflections from the rear face of the plate. It is interesting to point out that the straight lines passing at the vicinity of the center of the caustic, which are defined as the in~r~tion of the crack axis and the maximum transverse diameter of the caustic, present the larger deviations from their undeformed shape at the vicinity of the dimple around the crack tip. The maximum deviation appears, as is natural, for the straight line passing through the center of the caustic. The corresponding pseudocaustic presents a sharp refolding in a V-shape when it touches the respective rear face caustic and it then forms a large circle-like curve. However, this pseudocaustic is rather indistinguishable experimentally since it is created by sparse points of the generatrix line and it is therefore unusable in experiments. In order to determine the stress intensity factor of a mode I transverse crack a convenient genera&ix line, passing through the dimple surrounding the crack tip and, preferably, being close to the line coinciding with the maximum transverse diameter of the caustic at the reference screen, is traced or projected on the surface of the specimen from a conveniently ruled screen. The pair of p~ud~usti~ created from the rear and front captions present a symmet~ with respect to the crack axis. By measuring the distance between the intentions of the crack axis and the pair of pseudocaustics the respective SIFs can be determined. Indeed, relations (12) for y = 0 yield the equations: X = &{u - 2cKu-3’*}

(14a)

Y = 0.

(14b) Applying eq. (14a) for the rear and front pseudocaustics formed by the generatrix straight line x = u and subtracting these two equations we readily find that: a 312 dR K = 2&(Cr - C,) where dR = (Xr - X,), the abscissas of the intentions of the respective ~~~~ti~ symmetry axis of the crack. EFM 4iG-H

and the

712

P. S. THEOCARIS b=

. ,

f

l

.

,

30 mm

r,l” = 14.532

mm

. Fig. 4. The families of rear face p~~docaustics formed by a set of parallel equidistant lines, normal to the crack axis.

The global constants C, and C, correspond to reflections from the rear and front faces of the plate respectively, and they are well known for each material used in the tests. They are given by relations (3)-(5) for the various types of formation of the caustics and pseudocaustics. By measuring the distance dR = (KK’) in Fig. 1 and knowing the values of c,, a, A., and zO, we can readily determine the value for the mode I stress intensity factor Ki. Since the generatrix lines may be readily traced either with a very fine scratching or engraving procedure or projecting a very fine genera&ix straight line from a grating, the thickness of the pseudocaustic curve may be reduced considerably so that its position becomes very clear and distinguishable. The high definition of the pseudocaustic in & can be determined with a very high accuracy. Furthermore, by displacing the generatrix line at various distances Q from the crack tip, a series of pairs of pseudocaustics are formed, so that the value of K, may be multiply checked at a range of different positions. Indeed, it is clear from Fig. 4 that the maximum distance dR between two respective pseudocaustics is variable, depending on the value of distance a. It can be readily shown that for a certain optical set-up, this distance dR becomes maximum when the distance a is equal to a = 0.309 r, on the specimen, or a = 0.103 r, on the reference screen, where r, is the radius of the initial curve of the respective caustic[21]. This distance corresponds to a generatrix line coinciding with the maximum transverse diameter of the caustic. However, this line is ill-defined experimentally, since the amount of optical flux dist~buted along this p~udo~austic is a minimum, a fact which is clearly indicated by the sparse dots forming this curve and which correspond to equidistant markings on the generatrix line. Figure 5 presents the family of pseudocaustics created by the same bundle of parallel equidistant lines normal to the crack axis which, however, are now formed from reflections from the front face of the cracked plate. The pattern of pseudocaustics now becomes more complicated

Stress intensity factors evaluated by pseudocaustics

713

6 = 00

. . . . .. . .

fp

51

8.98 mm

,

Fig. 5. The families of front face pseudocaustics formed by a set of parallel equidistant lines, normal to the crack axis.

the pattern of Fig. 4 and it is therefore not suitable to use these complicated pseudocaustics for the evaluation of K,. The pseudocaustic corresponding to the generatrix line coinciding with the maximum transverse diameter of the internal caustic presents the characteristic feature that it is the only pseudocaustic refolding at its contact with the respective caustic under an acute angle disposing at this point a single tangent separating the two branches which are biforking there[l9]. In front of this typical pseudocaustic al1 other curves present smooth tangents with the caustic, remaining always on the one side of this tangent. These p~ud~ustics refold follo~ng smooth curves. However, a certain number of these pseudocaustics present smooth oval loops. All these curves should be avoided because of their complicated forms for the evaluation of K. Reference [22] presents a series of pseudocaustics formed by sets of parallel lines which are either parallel to the crack-axis or normal to it. Furthermore, concentric circles with the initial curves of the caustics are used for generating interesting pseudocaustics, which show the progressive evolution of these concentric circles to their limit, which is the respective caustic. Figures 4a, b and 5 of ref. [22] present such patterns. The analytic expressions for these pseudocaustics are very simple. The parametric equations of the concentric circles with center the crack tip and radii r, are given by: than

z = rmeie, 8 E[--K, n].

tw

When this relation is introduced into the expression for the ~uskhe~shvi~ stress function for a crack in an infinite plate made from an elastic and isotropic material it yields: W(z) = K(p + l)r&e’@.

(17)

Then the parametric equations for the pseudocaustics of these circles are expressed by: X = 1, (r, cos 0 + 4CK(p + l)rP, cosp8) Y=i,(r,sinB+4CK(p+l)rP,sinpO).

(18)

714

P. S. THEOCARIS

p=oo

1 (bl

,I

I

rin I:6.980mn

r,‘”I:14.532mm

Fig. 6. Caustics and pseudocaustics formed by families of concentric circles. (a) Rear face caustics and pseudocaustics. (b) Front face caustics and pseudocaustics.

Figure 6a, b presents the rear and the front face caustics and a pair of pseudocaustics corresponding to rm = 8 mm and 20 mm for the rear caustic and r, = 3 mm and 9 mm for the front caustic. The rear pseudocaustics resemble the respective caustics with longer or shorter tails depending on the radii of the generatrix circles, whereas the front pseudocaustics show a transition from pseudocaustics resembling the rear caustics but reversed to a caustic resembling the internal front caustic but with their cusp points, as the radii of the generatrix circles change from values close to rear initial curve radius to values close to the front initial curve. 3. THE PSEUDOCAUSTICS

OF A STRAIGHT LINE FOR A COMPLEX CRACK TIP STRESS HELD

MODE

We now consider the case of an oblique crack in an infinite plate of an elastic isotropic material under conditions of plane-stress and we study the form of the pair of pseudocaustics formed by a straight line traced parallel to the direction of the application of the external load at infinity. We assume further that the angle subtended by the axis of symmetry of the crack and the direction of loading is @, as indicated by Fig. lb. In this case the parametric equations of the caustic are given by: X = i, {Re(z) + 4C Re[@‘(z)]} Y = &{I&)

- 4CI,[@‘(z)]j.

(19)

Again, the stress field at the vicinity of the crack tip may be approximated by the Muskhelishvili stress function G(z) given by relation (6) and its first derivative with respect to z = (x + iy) given by relation (7). It is well known that the complex stress intensity factor is given by[l]: I(’ = I1Yle-”

(20)

K = (K: + K;,)‘:2

(21)

where

and K1= ~,(7tu)‘~~ sin 6,

K,, = cr,(x~)“~ sin fl cos fl

(22)

with w = tan-‘(~~,~~~)

(23)

and 2 = reHI.

(24)

Stress intensity factors evaluated by pseudocaustics

715

Then W(z) is expressed by: W(z) = lKl(p + I)rpe*@-@)

(25)

and the parametric equations of the caustics are given by: X = &,(a

COS 8 +

4C]K](p + l)fP

COS(@

-

0))

Y = J.m(Tiasin 8 - 4C]K](p + 1)P sin(p0 - 0))

(26)

where for the case of caustics it is valid that:

1

3ClKI IV5 Tin= [ L

(27)

and tano

= K,, /KI=cos/?.

(28)

For a mixed mode stress field at the vicinity of the crack tip the caustics (26) are similar to those for the mode I stress field with the difference that they are angularly displaced by an angle 20.1given by relation (28). For the parametric equations of the pseudocaustics formed by generatrix straight lines parallel to the direction of the external load and therefore subtending an angle /I with the crack axis, the equation of these linear segments AB in complex form is given by: z=(a+ycos&+Q~sin/3,

l.y]
(29)

where 26 is the length of the line segments. Setting: Z = rrLez

(30)

r,, = [(a + y cos /?)2 + y2 sin2 j?]“2

(31)

where

(32) and substituting into eq. (24) we obtain: G’(z) = IKl(p + l)&e”@‘ar -w).

(33)

Introducing now the derivative W(z) into the parametric relations (26) for the pseudocaustics where the radius ri, of the initial curve of the caustics is replaced by the radii r,, we obtain: x = Am{(a + Y cos

p) +

4CIKl(p + I)&

c0S(pe,, - 0))

Y = &,,{v sin /? - 4CIK](p + l)rfL sin( p8,, - a)}.

(34)

Figure 7a, b presents the pairs of pseudocaustics formed by lines parallel to the direction of loading when the oblique cracks subtend angles j3 with this direction equal to 30” and 45” respectively. According to the theory of caustics for an oblique crack in a purely tensile field the respective caustics are angularly displaced so that their axes of symmetry, defined by the polar radii passing through the cusps of the internal caustics, subtend an angle equal to 20, where the angle w is defined by relation (28). However, it is obvious that the pairs of pseudocaustics created by these generatrix lines do not present any kind of symmetry to any polar radius from the tip of the crack. This fact may be explained by eqs (34), where either pseudocaustic is derived from an overall constant C, or C,, which is different from the other one. Then, the second right-hand terms of eqs (34) are different for either caustic of the respective pair, whereas the first right-hand terms of these equations are the same. This fact creates this asymmetry in the pseudocaustics. Since any attempt to solve eqs (34) for particular values of the quantities involved is prone to large errors, we proceed for the evaluation of the complex stress intensity factor in mixed mode fields as follows: it is well known that for any kind of mixed mode deformation the caustics remain symmetrical to an axis subtending an angle 20 with the axis of the crack. In this case the pair of

716

P. S. THEOCARIS

\ $5300 r’“-b.077mm r).12.bSmm

I

r;“= b.59mm $:13.72 mm

(b)

Fig. 7. Caustics and pseudocaustics around cracks subjected to mixed mode loadings. (a) Angle of obliqueness ,9 = 30”. (b) Angle jI = 45”.

pseudocaustics can be created by projecting a fine straight line scribed on a reference transparent screen interposed in the optical bundle illuminating the specimen. This screen has the possibility of a fine rotation about a center of variable distance from the central point of the straight line segment. This distance may be selected to correspond to the chosen distance a for the position of the generatrix line. By rotating the reference screen about its center it is possible to define with a high accuracy its position for which both front and rear pseudocaustics are symmetrical. For this position the tangents of both pseudocaustics are parallel to each other and the points of their contact with the respective curves lie on the same polar radius from the crack tip. By measuring the angle subtended by this symmetric polar radius and the crack axis we define the angle 20 and therefore the angle B. By measuring the distance dR between the respective points K and K’ of the contact of the tangents with their pseudocaustics and using relation (15) we calculate the value of the mixed mode SIF, K. Then the components KI and K,, of the respective SIFs can be determined by using the relations: K = IK\exp( - io)

and

o = tan’(K,,/K,).

(35)

Figure 7 presents typical cases of oblique cracks with /I = 30” and 45” in a purely tensile field and the pseudocaustics created by a generatrix line subtending angles with the crack axis equal to fl, = (rc - 28) respectively. It is clear that both pseudocaustics are symmetric to this polar radius. By measuring the distance KK’ = dR, we readily define the value for (K(.

4. ILLUSTRATIVE

EXAMPLE

In order to check the theoretical values for the stress intensity factor K derived from the distances between the respective pairs of pseudocaustics formed by straight lines parallel to the axis of loading of the cracked plate, a series of tests were run with polymethylmethacrylate plates under conditions of plane-stress containing edge cracks whose axis was normal to the loading direction. For simplicity the plates were subjected to a pure tension load and the case with /I = 0” was considered. For the formation of the pairs of pseudocaustics a coarse grid of parallel lines with a frequency of two lines per millimeter was projected from different distances z, from the specimen. This grid was interposed between the specimen and the focus of a divergent light beam of a laser light source used to illuminate the specimen. Three different distances z, were selected yielding convenient arrays of pairs of pseudocaustics. Figure 8 presents one such array of pseudocaustics whose axes at the unloading state of the plate run normal to the crack axis. For each array of pseudocaustics three convenient pairs were selected presenting well defined pseudocaustics, whose distances dR were

Stress intensity factors evaluated by pseudocaustics

Fig. 8. Pseudocaustics formed by projecting a linear grating of a density of 2 lines/mm around the crack tip subjected to a mode I loading with the lines of the grating normal to the crack axis.

717

Stress intensity factors evaluated by pseudocaustics

719

measured, reduced to scale and averaged. The average values for the three mode I stress intensity factors were as follows: K, = 0.482a,,

K,, = 0.487a,,

& = 0.4720,

where the value given by applying the classical single-term approximation for the definition of the SIF from the respective caustic is equal to KC= 0.480,. The compa~son of the two sets of values for the Ki SIF from caustics and p~ud~austics indicates the validity and the accuracy of the proposed method of pseudocaustics. If one adds the great simplicity and versatility of the method, one may conclude that it constitutes a powerful method for evaluating stress intensity factors in plane elastic fields. The difference between estimating complex stress intensity factors and mode I factors lies only in the accurate definition of the angle u derived from the angular displacement of the caustic from the crack axis. This angle may be readily defined with high accuracy by the symmetric tangents to both curves for any pair of pseudocaustics. Therefore, the error introduced by the definition of angle o always remains infinitesimal and consequently the estimation of the complex stress intensity factor by the method of pseudocaustics may be achieved with the same accuracy and sensitivity as the respective case for the mode I SIF. Furthermore, the freedom existing in selecting the distance a between the tip of the crack and the generatrix straight line engendering the pair of pseud~austics gives the method a further versatility and presumably a higher accuracy. Indeed, by conveniently selecting a large enough that non-linear phenomena are negligible and, on the other hand, close enough to the crack tip so that the evaluation of K is executed inside the zone of influence of crack tip singularity, where the elastic stress field dominates, it is possible to assure a generally higher accuracy in estimating K in the elastic stress field. Furthermore, the method yields a sure experimental means to define the limits of validity of the simple elastic fracture mechanics theory and to indicate the eventual existence of a plastically deformed enclave around the crack tip in ductile materials.

REFERENCES [I] P. S. Theocaris, Elastic stress intensity factors evaluated by caustics, in ~~rirnent~ Determ~at~n of Crack-tip Stress Rttensity Factors, technics o~Fracfure, Vol. VII (Edited by G. C. Sih), Chapter 3, pp. 18%252. Martinus NijhotT, The Hague (1981). [2] P. S. Theocaris, Stress singularities at concentrated loads. Exp. Mech. 13(12), 511-518 (1973). [3] P. S. Theocaris, Stress singularities due to uniformly distributed loads along straight boundaries. Int. J. Solids Structures 9, 655670 (1973). [4] P. S. Theocaris and C. Razem, The end values of distributed loads in half-planes by caustics. J. appl. Mech. 45,313-319 (1978). [S] P. S. Theocaris and C. Razem, Deformed boundaries determined by the method of caustics. J. Strain Anal. 12.223-232

(1977). [6] P. S. Theocaris and C. Razem, Intensity, slope and curvature discontinuities in loading distributions at the contact of two plane bodies. Znt. J. mech. Sci. 21, 339-353 (1979). [7] P. S. Theocaris, The contact problem by the method‘of c&tics. Proc. 3rd Bulg. Nat. Congress Theor. Appl., Vama, Bulgaria, Vol. 1, pp. 263-268 (1977). [S] P. S. Theocaris, The load distribution in the generalized contact problem by caustics. Proc. 3rd ffulg. Nat. Congress Theor. Appl. Mech., Vama, Bulgaria, Vol. 3, pp. 105-143 (1978). [Q] P. S. Theocaris and C. Stassinakis, The elastic contact of two discs by the method of caustics Exp. Mech. 18(l), 409-416 (1978). [IO] P. S. Theocaris and C. Stassinakis, The determination of the coefficient of friction between a cylindrical indented and a plastically deformed body by caustics. ht. J. mech. Sci. 24, 717-727 (1982). [I I] P. S. Theocaris, C. Stassinakis and A. Mamalis, Roll resume distribution and coefficient of friction in hot rolling by caustics. Int. J. mech. Sci. 25, 833-844 (1983). [12] P. S. Theocaris, A. Mamalis and C. Stassinakis, On the pressure distribution and the friction coefficient between stock and rolls in ingot hot-rolling. Advanced Technology of Plasticity, Rot. First. hat. Con& Tokyo, Japan, Vol. 1, pp. 321-326 (1984). [13] P. S. Theocaris and C. Stassinakis, The contact problem studied by pseudocaustics formed at the vicinity of the contact zone. J. Engng Mater. Technol. 106,235-241 (1984). [14] P. S. Theocaris, A study of the contact zone and friction coefficient in hot-rolling, Anniversary Volume to Prof. W. Johnson (Edited by S. R. Reid), pp. 61-90 (1985). [15] P. S. Theocaris, Experimental study of plane elastic contact problems by the pseudocaustics method. J. Meeh. Phys. Solids 27, 15-32 (1979). [16] P. S. Theocaris, N. Ioacimides and C. Razem, The method of pseudocaustics for the experimental solution of simple elasticity problems. Int. J. mech. Sci. 23, 17-29 (1981).

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[17] P. S. Tbeocaris, Optical methods applied to composite materials research, in ~~e~~prne~z~ in Compasite ~a~eriais (Edited by Cr. S. Holister), Vol. 2, Chapter 6, pp. 165-201. Applied Science, Barking (1981). [18] P. S. Theocaris, Gauging the singularity in general stress fields by optical means. Proc. natn. Acad, Athens SO,437458 (I t--984). [19] P. S. Theocaris and G. Makrakis, Caustics and quasi~nformality: a new method for the evaluation of stress singularities. 2. ungew. Math. Phys. 40, 410-424 (1989). [20] P. S. Theocaris, Evaluation of the variable order of singul~ities by pseudocaustics. Exp. Mech. 33, 240-246 (lQQ0). [21] P. S. Theocaris and D. Pa&, Some further properties of caustics useful in mechanical applications. Appl. Optics 20, 4009-4018 (1981). [22] P. S. Theocaris, The caustic as a means to define the core region in brittle fracture. Engng Fracture Mech. 14,353-362 .I

(1981). (Received 29 November 1990)