Forman [5] have observed in tests with steel plates containing ... the elastic-plastic boundary around the crack tip is a circle, the radius of which was independent of work-hardening for small- ... (3). For the elastic plate under conditions of generalized plane stress, the sum of principal stresses defines the variation of the.
P. S. THEOCARIS Professor of Mechanics. Mem. ASME
E. GDOUTOS Senior Student, Department of Civil Engineering. The National Technical University, Athens, Greece
An Optical Method for Determining OpeningMode and Edge Slidinglode Stress-Intensity Factors Interference of the two partially reflected light beams from the front and back surfaces of a cracked thin plexiglas plate yielded a fringe pattern, which depicted the thickness variations of the plate due to loading. The interferogram at the vicinity of the crack tip consisted of a dense fringe pattern, which represented the constrained zone. The dense pattern was separated from the remaining sparce pattern by a caustic. The caustic was created by the reflected light rays emerging from the plate, which were twice refracted through the thickness and reflected on the back face of the plate and which were retarded according to Maxwell and Neumann's law. A general theory was developed where the dense fringe pattern was directly related to the complex intensity factor K * = Ki — iKu, combining the opening mode together with the edge-sliding mode of fracture. In this way, any combination of the two modes may be studied by this reflected shadow method and the contribution of each mode to fracture can be evaluated. A series of experiments in cracked plexiglas plates under longitudinal tension were executed. The internal symmetric crack in each plate was inclined to the axis of application of the load by a different angle 0, so that various combinations between Ki and i f i i were obtained and checked with theory.
Introduction I O B a cracked specimen under a state of plane stress, when the length of the crack is many times larger than its thickness, b u t comparatively small to the other dimensions of the specimen, so that the applied cross-sectional stress at infinity is sufficiently small, the plastic zone engendered at the crack tip may be regarded as negligibly small, and therefore not influencing the elastic stress singularity at the crack border. Such an elastic analysis was introduced by Irwin [ l ] , 1 who related the stress and displacement fields associated with each loading mode of the cracked plate at the vicinity of the crack tip with the corresponding stress-intensity factor. Williams [2] has shown that the radius of the embedded plastic zone at the crack tip m a y be directly related to K, b u t he did not estimate the plastic zone size. 1 Numbers in brackets designate References at end of paper. Contributed by the Applied Mechanics Division for publication
(without presentation) in the JOURNAL OF APPLIED MECHANICS.
Discussion on this paper should be addressed to the Editorial Department, ASME, United Engineering Center, 345 East 47th Street, New York, N. Y. 10017, and will be accepted until April 20, 1972. Discussion received after the closing date will be returned. Manuscript received by ASME Applied Mechanics Division, November 20, 1970; final revision, April 6, 1971. Paper No. 71-APM-QQ.
Journal of Applied Mechanics
Various shapes of plastic zones were observed during experiments. Thus Dugdale [3], Rosenfield, Dai, and Hahn [4], and Forman [5] have observed in tests with steel plates containing through the thickness cracks, t h a t the plastic zones have shapes and magnitudes, which were consistent with the wedge-shaped zone assumed by the Dugdale model. Hahn and Rosenfield [6] distinguished two different types of plastic zones, one associated with plane-strain conditions in the plate, and the other with plane stress. In the plane-strain type of plastic zone two yielded areas were spread out in approximately normal directions to the crack line (plastic-hinge type), while in the plane-stress type, large tapering wedges were projected forward, in front of the crack from the ends of the two spreads (plastic-wedge type). On the other hand, Ault and Spretnak [7] with sharp notches in molybdenum, and Gerberich [8] with cracks in several aluminum alloys, have detected plastic zones, which resemble the hinge type, with a plastic area approaching to a circular shape. Analytical work by Stimpson and Eaton [9] and by Swedlow, Williams, and Yang [10] revealed plastic zones of the hinge type. Furthermore, McClintock and coworkers [11] have studied the special case of cracked plates under an antiplane mode of loading, which occurs in torsion and longitudinal shear. In this case the 3e of plastic zone again resembles the hinge type. Other
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analytic elastic-plastic solutions have been developed for a limited type of model, which pertain to different simplified assumptions. Thus Hutchinson [12] expressed the form of the plane-stress singularities at the vicinity of the crack tip by using techniques identical to those of plane strain. Rice [13-16], in a series of papers, studied the case of sharply notched plates of a work-hardening or elastic-plastic material, loaded in antiplane longitudinal shear. I t has been established in these papers t h a t the elastic-plastic boundary around the crack tip is a circle, the radius of which was independent of work-hardening for smallscale yielding. Similar results were obtained by Rice [14], and by Irwin and Koskinen [17], for the case of a perfectly plastic material. Since the existing analytic elastic-plastic solutions are limited to cases of physical models corresponding to extensive simplifying assumptions, experimental techniques are appropriate for the study of constrained zones and comparison with theory. Oppel and Hill [18], and Bateman, Bradshaw, and Rooke [19] applied an interferometric technique for the study of plastic zones in aluminum alloy plates, which revealed hinge-type plastic zones. More recently, Dudderar [20] applied holographic methods in aluminum alloys with the same results. Theocaris [21-23], in a series of papers, studied the constrained zones in plexiglas plates elastically loaded under an opening mode of deformation and proved theoretically and experimentally that the shape of the constrained zone is a circle, which was directly connected with the corresponding stress intensity factor Ki. In this paper the theory contained in reference [21] was extended to include the study of constrained zones corresponding to any combination of the two principal modes of deformation (modes I and I I ) . Experimental evidence corroborated the theoretical results.
When both modes of deformation are superimposed the total sum of principal stresses equals \Ki
(8) For D = 0, it yields
iD U
(I)'
t)
K\ cos — — Ku sin -
+ [ Ki sin - + Ku cos
)•]
(9)
cos
-, 2'
- f r - ' A Kn sin - , f r - , / j Ku cos -
sin
-J
and
Q'R'
From these two vec-
tors, P ' Q ' subtends an angle i?/2 with r, while Q ' R ' is always normal to P ' Q ' . If w is referred to the system O'x'y', lying on the plane of the screen with its origin at the crack tip 0 ' , and which corresponds to ithe i, j system of the middle plane of the specimen, then w is expressed as w = fr" «A
Ki cos
3#
Ku sin
(14)
c = r|x*|
(15)
«-w
(16)
(10)
(2TT) 'A
fr-'/'Ki
Khi)l/i
we obtain
Zutc
Vector w is referred to the Cartesian system (u, v), which lies on the plane of the screen (Sc), with its origin at the projection P' of the point P on the screen, and which rotates together with angle $. . I t can be deduced from relation (9) that w, which is represented in Fig. 1 by P'R', consists of two component vectors
P'Q' ( - fr-'/'Ki
(X s +
Introducing the complex stress intensity factor K* = Ki — iKu, we have \K*\2 = (K\ + IOu). Introducing now a new constant C, so that
where the constant f is expressed as
r=
3# . 3#\ T^ cos — - Kn sm — \
which result was found in reference [21]. The radius r0 of the constrained zone depends on the modulus of the complex stressintensity factor, as well as on the distance 20, the material properties of the plate and the crack length 2a.
Constrained Zone Projected on Screen Sc When the constrained zone is projected on screen Sc, its boundary is described by equation (12). This expression for W represents a generalized epicycloid, the parametric equations of which on the plane O'x'y' are given by
3#
3# '30 r + ( K[ sm — + A n cos —
Relation (16) shows that the envelope of the constrained zone for the general case, where both the opening mode and the edgesliding mode of deformation are present at the crack tip, is a circle of radius r0. The same result was found earlier [21] for the case when only the opening mode is operative. Indeed, if Kn = /3fXi\2/i 0 it can be deduced from equation (14) that r = r0 = I —-— I ,
(11) x' = r0 cos •& + £Kir,mo:t and Sj" 101 yield the corresponding value for r0 which introduced into relation (18) determines Ki and Ku. Solving relation (18) for Ki yields 2r„V> #1 =
Experimental Evidence
6.0
40
+ tf
3f Vl
(21)
and
For the study of the development of the constrained zone around an oblique crack, a series of tests were undertaken with thin plexiglas plates subjected to tension. The plates contained a central slit of a length 2a = 9 -r 10 mm, forming an angle /3 with the longitudinal axis of the plate. Six cases were considered where angle /3 took the values 90, 75, 60, 45, 30, and 15 deg. The thickness of the plates was t = 1.81 mm while their width b = 50 mm. The replacement of the internal stationary crack by a central slit is in agreement with the simple linear approach of the stress distribution considered in this paper. The slit length 2a R* 10 mm was considered very small as compared to the width b of the plate, so t h a t it can be assumed that the crack is remote from any other boundary of the plate. The plexiglas plates were conveniently selected with almost flat surfaces, which were optically tested to yield sparce interferograms. The plates were interposed in a monochromatic light beam emitted from a He-Ne gas laser, Fig. 6. The light beam from the laser was widened by a lens and reflected on both surfaces of the slotted plate. The distance 20 between the midsurface of the specimen and the screen was z0 = 112-120 cm. A photographic camera Ca recorded the interference pattern, which was projected on Sc. The magnification ratio X was expressed as X =
Zj + Zp
(20)
,.»/2
2\ir Ku
^
Introducing into these relations the values for ro derived from the ratios D,/S, or D , - " 1 / ^ " 1 0 1 and D, mi "75, mi ' 1 we obtain the general relationships
Ku =
1.671 (D\h
1
z„tc \dj
V l + n* (22)
*
L5Z1 (*>Y' Zotc \Sj
V l + n*
where the ratio D/8 takes either of the values Dt/5„ Dim"/Simn, min min D, /
