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PHOTOELASTIC STRESS ANALYSIS OF CRACK PROPAGATION
WITHIN A COMPRESSIVE STRESS FIELD
by
EMANUEL G. BOMBOLAKIS
Geol. Eng., Colorado School of Mines(1952)
M.S., Colorado School of Mines(1959)
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMEXNTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February, 1963
Signature of Author .t. .-. ......... .* g *..-. ..- rnet**a0v&..... **.
Department of Geology and Geophysics, January 7, 1963
Certified by .......Thesis Supervisor
Accepted by ....................... . i tt........................... .
Chairman, Departmental Committee on Graduate Students
ABSTRACT
Title; Photoelastic Stress Analysis of Crack Propagation withina Compressive Stress Field.
Author: Emanuel G. Bombolakis
Submitted to the Department of Geology and Geophysics onJanuary 7, 1963 in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy at the MassachusettsInstitute of Technology.
Brittle shear fracture may not result from the growth of asingle flaw, as in the case of fracture produced by purely tensilestress. The coalescence of two or more flaws apparently isnecessary to form a macroscopic shear fracture surface or fault.When a critical Griffith crack, isolated within a compressivestress field, begins to grow, it curves out of its initial crackplane into a direction approaching that of the maximum-compression,r3. Propagation ceases when this orientation is attained, for thestress level required to initiate fracture is not sufficient tomaintain it; i.e., tensile stress concentrations at the heads ofthe fracture diminish below the theoretical tensile strength ofthe material at this stage of fracture. Photoelastic stressanalysis indicates that (1) a particular en echelon distributionof Griffith cracks and (2) a crack separation of somewhat less thantwice the crack length are required to permit coalescence of flawswhen the applied compression is of smaller magnitude than thetheoretical tensile strength of the material.
Thesis Supervisor: William F. Brace
Title: Associate Professor of Geology
TABLE OF O0NTENTS
Abat~ct * * * * * * * * * * * * * * * * * , * * * *. *99909....9 0~eOeOO 0O...0*O 0e
Introduction.*..........................*.........
Extension of the Griffith Theory.....................
Experimental Method.......*...*.*.***.....--.--
Single-Crack Problem.................. ......
Representation of crack shape by an ellipse.....
Fracture analysis of a critical "Griffith crack"
Many-Crack Problem..................... ..
Test of crack separation...............--....
Effect of en echelon distribution of "GriffithcracksR on tensile stress concentrations........
Coalescence of "Griffith cracks"............
Conclusions and Implications......................--
Acknowledgments...........................-----------
Suggestions for Further Investigation................
Biography of the Author..............................
Appendix A: Extension of the Griffith Theory........
Addendum...................------...-----------
Appendix B: Experimental Method.....................
Phn+nelastic method........* . .-----------..
Experimental technique...................
iii
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1
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6
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36
1-A
9-A
1-B
1-B-------. 9*.
- ANW-0- - _,__ - , __ -, -,- __ __ - ____
........-----.-- 8-B
Page
Appendix C: Experimental Results............. .. 1-C
Representation of crack shape by an ellipse............ 1-C
Fracture analysis of an isolated, critical triffith crack.lO-C
Test of crack separation........ .............
Analysis of en echelon distributions of *Griffith cracks..25-C
Appendix D: Error Analysis...... .*...*.*.*.*.**.* ........ 1-D
Bibliography.. .... ............................................ 37
ILLUSTRATIONS
Text
Page
Elliptic coordinates and definitionof crack boundary ................................ 3
Several of the successive stages in thedevelopment of "branch fractures" emanatingfrom a critical "Griffith crack".................. 15
Fracture produced at a critical *Griffithcrack" isolated within a compressive stress
........................................ 18
"Fracture" of "Griffith cracks" with a"crack separation" of twice the "crack length".... 21
Orientation of a shear fracture surface, asa function of overlap, crack separation, andinclination of en echelon Griffith cracks......... 24
Effect of varying overlap, between "Griffithcracks", on the "fracture" of en echelon"Griffith cracks* with three-fourth's "cracklength" separation............................. 25
Appendix BPage
Figure 1-B:
Figure 2-B:
Simplified, diagramnatic representationof a polariscope...............-.--.....--.
Loading system designed to produce a fieldof uniform compression within aphotoelastic plate..................
.. 2-B
..10-B
Appendix CPage
Figure 1-C: Isoclinic pattern produced by a dumbell-shaped hole in a plate subjected to uniaxialcompression perpendicular to the long axisof the hole..............................-----
Figure 1:
Figure 2:
Figure 3:
Figure 4:
Figure 5:
Figure 6:
-. 2-C
Pare
Figure 2-C:
Figure 3-C:
Figure 4-C:
Figure 5-C:
Figure 6-C:
Figure 7-C:
Figure 8-C:
Isoclinic pattern produced by a slot-shaped hole in a plate subjected touniaxial compression perpendicular tothe long axis of thehole........................ 3-C
Isoclinic pattern produced by an ellipticcavity in a plate subjected to uniaxialcompression perpendicular to the majoraxis of the cavity............................... 4-c
Isoclinic pattern produced by a slot-shaped hole, inclined at 404 to theaxis-of applied compression...................... 7-C
Isoclinic pattern produced by an ellipticcavity, with a major axis inclined at 404to the axis of applied compression............... 8-0
Comparison of the theoretical andphotoelastically measured variations ofstress at the boundary of a criticallyinclined elliptic cavity.........................ll-C
Comparison of theoretical and measuredstress variations at boundaries of inclinedelliptic cavities having a "crack separation"of twice the length of their major axes..........21-C
Comparison of the theoretical variation ofstress, at the boundary of a criticallyinclined cavity in an infinite plate subjectedto uniaxial compression, with the measuredvariations of stress at elliptic cavities ofthe same orientation and shape, but differingin their proximity to neighboring ellipticcavities..........*...* .**... -- - - .--O--- - .24-0
Plate I: Comnarison of interference fringe patternsproduced at cavities of varied shape................ 5-C
Plate II: Photoelastic stress analysis of theSingle Crack Problem.....*.. ...........------- *-.12-C
Plate III: Photoelastic stress analysis of theeffect of 'crack separation"......... ........... 20-C
Plate IV: Photoelastic stress analysis of theeffect of half "crack length" overlap............2
6-C
PHOTOELASTIC STRESS ANALYSIS OF CRACK PROPAGATION
WITHIN A C0MPRESSIVE STRESS FIELD
Introduction
The ordinary tensile strength of a brittle material is far
less than its theoretical tensile strength, often by as much as
a factor of 103. The Griffith theory of brittle fracture (Griffith,
1924) explains this great discrepancy by postulating that a natural
solid material contains flaws. The effect of such flaws, or
*Griffith cracks", is to provide local sources of high tensile stress
concentrations, even when the material is subjected to compressive
stresses. Fracture ensues when the value of the most critical ten-
sile stress concentration equals the theoretical tensile strength.
The Griffith theory has proven to be a fundamental criterion
of brittle fracture, under tensile loading conditions, for isotropic
materials such as glass (Griffith, 1921, p. 172). Whether it is
valid for brittle isotropic materials subjected to compression -
and, in suitably modified form, whether it is valid for rock
materials in compression -- remains to be determined. One step
in the right direction towards this end is to inquire as to the
directions in which flaws will propagate in consequence of the
conditions given in the Griffith theory. This already has been
done for the case of simple tension (Wells and Post, 1968'), but
not for the case of uniaxial compression.
We consider here the crack propagation of an elliptically-
shaped flaw isolated within a compressive stress field, developing
Griffith's theory to include two-dimensional elliptic flaws of
finite width and to predict their initial direction of crack
propagation, then proceeding with photoelastic stress analysis
to determine the subsequent path of propagation. The effect of
neighboring flaws on the crack propagation of a dangerous crack
also is considered.
Extension of the Griffith Theory
Griffith assumed that a crack can be represented two-dimensionally
by an eccentric ellipse. The state of stress at the elliptic boundary,
for the case of an elliptic cavity in a plate subjected to biaxial
plane stress (Ode, 1960, p. 295), is
whereCOILL a r. - CO.47I'
and are elliptical coordinates defined by x = C-coshl cost
and y = C'sinhy sin , with C being the focus (see
Figure 1);
is the member of the family of confocal ellipses that
defines the boundary of the elliptic cavity;
) represents the variation of principal stress acting
at T. , parallel to the crack boundary;
P and Q are the applied principal stresses, so defined that
Q(P algebraically, with tensile stresses positive
and compressive stresses negative; and
'9 is the angle of inclination of Q, measured as shown
in Figure 1.
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-
_ ~ -~
The conditions for extremals are . - = 0 and --
which respectively yield
(2) e~4~' e Dw~
and
Aw ree coo,a --2
Equation (2) implies that must be a small quantity of order r,when , is very small; i.e., when the elliptic cavity is highly
eccentric. As Ode' (1960, p. 296) has shown, this approximation
leads to the results of Griffith; e.g., when 3P / Q<O,
(4) cosa = -
Ode' also shows that
(5) Max.( 1 * (
Equation (4) determines the critical orientation of the
Griffith crack with respect to P and Q, and (5) is the tensile
stress concentrationacting somewhere near the ends of the crack.
These equations do not apply to an elliptic cavity unless it is
highly eccentric, nor can the initial direction of crack propaga-
tion be predicted adequately without knowing the locations of the
tensile stress concentrations.
However, the above-mentioned approximation need not be made.
It can be shown more generally (see Appendix A) that
(6) cos 2) =2 - - 2 sinh
and
(7) Max. (a/b)2 2
4ab PQ ,
with the positions of the maximum tensile and compressive stress
concentrations given by
(8) cos 2 = cosh 21 - 2 ( 2 . ,2. h 2 2 ,.
The magnitude of the maximum compressive stress concentrations is
shown in Appendix A to be three times that of the maximum tensile
stress concentrations. In (7), a and b resnectively represent the
semi-major and semi-minor axes of the elliptic cavity.
There are two criteria determining the most dangerous elliptic
flaw. Equations (6), (7), and (8) comprise one criterion. However,
if the applied stress is tensile, or if values of P, Q, and
cause (6) and (8) to take on values 9 fl or (-1, then (7)
does not apply. Instead, the condition for extremals implied by
(2) and (3) is A4' 2 :A ' 0 ,and the appropriate
stress concentrations are those given by Timoshenko and Goodier
(1951, Chapter 7). Griffith's expression, 3P / Q <0, for the
validity of (6) and (7), follows from these conditions if the
elliptic cavity is highly eccentric.
The initial direction of propagation for fracture now may be
calculated. If the crack boundary is defined alternatively by
2 2
2 b2a b
then the slope at any point on the boundary is
b2
dx) y
Referring to Figure 1, this may be expressed as
tank: = - - cot = -cotC.
Thus, the initial direction of propagation is given by ( , where
(9) tan* a tanD(
Once fracture begins to propagate, the problem becomes too
difficult to be treated analytically, for the boundary of the
fracture surface no longer can be represented by an ellipse, but
the problem can be investigated further by means of a photoelastic
stress analysis. The photoelastic method proposed in this
investigation analyzes the problem in a static step-by-step
manner, whereas actual fracture is a dynamic process. However,
the photoelastic experiments of Post (1951) and Wells and Post
(1959) have demonstrated that the results of a static analysis of
tensile fracture compare favorably with the results of a dynamic
analysis.
Experimental Method
Basically, the experimental method employed is to prepare a
model by cutting a suitably oriented elliptical cavity in a plate
of photoelastic material, to determine photoelastically the positions
of maximum tensile stress concentrations at the cavity when the plate
is subjected to plane stress, to cut away an increment of material in
a direction perpendicular to the "crack boundary" at each position of
the tensile stress concentrations, then to determine the new positions
of the maximum tensile stress concentrations photoelastically for the
same state of applied stress, and to remove another increment of
material in a direction perpendicular to the extended "crack boundary"
at each new position of the tensile stress concentrations. Proceeding
in this manner, the elliptical cavity gradually is extended to
simulate the geometry of fracture that should result as a critical
Griffith crack propagates.
Photoelasticity is based upon the phenomenon of temporary
double refraction produced in certain transparent materials when
they are subjected to stress. A model of a given problem is made
from photoelastic material, then loaded in the same manner as the
problem under consideration, and placed between Polaroid sheets,
normal to the incident light, in an instrument called a polariscope.
When the model is viewed in the field of polarized light, two
phenomena are observed. One is the isoclinic, a black band or
fringe denoting the locus of points of common inclination of the
principal stresses. The other is the interference fringe, along
which the principal stress difference, -Wfo , is of constant
magnitude. The relation between stress and optical effect is
linear, and is given in most texts on photoelasticity as
(10) a--Qr F.C.** J t
where F.C. is the fringe constant of the mate-rial, t is the
thickness of the model, and n is the order of the interference
fringe. At a free boundary, one of the principal stresses is zero;
the other is calculated directly from (10).
In the black-field arrangement of the polariscope, obtained
by crossing the planes of transmission of the Polaroid sheets,
integral-order interference fringes are produced with n = 0, 1, 2,
3, ... The light-field arrangement is obtained by rendering the
transmission planes parallel. In this case, half-order interval
interference fringes are produced with n = 1, l-, 4, ... Thus,
appropriate values of n in (10) determine, for example, the stress
along the boundary of an elliptic cavity in a plate subjected to
compression.
Axial compression loading of the sample was produced with
portable Blackhawk pumps and rams. The punp-ram-pressure gage
assembly was calibrated with dead-weight loads to determine
friction, as a function of load level, in the loading system.
Accuracy was 10 psi. The rams were braced in thc plane of the
loading frame to achieve and maintain axial loading.
Ram loads were distributed through a system of steel bars and
pins to produce a field of uniform compression within a 4
photoelastic plate (see Appendix B). Plexiglass models were made
to determine isoclinics; plexiglass has such a high fringe constant
that interference fringes do not obscure the isoclinics. Interference
fringes were obtained in models made from Columbia Resin #59. A
white-light source was employed in isoclinic determinations, whereas
a mercury-green-light source provided monochromatic light for the
study of interference fringes. Introduction of quarter-wave plates
in the polariscope permits isoclinics to be eliminated without
affecting interference fringes (see Frocht, 1941). Thus, isoclinics
are prevented from obscuring the interference fringe patterns.
Each model tested was held by a plexiglass retaining jig,
bolted to the loading frame, to prevent axial bending and to
maintain the plane of the model in the plane of the loading frame.
A check for bending is provided by a particular property of the
isoclinics formed in the model. Wherever points of zero stress
occur in a nonhomogeneous stress field, the directions of the
principal stresses are indeterminate. Four such points occur at
the boundary of an elliptic cavity in a plate subjected to plane
stress. Consequently, isoclinics in the vicinities of points of
zero stress must converge at these points. If bending were
involved, this condition would no longer hold. Thus, the observa-
tion that the isoclinice did converge at the points of zero stress
provided evidence that bending was negligible.
Single-Crack Problem
Representation of Crack Shape by an Ellipse:
Is it reasonable to represent the longitudinal cross-section
of a crack by an ellipse having the same length and radius of
curvature at the ends? According to Inglis's theoretical
- dwaii-m-
analysis (1913, p. 222), if the ends of a cavity are approximately
elliptic in form, then it is legitimate in calculating the stresses
at these points to replace the cavity by a complete ellipse having
the same total length and end formation. This is a basic tenet of
the Griffith theory.
An experimental test of Inglis's conclusion was performed by
photoelastic stress analyses (see Appendix c) on a dumbell-shaped
hole, a slot, and an elliptic cavity having a common length (i")
and radius of curvature (,4 = 1/16) at the ends. Disturbances of
the applied compressive stress field by the presence of each cavity
were remarkably similar, even with respect to positions of points
of zero stress on the boundaries of the cavities. Each cavity was
oriented with its long axis perpendicular to the applied load. The
stress concentrations at cavity ends were found to be the same, and
in satisfactory agreement with the stress concentration theoretically
predicted. Murray and Durelli (1993) made a photoelastic stress
analysis on an elliptic cavity of this orientation, for various
combinations of biaxial stress. Their experiments confirm equation (2).
Another test was made of the Inglis conclusion, this time on a
slot with its long axis inclined at 400 with respect to Q. The
angle of orientation was calculated from equation (6), setting
P = 0, for a critically-inclined "equivalent" ellipse, to be
circumscribed about the slot, having the slot's length (il") and
radius of curvature ( = 1/16") at the ends. Thus, after a photo-
elastic stress analysis was made of the slot in a field of uniaxial
compression, the slot was enlarged artificially into the form of
the ellipse circumscribed about the slot. Then, this elliptic
cavity' s disturbance of the superimposed stress field was analyzed
photoelastically.
In the case of the slot, the compressive and tensile stress
concentrations were found to be in the ratio of 3:1, precisely as
predicted for a critically-inclined elliptic cavity, but they were
approximately 33% greater in magnitude than those obtained at the
1equivalent" elliptic cavity for the same applied stress. In
equation (7), the factor 2 is related to the eccentricity
of the ellipse, or to departure of shape of an elliptically-shaped
cavity from the shape of a circular hole. Now, if an ellipse,
having the same length and width as the slot, instead is chosen
for comparative calculations, good agreement is found between the
stress concentrations obtained at the slot boundary and those
calculated for an ellipse inscribed in the slot. For example, the
maximum tensile stress concentration generated at the slot boundary
was (1.51 d .05)*4I , whereas equation (7) gives (1.56 .06).iQI
for the elliptic cavity just described vs. (1.12 .01).+41 for
the ellipse circumscribed about the slot. The probable errors
shown here were calculated by the method given by Topping (1957,
p. 20); see also Appendix D.
An alternative suggestion, made by J. Walsh of M.I.T., possibly
reconciles the two cases of orientation treated above. He suggests
that a more pertinent modification of the Inglis conclusion may be
to replace a flaw by an ellipse having the same length, but also
having the same radius of curvature at its point of maximum tension
as at the point of maximum tension on the flaw boundary. A check
on this may be made in the case of the inclined slot. The radius
of curvature at the point of maximum tension of the inclined slot
is approximately 0.063". The radius of curvature, calculated for
the theoretical point of maximum tension at the boundary of the
ellipse inscribed in the slot, is 0.065".
Fracture analysis of a critical "Griffith crack";
As mentioned above, a photoelastic strees analysis was made of
an elliptic cavity, U" long with a radius of curvature of 1/16" atits ends, critically inclined at 400 with respect to the axis of
uniaxial compression. The magnitude of the measured tensile stress
concentrat ions agrees with that predicted by equation (7), and
their measured positions on the cavity boundary ( T ~ 300 o 30,
2100 Z 30) also provided experimental confirmation of equation (8).
Thus, the initial direction of crack propagation predicted by
equation (9) would be C = 500.
A small circular cut was made in this direction at each of
the two points of maximum tension, normal to the plane of the
model, to simulate an initial stage of fracture. Once fracture
actually begins, the direction of propagation no longer is defined
by (9). Referring to Figure 2, imagine that each of two vectors
is drawn outward from, and perpendicular to, the cut at each new
position of maximum tension. Let each vector be defined by an
acute angle Od , measured between the major axis of the former
elliptic cavity and the vector. Then /d defines the new direction
of crack propagation.
The eft ect of varying the radius of curvature of the circular
cut was tested by increasing the radius of curvature from 1/52"
to 1116", equal to that at the ends of the elliptic cavity. The
new positions of maximum tension were unaffected; i.e., ,4' remained
constant with a value of approximately 55 . However, the 1/32"-cut
gave rise to a maximum tension of (1.55 .05).IQ as opposed to
(1.12 .05) 441 obtained when the radius of curvature of the cut
was enlarged to 1/16". But this latter value of maximum tension
is the same as that generated on the unmodified boundary of the
elliptic cavity. Evidently, the radius of curvature at the tips
of an initial fracture cannot be much greater than the radius of
curvature at the ends of the crack which gives rise to fracture,
otherwise the maximum tension developed would be below the
theoretical tensile strength of the material. If the radius of
curvature at the tips of the initial fracture is smaller than that
at the ends of the Griffith crack, then brittle fracture likely
proceeds initially with an accelerating velocity.
It seems reasonable to assume, on the basis of the structure
of matter, that at the tips of the propagating fracture, the radius
of curvature is of similar magnitude as the radius of curvature at
the ends of the crack that gives rise to fracture. Henceforth, all
artificial "fractures" will be cut with a radius of cufivature at
"fracture" tips equal to the radius of curvature at the ends of
the "Griffith crack" machined in the photoelastic model.
Figure 2 shows several of the successive stages in the
development of "branch fractures" having a radius of curvature
of 1/16" at their ends. The "branch fractures" were extended by
increments equal to the radius of curvature, with each step
analyzed photoelastically. Angle df gradually diminished from
550 to almost 400; i.e., the "branch fractures" arched into
positions semi-parallel to the direction of uniaxial compression.
During the transition of Af from 550 to 450, the maximum tension
at the heads of the "branch fractures" maintained the same
magnitude as the maximum tension of (1.12 .05)-4Qj that had
developed on the unmodified boundary of the stressed elliptic
cavity. However, as 4fddecreased from 450 to almost 400, the
tensile stress concentrations at the heads of the "branch fractures"
diminished to a constant value of (0.85 Z .05)4 Q . Further
artificial extension of the "branch fractures" did not alter this
value, nor the value of / .
I
-4--- -4
f -
The implication of these results is that a critically inclined
Griffith crack will produce branch fractures that may cease to
propagate when they approach an orientation parallel to the
direction of uniaxial compression, unless
(a) the radius of curvature at the heads of the branch
fractures is small enough to ensure that the tensile
stress concentrations there do not diminish below the
theoretical tensile strength of the material; or
(b) the kinetic energy associated with fracture contributes
to the crack extension force sufficiently to continue
propagation of the branch fractures; or
(c) the effect of other nearby flaws facilitates extension
of fracture.
Case (a) is unlikely because, if the branch fractures continue
to extend parallel to the direction of uniaxial compression, the
geometry of the fracture becomes less asymmetric and, according
to Inglis's analysis, it may be replaced by an ellipse with a major
axis parallel to the direction of uniaxial compression. For such
an oriented elliptical cavity (Timoshenko and Goodier, 1951, p. 203),
the magnitude of the tensile stress concentrations is the same as
the magnitude of the applied compression, regardless of the radius
of curvature at the ends. Therefore, the magnitude of the applied
compression actually would have to be increased until it equaled
the theoretical tensile strength of the material before the branch
fractures could continue to propagate.
Case (b) also is unlikely. Curvature of the "branch fractures"
takes place over only a modest distance in comparison to "crack
length". Apparently, there would be too little time for a
significant build-up of kinetic energy by the time the branch
fractures become parallel to the direction of uniaxial compression.
To test the results of the static photoelastic analyses, and
their implications, it was decided to perform a dynamic test of
fracture. A 2"-long slit was cut with a 1/32"-diameter end mill
in a 0.085"-thick plate of plexiglass, which measured almost 6" in
width and 4 1" in height. The critical inclination of the slit with
respect to Q was in the range 300 4 , based on
calculations of (6) for an ellipse circumscribed about the slit
and for an ellipse inscribed in the slit. The inclination chosen
was 320. This is very close to the theoretical inclination of the
most dangerous Griffith crack when the applied stress is uniaxial
compression. Assuming that the latter ellipse gives a better
estimate of the stress concentrations, equation (7) indicates that
the maximum tension should be approximately 16.5*)Qj during
loading.
The result of the experiment is shown in Figure 3. It
confirms the results and implications of the photoelastic stress
Figure 5: Fracture produced at a critical "Griffith crack"isolated within a compressive stress field. A2*-long slit, similating a critical Griffith crack,was cut in a thin plate of plexiglass by a 1/52"-diameter end mill. Branch fractures ceased propaga-tion under continued load when the direction ofpropagation attained a semi-parallel orientationwith the vertical axis of applied compression.Photograph taken of the plate under load, with awhite-light source in the light field arrangementof the polariscope.
analyses, and rules out cases (a) and (b). At a load of approximately
1000 pounds, contact was established between the loading pins and
two subsidiary thin plates (see Appendix C) designed to contribute
mechanical stability to the loading pins at high load levels.
Fracture in the test plate occurred at approximately a 1300 pound
load, with no noticeable drop in the gage reading. The implication
here is that much of the fracture history, in specimens such as
rocks, may go unnoticed during loading.
The tensile stress concentrations must have been greater than
30,000 psi at the moment of failure in the test plate. Continued
increase in the load from 1300 pounds to 1900 pounds could not
cause the fracture to propagate further. In view of these values,
the load would have to be greatly increased before the fracture
would again propagate. Evidently, the sharpness of a crack is not
a criterion of how far it will propagate. The "Griffith crack"
discussed here had an axial ratio of asb = 64:1, whereas the
elliptic cavity in the photoelastic model only had an axial ratio
of asb = 2:1. In both cases, lengths of branch fractures were
less than the 'track length" of the "Griffith crack" from which
they emanated.
Many-Crack Problem
Case (c), described above, remains in question; viz., the
effect of neighboring flaws on the crack propagation of a critical
Griffith crack. Now, certain restrictions must exist, placing limits
on the distribution of flaws in order that a critical Griffith
crack may continue to propagate or to coalesce with its neighbors.
It individual flaws are situated far enough apart from one another,
say by distances of 100 times the crack length, then each will
behave as if it were isolated within the applied compressive
stress field, for the disturbance of the superimposed stress
field -by a flaw diminishes rapidly within a distance of several
crack-lengths from the flaw. Therefore, one restriction would
seem to be concerned with the amount of crack separation between
individual flaws.
Test of Crack Separation:
As a first step in the analysis of this problem, a critical
"Griffith crack", surrounded by four elliptic cavities of the same
size and shape, was considered. The central cavity, 3/8" long with
a radius of curvature of 3/64" at its ends, the major axis inclined
at 400 with respect to Q, was cut in a 401"-square plate of G.R. 39.
The surrounding cavities were inclined at 600 in the same direction
with respect to Q. Each was placed so that boundaries of nearoet
neighbors were separated by 3/4"; i.e., with a %rack separation"
of twice the "crack length" (see Figure 4a).
Measured boundary stresses compared favorably with those
predicted for an isolated elliptical hole in an infinite plate
subjected to compression. The principal effect due to proximity
of the cavities was a mutual increase in the tensile stress
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concentrations. They differed by (20 1 4)% from the theoretical
tensile stress concentrations, whereas the measured compressive
stress concentrations were less than 7 smaller in magnitude than
the theoretical compressive stress concentrations. Since the
maximum tensile stress concentrations were essentially the same
at all cavities, "branch fractures" were extended from each in
the manner described previously. The result is shown in Figure 4b.
Surprisingly, "branch fractures" of the surrounding elliptic
cavities "propagated" faster than did the "branch fractures"
emanating from the central cavity. That is, a greater maximum
tension developed at "branch fractures" of the surrounding cavities
than at "branch fractures" of the central cavity, and this relation-
ship was maintained until all "branch fractures" had achieved the
same direction of "propagation". As before, as in the case of an
isolated elliptic cavity, "propagation" ceased when the "branch
fractures" had attained an inclination within 50 of the 4-axis,
for the tensile stress concentrations had diminished below the
value that would have been obtained on the unmodified boundaries
of the elliptic cavities. Evidently, a crack separation of twice
the crack length is insufficient for coalescence of Griffith cracks
when the magnitude of the applied compression is less than the
theoretical tensile strength of the material, even though the
branch fractures be on a collision course.
Effect of En Lchelon Distribution of "Grifcfith Cracks" on Tensile
Stress Concentrations:
Photoelastic stress analyses were made of three separate
groups of en echelon elliptic cavities identical in size, shape,
orientation, and "crack separation". Again, a "crack length"
of 3/8" was chosen with a radius of curvature of 3/64" at the
0ends of the cavities, their major axes inclined at 40 with
respect to Q. Since a "crack separation" of twice-the -"ciack
length" proved ineffective for coalescence of the flaws in the
preceding investigation, a "crack separation" of three-fourth's
"crack length" was employed between the three holes of each group.
"Crack separation" was measured in the manner shown in Figure 4 a;
specifically, between the two nearest boundary points of neighbor-
ing elliptic cavities.
Overlap also was held constant in each group. Therefore, the
groups differed from one another only in relative amount of overlap.
Overlap is defined diagrammatically in Figure 5. There, the over-
lap is k(2a), where 2a is the length of the "Griffith crack".
Thus, if k = 0, there is zero overlap; if k , then one-half
crack length" overlap exists; and if k = 1, we have a case of
full overlap. These were the three cases tested, with the end
results shown in Figure 6. If each of the initially elliptic
cavities were isolated within a strese field of uniaxial compression,
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LL
~
~ ~
<
a.t I
--%
A
z m
o
.Ile~~ta ... t
C
C4
W
) 24
-a
0 .
z >
Y.
z ..
\6
--
4t d 3 c -
a C
tv
(a
the maximum tensile stress generated would be (1.12 $ .01).IQI
at 300, 2100 at the cavity boundary.
The principal effect of zero overlap, with regard to those
cavity boundaries facing each other, was to increase the tensile
stress concentration to (1.64 Z .05).JQI , and to shift its
positions to 450 Z 30, 2250 30, further away from the
ends of the major axis. Proceeding to the case of one-half
"crack length" overlap, then to the case of full overlap, the
tensile stress concentrations exhibited a progressive decrease
in magnitude. The maximum tensile stress obtained for one-half
"crack length" overlap was (1.20 .05)-JQI at = 500 30,
2300 30; whereas, in full overlap relationship, a tensile stress
concentration of (0.94 .05)-IQI was produced at 25 130,
2050 Z 30. Thus, if such distributions of Griffith cracks were
present, the first to fail during loading would be those in zero
overlap relationship.
Coalescence of "Griffith Cracks"t
Figure 6a illustrates the end result for the case of zero
overlap; viz., no coalescence even though "branch fractures"
passed in close proximity -to neighboring flaws. As soon as the
direction of "propagation" became parallel to the Q-axis, the
tensile strese concentrations progressively diminished in magnitude
with further artificial extensions of the "branch fractures". The
"branch fractures" persisted on their course parallel to the axis
of applied compression until "propagation' ceased. Therefore,
overlap evidently is required if coalescence of Griffith cracks
is to occur.
Figure 6b represents a stage of impending coalescence of
"Griffith cracks" having half "crack length" overlap. Upon
initiating "branch fractures", the maximum tension showed a
sharp increase from (1.20 .05).jQ4 to (1.72 .05).ejQI. It
diminished to (1.62 .05)-IQI during "propagation" from
0( = 670 to /1 = 500. The maximum tension then remained
constant in magnitude during "propagation" in this direction,
even though the "branch fractures" were approaching imminent
collision with the hinds of neighboring elliptic cavities. The
points of approach on the elliptic cavities were points of
compressive stress concentration. The analogy here, perhaps,
is the case of fracture produced in the simple bending of a
beam of brittle material, in which the fracture propagates into
the region of maximum compression. Now, with regard to the
photoelastic experiment, collision probably would occur, but did
not occur because the remaining material, between "fracture tip"
and approached cavitywas becoming too thin for photoelastic
stress analysis.
In contrast to half "crack length" overlap, the effect of
full overlap of the elliptic cavities was to decrease the tensile
stress concentrations to (0.94 Z .0-)-IQI where individual cavity
boundaries faced one another. Thus, of the three cases tested,
this would be the last to fail. No significant change was observed
in the tensile stress concentrations when "branch fractures" were
initiated, nor during early stages of their extension. However,
as opposing "branch fractures" by-passed each other, they entered
into local regions of very low stress, and ceased "propagating".
As can be seen in Figure 6 c, only thin ligaments separate opposing
"branch fractures". Though the ligaments were too thin for
reasonable photoelastic stress analysis, the stress levels there
must have been rather small. One of the ligaments was less than
0.01" wide, yet an increase in the load to iton neither caused
the ligament to buckle or to break. Now, these results do not
necessarily imply that coalescence would not be obtained in a
case of full overlap of Griffith cracks, for the geometry of
Figure 6c may be changed considerably by altering axial ratios
and inclination of the elliptic cavities. However, of the three
cases tested, the case of half "crack length" overlap is the
most conducive for coalescence of Griffith cracks.
Brace (unpublished work) has performed carefully controlled
experiments leading to fracture of several crystalline rock
types. He was able to observe a number of steps in the fracture
process preceding the formation of a shear fracture surface. A
significant observation was that straight segments of grain
28
boundaries behave like Griffith cracks, and that those in en echelon
array app arently were activated preferentially to form the shear
fracture surface.
If we assume that Griffith cracks of approximately the same
length, orientation, crack senaration, and overlap relationship
are activated during compression, then the mean inclination of
their resulting shear fracture surface (see Figure 5) is obtained
from1-k 2
(11) tan9 = si
1 1' .- sing cosY
where 9 is measured with respect to 4, is the inclination of
the activated Griffith cracks with respect to Q, k is the overlap
factor, and a is the crack-separation factor. The principal K
and S values of interest are 0< k <l and 0 (<s 2. For example,
if Griffith cracks oriented at 4/ 450 are activated, and if
they have one-fourth crack-length separation (s = ) and half
crack-length overlap (k =4), then their resulting shear fracture
surface should form at 9 = 27r. Similarly, if k = 1, s = , and
= 450, then 9 310. By way of illustration, these examples
possibly explain why shear fracture surfaces of differing types
of brittle rock do not form consistently at precise angles to
the axis of applied compression. The en echelon distribution of
Griffith cracks also may explain why shear fracture surfaces are
irregular.
29
Conclusions and Implications
Griffith's theory applies only to cracks of practically
infinitesimal width, with each sufficiently separated from its
neighbors so that each behaves as if it were isolated within
the superimposed stress field. Friction between crack walls
has been considered by McClintock and Walsh (in the press).
The Griffith theory is extended here to isolated elliptically
shaped flaws of finite width, and the initial direction of
propagation of such flaws is predicted.
Photoelastic stress analysis indicates that the Inglis
conclusion, slightly modified, is valid; viz., if the ends of
a crack or of an elongate cavity are approximately elliptic in
form, then it is legitimate in calculating the stresses at such
points to replace the cavity by a complete ellipse having the
same total length and similar end formation. The analysis
further indicates that the radius of curvature at the tips of
an initiating fracture cannot be much greater than the radius of
curvature at the end of the flaw that gives rise to fracture,
otherwise the maximum tension at fracture tips would be less
than the theoretical tensile strength of the material.
A critical Griffith crack, isolated within a compressive
stress field, gives rise to branch fractures that propagate out
of the crack plane, arching into positions semi-parallel to the
axis of applied compression. Propagation ceases when this
orientation is attained, for the stress level required to initiate
fracture is not sufficient to maintain it; i.e., the tensile
stress concentrations at the fracture tips diminishes below
the theoretical tensile strength of the material at this stage
of fracture. The sharpness or eccentricity of such a crack
apparently is no criterion as to how far it will propagate,
for the length of an individual branch fracture is less than
the length of the flaw that gives rise to fracture. When
failure did occur in a particular case tested, there was no:
noticeable drop in the applied load. Therefore, in actual cases
of brittle shear fracture, much of the history of failure may
go unnoticed.
It follows that brittle shear fracture may not be the result
of growth of a single flaw, as in the case of fracture produced
by purely tensile stress. The coalescence of two or more flaws
apparently is necessary to form a macroscopic shear fracture
surface or fault. Photoelastic stress analysis indicates that
(1) en echelon overlap between Griffith cracks, and (2) a crack
separation of somewhat less than twice the crack length are
required to permit coalescence of flaws when the applied compression
is of smaller magnitude than the theoretical tensile strength of the
material. Artificially produced "branch fractures" exhibited a
remarkable persistence of path of "propagation", regardless of
the proximity of other flaws in the cases tested. If this
observation proves to be generally valid, then an additional
condition for coalescence of Griffith cracks is that their branch
fractures not propagate into local regions of low stress (e.g.,
see Figure 6c).
The Griffith equations no longer apply because close proximity
of flaws causes stress concentrations to differ in position and
magnitude from those calculated from the Griffith equations.
However, this does not in itself negate Griffith's theory when
the applied stress is compressive. The Griffith theory basically
is predicated on the law of conservation of energy and the
assumption that a natural solid material contains flaws. It
predicts which isolated flaw will be the first to propagate.
Therefore, it is the additional assumption of crack isolation
which must be modified.
Under the above conditions, the Griffith fracture criterion
cannot be equated with the fracture stress in compression. IS'
fracture produced by purely tensile stress, the critical Griffith
crack has its crack plane perpendicular to the axis of applied
tension, and so this crack propagates in its own plane with an
accelerating velocity; the geometry of fracture here produces
instability in the system and this type of fracture ordinarily
cannot be stopped. On the other hand, when the applied stress
is compressive, the critical Griffith crack is inclined,
propagation proceeds out.of the crack plane, and propagation
ceases unless neighboring flaws are in favorable proximity, because
the maximum tension at fracture tins diminishes below the theoretical
strength of the material when the direction of propagation approaches
parallelism with the axis of arplied compression. Since it is
unlikely that Griffith cracks initially are in optimum configuration
for the formation of a shear fracture surface, selective growth of
individual flaws or groups of flaws probably occurs during loading
in compression. Stages in the culmination of this process may
explain the curvature in the stress-strain curve just prior to the
formation of a shear fracture surface.
In the case of rocks, grain boundaries are a potential source
of Griffith cracks, but their branch fractures may cease to
propagate if they approach grain boundaries where the stress
locally may be small. Brace (unpublished work) has shown that,
in certain crystalline rocks tested, straight segments of grain
boundaries behave like Griffith cracks, and that those in en
echelon array apparently are activated preferentially to form a
shear fracture surface. It is particularly significant, therefore,
that photoelastic stress analyses indicate that en echelon Griffith
cracks of half crack-length overlap undergo crack propagation in
preference to those of full overlap relationship. Irregularities
of shear fracture surfaces, and their lack of consistent formation
at precise angles to stress axes, may be explained in part by
equation (11), in which the formation of a shear fracture surface
is shown to include overlap, crack separation, and crack orientation
as independent variables.
Considerable support is given to the above conclusions and
implications of the static photoelastic stress analyses by an
experiment carried to actual fracture. An inclined narrow slot
rounded at the ends, simulating a critical Griffith crack in a
thin plate subjected to uniaxial compression, gave rise to
branch fractures that propagated into an orientation approaching
the direction of maximum compression, and ceased propagation in
the manner predicted. In the addendum to Appendix A, an analysis
of displacements indicates that relatively eccentric, fracture-
prone Griffith cracks still may be open at depths of a few
kilometers in the earth. Thus, the present investigation may
be related to certain structural phenomena, such as thrust
faulting, occurring at shallow depths in the earth's crust.
Acknowledgments
Professor William Brace, of the Massachusetts Institute of
Technology, suggested the problem treated in this paper, and he
indicated the method by which the problem might be analyzed. The
suggestions and criticisms offered by Prof. Brace, Prof. Harry
Hughes, and J. Walsh -- also of M. I. T. -- were particularly
helpful.
m -
J. Walsh has suggested that the path of crack propagation
is' controlled by the stress trajectories, particularly the
direction of the compressive principal stress. This problem
may be analyzed by detailed studies of isoclinics during
artificial extensions of "branch fractures".
Prof. W. F. Brace suggests that the modified Griffith theory
of McClintock and Walsh (in press) might be analyzed photoelastically,
taking into account friction between crack walls. In regard to this
suggestion, it would be of interest to determine, both analytically
and photoelastically, that inclination of a Griffith crack that
separates those cracks which close from those which open due to
the applied stress.
A photoelastic stress analysis should be made to determine
critical values of crack separation permitting coalescence of
Griffith cracks. More complicated arrays of "Griffith cracks"
might also be studied to see if crack propagation and coalescence
still is preferred amongst those "Griffith cracks" in more or less
half "crack length" overlap. Since the Griffith equations do
not apply to Griffith cracks mutually in close proximity, the
orientation of a given array of elliptic cavities should be
analyzed to determine optimum orientation to obtain a maximized
tensile stress concentration, but still leading to coalescence
of flaws. Variation of orientation with respect to "crack
'5
Suggestions for Further Investigation
dimensions" may be checked by reverting successively to more
eccentric elliptic cavities. This, of course, is only one step
in finding the most optimum array conducive to crack propagation
and coalescence of flaws. The general problem may be treated
further for biaxial compressive stress, instead of the case of
uniaxial compression utilized exclusively in the present
investigation.
Biography of the Author
The author was born May 1, 1924, raised in New York and Colorado,
and married Adeline Marie Dezzutti in Denver in 1952. After service
in World War II, he attended the Colorado School of Mines, obtaining
the degrees of Geological Engineer in 1952 and Master of Science in
1959. During this time, he worked several years as a petroleum
geologist. His teaching experience includes being Instructor of
Mineralogy at the Colorado School of Mines, Lecturer in Geology
at Boston College, and an instructor in the Soils Division of the
Civil Engineering Dept. of M. I. T. during 1961-1962. He is
joining the Dept. of Geology at Boston College as an Assistant
Professor.
56
Bibliography
Brace, W. F. (in Press), Brittle Fracture of Rocks: Proc. ofSymposium on Stress in the Earth's Crust, SantaMonica, California, 1963.
Durelli, A.J. and Murray, W.M. (1945), Stress Distribution aroundan Elliptical Discontinuity in any Two-dimensionalUniform and Axial System of Combined Stress: Proc.Soc. Exp. Stress Analysis, Vol. I, No. 1, p. 19-32.
Frocht, M.M. (1941), Photoelasticity: Vols. I & II, John Wiley &Sons.
Griffith, A.A. (1921), The Phenomena of Rupture and Flow in Solids:Phil. Trans. Roy. Soc. London, Vol. CCXXI-A,p. 163-198.
Griff'ithA.A. (1924), The Theory of Rupture: Proc. First Int.Cong. Appl. Mech,, Delft, p. 5;-63.
Hetenyi, M. (1957), Handbook of Experimental Stress Analysis:John Wiley & Sons, N. Y., p. 1-1077.
Inglis, C.E. (1913), Stresses in a Plate due to the Presence ofCracks and Sharp Corners: Inst. Naval Arch. (London),vol. 55, p. 219-230.
Jessop, H.T. and Harris, F.C. (1949), Photoelasticity: Cleaver-Hume Press Ltd., London, p. 1-184.
Love, A.E.H. (1927), A Treatise on the Mathematical Theory ofElasticity: 4th Ed., Dover Publ., N.Y., p. 1-643.
McClintock, F.A., and Walsh, J.B. (in Press), Friction on GriffithCracks in Rocks under Pressure.
Ode, H. (1060), Faulting as a Velocity Discontinuity in PlasticDeformation: in Rock Deformation (edited by Griggsand Handin), Geol. Soc. America Memoir 79, p. 293-321.
Post, D. (1954), Photoelastic Stress Analysis for an Edge Crackin a Tensile Field: Proc. Soc. Exp. Stress Analysis,Vol. XII, No. 1, p. 99-116.
37
Tirmoshenko, S. and Goodier, J.i. (1951), Theory of Elasticity:McGraw-Hill Book Co., I.Y., p. 1-506.
Topping, J. (1957), Errors of Observation and their Treatment:Inst. Phys. Monographs for Students, Chapman andHall, Ltd., London, p. 1-119.
Wells, A. and Post, D. (1958), The Dynamic Stress DistributionSurrounding a Running Crack--a PhotoelasticAnalysis: Proc. Soc. Exp. Stress Analysis,Vol. XVI, No. 1, p. 69-92.
APPENDIX A
Extension of the Griffith Theory
Griffith assumed that a brittle, Hookean material contains
randomly oriented microscopic cracks situated sufficiently apart so
as not to interact with each other' s disturbance of the superimposed
stress field. The question is the distribution of stress around a
crack. If crack shape can be represented in two dimensions by an
eccentric ellipse, then an approximating mathematical formulation of
the problem in the theory of elasticity is one of an elliptical hole
in a thin, infinite plate subjected to a biaxial state of stress in
the plane of the plate. This is a boundary-value problem requiring
the use of elliptic coordinates and to describe mathema-
tically the shape of the crack. r and are related to the
cartesian coordinates x and y by
(1) x = C'cosh cos
y = C sinh sin
here, holding C fixed, with I as an arbitrary constant, a
family of confocal ellipses is generated in which r a O is
defined to be the all iptic boundary of the crack, and C its
focus. Holding C fixed at the same value, with as an
arbitrary constant, a family of confocal hyperbolae is generated.
1-A
Thus, coordinate points in the x-y plane are determined by
intersections of ellipses and hyperbolae. It follows that points
on the boundary of the crack are determined by I , called the
eccentric angle. If a circle is circumscribed about the elliptical
boundary, then the position of a point on the boundary is obtained
by constructing the eccentric angle, as shown in Figure 1 (p.
of the text). There, the eccentric angle is defined by an
aporopriate radius vector of the circumscribing circle. Dropping
a perpendicular to the major axis of the ellipse . from the end
of the radius vector, a point of intersection is produced between
the perpendicular and T,. The point of intersection represents
the position point, corresponding to a given , on the boundary
of the elliptically-shaped cavity.
The boundary conditions are that the principal stresses, P and
Q, act uniformly at infinity in the plane of the plate, and that the
surface of the elliptic cavity is free of stress. The solution for
the stresses around the cavity was found by Inglis (1913) and is
given by Timoshenko and Goodier (1951, Chapter 7). Accordingly, the
stresses acting at the crack boundary a are
(2) =0=a
(~) (p~..4t4 a. (Q)er. A'hICOV .
2-A
3-A
Following Griffith, we define that Q<P algebraically, with
tensile stresses positive and compressive stresses negative. Angle 9/
is the inclination of Q with respect to the major axis of the
"crack", measured counterclockwise from the positive x-axis to the
direction of Q. Figure 1 shows the stresses acting on an arbitrary
infinitesimal element. is the normal stress acting perpendicular
to a member of the family of ellipses, whereas U is the normal
stress acting perpendicular to a member of the family of hyperbolae.
is the shear stress. Thus,(I-)y represents the variation of
principal stress at , parallel to the crack boundary.
It may seem intuitively obvious on a physical basis that the
most dangerous crack in the case of simple tension is that crack
with its plane perpendicular to the axis of tension. Such actually
is the case if all the randomly oriented cracks are of the same size
and shape. Tensile stress concentrations occur at the ends of the
crack, and the crack will begin to propagate in its own plane when
the tensile strength is reached. For compression, however, the most
dangerous crack has its plane inclined to the principal stresses.
The Griffith theory, as developed thus far, indicates that the tensile
stress concentrations occur somewhere near the ends of this crack, not
at the ends (e.g., see Ode, 1960, p. 297), with the result that
fracture should begin by propagating out of the plane of the most
dangerous crack. To find the initial direction of propagation, the
Griffith theory nust be extended by determining analytically the
positions of the maximum tensile stress concentrations on the
boundary of the crack.
The state of stress at the crack boundary is given by (2).
For a given P, Q, and f,9 we have the functional relation ()= t.:()
Thus, the conditions for extremals are 0 and __
which respectively yield
and
(4) .|i. y= ge~ ces u -
Here, we have a case of relative maxima and minima, for these
equations are satisfied not only by : 0 , but for other
values of i and . Computations (Ode', 1960, p. 296) demonstrate
that when the applied stress is a tension, the most critical crack
(of the set of randomly oriented cracks having the same shape) is
that crack with its plane perpendicular to the applied tension, the
tensile stress concentrations occurring at the ends of the crack;
i.e., when V m. ( 0. Similarly, when the applied stress is
compressive, the most dangerous crack is inclined to the externally
applied principal stresses.
Equation (3) implies that one critical value of must be a
small quantity of order r when is very small. As Ode' (1960)
4-A
q
5-A
has shown, this approximation leads to the results of Griffith;
e.g., when 3P / Q (0, the critical orientation of the crack is
determined by
(5) COB = -A.
Ode' also shows that the maximum tension at the crack boundary is
(6) Max.(j4 P+Q
But (3) and (4) can be solved explicitly for cos al and cosaly
in terms of the other parameters without making the above-mentioned
approximation. Combining (3) and (4), noting that Ca& . T, - ce a >0
it follows that
(7) coae : cALg, - V p-2_ 0
and
(8) 'u~ae ~ _
The positive and negative signs in (8) are related to symmetry. For
example, if there is a most dangerous crack inclined to one side of
the stress axis in uniaxial compression, then there is potentially
another inclined at the same angle to the other side of the stress
axis. Thus, choosing the negative sign, (8) reduces to the Griffith
condition (5) for T,= 0. However, in virtue of (1), y,=0 implies
a slit having a radius of curvature equal to zero at the ends. This
is a physical impossibility because of the structure of matter. The
radius of curvature at the ends of actual cracks cannot be much less
than the spacing of atomic or molecular planes. Equation (5),
though, is a good approximation for very eccentric cracks,
provided the walls of the cracks are not forced into mutual
contact by the state of applied stress; this last point is given
further consideration in the addendum.
Since a = C-cosh? and b= C-sinhr, respectively represent
the semi-major and semi-minor axes of the elliptically-shaped
crack, (7) and (8) also may be expressed as
cos =a 2 b2 p 8a2b2
a2 - b2 (P -LQ (a 2 - b2) (a / b)2
and
cosaf - -Q.Ja/ b .. * l.4abCP AQ a -b P Q a2 - b2
for purposes of calculation.
Utilizing (7) and (8) in (2), we obtain for the stress
concentrations
(9) - (1 2) - ( 2 . (a / b) 2
~T ~(P / 44ab
If the applied stress is tensile, or if values of F, q, and
cause (7) and (8) to take on values greater than /1 or less
than -1, then (9) does not apply. Instead, the condition for
extremals implied by (3) and (4) is sina = sinty = 0, and
the appropriate stress concentrations are those given by
Timoshenko and Goodier (1951, Chapter 7). Griffith's expression,
3P / Q <0, for the validity of (5) and (6), follows from these
conditions when the elliptic crack is highly eccentric. Thus, there
are two criteria predicting the orientation of the most dangerous
elliptic cavity.
6-AI
7-A
The solution for the stress concentrations at a circular hole
in a plate subjected to uniaxial compression is well-known, and is
given in most texts on the theory of elasticity. Setting a = b
and P e 0, (9) reduces to this special case. As for the most
critically inclined elliptically-shaped cavity in a compressive
stress field, the tensile stress concentration is
Max. ( /)2 P q2\ / hab P Q
Equation (6) can be shown to be a special case of this result
when , and therefore b , is very small; e.g.,
e2 , a ba -b
.n ln(ab/ 2 = ln Z for Z)}.a -b] n Z
When b is small,
ln Z Z 1 1OO z
Thus,
1 21 (a / b)
4 4 ab .
Equations (7), (8), and (9) also have been given experimental
confirmation; this is discussed in Appendix C.
The form of (9) shows that both tensile and compressive
stresses are generated at the boundary of the inclined elliptic
cavity. Thus, four points of zero stress must exist somewhere
along the crack boundary. Their positions are determined from
(10) cos.? = e-2t-(cosegV - i ) cos 29
d 1 - e-2fe (cosalf- EAsinha sin2%0
obtained by equating the numerator of (2) with zero.
With (7) and (8) giving the inclination of the most dangerous
crack and the positions of its tensile stress concentrations, the
initial direction of propagation for fracture now may be calculated.
If the crack boundary is defined by
2 2A.. , L-- 1,a2 b2
then the slope at any point on the boundary is
dx a y.
Utilizing (1), and referring to Figure 1 (p. 3 of the text), this
may be expressed as
tan = -. cot = -cot a(
Thus, the initial direction of propagation is given by o( , where
(11) tan( a tan
8-A
Addendum
The equations of Appendix A may be anplied to isolated elliptic
cracks that are still open at the moment of fracture. Criteria, for
the case of friction between crack walls, were developed by McClintock
and Walsh (in the press). The question arises as to which cracks
have been closed before fracture commences. From the standpoint
of structural geology, a stress state of interest is F =4,
compressive. In terms of a two-dimensional analysis, a growing,
compressive, tectonic stress must pass through this stress state
before brittle fracture can occur at moderate depths in the earth,
because of initial stresses due to overburden.
Complex notentials are given by Timoshenko and Goodier (1951,
Chapter 7) that satisfy the Airy stress function for the problem
of an elliptic cavity in an infinite plate subjected to biaxial
stress in the plane of the plate. With P = Q, it can be shown
that the cartesian displacements, u and v, at the hole boundary
are given by
(12) u = xE a
v - -2-V E
where a is the semi-major axis of the elliptic cavity, b is the
semi-minor axis, and E is Young's modulus. An elliptic crack
9-A
U
may be said to close when, at y = b and x = 0, v becomes
equal to b. Therefore,
(13) b - _2a E
This equation shows that relatively eccentric, fracture-prone
cracks still may be open at depths of several kilometers. For
example, if the average specific gravity of rocks at moderate
depths is approximately 2.5, then the vertical normal stress is
approximately -250 bars/km. When P Q at a depth of 1 km,
Q =-250 bars. Assuming that Young's modulus is of the order
of 10) bars/unit-strain, equation (13) predicts that elliptically-
shaped cracks with axial ratio of aDproximately b:a = .005
are still onen.
10-A
APPENDIX B
Experimental Method
Photoelastic Method:
The photoelastic method employs the polariscope, shown
diagrammatically in Figure 1-B. In principle, the polariscope
is similar to the petrograhic microscope, one difference being
that collimated light is used in the polariscope instead of
convergent light. This anparatus incorporates a light source,
a collimating lens, a pair of crossed polarizing elements, two
quarter-wave plates, and a lens system to focus the model image
on a ground glass plate. A loading frame, containing the model,
is positioned between the quarter-wave plates, with the plane of
the model normal to the incident light. In the absence of load,
the transparent model does not refract the incident light, for
the material of the model is isotropic. Thus, if the polarizing
elements (polarizer and analyzer) are crossed with the quarter-
wave plates removed, no light can be transmitted through the
polariscope, and the field of view is dark.
When a photoelastic model is subjected to plane stress, each
infinitesimal element of the model behaves optically as a uniaxial
crystal with its polarizing planes coinciding with two planes of
principle stress. Thus, the fundamental laws of photoelasticity
are:
1-B
CollimatedLight
Polarizer First Model Second AnalyzerQuarter- Quarter-
wave wavePlate Plate
Figure 1-B: Simplified, diagrammatic representation of a
polariscope. The polarizer and analyzer comprise
the crossed polarizing elements (i.e., their
polarizing planes are mutually perpendicular
in the standard polariscope).
2-B
1. When light of normal incidence is transmitted through a
stressed photoelastic material, the light is polarized
in the directions of the principal stress axes, and it
is transmitted only on the planes of principal stress.
2. The velocity of transmission on each principal plane of
stress is dependent upon the magnitudes of both principal
stresses.
The phenomenological result of the first law is the formation
of isoclinics. Imagine that the quarter-wave plates are removed
from the polariscope in Figure 1-B. The instrument is now referred
to as a plane polariscope. Suppose that at some point in the
stressed model the planes of principal stress are inclined with
respect to the planes of transmission of the polarizer and
analyzer. Plane polarized light transmitted through the tnolarizer
is resolved into components that are transmitted through the
principal planes of stress at that point in the model. These
components, in turn, are resolved into new components transmitted
through the analyzer. Thus, light is transmitted through the
polariscope for such a point in the model. Now, in contrast,
suppose that at another point in the model, the princinal planes
of stress are respectively parallel to the transmission planes
of the polarizer and analyzer. Plane polarized light can be
transmitted through only one plane of principal stress at the
3-B
given point in the model, but this plane is perpendicular to the
transmission plane of the analyzer. Therefore, the apparent
result is extinction of light at the point in question. The
locus of such points produces a black fringe or band called an
isoclinic. Consequently, a given isoclinic denotes those points
in the model where the principal stresses have a particular
orientation.
The inclination of the principal stresses is defined to be
the acute angle 4g , measured between a horizontal axis and the
direction of the algebraically minimum principal stress. Angle L
is positive if measured counterclockwise from the horizontal axis,
negative if measured clockwise. Various isoclinics are obtained
by rotating the crossed polarizer-analyzer assembly about the
optical axis of the plane polariscope. The parameter %P of an
isoclinic readily is determined either by the angle of rotation
or by noting the point where the isoblinic intersects a free
boundary of the modrl, for the principal stresses must have the
same inclinations as the tangent and normal to the free boundary
at that point.
The phenomenological result of the second law is the formation
of interference fringes or isochromatics. Supoose that the principal
stresses S' and U differ in magnitude at a given point in the
rodel, and that the quarter-wave plates are removed from the
polariscope. Let be the solid acute angle measured between the
plane of transmission of the polarizer and the principal plane
of 07 . Then, in general, two component waves will be resolved
from the plane polarized wave incident to the model at the point
under consideration. In consequence of the second law, they will
be displaced relative to each other as they pass through the model.
In turn, these components will be resolved individually into
components transmitted through the polarizing plane of the
analyzer.
If we represent the incident plane polarized wave by the
light vector
(1) V = A'cos(pt),
where
A = amplitude of the incident light wave
p = 2 frf
f = frequency of the monochromatic light
t = time,
then it can be shown (Hetenyi, 1957, p.tf#) that the resultant
light vector emerging from the analyzer is given by
(2) R = Asin(2 j) - sin p t1- t2 - sin p t - t / t2 )
where
ti = time of transmission through the principal planeof 0
t2 = time of transmission through the principal planeof
5-B
From the form of the equation and the absence of the variable t,
the expression in the larger parentheses represents the amplitude of
the emergent light vector. Thus, the amplitude of the resultant
vibration is zero when either A = 0, sin(2 O) = 0p or sin p/tl-t2 0,
If A = 0, no light is being transmitted through the polarizer; e.g.,
the light source is turned off. The case of sin(2r) 0
corresponds to the formation of isoclinics. However, when
sin p 1-t2 = 0, there is retardation of the form
)p 1t2) =nil', for n ~ 0, 1, 2, 3, .... ,122where n is the order of interference. Therefore, where sinp !L_)
equals zero, black interference fringes or bands are produced with
a monochromatic light source, and isochromatics (interference color
bands) are produced with a white light source.
Most texts on the theory of photoelasticity (e.g., Frocht,
1943) show that the relation between the fringe order n and the
principal stress difference may be expressed as
(4) ( - ) ~ F.C. -h
where F.C. is the fringe constant of the photoelastic material,
and h is the thickness of the model. Where the interference
fringe meets a free boundary, one of the principal streeses is
zero; the value of the other is calculated directly from (4).
6-B
Isoclinics and interference fringes are superimposed unless
the quarter-wave plates are introduced into the polariscope. The
isoclinics are eliminated without altering the interference fringes
provided that the quarter-wave plates have their optic axes crossed,
oriented at 450 with respect to the transmission planes of the
polarizer and analyzer. For such an arrangement, the plane
polariscope is transformed into a circular polariscope. Mono-
chromatic plane polarized light transmitted through the polarizer
is converted to circularly polarized light by the first quarter-
wave plate. Thus, components of the light are transmitted through
every material point of the model, regardless of the orientation
of the principal stresses, with the result that no isoclinics can
form. The second quarter-wave plate reverts the circularly
polarized light back to plane polarized light. For further
details, see Murray's discussion of photoelastic theory in
Hetenyi's Handbook of Exoerimental Stress Analysis.
Whereas isoclinics may be eliminated so as to obtain a clear
representation of the interference fringes, the reverse cannot be
accomplished. As a result, the interference fringe pattern
ordinarily is determined more accurately in a photoelastic stress
analysis than the isoclinic pattern. To determine isoclinics
accurately, duplicate models were prepared from Columbia Resin 39
and plexiglass. C.R. 59 has a moderately low fringe constant, so
that interference fringes up to the tenth order are obtainable
7-B
with moderate loads. The fringe constant of plexiglass is so
high that few interference fringes are obtained under load,
permitting isoclinics to be readily observed.
The above description of the photoelastic method is based
on the dark field arrangement of the polariscope; viz., a polariscope
with the transmission planes of the polarizer and analyzer mutually
perpendicular. Additional detail of the stress distribution is
given by the light field arrangement, obtained by making the
transmission planes of the polarizer and analyzer parallel. In
this case, half-order interval interference fringes are produced
with n =, 11, 2-,
Experimental Technique:
Axial compression loading was produced with portable P-200
Blackhawk pumps and rams in the loading frame of the polariscope.
The pump-ram-pressure gage assembly was calibrated with dead-weight
loads to determine friction, as a function of load level, in the
loading system. Accuracy in the load was -5 pounds. Allowing
for an error of .002" in the caliper measurements of the thickness
and width of the photoelastic model, the probable error in the
applied compressive stress was within 10 psi.
The rams were braced in the plane of the loading frame to
achieve and maintain axial loading. Ram loads were distributed
through a set of steel bars and Dine to produce a field of
uniform compression within a 4j"I "X " photoelastic plate.
8-B
The arrangement is shown in Figure 2-B. Twelve loading pins,
in the form of circular cylinders spaced equally apart at
3/8"-intervals, were in contact with an edge of the plate.
Thus, each pin produced a concentrated load at the plate-edge
boundary. The same intensity of load at each pin was ensured
by directly adjusting the load at the pin. As indicated in
Figure 2-B, this is accomplished by adjusting screws, accessible
through keyholes in the superjacent steel bar. Actual adjustments
were made under load until interference fringes of the same order
had identical symmetry at each pin.
The fact that shear stress between a pin and the plate edge
was negligible is indicated by the orientation of the plane of
symmetry of the interference fringes produced at each pin. If
shear were significant, then the concentrated load at the affected
pins would be inclined instead of vertical or parallel to the
applied load, so that the planes of symmetry also would be inclined.
By virtue of St. Venant's Principle (Loveil927 p /7) the pin
arrangement described should produce a field of uniform compression
within the plate at some distance from the loading pins. Variations
in the stress were observed to extend approximately -" from the
pins into the plate. A sensitive test, determining that a
substantially uniform field of compression indeed was produced in
the plate, was performed by a "tint of passage" experiment. A
square plate of C.R. 39 was subjected to gradually increasing
9-B
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-11 - '.1 , - - - : 7 " - : - -. 1. - - : - _ , I ., - .' - -P -T ;_
0-, . --. - . _. - - . - _ ' , . -_ I I - , " - . . , . .: , . - , I -. - ' " .. - -, '-! *_' ;
,. 11 r .. I 1 71: ' I - , -- - , t I ..
,7., _:"' .-- I,. . . 1. I ' , - ;' I - I - - - "I L - . .
. - - - ' - - r . , ,- , , - ." "I . ; 2 1 - ! "
- I " . " I . - , , I I I T . - IL . , , . : -
compressive load. With a white light source, isochromatics are
observed to form. During the transition from red to blue, the
complimentary color is a dull purple which is very sensitive to
small changes in the principal stress difference. This color is
called the tint of passage (Frocht, 1941, vol. 1, p. 169). A
very small stress change causes this color to change into red
or blue. The color change produced in the plate was almost
uniform, except of course in the immediate vicinity of the pins.
A check for isoclinics, by removing the quarter-wave plates and
rotating the polarizer-analyzer assembly, confirmed that within
the plate the principal compressive stress was parallel to the
load.
The arrangement, shown in Figure 2-B, was contained in a
plexiglass retaining jig, bolted to the loading frame, to prevent
axial bending and to maintain the plane of the model in the plane
of the loading frame. Proper alignment of the model, with respect
to the collimated light, was checked by means of the light-field
arrangement of the polariscope. If the edges of the model are
accurately machined normal to the plane of the model, then the
edges appear sharply delineated in the light-field arrangement
provided that the model is properly aligned; otherwise, the
edges present a diffuse picture.
The model was sandwiched between i"-thick plexiglass plates
of the retaining jig, which was designed so that almost the entire
11-B
load would be transmitted only through the model. Since the
plate surfaces are polished, the coefficient of friction is
low. If stress had developed in the planes of these plates,
then tell-tale isoclinics (in addition to those in the model)
would have formed, but none were observed.
For the same reason, axial bending also was negligible.
An additional and independent check for axial bending is provided
by a particular property of the isoclinics formed in the model.
Wherever singular points occur (i.e., wherever G in a
nonhomogeneous stress field), the directions of the principal
stresses are indeterminate. Four such points occur at the boundary
of an elliptic cavity in a plate subjected to plane stress.
Consequently, isoclinics in the vicinities of the singular
points must converge at these points. This condition at singular
points would no longer hold if axial beading were involved. Thus,
the observation that the isoclinics did converge at the singular
points provided additional evidence that axial bending was
negligible.
All plastics show a certain amount of "time-creep" under
load, both in strain and in stress-optical effect. Jessup and
Harris (1949, p. 181) found that optical creep at room temnerature
is negligible in plexiglass and low in C.R. 39. The elastic
creep rate decreases progressively soon after load is applied.
To ensure consistent results, the practice was made of maintaining
12-B
the load for a period of fifteen minutes before recording data.
The principle decrease in a "fixed" load was due to slow leakage
of hydraulic fluid at the ram packing. It was decided to photo-
graph the model by focusing a Polaroid Land camera on the ground
glass plate of the polariscope, instead of taking time exposures
with the attached professional camera. Polaroid 3000 speed film
is sensitive to the monochromatic mercury green light used for
interference fringes, and the high speed of the film enabled
pictures to be taken with only several seconds time exposure.
15-B
TTY
APPENDIX C
Experimental Results
Representation of Crack Shape by an Ellipse:
Is it reasonable to represent the longitudinal cross-section
of a crack by an ellipse having the same length and radius of
curvature at the ends? According to Inglis's theoretical
analysis (1913, p. 222), if the ends of a cavity are approximately
elliptic in form, then it is legitimate in calculating the stresses
at these points to replace the cavity by a complete ellipse having
the same total length and end formation. This is a basic tenet
of the Griffith theory.
An experimental test of Inglis's conclusion was performed by
photoelastic stress analyses on a dumbell-shaped hole, a slot, and
an ellipse having a common length (") and radius of curvature
1/16") at the ends. The results are shown in Figures 1-C,
2-C, 3-C, and Plate I. Orientation of the major axis of each
cavity is perpendicular to the direction of applied compression.
For the same axial load, isoclinic patterns and the distribution
and order of the interference figures show that the disturbances
of the compression field are remarkably similar, even with respect
to positions of the four points of zero stress on the boundary of
each cavity. Moreover, the stress concentrations at cavity ends
1-C
li
Q
w
00 85 0 5* 00
so 100
50 85000 00
850 50
00 100 0 00
5* 850
Q
Boo
1
11
,11
Hes
L7me
ssAM
M
III1
(a) (b)
(c)
Comparison of interference fringe patterns producedby (a) a dumbell-shaped hole, (b) a slot-shaped hole,and (c) an elliptic cavity, each isolated in a *W-&thickplate of C.R. 39 subjected to a compressive stress of-455 psi perpendicular to the long axis of each cavity.Scale is approximately 1:1. Each cavity is -" long,
with /' = 1/16" at its ends, where a maximum fringeorder of 51 occurs. F.C. = 100 2. Light field
arrangement of the polariscope,
5-C
Plate I:
are the same, and are in satisfactory agreement with that
theoretically predicted. According to the data in Plate I,
the stress concentration generated at the ends of each cavity
is (4.84 ,L .05)Q, whereas the theoretical stress concentration
is (1 /,2a/b)Q or (4.95 _ .07)Q, which allows for an error of
.001" in the measurement of the semi-major and semi-minor axes
of the elliptic cavity. The points of zero stress at cavity
boundaries serve to separate compressive stresses from the
tensile stresses acting at the boundary of each cavity.
Another test was made of the Inglis conclusion, this time
on a slot with its long axis inclined at 400 with respect to Q.
The angle of orientation was calculated from equation (6) of
p. 5 of the text, setting P Z 0, for a critically-inclined
"equivalent" ellipse, to be circumscribed about the slot, having
the slot's length (i") and radius of curvature ( /9 1/16") at
the ends. Thus, after a photoelastic stress analysis was made
of the slot in a field of uniaxial compression, the slot was
enlarged artificially into the form of the ellipse circumscribed
about the slot. Then, this elliptic cavitIs disturbance of the
superimposed stress field was analyzed photoelastically; the
interference fringe pattern associated with this cavity is
shown in Plate II.
The isoclinic patterns of Figures 4-C and 5-C show that the
stress patterns of the slot and of the ellipse circumscribed about
it are very similar. In the case of the slot, the compressive and
6-CI
-Z -. IT - I
Qk--
~41
4-,
,.--J -I'-
kL F -M
" .irl
4~ ~77,
w
tensile stress concentrations were found to be in the ratio
of 3:1. precisely as predicted for a critically-inclined elliptic
cavity, but they were approximately 55% greater in magnitude than
those obtained at the "equivalent" elliptic cavity for the same
applied stress. In equation (7) on p. 5 of the text, the factor
( b2ab is related to the eccentricity of the ellipse, or to
departure of shape of an elliptically-shaped cavity from the
shape of a circular hole. Now, if thellipse, having the same
length and width as the slot, instead is chosen for comparative
calculations, good agreement is found between the stress
concentrations obtained at the slot boundary and those calculated
for an ellipse inscribed in the slot. For example, the maximum
tensile stress concentration generated at the slot boundary was
(1.51 Z .05)-IQl , whereas equation (7) gives (1.56 .06).IQI
for the elliptic cavity just described vs. (1.12 .0l)-IQJ for
the ellipse circumscribed about the slot.
An alternative suggestion, made by J. Walsh of M.I.T., possibly
reconciles the two cases of orientation treated above. He suggests
that a more pertinent modification of the Inglis conclusion may be
to replace a flaw by an ellipse having the same length, but also
having the same radius of curvature at its point of maximum tension
as at the point of maximum tension on the flaw boundary. A check
on this may be made in the case of the inclined slot. The radius
of curvature at the point of maximum tension of the inclined slot
9-0
is approximately 0.063". The radius of curvature, calculated for
the theoretical point of maximum tension at the boundary of the
ellipse inscribed in the slot, is 0.065.
Fracture analysis of an isolated, critical "Griffith crack":
As mentioned above, a photoelastic stress analysis was made of
the elliptic cavity shown in Figure 5-0 and Plate II. Figure 6-0
compares the theoretical variation of stress along the boundary
of the elliptic cavity, as determined from equation (2) on p. 2
of the text, with that determined by the photoelastic method. The
agreement is excellent, and provides experimental confirmation
of the derived equations (6), (7), and (8) given on p. 5 of the
text.
For vertical compression acting at 400 to the major axis of
the elliptic cavity shown in Plate II, the positions of maximum
tensile stress on the boundary are 300, 2100. Thus, the
initial direction of crack propagation predicted by equation (9),
on p. 6 of the text, would be a(, 500. A small circular cut was
made in this direction at each of the two points of maximum
tension, normal to the plane of the model, to simulate an initial
stage of fracture. A radius of curvature of 1/32" was chosen for
each cut to make the radius of curvature at the tips of the
"fracture" smaller than the radius of curvature (1/16") at the
ends of the elliptic cavity. Uniaxial compression of -740 psi
10-C
U ii.
.. 1
Q- i= (-72 10) psi
(c): At points of maximun tension,n =2, '/Q = 1.10 . .05, andsubsequent od = 550 .
Q = (-530 / 10) psi
(a): At points of maximum tension,fringe order n =1, 1 t Z 500,and ''/Q = 1.12 Z .05.
Plate II:
(d): At points of maximum tension,n=l. O'/ 0.83 .05,and terminal / =' 450.
Q = (-740 I 10) psi
(b) At points of maximum tension,n = 2j, "/Q = 1.35 Z .05,and subsequent /$ = 550.
Photoelastic stress analysis of the Single Crack Problem, showingseveral of the stages in the development, if artificial "branchfractures" emanating from an isolated, critical "Griffith crack".(a) is a critically inclined elliptic cavity (%V = 400, a/b = 2,/= 1/16" at the ends). (b) is an initial stage of "fracture",with /= 1/32" at "Fracture tips", which were extended in theinitial direction of >ropagation" of eW = 500; subsequent direc-tion would be 4 = 550. In (c), /0 at *fracture tips" isenlarged to 1/16". The terminal stage is shown in (d). Sc leis 1:1; C.R. 39 model thickness = .250" .002"; F.O.= 100 t 2.
12-C
Q = (-725 10) psi
produced an interference fringe of 2- order at the new positions
of maximum tension at the tips of the "fracture", as shown in
Plate II. Since the model is 0.250" thick, equation (10), on
p. 7 of the text, specifies that the ibaximur tensile stress
developed was approximately 1000 psi, resulting in a tensile
stress concentration factor of (1.35 .05) as opposed to the
tensile stress concentration factor of (1.12 .05) obtained
for the unmodified boundary of the stressed elliptic cavity.
The probable errors shown here were calculated by the method
given in Appendix D.
Once fracture begins, the direction of propagation no longer
is defined by equation (9). Referring to (b) in Plate II,
imagine that each of two vectors is drawn outward from, and
perpendicular to, the "fracture boundary" at each nosition of
maximum tension. Let each vector be defined by an acute angle /,
measured between the major axis of the former elliptic cavity and
the vector. Then / defines the new direction of crack
propagation. In the case just discussed, / = 550.Next, the radius of curvature at the tips of the "fracture"
was enlarged to 1/16", equal to the radius of curvature at the
ends of the elliptic cavity. As shown in (c) of Plate II, the
tensile stress concentration factor obtained was of the same
magnitude as that obtained for the unmodified boundary of the
stressed elliptic cavity. Since . = 550, the main effect of
15-C
varying the radius of curvature at the head of the "fracture" is
on the magnitude of the maximum tension, not on the ensuing
direction of propagation. Evidently, the radius of curvature
at the tips of an initial fracture cannot be much greater than
the radius of curvature at the ends of the crack which gives rise
to fracture, otherwise the maximum tension developed would be
less than the theoretical tensile strength of the material. If
the radius of curvature at the tips of the initial fracture is
much smaller than that at the ends of the Griffith crack, then
brittle fracture likely proceeds initially with an accelerating
velocity.
In proceeding from (c) to (d) in Plate II, the "branch
fractures", having a radius of curvature of 1/16" at their ends,
were extended by increments equal to the radius of curvature, with
each step analyzed photoelastically. Angle e gradually diminished
from 1550 to approximately 450; i.e., the "branch fractures" arched
into positions semi-narallel to the direction of uniaxial
compression. During the transition of /0 from 550 to 454, the
maximum tension at the tins of the "branch fractures" maintained
the same magnitude as that obtained on the unmodified boundary
of the stressed elliptic cavity. However, at a value of ,d close
to 450, the maximum tension diminished to (0.83 d .03)]Q j , as
shown in (d) in Plate II. Further artificial extensions of the
"branch fractures" did not alter this value, nor the value of /4.
14-C
The implication of these results is that a critical Griffith
crack will produce branch fractures that may cease to oropagate
when they approach an orientation parallel to the direction of
uniaxial compression, unless
(a) the radius of curvature at the heads of the branch
fractures is small enough to ensure that the tensile
stress concentrations there do not diminish below the
theoretical tensile strength of the material; or
(b) the kinetic energy associated with fracture contributes
to the crack extension force sufficiently to continue
propagation of the branch fractures; or
(c) the effect of other nearby flaws facilitates extension
of fracture.
Case (a) is unlikely because, if the branch fractures continue
to extend parallel to the direction of uniaxial comnression, the
geometry of the fracture becomes less asymmetric and, according
to Inglis's analysis, it may be replaced by an ellipse with a major
axis oarallel to the direction of uniaxial compression. For such
an oriented elliptical cavity (Timoshenko and Goodier, 195l, p. 203),
the magnitudes of the tensile stress concentrations are the same as
the magnitude of the applied compression, regardless of the radius
of curvature at the ends. Therefore, the magnitude of the applied
compression actually would have to be increased until it equaled
15-C
the theoretical tensile strength of the material before the branch
fractures could continue to propagate.
Case (b) also is unlikely. Curvature of the "branch fractures"
takes place over only a modest distance in comparison to "crack
length". Apparently, there would be too little time for a
significant build-up of kinetic energy by the time the branch
fractures become parallel to the direction of uniaxial compression.
To test the results of the static photoelastic analyses, and
their implications, it was decided to perform a dynamic test of
iracture. A 2"-long slit was cut with a 1/32"-diameter end mill
in a 0.085"-thick plate of plexiglass, which measurej almost 6" in
width and 4A1" in height. The critical inclination of the slit with
respect to Q was in the range 30-0 4J< ( 40, based on
calculations of equation (6), of the text, for an ellipse circum-
scribed about the slit and for an ellipse inscribed in the slit. The
inclination chosen was 320. This is very close to the theoretical
inclination of the most dangerous Griffith crack when the applied
stress is uniaxial compression. Assuming that the latter ellipse
gives a better estimate of the stress concentrations, the maximum
tension should be approximately 16.5+-QI during loading.
As may be anticipated, buckling presented serious problems:
axial bending of the plate into an arched surface, and rotation
of the plate edges in contact with the loading pins. The buckling
problem was solved in several steps. First, angle-iron braces were
16-C
bolted snugly, but not tightly, to the vertical edges of the test
plate. This would restrict arching of the plate. Bolt holes in
the test plate were enlarged to permit strain adjustment in the plate
at the vertical edges, thus avoiding fixed-grip conditions which
would have induced biaxial stress during loading. Second, the
braced test plate was sandwiched between two 0.089"-thick plexiglass
plates machined to dimensions a little less than 4 agx4n, the
composite thickness being 0.225". Since the loading pins are
in length, the two bounding plates would take up some of the load
during the later loading history. Thus, the test plate would carry
the majority of the load, whereas the bounding plates would lend
mechanical stability to the loading pins with which they come
in contact under increased load. Third, these three plates were
sandwiched snugly between 4"-thick plexiglass plates of the
retaining jig, which was bolted to the loading frame to maintain
the plane of the test plate in the plane of the loading frame.
The result of the experiment is shown in Figure 3 of the
text. It confirms the results and implications of the photoelastic
stress analyses, and rules out cases (a) and (b). At a load of
approximately 1000 pounds, contact was established between the
loading pins and the two subsidiary thin plates designed to
contribute mechanical stability to the loading pins at high load
levels. Fracture in the test plate occurred at approximately a
17-C
1300 pound load, with no noticeable drop in the gage reading. The
implication here is that much of the fracture history, in specimens
such as rocks, may go unnoticed during loading.
The tensile stress concentrations must have been greater than
30,000 psi at the moment of failure in the test plate. Continued
increase in the load from 1300 nounds to 1900 pounds could not
cause the fracture to propagate further. In view of these values,
the load would have to be greatly increased before the fracture
would again propagate. Evidently, the sharpness of a crack is
not a criterion of how far it will propagate. The "Griffith
crack" of Figure 3 had an axial ratio of asb = 64:1, whereas the
elliptic cavity in the photoelastic model only had an axial ratio
of a:b = 2:1. In both cases, lengths of branch fractures were
less than the "crack length" of the "Griffith crack" from which
they emanated.
Test of Crack Separation:
Case (c) described above, remains in question; viz., the
effect of neighboring flaws on the crack propagation of a critical
Griffith crack. Now, certain restrictions must exist, placing limits
on the distribution of flaws in order that a critical Griffith
crack may continue to propagate or to coalesce with its neighbors.
If individual flaws are situated far enough apart from one another,
say by distances of 100 times the crack length, then each will
18-C
behave as if it were isolated within the applied compressive stress
field, for the disturbance of the superimposed stress field by a
flaw diminishes rapidly within a distance of several crack-lengths
from the flaw. Therefore, one restriction would seem to be
concerned with the amount of crack separation between individual
flaws.
As a first step in the analysis of this problem, a critical
"Griffith crack", surrounded by four elliptic cavities of the
same size and shape, was considered. The central cavity, 3/8" long
with a radius of curvature of 3/64" at its ends, the major axis
inclined at 400 with respect to Q, was cut in a 4-"-square plate
of C.R. 59. The surrounding cavities were inclined at 600 in the
same direction with respect to 4. Each was placed so that boundaries
of nearest neighbors were separated by 3/4"; i.e., with a "crack
separation" of twice the "crack length" (see Figure 4a of the
text).
The interference fringe pattern associated with this set of
elliptic holes is given in (a) of Plate III. Figure 7-C shows
that measured boundary stresses compare favorably with those
predicted for an isolated elliptical hole in an infinite plate
subjected to compression. The principal effect due to proximity
of the cavities was a mutual increase in the tensile stress
concentrations. They differed by (20 4)% from the theoretical
tensile stress concentrations, whereas the measured comnressive
19-C
= (-63o 1 10) psi
(a): F.C. = 93 1 2. C.R. 39 modelthickness = 0.275" Z .002". Atpoints of maximum tension, fringeorder = 2J. Thus, 0'/Q =1.34 .05, C< = 500 for centralcavity inclined at 400 to Q, andWf = 670 for peripheral cavitieswith major axes inclined at 600to Q. Each cavity is 5/8" longwith /,* = 5/64" at the ends ofthe major axes. Light fieldarrangement of the polariscope.
= (-690 1 o) psi
(b): At points of maximumtension, n 2,
/Q = 1.01 1 .05,terminal,4 = 45o forcentral cavity, andterminal 4 = 650 forperipheral cavities.Dark field arrangementof the polariscope.
Plate III: Photoelastic stress analysis of the effect of "crack separation"of twice the "crack length" between "Griffith cracks". In (a),the minimum distance between closest elliptic cavities is twicethe length of their major axes, measured as shown in Fig. 4 a ofthe text. The terminal stage of "fracture" is represented in(b). "Branch fractures" were extended by increments of 3/64"with a radius of curvature of 3/64" at "fracture tips".
20-C
01.
1000
*A
*W
stress concentrations were less than 7% smaller in magnitude than
the theoretical compressive stress concentrations. Since the
maximum tensile stress concentrations were essentially the same
at all cavities, "branch fractures" were extended from each in
the manner described previously. The terminal stage of "fracture"
is shown in (b) of Plate III.
Surprisingly, "branch fractures" of the peripheral elliptic
cavities "propagated" faster than did the "branch fractures"
emanating from the central cavity. That is, a greater maximum
tension developed at "branch fractures" of the surrounding cavities
than at "branch fractures" of the central cavity, and this relation-
ship was maintained until all "branch fractures" had achieved the
same direction of "propagation". As before, as in the case of an
isolated elliptic cavity, "propagation" ceased when the "branch
fractures" had attained an inclination of 50 to the Q-axis, for
the tensile stress concentrations had diminished below the value
that would have been obtained on the unmodified boundaries of the
elliptic cavities. Evidently, a crack separation of twice the
crack length is insufficient for coalescence of Griffith cracks
when the magnitude of the applied compression is less than the
theoretical tensile strength of the material, even though the
branch fractures be on a collis&on course.
22-C
Analysis of En Echelon Distributions of "Griffith Cracks":
Photoelastic stress analyses were made of three separate
groups of en echelon elliptic cavities identical in size, shape,
orientation, and "crack separation". Again, a "crack length"
of 3/8" was chosen with a radius of curvature of 3/64" at the
ends of the cavities, their major axes inclined at 400 with
respect to Q. Since a "crack separation" of twice the "crack
length" proved ineffective for coalescence of the flaws in the
preceding investigation, a "crack separation" of three-fourth's
"crack length was employed between the three holes of each group.
"Crack separation" was measured in the manner shown in Figure 4 a
of the text; specifically, between the two nearest boundary
points of neighboring elliptic cavities.
Overlap also was held constant in each group. Therefore, the
groups differed from one another only in relative amount of overlap.
Overlap is defined diagrammatically in Figure 5 of the text. There,
the overlap is k(2a), where 2a is the length of the "Griffith crack".
Thus, if k = 0, there is zero overlap; if k =, then one-half
"crack length" overlap exists; and if k = 1, we have a case of
full overlap. These were the three cases tested, with the end
results shown in Figure 6 of the text.
Figure 8-C shows the theoretical variation of stress predicted
at the boundary of an isolated elliptic cavity, of axial ratio
a:b = 2, in an infinite plate, the major axis of the cavity
23-C
--V'Z.
-"-
-7
-
n -f
s -' -. -
*1~
-.
4-'-
9 . . --
-. - - -
4fl
A- --d -& ''-
- 49 - -' - dl i -4
-- Vk
t w
L -p
- s
* -' 4 - - .
@ :
if
-"
t4
-4 '
- --- -
- - " - lot
W' - - - - '
rx
Now- t
-t- -7-l -
- ' -' i : 9 .'-n ":' - -'''--r ' (1~
4&
h -5-g-t
-- -c
>-
-.- .--
-
.
inclined at 400 to the axis of uniaxial compression. Comparison also
is made with the measured stress variations at the boundary of the
central elliptic hole of each of the three groups described above.
The significant effect in the three cases tested is the alteration
in magnitudes and positions of maximum tension at cavity boundaries,
as compared to the case of an isolated elliptic cavity. As can be
seen in Figure 8-C, the first to fail would be the set of elliptic
cavities in zero overlap relationship. Figure 6a of the text
illustrates the end result for this case; viz., no coalescence even
though "branch fractures" passed in close proximity to neighboring
flaws. "Propagation" ceased when the "branch fractures" became
parallel to the axis of applied compression, because the maximum
tension at "fracture tips" had diminished below the stress required
to initiate the "branch fractures". Overlap evidently is required
if coalescence of Griffith cracks is to occur. Therefore, according
to Figure 8-C, the particular case of interest treated is that in
which the elliptic cavities are in half "crack length" overlap
relationship. The case of full overlap relationship is treated in
detail in the text.
In Plate IV, (b) represents a stage of impending coalescence of
"griffith cracks" having half "crack length" overlap. Upon
initiating "branch fractures", the maximum tension showed a sharp
increase from (1.20 Z .05).-Qj to (1.72 Z .05)-|QI. It diminished
to (1.62 .05)-IQI during "propagation" froi * -'67o to d= 500.
25-C
mb
Q = (-420 10) psi
0.R. 39 model thickness is0.280" Z .002". F.0. = 95 2.At points of maximum tension( = 500 Z 30, 2300 130),fringe order n = 2$. Thus,4-/Q = 1.20 Z .05, andWC = 670. Each ellipticcavity is 3/8" long with
= 3/64" at ends of majoraxes, inclined at 400 to q.Light field arrangement ofthe polariscope.
Plate IV:
(b): At points of maximum tension,n =2, O'/Q = 1.62 Z .05, andef = 500. Dark field arrange-ment of the polariscope.
Photoelastic stress analysis of the effect of half "crack length"overlap between "Griffith cracks" of three-fourth's "crack length"separation. (a) illustrates the distribution of "Griffith cracks"in a 4 j"-square plate subjected to uniaxial compression. (b)represents a stage of impending collision between neighboring"Griffith cracks" during their "crack propagation". "Branchfractures" were extended by increments of 3/64", with a radius ofcurvature of 3/64" at "fracture tips".
26-0
(a):
Q = (-705 10) psi
The maximum tension then remained constant in magnitude during
"propagation" in this direction, even though the "branch
fractures" were approaching imminent collision with the hinds
of neighboring elliptic cavities, as shown in (b) of Plate IV.
The analogy here, perhaps, is the case of fracture produced in
the simple bending of a beam of brittle material, in which the
fracture propagates into the region of maximum compression.
Now, with regard to the photoelastic experiment, collision
probably would occur, but did not occur because the remaining
material, between "fracture tip" and approached cavity, was
becoming too thin for photoelastic stress analysis.
27-C
AIPENDIX D
Error Analysis
The two equations used for comparative analysis incorporate
experimental error. They are equation (1) on p. 2 of the text,
and equation (10) on p. 7 of the text. Equation (10) is the
photoelastic equation determining stress values from experimental
observations, and is reproduced here in the form
(i) F.C. n Kt -Q
where j' is the measured stress, _Q is the applied compressive
stress, F.C. is the fringe constant of the photoelastic material
of the model, t is the thickAness of the model, and n is the fringe
order. According to the method of error analysis given by
Topping (1957, p. 20), the probable error 6K in the factor K
is obtained from
(ii) (SK) 2 = 9W.c.) 2 / (Sn)2 2 2
As an example, we analyze SK in the photoelastic stress
analysis of the elliptic cavity isolated within a compressive stress
field (see Single Crack Problem discussed in the text). The measure-
ments and their maximum errors are as follows:
F.C. = 100, 6 F.C. = 2
Q = -530 Dsi, $ 4 = 610 psi
1-D
t = 0.250", 5t = .002"
n = 1.50, Sn = Z0.05
Here, in was calculated to be ZO.05 on the basis of a 20
pound sensitivity of the applied load required to produce a
fringe order of n = 1.50, at the point of maximum tension at
the boundary of the elliptic cavity, in the light-field arrange-
ment of the polariscope. The thickness error was determined
from measurements with micrometer-type calipers, and the other
errors were determined from calibration tests. Substituting
the above values in (ii), the probable error in K is found to
be _K = Z0.048 | 0.05.
Equation (1) of the text represents the theoretical variation
of principal stress acting at the boundary of the elliptic cavity.
The uncertainty in (1) arises from errors in machining an elliptic
hole in a plate. The maximum dimensional error in lengths of the
semi-axes is 40. 00 2 ", and the maximum error of inclination of the
cavity's major axis, with respect to _, is $3 . Following the
above procedure, the probable error in (1), for the point of
maximum tension at an elliptic cavity (a = 0.250", b = 0.125")
inclined at 400 with respect to Q, is LK = Z0.01.
2-D
