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An electrical analogue solution for the stresses near a crack or hole in a flat plate
.J. Me&. Phys.Solids. 1!)60, Vol. 8, pp. 173 to 188. Pergamon Press Ltd., London. Printed in Great Britain
AN
ELECTRICAL
STRESSES
NEAR By
ANALOGUE A CRACK
FOR
OR I-IOLE IN A FLAT
S. C. REDSHAW
Department
SOLUTION
and
of Civil Engineering,
K.
R.
University
THE PLATE
RIJSHTON of Birmingham
SUMMARY AN ELECTRICAL analcgue for solving the biharmonic equation was used to determine the elastic stresses near a hole or crack in a fiat rectangular plate subjected to uniaxial tension. Problems concerning both simply and multiply connected regions were solved by the method and, where a comparison between the experimental results and existing analytical solutions was possible, the agreement was good. The advantages and limitations of the method are discussed and the desirability of extending the mH,hod for the determination of the stresses after plastic yielding has occurred, is emphasized.
1. THE
DETERMINATION
INTRO~U~~I~N
of the stresses in the region of a hole or crack in a stressed
plate is of great engineering importance, particularly but unfortunately
in components liable to fatigue,
theoretical solutions are difficult to obtain and, indeed, very few Experimental methods such as photo-elasticity, strain gauge
solutions exist. investigations and brittle lacquer techniques, have been used in an attempt to obtain reliable results but, although a hole can be made easily, a fine crack is ditllcult to manufacture.
lies in the inability
to
measure strains over a short length and at sufficient stations in the proximity
However,
the principal difficulty
of
the crack or hole. For this reason it was decided to attempt
a solution to the problem
by using
an electrical analogue method, a technique which had already yielded satisfactory solutions to other elastic problems.
2.
(i)
OUTLINE
OF
PROBLEM
General The cases investigated
all concerned the plane stress problem of a flat rectangular
plate subjected to uniaxial tension, and having either a central hole or a central crack disposed perpendicularly to the direction of applied stress, or external cracks centrally disposed in the same direction. The problems are summarized in Fig. 1. It will be seen that the problems considered concern either simply or multiply connected regions. PALMER. and REDSHAVV (1955) have established an electrical analogue for the solution of the biharmonic equation and, for their analogy to be used in the present 173
s. c.
174
RICDSIZAW
xad
Ii. lx. ItUSlZTOE
problem has to be expressed in terms of the iliry stress funct~ion and conditions formulated accordingly. When the boundary conditions have been represented on the electrical analoguc network the measurement of instance,
the
the boundary
the electrical potentials at various nodes supplies the ;liry stress fnnction at those
#=O
26 -p -z-
2x
(a) x+
-
FIG. 1. (i) I~:sternal cr;wks.
points.
Subsequent
0
(b)
(ii) Cirdm holes. (iii) Internal crac:ks. (a) Loading rmditions. (b) Airy stress functions.
double
differentiation
yields
the stresses but, additionally,
the sum of these stresses ran be measured directly from the network. For a simply connected region the boundary conditions may be expressed directly in terms of the Airs stress function, but for a mult.ipty connected region an additional
boundary
condition
is required.
An electrical analogue solution for the stresses near a crack or bole in a Aat plate
Determination
(ii)
qf Airy
stress fun&m
The stresses on any boundary, the components
of the surface x=1
175
see Fig. 2, can be expressed
stresses and the direction
az + m 7zyp
Y = m o,, i_
in terms of X and Y
cosines
t and m :
1 rxy.
(1)
1=cosa m=sina
FIG. 2. The stresses on any boundary.
Since
equations
(1) can
be written
in terms of the Airy stress function
4 and, on inte-
gration, -W = AZ i_ Urn -1 al + /3m. 3n
(3)
Hence
where A=.-s cc,
p
and y are constants
chosen arbitrarily. can be represented
a Y ds,
of integration
B =
“1Xc-Es. 1. 0
0
which for a simply connected
region can be
Therefore, when strains are specified on the boundary by the Airy stress function and its normal gradient.
they
In the problems considered, not only were the normal and shearing stresses known on the loaded bonndaries, y = + b (Fig. I), but the transverse stress was assumed to be zero. Hence, instead of having to specify 4 and &j,&n on these boundaries, V2 + and (b can be used, which is more convenient for the electrical analogne. If the body is multiply connected be chosen to ensure that the rotations
the constants are of importance and must and displacements are continuous. On one the constants are chosen arbitrarily boundary,
boundary, usually the external but their values on the internal boundaries tion.
are restricted
by the continuity
condi-
S. C. REDSHAWand K. R. RUSHTON
176
MICIIELI, (1899) first presented these conditions in terms of the Airy stress ; he deduced the equations for each internal boundary which must be
function
satisfied to ensure condition
and displacement
single valuedness
on the boundary.
These
are the rotation
conditions
IIygp!) -r$(v2+s=o, liy$(V2#++2+)
1
I
ds=O.
1
For a multiply ditions
connected
i
region the constants
must be chosen
(6)
so that these con-
are satisfied.
Twofold
symmetry
occurred
in all the problems
therefore c( and /3 become zero, also the displacement satisfied. There remains, methods
however,
the determination
exist : either a superposition
method,
described
conditions
in this paper
and
(6) are automatically
of y and for this two alternative described
by PRAGER (1946),
or
an iterative method dur to SO~TTHWEIJ, (1948). During the experiments both methods were tried, SOUTHWELL’S method being finally used as the superposition method
entailed
the summing
of two solutions,
with the attendant
inaccuracies.
Using Sou~~~w~~~~‘s~metl~od, various values of y were tried until the one giving the correct such that (iii)
solution
of (5) was found ; the nature
this presented
Boundary The boundary
conditions
of the electrical
analogue
is
no difficulty.
.for
conditions
cracks and holes for cracks
and holes require
special
a hole the normal stress and the shearing stress are both zero.
mention.
For
When substituted
in (3) and (4) this yields 4 L y and &$/An = 0. A fine line internal crack may be considered
as an ellipse whose minor axis is virtually
zero, and again the normal
and shearing st,resses are zero, giving the same boundary conditions as for a hole. For the external crack a somewhat similar boundary condition applies (see Section 4)
3. (i)
Descriptio,7
ofelectrical
ELECTRICLIT, ANUOGY
analogue
The electrical analogue in effect provides a solution to the biharmonic equation when expressed in finite difference form. The principle of the analogy has been previously described (Panmrc and REDSIIA~ 1955) and the electrical analogue used in the present instance is a very enlarged version of the prototype apparatus. The electrical resistance network is described diagrammatically in Fig. 3. The lower network consists of a square grid built with resistors of uniform resistance value, a similar network forms the middle network and the upper network comCorresponding nodes of the upper prises a similar grid but without resistors. and lower
nets are connected
to the intermediate
net through
resistors
whose
An electrical analogne solution for the stresses near a crack or hole in a flat plate resistance
value is large compared
the net permits an optimum of double
symmetry,
with that of the net resistors.
mesh separation
of &
x &
177
The fineness of
and, by taking advantage
the whole net was used to represent one quarter of the plate.
Upper net E 5 200V
Mlddle net V” f2V
Lower net X-c vn+20mV
FIG. 3. Resistance network.
On the network an axis of symmetry is automatically ensured by doubling the In the analogy a potential r) on the resistance value of the boundary resistors. lower net represents the Airy stress function and the potential V on the middle net represents (ii)
Satisfaction
the sum of the direct stresses (CT,+ uY). of boundary
The satisfaction
conditions
of a double
boundary
condition
is required
; for the loaded
edge a potential, representing a uniform stress, is applied to the middle net and the Airy stress function, as calculated by the method described in Section 2 (ii), is applied directly to the lower net, again in the form of a potential. For the free edge it: = f a the Airy stress function + and its normal @/bn
are specified, and this is analogous
to v and its normal
derivative
to Jv/‘>n.
to setting the lower net boundary To achieve
this, boundary
gradient potential
potentials
V
are set on the middle net and adjusted until v and &/&a are brought to the assigned values on the lower net. This involves a type of electrical iteration process but with a little care and experience the desired conditions can rapidly be achieved. For a hole or, a crack the conditions are again that v and 3v/&z are specified. A description of the technique for setting up the boundary conditions on the electrical analogue is to form the subject of a separate paper.
S. C. RI~DSIUW and K. R.
17X
4.
Three length
plane
stress problems
breadth
(i) (ii)
(iii)
ExPEI
ratio 3/2.
Two external
were investigated,
cracks ; i.e. a simply
with
a rectangular
connected
circular
hole ; i.e. a multiply
exists for a finite width of plate.
internal
crack ; i.e. a multiply
External
connected
solution
exist,
region
to which
region problem
greater experimental
but an analytical
an
similar
to
difficulty.
solution
for an
plate is available.
crack
The problem 3/2
connected
hole but presenting
analytical
of
region.
solution
An
plate
were as follows :
analytical
internal crack in an it/$&e
width.
selected
A central
No known
ratio
INVESTIGATION
The problems
that of the circular
(i)
RUSHTON
considered
and having The
was that of a Rat rectangular
two
geometry,
symmetrically
boundary
and
illustrated in Fig. 1. The half-length intervals, and therefore the half-width
disposed loading
plate of length/breadth
cracks
conditions
each & of
of the plate
the
problem
are
of the plate was represented by 48 mesh and each crack length by 32 and 12 units,
respectively. Using the concept stress function
of the slender ellipse, described
can bc calculated
stress and +5/3y = O. The electrically analogous and the potentials From
at
the electrical
the manner
measurements
are shown in Fig. 4. for comparison. It should be noted since an infinitely
condition
was applied
where
2 (iii), the Airy P is the applied
to the electrical
analogue,
on the middle and lower nets were measured as necessary.
M&S
described
in Section
to be + = Pa (a -x).
direct and shearing
in the Appendix. No known
theoretical
The
stresses were calculated
stress components
or experimental
results were available
that the stress at the end of the crack should
fine crack has been represented
in
a=, oY and T._
on the analogue,
be infinite,
but, owing to
the finite mesh used, the recorded stress at the end of the crack represents the average value of the stress over the area represented by the discrete interval. As a check 011 the experiment the average longitudinal stress across a number of sections was calculated from the experimental results and expressed as a percentage of the applied stress. in the section containing less than 2 per cent. (ii)
Circular
The agreement
the crack
was good, an error of about 5 per cent
being reduced
in subsequent
experiments
to
hole
A rectangular plate of the same size as that used in the last experiment was selected, but in place of the external cracks the plate was perforated with a centrally placed circular hole whose diameter was & the breadth of the plate. As it was not possible to represent the curved boundary of the hole 011 the square network mesh, the bourldary was considered to coincide with the nearest node. This technique has previously been found to be satisfactory and the result of the present experiment confirms this statement.
An electrical analogue solution for the stresses near a crack or hole in a fiat plate
I
-.-----._~ .__ -.-_.__.!-4
179
180
20
I
‘x ‘O a b E E
1
\
__- -f 1
’
____~
!\
7030 8057 8502 8582
I
8662
x
0
E -2 ? = -10
9486
Correct
VOlUe
Blichdl’s integral
Y
(
_ 17.7 - 7.7 - 2.1 - 14; + WI3 -t 8.5
>: -20
1 IO
9
8 7x103
7
6
FIG. 5. Michell’s integral
__
: circular hole.
FIc. 6. Circular hole : direct stresses. .\pplied stress : Direct stress ; - - Transverse stress 0,.
100.
An electrical analogue solution for the stresses near a crack or hole in a flat plate The loading function
conditions
are identical
2 (iii) the internal
on the external
to those obtaining
boundary
conditions
derivative to be zero. The correct The iterative process necessary consists in choosing electrical
a probable
analogue,
and
boundary
in terms of the Airy
in the last problem. require
measuring
the
y to be constant
potentials
and its normal condition
setting the condition on
stress
As shown in Section
value of the function satisfies (5). for setting up the internal boundary
value for the function,
181
the
middle
net
in the in close
proximity to the hole in order to evaluate the line integral in (5). Since V2 4 represents the middle net potential the line integral is simply the sum of the normal slope of the middle net potentials around the hole. It will thus be seen that only a few measurements
need be taken,
The process has to be repeated cally satisfied ; the iterative
the rest of the board
being left unscanned.
for several values of the function
until (5) it identi-
results are shown in Fig. 5. The experimental
process
was rapid and the correct value for y was obtained after about five trials, each trial taking less than one hour. The correct value for y was then set on the network The direct stresses oY and a, were then calculated and the field scanned completely. in the manner
described
The tangential experiment,
in the Bppendix,
the results being
stress on the circumference
has been compared
in Fig. 7 with the analytical
boundary
FIG. 7. (it) Tangential
stress on circumference
of circle.
shown
in Fig. 6.
of the hole, as determined
”
I
4
8
by the
result obtained
12
16
(b) Stress variation
20
24
28
by
32
on cross section.
HOLLAND (1930) for a plate of finite width and infinite length, containing
a circular
hole. On the same Figure a comparison between the experimental and theoretical stress variation on the central cross section is given. It will be noted that the It is interesting to observe that from a agreement in both cases is very good. collection of photo-elastic results compiled by Hevwoon (1952) the maximum tangential analogue (iii)
stress on the hole is given as 3.04, which lies between the present electrical result of 3.00 and I&WLAXD’S
Idernal
value of 3.14.
crack
With the same plate size as in the last two experiments lying along the horizontal axis of symmetry was considered. length from f to Q x the width of the plate were considered.
an internal crack Cracks varying in
s.
18%
The loading previous loading
conditions
experiments, condition
C‘. lllmslr‘~w
on the external
but
the
method
of
boundary
be used satisfactorily
were the same as for the
interpreting
varied from that for the circular
fact that the integral condition cannot
and I<. H. KusIrroN
hole.
the
internal
The difficulty
boundary lies in the
(5), which has to be npplicd to the internal boundary, because
the infinite stress which should arise at the
FIG. 8. Transverse stress for trial values of y. All values are expressed as fractions applied stress.
of the
Width of crack (mesh mtervals)
0
8 Wldlh
16
24
of crock
(mesh mfervals)
FIG. 9.
(a) Variation of y with width of crack. (b) Maximum stress. edge.
(c) Centre of unloaded
end of the crack cannot be represented in the analogue because of the finite mesh of the network. i2n alternative condition was sought and it was decided to use the condition that the transverse stress at the centre of the crack is a compressive would stress numerically equal to t.he applied tensile stress. This assmnption appear to be justified if the crack is considered to be an ellipse with an infinitely
I. ____ __.“lll______..-__. 2
__-..
A
s.
184
(‘.
1tI’:IJSliAW
and I<. IL
Hr:sArro~
small minor axis, in which case it is well known theoreticall>- that this stress condition is applicable. Apart from this, the iteration pror~ss for setting up the internal boundary csondition was identical to that used for the circrdar hole. The manner in which the transverse stress varies with y is shown in Fig. 8 ; it, should be noted that the correct value of y occurs when the transverse stress is unity. Six crack widths were investigated value of y, the maximum unloaded
by this method.
For each experimnlt
the
stress at the end of the crack, and the stress at the free
edge on t,he centre of the plate normal to the dire&ion
of the applied
stress, were determined and are plotted in Fig. 9. During the experiments it was found that a crack exceeding
$ of the plate
width could not be set up on the electrical analoguc because the stress at the end of the crack was \-cry large, and could not be represented by an equivalent
electrical
potential.
It will 1,~ apprec*iated that the htrcss at the end of tllc (*rack sholdd, for ever>crack lelrgth. be infinite but owing to the mesh size. as cxplaiiied previousI>,, finite Possit)l?- the network ma). be considered to be acting stresses were obtained. :~~~~lopusl~ to a metallic plate which will experience plastic yielding at the edge
of the craek. =\ detailed analysis of the stresses for a crack width of 2C ‘7; t,he plate width are shown in Fig. 10. It is of interest to note that the average direct stress at the
An electrical analogue solution for the stresses near a crack or hole in a flat plate
185
section containing the crack, as found from the experiment, was 98.5 per cent of the applied stress. Fig. 11 gives a comparison between the analogue results for a finite plate and the theoretical results for an infinite plate (FROST and DUGDALE 195q*. For the purpose of making a comparison the sum of the principal stresses have been presented in Fig. 11. It will be observed that a reasonable agreement is obtained in the region of the crack but, as would be anticipated, appreciable differences occur in the vicinity of the external boundary. The results of a photoelastic experiment made at the University of Washington (CHENG et. al. 1959) on a rectangular plate with a central crack, but of different proportions to the plate considered here, showed the same trend in the distribution of the principal stresses at points remote from the crack. The agreement in the region of the crack was poor since the crack was formed by making a saw cut, the thickness of the cut being & of the width of the crack as compare6 with the infinitely thin crack assumed for the electrical analogue. The mesh length used for the electrical analogue was f9 that used in the photo-elastic experiment.
5.
CONCLUSIONS
Within the limitations imposed by the size of the network the results obtained from the electrical analogue appear to be reasonable, and good agreement with the few analytical results available was obtained. In the case of a crack, the major limitations in the method lies in the inability to represent the infinite stress at the end of the crack, whereas for a circular hole this limitation does not apply and excellent results were obtained. It is probable that a closer approximation to the true results for a crack could be obtained if a graded mesh were constructed or if, in a future experiment, the results obtained for a portion of the field were reset to a finer mesh (PALMER and REDSHAW 1957). No attempt was made to introduce plastic yielding in the neighbourhood of the crack although, as mentioned in Section 4 (iii), this may be inherent in the electrical analogy. It is considered that an investigation into the possibility of introducing plasticity into the analogy is very desirable in view of the importance of plastic yielding in problems of crack propagation.
REFERENCES CHENG, Y. F., MILLS, D. B. and DAY, E. E. 1959 FROST, N. E. and 1958 DUGDALE, D. S. HEYWOOD, R. B. 1952 1930 HOWLAND, R. J. C. MICHELL, J. H. 1899 PALMER, P. J. and 1955 REDSHAW, S. C. 1957 1946 PRAGER, W. SOUTHWELL,R. V. 1948
Trend Engng Univ. Wash. 11, 15. J. Mech. Phys. Solids 6, 92. Designing by Photoelasticity, p, 268, (Chapman & Hall). Phil. Trans. Roy. Sot. A 229, 49. Proc. Lond. Math. Sot. 31, 100. Aero. Quart. 6, 13. J. Set. Inst. 34, 307. Quart. A&. Math. It 1, 377. PTOC. Roy. SOC. A 193, 147.
*Approximate formulae were derived by DUGDALE and quoted by FROST and DUGDALE. DUGDALE recorded that the formulae were previously given by T. POSCH~,Math. Zeit. 11, 89 (1912) and A. TIMEX, i?Zalh. Zeit. 17,189 (1923).
186
and K. R. HUSHTON
S. C. REDSHAW
APPENDIX Calculation
of Direct
and Shearing
Stresses from
the Stress Functiorl
Values of the stress function are obtained directly from the electrical analogue and these values have to be doubly differentiated to give the stresses. Numerous finite difference formulae exist for this purpose but it has been found that the following five point formula, written below, gives better results when applied to analogue readings than the more usual formulae, although the truncation error is actually greater. The purpose of the formula is to nrake the errors in the electrical readings of the same order as the truncation errors
where the suffices denote the nodes of the networks as defined in Fig. Al.
f
P 6
7
2
5
4
e
L h h -. J_
I2
/
FIG. Al.
In the neighbourhood of boundaries the above formula cannot be used but since the middle net potentials equal A2+/bx2 + P +/by2 and since a2 $/3y2 can still be calculated by the corresponding formula, Ss 4/bx2 may be obtained by subtraction. For the computation of the shearing stress the conventional formula
was used. On occasions, recourse to hwaphically smoothing of the differentials was found to be necessary.
AN
ELECTRICAL
STRESSES
NEAR By
ANALOGUE A CRACK
FOR
OR I-IOLE IN A FLAT
S. C. REDSHAW
Department
SOLUTION
and
of Civil Engineering,
K.
R.
University
THE PLATE
RIJSHTON of Birmingham
SUMMARY AN ELECTRICAL analcgue for solving the biharmonic equation was used to determine the elastic stresses near a hole or crack in a fiat rectangular plate subjected to uniaxial tension. Problems concerning both simply and multiply connected regions were solved by the method and, where a comparison between the experimental results and existing analytical solutions was possible, the agreement was good. The advantages and limitations of the method are discussed and the desirability of extending the mH,hod for the determination of the stresses after plastic yielding has occurred, is emphasized.
1. THE
DETERMINATION
INTRO~U~~I~N
of the stresses in the region of a hole or crack in a stressed
plate is of great engineering importance, particularly but unfortunately
in components liable to fatigue,
theoretical solutions are difficult to obtain and, indeed, very few Experimental methods such as photo-elasticity, strain gauge
solutions exist. investigations and brittle lacquer techniques, have been used in an attempt to obtain reliable results but, although a hole can be made easily, a fine crack is ditllcult to manufacture.
lies in the inability
to
measure strains over a short length and at sufficient stations in the proximity
However,
the principal difficulty
of
the crack or hole. For this reason it was decided to attempt
a solution to the problem
by using
an electrical analogue method, a technique which had already yielded satisfactory solutions to other elastic problems.
2.
(i)
OUTLINE
OF
PROBLEM
General The cases investigated
all concerned the plane stress problem of a flat rectangular
plate subjected to uniaxial tension, and having either a central hole or a central crack disposed perpendicularly to the direction of applied stress, or external cracks centrally disposed in the same direction. The problems are summarized in Fig. 1. It will be seen that the problems considered concern either simply or multiply connected regions. PALMER. and REDSHAVV (1955) have established an electrical analogue for the solution of the biharmonic equation and, for their analogy to be used in the present 173
s. c.
174
RICDSIZAW
xad
Ii. lx. ItUSlZTOE
problem has to be expressed in terms of the iliry stress funct~ion and conditions formulated accordingly. When the boundary conditions have been represented on the electrical analoguc network the measurement of instance,
the
the boundary
the electrical potentials at various nodes supplies the ;liry stress fnnction at those
#=O
26 -p -z-
2x
(a) x+
-
FIG. 1. (i) I~:sternal cr;wks.
points.
Subsequent
0
(b)
(ii) Cirdm holes. (iii) Internal crac:ks. (a) Loading rmditions. (b) Airy stress functions.
double
differentiation
yields
the stresses but, additionally,
the sum of these stresses ran be measured directly from the network. For a simply connected region the boundary conditions may be expressed directly in terms of the Airs stress function, but for a mult.ipty connected region an additional
boundary
condition
is required.
An electrical analogue solution for the stresses near a crack or bole in a Aat plate
Determination
(ii)
qf Airy
stress fun&m
The stresses on any boundary, the components
of the surface x=1
175
see Fig. 2, can be expressed
stresses and the direction
az + m 7zyp
Y = m o,, i_
in terms of X and Y
cosines
t and m :
1 rxy.
(1)
1=cosa m=sina
FIG. 2. The stresses on any boundary.
Since
equations
(1) can
be written
in terms of the Airy stress function
4 and, on inte-
gration, -W = AZ i_ Urn -1 al + /3m. 3n
(3)
Hence
where A=.-s cc,
p
and y are constants
chosen arbitrarily. can be represented
a Y ds,
of integration
B =
“1Xc-Es. 1. 0
0
which for a simply connected
region can be
Therefore, when strains are specified on the boundary by the Airy stress function and its normal gradient.
they
In the problems considered, not only were the normal and shearing stresses known on the loaded bonndaries, y = + b (Fig. I), but the transverse stress was assumed to be zero. Hence, instead of having to specify 4 and &j,&n on these boundaries, V2 + and (b can be used, which is more convenient for the electrical analogne. If the body is multiply connected be chosen to ensure that the rotations
the constants are of importance and must and displacements are continuous. On one the constants are chosen arbitrarily boundary,
boundary, usually the external but their values on the internal boundaries tion.
are restricted
by the continuity
condi-
S. C. REDSHAWand K. R. RUSHTON
176
MICIIELI, (1899) first presented these conditions in terms of the Airy stress ; he deduced the equations for each internal boundary which must be
function
satisfied to ensure condition
and displacement
single valuedness
on the boundary.
These
are the rotation
conditions
IIygp!) -r$(v2+s=o, liy$(V2#++2+)
1
I
ds=O.
1
For a multiply ditions
connected
i
region the constants
must be chosen
(6)
so that these con-
are satisfied.
Twofold
symmetry
occurred
in all the problems
therefore c( and /3 become zero, also the displacement satisfied. There remains, methods
however,
the determination
exist : either a superposition
method,
described
conditions
in this paper
and
(6) are automatically
of y and for this two alternative described
by PRAGER (1946),
or
an iterative method dur to SO~TTHWEIJ, (1948). During the experiments both methods were tried, SOUTHWELL’S method being finally used as the superposition method
entailed
the summing
of two solutions,
with the attendant
inaccuracies.
Using Sou~~~w~~~~‘s~metl~od, various values of y were tried until the one giving the correct such that (iii)
solution
of (5) was found ; the nature
this presented
Boundary The boundary
conditions
of the electrical
analogue
is
no difficulty.
.for
conditions
cracks and holes for cracks
and holes require
special
a hole the normal stress and the shearing stress are both zero.
mention.
For
When substituted
in (3) and (4) this yields 4 L y and &$/An = 0. A fine line internal crack may be considered
as an ellipse whose minor axis is virtually
zero, and again the normal
and shearing st,resses are zero, giving the same boundary conditions as for a hole. For the external crack a somewhat similar boundary condition applies (see Section 4)
3. (i)
Descriptio,7
ofelectrical
ELECTRICLIT, ANUOGY
analogue
The electrical analogue in effect provides a solution to the biharmonic equation when expressed in finite difference form. The principle of the analogy has been previously described (Panmrc and REDSIIA~ 1955) and the electrical analogue used in the present instance is a very enlarged version of the prototype apparatus. The electrical resistance network is described diagrammatically in Fig. 3. The lower network consists of a square grid built with resistors of uniform resistance value, a similar network forms the middle network and the upper network comCorresponding nodes of the upper prises a similar grid but without resistors. and lower
nets are connected
to the intermediate
net through
resistors
whose
An electrical analogne solution for the stresses near a crack or hole in a flat plate resistance
value is large compared
the net permits an optimum of double
symmetry,
with that of the net resistors.
mesh separation
of &
x &
177
The fineness of
and, by taking advantage
the whole net was used to represent one quarter of the plate.
Upper net E 5 200V
Mlddle net V” f2V
Lower net X-c vn+20mV
FIG. 3. Resistance network.
On the network an axis of symmetry is automatically ensured by doubling the In the analogy a potential r) on the resistance value of the boundary resistors. lower net represents the Airy stress function and the potential V on the middle net represents (ii)
Satisfaction
the sum of the direct stresses (CT,+ uY). of boundary
The satisfaction
conditions
of a double
boundary
condition
is required
; for the loaded
edge a potential, representing a uniform stress, is applied to the middle net and the Airy stress function, as calculated by the method described in Section 2 (ii), is applied directly to the lower net, again in the form of a potential. For the free edge it: = f a the Airy stress function + and its normal @/bn
are specified, and this is analogous
to v and its normal
derivative
to Jv/‘>n.
to setting the lower net boundary To achieve
this, boundary
gradient potential
potentials
V
are set on the middle net and adjusted until v and &/&a are brought to the assigned values on the lower net. This involves a type of electrical iteration process but with a little care and experience the desired conditions can rapidly be achieved. For a hole or, a crack the conditions are again that v and 3v/&z are specified. A description of the technique for setting up the boundary conditions on the electrical analogue is to form the subject of a separate paper.
S. C. RI~DSIUW and K. R.
17X
4.
Three length
plane
stress problems
breadth
(i) (ii)
(iii)
ExPEI
ratio 3/2.
Two external
were investigated,
cracks ; i.e. a simply
with
a rectangular
connected
circular
hole ; i.e. a multiply
exists for a finite width of plate.
internal
crack ; i.e. a multiply
External
connected
solution
exist,
region
to which
region problem
greater experimental
but an analytical
an
similar
to
difficulty.
solution
for an
plate is available.
crack
The problem 3/2
connected
hole but presenting
analytical
of
region.
solution
An
plate
were as follows :
analytical
internal crack in an it/$&e
width.
selected
A central
No known
ratio
INVESTIGATION
The problems
that of the circular
(i)
RUSHTON
considered
and having The
was that of a Rat rectangular
two
geometry,
symmetrically
boundary
and
illustrated in Fig. 1. The half-length intervals, and therefore the half-width
disposed loading
plate of length/breadth
cracks
conditions
each & of
of the plate
the
problem
are
of the plate was represented by 48 mesh and each crack length by 32 and 12 units,
respectively. Using the concept stress function
of the slender ellipse, described
can bc calculated
stress and +5/3y = O. The electrically analogous and the potentials From
at
the electrical
the manner
measurements
are shown in Fig. 4. for comparison. It should be noted since an infinitely
condition
was applied
where
2 (iii), the Airy P is the applied
to the electrical
analogue,
on the middle and lower nets were measured as necessary.
M&S
described
in Section
to be + = Pa (a -x).
direct and shearing
in the Appendix. No known
theoretical
The
stresses were calculated
stress components
or experimental
results were available
that the stress at the end of the crack should
fine crack has been represented
in
a=, oY and T._
on the analogue,
be infinite,
but, owing to
the finite mesh used, the recorded stress at the end of the crack represents the average value of the stress over the area represented by the discrete interval. As a check 011 the experiment the average longitudinal stress across a number of sections was calculated from the experimental results and expressed as a percentage of the applied stress. in the section containing less than 2 per cent. (ii)
Circular
The agreement
the crack
was good, an error of about 5 per cent
being reduced
in subsequent
experiments
to
hole
A rectangular plate of the same size as that used in the last experiment was selected, but in place of the external cracks the plate was perforated with a centrally placed circular hole whose diameter was & the breadth of the plate. As it was not possible to represent the curved boundary of the hole 011 the square network mesh, the bourldary was considered to coincide with the nearest node. This technique has previously been found to be satisfactory and the result of the present experiment confirms this statement.
An electrical analogue solution for the stresses near a crack or hole in a fiat plate
I
-.-----._~ .__ -.-_.__.!-4
179
180
20
I
‘x ‘O a b E E
1
\
__- -f 1
’
____~
!\
7030 8057 8502 8582
I
8662
x
0
E -2 ? = -10
9486
Correct
VOlUe
Blichdl’s integral
Y
(
_ 17.7 - 7.7 - 2.1 - 14; + WI3 -t 8.5
>: -20
1 IO
9
8 7x103
7
6
FIG. 5. Michell’s integral
__
: circular hole.
FIc. 6. Circular hole : direct stresses. .\pplied stress : Direct stress ; - - Transverse stress 0,.
100.
An electrical analogue solution for the stresses near a crack or hole in a flat plate The loading function
conditions
are identical
2 (iii) the internal
on the external
to those obtaining
boundary
conditions
derivative to be zero. The correct The iterative process necessary consists in choosing electrical
a probable
analogue,
and
boundary
in terms of the Airy
in the last problem. require
measuring
the
y to be constant
potentials
and its normal condition
setting the condition on
stress
As shown in Section
value of the function satisfies (5). for setting up the internal boundary
value for the function,
181
the
middle
net
in the in close
proximity to the hole in order to evaluate the line integral in (5). Since V2 4 represents the middle net potential the line integral is simply the sum of the normal slope of the middle net potentials around the hole. It will thus be seen that only a few measurements
need be taken,
The process has to be repeated cally satisfied ; the iterative
the rest of the board
being left unscanned.
for several values of the function
until (5) it identi-
results are shown in Fig. 5. The experimental
process
was rapid and the correct value for y was obtained after about five trials, each trial taking less than one hour. The correct value for y was then set on the network The direct stresses oY and a, were then calculated and the field scanned completely. in the manner
described
The tangential experiment,
in the Bppendix,
the results being
stress on the circumference
has been compared
in Fig. 7 with the analytical
boundary
FIG. 7. (it) Tangential
stress on circumference
of circle.
shown
in Fig. 6.
of the hole, as determined
”
I
4
8
by the
result obtained
12
16
(b) Stress variation
20
24
28
by
32
on cross section.
HOLLAND (1930) for a plate of finite width and infinite length, containing
a circular
hole. On the same Figure a comparison between the experimental and theoretical stress variation on the central cross section is given. It will be noted that the It is interesting to observe that from a agreement in both cases is very good. collection of photo-elastic results compiled by Hevwoon (1952) the maximum tangential analogue (iii)
stress on the hole is given as 3.04, which lies between the present electrical result of 3.00 and I&WLAXD’S
Idernal
value of 3.14.
crack
With the same plate size as in the last two experiments lying along the horizontal axis of symmetry was considered. length from f to Q x the width of the plate were considered.
an internal crack Cracks varying in
s.
18%
The loading previous loading
conditions
experiments, condition
C‘. lllmslr‘~w
on the external
but
the
method
of
boundary
be used satisfactorily
were the same as for the
interpreting
varied from that for the circular
fact that the integral condition cannot
and I<. H. KusIrroN
hole.
the
internal
The difficulty
boundary lies in the
(5), which has to be npplicd to the internal boundary, because
the infinite stress which should arise at the
FIG. 8. Transverse stress for trial values of y. All values are expressed as fractions applied stress.
of the
Width of crack (mesh mtervals)
0
8 Wldlh
16
24
of crock
(mesh mfervals)
FIG. 9.
(a) Variation of y with width of crack. (b) Maximum stress. edge.
(c) Centre of unloaded
end of the crack cannot be represented in the analogue because of the finite mesh of the network. i2n alternative condition was sought and it was decided to use the condition that the transverse stress at the centre of the crack is a compressive would stress numerically equal to t.he applied tensile stress. This assmnption appear to be justified if the crack is considered to be an ellipse with an infinitely
I. ____ __.“lll______..-__. 2
__-..
A
s.
184
(‘.
1tI’:IJSliAW
and I<. IL
Hr:sArro~
small minor axis, in which case it is well known theoreticall>- that this stress condition is applicable. Apart from this, the iteration pror~ss for setting up the internal boundary csondition was identical to that used for the circrdar hole. The manner in which the transverse stress varies with y is shown in Fig. 8 ; it, should be noted that the correct value of y occurs when the transverse stress is unity. Six crack widths were investigated value of y, the maximum unloaded
by this method.
For each experimnlt
the
stress at the end of the crack, and the stress at the free
edge on t,he centre of the plate normal to the dire&ion
of the applied
stress, were determined and are plotted in Fig. 9. During the experiments it was found that a crack exceeding
$ of the plate
width could not be set up on the electrical analoguc because the stress at the end of the crack was \-cry large, and could not be represented by an equivalent
electrical
potential.
It will 1,~ apprec*iated that the htrcss at the end of tllc (*rack sholdd, for ever>crack lelrgth. be infinite but owing to the mesh size. as cxplaiiied previousI>,, finite Possit)l?- the network ma). be considered to be acting stresses were obtained. :~~~~lopusl~ to a metallic plate which will experience plastic yielding at the edge
of the craek. =\ detailed analysis of the stresses for a crack width of 2C ‘7; t,he plate width are shown in Fig. 10. It is of interest to note that the average direct stress at the
An electrical analogue solution for the stresses near a crack or hole in a flat plate
185
section containing the crack, as found from the experiment, was 98.5 per cent of the applied stress. Fig. 11 gives a comparison between the analogue results for a finite plate and the theoretical results for an infinite plate (FROST and DUGDALE 195q*. For the purpose of making a comparison the sum of the principal stresses have been presented in Fig. 11. It will be observed that a reasonable agreement is obtained in the region of the crack but, as would be anticipated, appreciable differences occur in the vicinity of the external boundary. The results of a photoelastic experiment made at the University of Washington (CHENG et. al. 1959) on a rectangular plate with a central crack, but of different proportions to the plate considered here, showed the same trend in the distribution of the principal stresses at points remote from the crack. The agreement in the region of the crack was poor since the crack was formed by making a saw cut, the thickness of the cut being & of the width of the crack as compare6 with the infinitely thin crack assumed for the electrical analogue. The mesh length used for the electrical analogue was f9 that used in the photo-elastic experiment.
5.
CONCLUSIONS
Within the limitations imposed by the size of the network the results obtained from the electrical analogue appear to be reasonable, and good agreement with the few analytical results available was obtained. In the case of a crack, the major limitations in the method lies in the inability to represent the infinite stress at the end of the crack, whereas for a circular hole this limitation does not apply and excellent results were obtained. It is probable that a closer approximation to the true results for a crack could be obtained if a graded mesh were constructed or if, in a future experiment, the results obtained for a portion of the field were reset to a finer mesh (PALMER and REDSHAW 1957). No attempt was made to introduce plastic yielding in the neighbourhood of the crack although, as mentioned in Section 4 (iii), this may be inherent in the electrical analogy. It is considered that an investigation into the possibility of introducing plasticity into the analogy is very desirable in view of the importance of plastic yielding in problems of crack propagation.
REFERENCES CHENG, Y. F., MILLS, D. B. and DAY, E. E. 1959 FROST, N. E. and 1958 DUGDALE, D. S. HEYWOOD, R. B. 1952 1930 HOWLAND, R. J. C. MICHELL, J. H. 1899 PALMER, P. J. and 1955 REDSHAW, S. C. 1957 1946 PRAGER, W. SOUTHWELL,R. V. 1948
Trend Engng Univ. Wash. 11, 15. J. Mech. Phys. Solids 6, 92. Designing by Photoelasticity, p, 268, (Chapman & Hall). Phil. Trans. Roy. Sot. A 229, 49. Proc. Lond. Math. Sot. 31, 100. Aero. Quart. 6, 13. J. Set. Inst. 34, 307. Quart. A&. Math. It 1, 377. PTOC. Roy. SOC. A 193, 147.
*Approximate formulae were derived by DUGDALE and quoted by FROST and DUGDALE. DUGDALE recorded that the formulae were previously given by T. POSCH~,Math. Zeit. 11, 89 (1912) and A. TIMEX, i?Zalh. Zeit. 17,189 (1923).
186
and K. R. HUSHTON
S. C. REDSHAW
APPENDIX Calculation
of Direct
and Shearing
Stresses from
the Stress Functiorl
Values of the stress function are obtained directly from the electrical analogue and these values have to be doubly differentiated to give the stresses. Numerous finite difference formulae exist for this purpose but it has been found that the following five point formula, written below, gives better results when applied to analogue readings than the more usual formulae, although the truncation error is actually greater. The purpose of the formula is to nrake the errors in the electrical readings of the same order as the truncation errors
where the suffices denote the nodes of the networks as defined in Fig. Al.
f
P 6
7
2
5
4
e
L h h -. J_
I2
/
FIG. Al.
In the neighbourhood of boundaries the above formula cannot be used but since the middle net potentials equal A2+/bx2 + P +/by2 and since a2 $/3y2 can still be calculated by the corresponding formula, Ss 4/bx2 may be obtained by subtraction. For the computation of the shearing stress the conventional formula
was used. On occasions, recourse to hwaphically smoothing of the differentials was found to be necessary.