i, i pt,,(il becausie ol: the Large degree of ... I.,. 'uutip. 1i ic' Probl.emn of the Mathematical Theory of. 1:.>IL.! it(v, .... A l. I,. , II ?I /3. ..... ROCK ISLAND, IL 61299.
AD
TECHNICAL REPORT ARLCB-TR-82027
STRESS INTENSITY AND FATIGUE CRACK GROWTH IN A PRESSURIZED: AUTOFRETTAGED THICK CYLINDER A. J. J. C.
P. H. F. P.
Parker Underwood Throop Andrasic
September 1982
US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMAND LARGE CALIBER WEAPON SYSTEMS LABORATORY BENET WEAPONS LABORATORY WATERVLIET, N. Y. 12189
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S.
STRESS INTENSITY AND FATIGUE CRACK GROWTH IN A PRESSURIZED, AUTOFRETTAGED THICK CYLINDER
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7.
A. J. 9.
AUTHOR(e)
P. Parker 1 H. Underwood
J. F. C. P.
2
8.
Throop 2 Andrasic
3
PERFORMING ORGANIZATION NAME AND ADDRESS
i0.
US Army Armament Research & Development Command Benet Weapons Laboratory, DRDAR-LCB-TL Watervliet, NY 12189 I.
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SUPPLEMENTARY NOTES
Presented at Fourteenth National Symposium on Fracture-Mechanics, IUCLA, Los Angeles, CA, 29 June - I July 1981. Published in ASTM Special Technical Publication, I9.
KEY WORDS (Continue on reveree side If necoeary and identify by block ni mber)
Crack Growth Fatigue Crack Cylinders
Fracture (Materials) Residual Stress Stress Intensity Factor U
z2, A$r•IACT (Cmtli"ste d
,o ewwmm oft
If n
eeay and
It.enify' b-
Stress intensity factors are determined,
block noakber)
using the modified mapping collocation
(MMC) method for a single, radial, straight-fronted crack in a thick cylindrical tube which has been subjected to full autofrettage treatment (100 percent overstrain). By superposition of these results and existing solutions, stress intensity factors are datermined tor the same geometry with iaternal pressure and any amount of overstrain from zero to 100 percent.
(CONTID ON REVERSE) SI)
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7.
AUTHOR(S)
(CONT'D);
IGuest Scientist, US Army Materials and Mechanics Research Center, Watertown, MA, from The Royal Military College of Science, Shrivenham, SN6 8LA, England. 2
Research Engineers,
3
Research Scientist, The Royal Military College of Science, 8LA England.
20.
*
USA ARRADCOM,
Watervliet,
NY. Shrivenham,
SN6
ABSTRACT (CONT' D):
Correction factors for crack shape and non-ideal material yielding are determined from various sources for the pressurized, autofrettaged tubes containing semi-elliptical cracks. These results are employed in the life prediction of pressurized thick tubes with straight-fronted and semi-circular cracks, for various amounts of autofrettage. Experimentally determined lifetimes for tubes having zero and 30 percent nominal overstrain are significantly greater than the predictions for both straight fronted and semi-circular cracks. This is related to multiple initiation and early growth of cracks from the notch. Experimentally determined lifetimes for a tube with a 60 percent nominai overstrain are somewhat less than predicted. This effect is partially explained by additional experimental work which shows that the angle of opening of rings cut from autofrettaged tubes is somewhat less than the ideal predictions.t The latter effect is attributed to the Bauschinger effect and the associated reduced yield strength in compression during the unloading of tubes during the.'autofrettage process. \
II
w
AA
W
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laht. F.rno .a
TABLE OF CONTENTS Page ACKNOWLEDGEMENTS
iii
NOMENCLATURE
iv
INTRODUCTION
1
CALCULATION OF STRESS INTENSITY FACTORS
2
Loading A
5
Loading B
6
Loading C
7
CORRECTION FOR THE CRACK SHAPE EFFECTS
8
EFFECTS OF NONIDEAL RESIDUAL STRESS DISTRIBUTION
14
LIFE CALCULATIONS
17
DISCISSION AND CONCLUSIONS
21
REFERENCES
24
APPENDIX
A-I
APPENDIX REFERENCES
A-4 TABLES
I.
COMPARISON OF K VALUES FOR DEEPLY CRACKED CYLINDER AND, PLATE
12
LIST OF ILLUSTRATIONS 1.
Cracked thick cylinder geometry showing partitioning and mapping schemes. (a) z (pbyslcal) plane, (b) ý plane, (c) ý plane.
27
2.
Stress intensity factors for a single, straight-fronted, radial crack in a thick cylinder. (Inset) Short crack length convergence for 100 percent overstrain.
28
3.
Stress intensity factors for a single, straight-fronted crack in a pressurized thick cylinder with 0 percent, 30 percent, 60 percent and 100 percent overstrain.
29
Page 4.
Semi-elliptical
5.
(a) (b)
6.
(a) (b)
in
a thick cylinder.
Crack shape factors for a plate in after Reference 14. Crack shape factors for a plate in (, iT/2), after Reference 14.
-
0.8,
31
pure bending
32
12.
Crack shape factors a/c - 0.8, = 0.
(a)
Crack shape factors, Lp, for pressure in bore and cracks, S- r/2; two data points from Reference 14, a/c m 1.0. Crack shape factors, Lp, for pressure in bore and cracks, , = 0; two data points from Refnrence 14, a/c = 1.0. Crack shape factors, La, for 100 percent overstrain, 4 - n/2. Cr-ck shape factors, La, for 100 percent overstrain, 4, 0..
(c) (d)
for pressure
in
bore and cracks,
33
8.
Ratio of opening angle for partial overstrain to the theoretical value for 100 percent overstrain.
34
9.
Fatigue crack growth (a versus N) in thick cylinder with single, straight-fronted crack; experimental results and predictions.
35
10.
Fatigue crack growth (a versus N) in thick cylinder with single, semi-circular crack; experimental results and predictions.
36
V
0
V
7 !/2),
(c)
(b)
V
tension (4
30
Method of calculating proportion of tension and bending, pressure in bore and cra~ks. Crack shape factors for pressure in bore and cracks,
a/c
7.
crack
ii
ACKNOWLEDGMENTS The first author performed work on this paper during a TTCP attachment to the Engineering Mechanics Laboratory, US Army Materials and Mechanics Research Center.
The, fourth author acknowledges support from a UK Ministry of Defense
research contract as a Research Scientist at the Royal Military College of Science.
NOMENCLATURE
*
a
Crack de;th
A
Autofrettage
c
Surface crack length
C
Coefficient in
E
Modulus of elasticity
G
E/2(1+v)
HC
Correction factor defined
K
Stress intensity factor
KI
Opening mode stress intensity factor
KII
Sliding mode stress intensity factor
K
Non-dimensionalizing stress intensity factor
K0
Non-dimensionalizing
6K
Stress intensity factor range
Zn
Natural logarithm
m
Exponent
max
Maximum
min
Minimum
N
Number of loading cycles
p
Pressure
SPB
Pure bending
PT
Pure tension
Sr
Paris'
in Paris'
crack growth law
in equation (11)
stress intensity factor
crack growth law
Radius R1
Tube inner radius
iv
S
R2
Tube outer radius
RA
Tube autofrettage radius
RD
Tube reversed yielding radius
t
Theoretical
Y
Yield stress
z
Complex variable,
a
Proportion of tension
a
Proportion of bending
y
Tube opening angle (radians)
x + iy
Parameter plane 0
Angular coordinate
K•
3-4v (plane strain),
X
Yield stress in tension/yield stress in compression
v S
Poisson's ratio
(3-v)/(l+v)
(plane stress)
Parameter plane a
Direct stress
T
Shear stress
*
Complex stress function
*
Complex stress function
v
or
INTRODUCTION Fatigue crack growth arising from the cyclic pressurization of thick-wall A
cylinders tends to produce radial fatigue cracks emanating from the bore. K, is
knowledge of the crack tip stress intensity factor,
critical length and lifetime of such
to predict the fatigue crack growth rate, cracks.
It
common practice to produce an advantageous stress distribution
is
by autofrettage growth.
(overstrain) of the cylinder in order to slow or prevent crack
This autofrettage process may involve plastic strain throughout the
wall thickness,
V
necessary in order
or any lesser proportion of the wall thickness,
depending upon
the degree of overstrain applied to the cylinder by over pressure or by an oversized mandrel-swage process. An accurate stress intensity solution for pressurized,
a fundamental requirement for crack grcwth rate and life
thick cylinders is prediction. tions is
autofrettaged
Some of the wrk relating to two and three dimensional K soluOf the various types of solutions,
reviewed by Tan and Fenner,1
the
errors associated with collocation and integral equation solutions are of the order of 1 percent,
while 4 percent would be more typical of finite element
and boundary element methods. tainties in
However,
there may be more significant uncer-
crack growth and life prediction.
tors considered here, of the component,
Some key factors,
and the fac-
are the shape of.the crack during the fatigue lifetime
uncertainty over the exact proportion of overstrain in
the
tube, and a residual stress distribution which generally does not conform with
ITan, C. L. and Fenner, R. T., "Stress Intensity Factors for Semi-Elliptical Surface Cracks in the Pressurized Cyliuders Using the Boundary Integral Equation Method," Int.
J.
Fracture,
Vol.
16,
No.
3,
1980,
pp.
233-245.
the predictions of an iaealized elastic-plastic
analysis,
assumption of the same magnitude of yield strength in In this report it may be modified in -
-
growth rates,
is
proposed that accurate
particularly
tension and compression.
two dimensional K solutions
order to predict stress intensity factors,
for semi-elliptical cracks in
with residual stress distributions. experimental crack growth data,
the
and hence crack
pressurized thick-wall cylinders
Such predictions may be compared with
and some measure of the extent of overstrain
may be obtained by cutting autofrettaged
tubes radially,
and measuring the
angle of opening.
CALCULATION OF STRESS INTENSITY FACTORS Stress intensity factors for the plane (two dimensional)
V
illustrated (MMC)
geometry
in Figure 1(a) were obtained by the modified mapping collocation
method.
This method is
complex variable methods,
described in
detail in
due to Muskhelishvili
displacements within a body are given in
3
Reference 2.
are utilized.
Briefly,
Stresses and
terms of the complex stress functions
4(z) and 4(z) by: ax + ay
a-
x + 2i
=
4 Re J4'(z)}
(I)
2[z4"(z)
(2)
,y
2
* * *
+ 4'(z)]
Andrasic, C. P. and Parker, A. P., "Weight Functions for Cracked Curved Beams," Numerical Methods in Fracture Mechanics, D. R. J. Owen and A. R. Luxmoore (Eds.), Proceedings Second International Conference, Swansea, 1980, pp. 67-82. 3 Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, 1973.
Wp 2
2G(u+iv)
(z)
-K
where the complex variable z - x + iy, primes denote differentiation,
Z
-
-
(3)
-Tz7
x and y are the physical coordinates,
and bars represent the complex conjugate.
Also: E
G
-----2(1+v)
K
where G is
3
-
4v (plane strain),
the shear modulus,
K -
3-v -l+V (plane stress)
E the elaseic modulus,
and v is
Poisson's ratio,
while the resultant force over an arc s is:
fl + if2
(Xn+iYn)ds
i f
- 0(z) + z 4'(z) + *(z)
(4)
where Xn ds and Yn ds are the horizontal and vertical components of force acting on ds. The solution of the cracked, similar lines to that of Tracy. parallel to the real axis in plane, *
in
4
autofrettaged cylinde'r was carried out on A complex mapping transforms straight litnes
the c-plane to curved lines in
particular the real axis in
radius R1 , centered at the origin in is
the ý plane is the z-plane,
mapped to an arc of
Figure 1.
A further mapping
introduced which maps the unit semi-circle plus its e,ýterlor in
to the crack plus its exterior in
the t-plane,
see Figure 1(b,c).
analytic continuation arguments of Muskhelishvili
4
the physical (z)
are user
the ; plane The
to ensure traction-
Tracy, P. G., "Elastic Analysis of Radial ;racks Emanating From the Outer and Inner Surfaces of a Circ.lar Ring," Engng. Frac. Mech., Vol. 11. 1979, pp. 291-300.
3
S
free conditions Figure
along F'L'
and J'G
hence FL and JG in
the
physical
plane,
I.
For certain geometries dlesiied accuracy. general each there is region
and
it
necessary to use partitioning 2 to obtain the
is
Tne partitioning of
region has its
the cylinder
to
the right
partitioning
is
used it
of
shown in
own complex stress and mapping
symmetry about the imaginary axis, II
is
Figure 1.
function.
only region I and
that
the imaginary axis need be considered.
is
necessary
to "stitch"
along common
in
Since
pars
of
When
boundaries,
by
imposing equilibrium and compatibility of displacements. In
the MMC method
functions are
truncated
imposed at selected coefficients rows
in
in
the infinite to a finite
boundary
the stress
the main
,
number of terms.
Thus
the vector of
common boundary
point gives
A is
a matrix of
The common
four rows in
A and
X rows and in columns,
is
P.
A.
P.,
Each
zeros
respectively.
gene-:rally better when 2m < X < 2.5 m,
and Parker,
boundary
in
h).
where X and in depend upon
found that convergence
C.
("stitched")
four corresponding
NLnts and unknown coefficients
Andrasic,
the boundary
relating the unknown coefficients.
the number of boundary
2
and
It.w..: this
"Weight Functions For Cracked Curved Beams," Numerical Methods in Fracture Mechanics, D. R. J. Oweui and A. R. Luxmnore-ds.), Proceedings Second International Conference, Swansea, 1981), pp. 67-82.
4
W
elements in
two
b
unknown coefficients.
to obta'n conditions
general,
point produces
where:
points are used
In
the stress
Force conditions are
eý.ch boundary
and two corresponding
A x and x is
of
points which give conditions on the unknown
functions.
aatrix A,
conditions vector
series representations
compares well with other workers.
5
A least-square error minimization
procedure was used to solve the overdetermined
set of linear equations.
Knowing the coefficients for the stress function in crack tip stress intensity K - KI -
factor,
iKlI
-
the cracked region,
K, may be determined from:
2(2-n)1/2
Limit (z-zc)1/2 Z + Ze
6
(5)
.
*'(z)
where KI and KII are opening and sliding mode stress intensity respectively,
factors
and zc is the location of the crack tip.
We now consider the various loadings, will be necessary LoadingA:
and combinations
for the solution of the pressurized,
tube.
Internal pressure acting in bore and cracks.
for this configuration 7 and is
shown as the upper curve
solution was obtained by superposition of an all-round outer boundary.
of loadings which
autofrettaged
An accurate MMC opening mode stress intensity solution,
-2R2 p
the *uter radius,
the internal pressure actiing
in
ig Available This MMC
tension field on the form as KI/Kp,
2 ------
Rl the
KI,
in Figure 2.
The results are presented in dimensionless
where:
and R 2 is
the
Lnner radius,
a Is
bodh bre and crack.
5
(6)
n(ra)1/2
the crack depth and p is Also shown,
for the
Fasn, E. DI, "A Review of Least Squ:ares Methods for Solving Partial Differential Equations, Int. J. Num. Meth. in Engrg., Vol. 10, 1976, pp. 1021-1046. 6 Sih, G. C., Paris, P. C., and FErdogan, F., "Crack Tip Stress Intensity Factors for Plane Extension and Plate Bending Problems, J. Appl. Mech., Vol. 29, 1962, pp. 306-312. 7 Bowie, 0. L. and Freese, C, E., "Elastic Analysis For a Radial Crack in a Circular Ring," Engineering Fracture Mechanics, Vol. 4, No. 2, 1972, pp. 315-321.
5
purpnses
of comparison,
are points
from a solution du(
to Grandt
ipproximnate weight function method which was also employed of addii:ional solutions referred to in Loading_ B:
Ideal autofrettage
An MMC program, used to calculate
(o0)
in
tube is
tube.
9
was
for full autofrettage (100 Figure 2.
This was based on
where the distributiotn
of hoop stress
given by: R2 2
00 = -Y in (R2/Rj)
the uniaxial
section.
shown as the lower curve in
,R12
and Y is
the derivation
based on the formulation outlined in this section,
elastic-plastic solution,
the uncracked
based .)n an
residual stresses in uncracked
the stress intensity results
percent overstrain), the ideal,
the next
in
8
[I +- ------
(I
+
-r-)I
+ Y[l + Zn (r/Rj)I
yield strength of the material
1.15 x yield stength (vor
'ses'
criterion).
by Grandt are shown for comparison purposes.
(7)
(Tresca's criterion)
or
Points obtained from a solution The approximate results of
Grandt1 0 for a cracked tube of the same dimensions subjected to steady-state thermal loading were modified in
accordance with Reference
a comparison with the calculated autofrettage
.2•
results.
11 to make possible
The modification is
based on the fact that the residual stress distribution fof 100 percent over-
8Grandt, A. F., "Stress Intensity Factors For Cracked Holes and Rings lU)aded With Polynomial Crack Face Pressure Distributions," Lnt. J. Fracture, Vol. 14, 1•78, 'Iill., R., 195{U.
*
pp. R221-R229. fhie Mithematicil
Theory of
Plasticity,
1
Clarendon
Press,
oXto,-d,
')Grandt, A. F., "Two Dimensional Stress Intensity Factor Solutions For Radially Cracked Rings," AFML-TR-75-121, Air Force Materials Lab, Wright Patterson AFB, Ohio, 1975. 11 1parker, A. P. and Farrow, .1.R., "Stress Intensity Factors for Multiple Radial Cracks Emanating Form the Bore of an Autofrettaged or Thermally Stressed Thick Cylinder," Ei;gng. Frac. Mech., Vol. 14, 1981, pp. 237-241.
U
wl
6
strain
is
identical
to that
for
simple multiplying constant. A particularly beginning depths. 0.2, in
of
Many available
and are
Figure
range
therefore
2 s!.ows
1.12.
expended
in
at
and
use
it
accuracy.
results
results
of
The total
Ii -
percent I LOure
Internal stress
overstrain)
2,
Zin
that
for
autofrettage
tube
(
2 /K
pressure
intensity
where Kp is
is
in
I)
the
Linmiting,
lifetime may be
this solution are
results
provides
presented
the in
2 2
(-------)
Y (Ta)1/2
R7 -Rl2
and ideal
givenhb
(8)
autofrettage.
a pre.qsurized,
fully autofrettaged
the superposition
the stress
j~t,(,jrf~tj
e
field acting
pressure
intensity with pressure
stress intensity contribution due
stress
to the
in
Lnwt
of
the results
(100( in
thus: Kf111 ,
the
overstrain
to seek accurate
apparent full
The
KI/KA where:
KA
C:
gun tube
imprrtant
2R
Loading
percent
or
purpose,.
convergeiice
percent of
the
at shallow crack
prediction I'10
froin a
formulation outlined at
for a/(R 2 -R 1 )ven ) K ).1,
life
indicates good
apart
within 5 percent.
the results
for
clearly very
ft is
'he
form as
is
for
Loading,
generally
nut quoted
Since as much as 80
this range
dimensionless
limited
O.1,
shallow crack depths.
required
are
thermal
the MML
the accuracy
results of
is
feature of
the calculated
() < a/ (R 2 -RO) `
value of
is
state
Agreement
important
this section
steady
to
alone.
7
in
KP i-* KA bore and crack,
the 100 percent
In
and KA
overstrain
th
residual
In the event that the tube has been subjected to less than 100 percent overstrain, *
radius RA, !
--
the plastic flow during the autofrettage and the stress intensity factor in
SKpartal autofrettage
-
[1 + Y- Zn(R2/RA)
+ pressure
a P( RA
-
Y-.n(RA/Rl)
p
Results for 0, 30, =
2.0,
60,
+
this case is -
Y (R2 2 -RA 2 )/2R2 2 1 Kp + KA
(1IO)
12
(R22-RA2)
and 100 percent overstrain,
based on the superposition of results given in
Figure 3.
given by:12
P
Rj
where RA can be obtained from the expression:
process will extend to a
with Y/p
-
particular value Y/p
-
stress intensity (i.e.,
3.55,
R2 /Rl
Figure 2 are shown in
These curves indicate the very significant reduction in
intensity as a result of the ideal autofrettage
3.55,
process.
Indeed,
stress for the
100 percent overstrain causes a negative total
crack closure)
for crack depths up to a/(R2-Rl)
-
0.08.
SCORRECTrION FOR THE CRACK SHAPE EFFECTS Thus far we have ignored the effect of crack shape on stress intensity, assuming a through crack. semi-major axis c,
depth a,
We now consider the crack to be semi-elliptical, Figure 4.
In order to modify the two-dimensional
results to account for crack shape we employ two sets of work, extensive results for semi-elliptical
V
12Parker,
A. P.
and Farrow,
J.
cracks in
namely
the
a flat plate under tension or
R., "Stress Intensity Factors for Multiple
Radial Cracks Emanating From the Bore of an Aktofrettaged or Thermally Stressed Thick Cylinder,"Engng. Frac. Mech., Vol. 14, 1981, pp. 237-241.
V
W
8
i
bending,
due to Newman and Raju,1
elliptical crack in a pressurized Atluri and Kathiresan. First,
and the limited results for a semithick cylindeL,
due to Tan and FennerI
and
14
consider the flat plate containing a semi-elliptical crack.
the results given in plate in
3
Reference
13, we may define a correction factor
From
for the
pure tension HpT given by: Kr
Kpr where KPT is in
the stress intensity factor for semi-elliptical
tension and Kpr is the stress intensity
fronted
through crack given in
11pT are shown in
Figure
Similar correction
Reference 15.
crack in it plate
factor solution for a straightCurves of
the correction
facto)r
5(a). factors for the flat plate under pure bending HpB are
defined by:
KP 1 PB - ---
11
KpB where KpB is plate in crack.15
the stress Intensity factor for a semi-elliptical crack t.n a
pure bending and KpB is
the solution for a straight-fronted
Curves of the correction factor HpB are shown in Figure
through
')(h).
ITan, C. L. and Fenner, R. T., "Stress Intensity Factors for Semi-1Lliptical. Surface Cracks in Pressurized Cylinders Using the Boundary Integral Equation Method," Int. J. Fracture, Vol. 16, No. 3, 1980, pp. 233-245. l 3 Newman, J. C. and Raju, I. S., "Analyses of Surface Cracks in Finite Plates Under Tension or bending Loads," NASA TP 1578, 1979. 14 Atulri, N. and Kathiresan, K., "3D Analyses of Surface Flaws in tcTlhkWalled Reactor Pressure-Vessels Using Displacement-Hybrid Finite Element Method," Nuclear En ng. and Design, Vol. 51, 1979, pp. 163-176. 15 Rooke, D. P. and-Cartwright, D. J., Compedium of Stress Intensitv Factors, Her Majesty's Stationery Office, London, 1976.
9
It
is
proposed that,
at shallow crack depths,
the correction factors
applicable to the thick cylinder may be obtained by appropriate of HpT and HpB,
superpositions
given by, HC
alHPT + 511PB
(Ii)
where the multiplying factors a and ý are obtained by calculating the proportions of tension and bending prospective crack line.
in.the uncracked
tube which act over the
In order to test this hypothesis,
of a tube subjected to internal pressure.
consider the case
In this case the crack-line loading
comprises hoop stresses due to the internal pressure1 2 plus a contribution from the pressure, loading
SR12
p,
which infiltrates
the crack,
is:
thus the total crack line
R2 2 r00 - p[l + R22_RI2 (R + +-R7 2
as shown in 0 4 a/(R 2 -RI)
Figure 6(a). 4 0.2,
Using a straight line approximation,
the proportion of tension loading is
1.72/2.66 while the proportion of pure bending, correction factor H. is
B
-
given by a
0.94/2.66, 'hence the
HpB
Correction factors determined on this basis are plotted in
•
*
-
determined as HC -.. 645 HPT + .355
for the case a/c - 0.8.
for the range
Figure 6(b)
Also shown are the equivalent correction factors
obtained for the same configuration by Tan and Fenner,l using boundary
ITan, C. L. and Fenner, R. T., "Stress Intensity Factors for Semi-Elliptical Surface Cracks in Pressurized Cylinders Using the Boundary Integral Equation Method," Int. J. Fracture, Vol. 16, No. 3, 1980, pp. 233-245. 12Parker, A. P., "Stress Intensity and Fatigue Crack Growth in MultiplyCracked, Pressurized, Partially Autofrettaged Thick Cyltnders," Fatigue of Engng. Material'i and Structures (in press).
10
element
vvthods.
Agreement
is
deeper cracks the difference appreciable upcoming tabe
because of
paragraph.
within
between
2 percent
plate and cylinder
the differences
It
Is
important
in
constraint,
to emphasize
in which the vast ma1jority of compon.?nt
represented correction
by
the model,
factors
are shown in
i.e.,
for K at
Figure
6(c)
at a/(R 2 -R 1 )
for a/(R 2 -R
the point •
and exhibit
-
1
. 0.3.
0 (the
results
discussed
For
becomes in
the
that the proportion
lifetime )
- 0.2.
is
expended
is
of
the
well
The equivalent
free surface of
similar characteristics
to
the plate) those of
Figure 6(b). The significant in
Figure 6(b)
is
difference
attributed
cylinder
compared
cracks.
This difference can
straight-fronted,
particularly
be demonstrated -
0.8 crack
2.0 with the K for a straight-fronted, approximaýely
the same combination of
the pressurized cylinder. Reference 7 or with cylinder pure
bending'
5
is
in
constraint
for straight-fronted, by comparing
a pressurized
a/(R 2 -RI)
= 0.8
obtained
In Table
I.
using the expressions
ot a
deep
cylinder with R2 /RI
crack
tensien and bending
shown
factors
the K for a
in
a plate with
loading as that of
The K for the cylinder can be obtained
from Figure 2 and is loading
the cylinder and plate shape
to the significantly greater
vith a plate,
a/(R 2 -Rl)
between
from
The K for the plate for
poire tens i ,
and
as follows: Kplate -
KPT + KpB
7Bowie,
0. L. and Freese, C. E., "Elastic Analysis For a Radial Crack in a Circular Ring," Engineering Fracture Mechanics, Vol. 4, No. 2, 1972, pp. 315-321. 15 Rooke, D. P. and Cartwright, D. J., Compedium of Stress Intensity Factors, Her Majesty's Stationery Office, Wondor,, 1976.
11
The tension and bending stresses in ad
OpB
-
05 p,
several
Figure 6(a).
The K for the plate with a straight crack
times that of the cylinder,
less constrained.
2.0 p
by using a linear approximation of the entire cylinder
loading plot shown in is
KPT and KpB are determined as OPT
Therefore,
which indi.cates that the plate is
the increase in
constraint corresponding
much
to a
semi-elliptical rather than a straight crack will be much larger for a plate than for a cylinder. a cylinder,
This results
as observed in
in
the mu'ýh smaller Hc for a plate than for
Figure 6(b).
When the final comparison is
Table 1, K for deep semi-elliptical cracks in loaded plate are about the same, TABLE I.
plate with approx. cylinder
a cylinder and a similarly
as might be expected.
COMPARISON OF K VALUES FOR DEEPLY CRACKED CYLINDER AND, PLATE
K for a/(R2-RI) 0.8 straight crack
pressurized cylinder
made in
Hc from Fig. 6(b) for a/(R 2 -RI) 0.8, a/c - 0.8
K for a/(R2-Rl) = 0.8, a/c -0.8 semi-elliptical crack
6.99 p(a)1/2
0.42
2.9 p(a)1/2
p(a)1/2
0.07
3.3 p(a)I/2
46.7
-
loading *
In order to obtain a range of correction factors applicable to thick cylinders,
the procedure outlined by Eq.
(11),
Figure 6,
aau related
discussion was used to obtain correction factors from Newman's work for the case of internal pressure,
and full autofrettage
(100 percent overstrain).
These results for relatively short cracks were combined,
12
using engineering
judgment, factors Lp,
with those of Tan and Fenner for longer cracks for use over a wide range of crack length.
are shown in
for autofrettage, re'pectively.
Figure 7(a) and 7(b), LA,
are shown in
for
-
Figure 7(c)
In the case of pressure,
to obtain correction
The results for pressure,
ir/2 and 0.0 respectively, and 7(d)
for
we may compare
I.n each figure with the solutions due to Atluri and Kathiresan 1 -?oints, namely a/c 8 percent,
however,
1.0,
a/t - 0.5 and 0.8.
for 4
For
r/ /2
-
1/2 T and 0.0
-
Figure 7(a) and 7(b),
n,id
4
at two
agreement is
within
0 the correction factort. differ by 25 percent,
reflecting wide disagreement between workers on K solutions at the freesurface. It
13
is
,14 thus possible to calculate the stress intensity factor for any
combination of pressure, correcting with Eq.
ideal partial autofrettage and crack shape by
the stress intensity factors presented
(10)
in
Figure 2 in accordance
and correction factors Lp and LA to obtain:
Kpartial autofrettage + pressure (3D)
Y Lp [I + - Xn(R2/RA) P + LA KA
13
Y --
(R2 2 -RA
2
)/2R2 2 ]Kp
p (t2)
Newman, J. C. and Raju, I. S., "Analyses of Surface Cracks in Finite Plates Under Tension or Bending Loads," NASA TP 1578, 1979. 14 Atulri, N. and Kathiresan, K., "3D Analyses of Surface Flaws in Thick-WalLed Reactor Pressure-Vessels Using Displacement-Hybrid Finite Element Method," Nuclear Engng..and Design, Vol. 51, 1979, pp. 163-176.
13
EFFECTS OF NONIDEAL RESIDUAL STRESS DISTRIBUTION The residual stress distribution predicted by Eq. practice.
16
,17
(7)
may not occur in
Non-ideal Bauschinger effects and uncertainty over the exact
amount of autofrettage on unloading may produce a different stress distribuIn earlier work1 6 a constant reduction factor of 0.7 was applied to the
tion.
autofrettage contribution to stress intensity factor to account for primarily the Bauschinger effect. In the present work it in
a different manner.
radius,
it
If
is
proposed that these effects may be considered
an unflawed autofrettaged
will spring apart,
100%
Yt
in
an Appendix.
is
given by:
yield criterion as:18
8nY .
In the case of partial overstrain, reduced by a factor F.
cut along a
the theoretical angle of opening Yt being given
for 100 percent overstrain and using the von Mises'
ends is
tube is
... (RE
(13)
the total moment acting over the cut
Details of the calculation of F are contained
The theoretical angle of opening for partial overstrain yt%
100% Yt%
-
Fyt
16
(14)
Underwood, J. H. and Throop, J. F., "Residual Stress Effects on Fatigue Cracking of Pressurized Cylinders and Notched Bending Specimens." presented at Fourth SESA International Congress, Boston, MA, May 1980. 17 Milligan, R. V., Noo, W. H., Davidson, T. E., "The Bauschinger Effect in a High-Strength Steel," Trans. ASME, J. Bas. Engineering, Vol. 88, 1966, pp. 480-488. 18Parker, A. P. and Farrow, J. R., "On the Equivalence of Axisymmetric Bending, Thermal, and Autofrettage Residual Stress Fields," J. Strain Analysis, Vol. 15, No. 1, 1980, pp. 51-52.
14
A graphical representation of F for a tube having R2 /R 1 Figure 8.
The angle (and moment)
2.0 is shown in
ratio F between partial and 100 percent
ov-rstrain does not vary much with R2/RI.
In fact,
for tubes in the range 1.8
4 R2 /RI ( 2.2 the deviation from the curve in Figure 8 is only I percent, of course it
and
still goes asymptotically to the limit of 1.0 at 100 percent
overstrain, and to zero at 0 percent overstrain. We propose that a comparison of the ratio of measured opening angle to 100% Yt with the ratio F provides an indication of the non-ideality of the residual stress distribution in actual autofrettage cylinders. points shown in Figure 8 make this comparison.
Each point represents an auto-
frettaged steel cylinder of the type described in Reference 16. removed from each cylinder, optical comparator.
The data
A section wao
and the opening angle was measiared using an
The ratio of measured angle to 100 percent overstrain
theoretical angle was calculated using the measured value of yield strength, Y for each tube. The important features of the comparison between experiment and theory in Figure 8 are the following.
(a) The experimental results are generally near
or above the theoretical curve for relatively low overstrain and generally below the curve for high overstrain. effect since, yielding,
This can be explained by the Bauschinger
for high overstrain and the associated large amount of tensile
the Bauschinger effect would result in significant reverse yielding
and less than expected residual stress and opening angle.
(b)
The two sets of
1 6 Underwood, J. H. and Throop, J. F., "Residual Stress Effects on Fatigue Cracking of Pressurized Cylinders and Notched Bending Specimens," presented at Fourth SESA International Congress, Boston, MA, May 1980.
15
•*
,
-.
.
•- -- .
-
- .
-
*-
-
.
-
experimental results designated by dashed lines indicate less than expected opening angle and residual stress with increasing R2 /R 1 . consistent with the Bauschinger effect, *
that is,
larger R2 /Rl
This also is
using a similar rationale as with (a),
result in more tensile yielding, more reverse yielding,
and less than expected residual stress. Also shown in
Figure 8 are the measured-to-theoretical
angle ratios from
the 30 percent and 60 percent overstrained tubes nf the experimental wrk
*
described here.
Note that the average of the two angle measurements for 60
percent overstrain is that,
about 0.6 of the theoretical value.
at least as a first
approximation,
This would suggest
the contribution of overstrain to the
total K could be reduced by this 0.6 factor, and a shorter than expected fatigue life would result.
This is discussed further in the upcoming
comparison of exparimental and theoretical results. No attempt is made to relate the two sets of experimental results in
* ,
Figure 8, that is
-.-
"" .
the results designated by a dashed line for 50 percent to
100 percent overstrained tubes and the results from the 30 percent and 60 percent overstrntned tubes.
The reason for this is
that the methods used for
overstraining were quite different for the two sets of results. swaging process was used in process was used in
A mandrel
the former case and a hydraulic overpressure
the latter case.
The different overstraining methods
could result in different amounts of Bauschinger effect in the two sets of experimental results.
16
tV
LIFE CALCULATIONS The fatigue growth rate of cracks subjected to cyclic loading may be expressed
in
terms of Paris'
law:19 da --
-
(15)
C(AK~m
dN where da/dN is
the fatigue crack growth per loading cycle,
C and m
re
tla ringe of str,:ss intenseity defined by:
empirical constants and AK is
AK-
Kmax
AK
K.-ax
-
Kmin
(Kmin
> 0)
(Kmin < 0)
and Kmax and Kmin are respectively the maximum at,d minimum values of strus intensity during the loading cycle.
Note that the possibility of "over-
lapping" or touching of the crack surfaces at some point on the crack line remote from the crack tip
2
O is
not considered in
this report.
During the
lifetime of a particular cracked c.plinder the crack will propagate from some initial
depth ai to some final depth af, where af is
thickness of the cylinder, equation (15)
is
R2 -R1.
generally the total wall
In order to predict
the fatigue lif'ý,
rearranged to give: af
f-f
ai
da
-d
-NNi
(16)
C(hK)rl
19
Paris, P. C. and Erdogan, F., "A Critical Analysis of Crack Propagation Laws," Trans ASME, J. Bas. Engng., Vol. 85, 1963, pp. 528-534. 20 Parker, A. P., "Stress Intensity Factors, Crack. Profiles, and Fatigue Crack Growth Rates in Residual Stress Fields," presented at ASTM Symposium on Residual Stress Effects in Fatigue, Phoenix, AZ, May 1981.
17
UI
Crack growth (a versus N) predictions are made fcr two examples which are near the extremes of crack geometry encountered in
thick cylinders,
nearly straight-fronted crack and a single semi-circular crack.
a single
The
predictions are compared with ultrasonic crack growth measurements from cylinders in
which internal radius R 1 is
pressurization is
zero to 331 MPa.
area of 50 percent.
21
R2 /RI
is
2.0,
the cyclic
The cylinder material ie ASTM A723 forged -40 0 C Charpy impact energy of 34 J,
steel, with yield strength of 1175 MPA, reduction in
90 mm,
These properties can be slightly different
after overstrain, due to the plastic strain, which can be up to I percent at the inner radius of a 100 percent overstrained cylinder. small amount of plastic strain relative to reduction in
Considering this area,
no significant
elfect on fatigue irfe is expected as the result of the material property changes due to the overstrain process. In general, Simpson's rule.
the integral of Eq. For
(16)
was evaluatci numerically using
he particular steel employed in
the experimental crack
growth rate work, the measured constants are:
C - 6.52 x 10-12
m
,
for crack growth in metars per cycle and AK in (a)
Single,
-
3.0
MPa m1/2.
nearly straight-fronted crack
Figure 9 shows a versus N predictions based on the K results pecsented in
Figure 3 for a single,
The solid lines in
2 1
straight-fronted radial crack,
Figure 9 are predictionb
initial depth 6.4 mm.
for zero and 30 percent over-
Throop, J. F., "Fatigue Crack Growth in Thick-Walled Cylinders," Proceedings of the National Conference on Fluid Power, Fluid Power Society, Chicago, 1972, pp. 115-131.
W
18
strain.
The dashed line shows experimental
Reference 21,
results,
originally reported
in
for a tube with zero nomtnal overstrain in which a single notch
was cut using electro-discharge machining to a depth of 6.4 mm and a half surface length,
c,
of 254 mm.
For this example it simple,
possible 2 2 to integrate Eq.
shallow crack K expression is
constant value. choice,
is
From Figure 2,
used,
Kp/Kp -
that is,
1.05 ac a/R 2 -RI
Doing so and combining with Eqs.
following expression
Nf
- Ni -
Kp
analysis. (17),
a
- 0.2 is
a reasonable
expended at relatively
(6)
and (12)
gives the
C, R2 , Ri,
the dotted l.'.e in
I ---I
----------------2R2 2
and p in Figure 9,
2
2
C[=-(n)1/2 Kp
result is
if
for fatigue life of a tube with no overstrain: I 2[----
For values of ai,
directly
one with Kp/Kp at a
considering that most of the cylinder life is
low values of a.
(16)
Ri
-(17)
p p2
this example and setting Lp
1.0,
the
very close to the more general
For the conditions of this comparison,
although less rigorous and general,
is
the simpler analysis of Eq.
adequate and easier to use.
The lack of agreement between the two analytical p'-dictions experiment wuld be improved if
-
the shape factor,
and the
Lp was significantly
Less
21Throop, J. F., "Fatigue Crack Growth in Thick-Walled Cylinders," Proceedings of the National Conference cn Fluid Power, Fluid Power Society, Chicago, 1972, pp. 115-131. 22 Underwood, J. H. and Throop, J. F., "Surface Crack K-Estimates and Fatigue Life Calculations in Cannon Tubes," Part-Through Crack Fatigue Life Prediction, ASTM STP 687, J. B. Chang, Ed., American Society for Testing and Materials, 1979, pp. 195-210.
19
S
than unity. shape is
The initial shape is
(a/c)f
-
0.18.
of the cylinder life is is
estimated
to Figure 7(a) and considering that
Eq.
Single,
(17),
So even with this factor to the
cannot account for the differences
it
Figure 2,
in Figure 9.
semi-circular crack
Figure 10 shows a versus N predictions, in
most
expended with low values of both a/c and a/R 2 -R1 , LI
to be between 0.95 and 1.0.
third power in (b)
Referring
described by (a/c)i - 0.025 and tile final
based on the K values presented
as modified in accordance with Eq.
factors Lp and La from Figures 7(a) and 7(c)
(12),
utilizing correction
respectively,
for a/c -
1.0.
hl.,
K value3 are representative of the stress intensity at the deepest pol•nt o4 semi-cir-.lar
crack of initial depth b.4 mm in
a thick cylinder,
and therel.t-,,
implicitly assume that the crack retains its semi-circular shape during the life of the tube. experimental 40 percent,
"Ais assumption appeara
observations. 45 percent,
16
The predictions are made for zero,
and 50 percent overstrain,
indicate experimental resultsl tubes with zero,
30 percent,
For this example it
is
6
30 percent,
while the dashed lines
for the same initial crack depth for actual
and 60 percent nominal overstrain. also possible to obtain a closed form expression
for fatigue life, analogous to Eq. it
to be justified on the basis of
(17).
By combining Eqs.
(6),
(8),
and (IP11
can be shown that the ratio of the K expression for an autotrettagt-l
i•d
pressurized cylinder to that of a pressurized cylinder with no autofrett.;a1;, the following:
1
6Underwood, J. H. and Throop, J. F., "Residual Stress Effects on Fatigue Cracking of Pressurized Cylinders and Notched Bending Specimens," presentt,., at Fourth SESA International Congress, Boston, MA, May 1980
20
Kp+A M I + KP
-
p
RI
R2
Y
RA 2 -
[in (--) + 6Xn -- + RA R2
SR1 2
(1-6)R
-
2 2
(18)
-1
2R 2 2
KA/ KA wh;ere 6 is
Using Eq.
defined as ----Kp/kp (18)
in
,
the ratio of the K solutions in
a life expression in
1
K
2R2
Kc c[.2
2
Y
the form of Eq. 1
R2
R1
RA 2 -
(17)
Figure
gives Nf
6R1 2 -
(1-6)R
2.
-
Nj
22
-
_..
2R2 2 -- - ----------+ 6*.nn R2 n RAu) + . 2-R12) . . .( ( + ((.)/(---) / 2R2 + -p-[ [in(--) -I)Lp]~ (19)
which should give a good estimate of fatigue life for an autofrettaged, pressurized cylinder in which a shallow semi-elliptical crack dominates the
life.
Results from this shallow crack expression are shown in Figure 10 for
0 percent,
30 percent,
and 50 percent overstrain.
A mean L - 0.53 was used,
since Lp and LA are close to this value for a/R2-RI - 0.2, see Figures 7(a) and
7 (c).
The predictions of the shallow crack analysis,
to those of the more general analysis.
Eq.
(19),
are close
The lack of agreement between the two
analytical predictions and the experiments is discussed in .the upcoming section. DISCUSSION AND CONCLUSIONS
Life prediction for cracked tubes requires an accurate knowledge of stress intensity factor at short crack depths.
In this report two-dimensional
solutions for cracked tubee with various amounts of residual stress (overstrain) were obtained by use of the modified mapping collocation technique,
21
which gives good couvrovir,.
inalyt ic
t•,iltilon at very
small
crack depths. solulI an,; were
The cracks in
thick c..'Ilnde
in
the cylinder and
cracked three
plates
hi
lifetime is
,
hi
:pplyY'
,I
,ig
. !I
aold
,
offered
by
,.I
a.
U
This is
Inilnctes
. 1. I,
.
-,,.
i,
ipt,,(il
limited available
not
restricted
Fractore
at
to
Me;lchanics
lIUntLng, configuration
ol:
the
-i1 ! ,m-aritof rirtti);,ed,
I minlar crac(ks
,ii, I ,
(letermined for
Large degree
in
of
d body.
:•,,ritrr ,
iI
Pr
bra a sort. oi becausie
tension and bending
iw)st of the fatigue
whet-eiii
pipl, lcotIoin
the druhlv
thisc ,vi',,
,
semi-elliptical
good agreement
is however, iLed
Iy.
with nearly stra lglit-,fr(,i I
effect
-•
of
lacto•;r with
),TparisoTn
".
'Iiii' .hi,','
The calculatrns
this
..
.
which maximium error,; wrmla I It
one quarter of
i.- correction
V11T, rIders
ci-,c-,.l•
expenieidl
proportion
the
,iI"',
I...
Indeed , rho, thick
restraint
tlide three-diinnslonal
.iic
;,,rl ittr:.
thick cylinders,
Design.
rq
t-ns iian
dimensional
short to modiiii
1i ttr,,I
pressurized
predict
tubes
lives of about
iiLa,.t.Ily. r'
lost;ibLe explanations
,,
not inf iltrate
for
are:
The pos••
i ty
wiI
not coio;Ir dovro
designed to
avo•II
autofrettage
t
Ii,,
r
I i I. ,
iri
i
I
"I
r-,.
,.,.
the crack.
pe i irmo ii 1;i I work wao, deliberately
",
,I
ctset'
doer-:
II
,'
ilis
I.i" t
i[rwi.o not Ic',;ible ,oi,,),pres.slve r-
in
the
stress
closing the craok. b.
Residual. otr
considered
rv- ,
,,,'
,--.,,t.,frettage ,,
likely. *,h,,i
any tendency
to
3prtry,
,
-,iito
tibe. el.tat.,cd
j'f',e1 igihie gros•s
Again,
this is
not
tubes (to not exhibit residual
stresses.
c. This is
Multiple,
small,
semi-elliptical
cracks along the notch boundary.
considered a probable explanation.
Such a form of multiple crack
growth would result in slower overall crack growth than that of a single crack,
until the individual semi-elliptical cracks linked to form a single
continuous crack front. Lifetimes calculated on the basis of the 'ideal' indicate extreme sensitivity that an increase
to the amount of overstrain.
Note in
from 30 to 50 percent overstrain increases
by a factor of five,
the predicted
So there is
10( life
clearly much less
1ife due to increased overstrain than would be expected
calculations.
We believe
this effect is
expected opening angle measured
from the
directly related to the less than
from cylinders with 60 percent overstrain,
discussed in relation to Figure 8.
The preferred explanation
for thc
deviations of opening angle and lifetime from calculated values is reduction in
Figure
while an increase from 30 to 60 percent overstrain
increases measured life by a factor of 1.4. increase in
residual stress fteLd
the
the resieual stress field due to reversed yielding near the inner
radius caused by the Bauschinger effect. reduction in opening angle,
This would manifest itself as a
particularly at large overstrains,
relatively larger reduction in
lifetime,
and a
since most of the lifetime is
expended at shallow depths at which the hoop stresses are reduced significantly.
23
twst
R.i•;RtENCKS
TU,
C.
11
1,-
, tl',nwir, ;irtl.•o
ita I.
1NEW -.
,,,
.1,K
•
N
Oir
i;i i•n
tr Lo1 1
1:.>IL.!
it(v,
....A
*T'ri, Ac
.
.in
,P
i li1n4t.
,.
, l
7
U;.
'airis, 1 , 1, i
.. . 1:1 I
'uutip
,I,
II ?I/3.
,i.1.-I.ni
. ..
1i
:,I
,f
m,,
P. G.,
. Q! . n-
,ee,
Functions
I'r;.rtiire Mechanics,
No.
3,
1980,
for Cracked D.
R.
J.
of
pp.
Curved
Owen and
Lnterna tional Conference,
ic' Probl.emn
Ring,"
Ikensit Squares
..
I nL.
Nun.
Engng.
the Mathematical
Meth.
in
Mech.,
A.
Swanse,'
Theory of
for Solving
Engng.,
Vol.
Vol.
P Mlet F•ndltngr
Q. K.,
"'l.tc
Problems,"
11,
10,
1976,
Vol.
4,
AJ Appl.
No.
pp.
Intensity Mech.
Analysis For a Radial Crack
i'rature Mechanics,
1979,
Partial
"Crack Tip Stress
Wan
_;__ LneerLig
Frac.
Methods
and Erdogan , F.,
24
w•
the Boundary
Using
16,
Vol.
for Semi-
unlyls of Ra&dial Cracks Emanating From the Oute•r
"uvLiw 1
"Weight
-•, Se-c'olld
I.,
Cylinders
tlracture,
P.,
in
,d
y Factors
Intens.
r1zed
I.
a:i Circ(ularu
:iq.,
,,
LA;t.
, A.
iA
Di ll,,rIo"Li.;l
Nl~h,
in Prvs;q;
tIrQt'(,
rI"
.. ,"A. .
b•.
"Stress
Wk.thodn
.. I
I
'k:-
t ,nd Pldk,,r
.
h: L. .
T.
O.,,f hotdl,,"
H
,
Hic..m-;,"
R.
2,
1972,
in pp.
a
8.
A. F.,
Grandt,
Intensity Factors for Cracked Holes and Rings
"Stress
Int.
Loaded With Polynomial Crack Face Pressure Distributions," Vol.
Fracture, 9.
Hill,
14,
pp.
1978,
R221-R229. Clarendon Press,
The Mathematical Theory of Plasticity,
R.,
J.
Oxford,
1950. 10.
A. F.,
Grandt,
"Two
Dimensional. Stress
Intensity Factor Solutions For Air Force Materials
Radially Cracked Rings," AFMI,-TR-75-121, Patterson AFB, 11.
A.
Parker,
P.
Ohio,
Lab,
Wright
1975. ..
and Farrow,
R.,
for MultipLe
"Stress Intensity Factors
Radial Cracks Emanating From the. fkre of ait Autofrettaged or Thermal ly Engng.
Stressed Thick Cylinder," 12.
13.
A. P.,
Parker,
Pressuriz',d,
of Engng.
Materials J.
and Structures
C. aud Raju,
(in
15.
Rooke,
Nuclear Engng.
I). P.
Factors, 16.
Underwood,
237-241.
1).J.,
H. and Throop,
J.
F.,
Fatigue
Vol.
51,
1979,
GompeediLurn
Flaws in Thick-
pp.
Finite Element
163-176.
of StreL s
London,
Finite
1979.
Using Displacement-Hybrid
and Design,
anti Cartwright,
NASA TP 1578,
Her Majesty's Stationery Office, J.
pp.
press).
Analyses of Surface
"31)
K.,
Walled Rea-tor Pressure-Vessels Method,"
1981,
"Analyses of Surface Cracks in
1. S.,
N. and Kathiresan,
Atulri,
14,
Partially Autofrettaged Thick Cylinders,"
Pletes Under Tension or Bending Loads," 14.
Vol.
Mech.,
"Stress intensity and Fatigue Crack Growth in Multiply-
Cracked,
Newman,
Freac.
int0n-
ttY
1976.
"Residual Stress Effects on Fatigue
Cracking of Pressurized Cylind, -s and Notched Bending Specimens," presented at Fourth SESA International Congress,
25
Boston,
MA,
May
L98().
17.
MilLigan,
R. V.,
Koo,
W. H.,
Davidson,
a High-Strength Steel," Trans. pp. 18.
Parker,
A. P.
and Farrow,
Vol.
J.
15,
No.
Bas.
"The Bauschinger Effect Engineering,
Vol.
88,
P1.C. and Erdogan,
l.awo,"
Trans ASME, A. P.,
J.
F.,
Bas.
"Stress
Cu~l.k Growth IRites
Residual Stress Fields," J.
pp.
1966,
Strain
51-52.
"A Critical Analysis of Crack Propagation
Vol.
tnn_.,
85,
intensity Factors,
in Residual
1963,
in
pp.
528-534.
Crack Profiles,
Stress Fields,"
Symposium on Residual Stress Effects
and Fatigue
presented at ASTM
Fatigue,
Phoenix,
Arizona,
May
1981. 21.
Throop,
J.
F.,
"Fatigue Crack Growth in Thick-Walled Cylinders,"
"Proceedings of the National Conference on Fluid Power, Society, 24.
IIUderwood,
Chicago,
1972,
pp.
J. It. and Throop,
Fatigue Life Calculations in
Life Prediction,
1979,
Fluid Power
115-'31. J.
F.,
"Surface Crack K-Estimates and
Cannon Tubes,"
ASTM STP 687,
l's~tin';1 and MaterLals,
J.
pp.
B. Chang, 195-210. 2
V
V
26
w
in
"On the Equivalence of Axisymmetric
1, 1980,
Paris,
Parker,
R.,
and Autofrettage
Thermal,
Analysis,
20.
J.
E.,
480-488.
B('nding,
19.
ASME,
T.
Part-Through Crack Fatigue
Ed.,
American Society for
@'
A'
,\'
4B (,2
H'
.,['A
I.KJ
,
A"
U"
1
CB
H''
(a) C
Fr i u
a k sc em
F ig u re
. 1
d
t
k
i (
a
Ic l z
(p
ge
e
n y
i
a
l
s
y
m t n
)
ng
o b
F
l
n
,
(
)
a
n
n
n
i t i
r
p
C
p
a
p
1
e
lpllL, w ing p a rt it io n big• •tl nd mal try s i ho C ra c k ed th ic k cy l i nd e r g eo lne (c) • plane. (physical plane), (b) • plane, ()z scheme
27
-U
V
pA Pressure, 1.6
1p(7a2~
p
i.4 A
J
• Grandt
1.2
1.0
0.8 -
.0.4
0
100
9a
0
0
.R2120
-
0'
I,
0.0
0.9
overstrin,
R2.RI
1
a 0O.L 0'2
FI ure 2.
0,3
0.4
0. 0'5
0, 0.'
.9
1.
Stress intensity factors for a single, straight-fronted, radial (Inset) Short crack length convergence crack in a thick cylinder. for 100 percent overstrain.
28
,
5.0
itO •p(va)112
4.0
3.0
0A Overstrain
0%
2.0 -
'
RI
2.02 1.0%
0.1
0.'2
0.o3
0.6
0
0o7
0.'8 '
0.9
1.0 a
R2"R 1
IJ
Figure 3.
Stress intensity factors for a single, straight-fronted crack in a pressurized thick cylinder with 0 percent, 30 percent, 60 percint:, and 100 percent overstrain.
29
V%
Ia
t
•
Figure 4.
R2-RI
L a,c
Semi-elliptical crack in
30
S
a thick cylinder.
'lc
O1.0
1.0 HPB
0.0
a/c • 02
HPT
,
.
a/c • 0.2
0.5
0.5
0.51 0
1.0
0 0
02,
0.4
0.6
00
0.8 ant
0.4
0.6
alt
(b)
(a)
L
I
Figure 5.
(a) (b)
Crack after Crack after
shape factors for a plate in Reference 13. shape factors for a plate in Reference 13.
31
tension (
2)
=
pure bending
((
.
oTan
I ,
2.5
(b)
t l0
" I
0.4
02
Tan
HC
Li ner )reress!on • 1.5
0.6
0.8
Fenne,,
0.5 0.8
0.6
(,)
0 0
02
0.4
a ~R2-RI_.
•J LIPJ
(a) Method of calculatixi proportion of tension and bending, pressure in bore and cracks.
(b) ((')
C:a-k shape fact.rs for pressure in bore and cracks, n/I2. Crack shipe factors for pressure in bore and cracks, -=0.
32
w
08
1)
a R2 -Rj
Figure 6.
0.6
Lam6's stresses and j"..' crack pressure
/
2.0
0.4
02
0
0
&Fenner
a/c - 0.8, a/c - 0.8,
c
'U
0.52
1.0 (I(b) 0.4
0.2
0
0.6
0.1
0
0.2
0.4
a/c 0.6
0.0 0.8
aa LAR2
-ft
LA
1.0.
2R
1.0-
0.52 0.4
1____)
00
a/c 0.0
___1.0_____id)___
02
0.'4 ... 0. 6
'
0.8
0
.02
0.4
0.6
0.8
a R2-R1
U
R2 -R
Figure 7. (a) Crack shape factors, Lp, for pressure in bore and cracks, n/2; two data points from Reference 14, a/c -= 1.0. (b) Crack shape factors, Lp, for pressure in bore and cracks, S=0; two data points from Reference 14, a/c - 1.0. (c) Crack shape factors, La, fcr 100 percent overstrain, p~2 ~0 (d) Crack shape factors, La, for 100 percent overstrain,
33
.
1.82
R2/Rj
1. 1.0
"
0 o Experimental Results.
t'measured 10
0.5
Vt Theoretical Results: Eq. 14
0
10
20
30
40
50
6o
70
80
90
% Overstrain
U
rigure 8.
Ratio of opening angle for partial overstrain to the theoretical value for 100 percent overstrain.
34
,-r---r---
rr
--
50i ~40
•30
J30
Eq. 17
& :Result, "• ,
Experimental I (0)9 Overstrainfj
Results
109 10
0-
0% Overstrain e-t 0
0
-
30% Overstrain
IODD2000
N (Cycles)
Figure 9. Fatigtte crack growth (a versus N) in thick cylinder with single, straight-fronted crack; experimental results and predictions.
35
i
-
VI
*
A
h
90-
- .....
Eq. 19
3%I
60%1
0%1
W
Experiment
'I 7D"
"
I
"".50
I/
E
---
Prediction
/
445%
,~~
~
~30---
0
-
-
10,000
20,000
30,000
40,000
50,000
N (Cycles)
Figure 10.
Fatigue crack growth (a versus N) in thick cylinder with single, semi-circulir crack; experimental results and predictions.
36
APPENDIX
OPENING OF CUT TUBES WITH PARTIAL AUTOFRETTAGE The residual
an autofrettaged
radius R2 , autofrettage
external
*,•
,tresses in
-
-p
+ Y(l + £n(r/RI)
062
"where
p,
the autofrettage
R2 -R1l pressure
p
R1 ,
2 re:AJ
+ r2-1
R2-----ll
2R2 2
internal radius
,
R1
< r
4 RA
(Al)
R2 2
pR 1 2
YRA2 ----
rube,
R2'R2 -P(R2(_RI- [1 + -2-])
,
o1e
radius Rh.
1O REVERSED YIELDING
-
is
RA < r
,
< R(,
(A2)
given by:
Y
Y9n(RA/RI)
+ ---
j
(R2
2
2
-RA
)
(A3)
2R22 The total moment acting over any
i- -Jll cut is
M ,r R2
o
• r
*
given by: dr
•R1
or
RA GlR oe r
dr + f R2 O02 r
RI Since R 2 /R
1
the opening
, provided
reversed
the amount of opening
A]Hill,
R.,
1950. A2 Parker,
A.
angle of
is
P.
ful-ly autofrettaged
proportional
and Farrow,
R.,
"On
tubes of any
is
A-i
noment,
Clarendon Press,
the Equivalence of
Bending, Thermal, and Autofrettage Residual Analysis, Vol. 15, No. 1, 1980, pp. 51-52.
radius
given by 8[Y/f•E,A
to the applied bending
Theory of Plasticity, J.
(A4)
RA
yielding does not occur,
The Mathematical
• dr
it
J.
2
and is
Oxford,
Axisymmetric
Stress Fields,"
ratio,
Strain
possible to produce a non-dimensionalized
plot of theoretical angle of opening
for a cut tube with partial autofrettage,
for any radius ratio.
R2!Rj
-
2.0 is
shown in
Figure 8 of the main report.
The curve for
Deviations from this
curve for 1.8 & R2 /R 1 < 2.2 are less than 1 percent of the maximum opening. OPENING OF CUT TUBES WITH PARTIAL AUTOFRETTAGE - WITH REVERSED YIELDING If ol -XY, t.Ui
the material of the tube has a reduced yield strength in it
may undergo reversed yielding out to a radiua RD after removal of
autoftettage
p.
pressure,
063 z- -XY
004
-fp -
In this case the residual stresses are:A 3 (I + Xn(r/Rj))
RD (1+X) Ykn (--)}R
RI
,
YRA2 [--- - ---
p-
RD (1+X) Ygn (--)}
2R2 where RD is
R1
calculated in
equation (A4),
R2 -RD2
r
RD 4 r 4 RA R22 [I +-i-]
r
-
,
(A6)
RA • r
(
R2
(A7)
2
(0+X) [----.--2R 2 2
+ in(RD/Rl)]
the total moment acting over any radial cut is
RD fRD 003 r
RA dr + fR o4 r
(A8)
given by
R dr+f
RIRD
A.
I +
thus when reversed yielding occurs: M
A3 parker,
R24
R2 -RD2 ,
(A5)
an iterative fashion from:
Y
Once again,
r 4 RD
R-----D 2
RD2 ------
R2 2 -RD -
R11 RD2
-p + Y (I + Xn(r/R 1 ))
(• 5 -
compression
P.,
Sleeper,
K. A.,
dr
(A9)
RA
and Andrasic,
C. P.,
"Safe Life Design of Gun
Tubes, Some Numerical Methods and Results," US Army Numerical Analysis and Computers Conference, Huntsville, AL (1981) (in press).
A-2
qA
The reduction in opening arising from a Bauschinger effect equivalent to k X
0.5 can be calculated,
report.
leading to a plot similar to Figure 8 of the main
The maximum reduction in opening is
approximately 8 percent as a
result of this magnitude of Bauschinger effect. '
..
"
do not exceed I petcent for 1.8 < R2/RI
A
I
U
I
A- 3
2.2.
Deviations from this figure
UI APPENDIX REFERENCES Al.
Hill,
R.,
The Mathematical Theory of Plasticity,
Clarendon Press,
Oxford,
1950. A2.
Parker,
A. P. and Farrow,
Bending,
Thermal,
Analysis, A3.
Parker,
Vol. 15,
A. P.,
Gun Tubes,
J. R.,
"On the Equivalence of Axisymmetric
and Autofrettage Residual Stress Fields," J. Strain No.
Sleeper,
1, 1980, K.
A.,
pp.
51-52.
and Andrasic,
"Safe Life Design of
Some Numerical Methods and Results," US Army Numerical
Analysis and Computers Conference,
Huntsville,
6
A-4
q
C. P.,
AL (1981)
(in
press).
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