Download fracture analysis of surface- and through-cracked

En,wcerinl:
Frucruw
Mrchanics,
1973, Vol. 5, pp. 667-689.
Pergamon
Press.
Printed
in Great
Britain
FRACTURE
ANALYSIS
OF SURFACE- AND
THROUGH-CRACKED
SHEETS AND PLATES?
J. C. NEWMAN, Jr.
NASA Langley Research Center, Hampton, Virginia, U.S.A.
Abstract-The
Neuber stress-concentration
relation for notches in an elastic-plastic material subjected to
shear loading was generalized for a crack in a finite plate subjected to tensile loading, similar to the way in
which Kuhn modified the Hardrath-Ohman
notch equation for a cracked plate. An equation was derived
which related the linear elastic stress-intensity factor. the applied stress, and two material parameters. The
equation was then used as a two-parameter fracture criterion for surface- and through-cracked specimens.
Fracture data from the literature on surface- and through-cracked sheet and plate specimens of steel,
titanium alloy. titanium weldment. and aluminum alloy tested at room and cryogenic temperature were
analyzed according to the proposed equation. For surface cracks, wide ranges of crack-depth to crack-length
ratio and crack-depth to specimen-thickness ratio were considered. For through cracks, wide ranges of crack
length and specimen width were also considered. An empirical equation for the elastic magnification factors
on stress intensity for a surface crack in a unite-thickness plate was also developed. The fracture stress predictions computed from the two-parameter fracture criterion for both surface- and through-cracked sheet and
plate specimens are consistent with experimental failure stresses.
INTRODUCTION
FAILURES of numerous
aircraft and aerospace vehicle components have been traced to
surface cracks. These cracks initiate at structural discontinuities such as holes, material
defects, or other abrupt changes in configuration, and propagate to critical sizes at operating stress levels. In high-toughness materials and at stress levels below the yield stress
of the material, surface cracks may propagate to through cracks before failure. However, in low-toughness
(high strength) materials, unexpected failures have been caused
by small surface cracks. Such failures can occur at nominal stress levels well below the
yield stress of the material. In designing against these failures, the designer must be able
to predict the effects of crack size and shape on structural strength. The objective of
this paper is to present a method capable of correlating fracture data for a wide range of
materials, crack shapes, and crack sizes.
Irwin[l] derived the stress-intensity factor for an elliptical crack in an infinite
elastic solid subjected to uniaxial tension, based on an analysis by Green and Sneddon
[2]. Irwin also estimated the stress-intensity factor for a semielliptical surface crack in a
finite-thickness plate. His analysis is thought to be reasonably accurate for crack depths
less than one-half of the plate thickness, and for small-scale yielding. But in many situations, particularly in thin-walled pressure vessels, cracks are more than halfway through
the thickness and plastic deformation near the crack tip is large.
Many investigators have devised methods which account for the effects of the free
surfaces of the plate and plastic deformation near the crack tip on the fracture behavior
of surface-cracked specimens. Some of these investigators have deduced the effects of
the free surfaces on stress intensity magnification factors from experimental fracture
data, hence, the magnification factors obtained are generally applicable only to the
material tested. A failure criterion which is independent of crack size and specimen configuration would require detailed knowledge of the stress state near the crack tip. Several theoretical relations for calculating the elastic-plastic behavior at the root of
*Presented at the Symposium on Fracture and Fatigue at the School of Engineering and Applied Science.
George Washington University, Washington, D.C.. May 3-5. 1972.
667
668
J. C. NEWMAN,
Jr.
notches or cracks have been proposed. Among these are relations derived for notches
by Hardrath and Ohman[3], and by Neuber[4]; for cracks by Hutchinson[S], and by
Rice and Rosengren[6]. The Hardrath-Ohman
notch equation was generalized for a
through-cracked plate by Kuhn and Figge[7] and for a surface-cracked plate by Kuhn
[8]. In a similar way, the present paper uses Neuber’s relation to derive an equation
which relates the linear elastic stress-intensity factor, the applied stress, and two material parameters. The equation was similar to one proposed by Kuhn[9]. This equation
was used as a two-parameter fracture criterion to analyze fracture data from the literature for surface- and through-cracked
sheet and plate specimens of steel, titanium.
and aluminum alloy.
The elastic magnification factors on stress intensity for a surface crack in a finitethickness plate as a function of crack depth, crack length, and specimen thickness were
obtained from the analytical results of Smith and Alavi[ lo], Rice and Levy[ 1 l], and
Gross and Srawley [ 121 in their particular ranges of validity. An empirical equation was
chosen to fit these analytical results.
NOMENCLATURE
The physical quantities in this paper are given in both the International System of Units (Sl) and in the
U.S. Customary Units [13]. Appendix A presents factors relating these two systems of units.
gross-section area mm* (in*)
net-section area, mm2 (in2)
depth of surface crack, mm (in.)
haif-length of surface crack or through crack, mm (in.)
modulus of elasticitv, MN/m* (ksi)
secant modulus for the net section, MN/m’ (ksi)
fracture toughness computed by equation (9) MN/man (ksi-in”‘)
opening mode elastic stress-intensity factor, MN/m3’2 (ksi-in”‘)
elastic stress-concentration
factor
strain-concentration factor
stress-concentration
factor
elastic magnification factors on stress intensity
fracture-toughness parameter
elastic shape factor for an elliptical crack
gross-section stress, MN/m* (ksi)
calculated gross failure stress, MN/m2 (ksi)
experimental gross failure stress, MN/m” (ksi)
net-section stress, MN/m* (ksi)
temperature, “K (“F)
specimen thickness, mm (in.)
specimen width, mm (in.)
local strain
local fracture strain
strain corresponding to net-section stress
notch-root radius, mm (in.)
effective root radius for a crack, mm (in.)
local stress, MN/m2 (ksi)
local fracture strength, MN/m2 (ksi)
uniaxial ultimate tensile strength, MN/m2 (ksi)
uniaxial yield stress, MN/m2 (ksi)
complete elliptic integral of second kind
coordinate angle (Fig. 1 1)
THEORETICAL NOTCH ANALYSIS
Neuber [4] considered the problem of antiplane shear-strained bodies containing
notches with various nonlinear stress-strain laws and concluded that
K,K,
= KT2
(1)
Fracture
analysis
of surface-
and through-crackes
sheets and plates
669
where K, and K, are the stress- and strain-concentration
factors, respectively, and KT
is the elastic stress-concentration
factor. Crews [ 141 has indicated that Neuber’s equation represents the local stress-strain behavior of mildly notched sheet specimens better
than the Hardrath-Ohman notch equation [ 33.
In the present paper, Neuber’s notch equation was generalized for an elliptical crack
in a three-dimensional solid, similar to the way in which Kuhn and Figge [7] generalized
the Hardrath-Ohman
equation for a through crack in a plate. The present derivation
considers first an ellipsoidal cavity in an infinite elastic solid subjected to uniaxial stress
(see Fig. 1). The stress concentration at point A can be shown to be the greatest when a
is less than c. The expression for KT at point A was assumed to be analogous to that of
the two-dimensional case of an elliptical hole in an infinite plate[l5] and was given
approximately by
where
~=~~(sinzip+(~)?coszp)l”d~
(3)
is the complete elliptical integral of the second kind[l], p is the radius of the notch at
point A in the Y-Z plane, and (Yis a correction factor which accounts for the influence
of external boundaries. For an infinite solid (Y is equal to unity. Equation (2) is an
approximate expression which fits the numerical results of Sadowsky and Sternberg
[ 161 to within 3 per cent for a/b greater than 4. Equation (2) also reduces to the exact
plane-strain solution when c approaches infinity (0 = 1).
Combining equations (1) and (2), and expressing K, and K, in terms of the local
Fig. 1. Ellipsoidal
cavity
in an infinite solid subjected
to uniaxial
stress.
670
J. C. NEWMAN,
stress cr, local strain E, net-section
stress S,, and net-section
Jr.
strain E, gives
The net-section strain, E,, can be expressed as SJE, where E, is the secant modulus
obtained from the stress-strain curve for the net section. The stress-strain curve for the
net section is an ‘average’ curve which is influenced by the overall state of stress on the
net section. Generally, in the application of Neuber’s equation, the stress-strain curve
for the net section is assumed to be the uniaxial curve. Making the substitution for E,,
the following equation was obtained
Equation (5) is a general expression which relates the local notch-tip behavior (stress
and strain) to the net-section stress and notch configuration.
FRACTURE CRITERION
As a/b approaches infinity, the ellipsoidal cavity in Fig. 1 becomes a flat ellipsoidal
crack, and equation (5) relates the local crack-tip stress-strain behavior to the applied
stress and crack configuration. The fracture criterion for a given material (tensile properties and thickness) and test conditions (temperature, load rate, and environment) was
developed from equation (5) by substituting critical values for the local stress, local
strain, and notch-root radius as follows:
I. The local stress u was set equal to Us (local fracture strength)
2. The local strain E was set equal to ef (local fracture strain)
3. The notch was collapsed to a crack by setting the notch-root radius p equal to p*
(the effective root radius for a crack).
The effective root radius p* was assumed to be a material constant at failure because
the deformations at the crack tip should be the same for all tensile loading conditions
which cause local fracture under a given state of stress and test conditions. Substitution
of ai, E,, and p* into equation (5) gives
(6)
where K,, is the linear elastic stress-intensity factor. (Note that dn and c, have been
introduced for convenience in defining the stress-intensity factor and nondimensionalizing the net-section stress, respectively.) The linear elastic stress-intensity factor is given
by
K,,=S,6$=S
Jr$M,
(7)
where S is the remote stress, a is the crack depth, Q is the elastic shape factor, and M,
is the elastic magnification factor which accounts for the influence of external boundaries. An empirical equation for M, for surface cracks is developed in Appendix B.
Fracture analysis of surface- and through-cracked
sheets and plates
671
Because aj, es, and p* have been assumed to be constant at failure for a given material and test conditions, and are grouped together, the left-hand side of equation (6) was
set equal to & (the ‘fracture toughness’) and the term a,/was defined as m, a
fracture-toughness parameter. The resulting equation was
The denominator in equation (8) accounts for the nonlinearity of the material. An evaluation of the denominator for a given material is shown in Fig. 2. When the applied netsection stress is less than the uniaxial yield stress of the material, F is a linear function
of the net-section stress (solid line). The slope of the line is a negative m. If the applied
stress is greater than the uniaxial yield stress, the function F becomes non-linear and
is estimated by the dashed curve. Because the state of stress in the region of the crack
front is not uniaxial and the material may be strain-hardening, net-section yielding
(the plastic zone extends across the net section) is postponed until the applied stress
reaches a value higher than the yield stress of the material, that is, E, is still approximately equal to E. However, when net-section yielding does occur, the value of En,
and consequently F, is sharply reduced. For very small crack sizes, the state of stress
on the net section may also cp.use the cracked plate to carry a net-section stress larger
than the uniaxial tensile strength of the material (notch-strengthening),
but as the crack
size approaches zero the state of stress becomes uniaxial (if the remote stress is uniaxial) and the plate fails at the uniaxial tensile strength of the material (S, = ox). A complete evaluation of the function F above net-section yielding is quite complicated. For
simplicity, it was assumed that equation (8) can be applied up to the uniaxial tensile
strength of the material, in which case Kfwas given by
for S, < o,
(9)
The dash-dot line in Fig. 2 shows the function F for equation (9). The significance of
the vertical dash-dot line is explained later.
The fracture parameters K, and rrz are constant only in the same limited sense as the
ultimate tensile strength, that is, the parameters vary with material thickness, state of
stress, temperature, and rate of loading. To obtain fracture constants which are representative for the material and test conditions, the fracture data should be from a single
batch of material and should be over the widest range of crack size and shape possible.
At least two fracture tests are required, of course, because two constants must be evaluated.
In equation (9), if m is set equal to zero, K, is equivalent to the linear elastic stressintensity factor, K,,, and the equation is applicable to low-toughness materiais. If tn is
set equal to unity, the equation is similar to that obtained by Kuhn[& 91 and is applicable to high-toughness materials. It should be noted that reference[9] also presents a
two-parameter fracture criterion (the C;, method) which is similar to that presented
here. The fracture criterion of reference]91 and the present fracture criterion differ
mainly in the boundary-correction
factors on stress intensity (see Appendix B).
672
J. C. NEWMAN,
Jr.
F
Fig. 2. Evaluation of the function F.
The critical crack lengths (crack length at the failure load) should be used in equation (9), especially for materials which have substantial slow-crack growth before failure. However, most investigators do not record critical crack lengths because of the difficulties involved in defining and measuring crack length at the failure load. Therefore,
since few critical crack-length data are available, initial crack lengths are used in most
of the present fracture calculations.
FAILURE ANALYSIS
In designing against failures caused by cracks, the designer must be able to predict
the effects of crack size and shape on structural strength. This section gives the equations that were used to calculate the failure stress as a function of crack size and shape
in terms of the predetermined fracture constants, KI and m for surface- or throughcracked specimens (see Fig. 3).
Surface-cracked
con$gurations
For the surface crack, equations (7) and (9) were used to calculate the failure stress.
Substitution of equation (7) into equation (9) and solving for the gross failure stress
gives
where A, and A, are the gross- and net-section area, respectively. (See Appendix B for
the equation for M,.) Equation (10) was applied for net-section stresses up to the ultimate tensile strength of the material.
Through-cracked
configuration
For the through crack, the linear elastic stress-intensity
factor [ 171 is given by
Fracture analysis of surface- and through-cracked
t
t
t
sheets and plates
SECTION
673
A-A
+--I
4
L
(a)
SPECIMEN
(b)
SURFACE
CRACK
(c)
THROUGH
CRACK
CONFIGURATION
Fig. 3. Surface- and through-crack configuration.
K,,= S
dmset (5)
(11)
where S is the remote stress, c is the half length of the crack, and W is the plate width.
The secant term is a finite-width correction. Substitution of equation (11) into equation
(9) and solving for the gross failure stress gives
(12)
Again, equation (12) was applied for net-section stresses up to the ultimate tensile
strength of the material.
As the crack size approaches zero, equations (10) and ( 12) predict failure stresses
on the net section that exceed the ultimate tensile strength of the material. In these
cases the net-section stress at failure was assumed to be the ultimate tensile strength of
the material (vertical dash-dot line in Fig. 2) so that gross failure stress was given by
(13)
ANALYSIS OF TEST DATA
Fracture data from the literature [ 17-3 I] for a number of steel, titanium, and aluminum-alloy sheet and plate specimens containing either surface or through cracks were
analyzed according to equation (9). The surface- and through-crack configurations are
shown in Fig. 3. The fracture constants Kf and m from equation (9) were determined
from the fracture data by a best-fit procedure (see Appendix C). The tensile properties,
fracture constants, and test conditions for the materials analyzed are shown in Table 1.
E.F.M. Vol. 5, No. 3-K
674
J. C. NEWMAN,
Jr.
Fracture analysis of surface- and through-cracked
Surface-cracked
sheets and plates
675
specimens
Fracture data on surface-cracked sheet and plate specimens of steel, titanium ahoy,
titanium weldment, and aluminum alloy tested at room and cryogenic temperature were
analyzed according to equation (9). Wide ranges of crack-depth to crack-half-length
ratio (a/c = 0.08 to 1.4) and crack-depth to specimen-thickness ratio (a/t = 0.05 to 1.0)
were considered. The material thickness ranged from O-5 (0.02) to 23 mm (0.9 in.). The
initial crack sizes were used in the analysis for all surface crack data because critical
crack sizes (crack size at the failure load) are generally not reported. Slow crack extension before failure occurs in some surface-crack specimens, but is difficult to record.
Also, examination of the fracture surfaces from a number of fracture tests of 22 19-T87
aluminum alloy plate 1241 showed the material had delaminated in the short transversegrain direction ~delamination cracks were perpendicular to the surface crack). The test
results from these specimens were eliminated from the analysis.
References [l] and [32] have demonstrated that the stress-intensity factor varies
along the boundary of the surface crack. Furthermore, because the state of deformation
(hence, the fracture toughness) varies along the surface-crack boundary, the location
at which fracture initiates does not necessarily correspond to the location of maximum
K,,. These considerations raise a question as to the proper value of K,, to be employed
in the fracture criterion. The definitive answer to this question is not yet available.
Therefore, and for simplicity, equation (7) was assumed to give the value of K,, at which
fracture initiates, and the fracture-toughness parameter m in equation (9) was assumed
to account for any effects of the state of deformation.
Some results of the analysis on surface-cracked specimens are shown in Figs. 4
and 5. Figure 4(a) shows the data for a thin titanium-alloy (Ti-6Al-4V) sheet material
[21]. The linear elastic stress-intensity factor (Equation 7) at failure was plotted as a
function of the normalized crack dimension, (a/Q)Mp2. The normalized crack dimension accounts for the influence of both a/c and a/t ratios on stress intensity. The
symbols indicate the fracture data and the solid symbols are for tests in which the applied stress at failure (net section) was approximately equal to the uniaxial tensile
strength of the material. The ale ratios ranged from O-35 to 0.8 and the a/t ratios from
0.05 to 06. The solid curve shows the calculations from equation (9) using the predetermined fracture constants and an a/c ratio of 0%. The dash-dot curve was calculated from equation (7) and indicates tensile instability, that is, S, = (T,. The correlation
of theory with data is quite good, even for net-section stresses greater than the tensileyield stress of the material. The figure also shows that the linear elastic stress-intensity
factors (K,,) at failure are not constant.
Figure 4(b) shows K/ as a function of the normalized crack dimension for the same
data as shown on Fig. 4(a). In contrast to K,, at failure, K, was nearly constant. The Kf
values are computed from equation (9). The solid line is the average value of lu, (see
Appendix C). The dash-dot curve indicates tensile instability and was also computed
from equation (9). The data indicate that equation (9) cannot be used to calculate Kf
when the failure stress is near u,. In this region, failure was governed by tensile
instability and not crack instability.
Figure 4(c) shows results for tests of a 300-M steel[l7]. Again, K,, at failure was
plotted as a function of the normalized crack dimension. The a/c ratios ranged from
0.5 to 0.8 and the a/t ratios from O-15 to O-5. In contrast to the results shown in Figure
4(a), the K,, values at failure tend to be higher for smaller crack sizes. Two explanations
for this unusual behavior have been proposed. The increase in the critical value of EC,,
616
J. C. NEWMAN,
Jr.
87
k)
300
M
STEEL
Cd)
30*x+$
STEEL
Fig. 4. Critical elastic stress-intensity factor and K, as a function of the normalized crack dimension for surface cracks.
with smaller crack sizes may be related to the state of deformation in the crack-tip region. The state of defo~at~on on the surface crack boundary at which fracture initiates
may be essentially one of plane stress for shallow cracks and plane strain for deeper
cracks, Thus, the apparent fracture toughness of the material may be larger for the shallow cracks than for the deeper cracks. Second, a residual-stress field caused by the
manufacturing process could have a pronounced effect on stress intensity. A compressive residual stress near the surface would have been necessary to cause the behavior
shown in Fig. 4(c).
The solid curve in Fig. 4(c) shows the calculations from equation (9), in which an
a/c ratio of 0.6 was used. The dash-dot
curve indicates tensile instability (S, - a;).
The best fit of equation (9) to these results required that m be negative which is in viola-
Fracture analysis of surface- and through-cracked
sheets and plates
677
Fig. 5. Ratio of failure stress to ultimate tensile strength as a function of the normalized crack
dimension.
tion of the bounds imposed on m by the development of equation (9). A full explanation
of this inconsistency probably must await the evolution of accurate methods of threedimensional elastoplastic stress analysis. Residual stresses and variations in the state of
stress at the crack front (plane stress to plane strain) are likely to be especially important in the surface-crack specimen.
Figure 4(d) shows K, as a function of the normalized crack dimension for the same
data as shown on Fig. 4(c). Again, in contrast to K,, at failure, Kf was nearly constant.
The dash-dot curve indicates tensile instability.
In designing against structural failures caused by cracks, the designer must be able
to predict the effects of crack size and shape on residual strength. To demonstrate the
usefulness of equation (9) or (lo), failure stresses as a function of crack size and shape
for some of the materials analyzed are shown in Fig. 5. This figure presents the ratio of
J. C. NEWMAN,
678
Jr.
experimental failure stress to the uniaxial tensile strength of the material (symbols) and
the theoretical calculations (solid curves) from equation (10) as a function of the
normalized crack dimension.
Figures 5(a and b) show the test results for two grades of D6ac steel plate [18]
and Ti-6Al-4V
titanium alloy plate[l8], respectively, tested at room temperature.
For both the steels and titanium alloys, the a/c ratios ranged from approximately 0.5 to
1.2 and the a/t ratios from 0.2 to 0.5. The symbols indicate the test results. The solid
curves were calculated from equation (10) using an a/c ratio of 0.6, a specimen width
of 35.5 mm (1.4 in.), and the fracture constants, K, and m, for the respective materials.
The fracture constants were determined by the least-squares analysis presented in
Appendix C. The test results for the titanium alloys indicate that the ultimate tensile
strength was obtained on the net section at a finite crack size. The correlation of theory
with experimental data was good.
Figure 5(c) shows the test results for 4340 steel sheet specimens[26] tested at room
temperature. The a/c ratios ranged from 0.5 to 1.1 and the a/t ratios from 0.15 to 0.7.
The solid curve was calculated from equation (10) using an a/c ratio of 0.6 and a spec1 .‘I[-
i .+
/
I
?
:
I.0
$
R
.a
.6C
.Li -
.‘I
Oi0
__-A_.
---..-
_~~+__
/ HRcF
CRRCK
i
1 .i,
(a' CRRCK
DEPTH
LENGTH
(+)
STEELS
I .‘t
1.2 1
4
01
0
I
I
.zs
lb]
CRRCK
.50
DEPTH
/ PLATE
I
I
.75
THICKNESS
1 .oo
(+)
Fig. 6. Ratio of calculated to experimental failure stress as a function of crack size
for surface-cracked steel specimens.
and shape
Fracture analysis of surface- and through-cracked
sheets and plates
679
imen width of 38 mm (1.5 in.). The good agreement between the solid curve and the data
indicates that equation (10) can be used to up to the ultimate tensile strength of the
material.
Figure 5(d) shows the test results for 22 19-T87 aluminum alloy plate [24] tested at
cryogenic temperature. The a/c ratios ranged from 0.2 to 0.8 and the a/t ratios from 0.35
to 0.9. The solid curve was calculated from equation (10) for an a/c ratio of 0.2 and a
specimen width of 230 mm (9 in.). Specimen widths ranged from 140 (5.5) to 560 mm
(22 in.) but no consistent variation in test results with specimen width was observed.
Again, the correlation of theory with experimental data was good.
Figures 6-8 show the ratio of calculated failure stress to experimental failure stress
for all of the surface-crack data analyzed for steels, titanium alloys, and aluminum alloys, respectively, as a function of crack-depth to crack-half-length ratio a/c and also
crack-depth to specimen-thickness
ratio u/t. The crack-depth to specimen-thickness
ratio ranged from 0.05 to 1.0 and the crack-depth to crack-half-length ratio from O-08 to
1.4. Although very few fracture data were available for a/c ratios greater than unity, the
results indicate that equations (7) and (9) are adequate. The calculated failure stresses
1.2
I
e
Lo
\
UY
2
0
.o
.8
.6
.4
.2
0
I
0
(al
CRRCK
Cl;“,TH
I
/
HFlLF
OoU
fbl
Fig.
7.
.50
CRRCK
DEPTH
/
PLRTE
J
1.5
CRRdK”LENGTti
(3
I
.75
THICKNESS
i
1
.oo
(;’
Ratio of calculated to experimental failure stress as a function of crack size and shape
for surface-cracked titanium-alloy specimens.
680
J. C. NEWMAN,
Fig. 8. Ratio of calculated to experimental
for surface-cracked
Jr.
failure stress as a function of crack size and shape
aluminum-alloy specimens.
were consistent with the experimental failure stresses. For 95 per cent of the data analyzed, the difference between the calculated and experimental failure stresses was less
than 10 per cent, and the maximum error was approximately 22 per cent.
Through-cracked specimens
Fracture data from the literature on through-cracked
specimens of steel, titanium
alloy, and aluminum alloy tested at room and cryogenic temperature were also analyzed
according to equation (9). Wide ranges of crack-length to specimen-width ratio (2c/ W =
0.05 to 0.6) and specimen width [ W = 75 (3) to 1220 mm (48 in.)] were considered. The
material thickness ranged from 0.5 (0.02) to 3.6 mm (0.14 in.). For some of the data
analyzed, the critical crack lengths were used instead of the initial crack lengths. These
cases are indicated in Table 1. For example, in the case of the 4330 steel[30] the crack
lengths at failure were as much as four times the initial crack lengths. Slow-crack extensions of these magnitudes have to be accounted for in the analysis. In most cases, the
critical crack lengths were not reported.
Figure 9 shows the ratio of calculated failure stress to experimental failure stress as
Fracture analysis of surface- and through-cracked
681
sheets and plates
I
01
0
I
-25
I
-50
I
.75
CRRCK LENGTH / PLQTE WIDTH
I
1 -00
($)
Fig. 9. Ratio of calculated to experimental failure stress as a function of crack size for throughcracked steel, titanium, and aluminum-aIloy specimens.
a function of crack-length to specimen-width ratio for all of the through-crack fracture
data analyzed for steels, titanium, and aluminum alloys. Most of the data analyzed for a
given material and thickness had a wide range of crack lengths and specimen widths.
Again, the calculated failure stresses were consistent with the expe~mental failure
stresses. For 95 per cent of the data analyzed, the difference between the calculated
and experimental failure stresses was less than 10 per cent.
A review of equations (6) and (9) indicated that the parameters &and yn are related
by the effective root radius p” and the ultimate tensile strength of the material. If a relationship between the fracture parameters exists, then the analysis would be simplified
and the fracture toughness K, could be expressed directly in terms of failure stress,
crack configuration, and mechanical properties. To investigate this possibility, the ratio
of fracture toughness to Young’s modulus, K,/E, is plotted in Fig. 10 as a function of m
for all of the surface- and through-crack fracture data analyzed. Each data point indicates the Kf/E and m obtained for each set of data analyzed (material and specimen
configuration). The results suggest a common functional relationship between these two
fracture parameters for steels, titanium, and aIuminum alloys. Therefore, it appears
that a one-parameter fracture criterion can be obtained. The development of such a
criterion is currently under investigation.
CONCLUDING REMARKS
The Neuber stress-concentration
relation was generalized for a crack in a finite
plate. An equation was derived to relate the linear elastic stress-intensity factor, the applied stress, and two material parameters. This equation was used as a two-parameter
fracture criterion for both surface- and through-cracked sheet and plate specimens of
steel, titanium, and aluminum alloy tested at room and cryogenic temperature. Wide
682
J. C. NEWMAN,
o-
Jr.
0
-0.4
-0.2
0
0.2
0.4
06
0.8
1.0
m
Fig. 10. Ratio of fracture toughness to Young’s modulus as a function of m for steel, titanium,
and aluminum alloy surface- and through-cracked sheet and plate specimens.
ranges of crack depth, crack length, material thickness, and specimen width were considered. Also, an empirical equation for the elastic magnification factors on stress intensity for a surface crack in a finite-thickness plate was developed.
For both surface- and through-crack fracture data, 95 per cent of the calculated failure stresses were within 10 per cent of the experimental failure stresses. Data from
approximately
800 fracture tests were included in the analysis. The two-parameter
fracture criterion was able to relate fracture stress to crack size and shape for both lowand high-toughness materials. The results of the analysis also indicated that a common
functional relationship exists between the two fracture parameters (Kf and m) and the
modulus of elasticity.
REFERENCES
I I I G. R. Irwin, The crack extension force for a part through crack in a plate, Trans. ASME. J. appl. Mechs.
84,65 l-654 (1962).
[21 A. E. Green and I. N. Sneddon, The distribution of stress in the neighborhood
of a flat elliptical crack in
an elastic solid, Proc. Cambridgephil. Sot. 46, 159-164 (1950).
[3] H. F. Hardrath and Lachlan Ohman, A study of elastic and plastic stress concentration factors due to
notches and fillets in flat plates, NACA Rep. 1117, 1953. (Supersedes NACA TN 2566.)
[4] H. Neuber, Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress-strain law, Trans. ASME. Ser. E,J. uppl. Mechs. Z&544-550 (1961).
[5] J. W. Hutchinson, Singular behavior at the end of a tensile crack in a hardening material,./. Me&. Phys.
Solids, 16, 13-31 (1968).
[6] J. R. Rice and G. F. Rosengren, Plane strain deformation near a crack tip in a power-law hardening
material,/. Mech. Phys. So/ids 16, 1-12 (1968).
[7] P. Kuhn and I. E. Figge, Unified notch-strength
analysis for wrought aluminum alloys, NASA TN D1259 (1962).
[8] P. Kuhn, Residual tensile strength in the presence of through cracks or surface cracks, NASA TN
D-5432 (1970).
[9] P. Kuhn, Residual strength in the presence of fatigue cracks (Part I), Presented to the Structures and
Materials Panel of AGARD (Turin, Italy), April 1967.
Fracture analysis of surface- and through-cracked
sheets and plates
683
[lo] F. W. Smith and M. J. Alavi, Stress intensity factors for a penny shaped crack in a half space, Engng
Fracture Mech. Vol. 3, 1971, pp. 241-254.
[1 1] J. R. Rice and N. Levy, The part-through surface crack in an elastic plate, Tech. Rept. NASA NCL
40-002-080/3, Div. of Engr., Brown Univ., Trans. ASME, J. appl. Mech. Paper No. 71-APM-20
(1970).
[12] B. Gross and J. E. Srawley, Stress-intensity factors for single edge notch specimens in bending or
combined bending and tension by boundary collocation of a stress function, NASA TN D-2603
(1965).
[ 131 Committee on Metric Practice: ASTM Metric Practice Guide, NBS Handbook 102, U.S. Department
of Commerce, March 10, 1967.
[ 141 J. H. Crews, Jr. Elastoplastic stress-strain behavior at notch roots in sheet specimens under constantamplitude loading, NASA TN D-5253 (1969).
[ 151 S. Timoshenko and J. N. Goodier, Theory ofElasticity, McGraw-Hill (195 1).
[ 161 M. A. Sadowsky and E. Stemberg, Stress concentration around a triaxial ellipsoidal cavity, Trans.
ASME, J. appl. Mech. 16, 149- 157 (1949).
[ 171 Plane strain crack toughness testing of high strength metallic materials, ASTM STP 410, American
Society of Testing and Materials, 1966.
[ 181 P. N. Randall, Severity of natural flaws as fracture origins and a study of the surface-cracked specimen,
AFML-TR-66-204 (1966).
[ 191 D. W. Hoeppner, D. E. Pettit, C. E. Feddersen, and W. S. Hyler, Determination of flaw growth characteristics of Ti-6AI-4V sheet in the solution-treated and aged conditions, NASA CR-658 11, 1968.
1201 R. M. Bonesteel, Fracture behavior of $ inch thick 7075T651 Al containing semielliptical surface
flaws, Lockheed Missiles andSpace Company, Rept. 6-83-71-1, 1971.
[2 1] A. B. Smith, Missile motor cases, American Society of Metals, Metals Engng Qy, November 1963.
[22] C. F. Tiffany, P. M. Lorenz, and L. R. Hall, Investigation of plane-strain flaw growth in thick-walled
tanks, The Boeing Company, NASA Cr-54837,1966.
[23] L. R. Hall, Plane-strain cyclic flaw growth in 2014T62 aluminum and 6AI-4V (ELI) titanium, NASA
CR-72396 (1968).
[24] J, N. Masters, W. P. Haese. and R. W. Finger, Investigation of deep flaws in thin walled tanks. NASA
CR-72606 (1969).
1251 W. D. Bixler and W. L. Engstrom, Crack growth of D6 steel in air and high pressure oxygen, NASA
CR-114860(1971).
[26] V. Kerlins and S. L. Pendleberry, The effect of crack type and material thickness on fracture strength
of 4340 steel, Douglas Aircraft Company, Report SM-43 113, 1963.
1271 T. W. Orange, T. L. Sullivan, and F. D. Calfo, Fracture of thin sections containing through and partthrough cracks, NASATM X-52794 (1970).
[28] C. V. Thrash, Evaluation of high strength steel for DC-IO, Doug/as Aircraft Company. McDonnell
Douglas Corp., Engineering Technical Report No. ETR-DAC-67520,
1969.
~291 R. G. Forman, Experimental program to determine effect of crack buckling and specimen dimensions
on fracture toughness of thin sheet materials, Aerospace Res. Labs. Rept. No. AFFDL-TR-65-146.
Wright-Patterson Air Force Base (1966).
[30. T. W. Eichenberger, Fracture resistance data summary Rept. No. DA-20947, The Boeing Airplane
Company, Seattle, Washington, 1962.
[3)1 1. E. Figge, Residual static strength and slow crack growth behavior of duplex annealed Ti-8AI-IMo-IV
sheet, NASATN D-4358, (1968).
~321 F. W. Smith, Stresses near a semi-circular edge crack, University of Washington, Ph.D. Dissertation
(1966).
r331 A. S. Kobayashi and W. L. Moss, Stress intensity magnification factors for surface-flawed tension plate
and notched round tension bar, Second International Conf. on Fracture, Brighton, England (1969).
(Received
20 June 1972)
APPENDIX
CONVERSION
A
OF SI UNITS TO U.S. CUSTOMARY
UNITS
The International System of Units (SI) was adopted by the Eleventh General
Conference on Weights and Measures held in Paris in 1960[ 131. Conversion factors
required for units used herein are given in the following table:
684
Physical quantity
Force
Length
Stress
Stress intensity
J. C. NEWMAN,
Jr.
SI Unit
(+)
Conversion
factor
($)
Newtons(N)
Meters (m)
Newtons per sq. meter
(N/m’)
Newtons per mete?
U.S. Customary
Unit
0.2248
0.3937 x 10’
0.145 x 10-6
Ibf
in.
ksi = l@‘Ibf/in”
0.9099 x IO__”
ksi_in,“”
f K -- 459.67
“F
(N/m:mj
Temperature
degrees Kelvin (“K)
iPrefixed to indicate multiple of units are as follows:
Prefix
Multiple
mega@I)
10"
milli (m)
10 s
$Multiply value given in Sl Unit by conversion factor to obtain equivalent in U.S
Customary Unit.
APPENDIX B
ELASTIC MAGNIFICATION FACTORS FOR SURFACE CRACKS
IN A FINITE-THICKNESS PLATE
Several investigators have obtained the elastic magnification factors on stress
intensity for a surface crack in a finite-thickness plate [l], [lo], [ill, [24], [321, and
[33]. In the present paper, empirical equations are developed which relate these
magnification factors to crack size and plate thickness. This section also includes a
brief review of the stress intensity solutions for an elliptical crack and a semielliptical
surface crack in an elastic material.
Irwin[l] derived an expression for the stress-intensity factor around an elliptical
crack in an infinite elastic solid subjected to uniaxial tension based on an analysis by
Green and Sneddon[2]. The stress-intensity factor along the boundary of the elliptical
crack in an infinite solid subjected to a remote uniaxial stress S acting normal to the
plane of the crack was given by
(91)
where @ is given by equation (3).
The crack orientation and dimensions are shown in Fig. 1 l(a). For c > LI the maximum stress-intensity factor was at cp= r/2 and was given by
K,=S,
4,
(92)
Irwin [ l] also estimated the stress-intensity factor at cp= 9-r/2 for a semielliptical
crack in a finite-thickness plate. (See Fig. I l(b) for c < W.) The stress-intensity factor
Fracture analysis of surface- and through-cracked
sheets and plates
f
685
t
1
(hi Semielliptical crack in a fLnite plate.
Fig. 1I. Schematic of an elliptical crack in an infinite sofid and a semieltiptical surface crack in
a finite-thickness plate.
was also corrected for small-scale yielding at the crack tip and inciuded estimates for
the effects of both the front and back surfaces of the pfate. Irwin’s anafysis was restricted to crack depths less than one-half of the plate thickness. Smith[32], Kabayaski
and Moss [3 35, Smith and Alavi [ lo], and Rice and Levy [ 111 have also presented further
approximations for the effects of both piate surfaces on stress intensity.
The elastic magnification factors for a surface crack in a finite-thickness plate as a
function of u/t and nfc are shown in Fig. 12. These rna~~~~~t~on factors (symbofs)
were obtained from the analytical results of references [lo-l 23. Smith and Alavi] 101
obtained approximate solutions for a near penny-shaped crack in a finite-thickness
plate (tijc = 0.4 and 1.O in Fig. 12). Their approximations are fairly accurate for n/c
ratios near l-0, On the other hand, Rice and Levy 1I 1] obtained approximate solutions
for shallow cracks (a/c = 0.1 and O-2 in Fig. 12) Their approximations are fairly
accurate for luw u/c ratios but lose accuracy for a/c ratios near unity and n/r ratios
greater than 0.5. Gross and Srawley [ 121 have obtained an accurate solution for a/c = 0
(a singJe-edge cracked plate).
In the present paper, an expression for the elastic magni~~ation factors on stress
intensity for a surface crack in a finite-thickness plate was obtained by fitting an equation to the analytical results from references [lo- 121 in their particular ranges of
validity. For simplicity, the equation was chosen so that the through-crack equation
for stress intensity was obtained when the surface crack intersects the back surface
of the plate fa = t). The eIastic stress-intensity factor for the con~gurat~o~ shown in
Fig. 11(b) was given by
686
J. C. NEWMAN,
Jr.
. GROSS AND SRAWLEY (REF. 12)
0 RICE AND LEVY (REF. 11)
0 SMITH ANTI AIAVI (REF. 10)
---KUHN
(REF. 8)
__
EQUATION (B5) FOR w = co
3.or
2.5-
2.0 -
/
0.5 /
/
/
1’
/’
,A/’
/’
_---
_---
_---
_ __
10.
&:__----O/
0.2
0.4
0.8
0.6
a
t
1.0
Fig. 12. Elastic magnification factors on stress intensity for a surface crack in a finite-thickness
plate.
where Q = W. The expression for Q was approximated
by
Q = 1-t 1.47 ; “M for
0
E s 1.0
Q = 1 + 1.47 ; 1.64 for
0
:>I.0
034)
The maximum error in the stress-intensity factor by using these equations for Q was
approximately O-25 per cent. The elastic magnification factor, M,, was given by
where
p=2+g
0a_”
C
and Ml is the front-face correction. The secant term corrects the stress-intensity factor for the influence of the finite width. The equation for the magnification factor is
Fracture analysis of surface- and through-cracked
sheets and plates
687
an approximate equation which accounts for the influence of the front surface, the back
surface, and width of the plate on stress intensity. Figure 12 shows the magnification
factor (equation BS) divided by the square root of the shape factor as a function of
ale and a/t for W = cc).
The expression for M, was given by
M1z&(
1+0.03c
M, = 1.13-0.1
s
0
for
a)
for
a>l.O
(W
c
0.02 s z s 1.0
(B7)
For a/c > 1.0, the maximum stress-intensity factor is near p = 0 and 7~and the magnification factors are not influenced strongly by a/t ratios. For a/c < 0.02, the stressintensity factor for the single-edge crack plate, a/c = 0, subjected to uniaxial tension
[ 121 was assumed to apply and M, was given by
M,=
As a/t approaches
the stress-intensity
was given by
1.12-0.23
a_ +lO 55 a ‘-21
(tl
*
(t)
71 a 3+30 38 a 4
*
0
.
(W
Lb
unity for any value of a/c (except zero), equation (B3) reduces to
factor for a through crack of length 2c in a finite-width plate and
The magnification factors given by equation (BS) agree fairly well with the analytical
results of references [lo- 121.
The dashed lines in Fig. 12 show the estimates made by Kuhn[8]. Only the results
for a/c = O-1, 1.0, and IO.0 are shown. Kuhn approximated the surface crack by a
through crack having an ‘equivalent crack length.’ The equivalent crack length relationship was obtained by trial and error from an analysis of several sets of fracture
data. For a/t ratios less than approximately 0.6, the dashed lines do not agree well
with curves calculated by equation (B5).
APPENDIX
LEAST-SQUARES
PROCEDURE
FOR THE DETERMINATION
TWO FRACTURE
The two-parameter
C
OF THE
PARAMETERS
fracture criterion for cracked plates was given by
(Cl)
where K,, is the linear elastic stress-intensity factor for cracked plates subjected to
uniform remote loading. The fracture parameters Kf and m are assumed to be material
constants.
688
J. C. NEWMAN,
Jr.
For a given set of fracture data (material, thickness, and test conditions), the two
fracture parameters were obtained by minimizing the sum of the squares of the differences between the calculated failure stresses and the experimental stresses. The ratio
of the experimental failure stress to the calculated failure stress was
(C2)
where KI, and S, are obtained from the fracture-test results (crack size and failure
stress) and u, from tensile tests at the same test environment. The sum of the squares
of the errors in calculated failure stress was given by
(C3)
where N was the number of fracture
were then obtained by requiring
specimens
analyzed.
The parameters
K, and m
(C4)
Equations (C4) give the following two linear algebraic equations:
((25)
Solving equations (C5) form and K,gives
(C6)
and
5
Kf=
K;,
i=l
N
C KIe--m
i-
1
$ KIe
i-l
(C7)
Fracture
analysis
of surface-
and through-cracked
sheets and plates
689
In the analysis of fracture data, values of the fracture toughness parameter m
ranged from -0.35 to 1.1. The values of m greater than 1.0 lead to failure stresses less
than the uniaxial tensile strength of the material at zero crack length. To avoid this
situation, and because the failure stress calculations are not strongly affected, the value
of m was set equal to unity for these cases and K, was computed from equation (C7)
with m = 1. The values of K, and m for each material, thickness, and test conditions
are given in Table 1.
EFM.
Vd.
5, No. 3-L