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An introduction to fracture mechanics for engineers Part II: Using the stress intensity factor to characterise fracture and fatigue crack growth
An introduction to fracture mechanics for engineers Part II: Using the stress intensity factor to characterise fracture and fatigue crack growth R. A. SMITH Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ, England
This article continues the discussion o f the Stress Intensity Factor parameter by demonstrating how it can be used to characterise catastrophic (rapid) fracture and stable fatigue crack growth. Numerical values for the fracture toughness, crack growth 'constants' and threshold Stress Intensities are given for a wide range o f engineering materials. The principles o f the Fracture Mechanics approach to failure prevention are illustrated by worked examples. Introduction Part I of this series (Stresses due to notches and cracks, Mats. in Eng. Appl. 1, 2, pp 121-128, 1978) described how the stress fields near crack tips can be quantified by the stress intensity factor (S.I.F.) parameter, ao~, where o is the remote stress, a is the crack length and a, a geometrical factor which accounts for the shape of the body containing the crack and the type of loading to which it is subjected. The power and utility of this parameter is that it enables local conditions near the crack tip to be described by easily measured quantities remote from the complicated crack tip area - it allows us to step back from the fine detail at the crack tip. The S.I.F. as a fracture criterion The first useful engineering application of the S.I.F., as a fracture criterion, stems from this very ability. Imagine a component containing a crack of fixed length whilst the load on the
structure is continuously increased. The local stresses (and, hence, strains and displacements) near the crack tip will also increase, scaled in magnitude by the S.I.F. We make the hypothesis that rapid, unstable crack advance will occur when the local stresses reach some critical value, which will be constant for a given material. It immediately follows that these critical local conditions will correspond to a critical value of the S.I.F., which we can easily quantify and then use as a measure of a material's resistance to rapid crack advance; we call this resistance the material's fracture
toughness. Measurement o f fracture toughness This hypothesis has been tested experimentally and found to provide a consistent fracture criterion, providing the limitations of linear elastic fracture mechanics (LEFM), which were introduced in Part I, are not violated. These limitations stemmed from the need to restrict the volume of the plastically deformed material at the crack tip, so that the behaviour of the surrounding elastic region controlled the crack tip deformation behaviour. Within these conditions, the fracture behaviour can be satisfactorily quantified by the S.I.F. The experimental measurements are usually performed on one of the ASTM standard specimens, the single edge notch (SEN) because it is easy to load in bending or the compact tensile specimens (CTS) because it allows high
M A T E R I A L S IN E N G I N E E R I N G APPLICATIONS, Vol. 1, June 1979
values of SIF to be attained for relatively short crack lengths and is thus economical in the use of material. Both these specimen types have known SIF calibrations, see Part I, Appendix 2 for details. A crack is artificially introduced into the specimen by machining a slot which is further sharpened by fatigue loading under low alternating loads. A slowly increasing static load is then applied to the specimen until catastrophic fracture occurs. A tentative fracture toughness value can now be calculated by employing the measured values of crack length and fracture load in the SIF calibration equation. Checks must then be made to ensure that the limitations of LEFM have not been violated - in particular the thickness of the specimen must be greater than a value
which was introduced as Equation 11 in Part I. Additionally, to check that no extensive plasticity was introduced during the fatigue precracking process, the maximum SIF of the fatigue loading must be shown to be less than 0.67 of the tentative toughness value. If these conditions are fulfilled the tentative toughness value is confirmed as the plane strain fracture toughness of the material and designated, Klc. Further experimental details and more formal statements of the validity requirements can be found in the
227
Table 1. Typical values of plane strain fracture toughness, Klc
Material Steels Medium carbon Pressure vessel (ASTM A533B Q+T) High strength alloy Maraging steel AFC 77 Stainless
Modulus E, (MN/m 2)
2.1
X
Yield Stress Oy, (MN/m 2)
Toughness Klc, (MNm-3/2)
2.6 X 102
54
l0 s
Thickness 2.5~K1el 2 toy i mm
110
X 102 X 10~ X 102 X 102
208 98 76 83
487 11 4.4 7.9
4.2 X 102 5.4 X 102 5.6 X 102
27 30 23
10.4 7.9 4.2
1.08 X l0 s
10.6 x 102 11 x 102
73 38
12.6 3.1
~ 4 X 104 1 X l0 s 3 x 103
80 3 X 102 30
0.2-1.4 13 1
4.7 2.8
4.7 14.6 18 15.3
Aluminmmalloys
2024 T8 7075 T6 7178 T6
7.2
X
104
Titanium alloys
Ti-6A~--4V (High Yield)
For comparison
Concrete WC-Co composites PMMA
Note: Representative values only - not to be employed as design data.
standard test methods referenced at the end of this paper. Typical values of Klc for a variety of materials are presented in Table 1, together with values for the yield stress, (ry, thus enabling the thickness requirement (Equation 1) to be calculated. To establish orders of magnitude, we note that 100 MNm -3/2 is a reasonable K~c for steels, somewhat less for aluminium alloys, and that in general a trade-off occurs between values of yield stress and Klc for similar classes of metallic materials. It can be seen that for relatively low toughness, high yield materials the required thicknesses are modest, of the order of several mm, but for high toughness, low yield materials, where the thickness requirement may approach as much as 0.5m, both massive specimens and testing facilities would be required. This, coupled with the fact that such materials might not be used in such thick sections, is the reason for the current interest in elastic/plastic fracture mechanics which will be further discussed in the third and final part of this series. Use of the Klc fracture criterion If we know the value of the fracture
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toughness of the material from which a component containing a crack is made, we can ensure its survival if the applied SIF does not exceed Klc i.e. if a o V / - ~ < Klc
(2)
It is worth emphasising the physical origin of each side of the above inequality. The S.I.F. on the left hand side is a mechanics parameter, its value determined by the geometry, stress level and crack length in the component, and is a measure of the cracking effort being applied to the c o m ponent, whilst the right hand side is a material constant, a measure of the particular material's ability to resist rapid crack advance. A worthwhile analogy might be drawn with the criterion o < Oy
(3)
to avoid yield in simple tension. Here the applied stress o is a measure of the stretching effort being applied and the yield stress, Oy, is a material constant, a measure of the material's ability to resist stretching without permanent deformation. When an equality is inserted into
Equation (2), it is a straightforward task to calculate those combinations of crack length and stress level which will cause failure by rapid crack advance. We now illustrate this approach by a simple example: A tool similar to that shown in Fig.1 is used to dig up old roads. Its cross section is a rectangle of thickness 25mm. If a through thickness crack of length 5mm is found in the position shown, what would be the failure load? We first check if the LEFM approach would be suitable. For the values given the thickness criterion, Equ 1, leads to B>2.5
~
59 2
i.e. B > 3 . 9 X
10-3m
which is easily satisfied by the quoted thickness of 25mm. The cracked section is acted upon by both end load, P and a moment, M = P X 170 X 10-3Nm. Both contribute to the SIF, illustrating the important principle of superposition. For a crack length/ width ratio, a/w, of 5/40 i.e. 0.125, we obtain, by interpolation of the values given for beams in bending and tension in Part I, Appendix 2, f(a/w) bending = 1.03 f(a/w) tension = 1.27
MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979
hence, the combined SIF becomes:
Fig.l. Cracked tool: configuration and dimensions
6M V/'~. 1.03 Kz
= BW 2
P X/'~. BW
+
1.27
25ram
which reduces to
?.
K1 = P~/Tr5 X 10 -3 . (2.75 X 104 ) MNm-3/2
0
Setting Kl = Klc for fracture, p=
59 0.125 X 2.75 X 104- =
0.0172 MN or 17kN, rounded to the accuracy of the information given. We must now avoid being lulled into a false sense of security by the apparent ease of the above approach. Unfortunately, fracture avoidance, even where LEFM applies, is made difficult in practice by the problems of identification of the shape and length of any defects, by difficulties of the correct stress analysis of complex components and by the fact that cracks can extend in a sub-critical manner, which may mean that a component initially thought safe from fracture problems, might become dangerous after a period of service. It is to this last problem that we now turn; fortunately a modified form of the SIF turns out to be useful in the sub-critical crack growth area.
Sub-criticalcrackgrowth - mainlyby fatigue If a cracked component is subjected to varying loads, to a corrosive environment or to steady loads and a high temperature, crack extension can occur by fatigue, stress corrosion cracking or creep processes respectively. This kind of crack extension is termed stable in that crack length increases with applied cycles or time until the crack length• load combination reaches the critical value for catastrophic instability previously discussed. The useful life of a cracked component is therefore determined by the amount of allowable stable crack extension which can take place. Of these types of loading, fatigue is most common, certainly most understood, and has proved to be most amenable to quantification by fracture mechanics parameters. An idealised type of constant amplitude fatigue loading is shown in Fig.2. The applied stress cycles about a mean stress level, Omean , between maximum, Omax, and minimum, ffmin, stress levels. Recalling the
150 m m KIC=S9MN m- : ~ ¢~y = I S O O M N / m
Stress ~
" 2
&(~ = (~mo~-~min.)
I
~,n.
Jo'm,on
time
Fracture or Plastic Instability
tO-t
i 0 -=
10°3 Growth Rate,
Stable growth
do =C [ZIK)m dN m~4
,dOdN(mm/cycle]0-4
iO-S
iO-6 I
I
AKTH IO
!
I
50
IOO
=
Stress intensity factor, &K MN nT~2 Fig.2. Idealiscd relationship between growth rate and stress intensity factor range.
MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979
229
method of our calculation (Part I) of the plastic zone size ahead of a crack tip, we truncated the supposed elastic stress distribution ahead of the crack,
Ka
~ ,
at the value of the materials
yield stress Oy. Now by assuming an elastic unloading from this constant value, the effect of stress decrease from Omax to amin, can be estimated by applying a negative stress distribution due to the range, Ao, (=Oma x --Omin). The material will yield in
compression therefore when the sum of the stress distributions drops to a value
of-Oy.
i.e. when Oy
AK
---Oy ,
AK = o.Aox/Tra
(4)
which gives an estimate for the radius of the reversed plastic zone,
r = 2-~1~ ]
(5)
which depends only on the stress intensity factor range, AK, and not the mean value. Since this reversed plasticity at the crack tip will cause the material to stretch irreversibly on each application of a load cycle, the crack tip must advance through the material to accommodate this deformation. This type of advance leaves a series of marks on the eventual fracture surface, which can be clearly seen by an electron microscope, Fig.3. The spacing of these 'striations' is the amount by which the crack tip advances per cycle and correlates well with overall observed growth rates, d__a_a. dN Large volumes of data have been collected which support the view of fatigue crack propagation outlined above. Fig.2 generalises the type of relationship between growth rate and alternating stress intensity factor AK found for metallic materials. The bulk of the data can be represented by a power law of the type da = C(AK) m, C constant dN
(6)
where mean stress, microstructure and environment appear to have little effect and m is approximately 4. At very low levels of growth rate, where the amount of crack advance per cycle falls to the order of the atomic spacing "~ 5 X 10-Tmm, this continuum growth cannot occur across the whole of the
230
Direction of crack propagation Fig.3. Fatigue striations as seen in an electron microscope. crack front, causing the nett growth rate to drop very rapidly. The value of the applied stress intensity factor, below which no (measurable) amount of fatigue crack growth occurs, is termed the threshold stress intensity factor, AKTH , any particular value of which is however sensitive to the variables of mean stress, microstracture and environment. The implication for design is important since if in a cracked structure,
aox/-~ < AKTH
(7)
no crack extension by fatigue will occur which is a crucial requirement for components which experience a very large number of loading cycles in service.
Typical fatigue crack propagation properties Fig.4 illustrates values of fatigue crack propagation (FCP) data for mild steel. The scatter of about 4 or 5 times on growth rate for a given AK value is typical of the type of reproductibility which might be expected in FCP. It is worthwhile mentioning that the logarithmic scales on which da
~-~/AK
plots are presented tend to hide quite large scatter thus indicating the need for caution in deterministic FCP life
calculations! Fig.5 draws together data for widely differing metals, by emphasising the importance of strain in determining FCP rates; the parameter AK/E, E is Young's Modulus, may be (rather loosely) termed a strain intensity factor. Tables 2 and 3 give growth and threshold values for a wide variety of metallic materials under different conditions. For preliminary calculations (only!) this data may be summarised by da = 6.9 X 10 -12 (AK)3.°m/cycle for dN ferrite-pearlite steels, AK in MNm(-3a)
AKTH A 3 - 5 MNrn -3/2 for steels and 1 - 2 for aluminium & alloys
Fatigue crack propagation lives On substitution of the expression for the alternating SIF into the power law growth rate equation (6), we obtain d_.a_a= C(a~o,v/-~-)m dN Nf
_-foNf
dN =
1 ~"aF da c n m / : A o m J a ° t~m am/2
(8)
MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979,
,
iO-2
Fig.4. Fatigue crack growth rates for mild steel.
/
,/
/¢ I0 -2
~O-
Magnesium alloy (R=O)
D
I0°~ __
Mild steel i0-~
B
Titanium
da
I0-4 _
dR
(ram/cycle)
da dN
Aluminium
(ram/cycle) IO-s _
1o': -
io -4 _
"''--Copper io'6
_
I
~o ~, &K MN m 2
I00
I
iO-7 lO-S
10 4
iO-a
AK/E m'/2 Fig. 5. Crack growth rates for various materials compared on the basis of 'strain' intensity factor. where Nf is the n u m b e r o f cycles required to propagate the crack f r o m an initial length a o to a final length, a F. If the g e o m e t r y correction, a, is a strong f u n c t i o n o f crack length, then (8) is best integrated numerically. If however a is sensibly constant, we may write aQ(1 - m / 2 ) - a F ( 1 Nf = ~ m A a m z r m / 2 ( m / 2
- m/2) _ 1)
(9)
Experience shows that if a F is large
(10)
more complicated considerations ranging from the effect o f single overloads to completely r a n d o m loading, the reader's a t t e n t i o n is drawn to the list of suggested reading given at the end of the paper.
often proves to be useful for determining the effect of stress level on the FCP lives of a series o f similar cracked c o m p o n e n t s . To emphasise; we have assumed t h r o u g h o u t that the stress cycle is of constant amplitude. F o r
In principle therefore the prediction of FCP lives and hence for example, suitable inspection periods, suitable resolution of crack detection e q u i p m e n t or the effect of cracks i n t r o d u c e d during manufacture, is
c o m p a r e d with ao, its c o n t r i b u t i o n to (9) is o f t e n very small, hence the simplification:
Nf -
Constant Ao m
MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979
231
Table 2. Fatigue crack growth data for various materials
Material
Tensile strength (MN/m 2 )
0.1 or 0.2 per cent proof stress (MN/m 2)
R (= O min ) omax
m
AK for da/dN _- 10-~ mm/c
(MNm-S/2 )
195-255 95-125 180
0.06-0.74 0.64 0.07-0.43 0.54-0.76 0.75-0.92 0 -0.75 0.67 0.33-0.43 0.14-0.87 0.20-0.69
3.3 3.3 4.2 5.5 6.4 3.3 3.0 3.1 2.9 2.7
6.2 6.2 7.2 6.4 5.2 5.1 3.5 6.3 2.9 1.6
265
180
0.20-0.71
2.6
1.9
310
245-280
L71 Aluminium Alloy (4.5% Cu) L73 Aluminium Alloy (4.5% Cu) DTD 687A Aluminium AHoy (5.5% Zn)
480 435
415 } 370
0.25-0.43 0.50-0.78 0.14-0.46 0.50-0.88
3.9 4.1 3.7 4.4
2.6 2.15 2.4 2.1
540
495
ZW1 Magnesium Alloy (0.5% Zr) AM503 Magnesium AHoy (1.5% Mn)
250
165 i '
0.20-0.45 0.50-0.78 0.82-0.94 0
3.7 4.2 4.8 3.35
1.75 1.8 1.45 0.94
200
107
215-310 370 325
26-513
0.5 0.67 0.78 0.07--0.82 0.33-0.74 0 -0.33 0.51-0.72 0.08-0.94 0.17-0.86 0.28-0.71 0.81-0.94 0 -0.71 0 4) .67 0 -0.71
3.35 3.35 3.35 3.9 3.9 4.0 3.9 4.4 3.8 3.5 4.4 4.0 4.0 4.0
0.69 0.65 0.57 4.3 4.3 6.3 4.3 3.1 3.4 3.0 2.75 8.8 6.2 8.2
Mild steel Mild steel in brine* Cold rolled mild steel
Low alloy steel* Maraging steel* 18/8 Austenitic steel Aluminium 5% Mg-Aluminium AHoy HS30W Aluminium Alloy (1% Mg, 1% Si, 0.7% Mn) HS30WP Aluminium Alloy (1% Mg, 1% Si, 0.7% Mn)
Copper Phosphor bronze* 60/40 brass* Titanium 5% Titanium alloy 15% Mo Titanium ahoy Nickel* Monel* Iconel*
230
325 435 695
655
680 2010 665 125-155 310
555 835 1160 430 525 650
440 735 995
I'
*Data of limited accuracy obtained by an indirect method.
a straightforward m a t t e r using the approach just presented. In practice very careful consideration must be given to the difficulties o f correct stress analysis which have been mentioned in Part I; to the growth rate e q u a t i o n which can exhibit considerable scatter, the 'constants' are best determined in laboratory situations closely simulating those expected in service; and finally, to the evaluation of the initial crack length, a o. The direct experimental determination of crack lengths in c o m p o n e n t s is a study in itself for which readers are referred to the references given on non-destructive testing.
232
A practical example combining fatigue and fractu re An 8cm diameter e x t r u d e d rod of steel is machined into a hollow cylinder with a 7cm bore. If fluid is to be introduced into the bore and compressed repeated by a piston to a pressure of 100MN/m 2 , we are required to calculate h o w many cycles o f loading the cylinder can withstand if it is first p r o o f pressure tested to a pressure which corresponds to a m a x i m u m stress of 0.9 yield. A semi-circular flaw shape may be assumed. The relevant material properties are Klc = 30 MNM - a / 2 , yield stress, Oy = 1120 M N / m z and FCP rates typical of
a ferrite-pearlite steel. The p r o b l e m can be tackled in stages. First we require a stress analysis o f b o t h the c o m p o n e n t and the crack. In this case, Fig.6(a), the c o m p o n e n t is a cylinder with inner radius to wall thickness ratio o f 35 which is a little 5' too low for thin walled theory. However, if we use the mean wall radius in the thin walled t h e o r y equations we obtain an adequate a p p r o x i m a t i o n to the elastic stress distribution, O0 = p.R.R, of~ = p R t 2t
MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979
Table 3. Values of AKth for various materials
Material (Ferrous)
Tensile strength (MN/m 2
Mild steel
(=
430
-1 0.13 0.35 0A9 0.64 0.75 -1 0.23 0.33 -1 0.64
6.4 6.6 5.2 4.3 3.2 3.8 7.1 6.0 5.8 ~ 2.0 1.15
430
0.64
3.9
430 835 680
-1 -1 0 0.33 0.50 0.64 0.75 -1 0.23 0.33 0.64 0.67 -1 0 0.33 0.62 0.74 0 0.50
7.3 6.3 6.6 5.1 4.4 3.3 2.5 7.1 5.0 5.4 4.9 2.7 6.0 6.0 5.9 4.6 4.1 7.0 4.5
Mild steel at 300°C
480
Mild steel in brine
430
Mild steel in brine with cathodic protection Mild steel in tap water or SAE30 oil Low alloy steel
NiCrMoV steel at 300°C
Maraging steel 18/8 Austenitic steel
560
2010 685 665
Grey cast iron
255
Material (Non-ferrous)
R AKth omi._, n ) (MNm -3/2) - omax
Tensile strength (MN/m 2)
R (_ O min. - O~aax )
AKth (MNm-3/2 )
77
-1 0 0.33 0.53 -1 0 0.33 0.50 0.67 0.50
1.0 1.7 1.4 1.2 2.1 2.1 1.7 1.5 1.2 1.15
0 0.67 0 0.67 -1 0 0.33 0.60 -1 0 0.33 0.57 0.71 -1 0 0.33 0.50 0.67 -1 0 0.57 0.71
0.83 0.66 0.99 0.77 2.7 2.5 1.8 2.2 5.9 7.9 6.5 5.2 3.6 5.6 7.0 6.5 5.2 3.6 6.4 7.1 4.7 4.0
Aluminium
L65 Aluminium alloy (4.5% Cu)
L65 Aluminium alloy (4.5% Cu) in brine ZW1 Magnesium alloy (0.6% Zr) AM503 Magnesium alloy (1.6% Mn) Copper
450 495
250 165 225 215
Titanium Nickel
540 455 430
Monel
525
Iconel
655 650
PRESSURE,
YIELD
_ _
17"--
_
P
K, = 1.26% gaPo
I I I I
oF
°o
CRACK
LENGTH, a
Fig.6. (a) Thin walled cylinder-stress analysis. (b) Relationship between crack length and failure pressure for cracked cylinder.
MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979
233
Given that the flaw shape is semicircular, the worst orientation will be normal to the h o o p stress c o m p o n e n t , o 0. F r o m Part I, A p p e n d i x 2, the stress intensity factor for this shape of flaw is: Kl=
1.120 (rta) ½
=f~/2-t d~b = n'/2 for a semi-circle
p r o o f pressure Po, see Fig.6(b). Now under the action of the pulsating pressures we assume the vessel sees at the worst a cycle of 0 to 100 M N / m 2. Any flaw will therefore grow until its m a x i m u m stress intensity factor reaches the fracture toughness, Kxc, of the material. Hence, by putting p = 100 M N / m 2 into Equation (12), we calculate a crack length for failure as a F = 1.0mm. The FCP life can n o w be obtained by putting the SIF E q u a t i o n ( 1 1 ) i n t o the expression for the growth rate,
Therefore Kl = 1.26o0x/~" and o 0 = p_RR= 7.5p t
(ll)
with t as thickness, 5mm and R as mean radius, 37.5mm. We assume that the flaw shape remains semi-circular and the critical dimension is the depth, a, into the wall thickness. We may further note that this t r e a t m e n t is simplified for the purposes of illustration, because we have failed to account for the effect o f the fluid entering the crack, the crack depth/wall thickness ratio and the curvature o f the vessel walls. We n o w calculate the pressure at which vessel will first yield using the simple Tresca (max. shear stress) criterion, hence putting o0=oyi.e.7.5p=
1120
i.e. da = 6.9 X l0 -12 (1.26 X 7.5 dN Ap x/~) 3 m/cycle which yields the life
1 N f - 5.82 × 10 -3
{1 2 ("~o) - a o
= 3,800 cycles Because of our assumption that a o corresponds to the p r o o f pressure, our life is guaranteed, in the sense that in reality any initial defects are likely to be smaller than a o and would give correspondingly longer lives. As a check we ought to calculate the size of the plastic zone corresponding to the fracture pressure. Recalling E q u a t i o n (7), Part 1, the plastic zone radius is of the order r=l
~Klc~2 ~ 0.1mm
p = 150 M N / m 2 for yield. The relationship b e t w e e n crack depths a and failure pressures is obtained by, K = K l c i.e. 1.26 X 7.5px/~ = 3 0 ( 1 2 ) and plotted on the curved line on Fig.6(b), truncated by the constant yield pressure at small crack length values. By putting the p r o o f pressure, Po of 0.9 X 150 M N / m 2 into E q u a t i o n (12), it is seen that the vessel would fail by rapid fracture if a defect of depth, a ° = 1.26 X 7.5 × 135 = 0.55ram existed in the vessel wall. Given that the vessel survived the p r o o f loading, we are certain that no defect greater than this size existed, therefore we make the pessimistic assumption that the vessel only just survived, and set the initial crack length in our life calculation to ao, corresponding to
234
which is about 1/10 of the final crack size and 1/40 of the remaining ligam e n t size. Given that the biaxial stress state in the cylinder wall will somewhat restrict this plastic d e f o r m a t i o n , we are justified in using L.E.F.M. In practice, a major aim in the design of pressurised c o m p o n e n t s is to arrange for the critical crack length to be greater than the wall thickness. The vessel then fails when the crack reaches the outside wall in a stable 'leaking' manner rather than the catastrophic 'break'. This is an area of current research when studies o f elastic-plastic fracture mechanics are proving useful. It is hoped that this simplified treatment will afford readers an appreciation o f the p o w e r of fracture mechanics methods. Obvious industrial applications embrace a wide field. For example, in preliminary design the pressures of product liability legislation and the public 'right to k n o w ' have made quantitative assessments of the probable life and safety of structures of p a r a m o u n t importance. Always in the aircraft industry, and increasingly as a fuel conservation measure in the a u t o m o t i v e industry, weight saving has
lead to limited life replaceable components. Fracture mechanics m e t h o d s have enabled inspection and replacem e n t intervals to be calculated on a rational basis. The North Sea oil exploration has led to increasing attention being paid to the effect of random loads (waves) and corrosion (seawater) on large cracked (welded) structures. And finally, when the worst has sometimes c o m e to the worst, fracture mechanics techniques have proved to be extremely valuable in failure and accident investigation. Details of all these varied activities can be found in the cited reference. Summary 1. Fracture Mechanics methods can only be applied to cracked components. 2. The plane strain fracture toughness, Klc, of a material is a measure of that material's minimum (with respect to thickness) resistance to catastrophic (rapid) crack advance. 3. The cracking effort applied to a component is measured by the stress intensity factor, KI = (2ON/~. K1 must be less than the fracture toughness of the material in which the crack is contained if catastrophic crack advance is to be avoided. i.e.
K1 ( K l c
4. Fatigue crack growth resulting from
constant amplitude cyclic loads can be conveniently described by a relationship of the type da = C (/~K)m dN where C and m are material constants, m being typically in the order of 3--4 for metals. 5. Crack propagation lives can be obtained
by integrating the above equation between appropriate crack length limits. 6. A threshoM stress intensity factor range, Z~KTH, exists, below which no fatigue crack growth takes place. To avoid the possibility of fatigue crack growth therefore, the applied stress intensity factor must be less than the threshold. i.e. K 1 ( Z~KTH. Further reading Details of Experimental Technique Methods for Plane Strain Fracture Toughness (K zc) Testing, DD3, British Standards Institution, L o n d o n , 1971 Tentative Method of Test for ConstantLoad-Amplitude Fatigue Crack G r o w t h Rates above 10 - a m / c y c l e , ASTM E 6 4 7 - 7 8 T , Philadelphia, Pa,, 1978 Materials Data Damage Tolerant Design H a n d b o o k ,
MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979
Metals and Ceramics Information Centre, Battelle, Columbus, Ohio, 1975 Matthews, W. T. Plane Strain Fracture Toughness (Klc) Data Handbook for Metals, AMMRC MS 73-6, US Army Materials and Mechanics Research Centre, Watertown, Mass., 1973 Hudson, C. M. and Seward, S. K. Compendium of Sources of Fracture Toughness and Fatigue Crack Growth Data for Metallic Alloys, Int. J. Fracture, 14, 1978,4, pp R151-R184. Waterman, N. A. (Ed.) Fulmer Materials Optimizer, Fulmer Research Institute, Slough, 1974
Non-Destructive Testing
Birchon, D. Non-Destructive Testing, Eng. Design Guides, 09, O.U.P., 1975 McMaster, R. C. (Ed.) Non-Destructive Testing Handbook (2 Vols), Ronald Press, New York, 1959
Applications Rich, T. P. and Cartwright, D. J. (Eds.) Case studies in Fracture Mechanics, AMMRC MS 77-5, US Army Materials and Mechanics Research Centre, Watertown, Mass., 1975 Stewart, A. T. Fatigue Cracking in the Rotor of a 500-MW Alternator, CEGB
Research, 8, 1978, pp 37-44 The Markham Colliery Disaster, Case Study T351 8A-9A, Open University Milton Keynes, 1976 Stanley, P. (Ed.) Fracture Mechanics in Engineering Practice, Applied Science, London, 1977 Practical Application of Fracture Mechanics to Pressure-Vessel Technology, Inst. Mech. Engs., London, 1971 Tolerance of Flaws in Pressurised Components, Inst. Mech. Engs. Conference Publication 1978-10, London, 1978.
New high temperature insulating felts A range of refractory mixed-fibre felts for temperatures up to 1600°C is now being manufactured and marketed by Morganite Ceramic Fibres. The new materials, known as Unifelt, consist of vacuum-formed sheets, and are made from intimate blends of Morganite's own Triton Kaowool aluminosilicate fibres and ICI's Saffil alumina fibres with the addition of a flexible organic binding-medium. The presence of the binder is said to give the felts outstanding 'handleability' and resilience, as well as eliminating dust almost completely. Unifelt materials can be bent, cut, compressed, glued to themselves or cemented to other materials to form a variety of seals, joints and blocks.
Further information from: MORGANITE CERAMIC FIBRES LTD. Tebay Road, Bromborough, Wirral, Merseyside L62 3PH, England Telephone: 051-334 4030 Telex: 627846
or
MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979
MORGANITE CERAMIC FIBRES S.A. Parc Industriel des Hauts-Sarts B-4400 Herstal, Li6ge, Belgium Telephone: 646450 Telex: 41805 or main national d i s t r i b u t o r s - see list enclosed.
235
This article continues the discussion o f the Stress Intensity Factor parameter by demonstrating how it can be used to characterise catastrophic (rapid) fracture and stable fatigue crack growth. Numerical values for the fracture toughness, crack growth 'constants' and threshold Stress Intensities are given for a wide range o f engineering materials. The principles o f the Fracture Mechanics approach to failure prevention are illustrated by worked examples. Introduction Part I of this series (Stresses due to notches and cracks, Mats. in Eng. Appl. 1, 2, pp 121-128, 1978) described how the stress fields near crack tips can be quantified by the stress intensity factor (S.I.F.) parameter, ao~, where o is the remote stress, a is the crack length and a, a geometrical factor which accounts for the shape of the body containing the crack and the type of loading to which it is subjected. The power and utility of this parameter is that it enables local conditions near the crack tip to be described by easily measured quantities remote from the complicated crack tip area - it allows us to step back from the fine detail at the crack tip. The S.I.F. as a fracture criterion The first useful engineering application of the S.I.F., as a fracture criterion, stems from this very ability. Imagine a component containing a crack of fixed length whilst the load on the
structure is continuously increased. The local stresses (and, hence, strains and displacements) near the crack tip will also increase, scaled in magnitude by the S.I.F. We make the hypothesis that rapid, unstable crack advance will occur when the local stresses reach some critical value, which will be constant for a given material. It immediately follows that these critical local conditions will correspond to a critical value of the S.I.F., which we can easily quantify and then use as a measure of a material's resistance to rapid crack advance; we call this resistance the material's fracture
toughness. Measurement o f fracture toughness This hypothesis has been tested experimentally and found to provide a consistent fracture criterion, providing the limitations of linear elastic fracture mechanics (LEFM), which were introduced in Part I, are not violated. These limitations stemmed from the need to restrict the volume of the plastically deformed material at the crack tip, so that the behaviour of the surrounding elastic region controlled the crack tip deformation behaviour. Within these conditions, the fracture behaviour can be satisfactorily quantified by the S.I.F. The experimental measurements are usually performed on one of the ASTM standard specimens, the single edge notch (SEN) because it is easy to load in bending or the compact tensile specimens (CTS) because it allows high
M A T E R I A L S IN E N G I N E E R I N G APPLICATIONS, Vol. 1, June 1979
values of SIF to be attained for relatively short crack lengths and is thus economical in the use of material. Both these specimen types have known SIF calibrations, see Part I, Appendix 2 for details. A crack is artificially introduced into the specimen by machining a slot which is further sharpened by fatigue loading under low alternating loads. A slowly increasing static load is then applied to the specimen until catastrophic fracture occurs. A tentative fracture toughness value can now be calculated by employing the measured values of crack length and fracture load in the SIF calibration equation. Checks must then be made to ensure that the limitations of LEFM have not been violated - in particular the thickness of the specimen must be greater than a value
which was introduced as Equation 11 in Part I. Additionally, to check that no extensive plasticity was introduced during the fatigue precracking process, the maximum SIF of the fatigue loading must be shown to be less than 0.67 of the tentative toughness value. If these conditions are fulfilled the tentative toughness value is confirmed as the plane strain fracture toughness of the material and designated, Klc. Further experimental details and more formal statements of the validity requirements can be found in the
227
Table 1. Typical values of plane strain fracture toughness, Klc
Material Steels Medium carbon Pressure vessel (ASTM A533B Q+T) High strength alloy Maraging steel AFC 77 Stainless
Modulus E, (MN/m 2)
2.1
X
Yield Stress Oy, (MN/m 2)
Toughness Klc, (MNm-3/2)
2.6 X 102
54
l0 s
Thickness 2.5~K1el 2 toy i mm
110
X 102 X 10~ X 102 X 102
208 98 76 83
487 11 4.4 7.9
4.2 X 102 5.4 X 102 5.6 X 102
27 30 23
10.4 7.9 4.2
1.08 X l0 s
10.6 x 102 11 x 102
73 38
12.6 3.1
~ 4 X 104 1 X l0 s 3 x 103
80 3 X 102 30
0.2-1.4 13 1
4.7 2.8
4.7 14.6 18 15.3
Aluminmmalloys
2024 T8 7075 T6 7178 T6
7.2
X
104
Titanium alloys
Ti-6A~--4V (High Yield)
For comparison
Concrete WC-Co composites PMMA
Note: Representative values only - not to be employed as design data.
standard test methods referenced at the end of this paper. Typical values of Klc for a variety of materials are presented in Table 1, together with values for the yield stress, (ry, thus enabling the thickness requirement (Equation 1) to be calculated. To establish orders of magnitude, we note that 100 MNm -3/2 is a reasonable K~c for steels, somewhat less for aluminium alloys, and that in general a trade-off occurs between values of yield stress and Klc for similar classes of metallic materials. It can be seen that for relatively low toughness, high yield materials the required thicknesses are modest, of the order of several mm, but for high toughness, low yield materials, where the thickness requirement may approach as much as 0.5m, both massive specimens and testing facilities would be required. This, coupled with the fact that such materials might not be used in such thick sections, is the reason for the current interest in elastic/plastic fracture mechanics which will be further discussed in the third and final part of this series. Use of the Klc fracture criterion If we know the value of the fracture
228
toughness of the material from which a component containing a crack is made, we can ensure its survival if the applied SIF does not exceed Klc i.e. if a o V / - ~ < Klc
(2)
It is worth emphasising the physical origin of each side of the above inequality. The S.I.F. on the left hand side is a mechanics parameter, its value determined by the geometry, stress level and crack length in the component, and is a measure of the cracking effort being applied to the c o m ponent, whilst the right hand side is a material constant, a measure of the particular material's ability to resist rapid crack advance. A worthwhile analogy might be drawn with the criterion o < Oy
(3)
to avoid yield in simple tension. Here the applied stress o is a measure of the stretching effort being applied and the yield stress, Oy, is a material constant, a measure of the material's ability to resist stretching without permanent deformation. When an equality is inserted into
Equation (2), it is a straightforward task to calculate those combinations of crack length and stress level which will cause failure by rapid crack advance. We now illustrate this approach by a simple example: A tool similar to that shown in Fig.1 is used to dig up old roads. Its cross section is a rectangle of thickness 25mm. If a through thickness crack of length 5mm is found in the position shown, what would be the failure load? We first check if the LEFM approach would be suitable. For the values given the thickness criterion, Equ 1, leads to B>2.5
~
59 2
i.e. B > 3 . 9 X
10-3m
which is easily satisfied by the quoted thickness of 25mm. The cracked section is acted upon by both end load, P and a moment, M = P X 170 X 10-3Nm. Both contribute to the SIF, illustrating the important principle of superposition. For a crack length/ width ratio, a/w, of 5/40 i.e. 0.125, we obtain, by interpolation of the values given for beams in bending and tension in Part I, Appendix 2, f(a/w) bending = 1.03 f(a/w) tension = 1.27
MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979
hence, the combined SIF becomes:
Fig.l. Cracked tool: configuration and dimensions
6M V/'~. 1.03 Kz
= BW 2
P X/'~. BW
+
1.27
25ram
which reduces to
?.
K1 = P~/Tr5 X 10 -3 . (2.75 X 104 ) MNm-3/2
0
Setting Kl = Klc for fracture, p=
59 0.125 X 2.75 X 104- =
0.0172 MN or 17kN, rounded to the accuracy of the information given. We must now avoid being lulled into a false sense of security by the apparent ease of the above approach. Unfortunately, fracture avoidance, even where LEFM applies, is made difficult in practice by the problems of identification of the shape and length of any defects, by difficulties of the correct stress analysis of complex components and by the fact that cracks can extend in a sub-critical manner, which may mean that a component initially thought safe from fracture problems, might become dangerous after a period of service. It is to this last problem that we now turn; fortunately a modified form of the SIF turns out to be useful in the sub-critical crack growth area.
Sub-criticalcrackgrowth - mainlyby fatigue If a cracked component is subjected to varying loads, to a corrosive environment or to steady loads and a high temperature, crack extension can occur by fatigue, stress corrosion cracking or creep processes respectively. This kind of crack extension is termed stable in that crack length increases with applied cycles or time until the crack length• load combination reaches the critical value for catastrophic instability previously discussed. The useful life of a cracked component is therefore determined by the amount of allowable stable crack extension which can take place. Of these types of loading, fatigue is most common, certainly most understood, and has proved to be most amenable to quantification by fracture mechanics parameters. An idealised type of constant amplitude fatigue loading is shown in Fig.2. The applied stress cycles about a mean stress level, Omean , between maximum, Omax, and minimum, ffmin, stress levels. Recalling the
150 m m KIC=S9MN m- : ~ ¢~y = I S O O M N / m
Stress ~
" 2
&(~ = (~mo~-~min.)
I
~,n.
Jo'm,on
time
Fracture or Plastic Instability
tO-t
i 0 -=
10°3 Growth Rate,
Stable growth
do =C [ZIK)m dN m~4
,dOdN(mm/cycle]0-4
iO-S
iO-6 I
I
AKTH IO
!
I
50
IOO
=
Stress intensity factor, &K MN nT~2 Fig.2. Idealiscd relationship between growth rate and stress intensity factor range.
MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979
229
method of our calculation (Part I) of the plastic zone size ahead of a crack tip, we truncated the supposed elastic stress distribution ahead of the crack,
Ka
~ ,
at the value of the materials
yield stress Oy. Now by assuming an elastic unloading from this constant value, the effect of stress decrease from Omax to amin, can be estimated by applying a negative stress distribution due to the range, Ao, (=Oma x --Omin). The material will yield in
compression therefore when the sum of the stress distributions drops to a value
of-Oy.
i.e. when Oy
AK
---Oy ,
AK = o.Aox/Tra
(4)
which gives an estimate for the radius of the reversed plastic zone,
r = 2-~1~ ]
(5)
which depends only on the stress intensity factor range, AK, and not the mean value. Since this reversed plasticity at the crack tip will cause the material to stretch irreversibly on each application of a load cycle, the crack tip must advance through the material to accommodate this deformation. This type of advance leaves a series of marks on the eventual fracture surface, which can be clearly seen by an electron microscope, Fig.3. The spacing of these 'striations' is the amount by which the crack tip advances per cycle and correlates well with overall observed growth rates, d__a_a. dN Large volumes of data have been collected which support the view of fatigue crack propagation outlined above. Fig.2 generalises the type of relationship between growth rate and alternating stress intensity factor AK found for metallic materials. The bulk of the data can be represented by a power law of the type da = C(AK) m, C constant dN
(6)
where mean stress, microstructure and environment appear to have little effect and m is approximately 4. At very low levels of growth rate, where the amount of crack advance per cycle falls to the order of the atomic spacing "~ 5 X 10-Tmm, this continuum growth cannot occur across the whole of the
230
Direction of crack propagation Fig.3. Fatigue striations as seen in an electron microscope. crack front, causing the nett growth rate to drop very rapidly. The value of the applied stress intensity factor, below which no (measurable) amount of fatigue crack growth occurs, is termed the threshold stress intensity factor, AKTH , any particular value of which is however sensitive to the variables of mean stress, microstracture and environment. The implication for design is important since if in a cracked structure,
aox/-~ < AKTH
(7)
no crack extension by fatigue will occur which is a crucial requirement for components which experience a very large number of loading cycles in service.
Typical fatigue crack propagation properties Fig.4 illustrates values of fatigue crack propagation (FCP) data for mild steel. The scatter of about 4 or 5 times on growth rate for a given AK value is typical of the type of reproductibility which might be expected in FCP. It is worthwhile mentioning that the logarithmic scales on which da
~-~/AK
plots are presented tend to hide quite large scatter thus indicating the need for caution in deterministic FCP life
calculations! Fig.5 draws together data for widely differing metals, by emphasising the importance of strain in determining FCP rates; the parameter AK/E, E is Young's Modulus, may be (rather loosely) termed a strain intensity factor. Tables 2 and 3 give growth and threshold values for a wide variety of metallic materials under different conditions. For preliminary calculations (only!) this data may be summarised by da = 6.9 X 10 -12 (AK)3.°m/cycle for dN ferrite-pearlite steels, AK in MNm(-3a)
AKTH A 3 - 5 MNrn -3/2 for steels and 1 - 2 for aluminium & alloys
Fatigue crack propagation lives On substitution of the expression for the alternating SIF into the power law growth rate equation (6), we obtain d_.a_a= C(a~o,v/-~-)m dN Nf
_-foNf
dN =
1 ~"aF da c n m / : A o m J a ° t~m am/2
(8)
MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979,
,
iO-2
Fig.4. Fatigue crack growth rates for mild steel.
/
,/
/¢ I0 -2
~O-
Magnesium alloy (R=O)
D
I0°~ __
Mild steel i0-~
B
Titanium
da
I0-4 _
dR
(ram/cycle)
da dN
Aluminium
(ram/cycle) IO-s _
1o': -
io -4 _
"''--Copper io'6
_
I
~o ~, &K MN m 2
I00
I
iO-7 lO-S
10 4
iO-a
AK/E m'/2 Fig. 5. Crack growth rates for various materials compared on the basis of 'strain' intensity factor. where Nf is the n u m b e r o f cycles required to propagate the crack f r o m an initial length a o to a final length, a F. If the g e o m e t r y correction, a, is a strong f u n c t i o n o f crack length, then (8) is best integrated numerically. If however a is sensibly constant, we may write aQ(1 - m / 2 ) - a F ( 1 Nf = ~ m A a m z r m / 2 ( m / 2
- m/2) _ 1)
(9)
Experience shows that if a F is large
(10)
more complicated considerations ranging from the effect o f single overloads to completely r a n d o m loading, the reader's a t t e n t i o n is drawn to the list of suggested reading given at the end of the paper.
often proves to be useful for determining the effect of stress level on the FCP lives of a series o f similar cracked c o m p o n e n t s . To emphasise; we have assumed t h r o u g h o u t that the stress cycle is of constant amplitude. F o r
In principle therefore the prediction of FCP lives and hence for example, suitable inspection periods, suitable resolution of crack detection e q u i p m e n t or the effect of cracks i n t r o d u c e d during manufacture, is
c o m p a r e d with ao, its c o n t r i b u t i o n to (9) is o f t e n very small, hence the simplification:
Nf -
Constant Ao m
MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979
231
Table 2. Fatigue crack growth data for various materials
Material
Tensile strength (MN/m 2 )
0.1 or 0.2 per cent proof stress (MN/m 2)
R (= O min ) omax
m
AK for da/dN _- 10-~ mm/c
(MNm-S/2 )
195-255 95-125 180
0.06-0.74 0.64 0.07-0.43 0.54-0.76 0.75-0.92 0 -0.75 0.67 0.33-0.43 0.14-0.87 0.20-0.69
3.3 3.3 4.2 5.5 6.4 3.3 3.0 3.1 2.9 2.7
6.2 6.2 7.2 6.4 5.2 5.1 3.5 6.3 2.9 1.6
265
180
0.20-0.71
2.6
1.9
310
245-280
L71 Aluminium Alloy (4.5% Cu) L73 Aluminium Alloy (4.5% Cu) DTD 687A Aluminium AHoy (5.5% Zn)
480 435
415 } 370
0.25-0.43 0.50-0.78 0.14-0.46 0.50-0.88
3.9 4.1 3.7 4.4
2.6 2.15 2.4 2.1
540
495
ZW1 Magnesium Alloy (0.5% Zr) AM503 Magnesium AHoy (1.5% Mn)
250
165 i '
0.20-0.45 0.50-0.78 0.82-0.94 0
3.7 4.2 4.8 3.35
1.75 1.8 1.45 0.94
200
107
215-310 370 325
26-513
0.5 0.67 0.78 0.07--0.82 0.33-0.74 0 -0.33 0.51-0.72 0.08-0.94 0.17-0.86 0.28-0.71 0.81-0.94 0 -0.71 0 4) .67 0 -0.71
3.35 3.35 3.35 3.9 3.9 4.0 3.9 4.4 3.8 3.5 4.4 4.0 4.0 4.0
0.69 0.65 0.57 4.3 4.3 6.3 4.3 3.1 3.4 3.0 2.75 8.8 6.2 8.2
Mild steel Mild steel in brine* Cold rolled mild steel
Low alloy steel* Maraging steel* 18/8 Austenitic steel Aluminium 5% Mg-Aluminium AHoy HS30W Aluminium Alloy (1% Mg, 1% Si, 0.7% Mn) HS30WP Aluminium Alloy (1% Mg, 1% Si, 0.7% Mn)
Copper Phosphor bronze* 60/40 brass* Titanium 5% Titanium alloy 15% Mo Titanium ahoy Nickel* Monel* Iconel*
230
325 435 695
655
680 2010 665 125-155 310
555 835 1160 430 525 650
440 735 995
I'
*Data of limited accuracy obtained by an indirect method.
a straightforward m a t t e r using the approach just presented. In practice very careful consideration must be given to the difficulties o f correct stress analysis which have been mentioned in Part I; to the growth rate e q u a t i o n which can exhibit considerable scatter, the 'constants' are best determined in laboratory situations closely simulating those expected in service; and finally, to the evaluation of the initial crack length, a o. The direct experimental determination of crack lengths in c o m p o n e n t s is a study in itself for which readers are referred to the references given on non-destructive testing.
232
A practical example combining fatigue and fractu re An 8cm diameter e x t r u d e d rod of steel is machined into a hollow cylinder with a 7cm bore. If fluid is to be introduced into the bore and compressed repeated by a piston to a pressure of 100MN/m 2 , we are required to calculate h o w many cycles o f loading the cylinder can withstand if it is first p r o o f pressure tested to a pressure which corresponds to a m a x i m u m stress of 0.9 yield. A semi-circular flaw shape may be assumed. The relevant material properties are Klc = 30 MNM - a / 2 , yield stress, Oy = 1120 M N / m z and FCP rates typical of
a ferrite-pearlite steel. The p r o b l e m can be tackled in stages. First we require a stress analysis o f b o t h the c o m p o n e n t and the crack. In this case, Fig.6(a), the c o m p o n e n t is a cylinder with inner radius to wall thickness ratio o f 35 which is a little 5' too low for thin walled theory. However, if we use the mean wall radius in the thin walled t h e o r y equations we obtain an adequate a p p r o x i m a t i o n to the elastic stress distribution, O0 = p.R.R, of~ = p R t 2t
MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979
Table 3. Values of AKth for various materials
Material (Ferrous)
Tensile strength (MN/m 2
Mild steel
(=
430
-1 0.13 0.35 0A9 0.64 0.75 -1 0.23 0.33 -1 0.64
6.4 6.6 5.2 4.3 3.2 3.8 7.1 6.0 5.8 ~ 2.0 1.15
430
0.64
3.9
430 835 680
-1 -1 0 0.33 0.50 0.64 0.75 -1 0.23 0.33 0.64 0.67 -1 0 0.33 0.62 0.74 0 0.50
7.3 6.3 6.6 5.1 4.4 3.3 2.5 7.1 5.0 5.4 4.9 2.7 6.0 6.0 5.9 4.6 4.1 7.0 4.5
Mild steel at 300°C
480
Mild steel in brine
430
Mild steel in brine with cathodic protection Mild steel in tap water or SAE30 oil Low alloy steel
NiCrMoV steel at 300°C
Maraging steel 18/8 Austenitic steel
560
2010 685 665
Grey cast iron
255
Material (Non-ferrous)
R AKth omi._, n ) (MNm -3/2) - omax
Tensile strength (MN/m 2)
R (_ O min. - O~aax )
AKth (MNm-3/2 )
77
-1 0 0.33 0.53 -1 0 0.33 0.50 0.67 0.50
1.0 1.7 1.4 1.2 2.1 2.1 1.7 1.5 1.2 1.15
0 0.67 0 0.67 -1 0 0.33 0.60 -1 0 0.33 0.57 0.71 -1 0 0.33 0.50 0.67 -1 0 0.57 0.71
0.83 0.66 0.99 0.77 2.7 2.5 1.8 2.2 5.9 7.9 6.5 5.2 3.6 5.6 7.0 6.5 5.2 3.6 6.4 7.1 4.7 4.0
Aluminium
L65 Aluminium alloy (4.5% Cu)
L65 Aluminium alloy (4.5% Cu) in brine ZW1 Magnesium alloy (0.6% Zr) AM503 Magnesium alloy (1.6% Mn) Copper
450 495
250 165 225 215
Titanium Nickel
540 455 430
Monel
525
Iconel
655 650
PRESSURE,
YIELD
_ _
17"--
_
P
K, = 1.26% gaPo
I I I I
oF
°o
CRACK
LENGTH, a
Fig.6. (a) Thin walled cylinder-stress analysis. (b) Relationship between crack length and failure pressure for cracked cylinder.
MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979
233
Given that the flaw shape is semicircular, the worst orientation will be normal to the h o o p stress c o m p o n e n t , o 0. F r o m Part I, A p p e n d i x 2, the stress intensity factor for this shape of flaw is: Kl=
1.120 (rta) ½
=f~/2-t d~b = n'/2 for a semi-circle
p r o o f pressure Po, see Fig.6(b). Now under the action of the pulsating pressures we assume the vessel sees at the worst a cycle of 0 to 100 M N / m 2. Any flaw will therefore grow until its m a x i m u m stress intensity factor reaches the fracture toughness, Kxc, of the material. Hence, by putting p = 100 M N / m 2 into Equation (12), we calculate a crack length for failure as a F = 1.0mm. The FCP life can n o w be obtained by putting the SIF E q u a t i o n ( 1 1 ) i n t o the expression for the growth rate,
Therefore Kl = 1.26o0x/~" and o 0 = p_RR= 7.5p t
(ll)
with t as thickness, 5mm and R as mean radius, 37.5mm. We assume that the flaw shape remains semi-circular and the critical dimension is the depth, a, into the wall thickness. We may further note that this t r e a t m e n t is simplified for the purposes of illustration, because we have failed to account for the effect o f the fluid entering the crack, the crack depth/wall thickness ratio and the curvature o f the vessel walls. We n o w calculate the pressure at which vessel will first yield using the simple Tresca (max. shear stress) criterion, hence putting o0=oyi.e.7.5p=
1120
i.e. da = 6.9 X l0 -12 (1.26 X 7.5 dN Ap x/~) 3 m/cycle which yields the life
1 N f - 5.82 × 10 -3
{1 2 ("~o) - a o
= 3,800 cycles Because of our assumption that a o corresponds to the p r o o f pressure, our life is guaranteed, in the sense that in reality any initial defects are likely to be smaller than a o and would give correspondingly longer lives. As a check we ought to calculate the size of the plastic zone corresponding to the fracture pressure. Recalling E q u a t i o n (7), Part 1, the plastic zone radius is of the order r=l
~Klc~2 ~ 0.1mm
p = 150 M N / m 2 for yield. The relationship b e t w e e n crack depths a and failure pressures is obtained by, K = K l c i.e. 1.26 X 7.5px/~ = 3 0 ( 1 2 ) and plotted on the curved line on Fig.6(b), truncated by the constant yield pressure at small crack length values. By putting the p r o o f pressure, Po of 0.9 X 150 M N / m 2 into E q u a t i o n (12), it is seen that the vessel would fail by rapid fracture if a defect of depth, a ° = 1.26 X 7.5 × 135 = 0.55ram existed in the vessel wall. Given that the vessel survived the p r o o f loading, we are certain that no defect greater than this size existed, therefore we make the pessimistic assumption that the vessel only just survived, and set the initial crack length in our life calculation to ao, corresponding to
234
which is about 1/10 of the final crack size and 1/40 of the remaining ligam e n t size. Given that the biaxial stress state in the cylinder wall will somewhat restrict this plastic d e f o r m a t i o n , we are justified in using L.E.F.M. In practice, a major aim in the design of pressurised c o m p o n e n t s is to arrange for the critical crack length to be greater than the wall thickness. The vessel then fails when the crack reaches the outside wall in a stable 'leaking' manner rather than the catastrophic 'break'. This is an area of current research when studies o f elastic-plastic fracture mechanics are proving useful. It is hoped that this simplified treatment will afford readers an appreciation o f the p o w e r of fracture mechanics methods. Obvious industrial applications embrace a wide field. For example, in preliminary design the pressures of product liability legislation and the public 'right to k n o w ' have made quantitative assessments of the probable life and safety of structures of p a r a m o u n t importance. Always in the aircraft industry, and increasingly as a fuel conservation measure in the a u t o m o t i v e industry, weight saving has
lead to limited life replaceable components. Fracture mechanics m e t h o d s have enabled inspection and replacem e n t intervals to be calculated on a rational basis. The North Sea oil exploration has led to increasing attention being paid to the effect of random loads (waves) and corrosion (seawater) on large cracked (welded) structures. And finally, when the worst has sometimes c o m e to the worst, fracture mechanics techniques have proved to be extremely valuable in failure and accident investigation. Details of all these varied activities can be found in the cited reference. Summary 1. Fracture Mechanics methods can only be applied to cracked components. 2. The plane strain fracture toughness, Klc, of a material is a measure of that material's minimum (with respect to thickness) resistance to catastrophic (rapid) crack advance. 3. The cracking effort applied to a component is measured by the stress intensity factor, KI = (2ON/~. K1 must be less than the fracture toughness of the material in which the crack is contained if catastrophic crack advance is to be avoided. i.e.
K1 ( K l c
4. Fatigue crack growth resulting from
constant amplitude cyclic loads can be conveniently described by a relationship of the type da = C (/~K)m dN where C and m are material constants, m being typically in the order of 3--4 for metals. 5. Crack propagation lives can be obtained
by integrating the above equation between appropriate crack length limits. 6. A threshoM stress intensity factor range, Z~KTH, exists, below which no fatigue crack growth takes place. To avoid the possibility of fatigue crack growth therefore, the applied stress intensity factor must be less than the threshold. i.e. K 1 ( Z~KTH. Further reading Details of Experimental Technique Methods for Plane Strain Fracture Toughness (K zc) Testing, DD3, British Standards Institution, L o n d o n , 1971 Tentative Method of Test for ConstantLoad-Amplitude Fatigue Crack G r o w t h Rates above 10 - a m / c y c l e , ASTM E 6 4 7 - 7 8 T , Philadelphia, Pa,, 1978 Materials Data Damage Tolerant Design H a n d b o o k ,
MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979
Metals and Ceramics Information Centre, Battelle, Columbus, Ohio, 1975 Matthews, W. T. Plane Strain Fracture Toughness (Klc) Data Handbook for Metals, AMMRC MS 73-6, US Army Materials and Mechanics Research Centre, Watertown, Mass., 1973 Hudson, C. M. and Seward, S. K. Compendium of Sources of Fracture Toughness and Fatigue Crack Growth Data for Metallic Alloys, Int. J. Fracture, 14, 1978,4, pp R151-R184. Waterman, N. A. (Ed.) Fulmer Materials Optimizer, Fulmer Research Institute, Slough, 1974
Non-Destructive Testing
Birchon, D. Non-Destructive Testing, Eng. Design Guides, 09, O.U.P., 1975 McMaster, R. C. (Ed.) Non-Destructive Testing Handbook (2 Vols), Ronald Press, New York, 1959
Applications Rich, T. P. and Cartwright, D. J. (Eds.) Case studies in Fracture Mechanics, AMMRC MS 77-5, US Army Materials and Mechanics Research Centre, Watertown, Mass., 1975 Stewart, A. T. Fatigue Cracking in the Rotor of a 500-MW Alternator, CEGB
Research, 8, 1978, pp 37-44 The Markham Colliery Disaster, Case Study T351 8A-9A, Open University Milton Keynes, 1976 Stanley, P. (Ed.) Fracture Mechanics in Engineering Practice, Applied Science, London, 1977 Practical Application of Fracture Mechanics to Pressure-Vessel Technology, Inst. Mech. Engs., London, 1971 Tolerance of Flaws in Pressurised Components, Inst. Mech. Engs. Conference Publication 1978-10, London, 1978.
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MATERIALS IN ENGINEERING APPLICATIONS, Vol. 1, June 1979
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