Fatigue crack growth in flyash compacts

POWDER TECHNOLOGY ELSEVIER

Powder Technology 82 (1995) 169-175

Fatigue crack growth in flyash compacts L. X u a, R. Helstroom b, A.J. Chambers a, H.S. Kim

a

"Department of Mechanical Engineerin~ The University of Newcastle, Newcastle, N S W 2308, Australia b CSIRO Division of Wool Technology, 1-11 Anzac Avenue, North Ryde, N S W 2112, Australia

Received 1 February 1994; revised 1 August 1994

Abstract Fatigue characteristics of flyash compacts are studied within the framework of fracture mechanics. Also, a test method is proposed to investigate the fatigue behaviour of flyash compacts. The curves of stress versus number of cycles to failure for flyash with moisture contents of 0.1% and 0.3% indicate that flyash compacts possess fatigue characteristics. The da/dN-AK and a - N relationships have been deduced for moisture content 0.1% from the principles of fatigue crack growth. The results show that the rate of fatigue crack growth in flyash compacts obeys the Paris power law equation, and the fatigue properties of flyash compacts can be characterised as falling between those of metals and ceramics. Keywords: Flyash compacts; Fatigue; Crack growth

1. Introduction Considerable work has been done on the static failure of flyash compacts by using tensile and shear tests to characterise the mechanical properties of a flyash bed. However, little attention has been given to the dynamic failure, for example fatigue failure, which cannot be explored by these static methods alone. Fatigue is an important phenomenon of flyash bed failure in some applications, such as the shake cleaning of filter bags. In this particular process it has been observed that the dust cake built up on the surface of a filter is invariably never broken off by the first few strokes of the shaking motion and that dislodgment can be delayed by more than 100 strokes, or beyond the halfway point of the cleaning cycle [1]. In extreme cases, insufficient time has been provided for the shaking and the fall of dust never occurs. Further studies of dust adhesion [2,3] show that the dust cake could be released by the repeated application of a relatively low stress. This situation is similar to the fatigue failure of many materials undergoing cyclic stresses. The existence of fatigue phenomenon in flyash compacts subjected to vibration has been acknowledged and analysed statistically by Kamiya et al. [4,5]. Nonetheless, little progress has been made in understanding the basic mechanisms of fatigue failure of these materials. Recently, some interesting papers [6,7] have 0032-5910/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0032-5910(94)02911-7

focused the discussion onto the failure mechanism of particulate solids and have found that cracks in the solids are the major factor weakening the strength of compacts. If so, cracks may also play an important role in controlling the fatigue failure of flyash compacts. This work has the intention of contributing to this theme. The fracture mechanics concept of fatigue crack growth has been used to study the crack growth in a flyash compact. A test method is proposed to investigate the fatigue behaviour of a flyash compact characterised by weak inter-particle bonds. The results indicate that the fatigue characteristics of flyash compacts can be described using the Paris power law equation.

2. Theoretical background Fatigue is a common mode of failure when materials are subjected to repeated loading at loads lower than the ultimate strength. To determine the fatigue strength of a material, a sample is usually loaded repeatedly at a constant stress until failure occurs. After a series of tests at various stress levels, the S - N curve (stress (s) versus number of cycles to failure (N)) can be constructed, as illustrated in Fig. 1. This curve provides basic and useful information on the fatigue life for a structure. For some materials, the curve has a horizontal portion, known as the fatigue limit, below which failure does not occur.

L. Xu et al. / Powder Technology 82 (1995) 169-175

170

the first region, just above the threshold stress range, the crack growth rate rises rapidly with increasing AK before slowing as AK further increases. Stable growth of the fatigue crack, and often a linear log-log relation between da/dN and AK, is characteristically observed over a broad range of stress intensity in Region 2 before the growth of the crack accelerates catastrophically in the third region, just below the critical stress intensity factor Ko. Many attempts have been made to describe the crack growth rate using semi- or wholly empirical formulae fitted to a set of data. The most .widely known is the Paris equation [13], which describes Region 2 of the crack growth curve:

log N (Fatigue life)

Fig. 1. Schematic representation of an S--N curve.

i; /

da

, log AK

Kc

Fig. 2. Characteristics of a d a / d N - A K curve.

In general, the main concern with fatigue failure is not the presence of cracks, but the progression of crack growth; i.e. how long does it take for a crack to grow from an initial size to a critical size. Prediction of crack propagation behaviour is an important area of fracture mechanics [8]. It is possible to predict crack growth for any geometry for which the stress intensity factor is known. Linear elastic fracture mechanics (LEFM) has been used to relate residual strength to crack size for particulate solids [9-12]. LEFM might also adequately describe crack propagation in a flyash compact by relating the crack growth rate to a stress intensity factor which is determined by the loading condition and the crack size. In linear elastic fracture mechanics, the stress intensity factor is a sufficient parameter to describe the stress field at the tip of a crack. If two different cracks have the same stress intensity factor, they behave in the same manner and will show equal rates of propagation. A small crack at a high stress is equivalent in the stress field to a long crack at a low stress. The rate of fatigue crack propagation per stress cycle, da/dN, is expressed as a function of the stress intensity factor range AK. da ar

=f(AK) = Kmax - K . i n -- (O'max - - trmin)Yq r ' ~

= C(a/q"

(3)

in which C and m are material constants. Compressive stress (trmi, < 0) should not affect crack propagation [14]. Therefore, when a material is subjected to a repetitive stress that cycles from a tensile stress trm~,to a compressive stress O'min,the Paris equation is expressed only in terms of the magnitude of the tensile stress intensity factor as: da

=CKm~

(4)

Once the information of Fig. 2 is obtained for a material in a particular configuration, fatigue crack propagation can be predicted for any other configuration and loading history, provided the stress intensity factor can be specified for the geometry. For constant amplitude loading, the prediction of fatigue failure then becomes a fairly straightforward process of integrating the da/ dN-AK curve to yield: a¢

N,o=f car-da

(5)

ai

where Ni° is the number of stress cycles required to propagate a crack from an initial length al to the critical crack size ac, and hence fracture the material.

(1) (2)

The typical sigmoidal shape of a da/dN-AK fatigue crack growth rate curve is shown in Fig. 2. The curve may be divided into three regions according to the curve shape and the mechanism of crack growth. In

Ni Ni + 1 N u m b e r of cycles

Fig. 3. Schematic fatigue crack growth curve.

Nc

L. X u et al. / Powder Technology 82 (1995) 169-175

Eq. (5) can be used to calculate the number of stress cycles required for failure and to construct an a-N curve as well (Fig. 3).

3. Experimental The standard fatigue test used in fracture mechanics is not well suited to flyash specimens because the bond strength of these materials is inherently weak. An alternative means of investigating the fatigue behaviour of flyash compacts was therefore developed based on the cantilever test (Fig. 4), which has been described elsewhere [15]. In the cantilever-based test, specimens are prepared by consolidating sample flyash in a tube (O26 mm) under a controlled packing load (3.5 kg in this work). The tube is fitted in the manner of a shaft through the inner race of a bearing and clamped in position by two screws. The flyash compact can be pushed by a plunger and extruded from the open end to a certain unsupported length in the horizontal direction. A razor blade is used to create a straight-fronted edge notch on the surface of the flyash cantilever in the manner shown in Fig. 5. Notched specimen r/////////////////////,4

Bending stress

Rotating direction

~

Crack J

~ -tr

+~ o

A ~

o

~0

Rotating angle

of

Fig. 4. Cantilever test rig.

BearingUnit

Tube powder

I

HP5313A

Counter

Fig. 5. Stress distribution in a specimen.

]

sample

i

,

I ram teh oit

The extruded flyash assembly is rotated in the support bearing by a d.c.-gear motor at a set speed ranging from 10 to 60 rpm. In each test, the number of revolutions is continuously counted by an HP5313A universal counter. Counting is stopped by a microswitch underneath the dust-collecting tray that is actuated when the unsupported cylinder of dust collapses and drops onto the tray. The number of cycles to failure is then recorded as the fatigue life of the material at the stress imposed by the self-weight of the cantilever. The bending stress can be calculated by assuming that the flyash compacts behave elastically. As shown in Fig. 4, this bending or axial normal stress varies linearly from a maximum positive value (tension) at the top to a minimum negative value (compression) at the bottom of the cantilever. The maximum bending stress can be used for calculating K~ (subscript I indicates opening mode). In addition to KI, there would be a component of Ks (subscript II indicates shearing mode) due to the shearing force generated by the self-weight of the cantilever, thus resulting in mixed-mode fracture. However, it was observed that crack propagation direction is approximately perpendicular to the axis of the cantilever. This means that the Ku component is negligible in accordance with the maximum principal stress criteria [14]. Accordingly, only mode I fracture or, equivalently, the axial stress, was considered in the analysis of the data. When the extruded cantilever of flyash is rotated about the central axis, the direction of the bending or normal stresses at a point on a cross-section of the sample reverses at every half-rotation. If this stress cycle is repeated a number of times, the sample will finally break off. 3.1. Estimation of daMN

Bending stress variaLion on the surface the specimen.

P~lungers --~:i

171

Estimation of the crack growth rate requires construction of the da/dN-AK curve. To do this, a test sample with an initial crack or notch is loaded repeatedly on the fatigue test machine. During the test, the amplitude and frequency of loading are controlled as constants. The crack growth rate is defined as Aa/AN and, in the limit, is accepted as the differential da/dN. The stress intensity factor range AK may be calculated from the amplitude of the loading, the crack size and the geometry of the test sample. The relationship between da/dN and AK thus may be obtained using Eq. (3). Although several techniques have been proposed for measuring crack growth [16], most of these are not readily adaptable to the present work. An alternative approach is to use the correlation between the size of an artificially introduced initial crack al and the number of cycles to failure Nic, and then to find da/dN by differentiating the correlation curve a(N) over the appropriate range of crack lengths.

,~Microswi 0I teh Geax-motor /,

L. X u et al. / Powder Technology 82 (1995) 169-175

172 Table 1 Test conditions Dust Particle size Bulk density Moisture T u b e diameter Packing load N u m b e r of twists" Cycle frequency Temperature Relative humidity Razor blade

Ulan coal flyash 18.8 ,urn 900-930 kg/m3 0.1 wt.% and 0.3 wt.% 26 m m 3.5 kg 20 40 rpm or 0.67 Hz 2 2 + 2 °C 60% + 2% edge radius: 10-20 /xm thickness: 100 p.m

"90* twisting of the packing load was used to consolidate flyash evenly in the tube.

Table 2 Some parameters in the test to estimate fatigue crack g r o w t h " Test No.

1

2

3

Cantilever length (ram) Stress range (Pa) Average a~ ( m m ) Predicted a~ (ram) Stress ratio in a cycle, R

10 140 1.69 1.75 - 1

11 170 1.35 1.53 - 1

12 200 1.12 1.20 - 1

(d) Estimation of da/dN. Each N~c pertains to the growth of a crack from the initial crack ai to the critical size ac. The number of cycles to failure Nic is plotted against the initial crack size a~. The difference AN= (N~-N~c+~) between two adjacent points can be seen as the number of cycles required for a crack grows from ai to a~+~, if a~
" T h e specimens in this test were prepared with 0.1 wt.% moisture

content.

4. Results and discussions

In the cantilever test, the procedure to estimate the crack growth rate involves the following steps: (a) Determination of the critical crack size a~ for the specimen with a cantilever length 1. The flyash sample is extruded from the tubular support to an unsupported length l. A notch close to the edge of the tubular support is created on the top of the sample and extended into the sample until failure occurs under the weight of the unsupported length. The depth of the notch is considered as the critical crack size a~ where fracture occurs without rotation. (b) Calculation of the maximum bending stress and AK due to the cantilever length l from the following flexure formulas:

tr. . . . .

Md ,, = + ~

The data in Fig. 6 have been obtained without introducing a notch. The S-N curves (Fig. 6) indicate the numbers of loading cycles to failure at various stress levels for flyash compacts with moisture contents of 0.1% and 0.3%. It is noted that four orders of magnitude in the range of the fatigue life of the flyash compacts tested are due to the variation of stresses. This raw data has been redrawn in Fig. 7 where the failure stress is expressed as a percentage of the average tensile strength of the flyash in a linear scale. The average tensile strength is obtained by pushing the 1000

800

(6) 0

hence, A K = K ~ = o ' ~ Y ~ - a where I is the moment of inertia of the specimen, I = rrd4/64, M is the bending moment, M=ql2/2 and q=p~r(d/2) 2, d is the diameter of the specimen, and Y is a geometry factor which is a function of a and d. The values of Y used in this study were obtained from a numerical solution by Kiuchi et al. [17]. (c) Determination of the number of cycles to failure as N~. A number of fatigue tests on samples with various initial crack length a~ smaller than a~ can be performed.

[]

Moisture content: 0,3%

0

Moisture content: 0,1%

600 400

200

0

. . . . . .

10

,,I

. . . . . . . .

I

. . . . . . . .

[

100 1000 10000 Number of Cycles to Failure N c

,

,

i ~iiii

100000

Fig. 6. Fatigue life a n d its dependence on stress levels (coal flyash with moisture content 0.1% and 0.3%).

L. Xu et aL / Powder Technology 82 (1995) 169-175

173

1.6 1.0E-1 -

[2

1.4

Moisture content: 0.3%

0 ' M o i l t u m COntent: 0.1%

1.2

1.0E-2

1.0 o

1.0E-3

0.8 i

•~

0.6 ~

2

Z

J

1.0E4

0.4

1.0E-5

0.2

.~

O~ 140 P|

(3

0.0

100 1000 10000 Number of Cycles to Failure N c

10

F i g . 7. F a t i g u e stress to tensile

life and

its dependence

strength

10(3000

on ratio of fatigue

(coal flyash with moisture

content

failure

[]

,toP,

A

200Pa

1.0E-6 u.

5

6

7 8 9 10 Stress Intensity Factor Range ~xK N/mw2

20

F i g . 9. Fatigue crack growth rate (o',,=,: 140 Pa, 170 Pa, 200 Pa).

0.1%

and 0.3%).

2 0 - -0-

E 1.6

'o

1.2

~max

[

140Pa - 170Pa - 200Pa

ii i

Moisture Content 1.1%

i

----~ ......................................................... 4 ............................................................. ........ { ........................................................................

o

l

i

..:~:..:

----.-

O9

..q

/

~..!0.

o

0.8 ...............................................................i........................: " ~ O

:

.......................................................~. :..~...-. ~....................... .......................... oe .............~ .........................

I

~o

..................*L ~ ' " : " e : .........................o..... ~...........

*

~

tested stress levels of 140, 170 and 200 Pa respectively. These crack lengths are in good agreement with the average values of 1.69, 1.35 and 1.12 mm obtained from the static cantilever fracture tests. The fatigue crack growth rate can be obtained by differentiating Eq. (7) with respect to N. A log-log plot of da/dN versus z ~ is presented in Fig. 9. The results from the three stress levels show good correlation to a single line of the form:

...............................................................I.......................................................:....I ........................................................................

o.,

i

, ti

0

0

100

da

i

1

-

1000

Fig. 8. N u m b e r of cycles to failure vs. initial crack size ((rm,,: 140 Pa, 170 Pa, 200 Pa).

flyash compacts out until failure and using Eq. (6), as described in [15]. The results o f tests on the dusts at two different moisture contents show similar dependencies on N. There is scatter in both sets of data, due perhaps to the material properties, consolidation procedure, sample-to-sample variation and/or testing techniques. Results for relationship between the initial crack length and the n u m b e r of cycles to failure at three different stress levels are shown in Fig. 8 (moisture content 0.1%). These data all follow a convex shape on this semilogarithmic plot, and it is possible to fit the data by the equations: (r= 140 Pa: a -- 1.75 × 10 -3-5.45 × 10 -5 log2(N+ 1), r =0.874 ~r= 170 Pa: a = 1.53 × 10 -3 - 6.87 x 10-5 iog2(N+ 1), r = 0.916 10 -3 --

6.45 × 10-5

]og2(g+

(8)

10000

Number of Cycles to Failure Nc

(r= 200 Pa: a = 1.20 x

--- 1.37 X 10-17(MQ1°", r = 0.904

1), r = 0.864

(7) In these equations, the constants: 1.75, 1.53 and 1.2 mm represent the critical crack lengths at the three

In this representation of the fatigue crack propagation rate, the constant C and the exponent m vary in the range 7.74× 10-1s-2.43 × 10 -17 and 10.1-10.6 respectively from the linear regression. It is interesting to note that typical values of m for brittle materials such as ceramics and rocks range from 21 to 42, while for metals, rn is in the range 2 to 4 [18]. The value of m of the flyash compacts tested falls between these two categories. A n a - N curve for a constant amplitude loading can be constructed by integrating Eq. (8) for an appropriately chosen series of upper and lower limits (ai+l, al) in crack length: ai+l

1017

~QVi.i+l = 1.37(Yi.i+aO.m ,~)ao. 4

£

J a-52da

(9)

ai

where Y~.i+l = (Y~+ Y~÷ 1)/2. T h r e e a - N curves have been obtained as shown in Fig. 10. The solid lines plot the results calculated by Eq. (9) with the initial crack size 0.6 mm and the crack increment 0.05 mm for all three stress levels. The dash lines represent the variation limits of the a - N curve with respect to the variations of m and C. It can be seen that crack growth rate increases progressively with increasing crack size.

L. Xu et al. / Powder Technology 82 (1995) 169-175

174 2.0

140 Pa

E c5

1.6

~'

j

;

200 Pa

~,

1.2

¢0

.B e--

i~ jP

~

,,,

, ,

/

i

AK

K~

//

'

/

K~I

/

0.8

o) The inilial crack is I l N u r n e d to be 0.6 m m .

o

0.4

0.0

. . . . . . . .

100

i

. . . . . . . .

i

1000 10000 Nurnber of Cycles N¢

,

,

40000

Fig. 10. Prediction o f fatigue crack growth ( a - N curve) (~,,=~: 140

l M rn

N

N,c

Pa, 170 Pa, 200 Pa). ANi,i+

1

moment of inertia with respect to the neutral axis (m4) stress intensity factor range (N/m3r2) stress intensity factor for opening mode (N/m 3cz) stress intensity factor for shear mode (N/m31) maximum, minimum stress intensity factor in fatigue test (N/m 3/2) cantilever length (m) bending moment (Nm) constant in Eq. (3) number of cycles number of cycles for a crack growth from a i to ac number of cycles for a crack growth from a i to ai+ 1

Flyash compacts contain flaws or voids, and fatigue crack propagation could be initiated from the flaws. Eq. (8) would be useful in providing a first estimation of the crack growth behaviour in a flyash compact.

q R r

Y

5. Conclusions

Greek letters

The fatigue behaviour of flyash compacts has been investigated experimentally. The stress versus number of cycles to failure curves for flyash with moisture contents of 0.1% and 0.3% indicate that flyash compacts possess fatigue characteristics. Relationship between ratio of fatigue stress to tensile strength and number of cycles to failure for materials with two different moisture contents was found to be fairly independent of moisture content. A test method for flyash compacts has been proposed to estimate the crack extension over a period of loading cycles. The da/dN-AK and a - N relationships have been deduced from the principles of fatigue crack growth and linear elastic fracture mechanics. The results of this work indicate that the Paris equation is valid for flyash compacts and flyash compacts can be categorised as having a fatigue characteristic falling between metals and ceramics.

6. List of symbols a ai, ai+l a o

C d da/dN

intensity of distributed loading (N/m) ratio of the maximum and minimum stress in fatigue test correlation coefficient of fitting curve dimensionless factor in Eqs. (2), (6), (9) and (10)

crack length or notch depth (m) start and stop crack size in the calculation of AN in Eq. (9) (m) critical crack size (m) constant in Eq. (3) diameter of specimen (m) fatigue crack growth rate (m/cycle)

P

Or O'max, rnin

o-T

bulk density of testing specimen (kg/m3) fatigue stress range (Pa) stress (Pa) maximum, minimum stress in fatigue test (Pa) tensile strength (Pa)

References

[11

W. H u m p h r i e s et al., CSIRO 3rd Annual Report to ECNSW, 1988. [2] L. Xu, O J . Scott, A.J. C h a m b e r s a n d S. Keys, 6th World Filtration Congr., Nagoya, Japan, May 1993, pp. 448--454. [31 L. Xu, O J . Scott, A.J. C h a m b e r s a n d S. Keys, Bulk Materials Handling '93, Yeppoon, Australia, Sept. 1993, pp. 229-237. [41 H. Kamiya, J. Tsubaki and G. Jimbo, Kagaku Kogaku Ronbunshu, 11 (1985) 210-216. [51 H. Kamiya, J. Tsubaki a n d G. Jimbo, 1st World Congr. Powder Technology, Nurnber~ Germany, April 1986, pp. 173-185. [61 E.F. Hobbel and B. Scarlett, 2rid World Congr. Powder Technology, Kyoto, Japan, Sept. 1990, pp. 110-115. [71 K. Kendall, N. MeN. Alford a n d J.D. Birchall, Br. Ceram. Proc., 37 (1986) 255-265. [81 D. Brock, The Practical Use of Fracture Mechanics, Kluwer, Dordrecht, 1988. [91 M.J. A d a m s , J. Flyash Bulk Solid Technol., 9 (1985) 15-20. [101 J.D. Birchall, A.J. Howard a n d IC Kendall, Nature, 289 (1981) 388-389. [111 M.A. Mullier, J.P.K. Seville and M.J. A d a m s , Chem. Eng. Sci., 42 (1987) 667-677. [12l M.J. Adams, J. Mater. Sci., 24 (1989) 1772-1776. [131 C.P. Paris, M.P. G o m e z and W.E. Anderson, Trend Eng., 13 (1961) 9-14.

L. Xu et al. / Powder Technology 82 (1995) 169-175

[ 14] D. Broek, Elementary Engineenng Fracture Mechanics, Noordhoff, Groningen, 1974. [15] L. Xu, R. Helstroom, O.J. Scott and A J . Chambers, Powder Technol., accepted for publication. [16] S.A. Meguid, Engineering Fracture Mechanics, Elsevier, Amsterdam, 1989.

175

[17] A. Kiuchi, M. Aoki, M. Kobayashi and K. Ikeda, Z Iron Steel Inst. Jpn., 68 (1982) 1830-1838. [18] R.H. Dauskardt, D.B. Marshall and R.O. Ritchie, Z Am. Ceram. Soc., 73 (1990) 893-903. [19] H.S. Kim, Z ,4ppl. Poem. Sci., 50 (1993) 2223-2224.