Creep fracture of sandstones

$Page 1: Creep fracture of sandstones$
Theoretical and Applied Fracture Mechanics 47 (2007) 57–67

www.elsevier.com/locate/tafmec

Creep fracture of sandstones

Y.L. Chen a,b,c,*, R. Azzam c

a Department of Civil Engineering, University of Shanghai for Science and Technology, 516 Jungong Road, 200093 Shanghai, PR Chinab Department of Civil Engineering, Shanghai University, 149 Yanchang Road, 200072 Shanghai, PR China

c Department of Engineering Geology and Hydrogeology, RWTH Aachen University, Germany

Available online 21 November 2006

Abstract

Creep is one of the important properties of rocks and rock masses. But the mechanism and characteristic of rocks increep has not received the attention it deserves. Sandstones mined from Emei Mountain are investigated and tested by uti-lizing the three-point bending creep fracture test. A mathematical relation between stress intensity factors and durations ofsustained loading for the above specimen is obtained based on the gray prediction theory. The obtained results provide abetter understanding of rock fracture under creep.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Sandstone; Creep; Fracture; Three-point bending test; Gray prediction theory

1. Introduction

Creep behavior of rocks is important because ithas many applications to engineering geologicalproblems. Previous investigations on creep behaviorof rocks such as rock salt, sandstone, and even somehard rocks can be found in [1–4]. The subject hasbeen studied for over eighty years. Considerableamount of data can be found for rocks subjectedto different loading conditions in the laboratoryunder compression, bending and torsion and inmines [5,6]. Most of the results are more qualitative

0167-8442/$ - see front matter � 2006 Elsevier Ltd. All rights reserved

doi:10.1016/j.tafmec.2006.10.006

* Corresponding author. Address: Department of Civil Engi-neering, Shanghai University, 149 Yanchang Road, 200072Shanghai, PR China.

E-mail addresses: [email protected] (Y.L.Chen), [email protected] (R. Azzam).

than quantitative. Up to date, there are no standardtest procedures on rock creep [6].

Because the time-dependence of rock deforma-tion is very distinct in some Chinese rock projectssuch as the three Gorges hydro-power station pro-ject, the Humen bridge project over Zhujiang river,the Jiangyin bridge project and Yunyang bridgeproject over the Yangtse river, the Qingdao cityMetro project, the Nanjing city Metro project andso on, this work focuses attention on the creep char-acteristics of sandstones mined from Emei Moun-tain. Discussion of their behavior and tests followwith creep being defined as continued deformationdue to a constant applied load which will be usedfor studying rock specimens with prefabricatedcrack.

In some rocks, cracks may not initiate andpropagate at the beginning. They can initiate andpropagate after a time-interval of sustained loading

.

mailto:[email protected]

mailto:[email protected]

58 Y.L. Chen, R. Azzam / Theoretical and Applied Fracture Mechanics 47 (2007) 57–67

under creep condition. Up to now, the mechanismand criterion of crack initiation and propagationfor rocks under creep remain not understood. Thispaper is concerned with the study of sandstonesunder creep condition by using the three point bend-ing test. Also, the plane strain bent specimens areused. They are of the size: 5 cm · 5 cm · 25 cm.All specimens were fabricated from three rockmaterials

(i) Gray fine-grained sandstone without beddingplanes, as shown in Fig. 1(a).

(ii) Red fine-grained sandstone without beddingplanes, as shown in Fig. 1(a).

(iii) Red fine-grained sandstone with beddingplanes, as shown in Fig. 1(b) and (c).

The specimens are classified as:Specimen A – fabricated from gray fine-grained

sandstone without bedding planes, as shown inFig. 1(a); specimen B – fabricated from red fine-grained sandstone without bedding planes, asshown in Fig. 1(a); specimen C – fabricated fromred fine-grained sandstone, with pre-existing crackperpendicular to the bedding planes, as shown inFig. 1(b); specimen D – fabricated from red fine-grained sandstone, with pre-existing crack parallelto the bedding planes, as shown in Fig. 1(c).

The grain size seems to be a crucial factor in themeasurement of fracture toughness since it affectsthe size of crack tip FPZ and the fracture parame-ters (namely fracture energy, fracture toughness,etc). Hence, the influence of grain size can not beignored when fracture test is designed. Accordingto the obtained research results, the minimum speci-men dimension must be at least 20–40 times thegrain diameter in order to obtain the accurate KIC

for rock materials. Since all specimens were fabri-cated from fine-grained sandstone rocks with agrain size of about 1/16–1/8 mm, the grain size sat-isfies the above-mentioned condition. For metallicmaterials, creep behaviors may be classified intocreep-ductile and creep-brittle [7]. The rock materi-als adopted here are creep-brittle ones.

Three point bending test include two parts:traditional three point bending test and three pointbending creep test. The traditional test is performedon a servo-controlled testing machine (Instron test-ing machine). Load versus load-point displacement(LPD) curves and load versus crack mouth openingdisplacement (CMOD) curves are plotted automat-ically. In order to avoid the drift instability of servo

system under creep conditions, a rigid mechanicalthree point bending creep device is developed. Inthis device, the displacements are measured bymicrometers. It is loaded with weights. The experi-mental results show that due to creep deformation,rock crack will inevitably initiate and propagateunder a load of KI, which is less than KIC but notless than KIC2. KIC2 indicates the ability of rockmaterials to resist crack initiation and propagationunder creep condition and is less than fracturetoughness KIC, it is defined as creep fracture tough-ness in this paper. And KIC2 should be taken as animportant parameter on design and computationof rock engineering.

A rheological analysis on the criterion and mech-anism of crack development of rocks under creepconditions is presented, and the rheological meaningof KIC2 is given. When the applied stress intensityfactor (SIF) KI is more than KIC2 but less thanKIC, rock crack will initiate and propagate after atime interval of sustained loading under creep con-dition. In order to find the relation between dura-tion of sustained loading, which will lead to crackinitiation and propagation, and the SIF KI, sec-ondly, an unequal-interval time sequence forecast-ing model is introduced, and the relation betweenduration of sustained loading and SIF KI isobtained for homogeneous and isotropic red fine-grained sandstone. Finally, a modified fracturetoughness formula is presented, which fully consid-ers the influence of fracture process zone (FPZ).

2. Laboratory three point bending tests on traditional

fracture behavior and creep fracture of sandstones

The rock specimens used in the tests are fabri-cated from Cambrian fine-grained sandstones,which were mined from Emei Mountain, the Sich-uan Province of China. And single-crack specimensare adopted and illustrated in Fig. 1. As shown inFig. 1, specimen A and specimen B are fabricatedfrom homogeneous and isotropic fine-grained sand-stone materials, and specimen C and specimen Dare very close to plane-isotropic materials. Thecrack was pre-fabricated and includes two parts:the wide part and the thin one. The wide part waspre-made by saw blade and saw web, and the thinpart, whose width is about 0.2–0.3 mm, was madeby very thin diamond saw web. The labor fracturetests are performed on a servo-controlled testingmachine and a self-designed three-point bendingcreep test frame. The sketch map of three-point

Fig. 1. Illustration of the vertical longitudinal section of cuboidspecimens adopted for three point bending test: (a) fabricatedfrom fine-grained sandstone without bedding planes; (b) fabri-cated from red fine-grained sandstone, with pre-existing cracksperpendicular to the bedding planes; (c) fabricated from red-grained sandstone, with pre-existing cracks parallel to thebedding planes.

S

P

W

B

Crack length, a

P/2 P/2P/2

Fig. 2. Geometries of cuboid specimen in three-point bendingtest. a = Crack length, B = specimen thickness, P = applied load,S = the span between two supports, W = specimen width.

Table 1The quantity of rock samples used in three point bending test

Type of test SpecimenA

SpecimenB

SpecimenC

SpecimenD

Conventionalfracture test

2 0 3 3

Fracture testunder creepcondition

0 32 10 10

Notes: Specimen A – fabricated from gray fine-grained sandstonewithout bedding planes; specimen B – fabricated from red fine-grained sandstone without bedding planes; specimen C – fabri-cated from red fine-grained sandstone, with pre-existing cracksperpendicular to the bedding planes; specimen D – fabricatedfrom red fine-grained sandstone, with pre-existing cracks parallelto the bedding planes.

Y.L. Chen, R. Azzam / Theoretical and Applied Fracture Mechanics 47 (2007) 57–67 59

bending test is shown in Fig. 2. The tests includeconventional fracture test and creep fracture test.About 60 specimens are adopted. The quantitativedistribution of specimens is shown in Table 1.

2.1. Traditional fracture test

The conventional fracture test is performedadopting a servo-controlled testing machine (Instrontesting machine). The applied load is measured by apressducor. The load-point displacement (LPD) andthe crack mouth opening displacement (CMOD) aremonitored by linear transducers. Load versus LPDcurves and Load versus CMOD curves are plottedautomatically by testing machine. The sketch maps

of the specimens are shown in Figs. 1 and 2. Thetested result is shown in Table 2 and Figs. 3–5.

For specimens with middle-opened crack underthree point bending test, the formula for calculatingfracture toughness KIC are as the follows [7].

KIC ¼ f a=Wð ÞP max � S= BW 3=2� �

ð1Þ

where

f ða=W Þ ¼ 2:9ða=W 1=2Þ � 4:6ða=W 3=2Þþ 21:8ða=W 5=2Þ � 37:6ða=W 7=2Þþ 38:7ða=W 9=2Þ ð2Þ

Note that W is the specimen width, B the specimenthickness, S the span between two supports underthree point bending test, a the crack length, Pmax

the maximum load and KIC the stress intensity fac-tor for mode I. From Figs. 3–5 it can be found thatthe curves are approximately smooth, and the initialcompacting parts are not obvious in the curves, themain reason is that the rock specimens used in thetest do not contain obvious initial damage. FromTable 2 it can be concluded that the fracture tough-ness of the specimens with cracks perpendicular toits bedding planes is higher than that of the speci-mens with cracks parallel to the bedding planes.The crack extension develops more easily alongthe direction of bedding planes.

2.2. Creep fracture test

2.2.1. Test device and sample cutting

The creep fracture test is carried out on athree-point bending creep test frame, which was

Table 2The tested conventional fracture toughness and relative specimen dimension

Type ofspecimen

No. ofspecimen

W (cm) B (cm) S (cm) a (cm) Pmax (KN) KIC (MN m�3/2) Average KIC

(MN m�3/2)

Specimen A A1 5.09 4.98 20.0 1.78 0.76 0.493 0.494A2 5.09 5.04 20.0 1.91 0.77 0.495

Specimen C C1 5.11 5.01 20.0 2.00 0.44 0.475 0.425C2 5.10 5.07 20.0 2.14 0.47 0.518C3 4.98 5.09 20.0 1.95 0.41 0.281

Specimen D D1 4.98 5.19 20.0 1.97 0.17 0.113 0.243D2 5.06 5.12 20.0 2.05 0.23 0.250D3 5.00 4.87 20.0 1.95 0.32 0.365

Notes: W – specimen width; B – specimen thickness; S – the span between two supports under three point bending test; a – crack length;Pmax – maximum load; KIC – stress intensity factor for mode I.

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

000 .1 0.2 0.2 0.4 0.60.3 0.4 0.5

Load

, P (

kN)

Load

, P (

kN)

Crack mouth opening displacement, CMOD (mm) Load-point displacement, LPD (mm)

Fig. 3. Typical load versus CMOD curves and load versus load-point displacement curves for specimen A in three point bendingconventional fracture test.

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

Load

, P (

kN)

Load

, P (

kN)


Fig. 4. Typical load versus CMOD curves and load versus load-point displacement curves for specimen C in three point bendingconventional fracture test.


self-designed and self- made. In the test, three typesof specimens are adopted, namely, specimen B,specimen C, specimen D. All specimens are cut from

the above-mentioned red fine-grained sandstonematerials. The quantitative distribution of speci-mens is listed in Table 1. The duration of sustained

0

0.1

0.2

0.3

0. 4

0

0.1

0.2

0.3

0.4

Load

, P (

kN)

Load

, P (

kN)

0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5


Fig. 5. Typical load versus CMOD curves and load versus load-point displacement curves for specimen D in three point bendingconventional fracture test.


loading for each specimen is from several hours toseveral months or even more than a year.

2.2.2. Test result

The load-point deformation (LPD) versus timecurves for various sorts of rock specimens under dif-ferent levels of SIF KIC are obtained from the threepoint bending creep fracture test. And some of thecurves are shown in Fig. 6. It can be found fromFig. 6(a)

(i) when KI = 0.090 MN m�3/2, no obvious creepdeformation occurs in tested rock material;

(ii) when KI = 0.103 MN m�3/2 or KI = 0.130MN m�3/2, the creep deformation is apparentin tested rock material. However, after a timeinterval of sustained loading, it stopsdeveloping;

(iii) when KI = 0.179 MN m�3/2 or KI = 0.192MN m�3/2 or KI = 0.195 MN m�3/2, the creepdeformation is very obvious for the adoptedrock material, and it will develop continuouslyuntil the crack initiates and propagates, andthe specimen fractures;

(iv) when KI = 0.250 MN m�3/2,the specimen frac-tures immediately after being loaded.

Similar behavior can be observed from Fig. 6(b)–(c).

2.2.3. Suggested test conclusion

The results of creep fracture test suggests the fol-lowing conclusion:

There should exist three characteristic values inMode I creep fracture test for red fine-grained sand-stone materials under three point bending test,denoted as KIC1, KIC2, KIC3, when

(i) KI < KIC, no obvious creep deformationoccurs in red fine-grained sandstone materials;

(ii) KIC1 6 KI < KIC2, the creep deformation isapparent in red fine-grained sandstone materi-als, after a time interval of sustained loading,it will stop developing;

(iii) KIC2 6 KI < KIC3, the creep deformation isvery obvious for red fine-grained sandstonematerials, and it will develop continuouslyuntil the crack initiates and propagates, andthe specimen fractures;

(iv) KI P KIC3, the specimen fractures immediately.

The suggested characteristic values for redfine-grained sandstone materials used in the creepfracture test are shown in Table 3. The conclusionsobtained in the three point bending creep fracturetest agree with the results of the pervious direct ten-sile creep fracture test [2]. From Table 3 and Table 2it can be found that the crack initiation and propa-gation under a creep condition happens more easilyalong the direction of bedding planes. Due to thesandstone materials having obvious creep behavior,the stress intensity factor may be smaller than KIC

(but not less than KIC2)when a specimen fractures.KIC2 is a very important parameter for those rockprojects, in which the rocks have obvious creepbehavior. For convenience sake, here KIC2 is namedas creep fracture toughness.

3. Rheological explanation of creep fracture tests

According to the rheological theory of linearvisco-elastic materials [8], there prevails

GiðtÞ ¼ Gið0ÞfigðtÞ ði ¼ I; II; IIIÞ ð3Þ

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 1 2 3 4 5 6 7 8 9 101112131415161718192021

KI = 0.090MN.m-3/2

KI = 0.103MN.m-3/2

KI = 0.130MN.m-3/2

KI = 0.179MN.m-3/2

KI =0.250MN.m-3/2 KI =0.195MN.m-3/2

KI = 0.192MN.m-3/2

LPD

(m

m)

LPD

(m

m)

Time (month)

00.05

0.10.15

0.20.250.3

0.350.4

0.450.5

0 1 2 3 4 5 6 7 8 9 10 11 1213 14 15

KI = 0.127MN.m-3/2

KI = 0.169MN.m-3/2

KI = 0.198MN.m-3/2

KI = 0.250MN.m-3/2

KI = 0.370MN.m-3/2 KI = 0.266MN.m-3/2

KI = 0.258MN.m-3/2

Time (month)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

KI = 0.137MN.m-3/2

KI = 0.118MN.m-3/2

KI = 0.101MN.m-3/2

KI =0.180MN.m-3/2

KI = 0.280 MN.m-3/2

KI =0.196MN.m-3/2

KI =0.189MN.m-3/2

LPD

(m

m)

0 1 32 4 5 6 7 8 9 101112

a

b

c

Fig. 6. Some of LPD versus time curves of three sandstone materials adopted in three point bending creep fracture test. LPD = Load-point displacement. (a) LPD versus time curves for specimen B, (b) LPD versus time curves for specimen C, (c) LPD versus time curves forspecimen D.

Table 3The characteristic values of red fine-grained sandstone from EmeiMountain for crack initiation and propagation under creepconditions

Type ofspecimen

KIC1

(MN m�3/2)KIC2

(MN m�3/2)KIC3

(MN m�3/2)

Specimen B 0.103 0.179 0.250Specimen C 0.169 0.250 0.370Specimen D 0.118 0.180 0.280

Notes: KIC1, KIC2, KIC3 – characteristic values of crack initiationand propagation for rocks under creep conditions.


Here, i denotes I, II, III standing for Mode-I,Mode-II, Mode-III fracture, respectively,GI(t) for thestrain energy release rate for mode i; fig(t) for thetime factor of strain energy release rate for mode iand GI(0) for the initial value of GI(t). Now onlythe case i = I(also tensile fracture) is taken into con-sideration, then Eq. (3) becomes

GIðtÞ ¼ GIð0ÞfIgðtÞ ð4Þ

In Eq. (4), GI(0) is the initial value of the strainenergy release rate. It is a constant and does not


vary with time. However, fIg(t) is not a constant,due to creep deformation of microstructure zonearound crack tip, it increases with time monotoni-cally, and it satisfies the condition fIg(t) P 1. SoGI(t) will increase with time monotonically, too.Assuming that the rock used in the test is visco-elas-tic material and the effects of the shape and the sizeof specimens may be neglected. If for a rock speci-men, the crack initiates and propagates after sus-tained loading for a time interval tc under creepcondition, then it must satisfy the following formula[8]:

GIðtcÞ ¼ GIC ð5Þ

Note that GIC is the critical strain energy release ratefor mode I fracture test. From Eqs. (4) and (5) it canbe deduced

GIð0ÞfIgðtcÞ ¼ GIC ð6Þ

Namely

K2I

E0¼ K2

IC

E0fIgðtcÞð7Þ

KI ¼KICffiffiffiffiffiffiffiffiffiffiffiffifIgðtcÞ

p ð8Þ

In Eq. (5), E 0 is the effective Young’s modulus, KI

the stress intensity factor and KIC the fracturetoughness. Since fIg(t) increases monotonically withtime, and KIC is a constant, stress intensity factor KI

will not be smaller than KIC

limt!1ffiffiffiffiffiffiffiffifIgðtÞp when rock spec-

imens fracture under creep condition. From this it

can be concluded that KIC

limt!1ffiffiffiffiffiffiffiffifIgðtÞp is equal to the

above-mentioned KIC2.

KIC2 ¼KIC

limt!1

ffiffiffiffiffiffiffiffiffiffiffifIgðtÞ

p ð9Þ

In rheology, a method for solving fig(t) with Laplacetransformation and Laplace retransformation hasbeen given [8]. KIC2 can be obtain after giving fig(t).If creep deformation is very obvious, the computa-tions and designs should be performed basing onKIC2. From Tables 2 and 3 it can be found that thereexists a distinct difference between KIC and the sug-gested creep fracture toughness KIC2, for instancethe red fine-grained sandstone specimens with crackparallel to the bedding planes, the conventionalfracture and creep fracture characteristic valuesare KIC = 0.243, KIC2 = 0.180 and KIC3 = 0.280.

Theoretically, the engineering computation and de-sign should be conducted basing on KIC2.

4. Model for crack initiation and propagation of

intact rocks under creep

From the result of labor test, it can be found thatwhen KI < KIC2, the specimens will never fracture,and when KI > KIC2, the specimens will fracturesooner or later. When KIC2 < KI < KIC3, the speci-mens will fracture after a time interval of sustainedloading. Following a model is constructed toexpress the relation between time interval of sus-tained loading, over which the crack initiation andpropagation will happen, and the initial stress inten-sity factor KI basing on gray prediction theory.

4.1. Unequal-interval time sequence: prediction and

solution

The data occurring in engineering practice aresometimes unequal-interval time sequence, for con-venience sake, the equal-interval time sequence grayprediction theory can be extended to the status ofunequal-interval time sequence.

4.1.1. Gray prediction model [9]

Suppose that h groups of function arrays arewritten as fX ð0Þk ðtiÞgðk ¼ 1; 2; . . . ; h; i ¼ 1; 2; . . . ;NÞ,and assuming that all X ð0Þk ðtiÞ is not negative. In casethere exists X ð0Þk ðtiÞ < 0ð1 6 i0 6 NÞ; fX ð0Þk ðtiÞg is

rewritten as feX ð0Þk ðtiÞg, and defining

X ð0Þmin ¼ min1 6 k 6 h

1 6 i 6 N

eX ð0Þk ðtiÞð10Þ

Process the function array as

X ð0Þk ðtiÞ ¼ eX ð0Þk ðtiÞ þ X ð0Þmin

�� for k ¼ 1; 2; . . . ; h

and i ¼ 1; 2; . . . ;N ð11Þ

Then, the condition X ð0Þk ðtiÞP 0 ðk ¼ 1; 2; . . . ; h;i ¼ 1; 2; . . . ;NÞ is satisfied.

Assume

Dti ¼ ti � ti�1 for i ¼ 2; 3; . . . ;N ð12Þsuch that

Dt1 ¼ 1 ð13Þ

By summing the function arrays weightedly, the firstorder weighted sum arrays are obtained as follows:


X ð1Þk ðtiÞ ¼Xi

d¼1

DtdX ð0Þk ðtdÞ for k ¼ 1; 2; . . . ; h

and i ¼ 1; 2; . . . ;N ð14Þ

Assuming that fX ð1Þk ðtiÞg is fit for the systematicdynamical GM (n,m) model, thenXn

j¼0

ajdn�jðX ð1Þ1 Þ

dtn�j¼Xm�1

s¼1

bsXð1Þsþ1 ð15Þ

In Eq. (15), a0 is 1, n, the order of the systematicdynamical equation, m, the number of the systemfactors and X ð1Þ1 ;X ð1Þ2 ; . . . ;X ð1Þm , the system factors.Eq. (15) has n + m � 1 unknown coefficients. Ifthere are N samples (N P n + m), the unknowncoefficients can be calculated by the method of leastsquares

A ¼ ½a1; a2; . . . ; an; b1; b2; . . . ; bm�1�for m P 1; n P 1 ð16Þ

When there is only one group of function array(m = 1), and the order of systematic dynamicalequation is one (n = 1), the systematic dynamicalequation will be turned into

dX ð1ÞðtÞdt

þ aX ð1ÞðtÞ ¼ u ð17Þ

It is named the GM (1,1) model, and a ¼ ½a; u�T areunknown coefficients.

4.1.2. Solution of the systematic dynamical equation

Now the differential of Eq. (17) will be solved asfollows:

When t 2 [ti�1,ti]

dX ð1ÞðtÞdt

� X ð1ÞðtiÞ � X ð1Þðti�1ÞDti

ð18Þ

If

dX ð1ÞðtÞdt

� X ð0ÞðtiÞ ð19Þ

the interpolation formula is

X ð1ÞðtÞ ¼ nX ð1ÞðtiÞ þ ð1� nÞX ð1Þðti�1Þfor 0 6 n 6 1 ð20Þ

From Eqs. (17), (19) and (20), the unknown coeffi-cients can be calculated as follows by the methodof least squares:

a ¼ ½a; u�T ¼ ðBTBÞ�1BTY N ð21Þ

where

B ¼

�ðnX ð1Þðt2Þ þ ð1� nÞX ð1Þðt1ÞÞ 1

�ðnX ð1Þðt3Þ þ ð1� nÞX ð1Þðt2ÞÞ 1

..

. ...

�ðnX ð1ÞðtN Þ þ ð1� nÞX ð1ÞðtN�1ÞÞ 1

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;ð22Þ

Y N ¼

X ð0Þðt2Þ

X ð0Þðt3Þ

..

.

X ð0ÞðtNÞ

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;ð23Þ

In Eq. (22), n is the whitening parameter of grayprediction model, it can be calculated by the opti-mum identification method. The solution of Eq.(17) is

bX ð1ÞðtÞ ¼ X ð1Þðt1Þ �ua

� �e�aðt�t1Þ þ u

að24Þ

The fitting values of function array {X(0)(ti)} are asfollows:

bX ð0ÞðtiÞ ¼bX ð1ÞðtiÞ � bX ð1Þðti�1Þ

Dtið25Þ

If X(0)(ti) (i = 1,2, . . . ,N) had been processed by Eq.(10), then the fitting values of {X(0)(ti)} are calcu-lated by Eq. (26)

bX ð0ÞðtiÞ ¼bX ð1ÞðtiÞ � bX ð1Þðti�1Þ

Dti� jX 0

minj ð26Þ

Eq. (24) can be applied to creep fracture predictionthrough extrapolation.

4.1.3. Parameter n and the optimum prediction

function

In above calculating procedure, the selection ofparameter n is very important. The parameter ncan be obtained by adopting optimum identificationmethod. The error control formula is given in (27).

e ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNi¼1ðeð0ÞðiÞÞ

2

N

sð27Þ

where

eð0ÞðiÞ ¼ X ð0ÞðtiÞ � bX ð0ÞðtiÞ ði ¼ 1; 2; . . . ;NÞ ð28ÞThe prediction function corresponding to the abovecalculated n is called optimum prediction function.


4.2. Crack initiation and propagation of intact rocks

under creep condition

At room temperature of about 20 �C, three-pointbending creep fracture test is performed for a groupof specimens fabricated from red fine-grained sand-stone material without bedding planes, which isregarded as homogeneous and isotropic and meetsKelvin model. The sketch map of specimens isshown in Fig. 2. The quantity of specimens is listedin Table 1. The test result is compiled into Table 4.In Table 4, Ki and X(0)(ti) are the mean value of 6specimens in order to reduce the artificial error.Defining tI = 1/Ki, so that tI increases monotoni-cally. Using optimum identification method,n = 0.474 can be deduced approximately. It can beobtained from Eq. (21) that

a ¼ ½a; u�T ¼ ½�0:4181;�1:0862� ð29ÞSubstituting Eq. (29) into Eq. (24), the followingequation can be obtained:bX ð1ÞðtÞ ¼ 0:002009 exp½0:4181ðt � 0:004348Þ�

þ 2:5979 ð30Þ

Moreover, putting ti = 1/Ki into Eq. (30), thereyieldsbX ð1ÞðKIÞ ¼ 0:002009 exp½0:4181ð1=KI

� 0:004348Þ� þ 2:5979 ð31ÞThis is the relational expression between the firstorder weighted sum of time interval of sustainedloading, over which crack initiation and propaga-tion will happen, and the SIF Ki. By employing it,the time of crack initiation and propagation of redfine-grained sandstone material under creep condi-tion can be forecasted and predicted. SubstitutingEq. (31) or Eq. (30) into Eq. (25), the time intervalof sustained loading can be obtained. In the follow-ing part, the validity of Eqs. (30) and (31) will be

Table 4The time intervals of sustained loading and other relative data of red satest

Ordinal number KI ðkN m�32Þ ti ðkN m�

32Þ�1

1 230 0.0043482 215 0.0046513 206 0.0048544 196 0.0051025 187 0.005348

Notes: The type of specimens used in the test is specimen B. i = Ordiinterval of sustained loading for rock specimen under three point bepropagation; X(1)(ti) = the weighted summation of sequence {X(0)(ti)}.

verified. Substituting t5 = 0.005348 into Eq. (30), itcan be obtained thatbX ð1Þðt5Þ ¼ 2:7294 ð32Þ

From Eqs. (25) and (32), bX ð0Þðt5Þ ¼ 332:7 day can bederived, also when Ki = 187 kN m�3/2, the fittingvalue of time interval for sustained loading is332.7 day. Since the practical measured time intervalis 330.5 day, the relative error is less than 1%, thepredicted result is ideal. The measured curve andpredicted curve are shown in Fig. 7. It is verified thatthe obtained model can fit the experimental datavery well. It should be emphasized that the gray pre-diction model is ordinarily applicable for the systemswhose data are close to exponential functions. So itneeds to be tested whether the data are close to expo-nential functions before the gray prediction theory isapplied to the study of other rock materials.

5. Revised fracture toughness KIC for microcrack-

zone around the crack tip

The size of microcrack-zone around the crack tipcould be obtained from the theory of maximum nor-mal stress [10–12].

rI ¼ rt ð33Þ

Here, rt is the uniaxial tensile strength and rI repre-sents the maximum tensile stress selected from theprincipal stresses. From the theory of elasticity itcan be concluded that

rI ¼KIffiffiffiffiffiffiffi2prp cos

h2

1þ sinh2

� ð34Þ

Eq. (34) is satisfied around the crack tip. From Eqs.(33) and (34) it can obtained that

rðhÞ ¼ 1

2pKI

rt

� 2

cosh2

1þ sinh2

�� 2

ð35Þ

ndstone specimens adopted in three point bending creep fracture

X(0)(ti) (day) X ðlÞðtiÞ day kN m�32

2.6 2.610.3 2.603150.4 2.6134138.0 2.6476Unknown Unknown

nal number; Ki = stress intensity factor; ti = 1/Ki; X(0)(ti) = timending creep fracture test, which had led to crack initiation and

0 50 100 150 200 250 300 3504.2

4.4

4.6

4.8

5

5.2

5.4x 10-3

Time (day)

1/K

I ((k

N)-1

m3/

2)

* Predicted value

+ Monitored value

Fig. 7. The comparison between predicted and measured time intervals of sustained loading of red fine-grained sandstone specimensemployed in three point bending creep fracture test. Time = time intervals of sustained loading of red fine-grained sandstone specimensemployed in three point bending creep fracture test. KI = stress intensity factor.


The shape of microcrack-zone is shown in Fig. 8,where the boundary curve is plotted in a polar set-up. Assuming that the crack extends right forward,then h = 0 and Eq. (35) becomes

r ¼ 1

2pKI

rt

�2

ð36Þ

The microcrack-zone around the crack tip is alsonamed as fracture process zone (FPZ).

For the three-point bending fracture test, the ori-ginal formula of fracture toughness can be correctedas follows. When S = 4W

210

240

90

270

120

150

180

crack

Fig. 8. The shape of m

K1 ¼PS

BW32

�f

aW

� �ð37Þ

In which

faW

� �¼ 2:9

aW

� �12 � 4:6

aW

� �32 þ 21:8

aW

� �52

� 37:6aW

� �72 þ 38:7

aW

� �92 ð38Þ

The parameters P, S, W, a and B are defined in Eq.(2). Substituting P = Pc into Eq. (37) yields

0.5

1

1.5

2

30

60

300

330

0

θr

icrocrack zone.


KIC ¼P cS

BW32

�� f a

W

� �ð39Þ

where Pc is the maximum applied load. Substitutinga = a + r into Eq. (39), it can be obtained that

KIC ¼P cS

BW32

�� f aþ r

W

� �ð40Þ

This is the corrected fracture toughness formula,where r is the radius of the microcrack-zone. KIC,KIC1, KIC2 and KIC3 can be revised basing in Eq. (40).

6. Conclusions

The labor test of fracture and creep fracture forfine-grained sandstone rocks under three pointbending have been performed. The criterion andmechanism of crack initiation and propagationunder creep condition has been discussed basingon labor test and rheological theory. An unequal-interval time sequence gray prediction model hasbeen proposed and applied to construct a predictionmodel for crack initiation and propagation of intactred sandstone under creep condition. The influenceof fracture process zone on fracture toughness hasbeen investigated. The following conclusions canbe suggested through these studies:

(i) The parameter KIC2, which is defined as creepfracture toughness of rock materials and smal-ler than KIC, is a very important parameter forrock engineering. Due to occurrence of creepdeformation, rock material can fracture underthe condition KIC2 < Ki < KIC.

(ii) Gray prediction theory can be tried to forecastand predict the crack initiation and propaga-tion for some homogeneous and isotropicintact rocks under creep condition.

(iii) KIC2 has also a rheological meaning. It is equalto KIC

limt!1ffiffiffiffiffiffiffiffifIgðtÞp .

(iv) It should be emphasized that the tested datafor some rocks and rock masses are often verydiscrete. The homogeneity, continuity and

isotropy of many rock materials are also verypoor. So the applicability should be tested andexamined strictly before the conclusions andresults of current paper are applied to otherrock materials.

Acknowledgement

The financial support from the SRF for ROCSFoundation and the National Natural ScienceFoundation of China (59809005, 50279018) for thisstudy is gratefully acknowledged.

References

[1] V.G. Silberschmidt, V.V. Silberschmidt, Analysis of crackingin rock salt, Rock Mechanics and Rock Engineering 33 (1)(2000) 53–70.

[2] Y.L. Chen, Researches on Creep Fracture of Rock, Thesisfor the Doctorate, Tongji University, Shanghai, China, 1994(in Chinese).

[3] D.F. Malan, Time-dependent behaviour of deep level tabularexcavations in hard rock, Rock Mechanics and RockEngineering 32 (2) (1999) 123–155.

[4] D.F. Malan, Manuel rocha medal recipient simulating thetime-dependent behaviour of excavations in hard rock, RockMechanics and Rock Engineering 35 (4) (2002) 225–254.

[5] N.D. Cristescu, U. Hunsche, Time effects in rock mechanics,John Wiley & Sons, Chichester, 1998, pp. 1–30.

[6] A.D. Lama, V.S. Vutukuri, Handbook on MechanicalProperties of Rocks, Trans Tech Publications, 1978, pp.209–324.

[7] B.N. Whittaker, R.N. Singh, G. Sun, Rock FractureMechanics, Elsevier Science Publishers B.V., Amsterdam,The Netherlands, 1992, pp. 157–199.

[8] L.W. Yuan, Rheology, Science Press, Beijing, China, 1984,74–95 (in Chinese).

[9] Y.L. Chen, Gray prediction model GM(1,N) of unequal timeseries and its application to underground engineering,Journal of Computational Mechanics 13 (4) (1996) 483–488(in Chinese).

[10] B.R. Lawn, Fracture of brittle solids, second ed. CambridgeSolid State Science Series, Cambridge University Press, 1993.

[11] W.C. Zhu, C.A. Tang, Micromechanical model for simulat-ing the fracture process of rock, Rock Mechanics and RockEngineering 37 (1) (2004) 25–56.

[12] J.N. Tao, Several questions in testing fracture toughness ofrock (KIC), Chinese Journal of Rock Mechanics and RockEngineering 9 (4) (1990) 319–327 (in Chinese).

Documents

Creep fracture of sandstones