Stress intensity factor evaluation for cracked chevron notched brazilian disc specimens

K 2 = G "E'
E E'= .
(1 - v 2)
Where CBE' is the dimensionless compliance value for the CCNBD specimen and Y*(ct) is the dimensionless SIF values for the CCNBD specimens. Therefore if the compliance values for the CCNBD specimens can be derived, the corresponding SIF value will then be obtained. Generally two models can be used to derive the compliance value for a chevron notched geometry based on the compliance value of its corresponding straight through notched specimen. Namely these two models are: Hypothesis I--Munz's identical compliance changing rate model and Hypothesis II--Bluhm's slice compliance superimposition model.
HYPOTHESIS I AND Y*(~) EVALUATION
for plane stress condition, for plane strain condition.
(3) As a first approximation, Munz [9] assumed that the change in dimensionless compliance with a crack extension da (dot) for the chevron notch specimen is identical to that for a straight through specimen provided the other geometrical values are identical, i.e. d(CcN • BE') dot (4) d(Csv" BE') dot (8)
Technical Note
Stress Intensity Factor Evaluation for Cracked Chevron Notched Brazilian Disc Specimens
c. x u t
R. J. FOWELLt
INTRODUCTION
Recently cracked Brazilian discs, including Cracked Straight Through Brazilian Discs (CSTBD) and Cracked Chevron Notched Brazilian Discs (CCNBD) (Fig. 1), are being used in rock fracture research [1-3]. The advantages of using cracked Brazilian disc geometries for rock fracture investigation are being identified. It has been concluded [4] that these geometries can be developed to be simple and ideal configurations for rock fracture toughness measurement under Mode I, Mode II or mixed Mode I and Mode II fracture conditions, rock fracture strength locus test and rock fracture initiation and crack propagation investigations (single or mixed mode). The CSTBD fracture problem was first studied by Libatskii in 1967 [5] and then actively investigated by the following researchers [5-7]. A recent study [6] based both on theoretical evaluation by dislocation and complex stress function methods, and numerical calibration by finite element and boundary element methods has given accurate solutions to the CSTBD fracture problem for any fracture mode or mixed mode conditions for any crack length. This provides the foundations to solve the fracture problems for the CCNBD geometry.
for.
C O M P L I A N C E M E T H O D IN C C N B D S I F E V A L U A T I O N
For chevron notched specimens, the theoretical SIF soluபைடு நூலகம்ion method is normally based on the compliance method in combination with the solution of the
Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 31, No. 2, pp. 157-162, 1994
Pergamon
Copyright© 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0148-9062/94 $7.00 + 0.00
=
[-d(CBE')
•
2T.'2 ]
L
4'fl(ot)
B,/-R'Y*(~)
(7)
p2 dC 2_-~ - d---a
(2)
where b is the width of the crack front. Under linear elastic fracture mechanics (LEFM) conditions, the SIF value of a crack problem, K, can be related to G in a simple form:
Y P
/
/ N
\
I l
\ \ I I
(a)
(b)
Fig. 1. CSTBD and CCNBD specimen geometries. 157
(c)
158
XU and FOWELL:
TECHNICAL NOTE
corresponding straight through notched specimens. A specimen compliance is defined as the reverse of the specimen stiffness, i.e.
C =u/P
(1)
and ot(=a/R), O~o(=ao/R), otl(=az/R) and %(=B/R) are dimensionless expressions for the specimen geometrical dimensions. Substituting equation (6) back into equation (4) will yield:
where P is the external load, and u the related displacement, loading point displacement (LPD) for example, as shown in Fig. 2. Combined with the definition of strain energy release rate G, the relation between C and G can be derived. When the crack with length a propagates stably for an increment da, the change in the compliance value of the specimen is dC, then the relation can be expressed as: G
tDepartment of Mining and Mineral Engineering, University of Leeds, Leeds, LS2 9JT, U.K.
A single CCNBD specimen geometry was first used by Shetty in 1985 [7] for the Mode I fracture toughness measurement of ceramic materials. The stress intensity factor (SIF) solution for this geometry was solely obtained by numerical calibration. When CCNBD was introduced into rock fracture research in the late 1980s [3, 8], one single geometry was again proposed and the SIF solution was obtained in the same way. However it was soon found that in rock fracture research, one single geometry was far too limited for practical application as the availability of rock core and specimen preparation equipment vary greatly. Therefore the SIF solutions for a complete set of different CCNBD specimen geometries needed to be given so that CCNBD users can select their own geometry according to their practical situation. Obviously it is impossible to use numerical methods to calibrate all the possible CCNBD specimen geometries and therefore a theoretical evaluation method has to be derived. The correctness of the theoretical approach can be calibrated based on some typical CCNBD geometries and the error in the method can then be compensated