Theoretical Background and Formulations
3.5 Proposed modified Dally and Berger technique
In the present investigation Dally and Berger [51] strain gage technique (Section 3.4) is modified primarily to render the determination of the optimal radial locations definitively and consistently. A major difference between the present modified technique and that of original technique [51] is the inclusion of additional coefficients A2and C2 in Eqs. (3.33)-(3.35) which makes a five parameter strain series containing mode I coefficients A0, A1, A2, B0, B1 and shearing mode coefficients C0, C1, C2, D0, D1. It is assumed that these five parameter
representation can accurately describe the strain field within the zone II. With this, the rectangular strain components at any point within the zone II are given by
1/ 2 1/ 2
0 1
2 3/ 2
2
1/ 2
0 1 0
1/ 2 2
1
cos (1 ) (1 ) sin sin3 cos
2 2 2 2
1 3 3
(1 ) (1 ) sin 2(1 ) cos 3(1 ) sin sin
2 2 2 2
2 2 cos sin (1 ) cos cos3 2
2 2 2
sin 2 (1 ) cos
2 2
E xx A r A r
A r
B B r C r
C r
3/ 2 2
1
3 3
2 sin (1 ) sin cos
2 2 2
2 sin
C r D r
(3.47)
1/ 2 1/ 2
0 1
2 3/ 2
2
1/ 2
0 1 0
1/ 2 2
1
cos (1 ) (1 ) sin sin3 cos
2 2 2 2
1 3
(1 ) (1 ) sin 2(1 ) cos 3(1 ) sin sin
2 2 2 2
2 2 cos sin 2 (1 ) cos cos3
2 2 2
sin 2 (1 ) cos 2
E yy A r A r
A r
B B r C r
C r
2 3/ 2
1
2 sin3
2 2
3(1 ) sin cos 2 sin
2 2
C r D r
(3.48)
and
1/ 2 1/ 2 3/ 2
0 1 2
1/ 2 1/ 2 2
1 0 1
3/ 2 2
1 3 1 3
sin cos sin cos sin cos
2 2 2 2 2 2
2 sin cos 1 sin sin3 cos sin 1
2 2 2 2 2
3 3
cos sin sin
2 2 2
G xy A r A r A r
B r C r C r
C r
(3.49)
Once again referring to Fig. 3.6, the normal strain aa in the direction of an arbitrary angle with respect to crack axis is thus given by
1/ 2 0
1/ 2 2
1
3/ 2 2
0
1 3 3
(1 ) cos (1 ) sin cos 2 sin sin 2 cos
2 2 2 2
cos 1 sin (1 ) cos 2 1(1 ) sin sin 2
2 2 2
3 3
cos (1 ) (1 ) sin sin cos 2 cos sin 2
2 2 2 2
2 (co E aa A r
A r A r
B
2 2 2 2
1
1/ 2 2 2
0
1/ 2 2
1
s sin ) 2 cos cos cos sin
(1 ) sin sin 2 2 sin sin cos (3.50)
2
1 3 3
(1 ) cos sin 2 (1 ) sin cos cos 2 sin sin 2
2 2 2 2
cos (1 ) sin cos cos 2 1 sin sin
2 2 2 2
B r C r
C r
2 2
3/ 2 2 2 2
2
2 2
1
2 2 sin cos 2 sin sin
2 2
2 sin cos sin 3(1 ) sin cos cos 2
2 2 2
3 3
(1 ) cos sin sin sin 2 2 (cos sin )
2 2 2
C r
D r
It could be noted that the coefficient D0 is absent in the mixed mode strain field equation (3.50).
For
cos 2 1
1
(3.51)
the coefficients of B0 and D1 terms in Eq. (3.50) can be eliminated and if
tan cot 2
2
(3.52)
then, the coefficient of the A1 term also vanishes.
Referring to Fig. 3.6 for positive values of and , the strain component
aa and for negative values of both and (since for negative value of , the angle
1/ 2 0
3/ 2 2
2 2
1
1/ 2 2 2
0
1 3 3
(1 ) cos (1 ) sin cos 2 sin sin 2 cos
2 2 2 2
3 3
cos (1 ) (1 ) sin sin cos 2 cos sin 2
2 2 2 2
2 cos cos cos sin (1 ) sin sin 2
2 sin sin cos
2 E aa A r
A r B r C r
1/ 2 2
1
2 2 3/ 2 2 2 2
2
(1 ) cos sin 2 2
1 3 3
(1 ) sin cos cos 2 sin sin 2 (3.53)
2 2 2
cos (1 ) sin cos cos 2 1 sin sin 2
2 2 2 2
2 sin cos 2 sin sin 2 sin cos sin
2 2 2
3(1 ) sin cos 2
C r
C r
3 3
cos 2 (1 ) cos sin sin sin 2
2 2 2 2
and
1/ 2 0
3/ 2 2
2 2
1
1/ 2 2 2
0
1 3 3
(1 ) cos (1 ) sin cos 2 sin sin 2 cos
2 2 2 2
3 3
cos (1 ) (1 ) sin sin cos 2 cos sin 2
2 2 2 2
2 cos cos sin (1 ) sin sin 2
2 sin sin cos (1
2 E bb A r
A r B r C r
1/ 2 2
1
2 2 3/ 2 2 2 2
2
) cos sin 2 2
1 3 3
(1 ) sin cos cos 2 sin sin 2 (3.54)
2 2 2
cos (1 ) sin cos cos 2 1 sin sin 2
2 2 2 2
2 sin cos 2 sin sin 2 sin cos sin
2 2 2
3(1 ) sin cos co
2 2
C r
C r
3 3
s 2 (1 ) cos sin sin sin 2
2 2 2
Thus if a strain gage is placed at a radial distance r from the crack tip on a line making an angle of (Eq. (3.52)) and gage orientation of (Eq. (3.51)) with the crack axis then the measured strains contain only the terms A0, A2, B1, C0, C1, C2. Adding Eqs. (3.53) and (3.54) and multiplying with r results to
0
2 2
3/ 2 2 2
1
1 3
( ) 2 (1 ) cos (1 ) sin cos 2 sin
2 2 2
3 3
sin 2 cos 2 cos (1 )
2 2
3(1 ) sin sin cos 2 cos sin 2
2 2 2
4 cos cos sin (1 ) sin sin 2
aa bb
E r A
A r
B r
(3.55)
which contains only coefficients of mode I loading. Similarly subtraction of Eq. (3.54) from Eq. (3.53) and multiplying r results to
2 2
0
1
2 2 2
2 2 2
2
( ) 2 2 sin sin cos (1 ) cos sin 2
2 2
1 3 3
(1 ) sin cos cos 2 sin sin 2
2 2 2
2 cos (1 ) sin cos cos 2
2 2 2
1 sin sin 2 2 sin cos 2 sin sin
2 2 2
2 2 sin cos
2
aa bb
E r C
C r
C r
sin2
3(1 ) sin2
3 3
cos cos 2 (1 ) cos sin sin sin 2
2 2 2 2
(3.56)
which now contains only coefficients of mode II loading.
Let the coefficient of A0 in Eq. (3.55)
1
1 3 3
2 (1 ) cos (1 ) sin cos 2 sin sin 2 cos
2 2 2 2 I
(say)(3.57)
and the coefficient of C0 in Eq. (3.56) is
2 2
2
2 2 sin sin cos (1 ) cos sin 2
2 2
1 3 3
(1 ) sin cos cos 2 sin sin 2
2 2 2 I
(say) (3.58)
Eqs. (3.55) and (3.56) can now be rewritten as
2 2 0
1 1
3/ 2
2 2
1 1
( ) 2 3
cos (1 ) 2
3(1 ) sin sin cos 2 cos sin 2
2 2 2
4 cos cos sin (1 ) sin sin 2
aa bb
E r A r
I A I
B r I
(3.59)
1 0
2 2
2 2 2
2
2 2 2
2 2
2
( ) 2
cos (1 ) sin cos cos 2
2 2 2
1 sin sin 2 2 sin cos 2 sin sin
2 2 2
2 3
2 sin cos sin 1 sin cos cos 2
2 2 2
3 3
1 cos sin sin sin 2
2 2 2
aa bb
E r C r
I C I
C r I
(3.60)
Thus as r0 the LHS quantities of Eqs. (3.59) and (3.60) yield A0 and C0 respectively. Eqs. (3.59) and (3.60) can be rewritten in a simplified form as
2 3 2
0 1 2 2 1
1
( aa bb)
E r
A A r B r
I
(3.61)
and
2
0 1 1 2 2
2
( aa bb)
E r
C C r C r
I
(3.62)
where, 1, 2, 1 and 2 are constants for a given Poisson’s ratio and are given by
1
1
2 2
2
1
2 1
2
2 2
2 2
3 3
2 cos (1 ) (1 ) sin sin cos 2 cos sin 2
2 2 2 2
4 cos cos sin (1 ) sin sin 2
2 cos (1 ) sin cos cos 2 1 sin sin 2
2 2 2 2
2 sin cos 2 sin sin
2 2
2 2 sin
I
I I
I
2 2 2
2
cos sin 3 1 sin cos cos 2
2 2 2
3 3
1 cos sin sin sin 2
2 2 2
(3.63)
It is clear from Eqs. (3.59) and (3.60) that, the quantities on the LHS of these equations are to be measured on a cracked specimen using a minimum of three strain gages located at different radii on both the positive and negative gage lines as shown in Fig. 3.9 to determine all the six coefficients.
Figure 3.9 Proposed gage locations for a mixed mode cracked body
It should be noted that the radial distance from the crack tip to the corresponding strain gages on the positive and negative gage lines should be same (Fig. 3.9) for calculation of the LHS quantities of Eqs. (3.59) and (3.60).
In contrast to Dally and Berger [51] original approach, the present investigation proposes to determine the coefficients A0 and C0 by taking best-fit of the measured quantities of
1
( aa bb)
E r
I
and
2
( aa bb)
E r
I
to the curves of the form on the right hand sides (RHS) of Eqs. (3.61) and (3.62) respectively. Using the values of coefficients A0 and C0 from the best-fit regression, the mixed mode SIFs
KI and KII can be respectively determined as
0 0
2 and 2
I II
K A K C (3.64)
It is to be noted that equations corresponding to plane strain conditions can be obtained by replacing E by E
12
and by 1 1
in Eqs. (3.47)–(3.62). It should be noted that for the same Poisson’s ratio, plane stress and plane strain conditions will have different values of and hence . Considering =1 3 for which =600, Eqs. (3.53) and (3.54) and Eqs. (3.61) and (3.62) can be simplified to1/ 2 3/ 2 1/ 2 1/ 2 3/ 2
0 2 1 0 1 2
3 3 1 3
2 2 2 2 2
Eaa A r A r B r C r C r C r (3.65)
1/ 2 3/ 2 1/ 2 1/ 2 3/ 2
0 2 1 0 1 2
3 3 1 3
2 2 2 2 2
Ebb A r A r B r C r C r C r (3.66)
2 3/ 2
0 2 1
( ) 4
3 3
aa bb
E r A A r B r (3.67)
2
0 1 2
( aa bb) 2 3
E r C C r C r (3.68)
Eqs. (3.65)–(3.68) are extensively used in numerical examples of the present investigation and are also useful in understanding the subsequent discussions in the
present work. As can be observed from the description of the proposed modified DB technique, although the angular positions for all the strain gages such as and are clearly defined and easily determined but the approach for the optimal radial distances ri (i 1, 2, 3 . . .) for each of the three strain gages along both the gage lines which are necessarily to be placed is not available. In case of all or some of the strain gages are located very close to the crack tip, the strain measurements will be affected by strain gradients, 3D effects and plasticity effects. Though these effects can be avoided by pasting the gages quite away from the crack tip, but then Eqs. (3.53) and (3.54) may not be applicable for strain measurements at such locations. This is due to the reason that, the measured strain at such a large distances may also have contribution due to the coefficients other than A0, A1, A2, B0, B1 , C0, C1, C2, D0, and D1. It is clear that all strain gages should be located within the zone of domination of the above coefficients i.e. zone II and at the same time should not be located very close to the crack tip. It should be noted that the size of zone II for mixed mode is significantly larger than that in case of mode I due to the use of more number of parameters or coefficients in the strain field. As a consequence, more number of strain gages could be accommodated. This is a significant advantage of the proposed modified DB technique which justifies the increased number of gages. Therefore, prior knowledge of valid radial locations for strain gages is essential for successful application of the modified Dally and Berger technique for the accurate determination of KI and KII.
Similar to the case of determination of KI , the minimum radial distance rmin for strain measurements is governed by the presence of 3D state of stress near the crack tip. Accordingly, the rmin is given by [22]
min
thickness of the specimen
r 2 (3.69)
Defining rmax as the maximum radial distance from the crack tip or the upper bound for the valid radial locations for all the strain gages, clearly, rmax is also the extent of
validity of five parameter strain series represented by Eqs. (3.53) and (3.54) on positive and negative gage lines respectively. Conversely, Eqs. (3.53) and (3.54) can represent the strain field accurately along the positive and negative gage lines respectively upto a radial distance rmax. However, the extent of validity of Eq. (3.53) on positive gage line, say rmax need not necessarily be equal to the extent of validity of Eq. (3.54) on negative gage line say rmax . In order to take into account of Eqs.
(3.53) and (3.54), the maximum permissible (or upper bound) radial location or the extent of validity of both the Eqs. (3.53) and (3.54) (rmax), for a given specimen will then be
max minimum max, max
r r r (3.70)
Consequently, the optimal or valid radial locations ri (i 1, 2, 3, . .) for all strain gages both the gage lines in the proposed modified Dally and Berger technique can now be given as
min i max
r r r (3.71)