Theoretical Background and Formulations
3.6 Determination of maximum permissible radial location r max in mixed mode (I/II)
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)
3.6 Determination of maximum permissible radial location
along the positive and negative gage lines (OS and OT in Fig. 3.9) using FEA of a given cracked configuration. The quantities on the LHS of Eqs. (3.61) and (3.62) i.e.
1
( aa bb)
E r
I
and
2
( aa bb)
E r
I
are then computed at those points using the FE solutions. It should be noted that, the radial distances from the crack tip to the corresponding points on the positive and negative gage lines should be same for computation of the LHS quantities of Eqs.(3.61) and (3.62) and meshes for FEA should be designed accordingly.
Using linear regression models, a curve of the form A01A r2 22B r1 3/2 (RHS of Eq. (3.61)) is fitted to the computed values of
1
( aa bb)
E r
I
at all the points on the positive gage line. Initially the fit will not be good due to the fact that the form A01A r2 22B r1 3/2 (with only three parameters) could accurately represent the quantity
1
( aa bb)
E r
I
only up to a certain radial distance from the crack-tip. Since this radial distance is unknown, the computed values of
1
( aa bb)
E r
I
at larger values of r are then to be deleted gradually and continuously from the data set until a best-fit curve is obtained. The value of coefficients A A0, 2 and B1 for the best-fit regression is noted. The same procedure is repeated with Eq. (3.62) in order to obtain the best-fit coefficients C C0, 1 and C2.
Consistent and accurate values of the above six unknown coefficients can be obtained by ensuring that
(a) the corresponding plots of LHS (obtained using FEA) and RHS quantities (obtained using best-fit coefficients) of Eqs. (3.61) and (3.62) should be congruent to each other to the maximum possible radial distance from the crack tip
(b) the percentage relative error between the LHS and RHS quantities of Eqs.(3.61) and (3.62) should be 0.5% within the maximum possible radial distance and
(c) the quality of fit defined by the coefficient of determination R2 should be very close to 1.
Using the best-fit regression values of A A0, 2,B1, C C0, 1 and C2 and FE values of Eaa and Ebb, the LHS and RHS of quantities of Eqs. (3.53) and (3.54) can be compared graphically with respect to the radial distance from the crack-tip for all points on the positive and negative gage line respectively. It is clear that the RHS quantities of Eqs. (3.53) and (3.54) can only accurately represent the LHS quantities (i.e.Eaa and Ebb obtained using FEA) up to a certain radial distance because of a few number of coefficients (or parameters) A A0, 2,B1, C C0, 1 and C2 are present in RHS quantities. The point of deviation of the RHS of Eq. (3.53) from the finite element values of Eaa indicates the rmax value for the given configuration. Percent relative error between the LHS and RHS of Eq. (3.53) can be employed for computation of the rmax . In the present investigation an error 0.5% (as given in mode I) is employed for obtaining the rmax or the point of deviation in the above graphical analysis. Similarly the rmax can be obtained from the graphical comparison of LHS and RHS quantities of Eq. (3.54) and using the error criterion as mentioned above.
Thus, the maximum permissible radial location for the strain gages (rmax) for a given configuration, given state of stress and a given Poisson’s ratio is the minimum of rmax and rmax (Eq. 3.70), which satisfies both Eqs. (3.53) and (3.54). Thus, three strain gages are to be pasted on each gage line for the determination of KI and KII using the proposed technique such that the radial distance of each gage should be greater than half the thickness of the plate (to avoid 3D effects [21, 22]) and less than the rmax value (as it is the maximum permissible distance) of a given configuration. In order to avoid errors due to the crack tip complications such as strain gradient,
plasticity and 3D effects it is recommended in the present investigation to paste the strain gages as far as possible from the crack tip but not beyond the rmax.
It is worth mentioning that apart from the effect of mesh gradation, the consistency and accuracy of evaluation of the rmax (and hence unknown coefficients
0, 2
A A ,B1, C C0, 1 and C2) also depend on how the best-fit process is carried out and the field variables that are employed in the best-fit process. It has been noticed from extensive numerical investigation that highly erroneous and inconsistent coefficients and hence rmax are obtained if Eq. (3.53) or Eq. (3.54) is directly used for the best-fit process instead of using Eqs. (3.61) and (3.62) as suggested in the present investigation. The above observation is also noticed even in the case of highly refined finite element meshes.
The first and important advantage of using Eqs. (3.61) and (3.62) is that they contain less number of unknown coefficients as compared to Eqs. (3.53) and (3.54).
Further, Eq. (3.62) needs only a quadratic best-fit and Eq. (3.61) needs a nearly quadratic fit which can be easily and efficiently carried out using commercially available software.