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Research Papers

Three-Dimensional Constraint Effects on the Slitting Method for Measuring Residual Stress

[+] Author and Article Information
C. Can Aydıner

Assistant Professor
Department of Mechanical Engineering,
Bogazici University,
Bebek,
Istanbul, 34342, Turkey
e-mail: can.aydiner@boun.edu.tr

Michael B. Prime

Fellow ASME
R&D Engineer
Los Alamos National Laboratory,
Los Alamos, NM 87545
e-mail: prime@lanl.gov

Accommodating this precise L is often possible since the sample is usually cut out from larger plates. This operation is also related to how L = 4 is picked; it is a length sufficiently large for σy (x) in the original plate to be predominantly preserved in the experimental sample.

Equation (4) applies rigorously for linear elastic fracture mechanics conditions.

For the dead load problems in a linear elastic body, the elastic energy of the system (U defined in Sec. 2.5) is increased by the released amount during crack extension [15].

It is natural to ask whether the peak portion of the x = 0 curve is also characteristic. Looking at the results of immediate neighbor δ-loads at x = 0.005, 0.01,…, this part of the curve does not remain invariant, but drops steadily with increasing x.

Contributed by the Materials Division of ASME for publication in the Journal of Engineering Materials and Technology. Manuscript received June 11, 2012; final manuscript received February 10, 2013; published online May 6, 2013. Assoc. Editor: Marwan K. Khraisheh.

J. Eng. Mater. Technol 135(3), 031006 (May 06, 2013) (10 pages) Paper No: MATS-12-1140; doi: 10.1115/1.4023849 History: Received June 11, 2012; Revised February 10, 2013

The incremental slitting or crack compliance method determines a residual stress profile from strain measurements taken as a slit is incrementally extended into the material. To date, the inverse calculation of residual stress from strain data conveniently adopts a two-dimensional, plane strain approximation for the calibration coefficients. This study provides the first characterization of the errors caused by the 2D approximation, which is a concern since inverse analyses tend to magnify such errors. Three-dimensional finite element calculations are used to study the effect of the out-of-plane dimension through a large scale parametric study over the sample width, Poisson's ratio, and strain gauge width. Energy and strain response to point loads at every slit depth is calculated giving pointwise measures of the out-of-plane constraint level (the scale between plane strain and plane stress). It is shown that the pointwise level of constraint varies with slit depth, a factor that makes the effective constraint a function of the residual stress to be measured. Using a series expansion inverse solution, the 3D simulated data of a representative set of residual stress profiles are reduced with 2D calibration coefficients to yield the error in stress. The sample width below which it is better to use plane stress compliances than plane strain is shown to be about 0.7 times the sample thickness; however, even using the better approximation, the rms stress errors sometimes still exceed 3% with peak errors exceeding 6% for Poisson's ratio 0.3, and errors increase sharply for larger Poisson's ratios. The error is significant, yet, error magnification from the inverse analysis in this case is mild compared to, e.g., plasticity based errors. Finally, a scalar correction (effective constraint) over the plane-strain coefficients is derived to minimize the root-mean-square (rms) stress error. Using the posed scalar correction, the error can be further cut in half for all widths and Poisson's ratios.

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Copyright © 2013 by ASME
Topics: Stress , Errors
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References

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Figures

Grahic Jump Location
Fig. 1

The (a) 2D and (b) 3D finite element domains of the slitting sample depicted as shaded regions in the top sketches of the entire sample. O is the origin of the chosen x-y-z coordinate system; SG stands for strain gauge; δ is a point/line load in (a)/(b).

Grahic Jump Location
Fig. 3

A considered 3D finite element mesh composed of wedge and hexahedron elements. All parameters are at norm values (B = 1, L = 4, w = 0) and the shown slit depth, a, is arbitrary. O is the origin (refer to Fig. 1(b) for notation) and the front face is placed in a rectangle also revealing the corresponding 2D mesh. The x-y-z triad is displaced from O for clarity.

Grahic Jump Location
Fig. 2

(a) The first four elements of the stress series (Legendre polynomials 2 to 5), (b) the considered residual stress profiles detailed in Table 2

Grahic Jump Location
Fig. 4

For B = 1 and ν = 0.3, constraint ratios (Eq. (19)) of (a) strain at the gauge, (b) depth-derivative of the strain at the gauge, and (c) energy release rate. Each curve in parts (a), (b), or (c) corresponds to fixing the δ-load (see Fig. 1) at the x value shown in the legend and plots the corresponding ratio as a function slit depth a. The pl-σ level is at (1-ν2)-1=1.099.

Grahic Jump Location
Fig. 5

The ratio of the measured strain, γ, in the 3D analysis to plane strain analysis as a function of slit depth a. The particular loading is a δ-load (see Fig. 1) at the top (x = 0). This figure is at the norm point of the parameter space ν = 0.3, etc., except for each curve corresponding to different sample widths as indicated.

Grahic Jump Location
Fig. 6

Error in calculated stresses due to utilizing plane strain (solid markers) and plane stress (hollow markers) assumptions (Eq. (16)) as a function of the thickness-normalized width (B) of the actual three-dimensional body: (a) residual stress profile (Legendre) series elements, P1, P4, P7, P11, root-mean-square error; (b),(c) residual stress profiles S1S4 (Table 2), root-mean-square error in part (b) and maximum error in part (c). The intersection point of the plane stress and plane strain curves is shown specifically for profile S1 in part (b) with dashed lines.

Grahic Jump Location
Fig. 7

(a) The effective constraint Γ (Eq. (27)) versus thickness-normalized sample width B for stress profiles S1S4 (Table 2) and a representative Lorentzian fit to the combined results labeled eq. (b) rms error in each case using the individual Γ corrections in part (a). Fit order is m = 11 except the indicated additional curve for S3 profile with m = 3. (c) rms error in each case using the Γ-correction of profile eq for all, m = 11. Inset: S1 curve of this plot put in perspective of pl-σ/pl-ε errors from Fig. 6.

Grahic Jump Location
Fig. 8

For Poisson's ratios 0.2,…, 0.4, plots of (a) rms-error in reducing profile S1 with plane strain compliances, (b) remnant rms-error after one-parameter correction with Eq. (28), versus thickness-normalized sample width B

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