SCNU - Side Cut Non-Uniform residual stress measurement method

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Releases

24-JAN-2016: SCNURS 1.0, initial public release.

Publications and documentation

2016: H.K. Kim, M.J. Pavier, A. Shterenlikht, Measurement of Highly Non-uniform Residual Stress Fields in Thin Plate Using a New Side Cut Destructive Method, Proc. Residual Stresses 2016 – ICRS-10, Eds. Thomas M. Holden, Ondrej Muransky, Lyndon Edwards, Materials Research Proceedings 2(2016):115-120 DOI: 10.21741/9781945291173-20
2016: A. Shterenlikht, H. K. Kim, Forward and inverse solutions for 2D semi-infinite strip with self-equilibrated end loading, README, the most detailed explanation of the method and the code, work in progress.
2016: A. Shterenlikht, Finding roots of sinx+x=0 and sinx-x=0, short note, PDF
2016: H.K. Kim, A. Shterenlikht, M.J. Pavier, A. Velichko, N.A. Alexander, On stability of a new side cut destructive method for measuring non-uniform residual stress in thin plates, Int. J Solids Struct, 100-101(2016):223-233, DOI: 10.1016/j.ijsolstr.2016.08.019
2015: H. K. Kim, BSSM Young stress analysis competition talk.
2015: H. K. Kim, M. J. Pavier, A. Shterenlikht, Highly non-uniform residual stress measurement with reduced plastic error, Mech Eng Dept, The University of Bristol, UK, poster.
2015: H. K. Kim, H. E. Coules, M. J. Pavier, A. Shterenlikht, Measurement of Highly Non-Uniform Residual Stress Fields with Reduced Plastic Error, Exp. Mech. 55(7):1211-1224, DOI: 10.1007/s11340-015-0025-1, PDF.
2014: H. K. Kim, M. J. Pavier, A. Shterenlikht, Plasticity and stress heterogeneity influence on mechanical stress relaxation residual stress measurements, Proc. of 9th European Conference on Residual Stresses (ECRS), Troyes, France, 7-10 July 2014, Advanced Materials Research, 996:249-255, 2014, DOI: 10.4028/www.scientific.net/AMR.996.249, paper and talk.
2013: H. K. Kim, M. J. Pavier, A. Shterenlikht, Measuring locally non-uniform in-plane residual stress with straight cuts and DIC, Proc. 9th Int. Conf. Advances in Exp. Mechanics, 3-5 September 2013, Cardiff, Wales, UK, The British Society for Strain Measurements (BSSM), paper and talk.
2000: I. A. Razumovskii, A. L. Shterenlikht, Determining the locally-nonuniform residual-stress fields in plane parts by the sectioning method, J. Machinery Manufacture 4:40-45, translated from Russian, Allerton Press, USA, ISSN: 1052-6188.

The elastic problem

A 2D problem is assumed (x₁, x₂). Residual stress is assumed to be constant through thickness, x₃. The plate length is assumed to be much greater then width. The method is as follows:

Cut on x1=0 plane. Note that cut can be progressed along x2 or along x3, with different effects on plastic flow.
Measure relaxation disp. fields, u1 and u2, e.g. with DIC.
Use Mathieu series 2D elastic solution to the measured displacements.

See above our papers on the analysis of plastic flow on the measured RS fields.

Roots of the BC constraint equations

Enforcing the BC gives these 2 equations:

$\sin 2 x + 2 x = 0$

$\sin 2 x - 2 x = 0$

Each equation has an infinite number of complex roots, which can be plotted on the complex plane:

roots of the sin2x+2x=0, sin2x-2x=0

Simulated results with no error

Self-equlibrated loading is applied on the end. The forward problem is solved either analytically, or with FE. The result is u1 and u2 disp. fields. Then the inverse problem is solved and the series coefficients are calculated from the displacements using LLS or rank-deficient minimum norm LLS. We use Lapack.

In all plots the applied stress fields are shown with symbols. The reconstructed fields are shown with lines.

Symmetric step res. stress field with 2 discontinuities

50 terms

Anti-symm. step res. stress field with 3 discontinuities

49 terms, anti-symmetric loading

Generic step res. stress field with 3 discontinuities

49 terms, mixed loading

Slant res. stress field with 1 discontinuity

49 terms, slant loading

Smooth symmetric res. stress field - no discontinuities

80 terms, smooth symmetric loading

Smooth anti-symmetric res. stress field - no discontinuities

49 terms, smooth symmetric loading

Examples of instability

The inverse problem is ill-posed i.e. ill-conditioned. These are various manifestations.

5151 data points, 2 displacement values for each point, so 5151*2 = 10302 equations in total.

Symmetric loading

10 terms (40 unknowns) - too coarse!

The matrix is 10302 x 40.

10 terms

50 terms (200 unknowns) - just right!

The matrix is 10302 x 200.

50 terms

80 terms (320 unknowns) - getting worse!

The matrix is 10302 x 320.

80 terms

100 terms (400 unknowns) - too oscillatory!

The matrix is 10302 x 400.

100 terms

Anti-symmetric loading

10 terms (40 unknowns) - too coarse!

The matrix is 10302 x 40.

10 terms, anti-symmetric loading

40 terms (160 unknowns) - better!

The matrix is 10302 x 160.

40 terms, anti-symmetric loading

49 terms (196 unknowns) - best!

The matrix is 10302 x 196.

49 terms, anti-symmetric loading

50 terms (200 unknowns) - bad!

The matrix is 10302 x 200.

50 terms, anti-symmetric loading

100 terms (400 unknowns) - hopeless!

The matrix is 10302 x 400.

100 terms, anti-symmetric loading

Mixed loading

10 terms (40 unknowns) - too coarse!

The matrix is 10302 x 40.

10 terms, mixed loading

40 terms (160 unknowns) - better!

The matrix is 10302 x 160.

40 terms, mixed loading

49 terms (196 unknowns) - best!

The matrix is 10302 x 196.

49 terms, mixed loading

50 terms (200 unknowns) - bad!

The matrix is 10302 x 200.

50 terms, mixed loading

80 terms (320 unknowns) - hopeless!

The matrix is 10302 x 320.

80 terms, mixed loading

Slant loading

10 terms (40 unknowns) - too coarse!

The matrix is 10302 x 40.

10 terms, slant loading

40 terms (160 unknowns) - better!

The matrix is 10302 x 160.

40 terms, slant loading

49 terms (196 unknowns) - best!

The matrix is 10302 x 196.

49 terms, slant loading

50 terms (200 unknowns) - bad!

The matrix is 10302 x 200.

50 terms, slant loading

80 terms (320 unknowns) - hopeless!

The matrix is 10302 x 320.

80 terms, slant loading

100 terms (400 unknowns) - actually beautiful!

The matrix is 10302 x 400.

100 terms, slant loading

Smooth symmetric loading

5 terms (20 unknowns) - good!

The matrix is 10302 x 20.

5 terms, smooth symmetric loading

80 terms (320 unknowns) - very good!

The matrix is 10302 x 320.

80 terms, smooth symmetric loading

100 terms (400 unknowns) - getting worse!

The matrix is 10302 x 400.

100 terms, smooth symmetric loading

Smooth anti-symmetric loading

10 terms (40 unknowns) - good!

The matrix is 10302 x 40.

10 terms, smooth symmetric loading

40 terms (160 unknowns) - still good!

The matrix is 10302 x 160.

40 terms, smooth symmetric loading

49 terms (196 unknowns) - still fine!

The matrix is 10302 x 196.

49 terms, smooth symmetric loading

50 terms (200 unknowns) - bad!

The matrix is 10302 x 200.

50 terms, smooth symmetric loading

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