An Analytical Solution for Inverse Determination of Residual Stress Field

Journal of Solid Mechanics

مقاله 1، دوره 4، شماره 2، شهریور 2012، صفحه 114-127   

نوع مقاله: Research Paper

نویسندگان

S.A Faghidian^{* 1}؛ G.H Farrahi²؛ D.J Smith³

¹Department of Mechanical and Aerospace Engineering, Science and Research Branch, Islamic Azad University

²School of Mechanical Engineering, Sharif University of Technology

³Department of Mechanical Engineering, University of Bristol

چکیده

An analytical solution is presented that reconstructs residual stress field from limited and incomplete data.  The inverse problem of reconstructing residual stresses is solved using an appropriate form of the airy stress function.  This function is chosen to satisfy the stress equilibrium equations together with the boundary conditions for a domain within a convex polygon.  The analytical solution is demonstrated by developing a reference solution from which selected “measurement” points are used. An artificial error is then randomly added to “measurement” points for studying the stability of the reconstruction method utilizing Tikhonov-Morozov regularization technique. It is found that there is an excellent agreement between the model prediction and limited set of residual stress data in the sense of least-square approximation.

کلیدواژه‌ها

Residual stress؛ Incomplete discrete data؛ Inverse problems؛ Approximation theory؛ Convex polygon domain

مراجع

[1] Smith D.J., Farrahi G.H., Zhu W X., McMahon C.A., 2001, Obtaining multiaxial residual stress distributions from limited measurements, Materials Science and Engineering A 303: 281–291.

[2] Hill M.R., 1996, Determination of residual stress based on the estimation of eigenstrain, Ph.D. Thesis. Stanford University.

[3] Cao Y.P., Hu N., Lu J., Yao Z.H., 2002, An inverse approach for constructing residual stress field induced by welding, Journal of Strain Analysis for Engineering Design 37: 345–359.

[4] Qian X., Yao Z., Cao Y., Lu J., 2005, An inverse approach to construct residual stresses existing in axisymmetric structures using BEM, Journal of Engineering Analysis with Boundary Elements 29: 986–99.

[5] Korsunsky A.M., Regino G.M., Nowell D., 2007, Variational eigenstrain analysis of residual stresses in a welded plate, International Journal of Solids and Structures 44(13): 4574–4591.

[6] Jun T.S., Korsunsky A.M., 2010, Evaluation of residual stresses and strains using the eigenstrain reconstruction method, International Journal of Solids and Structures 47:1678–1686.

[7] Ballard P., Constantinescu A., 1994, On the inversion of subsurface residual stresses from surface stress measurements, Journal of the Mechanics and Physics of Solids 42(11): 1767-1787.

[8] Schajer G.S., Prime M.B., 2006, Use of inverse solutions for residual stress measurements, Journal of Engineering Materials and Technology 128: 375-382.

[9] Hoger A., 1986, On the determination of residual stress in an elastic body, Journal of Elasticity 16: 303-324.

[10] Mura T., Gao Z., 1991, Inverse method in micromechanics of defects in solids, Proceedings of the 3rd International Conference on Residual Stress 1157-1167.

[11] Robertson R.L., 1998, Determining residual stress from boundary measurements: a linearized approach, Journal of Elasticity 52: 63–73.

[12] Moore M.G., Evans W.P., 1958, Mathematical correction for stress in removed layers in X-ray diffraction residual stress analysis, S A E Transactions 66: 340-345.

[13] Faghidian S.A., 2010, Estimation of Residual Stress Field Using Limited Measurements, Ph.D. Thesis. Sharif University of Technology, Tehran, Iran.

[14] Farrahi G.H., Faghidian S.A., Smith D.J., 2009, An inverse approach to determination of residual stresses induced by shot peening, International Journal of Mechanical Science 51: 726–731.

[15] Farrahi G.H., Faghidian S.A., Smith D.J., 2009, Reconstruction of residual stresses in autofrettaged thick-walled tubes from limited measurements, International Journal of Pressure Vessel and Piping 86: 777-784.

[16] Farrahi G.H., Faghidian S.A., Smith D.J., 2009, A new analytical approach to reconstruct residual atresses due to turning Process, Proceeding of World Academy of Science Engineering and Technology 55: 453-457.

[17] Farrahi G.H., Faghidian S.A., Smith D.J., 2010, An inverse method for reconstruction of the residual stress field in welded plates, Journal of Pressure Vessel Technology - Transactions of the ASME 132: 061205-1:9.

[18] Faghidian S.A., Goudar D., Farrahi G.H., Smith D.J., 2011, Measurement analysis and reconstruction of residual stresses, Journal of Strain Analysis for Engineering Design 47(4): 254-264.

[19] Gurtin M.E., 1972, The Linear Theory of Elasticity, Handbuch der Physik, Vol. VIa/2. Springer-Verlag, Berlin.

[20] Cheney E.W., 1982, Introduction to Approximation Theory,American Mathematical Society, Chelsea Publishing, New York.

[21] Groetsch C.W., 2007, Stable Approximate Evaluation of Unbounded Operators, Springer, Berlin.

[22] Kandil F.A., Lord J.D., Fry A.T., Grant P.V., 2001, A Review of Residual Stress Measurement Methods a Guide to Technique Selection, MATC (A) 04, UK National Physical Laboratory (NPL) Materials Centre report .

[23] King A.C., Billingham J., Otto S.R., 2003, Differential Equations: Linear, Nonlinear, Ordinary, Partial. Cambridge University Press, Cambridge.

[24] Hirota R.Y., 2004,The Direct Method in Soliton Theory, Cambridge University Press, Cambridge.

[25] Demmel J.W., 1997, Applied Numerical Linear Algebra, SIAM (Society for Industrial and Applied Mathematics), Philadelphia.