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Boundary Effects in the Eigenstrain Method

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We present a comprehensive study of the effects of internal boundaries on the accuracy of residual stress values obtained from the eigenstrain method. In the experimental part of this effort, a composite specimen, consisting of an aluminum cylinder sandwiched between steel cylinders of the same diameter, was uniformly heated under axial displacement constraint. During the experiment, the sample temperature and the reaction stresses in the load frame in response to changes in sample temperature were monitored. In addition, the local (elastic) lattice strain distribution within the specimen was measured using neutron diffraction. The eigenstrain method, utilizing finite element modeling, was then used to predict the stress field existing within the sample in response to the constraint imposed by the load frame against axial thermal expansion. Our comparison of the computed and measured stress distributions showed that, while the eigenstrain method predicted acceptable stress values away from the cylinder interfaces, its predictions did not match experimentally measured values near them. These observations indicate that the eigenstrain method is not valid for sample geometries with this type of internal boundaries.

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  1. 1.

    The term “eigenstrain” stems from the word “eigen” in German which means “inherent, particular, characteristic or peculiar”. Thus, the term “eigenstrain” can also be termed “inherent strain”. Eigenstrain is not related to eigenvalues or eigenvectors commonly encountered in physical and mathematical analysis. In German literature residual stresses are termed “Eigenspannungen” [2].

  2. 2.

    During the loading operation, the composite sample was contained in an axially split Al tube, ½” (12.6 mm) inner diameter and 4″ (101.6 mm) in length, to keep all three cylinders in alignment. After the ends of the outer steel cylinders were captured in the Al spacers, the sample was loaded in compression and the alignment tube was removed. After this point the sample was kept together by the applied compressive load and friction at the cylinder surfaces.

  3. 3.

    This was ensured by monitoring the diffraction spectra as the beam position was stepped over the interface. The chosen locations, bracketing the respective interfaces at ±3 mm, yielded only Al or Fe spectra to avoid partially-buried gage-volumes; these cause large errors in the measured strain values (Spooner & Wang, 1997) [34].

  4. 4.

    For brevity this discussion assumes a uniformly heated crystalline material with isotropic thermal and mechanical properties, in which all eigenstrain terms, except thermal strains, are zero.

  5. 5.

    We note that, in the case of an isothermal uniaxial compression test, the temperature change ΔT is zero, and the boundary constraint term, B c , obtained from equation (5) would also be zero as long as there are no additional constraints imposed by buried interfaces.

  6. 6.

    We note that the steel and aluminum material volumes immediately bordering both interfaces-and containing the steep interaction strain gradients- could not be interrogated using neutron diffraction due to possible positioning errors and the attendant “unfilled gage volume” issues [34].

  7. 7.

    Based on the axial distribution of the boundary interaction coefficients, Bc(x1), (Fig. 10(c)) this might be a weak assumption.

  8. 8.

    A virtual interface in a quasi-homogeneous solid such as a polycrystalline sample larger than the representative volume, delineates regions of different hardness, yield stress, grain size, texture, etc. formed through heterogeneous plastic flow caused by boundary conditions. An example can be seen in reference [39].


  1. 1.

    Mura T (1987) Micromechanics of Defects in Solids, 2nd edn. Martinus Nijhoff Publishers, Dordrecht

  2. 2.

    Reissner H (1931) Eigenspannungen und Eigenspannungsquellen. Z Angew Math Mech 11:1–8

  3. 3.

    Eshelby JD (1957) The Determination of the Elastic Field of an Ellipsoidal Inclusion, And Related Problems. Proc Royal Soc London Series a-Math Phys Sci 241:376–396

  4. 4.

    Eshelby JD (1959) The Elastic Field outside an Ellipsoidal Inclusion. Proc Royal Soc London Series a-Math Phys Sci 252:561–569

  5. 5.

    Fujimoto T (1970) A Method for Analysis of Residual Welding Stresses and Deformations Based on the Inherent Strain - A Theoretical Study of Residual Welding Stresses and Deformations (Report 1). J Jpn Weld Soc 39:236–252

  6. 6.

    Ueda Y, Fukuda K, Nakacho K, Endo S (1975) A New Measuring Method of Residual Stresses with the Aid of Finite Element Method and Reliability of Estimated Values. Trans Jpn Weld Res Inst 4:123–131

  7. 7.

    Hill MR, Nelson DV (1995) The Inherent Strain Method for Residual Stress Determination and Its Application to a Long Welded Joint. ASME-Publications-PVP 318:343–352

  8. 8.

    Hill MR, Nelson DV (1998) The Localized Eigenstrain Method for Determination of Triaxial Residual Stress in Welds. ASME-Publications-PVP 373:397–404

  9. 9.

    Luckhoo HT, Jun TS, Korsunsky AM (2009) Inverse Eigenstrain Analysis of Residual Stresses in Friction Stir Welds. Proc Eng (Mesomechanics 2009) 1:213–216

  10. 10.

    Jun TS, Dragnevski K, Korsunsky AM (2010) Microstructure, Residual Strain, and Eigenstrain Analysis of Dissimilar Friction Stir Welds. Mater Des 31:S121–S125

  11. 11.

    Korsunsky AM (2005) On The Modelling of Residual Stresses due to Surface Peening Using Eigenstrain Distributions. J Strain Anal Eng Des 40:817–824

  12. 12.

    Jun TS, Venter AM, Korsunsky AM (2011) Inverse Eigenstrain Analysis of the Effect of Non-uniform Sample Shape on the Residual Stress Due to Shot Peening. Exp Mech 51:165–174

  13. 13.

    Song X, Liu WC, Belnoue JP, Dong J, Wu GH, Ding WJ, Kimber SAJ, Buslaps T, Lunt AJG, Korsunsky AM (2012) An Eigenstrain-Based Finite Element Model and the Evolution of Shot Peening Residual Stresses During Fatigue of GW103 Magnesium Alloy. Int J Fatigue 42:284–295

  14. 14.

    Korsunsky AM (2006) Residual Elastic Strain Due to Laser Shock Peening: Modelling by Eigenstrain Distribution. J Strain Anal Eng Des 41:195–204

  15. 15.

    Achintha M, Nowell D (2011) Eigenstrain Modelling of Residual Stresses Generated by Laser Shock Peening. J Mater Process Technol 211:1091–1101

  16. 16.

    Hu YX, Grandhi RV (2012) Efficient Numerical Prediction of Residual Stress and Deformation for Large-Scale Laser Shock Processing Using the Eigenstrain Methodology. Surf Coat Technol 206:3374–3385

  17. 17.

    Achintha M, Nowell D, Shapiro K, Withers PJ (2013) Eigenstrain Modelling of Residual Stress Generated by Arrays of Laser Shock Peening Shots and Determination of the Complete Stress Field Using Limited Strain Measurements. Surf Coat Technol 216:68–77

  18. 18.

    Correa C, Gil-Santos A, Porro JA, Diaz M, Ocana JL (2015) Eigenstrain Simulation of Residual Stresses Induced by Laser Shock Processing in a Ti6Al4V Hip Replacement. Mater Des 79:106–114

  19. 19.

    Coratella S, Sticchi M, Toparli MB, Fitzpatrick ME, Kashaev N (2015) Application of the Eigenstrain Approach to Predict the Residual Stress Distribution in Laser Shock Peened AA7050-T7451 Samples. Surf Coat Technol 273:39–49

  20. 20.

    Jun TS, Korsunsky AM (2010) Evaluation of Residual Stresses and Strains Using the Eigenstrain Reconstruction Method. Int J Solids Struct 47:1678–1686

  21. 21.

    Luzin V (2014) Use of the Eigenstrain Concept for Residual Stress Analysis. Materials Science Forum (International Conference on Residual Stresses 9) 768–769, 193–200

  22. 22.

    Noyan IC, Cohen JB (1985) An X-Ray-Diffraction Study of the Residual-Stress Strain Distributions In Shot-Peened 2-Phase Brass. Mater Sci Eng 75:179–193

  23. 23.

    Noyan IC, Cohen JB (1987) Residual Stress: Measurement by Diffraction and Interpretation. Springer, Berlin

  24. 24.

    Schajer GS (1988) Measurement of Non-uniform Residual-stresses using the Hole-drilling method .1. Stress Calculation Procedures. J Eng Mater Technol-Trans ASME 110:338–343

  25. 25.

    Schajer GS (1988) Measurement of Non-uniform Residual-stresses using the Hole-drilling method. 2. Practical Application of the Integral method. J Eng Mater Technol-Trans ASME 110:344–349

  26. 26.

    Prime MB (2001) Cross-Sectional Mapping of Residual Stresses by Measuring the Surface Contour after a Cut. J Eng Mater Technol-Trans ASME 123:162–168

  27. 27.

    Prime MB, Gnaupel-Herold T, Baumann JA, Lederich RJ, Bowden DM, Sebring RJ (2006) Residual Stress Measurements in a Thick, Dissimilar Aluminum Alloy Friction Stir Weld. Acta Mater 54:4013–4021

  28. 28.

    Schajer GS (2010) Relaxation Methods for Measuring Residual Stresses: Techniques and Opportunities. Exp Mech 50:1117–1127

  29. 29.

    Woo W, An GB, Kingston EJ, DeWaldd AT, Smith DJ, Hill MR (2013) Through-Thickness Distributions of Residual Stresses in Two Extreme Heat-Input Thick Welds: A Neutron Diffraction, Contour Method and Deep Hole Drilling Study. Acta Mater 61:3564–3574

  30. 30.

    Metals Handbook (1990) Vol. 1 - Properties and Selection: Irons, Steels, and High-Performance Alloys, 10th edn. ASM International, Russell Township

  31. 31.

    Metals Handbook (1990) Vol. 2 - Properties and Selection: Nonferrous Alloys and Special-Purpose Materials, 10th edn. ASM International, Russell Township

  32. 32.

    Noyan IC, Brügger A, Betti R, Clausen B (2010) Measurement of Strain/Load Transfer in Parallel Seven-wire Strands with Neutron Diffraction. Exp Mech 50:265–272

  33. 33.

    Lee SY, Skorpenske H, Stoica AD, An K, Wang XL, Noyan IC 2014. Measurement of Interface Thermal Resistance with Neutron Diffraction. J Heat Transfer-Trans ASME 136(3):031302-1–031302-12

  34. 34.

    Spooner S, Wang XL (1997) Diffraction Peak Displacement in Residual Stress Samples due to Partial Burial of the Sampling Volume. J Appl Crystallogr 30:449–455

  35. 35.

    Vondreele RB, Jorgensen JD, Windsor CG (1982) Rietveld Refinement with Spallation Neutron Powder Diffraction Data. J Appl Crystallogr 15:581–589

  36. 36.

    Clausen B (2004) SMARTSware Manual. LA-UR 04–6581. Los Alamos National Laboratory, Los Alamos

  37. 37.

    Mei F, Noyan IC, Brügger A, Betti R, Clausen B, Brown D, Sisneros T (2013) Neutron Diffraction Measurement of Stress Redistribution in Parallel Seven-Wire Strands after Local Fracture. Exp Mech 53:183–193

  38. 38.

    Brügger A, Lee S-Y, Mills JAA, Betti R, Noyan IC (2017) Partitioning of Clamping Strains in a Nineteen Parallel Wire Strand. Exp Mech 57:921–937

  39. 39.

    Noyan IC (1988) Plastic Deformation of Solid Spheres. Philos Mag A-Phys Condens Matter Struct Defect Mech Prop 57:127–141

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This research effort was sponsored by the Air Force Research Laboratory, Aerospace Systems Directorate, under contracts FA8650-10-D-3037 and FA8650-12-D-3212, and has benefited from the use of the Lujan Neutron Scattering Center at LANSCE. Los Alamos National Laboratory is operated by Los Alamos National Security LLC under DOE Contract DE-AC52-06NA25396. The samples used in the study were manufactured at the Carleton Laboratory of Columbia University. MEF is grateful for funding from the Lloyd’s Register Foundation, a charitable foundation helping to protect life and property by supporting engineering-related education, public engagement and the application of research.

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Correspondence to I. C. Noyan.

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Lee, S., Coratella, S., Brügger, A. et al. Boundary Effects in the Eigenstrain Method. Exp Mech 58, 799–814 (2018). https://doi.org/10.1007/s11340-018-0378-3

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  • Eigenstrain
  • Residual stress
  • Neutron diffraction
  • Finite element analysis
  • Mechanical constraint
  • Boundary condition