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Boundary Effects in the Eigenstrain Method

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Abstract

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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Notes

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

  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].

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Acknowledgements

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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Authors and Affiliations

  1. Applied Physics and Applied Mathematics, Columbia University, New York, NY, 10027, USA

    S.-Y. Lee & I. C. Noyan

  2. Aerospace Mechanics Division, University of Dayton Research Institute, Dayton, OH, 45469, USA

    S. Coratella

  3. Civil Engineering and Engineering Mechanics, Columbia University, New York, NY, 10027, USA

    A. Brügger

  4. MST-8, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    B. Clausen & D.W. Brown

  5. Air Force Research Laboratory, AFRL/RQVS, Wright-Patterson AFB, OH, 45433, USA

    K. Langer

  6. Faculty of Engineering and Computing, Coventry University, Gulson Rd, Coventry, CV1 2JH, UK

    M. E. Fitzpatrick

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  1. S.-Y. Lee
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Lee, SY., 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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