Characterization of the stiffness distribution in two and three dimensions using boundary deformations: a preliminary study


We present for the first time the feasibility to recover the stiffness (here shear modulus) distribution of a three-dimensional heterogeneous sample using measured surface displacements and inverse algorithms without making any assumptions about local homogeneities and the stiffness distribution. We simulate experiments to create measured displacements and augment them with noise, significantly higher than anticipated measurement noise. We also test two-dimensional problems in plane strain with multiple stiff inclusions. Our inverse strategy recovers the shear modulus values in the inclusions and background well, and reveals the shape of the inclusion clearly.

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  1. 1.

    H.A. Bruck, S.R. McNeill, M.A. Sutton, and W.H. Peters: Digital image correlation using Newton-Raphson method of partial differential correction. Exp. Mech. 29, 261–267 (1989).

    Article  Google Scholar 

  2. 2.

    A. IIiopoulos, J.G. Michopoulos, A.C. Orifici, and R.S. Thomson: Experimental validation of the 2D meshless random grid method. In Proc. ASME IDETC/CIE, 513–520 (2011).

  3. 3.

    J.G. Michopoulos and A. IIiopoulos: A computational workbench for remote full field 3D displacement and strain measurements. In Proc. ASME IDETC/CIE, 489–498 (2011).

    Google Scholar 

  4. 4.

    A. IIiopoulos and J.G. Michopoulos: Meshless methods for full field displacement and strain measurement. Adv. Comput. Inf. Eng. Res. 1, 28 (2014).

    Google Scholar 

  5. 5.

    M. Stamborska, M. Kvícala, and M. Losertova: Identification of the mechanical properties of high strength steel using digital image correlation, advanced materials research. Adv. Mater. Res. 980, 122–126 (2014).

    Article  Google Scholar 

  6. 6.

    A. Kato: Measurement of strain distribution in metals for tensile test using digital image correlation method and consideration of stress-strain relation. Mech. Eng. J. 3, 16–00141 (2016).

    Article  Google Scholar 

  7. 7.

    Y. Mei, R. Fulmer, V. Raja, S. Wang, and S. Goenezen: Estimating the non-homogeneous elastic modulus distribution from surface deformations. Int. J. Solids Struct. 83, 73–80 (2016).

    Article  Google Scholar 

  8. 8.

    Y. Mei, S. Wang, X. Shen, S. Rabke, and S. Goenezen: Mechanics based tomography: a preliminary feasibility study. Sensors 17, 1075 (2017).

    Article  Google Scholar 

  9. 9.

    H.T. Liu, L.Z. Sun, G. Wang, and M.W. Vannier: Analytic modeling of breast elastography. Med. Phys. 30, 2340–2349 (2003).

    CAS  Article  Google Scholar 

  10. 10.

    D.S. Schnur and N. Zabaras: An inverse method for determining elastic material properties and material interface. Int. J. Numer. Methods Eng. 33, 2039–2057 (1992).

    Article  Google Scholar 

  11. 11.

    E.W. Elijah, V. Houten, A. Peters, and J.G. Chase: Phantom elasticity reconstruction with digital image elasto-tomography. J. Mech. Behav. Biomed. Mater. 4, 1741–1754 (2011).

    Article  Google Scholar 

  12. 12.

    J.N. Reddy: An Introduction to the Finite Element Method, 3rd ed. (McGraw-Hill Education, New York, USA, 2005).

    Google Scholar 

  13. 13.

    K. Frick, D. Lorenz, and E. Resmerita: Morozov’s principle for the augmented Lagrangian method applied to linear inverse problems. Multiscale Model. Simul. 9, 1528–1548 (2011).

    Article  Google Scholar 

  14. 14.

    C. Zhu, R.H. Byrd, P. Lu, and J. Nocedal: L-BFGS-B: a limited memory FORTRAN code for solving bound constrained optimization problems Technical Report, NAM-11, EECS Department, Northwestern University (1994).

  15. 15.

    S. Goenezen: Inverse problems in finite elasticity: An application to imaging the nonlinear elastic properties of soft tissues. Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, NY, 2011).

    Google Scholar 

  16. 16.

    S. Goenezen, P. Barbone, and A.A. Oberai: Solution of the nonlinear elasticity imaging inverse problem: the incompressible case. Comput. Methods Appl. Mech. Eng. 200, 1406–1420 (2011).

    Article  Google Scholar 

  17. 17.

    Y. Mei, B. Stover, N.A. Kazerooni, A. Srinivasa, M. Hajhashemkhani, M.R. Hematiyan, and S. Goenezen: A comparative study of two constitutive models within an inverse approach to determine the spatial stiffness distribution in soft materials. Int. J. Mech. Sci. 140, 446–454 (2018).

    Article  Google Scholar 

  18. 18.

    Y. Mei, M. Tajderi, and S. Goenezen: Regularizing biomechanical maps for partially known material properties. Int. J. Appl. Mech. 9, 1750020 (2017).

    Article  Google Scholar 

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The authors would like to acknowledge funding from the National Science Foundation under Grant No. CMMI #1663435 and thank for their support. The authors would also like to thank the High Performance Research Computing at Texas A&M University for their computing resources.

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Luo, P., Mei, Y., Kotecha, M. et al. Characterization of the stiffness distribution in two and three dimensions using boundary deformations: a preliminary study. MRS Communications 8, 893–902 (2018).

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