Skip to main content
SpringerLink
Log in

Complex variable step method for sensitivity analysis of effective properties in multi-field micromechanics

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

This paper shows an approach to computing the effective properties of multi-field composite materials and their first-order sensitivities. The approach is based on the application of the complex variable step method for the micromechanical Mori–Tanaka scheme; hence, the first-order sensitivities can be computed in the same analysis. Numerical results are presented for magnetoelectroelastic properties of piezoelectric–piezomagnetic composite materials. A comparison of the results to those obtained by other methods shows that the presented sensitivity analysis gives highly accurate and stable results, but the values of the results are dependent on the applied micromechanical model. The presented approach may be used to solve ill-posed problems of optimal design or identification in coupled fields micromechanics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aboudi J.: Micromechanical analysis of fully coupled electro-magneto-thermo-elastic multiphase composites. Smart Mater. Struct. 10, 867–877 (2001)

    Article  Google Scholar 

  2. Abreu R., Stich D., Morales J.: On the generalization of the complex step method. J. Comput. Appl. Math. 241, 84–102 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Al-Mohy A.H., Higham N.J.: The complex step approximation to the Fréchet derivative of a matrix function. Numer. Algorithm 53, 133–148 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Atkinson K., Han W.: Elementary Numerical Analysis. Wiley, Hoboken (2004)

    Google Scholar 

  5. Benveniste Y.: A new approach to the application of Mori–Tanaka’s theory in composite materials. Mech. Mater. 6, 147–157 (1987)

    Article  Google Scholar 

  6. Budiansky B.: On the elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids 13, 223–227 (1965)

    Article  Google Scholar 

  7. Cacuci D.G.: Sensitivity and Uncertainty Analysis. Volume I: Theory. Chapman and Hall/CRC, Boca Raton (2003)

    Book  Google Scholar 

  8. Callahan J.J.: Advanced Calculus: A Geometric View. Springer, New York (2010)

    Book  Google Scholar 

  9. Dunn M.L., Taya M.: An analysis of piezoelectric composite materials containing ellipsoidal inhomogeneities. Proc. R. Soc. Lond. A443, 265–287 (1993)

    Article  Google Scholar 

  10. Dunn M.L., Taya M.: Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites. Int. J. Solids Struct. 30, 161–175 (1993)

    Article  MATH  Google Scholar 

  11. El Messiry M.: Theoretical analysis of natural fiber volume fraction of reinforced composites. Alex. Eng. J. 52, 301–306 (2013)

    Article  Google Scholar 

  12. Eshelby J.D.: The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. R. Soc. Lond. A241, 376–396 (1957)

    Article  MathSciNet  Google Scholar 

  13. Gao X.-W., He M.-C.: A new inverse analysis approach for multi-region heat conduction BEM using complex-variable-differentiation method. Eng. Anal. Bound. Elem. 29, 788–795 (2005)

    Article  MATH  Google Scholar 

  14. Gautschi W.: Numerical Analysis. Springer, New York (2012)

    Book  MATH  Google Scholar 

  15. Griewank A., Walther A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. SIAM, Philadelphia (2008)

    Book  Google Scholar 

  16. Hill R.: Theory of mechanical properties of fibre-strengthened materials—III. Self-consistent model. J. Mech. Phys. Solids 13, 189–198 (1965)

    Article  Google Scholar 

  17. Huang J.H., Liu H.-K., Dai W.-L.: The optimized fiber volume fraction for magnetoelectric coupling effect in piezoelectric–piezomagnetic continuous fiber reinforced composites. Int. J. Eng. Sci. 38, 1207–1217 (2000)

    Article  Google Scholar 

  18. Huang J.H., Chiu Y.-H., Liu K.-H.: Magneto-electro-elastic Eshelby tensors for a piezoelectric–piezomagnetic composite reinforced by ellipsoidal inclusions. J. Appl. Phys. 83, 5364–5370 (1998)

    Article  Google Scholar 

  19. Jin W., Dennis B.H., Wang B.P.: Improved sensitivity analysis using a complex variable semi-analytical method. Struct. Multidiscip. Optim. 41, 433–439 (2010)

    Article  Google Scholar 

  20. Kiran R., Khandelwal K.: Complex step derivative approximation for numerical evaluation of tangent moduli. Comput. Struct. 140, 1–13 (2014)

    Article  Google Scholar 

  21. Koutsawa Y., Belouettar S., Makradi A., Nasser H.: Sensitivities of effective properties computed using micromechanics differential schemes and high-order Taylor series: application to piezo-polymer composites. Mech. Res. Commun. 37, 489–494 (2010)

    Article  MATH  Google Scholar 

  22. Koutsawa Y., Belouettar S., Makradi A., Tiem S.: Automatic differentiation of micromechanics incremental schemes for coupled fields composite materials: effective properties and their sensitivities. Compos. Sci. Technol. 71, 113–121 (2011)

    Article  Google Scholar 

  23. Koutsawa Y., Belouetta S., Makradi A., Tiem S.: Generalization of the micromechanics multi-coating approach to coupled fields composite materials with eigenfields: effective properties. Mech. Res. Commun. 38, 45–51 (2011)

    Article  MATH  Google Scholar 

  24. Kröner E.: Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls. Z. Phys. 151, 504–518 (1958)

    Article  Google Scholar 

  25. Lai K.-L., Crassidis J.L.: Extensions of the first and second complex-step derivative approximations. J. Comput. Appl. Math. 219, 276–293 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Laporte E., Le Tallec P.: Numerical Methods in Sensitivity Analysis and Shape Optimization. Springer, New York (2003)

    Book  MATH  Google Scholar 

  27. Lin C.-H., Muliana A.: Micromechanics models for the effective nonlinear electro-mechanical responses of piezoelectric composites. Acta Mech. 224, 1471–1492 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lin X.-J., Zhou K.-C., Zhang X.-Y., Zhang D.: Development, modeling and application of piezoelectric fiber composites. Trans. Nonferr. Met. Soc. China 23, 98–107 (2013)

    Article  Google Scholar 

  29. Liu L.P., Kuo H.-Y.: Closed-form solutions to the effective properties of fibrous magnetoelectric composites and their applications. Int. J. Solids Struct. 49, 3055–3062 (2012)

    Article  Google Scholar 

  30. Lyness J., Moler C.: Numerical differentiation of analytic functions. J. Numer. Anal. 4, 202–210 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  31. Malakooti M.H., Sodano H.A.: Multi-inclusion modeling of multiphase piezoelectric composites. Compos. Part B 47, 181–189 (2013)

    Article  Google Scholar 

  32. Martins J., Kroo I., Alonso J.: An automated method for sensitivity analysis using complex variable. Am. Inst. Aeronaut. Astronaut. 38, 1–12 (2000)

    Article  Google Scholar 

  33. McLaughlin R.: A study of the differential scheme for composite materials. Int. J. Eng. Sci. 15, 237–244 (1977)

    Article  MATH  Google Scholar 

  34. Mori T., Tanaka K.: Average stress in matrix and average energy of materials with misfitting inclusions. Acta Metall. 21, 571–574 (1973)

    Article  Google Scholar 

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

  36. Nemat-Nasser S., Hori M.: Micromechanics: Overall Properties of Heterogeneous Materials. Elsevier, Amsterdam (1993)

    MATH  Google Scholar 

  37. Odegard G.M.: Constitutive modeling of piezoelectric polymer composites. Acta Mater. 52, 5315–5330 (2004)

    Article  Google Scholar 

  38. Ptaszny J., Dziatkiewicz G., Fedeliński P.: Boundary element method modelling of nanocomposites. Int. J. Multiscale Comput. Eng. 12, 33–43 (2014)

    Article  Google Scholar 

  39. Qu J., Cherkaoui M.: Fundamentals of Micromechanics of Solids. Wiley, Hoboken (2006)

    Book  Google Scholar 

  40. Rosen, C.Z., Hiremath, B.V., Newnham, R. (eds.): Piezoelectricity. American Institute of Physics, New York (1992)

    Google Scholar 

  41. Sakthivel M., Arockiarajan A.: Thermo-electro-mechanical response of 1–3–2 piezoelectric composites: effect of fiber orientations. Acta Mech. 223, 1353–1369 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  42. Salganik R.L.: Mechanics of bodies with many cracks. Mech. Solids (Engl. Transl.) 8, 135–143 (1973)

    Google Scholar 

  43. Silva E.C.N., Fonseca J.S.O., Kikuchi N.: Optimal design of piezoelectric microstructures. Comput. Mech. 19, 397–410 (1997)

    Article  MATH  Google Scholar 

  44. Squire W., Trapp G.: Using complex variables to estimate derivatives of real functions. SIAM 40, 110–112 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  45. Srinivas S., Li J.Y., Zhou Y.C., Soh A.K: The effective magnetoelectroelastic moduli of matrix-based multiferroic composites. J. Appl. Phys. 99, 043905 (2006)

    Article  Google Scholar 

  46. Stanley L.G., Stewart D.L.: Design Sensitivity Analysis: Computational Issues of Sensitivity Equation Methods. SIAM, Philadelphia (2002)

    Book  Google Scholar 

  47. Varma A., Morbidelli M., Wu H.: Parametric Sensitivity in Chemical Systems. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  48. Vieville P., Bonnet A.S., Lipinski P.: Modelling effective properties of composite materials using the inclusion concept. General considerations. Arch. Mech. 58, 207–239 (2006)

    MATH  MathSciNet  Google Scholar 

  49. Voorhees A., Millwater H., Bagley R.: Complex variable methods for shape sensitivity of finite element models. Finite Elem. Anal. Des. 47, 1146–1156 (2011)

    Article  Google Scholar 

  50. Voorhees A., Millwater H., Bagley R., Golden P.: Fatigue sensitivity analysis using complex variable methods. Int. J. Fatigue 40, 61–73 (2012)

    Article  Google Scholar 

  51. Wu T.-L., Huang H.L.: Closed-form solutions for the magnetoelectric coupling coefficients in fibrous composites with piezoelectric and piezomagnetic phases. Int. J. Solids Struct. 37, 2981–3009 (2000)

    Article  MATH  Google Scholar 

  52. Wu T.-L.: Micromechanics determination of electroelastic properties of piezoelectric materials containing voids. Mater. Sci. Eng. A 280, 320–327 (2000)

    Article  Google Scholar 

  53. Zouari R., Benhamida A., Dumontet H.: A micromechanical iterative approach for the behavior of polydispersed composites. Int. J. Solids Struct. 45, 3139–3152 (2008)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computational Mechanics and Engineering, Faculty of Mechanical Engineering, Silesian University of Technology, ul. Konarskiego 18A, 44-100, Gliwice, Poland

    Grzegorz Dziatkiewicz

Authors
  1. Grzegorz Dziatkiewicz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Grzegorz Dziatkiewicz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dziatkiewicz, G. Complex variable step method for sensitivity analysis of effective properties in multi-field micromechanics. Acta Mech 227, 11–28 (2016). https://doi.org/10.1007/s00707-015-1419-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-015-1419-y

Keywords

Search

Navigation