Effects of strain gradient and electromagnetic field gradient on potential and field distributions of multiferroic fibrous composites


The objective of this work is to study the effect of strain gradient, electric field gradient, and magnetic field gradient on the potential and field distributions of multiferroic fibrous composites subjected to generalized anti-plane shear deformation. A detailed energy variational formulation with strain, strain gradient, electric field, electric field gradient, magnetic field, and magnetic field gradient as independent variables is provided, and the equilibrium equations with the complete boundary conditions are simultaneously determined. Three internal characteristic lengths of the underlying microstructure are introduced in the constitutive equations. The general solutions in cylindrical polar coordinates consisting of the modified Bessel functions of first and second kind are derived. One inclusion problem’s solution is also obtained. The derived solutions are applied to a \(\hbox {CoFe}_{2}\hbox {O}_{4}\)-PZT-4 composite to demonstrate the effect of strain gradient, electric field gradient, and magnetic field gradient. Numerical results provide insights into the role of characteristic lengths on the potential and field distributions of a multiferroic fibrous composite.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14


a :

Radius of the inclusion, m

\(c_{ijkl}\) :

Elastic constant, N/m2

\(e_{ijk}\) :

Piezoelectric coefficient, C/m2

\(f_{i}^{0}\) :

Body force, N/m3

l :

Mechanical length scale parameter, m

\(m^{0}\) :

Body free magnetic pole density, Ns/m2C

\(n_{l}\) :

Outward unit vector, −

\(q^{0}\) :

Body free charge density, C/m3

\(q_{ijk}\) :

Piezomagnetic coefficient, N/Am

\(s_{i}^{0}\) :

Double traction, N/m

\(t_{i}^{0}\) :

Traction, N/m2

\(t_{ij}\) :

Linear elastic stress, N/m2

\(u_{i}\) :

Elastic displacement, m

\(B_{i}\) :

Total magnetic flux density, N/Am

\(D_{i}\) :

Total electric displacement, C/m2

\(E_{i}\) :

Electric field, V/m

\(F_{ijkl}\) :

Electric field gradient material constant, C2/N

\(G_{ijkl}\) :

Magnetic field gradient material constant, Nm2s2/C2

\(H_{i}\) :

Magnetic field, A/m

\(\mathcal {I}\) :

Energy functional, Nm

\(J_{ijmkln}\) :

Strain gradient material constant, N

\(P_{i}^{E}\) :

Total electric polarization density, C/m2

\(P_{i}^{M}\) :

Total magnetic polarization density, N/Am

W :

Internal energy density function, N/m2

\(\overline{\beta }_{i}^{0}\) :

Surface magnetic pole density, Ns/mC

\(\gamma _{zj}\) :

Engineering shear strain, −

\(\delta _{ij}\) :

Kronecker delta, −

\(\varepsilon _{ij}\) :

Elastic strain, −

\(\zeta \) :

Magnetic length scale parameter, m

\(\kappa ^{0}\) :

Vacuum permittivity, C2/Nm2

\(\kappa _{ij}\) :

Dielectric permittivity, C2/Nm2

\(\lambda _{ij}\) :

Magnetoelectric coupling coefficient, Ns/VC

\(\mu ^{0}\) :

Vacuum permeability, Ns2/C2

\(\mu _{ij}\) :

Magnetic permeability, Ns2/C2

\(\nu _{ijm}\) :

Double stress, N/m

\(\overline{\pi }_{i}^{0}\) :

Surface electric dipole polarization density, C/m

\(\pi _{i}\) :

Electric dipole polarization density, Cm2

\(\pi _{ik}\) :

Electric quadrupole polarization density, Cm3

\(\rho \) :

Electric length scale parameter, m

\(\overline{\sigma }^{0}\) :

Surface charge density, C/m2

\(\sigma _{ij}\) :

Total stress, N/m2

\(\varphi \) :

Electric potential, V

\(\chi _{ij}^{E}\) :

Electric susceptibility, −

\(\chi _{ij}^{M}\) :

Magnetic susceptibility, −

\(\psi \) :

Magnetic potential, C/s

\(\overline{\omega }_{i}^{0}\) :

Surface magnetic dipole polarization density, Ns/C

\(\omega _{i}\) :

Magnetic dipole polarization density, N/Am

\(\omega _{ik}\) :

Magnetic quadrupole polarization density, N/A

\(\Gamma _{mp}\) :

Electric second-order length scale tensor, m2

\(\Lambda _{mp}\) :

Magnetic second-order length scale tensor, m2

\(\Upsilon _{mp}\) :

Mechanical second-order length scale tensor, m2


  1. 1.

    Aifantis, E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30, 1279–1299 (1992)

    MATH  Article  Google Scholar 

  2. 2.

    Aifantis, E.C.: A concise review of gradient models in mechanics and physics. Front. Phys. 7, 239 (2020)

    Article  Google Scholar 

  3. 3.

    Arvanitakis, A.I., Kalpakides, V.K., Hadjigeorgiou, E.P.: Electric field gradients and spontaneous quadrupoles in elastic ferroelectrics. Acta Mech. 218, 269–294 (2011)

    MATH  Article  Google Scholar 

  4. 4.

    Askes, H., Aifantis, E.C.: Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct. 48, 1962–1990 (2011)

    Article  Google Scholar 

  5. 5.

    Altan, B.S., Aifantis, E.: On some aspects in the special theory of gradient elasticity. J. Mech. Behav. Mater. 8, 231–282 (1997)

    Article  Google Scholar 

  6. 6.

    Bai, X., Wen, Y., Yang, J., Li, P., Qiu, J., Zhu, Y.: A magnetoelectric energy harvester with the magnetic coupling to enhance the output performance. J. Appl. Phys. 111, 07A938 (2012)

    Article  Google Scholar 

  7. 7.

    Bühlmann, S., Dwir, B., Baborowski, J., Muralt, P.: Size effect in mesoscopic epitaxial ferroelectric structures: increase of piezoelectric response with decreasing feature size. Appl. Phys. Lett. 80, 3195–3197 (2002)

    Article  Google Scholar 

  8. 8.

    Cao, W., Yang, X., Tian, X.: Anti-plane problems of piezoelectric material with a micro-void or micro-inclusion based on micromorphic electroelastic theory. Int. J. Solids. Struct. 49, 3185–3200 (2012)

    Article  Google Scholar 

  9. 9.

    Cardano, G., Witmer, T.R., Ore, O.: The Rules of Algebra: Ars Magna. Dover, New York (2007)

    Google Scholar 

  10. 10.

    Cordero, N.M., Gaubert, A., Forest, S., Busso, E.P., Gallerneau, F., Kruch, S.: Size effects in generalized continuum crystal plasticity for two-phase laminates. J. Mech. Phys. Solids 58, 1963–1994 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Eerenstein, W., Mathur, N.D., Scott, J.F.: Multiferroic and magnetoelectric materials. Nature 442, 759–765 (2006)

    Article  Google Scholar 

  12. 12.

    Eringen, A.C., Suhubi, E.S.: Nonlinear theory of simple micro-elastic solids I. Int. J. Eng. Sci. 2, 159–203 (1964a)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Eringen, A.C., Suhubi, E.S.: Nonlinear theory of simple micro-elastic solids II. Int. J. Eng. Sci. 2, 389–404 (1964b)

    MATH  Article  Google Scholar 

  14. 14.

    Eringen, A.C.: Microcontinuum Field Theories. Springer, New York (1999)

    Google Scholar 

  15. 15.

    Eringen, A.C.: Continuum theory of micromorphic electromagnetic thermoelastic solids. Int. J. Eng. Sci. 41, 653–665 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Feng, W.J., Pan, E.: Dynamic fracture behavior of an internal interfacial crack between two dissimilar magneto-electro-elastic plates. Eng. Fracture Mech. 75, 1468–1487 (2008)

    Article  Google Scholar 

  17. 17.

    Fleck, N.A., Hutchinson, J.W.: A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41, 1825–1857 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Gao, X.-L., Ma, H.M.: Strain gradient solution for the Eshelby-type anti-plane strain inclusion problem. Acta Mech. 223, 1067–1080 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Gitman, I.M., Askes, H., Kuhl, E., Aifantis, E.C.: Stress concentrations in fractured compact bone simulated with a special class of anisotropic gradient elasticity. Int. J. Solids Struct. 47, 1099–1107 (2010)

    MATH  Article  Google Scholar 

  20. 20.

    Huang, J.H., Kuo, W.-S.: The analysis of piezoelectric/piezomagnetic composite materials containing ellipsoidal inclusions. J. Appl. Phys. 81, 1378–1386 (1997)

    Article  Google Scholar 

  21. 21.

    Kalpakides, V.K., Arvantitakis, A.I., Hadjigeorgiou, E.P.: The role of electric field gradient in modeling elastic ferroelectrics. Int. J. Fract. 166, 77–90 (2010)

    Article  Google Scholar 

  22. 22.

    Kuo, H.-Y.: Fibrous composites of piezoelectric and piezomagnetic phases: generalized plane strain with transverse electromagnetic fields. Mech. Mater. 75, 103–110 (2014)

    Article  Google Scholar 

  23. 23.

    Kuo, H.-Y., Bhattacharya, K.: Fibrous composites of piezoelectric and piezomagnetic phases. Mech. Mater. 60, 159–170 (2013)

    Article  Google Scholar 

  24. 24.

    Lee, J.D., Chen, Y.P., Eskandarian, A.: A micromorphic electromagnetic theory. Int. J. Solids Struct. 41, 2099–2110 (2004)

    MATH  Article  Google Scholar 

  25. 25.

    Li, J.Y., Dunn, M.L.: Micromechanics of magnetoelectroelastic composite materials: average fields and effective behavior. J. Intell. Mater. Syst. Struct. 9, 404–416 (1998a)

    Article  Google Scholar 

  26. 26.

    Li, J.Y., Dunn, M.L.: Anisotropic coupled-field inclusion and inhomogeneity problems. Phil. Mag. A. 77, 1341–1350 (1998b)

    Article  Google Scholar 

  27. 27.

    Lubarda, V.A.: Circular inclusions in anti-plane strain couple stress elasticity. Int. J. Solids Struct. 40, 3827–3851 (2003)

    MATH  Article  Google Scholar 

  28. 28.

    Lurie, S., Solyaev, Y., Shramko, K.: Comparison between the Mori-Tanaka and generalized self-consistent methods in the framework of anti-plane strain inclusion problem in strain gradient elasticity. Mech. Mater. 122, 133–144 (2018)

    Article  Google Scholar 

  29. 29.

    Lurie, S., Solyaev, Y.: Anti-plane inclusion problem in the second gradient electroelasticity theory. Int. J. Eng. Sci. 144, 103129 (2019)

    MATH  Article  Google Scholar 

  30. 30.

    Lurie, S., Volkov-Bogorodsky, D., Leontiev, A., Aifantis, E.: Eshelby’s inclusion problem in the gradient theory of elasticity: applications to composite materials. Int. J. Eng. Sci. 49, 1517–1525 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Ma, H.M., Gao, X.-L.: A new homogenization method based on a simplified strain gradient elasticity theory. Acta Mech. 225, 1075–1091 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Rati. Mech. Anal. 16, 51–78 (1964)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Rat. Mech. Anal. 11, 415–448 (1962)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Mishima, T., Fujioka, H., Nagakari, S., Kamigake, K., Nambu, S.: Lattice image observations of nanoscale ordered regions in Pb(Mag\( _{1/3}\)Nb\(_{2/3}\))O\(_{3}\). Jpn. J. Appl. Phys. 36, 6141–6144 (1997)

    Article  Google Scholar 

  35. 35.

    Multani, M.S., Palkar, V.R.: Morphotropic phase boundary in the PZT system. Mater. Rev. B 17, 101–104 (1982)

    Article  Google Scholar 

  36. 36.

    Nan, C.W., Bichurin, M.I., Dong, S., Viehland, D., Srinivasan, G.: Multiferroic magnetoelectric composites: Historical perspective, status, and future directions. J. Appl. Phys. 103, 031101 (2008)

    Article  Google Scholar 

  37. 37.

    Palneedi, H., Annapureddy, V., Priya, S., Ryu, J.: Status and perspectives of multiferroic magnetoelectric composite materials and applications. Actuators 5, 9 (2016)

    Article  Google Scholar 

  38. 38.

    Ru, C.Q., Aifantis, E.C.: A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech. 101, 59–68 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Rivera, J.P.: A short review of the magnetoelectric effect and related experimental techniques on single phase (multi-) ferroics. Eur. Phys. J. B 71, 299–313 (2009)

    Article  Google Scholar 

  40. 40.

    Sladek, J., Sladek, V., Stanak, P., Pan, E.: FEM formulation for size-dependent theory with application to micro coated piezoelectric and piezomagnetic fiber-composites. Comput. Mech. 59, 93–105 (2017a)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Sladek, J., Sladek, V., Repka, M., Kasala, J., Bishay, P.: Evaluation of effective material properties in magneto-electro-elastic composite materials. Comp. Struct. 174, 176–186 (2017b)

    Article  Google Scholar 

  42. 42.

    Solyaev, Y., Lurie, S.: Numerical predictions for the effective size-dependent properties of piezoelectric composites with spherical inclusions. Comp. Struct. 202, 1099–1108 (2018)

    Article  Google Scholar 

  43. 43.

    Vopson, M.M., Fetisov, Y.K., Caruntu, G., Srinivasan, G.: Measurement techniques of the magneto–electric coupling in multiferroics. Materials 10, 963 (2017)

    Article  Google Scholar 

  44. 44.

    Wang, X., Pan, E., Feng, W.J.: Anti-plane Green’s functions and cracks for piezoelectric material with couple stress and electric field gradient effects. Eur. J. Mech. A Solids 27, 478–486 (2008)

    MATH  Article  Google Scholar 

  45. 45.

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

    MATH  Article  Google Scholar 

  46. 46.

    Xue, C.-X., Pan, E.: On the longitudinal wave along a functionally graded magneto–electro–elastic rod. Int. J. Eng. Sci. 62, 48–55 (2013)

    Article  Google Scholar 

  47. 47.

    Yang, X.M., Hu, Y.T., Yang, J.S.: Electric field gradient effects in anti-plane problems of polarized ceramics. Int. J. Solids Struct. 41, 6801–6811 (2004)

    MATH  Article  Google Scholar 

  48. 48.

    Yang, J.S., Zhou, H.G., Li, J.Y.: Electric field gradient effects in an anti-plane circular inclusion in polarized ceramics. Proc. R. Soc. A 462, 3511–3522 (2006)

    MATH  Article  Google Scholar 

  49. 49.

    Yoo, K., Jeon, B.G., Chun, S.H., Patil, D.R., Lim, Y.J., Noh, S.H., Gil, J., Cheon, J., Kim, K.H.: Quantitative measurements of size-dependent magnetoelectric coupling in Fe\(_{3}\)O\(_{4}\) nanoparticles. Nano Lett. 16, 7408–7413 (2016)

    Article  Google Scholar 

  50. 50.

    Yue, Y.M., Xu, K.Y., Aifantis, E.C.: Microscale size effects on the electromechanical coupling in piezoelectric material for anti-plane problem. Smart Mater. Struct. 23, 125043 (2014)

    Article  Google Scholar 

  51. 51.

    Yue, Y.M., Xu, K.Y., Chen, T., Aifantis, E.C.: Size effects on magnetoelectric response of multiferroic composite with inhomogeneities. Physica B 478, 36–42 (2015)

    Article  Google Scholar 

Download references


The financial support from the Ministry of Science and Technology Taiwan under Grant No.: MOST 106-2628-E-009-MY3 is gratefully acknowledged.

Author information



Corresponding author

Correspondence to Hsin-Yi Kuo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.



In order to have nontrivial solutions of Eq. (3.6), we need

$$\begin{aligned} \left| {\mathbf {L}}_{d}\varvec{\eta -}k_{\Psi }^{2}{\mathbf {L}}\right| =0. \end{aligned}$$

The above relation gives

$$\begin{aligned} s^{3}+As^{2}+Bs+C=0, \quad s=k_{\Psi }^{2}, \end{aligned}$$

where ABC are given in Eq. (3.20). Further, let

$$\begin{aligned} s=y-\frac{A}{3}, \end{aligned}$$

Eq. (A2) becomes

$$\begin{aligned} y^{3}+py+q=0. \end{aligned}$$

Here, p and q are given in Eq. (3.19)\(_{1,2}.\) If we assume

$$\begin{aligned} y=u+v, \end{aligned}$$

we will have

$$\begin{aligned} u^{3}v^{3}=-\frac{p^{3}}{27}, \quad q=-\left( u^{3}+v^{3}\right) . \end{aligned}$$

In addition, \(u^{3}\) and \(v^{3}\) are also the roots of

$$\begin{aligned} Z^{2}+qZ-\frac{p^{3}}{27}=0, \end{aligned}$$


$$\begin{aligned} u^{3}= & {} -\frac{q}{2}+\sqrt{\frac{p^{3}}{27}+\frac{q^{2}}{4}}, \nonumber \\ v^{3}= & {} -\frac{q}{2}-\sqrt{\frac{p^{3}}{27}+\frac{q^{2}}{4}}. \end{aligned}$$

Substitute (A8) into (A5) and (A3), we obtain the solution (3.21) and the condition (3.18).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, GE., Kuo, HY. Effects of strain gradient and electromagnetic field gradient on potential and field distributions of multiferroic fibrous composites. Acta Mech (2021). https://doi.org/10.1007/s00707-020-02910-5

Download citation