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

Abstract

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.

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Abbreviations

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

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Acknowledgements

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

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Appendix

Appendix

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

$$\begin{aligned} \left| {\mathbf {L}}_{d}\varvec{\eta -}k_{\Psi }^{2}{\mathbf {L}}\right| =0. \end{aligned}$$
(A1)

The above relation gives

$$\begin{aligned} s^{3}+As^{2}+Bs+C=0, \quad s=k_{\Psi }^{2}, \end{aligned}$$
(A2)

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

$$\begin{aligned} s=y-\frac{A}{3}, \end{aligned}$$
(A3)

Eq. (A2) becomes

$$\begin{aligned} y^{3}+py+q=0. \end{aligned}$$
(A4)

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

$$\begin{aligned} y=u+v, \end{aligned}$$
(A5)

we will have

$$\begin{aligned} u^{3}v^{3}=-\frac{p^{3}}{27}, \quad q=-\left( u^{3}+v^{3}\right) . \end{aligned}$$
(A6)

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

$$\begin{aligned} Z^{2}+qZ-\frac{p^{3}}{27}=0, \end{aligned}$$
(A7)

i.e.,

$$\begin{aligned} u^{3}= & {} -\frac{q}{2}+\sqrt{\frac{p^{3}}{27}+\frac{q^{2}}{4}}, \nonumber \\ v^{3}= & {} -\frac{q}{2}-\sqrt{\frac{p^{3}}{27}+\frac{q^{2}}{4}}. \end{aligned}$$
(A8)

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

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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

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