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Effective multi-field properties of electro-magneto-thermoelastic composites estimated by finite element method approach
Acta Mechanica Solida Sinica, Vol. 28, No. 2, April, 2015 Published by AMSS Press, Wuhan, China
ISSN 0894-9166
EFFECTIVE MULTI-FIELD PROPERTIES OF ELECTRO-MAGNETO-THERMOELASTIC COMPOSITES ESTIMATED BY FINITE ELEMENT METHOD APPROACH Zhichao Zhang
Xingzhe Wang
(Key Laboratory of Mechanics on Environment and Disaster in Western China, The Ministry of Education of China; College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China)
Received 27 August 2013, revision received 18 March 2014
ABSTRACT A finite element approach based on the micromechanics was performed to estimate the multi-field properties of electro-magneto-thermoelastic composites. The thermal field and the involved pyroelectric and pyromagnetic effect of the multi-phase composite materials were taken into account in the investigation and implemented in the finite element modeling. The multifields related to the electric field, magnetic field, deformation and temperature field, as well as their coupling effects of the smart composites under periodic boundary conditions were obtained numerically. Especially, by means of the homogenization approximation, the effective thermal expansion coefficients, pyroelectric coefficients, pyromagnetic coefficients and other elastic, electric, and magnetic properties for the piezoelectric material, piezomagnetic material and magnetoelectric material were calculated, respectively. Some results are compared to the theoretical predictions by the well-known Mori-Tanaka method to show good agreements.
KEY WORDS multi-field property, electro-magneto-thermoelastic composite, pyroelectric and pyromagnetic effect, FEM of micromechanics
I. INTRODUCTION With wide potential applications in the aerospace, micro-electromechanical system, transportation and marine engineering, the smart materials and structures have evoked considerable interests in scientific and engineering communities. Due to the coupling effect between the material properties of the different constituents, the new unique effects that characterize the macroscopic composite but are absent from the constituents themselves can arise. The magnetoelectric, pyroelectric and pyromagnetic properties are known as the cross or product properties as examples of such properties. The contribution of second product effective due to the coupling of different phases plays a significant role even if the constituents of the composites exhibit intrinsic pyroelectricity properties[1]. Pyroelectric devices have been utilized for various applications such as infrared detection, imaging systems, and thermal-medical diagnostics[2] and have received increasing interest in recent years.
Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (No. 11172117), Doctoral Fund of Ministry of Education of China (No. 20120211110005) and the Foundation for Innovative Research Groups of the NNSFC (No. 11121202).
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Research on the modeling of predicting the effective property and response of smart composites can be classified into analytical and numerical approaches. Micromechanics methods are useful tool to predict effective properties of composites such as Green’s function[3, 4] . Based on the dilute, selfconsistent, Mori-Tanaka model (M-T) and differential micromechanics theories, Dunn and Taya[5–7] simplified the piezoelectric Eshelby’s tensor of elliptic fiber problem and extended analytical models to predict the effective electromechanical properties of piezoelectric composites. The Eshelby’s tensor was further utilized in the piezoelectric-magnetic composites and the Mori-Tanaka mean field approach was developed by Li and Dunn[8–10] to estimate effective properties of the two-phase magnetoelectric elliptic fiber composites. The expressions of effective thermal properties, including the coefficient of thermal expansion, pryoelectric coefficient and pryomagnetic coefficient, were given in their work. By the generalized Nemat-Nasser and Hori’s multi-inclusion model in elasticity[11] , solutions of multi-inclusion and inhomogeneity problems that serve as basis for an averaging scheme to model the effective magnetoelectroelastic moduli of heterogeneous materials were reported[12] . The generalized self-consistent method (GSCM) was also developed for predicting the effective properties of piezoelectric–magnetic fiber reinforced composites. The complex multi-field problem was reduced to a formal in-plane elasticity problem for which an exact closed form solution is available[13] . Challagulla and Georgiades[14] gave a general asymptotic homogenization model to analyze the longitudinally-layered composite material made of laminate of piezoelectric and piezomagnetic material. Even if some analytical methods have been attempted to describe the behavior of the smart composites, the researches reported in recent literature more and more focused on two or three phase composites and the inclusions, such as elliptic fiber and cylinder fiber. Numerical methods, such as finite element method seems to be a well suited approach to describe the behavior of smart materials, as there are arbitrary geometries, material properties, size and phases involved in their analysis. Berger et al.[15] used Representative Volume Element (RVE) method to predict the effective coefficients of periodic transversely isotropic piezoelectric fiber composites without the thermo-mechanics and thermo-electric couplings. Lee et al.[16] developed a finite element method to calculate the effective elastic composite materials as well as electrical and magnetic properties of the magneto-electric materials. By calculating the distribution of various physical fields in the RVE they obtained effective properties of composite materials directly. In particular, the analytical Mori-Tanaka method to estimate the effective properties of multiphase composites has also been developed without consideration of the thermal effective coupling with other fields. From the viewpoint of thermodynamic potential and variation principle, Tang and Yu[17, 18] proposed a micromechanical model to calculate effective properties of periodic smart composites with piezoelectric and piezomagnetic phases. Their theoretical derivation invoked two linearized basic assumptions associated with the micromechanics concept. In recent year, the finite element method is further utilized to deal with the functionally graded materials[19, 20] and multilayered structures[21] of smart composites. It is necessary to develop a simple approach that is applicable to a variety of smart structures in complex fields. Additionally, there always encounters difficulties and complexity when the influence of thermal field on the pryoelectric and pryomagnetic couplings is taken into account. Our work focused on the estimation of effective multi-field properties especially the thermal coefficients in multi-phase smart composite materials. The distribution of the displacement, electric potential and magnetic potential in the RVE at a uniform temperature boundary condition were calculated. We compared our prediction results with the results of the Mori-Tanaka method in two-phase piezoelectric composites to show good agreements.
II. FUNDAMENTAL EQUATIONS AND MODEL 2.1. Governing Equation and Model Description We consider the smart materials building up the composites which have linear coupled constitutive relations of magnetic electric elastic and temperature fields as below[22] : σij = Cijkl εkl + eijk (−Ek ) + qijk (−Hk ) − λij θ Di = eikl εkl − ηij (−Ej ) − aij (−Hj ) − pi θ Bi = qikl εkl − aij (Ej ) − μil (−Hj ) − mi θ
(1)
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in which σij , εkl , Di , Ei , Bi , Hi , θ are the stress tensor, strain tensor, electric displacement, electric filed, magnetic induction, magnetic field vector and the temperature change with respect to a referenced temperature, respectively; ail is the magnetoelectric tensor, Cijkl is the stiffness tensor, eijl , qijl are the piezoelectric and piezomagnetic tensor, and ηij , μij are the dielectric permittivity and magnetic permittivity tensor. The thermal stress tensor λij is obtained by λij = Cijkl αkl , in which αij is the thermal expansion coefficient. pi and mi are pyroelectric coefficients and pyromagnetic coefficients, respectively. The constitutive relations of Eqs.(1) for the transversely isotropic material can be written in a matrix form as follows: Σ = LZ − Πθ (2) where Σ is a 12 × 1 matrix representation of the strain, electric displacement and magnetic flux; L is a 12 × 12 matrix and Z is a 12 × 1 matrix representation of the strain, electric field and magnetic field; Π represents the thermal coefficients. The expressions of these matrixes are given in the Appendix. The effective moduli L∗ and the thermal coefficient matrix Π ∗ can be defined in the form
Σ = L∗ Z − Π ∗ θ
(3)
in which, A = (1/V ) V AdV denotes a volume average over the heterogeneous composite medium. The balance equation, Gauss’s low and the conservation of magnetic flux without magnetic poles are given by σij,j = 0 (4a) Di,i = 0
(4b)
Bi,i = 0
(4c)
For the thermal conduction in the composite, the temperature change is controlled by ∇2 θ = 0
(4d)
Usually, for a small thermal disturbance in an infinite material, the temperature is a constant after long enough time. A heat balance in the composite is assumed, in other words, the change of temperature θ is a constant in the domain. In the condition of small displacement gradients, the linearized strain-displacement relations are given by 1 εij = (ui,j + uj,i ) (5a) 2 e m The gradient of electric potential Φ and magnetic potential Φ are the electric field E and magnetic field H, respectively, Ei = −Φe,i (5b) Hi = −Φm ,i
(5c)
Combining Eqs.(1), (5a)-(5c) with (4a)-(4d), one can easily get the governing equations of displacement, electric and potentials, as well as the temperature. 2.2. Prediction of effective moduli Π ∗ To calculate the effective module Π ∗ , the zero condition of the far-field uniform general strain Z was applied. The displacement, electric potential, magnetic potential are periodic. The volume average of the general stress in this case is given as Σ = −Π ∗ θ
(6)
The general strain in an inclusion phase ‘i’ of the composite caused by temperature, Zθi , is a disturbed value. If there is no inclusion phase in an RVE, the general strain is zero after a temperature change due to the uniformity of material parameters inside. However, when there is an inclusion phase θ included in RVE, Zi will always be non-zero. For instance, for an infinite bar with thermal expansion coefficient of α, it is obviously that the displacement is zero at any point in the bar when the temperature change is uniform.
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Consequently, the stress, electric displacement, and the magnetic flux caused by temperature changing can be split in two parts, i.e., Σ = Σ I + Σ II . One part is from the difference of thermal expansion properties of the matrix phase and inclusion phase, denoted as Σ I ; the other one is caused by the influence between the matrix phase and inclusion phases, denoted as Σ II . For a given temperature change θ, then we have, N −1 Σ I = −cm Π m θ − ci Π i θ (7) i=1
θ
Σ II = cm Lm Zm +
N −1 i=1
θ
ci Li Zi
(8)
in which the subscript ‘m’ denotes the matrix, ‘i’ represents the inclusion phase, cm , ci are respectively the matrix volume fraction and inclusion phase volume fraction, and N − 1 is the total inclusion phases. θ If we obtain the general strain Zi caused by temperature change, the effective module Π ∗ is expressed as Π∗ = ΠV + Πθ (9) where N −1 V Π = cm Π m + ci Π i (10a) 1 Π =− θ θ
i=1 θ cm Lm Zm
+
N −1 i=1
θ ci Li Zi θ
(10b) θ
For the linear thermoelastic constitutive of materials, (1/θ) Zm and (1/θ) Zi always are constants related to the material properties which are denoted as B θm and B θi , and (10b) can be re-written as Π θ = −cm Lm B θm +
N −1
ci Li B θi
(11)
i=1
From the above formulas, one can calculate the effective moduli of the composite through calculating the disturbed displacement caused by the temperature change. 2.3. Periodic Boundary Conditions To estimate the effective properties, the displacement, the electric potential, the magnetic potential should be all periodic to meet the periodical demand. Due to the symmetry, we may use the periodic boundary conditions that provide the effective moduli. The periodic boundary condition given by Suquet[23] , was applied on the boundary of RVE. Each RVE in the composite has the same deformation mode, electric and magnetic potential mode. It means that the displacement, electric potential, magnetic potential at any point in the REV can be expressed in terms of an equivalent point in another RVE such as[15, 16] Fig. 1 The REV of two-phase composites. ∂ui dβ ui (xα + dα ) = ui (xα ) + ∂xβe ∂Φ e e dβ Φ (xα + dα ) = Φ (xα ) + (12) ∂xβ m ∂Φ dβ Φm (xα + dα ) = Φm (xα ) + ∂xβ Here, dβ are the components of the periodic vector at a point in one RVE to an equivalent point in a neighboring RVE. Eqs.(12) represent 15 boundary conditions when we chose a periodic unit cell as shown in Fig.1. Consider the uniform temperature field boundary condition, the temperature change will also be uniform in the composite. In the calculation, the temperature change θ = 0 is used to evaluate the effective moduli L∗ . The details to estimate the L∗ refer to the work by Lee[16] .
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III. NUMERICAL EXAMPLES AND RESULTS In this section, we will perform some numerical case studies on the composites consisting of matrix and filled single fiber in a parallel direction. An REV was chosen in the simulation, as shown in Fig.1. The region of cylinder denotes the inclusion fiber and the other region in the cube represents the matrix phase. The Cartesian coordinates (x1 , x2 , x3 ) are adopted, and the composites can be taken as plenty of REVs arranged periodically. The geometric size of the cuboid REV is characterized by the length a, the width b, and the height h. The cross section of the fiber is an ellipse with the semi-major axis β1 and semi-minor axis β2 . The displacement in the REV was computed using Comsol Multiphysics software[24], the periodic boundary conditions listed in Eqs.(12) were applied to the RVE. Comsol Multiphysics is a general finite-element modeling environment and allows users to develop complex numerical models with different geometries and multiple governing equations quickly using a GUI. The model defined by partial differential equations (PDEs) on spatially structured domains, are shown in Fig.1. The PDEs are discretized in space using the FE method. In the present work, the PDE module in Comsol is utilized to solve the governing equation and the volume average of strain, electric field and magnetic field of each phase are calculated using the post-processing module. In the following analysis, the related material phases of the piezoelectric (BaTiO3 ), piezomagntic (CoFe2 O4 ) and matrix phase (epoxy) in the multi-phase composites were used and the corresponding property coefficients were listed in Table 1. Table 1. Basic constants for relative materials[14, 16, 17]
Piezoelectric (BaTiO3 ) 166 77 78 162 43
Piezomagntic (CoFe2 O4 ) 286 173 170.5 269.5 45.3
Matrix phase (epoxy) 5.53 2.97 2.97 5.53 1.28
η11 (C2 /Nm2 ) η33 (C2 /Nm2 )
112 × 10−10 126 × 10−10
0.8 × 10−10 0.93 × 10−10
1.0 × 10−10 1.0 × 10−10
μ11 (Ns2 /C2 ) μ33 (Ns2 /C2 )
5 × 10−6 10 × 10−6
−590 × 10−6 157 × 10−6
1.0 × 10−6 1.0 × 10−6
e31 (C/m2 ) e33 (C/m2 ) e15 (C/m2 )
−4.4 18.6 11.6
0 0 0
0 0 0
q31 (N/Am) q33 (N/Am) q15 (N/Am)
0 0 0
580.3 699.7 550
0 0 0
α11 (×10−6 /K) α22 (×10−6 /K) α33 (×10−6 /K)
15.7 15.7 6.4
10 10 10
54 54 54
C11 C12 C13 C33 C44
(GPa) (GPa) (GPa) (GPa) (GPa)
3.1. Case 1. Piezoelectric-matrix Composite The two-phase piezoelectric-matrix composite is considered which is transversely isotropic. The ratio of semi-axes is taken as β1 /β2 = 1 (i.e., circular cross section), and the other material parameters are chosen as listed in Table 1. The effective piezoelectric and elastic coefficients were evaluated and plotted in Fig.2. For the purpose of comparison, the theoretical predictions from the Mori-Tanaka method were also presented in the figures[9, 12]. One can see that the present FEM predictions are quite close to the theoretical predictions by M-T method, especially for electric properties like piezoelectric coefficient as shown in Fig.2(a) and dielectric permittivity as shown in Fig.2(b). Figure 2(c) plots the elastic modulus components of the piezoelectric-matrix composite, C11 , C12 and C13 , C33 . One can see that there is quite good agreement between the theoretical predictions by M-T method and the present numerical results for small volume fraction of piezoelectric phase, while the obvious difference is present for the
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larger volume fraction. Figure 2(d) and Fig.2(e) plot the thermal expansion and pyroelectric coefficient varying with the volume fraction of piezoelectric phase for the composite. There is good agreement between the M-T predictions and FEM results. From Fig.2(e), it is found that the pyroelectric coefficient of the composite almost linearly increases with the volume fraction of the piezoelectric phase. There is a maximum value of the pyroelectric coefficient predicted by the theoretical M-T method for the volume fraction of about 0.8. As for the large volume fraction, the pyroelectric coefficient then decreases and reduces to zero when the volume fraction equals 1. It is reasonable since the pyroelectric coupling effect vanishes for the one-phase material. In addition, the maximum volume fraction of the inclusion fiber of the chosen RVE is about 0.785, and the volume fraction in our FEM simulation is less than that limit value.
Fig. 2. Effective properties of the piezoelectric composite dependence on the volume fraction of piezoelectric fiber.
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3.2. Case 2. Piezomagnetic-matrix Composite For the composite with piezomagnetic fiber inclusion, Fig.3 displays the effective properties by the FEM approach. The piezomagnetic coefficients of the composite are plotted in Fig.3(a). It is shown that the component of q33 with respect to the fiber direction has a larger value in comparison with the ones of components of q31 , q15 , which increases linearly with the volume fraction of piezomagnetic fiber. The thermal related properties of thermal expansion and pyromagnetic coefficient are depicted in Fig.3(b) and Fig.3(c). Form Fig.3(b), one can see that for a low volume fraction of piezomagnetic inclusion (e.g., vpm < 0.15) the thermal expansion component α33 initially rapidly decreases, while the component α11 increases slightly. For the volume fraction beyond vpm = 0.15, α33 linearly decreases and α11 decreases gently and keeps almost a constant. Similar to the pyroelectric coefficient of the piezoelectric-matrix composite, here we consider only the part of cross product resulting from the coupling of thermal field which indirectly affects the magnetic induction, as shown in Fig.3(c). The temperature change causes the thermal strain both in the matrix and piezomagnetic fiber, and the strain is then transferred to the piezomagnetic phase to result in the magnetic flux in the composite. In this sense, the elastic field in the composites plays the role of the bridge between the temperature field and magnetic field. In addition, the pyromagnetic coefficient of the piezomagnetic composite has negative value and its magnitude increases with the volume fraction.
Fig. 3. Effective property of piezomagnetic composites varying with the volume fraction of piezomagnetic fiber.
3.3. Case 3. Piezoelectric-piezomagnetic Composite In this study case, we concern the two-phase composite with piezoelectric and piezomagnetic constituents, and take the piezomagnetic fiber as the inclusion. Firstly, we pay attention to the electric and magnetic characteristics of the two-phase. Figure 4 displays the distribution of electric field Ex , the induced magnetic field Hx for an applied electric field Ex . Due to the periodic boundary conditions, the symmetry of electromagnetic feature is observed as shown in Fig.4(a) and Fig.4(c). The vector
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Fig. 4. The distribution of electromagnetic field of two phase magneto-electric composites (Ex = 100).
diagrams of electric field and magnetic field distribution on the cross section when z = 0 are also given in Fig.4(b) and Fig.4(d). Furthermore, the effective electric, magnetic, and elastic coefficients show the similar characteristics as the case studies of piezoelectric and piezomagnetic two-phase composites, and have not been presented here for space-saving. The thermal-related properties and the magnetoelectric coupling coefficient are predicted. In the simulations, the effect of the shape of cross-section of inclusion fiber (i.e., ratio of semi-axes β1 /β2 ) on the effective property is performed. Figure 5 plots the effective properties of the piezoelectric-piezomagnetic composite dependence on the volume fraction of the piezomagnetic fiber. It can be seen that, form Fig.5(a), the in-plane thermal expansions, α11 , α22 , linearly decrease while the fiber-direction component α33 increases with the volume fraction. The shape of inclusion fiber has obvious influence on the in-plane thermal expansions along x1 -axis and x2 -axis. With the increase of the ratio of semi-axes β1 /β2 , the component α11 increases and the component α22 decreases. However, for the fiber-direction component α33 , there is almost no difference for the different β1 /β2 . Figure 5(b) and Fig.5(c) respectively display the pyromagnetic and pyroelectric coefficients dependence on the volume fraction of piezomagnetic inclusion. The similar feature is shown that the magnitude of the pyromagnetic\pyroelectric coefficient along the fiber-direction increases with the volume fraction and obtains a maximum value at about vpm = 0.55. The shape of inclusion fiber exhibits little influence on the pyromagnetic\pyroelectric coefficient. As a common concern, the magnetoelctric coupling property is the important factor for the piezoelectricpiezomagnetic composite which is depicted in Fig.5(d) and Fig.5(e). Figure 5(d) plots the in-plane components of a11 , a22 which increase with the volume fraction of the piezomagnetic fiber. When the cross-section of inclusion fiber is circle, i.e., β1 /β2 = 1, the in-plane property is isotropic and a11 = a22 . With increase of the ratio of semi-axes, the magnetoelctric coefficient exhibits anisotropy. The fiber-
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Fig. 5. Effective property of the magneto-electric composite dependence on the volume fraction of piezomagnetic fiber.
direction component of magnetoelctric coefficient, a33 , is plotted in Fig.5(e). One can see that the magnetoelctric coefficient monotonously increases with the volume fraction until a maximum value is reached at about vpm = 0.55, and then decreases. That means an optimal volume fraction may exist so that a good performance of magnetoelectric transformation is achieved. In addition, for the different shape ratio of semi-axes, the predicted magnetoelectric values of fiber-direction are almost the same.
IV. CONCLUSION A numerical approach based on the micromechanics and FEM was developed to predict the effective multi-field properties of electro-magneto-thermoelastic composites. The thermal-related properties involving pyroelectric and pyromagnetic effects of the multiphase smart composites were incorporated in the modeling. The multi-fields related to the electric field, magnetic field, deformation and temperature field, as well as their coupling effects of the composites under periodic boundary conditions were obtained numerically. The effective thermal expansion coefficient, pyroelectric coefficient, pyromagnetic coefficient and other elastic, electric, and magnetic properties for the piezoelectric material,
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piezomagnetic material and magnetoelectric material were calculated by means of the homogenization approximation. Some of the predictions showed good agreement with the theoretical predictions by the well-known Mori-Tanaka method, and the Mori-Tanaka method always gives the good predictions on the electric and magnetic properties of the electromagnetic composite. The results demonstrated that the properties of fiber-direction for the composite are always dominant. The cross-section shape of inclusion fiber shows obvious influence on the in-plane thermal-related properties of piezoelectricpiezomagnetic composite but almost no effect on the fiber-direction ones. For a proper volume fraction of the inclusion fiber, there are optimal valued of the pyromagnetic and pyroelectric coefficients, as well as the magnetoelectric coupling coefficient.
References [1] Newnham,R.E., Skinner,D.P. and Cross,L.E., Connectivity and piezoelectric-pyroelectric composites. Materials Research Bulletin, 1978, 13: 525-536. [2] Satapathy,S., Gupta,P.K. and Varma,K.B.R., Enhancement of nonvolatile polarization and pyroelectric sensitivity in lithium tantalate (LT)/poly(vinylidene fluoride) (PVDF) nanocomposite. Journal of Physics D, 2009, 42: 055402. [3] Nan,C.W., Bichurin,M.I., Dong,S., Viehland,D. and Srinivasan,G., Multiferroic magnetoelectric composites: Historical perspective, status, and future directions. Journal of Applied Physics, 2008, 103: 031101. [4] Nan,C.W. and Clarke,D.R., Effective properties of ferroelectric and/or ferromagnetic composites: a unified approach and its application. Journal of the American Ceramic Society, 1997, 80: 1333-1340. [5] Dunn,M.L., Micromechanics of coupled electroelastic composites: effective thermal expansion and pyroelectric coefficients. Journal of Applied Physics, 1993, 73: 5131-5140. [6] Dunn,M.L. and Taya,M., An analysis of piezoelectric composite materials containing ellipsoidal inhomogeneities. Proceedings of the Royal Society of London, Series A: Physical and Engineering Sciences, 1993, 443: 265-287 [7] Dunn,M.L. and Taya,M., Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites. International Journal of Solids and Structures, 1993, 30: 161-175. [8] Li,J.Y. and Dunn,M.L., Anisotropic coupled-field inclusion and inhomogeneity problems. Philosophical magazine A, 1998, 77: 1341-1350. [9] Li,J.Y. and Dunn,M.L., Micromechanics of magnetoelectroelastic composite materials: average fields and effective behavior. Journal of Intelligent Material Systems and Structures, 1998, 9: 404-416. [10] Li,J.Y., The effective pyroelectric and thermal expansion coefficients of ferroelectric ceramics. Mechanics of Materials, 2004, 36: 949-958. [11] Hori,M. and Nemat-Nasser,S., Double-inclusion model and overall moduli of multi-phase composites. Journal of Engineering Materials and Technology, 1994, 116: 305. [12] Li,J.Y., Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials. International Journal of Engineering Science, 2000, 38: 1993-2011. [13] Tong,Z.H., Lo,S.H., Jiang,C.P. and Cheung,Y.K., An exact solution for the three-phase thermo-electromagneto-elastic cylinder model and its application to piezoelectric–magnetic fiber composites. International Journal of Solids and Structures, 2008, 45: 5205-5219. [14] Challagulla,K.S. and Georgiades,A.V., Micromechanical analysis of magneto- electro-thermo-elastic composite materials with applications to multilayered structures. International Journal of Engineering Science, 2011, 49: 85-104. [15] Berger,H., Kari,S., Gabbert,U., Ramos,R.R., Guinovart,R., Otero,J.A. and Castillero,J.B., An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites. International Journal of Solids and Structures, 2005, 42: 5692-5714. [16] Lee,J., Boyd,J.G. and Lagoudas,D,C., Effective properties of three-phase electro-magneto-elastic composites. International Journal of Engineering Science, 2005, 43: 790-825. [17] Tang,T. and Yu,W.Micromechanical modeling of the multiphysical behavior of smart materials using the variational asymptotic method. Smart Materials and Structures, 2009, 18: 125026-125040. [18] Tang,T. and Yu,W., Variational asymptotic homogenization of heterogeneous electromagnetoelastic materials. International Journal of Engineering Science, 2008, 46: 741-757. [19] Liu,W. and Zhong,Z., Three-dimensional thermoelastic analysis of functionally graded of plate. Acta Mechanica Solida Sinica, 2011, 24: 241-249. [20] Nabian,A., Rezazadeh,G., Almassi,M. and Borgheei,A., On the stability of a functionally graded rectangular micro-plate subjected to hydrostatic and nonlinear electrostatic pressures. Acta Mechanica Solida Sinica, 2013, 26: 205-220.
Vol. 28, No. 2 Zhichao Zhang et al.: Multi-Field Properties of Electro-Magneto-Thermoelastic Composites· 155 · [21] Wang,J.H., Guo,R., Bhalla,A.S., Finite element simulation of magnetostrictive and piezoelectric coupling in a layered structure. Ferroelectrics Letters, 2007, 34: 46-53. [22] Perez-Fernandez,L.D., Bravo-Castillero,J., Rodrı’guez-Ramos,R. and Sabina,F.J., On the constitutive relations and energy potentials of linear thermo-magneto-electro-elasticity. Mechanics Research Communication, 2009, 36, 343-350. [23] Suquet,P., Elements of homogenization theory for inelastic solid mechanics. In: Sanchez-Palencia, E., Zaoui, (Eds.), Homogenization Techniques for Composite Media. Berlin: Springer-Verlag, 1987, 194-275. [24] COMSOL, Comsol Multiphysics Modeling Guide, Version 4.3. www.comsol.com (2012).
APPENDIX In Eq.(2), the expression
Σ = σ11 σ22
Z = ε11 ε22 ⎡ C11 C12 ⎢ C12 C11 ⎢ ⎢ C13 C13 ⎢ ⎢ 0 0 ⎢ ⎢ 0 0 ⎢ ⎢ 0 0 L=⎢ ⎢ 0 0 ⎢ ⎢ 0 0 ⎢ ⎢ e31 e 31 ⎢ ⎢ 0 0 ⎢ ⎣ 0 0 q31 q31
Π = λ11 λ22
of each components: σ33 ε33
σ32 ε32
C13 C13 C13 0 0 0 0 0 e33 0 0 q33
0 0 0 C44 0 0 0 e15 0 0 q15 0
λ33
0
σ31 ε31
ε12
0 0 0 0 C44 0 e15 0 0 q15 0 0 0
σ12
0
D1 E1
0 0 0 0 0 C66 0 0 0 0 0 0 p1
D2 E2
0 0 0 0 e15 0 η11 0 0 a11 0 0 p2
p3
D3 E3
0 0 0 e15 0 0 0 η11 0 0 a11 0 m1
B1 H1
e31 e31 e33 0 0 0 0 0 η13 0 0 a33 m2
T B3 T
B2 H2
0 0 0 0 q15 0 a11 0 0 μ11 0 0 m3
H3 0 0 0 q15 0 0 0 a11 0 0 μ11 0
T
(13) ⎤
q31 q31 ⎥ ⎥ q33 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ a33 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ μ33
(14)
(15)
(16)
ISSN 0894-9166
EFFECTIVE MULTI-FIELD PROPERTIES OF ELECTRO-MAGNETO-THERMOELASTIC COMPOSITES ESTIMATED BY FINITE ELEMENT METHOD APPROACH Zhichao Zhang
Xingzhe Wang
(Key Laboratory of Mechanics on Environment and Disaster in Western China, The Ministry of Education of China; College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China)
Received 27 August 2013, revision received 18 March 2014
ABSTRACT A finite element approach based on the micromechanics was performed to estimate the multi-field properties of electro-magneto-thermoelastic composites. The thermal field and the involved pyroelectric and pyromagnetic effect of the multi-phase composite materials were taken into account in the investigation and implemented in the finite element modeling. The multifields related to the electric field, magnetic field, deformation and temperature field, as well as their coupling effects of the smart composites under periodic boundary conditions were obtained numerically. Especially, by means of the homogenization approximation, the effective thermal expansion coefficients, pyroelectric coefficients, pyromagnetic coefficients and other elastic, electric, and magnetic properties for the piezoelectric material, piezomagnetic material and magnetoelectric material were calculated, respectively. Some results are compared to the theoretical predictions by the well-known Mori-Tanaka method to show good agreements.
KEY WORDS multi-field property, electro-magneto-thermoelastic composite, pyroelectric and pyromagnetic effect, FEM of micromechanics
I. INTRODUCTION With wide potential applications in the aerospace, micro-electromechanical system, transportation and marine engineering, the smart materials and structures have evoked considerable interests in scientific and engineering communities. Due to the coupling effect between the material properties of the different constituents, the new unique effects that characterize the macroscopic composite but are absent from the constituents themselves can arise. The magnetoelectric, pyroelectric and pyromagnetic properties are known as the cross or product properties as examples of such properties. The contribution of second product effective due to the coupling of different phases plays a significant role even if the constituents of the composites exhibit intrinsic pyroelectricity properties[1]. Pyroelectric devices have been utilized for various applications such as infrared detection, imaging systems, and thermal-medical diagnostics[2] and have received increasing interest in recent years.
Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (No. 11172117), Doctoral Fund of Ministry of Education of China (No. 20120211110005) and the Foundation for Innovative Research Groups of the NNSFC (No. 11121202).
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Research on the modeling of predicting the effective property and response of smart composites can be classified into analytical and numerical approaches. Micromechanics methods are useful tool to predict effective properties of composites such as Green’s function[3, 4] . Based on the dilute, selfconsistent, Mori-Tanaka model (M-T) and differential micromechanics theories, Dunn and Taya[5–7] simplified the piezoelectric Eshelby’s tensor of elliptic fiber problem and extended analytical models to predict the effective electromechanical properties of piezoelectric composites. The Eshelby’s tensor was further utilized in the piezoelectric-magnetic composites and the Mori-Tanaka mean field approach was developed by Li and Dunn[8–10] to estimate effective properties of the two-phase magnetoelectric elliptic fiber composites. The expressions of effective thermal properties, including the coefficient of thermal expansion, pryoelectric coefficient and pryomagnetic coefficient, were given in their work. By the generalized Nemat-Nasser and Hori’s multi-inclusion model in elasticity[11] , solutions of multi-inclusion and inhomogeneity problems that serve as basis for an averaging scheme to model the effective magnetoelectroelastic moduli of heterogeneous materials were reported[12] . The generalized self-consistent method (GSCM) was also developed for predicting the effective properties of piezoelectric–magnetic fiber reinforced composites. The complex multi-field problem was reduced to a formal in-plane elasticity problem for which an exact closed form solution is available[13] . Challagulla and Georgiades[14] gave a general asymptotic homogenization model to analyze the longitudinally-layered composite material made of laminate of piezoelectric and piezomagnetic material. Even if some analytical methods have been attempted to describe the behavior of the smart composites, the researches reported in recent literature more and more focused on two or three phase composites and the inclusions, such as elliptic fiber and cylinder fiber. Numerical methods, such as finite element method seems to be a well suited approach to describe the behavior of smart materials, as there are arbitrary geometries, material properties, size and phases involved in their analysis. Berger et al.[15] used Representative Volume Element (RVE) method to predict the effective coefficients of periodic transversely isotropic piezoelectric fiber composites without the thermo-mechanics and thermo-electric couplings. Lee et al.[16] developed a finite element method to calculate the effective elastic composite materials as well as electrical and magnetic properties of the magneto-electric materials. By calculating the distribution of various physical fields in the RVE they obtained effective properties of composite materials directly. In particular, the analytical Mori-Tanaka method to estimate the effective properties of multiphase composites has also been developed without consideration of the thermal effective coupling with other fields. From the viewpoint of thermodynamic potential and variation principle, Tang and Yu[17, 18] proposed a micromechanical model to calculate effective properties of periodic smart composites with piezoelectric and piezomagnetic phases. Their theoretical derivation invoked two linearized basic assumptions associated with the micromechanics concept. In recent year, the finite element method is further utilized to deal with the functionally graded materials[19, 20] and multilayered structures[21] of smart composites. It is necessary to develop a simple approach that is applicable to a variety of smart structures in complex fields. Additionally, there always encounters difficulties and complexity when the influence of thermal field on the pryoelectric and pryomagnetic couplings is taken into account. Our work focused on the estimation of effective multi-field properties especially the thermal coefficients in multi-phase smart composite materials. The distribution of the displacement, electric potential and magnetic potential in the RVE at a uniform temperature boundary condition were calculated. We compared our prediction results with the results of the Mori-Tanaka method in two-phase piezoelectric composites to show good agreements.
II. FUNDAMENTAL EQUATIONS AND MODEL 2.1. Governing Equation and Model Description We consider the smart materials building up the composites which have linear coupled constitutive relations of magnetic electric elastic and temperature fields as below[22] : σij = Cijkl εkl + eijk (−Ek ) + qijk (−Hk ) − λij θ Di = eikl εkl − ηij (−Ej ) − aij (−Hj ) − pi θ Bi = qikl εkl − aij (Ej ) − μil (−Hj ) − mi θ
(1)
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in which σij , εkl , Di , Ei , Bi , Hi , θ are the stress tensor, strain tensor, electric displacement, electric filed, magnetic induction, magnetic field vector and the temperature change with respect to a referenced temperature, respectively; ail is the magnetoelectric tensor, Cijkl is the stiffness tensor, eijl , qijl are the piezoelectric and piezomagnetic tensor, and ηij , μij are the dielectric permittivity and magnetic permittivity tensor. The thermal stress tensor λij is obtained by λij = Cijkl αkl , in which αij is the thermal expansion coefficient. pi and mi are pyroelectric coefficients and pyromagnetic coefficients, respectively. The constitutive relations of Eqs.(1) for the transversely isotropic material can be written in a matrix form as follows: Σ = LZ − Πθ (2) where Σ is a 12 × 1 matrix representation of the strain, electric displacement and magnetic flux; L is a 12 × 12 matrix and Z is a 12 × 1 matrix representation of the strain, electric field and magnetic field; Π represents the thermal coefficients. The expressions of these matrixes are given in the Appendix. The effective moduli L∗ and the thermal coefficient matrix Π ∗ can be defined in the form
Σ = L∗ Z − Π ∗ θ
(3)
in which, A = (1/V ) V AdV denotes a volume average over the heterogeneous composite medium. The balance equation, Gauss’s low and the conservation of magnetic flux without magnetic poles are given by σij,j = 0 (4a) Di,i = 0
(4b)
Bi,i = 0
(4c)
For the thermal conduction in the composite, the temperature change is controlled by ∇2 θ = 0
(4d)
Usually, for a small thermal disturbance in an infinite material, the temperature is a constant after long enough time. A heat balance in the composite is assumed, in other words, the change of temperature θ is a constant in the domain. In the condition of small displacement gradients, the linearized strain-displacement relations are given by 1 εij = (ui,j + uj,i ) (5a) 2 e m The gradient of electric potential Φ and magnetic potential Φ are the electric field E and magnetic field H, respectively, Ei = −Φe,i (5b) Hi = −Φm ,i
(5c)
Combining Eqs.(1), (5a)-(5c) with (4a)-(4d), one can easily get the governing equations of displacement, electric and potentials, as well as the temperature. 2.2. Prediction of effective moduli Π ∗ To calculate the effective module Π ∗ , the zero condition of the far-field uniform general strain Z was applied. The displacement, electric potential, magnetic potential are periodic. The volume average of the general stress in this case is given as Σ = −Π ∗ θ
(6)
The general strain in an inclusion phase ‘i’ of the composite caused by temperature, Zθi , is a disturbed value. If there is no inclusion phase in an RVE, the general strain is zero after a temperature change due to the uniformity of material parameters inside. However, when there is an inclusion phase θ included in RVE, Zi will always be non-zero. For instance, for an infinite bar with thermal expansion coefficient of α, it is obviously that the displacement is zero at any point in the bar when the temperature change is uniform.
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Consequently, the stress, electric displacement, and the magnetic flux caused by temperature changing can be split in two parts, i.e., Σ = Σ I + Σ II . One part is from the difference of thermal expansion properties of the matrix phase and inclusion phase, denoted as Σ I ; the other one is caused by the influence between the matrix phase and inclusion phases, denoted as Σ II . For a given temperature change θ, then we have, N −1 Σ I = −cm Π m θ − ci Π i θ (7) i=1
θ
Σ II = cm Lm Zm +
N −1 i=1
θ
ci Li Zi
(8)
in which the subscript ‘m’ denotes the matrix, ‘i’ represents the inclusion phase, cm , ci are respectively the matrix volume fraction and inclusion phase volume fraction, and N − 1 is the total inclusion phases. θ If we obtain the general strain Zi caused by temperature change, the effective module Π ∗ is expressed as Π∗ = ΠV + Πθ (9) where N −1 V Π = cm Π m + ci Π i (10a) 1 Π =− θ θ
i=1 θ cm Lm Zm
+
N −1 i=1
θ ci Li Zi θ
(10b) θ
For the linear thermoelastic constitutive of materials, (1/θ) Zm and (1/θ) Zi always are constants related to the material properties which are denoted as B θm and B θi , and (10b) can be re-written as Π θ = −cm Lm B θm +
N −1
ci Li B θi
(11)
i=1
From the above formulas, one can calculate the effective moduli of the composite through calculating the disturbed displacement caused by the temperature change. 2.3. Periodic Boundary Conditions To estimate the effective properties, the displacement, the electric potential, the magnetic potential should be all periodic to meet the periodical demand. Due to the symmetry, we may use the periodic boundary conditions that provide the effective moduli. The periodic boundary condition given by Suquet[23] , was applied on the boundary of RVE. Each RVE in the composite has the same deformation mode, electric and magnetic potential mode. It means that the displacement, electric potential, magnetic potential at any point in the REV can be expressed in terms of an equivalent point in another RVE such as[15, 16] Fig. 1 The REV of two-phase composites. ∂ui dβ ui (xα + dα ) = ui (xα ) + ∂xβe ∂Φ e e dβ Φ (xα + dα ) = Φ (xα ) + (12) ∂xβ m ∂Φ dβ Φm (xα + dα ) = Φm (xα ) + ∂xβ Here, dβ are the components of the periodic vector at a point in one RVE to an equivalent point in a neighboring RVE. Eqs.(12) represent 15 boundary conditions when we chose a periodic unit cell as shown in Fig.1. Consider the uniform temperature field boundary condition, the temperature change will also be uniform in the composite. In the calculation, the temperature change θ = 0 is used to evaluate the effective moduli L∗ . The details to estimate the L∗ refer to the work by Lee[16] .
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III. NUMERICAL EXAMPLES AND RESULTS In this section, we will perform some numerical case studies on the composites consisting of matrix and filled single fiber in a parallel direction. An REV was chosen in the simulation, as shown in Fig.1. The region of cylinder denotes the inclusion fiber and the other region in the cube represents the matrix phase. The Cartesian coordinates (x1 , x2 , x3 ) are adopted, and the composites can be taken as plenty of REVs arranged periodically. The geometric size of the cuboid REV is characterized by the length a, the width b, and the height h. The cross section of the fiber is an ellipse with the semi-major axis β1 and semi-minor axis β2 . The displacement in the REV was computed using Comsol Multiphysics software[24], the periodic boundary conditions listed in Eqs.(12) were applied to the RVE. Comsol Multiphysics is a general finite-element modeling environment and allows users to develop complex numerical models with different geometries and multiple governing equations quickly using a GUI. The model defined by partial differential equations (PDEs) on spatially structured domains, are shown in Fig.1. The PDEs are discretized in space using the FE method. In the present work, the PDE module in Comsol is utilized to solve the governing equation and the volume average of strain, electric field and magnetic field of each phase are calculated using the post-processing module. In the following analysis, the related material phases of the piezoelectric (BaTiO3 ), piezomagntic (CoFe2 O4 ) and matrix phase (epoxy) in the multi-phase composites were used and the corresponding property coefficients were listed in Table 1. Table 1. Basic constants for relative materials[14, 16, 17]
Piezoelectric (BaTiO3 ) 166 77 78 162 43
Piezomagntic (CoFe2 O4 ) 286 173 170.5 269.5 45.3
Matrix phase (epoxy) 5.53 2.97 2.97 5.53 1.28
η11 (C2 /Nm2 ) η33 (C2 /Nm2 )
112 × 10−10 126 × 10−10
0.8 × 10−10 0.93 × 10−10
1.0 × 10−10 1.0 × 10−10
μ11 (Ns2 /C2 ) μ33 (Ns2 /C2 )
5 × 10−6 10 × 10−6
−590 × 10−6 157 × 10−6
1.0 × 10−6 1.0 × 10−6
e31 (C/m2 ) e33 (C/m2 ) e15 (C/m2 )
−4.4 18.6 11.6
0 0 0
0 0 0
q31 (N/Am) q33 (N/Am) q15 (N/Am)
0 0 0
580.3 699.7 550
0 0 0
α11 (×10−6 /K) α22 (×10−6 /K) α33 (×10−6 /K)
15.7 15.7 6.4
10 10 10
54 54 54
C11 C12 C13 C33 C44
(GPa) (GPa) (GPa) (GPa) (GPa)
3.1. Case 1. Piezoelectric-matrix Composite The two-phase piezoelectric-matrix composite is considered which is transversely isotropic. The ratio of semi-axes is taken as β1 /β2 = 1 (i.e., circular cross section), and the other material parameters are chosen as listed in Table 1. The effective piezoelectric and elastic coefficients were evaluated and plotted in Fig.2. For the purpose of comparison, the theoretical predictions from the Mori-Tanaka method were also presented in the figures[9, 12]. One can see that the present FEM predictions are quite close to the theoretical predictions by M-T method, especially for electric properties like piezoelectric coefficient as shown in Fig.2(a) and dielectric permittivity as shown in Fig.2(b). Figure 2(c) plots the elastic modulus components of the piezoelectric-matrix composite, C11 , C12 and C13 , C33 . One can see that there is quite good agreement between the theoretical predictions by M-T method and the present numerical results for small volume fraction of piezoelectric phase, while the obvious difference is present for the
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larger volume fraction. Figure 2(d) and Fig.2(e) plot the thermal expansion and pyroelectric coefficient varying with the volume fraction of piezoelectric phase for the composite. There is good agreement between the M-T predictions and FEM results. From Fig.2(e), it is found that the pyroelectric coefficient of the composite almost linearly increases with the volume fraction of the piezoelectric phase. There is a maximum value of the pyroelectric coefficient predicted by the theoretical M-T method for the volume fraction of about 0.8. As for the large volume fraction, the pyroelectric coefficient then decreases and reduces to zero when the volume fraction equals 1. It is reasonable since the pyroelectric coupling effect vanishes for the one-phase material. In addition, the maximum volume fraction of the inclusion fiber of the chosen RVE is about 0.785, and the volume fraction in our FEM simulation is less than that limit value.
Fig. 2. Effective properties of the piezoelectric composite dependence on the volume fraction of piezoelectric fiber.
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3.2. Case 2. Piezomagnetic-matrix Composite For the composite with piezomagnetic fiber inclusion, Fig.3 displays the effective properties by the FEM approach. The piezomagnetic coefficients of the composite are plotted in Fig.3(a). It is shown that the component of q33 with respect to the fiber direction has a larger value in comparison with the ones of components of q31 , q15 , which increases linearly with the volume fraction of piezomagnetic fiber. The thermal related properties of thermal expansion and pyromagnetic coefficient are depicted in Fig.3(b) and Fig.3(c). Form Fig.3(b), one can see that for a low volume fraction of piezomagnetic inclusion (e.g., vpm < 0.15) the thermal expansion component α33 initially rapidly decreases, while the component α11 increases slightly. For the volume fraction beyond vpm = 0.15, α33 linearly decreases and α11 decreases gently and keeps almost a constant. Similar to the pyroelectric coefficient of the piezoelectric-matrix composite, here we consider only the part of cross product resulting from the coupling of thermal field which indirectly affects the magnetic induction, as shown in Fig.3(c). The temperature change causes the thermal strain both in the matrix and piezomagnetic fiber, and the strain is then transferred to the piezomagnetic phase to result in the magnetic flux in the composite. In this sense, the elastic field in the composites plays the role of the bridge between the temperature field and magnetic field. In addition, the pyromagnetic coefficient of the piezomagnetic composite has negative value and its magnitude increases with the volume fraction.
Fig. 3. Effective property of piezomagnetic composites varying with the volume fraction of piezomagnetic fiber.
3.3. Case 3. Piezoelectric-piezomagnetic Composite In this study case, we concern the two-phase composite with piezoelectric and piezomagnetic constituents, and take the piezomagnetic fiber as the inclusion. Firstly, we pay attention to the electric and magnetic characteristics of the two-phase. Figure 4 displays the distribution of electric field Ex , the induced magnetic field Hx for an applied electric field Ex . Due to the periodic boundary conditions, the symmetry of electromagnetic feature is observed as shown in Fig.4(a) and Fig.4(c). The vector
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Fig. 4. The distribution of electromagnetic field of two phase magneto-electric composites (Ex = 100).
diagrams of electric field and magnetic field distribution on the cross section when z = 0 are also given in Fig.4(b) and Fig.4(d). Furthermore, the effective electric, magnetic, and elastic coefficients show the similar characteristics as the case studies of piezoelectric and piezomagnetic two-phase composites, and have not been presented here for space-saving. The thermal-related properties and the magnetoelectric coupling coefficient are predicted. In the simulations, the effect of the shape of cross-section of inclusion fiber (i.e., ratio of semi-axes β1 /β2 ) on the effective property is performed. Figure 5 plots the effective properties of the piezoelectric-piezomagnetic composite dependence on the volume fraction of the piezomagnetic fiber. It can be seen that, form Fig.5(a), the in-plane thermal expansions, α11 , α22 , linearly decrease while the fiber-direction component α33 increases with the volume fraction. The shape of inclusion fiber has obvious influence on the in-plane thermal expansions along x1 -axis and x2 -axis. With the increase of the ratio of semi-axes β1 /β2 , the component α11 increases and the component α22 decreases. However, for the fiber-direction component α33 , there is almost no difference for the different β1 /β2 . Figure 5(b) and Fig.5(c) respectively display the pyromagnetic and pyroelectric coefficients dependence on the volume fraction of piezomagnetic inclusion. The similar feature is shown that the magnitude of the pyromagnetic\pyroelectric coefficient along the fiber-direction increases with the volume fraction and obtains a maximum value at about vpm = 0.55. The shape of inclusion fiber exhibits little influence on the pyromagnetic\pyroelectric coefficient. As a common concern, the magnetoelctric coupling property is the important factor for the piezoelectricpiezomagnetic composite which is depicted in Fig.5(d) and Fig.5(e). Figure 5(d) plots the in-plane components of a11 , a22 which increase with the volume fraction of the piezomagnetic fiber. When the cross-section of inclusion fiber is circle, i.e., β1 /β2 = 1, the in-plane property is isotropic and a11 = a22 . With increase of the ratio of semi-axes, the magnetoelctric coefficient exhibits anisotropy. The fiber-
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Fig. 5. Effective property of the magneto-electric composite dependence on the volume fraction of piezomagnetic fiber.
direction component of magnetoelctric coefficient, a33 , is plotted in Fig.5(e). One can see that the magnetoelctric coefficient monotonously increases with the volume fraction until a maximum value is reached at about vpm = 0.55, and then decreases. That means an optimal volume fraction may exist so that a good performance of magnetoelectric transformation is achieved. In addition, for the different shape ratio of semi-axes, the predicted magnetoelectric values of fiber-direction are almost the same.
IV. CONCLUSION A numerical approach based on the micromechanics and FEM was developed to predict the effective multi-field properties of electro-magneto-thermoelastic composites. The thermal-related properties involving pyroelectric and pyromagnetic effects of the multiphase smart composites were incorporated in the modeling. The multi-fields related to the electric field, magnetic field, deformation and temperature field, as well as their coupling effects of the composites under periodic boundary conditions were obtained numerically. The effective thermal expansion coefficient, pyroelectric coefficient, pyromagnetic coefficient and other elastic, electric, and magnetic properties for the piezoelectric material,
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piezomagnetic material and magnetoelectric material were calculated by means of the homogenization approximation. Some of the predictions showed good agreement with the theoretical predictions by the well-known Mori-Tanaka method, and the Mori-Tanaka method always gives the good predictions on the electric and magnetic properties of the electromagnetic composite. The results demonstrated that the properties of fiber-direction for the composite are always dominant. The cross-section shape of inclusion fiber shows obvious influence on the in-plane thermal-related properties of piezoelectricpiezomagnetic composite but almost no effect on the fiber-direction ones. For a proper volume fraction of the inclusion fiber, there are optimal valued of the pyromagnetic and pyroelectric coefficients, as well as the magnetoelectric coupling coefficient.
References [1] Newnham,R.E., Skinner,D.P. and Cross,L.E., Connectivity and piezoelectric-pyroelectric composites. Materials Research Bulletin, 1978, 13: 525-536. [2] Satapathy,S., Gupta,P.K. and Varma,K.B.R., Enhancement of nonvolatile polarization and pyroelectric sensitivity in lithium tantalate (LT)/poly(vinylidene fluoride) (PVDF) nanocomposite. Journal of Physics D, 2009, 42: 055402. [3] Nan,C.W., Bichurin,M.I., Dong,S., Viehland,D. and Srinivasan,G., Multiferroic magnetoelectric composites: Historical perspective, status, and future directions. Journal of Applied Physics, 2008, 103: 031101. [4] Nan,C.W. and Clarke,D.R., Effective properties of ferroelectric and/or ferromagnetic composites: a unified approach and its application. Journal of the American Ceramic Society, 1997, 80: 1333-1340. [5] Dunn,M.L., Micromechanics of coupled electroelastic composites: effective thermal expansion and pyroelectric coefficients. Journal of Applied Physics, 1993, 73: 5131-5140. [6] Dunn,M.L. and Taya,M., An analysis of piezoelectric composite materials containing ellipsoidal inhomogeneities. Proceedings of the Royal Society of London, Series A: Physical and Engineering Sciences, 1993, 443: 265-287 [7] Dunn,M.L. and Taya,M., Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites. International Journal of Solids and Structures, 1993, 30: 161-175. [8] Li,J.Y. and Dunn,M.L., Anisotropic coupled-field inclusion and inhomogeneity problems. Philosophical magazine A, 1998, 77: 1341-1350. [9] Li,J.Y. and Dunn,M.L., Micromechanics of magnetoelectroelastic composite materials: average fields and effective behavior. Journal of Intelligent Material Systems and Structures, 1998, 9: 404-416. [10] Li,J.Y., The effective pyroelectric and thermal expansion coefficients of ferroelectric ceramics. Mechanics of Materials, 2004, 36: 949-958. [11] Hori,M. and Nemat-Nasser,S., Double-inclusion model and overall moduli of multi-phase composites. Journal of Engineering Materials and Technology, 1994, 116: 305. [12] Li,J.Y., Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials. International Journal of Engineering Science, 2000, 38: 1993-2011. [13] Tong,Z.H., Lo,S.H., Jiang,C.P. and Cheung,Y.K., An exact solution for the three-phase thermo-electromagneto-elastic cylinder model and its application to piezoelectric–magnetic fiber composites. International Journal of Solids and Structures, 2008, 45: 5205-5219. [14] Challagulla,K.S. and Georgiades,A.V., Micromechanical analysis of magneto- electro-thermo-elastic composite materials with applications to multilayered structures. International Journal of Engineering Science, 2011, 49: 85-104. [15] Berger,H., Kari,S., Gabbert,U., Ramos,R.R., Guinovart,R., Otero,J.A. and Castillero,J.B., An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites. International Journal of Solids and Structures, 2005, 42: 5692-5714. [16] Lee,J., Boyd,J.G. and Lagoudas,D,C., Effective properties of three-phase electro-magneto-elastic composites. International Journal of Engineering Science, 2005, 43: 790-825. [17] Tang,T. and Yu,W.Micromechanical modeling of the multiphysical behavior of smart materials using the variational asymptotic method. Smart Materials and Structures, 2009, 18: 125026-125040. [18] Tang,T. and Yu,W., Variational asymptotic homogenization of heterogeneous electromagnetoelastic materials. International Journal of Engineering Science, 2008, 46: 741-757. [19] Liu,W. and Zhong,Z., Three-dimensional thermoelastic analysis of functionally graded of plate. Acta Mechanica Solida Sinica, 2011, 24: 241-249. [20] Nabian,A., Rezazadeh,G., Almassi,M. and Borgheei,A., On the stability of a functionally graded rectangular micro-plate subjected to hydrostatic and nonlinear electrostatic pressures. Acta Mechanica Solida Sinica, 2013, 26: 205-220.
Vol. 28, No. 2 Zhichao Zhang et al.: Multi-Field Properties of Electro-Magneto-Thermoelastic Composites· 155 · [21] Wang,J.H., Guo,R., Bhalla,A.S., Finite element simulation of magnetostrictive and piezoelectric coupling in a layered structure. Ferroelectrics Letters, 2007, 34: 46-53. [22] Perez-Fernandez,L.D., Bravo-Castillero,J., Rodrı’guez-Ramos,R. and Sabina,F.J., On the constitutive relations and energy potentials of linear thermo-magneto-electro-elasticity. Mechanics Research Communication, 2009, 36, 343-350. [23] Suquet,P., Elements of homogenization theory for inelastic solid mechanics. In: Sanchez-Palencia, E., Zaoui, (Eds.), Homogenization Techniques for Composite Media. Berlin: Springer-Verlag, 1987, 194-275. [24] COMSOL, Comsol Multiphysics Modeling Guide, Version 4.3. www.comsol.com (2012).
APPENDIX In Eq.(2), the expression
Σ = σ11 σ22
Z = ε11 ε22 ⎡ C11 C12 ⎢ C12 C11 ⎢ ⎢ C13 C13 ⎢ ⎢ 0 0 ⎢ ⎢ 0 0 ⎢ ⎢ 0 0 L=⎢ ⎢ 0 0 ⎢ ⎢ 0 0 ⎢ ⎢ e31 e 31 ⎢ ⎢ 0 0 ⎢ ⎣ 0 0 q31 q31
Π = λ11 λ22
of each components: σ33 ε33
σ32 ε32
C13 C13 C13 0 0 0 0 0 e33 0 0 q33
0 0 0 C44 0 0 0 e15 0 0 q15 0
λ33
0
σ31 ε31
ε12
0 0 0 0 C44 0 e15 0 0 q15 0 0 0
σ12
0
D1 E1
0 0 0 0 0 C66 0 0 0 0 0 0 p1
D2 E2
0 0 0 0 e15 0 η11 0 0 a11 0 0 p2
p3
D3 E3
0 0 0 e15 0 0 0 η11 0 0 a11 0 m1
B1 H1
e31 e31 e33 0 0 0 0 0 η13 0 0 a33 m2
T B3 T
B2 H2
0 0 0 0 q15 0 a11 0 0 μ11 0 0 m3
H3 0 0 0 q15 0 0 0 a11 0 0 μ11 0
T
(13) ⎤
q31 q31 ⎥ ⎥ q33 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ a33 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ μ33
(14)
(15)
(16)