Theory of magnetoelectric effect in multilayer

Theory of magnetoelectric effect in multilayer nanocomposites on a substrate: Static bending-mode response Matthias C. Krantz and Martina Gerken Citation: AIP Advances 3, 022103 (2013); doi: 10.1063/1.4790630 View online: http://dx.doi.org/10.1063/1.4790630 View Table of Contents: http://aipadvances.aip.org/resource/1/AAIDBI/v3/i2 Published by the AIP Publishing LLC.

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AIP ADVANCES 3, 022103 (2013)

Theory of magnetoelectric effect in multilayer nanocomposites on a substrate: Static bending-mode response Matthias C. Krantza and Martina Gerken Institute of Electrical and Information Engineering, Christian-Albrechts-Universitaet zu Kiel, Germany (Received 15 November 2012; accepted 16 January 2013; published online 1 February 2013)

Magnetoelectric (ME) coefficients for bending excitation in static magnetic fields and the bending response of multilayer composites with alternating magnetostrictive (MS) and piezoelectric (PE) layers on a substrate are investigated systematically. Theory and closed-form analytic solutions for the static magnetoelectric and the bending response coefficients are presented. Results of systematic variation of layer numbers, layer sequences, PE volume fractions, substrate thicknesses, and four different material systems (employing FeCoBSi, Terfenol-D, AlN, PZT, and Si) are given for a fixed total composite thickness of 5μm. Among more than 105 structures investigated the greatest static ME coefficient of 62.3 V/cmOe is predicted for all odd layer number FeCoBSi-AlN multilayer composites on a Si substrate at vanishing substrate thickness and a PE material fraction of 38%. Varying the substrate thickness from 0μm to 20μm and the PE fraction from 0% to 100%, broad parameter regions of high ME coefficients are found for odd and large layer number nanocomposites. These regions are further enhanced to narrow maxima at vanishing substrate thickness, which correspond to structures of vanishing static bending response. For bilayers and even layer number cases broad maxima of the ME coefficient are observed at nonzero substrates and bending response. The optimal layer sequence and PE fraction depend on the material system. Bending response maxima occur at zero Si substrate thickness and nonzero PE fractions for bilayers. For multilayers nonzero Si substrates and zero PE fractions are found to be optimal. Structures of even ME layer numbers of PE-MS...Sub layer sequence display regions of vanishing bending response with large ME coefficients, i.e., produced by longitudinal excitation. Copyright 2013 Author(s). This article is distributed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4790630]

I. INTRODUCTION

Magnetoelectric (ME) nanocomposites of piezo-/ferromagnetic and piezo-/ferroelectric materials have recently attracted significant attention due to their interesting interactions and their application potential as multifunctional devices.1–3 Multiferroic coupling of different orders in the constituent phases or linear magnetoelectric coupling via strain has yielded among others sensitive magnetic field sensors4–7 and energy harvesting devices.8 Here, we investigate the static ME coefficient as the device performance measure at low, non-resonant frequencies, which are, e.g., relevant for the detection of biosignals. Many of the magnetoelectric interactions are inherently nonlinear. However, we are concerned with the linear magnetoelectric effect here, which is produced by stress-mediated coupling between different phases of the nanocomposite. Among the different nanocomposite structures,2 i.e., particles in a matrix (0-3), columns in a matrix (1-3),

a [email protected]

2158-3226/2013/3(2)/022103/26

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C Author(s) 2013


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FIG. 1. Layered magnetoelectric (ME) nanocomposite with magnetostrictive (MS) and piezoelectric (PE) layers on a substrate (Sub) with excitation by static magnetic or electric fields producing a static bending response and electric potential across the multilayer. Also shown are notations for different layers, the coordinate system displaying the displacement z(x) in z-direction with the z-coordinate originating at the neutral plane, and the neutral plane position d from the bottom of the multilayer stack (white line, zero strain position without external fields). Not shown is the field-dependent zero strain position zN . which may be inside or outside of the nanocomposite (a). Examples of ME-nanocomposite cross sections of different layer number, layer sequence, PE-volume fraction of the nanocomposite, and substrate thickness (b).

and layered systems (2-2), bilayer composites on a substrate currently produce record dynamic ME coefficients of several 100 V/cmOe.4, 5 These sensors operate in bending mode at resonance, whereby resonance-enhancement contributes significantly to the sensitivity.9–11 Layered systems have the additional benefit to be compatible with MEMS production techniques.12 Magnetoelectric multilayer nancomposites have been pursued for improved interface coupling.13–16 Further reasons for pursuing multilayers are, e.g., enhanced charge collection from piezoelectric layers,15 enhanced ferroic coupling,17–19 magnetic noise reduction, or simply enhanced mechanical durability of thin cantilevers. Concerning the excitation mode most experimental effort has concentrated on the longitudinal mode9, 20, 21 with theoretical work in the area of analytic or finite element models20 for bilayers, trilayers, or effective medium models22, 23 for multilayers. Magnetoelectric nanocomposites excited in bending modes24–26 have been investigated to a lesser degree. Furthermore, bilayer ME composites on a substrate excited in bending mode have been investigated.27 Here, we present a systematic analytic investigation of the static ME coefficient of multilayer nanocomposites on a substrate. The paper is structured as follows. In section II we present the theory for the static bending mode of magnetoelectric multilayer composites on a substrate In addition to readily usable analytic expressions for the static magnetoelectric coefficient we also derive equations for the bending radii of multilayer ME nanocomposites on a substrate in external fields and the magnetic bending response coefficient. In section III we employ the derived expressions to calculate the ME coefficients and bending response coefficients for multilayer nanocomposites on a substrate, whereby we systematically investigate the five parameter space of layer number, layer sequence, PE fraction, substrate thickness, and material combination. The results are summarized in section IV.

II. STATIC ME COEFFICIENT AND BENDING RESPONSE FOR MS-PE MULTILAYERS ON A SUBSTRATE

The theory of the static bending mode magnetoelectric coefficient for MS-PE multilayers on a substrate and the corresponding bending behavior is derived as follows. We calculate the bending response of the multilayer nanocomposite and the resulting voltage across the multilayer stack induced by a magnetic field parallel to the long axis of the ME nanocomposite (Fig. 1). The magnetostrictive layers are considered electrically conducting and potential differences across them are neglected. Fig. 1 shows the general layout of the magnetoelectric multilayer stack on a substrate.


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Our model considers even and odd ME layer numbers, different layer sequences, i.e., MS-PEMS...PE-Sub, MS-PE-MS...MS-Sub, PE-MS-PE...PE-Sub, and PE-MS-PE....MS-Sub, as well as varying PE volume fractions in the ME nanocomposite, and arbitrary substrate thicknesses. Please note that all layers of the same type have the same layer thickness. The coordinate system is shown with z(x) denoting the bending displacement perpendicular to the long axis of the nanocomposite, which results from applied magnetic fields parallel to the cantilever. Furthermore, different local coordinate systems are used in the MS, PE, and substrate layers in the constitutive equations below with the x-direction corresponding to 3 in the MS layers and to 1 in the PE, and substrate layers. Here, we consider cantilevers with aspect ratios obeying length width height. Thus, y-direction dependencies may be neglected and all calculations are performed in the xz-plane. Considering free boundary conditions at each end of the multilayer nanocomposite, there is no xdependency of the results for the static case. The ME coefficient and the bending radius are calculated by evaluation of the force and bending moment balance over the crosssection of the multilayer system on a substrate. Both, the integrated stress in x-direction and the net bending moment about the y-axis contributed from x-direction stresses must vanish in the static case. The results obtained for the static ME coefficient and the bending radius are also applicable to long cantilever structures with one fixed end. Constitutive equations describe the relation between the stress, and external field-induced stains for each layer. Strictly linear constitutive equations are used, thus the theory is valid, where these linear effects dominate the many known nonlinear interactions. The constitutive equation describing the piezomagnetic effect in the magnetostrictive layers is given as: S3m = s33m T3m + d33m H3

(1)

Here S, T, and H denote the strain, stress and magnetic field respectively, while s33m is the elastic compliance at constant H-field and d33m is the longitudinal piezomagnetic constant. The corresponding equation involving the effect of stress on the magnetic induction is not required here as we are considering external magnetic fields only. The constitutive equations for the piezoelectric layers are S1 p = s11 p T1 p + g31 p D3

(2)

E 3 = −g31 p T1 p + β33 D3

(3)

and

Here D and E denote the electric displacement and field respectively and s11p , g31p , and β 33 are the elastic compliance at constant electric displacement, the transverse piezoelectric constant, and the inverse dielectric constant at constant stress. For the substrate the electronic properties are neglected and the constitutive equation is limited to Hooks law, i.e., S1s = s11s T1s

(4)

with S, T, and s11s denoting strain, elastic compliance, and stress respectively. We now consider the force balance over the crosssection of the multilayer system on a substrate. The total stress in x-direction contributed from all surface elements of a cross section of the multilayer and the substrate in the yz plane must vanish, i.e., ∫ Tx x dz = 0, whereby any variations in y-direction have been neglected. Furthermore, ideal interface lamination between different layers is assumed. For the static stress induced bending of the nanocomposite cantilever by external fields the strain depends linearly on the z-coordinate25, 30 as Sx =

zN − z R

(5)

whereby R is the bending radius and zN denotes the z-position of vanishing strain in the general case of external fields being present. This implies the above mentioned coordinate system with the origin of the z-coordinate located at the neutral plane. Using the constitutive equations for the stress of the


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PE, MS, and substrate layers and the above strain linearity with z this yields n p −1 1 j (h m +h p )+h p +(n m −n p +na )h m +h s −d z N − z − g31 p D3 dz s11 p j=0 j (h m +h p )+(n m −n p +na )h m +h s −d R +

1 s33m

n m −1 k (h m +h p )+h m +(n p −n m +1−n a )h p +h s −d k=0

k (h m +h p )+(n p −n m +1−n a )h p +h s −d

z N −z 1 h s −d z N −z − d33m H3 dz + dz = 0 R ss −d R (6)

Here the sum and integral in the first term perform the integration over all PE layers of the ME multilayer stack, while the second and third terms integrate over the MS and substrate layers, respectively. Above h denotes layer thickness and n layer number with the subscripts m, p, and s referring to the MS, PE, and substrate layers. The parameter na determines the layer sequence, i.e., na = 0 corresponds to the MS-PE...Sub sequence while na = 1 produces the PE-MS...Sub sequence. Odd magnetoelectric layer numbers of the nanocomposite, i.e., not counting the substrate, are determined by the values for nm and np . The integration boundaries in (6) are counted from the bottom of the multilayer stack and d denotes the neutral plane position. Thus we use a coordinate system centered at the neutral plane. For vanishing external D and H fields zN = 0 in the neutral plane centered coordinate system used here. The neutral plane position d can be calculated directly by setting zN = 0, D3 = 0, and H3 = 0 and solving (6). This yields the neutral plane position d:

h m n m h m +2h p −2h p n m +2h p n p +2h s − 2h p n a +(n m − 1) h m +h p 1 d= h n h n m m 2s33m + p p + hs s33m

s11 p

ssub

h p n p h p + 2h m n m − 2h m n p + 2h s + 2h m n a + n p − 1 h m + h p hs 2 + + 2s11 p 2s Sub (7)

Next, we consider the bending moment. The net bending moment about the y-axis contributed from x-direction stresses of surface elements in the yz-plane for all layers of the multilayer and substrate stack must vanish, i.e., ∫ zTx x dz = 0. In detail this yields the following integral equation for all layers of the stack: n p −1 1 j (h m +h p )+h p +(n m −n p +na )h m +h s −d z N − z − g31 p D3 )zdz ( s11 p j=0 j (h m +h p )+(n m −n p +na )h m +h s −d R

+

n m −1 k (h m +h p )+h m +(n p −n m +1−n a )h p +h s −d zN − z 1 h s −d z N − z 1 − d33m H3 )zdz + zdz = 0 ( s33m k=0 k (h m +h p )+(n p −n m +1−na )h p +h s −d R ss −d R (8)

Here again the above constitutive equations and strain linearity were used. Equations (6) and (8) can be solved for the zero strain position in the presence of nonzero external fields zN and and the bending radius R. This yields the closed form solutions for zN and R in terms of external fields, material constants, and multilayer nanocomposite and substrate geometries. In order to present the analytic results in compact form we have defined numbered terms aj , bj , cj , dj , ej , fj , and gj with j as the numbering index (not to be confused with materials properties in the constitutive equations, e.g., d33m ). The bending radius solution in external fields is 36 j=1 a j R = 14 (9) 14 j=1 b j H3 + j=1 c j D3 whereby the aj , bj , and cj denote numbered terms given in the appendix. The solution for the general position of the zero strain plane of the multilayer ME nanocomposite in the presence of H and


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D-fields is given as

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53 zN =

j=1 14 j=1

d j H3 + f j H3 +

53 j=1

e j D3

j=1

g j D3

14

(10)

Again dj , ej , fj , and gj denote numbered terms of geometry parameters and material properties given in the appendix. The results given in (9) and (10) enable a discussion of the general behavior of R and zN . The inverse bending radius is linear in the magnetic and displacement fields H3 and D3 whereby the respective prefactors depend on layer number, geometry, and material properties. Thus 1/R behaves as a tilted plane in the H3 -D3 parameter space with the prefactors determining the slopes in H3 and D3 direction. The ratio of the fields versus the ratio of the corresponding prefactors determine which field dominates the radius behavior and which contributes a small offset. The bending radius changes sign with the dominating field. The behavior of the zero strain position zN also greatly depends on the ratio of the external H3 and D3 fields as determined by the ratio of the respective prefactors in (10). If any of the fields vanish zN is constant. For nonzero fields zN is a completely antisymmetric function of the fields but offset by the zero field constant value. Depending on the external field strengths, directions, geometry and materials parameters, the zero strain position zN may be located inside the nanocomposite or outside, the latter case corresponding to a net static strain of the whole cantilever. As opposed to the bending radius the zero strain location zN is determined by field strength ratios not magnitude. The field ratios determining the zN behavior are given by the prefactors in (10). If the H3 /D3 ratio is smaller than the characteristic ratio of the prefactors, D3 dominated behavior results and zN is similar to the neutral plane position d, i.e. the zero strain location for bending oscillations without external fields. For greater H3 /D3 ratios and H3 dominated behavior zN moves away from d. Thus the field dependent zero strain behavior zN generally differs greatly from the field independent neutral plane behavior. This reflects the fact that the neutral plane position corresponds to the dynamic bending mode behavior driven by inertia whereas zN is determined by static field-induced deformations of the nanocomposite, which is different even in the limit of vanishing fields. We now proceed by integrating the electric field across the ME multilayer stack to obtain the potential produced by static external field-induced strains. The magnetostrictive layers and the substrate are considered conducting, thus the E-field integration is limited to the piezoelectric layers, i.e., n p −1 j h +h +h + n −n +n h +h −d ( m p) p ( m p a) m s −g31 p z N − z g31 p 2 + + β33 D3 dz = V (11) s11 p R s11 p j=0 j (h m +h p )+(n m −n p +n a )h m +h s −d Again equations (2), (3), and (5) have been used here and the following solution for the external voltage is produced. n p hm n ph p n p h p g31 p hm − + + hs + hm na V = nm hm − z N − d − 2 2 2 s11 p R n p h p g31 p 2 + n p h p β33 + D3 (12) s11 p The static magnetoelectric voltage coefficient normalized by the total ME nanocomposite thickness, 1 dV is derived by inserting (9) and (10), i.e., the field dependent results for R i.e. αme = n m h m +n p h p d H3 and zN , and differentiation. In the most general case of nonzero external magnetic and displacement fields this yields the static ME-coefficient solution 14 n p hm n p h p hm 1 j=1 b j g31 p αme = − n ph p d + +z N − n m h m + − −h s −h m n a 36 n m h m +n p h p 2 2 2 j=1 a j s11 p

⎤ 14 14 14 53 53 D3 g31 p n p h p D3 14 j=1 c j + H3 j=1 b j j=1 d j j=1 g j − j=1 e j j=1 f j ⎥ + ⎦

2 36 14 14 j=1 a j s11 p D3 j=1 g j + H3 j=1 f j (13)


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FIG. 2. Calculated static bending mode magnetoelectric coefficient for Terfenol-D - PZT nanocomposites without substrate displaying the variation with PE volume fraction for N=2, 3, 4, and 100 layers. The results agree with published bilayer and trilayer results given in Refs. 25 and 29.

Here the field dependent zN was resubstituted from (10) after differentiation. In the following we assume the open-circuit condition D3 = 0 resulting in Eq. (14). 14 1 j=1 b j g31 p n ph p αme = − 36 nm hm + n p h p j=1 a j s11 p 53 n ph p n p hm hm j=1 d j + 14 − − hs − hm na × d+ − nm hm + (14) 2 2 2 j=1 f j The field independence of the ME coefficient is expected in a linear model. In order to understand the corresponding bending response of ME multilayer nanocomposites on substrates we also calculate d R1 , d H3

i.e., the change in cantilever curvature in an applied H-field. This is denoted the magnetic bending response coefficient subsequently. Using (9) it is obtained as: 14 d R1 j=1 b j = 36 (15) d H3 j=1 a j The bending response coefficient behavior is of interest for energy harvesters, optical detection of cantilever bending, and interpretation of static and dynamic ME coefficient findings. Results for dV d H3

d

1

and d HR3 will be investigated systematically for the 5 parameter space of varying layer numbers, layer sequences, PE fraction, substrate thickness and materials in the following section. To validate the results of the analytic multilayer nanocomposite model we conducted numerous tests: (1)

(2)

The zero-substrate results were compared to available static ME coefficient calculations for Terfenol-D-PZT bilayers and excellent quantitative agreement was found with Ref. 25. Qualitative agreement is obtained for N=2 and N=3 with the results given in Ref. 29. Our results are given in Fig. 2. The symmetry of arbitrary-thickness bilayers without substrates with respect to a layer sequence change requires identical results in each case except for sign changes for R, zN , ddHV3 , d

(3) (4)

1

and d HR3 . This was verified for different PE fractions. Correspondingly the bending response of 3-layer nanocomposites without substrates is predicted to vanish by symmetry independent of PE fraction. This was also confirmed. For arbitrary even layer number nanocomposites of equal MS and PE layer thickness on a substrate the results for R, zN , and ddHV3 are the same by symmetry if the numerical values for


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FIG. 3. Calculated static bending mode magnetoelectric coefficient for multilayer ME nanocomposite for different layer numbers, layer sequences, piezoelectric volume fraction, and substrate thickness for the FeCoBSi-AlN-Si system. The ME voltage coefficient is given for a 5μm fixed total thickness of the ME nanocomposite. Fig. 3(a)-(d) display results for the MS-PE-MS...Sub sequence and Fig. 3(e)-(h) give the PE-MS-PE...Sub sequence results. Each surface plot displays the MEcoefficient behavior as function of PE fraction and substrate thickness with contour lines in 5 V/cmOe steps. The greatest static ME coefficients are observed for odd or large layer numbers over a wide substrate layer thickness range with a small enhancement near vanishing substrates. Maxima occur, where the cantilever bending response in H-fields vanishes, i.e., at longitudinal mode excitation.

(5)

(6)

s11p ↔s33m , d33m ↔g31p , and the layer sequence na = 0 ↔ na = 1 are exchanged simultaneously. This was also successfully confirmed. For bilayers with substrates the static ME coefficient was calculated using commercial finite element software for the FeCoBSi-AlN-Si system and close quantitative agreement was found with the results given in Fig. 3(a). Finally, to exclude the possibility of typographical errors, the model with all equations was d

1

reconstructed from a printed copy and the R, zN , ddHV3 , and d HR3 results were found to agree with the results of the original calculation program for numerous nontrivial cases.

III. RESULTS AND DISCUSSION A. Variation of layer number and layer sequence

First, we evaluate the influence of the number of layers N in the multilayer composite and the layer sequence (MS-PE-MS...Sub versus PE-MS-PE...Sub) on the ME coefficient. For this purpose we evaluate FeCoBSi-AlN multilayer nanocomposites on a Si substrate. We use the material parameters given in Table I. Here, we are interested in geometry-dependent effects. Therefore, we assume constant thickness-independent material parameters. Results of static ME coefficient calculations using (14) are given in Fig. 3. For each layer number N and layer sequence the


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TABLE I. Materials Parameters.

Kg Density ( m 3 )

FeCoBSia Terfenol-Db AlNa PZTb Si a Parameters b Parameters

7250 9200 3268 7700 2328

Elastic compliance s33m , s11p , sSub (Pa−1 ) s33m = 7.738x10−12 s33m = 3.64x10−11 s11p = 2.899x10−12 s11p = 1.787x10−11 sSub = 5.974x10−12

Piezoelectric constant Piezomagnetic constant Dielectric constant 2

g31p ( mAs )

d33m ( mA )

β 33 ( VAsm )

6.577x10−8 6.548x10−9 0.024 7.724x10−3

1.27x1010 4.416x107

from Ref. 4. from Ref. 20.

ME coefficient is calculated for variation of the PE volume fraction in a 5μm fixed thickness ME nanocomposite and the separately varied Si substrate thickness. In the PE fraction- substrate thickness parameter space we observe broad ME coefficient maxima at PE fractions in the 0.3 - 0.5 range and substrate thicknesses below 8 μm. As expected based on the product property of the ME coefficient in MS-PE composites, the ME coefficient vanishes, if either the PE fraction vanishes or the MS fraction vanishes. For even layer numbers of both layer sequences the ME coefficients decrease towards vanishing substrate thickness. However, for odd or large layer numbers of both sequences the broad maxima are further enhanced towards vanishing substrate thickness. This behavior is essentially opposite to the corresponding findings for the dynamic ME coefficient for the lowest resonant bending mode, where greatest ME response was observed for bilayers without substrates and lower ME coefficients for higher layer numbers at nonzero substrate thickness.28 Investigating layer sequence changes on the static bending mode ME coefficient we observe a shift of the maxima to different PE fractions primarily for bilayers and less so for multilayers. At large layer numbers even-odd differences vanish. For the FeCoBSi-AlN-Si bilayer sequence we predict a 16% higher static ME coefficient than for the AlN-FeCoBSi-Si bilayer sequence. The highest ME coefficient of 56.5 V/cmOe is obtained for 49%PE fraction and 5.6 μm substrate thickness. In order to gain a better understanding of the cantilever behavior, we next evaluate the bending response of the multilayer nanocomposites in static magnetic fields. According to (9) the inverse d

1

bending radius R1 is linear in the external fields H3 and D3 . The bending response coefficient d HR3 is thus independent of external fields and depends only on nanocomposite properties, i.e., PE fraction, substrate thickness, layer number, layer sequence, and material properties. The surface plots in Fig. 4 display the magnetic bending response coefficient as a function of substrate thickness and PE fraction for different layer numbers (top to bottom) and layer sequence (left to right). In correspondence to Fig. 3 the FeCoBSi-AlN-Si system is chosen with a fixed nanocomposite thickness of 5μm. For the MS-PE-Sub bilayers in Fig. 4(a) the greatest bending response to static magnetic fields of 1.6 cm1Oe is observed for vanishing substrate thickness near 40% PE fraction. With increasing substrate thickness the bending response decreases rapidly and the maximum shifts to lower PE fractions. For the N = 4 layer nanocomposite in Fig. 4(c) the behavior is quite similar, but the zero-substrate bending response is lower and the maximum occurs at about 2.5μm substrate thickness and vanishing PE fraction. The behavior for the N = 3 and N = 100 layer nanocomposites are essentially identical with maximum bending at small substrate thickness and vanishing PE fractions (Fig. 4(b) and 4(d)). However, here the bending response at zero substrate thickness also vanishes. For the N = 3 and infinite layer number systems this can be strictly attributed to inversion symmetry in z-direction, whereby the N = 100 layer system approximates the infinite layer number behavior. The maxima of the bending response coefficients for all except bilayers are equal at 1.0 cm1Oe as they occur at vanishing PE fractions and thus correspond to identical structures. In the PE-MS...Sub sequence the results differ greatly for the even layer number nanocomposites (Fig. 4(e) and 4(g)). For the bilayer case bending occurs in opposite directions for vanishing substrates and vanishing PE fractions with a sign reversal of the bending radius in between. The sign reversal


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FIG. 4. Static magnetic bending response coefficient for different layer numbers N and sequences for the FeCoBSi-AlN-Si nanocomposite. Displayed are the respective variations with PE volume fraction and substrate thickness in each surface plot. Fig. 4(a)-(d) display results for the MS-PE-MS...Sub sequence and Fig. 4(e)-(h) give the PE-MS-PE...Sub sequence results. Please note the different scales in the different plots. Negative numbers correspond to opposite bending direction.

of the bending or vanishing magnetic bending response indicates regions without curvature and corresponds to purely longitudinal excitation of the ME bilayer on the substrate. It is realized in a linear area in PE fraction substrate thickness space. This area also coincides with the vanishing dynamic bending mode ME coefficient for this layer sequence.28 For the static ME coefficient, however, no corresponding behavior is observed (Fig. 3(e)). The N = 4 layer bending response (Fig. 4(g)) is similar to the N = 2 case with the difference that the linear zero-bending zone is shifted to smaller substrate thickness. The behavior of the N = 3 layer system with PE-MS-PE-Sub sequence (Fig. 4(f)) is found to be similar to the corresponding MS-PE-MS-Sub nanocomposite. This is only trivial at vanishing PE fraction, where there is no difference between both sequences. All bending response minima of the PE-MS...Sub sequence are equal to the MS-PE..Sub results except for reversed signs for bilayers at vanishing substrate thickness. Comparison of Fig. 3 and Fig. 4 shows that the enhanced maxima of the ME coefficient at vanishing substrate thickness observed only for the odd (N=3) or high (N=100) layer number systems correspond to ME cantilevers, which do not change curvature in applied magnetic fields. Hence these maxima correspond to excitation in a longitudinal mode. Their occurrence at odd and large layer numbers only is consistent with symmetry considerations, whereby inversion symmetry in z-direction prohibits bending response for odd ME layer numbers and vanishing substrate thickness. For other ME nanocomposites in this vast parameter space no direct correlation between static ME coefficient and static bending response to magnetic fields is observed. This is due to the product property of the magnetoelectric effect in MS-PE-composites. Largest bending is observed for vanishing PE fraction, but in this case no electric voltage is produced and the ME coefficient vanishes.


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FIG. 5. Static bending mode magnetoelectric coefficient for different multilayer nancomposite material systems. Shown are the bilayer and 100 layer results for FeCoBSi-AlN-Si (a-b) scale 0-65 V/cmOe, FeCoBSi-PZT-Si (c-d) scale 0-8.5 V/cmOe , Terfenol-D-AlN-Si (e-f) scale 0-2,3 V/cmOe, and Terfenol-D-PZT-Si (g-h) scale 0-0.4 V/cmOe with layer sequence MS-PE...Sub. Each surface plot displays the ME coefficient as function of PE volume fraction and substrate thickness with contour line steps of 5 V/cmOe (a-b), 0.5 V/cmOe (c-d), 0.1 V/cmOe (e-f), and 0.05 V/cmOe (g-h). For bilayers the highest ME coefficient is observed with a thin substrate, while for many-layer composites a substrate is disadvantageous. The PE fraction needs to be optimized depending on the material system.

B. Variation of materials

Here, we investigate the influence of the material choice on the ME coefficient. Fig. 5 compares the static bending-mode ME coefficient results of bilayers and large layer numbers for the four material systems FeCoBSi-AlN-Si, FeCoBSi-PZT-Si, Terfenol-D-AlN-Si, and Terfenol-D-PZT-Si. Shown is the ME coefficient behavior for N=2 and N=100 ME nanocomposites of MS-PE...Sub layer sequence on a substrate as a function of PE fraction and substrate thickness. Please note that we have calculated the ME coefficients for different materials systems without considering the severe issues of possible conflicting deposition, poling, or field-annealing environments, which may preclude producing some of these nanocomposites. The results for all material combinations display the above described broad maxima at small but nonzero substrate thickness for bilayers and a similar 100-layer behavior with enhanced response at vanishing substrates. The magnitude of the ME coefficient maxima differs greatly for the nanocomposites investigated. The complex behavior of (14) cannot be reduced to a simple scaling behavior and the scaling ratios suggested by the piezomagnetic piezoelectric product property behavior, i.e., d33m g31p , produce ratios are two to three times too small, or vastly too large if normalized by elastic compliances. Furthermore, different from the other investigated materials we find a 9% greater ME coefficient maximum for bilayers than for tri- or N = 100 multilayers in Terfenol-D-AlN (Fig. 5(e) and 5(f), which is possibly attributable to the great elastic compliance mismatch. Aside from the magnitude effect of the static ME coefficient for different materials we find systematic shifts of the ME coefficient maxima towards different PE


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FIG. 6. Static magnetic bending response coefficient for bilayers and N = 100 layers and the MS-PE..Sub sequence for material systems FeCoBSi-AlN-Si (a-b), FeCoBSi-PZT-Si (c-d), Terfenol-D-AlN-Si (e-f), and Terfenol-D-PZT-Si (g-h).

fractions. These effects are attributed to different elastic compliances. i.e., there is a low correlation but no scaling of the PE fraction of the multilayer ME coefficient maxima with the s11p /s33m ratio. Fig. 6 shows the magnetic bending response coefficient for the different material combinations corresponding to Fig. 5. We find the above described general bilayer and N = 100 layer behavior with maxima at vanishing substrates (N = 2) and at vanishing PE fraction (N = 100). The bending response maxima of systems with the same magnetostrictive materials are equal, whereby the bending response with FeCoBSi layers is about 10x higher than with Terfenol-D. The behavior with variation of PE-fraction and substrate thickness is the same in all material cases except for pronounced PE fraction and substrate thickness shifts observed. Compared to the FeCoBSi-AlN-Si results the FeCoBSi-PZT-Si behavior is shifted to higher and the Terfenol-D-AlN-Si and TerfenolD-PZT-Si behavior to lower PE fractions. These shifts are similar to the shifts of the static ME coefficient results (Fig. 5), however, the maxima of the static ME coefficient and the static bending response do not coincide in PE fraction-substrate thickness space. The presented results show that for all studied material systems a thin substrate is advantageous in the bilayer case, while it is disadvantageous for many-layer nanocomposites. The PE fraction needs to be optimized depending on the material system.

C. Sequence variation in bilayers on a substrate

In Fig. 7 we compare the static ME coefficient behavior for bilayers of layer sequences MS-PESub and PE-MS-Sub for the four material systems FeCoBSi-AlN-Si, FeCoBSi-PZT-Si, Terfenol-DAlN-Si, and Terfenol-D-PZT-Si. We observe a change of the optimal PE fraction depending on the layer sequence. We find a shift of the ME coefficient maxima towards lower PE fractions for the


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FIG. 7. Static bending-mode magnetoelectric coefficient for bilayers on Si substrates for the nancomposite material systems of Fig. 5. Shown are the results for different layer sequences (left vs.right), i.e., MS-PE-Sub sequence in Fig. 7(a),(c),(e),(g), PE-MS-Sub sequence in Fig. 7(b),(d),(f),(h), and different materials (top-bottom), i.e., FeCoBSi-AlN-Si (a)-(b) scale 0-60 V/cmOe, FeCoBSi-PZT-Si (c)-(d) scale 0-7,5 V/cmOe, Terfenol-D-AlN-Si (e)-(f) scale 0-2.3 V/cmOe, and Terfenol-D-PZTSi (g)-(h) scale 0-0.35 V/cmOe. Aside from great magnitude differences for the different material systems, also the optimal PE fraction, substrate thickness, and bilayer sequence depend on the material system.

PE-MS-Sub sequence as compared to the MS-PE-Sub sequence. Furthermore, the material system determines if the MS-PE-Sub sequence or the PE-MS-Sub sequence is optimal. Fig. 8 compares the bilayer bending response of the MS-PE-Sub and PE-MS-Sub sequence for the different material combinations. The PE-MS-Sub results (Fig. 8(b), 8(d), 8(f), and 8(h)) all display the sign reversal of the bending radius. The location of the linear region of vanishing bending response in PE-fraction - substrate thickness space is found to shift between different materials systems. This behavior occurs in addition to the layer number effect (Fig. 4(e) and 4(g)) and is attributed to the different elastic compliances of the MS layers involved. IV. SUMMARY

In summary we have developed an analytic model for the bending mode static magnetoelectric coefficient, the magnetic bending response coefficient, the bending radius, and the zero strain position in the presence of external fields for multilayer nanocomposites on a substrate. All equations necessary to reproduce the model are given explicitly (and we checked that a student needs less than one week to copy them into a mathematics software and start calculations). Using the closed form solutions yielded by the model, we systematically investigated the effect of varying layer number, layer sequence, piezoelectric volume fraction, substrate thickness, and using different piezoelectric and magnetostrictive materials in the layered ME nanocomposite. In contrast to the dynamic bending mode results presented in the companion paper28 we find the greatest static bending mode ME coefficients for odd layer number in most ME nanocomposites at vanishing substrates. This


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FIG. 8. Comparison of the static magnetic bending response for bilayers with different layer sequence, i.e., MS-PE-Sub (Fig. 8(a), (c), (e), and (g)) and PE-MS-Sub (Fig. 8(b), (d), (f), and (h)). Given are the results for the materials systems FeCoBSi-AlN-Si (a)-(b), FeCoBSi-PZT-Si (c)-(d), Terfenol-D-AlN-Si (e)-(f), and Terfenol-D-PZT-Si (g)-(h).

corresponds to a vanishing bending response, i.e., an effective longitudinal excitation. Assuming a fixed total nanocomposite thickness as well as thickness-independent material parameters, the ME coefficient is independent of the number of layers in the nanocomposite for odd layer numbers and large layer numbers. Lower maximal ME coefficients are observed for even layer number ME nanocomposites at nonzero substrates. Optimized static bending mode ME coefficients of FeCoBSi-AlN-Si nanocomposites are about 8x, 28x, and 160x greater than for FeCoBSi-PZT-Si, Terfenol-D-AlN-Si, and Terfenol-D-PZT-Si, respectively. ACKNOWLEDGMENTS

The authors thank Jascha Lukas Gugat for confirming selected results of this paper by finite element simulations and Adrian Zaman for reconstructing the model with all equations from a printed copy. This work was supported by the German Science Foundation (DFG) within the Collaborative Research Centre SFB 855 “Magnetoelectric Composite Materials Biomagnetic Interfaces of the Future”. APPENDIX: DEFINITIONS OF NUMBERED TERMS

a1 = n m 4 h m 4 s11 p 2 s Sub 2

(A.1)

a2 = 2(n m 4 − n m 2 )h m 3 h p s11 p 2 s Sub 2

(A.2)


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a3 = [2n m n p (2n m 2 + 2n p 2 − 3n m n p − 3n m + 3n p + 1)]h m 3 h p s11 p s33m s Sub 2

(A.3)

a4 = 4n m 3 h m 3 h s s11 p 2 s33m s Sub

(A.4)

a5 = (n m 4 − n m 2 )h m 2 h p 2 s11 p 2 s Sub 2

(A.5)

a6 = [2n m n p (4n m 2 + 4n p 2 − 6n m n p − 6n m + 6n p + 1)]h m 2 h p 2 s11 p s33m s Sub 2

(A.6)

a7 = (n p 4 − n p 2 )h m 2 h p 2 s33m 2 s Sub 2

(A.7)

a8 = (6n m 2 − 4n m 3 − 2n m + 12n m 2 n p ) h m 2 h p h s s11 p 2 s33m s Sub

(A.8)

a9 = (4n p 3 + 6n p 2 + 2n p − 12n m n p −12n m n p 2 + 12n m 2 n p )h m 2 h p h s s11 p s33m 2 s Sub a10 = 6n m 2 h m 2 h s 2 s11 p 2 s33m s Sub

(A.9)

(A.10)

a11 = [2n m n p (2n m 2 + 2n p 2 − 3n m n p − 3n m + 3n p + 1)]h m h p 3 s11 p s33m s Sub 2 a12 = 2(n p 4 − n p 2 )h m h p 3 s33m 2 s Sub 2

(A.11)

(A.12)

a13 = (4n m 3 − 6n m 2 + 2n m + 12n m n p + 12n m n p 2 − 12n m 2 n p )h m h p 2 h s s11 p 2 s33m s Sub

(A.13)

a14 = (12n m n p 2 − 2n p − 6n p 2 − 4n p 3 ) h m h p 2 h s s11 p s33m 2 s Sub

(A.14)

a15 = 6(2n m n p + n m − n m 2 ) h m h p h s 2 s11 p 2 s33m s Sub

(A.15)

a16 = 6(2n m n p − n p − n p 2 ) h m h p h s 2 s11 p s33m 2 s Sub

(A.16)


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a17 = 4n m h m h s 3 s11 p 2 s33m s Sub

(A.17)

a18 = n p 4 h p 4 s33m 2 s Sub 2

(A.18)

a19 = 4n p 3 h p 3 h s s11 p s33m 2 s Sub

(A.19)

a20 = 6n p 2 h p 2 h s 2 s11 p s33m 2 s Sub

(A.20)

a21 = 4n p h p h s 3 s11 p s33m 2 s Sub

(A.21)

a22 = h s 4 s11 p 2 s33m 2

(A.22)

a23 = a37 h m 2 n m 2 n p s Sub

(A.23)

a24 = −a37 h m 2 n m n p 2 s Sub

(A.24)

a25 = 2a37 h m h p n m 2 n p s Sub

(A.25)

a26 = −2a37 h m h p n m n p 2 s Sub

(A.26)

a27 = −a37 h m h s n m 2 s11 p

(A.27)

a28 = 2a37 h m h s n m n p s33m

(A.28)

a29 = −a37 h m h s n p 2 s33m

(A.29)

a30 = a37 n m 2 h p 2 n p s Sub

(A.30)

a31 = −a37 h p 2 n m n p 2 s Sub

(A.31)

a32 = a37 h p h s n m 2 s11 p

(A.32)

a33 = −2a37 h p h s n m n p s11 p

(A.33)

a34 = a37 h p h s n p 2 s33m

(A.34)


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a35 = −a37 h s 2 n m s11 p

(A.35)

a36 = a37 h s 2 n p s33m

(A.36)

a37 = 12n a h m h p s11 p s33m s Sub

(A.37)

b1 = b15 h m h p h s n m 2 s11 p

(A.38)

b2 = −b15 h m 2 h s n m 2 s11 p

(A.39)

b3 = −b15 h m h p h s n m s11 p

(A.40)

b4 = −b15 h m h s 2 n m s11 p

(A.41)

b5 = −b15 h m h p 2 n m n p s Sub

(A.42)

b6 = −b15 h m 2 h p n m n p s Sub

(A.43)

b7 = −b15 h m h p 2 n m n p 2 s Sub

(A.44)

b8 = b15 h m h p 2 n m 2 n p s Sub

(A.45)

b9 = −b15 h m 2 h p n m n p 2 s Sub

(A.46)

b10 = b15 h m 2 h p n m 2 n p s Sub

(A.47)

b11 = −2b15 h m h p h s n m n p s11 p

(A.48)

b12 = 2b15 h m h p h s n m n a s11 p

(A.49)

b13 = 2b15 h m h p 2 n m n p n a s Sub

(A.50)

b14 = 2b15 h m 2 h p n m n p n a s Sub

(A.51)

b15 = −6s11 p s33m s Sub d33m

(A.52)


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c1 = c15 h p h s 2 n p s33m

(A.53)

c2 = c15 h p 2 h s n p 2 s33m

(A.54)

c3 = −c15 h m h p h s n p s33m

(A.55)

c4 = −c15 h m h p h s n p 2 s33m

(A.56)

c5 = −c15 h m h p 2 n m n p s Sub

(A.57)

c6 = −c15 h m 2 h p n m n p s Sub

(A.58)

c7 = −c15 h m h p 2 n m n p 2 s Sub

(A.59)

c8 = c15 h m h p 2 n m 2 n p s Sub

(A.60)

c9 = −c15 h m 2 h p n m n p 2 s Sub

(A.61)

c10 = c15 h m 2 h p n m 2 n p s Sub

(A.62)

c11 = 2c15 h m h p h s n m n p s33m

(A.63)

c12 = 2c15 h m h p h s n p n a s33m

(A.64)

c13 = 2c15 h m h p 2 n m n p n a s Sub

(A.65)

c14 = 2c15 h m 2 h p n m n p n a s Sub

(A.66)

c15 = 6s11 p s33m s Sub g31 p

(A.67)

d1 = 6d54 h p 2 h s n p s33m s Sub

(A.68)

d2 = 6d54 h p 2 h s n p 2 s33m s Sub

(A.69)

d3 = −6d54 h p 2 h s n m n p s33m s Sub

(A.70)


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d4 = −6d54 h p 2 n p 2 ds33m s Sub

(A.71)

d5 = −6d54 h p 2 n p ds33m s Sub

(A.72)

d6 = 6d54 h p 2 n m n p ds33m s Sub

(A.73)

d7 = 3d54 h p 3 n p 2 s33m s Sub

(A.74)

d8 = 2d54 h p 3 n p 3 s33m s Sub

(A.75)

d9 = −3d54 h p 3 n m n p 2 s33m s Sub

(A.76)

d10 = 6d54 h m h p h s n p s33m s Sub

(A.77)

d11 = 6d54 h m h p h s n p 2 s33m s Sub

(A.78)

d12 = −6d54 h m h p h s n m n p s33m s Sub

(A.79)

d13 = −6d54 h m h p dn p 2 s33m s Sub

(A.80)

d14 = −6d54 h m h p dn p s33m s Sub

(A.81)

d15 = 6d54 h m h p dn m n p s33m s Sub

(A.82)

d16 = −d54 h m h p 2 n m 3 s11 p s Sub

(A.83)

d17 = −6d54 h m h p 2 n m 2 n p s33m s Sub

(A.84)

d18 = 6d54 h m h p 2 n m n p 2 s33m s Sub

(A.85)

d19 = 9d54 h m h p 2 n m n p s33m s Sub

(A.86)

d20 = d54 h m h p 2 n m s11 p s Sub

(A.87)

d21 = −2d54 h m h p 2 n p 3 s33m s Sub

(A.88)


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d22 = −3d54 h m h p 2 n p 2 s33m s Sub

(A.89)

d23 = −d54 h m h p 2 n p s33m s Sub

(A.90)

d24 = 3d54 h m h s 2 n m s11 p s33m

(A.91)

d25 = −6d54 h m h s n m ds11 p s33m

(A.92)

d26 = 3d54 h p h s 2 s11 p s33m

(A.93)

d27 = −3d54 h p h s 2 n m s11 p s33m

(A.94)

d28 = 6d54 h p h s 2 n p s11 p s33m

(A.95)

d29 = −6d54 h p h s ds11 p s33m

(A.96)

d30 = 6d54 h p h s n m ds11 p s33m

(A.97)

d31 = −12d54 h p h s n p ds11 p s33m

(A.98)

d32 = −2d54 h m 2 h p n m 3 s11 p s Sub

(A.99)

d33 = −6d54 h m 2 h p n m 2 n p s33m s Sub

(A.100)

d34 = 9d54 h m 2 h p n m n p 2 s33m s Sub

(A.101)

d35 = 9d54 h m 2 h p n m n p s33m s Sub

(A.102)

d36 = 2d54 h m 2 h p n m s11 p s Sub

(A.103)

d37 = −4d54 h m 2 h p n p 3 s33m s Sub

(A.104)

d38 = −6d54 h m 2 h p n p 2 s33m s Sub

(A.105)

d39 = −2d54 h m 2 h p n p s33m s Sub

(A.106)


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d40 = 2d54 h s 3 s11 p s33m

(A.107)

d41 = −6d54 h s 2 ds11 p s33m

(A.108)

d42 = −d54 h m 3 n m 3 s11 p s Sub

(A.109)

d43 = 3d55 h m 2 n m 2 n p d33m s Sub

(A.110)

d44 = −2d55 h m 2 n m n p 2 d33m s Sub

(A.111)

d45 = 3d55 h m h p n m 2 n p d33m s Sub

(A.112)

d46 = −d55 h m h p n m n p 2 d33m s Sub

(A.113)

d47 = 2d55 h m h s n m n p d33m s Sub

(A.114)

d48 = −2d55 h m n m n p dd33m s Sub

(A.115)

d49 = d55 h p 2 n m n p 2 d33m s Sub

(A.116)

d50 = 2d55 h p h s n m n p d33m s Sub

(A.117)

d51 = −2d55 h p n m n p dd33m s Sub

(A.118)

d52 = d55 h s 2 n m d33m s11 p

(A.119)

d53 = −2d55 h s n m dd33m s11 p

(A.120)

d54 = −h m n m s11 p d33m

(A.121)

d55 = 6n a h m h p s11 p s33m

(A.122)

e1 = 2e54 h m 2 h p n m 3 s11 p s Sub

(A.123)

e2 = −6e54 h m 2 h p n m 2 n p s11 p s Sub

(A.124)


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e3 = −3e54 h m 2 h p n m 2 s11 p s Sub

(A.125)

e4 = 6e54 h m 2 h p n m n p 2 s11 p s Sub

(A.126)

e5 = 9e54 h m 2 h p n m n p s11 p s Sub

(A.127)

e6 = e54 h m 2 h p n m s11 p s Sub

(A.128)

e7 = e54 h m 2 h p n p 3 s33m s Sub

(A.129)

e8 = −e54 h m 2 h p n p s33m s Sub

(A.130)

e9 = 6e54 h m 2 h s n m s11 p s Sub

(A.131)

e10 = −6e54 h m 2 h s n m 2 s11 p s Sub

(A.132)

e11 = 6e54 h m 2 h s n m n p s11 p s Sub

(A.133)

e12 = 6e54 h m 2 n m 2 ds11 p s Sub

(A.134)

e13 = −6e54 h m 2 n m ds11 p s Sub

(A.135)

e14 = −6e54 h m 2 n m n p ds11 p s Sub

(A.136)

e15 = 3e54 h m h s 2 s11 p s33m

(A.137)

e16 = −6e54 h m h s 2 n m s11 p s33m

(A.138)

e17 = 3e54 h m h s 2 n p s11 p s33m

(A.139)

e18 = 6e54 h m h p h s n m s11 p s Sub

(A.140)

e19 = −6e54 h m h p h s n m 2 s11 p s Sub

(A.141)

e20 = 6e54 h m h p h s n m n p s11 p s Sub

(A.142)


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e21 = 6e54 h m h p n m 2 ds11 p s Sub

(A.143)

e22 = −6e54 h m h p n m ds11 p s Sub

(A.144)

e23 = −6e54 h m h p n m n p ds11 p s Sub

(A.145)

e24 = −6e54 h m h s ds11 p s33m

(A.146)

e25 = 12e54 h m h s n m ds11 p s33m

(A.147)

e26 = −6e54 h m h s n p ds11 p s33m

(A.148)

e27 = 4e54 h m h p 2 n m 3 s11 p s Sub

(A.149)

e28 = −9e54 h m h p 2 n m 2 n p s11 p s Sub

(A.150)

e29 = −6e54 h m h p 2 n m 2 s11 p s Sub

(A.151)

e30 = 6e54 h m h p 2 n m n p 2 s11 p s Sub

(A.152)

e31 = 9e54 h m h p 2 n m n p s11 p s Sub

(A.153)

e32 = 2e54 h m h p 2 n m s11 p s Sub

(A.154)

e33 = 2e54 h m h p 2 n p 3 s33m s Sub

(A.155)

e34 = −2e54 h m h p 2 n p s33m s Sub

(A.156)

e35 = 3e54 h m 3 n m 2 s11 p s Sub

(A.157)

e36 = −2e54 h m 3 n m 3 s11 p s Sub

(A.158)

e37 = 3e54 h m 3 n m 2 n p s11 p s Sub

(A.159)

e38 = −3e54 h p h s 2 n p s11 p s33m

(A.160)


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e39 = 6e54 h p h s n p ds11 p s33m

(A.161)

e40 = −2e54 h s 3 s11 p s33m

(A.162)

e41 = 6e54 h s 2 ds11 p s33m

(A.163)

e42 = e54 h p 3 n p 3 s33m s Sub

(A.164)

e43 = e55 h m 2 n m 2 n p g31 p s Sub

(A.165)

e44 = −e55 h m h p n m 2 n p g31 p s Sub

(A.166)

e45 = 3e55 h m h p n m n p 2 g31 p s Sub

(A.167)

e46 = 2e55 h m h s n m n p g31 p s Sub

(A.168)

e47 = −2e55 h m n m n p dg31 p s Sub

(A.169)

e48 = −2e55 h p 2 n m 2 n p g31 p s Sub

(A.170)

e49 = 3e55 h p 2 n m n p 2 g31 p s Sub

(A.171)

e50 = 2e55 h p h s n m n p g31 p s Sub

(A.172)

e51 = −2e55 h p n m n p dg31 p s Sub

(A.173)

e52 = e55 h s 2 n p g31 p s33m

(A.174)

e53 = −2e55 h s n p dg31 p s33m

(A.175)

e54 = h p n p g31 p s33m

(A.176)

e55 = −6n a h m h p s11 p s33m

(A.177)

f 1 = f 15 h p 2 n m n p s Sub

(A.178)


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f 2 = − f 15 h p 2 n p 2 s Sub

(A.179)

f 3 = − f 15 h p h s s11 p

(A.180)

f 4 = − f 15 h p 2 n p s Sub

(A.181)

f 5 = − f 15 h m h p n p 2 s Sub

(A.182)

f 6 = − f 15 h s 2 s11 p

(A.183)

f 7 = − f 15 h m h s n m s11 p

(A.184)

f 8 = f 15 h p h s n m s11 p

(A.185)

f 9 = −2 f 15 h p h s n p s11 p

(A.186)

f 10 = − f 15 h m h p n p s Sub

(A.187)

f 11 = f 15 h m h p n m n p s Sub

(A.188)

f 12 = 2 f 15 h p 2 n p n a s Sub

(A.189)

f 13 = 2 f 15 h p h s n a s11 p

(A.190)

f 14 = 2 f 15 h m h p n p n a s Sub

(A.191)

f 15 = 6h m n m d33m s11 p s33m

(A.192)

g1 = g15 h s 2 s33m

(A.193)

g2 = g15 h m 2 n m 2 s Sub

(A.194)

g3 = −g15 h m h s s33m

(A.195)

g4 = −g15 h m 2 n m s Sub

(A.196)


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g5 = g15 h m h p n m 2 s Sub

(A.197)

g6 = −g15 h m 2 n m n p s Sub

(A.198)

g7 = 2g15 h m h s n m s33m

(A.199)

g8 = −g15 h m h s n p s33m

(A.200)

g9 = g15 h p h s n p s33m

(A.201)

g10 = −g15 h m h p n m s Sub

(A.202)

g11 = −g15 h m h p n m n p s Sub

(A.203)

g12 = 2g15 h m 2 n m n a s Sub

(A.204)

g13 = 2g15 h m h s n a s33m

(A.205)

g14 = 2g15 h m h p n m n a s Sub

(A.206)

g15 = −6h p n p g31 p s11 p s33m

(A.207)

1 M.

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