A finite element formulation for large deflection of multilayered magneto-electro-elastic plates

Accepted Manuscript A finite element formulation for large deflection of multilayered magneto-elec‐ tro-elastic plates A. Alaimo, I. Benedetti, A. Milazzo PII: DOI: Reference:

S0263-8223(13)00439-X http://dx.doi.org/10.1016/j.compstruct.2013.08.032 COST 5324

To appear in:

Composite Structures

Received Date: Accepted Date:

7 July 2013 22 August 2013

Please cite this article as: Alaimo, A., Benedetti, I., Milazzo, A., A finite element formulation for large deflection of multilayered magneto-electro-elastic plates, Composite Structures (2013), doi: http://dx.doi.org/10.1016/ j.compstruct.2013.08.032

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A finite element formulation for large deflection of multilayered magneto-electro-elastic plates A. Alaimoa , I. Benedettib , A. Milazzob,∗ a

Facolt´ a di Ingegneria, Architettura e delle Scienze Motorie, University of Enna Kore, Cittadella Universitaria, 94100 Enna - Italy b Dipartimento di Ingegneria Civile Ambientale, Aerospaziale, dei Materiali Universit´ a di Palermo, Viale delle Scienze, Edificio 8, 90128 Palermo - Italy

Abstract An original finite element formulation for the analysis of large deflections in magneto-electro-elastic multilayered plates is presented. The formulation is based on an equivalent single-layer model in which first order shear deformation theory with von Karman strains and quasi-static behavior for the electric and magnetic fields are assumed. To obtain the plate model, the electro-magnetic state is firstly determined and condensed to the the mechanical primary variables, namely the generalized displacements. In turn, this result is used to obtain laminate effective stiffness coefficients that allow to express the plate mechanical stress resultants in terms of the generalized displacements and applied electro-magnetic boundary conditions only, taking the magneto-electro-elastic couplings into account. The weak form of the mechanical equilibrium equations is then written and used to express the finite element matrices. Numerical results obtained by implementing an ∗

Corresponding author. Tel.: +39 09123896748 FAX: +39 091485439 URL: [email protected] (A. Alaimo), [email protected] (I. Benedetti), [email protected] (A. Milazzo)

Preprint submitted to Composite Structures

August 29, 2013

isoparametric four-node MITC finite element are presented to validate the approach and show its features and potentiality. The proposed modeling strategy and the corresponding finite element formulation can take advantage of the solution tools available for the mechanics of classical multilayered plates leading to the straightforward integration of magneto-electro-elastic plate finite elements into existing codes. Keywords: Magneto-electro-elastic structures, Multilayered plates, MITC Finite Elements, 1. Introduction In recent years, the employment of smart materials able to provide multifunctional capabilities, besides the traditional structural functions, has been gaining attention in several technological fields (automotive, aerospace, biomedical, robotics, etc.). In general these materials exhibit some interaction between the mechanical state and some other field, e.g. electrical, activated by the mechanical field itself by virtue of some microstructural features: a classical example is provided by piezoelectric materials. The possibility of coupling the different fields can be and it has been exploited in transducer applications, structural health monitoring, vibration control, energy harvesting and other applications. In this framework, magneto-electro-elastic (MEE) materials are attracting increasing consideration from academic and industrial audiences: MEE materials have the ability to couple mechanical, electrical and magnetic fields and this makes them particularly suitable for smart applications [1, 2]. Generally, besides few exceptions [3], single-phase materials exhibit either piezoelectric or piezomagnetic behavior, but no di2

rect magneto-electric coupling is observed. The full magneto-electro-elastic coupling is actually obtained by employing simultaneously piezoelectric and piezomagnetic phases, that then provide a magneto-electric effect through the elastic field [4, 5]. MEE composites are obtained in the form of multiphase materials, i.e. piezoelectric and piezomagnetic particles and/or fibers [6], or in the form of laminated structures, with piezoelectric and piezomagnetic layers stacked to achieve the desired coupling effects [7]. It is worth observing that multilayered configurations appear to be more effective than bulk composites. Thus, reliable and efficient modeling tools are required for the analysis and design of smart magneto-electro-elastic laminated plates. Starting from the pioneering 3-D analytical exact solution by Pan [8], different approaches have been proposed to tackle the modelization of MEE plates. Pan and co-workers developed 3-D exact solution for free vibrations [9] and cylindrical bending [10] of MEE laminates. Wang et al. approached the problem by the state space formulation [11]. These exact solutions describe the laminate response without any assumption on the involved fields; however they are able to deal only with specific geometries and boundary conditions. For more complex configurations numerical solutions are needed and considering that fully-coupled 3-D finite element solutions for multilayered plates and shells present very high computational costs, 2-D efficient laminated plates theories and the corresponding finite element solutions have been developed with the aim of reducing the analysis effort while preserving a suitable level of accuracy. In the framework of 2-D plate theories, finite elements solutions based on layer-wise modeling have been proposed (e.g Lage et al. [12], Phoenix et al. [13], Carrera and Nali [14]). The layer-

3

wise approach enables high accuracy; however its computational cost grows as the number of layers increases. Additionally, layer-wise modeling of laminated plates requires the development of ad hoc procedures and elements which make somehow difficult to integrate them into finite element commercial codes. On the other hand, equivalent single-layer plate theories do not present these drawbacks as their solution complexity is independent from the number of layers although they are generally less accurate than the layer-wise ones. Finite element solutions based on these theories have been investigated by Moita et al. [15] and Carrera and Nali [14], whose models take the electric and magnetic primary variables as independent unknowns. More recently, an equivalent single-layer approach for multilayered MEE plates [16, 17] and its finite element solution [18] have been proposed by Milazzo and co-workers, who developed an effective purely mechanical plate model as result of the condensation of the electro-magnetic state to the mechanical variables. It is worth noting that such a modeling strategy could take advantage of the solution tools available for the mechanics of classical multi-layered plates and could lead to the straightforward integration of MEE plate elements into available codes. The literature survey evidences that the proposed approaches generally refer to the case of plate small displacements, whereas the nonlinear large displacements behaviour has been poorly addressed, although its influence can be important for sensors and actuators applications. Recently Xue et al. [19] presented a large deflection solution for homogeneous single-layer rectangular magneto-electro-elastic thin plates based on the von Karman’s plate assumptions, and more recently Sladek et al. [20] employed the meshless

4

local Petrov-galerkin method for the solution of the same problem. However, to the best of the author’s knowledge, no solutions for large deflection of multilayered magneto-electro-elastic laminates have been proposed. Considering the rationale behind the 2-D plate theories and their features and focusing on the state of the art of the modeling and simulation of multilayered MEE plates, this paper proposes an equivalent single-layer model for the large deflection analysis of multilayered magneto-electro-elastic laminates and its solution by finite elements. The model is based on the one previously presented by the authors for the small displacements case [16], which is here reformulated for large deflections by using the von Karman’s plate assumptions. It is then used to develop finite elements belonging to the well-established mixed interpolation of tensorial components (MITC) family, developed for conventional [21] and piezoelectric materials [22, 23]. The outline of the paper is as follows. The basic assumptions and the governing equations are summarized in Section 2. Then, the proposed equivalent single-layer model for MEE plates is discussed in Section 3 and the corresponding finite element formulation is developed in Section 4. Finally, in Section 5, the validation of the approach is carried out for homogeneous plates and some original results are given for multilayered MEE plates to: (i) illustrate the potential of the present approach, (ii) provide reference solutions for further developments, (iii) highlight the effect of large deflections. 2. Governing equations In this Section the basic assumptions and the governing equations for the mechanical and electro-magnetic states of MEE plates are presented ac5

counting for large deflection behavior. 2.1. Basic assumptions Consider a multilayered plate referred to a coordinate system with the x and y coordinates spanning its mid-plane and the z axis directed along the thickness. The plate consists of N -layers of homogeneous and orthotropic magneto-electro-elastic materials having poling direction parallel to the zaxis and principal directions rotated by an angle θ with respect to the x- and y-axes. The plate has thickness H; the k-th layer of the laminate has constant thickness with z = hk−1 and z = hk representing the quote of its bottom and top faces, respectively (see Fig. 1). The bottom and top surfaces of the plate are identified by the coordinate z = −H/2 and z = H/2, respectively. The proposed magneto-electro-elastic multilayered plate model is formulated under the following mechanical and electro-magnetic assumptions. From the mechanical point of view, each segment perpendicular to the midplane of the laminate is considered to be inextensible along the thickness direction and subjected to linear distortion. Thus, the displacement field of the plate corresponds to that of mechanical first order shear deformation theory (FSDT) [24]. Additionally, since the stress normal to the mid plane is small when compared to the other stress components it is neglected [25]. For the electro-magnetic state, as the elastic waves propagate several order of magnitude slower than the electro-magnetic ones, the quasi-static approximation of the corresponding fields is considered together with the assumption of zero applied electric charge density. Moreover, the widely employed assumption of negligible in-plane components of the electric and magnetic field with respect to the corresponding component along the thick6

ness direction is introduced. The last assumption can determine a loss in accuracy as the plate thickness-to-length ratio increases [26, 27], although the comparison of solutions, obtained under these hypotheses, with 3-D exact solution shows that they can be considered sufficiently accurate for thin to moderately thick plates [15, 17]. 2.2. Mechanical state governing equations According to the assumptions introduced in the previous section, the displacement field of the plate is given by the mechanical first order shear deformation theory as u (x, y, z) = u0 (x, y) + zϑx (x, y)

(1a)

v (x, y, z) = v0 (x, y) + zϑy (x, y)

(1b)

w (x, y, z) = w0 (x, y)

(1c)

where u0 , v0 and w0 are the displacements of the midplane points and ϑx and ϑy are the plate section rotations. These correspond to the problem mechanical primary variables [24]. To exploit the advantages of the matrix notation let us collect and partition the kinematical variables in the following vector

n U=

oT u0 v0 ϑx ϑy w0

n =

oT U 0 Θ w0

(2)

and let us introduce the in-plane strain vector ε0 , the curvatures vector κ and the shear strain vector γ. By using the von Karman’s theory for large deflection of plates, the strain-displacement relations are written as    T       ε  D 03×2 D3×2 (w0 ) D2×1  U    0    3×2   0     = = DU   κ 0 D3×2 03×1 Θ     3×2         γ    w   0 I D 2×2

2×2

2×1

7

0

(3)

where the definitions and notation for the employed operators are given in Appendix A

n oT The plate in-plane forces per unit length N = , Nxx Nyy Nxy n oT and moments per unit length M = Mxx Myy Mxy are defined as the n oT through-the-thickness resultants of the in-plane stresses σ = σxx σyy σxy , n oT whereas the shear forces per unit length Q = Qxz Qyz are the throughn oT the-thickness resultants of the shear stresses τ = σxz σyz . Accordingly, the mechanical state of the plate is governed by the following equations of motion holding for the plate as a whole DT3×2 N

∂ 2U 0 ∂ 2Θ = −p + J0 + J1 2 ∂t2 ∂t

(4a)

∂ 2 w0 c D2×1 w0 − DT2×1 N (4b) ∂t2 ∂2U 0 ∂ 2Θ T D3×2 M − Q = −m + J1 + J2 2 (4c) ∂t2 ∂t n oT where the vector p = px py collects the components of the in-plane DT2×1 Q = −q + J0

applied mechanical load per unit area, q is the applied transverse mechanical n oT load per unit area and m = is the vector of the applied mx my moments per unit area. The plate inertia properties involved in Eqs. (4) are defined as

Z Jm =

H 2

ρz m dz

(5)

−H 2

being ρ the material density, whereas the in-plane forces per unit length c is given by matrix N

 c= N

 Nxx Nxy Nxy Nyy 8



(6)

The essential and natural boundary conditions associated with these equilibrium equations are those of the corresponding mechanical first order shear plate theory[24]. It is worth noting that the assumed displacement field inherently enforces the kinematic continuity between contiguous layers, whereas the interface stress continuity cannot be guaranteed. This is common to all of the equivalent single-layer models of laminated plates, for which the outof-plane stress components with through thickness continuous distribution need to be computed in the postprocessing step by integrating the threedimensional equilibrium equations. A comprehensive treatment of this topic specifically pertaining to single-layer equivalent theories can be found in Refs. [24] and [28]. 2.3. Electro-magnetic state governing equations The electro-magnetic state can be described in terms of the electric and magnetic scalar potentials that are denoted by Φ and Ψ , respectively [29]. Thus, according to the stated assumptions, the following expressions for the electric field E and magnetic field H hold       0  E xy    0 E= =  E    z    − ∂Φ ∂z       0  H xy    0 H= =  H    z    − ∂Ψ ∂z

9

          

(7a)

          

(7b)

Correspondingly, the electric displacement D and magnetic induction B vectors, suitably partitioned, can be written as        D    x    D xy  D= = Dy    D    z   D   z        B    x    B xy  B= = By    B    z   B  

(8a)

(8b)

z

The electro-magnetic governing equations are the Gauss’ laws for electrostatics and magnetostatics which holds inside each single laminate layer: DT2×1 D xy +

∂Dz =0 ∂z

(9a)

DT2×1 B xy +

∂Bz =0 ∂z

(9b)

Differently from the mechanical state, since no electric and magnetic potentials distribution is a priori assumed along the plate thickness, for multilayered plates Eqs. (9) have to be supplemented by interface equations, which ensure the continuity of the electric and magnetic potential and normal components of the electric displacement and magnetic induction. For the interface located at z = hk , these are Φhki (x, y, hk , t) = Φhk+1i (x, y, hk , t)

(10a)

Dzhki (x, y, hk , t) = Dzhk+1i (x, y, hk , t)

(10b)

Ψhki (x, y, hk , t) = Ψhk+1i (x, y, hk , t)

(10c)

Bzhki (x, y, hk , t) = Bzhk+1i (x, y, hk , t)

(10d)

10

where the notation (·)hki means quantities pertaining the k-th layer. Finally, the corresponding electro-magnetic boundary conditions are applied at the bottom and top surfaces of the plate. They are given by (i) prescribed values of either the electric potential or normal electric displacements and (ii) prescribed values of either the magnetic potential or normal magnetic induction. 2.4. Constitutive equations The mechanical and magneto-electric states are coupled by the magnetoelectro-elastic constitutive equations. Under the stated assumptions, for the k-th layer, these are conveniently written as hki σ hki = C hki xy (ε0 + zκ) + ez

∂Φ ∂Ψ + q hki z ∂z ∂z

τ hki = C hki z γ

(11b)

hki D hki xy = exy γ

(11c)

hki B hki xy = q xy γ

(11d)

hki ∂Φ

T

Dzhki = ehki (ε0 + zκ) − ²33 z Bzhki = q hki z where

T

(11a)

hki ∂Ψ

− d33

∂z ∂z hki ∂Φ hki ∂Ψ (ε0 + zκ) − d33 − µ33 ∂z ∂z 

(11e) (11f)



C C12 C16  11  C xy =  C12 C22 C26  C16 C26 C66   C44 C45  Cz =  C45 C55 11

   

(12a)

(12b)

 exy =   q xy = 

 exy1 exy2 q xy1



= 

 e14 e15 e24 e25



=



(12c)

 q14 q15

q xy2 q24 q25 h i T ez = e31 e32 e36 h i T q z = q31 q32 q36



(12d) (12e) (12f)

being Cij the elastic stiffness coefficients, ²ij and µij the dielectric constants and magnetic permeabilities, respectively, eij and qij the piezoelectric and piezomagnetic coefficients, respectively, and dij the magnetoelectric coupling coefficients. It is understood that, according with the stated basic assumptions and the common practice for 2-D plate theories, the constitutive law is written by using the suitable reduced coefficients, which take the σzz = 0 approximation into account. 3. Equivalent single-layer model for magneto-electro-elastic plates The equivalent single-layer model for magneto-electro-elastic plates is built in two steps. First, the electro-magnetic state is determined in terms of the mechanical primary variables, by solving the strong form of the magnetoelectric governing equations. In turn the weak form of the mechanical equilibrium equations is written by taking the electro-magnetic contributions into account and it is used to obtain the set of the solving equations for the mechanical primary variables, namely u0 , v0 , w0 , ϑx and ϑy . Eventually, a post processing procedure allows to obtain the magneto-electric field, defined by the electric and magnetic potentials Φ and Ψ , in terms of the mechanical 12

primary variables. In the following of this section, the essential characteristics and resulting relations are illustrated focusing on the large deflection contribution and referring the reader to Refs [16, 17] for further details on the applied procedure. 3.1. Solution of the electro-magnetic governing equations Substituting the constitutive equations, namely Eqs. (11), into the Gauss’ equations for electrostatics and magnetostatics, namely Eqs. (9a) and (9b), leads to the following system of equations holding for each k-th layer 2 Φ ∂γ ∂γ T hki ∂ Ψ + d = ehki κ + ehki + ehki 33 z xy 1 xy 2 2 2 ∂z ∂z ∂x ∂y

hki ∂

²33

2 Φ ∂γ ∂γ T hki ∂ Ψ + µ = q hki κ + q hki + q hki 33 z xy 1 xy 2 2 2 ∂z ∂z ∂x ∂y

hki ∂

d33

2

(13a)

2

(13b)

Integration of the Eqs. (13) provides the expressions of the electric and magnetic potentials inside the k-th layer µ ¶ 2 z hki hki ∂γ hki ∂γ hki hki hki Φ = AΦ κ + B Φ + CΦ + a Φ z + bΦ ∂x ∂y 2 µ ¶ 2 z hki hki ∂γ hki ∂γ hki hki hki Ψ = AΨ κ + B Ψ + CΨ + aΨ z + bΨ ∂x ∂y 2

(14a) (14b)

hki hki where the row vectors Ahki α , Bα and Cα collect constants depending on the

dielectric, magnetic permeability, piezoelectric, piezomagnetic and magnetoelectric coupling coefficients of the considered layer, which are defined in Aphki

hki

hki

hki

pendix B, whereas aΦ , bΦ , aΨ and bΨ are integration constants, functions of x and y. These are determined by enforcing the electro-magnetic interface continuity conditions at each of the N − 1 laminate interfaces and the electro-magnetic boundary conditions at the bottom and top surfaces of the 13

laminate. All of these conditions provide a linear algebraic system whose solution determines the integration constants in terms of the mechanical states and the applied electro-magnetic boundary conditions [16]. Compactly the integration constants can be written in the form hki

hki

hki

∂γ hki ∂γ hki + ζ λγ + ζ λξ ξ x ∂x y ∂y

hki

ζλ = ζ λε ε0 + ζ λϑ κ + ζ λγ 0

(15)

In the preceding general relationship, ζ stands for a or b, λ stands for Φ or Ψ and the row vectors ζ αβ contains the solution coefficients; the components ξi of the vector ξ are the prescribed values of the electro-magnetic boundary conditions at the plate bottom surface (i = 1, 2) and those pertaining to the top surface (i = 3, 4). 3.2. Magneto-electro-elastic laminate equivalent single-layer constitutive law Once the electro-magnetic state has been determined in terms of the mechanical primary variables and electro-magnetic boundary conditions, the smart laminate forces and moments resultants per unit length are expressed in terms of the mechanical primary variables by substituting Eqs. (11) into their definition and taking Eqs. (14) into account. One obtains ∂γ ∂γ +F +V ξ ∂x ∂y ∂γ ∂γ M = C ε0 + D κ + G +H +Y ξ ∂x ∂y fγ Q=A N = A ε0 + B κ + E

(16)

where the involved magneto-electro-elastic effective stiffness matrices are defined as N ³ ´ X hki hki hki hki hki hki hki A, Ci = hA C xy + ez aΦε + q z aΨε hI0 , I1 i 0

k=1

14

0

(17)

B, D i = hB

N h³ X

hki

k=1

E, Gi = hE

hki

hki hki C hki xy + ez A Φ + q z A Ψ

³ N h³ X k=1

³

hki

hki ehki z BΦ

hki

+

hki q hki z BΨ

hki

k=1

³

hki

´

hki

hki

hki

´

+

hki q hki z CΨ

hki

´

hki

hki ehki z aΦγ + q z aΨγ

(18)

hki

hki

hki

hki

hki

i

(19)

hI0 , I1 i

x

hki ehki z CΦ

i

hI1 , I2 i+

hki

y

V,Yi= hV

´

hki ehki z aΦγ + q z aΨγ

N h³ X

hki

hI1 , I2 i+

hki ehki z aΦϑ + q z aΨϑ hI0 , I1 i

x

F, H i = hF

´

y

hI1 , I2 i+ ´

hki

hki

i

(20)

hI0 , I1 i

N ³ ´ X hki hki hki hki hki ehki a + q a z z Φξ Ψξ hI0 , I1 i

(21)

k=1

f= A

N X

hki

(22)

z m dz

(23)

C hki z I0

k=1

In the previous equations one sets Z hki Im

hk

= hk−1

3.3. Weak form of the mechanical equilibrium governing equation The plate equilibrium equations Eqs. (4) are multiplied with the dummy variations δU 0 , δw0 , and δΘ, respectively and integrated over the domain. After integration by parts, one gets the weak form of the mechanical equilibrium equations, which corresponds to the stationarity of the mechanical energy functional. It writes as ¸ Z · Z 2 T T ∂ U T (DδU ) S + δU J 2 − δU q dΩ − δU T P d∂Ω ∂t Ω ∂Ω 15

(24)

n

oT

where the vector of the generalized forces per unit length S = N M Q n oT the vector of the external loads per unit area q = px py mx my q and the inertia matrix   J0 I 2×2 J1 I 2×2 01×2     J = J1 I 2×2 J2 I 2×2 01×2    01×2 01×2 J0

(25)

are introduced. In the Eq. (24), P is the vector of the secondary mechanical variables defined as   Px       Py   P = Mx      M y     Qn

                 

                  

=

Nxx α1 + Nxy α2 Nxy α1 + Nyy α2 Mxx α1 + Mxy α2

Mxy α1 + Myy α2    Qxz α1 + Qyz α2 +     ∂w0    (Nxx α1 + Nxy α2 ) +   ∂x    ∂w0   (Nxy α1 + Nxy α2 ) ∂y

                                  

     =

D2×3 N

    

D2×3 M       D1×2 Q + (D1×2 w0 )T D2×3 N  

(26)

where the matrices D2×3 and D1×2 are obtained from the transpose of the operators D3×2 and D2×1 by substituting the derivatives with the corresponding direction cosines αi of the boundary outer normal. The weak form of the equations of motion, namely Eq. (24), constitutes the basis for the development of solutions for the MEE plate model and it is employed in the following to formulate finite elements. 4. Finite element formulation In this section a finite element formulation of the proposed magnetoelectro-elastic plate model is developed. Let us consider a quadrilateral finite

16

,

element, occupying the region Ωe with contour ∂Ωe , over which the ξ − η natn oT ural coordinate system is defined. Denoting by ∆i = ui0 v0i ϑix ϑiy woi the vector containing the values of the mechanical primary variable at the i-th node of the element, the mechanical primary variables are interpolated by using standard lagrange shape functions      u0               v Nu  M  0  X   U = ϑx = Ni ∆i = N ∆e  N θ      i=1       ϑ Nw y       w   0

    ∆e 

(27)

where M is the number of the element nodes, ∆e is the vector collecting the element nodal values and N is a 5 × 5M matrix containing the shape functions Ni , suitably partitioned for the following developments. According to the MITC formulation for the Mindlin/Reissner plate bending finite element, the shear strains γ are differently interpolated as γ = N γ ∆e where N

γ

(28)

are interpolation functions whose derivation details can be found

in Ref. [30]. Preliminary, it  D  3×2  DδU =  03×2  02×2

is observed that the following identity holds       03×2 D3×2 (w0 ) D2×1  δU 0   0   3×2    δΘ +  03×2 D3×2 03×1          02×2 02×1 δw0 I 2×2

 n o   δγ = 

DU δU + Dγ δγ = (DUL + DUN L ) δU + Dγ δγ (29) 17

where the operator definitions are given in Appendix A. Accordingly, one can writes

       U 0         AB 0 C E V   DU + 1 DU Dγ  Θ    L NL      2 S = M =F D 0 G H  + Y  ξ =           0 D∂    w0   Q    e   0 0 A 0 0 0  γ         U 0         V Θ    h i DU + 1 DU Dγ  L NL    2 + Y  ξ = Z 11 Z 12      w0  0 D∂        0  γ         U 0       V · µ ¶ ¸ Θ    1   Z 11 DUL + DUN L Z 11 Dγ + Z 12 D∂ + Y  ξ   2    w0        0  γ  (30)      N    





where D∂ is defined in Appendix A By using the introduced interpolations and defining B UL = DUL N

(31a)

B UN L = DUN L N

(31b)

B γ = Dγ N

γ

(31c)

B ∂ = D∂ N

γ

(31d)

one gets B UL + B γ + B UN L ) δ∆e DδU = (B µ ¶ ¸ 1 S = Z 11 B UL + B γ + B UN L + Z 12B ∂ ∆e + W ξ 2 ·

18

(32) (33)

Upon substituting Eq. (32) into the weak form of the mechanical equilibrium equations, Eq. (24), one is able to obtain the expression of the internal and external forces for the considered element, which are given by Z e B UL + B γ + B UN L )T SdΩ F int = (B

(34a)

Ωe

Z

F eext

Z

∂ 2U = N qdΩ + N J 2 dΩ + ∂t Ωe Ωe T

Z

T

N T P d∂Ω

(34b)

∂Ωe

In order to provide for the problem nonlinear incremental solution, let us introduce the internal force increment dF eint , which is obtained by differentiation of Eq. (34a) and is given by Z Z T e B UL + B γ + B UN L ) dSdΩ + dF int = (B Ωe

Ωe

B UL + B γ + B UN L )T SdΩ d (B (35)

From Eq. (33) one has µ B UL + B γ ) + Z 12B ∂ ] d∆e + Z 11 d dS = [Z 11 (B

1 B UN L ∆e 2

¶ + W dξ (36)

and observing that due to the properties of the operator B UN L µ ¶ 1 d B UN L ∆e = B UN L d∆e 2

(37)

it results B UL + B γ ) + Z 12B ∂ + Z 11B UN L ] d∆e + W dξ dS = [Z 11 (B

(38)

Additionally, one straightforwardly recognizes that cG d∆e B UL + B γ + B UN L )T S = d (B B UN L )T S = G T NG d (B

(39)

where G = D2×1N 19

w

(40)

Finally, substituting Eqs. (38) and (39) into Eq. (35) one obtains dF eint = K eT d∆e + K eEM dξ

(41)

where K eT is the elemental magneto-electro-elastic effective tangent stiffness matrix and K eEM is the matrix providing for the contribution of the applied magneto-electric loads increment. The tangent stiffness matrix definition consists as usual of three contributions, namely the linear stiffness matrix K el , the large displacement stiffness matrix K enl and the geometric stiffness matrix K eσ . Thus, one has K eT = K el + K enl + K eσ

(42)

where Z K el

= Ωe

B UL + B γ +)T [Z 11 (B B UL + B γ ) + Z 12B ∂ ] dΩ (B

K enl

(43a)

Z = ZΩe Z

Ωe

Ωe

B UL + B γ )T Z 11B UN L dΩ+ (B B UL + B γ ) + Z 12B ∂ ] dΩ+ B TUN L [Z 11 (B

(43b)

B TUN L Z 11B UN L dΩ Z K eσ

cG dΩ G T NG

=

(43c)

Ωe

The matrix K eEM is defined as Z e B UL + B γ + B UN L )T W dΩ K EM = (B

(44)

Ωe

In conclusion, the elemental incremental equilibrium equation is thus written as K eT d∆e = dF eext − K eEM dξ 20

(45)

where dF eext is the increment of the external loads defined according with Eq. (34b). Eq. (45) is the basis for the incremental solution of the proposed finite element model for magneto-electro-elastic multilayered plates nonlinear problems. 5. Numerical results In this section, some numerical results are presented in order to validate the proposed finite element for MEE plates undergoing large deflections and to highlight the effects of large deflection with respect to linear formulation under both sensing and actuating functions. To this purpose the proposed formulation has been implemented for 4-noded quadrilateral isoparametric elements, whose general behavior for the linear case has been previously ascertained [18]. The solution is obtained by the incremental-iterative technique proposed in Ref. [31]. 5.1. Validation To validate the proposed finite element for large deflection of MEE plates, a square plate with side-length Lx = 0.254m and thickness H = 0.012m is analyzed. The plate has a single layer of a MEE composite, labeled by BF50% , obtained through the dispersion of a 50% volume fraction of fibrous piezoelectric phase of BaT iO3 in a magnetostrictive matrix of CoF e2 O4 . This material exhibits a fully coupled magneto-electro-elastic behavior and its effective piezoelectric, piezomagnetic and magnetoelectric moduli are given in Ref. [20] The plate is loaded by a uniform transverse load q and closed circuit electro-magnetic boundary conditions, corresponding to zero electric and magnetic potentials on the top and bottom plate surfaces, are imposed. 21

Fig. 2 shows the nondimensional vertical displacement w/H at the center of the plate as function of the nondimensional load parameter qd = qL4x /C11 H 4 for the case of simply-supported edges. Results are given for three different meshes and they are compared with those recently obtained in Ref. [20] by means of a meshless local Petrov-Galerkin approach. Linear solution is also plotted to highlight the nonlinear stiffening. Similar results are shown in Fig. 3 for the same plate with clamped edges. Good agreement is obtained between the present results and those found in the literature proving the accuracy of the approach. Good convergence properties have been also observed. 5.2. Magneto-electro-elastic multilayered plates subjected to mechanical load To show the effect of large deflection on MEE multilayered plates subjected to mechanical load, a square simply-supported thin plate with side length Lx = 0.254m and thickness H = 0.012m is analyzed when subjected to an uniform transverse load q. According to the most investigated configurations found in the literature, the plate considered in the present analysis has a three-layered stacking sequence of equal thickness plies of piezoelectric barium titanate BaT iO3 and piezomagnetic cobalt ferrite CoF e2 O4 , whose material properties are given in [8]; taking B labels for BaT iO3 and F labels for CoF e2 O4 , the investigated lay-up is B/F/B. The analyses have been carried out considering electro-magnetic boundary conditions corresponding to closed circuit. Fig. 4 shows the variation of displacements and potentials with respect to the load intensity for the point of coordinates (0.67Lx , 0.67Ly , 0). The results are given in terms of non-dimensional quantities whose definition is given in 22

the plot axes, provided that material properties denoted by a superimposed tilde have to be chosen as the maximum value among the corresponding properties of the plate materials. The presented results have been obtained by using a 12×12 mesh of regular elements and they are presented in comparison with the linear solution in order to point out the large displacement effect. It is worth noting that no comparison is presented with results from other solutions since, to the best of the authors’ knowledge, no solutions have been proposed in the literature for large deflection of multilayered MEE plates. The through the thickness variation of both the mechanical displacements and electric and magnetic potentials at coordinates x = 0.67Lx and y = 0.67Ly are plotted in Figs 5 and 6 for two different load intensities,respectively, whereas the corresponding through-the-thickness distributions of the stress components, electric displacements and magnetic inductions are plotted in Figs. 7 and 8. Regarding the computation of the through-the thickness distributions of the stresses, electric displacement and magnetic induction, it is pointed out that the transverse shear field and the in-plane components of the electric displacement and magnetic induction computed through the constitutive law result constant so for a more accurate appraisal they have to be computed in postprocessing by integrating the three-dimensional equilibrium equations, as common in the equivalent single-layer models of laminated plates. The proposed results clearly show the effect of large displacements in MEE laminated plates for which the classical stiffening effect is found, as expected, and a meaningful influence on the electro-magnetic state variable distribution is observed. This effect have to be adequately considered in

23

sensing devices analysis and design to avoid unwanted behaviors or ineffective performances. 5.3. Magneto-electro-elastic multilayered plates subjected to electric and magnetic load A rectangular plate with side lengths Lx = 0.254m and Ly = 0.0508m and overall thickness H = 2.54mm is considered. The plate is simply supported on the short sides while the other edges are free. The same constituents as those for the previous analyses are used, i.e. piezoelectric barium titanate BaT iO3 and piezomagnetic cobalt ferrite CoF e2 O4 , here arranged in the bimorph configuration B/F . To analyze the effect of the large displacements on the actuating behavior of MEE plates, no mechanical load is applied on the structure and electric or magnetic potentials are alternatively applied on the top surface with the bottom one grounded. Figures 9 and 10 show the transverse displacement of the central point when the electric potential and magnetic potential are applied, respectively. These results show the difference between linear and nonlinear plate response and evidence that the stiffening effect depends on the sign of the applied potential, which is a remarkable effect to consider in design and performance analysis of smart actuators. 6. Conclusions A finite element formulation and solution for multilayered magneto-electroelastic plates undergoing large deflections has been presented. It relies on an equivalent single-layer model, which involves mechanical variables only as the result of the condensation of the magneto-electric state to the mechanical one. 24

The employed approach allows to achieve the computational advantages of a mechanical equivalent single-layer model, providing also the opportunity to develop finite elements which can be straightforwardly incorporated in general purpose finite element codes. Postprocessing allows the layer-wise determination of the magneto-electric and mechanical quantities. To validate the proposed solution, an isoparametric four-noded element has been formulated for the large displacement case by using the well-established mixed interpolation of tensorial components (MITC) approach. Numerical results are in in good agreement with those available in the literature for homogeneous plates showing the accuracy and effectiveness of the present solutions. Original results for multilayered configurations are provided to show the capability of the proposed solution, provide reference data for future works and highlight the effect of large deflection on the magneto-electro-elastic plate response. In conclusion, the proposed finite element solution for large deflection of magneto-electro-elastic multilayered thin to moderately thick plates appears to be reliable and efficient and it can be used in the modeling of a variety of smart structures problems. Appendix A. Operators definition and notation In the following the definition and notation for the operators employed in the formulation of the plate model and finite element are given.   ∂  ∂x 0   ∂   0  D3×2 =   ∂y   ∂ ∂  ∂y ∂x 25

(A.1)



D2×1  D UL

 ∂   =  ∂x ∂  ∂y

(A.2) 

D 03×2 03×1  3×2  =  03×2 D3×2 03×1  02×2 02×2 02×1

   



(A.3) 

03×2 03×2 D3×2 (w0 ) D2×1

  DUN L =  03×2 03×2  02×2 02×2 

03×1

   

(A.4)

02×1  03×2

    Dγ =  03×2    I 2×2   ∂ 0  ∂x    ∂  0    ∂x   D∂ =  ∂  0    ∂y   ∂  0 ∂y

(A.5)

(A.6)

The symbol 0m×n denotes the zero matrix of order m×n and the symbol I n×n indicates the identity matrix of order n. The notation Dα×β (f ) means the application of the operator to the scalar function f , whereas the notation Dα×β f means the application of the operator following the formal rule of matrix multiplication.

26

Appendix B. Electric and magnetic potential coefficients

hki AΦ

hki hki T

=

µ33 ez

hki A Ψi

hki 2

hki hki

(B.1a)

²33 µ33 − d33

hki B Φi =

hki C Φi

hki hki T

− d33 q z

hki hki

hki hki

µ33 exy1 − d33 q xy1 hki 2

hki hki

hki hki

=

(B.1b)

²33 µ33 − d33

hki hki

µ33 exy2 − d33 q xy2 hki 2

hki hki

(B.1c)

²33 µ33 − d33 hki hki T

=

hki B Ψi =

hki C Ψi =

²33 q z

hki hki T

− d33 ez

hki hki ²33 µ33

−

hki 2 d33

hki hki hki hki ²33 q xy1 − d33 exy1 hki hki hki 2 ²33 µ33 − d33 hki hki

(B.2b)

hki hki

²33 q xy2 − d33 exy2 hki hki ²33 µ33

(B.2a)

−

hki 2 d33

(B.2c)

References [1] S. Priya, R. Islam, S. Dong, D. Viehland, Recent advancements in magnetoelectric particulate and laminate composites, Journal of Electroceramics 19 (1) (2007) 149–166. [2] C.-W. Nan, M. Bichurin, S. Dong, D. Viehland, G. Srinivasan, Multiferroic magnetoelectric composites: Historical perspective, status, and future directions, Journal of Applied Physics 103 (3) (2008) 031101– 031101. [3] W. Eerenstein, N. Mathur, J. F. Scott, Multiferroic and magnetoelectric materials, Nature 442 (7104) (2006) 759–765. 27

[4] M. Fiebig, Revival of the magnetoelectric effect, Journal of Physics D: Applied Physics 38 (8) (2005) R123. [5] Y. Fetisov,

Magnetoelectric effect in multilayer ferromagnetic-

piezoelectric structures and its application in electronics, Bulletin of the Russian Academy of Sciences: Physics 71 (11) (2007) 1626–1628. [6] C.-W. Nan, Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases, Physical Review B 50 (9) (1994) 6082. [7] J. Zhai, Z. Xing, S. Dong, J. Li, D. Viehland, Magnetoelectric laminate composites: an overview, Journal of the American Ceramic Society 91 (2) (2008) 351–358. [8] E. Pan, Exact solution for simply supported and multilayered magnetoelectro-elastic plates, Journal of Applied Mechanics, Transactions ASME 68 (4) (2001) 608–618. [9] E. Pan, P. R. Heyliger, Free vibrations of simply supported and multilayered magneto-electro-elastic plates, Journal of Sound and Vibration 252 (3) (2002) 429 – 442. [10] E. Pan, P. Heyliger, Exact solutions for magneto-electro-elastic laminates in cylindrical bending, International Journal of Solids and Structures 40 (24) (2003) 6859–6876. [11] J. Wang, L. Chen, S. Fang, State vector approach to analysis of multilayered magneto-electro-elastic plates, International Journal of Solids and Structures 40 (7) (2003) 1669–1680. 28

[12] R. Garcia Lage, C. M. Mota Soares, C. A. Mota Soares, J. N. Reddy, Layerwise partial mixed finite element analysis of magneto-electroelastic plates, Computers & Structures 82 (17-19) (2004) 1293–1301. [13] S. Phoenix, S. Satsangi, B. Singh, Layer-wise modelling of magnetoelectro-elastic plates, Journal of Sound and Vibration 324 (3-5) (2009) 798–815. [14] E. Carrera, P. Nali, Multilayered plate elements for the analysis of multifield problems, Finite Elements in Analysis and Design 46 (9) (2010) 732 – 742. [15] J. Sim˜oes Moita, C. Mota Soares, C. Mota Soares, Analyses of magnetoelectro-elastic plates using a higher order finite element model, Composite Structures 91 (4) (2009) 421 – 426. [16] A. Milazzo, An equivalent single-layer model for magnetoelectroelastic multilayered plate dynamics, Composite Structures 94 (6) (2012) 2078– 2086. [17] A. Milazzo, C. Orlando, An equivalent single-layer approach for free vibration analysis of smart laminated thick composite plates, Smart Materials and Structures 21 (7). [18] A. Alaimo, A. Milazzo, C. Orlando, A four-node mitc finite element for magneto-electro-elastic multilayered plates, Computers and Structuresdoi:10.1016/j.compstruc.2013.04.014. [19] C. Xue, E. Pan, S. Zhang, H. Chu, Large deflection of a rectangular 29

magnetoelectroelastic thin plate, Mechanics Research Communications 38 (7) (2011) 518 – 523. [20] D. Sladek, V. Sladek, S. Krahulec, E. Pan, The mlpg analyses of large deflections of magnetoelectroelastic plates, Engineering Analysis with Boundary Elements 37 (2013) 673–682. [21] F. Brezzi, K.-J. Bathe, M. Fortin, Mixed-interpolated elements for reissner-mindlin plates, International Journal for Numerical Methods in Engineering 28 (8) (1989) 1787–1801. [22] M. Kogl, M. Bucalem, Analysis of smart laminates using piezoelectric mitc plate and shell elements, Computers and Structures 83 (15-16) (2005) 1153–1163. [23] M. Kogl, M. Bucalem, A family of piezoelectric mitc plate elements, Computers and Structures 83 (15-16) (2005) 1277–1297. [24] J. Reddy, Mechanics of Laminated Composite Plates and Shells. Theory and analysis, CRC Press, 2004. [25] J. Mitchell, J. Reddy, A refined hybrid plate theory for composite laminates with piezoelectric laminae, International Journal of Solids and Structures 32 (16) (1995) 2345–2367. [26] V. Z. Parton, B. A. Kudryavtsev, Electromagnetoelasticity: piezoelectrics and electrically conductive solids, Gordon and Breach Science Publishers, 1988.

30

[27] H. Tzou, Piezoelectric Shells, Distributed Sensing and Control of Continua, Kluwer Academic Publishers, 1993. [28] E. Carrera, Single- vs multilayer plate modelings on the basis of reissner’s mixed theorem, AIAA Journal 8 (2) (2000) 342–352. [29] B. S. Guru, H. R. Hiziroglu, Electromagnetic Field Theory Fundamentals, Cambridge University Press, 2004. [30] K.-J. Bathe, E. N. Dvorkin, Four-node plate bending element based on mindlin/reissner plate theory and a mixed interpolation., International Journal for Numerical Methods in Engineering 21 (2) (1985) 367–383. [31] G. E. Blandford, Progressive failure analysis of inelastic space truss structures, Computers & Str 58 (5) (1996) 981–990.

31















































Figure 1: Multilayered MEE plate geometry.

32



0.5

MLPG solution Linear FEM - 12 × 12 mesh Nonlinear FEM - 12 × 12 mesh Nonlinear FEM - 8 × 8 mesh Nonlinear FEM - 4 × 4 mesh

0.45 0.4 0.35

w/H

0.3 0.25 0.2 0.15 0.1 0.05 0 0

1

2

3

4

5

6

7

8

9

10

qL4x /C11 H 4 Figure 2: Central plate deflection vs. transverse load intensity for the BF50% simply supported plate.

33

0.9

MLPG solution Linear FEM - 16 × 16 mesh Nonlinear FEM - 16 × 16 mesh Nonlinear FEM - 12 × 12 mesh Nonlinear FEM - 8 × 8 mesh

0.8

0.7

w/H

0.6

0.5

0.4

0.3

0.2

0.1

0 0

10

20

30

40

50

60

qL4x /C11 H 4 Figure 3: Central plate deflection vs. transverse load intensity for the BF50% clamped plate.

34

0.05

10

0.04

8

wLx /H 2

uLx /H 2

Nonlinear FEM solution Linear FEM solution

0.03 0.02 0.01

6 4 2

0 −0.01

0 0

1

2

3

4

5

0

1

qL4x / C˜1 1H 4

3

4

5

4

5

qL4x / C˜1 1H 4

0

Ψ˜ µ 1 1L2x / q˜3 3H 3

0.8

Φ˜²1 1L2x / e˜3 3H 3

2

0.6 0.4 0.2 0

−0.02 −0.04 −0.06 −0.08 −0.1

0

1

2

3

4

5

0

qL4x / C˜1 1H 4

1

2

3

qL4x / C˜1 1H 4

Figure 4: Load-displacement and load-potential curves for the B/F/B laminated square plate subjected to uniform mechanical transverse load

35

Nonlinear FEM solution Linear FEM solution 0.5

z/H

z/H

0.5

0

−0.5 −0.05

0

uLx /H

−0.5 1.075

0.05

1.08

2

1.085

wLx /H

1.09

2

0.5

z/H

0.5

z/H

0

0

−0.5 0

0.02

0.04

0.06

0

−0.5 −0.3

0.08

Φ˜ ²1 1L2x / e˜3 3H 3

−0.2

−0.1

0

0.1

Ψ˜ µ 1 1L3x / q˜3 3H 4

Figure 5: Through-the-thickness distribution of displacements and potentials at coordinates x = 0.67Lx and y = 0.67Ly for the B/F/B laminated square plate subjected to constant mechanical transverse load with qd = 0.56.

36

Nonlinear FEM solution Linear FEM solution 0.5

z/H

z/H

0.5

0

−0.5 −0.4

−0.2

0

uLx /H

0.2

−0.5 6.5

0.4

7

7.5

2

8

wLx /H

8.5

9

2

0.5

z/H

0.5

z/H

0

0

−0.5 0

0.2

0.4

0.6

0

−0.5 −2

0.8

Φ˜ ²1 1L2x / e˜3 3H 3

−1

0

1

2

Ψ˜ µ 1 1L3x / q˜3 3H 4

Figure 6: Through-the-thickness distribution of displacements and potentials at coordinates x = 0.67Lx and y = 0.67Ly for the B/F/B laminated square plate subjected to constant mechanical transverse load with qd = 4.5.

37

Nonlinear FEM solution - Constitutive equations Linear FEM solution - Constitutive equations Nonlinear FEM solution - Equilibrium equations Linear FEM solution - Equilibrium equations 0.5

z/H

z/H

0.5

0

−0.5 −0.2

−0.1

0

0.1

0.2

0

−0.5 −0.02

σxx L2x / C˜ 1 1H 2

z/H

z/H −0.06

−0.04

−0.02

−0.5 −1

0

−0.5

0

0.5

1

σzz L2x / C˜ 1 1H 2 0.5

z/H

0.5

z/H

0.02

0

σxz L3x / C˜ 1 1H 3

0

−0.5

0

−0.5 −0.2

−0.1

0

0.1

−0.4

Dx L3x / e˜3 3H 3

−0.2

0

0.2

Bx L3x / q˜3 3H 3 0.5

z/H

0.5

z/H

0.01

0.5

0

0

−0.5 −3

0

σxy L2x / C˜ 1 1H 2

0.5

−0.5 −0.08

−0.01

−2

−1

0

1

Dz L3x / e˜3 3H 3

−3 x 10

0

−0.5 −1

0

1

Bz L3x / q˜3 3H 3

2 −3 x 10

Figure 7: Through-the-thickness distribution of stresses, electric displacements and magnetic induction at coordinates x = 0.67Lx and y = 0.67Ly for the B/F/B laminated square plate subjected to constant mechanical transverse load with qd = 0.56.

38

Nonlinear FEM solution - Constitutive equations Linear FEM solution - Constitutive equations Nonlinear FEM solution - Equilibrium equations Linear FEM solution - Equilibrium equations 0.5

z/H

z/H

0.5

0

−0.5 −2

−1

0

1

0

−0.5 −0.2

2

σxx L2x / C˜ 1 1H 2

z/H

z/H −0.6

−0.4

−0.2

−0.5 −1

0

0

0.5

1

0.5

z/H

z/H

−0.5

σzz L2x / C˜ 1 1H 2

0.5

0

−1.5

−1

−0.5

0

0.5

0

−0.5 −3

−2

−1

0

Bx L3x / q˜3 3H 3

Dx L3x / e˜3 3H 3 0.5

z/H

0.5

z/H

0.2

0

σxz L3x / C˜ 1 1H 3

0

−0.5 −0.15

0.1

0.5

0

−0.5 −2

0

σxy L2x / C˜ 1 1H 2

0.5

−0.5 −0.8

−0.1

−0.1

−0.05

0

0.05

0

−0.5 −0.05

Dz L3x / e˜3 3H 3

0

0.05

0.1

Bz L3x / q˜3 3H 3

Figure 8: Through-the-thickness distribution of stresses, electric displacements and magnetic induction at coordinates x = 0.67Lx and y = 0.67Ly for the B/F/B laminated square plate subjected to constant mechanical transverse load with qd = 4.5.

39

0.16

Nonlinear FEM solution Linear FEM solution

0.14 0.12

|w|/H

0.1 0.08 0.06 0.04 0.02 0 −15

−10

−5

0

Φ˜ ²1 1Lx /˜ e3 3H

5

10

15

2

Figure 9: Variation of the bimorph plate center deflection with the applied electric potential.

40

0.08

Nonlinear FEM solution Linear FEM solution

0.07 0.06

|w|/H

0.05 0.04 0.03 0.02 0.01 0 −150

−100

−50

0

50

100

150

Ψ˜ µ 1 1Lx / q˜3 3H 2 Figure 10: Variation of the bimorph plate center deflection with the applied magnetic potential.

41