Composite Structures 132 (2015) 737–745

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Two-dimensional dynamic model for composite laminateswith embedded magnetostrictive materials

http://dx.doi.org/10.1016/j.compstruct.2015.04.0620263-8223/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: ssantapuri@pupr.edu (S. Santapuri), scheidler.8@osu.edu

(J.J. Scheidler), dapino.1@osu.edu (M.J. Dapino).

S. Santapuri b, J.J. Scheidler a, M.J. Dapino a,⇑a Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USAb Department of Mechanical Engineering, Polytechnic University of Puerto Rico, San Juan 00918, Puerto Rico

a r t i c l e i n f o

Article history:Available online 14 May 2015

Keywords:Magnetostrictive materialsSmart composites2-D plate theoriesNonlinear modelingBending actuators

a b s t r a c t

A nonlinear, two-dimensional plate model is presented that describes the dynamic response of compositelaminate structures with embedded magnetostrictive materials. The model consists of Navier’s equationcoupled with a discrete energy-averaged constitutive model for magnetostrictive materials. Assuming athin composite geometry, the complete three-dimensional model is reduced to a two-dimensionalequivalent single layer plate model by assuming a functional form for displacements. The resultingtwo-dimensional variational form, solved using finite element software, is applied to analyze thedisplacements of a Galfenol–aluminum composite actuator, wherein Galfenol sheets are embedded intoan aluminum substrate. The model is validated at lower frequencies using time domain measurements ofthe dynamic actuation response of the actuator. This framework is subsequently utilized to perform aparametric study to maximize the tip displacements by varying the geometric parameters: Galfenol loca-tion, thickness ratio, and substrate properties. The general finite element framework presented in thispaper is applicable to a wider range of magnetostrictive materials and complex composite geometries.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Magnetostrictive materials exhibit the Joule effect, a shapechange in response to externally applied magnetic fields, and theVillari effect, a change in magnetic susceptibility of the materialwhen subjected to external stress [1]. These properties make mag-netostrictive materials useful in sensing and actuation applicationssuch as active vibration control, energy harvesting, stress and tor-que sensing, and machining [2,3]. Terfenol-D (Tb0.3Dy0.7Fe1.9) exhi-bits the largest known room-temperature magnetostriction ofabout 1600 ppm [4], but needs to be operated in compressiondue to its brittleness. This limitation can be overcome, at theexpense of magnetostriction, by embedding Terfenol-D particlesin epoxy, polymer, or metal matrices. Galfenol (Fe81.6Ga18.4) is arecent magnetostrictive material that exhibits moderate magne-tostriction (�350 ppm), but has structural properties comparableto low-carbon steels, allowing its use in mechanically-harsh envi-ronments [2]. Galfenol can be integrated with passive componentsthrough fusion welding, machining, or other traditional processesto design multifunctional, load-bearing structures.

Magnetostrictive materials have been used to build active com-posite structures (consisting of active magnetostrictive and passivemetallic components) for a variety of applications. Some examplesinclude a strain sensor [5], torque sensor [6], bioMEMS sensor [7],and a MEMS actuator [8]. The recent development of ultrasonicadditive manufacturing (UAM), a solid-state welding process thatadditively welds thin metal tapes to create complex, metallic com-posites without melting the parent materials, has the potential torevolutionize this field. It enables the manufacture of complex,3-D composite geometries that can be composed of internal chan-nels, dissimilar metals, electronics, heat-wicking materials, orsmart materials [9–11].

To permit the design, optimization, and control of theseadvanced composites, transducer models need to be developed inparallel with advancements in the manufacturing technology. Akey challenge in developing mathematical models for magne-tostrictive materials is their highly nonlinear, anisotropic constitu-tive behavior. Thus, a transducer model for magnetostrictivecomposites should be formulated such that it can predict the com-plex, dynamic response of magnetostrictive materials whileaccommodating the sophisticated geometries made possible bynew manufacturing capabilities.

In the past, composites with thin geometries were modeledusing equivalent single layer (ESL) theories that were derived from

Fig. 1. Schematic of an active composite plate consisting of an embeddedmagnetostrictive layer of volume Vm and a passive metallic layer of volume Vp .

738 S. Santapuri et al. / Composite Structures 132 (2015) 737–745

3-D continuum theories by making suitable assumptions about thekinematics of deformation through the thickness of the laminate[12]. These theories allowed the reduction of a 3-D problem to a2-D problem by assuming a functional form for displacement fields[13]. Early developments of classical plate theories for materialsexhibiting coupled behavior were limited by linear constitutivemodels (e.g., piezoelectric, piezomagnetic). One of the earliestdevelopments includes Mindlin’s first-order plate theory for highfrequency piezoelectric crystals [12]. Reddy enhanced these mod-els by using a third-order plate theory for thick laminated compos-ites with integrated sensors and actuators [14]. These linearmodels, although sufficient for some regimes of coupled behavior,are often insufficient to model the nonlinear constitutive responseof magnetostrictive materials.

One of the earlier attempts to model magnetostrictive materialswith nonlinear constitutive behavior was developed by Kannanand Dasgupta [15]. This paper presented a two-dimensional,quasi-static, finite-element scheme to model the nonlinear interac-tions between mechanical and magnetic fields in Terfenol-D. Dattaet al. [16,17] used classical laminated plate theory with theArmstrong magnetomechanical model to characterize laminatedsensors and actuators in the absence of current-induced magneticfields. Neither of these models accounted for dynamic behavior.More recently, Shu et al. [18] utilized a nonlinear, discreteenergy-averaged (DEA) model [19] to simulate the 1-D dynamicresponse of Galfenol-driven unimorph actuators.

In this work, we present the development of a dynamic,two-dimensional plate model describing the nonlinear magne-tomechanical behavior of composite plates containing embeddedmagnetostrictive materials. The modeling approach is based uponvariational principles and the finite element method. Starting froma coupled 3-D formulation, the problem is reduced to a 2-D ESLtheory. To model the nonlinear, anisotropic behavior of magne-tostrictive materials, an anhysteretic, DEA model developed byEvans and Dapino [19] is utilized. The model is then validatedusing measurements of the dynamic, bending actuation responseof a Galfenol–aluminum composite plate (manufactured usingUAM). To illustrate its usefulness for design, the model is appliedto a prototype bending actuator composed of an aluminum platewith embedded Galfenol sheets.

The paper is structured as follows: Section 2 presents a 3-Ddescription of a smart composite plate. In Section 3 the 2-Dmodel is developed for thin composite plates starting from thecomplete 3-D formulation. Section 3.2 summarizes the DEAmodel developed by Evans and Dapino [19] for fully-coupledmagnetostrictive materials with cubic symmetry. The frameworkoutlined in Sections 2 and 3 is applied to a Galfenol–aluminumplate in Section 3.4 and experimentally validated in Section 4.In Section 5, the framework is utilized to perform a parametricstudy of key geometric parameters to optimize the plate’s tip dis-placement. Finally, the results and contributions of this work arediscussed in Section 6.

2. 3-D magnetomechanical model for active composites withembedded magnetostrictive layers

This work considers an active composite plate composed of ametal matrix and embedded magnetostrictive laminae of arbitrarygeometry. Fig. 1 shows a schematic of an example composite platecontaining embedded magnetostrictive layers.

A 3-D model of the composite plate can be formulated bycoupling Navier’s equation with a nonlinear constitutive law forthe magnetostrictive material and linear Hooke’s law for thepassive material. In the 3-D weak form, Navier’s equation can bewritten as

ZVtot

q@2u@t2 �duþ c

@u@t�duþ T �dS

" #dV ¼

Z@Vtot

t �dud@V þZ

Vtot

fB �dudV ;

ð1Þ

where q is the density, c represents the viscous damping factor, andT and fB denote, respectively, the stress tensor and external bodyforce acting on the domain Vtot . The traction vector t acts on theboundary @Vtot . Also, S and u represent the strain tensor and dis-placements at each point in the domain Vtot . The strain is definedin terms of the displacements as

S ¼ 12ruþruT� �

; ð2Þ

where T denotes the transpose operator. The constitutive equationsin the passive and magnetostrictive material domains, respectively,are

T ¼ C � S ðPassive materialÞ ð3Þ

and

T ¼ TmðS;HÞ ðMagnetostrictive materialÞ; ð4Þ

where C is the stiffness matrix, H is the magnetic field, and the func-tion TmðS;HÞ represents the nonlinear constitutive relationship formagnetostrictive materials. In this paper, TmðS;HÞ is a DEA modeldeveloped by Evans and Dapino [19], which will be summarizedin Section 3.2. It is assumed that the magnetic field (induced by aconductive coil) is uniform inside the magnetostrictive materialand can be approximated using Ampére’s Law H ¼ nI, where n isthe coil constant and I is the current in the coil. By assuming acurrent-field relationship, the mechanical response of the compos-ite plate can be modeled without considering magnetic circuitequations. This is a good approximation of the field inside a longcoil that is tightly wound around a material with a high magneticpermeability, where coil fringing effects can be neglected.

3. Dimensional reduction using classical plate theory

For the analysis of thin composite plates, the conventional mod-eling approach is based on ESL theories, which are derived from3-D continuum theories by making suitable assumptions on thekinematics of deformation or the stress state through the thicknessof the laminate. These theories allow the reduction of a 3-Dproblem to a 2-D problem [20].

An ESL plate theory for active composite structures that containembedded magnetostrictive materials is developed by assumingthe following expansion for the displacement field u:

uðx; y; zÞ ¼ u;v ;w½ �T ¼Xn¼1n¼0

zn uðnÞðx; yÞ: ð5Þ

S. Santapuri et al. / Composite Structures 132 (2015) 737–745 739

The simplest form of laminated plate theory is the classical platetheory, where the displacement field takes the form:

uðx; y; z; tÞ ¼ u0ðx; y; tÞ þ z/xðx; y; tÞ; ð6Þvðx; y; z; tÞ ¼ v0ðx; y; tÞ þ z/yðx; y; tÞ; ð7Þwðx; y; z; tÞ ¼ w0ðx; y; tÞ; ð8Þ

where /x and /y denote rotations about the y and x axes, respec-tively. Assuming that the deformation has only bending andin-plane stretching components (i.e., transverse normal and trans-verse shear effects are negligible), these rotations are represented as

/x ¼ �@wo

@x; /y ¼ �

@wo

@y: ð9Þ

Classical plate theories work well for thin composite plates withthickness ratio (ratio of thickness to length or width) less than0.1. As the thickness increases, higher order theories need to be con-sidered [20]. The displacement forms described in (6)–(9) allow usto ignore the strain components Szz; Sxz, and Syz, i.e., the problemreduces to a plane strain problem. The remaining strain compo-nents can be written as

Sxx

Syy

Sxy

264

375 ¼

Sð0Þxx

Sð0Þyy

Sð0Þxy

2664

3775þ z

Sð1Þxx

Sð1Þyy

Sð1Þxy

2664

3775; ð10Þ

where

Sð0Þxx ¼@u0

@x; Sð0Þyy ¼

@v0

@y; Sð0Þxy ¼

12

@u0

@yþ @v0

@x

� �;

Sð1Þxx ¼@/x

@x; Sð1Þyy ¼

@/y

@y; Sð1Þxy ¼

12

@/x

@yþ@/y

@x

� �:

ð11Þ

To define the stresses in the passive and active subdomains,constitutive equations for each subdomain must be defined.

3.1. Stress components: incorporation of constitutive equations

The composite plate is split into passive and active (magne-tostrictive) subdomains, represented by subscripts p and m,respectively.

3.1.1. Passive subdomain constitutive equationsFor the passive layers, linear Hooke’s law is used, which

assumes linear isotropic behavior.1 The corresponding constitutiveequations are

Txx

Tyy

Txy

264

375ðpÞ

¼ Ep

ð1þ mpÞð1� 2mpÞ

1� mp mp 0mp 1� mp 00 0 1� 2mp

264

375

Sxx

Syy

Sxy

264

375ðpÞ

:

ð12Þ

The following material parameters are used for the aluminummatrix: Young’s modulus Ep ¼ 69 GPa, Poisson’s ratio mp ¼ 0:33,and density qp ¼ 2700 kg/m3.

3.1.2. Magnetostrictive subdomain constitutive equationsThe inputs for the magnetostrictive constitutive model are mag-

netic field and stress, and the outputs are magnetization and strain,which includes magnetoelastic and purely mechanical compo-nents. Consequently, the stress–strain constitutive law can be writ-ten as

1 This assumption is valid for the aluminum matrix considered in this paper.However, anisotropic materials can be easily considered by using a stiffness matrixwith fewer restrictions on material symmetry.

Txx

Tyy

Txy

264

375ðmÞ

¼ Em

ð1þ mmÞð1� 2mmÞ

1� mm mm 0mm 1� mm 00 0 1� 2mm

264

375

�Sxx

Syy

Sxy

264

375ðmÞ

� k TðmÞ;H� �8><

>:9>=>;;

ð13Þ

where k denotes the macroscopic magnetoelastic strain (magne-tostriction) in the magnetostrictive material. Due to the dependenceof k on TðmÞ, (13) is implicit and must be calculated by inverting theconstitutive model of the magnetostrictive material after solving forthe total strain. This work utilizes a fully-coupled, nonlinear DEAmodel for cubic magnetostrictive materials developed by Evansand Dapino [19], which is accurate yet efficient and has been suc-cessfully used for the modeling of Galfenol-based systems[21,18,22,23]. This model is summarized in the following section.To model composite structures containing Terfenol-D constituents,the analogous DEA model for Terfenol-D developed by Chakrabartiand Dapino [24] can be used instead, with minor changes to theproposed framework.

3.2. Review of the two-dimensional discrete energy-averaged modelfor cubic magnetostrictive materials

The energy-averaged class of models calculates the macroscopicconstitutive response as an energy-weighted summation of theresponse due to domains aligned along different crystallographicdirections. With homogeneously-distributed, fixed orientations(as in Armstrong’s model [25]), obtaining high accuracy requiresa large number of possible orientations, which results in significantcomputational effort.

To improve the computational efficiency while preserving accu-racy, Evans and Dapino [19] restricted the number of possible ori-entations to six by considering only the directions that minimizean energy functional that is locally-defined around each of the

six easy crystallographic directions. The total free energy Gk

of amagnetic moment oriented in the vicinity of the kth easy direction

ck

is formulated as the sum of the local magnetocrystalline(anisotropy),2 magnetoelastic (magnetomechanical coupling) andmagnetic field (Zeeman) energies. This energy can be written inmatrix form as

Gk¼ 1

2mk�K m

k�m

k� B

kþK0

kðk ¼ 1;2; . . . ;6Þ ð14Þ

where mk

is the magnetic moment direction,

K ¼K � 3k100Txx �3k111Txy 0�3k111Txy K � 3k100Tyy 0

0 0 K

264

375; ð15Þ

Bk¼ c1

kK þ loMsH1 c2

kK þ loMsH2 c3

kK

h iT; ð16Þ

where K and K0

kare anisotropy energy constants, Ms is the satura-

tion magnetization, k100 and k111 are magnetostriction constants,and lo is the permeability of free space. The minimization of (14)

is constrained (kmkk ¼ 1) and solved, through the use of Lagrange

multipliers, as an inhomogeneous eigenvalue problem [19]. By

approximating the constraint using kmkk �m

k� c

k, the minimization

problem has the explicit solution

2 The improved anisotropy energy given by Chakrabarti [26] is used.

740 S. Santapuri et al. / Composite Structures 132 (2015) 737–745

mk¼ K�1 B

kþ1� c

k�K�1 B

k

ck�K�1 c

kck

24

35: ð17Þ

The anhysteretic volume fraction nk

an of magnetic domains orientedalong the kth minimum energy direction is calculated usingBoltzmann-type averaging,

nk

an ¼exp �G

k=X

� �Pr

j¼1 exp �Gj

=X� � ; ð18Þ

where X is a smoothing constant. Macroscopic anhysteretic mate-rial behavior is obtained by summing the contributions due to each

minimum energy magnetization direction mk

weighted by their cor-responding volume fraction. Thus, the strain tensor S is given by thesum of the elastic and magnetoelastic strains,

S ¼ sTþ k ¼ sTþXr

k¼1

nk

ankk

; ð19Þ

where s is the 3� 3 compliance tensor for Galfenol and kk

representsthe magnetostriction of a ferromagnetic material with cubic sym-metry due to the kth minimum energy direction,

kk

¼3=2k100m1

k 2

3=2k100m2k 2

3k111 m1k

m2k

26664

37775: ð20Þ

Material constants for Galfenol used in this model are3:Em ¼ 57 GPa, mm ¼ 0:3; qm ¼ 7870 kg/m3, K ¼ 27:1 kJ/m3, K0 ¼0:021 kJ/m3, l0Ms ¼ 1:3 T, k100¼156:56 ppm; k111¼1=3� �20ð Þ ppm,and X¼1300 J/m3.

As detailed in Section 3.1, to integrate the constitutive Eq. (19)into Navier’s equation (1), inversion of the constitutive model isrequired, wherein stress is the output (dependent variable) andstrain is the input (independent variable). Owing to the nonlineardependence of S on T and H and the homogenization equation(18), an analytical inverse does not exist. Following the approachof Chakrabarti and Dapino [21], the inversion is performed usingthe quasi-Newton SR1 formula. By also utilizing the Sherman-Morrison formula, the need for calculating and inverting thematerial Jacobian within the iteration loop is avoided. The use ofan iterative inversion algorithm adds computational cost, butallows the solution procedure to retain the full nonlinearity ofthe constitutive model, unlike inversions based upon linearizationof the constitutive model. Having presented the constitutive modelfor the passive and active subdomains, we now proceed to derivethe 2-D ESL plate theory for the magnetostrictive composite plate.

3.3. Derivation of equivalent two-dimensional variational principle

To derive the 2-D weak form starting from the 3-D compositeplate model (1), the 3-D weak form is recalled,Z

Vtot

q@2u@t2 � du þ c

@u@t� du þ T � dS

" #dV

�Z@Vtot

t � dud@V �Z

Vtot

fB � dudV ¼ 0: ð21Þ

3 The Galfenol material parameters (including damping coefficient) used in thisframework are optimized for a 2-D problem to accommodate for the plane strainassumption.

The variational form of 2-D plate theory is derived by substitutingstress, displacement, and strain expressions (6)–(13) into the 3-Dformulation of Navier’s equation (21), which decouples thethrough-thickness integration from the in-plane integration. WithXe as the ESL area and ttot as the plate’s thickness, the first termon the left-hand side of (21) is reduced to 2-D as follows:Z

Vtot

q@2u@t2 �dudV ¼

Zttot

ZXe

q@2u@t2 duþq

@2v@t2 dvþq

@2w@t2 dw

" #dxdydz

¼Z

Xe

Zttot

qdz� �

@2uo

@t2 duoþZ

ttot

qzdz� �

@2/x

@t2 duo

(

þZ

ttot

qzdz� �

@2uo

@t2 d/xþZ

ttot

qz2dz� �

@2/x

@t2 d/x

þZ

ttot

qdz� �

@2vo

@t2 dvoþZ

ttot

qzdz� �

@2/y

@t2 dvo

þZ

ttot

qzdz� �

@2vo

@t2 d/yþZ

ttot

qz2dz� �

@2/y

@t2 d/y

þZ

ttot

qdz� �

@2wo

@t2 dwo

)dxdy: ð22Þ

The remaining terms in (21) are reduced to 2-D in a similar manner.The density integrals in (22) depend only on the geometry andconstituent materials and must be evaluated only once. Similarintegrals will arise for the viscous damping factor c and the stressterms Txx; Txy, and Tyy. In the interest of simplifying the presenta-tion of equations, the following equivalent properties for the 2-Dplate are defined:

�qk ¼Z

ttot

qzk dz; �ck ¼Z

ttot

c zk dz ðk ¼ 0;1;2Þ ð23Þ

Nxx

Nyy

Nxy

8><>:

9>=>; ¼

Zttot

Txx

Tyy

Txy

8><>:

9>=>;dz;

Mxx

Myy

Mxy

8><>:

9>=>; ¼

Zttot

Txx

Tyy

Txy

8><>:

9>=>;zdz: ð24Þ

Using these representations, the weak form of the equations reducetoZ

s

ZXe

"�qo

@2uo

@t2 duoþ@2vo

@t2 dvoþ@2wo

@t2 dwo

!(

þ�co@uo

@tduoþ

@vo

@tdvoþ

@wo

@tdwo

� �

þ �q1@2/x

@t2 duoþ@2/y

@t2 dvoþ@2uo

@t2 d/xþ@2vo

@t2 d/y

!

þ�c1@/x

@tduoþ

@/y

@tdvoþ

@uo

@td/xþ

@vo

@td/y

� �

þ �q2@2/x

@t2 d/xþ@2/y

@t2 d/y

!þ�c2

@/x

@td/xþ

@/y

@td/y

� �

þNxxdSð0Þxx þNxydSð0Þxy þNyydSð0Þyy þMxxdSð1Þxx þMxydSð1Þxy þMyydSð1Þyy

#dxdy

�Z@Xe

N̂nnduonþ N̂nsduos�M̂nn@dwo

@n�M̂ns

@dwo

@sþ Q̂ndwo

� �ds

)dt¼0;

ð25Þwhere @Xe represents the boundary to Xe and s represents the timeover which the dynamic system is studied. Also, the boundary

terms N̂nn; N̂ns; M̂nn; M̂ns, and Q̂n are defined as

N̂nn

N̂ns

( )¼Z

ttot

r̂nn

r̂ns

� dz;

M̂nn

M̂ns

( )¼Z

ttot

r̂nn

r̂ns

� zdz; Q̂n¼

Zttot

r̂nzdz;

ð26Þwhere r̂nn; r̂ns, and r̂nz are the specified stress components on theportion of the boundary @Xe. Through additional knowledge of the

S. Santapuri et al. / Composite Structures 132 (2015) 737–745 741

composite geometry and the constituent material properties, theintegrals listed in (23) and (24) can be evaluated. These expressions,when substituted into the weak form (25), will provide the com-plete system model in 2-D.

The model presented above is general and applies to thin com-posite plates of arbitrary construction that contain magnetostric-tive materials. In what follows, we specialize this 2-D variationalform to a composite plate actuator consisting of a Galfenol layerembedded into an aluminum matrix.

3.4. 2-D variational form

To develop the equivalent 2-D system corresponding to the 3-Dcomposite plate described in Fig. 1, the integral terms that appearin the 2-D variational form (25) are evaluated. For this application,it is assumed that the plate is only excited by an external magneticfield (i.e., there are no external forces and the integrals in Eq. (26)equate to zero). This excitation enters the model through the mag-netostriction of the Galfenol sheets, which loads and deforms thepassive volume. The electromagnetic body forces and rotary inertiaterms are also ignored. This translates to ignoring the dynamicterms with coefficients �q1; �q2; �c1, and �c2.

The plate is divided into two equivalent 2-D subdomains(Fig. 2), i.e., (i) the passive subdomain Ap and (ii) the active subdo-main Am. The integrals (23) and (24), which are evaluated usingthe composite’s dimensions and its material properties, are substi-tuted into (25) to deduce the 2-D variational form. The densityintegrals can be calculated as

Passive Ap : �qk ¼Z

ttot

qzk dz ¼qp

kþ 1zkþ1

1 � zkþ14

� �;

Active Am : �qk ¼qp

kþ 1zkþ1

1 � zkþ12

� �þ qm

kþ 1zkþ1

2 � zkþ13

� �þ

qp

kþ 1zkþ1

3 � zkþ14

� �; ð27Þ

where qp and qm are the densities of aluminum and Galfenol,respectively, and z1; z2; z3, and z4 are the z-coordinates at eachthrough-thickness material boundary. The equivalent dampingcoefficients �ck are calculated in a similar manner. The stressresultant Nxx is calculated as

Passive Ap : Nxx ¼Z

ttot

Txx dz ¼Z

ttot

CðpÞ11 Sxx þ CðpÞ12 Syy

� �dz

¼Z

ttot

CðpÞ11 ðSð0Þxx þ zSð1Þxx Þ þ CðpÞ12 ðS

ð0Þyy þ zSð1Þyy Þ

h idz

¼ CðpÞ11 Sð0Þxx þ CðpÞ12 Sð0Þyy

� �ðz1 � z4Þ

þ 12

CðpÞ11 Sð1Þxx þ CðpÞ12 Sð1Þyy

� �ðz2

1 � z24Þ; ð28Þ

Fig. 2. Equivalent 2-D formulation of the embedded magnetostrictive composite inFig. 1.

Active Am : Nxx ¼Z

ttot

Txx dz ¼Z

tp

Txx dzþZ

tm

Txx dz

¼Z

tp

CðpÞ11 Sxx þ CðpÞ12 Syy

� �dz

þZ

tm

CðmÞ11 Sxx � kxxð Þ þ CðmÞ12 Syy � kyy� �n o

dz

¼ CðpÞ11 Sð0Þxx þ CðpÞ12 Sð0Þyy

� �ððz1 � z2Þ þ ðz3 � z4ÞÞ

þ CðmÞ11 Sð0Þxx � kxx

� �þ CðmÞ12 Sð0Þyy � kyy

� �n oðz2 � z3Þ

þ 12

CðpÞ11 Sð1Þxx þ CðpÞ12 Sð1Þyy

� �ððz2

1 � z22Þ þ ðz2

3 � z24ÞÞ

þ 12

CðmÞ11 Sð1Þxx þ CðmÞ12 Sð1Þyy

� �z2

2 � z23

� �: ð29Þ

A similar procedure is applied to calculate the remaining stressresultants. Due to the Galfenol sheet’s small thickness, the magne-tostrictive strain components kxx; kyy, and kxy are assumed constantthrough the thickness of each Galfenol layer.

3.5. Computational methodology

The weak form equations are discretized and solved in COMSOLMultiphysics finite element software. The anhysteretic constitutivemodel for Galfenol is supplied to COMSOL through MATLAB func-tions to retain the model’s full nonlinearity. This work considerscantilever boundary conditions, for which the field-excited com-posite plate acts as a bending actuator. Thus, the primary outputof the model is the tip displacement of the plate.

It has been observed that the 2-D framework has an improvedcomputational efficiency compared to a full 3-D simulation of thesame composite structure. The 2-D simulations are approximatelyfive times faster than the 3-D framework.

4. Model validation

To validate the model, the tip displacement of a Galfenol–alu-minum composite plate was measured in response to quasi-staticand dynamic currents produced in an excitation coil. The compos-ite plate (Fig. 4) was manufactured using ultrasonic additive man-ufacturing (UAM). The welding was performed using a 9 kWFabrisonic SonicLayer 4000 UAM machine. A single sheet ofGalfenol was embedded into the center of the aluminum plate.As shown in Fig. 3, the experimental setup consisted of a magneticcircuit composed of a U-shaped Metglas core upon which the exci-tation coils were wound, a cantilevering fixture, and the compositeplate, which completed the magnetic flux path. The air gap

Fig. 3. Experimental setup: cantilevered Galfenol–aluminum composite clamped toU-shaped magnetic circuit made of Metglas; cantilever geometry (also used invalidation simulations): W ¼ 22:9 mm, ttot ¼ 1:65 mm, bm ¼ 6:35 mm, tm ¼ 0:965 mmand L ¼ 25:1 mm.

Fig. 4. Geometry of the Galfenol–aluminum composite plate manufactured using ultrasonic additive manufacturing.

(i)0 5 10 15 20

0

2

4

6

8

10

12

14

Time (s)

Dis

plac

emen

t (µm

)

0.1 Hz

Experimental dataProposed model

(ii)0 0.02 0.04 0.06 0.08 0.1

0

2

4

6

8

10

12

14

Time (s)

Dis

plac

emen

t (µ m

)

50 Hz

Experimental dataProposed model

(iii)0 0.005 0.01 0.015 0.02 0.025 0.03

0

2

4

6

8

10

12

14

Time (s)

Dis

plac

emen

t (µm

)

150 Hz

Experimental dataProposed model

(iv)0 0.005 0.01 0.015 0.02 0.025

0

2

4

6

8

10

12

14

Time (s)

Dis

plac

emen

t (µm

)

200 Hz

Experimental dataProposed model

Fig. 5. Comparison of the experimental and simulated dynamic actuation of the Galfenol–aluminum plate at different frequencies: (i) 0.1 Hz, (ii) 50 Hz, (iii) 150 Hz, (iv)200 Hz.

742 S. Santapuri et al. / Composite Structures 132 (2015) 737–745

between the magnetic flux return path and the end of the compos-ite was approximately 0.0254 mm.

The tip displacement of the composite was measured using aKeyence LK-G32 laser displacement sensor having a repeatabilityof 0.05 micron. A National Instruments cDAQ-9178 was used toacquire data and generate the excitation voltage, which was ampli-fied using a Techron 7782 linear power amplifier. For dynamictests, sinusoidal excitations of amplitude � 3:25 kA/m were super-imposed on a bias field of � 3:50 kA/m, which was used to maxi-mize the magnetic field-induced strain and provide two-wayactuation. A comparison between the measured and calculated

tip displacement for frequencies up to 200 Hz is shown in Fig. 5.Due to the relatively high stiffness of the plate and low Galfenolvolume fraction, the tip displacements are small.

Overall, the model accurately calculates the steady-state,dynamic actuation response of the composite plate up to 200 Hz.However, the model does not capture the transient, wherein aslight decrease in tip displacement is observed as the actuation fre-quency increases. This can be explained by the lack of electromag-netic dynamics, namely eddy currents, in the model. The inclusionof eddy currents and magnetic field nonuniformity can improvethe model’s performance.

−10 −5 0 5 107

7.5

8

8.5

9

9.5

10

10.5

x−coordinate (mm)

Tip

Dis

plac

emen

t (µm

)

yo = 1 mmyo = 3 mmyo = 5 mmyo = 7 mmyo = 8 mm

Fig. 7. Variation of tip displacement (lm) along the free end of the composite platefor varying distances (along the y-axis) of the Galfenol sheets from the center axis.

Fig. 6. Galfenol–aluminum composite plate for actuator design study.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

2

4

6

8

10

12

14

16

Galfenol distance from the top surface (mm)

Tip

Dis

plac

emen

t (µm

)

a = 0.8a = 1a = 1.2

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

5

10

15

20

25

Galfenol strip distance from the top surface (mm)

Tip

Dis

plac

emen

t (µm

)

Es = 30 GPaEs = 50 GPaEs = 69 GPaEs = 90 GPaEs = 120 GPa

(b)Fig. 8. Variation of tip displacement (lm) as a function of Galfenol sheet’s distance from the top surface for different (a) plate aspect ratios (a = width/length), and (b)substrate Young’s moduli.

S. Santapuri et al. / Composite Structures 132 (2015) 737–745 743

5. Actuator design study: maximizing tip displacements

In this section, we utilize the validated 2-D plate model to max-imize the displacements of the Galfenol actuator shown in Fig. 6.Actuator displacements are studied by changing several designparameters while keeping the volume of Galfenol constant.

The actuator is described as follows. The x–y plane of the coor-dinate system is coincident with the midplane of the compositeplate, and the x–z plane is a plane of symmetry. The plate dimen-sions are: length L ¼ 25:4 mm, width W ¼ 25:4 mm, and thicknessttot ¼ 1:45 mm. The dimensions of each Galfenol patch are: lengthL ¼ 25:4 mm, width bm ¼ 3:18 mm, and thickness tm ¼ 0:381 mm.These sheets are located a distance yo from the x–z symmetryplane. Finally, the plate is actuated using a conductive coil,4 whichgenerates sinusoidal and bias magnetic fields that are uniform andoriented along the x-axis.

Using the proposed model, we aim to maximize the composite’stip displacement by modifying: (i) the distance (along y- andz-axes) of the Galfenol sheets from center axis, (ii) the thicknessratio, and (iii) the substrate material properties. Parametric studiesare performed by separately varying each of these quantities and,in some cases, for different plate aspect ratios.

4 The coil and accompanying magnetic circuit need not be modeled due to theassumptions detailed in Section 2.

5.1. Varying Galfenol location along the y-axis

We first study the displacements at the free end of the plate asthe Galfenol sheets move away from the x–z axis along the widthdirection (i.e., as yo varies). The displacements at every point alongthe free edge are plotted for five different yo values in Fig. 7. Theresults show that the actuator generates highest displacement atthe tip’s center (x ¼ L; y ¼ 0) when both of the Galfenol sheetsare located at the x–z axis, i.e., when yo ¼ 0. As yo increases, the dis-placement at the tip’s center decreases and local maxima appear atthe x-coordinates about which the Galfenol sheets are centered.

5.2. Varying Galfenol location along the z-axis

Now, the tip displacements of an actuator consisting of a singlecentered sheet of double the width is considered (i.e., y0 ¼ 0 isfixed). In particular, the tip displacements at y ¼ 0 are calculatedas the Galfenol layer moves along the thickness (z-axis). The resultsare plotted for different plate aspect ratios a and substrate moduliEs in Fig. 8. The results demonstrate that the maximum displace-ment is obtained when the Galfenol sheet is located away fromthe neutral bending axis. Also, the displacement is more sensitiveto changes in aspect ratio and substrate modulus when theGalfenol layer is located away from the neutral axis. The displace-ment varies linearly with z-coordinate, as assumed in (6)–(8).

0 5 10 15 200

2

4

6

8

10

12

14

Galfenol/substrate thickness ratio

Tip

Dis

plac

emen

t (µm

)a = 0.8a = 1a = 1.2

(a)

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

Galfenol to substrate thickness ratio

Tip

Dis

plac

emen

t (µm

)

Es = 30 GPaEs = 50 GPaEs = 69 GPaEs = 90 GPaEs = 120 GPa

(b)

Fig. 9. Variation of tip displacement (lm) as a function of Galfenol/substrate thickness ratio for different (a) plate aspect ratios (a = width/length), and (b) substrate Young’smoduli.

744 S. Santapuri et al. / Composite Structures 132 (2015) 737–745

5.3. Varying the Galfenol to substrate thickness ratio

According to the results presented in Figs. 7 and 8, the Galfenollayer is fixed at the top of the plate with y0 ¼ 0. The tip displace-ment is now studied as a function of substrate thickness. Theresults in Fig. 9 show the existence of a critical thickness ratiobeyond which the tip displacements decrease with increasingGalfenol-to-substrate thickness ratio.

The tip displacement is maximized by simultaneously maximiz-ing the Galfenol volume fraction (up to the critical thickness ratio)and the Galfenol sheet’s offset from the neutral axis. As the sheet’sthickness increases above the critical value, the neutral axis moveswithin the sheet, thus reducing the bending displacement.Interestingly, a slight increase in the critical thickness ratio isobserved with an increase in aspect ratio and substrate modulus.At very high substrate stiffness values, the shape of the curvebecomes flatter and the shift in the peak is minimal.

The elastic modulus of the substrate is seen to have a significantimpact on the tip displacement. As the substrate’s modulusdecreases, the flexural rigidity of the beam decreases, thereby pro-viding less resistance to bending deformation for a givenmagnetostriction-induced force. This result is consistent with themodeling results of Scheidler and Dapino [27], who modeledchanges in the stiffness of Galfenol composite beams.

6. Conclusions

A two-dimensional finite element framework for compositeplates consisting of magnetostrictive laminae has been developedfor transducer applications. The model was developed to accom-modate complex geometries, nonlinear magnetostrictive materialbehavior, and mechanical dynamics. To validate the model andillustrate its usefulness, the model was applied to a prototypebending actuator composed of an aluminum plate with embeddedGalfenol sheets. Validation was performed by comparingtime-domain tip displacement measurements to simulations fordifferent actuation frequencies.

The nonlinear finite element framework was utilized to performa design study for a prototype actuator. Parametric studies wereperformed on key geometric parameters to maximize the actua-tor’s tip displacement. A critical value of the Galfenol/aluminumthickness ratio was observed, above which the tip displacementdecreases. It was also observed that the tip displacement increases

as the Galfenol sheets move away from the neutral axis.Additionally, the elastic modulus of the substrate had a significanteffect on the results of the parametric studies, where a softer mod-ulus resulted in larger displacements.

Acknowledgments

This work was supported by the NASA Aeronautics ScholarshipProgram (grant # NNX14AE24H). Additional support was providedby the Smart Vehicle Concepts Center (www.SmartVehicleCenter.org), a National Science Foundation Industry/University CooperativeResearch Center.

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