ISSN: 1537-6494 print / 1537-6532 online DOI: 10.1080 ... · ISSN: 1537-6494 print / 1537-6532 online DOI: 10.1080/15376490600777624 Numerically Efﬁcient Finite Element Formulation

Mechanics of Advanced Materials and Structures, 13:379–392, 2006Copyright c© Taylor & Francis Group, LLCISSN: 1537-6494 print / 1537-6532 onlineDOI: 10.1080/15376490600777624

Numerically Efficient Finite Element Formulationfor Modeling Active Composite Laminates

Dragan Marinkovic, Heinz Koppe, and Ulrich GabbertInstitut fur Mechanik, Otto-von-Guericke-Universitat Magdeburg, Magdeburg, Germany

Active systems have attracted a great deal of attention in thelast few decades due to the potential benefits they offer over theconventional passive systems in various applications. Dealing withactive systems requires the possibility of modeling and simulationof their behavior. The paper considers thin-walled active structureswith laminate architecture featuring fiber reinforced composite asa passive material and utilizing piezoelectric patches as both sensorand actuator components. The objective is the development of nu-merically effective finite element tool for their modeling. A 9-nodedegenerate shell element is described in the paper and the mainaspects of the application of the element are discussed through aset of numerical examples.

1. INTRODUCTIONThe science and technology have made amazing develop-

ments in the design of structures using advanced materials, suchas multifunctional fiber reinforced composites including smartmaterial components, which offer the possibility of redefiningthe concept of structures from a conventional passive elastic sys-tem to an active controllable system with inherent self-sensing,diagnosis, actuation and control capabilities [1, 2]. Such newmaterials and structural concepts also require advanced com-putational methods for modeling and design purposes. The pa-per considers modeling thin-walled smart laminate structures,the smartness of which is provided by the inherent propertyof embedded piezoelectric material to couple mechanical andelectrical fields. Dealing with active systems requires adequatemodeling tools in order to simulate their behavior. Nowadaysthe numerically efficient formulations are practically by defaultaddressed to the finite element method. Since the pioneer workof Allik and Hughes [3], various types of finite elements forthe coupled electro-mechanical field have been developed, in-cluding beam, plate, solid and shell elements [4]. Applicationof solid elements for the analysis of thin-walled structures is

Received 15 March 2006; accepted 5 April 2006.Address correspondence to Dragan Marinkovic, Institut fur

Mechanik, Otto-von-Guericke-Universitat Magdeburg, Univer-sitatsplatz 2, Magdeburg, D-39106, Germany. E-mail: [email protected]

numerically very time-consuming, especially in large-scale ap-plications of laminated structures and, therefore, attention hasturned to shell type finite elements, which can provide a satis-fying accuracy with acceptable numerical effort. Piefort [5] hasextended a layerwise facet element available in the commercialfinite element package SAMCEF (Samtech s.a.) by adding anarbitrary number of piezoelectric layers. The 4-node elementis based on the Mindlin-Reissner kinematics, has a plane formand takes the full piezoelectric coupling into account. Tzou andYe [6] have developed a triangular solid-shell element with bi-quadratic shape functions for the in-plane mechanical degreesof freedom and linear shape functions for both mechanical andelectrical degrees of freedom in the thickness direction. Gabbertet al. [7] have extended a Semi-Loof shell element to an electro-mechanical coupled element for modeling active composite lam-inates. The original mechanical Semi-Loof element was first pub-lished by Irons [8]. The formulation is based on the Kirchhoffkinematics and does not take the transverse shear strains intoaccount. Hence, it is free of shear locking effect, but the mem-brane locking acts with this element. This element has proven togive good results in modeling thin piezoelectric composite lam-inates and will be used in the present work for the comparisonpurpose. Lammering [9] has described a bilinear shell elementwith the electrical degrees of freedom for the upper and lowerpiezoelectric layer at each node. The selectively reduced inte-gration is included in the formulation. A similar approach hasbeen presented by Koegl and Bucalem [10] but the locking ef-fects were treated by means of Mixed Interpolation of TensorialComponents method. Mesecke-Rischmann [11] has formulateda shallow shell element with a quadratic distribution of electricpotential over the thickness of piezoelectric layers.

A variety of higher order lamination theories has been pro-posed during the last decade in order to improve the transverseshear stress calculation. Nevertheless, an analytical trade-off ofhigher order theories carried out by Rohwer [12] showed, that forfinite element analysis the First-order Shear Deformation The-ory (FSDT) is the best compromise between the accuracy andeffort. More recently developed are the so-called layerwise the-ories based on the polynomial distributions (zig-zag functions)of the membrane displacements. If no constraints for the zig-zag functions are introduced, the number of functional degrees

379

380 D. MARINKOVIC ET AL.

of freedom depends on the number of layers leading to a finiteelement formulation that requires a computational effort in therange of a full 3D-analysis. The number of functional degreesof freedom can be significantly reduced by a priori ensuring thecontinuity of the transverse shear stresses. Corresponding finiteelement formulations either need C1-continuity, based on mixedformulations [13] or a weak form of the Hook’s law permittingto eliminate the stress variables at the layer level [14].

This paper takes advantage of a relatively simple equivalentsingle-layer formulation. As a new feature it offers the exten-sion of the degenerate shell approach based on the kinematics ofthe FSDT to modeling multilayered composite structures (direc-tionally dependent properties) of arbitrary shape with embeddedpiezoelectric active layers polarized in the thickness direction.A similar approach was recently used by Zemcık et al. [15],who developed a bilinear (4-noded) degenerate shell elementwith the discrete shear gap (DSG) method and the enhancedassumed strain (EAS) method used to resolve the shear lock-ing and membrane locking problems, respectively. The presentwork deals with a full biquadratic (9-noded) formulation of thedegenerate shell element, which is more suitable for generallycurved structures and less prone to the mentioned locking prob-lems. The paper also investigates the possibility of alleviatinglocking problems by means of a simpler and numerically moreefficient technique—uniformly reduced integration.

2. FORMULATION OF THE ELEMENT

2.1. Coordinate Systems and Geometry of the ElementThe degenerate shell element was first developed by Ahmad

[16] from a 3D solid element by a degeneration process whichdirectly reduced the 3D field approach to a 2D one in terms ofmid-surface nodal variables. The most significant advantagesachieved in this manner are that the element is not based on theclassical shell theories and is applicable over a wide range ofthickness and curvatures.

The inherent complexity of the degenerate shell element re-quires the usage of several coordinate systems in order to de-scribe the element geometry, displacement field and to developthe strain field (Figure 1b). Besides the global (x, y, z) and thenatural (r, s, t) coordinate systems it is necessary to introducethe local-running (co-rotational) coordinate system (x′, y′, z′).

FIG. 1. Element geometry and coordinate systems.

The local coordinate system is defined so as to have one of itsaxes (say z′-axis) perpendicular to the mid-surface, while theother two axes form the tangential plane. In the case of in-planeisotropic material properties the orientation of in-plane axes maybe arbitrary. However, the kind of non-isotropy exhibited by thefiber reinforced composite laminates requires the introduction ofa structure reference direction (defined by the user), with respectto which the fiber orientation in the layers is given. In this case itis reasonable to fix the orientation of the local in-plane axes withrespect to the structure reference direction. The simplest way isoverlapping one of the axes (say x′-axis) with it (Figure 1a).

Using the full biquadratic Lagrange shape functions Ni [17],the coordinates of a mid-surface point are given by:

x

y

z

=

9∑i=1

Ni

xi

yi

zi

(1)

with xi, yi and zi denoting global coordinates of the nine nodes.The thickness of the shell is assumed to be in the direction nor-mal to the mid-surface. Denoting the unity vectors of the localcoordinate system with respect to the global coordinate systemby �eai, where a stands for x′, y′ or z′ depending on the axis andi denotes the node, the 3D shell geometry may be regeneratedfrom its mid-surface in the following way:

x

y

z

=

9∑i=1

Ni

xi

yi

zi

+

9∑i=1

hi

2Nit �ez′i (2)

where hi denotes the shell thickness at node i and −1 < t < +1.

2.2. Element Displacement and Strain FieldThe degeneration process performed by Ahmad is based on

the assumption that the thickness direction line of the shell re-mains straight after deformation but not necessarily perpendic-ular to the mid-surface (the Mindlin kinematical assumption).Therefore, the displacement of any point within the volume ofthe shell is given as a superposition of the corresponding mid-surface point displacement and a linear function of the rotations

NUMERICALLY EFFICIENT FINITE ELEMENT FORMULATION 381

about the local x′- and y′-axis through the mid-surface point, θx′

and θy′ :

u

v

w

=

9∑i=1

Ni

ui

vi

wi

+

9∑i=1

hi

2Nit [ −�ey′i�ex′i]

{θx′

θy′

}(3)

It is to be noted that the rotations are in the local coordinatesystem and upon transformation to the global coordinate systemone gets:

uvw

=

9∑i=1

Ni

ui

vi

wi

+

9∑i=1

hi

2Nit[ −�ey′i�ex′i] [T′]T

θx

θy

θz

(4)

where [T′] is the modified form of the transformation matrixrelating the local and the global coordinate systems, [T], and isgiven in the form [T′] = ��ex′ �ey′ �.

Now, due to the directionally dependent material properties,it is of crucial importance to develop the strain field in the localcoordinate system (Figure 2). This allows direct application ofthe composite laminates constitutive matrix, the so-called ABDmatrix. The advantage of having the strain field with respect tothe local coordinate system is also obvious when the piezoelec-tric coupling within the thickness polarized piezopatch using thee31 effect is considered.

The interpolations are performed in the natural coordinatesystem. Hence, the displacement derivatives with respect to thenatural coordinates are directly obtained from Eq. (4). The trans-formation of derivatives from the natural to the global coordinatesystem is achieved by means of Jacobian inverse:

[J] =

x,r y,r z,r

x,s y,s z,s

x,t y,t z,t

⇒

u,x v,x w,x

u,y v,y w,y

u,z v,z w,z

= [J]−1

u,r v,r w,r

u,s v,s w,s

u,t v,t w,t

(5)

FIG. 2. Mappings and transformations for the geometry and strain description.

where, for example, u,x = ∂u/∂x. The global derivatives areafterwards transformed to the local derivatives by means of thetransformation matrix [T] = ��ex′ �ey′ �ez′ �:

u′,x′ v′,x′ w′,x′

u′,y′ v′,y′ w′,y′

u′,z′ v′,z′ w′,z′

= [T]T

u,x v,x w,x

u,y v,y w,y

u,z v,z w,z

[T] (6)

The derivatives in Eq. (6) can be divided into a part related toglobal displacements and a part related to global rotations only.For example, observing the term u′,x ′ one obtains:

(∂u′

∂x′

)T

=9∑

i=1

[l1

(∂Ni

∂xl1ui + ∂Nr

∂xm1vi + ∂Ni

∂xn1wi

)

+ m1

(∂Ni

∂yl1ui + ∂Ni

∂ym1vi + ∂Ni

∂yn1wi

)

+ n1

(∂Ni

∂zl1ui + ∂Ni

∂zm1vi + ∂Ni

∂zn1wi

)](7)

and

(∂u′

∂x′

)R

=9∑

i=1

hi

2

[l1

(t∂Ni

∂x+ J∗

13Ni

)+ m1

(t∂Ni

∂y+ J∗

23Ni

)

+ n1

(t∂Ni

∂z+ J∗

33Ni

)]· �(n3iθyi − m3iθzi)l1

+ (l3iθzi − n3iθxi)m1 + (m3iθxi − l3iθyi)n1� (8)

where l1, m1, n1 and l3, m3, n3 are the cosine directions of theunit vectors �ex′ and �ez′ with respect to the global coordinates,respectively, i.e., �ex′ = {l1 m1 n1}T and �ez′ = {l3 m3 n3}T, thosewith subscripting i are to be calculated at the i th node, andthose without i at integration points, J∗

ab (a, b = 1, 2, 3) are theconstants of the Jacobian inverse, and:

∂Ni

∂x= J∗

11∂Ni

∂r+ J∗

12∂Ni

∂s;

∂Ni

∂y= J∗

21∂Ni

∂r+ J∗

22∂Ni

∂s;

∂Ni

∂z= J∗

31∂Ni

∂r+ J∗

32∂Ni

∂s; (9)


It is quite common within a 2D formulation to give the strainfield in the form that makes a distinction between the in-planecomponents and the out-of-plane components.

{ε′} =

εx′x′

εy′y′

γx′y′

− −γy′z′

γx ′z′

=

∂u′∂x′∂v′∂y′

∂u′∂y′ + ∂v′

∂x′

− − − − −∂v′∂z′ + ∂w′

∂y′∂u′∂z′ + ∂w′

∂x′

(10)

The formulation adopts the plane-stress state assumption, i.e.,σz′z′ = 0, and consequently, the normal transverse strain com-ponent εz′z′ is not included in Eq. (10), since the product of thementioned strain and stress components does not contribute tothe strain energy. After determining all partial derivatives in thelocal coordinate system (Eq. (6)), the discretized strain field canbe suitably given in the following form:

{ε′} =

{ε′mf}

−−{ε′

s}

=

[Bmf]

− −[Bs]

{d}

=

[BTm] t[BRlf]

− − − + − − − − − − − − −[BTs] [BR0s] + t[BRls]

{d}

= [Bu]{d} (11)

where {ε′mf} = {εx′x′εy′y′γx′y′ }T is the membrane-flexural (in-

plane) strain field, {ε′s} = {γy′z′γx′z′ }T comprises transverse

shear strains, [Bmf] and [Bs] are the corresponding strain-displacement matrices further suitably represented in terms ofthe B-matrices having “m”, “f” and “s” in the subscript depend-ing whether they contribute to the definition of the membrane,flexural or shear strains, respectively, those with subscripting“T” are related to the nodal translations and with “R” are relatedto the nodal rotations, and finally, “0” denotes constant termswhile “1” denotes linear terms with respect to the natural thick-ness coordinate t. The vector {d} comprises nodal displacements(translations and rotations).

2.3. Piezoelectric LayerThe constitutive equations of the piezoelectric material de-

pend on the choice of the independent variables. Since the aimis vibration suppression of the considered structures it is suit-able to choose the mechanical strain and the electric field [18],yielding:

{σ} = [CE]{ε} − [e]T{E}(12){D} = [e]{ε} + [dε]{E}

where {σ} is the mechanical stress in vector (Voigt) notation,{D} is the electric displacement vector, [CE] is the piezoelec-tric material Hook’s matrix at constant electric field E, [dε] isthe dielectric permittivity matrix at ε constant, and [e] is thepiezoelectric coupling matrix.

This paper considers piezoceramic elements with electrodeson the top and bottom surfaces and poled in the thickness direc-tion, where the in-plane strains are coupled with the perpendic-ularly applied electric field through the piezoelectric e31 effect.The authors of the paper have made an investigation about theaccurate description of the electric potential distribution acrossthe thickness of the piezolayer [19]. The Gauss’ law for di-electrics (no free electric charge density) given as div{D} = 0together with the here presented kinematical assumptions leadto a quadratic distribution of the electric potential. Neverthe-less, the investigation has shown a negligible difference in theobtained results when the linear and the quadratic distributionof the electric potential is accounted for. This is valid for mostof the typical piezoelectric laminate structures with relativelysmall thickness of the piezolayers in comparison to the overalllaminate thickness. Therefore, the numerically more efficientformulation with linear approximation is adopted here, thus:

E = −∂ϕ

∂z′ ⇒ Ek = −��k

hk(13)

where ϕ is the electric potential, ��k is the difference of theelectric potentials between the electrodes of the kth layer andhk is the thickness of the piezolayer. The approximation in theEq. (13) defines a diagonal electric field—electric potential ma-trix [Bφ] with typical term 1/hk on the main diagonal. The di-agonal form of the matrix [Bφ] results from the fact that thedifference of the electric potentials of a layer affects only theelectric field within the very same layer.

3. FINITE ELEMENT EQUATIONSThe governing equation of the structure’s dynamical behav-

ior is given by the Hamilton’s principle, which says that thesystem takes the path of the least action. Since it is dealt withthe piezoelectric continuum, the Lagrangian is properly adaptedin order to include the contribution from the electrical field be-sides the contribution from the mechanical field. The governingthermodynamic equation has to correspond to the chosen inde-pendent variables (the mechanical strain and the electric field,see Eq. (12)) and it is the electric enthalpy. Upon integrationby parts of the kinetic energy, the Hamilton’s principle for thepiezoelectric continuum can be given in the following developedform:

∫v[ρ{δu}T{u} + {δε}T[CE]{ε} − {δε}T[e]T{E}

−{δE}T[e]{ε} − {δE}T[dε]{E}] dV


=∫

v{δu}T{δFv} dV +

∫S1

{δu}T{δFSi

}dS1

+{δu}T{δFP} −∫

S2

δφqdS2 − δφQ (14)

where Fv, FS1 and FP are the external volume, surface (actingon surface S1) and point loads, respectively, q and Q are the sur-face electric charge (acting on surface S2) and the point electriccharges, respectively. Due to the assumption of a constant differ-ence of the electric potential over the surface of each piezoelec-tric layer, only the corresponding electrical loads are consideredin the sequel and those are the uniformly distributed surfaceelectric charges.

Performing the discretization of the structure one comes upwith the well-known semi-discrete form of the finite elementequations:

[M]{d} + [C]{d} + [Kuu]{d} + [Kuφ{φ} = {Fext} (15)

�Kφu�{d} + �Kφφ�{φ} = {Qext} (16)

with vector {φ} comprising piezolayers electrical degrees offreedom (layer-wise differences of electric potentials, assumedto be constant per single piezoelectric layer within an element)and the following matrices and vectors are introduced:

-element mass matrix:

[Mu] =∫

Ve

[Nu]Tρ[Nu] dVe (17)

-element mechanical stiffness matrix:

[Kuu] = [Kmf] + [Ks] + [Kt]

-element piezoelectric coupling matrix:

[Kuφ] =∫

Ve

([Bu]T[e]T[Bφ]) dV = [Kφu]T (18)

-element dielectric stiffness matrix:

[Kφφ] = −∫

Ve

[Bφ]T[dε][Bφ] dVe (19)

-element damping matrix (Rayleigh):

[C] = α[M] + β[Kuu] (20)

-nodal mechanical loads:

{Fext} =∫

Ve

[Nu]T{Fv} dVe +∫

S1

[Nu]T{FS1

}dS1 + [Nu]T{Fp}

(21)

-piezolayer electric charges:

{Qext} = −∫

S2

{q} dS2 (22)

where the matrix [Nu] is the element shape-functions matrix, Ve

is the volume of the element and {q} is the vector comprisingsurface electric charges for all piezoelectric layers within theelement (each component of the vector corresponding to onepiezolayer). It should be noted that Eq. (22) takes advantage ofthe assumption of uniform electric charge distribution over thesurface of the element, for each piezolayer (i.e., Nq ≡ 1).

The element stiffness matrix comprises the contribution fromthe stiffness matrix related to the membrane-flexural (in-plane)strains [Kmf], the stiffness matrix related to the shear strains[Ks] and the so-called torsional stiffness [Kt]. The former twoare determined by taking advantage of the developed strain field,hence:

[Kmf] =∫

Ve

[BTm]T

− − −−t[BR1f]T

[Cm][[BTm] t[BR1f]] dVe (23)

[Ks] =∫

Ve

[BTs]T

− − − − − − −−[BR0s]T + t[BR1s]T

[Cs][[BTs] [BR0s]

T

+ t[BR1s]] dVe (24)

where [Cm] is the part of the Hook’s matrix relating the in-plane stresses and strains, while [Cs] relates the transverse shearstresses and strains.

The element has five degrees of freedom in the local-runningc.s. (only two rotations, Eq. (3)), but due to their transformationthere are in general six degrees of freedom in the global c.s. (allthree rotations, Eq. (4)). Consequently, for a special position ofthe element, when the z′-axis of the local-running c.s. coincideswith one of the global coordinate axes, the stiffness constantcorresponding to the rotational degree of freedom around thataxis is equal to zero. The element will not be constraint enough,and any slight disturbance in the load corresponding to this de-gree of freedom could cause an erratic behavior of the element.The solution of the problem is achieved following Zienkiewiczand Taylor [20] through the introduction of additional torsionalstiffness, by defining the governing torsional strain energy, Et,that behaves as a penalty function forcing the local rotation θz′

be approximately equal to 0.5·(∂v′/∂x′−∂u′/∂y′) at integrationpoints:

Et = 1

2αnYhn

∫A

[θz′ − 1

2

(∂v′

∂x′ − ∂u′

∂y′

)]2

(r,s,o)

dA, (25)

where Y is the Young’s elasticity module and αn is a fictitiouselastic parameter (torsional coefficient) that can be determined


following the suggestions of Krishnamoorthy [21]. More detailsregarding the calculation of the torsional stiffness matrix can befound in [22].

As pointed out earlier, the formulation covers the piezoelec-tric coupling achieved between the in-plane strains and the elec-tric field in the thickness direction. Thus, only the matrix [Bmf]contributes in [Kuφ], which is therefore calculated by the fol-lowing integral:

[Kuφ] =∫

Vae

[BTm]T

− − −−t[BR1f]T

[e][Bφ] dVae (26)

where Vae denotes that the integration in Eq. (26) involves onlythe active, i.e., piezoelectric layers of the element and the di-mension of the matrix [Kuφ] is 54 × npe, where npe denotes thenumber of piezoelectric layers.

Finally, the matrix [Kφφ] has the dimension npe × npe and isof a diagonal form, because the piezolayers are electrically notcoupled and there is only one electrical degree of freedom perlayer.

Altogether the element has 54 mechanical and npe electricaldegrees of freedom.

4. NUMERICAL INTEGRATION AND ITS ORDERThe calculation of the previously given matrices and vectors

requires an integration of the respective terms. The integration inthe thickness direction is performed analytically in a layerwisemanner since the mechanical, piezoelectric and dielectric prop-erties are generally different for different layers. The advantageof differentiating between the B matrices related to the constantand linear strain terms with respect to the thickness coordinate(Eq. (11)) becomes obvious when the mentioned analytical inte-gration in the thickness direction is considered. The integrationin the in-plane directions, r- and s-direction, is to be performedby means of numerical integration procedures. Among differentprocedures for the numerical integration the Gauss quadratureformulas are very attractive from the finite element method pointof view due to their efficiency (n sampling points required forthe exact integration of the polynomial of the order (2n − 1)).

The exact integration of the matrices and vectors for the fullbiquadratic degenerate shell element requires 3 × 3 integrationrule (3 Gauss integration points for both in-plane directions).Nevertheless, the susceptibility of the degenerate shell elementto both shear and membrane locking is a well-known fact. AsPrathap [23] has pointed out, referring to the purely mechanicalsingle-layer formulation of the 9-node degenerate shell element,it “does not lock severely,” which is due to the full biquadraticshape functions. This means that the refinement of the meshalways leads to a converged solution (which might not be al-ways the case with a bilinear element), but the element suffers asuboptimal convergence. A number of researchers have made in-vestigations regarding the use of uniformly reduced integration,

selectively reduced integration, assumed and enhanced strainmethods, addition of incompatible (bubble) modes etc. All of theoffered methods are based on the experience and are not withoutcertain drawbacks. It is the inherent complexity of the elementthat inhibits a solution for the problem that would be variation-ally consistent and this represents one of the most importantdrawbacks of the mentioned methods. Prathap has argued thata quite simple, satisfactory and numerically very efficient so-lution is given by the reduced integration techniques, since itdoes not require reconstitution of either displacement nor strainfields. For a rectangular plane form of the element a selectiveintegration technique (the 2 × 3 rule for ε′

x term and the 3 × 2rule for ε′

y term) would lead to a locking free solution. However,for a general quadrilateral double-curved geometry of the ele-ment the uniformly reduced integration technique introduced byZienkiewicz et al. [24] (the 2 × 2 rule) is the optimal one. Animportant drawback of the technique is that it might cause theunwanted zero-energy (“hour-glass”) mechanisms. Except for afew cases where they act up, the uniformly reduced 9-node de-generate shell element represents a general shell element accept-ably free of shear and membrane locking that can be gainfullyused for the analysis of doubly curved “passive” as well as piezo-electric active laminate structures. This will be demonstrated ona few examples covering both purely mechanical cases and thecases involving the piezoelectric coupling.

5. NUMERICAL EXAMPLESLiterature provides a great number of examples successfully

demonstrating the application of the reduced integration to re-lieve the locking effects for purely mechanical field. The aimhere is to show that the same technique may be just as wellused for the coupled electro-mechanical formulation given pre-viously. A special emphasis is put on the simplicity and highnumerical efficiency as an advantage of this method.

The examples are calculated by means of COSAR (www.femcos.de), a general purpose finite element package devel-oped in cooperation between the Institute of Mechanics of the

TABLE 1Material properties of the layers in the principal

material directions

Material propertiesPZT G1195

piezoceramic layerT300/976

graphite/epoxy

Elastic propertiesE11 (GPa) 63 150E22 (GPa) 63 9ν12 0.3 0.3G12 (GPa) 24.2 7.1

Piezoelectric propertiese31 (10−5 C/mm2) 2.286 0e32 (10−5 C/mm2) 2.286 0


FIG. 3. Shape control of a simply supported composite plate.

Otto-von-Guericke University of Magdeburg and the engineer-ing company FEMCOS mbH. Two different shell type elementsare compared—the here presented piezoelectric Shell9 elementand the previously developed and in praxis already proven tobe accurate [7, 25–27] piezoelectric 8-node Semi-Loof element.The obtained results are compared to each other and to solutionsfrom other authors, for the cases where they are available.

All the below given examples consider composite structureswith two types of layers—the graphite/epoxy layer and the PZTlayer. The properties of the layers are given in Table 1, which isin full (except for thermo-elastic properties) taken from the ref-erence [28]. Not specified piezoelectric constants are consideredto be equal to zero. The paper is focused on the actuator functionof the piezoelectric layers, hence the dielectric constants are notneeded and are not given in the table. It should be noted that thesequence and the thickness of layers differs in the consideredexamples and will be specified separately.

5.1. Shape Control of Adaptive Composite PlateThe first example demonstrates the shape control of structures

by means of piezoelectric actuation. A simply supported plate(a × b = 254 × 254 mm) has a cross-ply sequence of graphite

FIG. 4. Deformed shape of a composite plate under uniform pressure and piezoactuation.

epoxy [0/90/0]s and two piezolayers bonded to the top and bot-tom surfaces (material properties summarized in Table 1). It isinitially subjected to a uniformly distributed load of 200 Nm−2.The thickness of each graphite/epoxy layer is 0.138 mm, and ofthe piezolayer 0.254 mm. The piezolayers are oppositely polar-ized and a constant voltage is supplied to both layers resulting inbending moments which tend to recover the initial flat shape ofthe plate. The voltage giving the shape that corresponds mostlyto the initial undeformed shape is found. This example is orig-inally proposed by Kioua and Mirza [28] whose approximatesolution is obtained using the conventional Ritz analysis basedon the shallow-shell theory.

The structure was discretized by an 8 × 8 finite element meshand the normalized centerline deflection (deflection/plate side)was calculated for the same voltages as given in the previousreference, i.e., 0 V (initial deformed shape under distributedload), 15 V and 27 V. Only the full integration of the elementswas used in this case, since no locking effects were excited.Figure 3 shows the results obtained with the Shell9 element andthe 8-node Semi-Loof element (solid lines, nearly congruent dueto the high agreement of the results) and the results from Kiouaand Mirza (dashed lines).


FIG. 5. Cantilevered composite shell—geometry and finite element meshes.

According to the Ritz solution, the structure recovers the flatshape for the last applied voltage (Figure 3, 27 V). However,the finite element results show that the structure subjected to thevoltage of 27 V (and the uniform load) is not exactly flat, al-though very close to it (Figures 3 and 4b). It should be noted thatthe action of the moments uniformly distributed over the plateedges, such as those obtained by the actuation of the piezolayers,certainly cannot recover the original (unloaded) flat geometry inthis case. This points out the deficiency of the assumed shapefunctions within the Ritz solution, which are obviously not of ahigh enough order (only up to the 2nd order polynomials used,while at least the 4th order polynomials would be required inthis case). Actually, the Ritz solution represents the best fit ofthe actual solution yielding a flat line in the case of the aboveconsidered plate for the voltage of 27 V.

The here considered example was also calculated by a num-ber of authors. The here presented results were compared withthe results published by Lee et al. [29] and Lim et al. [30].The curves of the graphical representations published in bothpapers coincide with the results in Figure 3 in the limit of ac-curacy of the pictures. Therefore these curves are not includedin Figure 3. Lee et al. [29] used their 9-node assumed strainshell element and also an 8 × 8 mesh. Lim at al. [30] applied apiezoelectric 18-node assumed strain solid element and, addi-tionally, the NASTRAN HEXA8 element with the piezoelectricstrain equivalently replaced by thermally induced strain (thermalanalogy).

TABLE 2Transverse deflection of the characteristic points of the composite shell subjected to a uniformly distributed edge force

q (R = 10 × b)

Edge Force q = 2.0 10−2 N/mm

(w3/b) × 10−3 (w2/b) × 10−3 (w1/b) × 10−3

Meshb = 254 mm Shell9 Shell9 Semi-Loof Shell9 Shell9 Semi-Loof Shell9 Shell9 Semi-Loof

Int. rule 3 × 3 2 × 2 3 × 3 3 × 3 2 × 2 3 × 3 3 × 3 2 × 2 3 × 31 × 2 −4,894 −6,957 −6,319 −5,154 −6,740 −5,484 −5,780 −7,945 −5,5562×4 −6,095 −6,598 −6,520 −6,110 −6,512 −6,232 −7,268 −7,701 −7,2604×4 −6,386 −6,594 −6,528 −6,370 −6,506 −6,378 −7,461 −7,697 −7,4928×8 −6,520 −6,571 −6,535 −6,453 −6,486 −6,457 −7,618 −7,671 −7,638

5.2. Cantilevered Curved Composite ShellThe second example considers a cylindrical composite shell

of dimensions a × b = 254 × 254 mm (Figure 5). The radiusof the shell will vary throughout the considered cases. Materialof the shell is defined by the following stacking sequence ofthe graphite/epoxy [302/0]s and two PZT layers are bonded tothe top and bottom surfaces. The structure reference direction istaken to be the global x-axis. The thickness of the graphite/epoxylayers is 0.138 mm and of the PZT layers is 0.254 mm.

At first this structure is subjected to a uniformly distributedtransverse force over the free edge of the shell (Figure 5). ThePZT layers are short-circuited so that a zero voltage is im-posed and the layers act only as passive layers. The radius istaken to be R = 10 × b. Due to the unbalanced stacking se-quence of the shell, the load causes both bending and twisting ofthe shell. Therefore, the transverse deflection is observed at threecharacteristic points, two end-points and the mid-point of thefree edge, denoted in the Figure 5 as 1, 3 and 2, respectively.

The structure is discretized by 4 different meshes given inFigure 5 and the results for the above mentioned characteristicpoints are summarized in Table 2. The Shell9 element is usedwith both full and reduced integrations, while for the applica-tion of the Semi-Loof element the full integration is availableonly. Due to the large span to thickness ratio of approximately190, the locking effects are obvious when a rough mesh is usedwith exactly integrated Shell9 element. The diagram in Figure 6shows the results for the shell free edge transverse deflection


FIG. 6. Cantilever composite shell under uniform edge force—free edge transverse deflection, convergence of the fully (FI) and reduced (RI) integrated Shell9element.

obtained with the fully and reduced integrated Shell9 element.As the diagram and Table 2 show, the refinement of the mesh withthe fully integrated Shell9 element (dotted lines on the diagram,in the legend denoted with FI) leads to a converged solution. Onthe other hand, a much better convergence rate may be noticedwith the uniformly reduced Shell9 element (solid lines on thediagram, in the legend denoted with RI), pointing out that it isfree of locking effects. It should also be noticed that the conver-gence is monotonic, but the results of the fully integrated elementconverge from below, while those from the reduced integratedelement from above.

In the second case the same structure under the influence ofthe electrical load is observed. The two piezolayers are oppo-sitely polarized and subjected to the same voltage of 100 V. This

TABLE 3Transverse deflection of the characteristic points of the active composite shell subjected to electric voltage ϕ(R = 10 × b)

Electric Potential ϕ = 100 V

(w3/b) × 10−3 (w2/b) × 10−3 (w1/b) × 10−3

Meshb = 254 mm Shell9 Shell9 Semi-Loof Shell9 Shell9 Semi-Loof Shell9 Shell9 Semi-Loof

Int. rule 3 × 3 2 × 2 3 × 3 3 × 3 2 × 2 3 × 3 3 × 3 2 × 2 3 × 31 × 2 −6,394 −6,299 −5,682 −4,241 −2,775 −3,380 −8,464 −10,385 −7,2872 × 4 −6,840 −6,431 −6,163 −3,675 −3,359 −3,180 −10,439 −10,741 −10,1934 × 4 −6,507 −6,423 −6,181 −3,603 −3,354 −3,343 −10,514 −10,762 −10,3948 × 8 −6,450 −6,397 −6,347 −3,420 −3,367 −3,354 −10,732 −10,730 −10,686

results in bending moments uniformly distributed over the shelledges. This case is also proposed by Kioua and Mirza [28] andsolved by already mentioned conventional Ritz analysis basedon the shallow-shell theory, whereby variable radius of the shellwas considered. The results obtained with the Shell9 and theSemi-Loof element for the cylindrical shell with R = 10 × b arecompared in Table 3, which has the same structure as Table 2.

As in the previous case, the diagrams in Figures 7 and 8 givethe results for the shell free edge transverse deflection obtainedwith the fully and reduced integrated Shell9 element, respec-tively. The faster convergence rate of the reduced integratedShell9 element is obvious once again and except for the coarsestmesh, the results are nearly congruent in Figure 8. It should alsobe noticed that the convergence progresses in a different manner


FIG. 7. Cantilever composite shell under piezoelectric excitation—free edge transverse deflection, convergence of the fully integrated Shell9 element.

in comparison to the case of the purely mechanical field. Ac-tually, it is non-monotonic. This is due to the loads originatingfrom the piezoelectric coupling effect. The refinement of themesh does not affect the mechanical stiffness matrix only, butalso the piezoelectric stiffness matrix, and, in further instance,the mechanical loads originating from the piezoelectric cou-pling. Furthermore, the description of the piezoelectric coupling

FIG. 8. Cantilever composite shell under piezoelectric excitation—free edge transverse deflection, convergence of the reduced integrated Shell9 element.

is directly affected by the locking effects and their alleviatingby means of the reduced integration technique since it couplesa part of the strain field affected by the locking effects to theelectric field.

The diagram in Figure 9 compares the solution from theShell9 and Semi-Loof element obtained by the 8 × 8 mesh (fullintegration) to the above mentioned approximate Ritz solution


FIG. 9. Bending and twisting of active cantilevered composite shell.

FIG. 10. Simply supported composite cylindrical arch subjected to uniformly distributed vertical force.

FIG. 11. Radial deflection of the composite arch, force excitation—convergence of the Shell9 element.


FIG. 12. Radial deflection of the composite arch, force excitation—convergence of the Semi-Loof element.

obtained by the shallow shell assumptions [28]. The indicatorof bending deformation is taken similarly to the previous case,i.e., the ratio (w2/b) is observed, while the ratio (w3 − w1)/b istaken as an indicator of twisting. The solutions from both ele-ments show very high agreement. Also, a good agreement withthe Ritz solution can be noticed, especially for higher valuesof radius to span ratio, which is expected. As this ratio takeslower values, the deformation of the shell becomes more com-plex and the previously mentioned deficiency of the assumedshape functions of the Ritz solution becomes more pronounced

FIG. 13. Radial deflection of the piezoelectric active composite arch—convergence of the Shell9 element.

resulting in higher discrepancy between the finite element andRitz results observable in the Figure 9. It should also be notedthat the shallow shell assumptions have higher (negative) impacton the accuracy of the Ritz solution in the area of smaller radii,which is, however, not expected to be crucial for the in this caseconsidered radius to span ratios.

5.3. Simply Supported Composite Cylindrical ArchIn the third example a simply supported cylindrical arch

of radius R = 100 mm and width b = 60 mm is considered.


The stacking sequence of the composite is now [45/–45/0]s

and two PZT layers are also added to the top and bottom sur-faces. The thickness of the graphite/epoxy is 0.12 mm and ofthe PZT layer 0.24 mm. It should be noted that the stackingsequence is now balanced and results in orthotropic materialproperties.

Following the same pattern from the first example, the struc-ture is at first subjected to the vertical force uniformly dis-tributed over the width line according to Figure 10, while thepiezoelectric layers are short-circuited imposing zero electricvoltage.

The structure is discretized by 3 different meshes. Each meshhas 4 elements in the axial direction (width), but the number ofelements in the circumferential direction varies from 10, 20 to40 elements. The calculated radial deflection of the width mid-line is presented in Figures 11 and 12. Again, the Shell9 elementin combination with the reduced integration technique yieldssatisfactory results already with a very rough mesh (Figure 11).The refinement of the mesh with fully integrated Shell9 elementleads to the converged solution, but a much larger number ofelements is needed for a satisfying accuracy.

The conclusions from the purely mechanical case extend aswell to the case when the considered cylindrical arch is excitedby the actuation of the piezolayers. The oppositely polarizedlayers are subjected to the voltage of 100 V. As can be seen inFigure 13, already the mesh with 10 uniformly reduced Shell9elements in the circumferential direction yields very accurateresults. Fully integrated element needed a mesh with 40 elementsin the circumferential direction to provide the same accuracy. Asimilar behavior is observed with the fully integrated Semi-Loofelement in the Figure 14.

FIG. 14. Radial deflection of the piezoelectric active composite arch—convergence of the Semi-Loof element.

6. CONCLUSIONSDevelopment of active structures requires adequate modeling

tools in order to simulate and analyze their behavior. Such amodeling tool should provide a satisfying accuracy, but at thesame time the numerical effort is supposed to be acceptable—the two requirements that are not easy to conciliate. The presentpaper tried to achieve that by extending the degenerate shellapproach to modeling thin-walled piezoelectric active compositelaminates.

The originally formulated 9-node degenerate shell element,although numerically more efficient than a 3D solid element,still exhibits numerical problems recognized as locking effects.It can be argued that the element is reliable since the lockingis not severe, meaning that the refinement of the mesh leads toa converged solution but at a suboptimal rate. Resolving thisproblem in a variationally consistent manner still remains anopen and attractive task. Among several empirical solutions de-veloped for the “passive” structures, the authors have chosen theuniformly reduced integration technique as a numerically veryefficient one. The use of this technique is successfully demon-strated through several numerical tests, which prove that a sat-isfying accuracy may be achieved with a relatively rough meshand by using only 4 (2 × 2 integration rule) instead of 9 integra-tion points (3 × 3 integration rule). A multiple reduction of thenumerical effort is obvious, which offers a good basis to han-dle large-scale problems, geometrically nonlinear analysis, etc.The most important drawback of the method, although only inrare cases, is the possible appearance of the zero-energy modes.The possible solution to this problem could be an hour-glasscontrol [31]. The presented Shell9 element has noteworthy ad-vantages in geometrically non-linear applications and also to


include a material non-linear behavior. These extensions of theelement are under progress.

ACKNOWLEDGEMENTSThis work has been partially supported by the postgraduate

program of the German Federal State of Saxony-Anhalt. Thissupport is gratefully acknowledged.

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ISSN: 1537-6494 print / 1537-6532 online DOI: 10.1080 ... · ISSN: 1537-6494 print / 1537-6532 online DOI: 10.1080/15376490600777624 Numerically Efﬁcient Finite Element Formulation