Generalized layerwise mechanics for the static and modal response of delaminated composite beams with active piezoelectric sensors

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International Journal of Solids and Structures 44 (2007) 8751–8768 www.elsevier.com/locate/ijsolstr

Generalized layerwise mechanics for the static and modal response of delaminated composite beams with active piezoelectric sensors Nikolaos A. Chrysochoidis, Dimitris A. Saravanos

*

Department of Mechanical Engineering and Aeronautics, University of Patras, Patras GR26500, Greece Received 16 October 2006; received in revised form 14 June 2007 Available online 3 August 2007

Abstract A coupled linear layerwise laminate theory and a beam FE are formulated for analyzing delaminated composite beams with piezoactuators and sensors. The model assumes zig-zag fields for the axial displacements and the electric potential and it treats the discontinuities in the displacement fields due to the delaminations as additional degrees of freedom. The formulation naturally includes the excitation of piezoelectric actuators, their interactions with the composite laminate, and the effect of delamination on the predicted sensory voltage. The quasistatic and modal response of laminated composite Gr/ Epoxy beams with active or sensory layers having various delamination sizes is predicted. The numerical results illustrate the strong effect of delamination on the sensor voltage, on through the thickness displacement and on the stress fields. Finally, the effect of delamination on modal frequencies and shapes are predicted and compared with previously obtained experimental results.  2007 Elsevier Ltd. All rights reserved. Keywords: Composite materials; Laminate; Delamination; Modelling; Finite element; Beams; Piezoelectrics

1. Introduction The development of active structural health monitoring systems and techniques using piezoelectric actuator and sensor wafers or films is an area experiencing significant technical activity. Particular emphasis is placed on the active structural health monitoring methods for composite structures, as the likelihood of internal defects, and the evolution of invisible damage in the composite material during service life remains high. One candidate type of damage is delamination cracks, which are usually induced during low-velocity impact and fatigue, remain hidden and can propagate quickly leading to catastrophic failure. Among the many open issues in this field is the development of analytical and numerical models capable of capturing the effect of delaminations on the structural response, particularly on the electromechanical response components associated with the piezoelectric actuators and sensors, as such models may help understand the sensitivity of *

Corresponding author. Tel.: +30 2610996191; fax: +30 2610997234. E-mail address: [email protected] (D.A. Saravanos).

0020-7683/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2007.07.004

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N.A. Chrysochoidis, D.A. Saravanos / International Journal of Solids and Structures 44 (2007) 8751–8768

electromechanical response on damage parameters, and thus provide a basis for the development of damage detection and localization techniques and the design of the smart composite system. To this end, this paper presents a layerwise mechanics theory and a finite element for analyzing laminated beams with delamination cracks and active piezoelectric sensors. Early work in the area was focused on the effects of a single delamination on the natural frequencies of composite beams (Tracy and Pardoen, 1989; Nagesh Babu and Hanagud, 1990; Paolozzi and Peroni, 1990; Shen and Grady, 1992) and plates (Tenek et al., 1993; Rinderknecht and Kroplin, 1995) using experimental results and simple analytical beam and plate models of the so-called ‘‘four-region’’ approach. Keilers and Chang (1995) presented experimental work for identifying a delamination in composite beams using built-in piezoelectric sensors. A generalized composite beam model for predicting the effect of multiple delaminations on the modal damping and modal frequencies of composite laminates was reported together with experimental studies (Saravanos and Hopkins, 1996), which treated the opening and sliding at the delamination interfaces as additional degrees of freedom. Barbero and Reddy (1991) used a layerwise laminate theory to describe plates with multiple delamination cracks between the layers. Di Sciuva and Librescu (2000) presented a geometrically non-linear theory of multilayered composite plates and shells with damaged interfaces, Luo and Hanagud (2000) presented a model for describing the static, modal and dynamic response of delaminated beams using a nonlinear approach based on piecewise linear spring models between the delaminated sublaminates; Thornburgh and Chattopadhyay (2001) used a higher-order theory to model matrix cracking and delamination in laminated composite structures; Hu et al. (2002) analyzed the vibration response of delaminated composite beams using a higher-order finite element with a C0 type finite element materials; and Diaz Valdes and Soutis (1999) presented experimental work on the detection of delaminations in composite beams using piezoelectric actuator and sensor. Chrysochoidis and Saravanos (2004) investigated mainly experimentally effects of delamination on the modal parameters and frequency response functions of artificially delaminated composite beams obtained via attached piezoceramic actuators and sensors pairs and demonstrated the potential of the latter compared to other traditional sensors. Chattopadhyay et al. (1999) studied the dynamics of delaminated composite plates with piezoelectric actuators using a third-order shear theory and more recently (Chattopadhyay et al., 2004) reported a refined layerwise theory for the prediction of the vibration of delaminated plates considering the crack faces interface contact using a system of nonlinear springs. The previous studies have shown that vibration methods based on lower vibrational modes are insensitive to detecting small size delaminations, hence, pointing to the excitation and monitoring of more localized response and higher frequency modes. The later also suggests the development of analytical and numerical models, which can capture local effects of delaminations. In the present paper, an electromechanically coupled layerwise theory is described for composite beams with piezoelectric actuators and sensors, treating interfacial sliding, crack opening and slope discontinuity across a delamination crack, as additional degrees of freedom. The generalized stiffness, mass piezoelectric, and permittivity matrices are formulated and a 2-node finite element is further developed. The new finite element capabilities are evaluated by predicting the effect of a single delamination on the modal and quasistatic response of composite beams with passive or active piezoelectric layers. The analytical predictions of the modal frequencies are further correlated with available measurements. Also the delamination mode shapes are presented showing the delamination ‘‘breathing’’. Through these model predictions, some mechanisms capable of revealing the delamination presence are further discussed with an eye toward active delamination detection. 2. Theoretical formulation This section presents the formulated mechanics for piezoelectric laminated beams with interlaminar delamination cracks. 2.1. Governing material equations Each ply is assumed to exhibit linear piezoelectric behavior. In the current beam case, in-plane and interlaminar shear strains are considered in the elastic field. The piezoelectric materials considered are monoclinic class two crystals with the poling direction coincident with the z-axis. The constitutive equations have the form

N.A. Chrysochoidis, D.A. Saravanos / International Journal of Solids and Structures 44 (2007) 8751–8768

 

r1 r5 D1 D3

"

 ¼ 

 ¼

#       S1 0 e31 E1 0    E S5 e15 0 E3 C 55        S e15 E1 S1 0 e þ 11 S   0 S5 E3 0 e33

C E11 0 0 e31

8753

ð1Þ

r1, r5 and S1, S5 are the axial and shear mechanical stresses and strains, respectively; E1 and E3 are the electric field vectors; D1 and D3 are the electric displacement vectors; C11 and C55 are the elastic stiffness coefficients; e15 and e31 are the piezoelectric coefficients; and e11, e33 are the electric permittivity coefficients of the material. Superscripts E and S indicate constant electric field and strain conditions, respectively. The above system of equations encompasses the behavior of a beam’s piezocomposite layer. The electric field vector is the negative gradient of the electric potential u Ek ¼ ou=oxk ;

k ¼ 1; 3

ð2Þ

2.2. Equations of equilibrium The variational statement of the equations of equilibrium for the piezoelectric structure is Z Z I I  dC ¼ 0 dH dA þ d uðq € uÞdA þ d uTs dC þ d uD A

A





Cs



ð3Þ

CD

 are, where A is the x–z surface of the beam; HL is the electric enthalpy of the piezoelectric laminate; s and D u respectively, the surface fractions and charge, acting on the boundary surface Cs, CD; u ¼ f g is the displacew  ment vector. The electric enthalpy includes the components of elastic strain energy and electric energy. Thus, the electric enthalpy in the mth layer is expressed in variational form as Z zmþ1   dHl ¼ dST ½C S dET ½e S dET ½e E dz ð4Þ zm













The vectors and matrices appearing in the previous equation imply the respective vectors and matrices appearing in constitutive equations (1). zm and zm+1 indicate bottom and top surface location of the layer, respectively.

2.3. Kinematic assumptions Kinematic hypotheses of a layerwise theory (Saravanos and Heyliger, 1995) are adopted, admitting piecewise linear (zig-zag) fields through the thickness. A typical laminate is assumed to be subdivided into N discrete layers, where each discrete-layer may contain a single ply, a sub-laminate, or a sub-ply. Linear fields are assumed in each discrete layer for the in-plane and the electric potential field through the laminate thickness. Multiple delamination cracks are also considered at the interfaces of two adjacent plies. In the later case, additional terms are included in the displacement field to represent the interlaminar discontinuities in displacement fields due to delaminations. The resultant zig-zag field maintains continuity across the discrete layer boundaries, yet, allows for different in-plane displacement slopes within each discrete layer, and admit sliding and opening across a delamination crack (see Fig. 1). The displacement field for the laminate within a discrete layer, taking into account Nd delaminations, takes the form:

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N.A. Chrysochoidis, D.A. Saravanos / International Journal of Solids and Structures 44 (2007) 8751–8768 LAMINATE

k-th delamination

Electric potential

Displacements uN

φ j j-1 φ

zj uj

u~ k u1

(a) Smart Laminate

W o

φN φN-1

~ k w

φ2 φ1

(b) Discrete-layer laminate theory assumptions

Fig. 1. Illustration of the assumed displacement and electrical fields through the thickness of a delaminated composite laminate.

uðx; zÞ ¼

N X 

Nd X  k ~ ui ðxÞwi ðzÞ þ u ðxÞwk ðzk ÞH ðz  zk Þ

w ¼ wo ðxÞ þ

ð5Þ

k¼1

i¼1 Nd X

~ k H ðz  zk Þ w

ð6Þ

k¼1

uðx; zÞ ¼

N X 

Ui ðxÞwi ðzÞ



ð7Þ

i¼1

where superscripts i = 1, . . . , N indicate the discrete layers, k = 1, . . . , Nd the number of the delaminations through the thickness, and o the mid-plane. In (5), terms ui, ui+1 are the in-plane displacements at the inter~ k are faces of each discrete layer. Both effectively describe the extension and rotation of the layer. Also ~uk and w new degrees of freedom describing the sliding and opening across the faces of the kth delamination. H(z) is Heaviside’s step function and zk is the distance of the kth delamination from the mid-plane. Finally, wi and wk are the linear interpolation functions through the laminate thickness for the ith layer and the kth delamination, given8by: i1 > < zi  z i1 ; z 6 zi i z  z ð8Þ w ðzÞ ¼ iþ1 > : ziþ1  zi ; z > zi z z kþ1 z z wk ðzÞ ¼ kþ1 ð9Þ z  zk 2.4. Strain–displacement relations and electrical field In the context of kinematic assumptions (5, 6), the axial and shear strains S1 and S5 of a laminate consisting of N layers and Nd delamination cracks take the form: Nd N X X S1 ¼ S i1 wi þ ð10Þ S~k1 wk H ðz  zk Þ k¼1

i¼1

S 5 ¼ wo;x þ

N X

ui wi;z þ

Nd h i X ð~ wk;x þ ~ uk wk;z ÞH ðz  zk Þ þ ~uk wk ^dðz  zk Þ

ð11Þ

k¼1

i¼1

where ^ dðzÞ is the Dirac impulse function. In the axial strain equation (10), S i1 ¼ ui;x represents generalized axial strains of a healthy laminate, whereas S~k1 ¼ ~ uk ;x is a new strain component expressing the effect of the kth delamination on the axial strain above the crack. In the shear strain equation (11), the generalized strain term S i5 ¼ wo;x þ

N X i¼1

ui wi;z

ð12Þ

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yields a constant shear strain term through the respective discrete layer, the term ~ k;x þ ~ uk wk;zk S~k5 ¼ w

ð13Þ

represents a constant shear strain added to all layers above the crack due to the presence of the kth delamination, and the last generalized strain term S^k5 ¼ ~ uk

ð14Þ

represents the relative sliding at delamination faces. Finally, combining Eqs. (2) and (7) the electric field is derived Ei3 ¼ 

N   X Ui wi;z

ð15Þ

i¼1

The above equation yields a constant approximation of the electric field in each discrete layer. The previous strain and electrical field equations were included into the variational form of equilibrium equations, and the equivalent stiffness and mass matrices were derived. 2.5. Laminate electric enthalpy and laminate matrices Assuming the limitations of this work and combining Eqs. (4), (10), (11), and (15), the electric enthalpy of the laminate in each discrete layer i takes the form i

dH ¼

N ip Z X k¼1

zkþ1

  dS ki C kij S kj  dEk3 ek31 S k1  dS k1 ek31 Ek3  dEk3 ek33 Ek3 dz

ð16Þ

zk

where i,j = 1,5 and k denotes the kth ply in the ith discrete layer; N ip denotes the plies of the ith discrete layer. I was further assumed, Because in most cases continuous electrodes exist on the surface of a piezoceramic layer or wafer reinforcing a uniform axial voltage distribution, the axial component of the electrical field in Eq. (2) will be negligible and was ignored. Moreover, interfacial contact and friction between the delamination faces were neglected. Thus, the corresponding terms in the variation of electric enthalpy equation (16) were assumed to be negligible. Combining Eqs. (10), (11), (15) and (16), integrating through the thickness of each discrete layer and collecting the common terms, the variation of laminate enthalpy is related to the resultant laminate stiffness, piezoelectric and permittivity matrices as dHL ¼ dw;x A55 w;x þ

N X  m¼1

þ

Nd  X

N X N   X n m mn n dw;x Bn55 un þ dum Bm55 w;x þ dum;x Dmn 11 u;x þ du D55 u m¼1 n¼1

k k w  k w;x þ d~uk B  k uk þ d~  k w;x dw;x A wk;x A 55 ~ ;x þ dw;x B55 ~ 55 55



k¼1

þ

Nd  N X X

k m  mk k m  mk k m  mk ~  mk um þ d~  mk um u þ d~uk;x D dum;x D wk;x F mk uk D 11 u;x þ du F 55 w;x þ du D55 ~ 11 ;x 55 u þ d~ 55



ð17Þ

m¼1 k¼1

þ

Nd X Nd  X

k2 ~ k1 k2 w ~ k111 k2 ~ ~ k551 k2 ~uk2 þ d~uk1 B ~ k551 k2 w ~ k551 k2 ~uk2 ~ k;x2 þ d~uk1 D d~ uk;x1 D wk;x1 A wk;x1 B uk;x2 þ d~ 55 ~ ;x þ d~



k 1 ¼1 k 2 ¼1

þ

N X N  X m¼1 n¼1

Nd  N X  X  n m mn n dUm P mn dUm P mk uk;x þ 31 u;x þ dU G33 U 31 ~ m¼1 k¼1

In the previous equation Dmn 11 is the in-plane generalized laminate matrix given by Z N p zlþ1 X Dmn C l11 wm ðzÞwn ðzÞdz 11 ¼ l¼1

zl

ð18Þ

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N.A. Chrysochoidis, D.A. Saravanos / International Journal of Solids and Structures 44 (2007) 8751–8768

A55, Bm55 and Dmn 55 are the interlaminar generalized laminate matrices defined as N p Z zlþ1 X C l55 dz A55 ¼ l¼1

Bm55 ¼ Dmn 55 ¼

Np Z X

zl zlþ1

zl l¼1 N p Z zlþ1 X

C l55 wm;z ðzÞdz

ð19Þ

C l55 wm;z ðzÞwn;z ðzÞdz

zl

l¼1

 mk is the generalized in plane laminate matrix D 11 N p Z zlþ1 X Dmk ¼ C l11 wm ðzÞwk ðzk ÞH ðz  zk Þdz 11

ð20Þ

zl

l¼1

k ; B  k ; F mk and D  mk are the generalized shear laminate matrices defined as and A 55 55 55 55 Z N p z lþ1 X Ak55 ¼ C l55 H ðz  zk Þdz l¼1

Bk55 ¼ F mk 55 Dmk 55

¼ ¼

Np Z X l¼1 Np X l¼1 Np X

zl zlþ1

C l55 wk;z ðzÞH ðz  zk Þdz

zl

Z

zlþ1

 zk Þdz

zl

Z

l¼1

ð21Þ C l55 wm;z ðzÞH ðz

zlþ1

C l55 wm;z ðzÞwk;z ðzÞH ðz  zk Þdz

zl

Each one of the overbar terms represents stiffness coupling between pristine and delaminated degrees of freedom. ~ k1 k2 ; B ~ k111 k2 ,A ~ k551 k2 and D ~ k551 k2 are laminate matrices describing additional stiffness terms introduced Continuing, D 55 by the delamination degrees of freedom N p Z zlþ1 X e k111 k2 ¼ D C l11 wk1 ðzÞwk2 ðzÞH ðz  zk2 Þdz zl

l¼1

e k1 k2 ¼ A 55

Np X

Z

e k551 k2 ¼ B

Z

zlþ1

ð22Þ C l55 wk;z1 ðzÞH ðz

 zk1 ÞH ðz  zk2 Þdz

zl

l¼1

e k551 k2 ¼ D

C l55 H ðz  zk1 ÞH ðz  zk2 Þdz

zl

l¼1 Np X

zlþ1

Np Z X

zlþ1

C l55 wk;z1 ðzÞwk;z2 ðzÞH ðz  zk1 ÞH ðz  zk2 Þdz

zl

l¼1

P mn 31

Similarly, is the generalized piezoelectric laminate matrix describing effective piezoelectric coefficients of the pristine laminate N p Z zlþ1 X ¼ el31 wm;z ðzÞwn ðzÞdz ð23Þ P mn 31 l¼1

zl

and P mk 31 represents the effect of delamination on the effective piezoelectric coefficients of the laminate. N p Z zlþ1 X mk P 31 ¼ el31 wm;z ðzÞwk ðzk ÞH ðz  zk Þdz ð24Þ l¼1

zl

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8757

Finally, Gmn 33 is the generalized permittivity laminate matrix given by Gmn 33

¼

Np Z X

zlþ1

el33 wm;z ðzÞwn;z ðzÞdz

ð25Þ

zl

l¼1

where Np is the number of plies in the laminate. 2.6. Finite element formulation A beam finite element was formulated based on the previous laminate mechanics, implementing C1 continuous Hermitian polynomials Hi(x) for the local approximation of the transverse displacements and C0 shape functions Ni(x) for the remaining axial and electric DOFs. In this manner, the local approximation of the generalized state variables in the element take the form L X

ui ðx; tÞ ¼

uim ðtÞN m ðxÞ

m¼1

~ uk ðx; tÞ ¼

L X

~ ukm ðtÞN m ðxÞ

m¼1

 Le om w;x ðtÞH m2 ðxÞ 2 m¼1

 L X Le km ~ k ðx; tÞ ¼ ~ km ðtÞH m1 ðxÞ þ w ~ ;x ðtÞH m2 ðxÞ w w 2 m¼1 wo ðx; tÞ ¼

Uk ðxÞ ¼

L

X

L X

wom ðtÞH m1 ðxÞ þ

ð26Þ

Uki N i ðxÞ

i¼1

where i = 1, . . . , N + 1 and k = 1, . . . , Nd indicate discrete layer and delamination; m = 1, . . . , L the node number; Le denotes the element length. A 2-node (L = 2) beam element was further developed and encoded in a prototype research code employing linear interpolation functions Ni and cubic Hermitian polynomials H i1 , H i2 . Substituting Eqs. (17) and (26) into the governing equations of equilibrium (3), then collecting the common coefficients, the coupled piezoelectric system was expressed in the standard discrete matrix form 9 8 2 h i3 h i FF FA A >    >   = < ½  K u K P  K f g f g uu uu 7 uu ½M uu  0 ug f€ fug 6 h i h i h i ð27Þ þ ¼ 4 5

> A > €Sg 0 0 fu fuF g ; : QF  K FA K FF K FF uu uu uu fu g Submatrices [Kuu], [Kuu] and [Kuu] indicate the elastic, piezoelectric and permittivity matrices of the structure; [Muu] indicates the mass matrix. It is worth pointing out that the previous system matrices implicitly introduce the delamination parameters into the system response. The piezoelectric and capacitance matrices have been partitioned into active and sensory parts. Superscripts F and A indicate, respectively, sensory (free) and active (applied) electric potential components; {P} is the applied mechanical forces vector and {QF} is the applied electric charge at the sensors. Solution of the above system yields the coupled electrostatic response of the piezoelectric beam structure in active and/or sensory configuration. 3. Applications and discussion € ¼ 0Þ and free dynamic response In the present paper, we focus on the quasistatic response ð€u ¼ u (P(t) = Q(t) = uA(t) = 0) of the delaminated beam.

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3.1. Materials, geometry, and assumptions This section presents numerical results from several representative problems. The accuracy of the developed mechanics is first validated for the case of pristine beams with previous models. Predicted modal frequencies of delaminated beams are also compared with previously contacted experimental results (Chrysochoidis and Saravanos, 2004). All applications were focused on composite Gr/Epoxy T300/934 beams with quasi-isotropic laminations [0/90/45/45]s, 0.127 mm nominal ply thickness, 280 mm long and 25 mm wide. Four different full-width delamination crack sizes were used (see Fig. 2) covering the 10.9%, 21.8% and 43.6% of the total specimen’s length, respectively. In the composite beams two piezoceramic layers were considered each one covering the top and bottom surface of the specimen having thickness equal to the nominal ply thickness. The piezoceramic layers were used either as actuators or sensors. The properties of the composite and the piezoceramic layers are illustrated in Table 1. Unless otherwise stated, the through thickness fields in the composite piezoceramic beams were modeled using 4/24/4 discrete layers uniformly spaced in the bottom piezoceramic layer, the composite laminate (three layers for each ply), and the top piezoceramic layer, respectively. In all cases, the beams were modeled in the axial direction using 50 finite elements uniformly spaced along the pristine and delaminated sections, respectively, as follows: in all pristine beams all 50 elements were uniformly spaced; in the beams with 10.9% delamination, 42 elements were uniformly spaced in the pristine section and 8 elements in the delaminated section; in the beams with 21.8% delamination, 32 elements were used in the pristine and 18 in the delaminated part; finally, in the beam with 43.6% delamination, 24 elements were used in the pristine and 26 in the delaminated part. 3.2. Verification cases The developed beam finite element for delaminated composite beams was correlated with a previously developed beam finite element code and experiments:

z

L x Delamination

x y

Fig. 2. Typical composite beam specimen with delamination.

Table 1 Material properties for Gr/Epoxy [0/90/45/45]s composite beams and Pz27 (Ferroperm) piezoceramic actuators and sensors Property

T300/934

Pz27 (Ferroperm)

E11 (GPa) E33 (GPa) v13 G13 (GPa) d31 (m/V) d33 (m/V) d15 (m/V) e31 (farad/m) e11 (farad/m) q (kg/m3)

127 7.9 0.275 3.4 0 0 0 0 0 1578

58.82 43.1 0.371 22.98 170 · 1012 425 · 1012 506 · 1012 15.94 · 109 15.94 · 109 7700

N.A. Chrysochoidis, D.A. Saravanos / International Journal of Solids and Structures 44 (2007) 8751–8768

8759

3.2.1. Pristine piezocomposite beams The developed layerwise theory and beam finite element was initially correlated with a high-order layerwise (HOLT) piezoelectric beam finite element (Plagianakos and Saravanos, 2005) for a [0/90/45/45]s simply supported pristine beam having two piezoceramic layers as shown in Fig. 3a. The beam is loaded quasistaticly with applied electric potential 100 V. Normal displacements, sensory voltage, axial and shear stresses are shown in Figs. 4a–d. The agreement between the two layerwise models and finite elements is excellent and validates the capability of the new linear layerwise finite element to predict the response of a pristine beam with actuator and sensor layers. Additionally, predictions of modal frequencies for the previous beams are shown in Table 2, and correlate excellently with the predictions of the HOLT finite element. 3.2.2. Correlation with an exact solution Numerical results of the developed beam finite element were correlated with an exact solution for a thick simply supported hybrid laminated beam of aspect ratio L/h = 4 in cylindrical bending (Heyliger and Brooks, 1995), having a total thickness of 10 mm. The beam has two surface attached piezoelectric layers, of 1 mm thickness each, where electrical voltages were set equal to zero at the outer surfaces and remained free in their inner surfaces. Mechanical and electrical properties are illustrated in Table 3. The fundamental modal frequency of the beam was predicted to be 313 kHz, a 3.2% difference with literature predictions (Heyliger and Brooks, 1995), using a discretization of 50 elements and 24 layers. The

ΦS Sensor

Φ =0 Φ =0

Actuator

ΦA Φs

ΦS Φs

ΦA

- ΦA

Fig. 3. Configurations of simply supported composite beams with piezoelectric layers: (a) pristine active–sensory beam; (b) beam with central delamination and two distributed piezoceramic sensor layers; (c) beam with one sensor layer and two equal size actuators loaded by fields of opposite polarity.

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N.A. Chrysochoidis, D.A. Saravanos / International Journal of Solids and Structures 44 (2007) 8751–8768

a

0.5

b -1.00 -1.02

0.4

-1.04

(Volt)

W (mm)

-1.06 0.3

0.2

-1.08 -1.10 -1.12 -1.14 -1.16

0.1

Present HOLT 0.0 0.0

c

0.2

0.4

x/L

0.6

1.0

0.8

1.0

-1.20 0.0

d 1.0

Sensor Present HOLT

0.0

0.2

0.4

x/L

0.6

0.8

1.0

PZT Present HOLT

0.5

z/h

0.5

z/h

Present HOLT

-1.18

0.0

-0.5

-0.5

Actuator

-1.0 -6

-4

PZT

-2

0

2

4

6

-1.0 -35-

(MPa) 1

30

-25

-20

-15 5

-10

-5

0

5

10

(Pa)

Fig. 4. Predicted quasistatic response of an active–sensory beam [0/90/45/45]s with one piezoceramic layer used as actuator and the other as sensor.

Table 2 Predicted modal frequencies of the pristine Gr/Epoxy [0/90/45/45]s composite beam under simple supporting conditions Modal frequency

Present FE

HOLT

1 2 3 4 5 6 7 8 9 10

37.9 151.6 340.8 605.2 944.4 1357.7 1844.6 2404.1 3035.3 3737.1

37.9 151.6 340.8 605.2 944.4 1357.8 1844.6 2404 3035.1 3736.8

corresponding through-thickness mode shapes of the axial displacements and electrical voltages are illustrated in Figs. 5a and b, respectively. There is excellent agreement in the axial displacements field between FEM and analytical solutions, and excellent agreement in the prediction of the electric field within the piezoelectric layers. The deviation in the predicted electrical field through the passive composite plies in the laminate core is

N.A. Chrysochoidis, D.A. Saravanos / International Journal of Solids and Structures 44 (2007) 8751–8768

8761

Table 3 Material properties for the Hybrid three layer laminate [A/C/A] Property

C

A

E11(GPa) E33(GPa) v13 G13(GPa) e31 (C/m2) e33/e0 e11/e0 q (kg/m3)

132.38 10.756 0.24 5.65 0 0 0 1

81.3 64.5 0.432 25.6 5.20 1300 1475 1

Heyliger and Brooks (1995).

1.0

1.0

PZT

0.8 0.6 0.4

0.6

L/H=4

z/h

0.2

0.0

0.0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8 PZT

-1.0 -1.0

L/H=4

Present Heyliger and Brooks (1995)

0.4

0.2

z/h

PZT

0.8 Heyliger and Brooks (1995) Present

-0.5

0.0

0.5

Normalized Axial Displacements

1.0

-0.8 -1.0 0.0

PZT

0.2

0.4

0.6

0.8

1.0

Normalized Voltage

Fig. 5. Prediction and comparisons of the through the thickness axial displacements and voltages of the 1st mode shape for a hybrid piezoelectric laminated beam.

rather superficial, and is attributed to a two-dimensional distribution of the electric field in the composite core due to the short thickness aspect ratio of the beam and the axial permittivity of the composite plies, which the present model may not capture as it neglects axial electric field components. Overall, the good correlation between the present model and exact solution enforces the ability of our model to predict the modal response of healthy piezoelectric laminates. 3.2.3. Delaminated composite beams Predicted modal frequencies of delaminated composite beams under free–free supporting conditions were correlated with measured experimental values (Chrysochoidis and Saravanos, 2004). In the experiments, the delaminated specimens were excited by a surface bonded piezoceramic wafer and their response was measured by a piezopolymer sensor, which were both included in the FE model. Predicted and measured modal frequencies of the delaminated beams are shown in Fig. 6. The solid lines represent predicted resultant frequencies of bending modes and the dashed line represents the resultant frequency of the opening mode of the delaminated beams. Very good agreement is observed between the measurements and the predictions for all crack sizes and for at least the first four bending modes. This lends credence to the ability of the developed analytical model to capture the effect of delamination on the modal frequencies. 3.3. Effect on active quasistatic response The effect of delamination on the quasistatic response of a simply supported beam with a distributed surface attached actuator and sensor layer (Fig. 3c) is further investigated. The inner surfaces of the piezoelectric

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1400 Predicted 1st 2nd 3rd 4th 5th opening

Frequency (Hz)

1200 1000 800 600

Measured 1st 2nd 3rd 4th 5th

400 200 0.0

0.1

0.2

0.3

0.4

0.5

Delamination Length dL/L Fig. 6. Effect of delamination size on modal frequencies.

layers were grounded, and at the outer faces of the two actuators of uA = 100 V were applied of opposite polarity. The axial variations of the transverse displacements and sensor potential are presented in Fig. 7a and b. The introduction of delamination cracks causes jumps in the voltage distribution. The jumps increase with the damage size and their axial peak position coincides with the delamination ends. These jumps may provide a good damage indicator for the detection and localization of delamination cracks; however, they may need a dense network of sensors. The axial and shear stresses fields at the middle of the specimens are presented in Figs. 8a and b. Both fields are significantly affected by the delamination. In any case, the present case outlines the ability to exemplify the existence and location of the delamination by proper actuator and excitation. 3.4. Effect of delamination on the local modal response The effect of delamination on the shape, sensor signal and local displacement and stress fields were further predicted for the composite beams under simple supports with two distributed piezoceramic sensor layers (Fig. 3b). 3.4.1. 1st bending mode The 1st modal frequency of the pristine beam was predicted to be 37.9 Hz and remains mostly insensitive to the existence and size of damage. Figs. 9a and b present the transverse displacement mode shape and the sen150

2.5 Delamination size 0% 10.9% 21.8%

2.0 1.5

50

1.0

(Volt)

W/ Wmax x1e-6

100

0 -50

0.5 0.0 -0.5 Delamination size 0% 10.9% 21.8%

-1.0 -100 -1.5 -150 0.0

0.2

0.4

0.6

x/L

0.8

1.0

-2.0 0.0

0.2

0.4

0.6

0.8

1.0

x/L

Fig. 7. Effect of delamination on the response of the active sensory beam (a) transverse displacement w; (b) sensory electric potential u.

N.A. Chrysochoidis, D.A. Saravanos / International Journal of Solids and Structures 44 (2007) 8751–8768 1.0

1.0

PZT 0.8

0.8

0.6

0.6

0.4

0.4

PZT

0.2 DELAMINATION

0.0 -0.2

-0.6 -0.8

Delamination size 0% 10.9% 21.8%

-0.4 -0.6 -0.8

PZT

-1.0 -0.3

DELAMINATION

0.0 -0.2

Delamination size 0% 10.9% 21.8%

-0.4

Z /h

0.2

z/h

8763

PZT

-0.2

-0.1

0.0

0.1

-1.0 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

0.2

1(MPa)

5

(MPa)

Fig. 8. Effect of delamination on through the thickness distributions at the midspan of the active sensory beam of (a) axial and (b) interlaminar stresses.

7 0.0

6 5

(KVolt)

W/ Wmax

-0.2

Delamination size 0% 10.9% 21.8%

-0.4 -0.6

4 3

Delamination size 0% 10.9% 21.8%

2

-0.8

1 -1.0 0 0.0

0.2

0.4

0.6

x/L

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

x/L

Fig. 9. Effect of delamination on the 1st bending mode characteristics. (a) Normal displacement w; (b) electric potential u.

sor voltage predicted along the beam axis, respectively, for three specimens having a delamination crack equal to 0%, 10.9% and 21.8% of their length. The mode shape changes (Fig. 9a) appear negligible. The predicted sensory voltage distribution (Fig. 9b) shows small jumps at the points which coincide with the tips of crack. However, they seem to be insufficient to reveal the damage existence. Fig. 10a illustrates the axial strain field through the thickness at the mid-span, where the delamination existence causes a jump at the strain’s distribution, which increases with the damage size. Predicted interlaminar stress r5 values are shown in Fig. 10b at the same axial position. The values of shear stress seems to change drastically in the presence of the crack and may provide the basis for a sensitive damage indicator. It is worth pointing out, that the present formulation provides accurate shear stress and strain predictions at the center of each discrete layer, as depicted by the symbols in Fig. 10b. Moreover, the predicted shear stresses weakly satisfy the surface traction conditions at both free surfaces and the delamination surfaces, as well as the continuity conditions at ply interfaces, through the variational statement of the equilibrium equations (3, 17). Thus, the predicted shear stress values will asymptotically converge to the surface traction values by increasing the density of discrete layers. The latter is demonstrated in Fig. 10b, as the calculated shear stress seem to correctly approach to the zero shear traction values assumed on the free surfaces and on the freely sliding delamination faces. Finally Figs. 11a and b present the predicted relative sliding of the top and bottom surface of the crack. Top and bottom delamination surfaces are shown partially to move toward the positive and negative x-axis direction due to the symmetric

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1.0

PZT

0.8

0.4

Z /h

0.2

Delamination size 0% 10.9% 21.8%

0.6 0.4 DELAMINATION

0.0

DELAMINATION

0.0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

Delamination size 0% 10.9% 21.8%

0.2

Z /h

0.6

PZT

0.8

-0.8

PZT

PZT

-1.0

-1.0 -0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 0.075 0.100

0.0

0.2

0.4

S1

0.6 5

0.8

1.0

(MPa)

Fig. 10. Through the thickness modal distributions predicted at the midspan of the beam (1st mode). (a) Axial strain; (b) interlaminar shear stress.

80

10.9% Delamination

8

21.8% Delamination 60

6

40 20

2

U( m)

U( m)

4

0 -2

-20

-4

-40 Faces Top Bottom

-6 -8 0.0

0

0.2

0.4

0.6

x/L

0.8

1.0

Faces Top Bottom

-60 -80 0.0

0.2

0.4

x/L

0.6

0.8

1.0

Fig. 11. Relative axial displacements between top and bottom crack faces for the 1st bending mode. (a) Small delamination; (b) medium delamination.

shape of the first mode. The results illustrate the ability of the model to describe the relative slip at delamination faces through the considered degrees of freedom. 3.4.2. 2nd bending mode The second bending frequency was more sensitive to delamination size with predicted values of 152, 149, and 132 Hz for 0%, 10.9%, and 21.8% delamination crack size, respectively. Figs. 12–14 present the predicted deflection and sensory voltage, stress fields, and interfacial sliding for the second bending mode. However, the predicted sensor voltage and stress field of this mode seem to be much more sensitive to the delamination existence. Interest is focused on the changes induced in sensor voltage variations across the length and the stress distributions through the thickness of the composite beams. In any case, the results illustrate that for some mode shapes and possibly delamination locations, the sensor voltage and stress response can be more sensitive to the damage existence and size. The value of the present FE in identifying these mode shapes and proper sensor location and placement is apparent. Finally, Figs. 14a and b present the relative slip for various delamination sizes.

N.A. Chrysochoidis, D.A. Saravanos / International Journal of Solids and Structures 44 (2007) 8751–8768 30

1.0

Delamination size 0% 10.9% 21.8%

0.8 0.6 0.4

20 10

0.2

(KVolt)

W/ Wmax

8765

0.0 -0.2

0 -10

-0.4

Delamination size 0% 10.9% 21.8%

-0.6 -20

-0.8 -1.0

-30 0.0

0.2

0.4

x/L

0.6

0.8

0.0

1.0

0.2

0.4

0.6

0.8

1.0

x/L

Fig. 12. Effect of delamination on the 2nd bending mode characteristics. (a) Normal displacement w; (b) electric potential u.

1.0

1.0

PZT

0.8

0.8

0.6

0.6

0.4

0.4 0.2 DELAMINATION

0.0 Delamination size 0% 10.9% 21.8%

-0.2 -0.4 -0.6 -0.8

Z /h

z/h

0.2

PZT

DELAMINATION

0.0 -0.2

Delamination size 0% 10.9% 21.8%

-0.4 -0.6 -0.8

PZT

-1.0

PZT

-1.0 -3

-2

-1

0

1

2

0

3

20

40

60

80

1(GPa)

100

120

140

160

180

(MPa) 5

Fig. 13. Through the thickness modal distributions predicted at the midspan of the beam (2nd mode). (a) Axial strain; (b) interlaminar shear stress.

3.0

10

10.9% Delamination

2.5

21.8% Delamination

8

2.0

6

1.5 4

0.5

U(mm)

U(mm)

1.0 0.0 -0.5 -1.0

0 -2 -4

-1.5

-6

-2.0

Faces Top Bottom

-2.5 -3.0 0.0

2

0.2

0.4

x/L

0.6

0.8

Faces Top Bottom

-8 -10 1.0

0.0

0.2

0.4

x/L

0.6

0.8

1.0

Fig. 14. Relative axial displacements between top and bottom crack faces for the 2nd bending mode. (a) Small delamination; (b) medium delamination.

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3.5. Opening mode shapes The present method and finite element offers the ability to predict ‘‘breathing’’ delamination mode shapes and modal frequencies. Figs. 15a–c present interfacial crack ‘‘breathing’’ for three different sizes of delamination. Figs. 16a–c illustrate sensory voltage, axial strain, and shear stresses, respectively, for a delamination crack covering the 21.8% of the total length. Sensory voltage is the more significant indicator of the breathing mode as it appears to be very sensitive to the damage existence. Jumps in the sensory voltage distribution of a network of sensors covering the beam’s surface will indicate the delamination boundaries (Fig. 16a). Axial strains offer symmetric and linear variation through the thickness with maximum strain at the delamination interface. Finally, the interlaminar stresses distribution is antisymmetric in both sublaminates, thus validating the antisymmetric mode shape. Although the present method neglects contact, the breathing modes may in practice appear in different nonlinear forms; there are works suggesting that they can be used in the detection of small cracks. In this context, the previous results may provide some insight, for their characteristics. 4. Summary Layerwise mechanics were formulated and described for analyzing the coupled electromechanical response of delaminated composite beams with embedded passive or active piezoelectric layers. Delamination crack movements (relative slip and opening) were included in the mechanics as additional degrees of freedom. 0.6

f=2960 Hz

0.6

f=752Hz

0.4

Normalized W

Normalized W

0.4 0.2 0.0 -0.2 Faces Top Bottom

-0.4 -0.6 0.0

0.2

0.4

x/L

0.6

0.8

0.0

-0.2 Faces Top Bottom

-0.4

-0.6 0.0

1.0

0.6

0.2

0.2

0.4

x/L

0.6

0.8

1.0

f=198 Hz

Normalized W

0.4

0.2

0.0

-0.2 Faces Top Bottom

-0.4

-0.6 0.0

0.2

0.4

x/L

0.6

0.8

1.0

Fig. 15. Opening (breathing) mode shapes of beams with various delamination sizes: (a) 10.9%, (b) 21.8% and (c) 43.6% of the total length.

N.A. Chrysochoidis, D.A. Saravanos / International Journal of Solids and Structures 44 (2007) 8751–8768 60

8767

1.0

f=752Hz

40

0.5

z/h

(KVolt)

20

0

-20

Delamination

-0.5

-40 0.0

0.0

Delamination Size: 21.8% 0.2

0.4

x/L

0.6

0.8

-1.0 -1.0

1.0

-0.5

S1

0.0

0.5

1.0 0.8 0.6 0.4

z/h

0.2 0.0

Delamination

-0.2 -0.4 -0.6 -0.8 -1.0 -0.15-

0.10

-0.05

0.00 5

0.05

0.10

0.15

(GPa)

Fig. 16. Modal response at the midspan of the beam with 21.8% delamination (1st breathing mode). (a) Electric potential; (b) axial strain; (c) shear stresses.

The capability and versatility of the new finite element to model the effect of delamination on the modal and quasistatic response was evaluated on Gr/Epoxy T300 composite beams with various sizes of delamination cracks. Modal frequency predictions were favorably correlated with previously contacted experimental measurements. Numerical studies of the effect of delamination on local fields evaluated the capabilities of the method and pointed out some promising damage indicators. Usage of a network of piezoceramic or piezopolymer sensors covering the surface of a critical structural area appears promising in the low-frequency region, provided that proper modes are excited and monitored. Voltage variations across the length of the beam may offer information about the axial position and the boundaries of delamination, which can be exemplified by proper actuator configurations. On the other hand, axial and shear strain or stress distribution appear far more sensitive to the vertical and horizontal position of debonding. Combination of the information offered from the sensory voltage and the axial or shear strains may effectively reveal the position and size of a delamination. An additional damage index may be the sensor voltage distribution of a beam actuated near a breathing modal frequency. Overall, the developed FE offers many advantages toward the simulation and design of active delamination monitoring structures. Future work has been directed toward the inclusion of interfacial contact into the mechanics and FE, and the prediction of the non-linear dynamic response.

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Acknowledgements This work was funded by the European Social Fund (ESF), Operational Program for Educational and Vocational Training II (EPEAEK II), and the Program HERAKLITOS. This funding is gratefully acknowledged by the authors. References Barbero, E.J., Reddy, J.N., 1991. Modeling of delamination in composite laminates using a layerwise plate theory. International Journal of Solids and Structures 28 (3), 373–388. Chattopadhyay, A., Gu, H., Dragomir-Daescu, D., 1999. Dynamics of delaminated composite plates with piezoelectric actuators. AIAA Journal 37 (2), 248–254. Chattopadhyay, A., Kim, H.S., Ghoshal, A., 2004. Non-linear vibration analysis of smart composite structures with discrete delamination using a refined layerwise theory. Journal of Sound and vibration 273 (1–2), 387–407. Chrysochoidis, N.A., Saravanos, D.A., 2004. Assessing the effects of delamination on the damped dynamic response of composite beams with piezoelectric actuators and sensors. Smart materials and Structures 13 (4), 733–742. Diaz Valdes, S.H., Soutis, C., 1999. Delamination detection in composite laminates from variation of their modal characteristics. Journal of Sound and Vibration 228 (1), 1–9. Di Sciuva, M., Librescu, L., 2000. A dynamic nonlinear model of multilayered composite shells featuring damage interfaces. In: Proceedings of the 41st AIAA/ASME/ASCE/ AHS/ASC SDM, AIAA-2000-1384, Atlanta, GA, April 3–6. Heyliger, P., Brooks, S., 1995. free vibration of piezoelectric laminates in cylindrical bending. International Journal of Solids and Structures 32 (20), 2945–2960. Hu, N., Fukunaga, H., Kameyama, M., Aramaki, Y., Chang, F.K., 2002. Vibration analysis of delaminated composite beams and plates using a higher-order finite element. International Journal of Mechanical Science 44 (7), 1479–1503. Keilers, C.H., Chang, F.K., 1995. Identifying delamination in composite beams using built-in piezoelectrics. 1. Experiments and analysis. Journal of Intelligent Material Systems and Structures 6 (5), 649–663. Luo, H., Hanagud, S., 2000. Dynamics of delaminated beams. International Journal of Solids and Structures 37, 1501–1519. Nagesh Babu, G.L., Hanagud, S., 1990. Delaminations in smart composite structures: a parametric study on vibrations. In: Proceedings of the 31st AIAA/ASME/ASCE/AHS/ASC SDM Conference, AIAA paper 90-1173-CP, pp. 2417–2426. Paolozzi, A., Peroni, I., 1990. Detection of debonding damage in a composite plate through natural frequency variations. Journal of Reinforced Plastics and Composites 9, 369–389. Plagianakos, T.S., Saravanos, D.A., 2005. Coupled high-order shear layerwise analysis of adaptive sandwich piezoelectric composite beams. AIAA Journal 43 (4), 885–894. Rinderknecht, S., Kroplin, B., 1995. A finite element model for delamination in composite plates. Mechanics of Composite Material and Structures 2, 19–47. Saravanos, D.A., Heyliger, P.R., 1995. Coupled layerwise analysis of composite beams with embedded piezoelectric sensors and actuators. Journal of Intelligent Material Systems and Structures 6 (3), 350–363. Saravanos, D.A., Hopkins, D.A., 1996. Effects of delaminations on the damped dynamic characteristics of composite laminates: analysis and experiments. Journal of Sound and Vibration 192 (5), 977–993. Shen, M.H.H., Grady, I., 1992. Free vibrations of delaminated beams. AIAA Journal 30 (5), 1361–1370. Tenek, L.H., Henneke, E.G., Gunzburger, M.D., 1993. Vibration of delaminated composite plates and some applications to nondestructive testing. Composite Structures 23, 253–262. Thornburgh, R., Chattopadhyay, A., 2001. Unified approach to modeling matrix cracking and delamination in laminated composite structures. AIAA Journal 39 (1), 153–160. Tracy, J.J., Pardoen, G.C., 1989. Effect of delamination on the natural frequencies of composite laminates. Journal of Composite Materials 23, 1200–1215.