Moodle generalised formulation of laminate theory using beam fe for delaminated composite beams with pie
GENERALISED FORMULATION OFLAMINATE THEORY USING BEAM FE FOR DELAMINATEDCOMPOSITE BEAMS WITH PIEZO ACTUATORS AND PIEZO SENSORS B. Kavi, B. K. Nanda Research ScholarDepartment of Mechanical Engineering NIT, Rourkela 769008 ProfessorDepartment of Mechanical Engineering NIT, Rourkela 769008 E-mail: firstname.lastname@example.org E-mail: email@example.comABSTRACTA coupled linear layer wiselaminate theory and a beam FE areformulated for analyzingdelaminated composite beams withpiezoactuators and sensors. Themodel assumes zigzag fields for theaxial displacements and the electricpotential and it treats thediscontinuities in the displacementfields due to the delaminations asadditional degrees of freedom. Theformulation naturally includes theexcitation of piezoelectric actuators,their interactions with the compositelaminate, and the effect ofdelamination on the predictedsensory voltage. The quasistaticand modal response of laminatedcomposite Gr/Epoxy beams withactive or sensory layers havingvarious delamination sizes ispredicted. The numerical resultsillustrate the strong effect ofdelamination on the sensorvoltage, on through the thicknessdisplacement and on the stressfields. Finally, the effect ofdelamination on modal frequenciesand shapes are predicted andcompared with previouslyobtained experimental results. Keywords: Composite materials; Laminate;
Delamination; Modelling; Finite element; Beams; Piezoelectrics1.0 INTRODUCTION1.1. General informationThe development of active structuralhealth monitoring systems andtechniques using piezoelectric actuatorand sensor wafers or films is an areaexperiencing significant technicalactivity[1 a]. Particular emphasis isplaced on the active structural healthmonitoring methods for compositestructures, as the likelihood of internaldefects, and the evolution of invisibledamage in the composite materialduring service life remains high. Onecandidate type of damage isdelamination cracks, which areusually induced during low-velocityimpact and fatigue, remain hidden andcan propagate quickly leading tocatastrophic failure. Among the manyopen issues in this field is thedevelopment of analytical andnumerical models capable of capturingthe effect of delaminations on thestructural response, particularly onthe electromechanical responsecomponents associated with thepiezoelectric actuators and sensors, assuch models may help understand thesensitivity of electromechanicalresponse on damage parameters, andthus provide a basis for thedevelopment of damage detection andlocalization techniques and the design ofthe smart composite system. To thisend, this paper presents a layerwisemechanics theory and a finite elementfor analyzing laminated beams withdelamination cracks and activepiezoelectric sensors. In the present paper, anelectromechanically coupledlayerwise theory is described forcomposite beams with piezoelectricactuators and sensors, treatinginterfacial sliding, crack opening andslope discontinuity across adelamination crack, as additionaldegrees of freedom. The generalizedstiffness, mass piezoelectric, andpermittivity matrices are formulated anda 2-node finite element is furtherdeveloped. The new finite elementcapabilities are evaluated by predictingthe effect of a single delamination onthe modal and quasistatic response ofcomposite beams with passive oractive piezoelectric layers. The
analytical predictions of the modalfrequencies are further correlated withavailable measurements. Also thedelamination mode shapes are presentedshowing the delamination „„breathing‟‟.Through these model predictions, somemechanisms capable of revealing thedelamination presence are furtherdiscussed with an eye toward activedelamination detection.2.0 EQUATIONS AND SYMBOLSThis section presents the formulatedmechanics for piezoelectric laminatedbeams with interlaminar delaminationcracks.2.1. Governing material equationsEach ply is assumed to exhibit linearpiezoelectric behavior. In the currentbeam case, in-plane and interlaminarshear strains are considered in the elasticfield. The piezoelectric materialsconsidered are monoclinic class twocrystals with the poling directioncoincident with the z-axis. Theconstitutive equations have the form 1,5 and S1, S5 are the axial and shearmechanical stresses and strains,respectively; E1 and E3 are the electricfield vectors; D1 and D3 are the electricdisplacement vectors; C11 and C55 arethe elastic stiffness coefficients; e15 ande31 are the piezoelectric coefficients;and 11, 33 are the electric permittivitycoefficients of the material. SuperscriptsE and S indicate constant electric fieldand strain conditions, respectively. Theabove system of equations encompassesthe behavior of a beam‟spiezocomposite layer. The electric fieldvector is the negative gradient of theelectric potential 2.2. Equations of equilibrium
2.3. Kinematic assumptions2.4. Strain–displacement relations andelectrical field2.5. Laminate electric enthalpy andlaminate matrices2.6. Finite element formulation3.0 APPLICATIONS AND DISCUSSIONIn the present paper, we focus on thequasistatic response ( =0)and freedynamic response (P(t) = Q(t) = θ A(t) =0) of the delaminated beam.
3.1.Materials, geometry, andassumptions:This section presents numerical resultsfrom several representative problems.The accuracy of the developedmechanics is first validated for the caseof pristine beams with previous models.Predicted modal frequencies ofdelaminated beams are also comparedwith previously contacted experimentalresults [1 b]. All applications werefocused on composite Gr/EpoxyT300/934 beams with quasi-isotropiclaminations [0/90/45/_45]s, 0.127 mmnominal ply thickness, 280 mm long and25 mm wide. Four different full-widthdelamination crack sizes were used (seeFig. 2) covering the 10.9%, 21.8%and 43.6% of the total specimen‟slength, respectively. In the compositebeams two piezoceramic layers wereconsidered each one covering the topand bottom surface of the specimenhaving thickness equal to the nominalply thickness. The piezoceramic layerswere used either as actuators or sensors.4.0 ILLUSTRATION
Layerwise mechanics wereformulated and described for analyzingthe coupled electromechanical responseof delaminated composite beams withembedded passive or activepiezoelectric layers. Delamination crackmovements (relative slip and opening)were included in the mechanics asadditional degrees of freedom. Thecapability and versatility of the newfinite element to model the effect ofdelamination on the modal andquasistatic response was evaluated onGr/Epoxy T300 composite beams withvarious sizes of delamination cracks.Modal frequency predictions werefavorably correlated with previouslycontacted experimental measurements. Numerical studies of the effect ofdelamination on local fields evaluatedthe capabilities of the method andpointed out some promising damageindicators. Usage of a network ofpiezoceramic or piezopolymer sensorscovering the surface of a criticalstructural area appears promising in thelow-frequency region, provided thatproper modes are excited andmonitored. Voltage variations across thelength of the beam may offerinformation about the axial position andthe boundaries of delamination, whichcan be exemplified by proper actuatorconfigurations. On the other hand, axialand shear strain or stress distributionappear far more sensitive to the verticaland horizontal position of debonding.Combination of the information offeredfrom the sensory voltage and the axialor shear strains may effectively revealthe position and size of a delamination.An additional damage index may be thesensor voltage distribution of a beamactuated near a breathing modalfrequency. Overall, the developed FEoffers many advantages toward thesimulation and design of activedelamination monitoring structures.Future work has been directed towardthe inclusion of interfacial contact intothe mechanics and FE, and theprediction of the non-linear dynamicresponse.5.0 REFERENCES1. a. Nikolaos, Chrysochoidis, Dimitris,Saravanos, “Generalized layerwisemechanics for the static and modal response of delaminated composite beams with active piezoelectric sensors”, International Journal of Solids and Structures 44, 2007, pp. 8751–8768.
b. Chrysochoidis, Saravanos,“Assessing the effects of delaminationon the damped dynamic response ofcomposite beams with piezoelectricactuators and sensors”, Smartmaterials and Structures 13 (4), 2004,pp. 733–742.2. a. Jinquan Cheng, Farid Taheri, “Asmart single-lap adhesive jointintegrated withpartially distributed piezoelectricpatches”, International Journal ofSolids and Structures 43, 2006, pp.1079–1092.b. Jinquan Cheng, Xiaoxia Wu ,Guoqiang Li, Farid Taheri, Su-SengPang, ” Development of a smartcomposite pipe joint integrated withpiezoelectric layers under tensileloading”, International Journal ofSolids and Structures 43, 2006, pp5370–5385.3. G. Song, V. Sethi, H.-N.Li,“Vibration control of civilstructures using piezoceramic smartmaterials: Areview”, Engineering Structures 28,2006, pp. 1513–1524.