Generalised formulation of laminatetheory using Beam Finite Element fordelaminated composite beams withpiezoactuators and ...
displacements and electric potential and it treats thediscontinuities in the displacement fields due to thedelaminations a...
straight, normal and inextensional during thedeformation so that transverse normal and shearstrains are neglected in deriv...
representation of general electromechanical boundaryconditions. Approximate finite element solutions forthe static and fre...
with piezoelectric     actuators and sensors, treatinginterfacial sliding, crack opening and            slopediscontinuity...
degrees of freedom. The quasi-static and modalresponse of laminated composite Gr/Epoxy beamswith active or sensory layers ...
T300/934 beams with quasi-isotropic laminations[0/90/45/_45]s, 0.127 mm nominal ply thickness, 280mm long and 25 mm wide. ...
piezopolymer sensors covering the surface of acritical structural area appears promising in the low-frequency region, prov...
up; Hermitian polynomial; Laminate;Delamination; Modelling; Finite element; Beams;Piezoelectrics
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Generalised_formulation_of_laminate_theory_using_beam_fe_for_delaminated_composite_beams_with_piezoactuators_and_sensors

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Technical write up report for extended abstract submitted to Prof. S K Patel, Mechanical Department, NIT Rourkela.

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Generalised_formulation_of_laminate_theory_using_beam_fe_for_delaminated_composite_beams_with_piezoactuators_and_sensors

  1. 1. Generalised formulation of laminatetheory using Beam Finite Element fordelaminated composite beams withpiezoactuators and sensors B. Kavi, B. K. NandaResearch Scholar ProfessorDepartment of Mechanical Engineering Department of Mechanical EngineeringN. I. T., Rourkela 769008, Odisha N. I. T., Rourkela 769008, OdishaE-mail: binit.kavi@gmail.com E-mail: bknanda@nitrkl.ac.inAbstract: The application of fibre-reinforced compositematerials in the aerospace industry extends fromcommercial to military aircraft, such as the BoeingF18, B2 Stealth Bomber, AV 8B Harrier. Theattractiveness of composite lies in their mechanicalproperties such as damping characteristics, weight,strength, stiffness, corrosion resistance, fatigue life,etc. Composites are widely used for controllingsurfaces such as ailerons, flaps, stabilizers, rudders,as well as rotary and fixed wings. That is why theanalysis of composite structures is imperative foraerospace industry. The main advantage ofcomposites is their flexibility in design. Mechanicalproperties of the laminates can be altered simply bychanging the stacking sequence, fibre lay-up andthickness of each ply which leads to optimization in adesign process. A coupled linear layer wise laminatetheory and a beam Finite Element theory areformulated for analyzing delaminated composite (Gr/Epoxy T300) beams with piezoactuators andsensors. The model assumes zigzag fields for axial
  2. 2. displacements and electric potential and it treats thediscontinuities in the displacement fields due to thedelaminations as additional degrees of freedom.Thedevelopment of active structural health monitoringsystems and techniques using piezoelectric actuatorand sensor wafers or films is an area experiencingsignificant technical activity. Particular emphasis isplaced on the active structural health monitoringmethods for composite structures, as the likelihoodof internal defects, and the evolution of invisibledamage in the composite material during service liferemains high. One candidate type of damage isdelamination cracks, which are usually inducedduring low-velocity impact and fatigue, remainhidden and can propagate quickly leading tocatastrophic failure. The small deflection bending theory for a thinlaminate composite beam is developed based onBernoullis assumptions for bending of an isotropicthin beam. The development of the classical bendingtheory for a thin laminated composite plate followsKirchhoffs assumptions for the bending of an isotropicplate. Kirchhoffs main suppositions are asfollows:The material behaviour is linear and elastic.The plate is initially flat. The thickness of the plate issmall compared to other dimensions. Thetranslational displacements are small compared to theplate thickness, and the rotational displacements arevery small compared to unity. The normals to theundeformed middle plane are assumed to remain
  3. 3. straight, normal and inextensional during thedeformation so that transverse normal and shearstrains are neglected in deriving the plate kinematicrelations. The transverse normal stresses areassumed to be small compared with other normalstress components . So that they may be neglected inthe constitutive relations. The relations developedearlier are essentially based on the above Kirchhoffsbasic assumptions. Some of these relations will beutilized to derive the governing equations for thincomposite plates. It may be noted that Kirchhoffsassumptions are merely an extension of Bernoullisfrom one-dimensional beam to two-dimensional plateproblems. Hence a classical plate bending theory sodeveloped can be reduced to a classical beambending theory. Here, also, the governing plateequations are derived first, and the beam equationsare subsequently obtained from the plate equations. Unified mechanics are developed with thecapability to model both sensory and active compositelaminates with embedded piezoelectric layers. Twodiscretelayer (or layerwise) formulations enableanalysis of both global and local electromechanicalresponse. The first assumes constant through-the-thickness displacement, while the second permitspiecewise continuous variation. The mechanicsinclude the contributions from elastic, piezoelectricand dielectric components. The incorporation ofelectric potential into the state variables permits
  4. 4. representation of general electromechanical boundaryconditions. Approximate finite element solutions forthe static and freevibration analysis of beams arepresented. Applications on composite beamsdemonstrate the capability to represent either sensoryor active structures, and to model the complicatedstressstrain fields, the interactions betweenpassive/active layers and interfacial phenomenabetween sensors and composite plies. The capabilityto predict the dynamic characteristics under variouselectrical boundary conditions is demonstrated. Someadvantages of the variable transverse displacementformulation on the freevibration response of sensorystructures are also shown. Among the many open issues in this field isthe development of analytical and numericalmodels capable of capturing the effect ofdelaminations on the structural response,particularly on the electromechanical responsecomponents associated with the piezoelectricactuators and sensors, as such models may helpunderstand the sensitivity of electromechanicalresponse on damage parameters, and thusprovide a basis for the development of damagedetection and localization techniques and the designof the smart composite system. To this end, thispaper presents a layerwise mechanics theory and afinite element for analyzing laminated beams withdelamination cracks and active piezoelectric sensors.In the present paper, an electromechanically coupledlayerwise theory is described for composite beams
  5. 5. with piezoelectric actuators and sensors, treatinginterfacial sliding, crack opening and slopediscontinuity across a delamination crack, asadditional degrees of freedom. The generalizedstiffness, mass piezoelectric, and permittivity matricesare formulated and a 2-node finite element is furtherdeveloped. The new finite element capabilities areevaluated by predicting the effect of a singledelamination on the modal and quasistatic responseof composite beams with passive or activepiezoelectric layers. The analytical predictions of themodal frequencies are further correlated withavailable measurements. Also the delamination modeshapes are presented showing the delamination‘‘breathing’’. Through these model predictions, somemechanisms capable of revealing the delaminationpresence are further discussed with an eye towardactive delamination detection. The formulation naturally includes the excitation ofpiezoelectric actuators, their interactions with thecomposite laminate, and the effect of delamination onthe predicted sensory voltage. A beam finite elementhas been formulated based on the previous laminatemechanics, implementing continuous Hermitianpolynomials for the local approximation of thetransverse displacements and shape functions for theremaining axial and electric Degree of Freedoms.Delamination crack movements (relative slip andopening) are included in the mechanics as additional
  6. 6. degrees of freedom. The quasi-static and modalresponse of laminated composite Gr/Epoxy beamswith active or sensory layers having variousdelamination sizes is predicted. The numerical resultsillustrate the strong effect of delamination on thesensor voltage through the thickness displacementand the stress fields. Finally, the effect ofdelamination on modal frequencies and shapes arepredicted and compared with the previously obtainedexperimental results. Modal frequency predictions arefavourably correlated with previously conductedexperimental measurements. Numerical studies of theeffect of delamination on local fields evaluated thecapabilities of the method and pointed out somepromising damage indicators. Use of a network ofpiezoceramic or piezopolymer sensors covering thesurface of a critical structural area appears promisingin the low-frequency region, provided that propermodes are excited and monitored. Voltage variationsacross the length of the beam may offer informationabout the axial position and boundaries ofdelamination which can be exemplified by properactuator configurations. The accuracy of thedeveloped mechanics is first validated for the case ofpristine beams with previous models. Predicted modalfrequencies of delaminated beams are also comparedwith previously contacted experimental results. Allapplications were focused on composite Gr/Epoxy
  7. 7. T300/934 beams with quasi-isotropic laminations[0/90/45/_45]s, 0.127 mm nominal ply thickness, 280mm long and 25 mm wide. Four different full-widthdelamination crack sizes were used covering the10.9%, 21.8% and 43.6% of the total specimen’slength, respectively. In the composite beams twopiezoceramic layers were considered each onecovering the top and bottom surface of the specimenhaving thickness equal to the nominal ply thickness.The piezoceramic layers were used either asactuators or sensors. Layerwisemechanics were formulated and described foranalyzing the coupled electromechanical responseof delaminated composite beams with embeddedpassive or active piezoelectric layers. Delaminationcrack movements (relative slip and opening) wereincluded in the mechanics as additional degrees offreedom. The capability and versatility of the newfinite element to model the effect of delamination onthe modal and quasistatic response was evaluated onGr/Epoxy T300 composite beams with various sizesof delamination cracks. Modal frequency predictionswere favorably correlated with previously contactedexperimental measurements. Numerical studies of the effect of delaminationon local fields evaluated the capabilities of themethod and pointed out some promising damageindicators. Usage of a network of piezoceramic or
  8. 8. piezopolymer sensors covering the surface of acritical structural area appears promising in the low-frequency region, provided that proper modes areexcited and monitored. Voltage variations across thelength of the beam may offer information about theaxial position and the boundaries of delamination,which can be exemplified by proper actuatorconfigurations. On the other hand, axial and shearstrain or stress distribution appear far more sensitiveto the vertical and horizontal position of debonding.Combination of the information offered from thesensory voltage and the axial or shear strains mayeffectively reveal the position and size of adelamination. An additional damage index may bethe sensor voltage distribution of a beam actuatednear a breathing modal frequency. Overall, thedeveloped FE offers many advantages toward thesimulation and design of active delaminationmonitoring structures. Future work has been directedtoward the inclusion of interfacial contact into themechanics and FE, and the prediction of the non-linear dynamic response. Keywords: Fibre-reinforced composites; corrosion resistance; fatigue life; ailerons; rudders; fibre lay-
  9. 9. up; Hermitian polynomial; Laminate;Delamination; Modelling; Finite element; Beams;Piezoelectrics

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