passively

Static and dynamic analyses of actively/passively controlled composite structures

J. Njuguna* Department of Materials, School of Industrial and Manufacturing Science, Cranfield University, Cranfield, Bedfordshire MK43 0AL, England, UK

Abstract Due to large potential applications in the fields of aerospace, civil engineering, shipbuilding, automobile, precision instruments, and machines, the field of adaptronics have developed rapidly. The active elements in smart structures can be embedded in or attached to the structure either discretely or distributed to provide built-in structural quality assessment capabilities, both during material processing and vehicle operation. In addition, recent progress in informatics and high-capability computing devices has offered a brand-new springboard for the aerospace community to reshuffle its traditional research and development criteria for functionalized composite structures. Particularly, artificial intelligence, an intriguing information processing technique, exhibits outstanding effectiveness in accommodating the highly demanding requirements of new generation airframes. Appropriate utilization of artificial intelligence techniques in functionalized composite structures design is expected to contribute to the realization of high-capability intelligent systems. This review work gives a detailed overview on analytical approaches, static and dynamic analyses of actively/passively controlled composite structures. This work concentrates on aerospace composite structures’ analyses including piezoelectric actuation and sensing, surface layer treated structures and shape memory alloy embedded structures. The constitutive relations and modelling issues for adaptive materials is presented. Further insight is given on nanomaterials, micro- and nano-electromechanical devices. Useful discussions are given on ageing and lifetime predictions of polymer composite structures.

Various active and passive vibration suppression schemes found in the literature are

reviewed and significant laboratory and real-world applications are discussed.

Keywords: Structural analysis, active/passive control, composites, adaptronics

*

Corresponding author. [email protected], Tel. +44 1234 754186, Fax: +44 1234 752473 (J. Njuguna).

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1

Introduction

Perhaps the most challenging task today is the proper prediction of the flight shape and its impacts on the induced drag of transport aircraft wings during the initial design phases. Because this shape is not the same for different payload conditions, and does not remain constant during the flight due to changing fuel mass and flight conditions, active concepts are required to adjust the shape. Aerospace demand sophisticated equipment, structural and propulsive systems with almost guaranteed safety level as the loss is often great, costly and sometimes even catastrophic. There is therefore the need for lighter, cheaper and more reliable technologies in day-to-day demands. The development of smart composites offers great potential for use in advanced aerospace structures because they are light in weight and possess adaptive control capabilities. Such systems exploit the aerodynamic forces in such a way that the elastic deformation of the complete aerodynamic surface is used to create desired control forces or enhance stabilizer effectiveness thus reinforcing the structure in order to reduce negative impacts from the deformations [1,2]. Actively shaping flexible devices of an aircraft during flight via such control elements creates aeroservoelastic interactions, which can be much more power and energy efficient than traditional means of flight control. Energy and power efficient are such concepts, which exploit aeroelastic effects by taking the needed energy out of the flow past the aircraft. The advantage of smart materials is their high power density, compared to hydraulic actuators. The disadvantage is that only small deformations can be achieved. One example is the use of active materials such as (thermo) ferroelectric elements and/or shape-memory alloys to construct a so-called smart, i.e., adaptive wing. The camber of a wing, constructed in this active material, can be changed without using a hinged control surface. The hinge point in conventional control surfaces induces flow separation and increases drag, and preliminary wind tunnel tests indicate that elimination of the hinge does indeed significantly reduce drag. Further more, studies have indicated that the smart-wing concept (Figure 1) can improve the performance by extending the perimeter of the full flight [2,3].

Figure 1

More so, the advancement of technology in the search for multifunctional materials has resulted in the concept of adaptive laminated composites. These electro-magneto-thermo-mechano-rheological materials have presented an exceptional promise when compared to conventional ones. In this respect, it is acknowledgeable that structural failures often occur as a consequence of aeroelastically-induced motion. Thus, the analysis and understanding of aeroelastic behaviour of structural component is of primary importance in the design of structures especially in aerospace industry. For the analysis of

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these structures, it is essential to accurately model both the strain field and the electric field. Crawley [4] and Chopra [5] offers excellent reviews of the current status of smart composite structures. The following subchapters give a detailed overview on analytical approaches, static and dynamic analyses of actively/passively controlled composite structures. In particular, this work briefly reviews past research in adaptive technology and discuss barriers to technology readiness. The constitutive relations and modelling issues for adaptive materials is presented. The various active and passive vibration suppression schemes found in the literature are reviewed and the significant laboratory and real-world applications discussed. Then, on each subchapter the work also concludes with a discussion of structural integrity of adaptive systems, the largest perceived barriers to technology readiness and presents some recommendations for future research. The work admits that given numerous number of papers published in this field over decades, a complete review of the literature is not possible. Instead the works discuss significant published works which form the foundation of the field. Omission of significant publications is inevitable and the author wish to apologize in advance for

anyone

omitted.

The

works

covered

are

subdivided

into

experimental

and

computational/numerical studies such that the reader can be sure of the area of interest and can concentrate on it. Several review articles are mentioned on each subsection for further reading into specialised areas. And finally, a complete discussion and conclusions are given too on each subsection.

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Piezoelectric Materials

Smart structures/materials can be defined in many ways depending upon the context and situation. It can be succinctly defined, in the context of this review paper, as the structure, which has built-in capabilities, or with intrinsic sensors that perceive and process the operating environments, and take effective actions to fulfil the intended mission. Smart structures/materials, in fact, are consistently identified as one of the enabling technologies for the twenty-first century. There are many technologies that can be feasible or achieved with the development of smart materials/structures. For example, in aerospace arena, the few key technologies which would be possible only through the smart materials/structures are vibration suppression/control, health/load monitoring, flow/shape control, smart skins in aerospace structures, acoustic control, stealth enhancement, selfreconfigurability, self-repairabilty, etc. Recent military aircraft have demonstrated piezoelectric vibration suppression on full-scale aircraft structures e.g. significant vibration reduction have been demonstrated on both F-18 tails and B-1 aft fuselage skin. Although performance has been validated,

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several key issues remain before piezoelectric vibration suppression can be transitioned to real-world systems.

2.1 Piezoelectric actuation and sensing The integration of piezoceramic (PZT) fibres within composite materials represents a new type of structural materials. Tiny PZT fibres of 30 m in diameter can be aligned in an array, electrodised and then integrated into planar architectures. Such architectures are embedded within glass or graphite fibre-reinforced polymers and become piezoelectric after being poled [6].

Piezoelectric fibre

composites (PFCs) have a large potential for active control, underlined that matrix and ceramic combinations, volume fractions, and ply angles contribute to the tailorability of PFCs, which make them applicable to structures requiring highly distributed actuation and sensing [7]. In the long run, manufacturing technologies of PFCs have been adopted from graphite/epoxy manufacturing methods and to date PFCs are being equipped with an interdigitated electrode pattern (IDEPFCs). Regardless of the electrode arrangement the piezoelectric composites create a class of active materials that can cover entire structures - the actuators that are conformable to curved elements such as shafts, tubes or shells [8]. Piezoelectric actuators and sensors associated with appropriate control systems are known to be effective to control low frequency and amplitude vibrations. The main difficulties when associating such active and passive damping mechanisms are that active controllers are generally very sensitive to system changes and the viscoelastic materials properties are frequency- and temperaturedependent [9]. A typical smart composite plate incorporating piezoelectric layers is shown on Figure 2.

Figure 2

Hence, to design a sufficiently reliable and robust control system, both piezoelectric and viscoelastic materials must be well modelled. Recently, some methods, such as fractional derivatives, anelastic displacements fields (ADF), Golla-Hughes-McTavish (GHM) and Yiu's, were proposed to model the frequency dependence of stiffness and damping properties of viscoelastically damped structures [10]. Comparison between GHM, ADF and iterative modal strain energy (MSE) models has shown that both GHM and ADF, though increasing the system dimension, are superior to MSE for time-domain analyses of highly damped structures [11]. Also, while GHM and ADF models are equivalent and lead to similar results, ADF yields to accurate damping prediction while minimizing additional degree of freedom (DOF) and the number of parameters to be curve-fitted to material master curves [11]. More so, operational temperature variations can lead to high changes in the active control performance [12,13].

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2.2 Static and dynamic application of piezoelectric smart materials

2.2.1

Static behaviour of piezoelectric actuated beams

Static testing of piezoceramic actuators is receiving very little attention in favour of dynamic piezoactuator uses (vibration damping, noise suppression, etc.). It is worthwhile to mention that much more needs to be learned about the static behaviour of piezoactuators if they are to be used for structural applications. In the past, many efforts have been done to model piezoelectric actuation using finite elements; however these were generally performed with in-house written codes. Although their formulations were analytic rather than FE, Crawley and de Luis [14] analytical formulations are often regarded as the starting point for most subsequent works. They provided the beginnings for the development of smart structures modelling using both surface bonded and embedded piezoactuators in beams. For surface bonded actuators, they assumed a linear strain distribution in the beam substructure and constant in the actuator. For embedded actuators, they assumed a linear EulerBernoulli-type strain distribution across the beam and actuator. Shen [15] developed a one-dimensional FE formulation that included the effect of coupling between longitudinal deflection and bending deflection. The work explained that the automatic meshing procedure of general-purpose FE codes might not be applicable in the case of adaptive structures containing discretized piezoelectrics. It was indicated that a manual meshing or semi-automatic meshing was necessary, which can be cumbersome and expensive in modelling practical structures, so the author developed a systematic modelling technique to predict actuation mechanisms of integrated active beams. Also, Crawley and Anderson [16] performed several techniques for modelling-induced strain actuation of beam and plate-like components of piezoceramic actuated structures. In another work, Crawley and Lazarus [17] performed techniques for modelling-induced strain actuation of beam and plate-like (isotropic and anisotropic) components of piezoceramic actuated structures. Plate strain energy relations were developed, and exact solutions were found for simple actuator/substrate systems - It is known that the physical behaviour of a piezoelectric material follows a nonlinear constitutive mathematical model. Notably, Akpan et al. [18] took a unique and different approach than everyone else in their FEM model in that they used a fuzzy FEM-based approach for modelling smart structures with vague or imprecise uncertainties. The finite element equations for a structural system can be obtained using any standard procedure [19].The most general form of the resulting equation of motion is given by

MU  CU  KU  F where M is the mass matrix, C is the damping matrix, and K is the

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stiffness matrix. U , U and U are the displacement, velocity and acceleration vectors, respectively, and F is the load vector. For a static analysis, this simplifies to

KU  F

(1)

and for natural frequency analysis it simplifies to

MU  KU  0

(2)

The coefficients of the matrices and vectors may contain fuzzy variables, which requires that these equations be solved using a fuzzy logic framework. Fuzzy FEM can be broadly classified into two groups, namely, explicit and implicit formulations. In the explicit formulations, the problem and solution strategies are explicitly developed in terms of interval operations at an α-level representation. For example, for the quadratic approximation, explicit functional relationship takes the form N

N

i 1

i , j 1

y ( x)  a   bi xi   cij xi x j

(3)

where N is the number of the fuzzy variables x , y (x) is the approximation of the response function, and the coefficients a , b , and c are determined by application of regression analysis on results from numerical experiments. The advantage of this approach is that the bounds of the fuzzy response at αlevel can be obtained by solving the resulting set of interval equations once. This approach is mathematically elegant and not computationally expensive. However, it is plagued with practical problems. In the implicit formulation, all the binary combinations of the extreme values of the fuzzy variables at α-level are fed into a deterministic finite element model. The bounds of the fuzzy response at the α-level are then obtained by choosing the maximum and minimum responses. This procedure is repeated for all the α-levels of interest. The higher the number of α-levels under consideration means the greater the accuracy of the possibility distribution of the response. The advantages of the implicit formulation include the ability to use existing deterministic finite element codes and the ability to give accurate and bounded results for realistic and bounded fuzzy input parameters. However, the amount of computational effort that is required with this method grows exponentially with the number of fuzzy input variables that are used in the analysis. Because of this reason, most of the reported studies have restricted the number of fuzzy variables that are used in analysis. A typical flow chart for the solution strategy that has been developed is illustrated on Figure 3 [18].

Figure 3

Fuzzy sets were used to represent the uncertainties in the piezoelectric, mechanical, thermal, and physical properties of the smart structure. They found the methodology to be accurate and efficient for

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solving practical problems, and packaged it in a computation tool called SMARTCOM [18]. Gaudenzi and Bathe [20] developed a simple FE approach modelling the electromechanical coupled behaviour of piezoelectric two- and three-dimensional continua that could be used in the general nonlinear incremental FE solution. They based the technique on separating the FE equations for the mechanical response and the electrical field. The response for the direct and converse effect could then be solved, and an iterative process used to account for the electromechanical coupling. At the end of their report, they were able to compare their numerical results using the FE technique with the experimental aluminium cantilever results from Anderson and Crawley [17]. Tzou and Tseng [21] developed a piezoelectric FE model with internal degrees of freedom. They used their FE technique to evaluate the dynamic performance of a plate with an integrated, distributed piezoelectric sensor/actuator. They derived a thin piezoelectric solid element with internal degrees of freedom by using a variational principle, and the dynamic system equation was formulated using Hamilton’s principle. In the meantime, Ha et al. [22] modelled the dynamic as well as static response of laminated composites containing distributed piezoelectric ceramics subjected to both mechanical and electrical loadings. Their formulation used the variational principle, considering both the total potential energy of the structures and the electrical potential energy of the piezoceramics. They devised an eight-noded, 3-D brick element, and 3-D incompatible modes were introduced to account for the global bending that results from the local deformations of the piezoceramics.

2.2.2

Dynamic behaviour of piezoelectric actuated beams

The finite element formulation has been proposed for the dynamic analysis of smart composite structures [23]. Based on visco-piezoelectricity, the numerical procedures were developed to analyse dynamic responses of composite structures with piezoelectric sensors and actuators as well as viscoelastic damping layers. A solid element with Allman's rotation was used since it is more efficient for structures of arbitrary shape and can avoid compatibility discrepancies between plate and solid elements. In the meantime, Chattopadhyay et al. [24,25] developed a hybrid displacement field theory to model the deformation in the various layers for a rotor blade modelled into a composite box with surface controlled layers. PZT actuators were first surface bonded to all walls of the box beam to provide both lead-lag and flap actuation and to enhance control through structural coupling. Further, surface controlled layers were bonded to the upper and lower surfaces to provide passive damping in follow-up investigations. The investigations entailed the use of a quasi-steady aerodynamic model to calculate the aerodynamic loads along the span of the blade. The linear quadratic regulator (LQR) controller method was used to design the control system for vibratory load reduction with PZTs. The non-linear aerodynamic model was linearized for control system design. The control system was

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coupled with a hybrid optimisation technique to investigate the structures/control interaction problem associated with vibration reduction. Ground and air resonance models were implemented in the rotor blade built around the composite box beam with segmented SCLs to investigate the stability issues. Despite the usefulness of analytical solution of the forced vibration for the study of vibration suppression, very few analytical works about the forced vibration of composite sandwich beams can be found in the literature despite tremendous analytical work on buckling and free vibration of composite sandwich beams. Unlike the usual orthogonality relation for shallow beams in which the modal shapes are the shapes of deflections corresponding to the natural frequencies, the relation for the composite sandwich beams cannot stand without including the rotation angles. Through the establishment of the orthogonality relation, the forced vibration problems can then be solved by modal analysis. Hwu et al. [26] investigated smart composite sandwich beams with surface bonded piezoelectric sensors and actuators, and even went further and obtained the closed-form solution for the free vibration problems and derived an orthogonality relation consisting of the effects of rotary inertia and shear deformation. Based upon the analytical results for the forced vibration, the sensor equation and actuator equation associated with the surface bonded piezoelectric sensors and actuators were also derived based upon analytical expressions of forced vibration. A linear quadratic Gaussian with loop transfer recovery (LQG/LTR) controller (Figure 4) was then designed on the basis of these analytical solutions, in which the Kalman filter was used as an observer, and the control gain was determined to minimize a linear quadratic performance index.

Figure 4

The parametric studies showed the first few vibration modes were successfully controlled. On the other hand, Trindade et al. [27] conducted pioneer work on the analysis of LQG algorithm and the state space real representation of complex modal reduced models for hybrid piezoelectric-active viscoelastic-passive vibration control. The FE model assumed Lagrange linear shape functions for the mean u and relative u~ axial displacements of the surface layers and Hermite cubic ones for the transverse deflection w . Electrical difference of potentials Vkj

k  a, b; j  1,..., (nˆmˆ )

were

ˆ piezoelectric sub-layers of the surface layer, k , and were assumed considered only in the nˆ , m constant and uniform in the element (Figure 5).

Figure 5 With these assumptions, the following elementary DOF column vector qˆ e may be written:

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qˆe  col uˆ1 , w1 , w1, u~1 , u~2 , w2 , w2 , u~2Va1e ,...Vaiiˆe ,Vb1e ,...Vaiiiˆe  where

(4)

  implies the x-space derivative, u  ua  ub  2 and u~  ua  ub  .

Discretization of the inertial, electromechanical and external virtual works of d'Alembert's variational formulation, leads to the equations of motion

    k where

 Mˆ j

e kj

   Mˆ ce qˆe      k

 Kˆ

e kj

j

  Kˆ ce qˆ e  Fˆme , k  a, b; j  1,..., (nm)  

 stands for the second time-derivative. Mˆ ..

e kj

(5)

and Mˆ ce are the elementary mass matrices of the

k j th face sub-layer and core respectively. They are due to contributions related to translations in the x and z directions, and rotations. Lamination of the surface layers also induce translation-rotation e coupling terms. The stiffness matrices of the face sub-layers Kˆ kje decomposed into mechanical Kˆ kjm , e e piezoelectric Kˆ kjme and dielectric Kˆ kje terms, such that e e e T e Kˆ kje  Kˆ kjm  Kˆ kjme  Kˆ kjme  Kˆ kje

(6)

whereas that of the core Kˆ ce is composed only of a mechanical contribution. These matrices are defined in terms of the generalized displacements, u , u~ and w . Fˆme is a mechanical point forces vector added a posteriori to the discretized system. Since the mechanical and electrical DOF are coupled statically only and, decomposing the element DOF vector qˆ e in mechanical DOF qe and unknown (sensor) VSe and applied (actuator) V Ae voltages





so that qˆe  col qe , VSe , VAe , system may be condensed as



1

T



M e qe  K efm  K efmeS K efmeS  K ce qe  Fme  Fee

(7)

where K ef represents the sum of all the stiffness matrices of faces sub-layers. The applied voltages

V Ae provide an equivalent electrical load vector Fme  K efmeAVAe , and the unknown potentials VSe are 1

T

related to the mechanical DOF VSe  K efeS K efmeS qe . Assembling the condensed system for all elements produces

Mq  Dq  K f  K c q  Fm  Fe

(8)

where D is a viscous damping matrix added a posteriori and q is a velocity vector. The work evaluated the control design and performance of the piezoelectric active control of damped sandwich beams using three control algorithms applied to the reduced-order model i.e. linear quadratic regulator (LQR), linear quadratic Gaussian (LQG) and derivative feedback. The drawback

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of the LQR control algorithm is that it requires the measurement of all state variables. This may be remedied for by considering an optimal state observer, leading to an LQG regulator. To guarantee control feasibility and prevent piezoelectric material depoling, Trindade et al. [27] used these algorithms in an iterative form to account for maximum control voltage. Parametric analyses of an actively controlled damped sandwich beam (Figure 5) indicated that LQR controllers improve some selected modal dampings, while retaining the passive damping of uncontrolled modes. Derivative feedback controllers were less effective than an LQR one, but their well-known spillover destabilizing effects were attenuated by the increase of stability margins provided by the viscoelastic damping. It was also shown that LQG controllers perform as well as LQR ones. Moreover, the delay effect induced by the state estimation of LQG associated with the passive attenuation lead to a damping performance similar to that of LQR with less control voltage. Unfortunately, the work ignored the temperature dependence on the grounds such that it was considered less important, since temperature changes are slow compared to the structural dynamics. Hence, the temperature was considered known but constant.

2.3 Experimental studies The investigations by Crawley and Anderson [17] involved performing several techniques for modelling-induced strain actuation of beam and plate-like components of piezoceramic actuated structures. They specifically used ADINA [28] to model their structures, and performed experiments to verify their FE solutions. Their models included the extension, bending, and localized shearing deformations induced in these structures. Works by Bernhard and Chopra [29,30] investigated on the helicopter applications including static deflection tests. In the meantime, Barrett [31-34] on missiles conducted a series of static deflection tests related to missile wings and missile fins, beginning with directionally attached piezoelectric (DAP) elements and enhanced DAP elements and eventually focusing on active flexspar actuators. In each case, various static and dynamic tests were performed to determine the deflection characteristics of the structures and to evaluate the missile flight performance enhancement. Chandra and Chopra [35] also performed static deflection experiments. They were verifying the results of their analytical structural modelling of couple composite beams with distributed induced-strain actuators. They fabricated several bending-torsion and extension-torsion couple graphite-epoxy solid beams, surface mounted with piezoelectric actuators. The actuators produced the local bending moment and axial force on the beam, and they measured the bending slope, induced twist, and surface strain. In another development, Yocum and Abramovich [36] presented a colleration studies on the static deflection behaviour of a piezoactuated structure. The study detailed the static experimental behaviour

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of a cantilever beam actuated using piezoceramic patches. FE modelling was used to compare with experimental results - the FE model started at its most basic and increased in complexity throughout the study from linear to nonlinear. It was noted that experimentally, it is important to properly prepare a piezoelectric material by running a DC voltage at or higher than the test voltage to eliminate the problem of drift, which is a slow increase of the free strain with time after the application of a DC field. Another important consideration noted was the collection of data in one pass as voltage was increased. The researchers highlighted that hysteresis and residual strains can complicate data collection if the test is cycled through the various voltages. At lower voltage applications, FE program ADINA [28] (commercially available) predicted the deflection better if the constant manufacturersupplied value for the piezoelectric coefficients (d31) was used, and for higher voltages, the nonlinear form of d31 (piezoelectric manufacturers supply one constant value denoted as d31) must be used; although nowadays there are FE techniques that can be used now to better model the nonlinear electromechanical coupling in piezoelectric materials. Hagood and von Flotow [37] produced the first analytical model of an L  R shunted piezoelectric vibration absorber. They found that the key performance parameter was the generalized piezoelectric coupling coefficient (K ) which is the ratio of mechanical energy converted to electrical energy by the piezoelectric material. This is a modal quantity that varies with the vibration mode in question. They were able to experimentally (or analytically) determine this quantity as the difference between the open-circuit natural frequency and short-circuit natural frequency:

     K   D 2

E 2

(9)

D 2

To maximize performance of the piezoelectric vibration absorber, this quantity must be maximized. Hagood and von Flotow also determined the optimal tuning for the damped vibration absorber in S terms of the shunt frequency, electrical , and damping ratio r , as

S electrical 

1 LC

s



1  K

E r  RC S mechanical  2

S

E  mechanical

K 1 K 2



(10)



(11)

The analysis involved in determining these factors are analogous to that used for damped mechanical vibration absorbers. They experimentally verified all of their results. Many researchers have since then extended the initial results of Hagood and von Flotow [37].

Edberg and Bicos [38]

experimentally, and Hollkamp [39] analytically, demonstrated the use of a single monolithic piezoelectric patch to damp multiple modes. Hollkamp [39] also determined a novel method of

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determining the generalized piezoelectric coupling coefficient by using the standard finite element software, which is given as:

     k     1  k  

  E 2  * K2  2  E 

2

2

(12)

2

The quantity  E is the short-circuited natural frequency of the mode, k is a material parameter, and

 * is the natural frequency of the mode with the mass of the piezoelectric material included, but the stiffness neglected. Hollkamp and Starchville [40] also demonstrated a self-tuning shunt. Using a tuning criterion from the literature, they were able to react to changes in the tip mass of a cantilever beam, adjusting the frequency of the shunt using a motorized resistor. Note that due to the high inductance required most researchers have utilized synthetic inductors.

2.4 Comments The piezoelectric materials generate an electric charge in response to mechanical deformations. Conversely, these materials produce mechanical strains under an applied electric field. Piezoelectric materials are available in a wide variety of shapes and sizes and can be distributed along a structure without greatly increasing its mass. A piezoelectric actuator in an adaptive structure is a thin rectangular element that is generally poled in the thickness direction and is usually bonded to the surfaces of the host structure. The application of an electric field in the thickness direction causes the actuator's lateral dimensions to increase or decrease. The lateral deformations of the actuator force the host structure to deform. In order to successfully incorporate piezoelectric actuators into a structure, the mechanical interaction between the actuators and the host structure must be fully understood [41]. Several mechanical models and finite-element studies have already been presented for hybrid beams and plates with thickness-poled actuators. Unfortunately, brittle behaviour and electric fatigue are two major concerns that limit industrial applications of piezoelectric materials. Excellent review papers on piezoelectric elements commonly used as sensors and actuators in adaptive structures have been provided by Rao and Sunar [42] on piezoelectricity and its use in disturbance sensing and control of flexible structures: Chopra [4] on state-of-the-art of smart structures and integrated systems, Tani et al. [43] on intelligent materials systems: application of functional materials, Ballhause et al. [44] on modelling of smart structures and Sunar and Rao [45] on sensing and control of flexible structures via piezoelectric materials technology, and Ballhause et al. [44] has provided a discussion on theories addressing ZZ and IC. Basing on reviews mentioned there is enough evidence that piezoelectric ceramics have become preferred materials for a wide variety of electronic and mechatronic devices due to their pronounced piezoelectric, dielectric, and pyroelectric

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properties. However, as mentioned earlier on, piezoelectric ceramics are brittle and susceptible to cracking at all scales ranging from electric domains to devices. Various defects, such as domain walls, grain boundaries, flaws and pores, impurities and inclusions, etc., exist in piezoelectric ceramics. The defects cause geometric, electric, thermal, and mechanical discontinuities and thus induce high stress and/or electric field concentrations, which may induce crack initiation, crack growth, partial discharge, and cause dielectric breakdown, fracture and failure. Due to the importance of the reliability of these devices, there has been tremendous interest in studying the fracture and failure behaviours of such materials. Among the many types of piezoelectric elements, the cylindrical (solid and hollow) shape is used in a broad range of practical applications such as resonators, actuators, fuel injectors, atomic force microscopes, high-precision telescopes, etc. The study of electroelastic field of a piezoelectric cylinder under combined electromechanical loading is therefore one of the fundamental problems of adaptive structures technology. Electric field and stress concentration in a cylindrical element could lead to dielectric breakdown, electrode delamination and fracture. Furthermore, tensile stresses due to applied loading could lead to tensile fracture as the tensile strength of these materials is relatively low. And because of their self-monitoring and self-adaptive capability, so-called advanced intelligent structures have attracted considerable research over the past few years. These structures have some distinct advantages over conventional actively controlled structures. Since intelligent structures are characterized by distributed actuation and sensing systems, more accurate response monitoring and control are possible. Some of the most significant work has concentrated on the development and implementation of actuators and sensors made of piezoelectric materials. In most applications, the piezoelectric materials are used as sensor/actuator layers by embedding them in a multilayered structure, which is often made by, advanced anisotropic composite materials.

Although

piezoelectricity has a long history, its use in actuation and control of light flexible structures for aerospace applications is relatively new. The most important problems arising in the design of intelligent structures embedding piezo-layers are manufacturing, electro-mechanical modelling, optimization and control. Furthermore, the accurate description of mechanical and electrical fields in the layers is, in fact, essential in order to both perform a realistic simulation of direct/converse effect and prevent failure mechanisms of the structure. The main issue of multilayered piezoelectric constructions is related to the possibility of exhibiting different mechanical–electrical properties in the thickness direction. In addition, anisotropic multilayered composites often exhibit both higher transverse shear and transverse normal flexibilities, with respect to in-plane deformability, than traditional isotropic one-layered ones. As a consequence, plate theories that are based on the extension of so-called Kirchhoff (classical lamination theory, CLT) and Reissner–Mindlin (first order shear deformation theory, FSDT) hypotheses can be

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ineffective to trace static and dynamic response of piezoelectric plates. Higher transverse deformability demands the inclusion of transverse shear and normal stresses which are discarded in classical analyses. Furthermore, transverse discontinuous mechanical properties cause displacement





fields u  u x , u y , u z which can exhibit a rapid change of their slopes in the thickness direction in correspondence to each layer interface. This is known as the zig-zag (ZZ) effect. Nevertheless,





equilibrium reasons require interlaminar continuity for transverse stresses,  n   xz ,  yz ,  zz . As far as electrical variables are concerned, it should be noticed that in order to include a correct description of electrical stiffness, the electric field should have at least a linear distribution in thickness direction of the piezo-layers. As a consequence, at least a parabolic assumption for the electric potential into the layers is required. However, most of the articles proposing refined theories for piezoelectric plates restricted the numerical comparison to classical plate theories, such as CLT and FSDT, and to available 3D solutions. Only in some cases other refined theories were included in the verification of the new method. Such a restriction does not permit to give a complete overview and assessment of available theories. In the study of piezoelectric fractures and failures, available solutions in purely elastic media have been extended to the corresponding problems in piezoelectric materials, for example, crack problems, inclusion problems, Green's function problems, problems concerning interactions between dislocation and cracks and/or inclusions in piezoelectric media, thermo-electro-elastic problems, dynamic fracture problems, and problems related to failure criteria. However, the fracture behaviours of piezoelectric ceramics are more complex than that in conventional materials, because of the non-linear nature of the mechanical and electrical behaviours and the complicated coupling relationship between mechanical and electric fields. There are new challenges in studying the fracture behaviour of piezoelectric ceramics, e.g., the determination of the electric boundary conditions on crack faces, the effect of electric fields on the piezoelectric fracture behaviour, the calculation of the global and local energy release rates. More challenging tasks include non-linear simulations of the fracture behaviour of piezoelectric ceramics because domain switching causes the non-linearity between the polarization and the electric field strength, the non-linearity of the strain versus the electric field strength, and the non-linearity between stress and strain. Theoretically, mechanism-based micromechanical approaches and phenomenological methods are all adopted to solve the non-linear fracture problems of piezoelectric ceramics and much effort has been dedicated to the modelling of the non-linear failure behaviour of ferroelectric and ferroelastic ceramics. At the moment, although there are voluminous theoretical studies on the fracture of piezoelectric materials, the number of experimental studies is still limited. Due to the complexity of the failure behaviour of piezoelectric ceramics, many failure and/or fracture criteria and mechanisms have been proposed, each of which can successfully explain and predict a failure mode under a certain circumstance.

14

In recent years several new shell elements have been proposed, where piezoelectric constitutive relations have been included in the underlying element formulation. Some of these elements model a reference surface of the shell structure. Several of these element formulations are restricted to shallow shell structures, where the initial shell curvature is assumed to be small. Due to the fact that the piezoelectric devices have traditionally laminate forms, the above-mentioned shell formulations include a more or less sophisticated laminate theory. A survey of different laminate theories is given in Saravanos and Heyliger [46]. The so-called solid shell elements circumvent complicated laminate theories by modelling the top and the bottom surface of shell structures - for each ply in a laminate one element is employed in thickness direction. Benjeddou [47] conducted a survey on advances in piezoelectric finite element modeling of adaptive structural elements. And Pines and Hiraishi [48] critically looked in to the special issues on smart materials and structures technology. A specific field in which piezoelectric materials show an increasing importance is that of piezoelectric motors. The travelling wave ultrasonic motors represent a new generation of motors, suited for many applications and their technology has experienced growing interest in the last decade due to the particular advantages that new piezoelectric materials offer. Nakamura and Uhea [49] offers an excellent review and discussion on potential ability of ultrasonic motors. Ultrasonic motors have been commercially employed for automated focusing systems of cameras, for special instruments and

X  Y positioning systems [50], while practical researches aiming at producing systems for transport and office automation are now being undertaken. Nevertheless, a large-scale employment has been somehow hindered so far by some problems related to their long-term reliability and high cost. In short, application of smart structure technologies attracts increasing attention in engineering and industrial applications, especially in the wide field of noise and vibration reduction of thin walled structures. The development and industrial realization of piezoelectric smart structures require effective and reliable simulation and design tools. The finite element method provides an excellent basis for such software tools, which meet engineering requirements. And the following points need to be noted: 

One of research interest can be focused on the relation between the controller design and the structure, which consists of the passive base structure, actuators and sensors as well as the control electronics. The quality of a smart structure depends decisively on the amount, the shape and the distribution of the active material at the passive structure. The optimal design of the actuator/sensor configuration is a very complex problem, which has not been fully solved.



Although piezoelectricity has a long history, its use in actuation and control of light flexible structures for aerospace applications is relatively new. The most important problems arising

15

in the design of intelligent structures embedding piezo-layers are manufacturing, electromechanical modelling, optimization and control. 

The field of smart or adaptive structures has progressed over the past decade from simple cantilever beam experiments to complex real-world aircraft structures. Enabled by piezoelectric materials, vibration suppression/control has led the way. These techniques were recently applied to several full-scale aircraft applications.



A specific field in which piezoelectric materials show an increasing importance is that of piezoelectric motors. At present, a large-scale employment has been somehow hindered so far by some problems related to their long-term reliability and high cost.



Challenging tasks include non-linear simulations of the fracture behaviour of piezoelectric ceramics because domain switching causes the non-linearity between the polarization and the electric field strength, the non-linearity of the strain versus the electric field strength, and the non-linearity between stress and strain.



An area of special interest for future vehicles will be the suppression of vibrations with smart materials.



One of the major areas, that still need investigation, is the integrity of smart/adaptive structures as the whole system. Even though the future for the smart structure technology is extremely promising, its application to the real-life applications would not be possible until its integrity issues are fully understood. Mechanical response and integrity of smart structures/materials under various operating conditions, such as static loads, cyclic loads, temperature variations, radiation, corrosion, humidity, impact etc. is therefore needed. It should be reiterated that the author is emphasizing here the issues related to fully integrated adaptive structures, not pertaining to the components such as characterization, performance and failure of PZT itself (i.e. when not integrated with the structure). There has been a reasonable amount of the research in the latter part, but a very little investigation has been reported pertaining to the adaptive structures integrated with PZT.

3

Surface layer treated structures

Undesirable large-amplitude vibrations and radiated noise often impede the effective operation of various types of dynamic civilian and military systems such as rotorcrafts, missiles, land vehicles, and weapon systems. It is prudent to introduce structural damping into a dynamic system to achieve a more satisfactory response. The development of effective and economical structural damping approaches that can suitably adjust mechanical properties to appropriate specifications could be beneficial in designing future systems [51]. Recently, new concepts for enhancing the structural

16

damping characteristics were introduced in the study of adaptive structures. Such active damping techniques, based on combinations of viscoelastic, magnetic, and/or piezoelectric materials, magnetorheological (MR) fluids, shunted electric circuits, and active non-linear control strategies, integrated methods using smart materials and passive circuits to dissipate energy, have emerged as several likely candidates for improving structural performance and reliability.

3.1 Finite element formulations in conjunction with smart materials Finite element based multibody dynamics formulations extend the applicability of classical finite element methods to the modelling of flexible mechanisms. Such numerical procedures implement a number of generalized elements, each providing a basic functional modelling capability. As multibody formulations become more widely accepted, the need to model a wider array of phenomena increases. Constrained layer treatments have been used for years to reduce structural and machine vibration. For simple 1-D structural components, for example, beams, closed form solutions are still possible. For two-dimensional structural components, such as plates and shells, it is more practical to obtain meaningful solutions through finite element calculations [52]. In the past, researchers have developed various finite element formulations for constrained layer structures. Basically, there are two different approaches. The first approach is to use existing finite element codes, such as NASTRAN, to analyse constrained layer plates and shells. Several works have been dedicated to this approach, e.g., Lu et al. [53] analysed constrained layer plates with NASTRAN whereby the plates and the constraining layers were modelled by plate elements, and the viscoelastic layers were modelled by brick elements. Soni and Bogner [54] used MAGNA to analyse constrained layer treatments but in this case the constraining layers and the base structures were modelled by eight-node thin shell elements or penalty function shell elements. The viscoelastic layer was modelled by eightnode solid elements with reduce integration. Work reported by Johnson and Kienholz [55] applied NASTRAN to analyse vibration of beams, rings, and plates with constrained layer damping. For constrained layer plates, they modelled the constraining layers and the base structures through plate elements and modelled the viscoelastic cores with solid elements. Mace [56] used NASTRAN to verify a new theory developed for constrained layer beams. Imaino and Harrison [57] used the P-code, which uses higher-order base function and could model systems with high aspect ratios with relatively few elements, to analyse vibration of partially treated beam. And lastly, Lu and Killian [58] analysed damped plate composites by using NASTRAN. The constraining layers and the base structures were modelled with homogeneous fournode-quadrilateral plate elements. The viscoelasticcores were modelled with linear isoparametric

17

eight-node brick elements. The numerical results were obtained in the form of driving point mechanical impedance. The second approach is to develop new elements for structural components with constrained layer treatments. These new elements can vary substantially because different elements have different assumptions. For example, Rikards et al. [59] developed a six-node triangular plate element. Three nodes were on the constraining layer and three on the base plate. The displacement field of the viscoelastic layer was determined through kinematic compatibility, that is, no-slip conditions. Each node had six degrees of freedom, i.e., three translational and three rotational. They performed numerical tests on rectangular plates. Rao et al. [60] developed offset beam elements and shear deformation viscoelastic elements to model constrained layer damping. Elsewhere, Ramesh and Ganesan [61,62] analysed vibration and damping of cylindrical shells with a viscoelastic core. They used two-node finite elements based on different shell theories. In a more recent work, 18-node degenerate constrained layer element with nine nodes located on the base shell (or plate) structure and nine nodes (each node with five degrees of freedom) on the constraining layer was developed for plate and shell structures [52]. For thin plate structures, numerical results showed that the isoparametric element could predict natural frequencies, loss factors, and mechanical impedances that are as accurate as NASTRAN with substantially fewer elements. Further, it was also shown that for thin shell structures, applications of the isoparametric formulation are possible, if spurious modes control can be implemented. And lastly, Ioannides and Grootenhuis [63] developed a triangular plate element. They assumed that all three layers had the same deflection and slopes. Therefore, the plate element had two in-plane displacements for the constraining layer, two in-plane displacements for the base plate, and one out-of-plane deflection and two slopes for all three layers. They also performed experiments on circular plates and beams with different boundary conditions.

3.2 Dynamic behaviour of structures treated with damping layers A method based on the transition matrix approach to analyse the static and dynamic characteristics of structures doped ‘distributed-parameter finite element method’ (DPFEM) has been developed [64,65]. DPFEM was targeted on dynamic interaction between the viscoelastic damping layer, magnets, the piezoelectric sensor, the piezoelectric constraining layer, and the base structure as well as interaction between the vibration and sound radiation from the smart structural systems, untreated structures as well as structures treated with passive constrained layer damping (PCLD), active constrained layer damping (ACLD), magnetic constrained layer damping (MCLD) and piezo-magnetic constrained layer damping (PMCLD) treatments. Figure 6 displays the main parameters of a beam treated with the ACLD when considered undeflected and deflected [66].

18

Figure 6

The shear strain was considered as

  hw  u1  u2  h2 ,

h  h2  h1 2  h3 2

(13)

where u1 and u3 are longitudinal deflections of the piezoactuaor layer and beam/sensor layer respectively. w denotes the transverse deflection of the beam system, x is denotes partial differentiation with respect to x and h1 , h2 and h3 defines the thicknesses of the piezoactuator, the viscoelastic layer, the piezosensor/base beam system respectively. The equation and boundary conditions governing the operation of the beam/ACLD system were obtained applying Hamilton’s principle as t1



t1



3



t1

i 1



t1



2





j 1



   KE  U i dt     KE  W j dt  0

(14)

where  (. ) denotes the first variation in the quantity inside the parentheses. Also, the following were considered: kinetic energy, KE  bending energy, U 2  piezoforces, W1  K1



L

0

L L 1 l 2 1 1 m wt dx , extension energy U1  K1  ut2 dx  K 3  u32 dx , 0 0 2 0 2 2

L L 1 1 Dt  w2x dx , shear energy, U 3  G2 h2   2 dx , work done by 0 0 2 2

L

 p u1 dx and work dissipated in the viscoelastic core, W2  h2   d  dx x

0

where m is the mass/ unit width and unit length of the sandwiched beam, L is the beam length,

K1  E1h1 and K 3  E3 h3 with E1 and E3 denote Young’s modulus of the piezoactuator layer and beam/sensor system. Also, Dt  E1 I1  E3 I 3  per unit width, with E1 I1 and E3 I 3 denoting the flexural rigidity of the piezoactuator and the beam/sensor layer respectively. The storage of the viscoelastic layer is G2 and  p is the strain induced in the piezoelectric constraining layer. In the study,  p was assumed constant over the entire constraining layer.  d is the dissipative shear stress developed by the viscoelastic core, and was given as  d  G2 /   t  iG2  where  ,  and

i denote the loss factor of the viscoelastic core, the frequency and

 1 , respectively. The DPFEM

method provides an exact model of the structures without any mode truncation or without the need for assuming shape or interpolation functions while using the smallest number of distributed-parameter elements (DPE). Using the DPFEM approach, the frequency domain description of the dynamics of the smart structures/PMCLD treatment lends itself to the use of the H2/H robust control theory where

19

constraints can be imposed to ensure disturbance rejection and to the accommodation of parametric uncertainty. The distributed and modal parameters of the smart structural systems (stationary and rotating beams, plates and cylinders) treated with the PMCLD identified using time and frequency domain methods. Arafa and Baz [67] developed a FE model to investigate the energy dissipation characteristics of active piezoelectric damping composites (APDC). The APDC considered consisted of piezoelectric rods that were obliquely embedded in a viscoelastic matrix to provide active control of its shear and compression damping characteristics. A derivative controller was considered in the work such that the electric field was expressed as,

Ez  K g . 

(15)

where

K  g sC g

(16)

d



and g d is the derivative controller gain, s is the Laplace operator, and C is a vector denoting the degree(s) of freedom upon which control effort is based. The system equations of motion may then be written as,

M  K     f g d i C 

(17)

and hence the effective system stiffness becomes,

K   K   f g i C eff

(18)

d

Note that the second term on the RHS of the above equation represents the added stiffness due to the APDC piezoelectric effect. The system total energy is then given by,

U

 

1 T   K eff   2

(19)

where   is the vector of nodal displacements corresponding to the externally imposed deformation pattern. The loss factor, defined as the ratio of the dissipated to the stored energy, may then be expressed as follows,



Im(U ) Re(U )

(20)

For any APDC configuration, the constitutive relationships were used, along with Eqns. (15), and Eqns. (17) to (19) to determine the effect of various design parameters of the APDC on the loss factor. In this manner, the optimal design parameters were determined to maximize the energy dissipation characteristics of the APDC treatment. It then follows that discrete APDC patches may be bonded to or embedded within structural systems to control their vibrations. The APDC, with their inherent active and passive control capabilities, provide effective means of enhancing the energy dissipation

20

mechanism of the viscoelastic layer and hence the overall dynamic behaviour of the system can be improved. The effect of the inclination angle of the piezoelectric rods and the control gains on the energy-dissipation characteristics of APDC was presented. The results obtained indicated that effectively high loss factors might be attained by proper selection of the design parameters of the APDC patches, which in turn makes the APDC suitable for controlling structural vibrations and noise over a broad frequency band. Also, amplitude attenuation of about 54% for the first vibration mode and almost 95% for the second mode were achieved with control voltages not exceeding 350 V. Attenuation of higher modes were also observed. Comparison with PCLD indicated the effectiveness of APDC in attenuating the resonance peaks while having slightly higher response values at off-peaks. However, further investigations were recommended for optimal placement and sizing of the APDC patches in order to study plate and shell vibrations as well as to attain an optimal balance between the vibration damping, control effort and treatment weight. In another development, Yellin et al. [68] developed a mathematical model for 1-D PSOL damping treatments in which the standoff layer was non-ideal and therefore had finite shear stiffness and a nonzero bending stiffness. This mathematical model considered the base beam and standoff layers as an asymmetric composite, which allowed the standoff layer to contribute a bending stiffness to the overall damping treatment. In addition, this model gave the standoff layer finite shear stiffness and the capability to dissipate energy internally. Numerical study showed that the ideal standoff layer assumption was reasonably accurate when the standoff layer had relatively high shear stiffness and low internal damping. However, for standoff layers with significant internal damping, this assumption became much less accurate. Verification experiments measured the frequency responses of two beams treated with PSOL damping treatments and showed that the analytical model was able to predict very accurately the frequency responses of the beams treated with PSOL.

3.3 Experimental studies Fundamental studies on surface damping treatments for beams, plates, and shells have been vigorously pursed primary due to problems encountered in non-linear vibration control problems such as in helicopter rotor blades and acoustic excitations from the jet engine right from the design stage. In recent years, active standoff constrained layer (ASCL) treatments have been proposed to address such problems [52,68,69]. The following existing studies of passive standoff layer (PSOL) damping treatments assumed ideal physical conditions for the standoff layer. In the ideal case, the standoff layer simultaneously has infinite stiffness in shear and zero stiffness in bending. With this assumption, the standoff layer itself does not dissipate any additional energy internally. Such studies include experimental work and theoretical predictions by Falugi [70,71]and Parin et al. [72] on plates and

21

airplane wings partially treated with a PSOL treatment in which the standoff layer was slotted. This early model for a discontinuous standoff layer qualitatively predicted the trend of the experimentally measured loss factors. Garrison and Miles [73] also presented an analytical model for the random response of a plate partially treated with PSOL. Using commercially available slotted standoff layer (SSOL) damping treatments, Rogers and Parin [74] demonstrated experimentally that these treatments provided significant damping in aeronautical structures such as fuselages and wing skins. Experimental and finite element studies by Tao et al. [75] confirmed that these commercially available SSOL damping treatments were effective in reducing low frequency vibration in aerospace structures. An analytical study of passive stand-off layer damping treatments by Mead [76] considered finite shear stiffness and internal loss factor in the stand-off layer by assuming that the viscoelastic and stand-off layers act as dampers in parallel deforming in pure shear. However, this method does not allow the standoff layer to have a bending moment and therefore makes the total contribution of the standoff layer more difficult to quantify. In another development, the researchers corrected the Mead-Markus model by posing more accurate boundary conditions by incorporating the existence of thickness deformation in constrained layer treatments, and then verifying the improvements through experiments [52,68]. The experimental results indicated that the nitinol-aluminum metal matrix composite presents important passive damping characteristics compared with other structural materials (e.g., 1% at 200C and 2% at 200C). In addition, the damping could be turned on or off via temperature control.

3.4 Comments In an active composite plate/shell/structure structure, distributed sensors and actuators are embedded to control the response of the structure by means of a control algorithm. The performance of smart structures depends on the magnitude of the piezoelectric stress/strain coefficients. The magnitudes of the piezoelectric coefficients of the existing monolithic piezoelectric materials being used in smart structures are very low. Hence, large control voltage is necessary for achieving significant amount of active damping in smart structures if the piezoelectric actuators are directly bonded to the structures. Piezoelectric materials are also being used as the component of ACLD treatment for potential improvement of the damping characteristics of smart structures. The ACLD treatment consists of a viscoelastic constrained layer and a piezoelectric layer acting as the active constraining layer. When the treatment is integrated with a base structure (substrate) and is augmented with an appropriate control strategy, the vibration of the base structure can be substantially damped out leading to the ACLD of this structure. It is well known that the flexural vibration control by the constrained layer damping treatment is attributed to the dissipation of energy in the viscoelastic core undergoing

22

transverse shear deformation. As the constraining layer of the activated ACLD treatment increases the passive transverse shear deformation of the viscoelastic constrained layer, the ACLD treatment improves the overall damping characteristics of the flexible structures over its passive counterpart. Since the control effort necessary to increase the shear deformation of viscoelastic layer is compatible with the low control authority of the monolithic piezoelectric materials, the piezoelectric materials perform much better to attenuate the vibration of smart structures when they are used as active constraining layer of the ACLD treatment than when they are used alone as distributed actuators. Also, ACLD treatment provides the attributes of both passive and active damping occurring simultaneously because of the fact that passive damping mechanism is integral to this treatment. Hence, since its inception ACLD treatment has been extensively used for efficient and reliable active control of flexible structures. However, very little work is done on the use of ACLD treatment for active control of laminated composite shells which are being most frequently used as light weight structures. For instance, design of space structures, robotic manipulators and the like need the development of high performance lightweight structures because of the stringent considerations as regards weight. The lightweight structures inherently possess low internal damping, higher flexibility and are susceptible to large vibrations with long decay time. Such structures require suitable integration of active means of control to show better performance in operation. The quest for the development of high performance lightweight structures has led to the generation of the emerging idea of providing these structures with self-monitoring and self-controlling capabilities. Consequently, the use of piezoelectric materials as distributed sensors and actuators has attained a great deal of importance in active control of vibrations of beams, plates and shells. Further efforts to improve the active damping have led to the development of ACLD treatment, which combines the attributes of both passive and active damping. Also in ACLD, the active part of the damping is due to the use of a piezoelectric actuator as a constraining layer. Thus piezoelectric materials are playing a major role in achieving active damping in smart structures. However, existing monolithic piezoelectric materials have low control authority as their piezoelectric constants are of very small magnitude. As the active damping of smart structures is dependent on piezoelectric material properties such as piezoelectric stress/strain constants, tailoring of these properties may further improve the damping characteristics of flexible structures. Applications of active structures range from vibration and buckling control, to shape control and noise suppression. Active structures have great potential in the design of light-weight and high-strength structures that are widely used in areas such as aerospace and automotive industries. In recent years, considerable effort has been devoted to the modelling and controlling of active piezoelectric shell structures. The coupled electromechanical properties of piezoelectric ceramics and their availability in thin shell form make them increasingly popular for the use as distributed sensors and actuator. The

23

direct and converse piezoelectric effects govern the electromechanical interaction in these materials. The direct piezoelectric effect states that a mechanical strain applied to the material is converted to an electric charge, while the converse piezoelectric effect states that an electric potential applied to the material is converted to a mechanical strain. Finite element modelling based on the classical laminate plate theory and first-order shear deformation theories has certain limitations due to their improper modelling of the piezoelectric structure. For instance, in the shell structures investigations, improvements have been made by using high-order shear deformation theories and layerwise shell element formulations, while some shortcomings still remain in that (i) they do not consider the transverse normal stress in the element formulation, which may affect the behaviour of multilayer structure; (ii) for the large deformation analysis, the finite rotation update associated with rotational degrees of freedom in shell formulations is complex to handle; and (iii) it is troublesome, if not impossible, to incorporate non-linear piezoelectric material models associated with large input signals into shell elements based on the plane-stress assumption. On the other hand, due to the complex geometry, the material anisotropy, the coupling of electric field and mechanical field, and the need to satisfy boundary conditions of both electric field and mechanical field, 3-D detailed modelling for piezoelectric shell structures is used extensively. For example, twenty-node solid element in and eight-node solid element with incompatible modes may used to model such structures. The eight node solid element with incompatible modes and the twenty-node solid element (with reduced integration) are not as accurate as shell elements for thin plate/shell problems. To achieve better accuracy, excessive numbers of solid elements are needed to be computationally economical; some works have proposed the use of 3-D solid elements in modelling piezoelectric devices, shell elements for the host structure, and transition elements to connect 3-D solid elements in the piezoelectric region with shell elements used for the structure. Such modelling is complex to handle because of the mixing of several types of elements and the need for tuning of the aspect ratio of the transition elements. Moreover, the use of 3-D solid elements leads to unnatural stiffening of piezoelectric devices, and as a result, artificially high natural frequencies. Alternatively, an assumed-stress piezoelectric solid shell element which have coarse mesh accuracy and could give accurate results for linear piezoelectric shell analysis may be used; by construction, however; the assumed stress method, obtained from the Hellinger–Reissner variational principle, encounters the difficulty when incorporating the classical strain-driven non-linear lateral models.

24

4

Structures embedded with shape memory alloy (SMA)

Rapid development in smart structures has enhanced the great efforts in understanding the mechanical and thermo-mechanical behaviour of shape-memory alloy (SMA) materials. It is well recognised that the mechanical properties of embedded SMA (Figure 7) are highly dependent on the integrity of the interface, particularly by the existence of high thermal-induced shear stress at the SMA wire/epoxy interface.

Figure 7

The unique shape memory effect in an equiatomic Ti–Ni alloy was first found by Buehler et al. [77] in 1963. Although Chang and Read [78] had found the same effect was found in Au–47.5 at. %Cd, and Burkart and Read [79] [80]and Basinski and Christian in In–Tl alloys earlier, it had not attracted much attention of researchers. In contrast, the Ti–Ni alloy became quite popular soon after the discovery, partly by the worldwide publicity by the people in Naval Ordinance Laboratory who found it suitable for specialised applications thanks to the alloy good mechanical properties. Despite the fact, the understanding of the phenomena and the martensitic transformation, from which the phenomena originate, did not develop rapidly. This is because the Ti–Ni alloy system is quite a complicated system, as it turned out later. The phase diagram of the system had been controversial until the end of 1980s. Various precipitates, which appear under certain heat-treatments, had not been understood well from the proposed phase diagram until then. Furthermore, the R-phase transformation which were thought to be a pre-martensitic phenomenon at one time, and is characterized by the 1/3 reflections along 110 * - the symbol * associated with u, v, r

represents that it is the direction in the

reciprocal space direction in the reciprocal space, appears under certain conditions prior to the martensitic transformation. These complexities may often appear all at once, which make them difficult to understand. Thus, it took a long time to separate each factor and to make them understandable. In fact, it took much longer time for the understanding of the martensitic transformation and the shape memory effect in the present alloy compared to those in other shape memory alloys. However, most of the difficulties including the structure determination of martensites are cleared by now by many systematic and extensive researches in 1970s, 1980s and thereafter after the incubation period in 1960s. Although Ti–Ni-based alloys have many common characters with other shape memory alloys by exhibiting shape memory effect, superelasticity, two-way shape memory effect (known as the all round shape memory effect) etc., they also exhibit many other characteristics, which are quite unique compared to other shape memory alloys. For instance, they exhibit quite a low elastic anisotropy ( A  c44 / c ) as low as nearly 2, although most of other shape memory alloys exhibit the value of

25

about 10 or more. Here c represents resistance for 110 110 shear, and c44 resistance for 0 0 1

110 shear. The elastic constant c44 decreases with decreasing temperature, which is just the opposite behaviour in most of other shape memory and normal alloys. Besides, the structure of martensite B19 (monoclinic) appears only in Ti-Ni-based alloys, and the structure of R-phase has similarity only with  2 martensite in Au–Cd alloys. Amorphisation of the alloys by sputtering is another advantage for the applications of thin films, since the process of amorphisation leads to small grain sizes, which are useful for mechanical properties. In addition to the above, Ti–Ni-based alloys have other good properties. Although it is a kind of intermetallic compound, it is quite ductile, under certain conditions 60% cold working being possible. One of the reasons for such an incredible high ductility probably lies in its low elastic anisotropy described above. Corrosion resistance and abrasion resistance are also superb. Because of these excellent properties, most of the commercial applications have been done for Ti–Ni alloys among many shape memory alloys, such as a flap in air-conditioner, coffee maker, brassiere, and antenna for mobile phones, medical applications such as orthodontic wire, guide wire and stent.

4.1 Analytical and FEM modelling The use of embedded SMA actuators for composite beams deflection and buckling controls provide very promising results in which the composite beams could sustain further external loads without failures by high deflection or buckling. Lee and Choi [81] work assumed that the critical buckling load, at which the thermal buckling of the beam occurs, is equal to the Euler load of a composite beam without any embedded wires, the critical buckling load, Pcr , was expressed as:

Pcr   com Tcr EA 

 2 EI

(21)

lc2

where Tcr is the changed quantity of the critical buckling temperature, A , EI ,  com and lc are the crosssectional area, the flexural rigidity, the effective thermal expansion coeficient and the effective length of the composite beam, respectively. The critical buckling temperature change, Tcr , was given by

Tcr 

2

(22)

 com S 2

where S 2  lc r is the slenderness ratio and r is the radius of gyration. The effective thermal expansion,  com , was calculated based on the classical lamination theory – the critical buckling

26

temperature of the beam is independent of Young’s modulus and inversely proportional to  com . Park et al. [82] have recently investigated the vibration behaviour of the thermally buckled composite plate embedded with shape memory alloy (SMA) fibres. The nonlinear finite element equations based on the first-order shear deformation plate theory (FSDT) were formulated for the laminated composite plate embedded with SMA fibres. To account for the large deflection, the von Karman straindisplacement relation for the FSDT was determined as follows

    m  zk   lin      0  zk

(23)

where  lin  ,   , and  0  are the incremental inplane linear strain vector, the incremental inplane nonlinear strain vector, the incremental inplane strain vector due to the initial deflection and the incremental curvature strain vector, respectively. In addition, u , v and w are the incremental displacements in the x , y and z directions, respectively, and w0 is the total initial deflection.  x and  y are the rotations in the xz - and yz -planes, respectively. The incremental transverse shear strain vector was expressed as

w    y  yz   y      y   xz  w x   x 

(24)

The incremental method considering the influence of the initial deflections and initial stresses was adopted for the temperature-dependent material properties of SMA fibres and composite matrix. In addition, the marching method was used for determining the thermal large deflections and predicting the critical temperatures. The marching method can be summarized as follows: (i) The temperature change consists of many small temperature increments. (ii) The first increment of temperature starts from the reference temperature, Tref . At the reference temperature, the plate is assumed to be a flat state without the initial stresses. (iii) If the initial displacement is zero, that is, the pre-buckled state, an artificial displacement is imposed for the Newton–Rapson method. (iv) Newton–Rapson method is performed until the convergence criterion is satisfied. (v) The procedure from (ii) to (iv) is performed using the same temperature increment until the maximum deflection converges to be non-zero value.

In the general thermal post-buckling analysis, the initial estimated displacement can be obtained from the mode shape vector through Euler buckling analysis. However, the Euler buckling analysis cannot be adopted in the study because of the temperature-dependent material properties with the high nonlinearity, thus the adoption of the marching method. The numerical results showed the vibration characteristic of the laminated composite plate embedded with SMA fibres in the pre-buckled and

27

post-buckled regions. Most recently, Park and co-workers [83] have used the nonlinear finite element formulations based on the FSDT and solution procedure and presented an analysis of the thermal postbuckling, vibration and flutter of the SMA composite plate. The numerical results showed that the higher the volume fraction and the initial strain of the SMA fibre are, the stiffer the plate is. It was deduced that the critical temperature was increased and also that the use of SMA fibre decreased the thermal large deflection. In the pre-buckled region, the natural frequencies were increased due to the recovery stress of the SMA. However, in the post-buckled region, the natural frequencies of the plate with SMA were lower than those of the plate without SMA. This was attributed to the increase of the weight of plate and decrease of the thermal large deflection using the SMA fibre. It was noted that the flat region under the combined aerodynamic and thermal loading was expanded significantly while using SMA increased the critical dynamic pressure. Similarly, Zhong et al. [84] studied the thermal buckling and post-buckling of the laminated composite plate with SMA fibres using the analytical and finite element method. The iterative scheme was employed to determine the critical temperature. Duan et al. [85] investigated the control of the thermal post-buckling deflection of the laminated composite plate using the SMA. For the temperature-dependent material properties, the incremental method was used, and the finite element method based on the classical plate theory was adopted.

4.2 Experimental studies Lau et al. [86] embedded SMAs, in the form of wires and strips into advanced composite structures to control the shape and residual stress of the structures. SEM micrographs showed that excellent bonding properties are achievable when the pre-strained level of the SMA wires is lower than 8%. However, an increase in the pre-strained level resulted in increased recovery force in the SMA wire, which generated sufficient compressive strain in the wire to cause debonding. Many circumferential cracks were found on the matrix surface leading to suspection that the cracks formed on the matrix surface due to a high relative tensile strain on the matrix surface close to the wire-end region since the wire contracted upon heating, while the surrounding matrix expanded. The difference between these geometrical changes in shape caused a high tensile stress on the matrix surface, a fact denoted by several other reports [87,88]. The report went on to claim that the magnitude of the stress increased with increasing recovery strain. It was further suggested that this problem might be less serious if the SMA wires are embedded into carbon or Kevlar composites as these composites generally give negative thermal strain with increasing temperature. Also, high air bubble-content inside the matrix may generate another type of crack in the radial direction during the interfacial debonding process. This crack may propagate in the transverse direction and influence neighbouring fibre systems in composite materials. The surface film of the SMA wire under a high annealing temperature may then

28

peel off resulting in poor bonding characteristics between the embedded SMA wire and the surrounding matrix material. Many studies related to the uses of embedded SMA actuators for noise reduction in rotor blade systems have been found recently too [87,89-91]. The major achievements of those works were to control the stiffness, angle of blade twist, tip-configuration, natural frequency and damping property of composite rotor blades in order to improve the unstable flutter and noise of the blades due to blade vortex interaction e.g., coupled conventional constitutive law for shape memory materials, non-linear beam deflection analysis and numerical approach to simulate the deflection of simple SMA composite beams with and without actuations by embedded SMA actuators are beginning to appear [89]. Additionally, since the SMA materials possess super-elastic property at a temperature greater than Af, the superelasticity of SMA fibres is able to suppress or damp structural vibration by applying internal forces (distributed and/or localised) to structures in such a way as to dissipate the energy within the structures. Therefore, the superelastic SMA fibres could reduce vibration amplitudes and increase the damping ratio of the structure [90]. In an excellent work, Ostachowicz and Kaczmarczyk [91] demonstrated that a SMA-based control system can be designed to reduce the adverse dynamic response of composite structures due to flutter phenomena. Specifically, the natural frequencies could be modified via a direct SMA heating and this way, one can control the flutter instability regions. A finite element model to predict the dynamic response of the system with embedded SMA fibres was proposed and the flutter boundaries identified with and without delamination damage. It was assumed that the plate material properties are a function of temperature expect density, the angle of the graphite fibres and the coefficient of expansion are functions of temperature. It was further shown that for the delamination case the flutter boundaries exceeded at lower critical aerodynamic loads, as compared with the undamaged case. Since in the FEM formulation there are no restrictions as far as the panel geometry is concerned, the computer model proposed by Ostachowicz and Kaczmarczyk [91] can be applied to design efficient SMA fibre-based control mechanism to avoid flutter instability phenomenon in composite structures with delamination.

4.3 Future outlook and conclusions Since the discovery of the SMAs in the 1960s researchers have been investigating both experimental aspects of their behaviour as well as their constitutive modelling. The Ti–Ni based alloys are the most important practical SMA with excellent mechanical properties. There are many phase transformations in Ti–Ni-based alloys system, which include not only diffusionless/martensitic transformations, from which shape memory and superelastic effects arise, but also diffusional transformations. Thus even the latter transformations have been used effectively to improve shape memory characteristics. Furthermore, the alloy system will serve as an excellent case study of physical metallurgy, as is the

29

case for steels where all kinds of phase transformations are utilized to improve the physical properties. Various aspects of Ti–Ni-based alloys include crystallographic mechanism of martensitic transformations, mechanisms of shape memory and superelastic effects, effects of aging on martensitic transformations and on shape memory/superelastic characteristics etc. It is also evident that Landau theory is useful to understand the relation between the parent phase and martensite through their symmetry relation and to understand premartensitic behaviour. Since this work is devoted to the unified understanding of fundamentals of SMA, applications of these alloys is not full represented, but there is a common believe that these basics are useful for applications in how the aging and thermomechanical treatment can be effectively utilized to change transformation temperatures and to improve shape memory characteristics of the alloys concerned. For readers interested in further reading on Ti–Ni-based alloys they are referred to the most recent work by Otsuka and Ren [92] on physical metallurgy of Ti–Ni-based shape memory alloys, Lagoudas and Entchev [93] on modelling of transformation-induced plasticity and its effect on the behaviour of porous shape memory alloys, Miyazaki [94,95] on fatigue of shape memory alloys and Matsuzaki on SMA and aeroelastic stability prediction. Attempts have also been made recently to develop Ti-Ni matrix tribo composites. Composites have been widely used in industry for wear applications. However, conventional composites may not always be effective under different wear conditions. For example, WC–NiCrBSi, a well-known hard facing material, is still not sufficiently strong to resist severe wear attack by abrasive sand. The NiCrBSi matrix is relatively weak and this decreases the integrated resistance of the composite to wear. In addition, this composite has a low impact resistance and performs poorly when subjected to impact wear. Under impact, carbide particles fail by brittle fracture and are broken away from the matrix. Attempts have been made to improve such type of conventional materials e.g., by modifying the matrix composition. However, the improvement is rather limited, since such modification is somewhat contradictory. It is known that the advantage of a composite is the combination of hard particles and a ductile matrix. The hard particles are generally brittle. A ductile matrix is therefore required to reduce cracking of the hard particles by accommodating large deformation or absorbing impact. Thus, the following problem arises when one attempts to improve the composite. If the hardness of the matrix is increased, its ductility will decrease, thus reducing its ability to absorb impact and to accommodate deformation. However, if the hardness of the matrix remains low, it will become the weak region and quick wearing of the matrix will result in rapid removal of the reinforcing particles. The difficulty in solving this problem greatly retards progress in improving existing tribo composites. Materials engineers’ have performed numerous tests to modify existing matrix. However, the improvement based on improving a composite by modifying its matrix

30

composition through trial-and-error tests is rather limited and costly. Significant improvement may not be anticipated if the modification is based on conventional concepts. The recent research on the influence of temperature on the fatigue performance of SMAs have shown that microstructure is important in governing the stability of the stress–strain hysteresis during strain controlled fatigue loading, and secondly, in the localization of strain/transformation. Functional fatigue is associated with an increase of residual strain (with plastic and pseudo plastic components), which corresponds to an incomplete reverse transformation. Moreover, it have been demonstrated that structural fatigue of pseudo-elastic SMAs is governed by the initiation and growth of surface cracks. Pseudo-elastic shape memory alloys are damage tolerant because the stress induced transformation limits stresses and thus stress intensity factors, which drive crack growth. In case of SMA, structural and functional fatigue of Ni-Ti shape memory alloys can be categorised as: 

The evolution of the stress–strain hysteresis in low cycle pull–pull fatigue of pseudo-elastic Ni-Ti wires



Bending–rotation fatigue rupture of pseudo-elastic Ni-Ti wires



Strain localization during the stress induced formation of martensite



Generic features of functional fatigue in Ni-Ti shape memory actuator springs.

The functional properties of SMAs clearly represent the reason for their success; but there are also many applications where structural properties are very important. This particularly holds for cases where SMAs are cyclically loaded. In terms of structural fatigue, SMAs subjected to high cyclic loads can fail like any other engineering material, but unlike normal engineering materials SMAs show different properties in different temperature ranges that affect fatigue characteristics. It also has to be established how a stress induced martensitic transformation interacts with cyclic strain accumulation and fatigue damage accumulation. The term functional fatigue indicates that during cyclic loading, SMAs generally suffer a decrease in functional properties. It is then admissible that there is a need to understand the underlying microstructural processes for safe design of SMA components.

In conclusion, the following additional points have been highlighted: 

The SMAs can be fabricated either as single crystals, such as some of the copper-based SMAs, or polycrystal, such as NiTi-based, iron-based and copper-based SMAs. While single crystal SMAs can be used for detailed analysis of the nature of the martensitic phase transformation, their use in applications is limited by the high manufacturing costs. The most practical applications involve polycrystalline SMAs.

31



Experimental research on SMAs undergoing cyclic loading in the pseudoelastic regime has shown that a significant part of the developed strain is not recovered upon unloading. This effect has been attributed to the development of plastic strains during the thermomechanical cycling of dense SMAs undergoing phase transformation. Similar effects have also been observed during cyclic mechanical loading of porous SMAs.



The problem associated with the development of plastic strains during cyclic phase transformation has been addressed for the case of one-dimensional SMA actuators (wires) undergoing cyclic temperature-induced phase transformation. However, a three-dimensional formulation is still lacking.

5

Microelectromechanical systems (MEMS), Nanoelectromechanical systems (NEMS) and nanomaterials

5.1 Microelectromechanical systems (MEMS) An increasingly strong interest for the development of micromachining technology has driven a rapidly growing research effort in microelectromechanical systems (MEMS) in the last decade. These microsystems involve integrated sensing and actuating elements with sophisticated shape and functions. The need to manufacture such diverse elements oftentimes necessitates utilizing materials that are beyond conventional integrated circuit IC) fabrication, and consequently compels alternative microfabrication techniques. MEMS are an integrated circuit (IC) derived fabrication technology developed during the 1980s that enables large, batch scale production of micron scale mechanical devices, either as microactuators or microsensors. Some examples of MEMS products are micropressure sensors, accelerometers, inkjet printer heads, digital mirror devices for projection systems, optical switches, and lab-on-a-chip systems for separation, preparation, and detection of DNA or pathogens. Additionally, since the same processes are often used to create both MEMS devices and traditional IC circuits, by carefully designing the fabrication process flow it becomes possible to integrate transducers and microelectronics on the same wafer chip. This normally results in both cost savings and better performance. Basic MEMS fabrication techniques include bulk micromachining, surface micromachining, and wafer bonding. Microstereolithography (SL), for example, adopted from the stereolithographic process in rapid prototyping, turns out to be a viable candidate in this area, both for polymeric and ceramic materials. SL enables the manufacturing of complex 3D shapes, by means of localized photopolymerization via a sharply focused laser beam, with a lateral resolution of 1–1.2 m on the defined structures. The 32

mechanical properties of microstructures are a critical factor in the performance and reliability of MEMS devices, and thus important for microfabrication technology. In numerous cases, catastrophic failure of microactuated devices occurs when surfaces impact—either unintentionally or as a part of the normal device operation—and components suffer permanent damage. Reducing the stiffness of such components will facilitate their survival. At the same time, the same MEMS components should have high enough modulus to enable the efficient transfer of forces from one element to another. Consequently, it would be of great importance to develop a method that can produce well-defined MEMS elements with controlled stiffness and strength, as dictated by design principles. In the study of membrane like devices, the most frequently used geometries are beam or membranelike which requires that the models be reduced to a dimensionality that properly describes the physical behaviour of membrane like devices. To accurately simulate such thin structures, Muller and Korvink [96] have developed an accurate and stable Kirchhoff–Love multi-layer plate model implemented as an Argyris finite element. Primarily targeted for MEMS designers, the model enables functional optimisation at an early stage of device development, saving resources. The model also encompasses effects that are crucial to the functionality of MEMS.

By extending the primarily mechanical

Kirchhoff–Love plate model, first to thermal effects and then to piezoelectric effects, they were able to simulate a wide range of typical MEMS behaviour. Hierarchically, the model allowed simulation of thin structure behaviour starting from structural mechanics to effects comprising more complex coupling effects. The basic plate model considered the bending field of thin structures only. But since multi-functional thin structures generally consist of several layers the coupling of the membrane field to the bending field has also to be taken into account. This is done by extending the bending equations by the in-plane or membrane field terms. Only in the rare case that the material stacks are perfectly symmetric to the plate's midplane do the in-plane and the out-of-plane displacement fields act independently.

5.2 Nanoelectromechanical systems (NEMS) and nanomaterials Polymer micro- and nano-particles are fundamental to a number of modern technological applications, including polymer blends or alloys, biomaterials for drug delivery systems, electro-optic and luminescent devices, coatings, polymer powder impregnation of inorganic fibres in composites, and are also critical in polymer-supported heterogeneous catalysis. By using instrumentation developed for probing single fluorescent molecules in micron-sized liquid droplets, polymer particles of nearly arbitrary size and composition can be made with a size dispersion that is ultimately limited by the chain length and number distribution within the droplets. Depending on the time scale for solvent evaporation a ‘tunable parameter’ in phase separation of otherwise immiscible polymers can be avoided by confinement effects, producing homogeneous polymer blend micro- or nano-particles.

33

These particles have tunable properties that can be controlled simply by adjusting the size of the particle, or the relative mass fractions of the polymer components in solution [97,98]. Nanomechanical devices promise to revolutionize measurements of extremely small displacements and extremely weak forces, particularly at the molecular scale. Indeed with surface and bulk nanomachining techniques, NEMS can now be built with masses approaching a few attograms (10 -18 g) and with cross-sections of about 10 nm. The small mass and size of NEMS gives them a number of unique attributes that offer immense potential for new applications and fundamental measurements. The mechanical element either deflects or vibrates in response to an applied force. To measure quasistatic forces, the element typically has a weak spring constant so that a small force can deflect it by a large amount. Time-varying forces are best measured using low-loss mechanical resonators that have a large response to oscillating signals with small amplitudes.

In general, the output of an

electromechanical device is the movement of the mechanical element. There are two main types of response: the element can simply deflect under the applied force or its amplitude of oscillation can change. Detecting either type of response requires an output or readout transducer, which is often distinct from the input one. Today mechanical devices contain transducers that are based on a host of physical mechanisms involving piezoelectric and magnetomotive effects, nanomagnets and electron tunnelling, as well as electrostatics and optics. Mechanical systems vibrate at a natural angular frequency, w0, which can be approximated by

 keff w0   m  eff

  

1

2

(25)

where keff is an effective spring constant and meff is an effective mass. Underlying these simplified ‘effective’ terms is a complex set of elasticity equations that govern the mechanical response of these objects. By reducing the size of the mechanical device while preserving its overall shape, then the fundamental frequency, w0, increases as the linear dimension (l) decreases. Underlying this behaviour is the fact that the effective mass is proportional to l3, while the effective spring constant is proportional to l. This is important because a high response frequency translates directly to a fast response time to applied forces. It also means that a fast response can be achieved without the expense of making stiff structures. Resonators with fundamental frequencies above 10 GHz (1010 Hz) can now be built using surface nanomachining processes involving state-of-the-art nanolithography at the 10 nm scale. Such high-frequency mechanical devices are unprecedented and open up many new and exciting possibilities. Among these are ultralow-power mechanical signal processing at microwave frequencies and new types of fast scanning probe microscopes that could be used in fundamental research or perhaps even as the basis of new forms of mechanical computers. A second important attribute of NEMS is that they dissipate very little energy, a feature that is characterized by the high quality or Q factor of resonance. As a result, NEMS are extremely sensitive

34

to external damping mechanisms, which is crucial for building many types of sensors. In addition, the thermomechanical noise, which is analogous to Johnson noise in electrical resistors, is inversely proportional to Q. High Q values are therefore an important attribute for both resonant and deflection sensors, suppressing random mechanical fluctuations and thus making these devices highly sensitive to applied forces. Indeed, this sensitivity appears destined to reach the quantum limit. The small size of NEMS also implies that they have a highly localized spatial response. Moreover, the geometry of a NEMS device can be tailored so that the vibrating element reacts only to external forces in a specific direction. This flexibility is extremely useful for designing new types of scanning probe microscopes. NEMS are also intrinsically ultralow-power devices. Their fundamental power scale is defined by the thermal energy divided by the response time, set by Q/wo. At 270C, NEMS are only overwhelmed by thermal fluctuations when they are operated at the attowatt (10-18 W) level. Thus driving a NEMS device at the picowatt (10-12 W) scale provides signal-to-noise ratios of up to 106. Even if a million such devices can be operated simultaneously in a NEMS signal processor, the total power dissipated by the entire system would still only be about a microwatt. This is a three or four order of magnitude less than the power consumed by conventional electronic processors that operate by shuttling packets of electronic charge rather than relying on mechanical elements. One more advantage of MEMS and NEMS is that they can be fabricated from silicon, gallium arsenide and indium arsenide - the cornerstones of the electronics industry - or other compatible materials. As a result, any auxiliary electronic components, such as transducers and transistors, can be fabricated on the same chip as the mechanical elements. However, patterning NEMS (so that all the main internal components are on the same chip) means that the circuits can be immensely complex. It also completely circumvents the insurmountable problem of aligning different components at the nanometre scale.

5.3 Examples of MEMS applications Although the research and development on ceramics and dipolar polymers started in late 1960s, the work on MEMS based piezo-restrictive, piezo-capacitive and pyro-electric materials has begun in 1990s. Research trends are; study of polarization dynamics, optical switching and non-uniform charge distribution as well as integration with microelectronics for application in devices. Arshak et al. [99] have reviewed the potential of a wireless MEMS and TFT microsystems, Goel provide a review on recent developments in electroceramics [100], and Baltes et al. [101] have recently written a great textbook on enabling technologies for MEMS and Nanodevices. A recent summary of existing error estimation techniques can be found in the textbook of Ainsworth and Oden [102].

35

To start with, the similarities between IC and MEMS manufacturing technologies offer some advantages, particularly in the case of surface micromachining where there is a strong data base of knowledge in terms of the tools and manufacturing process. However, the differences between the two technologies present challenges that must be dealt with. The most important of these are reliability, packaging and cost, research and development in order to overcome such issues is costing millions. For instance, the difficulty in MEMS failure analysis arises when structures of interest are not readily exposed for direct observation. Optical micromirrors, contact switches, and micro-fluidic MEMS are just a few components where structures that provide the stimulus for motion or actuation are obscured from view. This obstruction will be found at both the wafer/die level and at the package level. Although physical obstructions will be observed for many MEMS analyses, it represents one area of difficulty for the MEMS failure analyst. Further difficulty is found in packaged and capped structures. Here, the difficulty lies in non-destructively removing the cap or depackaging the device to gain access to the MEMS structure. Capped and packaged devices are assembled to contain the device within a sealed environment to perform its designed function. Destructive methods employed to gain access to the MEMS device will compromise the sealed environment and risk compromising the failure mechanism. This will lead to erroneous conclusions in determining the root cause of failure. Although destructive analysis may be needed at times to structurally and chemically characterize a MEMS component, the failure analyst should ensure that he/she has all the pertinent data prior to employing destructive methods. Assessing the failure mechanism while minimizing perturbing the device or failure mechanism is of the utmost importance. There are many issues a failure analyst confronts when performing analysis on MEMS components. These include on if IC failure analysis tools can be applied to MEMS technology, and if yes, issues arising include how to prepare of the sample for analysis without degrading the failure mechanism, and on the accessibility to the device. Some failure analysis tools and techniques can be readily applied to MEMS technology while others require more design or significant sample preparation. If the analysis can be performed at the wafer/die level, the big hurdle for package failure analysis is gaining access to the die. After performing failure analysis on several MEMS components of the device and package level, experience has shown that there is little consideration during design for testing and failure analysis. It has been shown that over 80% of the costs associated with manufacturing MEMS devices are associated with packaging. It is not surprising that some of the more difficult failure mechanisms to assess occur during packaging or post package testing. A major problem observed in doing package level failure analysis on MEMS technology is the difficulty accessing the die. By implementing a design for testing and failure analysis approach, transparent lids can be used instead of metal or ceramic ones to allow rudimentary inspection of the device after test. This would in turn provide very

36

important feedback from testing and would decrease the time needed to determine the root cause of failure. On the other hand, colloid thrusters were first studied in 1960s, but further developments were affected by packaging problems caused by high starting and working voltages in the range of 5– 10 kV. The urgent demand for micro spacecrafts and the emergence of MEMS technology triggered the rebirth of the colloid thruster in recent years. Due to lower cost, smaller risk in launch, and the ability of formation flying, there exists an increasing demand for micro size and high efficient propulsion systems for micro spacecrafts. If a large and costly satellite can be replaced by a number of distributed micro satellites in the future, the failure of a major component no longer threatens the whole mission. Dramatically reducing the size and mass of a satellite requires substantial miniaturization of subsystems. One of the subsystems impacted by the need of miniaturization is propulsion. In addition, the value of impulse bit micro propulsion must be as low as 10 −6 N s for the attitude control of a micro satellite. More propulsion systems are needed to meet the needs of the micro satellite. For instance, a MEMS based micro thruster provides a solution to meet the needs of micro spacecrafts. Reasons for exploring MEMS technologies for fabricating micro propulsion components and systems may be related to size, performance and integration ability. There are several ongoing works involving the fabrication of MEMS based micro thrusters, such as resistor jet, digital thruster, vaporizing liquid micro thruster, and micro ion thruster. All of these thrusters (except vaporizing liquid micro thrusters) operate in pulse mode with output thrust in the order of μN. A micro colloid thruster can output a constant thrust in the μN range. Other advantages of the colloid thruster lie in high specific impulse, high-energy conversion efficiency, and control flexibility. Recent studies have suggested that via postfabrication exposure to UV radiation, polymeric MEMS fabricated by microstereolithography can have their stiffness increased up to the bulk modulus of feasible moduli from 50 MPa to 20 Gpa [103]. Other examples of MEMS fluidic sensors now available include piezoresistive pressure sensors, shear stress sensors, and micromachined hotwires. In aerodynamics, flexible MEMS bubble actuators have been used to affect the rolling moment of a delta wing [104]. Flexible shear stress sensors have also been used to detect the separation line on a rounded leading edge of a delta wing as well as on a cylinder [105]. MEMS actuators are known to be relatively power thrifty and can interact with and manipulate the relevant flow structures to effect global flow property changes from local actuation. This ability is due to the length scale of the actuator (anywhere from hundreds of microns to a few millimeters) being comparable to the flow structure, thus allowing the actuator to directly excite flow instabilities at their origin. A distributed field of such actuators can therefore efficiently achieve large aerodynamic performance improvements. Of equal importance is also the ability to batch fabricate these devices on thin films and distribute them on the aerodynamic surface of interest to form a distributed control system.

37

Micro/nano satellites are drawing considerable attention of space technologists in recent years. This new range of satellites requires micro-propulsion units with an extremely high precision of control. For miniaturizing propulsion systems, the conventional fabrication technology can hardly be used to scale down the size below several inches. The microfabrication technology of MEMS has been successfully employed to batch-fabricate micro-propulsion systems or microthrusters with the dimension in the range of millimetres to sub millimetres, capable of producing extremely small thrusts from micro-Newtons to milli-Newtons. Of the various types of MEMS microthrusters developed so far, the vaporizing liquid microthruster (VLM) is simple and has been widely investigated. VLM can produce continuously variable microthrusts using a non-toxic liquid propellant. The work on VLM reported so far is mostly experimental with little theoretical content. The structure of a VLM realised by silicon bulk micromachining is rather complex from the viewpoint of theoretical simulation. It involves rigorous numerical simulations using 3D microfluidic, thermodynamic and electrothermal solvers. Detailed simulation results are scarce in published literature to date. The 3D numerical simulations are computationally cumbersome and not convenient in the initial phase of development. Radio

frequency

(RF)

components

have

been

viewed

as

an

ideal

opportunity

for

microelectromechanical systems (MEMS). One reason for this is that high frequency and especially high power RF products are still based on discrete components and selected based on performance and MEMS have already demonstrated superior performance in a number of critical device metrics. Moreover, these products often use hermetic or quasi-hermetic packaging and chip and wire assembly, as do MEMS components. As reliability and manufacturing yield for MEMS become more stable, it is expected that MEMS will play an important role in these products. Most recently, Singh et al. [106] has shown that follower forces, as well as axial forces, can be easily implemented in MEMS, and these end loads provide the scope for performance enhancement of certain MEMS devices. While axial end loads can lead to buckling, follower forces are a standard means of introducing flutter. Other types of end loads are also possible, for example, those that are combinations of follower and axial loads, as well as variable-orientation bucking-type loads. It is quite simple to experimentally apply axial forces on beams by making use of dead loads, but it is quite challenging to realize follower forces in the laboratory. To date, the only experimental implementations of follower forces make use of a fluid stream that is ejected from a nozzle fixed to the free end of a cantilever beam [107]. The convection of fluid along the beam, however, adds a term to the equation of motion [108], and leads to a complex version of a follower force. Nowadays, nanomaterials encompass nanotubes, fullerenes, polyhedral oligomeric silsesquioxanes (POSS), organic or inorganic clays. On a few words on nanotubes; ever since their discovery carbon nanotubes (CNTs) have been attracting major interest in the scientific community e.g. see Ekinci et al. [109], Desquenes et al. [110], Njuguna et al. [97,98,111-113] and Fennimore et al. [114]. In the last

38

decade, the mechanical and electronic properties of nanotubes have been investigated. Small size, low density, high stiffness, flexibility and strength, as well as excellent electronic properties and unique coupled electromechanical behaviours suggest that nanotubes have the potential to impact the development of novel composites, electronic devices and NEMS. Nanotubes (as well as nanoropes— composed of several nanotubes—and nanowires—having different shaped compact cross-sections) are envisioned as the ultimate fibre reinforcements as a consequence of their extremely high stiffness (Young's modulus of the order of 1 TPa) and flexibility (strain at tensile failure of the order of 30%). Recently some research groups have been able to manufacture NEMS devices e.g. nanotweezers, a carbon nanotube-based nonvolatile random access memory, etc. The viability of the concept has been demonstrated by the experimental realization of a reversible bistable nanotube-based bit. Furthermore, the first really true nanotube-based NEMS, fully integrating electronic control and mechanical response by realization of an electromechanical motor incorporating a rotational metal plate, with a multi-walled carbon nanotube serving as the key motion-enabling element. With the advent of NEMS, the realization of a single-electron transistor, which is shuttling charge between the two contacts by mechanical motion, became possible. One of the most interesting opportunities for future application of a nanoelectromechanical electron shuttle seems to be the possibility of defining an extremely accurate current source. Moreover, carbon nanotubes (CNTs) are of tremendous current interest in both fundamental research and for nanoelectronics applications including flat panel displays, chemical sensors and field emitters. The development of carbonmicroelectromechanical systems (C-MEMS) and carbon-nanoelectromechanical systems (C-NEMS) depends on lithographic patterning of resists (using any type of lithography) and their subsequent pyrolysis. On the other hand, since thin films with nanometer grains (about 10-100s of nanometers) still show shape memory effect, it is promising to fabricate nanoscale SMA thin film structures with the aid of precision tools such as focused ion milling. These structures may be able to perform physical actuation (push, pull, etc.) at nano-scale. Possible difficulties of Ti-Ni films in nanoscale structures may include: (i) large amount of oxygen and carbon adsorption on Ti-Ni surface due to the extremely reactive nature of Ti elements, and the oxide and oxygen diffusion depth could be as large as tens of nanometers; (ii) the difficulty in fabrication and manipulation of these nanostructure, although laser beam, or focused ion milling is promising. Nanotechnology methods offer new challenges for materials integration as well as properties (optical, electrical and mechanical) and modelling studies. New domain configurations would lead to development of micro-optic-electro-mechanical systems (MOEMS) with unprecedented applications. The work on combining desirable properties of PZT and different piezoelectric polymers to forms electro active materials for advance sensors was started 15 years ago. With the current research focus on new compositions, new processes and new device concepts in ceramic: polymer and silicon

39

integration using micro machining, material integration approaches in ceramics–polymers–silicon have become the centre of attraction for emergence of niche applications in actuators and smart sensors. Finally, while this trend will continue, it would be desirable to have state-of-the-art MEMS fabrication facility at more academic and industrial institutions across the world for technological advancement and indigenous capability building. Materials are assumed to be homogeneous and continuous in classical continuum mechanics, generally, their intrinsic microstructures are neglected. However, any macro-medium consists of mass of grains at microscopic scale. Furthermore, materials always contain tremendous defects, such as dislocations, grain boundaries and micro-flaws. These micro-structures and defects have great effects on materials’ mechanical behaviours at microscopic scale. Recent experiments have shown that materials will display strong scale effect when the scale of deformation field associated their characteristic length scale are in the order of microns. The trend of miniaturization in engineering, such as MEMS, electronic materials, ultra-thin films, etc., requires a comprehensive understanding of the effects of the intrinsic micro-structures on the material's macroscopic responses. In conclusion, the following points are also noted: 

The biggest challenge for capacitive devices lies in its packaging. Knowledge in relation to packaging of MEMS devices is at an infant stage and is made more complex with regard to what is required for microelectronics. The main reason for this is that MEMS devices often contain 3D structures with moving parts that are often fabricated using unique manufacturing techniques. This means that there is no knowledge base to draw from when designing suitable packaging.



It can be seen from the above examples that silicon micromachining is the application of silicon planar IC processing techniques to the selective etching of silicon and other films in order to fabricate microstructures. However, MEMS devices often require a variety of materials that would not normally be allowed near an IC foundry. Examples of some metals commonly used include titanium, tungsten, molybdenum, ruthenium and chromium. Depositing these materials in a controlled fashion is difficult for engineers and is the source of some reliability problems including adhesion, friction, toughness, impact tolerance, wear resistance, fatigue and creep.



It is particularly important that reliability issues be addressed before implantable devices can be made commercially available. Implantable devices must be highly reliable with a long mean time to failure; otherwise it is likely that they may have to be surgically removed, incurring a risk of infection. Such issues are becoming increasingly important as the requirements for device complexity and reliability rise. It is also argued that attempts to

40

fabricate a single chip system result in forcing elements of the system together. This may compromise the performance and reliability of the device and perhaps developing a two-chip system would yield a better result. 

The establishment of a cost effective and reliable process suitable for use in a silicon foundry is delayed by the fact that no one technology has become the established method of fabricating MEMS devices. While this is the case, very large-scale production of MEMS devices may be the only cost effective option, leaving other technologies to dominate for medium scale production requirements. Thick film technology is one, which could easily dominate at the medium scale level. However, while the technology has been used to produce robust devices with small sensor geometries, little research have been undertaken in this direction.



Despite these concerns, researchers have undertaken the development of wireless pressure measurement systems based on MEMS technology. Until recently, such systems were principally developed for the measurement of intra cranial or intro ocular pressure. However, it is thought that large start up costs will be involved.



The powerful combination of nanomaterials and MEMS technology will likely stimulate the creation of new and unique devices useful for a variety of other applications. For instance, Ti-Ni thin film SMA in its super-elastic state is promising for some compliant elements in MEMS devices.



In spite of this fast acceleration in the development of NEMS structures, the amount of experimental data is extremely limited due to the complexity involved in the realization of nanodevices. Likewise, accurate analyses and formulas needed in the design of NEMS are still lacking.

6

Ageing and lifetime predictions

Polymers and their composites are subjected to destructive factors such as mechanical stress, the presence of different chemicals, UV-light, ablation and high temperatures throughout shelf- and service-life. As polymers are takes a central role in modern structures, degradation ageing, and life prediction concerns must be emphasized here and readers are directed to Ref. [115] for the latest in this field. These factors cause degradation and ultimately affect performance and lifetime of the polymers and the end-products (e.g. polymer fibre reinforced composites) that are sometimes stored and/or used for long periods of time. Therefore it is important to know how long and under which conditions the polymers may best be stored with minimum deterioration of the properties. According to the lifetime stages of polymers, the relevant processes are classified as melt degradation; long-term

41

heat ageing and weathering based on the mechanisms involved, i.e. thermomechanical, thermal, catalytic and radiation-induced oxidations and environmental biodegradation. The products are different low molecular weight additives or degradation products from the additives or the polymer itself. The diffusion of low molecular weight products changes the properties of the material and shortens the lifetime. Diffusion of the additives and degradation products from the material further changes the properties and results in bad air in the storage and/or while in use. For safety reasons it is necessary to have a good understanding of their thermal resistance and to precisely identify the products likely to be formed. In addition to temperature, the induction time and therefore the durability of polymers and thus polymer composites depend upon the physical and chemical structure of the polymer, the efficacy of the stabilizing additives, the presence of metal catalysts, the presence of stress and the power of oxidizing agent. Forecasting changes in the properties of polymers and polymer composites’ properties with time is the task of predicting the performance. The forecasting can either be of determining the service life of the material in a given set of conditions or by determining the guaranteed period of required performance by the products of a given type. The prediction can be approached at three levels [115]: 

Empirical predicting results from testing a given material



Semi-emphirical predicting - based on the assumption that the mechanism of degradation can be presented in the form of a simplified model and the parameters have a physical meaning, and



Non-emphirical predicting - based on the chemical physics of the polymeric material.

The above points specify the principles and procedures for evaluating the thermal endurance properties of polymeric composites exposed to elevated temperature for long periods. The study of the thermal ageing is based solely on the change in certain properties resulting from a period of exposure to elevated temperature. The properties studied are always measured after the temperature has returned to ambient. For industrial practice, ISO standards have been developed covering thermal ageing and environmental degradation [116]. Thus, while aging effects in composites are beginning to receive serious analytical and experimental attention; experimental data on aging of adaptive materials and structures is only starting to appear. For example, Hilton et al. [117-119] have formulated an analytical nonlinear theory of anisotropic piezoelectric-thermo-viscoelasticity and even went on to apply the theory to a number of linear viscoelastic bending and aero-viscoelastic problems. Bending of nonlinear viscoelastic beams with small or large deformations was analyzed in the presence of viscoelastic piezoelectric devices placed on the upper and lower outer beam fibres. Piezo-viscoelastic control was applied through impositions

42

of piezoelectric voltages to limit and thus reduce deformations, stresses and probabilities of delamination onset and to delay failures in time. Nonlinear viscoelastic small and large deformation beam analyses indicated that lightweight piezo devices could be used to reduce stress and deformations. The work also deduced that failure probabilities and ultimate times to failure can also be reduced significantly by the imposition of elastic and/or viscoelastic peizoelectric structural control in the form of applied voltages to piezo strips bonded to the upper and lower beam surfaces.

7

Conclusions

Due to large potential applications in the fields of aerospace, civil engineering, shipbuilding, automobile, precision instruments, and machines, the field of adaptronics have developed rapidly. The active elements in smart structures can be embedded in or attached to the structure. Typical sensors include fibre optics, piezoelectric ceramics and polymers. Embedded sensors can be either discrete or distributed to provide built-in structural quality assessment capabilities, both during material processing and vehicle operation. For example, smart wings can provide more lift and/or better aeroelastic dynamic performance by driving integrated actuators to change the curvature of a profile or the leading- and trailing edge angles of an airfoil. Smart rotors can have less vibration load and longer fatigue-life. Sensors can also be used for monitoring in-service or environmental loading, and for shape sensing. However, much more needs to be learned about the static and dynamic behaviour of structures embedded with adaptronics if they are to be used for structural applications. Such issues have been discussed and their importance and developments in structural perspective have been commented on. To add a few more words, it is worthwhile mentioning that the widespread research over the last decade has investigated the many possible applications of adaptive structures. One of the most promising applications of adaptive structures’ technology is vibration suppression - see works by Noor et al. [120], and Agnes and Mall [121] who have addressed on structures technology for future systems. On this note, owing to their mechanical and electrical coupling effects, adaptive materials (especially piezoelectric materials) have found a wide range of applications in monitoring and controlling the responses of structures. Smart composite structures with integrated piezoelectric sensors and actuators have been investigated extensively in past decades. Important challenges in structural materials design include predicting the formation of large-scale self-assembled structures based on local atomic-level interactions and then endowing such structures with the ability to respond sensitively to environmental cues.

Adaptive structures, a subset of smart or intelligent structures,

combine the traditional load-carrying structures with the smart materials in order to perform the

43

traditional load-carrying function as well as to sense their environment and respond to it with appropriate action. For instance, piezoelectric materials create an electric charge when deformed or, conversely, deform when placed in an electric field. While in use for various applications for years, recent developments have brought smart materials, most notably ceramics such as PZT to the forefront of adaptive structures technology. In the field of smart structures, there have been few investigations of the integrity of passive systems, primarily structures with embedded fibre optics. However, adaptive structures are active not passive, and therefore investigation of integrity of the active smart structures becomes very complex since different loads/environments are applied to the host structure and integrated or embedded devices (i.e. PZT). First, as mentioned earlier, there have been a very limited number of studies in this area. Second, the most of these studies are for cases where the embedded device is an inert (or dummy) piece to simulate the active device, or subjected to a simple loading such as mechanical loading of the host structure where the embedded device is almost inactive. Vibration suppression devices such as PZT can be either bonded on the surface of the structure or embedded within a structure. The integration of PZT in the structure depends upon the requirements of the smart structure. However, the laminated fibre-reinforced composites are highly preferred host structures due to ease of embedment of sensors and/or actuators and due to inherent advantages of the composite materials. Nowadays composite materials are used in nearly all phases of structure work, from space craft to marine vessels, from bridges and domes on civic buildings to sporting goods. The significant increase in the use of composite materials and structure, demands for the development of rigorous mathematical methods capable of modelling, designing and optimising of the composite under any given set of conditions. One of the major challenges in computational structural mechanics is the development of the advanced models and numerical techniques in order to provide efficient tools exhibiting good interior and edge solutions. It can be seen in this review that the finite-element method is an almost indispensable tool in the analysis of adaptive composites, either for determining interlaminar stresses and strain-energy-release rate or for verifying simple formulations. Both hybrid and displacement elements are widely employed, with the former providing more accurate stresses than the latter. Moreover, the overwhelming dependence of finite elements on modelling the beams/plates would require huge computational resources for accurate analysis. Because finite element modelling is carried out subject to personal experience and the methodology chosen, a variation of results would be expected, even for the same problem. Non-FEM approaches that are formed based on physical concepts, and that can provide equivalent accuracy to the FEM approach, deserve special attention. Fracture mechanics and damage mechanics may provide e.g. an effective way for delamination analysis of laminated tapers according to their structural characteristics.

44

Stringent requirements from new generation cost-efficient applications have imposed unprecedented challenges to the scientific community, and have dramatically motivated research and development for novel and practical techniques. In particular, the introduction of microtechnology-based techniques shows encouraging prospects in this field and claims an essential role in this millennium. Multi-functional

structural

systems,

comprising

advanced

composites

and

incorporating

microtechnology, will have a great impact on performance enhancement, cost reduction and life extension. In this respect, the uncontroversial superiority of functionalized composite structures for new generation airframes has been well acknowledged by the research community. Such an approach has the potential to substantially enhance system performance and reduce overall manufacture– operation–maintenance expenditure. Recent progress in informatics and high-capability computing devices has offered a brand-new springboard for the aerospace community to reshuffle its traditional research and development criteria for functionalized composite structures. Particularly, artificial intelligence, an intriguing information processing technique, exhibits outstanding effectiveness in accommodating the highly demanding requirements of new generation airframes. Appropriate utilization of artificial intelligence techniques in functionalized composite structures design will contribute to the realization of high-capability intelligent systems.

45

8

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Caption of Figures

Figures Figure 1

Adaptive wing concept [3]

Figure 2

Smart composite plate incorporating piezoelectric layers [9]

Figure 3

A block diagram of closed-loop LQG/LTR system with a state estimator [27]

Figure 4

Response surface based solution strategy for fuzzy analysis [18]

Figure 5

Piezoelectric laminated sandwich beam finite element [27]

Figure 6

(a) Main parameters of a beam treated with the ACLD; (b) undeflected, (c) deflected [66].

Figure 7

The laminated composite plates embedded with SMA fibres [104]

54

Figure 1

Adaptive wing concept [3]

55

Figure 2

Smart composite plate incorporating piezoelectric layers [9]

56

Figure 3

A block diagram of closed-loop LQG/LTR system with a state estimator [27]

57

Figure 4

Response surface based solution strategy for fuzzy analysis [18]

58

Figure 5

Piezoelectric laminated sandwich beam finite element [27]

59

Figure 6

(a) Main parameters of a beam treated with the ACLD; (b) undeflected, (c) deflected

[66].

60

Figure 7

The laminated composite plates embedded with SMA fibres [104]

61