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A beam element for piezoelectric planar frames under small strains but large rotations using the total Lagrangian description

Abstract

Small strains are consistently incorporated into a large rotation piezoelectric beam theory, where the displacement is assumed to vary in accordance with the Timoshenko assumption and the electric potential has linear variation through each piezoelectric layer thickness. A finite element for planar frames, based on the total Lagrangian description, is then developed. The displacements and rotation are linearly approximated over the element and the voltage in a piezoelectric sensor layer is taken to be constant. In addition to the well-known membrane and shear lockings, it is disclosed the presence of locking in the application of Gauss law to the piezoelectric sensor layers. All the mechanical and electrical contributions to the internal load vectors are evaluated using a reduced one-point Gaussian quadrature to make this simple element efficient. No spurious modes are introduced in this process. An incremental-iterative approach, based on the Newton-Raphson algorithm, is employed in the solutions of the numerical examples to illustrate the large rotation capability of the developed model.

Keywords:
Piezoelectric; Large rotation; Total Lagrangian description; Finite element; Actuator; Sensor

Graphical Abstract

1 INTRODUCTION

Sensors and actuators attached or embedded into flexible host structures are two of the most basic elements of a smart structural system. In the sensor application, strains can be determined from measurement of the induced electrical potential, whereas in actuator applications strains can be controlled through the input of appropriate electric potential. Smart structures has been used in aerospace, hydrospace, nuclear, defense, automotive and civil structural applications. A vast literature is already available mainly focused on the linear analysis of beams, plates and shells, whereas less attention has been given to the more sophisticated nonlinear analysis (Cardoso and Fonseca, 2004Cardoso, E.L., Fonseca, J.S.O. (2004). An incremental Lagrangian formulation to the analysis of piezoelectric bodies subjected to geometric non-linearities, International Journal for Numerical Methods in Engineering 59(7):963-987.; Chróścielewski et al., 2019Chróścielewski, J., Schmidt, R., Eremeyev, V.A. (2019). Nonlinear finite element modeling of vibration control of plane rod-type structural members with integrated piezoelectric patches, Continuum Mechanics and Thermodynamics 31(1):147-188.). However, consideration of geometric nonlinearities can be crucial because external excitations can cause large rotations of smart structures, still in the range of small strains, due to their inherent flexibility (Zhang et al., 2015Zhang, S.Q., Li, Y.X., Schmidt, R. (2015). Active shape and vibration control for piezoelectric bonded composite structures using various geometric nonlinearities, Composite Structures 122:239-249.). A linear analysis may thus be insufficient to provide accurate estimations of the voltage sensed by piezoelectric sensors in light-weight and highly flexible structures (Zhang et al., 2015; Mukherjee and Chaudhuri, 2002Mukherjee, A., Chaudhuri, A.S. (2002). Piezolaminated beams with large deformations, International Journal of Solids and Structures 39(17):4567-4582., 2005), which are important data for their control by actuators. It seems to be quite realistic to assume that such structures may be subjected to large rotations, but small strains. Moreover, it must be considered that the piezoceramics like the PZTs, which are the most commonly used piezoelectric material, are fragile and unable to bear large strains (Ramya et al., 2015Ramya, R., Saravanakumar, G., Ravi, S. (2015). Energy harvesting in wireless sensor networks, Proceedings of the 1st International Conference on Artificial Intelligence and Evolutionary Computations in Engineering Systems, Chennai, India.).

In finite element analysis, the phenomenon of locking is quite well-known for linear formulations. Various types of locking, such as membrane locking, shear locking, volumetric locking, and so on, have been identified (Cook et al., 2002Cook, R.D., Malkus, D.S., Plesha, M.E., Witt, R.J. (2002). Concepts and Applications of Finite Element Analysis, 4th edn., John Wiley (New York).; Zienkiewicz et al, 2013Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z. (2013). The Finite Element Method: Its Basis and Fundamentals, 7th edn., Elsevier (Amsterdam).). In particular, linear straight Timoshenko beam finite elements are likely to suffer only of shear locking. A number of such elements have appeared in the literature (Nickel and Secor, 1972Nickel, R.E., Secor, G.A. (1972). Convergence of consistently derived Timoshenko beam finite elements, International Journal for Numerical Methods in Engineering, 5(2), 243-252.; Tessler and Dong, 1981Tessler, A., Dong, S.B. (1981). On a hierarchy of conforming Timoshenko beam elements, Computers and Structures 14(3-4):335-344.; Prathap and Bhashyam, 1982Prathap, G., Bhashyam, G.R. (1982). Reduced integration and the shear-flexible beam element, International Journal for Numerical Methods in Engineering 18(2), 195-210.; Reddy, 2006Reddy, J.N. (2006), An Introduction to the Finite Element Method, 3rd edn., McGraw-Hill (New York).). They differ from each other in the choice of interpolation functions used for the transverse displacement and rotation. The 2-node Timoshenko element with linear interpolation for the axial and transverse displacements, as well as for the rotation, is the simplest one. However, the shear locking makes this element very stiff in slender beams, i.e., as the length-to-thickness ratio becomes large. Detection of locking in a nonlinear formulation seems to have appeared for the first time in Kikuchi and Aizawa (1982).

Lanzoni and Tarantino (2018Lanzoni, L., Tarantino, A.M. (2018). Finite anticlastic bending of hyperelastic solids and beams, Journal of Elasticity 131(2):137-170.) propose a three-dimensional displacement field to describe the plane bending of solids under large strains that takes into account anticlastic effect. The solid cross section is assumed to remain plane during deformation, with no restriction placed on the magnitude of its rotations. Some geometrical quantities of the displacement field lose their physical meaning for slender beams (Falope et al., 2019Falope, F.O., Lanzoni, L., Tarantino, A.M. (2019). The bending of fully nonlinear beams. Theoretical, numerical and experimental analyses, International Journal of Engineering Science 145:103167.; Lanzoni and Tarantino 2020).

Lucena Neto et al. (2019Lucena Neto, E., Carvalho Neto, A.L., Monteiro, F.A.C., Santos, R.R.F. (2019). A consistent Timoshenko element beam for small strain but large rotation, Proceedings of the 7th International Symposium on Solid Mechanics, São Carlos, Brazil.) describe the behavior of Timoshenko beams under large rotations, for which small strains are consistently incorporated into a total Lagrangian formulation with the help of the polar decomposition of the deformation gradient (Malvern, 1969Malvern, L.E. (1969). Introduction to the Mechanics of a Continuous Medium, Prentice-Hall (Upper Saddle River).; Chandrasekharaiah and Debnath, 1994Chandrasekharaiah, D.S., Debnath, L. (1994). Continuum Mechanics, Academic Press (Boston).). In addition to shear locking, the use of the simple 2-node Timoshenko element in this highly nonlinear formulation also activates membrane locking. Both locking are alleviated by evaluating all the internal load vectors using a reduced one-point Gaussian quadrature.

Nonlinear models of piezoelectric beams subjected to large rotations with some restriction of magnitude (Mukherjee and Chaudhuri, 2002Mukherjee, A., Chaudhuri, A.S. (2002). Piezolaminated beams with large deformations, International Journal of Solids and Structures 39(17):4567-4582.; 2005; Kapuria and Alam, 2004Kapuria, S., Alam, N. (2004). Zigzag theory for buckling of hybrid piezoelectric beams under electromechanical loads, International Journal of Mechanical Sciences, 46(1):1-25.) are more common in the literature than those with rotations free of any restrictions (Cardoso and Fonseca, 2004Cardoso, E.L., Fonseca, J.S.O. (2004). An incremental Lagrangian formulation to the analysis of piezoelectric bodies subjected to geometric non-linearities, International Journal for Numerical Methods in Engineering 59(7):963-987.). In this paper, the work of Lucena Neto et al. (2019Lucena Neto, E., Carvalho Neto, A.L., Monteiro, F.A.C., Santos, R.R.F. (2019). A consistent Timoshenko element beam for small strain but large rotation, Proceedings of the 7th International Symposium on Solid Mechanics, São Carlos, Brazil.) is modified to account for piezoelectric patches attached to the beam surface, assuming electric potential with linear variation through each patch thickness (Marinković et al., 2009Marinković, D., Köppe, H., Gabbert, U. (2009). Aspects of modeling piezoelectric active thin-walled structures, Journal of Intelligent Material Systems and Structures 20(15):1835-1844.). In the finite element development, the voltage in a piezoelectric sensor patch is taken to be constant. This work reveals the existence of locking associated with the application of Gauss law to piezoelectric sensor layers. Numerical results demonstrate the accuracy of this simple element for nonlinear analysis of piezoelectric planar frames.

2 FUNDAMENTALS

In the development of the mathematical model, a beam with attached piezoelectric layers is considered. While the displacement field is assumed to vary in accordance with the Timoshenko assumption, the electric potential has linear variation through each piezoelectric layer thickness. Small strains are consistently incorporated.

2.1 Displacement, strain, electric potential and electric field

Suppose that a point B(X,Y,Z) of the beam shown in Figure 1 in the initial configuration moves to the point b(x,y,z) in the current configuration with no motion in the Z direction. Under the Timoshenko assumption (Reddy, 2002Reddy, J.N. (2002). Energy Principles and Variational Methods in Applied Mechanics, 2nd edn., John Wiley (Hoboken).), the cross section is taken to be infinitely rigid in its own plane, remaining plane but not necessarily perpendicular to the beam axis during the motion. The points have their orthogonal Cartesian coordinates related by

x = X + u - Y sin θ y = v + Y cos θ z = Z . (1)

The displacement components u(X) and v(X) refer to the point A(X,0,0) on the beam axis, which moves to the point a(x,0,0), and θ(X) is the cross-section rotation about the Z axis.

The polar decomposition theorem (Malvern, 1969Malvern, L.E. (1969). Introduction to the Mechanics of a Continuous Medium, Prentice-Hall (Upper Saddle River).; Chandrasekharaiah and Debnath, 1994Chandrasekharaiah, D.S., Debnath, L. (1994). Continuum Mechanics, Academic Press (Boston).) ensures that the deformation gradient

Figure 1:
Timoshenko beam kinematics.

F = x X x Y x Z y X y Y y Z z X z Y z Z = 1 + u ' - Y θ ' cos θ - sin θ 0 v ' - Y θ ' sin θ cos θ 0 0 0 1 (2)

can be uniquely written in the form

F = R U (3)

where R is orthogonal and fully responsible for the rigid body rotation, U is symmetric and fully responsible for the strain and primes denote derivatives with respect to X. If ω is the angle of rigid body rotation about the Z axis,

R = cos ω - sin ω 0 sin ω cos ω 0 0 0 1 (4)

and

U = R T F = 1 + u ' cos ω + v ' sin ω - Y θ ' cos ω - θ sin ω - θ 0 - 1 + u ' sin ω + v ' cos ω + Y θ ' sin ω - θ cos ω - θ 0 0 0 1 . (5)

The symmetry of U must be imposed to identify ω and complete the decomposition (3). However, the explicit value of ω to be derived from such an operation is not of interest here and, thus, will not be addressed.

One may define the displacement gradient associated with U by means of

H = U - I = 1 + u ' cos ω + v ' sin ω - Y θ ' cos ω - θ - 1 sin ω - θ 0 cos ω - θ - 1 0 s y m . 0 (6)

to rewrite the Green strain tensor in the form

Figure 2:
Beam with bottom and top piezoelectric layers.

ϵ = 1 2 ( F T F - I ) = 1 2 H + H T + H T H . (7)

Since the components of H are necessarily small for small strains, the following simplified expression may be adopted in this case:

ϵ 1 2 H + H T = ϵ X 1 2 γ X Y 0 ϵ Y 0 s y m . 0 . (8)

Moreover, the smallness of the component sinω-θ of H implies ω θ and thus leads to

ϵ X ϵ 0 - Y θ ' ϵ Y 0 γ X Y - 1 + u ' sin θ + v ' cos θ (9)

where

ϵ 0 = 1 + u ' cos θ + v ' sin θ - 1 . (10)

Figure 2 shows a host beam of length L, having a rectangular cross section of width b and thickness 2h, with two piezoelectric layers bonded on its bottom and top surfaces. The lower layer has width b1 and thickness t1, and the upper layer has width b3 and thickness t3. In electrostatics, the electric field vector E is irrotational and can therefore be derived from a scalar function ϕ(X,Y,Z), known as electric potential, in accordance with

E = E X E Y E Z = - ϕ X ϕ Y ϕ Z T . (11)

The electric potential is here assumed to be independent of Z (beam with no motion in the Z direction) and the gradient ϕ/Y through the thickness of a piezoelectric layer is the only to be considered (Lammering, 1991Lammering, R. (1991). The application of a finite shell element for composites containing piezo-electric polymers in vibration control, Computers and Structures 41(5):1101-1109.; Almajid et al., 2001Almajid, A., Taya, M., Hudnut, S. (2001). Analysis of out-of-plane displacement and stress field in a piezocomposite plate with functionally graded microstructure, International Journal of Solids Structures 38(19):3377-3391.; Trindade et al., 2001Trindade, M.A., Benjeddou, A., Ohayon, R. (2001). Finite element modelling of hybrid active-passive vibration damping of multilayer piezoelectric sandwich beams - Part I: formulation, International Journal for Numerical Methods in Engineering 51(7):835-854.; Sze et al., 2004Sze, K.Y., Yang, X.M., Fan, H. (2004). Electric assumptions for piezoelectric laminate analysis, International Journal of Solids Structures 41(9-10):2363-2382.):

E X = - ϕ X 0 E Y = - ϕ Y E Z = - ϕ Z = 0 . (12)

On the assumption that the electric potential varies linearly through the thickness of a piezoelectric layer k, between the values ϕk at the bottom (Y=Yk) and ϕk+1 at the top (Y=Yk+tk) of the layer,

Figure 3:
Free-body diagram of a piezoelectric beam showing all the applied loads and the end displacements.

ϕ X , Y = 1 - Y - Y k t k ϕ k X + Y - Y k t k ϕ k + 1 X k = 1 , 3 . (13)

The nonzero component of E reduces to

E Y = - ϕ - k X t k (14)

where ϕ-k=ϕk+1-ϕk is the voltage (difference of electric potential) in the piezoelectric layer k. Marinković et al. (2007Marinković, D., Köppe, H., Gabbert, U. (2007). Accurate modeling of the electric field within piezoelectric layers for active composite structures, Journal of Intelligent Material Systems and Structures 18(5):503-513.) and Sulbhewar and Raveendranath (2014Sulbhewar, L.N., Raveendranath, P. (2014). An accurate novel coupled field Timoshenko piezoelectric beam finite element with induced potential effects, Latin American Journal of Solids and Structures 11(9):1628-1650.) show that an electric potential consistent with a first-order shear deformation theory, as the one based on the Timoshenko assumption, should have an additional quadratic term in Y related to the curvature. Marinković et al. (2007; 2009) also show that the use of (13) is sufficiently accurate for typically thin piezoelectric layers.

When a sensor (actuator) layer k has the bottom surface grounded, the quantity ϕk stands for the electric potential read at (applied to) the sensor (actuator) top layer surface. On the other hand, when the sensor (actuator) layer has the top surface grounded, ϕk stands for the opposite electric potential read at (applied to) the sensor (actuator) bottom layer surface.

2.2 Principle of virtual displacements

The principle of virtual displacements for the piezoelectric beam, shown schematically in Figure 3, takes the form (Cardoso and Fonseca, 2004Cardoso, E.L., Fonseca, J.S.O. (2004). An incremental Lagrangian formulation to the analysis of piezoelectric bodies subjected to geometric non-linearities, International Journal for Numerical Methods in Engineering 59(7):963-987.):

- S X δ ϵ X + S X Y δ ã X Y d A d X + D Y δ E Y d A d X + 0 L q X δ u + q Y δ v d X + i = 1 2 F X i δ u i + F Y i δ v i + M Z i δ θ i = 0 . (15)

The normal stress SX and transverse shear stress SXY refer to the second Piola-Kirchhoff stress tensor; DY is the transverse electric displacement; qX and qY are the distributed axial and transverse forces (force per unit length in the initial configuration); FXi, FYi and MZi are the end loads; ui, vi and θi are the end displacements; and A is the piezoelectric beam cross-section area.

Substitution of (9) and (14) into (15) gives

- 0 L N δ ϵ 0 + M δ θ ' + Q δ ã X Y + L 1 δ ϕ - 1 + L 3 δ ϕ - 3 d X + 0 L q X δ u + q Y δ v d X + i = 1 2 F X i δ u i + F Y i δ v i + M Z i δ θ i = 0 (16)

where

N = S X d A M = - S X Y d A Q = S X Y d A L 1 = b 1 t 1 - h - t 1 - h D Y d Y L 3 = b 3 t 3 h h + t 3 D Y d Y . (17)

2.3 Constitutive relations

With polarization in the positive Y-axis and the remaining material principal directions aligned with the X- and Z-axes, the following constitutive relations cover a wide range of piezoelectrics (Yang, 2018Yang, J. (2018). An Introduction to the Theory of Piezoelectricity, 2nd edn., Springer (Cham).):

S X S Y S Z S Y Z S X Z S X Y D X D Y D Z = C 11 C 13 C 12 0 0 0 0 e 31 0 C 33 C 23 0 0 0 0 e 33 0 C 22 0 0 0 0 e 32 0 C 44 0 0 0 0 e 24 C 66 0 0 0 0 C 55 e 15 0 0 - ξ 1 0 0 s y m . - ξ 3 0 - ξ 2 ϵ X ϵ Y ϵ Z γ Y Z γ X Z γ X Y - E X - E Y - E Z . (18)

Here S’s and D’s refer to the components of the second Piola-Kirchhoff stress tensor and electric displacement vector; ϵ’s and ã’s refer to the components of the Green strain tensor; and Cij, eij and ξi are the elastic, piezoelectric and dielectric coefficients. It is timely to say that the adopted measure for the electric displacement and electric field vectors, like the second Piola-Kirchhoff stress and Green strain tensors, refer to the initial configuration and are unaffected by rigid-body motion (Cardoso and Fonseca, 2004Cardoso, E.L., Fonseca, J.S.O. (2004). An incremental Lagrangian formulation to the analysis of piezoelectric bodies subjected to geometric non-linearities, International Journal for Numerical Methods in Engineering 59(7):963-987.). This latter feature justifies the use of the relations (18), identical to those of the corresponding linear theory, as explained in Bathe (1996Bathe, K.J. (1996). Finite Element Procedures, Prentice-Hall (Englewood Cliffs).).

Special stress assumptions, commonly made in beam theory, require modifications of relations (18) to reduce them to the simple form

S X S X Y D Y = Q ~ 11 0 e ~ 31 K C 55 0 s y m . - ξ ~ 3 ϵ X γ X Y - E Y (19)

where K is the shear correction factor (Hutchinson, 2001Hutchinson, J.R. (2001). Shear coefficients for Timoshenko beam theory, Journal of Applied Mechanics 68(1):87-92.; Puchegger et al., 2003Puchegger, S., Bauer, S., Loidl, D., Kromp, K., Peterlik, H. (2003). Experimental validation of the shear correction factor, Journal of Sound and Vibration 261(1):177-184.). The coefficients Q~11, e~31 and ξ~3 depend on the type of special assumption made. Following Butz et al. (2008Butz, A., Klinkel, S., Wagner, W. (2008). A geometrically and materially non-linear piezoelectric three-dimensional-beam finite element formulation including warping effects, International Journal for Numerical Methods in Engineering 76(5):601-635.), Khdeir et al. (2012Khdeir, A.A., Darraj, E., Aldraihem, O.J. (2012). Free vibration of cross ply laminated beams with multiple distributed piezoelectric actuators, Journal of Mechanics 28(1):217-227.), Elshafei et al. (2014Elshafei, M.A., Ajala, M.R., Riad, A.M. (2014). Modeling and analysis of smart Timoshenko beams with piezoelectric materials, International Journal of Engineering and Innovative Technology 3(11):21-33.) and Sulbhewar and Raveendranath (2014Sulbhewar, L.N., Raveendranath, P. (2014). An accurate novel coupled field Timoshenko piezoelectric beam finite element with induced potential effects, Latin American Journal of Solids and Structures 11(9):1628-1650.), one adopts

Q ~ 11 = Q 11 - Q 12 2 Q 22 e ~ 31 = e - 31 - Q 12 e - 32 Q 22 ξ ~ 3 = ξ - 3 + e - 32 2 Q 22 (20)

where

Q i j = C i j - C i 3 C j 3 C 33 e - 3 i = e 3 i - C i 3 e 33 C 33 ξ - 3 = ξ 3 + e 33 2 C 33 . (21)

Substitution of (9), (14) and (19) into (17) gives

N M Q L 1 L 3 = A 11 A 12 0 A 14 A 15 A 22 0 A 24 A 25 A 33 0 0 s y m . A 44 0 A 55 ϵ 0 θ ' γ X Y ϕ - 1 ϕ - 3 (22)

where

A 11 = Q ~ 111 A 1 + Q ~ 11 b A b + Q ~ 113 A 3 A 12 = Q ~ 111 A 1 t 1 2 + h - Q ~ 113 A 3 t 3 2 + h A 14 = e ~ 311 A 1 t 1 A 15 = e ~ 313 A 3 t 3 A 22 = Q ~ 111 A 1 t 1 2 3 + h t 1 + h 2 + 1 3 Q ~ 11 b A b h 2 + Q ~ 113 A 3 t 3 2 3 + h t 3 + h 2 A 24 = e ~ 311 A 1 t 1 t 1 2 + h A 25 = - e ~ 313 A 3 t 3 t 3 2 + h A 33 = K C 551 A 1 + C 55 b A b + C 553 A 3 A 44 = - ξ ~ 31 A 1 t 1 2 A 55 = - ξ ~ 33 A 3 t 3 2 . (23)

Here Q~111 and A1 refer to the coefficient Q~11 and the cross-section area of the bottom piezoelectric layer; Q~11b and Ab refer to the coefficient Q~11 and the cross-section area of the host beam; and so on.

3 FINITE ELEMENT MODEL

In view of the constitutive relations for N and M in (22), expression (16) reads

- 0 L A 11 ϵ 0 + A 12 θ ' + A 14 ϕ - 1 + A 15 ϕ - 3 δ ϵ 0 + A 12 ϵ 0 + A 22 θ ' + A 24 ϕ - 1 + A 25 ϕ - 3 δ θ ' + Q δ ã X Y + L 1 δ ϕ - 1 + L 3 δ ϕ - 3 d X + 0 L q X δ u + q Y δ v d X + i = 1 2 F X i δ u i + F Y i δ v i + M Z i δ θ i = 0 . (24)

Introducing the constant Ci=0 or Ci=1 in order to identify the piezoelectric layer i as actuator or sensor, respectively, the above expression is rewritten as

- 0 L A 11 ϵ 0 + A 12 θ ' + C 1 A 14 ϕ - 1 + C 3 A 15 ϕ - 3 δ ϵ 0 + A 12 ϵ 0 + A 22 θ ' + C 1 A 24 ϕ - 1 + C 3 A 25 ϕ - 3 δ θ ' + Q δ ã X Y + C 1 L 1 δ ϕ - 1 + C 3 L 3 δ ϕ - 3 d X + 0 L q X δ u + q Y δ v - 1 - C 1 A 14 ϕ - 1 + 1 - C 3 A 15 ϕ - 3 δ ϵ 0 - 1 - C 1 A 24 ϕ - 1 + 1 - C 3 A 25 ϕ - 3 δ θ ' d X + i = 1 2 F X i δ u i + F Y i δ v i + M Z i δ θ i = 0 (25)

or

- 0 L N - δ ϵ 0 + M - δ θ ' + Q δ ã X Y + C 1 L 1 δ ϕ - 1 + C 3 L 3 δ ϕ - 3 d X + 0 L q X δ u + q Y δ v - ϕ ϵ δ ϵ 0 - ϕ θ δ θ ' d X + i = 1 2 F X i δ u i + F Y i δ v i + M Z i δ θ i = 0 (26)

with

N - = A 11 ϵ 0 + A 12 θ ' + C 1 A 14 ϕ - 1 + C 3 A 15 ϕ - 3 M - = A 12 ϵ 0 + A 22 θ ' + C 1 A 24 ϕ - 1 + C 3 A 25 ϕ - 3 ϕ ϵ = 1 - C 1 A 14 ϕ - 1 + 1 - C 3 A 15 ϕ - 3 ϕ θ = 1 - C 1 A 24 ϕ - 1 + 1 - C 3 A 25 ϕ - 3 . (27)

The displacements and rotation are linearly approximated over the element, whereas the voltage is taken to be constant in a piezoelectric sensor layer:

u = N u T d v = N v T d θ = N θ T d ϕ - 1 = N 1 T d ϕ - 3 = N 3 T d (28)

where

d = u 1 v 1 θ 1 u 2 v 2 θ 2 ϕ - 1 c ϕ - 3 c T (29)

collects the nodal values of u, v and θ, and the element constant values assumed for ϕ-1 and ϕ-3. The shape functions are given by

N u = 1 - X L 0 0 X L 0 0 0 0 T N v = 0 1 - X L 0 0 X L 0 0 0 T N θ = 0 0 1 - X L 0 0 X L 0 0 T N 1 = 0 0 0 0 0 0 1 0 T N 3 = 0 0 0 0 0 0 0 1 T . (30)

The adoption of constant voltages ϕ-1 and ϕ-3 is consistent with the fact that a piezoelectric layer contains metallic electrodes on its top and bottom surfaces so that each electrode forms an equipotential region (Yasin et al., 2010Yasin, M.Y., Ahmad, N., Alam, M.N. (2010). Finite element analysis of actively controlled smart plate with patched actuators and sensors, Latin American Journal of Solids and Structures 7(3):227-247.).

Substitution of (28) into (25) gives

δ d T f N + f M + f Q + f L 1 + f L 3 + p + r = 0 (31)

with

f N d = - 0 L N - cos θ N u ' + sin θ N v ' + ã X Y N θ d X f M d = - 0 L M - N θ ' d X

f Q d = 0 L Q sin θ N u ' - cos θ N v ' + 1 + ϵ 0 N θ d X

Figure 4:
(a) Components of Äd in the local system XY; (b) components of Ädg in the global system X-Y-.

f L 1 ( d ) = - C 1 0 L L 1 N 1 d X f L 3 ( d ) = - C 3 0 L L 3 N 3 d X (32)

p ( d ) = 0 L q X N u + q Y N v - ϕ ϵ cos θ N u ' + sin θ N v ' + γ X Y N θ - ϕ θ N θ ' d X

r = F X 1 F Y 1 M Z 1 F X 2 F Y 2 M Z 2 0 0 T .

The internal load vectors fN, fM, fQ, fL1 and fL3 are associated with the axial force N, bending moment M, shear force Q, and the electrical quantities L1 and L3, respectively. The external load vector p is work equivalent to the distributed forces qX and qY, and the voltages ϕ-1 and ϕ-3 applied to the piezoelectric actuator layers. The unknown reactions are denoted by r. As (31) must hold for arbitrary δd, the term in parentheses must vanhish to result in the eight element equations

ψ d = f N + f M + f Q + f L 1 + f L 3 + p + r = 0 . (33)

The first six equations represent the element nodal equilibrium equations, while the last two equations are the application of Gauss law to the piezoelectric layers.

4 COMPUTATIONAL ASPECTS

If d does not satisfy the system of nonlinear equations (33) for a given externally applied load or voltage, i.e., ψd0, corrections Δd is found using the Newton-Raphson algorithm (Zienkiewicz et al, 2013Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z. (2013). The Finite Element Method: Its Basis and Fundamentals, 7th edn., Elsevier (Amsterdam).) replacing (33) by the linear system

k Δ d = Δ f (34)

where Δf=ψd is the unbalanced load vector. In this iterative process, the element tangent stiffness matrix

k = - ψ d (35)

is updated at each iteration, replacing d by d+Δd, until a state of convergence is achieved.

The linear system (34) of each element must be written first in the global system XgYg to then obtain the assembled linear system of the entire structure from all element contributions. In the notation of Figure 4, the increments Δd and Δf in the local system XY are related to the increments Δdg and Δfg in the global system XgYg by

Table 1
Properties of PZT-5H.

Δ d = T Δ d g Δ f = T Δ f g . (36)

The fact that

T = cos α sin α 0 0 0 0 0 0 - sin α cos α 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 cos α sin α 0 0 0 0 0 0 - sin α cos α 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 (37)

is orthogonal implies that the element tangent stiffness matrix in both systems are related by

k g = T T k T . (38)

The contribution of ϕϵ to the vector p is clearly of follower nature because it is dependent on the structure displacements (Hibbitt, 1979Hibbitt, H.D. (1979). Some follower forces and load stiffness, International Journal for Numerical Methods in Engineering 14(6):937-941.). Marinković et al. (2009Marinković, D., Köppe, H., Gabbert, U. (2009). Aspects of modeling piezoelectric active thin-walled structures, Journal of Intelligent Material Systems and Structures 20(15):1835-1844.), Rama et al. (2016Rama, G., Marinkoviז, D., Zehn, M. (2016). Piezoelectric co-rotational 3-node shell element, American Journal of Engineering and Applied Sciences 9(4):902-912.) and Marinković and Rama (2017) devote special attention in the analysis of similar contribution present in their models of piezoelectric shells. One must point out here,however, that the work done by the contribution of ϕϵ is not path dependent because it does not affect the symmetry of k and, consequently, it is definitely conservative (Bažant and Cedolin, 2010Bažant, Z.P., Cedolin, L. (2010). Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories, 3rd edn., World Scientific (Hackensack).).

Because the magnitudes of the coefficients Cij, eij and ξi are quite different, the condition number of the global tangent stiffness matrix can be very high. To overcome this problem, Qi et al. (1997Qi, H., Fang, D., Yao, Z. (1997). FEM analysis of electro-mechanical coupling effects of piezoelectric materials, Computational Materials Science 8(4):283-290.) proposed to balance the magnitudes of the constitutive coefficients where a multiple of the force unity 1 N'10pN is used with p. Doing this, Cij [N/m2] and i [C2/Nm2] in (18) are changed to Cij'Cij×10-p and i'i×10p. Also, the voltage ϕ-k [Nm/C] is changed to ϕ-k'ϕ-k×10-p. For the PZT-5H given in Table 1, the value p=9 is chosen so that the orders of 101 exhibited by Cij' and 100 exhibited by i' are well balanced with the order of 100 exhibited by eij.

The internal load vectors fN and fM may be rewritten as

f N = f N m + f N e f M = f M m + f M e (39)

where

f N m = - 0 L A 11 ϵ 0 + A 12 θ ' cos θ N u ' + sin θ N v ' + γ X Y N θ d X f M m = - 0 L A 12 ϵ 0 + A 22 θ ' N θ ' d X (40)

are mechanical contributions and

f N e = - 0 L C 1 A 14 ϕ - 1 + C 3 A 15 ϕ - 3 cos θ N u ' + sin θ N v ' + γ X Y N θ d X f M e = - 0 L C 1 A 24 ϕ - 1 + C 3 A 25 ϕ - 3 N θ ' d X (41)

Figure 5:
Cantilever beam with transverse tip force and distributed sensing.

are electrical contributions from piezoelectric sensor layers. The magnitudes of fNe and fMe are very small in comparison with those of fNm and fMm, respectively.

As reported by Laulusa and Reddy (2004Laulusa, A., Reddy, J.N. (2004). On shear and extensional locking in nonlinear composite beams, Engineering Structures 26(2):151-170.), locking generates large internal load vectors that in turn may stiffen the structure. Lucena Neto et al. (2019Lucena Neto, E., Carvalho Neto, A.L., Monteiro, F.A.C., Santos, R.R.F. (2019). A consistent Timoshenko element beam for small strain but large rotation, Proceedings of the 7th International Symposium on Solid Mechanics, São Carlos, Brazil.) show the importance of evaluating fNm and fQ using a reduced one-point Gaussian quadrature to alleviate membrane and shear locking in purely mechanical beams. The contributions of fNe, fMm and fMe are also here evaluated in this manner. While fMe is always exactly obtained with just one Gauss point, fMm would require more points for nonsymmetric cross sections. Reduced one-point Gaussian quadrature is also important in the evaluation of fL1 and fL3 because of locking, and in the evaluation of the contribution of ϕϵ to the vector p because of computational efficiency. The use of reduced integration may introduce spurious modes into the element stiffness matrix (Cook et al., 2002Cook, R.D., Malkus, D.S., Plesha, M.E., Witt, R.J. (2002). Concepts and Applications of Finite Element Analysis, 4th edn., John Wiley (New York).; Zienkiewicz et al., 2013Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z. (2013). The Finite Element Method: Its Basis and Fundamentals, 7th edn., Elsevier (Amsterdam).). Fortunately, an eigenvalue analysis of the stiffness matrix (35) in the initial configuration and obtained with just one Gauss point reveals that the element possesses only three null eigenvalues associated with the expected physical rigid body motions.

5 RESULTS

According to the mathematical model, the piezoelectric layers solely operate in extension mode since both polarization and electric field are in the Y- direction. In the numerical evaluation of the constitutive parameters Q~11, e~31 and ξ~3 in (20), the host structure is assumed to be made of aluminum with Young´s modulus E=70.3 GPa and Poisson´s ratio ν=0.345, and the piezoelectric layers are assumed to be made of PZT-5H with properties indicated in Table 1 (Yang, 2018Yang, J. (2018). An Introduction to the Theory of Piezoelectricity, 2nd edn., Springer (Cham).). As a transversely isotropic material (isotropic in the plane normal to Y-axis), the PZT-5H presents C23=C13, C22=C11, C55=C44, C66=C11-C12/2, e32=e31, e24=e15 and ξ2=ξ1.

The accuracy and effectiveness of the proposed element under mechanical or electrical loading is demonstrated by means of nonlinear static problems. A code in MATLAB language has been written to carry out the numerical tests. For convergence control, displacement and unbalanced load criteria are selected (Cook et al., 2002Cook, R.D., Malkus, D.S., Plesha, M.E., Witt, R.J. (2002). Concepts and Applications of Finite Element Analysis, 4th edn., John Wiley (New York).). The Newton-Raphson iterations cease when

m a x e D ; e F 10 - 3 (42)

where

e D = D k T D k D k + 1 T D k + 1 e F = F k T F k P + R . (43)

The quantity eD represents the ratio between the Euclidean norms of the displacement increment vector Dk at the end of iteration k and the accumulated displacement vector Dk+1. The quantity eF represents the ratio between the Euclidean norms of the unbalanced load vector Fk at the end of iteration k and the sum of the equivalent external load vector P and the load vector R applied directly to the nodes.

Table 2
Values of U, V, Θ, Φ for the cantilever beam with transverse tip force and distributed sensing according to Santos (2018Santos, R.R.F. (2018). Vigas Piezelétricas sob Pequenas Deformações mas Grandes Rotações: uma Formulação Lagrangiana Total, MSc Thesis, Instituto Tecnológico de Aeronáutica, São Paulo, Brazil.).

5.1 Cantilever beam with transverse tip force and distributed sensing

The cantilever beam of Figure 5 is subjected to a transverse tip force, with two symmetrical piezoelectric layers continuously distributed as sensors. The following geometry is adopted (units in mm):

L = 200 b = b 1 = b 3 = 25 2 h = 2 t 1 = t 3 = 1 . (44)

Table 2 shows the solutions Us, Vs and Θs found in Santos (2018Santos, R.R.F. (2018). Vigas Piezelétricas sob Pequenas Deformações mas Grandes Rotações: uma Formulação Lagrangiana Total, MSc Thesis, Instituto Tecnológico de Aeronáutica, São Paulo, Brazil.) for U=-uL, V=-vL and Θ=-θL, respectively. Also, the table shows the solution s that represents both

1 = A 44 L A 24 ϕ - 1 0 t 1 3 = A 55 L A 25 ϕ - 3 0 t 3 (45)

in Santos’ model. All the quantities are evaluated for PL2/K3=1, 4, 10 where

K 3 = A 22 - A 24 2 A 44 - A 25 2 A 55 . (46)

For γXY=0 (no transverse shear strain) and ϵ0=0 (inextensibility of the beam axis), the Santos’ solutions are “exact” in the sense that they are analytically stated in a closed form. Moreover, these solutions can be widely used for a fixed value of PL2/K3. In the sequel, the performance of the proposed element is accessed in terms of the parameters

k u = U U s k v = V V s k θ = Θ Θ s k ϕ 1 = 1 s k ϕ 3 = 3 s . (47)

Lucena Neto et al. (2019Lucena Neto, E., Carvalho Neto, A.L., Monteiro, F.A.C., Santos, R.R.F. (2019). A consistent Timoshenko element beam for small strain but large rotation, Proceedings of the 7th International Symposium on Solid Mechanics, São Carlos, Brazil.) have shown that a reduced one-point Gaussian quadrature must be employed in the evaluation of mechanical contributions to the internal load vector to properly alleviate the membrane and shear locking effects. One should now investigate the importance of a reduced quadrature in the evaluation of fL1 and fL3 in order to detect the presence of locking in the application of Gauss law to the piezoelectric sensors.

Taking PL2/K3=4 at 10 increments and a coarse regular mesh of 4 elements, Figure 6 plots kϕ1 and kϕ3 against the length-to-thickness ratio L/H, where H=2h+t1+t3, varying the number of Gauss points in the evaluation of fL1 and fL3. Results of ku, kv, and kθ are not shown here as they are indistinguishable from those of Santos (2018Santos, R.R.F. (2018). Vigas Piezelétricas sob Pequenas Deformações mas Grandes Rotações: uma Formulação Lagrangiana Total, MSc Thesis, Instituto Tecnológico de Aeronáutica, São Paulo, Brazil.) at the scale of the figure. Whenever the quadrature rule includes two or more points, the locking effect is clearly triggered, having almost the same intensity for a given ratio L/H and being particularly pronounced for slender beams. A reduced one-point Gaussian quadrature alleviates the locking effects regardless of the value of L/H. Therefore, to make this simple element efficient, just one Gauss point will be adopted from now on in the numerical integrations of the internal load vectors.

A convergence analysis with respect to the mesh refinement is now carried out. Regular meshes of 4, 8, 16, 32 and 64 elements and tip forces of PL2/K3=1,4,10, each applied at 10 increments, are considered. Table 3 shows that the obtained results converge quickly to the solutions given by Santos (2018Santos, R.R.F. (2018). Vigas Piezelétricas sob Pequenas Deformações mas Grandes Rotações: uma Formulação Lagrangiana Total, MSc Thesis, Instituto Tecnológico de Aeronáutica, São Paulo, Brazil.). It should be noted that: (a) the tip displacements and rotation converge faster than the voltages at the clamped end of the beam. In fact, the voltages ϕ-10 and ϕ-3(0) given by Santos (2018) are compared with the constant values measured over the sensor patches attached to the element placed next to the clamped end; (b) the convergences of the displacements and sensed

Figure 6:
Cantilever beam with a transverse tip force of PL2/K3=4 and distributed sensing. The internal load vectors fN, fM and fQ are evaluated using a reduced one-point Gaussian quadrature, while fL1 and fL3 are evaluated using different numbers of Gauss points. A coarse regular mesh of 4 elements is used in the beam model.

Table 3
Cantilever beam with transverse tip force and distributed sensing.

Figure 7:
Deformed shapes of the cantilever beam with distributed sensing at some values of PL2/K3.

voltage are monotonic from below, while the convergence of the rotation is monotonic from above; (c) the induced voltage measured by the bottom sensor patch is, in general, slightly smaller than that measured by the top one. This can be physically understood if one imagines that the contraction of the bottom patch is also, in general, slightly smaller than the extension of the top patch.

Although the table results are obtained using L/H=50, nearly no changes are observed for L/H=400 which reveals the effective locking alleviation given by the one-point reduced integration scheme. However, when the code is running for L/H=4 (see tabled results in parentheses) larger differences are observed. This is a clear consequence of the relevant contribution given by the transverse shear strain. It is worth mentioning that for all ratios L/H the obtained results are in close agreement with those found by Meireles (2019Meireles, J.G.S.A. (2019). Um Modelo Corrotacional de Elementos Finitos para Pórticos Planos Piezelétricos, MSc Thesis, Instituto Tecnológico de Aeronaútica, São Paulo, Brazil.), which uses finite element corotational models based on a first-order shear deformation theory.

Deformed shapes of the cantilever beam at successive load levels PL2/K3=1, 4, 10 are depicted in Figure 7. As shown, excellent results are quickly reached with the mesh refinement. This same comment also applies to the results ploted in Figure 8, where the distributions of ϕ-1 and ϕ-3 along the beam are obtained for PL2/K3=10 using a mesh of 8 elements.

5.2 Cantilever beam with distributed actuation

With respect to the previous example, the following changes are carried out: (a) the applied tip load is removed; (b) the electric potentials ϕ-1=10 and ϕ-3=0 (units in volt) are applied to the bottom and top piezoelectric layers,respectively; (c) the thickness of the bottom piezoelectric layer is set at a higher value of t1=2.5 mm. Santos (2018Santos, R.R.F. (2018). Vigas Piezelétricas sob Pequenas Deformações mas Grandes Rotações: uma Formulação Lagrangiana Total, MSc Thesis, Instituto Tecnológico de Aeronáutica, São Paulo, Brazil.) has reported the “exact” results

U s = - 5.1638 × 10 - 8 m m V s = - 5.9929 × 10 - 6 m m Θ s = - 5.9929 × 10 - 5 r a d (48)

for this problem.

Changes in the number of Gauss points in the evaluation of the contribution of ϕϵ to the vector p have led to nearly the same results. Adopting for simplicity a reduced one-point Gaussian quadrature in that evaluation, the convergence in terms of ku, kv and kθ is attained, with at least 3 significant figures, even for a coarse mesh of 2 elements and 2 load increments. This remarkably result with a reduced number of increments is mainly due to the properly account of p/d in the tangent stiffness matrix, whose approach is apparently not followed by Marinković et al. (2009Marinković, D., Köppe, H., Gabbert, U. (2009). Aspects of modeling piezoelectric active thin-walled structures, Journal of Intelligent Material Systems and Structures 20(15):1835-1844.), Rama et al. (2016Rama, G., Marinkoviז, D., Zehn, M. (2016). Piezoelectric co-rotational 3-node shell element, American Journal of Engineering and Applied Sciences 9(4):902-912.) and Marinković and Rama (2017) in their nonlinear analyses of piezoelectric shells.

Figure 8:
Distribution of the sensed voltage Φ1=A44Lϕ-1/A24t1 and Φ3=A55Lϕ-3/A25t3 for PL2/K3=10 using 8 elements.

5.3 Modified Williams toggle frame

The toggle frame of Williams (1964Williams, F.W. (1964). An approach to the non-linear behaviour of the members of a rigid jointed plane framework with finite deflections, The Quarterly Journal of Mechanics and Applied Mathematics 17(4):451-469.) has been used by several authors (Wood and Zienkiewicz, 1977Wood, R.D., Zienkiewicz, O.C. (1977). Geometrically nonlinear finite element analysis of beams, frames, arches and axisymmetric shells, Computers and Structures 7(6):725-735.; Nanakorn and Vu, 2006Nanakorn, P., Vu, L.N. (2006). A 2D field-consistent beam element for large displacement analysis using the total Lagrangian formulation, Finite Elements in Analysis and Design 42(14-15):1240-1247.; Lee et al., 2014Lee, K., Han, S., Hong, J.W. (2014). Post-buckling analysis of space frames using concept of hybrid arc-length methods, International Journal of Non-Linear Mechanics 58:76-88.) to verify the performance and efficiency of nonlinear finite elements. The nonlinear response of the frame is dictated by the geometry and inclination of the members. The original version of the toggle frame involves clamped supports. Herein, in order to intensify the nonlinear effects, one has modified the original geometry and changed the support conditions to simply-supported ones (see Figure 9). Moreover, two piezoelectric sensor patches, with 25 mm of width and 1 mm of thichness, are bonded to the top surface of each frame leg which has 25 mm of width and 2 mm of thichness. Due to the presence of snap through, the vertical displacement =6 mm applied at 10 increments, rather than the load P, is prescribed at the apex (displacement control). In a more sophisticated manner, the arc-length technique could also be used to trace the whole equilibrium path in a slightly expensive way (Crisfield, 1991Crisfield, M.A. (1991). Non-linear Finite Element Analysis of Solids and Structures, vol. 1, John Wiley (Chichester).).

Assuming that the structure deformation takes place symmetrically, only half of the frame needs to be meshed. Figure 10 shows the responses P(), obtained with a regular mesh of eight elements per frame leg, for the structure with and without sensor patches. For the purely mechanical case, converged results obtained from a Nastran model of CBEAM elements are also included in the figure (marked with small circle), which are in very good agreement with the developed finite element. Finally, the voltages ϕ-3 predicted by the sensor patches 1 and 2 along the loading history are plotted in Figure 11. Both patches exhibit highly nonlinear responses which can be useful to access the limit point of the structure. It is worth mentioning that the presence of turning points in the space ϕ-3P, as shown in the figure, does not bring any numerical difficulties because the solution control is guided by the vertical displacement at the apex (see space P in Figure 10).

Figure 9:
Modified Williams toggle frame (units in mm).

Figure 10:
Apex load P versus the applied vertical displacement (Nastran results are indicated by small circles).

Figure 11:
Voltage ϕ-3 predicted by the sensors 1 and 2.

6 CONCLUSION

The developed finite element for piezoelectric planar frames is simple and remarkably accurate. This has been confirmed by comparing its results with analytical solutions, when available, and with converged results obtained from developed finite element models. The adopted methodology is also versatile because it can accommodate both continuously and discretely attached piezoelectric layers in sensing or actuating arrangements.

Some points deserve to be highlighted:

  • (a) small strains are consistently incorporated, where the displacement field is assumed to be in accordance with the Timoshenko beam theory and the electric potential varies linearly through each piezoelectric layer thickness. No restrictions are imposed on the amount of rotation.

  • (b) the discretized problem is written in terms of two flags C1 and C3 that properly identify the piezoelectric layers as actuator or sensor;

  • (c) the contribution of ϕϵ to the vector p, which is of follower nature, is identified as being conservative;

  • (d) the locking phenomenon revealed in the application of Gauss law to the piezoelectric sensor layers are removed by evaluating the internal load vectors fL1 and fL3 by a reduced one-point Gaussian quadrature;

  • (e) it seems that limit points of a structure may be easily accessed by reading a sensor patch response;

  • (f) the evaluation of all internal load vectors fN, fM, fQ, fL1 and fL3, and also the evaluation of the contribution of ϕϵ to the vector p, by means of just one Gauss point suffice to make this simple element efficient. No spurious modes are introduced in this process.

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Edited by

Guest Editors:

Volnei Tita and Nicholas Fantuzzi.

Publication Dates

  • Publication in this collection
    09 Nov 2020
  • Date of issue
    2020

History

  • Received
    31 Jan 2020
  • Reviewed
    16 July 2020
  • Accepted
    28 Aug 2020
  • Published
    02 Sept 2020
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