Static and Dynamic Analysis of Piezoelectric Laminated Composite Beams and Plates

In this chapter, the mechanical behavior analysis of piezoelectric laminated composite beams and plates is influenced subjected to static, dynamic, and aerodynamic loads. Algorithm for dynamic, stability problem analysis and vibration control of laminated composite beams and plates with piezoelectric layers is presented. In addition, numerical calculations, considering the effect of factors on static, dynamic, and stability response of piezoelectric laminated composite beams and plates are also clearly presented. The content of this chapter can equip readers with the knowledge used to calculate the static, dynamic, and vibration control of composite beams, panels made of piezoelectric layers applied in the field different techniques.


Introduction
The content of this chapter is the inheritance and development of the research results of the authors and other authors by published scientific works on composite materials, piezoelectric and structural calculation by piezoelectric composite materials.

Electromechanical interaction of piezoelectric materials 2.1 Mechanical-electrical behavior relations
Let us consider a block of elastic material in an environment with an electric field of zero, the relationship between stress and strain is followed Hooke's law, and written as follows [1,2]: where {σ} is the mechanical stress vector, {ε} is the mechanical strain vector, and [c] is the material stiffness matrix of beam.
Mechanical-electrical relations in piezoelectric materials have an interactive relationship, strain {ε} will produce eε -polarization, where e is the voltage stress factor when there is no mechanical strain. The imposed electric field E produces the -eE stress in the piezoelectric material according to the reverse voltage effect. Therefore, we have a mathematical model that describes the mechanical-electrical interaction relationship in piezoelectric materials as follows [3][4][5][6]: where [e] is the piezoelectric stress coefficient matrix, [p] is the dielectric constant matrix, {E} is the vector of applied electric field (V/m), and {D} is the vector of electric displacement (C/m 2 ).
For the linear problem and small strain, strain vector in the piezoelectric structures can be defined as follows: in which [s] is the matrix of compliance coefficients (m 2 /N), [d] is the matrix of piezoelectric strain constants (m/V).
In the field of engineering, piezoelectric materials are used by two types. The first type, the piezoelectric layers or the piezoelectric patches act as actuators, called the piezoelectric actuators. In this case, the piezoelectric layers are strained when imposing an electric field on it. The second type, the piezoelectric layers or piezoelectric patches act as sensors, called piezoelectric sensors. In this case, the voltage is generated in piezoelectric layers when there is mechanical strain.

Piezoelectric actuators
Eq. (5) can be written in the matrix form as follows [4,6]: Finally, Eq. (6) becomes: 3. Static and dynamic analysis of laminated composite beams with piezoelectric layers

Displacement and strain
Based on the first-order shear deformation theory (FSDT), the displacement field at any point of the beam is defined as [1,2]: where u, w denotes the displacements of a point (x, z) in the beam; u 0 , w 0 are the displacements of a point at the beam neutral axis, and θ y is the rotation of the transverse normal about the y axis. The bending and shear strains associated with the displacement field in Eq. (10) are defined as: in which ε x , γ xz are the normal strain, and shear strain, respectively. Using finite element method, we consider 2-node bending elements with 3 degrees of freedom per node ( Figure 1).
The displacements of the beam neutral axis are expressed in local coordinate system in the form: where {q b } e is the vector of vector of nodal displacements of element, [N M ] is the matrix mechanical shape functions:  Substituting Eq. (12) into Eq. (11), we obtain: The electric potential is constant over the element surface: where n is the element node number. A voltage ϕ is applied across an actuator of layer thickness t p generates an electric field vector {E}, such that [4,[8][9][10]: in which where t pk is the thickness of the k th piezoelectric layer. Substituting Eq. (19) into Eq. (18), the electric field vector {E} can also be defined in terms of nodal variables as: Using Eqs. (15), and (20), the linear piezoelectric constitutive equations coupling the elastic and electric fields will be completely determined by Eqs. (2) and (3).

Finite element equations
Using Hamilton's principle, we have [11][12][13]: where T e , U e are the kinetic and potential energy, respectively and W e is the work done by external forces. They are determined by: where Element mass matrix: Element mechanical stiffness matrix: Element mechanical-electrical coupling stiffness matrix: Element electrical-mechanical coupling stiffness matrix: Element piezoelectric permittivity matrix: where { f} e , {Q } e are the applied external load and charge, respectively.

Static analysis
In the case of beams subjected to static loads, zero acceleration, from Eqs. (25) and (26), we obtain the static equations of the beam as follows: Assembling the element equations yields general static equation: where [K bb ], [K ϕϕ ] are the overall mechanical stiffness and piezoelectric permittivity matrices respectively; [K bϕ ] and [K ϕb ] are the overall mechanicalelectrical and electrical -mechanical coupling stiffness matrices, respectively, and {q}, {ϕ} are respectively the overall mechanical displacement, and electric potential vector.

Free vibration analysis
For free vibrations, from Eq. (40), the governing equation is: The beam vibrations induce charges and electric potentials in sensor layers. Therefore, the control system allows current to flow and feeds back to the actuators. In this case, if we apply no external charge Q to a sensor, from Eq. (39), we will have: and Q f g s ¼ K ϕb Â Ã s q f g s is the induced charge due to strain. The operation of the amplified control loop implies, the actuating voltage is determined by the following relationship [1,10,14]: where G d and G v are the feedback control gains for displacement and velocity. Substituting Eq. (43) into Eq. (39), the charge in the actuator due to actuator strain in response to the beam vibration modified by control system feedback is: Substituting (42) into (44) leads to: Substituting Eq. (45) into (40), we obtain: In the case of considering the structural damping, the equation of motion of the beam is: is the active damping matrix, is the total of mechanical stiffness matrix and piezoelectric, is the overall structural damping matrix, α R , and β R are respectively the Rayleigh damping coefficients, which are generally determined by the first and second natural frequencies (ω 1 , ω 2 ) and ratio of damping ξ, {f} is the overall mechanical force vector.
Eq. (48) can be solved by the direct integration Newmark's method.

Numerical analysis
An example for free vibration of laminated beam affected by piezoelectric layers is presented here. The beam is made of four layers symmetrically (0°/90°/90°/0°) of epoxy-T300/976 graphite material with 2.5 mm thickness per layer, and with one layer piezo ceramic materials bonded to the top and bottom surfaces, 2.0 mm thickness per layer as shown in Figure 2 is considered (a = 0.254 m, b = 0.0254 m). The material properties of the piezo ceramic layers and graphite-epoxy are shown in Table 1.
The direct integration Newmark's method is used with parameters α R = 0.5, β R = 0.25; integral time step Δt = 0.005 s with total time calculated t = 15 s.

Dynamic analysis of laminated piezoelectric composite plates 4.1 The electromechanical behavioral relations in the plate
Consider laminated composite plates with general coordinate system (x, y, z), in which the x, y plane coincides with the neutral plane of the plate. The top and bottom surfaces of the plate are bonded to the piezoelectric patches or piezoelectric layers (actuator and sensor). The plate under the load acting on its neutral plane has any temporal variation rule ( Figure 5).
Hypothesis: The piezoelectric composite plate corresponds with Reissner-Mindlin theory. The material layers are arranged symmetrically through the neutral plane of the plate, ideally adhesive with each other.

Strain -displacement relations
Based on the first-order shear deformation theory, the displacement fields at any point in the plate are [7,8]: where u, v and w are the displacements of a general point (x, y, z) in the laminate along x, y and z directions, respectively. u 0 , v 0 , w 0 , θ x and θ y are the displacements and rotations of a midplane transverse normal about the y-and xaxes respectively. The components of the strain vector corresponding to the displacement field (49) are defined as: For the linear strain: or in the vector form: and for the nonlinear strain:

Stress-strain relations
The equation system describing the stress-strain relations and mechanicalelectrical quantities is respectively written as [8,14]: where the shear stress vector, [Q] is the ply in-plane stiffness coefficient matrix in the structural coordinate system, [Q s ] is the ply out-of-plane shear stiffness coefficient matrix in the structural coordinate system. Notice that {τ b } is free from piezoelectric effects. The in-plane force vector at the state pre-buckling: σ 0

Total potential energy
The total potential energy of the system is given by: where W is the energy of external forces, V p is the entire domain including composite and piezoelectric materials.

Introducing [A], [B], [D], [A s ], and vectors {N p }, {M p } as [8]:
where h is the total laminated thickness and combining with (5), (6) the total potential energy equation (8) can be written where Ω is the plane xy domain of the plate.

Finite element models
Nine-node Lagrangian finite elements are used with the displacement and strain fields represented by Eqs. (49), (53), and (54). In the developed models, there is one electric potential degree of freedom for each piezoelectric layer to represent the piezoelectric behavior and thus the vector of electrical degrees of freedom is [6,14]: in which NPL e is the number of piezoelectric layers in a given element. The vector of degrees of freedom for the element {q e } is: where q e is the mechanical displacement vector for node i.

Dynamic equations
The dynamic equations of piezoelectric composite plate can be derived by using Hamilton's principle, accordingly, the vibration equation of the membrane (without damping) with in-plane loads is: The equation of bending vibrations with out-of-plane loads is: where [M ss ], [K ss ] are the overall mass, membrane elastic stiffness matrix respectively, and q ss È É , _ q ss È É , € q ss È É are respectively the membrane displacement, velocity, acceleration vector. [M bb ], [K bb ] and q bb È É , _ q bb È É , € q bb È É are the overall mass, bending elastic stiffness matrix and the bending displacement, velocity, acceleration vector; [K G ] is the overall geometric stiffness matrix; ([K G ] is a function of external in-plane loads); {F(t)} is the in-plane load vector, {R} is the normal load vector, {Q el } is the vector containing the nodal charges and in-balance charges.
The element coefficient matrices are: where in which N 0 where [A b ] and [B s ] are the stiffness coefficient matrix and strain-displacement matrix of the plane bending problem.
Stability criteria [14]: • In the case of plate subjected to periodic in-plane loads and without damping, the elastic stability problems become simple only by solving the linear equations to determine the eigenvalues.
• In case of the plate under any in-plane dynamic load and with damping, the elastic stability problems become very complex. This iterative method can be proved effectively and the following dynamic stability criteria are used: • Plate is considered to be stable if the maximum bending deflection is three times smaller than the plate's thickness: Eq. (71) has the solution (w i ) max satisfying the condition 0 ≤ w i j j max < 3h, where w i is the deflection of the plate at node number i.
• Plate is called to be in critical status if the maximum bending deflection of the plate is three times equal to the plate's thickness. Eq. (71) has the solution (w i ) max satisfying the condition w i j j max ¼ 3h.
• Plate is called to be at buckling if the maximum deflection of the plate is three times larger than the plate's thickness: Eq. (71) has the solution (w i ) max satisfying the condition w i j j max >3h.
The identification of critical forces is carried out by the iterative method.

Iterative algorithm
Step 1. Defining the matrices, the external load vector and errors of load iterations.
Step 2. Solving Eq. (71) to present unknown displacement vector, {q ss } and the stress vector is defined by (72), updating the geometric stiffness matrix [K G ].
Step 3. Solving Eq. (71) to present unknown bending displacement vector {q bb }, and then testing stability conditions. À If for all w i j j ¼ 0: increase load, recalculate from step 2; À If at least one value w i j j 6 ¼ 0: + In case: 0 < w i j j max < 3h: Define stress vector by Eq. (73), update the geometric stiffness matrix [K G ]. Increase load, recalculate from step 2; ≤ ε D : Critical load p = p cr . End.

Numerical analysis
Stability analysis of piezoelectric composite plate with dimensions a Â b Â h, where a = 0.25 m, b = 0.30 m, h = 0.002 m. Piezoelectric composite plate is composed of three layers, in which two layers of piezoelectric PZT-5A at its top and bottom are considered, each layer thickness h p = 0.00075 m; the middle layer material is Graphite/Epoxy material, with thickness h 1 = 0.0005 m. The material properties for graphite/epoxy and PZT-5A are shown in Section 5.1 above. One short edge of the plate is clamped, the other three edges are free. The in-plane half-sine load is evenly distributed on the short edge of the plate: p(t) = p 0 sin(2πft), where p 0 is the amplitude of load, f = 1/T = 1/0.01 = 100 Hz (0 ≤ t ≤ T/2 = 0.005 s) is the excitation frequency, voltage applied V = 50 V. The iterative error of the load ε D = 0.02% is chosen.
Consider two cases: with damping (ξ = 0.05, G v = 0.5, G d = 15) and without damping (ξ = 0.0, G v = 0.0, G d = 15). The response of vertical displacement at the plate centroid over the plate thickness for the two cases is shown in Figure 6.
The results show that the critical load of the plate with damping is larger than that without damping. In the two cases above, the critical load rises by 6.8%.
Analyze the stability of the plate with damping when a voltage of À200, À150, À100, À50, 0, 50, 100, 150 and 200 V is applied to the actuator layer of the piezoelectric composite plate.   Figure 7 shows the time history of the vertical displacement at the plate centroid over the plate thickness when a voltage of 0, 50, 100, 150 and 200 V is applied. The relation between critical load and voltages is shown in Figure 8.
The results show that the voltage applied to the piezoelectric layers affects the stability of the plate. As the voltage increases, the critical load of the plate also increases.
When the amplitude of the load changes from 0.25p cr to 1.5p cr (where p cr is the amplitude of the critical load), a voltage of 50 V is applied to the actuator layer of the plate.
The results show the time history response of the vertical displacement at the plate centroid over the plate thickness as seen in Figure 9.

M xs
with [B xs ] is the strain-displacement relations matrix, [D xs ] is the stress-strain relations matrix and l e is the element length, N u 0 ½ , N w ½ and N θ x ½ are the shape function matrices relating the primary variables u 0 , w, x, in terms of nodal unknowns, I y is the area moment of inertia related to the y-axis and P ¼ P n with ρ k is density of k th layer.

Formulation of y-Stiffener
The same as for x-stiffener, the element stiffness and mass matrices of the ystiffener are defined as follows:

Modeling the effect of aerodynamic pressure and motion equations of the smart composite plate-stiffeners element
Based on the first order theory, the aerodynamic pressure l h and moment m θ , can be described as [15][16][17]: where k ¼ bω=U is defined as the reduced frequency, ω is the circular frequency of oscillation of the airfoil, U is the wind velocity, B is the half-chord length of the airfoil or half-width of the plate, ρ a is the air density and α is the angle of attack. where The solution of nonlinear Eq. (89) is carried out by using Newmark direct and Newton-Raphson iteration method.

Numerical applications
A rectangle cantilever laminated composite plate is assumed to be [0°/90°] s with total thickness 4 mm, length of 600 mm and width of 400 mm with three stiffeners along each direction x and y. The geometrical dimension of the stiffener is 5 mm of high and 10 mm of width. The plate and stiffeners are made of graphite/epoxy with mechanical properties: E 11 = 181 GPa, E 22 = E 33 = 10.3 GPa, E 12 = 7.17 GPa, ν 12 = 0.35, ν 23 = ν 32 = 0.38, ρ = 1600 kgÁm À3 . Material properties for piezoelectric layer made of PZT-5A are: d 31 = d 32 = À171 Â 10 À12 m/V, d 33 = 374 Â 10 À12 m/V, d 15 = d 24 = À584 Â 10 À12 m/V, G 12 = 7.17 GPa, G 23 = 2.87 GPa, G 32 = 7.17 GPa, ν PZT = 0.3, ρ PZT = 7600 kgÁm À3 and thickness t PZT = 0.15876 mm, ξ = 0.05, G v = 0.5, G d = 15. The effects of the excitation frequency and location of the actuators are presented through a parametric study to examine the vibration shape of the composite plate activated by the surface bonded piezoelectric actuators. The iterative error of the load ε D = 0.02% is chosen. The piezoelectric stiffened composite plate is subjected to the airflow in the positive x direction as shown in Figure 10a.
Dynamic response of the piezoelectric stiffened composite plate is shown in Figure 12.