Thermal buckling analysis of porous circular plate with piezoelectric sensor-actuator layers under uniform thermal load

Abstract

This study presents the thermal buckling analysis of solid circular plate made of porous material bounded with piezoelectric sensor-actuator patches. The porous material properties vary through the thickness with specific function. The general mechanical nonlinear equilibrium and linear stability equations are derived using the variational formulations to obtain the governing equations of piezoelectric porous plate. Thermal buckling load is derived for solid circular plates under uniform temperature load for the clamped edge condition. In recent paper the effects of porous plate’s thickness, porosity, porous thermal expansion coefficient, piezoelectric thickness, piezoelectric thermal expansion coefficient, and feedback gain on thermal stability of the plate are investigated.

Keywords

Thermal buckling circular plate porous material piezoelectric sensor-actuator patches

Introduction

The poro material structures, such as beams, plates, and shells, are widely used in structural design problems. Biot is the pioneer to study the field of poroelasticity interaction. In his landmark paper on poroelasticity, Biot introduced the following bulk dynamic and kinematic variables (in the current notation); the total stress tensor σ_ij, the fluid pressure P_p, the solid strain tensor $ɛ ij$ , and the variation of fluid volume content ζ. This theory was reformulated using the concept of partial stresses. In the Biot model, the deformable porous medium is viewed as a continuum consisting of a solid phase and fluid phase. The medium stress–strain relations are generally expressed in terms of the elastic constants of the solid and fluid phases by Detournay and Cheng [1].

During the last several years problem of deflection and buckling of the poro plates with varying properties has been developed by many authors. For example, buckling of porous beams with varying properties was described by Magnucki and Stasiewicz [2]. They used shear deformation theory for solving the critical buckling load. In this work, the effect of porosity on the strength and buckling load of the beam is investigated. Buckling of circular porous plate with varying properties and simply supported boundary conditions are described by Magnucka-Blandzi [3]. Dynamic stability of a metal foam circular plate with varying properties is described by Magnucka-Blandzi [4]. Zimmerman [5] studied on thermoelastic and poroelastic coupling parameters for a linear poroelastic saturated rock. He concluded that poroelastic coupling parameter has a stronger influence than thermoelastic one.

Buckling analysis of poro plates with functional properties has similarities with the FGM plates to some extent. Ma and Wang [6 –8] investigated nonlinear bending and post-buckling of circular functionally graded plates subjected to mechanical and thermal loads based on the first-order and the third-order shear deformation theories. Zenkour and Sobhy [9] studied on thermal buckling of various types of FGM sandwich plates. They considered the shear deformation, the higher and first-order shear deformation theories, and classical plate theory (CPT) in their investigation. Wu [10] obtained the closed form solution for the thermal buckling of functionally graded rectangular simply supported plates subjected to two types of temperature fields; uniform temperature rise and gradient across the thickness of the plate, employing the first-order shear deformation theory. Woo and Meguid [11] derived an analytical solution expressed in terms of the Fourier series for large displacement of FG plates and shallow shells under both transverse mechanical loading and temperature rise fields. Ghiasian et al. [12] investigated on thermal buckling of shear deformable temperature-dependent circular/annular FGM plates. They find out general nonlinear equilibrium equations based on the first-order shear deformation plate theory. Thermoelastic buckling analysis of orthotropic solid circular plates based on first-order plate theory and the Sanders assumption is given by Najafizadeh and Heydari [13]. Javaheri and Eslami [14,15] presented the thermal buckling of rectangular FGM plates based on the first- and higher-order plate theories. The thermal buckling analysis of circular FGM plates based on first-order-theory is given by Najafizadeh and Eslami [16].

One of the techniques to increase the buckling load capacity is equipping a structure with the smart materials. Piezoelectric materials, as sensors and actuators, are one of the most common sub-group of smart materials and are used in solid structures to control the deformation, vibration, and buckling of structures. Among different types of piezoelectric materials, PZT has higher piezoelectric, dielectric, and elasticity coefficients. The active buckling control of smart functionally graded plates using sensor-actuator patches is studied by Viliani et al. [17]. Mirzavand and Eslami [18] obtained closed-form solutions for the critical temperatures of simply-supported piezoelectric FGM cylindrical shells based on the higher order displacement field. Their study includes three types of temperature loadings combined with constant applied voltage. Jabbari et al. [19] studied the buckling of porous circular plate with piezoelectric layers based on CPT. They investigated the effect of porosity and piezoelectric plate on stability of the plate under uniform compression load. Khorshidvand et al. [20] presented thermoelastic buckling analysis of functionally graded circular plates integrated with the piezoelectric layers based on the CPT. They also investigated on buckling analysis of porous plate with sensor actuator patches under uniform compression load [21]. Wang et al. [22] presented an analytical solution for free vibration analysis of a piezoelectric coupled circular plate with simply supported and clamped boundary conditions based on the CPT. They showed that the CPT cannot furnish a reliable solution for the vibration response when the plate thickness increases, where the reason is the neglect of transverse shear deformation and rotary inertia effects. Thermo-electro-mechanical buckling and post-buckling of FGM plates with piezoelectric actuators is reported by Shen [23 –25] based on the singular perturbation method. Liew et al. [26] presented post-buckling of piezoelectric FGM plates subjected to the thermo-electro-mechanical loading. They used a semi-analytical iteration to determine the post-buckling response of the plate. Seifi et al. [27] investigated on critical buckling loads and modes of cross-ply laminated annular plates. They studied on buckling of composite annular plates under uniform internal and external radial edge loads by using energy method. Vosoughi et al. [28] investigated on thermal buckling and post-buckling of laminated composite beams with temperature-dependent properties. The governing equations are based on the first-order shear deformation beam theory (FSDT) and the geometrical nonlinearity is modelled using Green’s strain tensor in conjunction with the von Karman assumptions in their investigation. Padmanav [29] studied on buckling and post-buckling of laminated composite plates using higher order shear deformation theory associated with GreenLagrange nonlinear strain-displacement relationships.

The present paper provides an analysis to obtain the critical buckling temperature of porous circular plates integrated with piezoelectric sensor-actuator patches, subjected to uniform thermal load. The energy method, using calculus of variations, is employed to obtain the governing equations based on the CPT. Eigenvalue solution is carried out for the plates with clamped edge. Material properties of porous plate are assumed to vary by a power law distribution through the thickness of plate. The input voltage to the actuator is supplied with the output voltage of the sensor.

Derivation of the governing equations

Strain-displacement relations

The displacement field is based on classical laminated composite plate theory and is expressed as follows [30]

u = u 0 - z \frac{\partial w 0}{\partial r} w = w 0

(1)

where u₀, w₀ are the displacements along the r and z coordinate directions, respectively, for a point on the midplane (z = 0).

The plate is assumed to be comparatively thin, and according to the Love-Kirchhoff assumptions, planes normal to the median surface are assumed to remain plane after deformation. Thus, shear deformations normal to the plate are disregarded. Using the CPT, strain components at distance z from the middle plane are given in matrix form as Reddy [30] demonstrated

ɛ rr = \bar{ɛ} rr + zk rr ɛ θ θ = \bar{ɛ} θ θ + zk θ θ γ r θ = \bar{γ} r θ + 2 zk r θ

(2)

where

ɛ rr, ɛ θ θ

, and γ_rθ are the strain components along the r-, θ-, and z- directions, respectively. Here,

\bar{ɛ} rr

\bar{ɛ} θ θ

, and

\bar{γ} r θ

are the middle plane strain components and k_rr, k_θθ, and k_rθ are bending and twisting curvatures with respect to the r- and θ- axes, respectively.

The relations between the middle plane strains and curvatures with the displacement components according to the Sanders assumption are obtained as Brush and Almorth [31] demonstrated

\bar{ɛ} rr = u, r + \frac{1}{2} (w, r) 2 \bar{ɛ} θ θ = \frac{1}{r} v, θ + \frac{u}{r} + \frac{1}{2} (\frac{1}{r} w, θ) 2 \bar{γ} r θ = \frac{1}{r} u, θ + v, r - \frac{v}{r} + \frac{1}{r} (w, r) (w, θ) k rr = - w, rr k θ θ = - \frac{1}{r} w, r - \frac{1}{r 2} w, θ θ k r θ = - \frac{1}{r} w, r θ + \frac{1}{r 2} w, θ

(3)

where a comma in subscript indicates partial differentiation.

Consider a uniform thin circular plate made of porous material with radius a and thickness h in the middle with sensor-actuator patches with thicknesses h_a bonded to its upper and lower surfaces, as shown in Figure 1. To derive the formulations, a cylindrical coordinates system is taken in the center of plate middle plane. The functional relationship between E, G, and z for poro plate are assumed as [3,4]

G (z) = G 0 [1 - e 1 \cos ((\frac{π}{2 h}) (z + \frac{h}{2}))] E (z) = E 0 [1 - e 1 \cos ((\frac{π}{2 h}) (z + \frac{h}{2}))] e 1 = 1 - \frac{G 1}{G 0} = 1 - \frac{E 1}{E 0}

(4)

where e₁ is the coefficient of plate porosity

0 < e 1 < 1

, E₁ and E₀ Young’s modulus of elasticity at

z = - h / 2

and z = h/2, respectively, and G₁ and G₀ are the shear modulus at

z = - h / 2

and z = h/2, respectively. The relationship between the modulus of elasticity and shear modulus for j = 0 and 1 is E_j = 2G_j(1 + ν) and ν is Poisson’s ratio, which is assumed to be constant across the plate thickness. Mechanical properties of the porous material vary across the thickness of the plate (

G 0 \geq G 1

Figure 1.

Geometry of a piezoelectric coupled porous circular plate.

Stress–strain relation in porous plate

The linear poroelasticity theory of Biot has two features as Detournay and Cheng [1] presented:

An increase of pore pressure induces a dilation of pore.

Compression of the pore causes a rise of pore pressure.

The stress–strain relations for porous material [32] and piezoelectric parts [33] of the plate are written as following

σ ij h = 2 G ɛ ij + λ ɛ kk δ ij - P p α δ ij - 3 K β s' T δ ij

(5)

Δ ɛ kk = \frac{Δ σ kk}{3 K} + \frac{α Δ P p}{K} + β s' Δ T

(6)

Δ ɛ ij = \frac{Δ σ ij}{2 G}, i \neq j

(7)

where

P p = M [ζ - α ɛ + e 1 (β f - β s'') T]

(8)

M = \frac{2 G (ν u - ν)}{α 2 (1 - 2 ν u) (1 - 2 ν)}

(9)

ν u = \frac{ν + α B (1 - 2 ν) / 3}{1 - α B (1 - 2 ν) / 3}

(10)

Here, the suffix h is used to denote the porous plate. P_p is pore fluid pressure, λ and G are Lame’s parameters, K is bulk modulus $(K = 2 G \frac{1 + ν}{3 (1 - 2 ν)})$ , M is Biot’s modulus, ζ is variation of fluid volume content, ν_u is undrained Poisson’s ratio $ν < ν u < 0.5$ , α is Biot coefficient of effective stress $0 < α < 1, β s'$ is the volumetric thermal expansion coefficient of the bulk solid under constant pore pressure and stress, $β s''$ and β_f represent volumetric thermal expansion coefficients of the solid matrix and the pore fluid, respectively. $β s'$ may be considered equal to $β s''$ if the change in temperature is not expected to change porosity [32], $ɛ kk$ is the volumetric strain, and B is Skempton coefficient, the pore fluid properties are introduced by the Skempton coefficient. The two-dimensional stress–strain law for plane-stress condition in the cylindrical coordinate is given by in the undrained condition (ζ = 0)

σ rr h = A 1 ɛ rr + B 1 ɛ θ θ - C 1 T σ θ θ h = A 1 ɛ θ θ + B 1 ɛ rr - C 1 T σ r θ h = G γ r θ P p = M [α (β f - β s'') T - α ɛ kk]

(11)

Taking n₁, n₂, n₃, A₁, B₁, C₁ as follows

A 1 = (\frac{2}{1 - ν u 2}) [1 + ν u + \frac{(ν u - ν) (1 + ν u)}{1 - 2 ν} (1 - \frac{n 3}{n 2})] G (z) B 1 = (\frac{2}{1 - ν u 2}) [(1 + ν u) ν u + \frac{(ν u - ν) (1 + ν u)}{1 - 2 ν} (1 - \frac{n 3}{n 2})] G (z) C 1 = (\frac{2 n 1}{1 - ν u 2}) [\frac{(1 + ν u) 2}{3} - \frac{(ν u - ν) (1 + ν u)}{1 - 2 ν} (1 - \frac{K}{n 2})] G (z) n 1 = 3 β s' + B α (β f - β s') n 2 = 2 [1 + \frac{ν u}{1 - 2 ν u} + \frac{ν u - ν}{(1 - 2 ν u) (1 - 2 ν)}] G (z) n 3 = n 2 - 2 G (z)

(12)

Stress–strain relation in Piezoelectric plate

The stress–strain relations for attached piezoelectric patches are as follows that Lien et al. [34] presented

{σ} p = {σ rr p σ θ θ p σ r θ p} = [c 11 c 12 0 c 12 c 22 000 c 44] {ɛ rr - α p T (z) ɛ θ θ - α p T (z) γ r θ} - [00 e 31 00 e 32 000] {E r E θ E z}

(13)

{D r D θ D z} = [000000 e 31 e 32 0] {ɛ rr ɛ θ θ γ r θ} + [k 11 000 k 22 000 k 33] {E r E θ E z}

(14)

Here, the suffix p is used to denote the piezoelectric plate. ${σ}$ , {c}, ${ɛ}, {e ij}$ , {D}, {k}, and {E} are the stress component, component of the plane-stress-reduced stiffness, strain component, piezoelectric stiffness coefficients, the electrical displacement components, dielectric permeability, and electrical components, respectively, and are determined from the piezoelectric material properties. Here α_p is the thermal expansion of piezoelectric plate. Assuming that the actuator is poled along the z, and viewing the piezoelectric material as a transversely isotropic material which is true for piezoelectric ceramics, many of the parameters in the mentioned matrices will be either zero or can be expressed in terms of the other parameters. In particular, the non-zero coefficients of piezoelectric properties may be written as c₁₁ = c₂₂, c₁₂, c₄₄, e₃₁ = e₃₂, and the non-zero dielectric coefficients are k₁₁ = k₂₂, k₃₃.

Actuator and sensor’s constitutive relations

Actuator’s constitutive relation

For a piezoelectric patch the Maxwell equations are as follows that being presented by Halliday et al. [35]

\nabla . D = 0

(15)

For thin piezoelectric plates the charge equation (15) reduces to

\frac{\partial (rD r)}{r \partial r} + \frac{\partial D θ}{r \partial θ} + \frac{\partial D z}{\partial z} = 0 \to \frac{\partial D z}{\partial z} = 0

(16)

The electric potential inside the actuator patch is assumed as

φ a = φ 0 + z φ 1 + z 2 φ 2

(17)

Here, the suffix a is used to denote the actuator layer. To find $φ 1$ and $φ 2$ , the electric boundary condition is assumed as

φ = V a at z = z 0 = h / 2 + h a φ = 0 at z = z 1 = h / 2

(18)

By substituting equation (17) into electric boundary condition we have

φ 1 = \frac{V a}{h a} - (h a + h) φ 2

(19)

Substituting equation (19) into equation (17) gives

φ a = φ 0 + z (\frac{V a}{h a} - (h a + h) φ 2 a) + z 2 φ 2

(20)

By substituting equation (14) into (16) and considering $E z = - \frac{\partial φ}{\partial z}$ , we have

φ 2 = \frac{e 13}{2 k 33} (k rr + k θ θ)

(21)

Thus

φ a = φ 0 + z (\frac{V a}{h a} - (h a + h) φ 2 a) + z 2 (\frac{e 13}{2 k 33} (k rr + k θ θ))

(22)

Then the electric field inside the actuator can be obtained as

E z a = - \frac{\partial φ a}{\partial z} = - \frac{V a}{h a} + e 13 (\frac{h m a - z}{k 33}) (k rr + k θ θ)

(23)

where h_m is

(h a + h) / 2

Sensor’s constitutive relation

According to open circuit condition for sensor patch, the dielectric displacement is zero (D_z = 0). Then, the electric field in the sensor is as follows that is presented by Jalili [36]

E z s = - \frac{e 13}{k 33} ((\bar{ɛ} rr + \bar{ɛ} θ θ) + z (k rr + k θ θ))

(24)

Here, the suffix s is used to denote the sensor layer. Substituting equation (24) into equation (13), the stress–strain relations for the sensor patch can be obtained as

σ rr s = (c 11 + \frac{e 312}{k 33}) ɛ rr + (c 12 + \frac{e 312}{k 33}) ɛ θ θ - (c 11 + c 12) α p T (z)

(25)

σ θ θ s = (c 12 + \frac{e 312}{k 33}) ɛ rr + (c 22 + \frac{e 312}{k 33}) ɛ θ θ - (c 12 + c 22) α p T (z)

(26)

σ r θ s = c 44 γ r θ

(27)

The average sensor potential (the voltage that appears between sensor electrodes) is

V s = \frac{1}{A s} \int (φ s | z = z 2 - φ s | z = z 3) d A = \frac{e 31 h a}{k 33} ((\bar{ɛ} rr + \bar{ɛ} θ θ) + h m (k rr + k θ θ))

(28)

The output voltage of sensor (V^s) is fed into the control algorithm and V^a is obtained as follows

V a = GV s, V a = \frac{Ge 31 h a}{k 33} ((\bar{ɛ} rr + \bar{ɛ} θ θ) + h m (k rr + k θ θ))

(29)

Here G is feedback gain of piezoelectric patch. Substituting equation (29) into equation (23), then substituting into equation (13), the stress–strain relations for the actuator patch can be obtained as

σ rr a = (c 11 + G \frac{e 312}{k 33}) \bar{ɛ} rr + (c 12 + \frac{Ge 312}{k 33}) \bar{ɛ} θ θ + [(c 11 + \frac{e 312}{k 33}) z + \frac{(G - 1)}{k 33} h m e 312] k rr + [(c 12 + \frac{e 312}{k 33}) z + \frac{(G - 1)}{k 33} h m e 312] k θ θ - (c 11 + c 12) α p T (z) σ θ θ a = (c 11 + G \frac{e 312}{k 33}) \bar{ɛ} θ θ + (c 12 + \frac{Ge 312}{k 33}) \bar{ɛ} rr + [(c 11 + \frac{e 312}{k 33}) z + \frac{(G - 1)}{k 33} h m e 312] k θ θ + [(c 12 + \frac{e 312}{k 33}) z + \frac{(G - 1)}{k 33} h m e 312] k rr - (c 12 + c 22) α p T (z) σ r θ a = c 44 γ r θ

(30)

Variational formulation

The total potential energy for porous circular plate integrated with two piezoelectric layers can be written as follows

U = U h + U p

(31)

In which the potential energy for porous circular plate (U^h) is as

U h = \frac{1}{2} \int σ ij h d ɛ ij E d V

(32)

where

ɛ ij E

is elastic strain. Elastic strain energy for porous materials is comprised of elastic strain energy for solid body and fluid in pores. Substituting equation (11) into equation (6) in the undrained condition, strain relation is obtained as

ɛ ii = \frac{(1 - α B)}{3 K} σ ii + [\frac{α 2 M}{K} (β f - β s') + β s'] T, i = r, θ

(33)

Here, $(1 - α B)$ is relation between drained bulk modulus and undrained bulk modulus, αB is coupling between pore fluid effects and macroscopic deformation. Substituting $σ kk = 3 K ɛ kk E$ in the equation (33) elastic strain gives

ɛ ii E = \frac{1}{(1 - α B)} [ɛ ii - n 4 T], i = r, θ

(34)

where

n 4 = β s' + B α (β f - β s')

(35)

where σ_kk and

ɛ kk E

are volumetric stress and elastic volumetric strain, respectively. Substituting equation (34), which calculates strain energy for porous materials, into equation (32) gives strain energy for the undrained condition

U h = \frac{1}{2} \int (\frac{(ɛ rr - n 4 T)}{(1 - α B)} σ rr h + \frac{(ɛ θ θ - n 4 T)}{(1 - α B)} σ θ θ h + γ r θ σ r θ h) d V

(36)

Substituting equations (11) into equation (36), the following expression for U^h is obtained

U h = \frac{1}{2} \int_r \int_θ \int_z [\frac{A 1}{(1 - α B)} (ɛ rr 2 + ɛ θ θ 2) + \frac{2 B 1}{(1 - α B)} (ɛ rr ɛ θ θ) \times 2 H 1 (1 - α B) (ɛ rr + ɛ θ θ) T + 2 C 1 n 4 T 2 + 2 G γ r θ 2] r d r d θ d z

(37)

where

H 1 = \frac{1}{2} (n 4 A 1 + n 4 B 1 + C 1)

(38)

and the strain energy for piezoelectric layers (U^p) with considering the thermal effects is as

U p = \frac{1}{2} \int r \int θ \int z ({ɛ - α p Δ T} T ([C]) {ɛ - α p Δ T} - {E} T ([k]) {E} - 2 {ɛ - α p Δ T} T ([e]) {E}) r d z d θ d r

(39)

Substituting relations (37) and (39) into relation (31), it becomes

V = U + Ω = \int 0 r \int 0 2 π F d r d θ

(40)

where

F = f (u, u, r, u, θ, v, v, r, v, θ, w, w, r, w, θ, w, rr, w, θ θ, w, r θ) \times r

(41)

Applying the Euler equations for total functional of V in equation (41), we obtain

N rr, r + \frac{(N rr - N θ θ)}{r} + \frac{1}{r} N r θ, θ = 0 \frac{2}{r} N r θ + \frac{1}{r} N θ θ, θ + N r θ, r = 0 (rN rr w, r + N r θ w, θ - M θ θ), r + (\frac{1}{r} N θ θ w, θ + N r θ w, r + \frac{2}{r} M r θ), θ + (rM rr), rr + (2 M r θ), r θ + (\frac{1}{r} M θ θ), θ θ = 0

(42)

The stability equations of the circular plate are derived using the adjacent equilibrium criterion. We assume u₀, v₀, and w₀ as the displacement components of the equilibrium state and u₁, v₁, and w₁ as the virtual displacements corresponding to a neighboring state. The displacement components of the neighboring state are

u = u 0 + u 1 v = v 0 + v 1 w = w 0 + w 1

(43)

According to the adjacent equilibrium criterion in the neighboring state of equilibrium, the stability equations are found. Similar to equations (43), the stress and moment resultants are found to be the sum of those related to the equilibrium and neighboring states as

N rr = N rr 0 + N rr 1 M rr = M rr 0 + M rr 1 N θ θ = N θ θ 0 + N θ θ 1 M θ θ = M θ θ 0 + M θ θ 1 N r θ = N r θ 0 + N r θ 1 M r θ = M r θ 0 + M r θ 1

(44)

Substituting relations (43) and (44) in equations (42) and collecting the second-order terms, the stability equations are obtained as

N rr 1, r + \frac{N rr 1 - N θ θ 1}{r} + \frac{1}{r} N r θ 1, θ = 0 \frac{2}{r} N r θ 1 + \frac{1}{r} N θ θ 1, θ + N r θ 1, r = 0 (rN rr 0 w 1, r + N r θ 0 w 1, θ - M θ θ 1), r + (\frac{1}{r} N θ θ 0 w 1, θ + N r θ 0 w 1, r + \frac{2}{r} M r θ 1), θ + (rM rr 1), rr + (2 M r θ 1), r θ + (\frac{1}{r} M θ θ 1), θ θ = 0

(45)

The force and moment resultants of plate are expressed in terms of the stress components through the thickness as

{N} = {N rr N θ θ N r θ} = \int - h / 2 + h / 2 {σ} h d z + \int - (h / 2 + h a) - h / 2 {σ} p d z + \int h / 2 h / 2 + h a {σ} p d z

(46)

{M} = {M rr M θ θ M r θ} = \int - h / 2 + h / 2 z {σ} h d z + \int - (h / 2 + h a) - h / 2 z {σ} p d z + \int h / 2 h / 2 + h a z {σ} p d z

(47)

The stress resultant can be simplified in the matrix form as

{N rr N θ θ N r θ M rr M θ θ M r θ} = [2 A 2 * B 2 * 0 A 3 B 3 0 B 2 * 2 A 2 * 0 B 3 A 3 000 2 C 2 * 00 C 3 A 3 B 3 0 2 A 4 * B 4 * 0 B 3 A 3 0 B 4 * 2 A 4 * 000 C 3 00 2 C 4 *] {\bar{ɛ} rr \bar{ɛ} θ θ \bar{γ} r θ k rr k θ θ k r θ} + {N rr N θ θ N r θ M rr M θ θ M r θ} T

(48)

where

A 2 = \frac{1}{2} \int - h / 2 + h / 2 A 1 d z, B 2 = \frac{1}{2} \int - h / 2 + h / 2 2 B 1 d z, C 2 = \frac{1}{2} \int - h / 2 + h / 2 G d z A 3 = \frac{1}{2} \int - h / 2 + h / 2 2 A 1 z d z, B 3 = \frac{1}{2} \int - h / 2 + h / 2 2 B 1 z d z, C 3 = \frac{1}{2} \int - h / 2 + h / 2 4 Gz d z A 4 = \frac{1}{2} \int - h / 2 + h / 2 A 1 z 2 d z, B 4 = \frac{1}{2} \int - h / 2 + h / 2 2 B 1 z 2 d z, C 4 = \frac{1}{2} \int - h / 2 + h / 2 4 Gz 2 d z A 2 * = A 2 + \frac{2 C 11 h a + \frac{e 312}{k 33} h a + G \frac{e 312}{k 33} h a}{2}, B 2 * = B 2 + \frac{4 C 11 h a + 2 \frac{e 312}{k 33} h a + 2 G \frac{e 312}{k 33} h a}{2}, C 2 * = C 2 + C 44 h a A 3 * = A 3 + \frac{2 (G - 1) \frac{e 312}{k 33} h m h a}{2}, B 3 * = B 3 + \frac{2 (G - 1) \frac{e 312}{k 33} h m h a}{2}, A 4 * = A 4 + \frac{(G - 1) \frac{e 312}{k 33} h m 2 h a + 2 (c 11 + \frac{e 312}{k 33}) (\frac{h 2 h a}{4} + \frac{hh a 2}{2} + \frac{h a 3}{3})}{2}, B 4 * = B 4 + \frac{2 (G - 1) \frac{e 312}{k 33} h m 2 h a + 4 (c 11 + \frac{e 312}{k 33}) (\frac{h 2 h a}{4} + \frac{hh a 2}{2} + \frac{h a 3}{3})}{2}, C 4 * = C 4 + 2 C 44 (\frac{h 2 h a}{4} + \frac{hh a 2}{2} + \frac{h a 3}{3})

(49)

Here, the quantities N⁽ ^T ⁾ and M⁽ ^T ⁾ are the resultants due to the applied temperature on the plate, respectively, and they can be computed as

{N i, M i} T = {N i, M i} Porous T + {N i, M i} Piezo T, i = rr, θ θ

(50)

{N i, M i} Porous T = \int - h / 2 + h / 2 H 1 \frac{(1, z)}{(1 - α B)} T (z) d z, i = rr, θ θ

(51)

{N i, M i} Piezo T = \int - h / 2 - h a - h / 2 α p T (z) (c 11 + c 12) (1, z) d z + \int h / 2 h / 2 + h a α p T (z) (c 11 + c 12) (1, z) d z, i = rr, θ θ

(52)

Thermal axisymmetric buckling analysis

Consider a circular plate subjected to a uniform Thermal load. Here, the polar symmetry condition is considered. Thus, for this case of discussion the second of stability equations will be satisfied and the first and third equations (45), based on the displacement components and change of variable as k = βr, lead to

2 A 2 * [(\frac{d 2 u 1}{dk 2}) + \frac{1}{k} (\frac{du 1}{dk}) - \frac{1}{k 2} (u 1)] - A 3 * β [(\frac{d 3 w 1}{dk 3}) + \frac{1}{k} (\frac{d 2 w 1}{dk 2}) - \frac{1}{k 2} (\frac{dw 1}{dk})] = 0

(53)

- 2 A 4 * β 3 [k (\frac{d 4 w 1}{dk 4}) + 2 (\frac{d 3 w 1}{dk 3}) - \frac{1}{k} (\frac{d 2 w 1}{dk 2}) + \frac{1}{k 2} (\frac{dw 1}{dk})] + A 3 * β 2 [k (\frac{d 3 u 1}{dk 3}) + 2 (\frac{d 2 u 1}{dk 2}) - \frac{1}{k} (\frac{du 1}{dk}) + \frac{1}{k 2} (u 1)] + β \frac{d}{dk} (kN rr 0 \frac{dw 1}{dk}) = 0

(54)

The functions for displacements that satisfy governing equations and boundary conditions are

u 1 = D 1 J 1 (k) + D 2 Y 1 (k) + D 3 k + \frac{D 4}{k}

(55)

w 1 = R 1 J 0 (k) + R 2 Y 0 (k) + R 3 \ln (k) + R 4

(56)

where

J 1, J 0, Y 0, Y 1

are the Bessel functions of first, and zero orders and first and second kinds, respectively. Also, D₁, D₂, D₃, D₄, R₁, R₂, R₃, and R₄ are the integration constants. For solid circular plate, D₂ and R₂ are considered equal to zero.

Substitution of equations (55) and (56) into equations (53) and (54) yield

[2 A 2 * D 2 + β A 3 * D 5] kJ 0 (k) + [- 2 A 2 * D 1 - β A 3 * D 4 + 2 β A 3 * D 5] J 1 (k) = 0

(57)

[- 2 A 4 * β 3 D 4 - A 3 * β 2 D 1 - β N rr 0 D 4 + 8 A 4 * β 3 D 5 + 2 A 3 * β 2 D 2 + 2 β N rr 0 D 5] kJ 0 (k) + [- 2 A 4 * β 3 D 5 - A 3 * β 2 D 2 - β N rr 0 D 5] k 2 J 1 (k) = 0

(58)

For a nontrivial solution of these equations, the coefficients of functions must be set to zero

2 A 2 * D 2 + β A 3 * D 5 = 0 - 2 A 2 * D 1 - β A 3 * D 4 + 2 β A 3 * D 5 = 0 - 2 A 4 * β 3 D 4 - A 3 * β 2 D 1 - β N rr 0 D 4 + 8 A 4 * β 3 D 5 + 2 A 3 * β 2 D 2 + 2 β N rr 0 D 5 = 0 - 2 A 4 * β 3 D 5 - A 3 * β 2 D 2 - β N rr 0 D 5 = 0

(59)

The second and third equations give relations between integration constants and combination of the first and fourth equations give

[- 2 A 2 * - A 3 * β - A 3 * β - 2 A 4 * β 2 - N rr 0] [D 2 D 5] = 0

(60)

By setting the determinant equal to zero, yields

β 2 (4 A 2 * A 4 * - A 32) + (2 A 2 *) (N rr 0) = 0

(61)

Equation (61) is the characteristic equation. Total pre-buckling force is

N rr 0 = N rr 0 (T) = - \int - h / 2 + h / 2 \frac{H 1}{(1 - α B)} T (z) d z - \int - h / 2 - h a - h / 2 α p T (z) (c 11 + c 12) d z - \int h / 2 h / 2 + h a α p T (z) (c 11 + c 12) d z

(62)

Consider porous and piezoelectric plates at reference temperature. In such a case, the uniform temperature may be raised with ΔT to such that the plates buckle. Introducing dimensionless form for T_cr as $T * = T cr β s'$ and substituting in equations (61) and (62), critical temperature concludes as follows

T * = \frac{β 2 (2 A 4 * - A 32 / 2 A 2 *)}{{\int - h / 2 + h / 2 \frac{H 1}{(1 - α B) β s'} T (z) d z + \int - h / 2 - h a - h / 2 \frac{α p}{β' s} T (z) (c 11 + c 12) d z + \int h / 2 h / 2 + h a \frac{α p}{β s'} T (z) (c 11 + c 12) d z}}

(63)

Boundary condition

The continuous and symmetric conditions at the center of the plate (r = 0)

w 1 (r = 0) = Finite, \frac{d}{dk} w 1 (r = 0) = u 1 (r = 0) = 0

(64)

For the clamped boundary condition at the edge of plate (r = a) is expressed as

w 1 (r = a) = \frac{d}{dk} w 1 (r = a) = 0, u 1 (r = a) = 0

(65)

Substituting from equations (55) and (56) into equations (64) and (65) yields

R 1 J 1 (β a) = 0

(66)

The first zero of the Bessel function of first kind and order one yield

β a = 3.832

(67)

Substitution equation (67) into equation (63) yields

T * = 3.8322 (2 A 4 * - A 32 / 2 A 2 *) {\int - h / 2 + h / 2 \frac{H 1}{(1 - α B) β s'} T (z) d z + \int - h / 2 - h a - h / 2 \frac{α p}{β s'} T (z) (c 11 + c 12) d z + \int h / 2 h / 2 + h a \frac{α p}{β s'} T (z

(68)

Results and discussion

For the problem under consideration, the axisymmetric stability and thermal buckling load of a porous circular plate integrated with sensor-actuator patches subjected to uniform thermal load is derived and summarized in the preceding section. The material properties of porous and piezoelectric plates are given in Table 1.

Table 1.

Material properties of saturated porous plate (Tennessee marble) presented by Detournay and Cheng [1] piezoelectric (PZT4) constituents presented by Yang [33].

Tennessee marble	Piezoelectric: PZT4
G₀ = 24 (GPa)	c₁₁ = 138 (Gpa)
ν = 0.3	c₁₂ = 77.8 (Gpa), ν = 0.3, $α p = 2 * 106 (1 / \circ C)$
B = 1	$k 33 = 0.562 * 10 - 8$ (C/vm), $e 31 = e 32$ = −5.2 (C/m²)

For this example the results for thermal buckling load are plotted in Figures 2 –6. Figure 2 presents thermal buckling load for a porous plate without piezoelectric patches versus h/a for different values of porosity. It can be seen that by increasing the thickness of porous plate the plate tends to be stable. This stability decreases by increasing the porosity too. Figure 3 shows the thermal stability of porous plate versus h/a for different values of $\bar{β}$ . As Figure 2 shows, for a porous plate without sensor actuator patches it can be seen that by increasing the thickness of porous plate, thermal stability of plate increases, and as it is clear by increasing the $\bar{β}$ the plate tends to be unstable. Because when the temperature increases, the porous plate with high ratio of pore fluid thermal expansion to solid matrix thermal expansion $(\bar{β})$ tends to expand more and therefore the plate tends to be more instable. In Figure 4 different behavior for porous circular plate with piezoelectric sensor actuator patches can be seen. In this figure porous and piezoelectric circular plate has these two features: Porous plate with $(\bar{β} = 10)$ and piezoelectric plate with (β_p = 0.5). These two values clearly show that thermal expansion coefficient of porous plate is much more than piezoelectric patches. Therefore it has significant effect on behavior of porous piezoelectric plate. By increasing the porous plate thickness, first the plate tends to be instable, because by increasing of the porous plate’s thickness we are going to increase the proportion of the plate with more thermal expansion coefficient (Porous plate) with respect to the plate with less thermal expansion (Piezoelectric patches). Therefore increasing the thickness of porous plate to specific values increases the total thermal expansion coefficient of plates and decreases the stability. Then increasing the thickness of the plate to more than specific value has other effect on stability that increases the total thickness and therefore it increases the plate’s stiffness and the plate tends to be more stable. In Figure 5 it can be seen that by adding the piezoelectric patches the plate’s stability increases significantly with respect to the porous plate without piezoelectric patches, because the total stiffness of plates increases by adding stiffer patches. In this figure it is also seen that by adding the positive feedback gain the stability of plate increases and by applying the negative one the stability of plate decreases. It is related to the effect of feedback gain on coefficients $A 2 *, A 3 *$ , and $A 4 *$ . In Figure 6 variations of buckling load with respect to feedback gain for specific values of porosity is presented. It shows that positive feedback gains increase the buckling load and the negative ones decrease the buckling load. As mentioned before, positive feedback gain increases the stiffness coefficients and the plate will be more stable.

Figure 2.

None dimensional thermal load vs. $\frac{h}{a}$ , for clamped plate, and for the cases [e₁ = 0.1, 0.3, 0.5, 0.7], $\bar{β} = 10, \frac{ha}{a} = 0$ .

Figure 3.

None dimensional thermal load vs. $\frac{h}{a}$ , for clamped plate, and for the cases $[\bar{β} = 3, 5, 7, 10], e 1 = 0.5,$ $\frac{ha}{a} = 0$ .

Figure 4.

None dimensional thermal load vs. $\frac{h}{a}$ , for clamped plate, and for the cases [e₁ = 0.1, 0.3, 0.5, 0.7, 0.9], $\bar{β} = 10, \frac{ha}{a} = 0.0034, β p = 0.5$ and for G = 0.

Figure 5.

None dimensional thermal load vs. $\frac{h}{a}$ , for clamped plate, and for the cases $e 1 = 0.7, [\bar{β} = 10],$ $\frac{ha}{a} = 0.0034, β p = 0.5$ , and for different feedback gains.

Figure 6.

None dimensional thermal load vs. G, for clamped plate and for the cases [e₁ = 0.4, 0.5, 0.6]. $\bar{β} = 10, \frac{ha}{a} = 0.0034, β p = 0.5$ .

Conclusions

In the present article, the energy method is used for the thermal buckling analysis of plate and derivation is based on the CPT with the assumption that the porosity of the material changes as a specific function. The equilibrium and stability equations for a porous circular plate bonded with sensor-actuator patches are obtained. The boundary conditions of the plate are taken to be clamped. Plate is subjected to uniform thermal load. The effects of porosity and piezoelectric layers on thermal buckling capacity of circular plates as closed-form solution are presented. It is concluded that

The critical thermal load (T*) decreases and the plate will be unstable by increasing the porosity.

For the porous plate without piezoelectric patches the critical thermal load (T*) will increase when the ratios h/a increase.

The critical thermal load (T*) for circular plate with piezoelectric patches has different behavior with respect to the porous plate without piezoelectric patches. In analysis of porous plate with piezoelectric layers it can be seen that increases the porous plate thickness first increases the total thermal expansion and therefore decreases the stability of plate, then by increases more thickness the total thickness and therefore total stiffness of the plates increases and therefore plate tends to be more stable.

The critical thermal load (T*) will increase by the positive feedback gain and will decrease by the negative ones.

Footnotes

Funding

This research is supported by Islamic Azad University, South Tehran Branch, Tehran, Iran.

Conflict of interest

None declared.

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