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Уточненная квазитрехмерная теория функционально-градиентных пористых пластин: смещения и напряжения
A.M. Zenkour
Университет короля Абдул-Азиза, Джидда, 21589, Саудовская Аравия Университет Кафрельшейха, Кафрельшейх, 33516, Египет
В работе представлена теория сдвиговой и нормальной деформации высокого порядка для решения статической задачи функционально-градиентных пористых толстых прямоугольныж пластин. Учитывается влияние растяжения пористыж пластин в продольном направлении. Свойства функционально-градиентного пористого материала варьируются по толщине пластин. С использованием принципа виртуальныж перемещений получены определяющие уравнения. Статическая задача решается для свободно опертой пластины при синусоидальной нагрузке. Получены точные выражения для смещений и напряжений. Обсуждается влияние градиентного параметра в экспоненциальной форме и коэффициента пористости на смещения и напряжения пористыж пластин. Даны примеры, обосновывающие точность данной 3D теории в предсказании отклика на изгиб пористых пластин. Эффективность представленной модели оценивается на основе численныж результатов для смещений и напряжений функционально-градиентных пористых пластин. Полевые переменные функционально-градиентных пластин очень чувствительны к изменению коэффициента пористости.
Ключевые слова: пористые пластины, функционально-градиентный материал, уточненная теория, смещения, напряжения
DOI 10.24411/1683-805X-2019-11003
Quasi-3D refined theory for functionally graded porous plates: Displacements and stresses
A.M. Zenkour
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh, 33516, Egypt
This paper presents a higher-order shear and normal deformation theory for the static problem of functionally graded porous thick rectangular plates. The effect of thickness stretching in the functionally graded porous plates is taken into consideration. The functionally graded porous material properties vary through the plate thickness with a specific function. The governing equations are obtained via the virtual displacement principle. The static problem is solved for a simply supported plate under a sinusoidal load. The exact expressions for displacements and stresses are obtained. The influences of the functionally graded and porosity factors on the displacements and stresses of porous plates are discussed. Some validation examples are presented to show the accuracy of the present quasi-3D theory in predicting the bending response of porous plates. The effectiveness of the present model is evaluated by numerical results that include displacements and stresses of functionally graded porous plates. The field variables of functionally graded plates are very sensitive to the variation of the porosity factor.
Keywords: porous plates, functionally graded materials, refined theory, displacements, stresses
1. Introduction
Porous material beams, plates, and/or shells are widely used in structural design problems. The permeability, tensile strength, and electrical conductivity depend on the properties of the matrix and fluid in pores, in so soing porosity is regarded as the main property of the porous material. Numerous theoretical and experimental studies have been carried out on porous structures, which operate as novel multifunctional ultralight structural forms with new mechanical properties.
Biot was the first who developed a 3D model of wave propagation in fluid-saturated porous elastic structures [1]. In the meantime, poroelasticity has been successfully used in studies of various problems in engineering structures [2, 3]. Taber performed the quasi-static analysis of a poroelastic plate by applying the classical Kirchhoff plate theory [4]. As it is well known, Kirchhoff's assumptions are invalid for thick plates. Therefore, Busse et al. used the first-order Mindlin plate theory to model moderately thick plates based on Biot's poroelastic theory without presenting numerical
© Zenkour A.M., 2019
results [5]. Borsan used the porous plate theory to propose field equations for bending of thermoelastic plates [6]. Magnucki and Stasiewicz investigated elastic bending of isotropic porous beams with nonlinear hypotheses [7]. Nappa and Iesan presented thermoelastic problems of porous circular pipes and cylinders [8]. Magnucki et al. discussed bending and buckling of a rectangular porous plate taking into account the effect of shear deformation [9]. Magnucka-Blandzi discussed the nonlinear dynamic stability, deflection, and buckling of porous plates [10]. Yang et al. discussed quasi-static and dynamic bending of porous beams under step loads at their free ends [11]. Borsan and Altenbach presented the theory of porous elastic and thermoelastic thin rods modeled by the direct approach [12, 13]. Kumar and Devi presented transient problems for thermoelastic porous materials taking into account the temperature dependence of material properties [14]. Ghiba considered bending of thermoelastic porous plates of the Mindlin type [15]. Sladek et al. presented the first-order Mindlin theory for the bending response of porous plates according to Biot's poroelastic theory [16]. Lyapin and Vatulyan developed a mathematical model to study deformation of porous plates [17].
In recent years, functionally graded materials (FGMs) have found other applications in electrical appliances, energy transformation, biomedical engineering, optics, etc. However, in the manufacture of FGMs, porosities may occur in the materials during the sintering process. Chen et al. performed the buckling and bending analyses of functionally graded porous beams using the first-order beam theory [18]. Bensaid and Guenanou [19] applied the nonlocal Timoshenko beam model to present deflection and buckling of functionally graded nanoscale beams with porosity. Akbas dealt with nonlinear static deflections of functionally graded porous beams under the thermal effect with position- and temperature-dependent material properties [20]. Behravan Rad presented the static response of porous multidirectional heterogeneous structures resting on developed gradient elastic foundations [21]. Barati et al. [22] and Barati and Zenkour [23] discussed the electromechanical vibration of smart piezoelectric functionally graded plates with porosities. Barati and Zenkour presented the
postbuckling behavior of beams reinforced with graphene platelets with porosities and geometrical imperfection [24]. Barati and Zenkour discussed the nonlocal strain gradient elasticity for wave propagation in graded nanoporous double nanobeam systems on the elastic foundation [25, 26].
The present paper obtains displacements and stresses in functionally graded porous plates subjected to a sinusoidal distributed load. It is assumed that the material properties of the porous plate are changed through the plate thickness. The plate is graded according to a novel type of the polynomial law. The plate faces are perfectly homogeneous while the whole plate has a perfect porous homogeneous shape according to the volume fraction of voids (porosity) or the graded factors. The variational principle is used to derive the governing equations based on the quasi-3D theory. Several important aspects that affect displacements and stresses are discussed in details.
2. Formulation of the problem
2.1. Structural model
Consider an functionally graded thick rectangular plate with thickness h, length a, and width b, as depicted in Fig. 1. The Cartesian coordinate system is established so that 0 < x < a, 0 < y < b, and -h/2 < z < h/2. Let the plate be under a distributed load q(x, y) at the upper face z = + h/ 2.
Young's modulus E(z) may vary continuously through the plate thickness by means of the polynomial material law [27]:
E = Ec, p = 0,
p > 0, 0 <a<< 1,
E = Em, P ^^ where a represents the porosity volume function and p is the scalar parameter that defines gradation of material properties in the through-thickness direction, Em and Ec are Young's moduli of the lower (as metal) and upper (as ceramic) faces of a functionally graded plate, respectively. In the case of p = 0 or p ^ the plate becomes purely homogeneous (ceramic or metal) without any porosity.
+ Ec
z+IJ _a
h + 2 I 2
(1)
Fig. 1. Geometry and coordinates of the functionally graded porous plate
2.2. Quasi-3D theory
The quasi-3D shear deformation theory for plates is suitable for the following displacements [27-32]:
dw
u(x, y, z) = uo (x, y) - z— + y(z)Uj (x, y), dx
dw
v(x, y, z) = Vo (x, y) - z— + z)Vi (x, y),
dy
(2)
w(x, y, z) = Wo (x, y) + y (z)w (x, y), where (u, v, w) denote the displacement field of a general point (x, y, z) in the functionally graded plate, (u0, v0, w0) denote the displacement projections on the midplane, uj and Vj are the rotations about the y and x axes, and w1 represents the additional unknown variable. The variables u0, v0, w0, uj, vj; and w1 are the functions of x and y. The prime ' represents differentiation with respect to z. The present theory considers a new perception of ^(z) as
¥( z) =
zh2
5h2+4z2
4z_
27h 2
(3)
which comprises only one additional dependent unknown (w1 ^ 0) in contrast to other theories but takes account of the effect of transverse normal strain through the plate thickness. No shear correction factors are needed for the present theory. According to the displacements in Eq. (2), the linear strains ei (ej = en, e2 = £ 22,e3 = e 33,e4 = e 23, e5 = e13, e6 = e12) are presented as
(4)
£1 ip(0) £1 IV1)] £1 i£(2) £1
£2 = £(0) e2 • + z £(1) e2 + ¥( z)< £(2) £2
£6_ £(0) £6 £(1) £6 £(2) £6
{£4, £5} = V(z) {ef, 4°>}, £3 = V(z)ef. Here £(0), e®, and e(2) are expressed as
£(0) = duo pV-, =
£1 = , £2 = a
dx dy
(0) = dv0
£(°) = n
e(0) =
= u, +
dw1 dx
(0) =
nvu; — —
dvo
du.
e(0) =
u0
£(1) =-d 2 w0 £2 = -, 2 '
dy2
£(1) =-2 d
dx dy
w
dw,
= v, +—1. 1 dy
e(i) = d 2 w0 £1 =
dx2
dxdy
£12) = du1 1 dx
(5)
e22) =
= dv1 e(2) =_dvi ! du1
dy
e^ =—-+-
dx dy
The stress-strain constitutive equations for the present porous plate are represented as
= M z)[e1 + e2 + e3]8i + 2^(z )ei, (6)
where 8i = 1 for i = 1, 2, 3, 8i = 0 for i = 4, 5, 6, X(z) and ^(z) are the Lame coefficients given by
M z) =-
vE ( z)
-, z) =
E ( z)
(7)
(1 + v)(1 - 2 v) 2(1 + v)
and E(z) varies in the through-thickness direction of the porous plate as discussed previously in Eq. (1) while v represents a fixed value of the Poisson ratio.
3. Equilibrium equations
The static equations can be obtained by applying the principle of virtual displacements. It can be stated in its analytical form as
f v 2 f
jfjN J ^i dz-q[wo + V^Su^-^ fdQ = 0 (8)
Q [-h/2 J
or
N1
1 ^L+n2 dx
dSv0
+ N(
' dSu0 + d8v0 ^ dy dx
„, d28w0 d28w0 + N38w1 - M1-^ - M2 0
dx2
dy2
- 2M6 ^ + + , +
dxdy 1 dx
dy
+ S(
' d8v1 + d8u1 dx dy
+ Qs I 8«1 +
d8w!
+ Q41 8V1 + |-q8w0
dx dQ = 0,
(9)
where Nj and Mj (j = 1, 2, 6) are the stress resultants and stress couples, Sj is the additional stress couples, and Ql (l = 4, 5) and N3 are the transverse and normal shear stress resultants. They are introduced in Eq. (6) as
V 2
{Nj, Mj, Sj} = J {1, z, V(z)}ojdz, j = 1, 2, 6,
-V2 (10)
A/2 A/2
Ql = J z)ajdz, N3 = J z)o3dz. -h/2 -h/ 2 Then, by integrating Eq. (9) by parts and putting the coefficients of 8u0, 8c0, 8w0, 8u1,8^1, and 8w1 to zero, separately. Thus, according to the present theory we have
s dN1 dN6 n
8mo: --1 + —6 = 0, dx
8w0:
dy
■ + 2 d2 M 6 1 d 2 M2
dx2 dxdy dy2
8V0: + = 0,
dx dy
+ q = 0,
(11)
„ dS, dS6 ^ 8"1: IT1 + - Qs = 0, dx dy
„ dS6 dS2
8^1: ^^- Q4 = o,
dx dy
8w1: + - N3 = 0. dx dy
In addition, the natural boundary conditions may be represented by
8"o: Nn + N6 ny, 8^o: N6 nx + N2 ny,
8u1: S1nx + S6ny, 8v1: S6 nx + S2n
8w0:
I
dM11 + dM 6 dx dy
\
I dMe +_dMi ^
dx dy
nx +
dM„
(12)
„ 38w0
W Q5nx + Q4 n , Mn,
on
where
Mns = (M2 - Ml)«x«y + M6( nl - «y2),
Mn = M1n2x + M 2«, + 2 M6 «x«y, (13)
a a a -= nx--+ ny-•
dn dx dy The stress resultants can be obtained in terms of the total strains by substituting Eq. (6) into Eq. (10):
where [k] represents the symmetric matrix of differential operators and {f} = {0, 0, q, 0, 0,0} denotes the generalized force vector while {A} = {u0, v0, w0, u1, v1, w1}t. The elements kj = of the symmetric matrix [k] are written in Appendix A.
4. Two-dimensional solution
The following simply supported boundary conditions are set at the side edges of the functionally graded plate:
N1
N2 N6 M1 M 2 M6 S1
N3
A11 A12 0 B11 B12 0 BS1 B12 0 gB
A12 A11 0 B12 B11 0 B12 B1s1 0 gB
0 0 A66 0 0 B66 0 0 B66 0
B11 B12 0 B1s1 B12 0 D11 Ds D12 0 H1s3
B12 B11 0 B12 B1s1 0 D12 D11 0 H1s3
0 0 B66 0 0 B66 0 0 D66 0
Bs1 B\2 0 D1s1 Dn 0 F1s1 Fu 0 Js J13
B12 B1s1 0 D1s2 D1s1 0 F2 F\1 0 Js J13
0 0 B66 0 0 D66 0 0 F6s6 0
G13 gB 0 H1s3 H1s3 0 Js J13 Js J13 0 Ps P33
du0 / dx
dv0/dy
dv0/ dx + 3u0/ dy 3/ dx2
J dy2
0/dxdy
-a2^ -2 a2!
duj dx dvil dy dv1/ dx + du1/ dy
w
(14)
'4, Q5} = A44 >2 + +
dy ' T1 dx }' ( )
where
{Aii, Bii, Bisi, Dii, Dii, Fn} =
V 2
= J [X(z) + 2^(z)]{1, z, y(z), z2, zy(z),[y(z)]2 }dz,
-h 2
{A12, B12, B12, D12, Df2, F1S2 } = h/ 2
= J X(z){1, z, y(z), z2, zy(z),[y(z)]2}dz, -h/ 2
{A66, B66, B66, D66, D66, F66 } = h2
= J z){1, z, y(z), z2, zy(z),[(z)]2}dz, (16)
-h 2
h2
A44 = J Kz)[V(z)]2}dz,
-h 2 h2
P3 = J [X(z) + 2^(z)][V(z)]2dz,
-h 2
h2
{G13, H3, J3} = J X(z){1, z, V(z)}V(z)dz.
-h 2
Substituting Eqs. (14) and (15) into Eq. (11) yields a system of algebraic equations expressed in a compact form as [k]{A} = {/}, (17)
v0 = w0 = v1 = w1 = N1 = M1 = S1 = 0 at x = 0, a,
(18)
(u0, u1)
w w1)
^ v1).
u0 = w0 = u1 = w1 = N2 = M 2 = S2 = 0 at j = 0, The external force according to the Navier solution can be expressed in the sinusoidal form as
q( x, y) = q0 sin (tx) sin (ny), (19)
where t = n/fl, n = n/b, and q0 denotes the load intensity in the plate centre. According to the Navier technique, one assumes the following forms for u0, v0, w0, u1, v1, and w1 that satisfy the above conditions:
(U, X) cos (tx) sin (ny) (W, Z)sin(^x)sm(ny) -, (20)
(V, Y) sin (tx) cos (ny)
where U, V, W, X, Y, and Z are the arbitrary parameters. Equation (11) can be combined with Eqs. (14) and (15) into the system of first-order equations
[[ ]{A} = {F}, (21)
where {A} and {F} denote the columns {A} = {U, V, W, X, Y, Z}1, {F} = {0, 0, q0, 0, 0, 0}, (22)
and the elements Kj = Kjt of the coefficient matrix [K] are written in Appendix B.
5. Numerical results and discussion
The bending response of simply-supported functionally graded porous thick rectangular plates subjected to a sinu-
soidally distributed load is investigated. The effect of thickness stretching e3 of the functionally graded porous plates is taken into account in the present quasi-3D theory. The displacements and stresses determined here are reported according to the following dimensionless forms:
10.ECh3 (a b 1 a qo ^22 )
100Ech3 (a b 1 u(z) =-4-ul T>T> z I >
a q0 122 )
04 (z) = — 04(a,0, z 1, (23)
aq0 ^2 )
Table 1
Comparisons of deflection w(0) of functionally graded porous square plates
Methods ah
p 4 10 100
Ref. [36] 0.366500 0.294200 0.280300
0 Present a = 0.0 0.366044 0.293938 0.280089
a = 0.1 0.389082 0.312438 0.297717
Ref. [36] 0.549300 0.454800 0.436500
0.5 Present a = 0.0 0.541750 0.445172 0.426618
a = 0.1 0.596922 0.491987 0.471828
CPT [36] 0.562300 0.562300 0.562300
FSDT [36] 0.729100 0.588900 0.562500
Ref. [33] 0.717100 0.587500 0.562500
1 Ref. [34] 0.717100 0.587500 0.562500
Ref. [35] 0.699700 0.584500 0.562400
Ref. [36] 0.702000 0.586800 0.564700
Present a = 0.0 0.690817 0.569038 0.545649
a = 0.1 0.791092 0.655782 0.629794
CPT [36] 0.828100 0.828100 0.828100
FSDT [36] 1.112500 0.873600 0.828100
Ref. [33] 1.158500 0.882100 0.828600
Ref. [34] 1.158500 0.882100 0.828600
4 Ref. [35] 1.117800 0.875000 0.828600
Ref. [36] 1.110800 0.870000 0.824000
Present a = 0.0 1.098263 0.841682 0.792489
a = 0.1 1.405700 1.082177 1.020157
CPT [36] 0.935400 0.935400 0.935400
FSDT [36] 1.317800 0.996600 0.936000
Ref. [33] 1.374500 1.007200 0.936100
10 Ref. [34] 1.374500 1.007200 0.936100
Ref. [35] 1.349000 0.875000 0.828600
Ref. [36] 1.333400 0.988800 0.922700
Present a = 0.0 1.335130 0.981809 0.914068
a = 0.1 1.784337 1.284910 1.189183
05(z) = — 05 ( 0 b z|> z = 7 >
aq0 ^ 2 ) h
06(z) = — 06 0 z)
aq0
0i (z) = — 0i | a,b, z|, i =1,2,3
aq0 ^22 )
The lower surface of the functionally graded plate is metal (Em = 70 GPa) while the upper surface is ceramic (Ec = 380 GPa). Poisson's ratio is fixed at v = 0.3. As a validation example, the present deflection and in-plane
Comparisons of in-plane normal stress Oj(1/3) of functionally graded porous square plates
Table 2
Methods a/h
p 4 10 100
Ref. [36] 0.527800 1.317600 13.161000
0 Present a = 0.0 0.522946 1.303085 13.019919
a = 0.1 0.522946 1.303085 13.019919
Ref. [36] 0.586000 1.468000 14.673000
0.5 Present a = 0.0 0.571960 1.419381 14.170342
a = 0.1 0.577905 1.433585 14.311072
CPT [36] 0.806000 2.015000 20.150000
FSDT [36] 0.806000 2.015000 20.150000
Ref. [33] 0.622100 1.506400 14.969000
1 Ref. [34] 0.622100 1.506400 14.969000
Ref. [35] 0.592500 1.494500 14.969000
Ref. [36] 0.591100 1.491700 14.945000
Present a = 0.0 0.576180 1.434203 14.326085
a = 0.1 0.586141 1.458887 14.572361
CPT [36] 0.642000 1.604900 16.049000
FSDT [36] 0.642000 1.604900 16.049000
Ref. [33] 0.487700 1.197100 11.923000
Ref. [34] 0.487700 1.197100 11.923000
4 Ref. [35] 0.440400 1.178300 11.932000
Ref. [36] 0.433000 1.158800 11.737000
Present a = 0.0 0.425573 1.103477 11.104869
a = 0.1 0.396489 1.043439 10.528310
CPT [36] 0.479600 1.199000 11.990000
FSDT [36] 0.479600 1.199000 11.990000
Ref. [33] 0.369500 0.896500 8.907700
10 Ref. [34] 0.347800 0.896500 8.907700
Ref. [35] 0.322700 1.178300 11.93200
Ref. [36] 0.309700 0.846200 8.601000
Present a = 0.0 0.310215 0.824260 8.330706
a = 0.1 0.259359 0.716567 7.290624
stress will be compared with the classical plate theory (CPT), the first-order shear deformation theory (FSDT), that uses a shear correction factor K = 5/6 and those from Carrera et al. [33, 34] and Neves et al. [35, 36], which account for e3 ^ 0 and use the Carrera unified formulation (CUF). Note that displacement fields of the classical plate theory and first-order shear deformation theory are given from Eq. (2), respectively, by setting ^(z) = 0 and ^(z) = z, w1 = 0. No shear correction factor is needed for the mentioned theories, except for the first-order shear deformation theory.
Tables 1 and 2 contain dimensionless deflection and inplane stress in nonporous (a = 0) functionally graded square plates for different side-to-thickness ratio ajh and graded parameter p. The inclusion of the porosity factor a is represented in these tables. The present nonporous results (a = 0) are more accurate than those generated by the classical plate theory and first-order shear deformation theory. The present results are comparable with those of other quasi-3D theories even for thicker plates. This points to the use of the new assumption given in Eq. (3), which has a maximal effect on the accuracy of the results. Table 1 shows
Table 3
Comparisons of dimensionless stresses and displacements of functionally graded porous square plates (a/h = 10)
p Methods U (-1/4) w(0) ôi (1/3) 05(1/6) °6(-i/3)
0 Present a = 0.0 0.216337 0.293938 1.303085 0.212331 0.700122
a = 0.1 0.229953 0.312438 1.303085 0.212331 0.700121
1 ITST [41] 0.641300 0.589000 1.489000 0.261100 0.611100
Quasi-3D [33] 0.643600 0.587500 1.506200 0.251000 0.608100
Quasi-3D [38] 0.643600 0.587600 1.506100 0.251100 0.611200
SSDT [37] 0.662600 0.588900 1.489400 0.262200 0.611000
HSDT [39] 0.639800 0.588000 1.488800 0.256600 0.610900
TSDT [40] 0.641400 0.589000 1.489800 0.260800 0.611100
Present a = 0.0 0.587959 0.569038 1.434203 0.261088 0.565518
a = 0.1 0.701623 0.655782 1.458887 0.266506 0.546517
2 ITST [41] 0.898200 0.757300 1.395900 0.274200 0.544200
Quasi-3D [33] 0.901200 0.757000 1.414700 0.249600 0.542100
Quasi-3D [38] 0.901300 0.757100 1.413300 0.249500 0.543600
SSDT [37] 0.928100 0.757300 1.395400 0.276300 0.544100
HSDT [39] 0.895700 0.756400 1.394000 0.274100 0.543800
TSDT [40] 0.898400 0.757300 1.396000 0.273700 0.544200
Present a = 0.0 0.807013 0.721361 1.325404 0.274742 0.493683
a = 0.1 1.049624 0.885498 1.332625 0.284519 0.453813
4 ITST [41] 1.050000 0.881600 1.179200 0.254600 0.566900
Quasi-3D [33] 1.054100 0.882300 1.198500 0.236200 0.566600
Quasi-3D [38] 1.054100 0.882300 1.184100 0.236200 0.567100
SSDT [37] 1.094100 0.881900 1.178300 0.258000 0.566700
HSDT [39] 1.045700 0.881400 1.175500 0.262300 0.566200
TSDT [40] 1.050200 0.881500 1.179400 0.253700 0.566900
Present a = 0.0 0.942715 0.841682 1.103477 0.255831 0.514100
a = 0.1 1.310574 1.082177 1.043439 0.267216 0.479271
8 ITST [41] 1.075900 0.974600 0.947300 0.209400 0.585700
Quasi-3D [33] 1.083000 0.973800 0.968700 0.226200 0.587900
Quasi-3D [38] 1.083000 0.973900 0.962200 0.226100 0.588300
SSDT [37] 1.134000 0.975000 0.946600 0.212100 0.585600
HSDT [39] 1.070900 0.973700 0.943100 0.214000 0.585000
TSDT [40] 1.076300 0.974600 0.947700 0.208800 0.585800
Present a = 0.0 0.982899 0.944138 0.886117 0.210366 0.540509
a = 0.1 1.393026 1.232257 0.785381 0.210846 0.511665
that w decreases as a/h increases and the graded parameter p decreases. Table 2 shows that a increases as a/h increases. Note that a no longer increases as p increases and CTj has its minimum values for homogeneous ceramic and metal materials.
In another validation example, Table 3 presents a comparison of the dimensionless displacements u and w and stresses CTj, a5 and a6 of nonporous (a = 0) functionally
graded square plates (a/h = 10). The present displacements and stresses will be compared with the sinusoidal shear deformation theory (SSDT) [37], quasi-3D elasticity theory [28, 38], higher-order shear deformation theory (HSDT) [39], third-order shear deformation theory (TSDT) [40], and a new inverse trigonometric shear deformation theory (ITST) [41].
Note that displacements of the sinusoidal shear deformation theory [37] are given from Eq. (2) by putting
II
a = 0.20
^ ^ ^ ^
0.15
* s , a = 0.10
a = 0.00
0
~i i i i i i i r 8 12 16 a/h
. p = 5 -a = 0.00
\ ----a = 0.10
% -----a = 0.15
--a = 0.20
A ^ - ■ ■ - a = 0.25
\
-r ,■■_
0.5
1.5
0
2.5 alb
Fig. 2. Change of deflection w versus different parameters of functionally graded porous plates: w(z) across the plate thickness for different p (a) and a (b); w(0) versus a/h (c) and a/b (d) for different a
y (z) = — sin n
and w1 = 0.
(24)
Displacements of the higher-order shear deformation theory [39] are given by putting
y( z ) = —z + sin 2 h
nz ~h
V2cos( nz/h)
w1 = 0.
(25)
Displacements of the third-order shear deformation theory [40] are given by putting
4 z 3
y(z) = z _777' w0 = Wb + Ws
3h
(26)
w
0.
3ws 3ws «1 = ^JL, ^
dx dy
Finally, displacements of the inverse trigonometric shear deformation theory [41] are given by putting
y (z) = h arctg
rz h
16rz3
3h2(r2+4)
W1 = 0,
(27)
where r is the arbitrary parameter.
The present results reported in Table 3 are in close agreement with various shear deformation theories. We can also say that the present theory is more accurate than those theories since it includes the effect of transverse normal strain (e3 ^ 0). Table 3 shows that the displacements u and w increase as the graded parameterp increases. The in-plane normal stress a1 and transverse shear stress g5 no longer
increase as p increases. In addition, the tangential stress a6 no longer decreases as p increases.
In fact, the present quasi-3D theory gives results comparable with the inclusion of the porosity factor (a = 0.1). It is clear that this factor has a significant effect on displacements and stresses. The difference in deflections w obtained by the present quasi-3D theory with porosity (a = = 0.1) and nonporosity (a = 0) may increase as p increases and a/h decreases. The porosity normal stress a1 is greater than the nonporosity stress for functionally graded plates with 0 < p < 4 and vice versa with 4 < p < In addition, the porosity in-plane displacement U is greater than the nonporosity in-plane displacement U for functionally graded plates for all p. Finally, the porosity shear stress a5 {tangential stress a6} is greater {smaller} than the nonporosity shear stress a5 {tangential stress a6} for functionally graded plates for all p.
The inclusion of the porosity factor will be investigated in Figs. 2-9. In these figures we assume, unless otherwise stated, that a = 0.1, p = 5, a/h = 5, and b/a = 2. All figures illustrate the variable quantities through the plate thickness versus the thickness ratio a/h and the aspect ratio a/b.
Figure 2, a shows that the deflection W increases as p decreases for the fixed a = 0.1, while Fig. 2b shows that the deflection increases as a increases for the fixed p = 0.1. A porosity deflection value at a = 0.25 may triple as com-
0.3-
0.1-
-o.i-
-0.3-
-0.5
____p = 1
\ V I -----p = 2
MetalN.. A--p = 5
\ \
vVjS
a = 0.1 \\>A
\ x \ \ Ceramic! \ . '
0
15
10
5-
0-
-a = 0.00
\ ----a = 0.10
\ -----a = 0.15
--a = 0.20
\ - ■ ■ - a = 0.25
x\ \
OsW\
\ss. \
0
0.5
1.5
2.5 alb
Fig. 3. Variation of in-plane displacement u versus different parameters of functionally graded porous plates: u (z) across the plate thickness for differentp (a) and a (b); u (-1/4) versus a/h (c) and a/b (d) for different a
z- o = 5 -___ —
0.30.1- ¡f in W
-0.1- -a = 0.00 0.10
-0.3- n __^ = 0.15 0.20
- ------a = 0.25
-0.5" /'#
1 1 1 -1 1 i i i i i 3 5 i i 7 g,
76 54 321
•. p = 5 \ -a = 0.00
----a = 0.10
A'\ -----a = 0.15
- ■ ■ - a = 0.20 = 0.25
v\v
W-
0.5
~r 1.5
0
2.5 alb
Fig. 4. Change of in-plane normal stress gx versus different parameters of functionally graded porous plates: gx(z) across the plate thickness for differentp (a) and a (b); cx(l/2) versus a/h (c) and a/b (d) for different a
Fig. 5. Variation of the in-plane longitudinal stress a2 versus different parameters of functionally graded porous plates: a2(z) across the plate thickness for differentp (a) and a (b); a2(1/2) versus a/h (c) and a/b (J) for different a
Fig. 6. Variation of the out-of-plane normal stress a3 versus different parameters of functionally graded porous plates: a3(z) across the plate thickness for differentp (a) and a (b); a3(1/2) versus a/h (c) and a/b (d) for different a
ct4
0.15 0.14 0.13 0.12 0.11
a = 0.00
a = 0.10
a = 0.15
a = 0.20 p = 5
a = 0.25
1-1-1-1-1-1-1-1-r
0
g4" 0.18 0.16 0.14 0.12 0.10 0.08
\ P = 5
/ x* / / x \
// r'* •oX \ \ \
r
* •x N
/_ a = 0.00 %
--- ■a = 0.10 ^ 0.15 % ^ .
— . — a =
— - ■a = 0.20
— ■ ■ a = : 0.25
m
12
16
a/h
0.5
1.5
2.5 alb
Fig. 7. Change of transverse shear stress g4 versus different parameters of functionally graded porous plates: a4(z) across the plate thickness for differentp (a) and a (b); a4(0) versus a/h (c) and a/b (d) for different a
a = 0.00 a = 0.10 a = 0.15
a = 0.20 a = 0.25 p = 5
0
12
16
a/h
Fig. 8. Change of transverse shear stress a5 versus different parameters of functionally graded porous plates: a5(z) across the plate thickness for differentp (a) and a (b); a5(0) versus a/h (c) and a/b (d) for different a
0.3 0.1 -0.1 -0.3 H -0.5
—- v % a = 0.1
X A
l
\ i -Ceramic \\
----p=\ V
-----p = 2 \
--p = 5
------Metal i \ /i\ \
-1.6 -1.2 -0.8 -0.4 0.0 0.4 a6
ii
-a = 0.00
----a = 0.10
-----a = 0.15
--a = 0.20 ll
- ■ ■ - a = 0.25 \\
b
Fig. 9. Variation of the in-plane tangential stress g6 versus different parameters of functionally graded porous plates: g6(z) across the plate thickness for differentp (a) and a (b); G6(-l/2) versus a/h (c) and a/b (d) for different a
pared to nonporosity one. Figure 2, c shows that the deflection decreases slightly as a/h increases, while Fig. 2, d shows that the deflection decreases rapidly as a/b increases. The differences between deflections may be fixed at the variation of a/h while these differences are larger for small aspect ratios.
Figure 3, a shows that the porosity metallic in-plane displacement u at the lower surface z = -0.5 is the greatest one while the porosity ceramic in-plane displacement u is the smallest one. The porosity FG in-plane displacement u increases asp increases. The inverse occurs at the upper surface z = 0.5. The same occurs in Fig. 3, b. It is seen from Fig. 3, b that the variation of the porosity factor has no effect on the in-plane displacement u at the level of z = 1/3. Figures 3, c and 3, d show that the in-plane displacement u decreases as a/h and a/b increase. The differences between the in-plane displacements u are larger for small thickness a/h and aspect a/b ratios. The variation of the porosity factor may have no effect on u for larger aspect ratios (a/b > 3).
Figure 4, a shows that the porosity metallic in-plane normal stress g1 is the same as the porosity ceramic in-plane normal stress. Figure 4, b shows that the porosity functionally graded normal stress g1 increases as a decreases in the interval -0.16<z <0.38. Otherwise, g1 increases asain-creases. Figures 4, c and 4, d show that g1 increases as
a/h increases and a/b decreases. The normal stress g1 increases as the porosity factor a increases. The differences between the in-plane normal stresses g1 are larger for a large thickness ratio a/h and a small aspect ratio a/b. Figure 5 shows that the in-plane longitudinal stress g2 behaves similarly to the normal stress g1. The maximum in-plane longitudinal stress g2 occurs at a/b = 0.6 as shown in Fig. 5, d.
Once again Fig. 6, a shows that the porosity metallic out-of-plane transverse normal stress g3 is the same as the porosity ceramic one. Figure 6, b shows that the porosity functionally graded transverse normal stress g3 is very sensitive to the variation of the porosity factor a at the level z = 0.38. Figure 6, c shows that g3 increases as a/h increases for a = 0.25. At a = 0.2, g3 no longer decreases as a/h increases and the minimum value of g3 occurs at ah = 6.8. However, for a < 0.15 the out-of-plane transverse normal stress g3 directly decreases as a/h increases. Figure 6, d shows that g3 slightly increases as a/b increases only for nonporous plates (a = 0). When a = 0.1, g3 is not affected by the variation of the aspect ratio a/b. However, at a > 0.15 g3 decreases as a/b increases.
Figures 7 and 8 show that the transverse shear stresses g 4 and g5 have the same behavior through the thickness of the functionally graded porous and nonporous plates. The shear stresses are the same for ceramic and metallic
homogeneous plates. The maximum shear stresses have different positions according the value of the graded parameter p. However, the maximum shear stresses occur at the position (z = 1/3) for different porosity factors a. In this position, the maximum shear stress increases as a increases. The transverse shear stresses are affected by the variation of a/h. Finally, a5 directly decreases as a/b increases while a4 no longer increases as a/b increases and has its maximum value at a/b = 1. In the meantime, the transverse shear stresses decrease as a increases.
Figure 9, a shows that the porosity metallic in-plane tangential stress a6 is the same as the porosity ceramic one. Figure 9, b shows that the porosity functionally graded tangential stress a6 increases as a increases in the interval -0.16 < z < 0.38. Otherwise, a6 increases as a decreases. Figure 4, c shows that a6 increases as a/h increases. In addition, the tangential stress a1 increases as the porosity factor a decreases. The differences between the in-plane tangential stresses a6 are larger for a large thickness ratio afh and small aspect ratio a/b. Once again, the maximum in-plane tangential stress a6 occurs at a/b = 0.6, as shown in Fig. 9, d.
6. Conclusions
A refined quasi-3D shear and normal deformation theory is developed for functionally graded porous plates. The effect of thickness stretching in functionally graded porous plates is presented. The governing equations are derived and analytical solutions for simply supported functionally graded porous rectangular plates are obtained. The inclusion of the graded and porosity parameters is investigated. Many validation examples are reported, and numerical results of the present quasi-3D theory are accurate in predicting the bending response of nonporous plates. All stresses for porous or nonporous homogeneous plates are the same. That is because the stresses are independent of Young's modulus. The displacements and stresses are very sensitive to the variation of the porosity factor for functionally graded plates. It should be noted that the inclusion of the porosity factor is the main property of the porous materials, especially for functionally graded structures.
7. Appendices
7.1. Appendix A
The elements of the symmetric matrix [k] presented in Eq. (17) are given by
d2 . d2 , , d2
k11 = A11TT + k12 = (A11 + A66)
dx dy
dxdy'
d 3
k13 = —Bii —t -(B12 +2 B66)-t-,
13 11 dx dxdy2
k14 = B1S1 + B66, k15 = (B1S2 + B66)-^_,
dx dy dxdy
d_
' dx'
k16 = G13~, k22 = A€6T~T + A11TT, dx dy
k23 = —(B12 +2 B66)
k24 = k15, k25 = B66
dx dy d2
-B
a7+B1 ^
jUL
dy 3
dL
dy7
k26 = G13~,
d_
'dy'
(
k33 = B1S1
i4 >
dx4 + dy4
+ 2(BS2 + 2B66)
dx dy2
k34 =—^1s1 drr — (D2 +2 D66) d 4 dx
k35 =—(D1s2 +2 D^6)
■-D
dxdy
a3
dx dy dy
(
k36 = —H13
d2 d
dx2 dy2
2 ^
k44 = Fn TT + F66rT A44,
dx dy
d 2 d k45 = (F12 + k46 = (J13 — A44)3_,
dxdy dx
d 2 d 2 d k55 = F66T~T + F11TT — A44, k56 = (J13 — A44
dx dy dy
(
k66 = K3s3 A44
i2 >
d2 d
dx2 + dy2
V J
7.2. Appendix B
The elements of the symmetric matrix [K] presented in Eq. (21) are given by
K11 =VA11 + VA66, K12 =ln(A12 + A66^
K13 =—l[V B11 +n2(B12 + 2B66)],
K14 = l2 B1S1 +n2в66,
P15 =ln(BS12 +B66), P16 = —G3,
P22 = ?2 A66 + V A11,
P23 =—n[V(B12 +2 B66) + V B11],
k24 = K15,
K 26 =-nG]
K 25 = ?2 B66 +n2 B
12,
13,
K33 = (14 +n4)B66 + 2n2(B12 + B66),
K34 =—m2 Ds1 +n2(D1S2 +2 D66)],
K35 =—n[I2(D1S2 +2D66) + V DU
K36 = (?2 + n2) H3,
K44 =?2 F1s1 +n2 F6s6 + A44,
K45 =ln(F1S2 +F6s6), K46 =l(A44 — J3), K55 =?2F6s6 +n2F1S1 + A44, K56 =n(A44 — J3),
K66_= (?2 +n2) A44 + P3s3, where ? = £/ h and n = V h.
References
1. Biot M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range // J. Acoust. Soc. Am. - 1956. -V. 28. - P. 168-178.
2. Biot M.A.Theory of elasticity and consolidation for a porous anisotropic solid // J. Appl. Phys. - 1955. - V. 26. - P. 182-185.
3. SelvaduraiA.P.S. Mechanics of Poroelastic Media. - Dordrecht: Kluwer
Academic, 1996.
4. Taber L.A. A theory for transverse deflection of poroelastic plates // J. Appl. Mech. ASME. - 1992. - V. 59. - P. 628-634.
5. Busse A., Schanz M., Antes H. A poroelastic Mindlin plates // Proc. Appl. Math. Mech. - 2003. - V. 3. - P. 260-261.
6. Borsan M. A bending theory of porous thermoelastic plates // J. Therm.
Stresses. - 2003. - V. 26. - P. 67-90.
7. Magnucki K., Stasiewicz P. Elastic bending of an isotropic porous beam // Int. J. Appl. Mech. Eng. - 2004. - V. 9. - No. 2. - P. 351-360.
8. Nappa L., Iesan D. Thermal stresses in plane strain of porous elastic solid // Mecc. - 2004. - V. 39. - No. 2. - P. 125-138.
9. Magnucki K., Malinowski M, Kasprzak J. Bending and buckling of a rectangular porous plate // Steel Compos. Struct. - 2006. - V. 6. -No. 4. - P. 319-333.
10. Magnucka-Blandzi E. Axi-symmetrical deflection and buckling of circular porous-cellular plate // Thin-Walled Struct. - 2008. - V. 46. -No. 3. - P. 333-337.
11. Yang Y., Li L., Yang X. Quasi-static and dynamical bending of a cantilever poroelastic beam // J. Shanghai Univ. - 2009. - V. 13. - No. 3. -P. 189-196.
12. Borsan M., Altenbach H. On the theory of porous elastic rods // Int. J. Solids Struct. - 2011. - V. 48. - P. 910-924.
13. Borsan M., Altenbach H. Theory of thin thermoelastic rods made of porous materials // Arch. Appl. Mech. - 2011. - V. 81. - P. 13651391.
14. Kumar R., Devi S. Deformation in porous thermoelastic material with temperature dependent properties // Appl. Math. Inform. Sci. - 2011. -V. 5. - P. 132-147.
15. Ghiba I-D. On the spatial behaviour in the bending theory of porous thermoelastic plates // J. Math. Anal. Appl. - 2013. - V. 403. - P. 129142.
16. Sladek J., Sladek V., Gfrerer M., Schanz M. Mindlin theory for the bending of porous plates // Acta Mech. - 2015. - V. 226. - P. 19091928.
17. Lyapin A.A., Vatulyan A.O. On deformation of porous plates // Z. Angew. Math. Mech. - 2018. - V. 98. - No. 2. - P. 330-340.
18. Chen D., Yang J., Kitipornchai S. Elastic buckling and static bending of shear deformable functionally graded porous beam // Compos. Struct. - 2015. - V. 133. - P. 54-61.
19. Bensaid I., Guenanou A. Bending and stability analysis of size-dependent compositionally graded Timoshenko nanobeams with porosities // Adv. Mater. Res. - 2017. - V. 6. - No. 1. - P. 45-63.
20. Akbas S.D. Nonlinear static analysis of functionally graded porous beams under thermal effect // Coupl. Sys. Mech. - 2017. - V. 6. -No. 4. - P. 399-415.
21. Behravan Rad A. Static analysis of non-uniform 2D functionally graded auxetic porous circular plates interacting with the gradient elastic foundations involving friction force // Aeros. Sci. Tech. - 2018. -V. 76. - P. 315-339.
22. Barati M.R., Shahverdi H., Zenkour A.M. Electro-mechanical vibration of smart piezoelectric FG plates with porosities according to a refined four-variable theory // Mech. Adv. Mater. Struct. - 2017. - V. 24. - No. 12. - P. 987-998.
23. Barati M.R., Zenkour A.M. Electro-thermoelastic vibration of plates made of porous functionally graded piezoelectric materials under various boundary conditions // J. Vib. Control. - 2018. - V. 24. - No. 10. -P. 1910-1926.
24. Barati M.R., Zenkour A.M. Post-buckling analysis of refined shear deformable graphene platelet reinforced beams with porosities and geometrical imperfection // Compos. Struct. - 2017. - V. 181. - P. 194202.
25. Barati M.R., Zenkour A.M. A general bi-Helmholtz nonlocal strain-gradient elasticity for wave propagation in nanoporous graded double-nanobeam systems on elastic substrate // Compos. Struct. - 2017. -V. 168. - P. 885-892.
26. Zenkour A.M. A quasi-3D refined theory for functionally graded single-layered and sandwich plates with porosities // Compos. Struct. -2018.- V. 201. - P. 38-48.
27. Zenkour A.M. A comparative study for bending of cross-ply laminated plates resting on elastic foundations // Smart Struct. Sys. -2015. - V. 15. - No. 6. - P. 1569-1582.
28. Zenkour A.M. Benchmark trigonometric and 3-D elasticity solutions for an exponentially graded thick rectangular plate // Arch. Appl. Mech. - 2007. - V. 77. - No. 4. - P. 197-214.
29. Zenkour A.M. The refined sinusoidal theory for FGM plates on elastic foundations // Int. J. Mech. Sci. - 2009. - V. 51. - No. 11-12. -P. 869-880.
30. Zenkour A.M. The effect of transverse shear and normal deformations on the thermomechanical bending of functionally graded sandwich plates // Int. J. Appl. Mech. - 2009. - V. 1. - No. 4. - P. 667707.
31. Zenkour A.M. Hygro-thermo-mechanical effects on FGM plates resting on elastic foundations // Compos. Struct. - 2010. - V. 93. - No. 1. -P. 234-238.
32. Zenkour A.M. Exact relationships between the classical and sinusoidal plate theories for FGM plates // Mech. Adv. Mater. Struct. - 2012.-V. 19. - No. 7. - P. 551-567.
33. Carrera E., Brischetto S., Robaldo A. Variable kinematic model for the analysis of functionally graded material plates // AIAA J. - 2008. -V. 46. - P. 194-203.
34. Carrera E., Brischetto S., Cinefra M., Soave M. Effects of thickness stretching in functionally graded plates and shells // Compos. B. -2011. - V. 42. - P. 123-133.
35. Neves A.M.A., Ferreira A.J.M., Carrera E., Roque C.M.C., Ci-nefra M., Jorge R.M.N., Soares C.M.M. A quasi-3D sinusoidal shear deformation theory for the static and free vibration analysis of functionally graded plates // Compos. B. - 2012. - V. 43. - No. 2. - P. 711725.
36. Neves A.M.A., Ferreira A.J.M., Carrera E., Cinefra M., Roque C.M.C., Jorge R.M.N., Soares C.M.M. Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique // Compos. B. - 2013. - V. 44. - No. 1. - P. 657-674.
37. Zenkour A.M. Generalized shear deformation theory for bending analysis of functionally graded materials // Appl. Math. Model. - 2006. -V. 30. - P. 67-84.
38. Wu C-P., Chiu K-H. RMVT-based meshless collocation and elementfree Galerkin methods for the quasi-3D free vibration analysis of multilayered composite and FGM plates // Compos. Struct. - 2011. -V. 93. - No. 5. - P. 1433-1448.
39. Mantari J.L., Oktem A.S., Soares O.G. Bending response of functionally graded plates by using a new higher order shear deformation theory // Compos. Struct. - 2012. - V. 94. - P. 714-723.
40. Thai H-T., Kim S.E. A simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates // Compos. Struct. - 2013. - V. 96. - P. 165-173.
41. Nguyen V-H., Nguyen T-K., Thai H-T., Vo T.P. A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates // Compos. B. - 2014. - V. 66. - P. 233-246.
Received 16.05.2018, revised 15.11.2018, accepted 22.11.2018
Сведения об авторе
Ashraf M. Zenkour, PhD, Prof., King Abdulaziz University, Saudi Arabia, [email protected], [email protected]
