Bending examination of advanced generation of composite structures with specific properties exposed to different loads Текст научной статьи по специальности «Механика и машиностроение»

e-mail: b_hamza_2005@yahoo.fr,

ORCID iD: ©https://orcid.org/0000-0002-7871-4017

DOI: https://doi.org/10.5937/vojtehg72-47852

stress conditions on both the upper and lower surfaces. Numerical investigations are introduced to interpret the influences of loading conditions and variations of power of functionally graded material, modulus ratio, aspect ratio, and thickness ratio on the bending behavior of FGPs. These analyzes are then compared to the results available in the literature.

Bending examination of advanced %

generation of composite structures with specific properties exposed to different loads

Ahmed Zitouni3, Bachir Bouderbab, Abdelkader Dellalc, Hamza Madjid Berrabahd

ä Tissemsilt University, Department of Science and Technology, -g

Mechanical Engineering Materials and Structures Laboratory, g Tissemsilt, People's Democratic Republic of Algeria,

e-mail: ahmed.zitouni@univ-tissemsilt.dz,corresponding author, ^

ORCID iD: ©https://orcid.org/0000-0003-1627-6020 -e

b Tissemsilt University, Department of Science and Technology, g

Mechanical Engineering Materials and Structures Laboratory, p

Tissemsilt, People's Democratic Republic of Algeria, e-mail: bouderba.bachir@univ-tissemsilt.dz, ORCID iD: https://orcid.org/0000-0003-4668-122X

: Tissemsilt University, Department of Science and Technology; Tissemsilt, People's Democratic Republic of Algeria, e-mail: dellal.abdelkader@univ-tissemsilt.dz,

ORCID iD: https://orcid.org/0009-0003-2305-4845 '

d University of Relizane, Department of Civil Engineering, Mechanical Engineering Materials and Structures Laboratory, Relizane, People's Democratic Republic of Algeria, |

'S

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FIELD: mechanics

ARTICLE TYPE: original scientific paper

Abstract: J

Introduction/purpose: This article presents the bending examination of advanced-generation composite structures with specific properties exposed to different loads.

Methods: This paper thus proposes and introduces a new generalized five- | variable shear strain theory for calculating the static response of functionally graded rectangular plates made of ceramic and metal. Notably, our theory eliminates the need for a shear correction factor and ensures zero-shear ne

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Results: Preliminary results include a comparative analysis with standard higher-order shear deformation theories (PSDPT, ESDPT, SSDPT), as well as Mindlin and Kirchhoff theories (FSDPT and CPT). Conclusion: Our theory contributes alongside established theories in the field, providing valuable insights into the static thermomechanical response of functionally graded rectangular plates. This encompasses the influence of volume fraction exponent values on nondimensional displacements and stresses, the impact of aspect ratios on deflection, and the effects of the thermal field on deflection and stresses. Numerical examples of the bending examination of advanced-generation composite structures with specific properties exposed to different loads demonstrate the accuracy of the present theory.

Key words: functionally graded materials, bending, higher-order shear deformation theories, thermomechanical.

Introduction

Composites are materials formed by combining two or more constituent materials to create superior properties, defying traditional material constraints.

They are commonly used in aerospace, automotive, construction, and many other industries due to their exceptional strength-to-weight ratio (where the quest for ever-lighter and stronger materials is a perpetual challenge) as well as good performance.

Despite these good properties, however, there is a negative aspect that worries industrialists and researchers; it is the failure in difficult operational environments. It shows signs of failure and disintegration.

Composite materials, while versatile, can deteriorate over time due to weakening interfaces between their layers (Pindera et al, 1998; Boggarapu et al, 2021), leading to performance issues and failures. In response to these challenges, functionally graded materials (FGMs) have emerged as a progressive development in material science.

Functionally graded materials (FGMs) indeed represent an innovative and sophisticated approach to addressing some of the challenges associated with conventional composite materials. FGMs are designed to provide tailored material properties by gradually changing, or grading, the composition, structure, and properties of the material in a specific direction. This gradient-based approach offers several advantages for various engineering and industrial applications.

In 1984, a group of researchers in Sendai, Japan (Koizumi, 1993; Koizumi & Niino, 1995; Koizumi, 1997), introduced the concept of Functionally Graded Materials (FGMs). These materials are characterized

by their uninterrupted variation in properties, distinguishing them from conventional materials.

Functionally graded materials (FGMs) find application across diverse industries (Kieback et al, 2003), and recent research illuminates their behavior under thermal and mechanical loads. Classical plate theories, like CPT, lack accuracy for thicker structures (Bouazza et al, 2011). The First-Order Shear Deformation Theory (FSDT) (Reissner, 1945; Mindlin, 1951; Timoshenko & Woinowsky-Krieger, 1959) addressed this but needs a corrective factor. Higher-order shear deformation theories (HSDT) excel, showing improved accuracy without requiring a correction factor, unlike sop previous models.

In this context, Reddy (2000) studied the static behavior of FGM plates using the third-order shear deformation theory. Zenkour & Alghamdi (2010) explored the bending behavior of sandwich plates, investigating the impacts of thermomechanical loads on stresses and deflections. Additionally, Bouderba & Benyamina (2018) introduced a model for analyzing the thermal-mechanical behavior of thick metal/ceramic FGM plates.

Li et al. (2020) introduced a new five-variable shear deformation theory for predicting the static response of functionally graded plates. Daikh et al. (2020) studied the thermomechanical bending behavior of functionally graded sandwich plates under varied temperatures. Brischetto & Carrera (2010) proposed enhanced mixed theories, while Benyamina et 8 al (2018) examined composite material plates in thermal settings, ° contributing to a deeper understanding of their performance. n

Bouderba et al. (2016) used a simple shear deformation theory to examine the thermal stability of FGM sandwich plates. Shinde et al. (2015) introduced a refined trigonometric shear deformation theory for the analysis of bending in both isotropic and orthotropic plates under a variety of loading conditions. In a related vein, Bouderba & Berrabah (2022) delved into the bending response of porous advanced composite functionally graded material (FGM) plates subjected to thermomechanical loads. m

Zenkour & Hafed (2020) conducted a study on the bending analysis of functionally graded piezoelectric (FGP) material plates under simply supported edge conditions. They employed a simple quasi-3D sinusoidal shear deformation theory for their analysis. In another study, Brischetto et al (2008) examined the deformations of a simply supported rectangular plate composed of functionally graded material (FG) subjected to both thermal and mechanical loads.

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On the other hand, Berrabah & Bouderba (2023) employed an accurate shear deformation theory to investigate the mechanical buckling behavior of FG plates. Furthermore, higher-order plate theories have been introduced in the literature to address the mechanical and thermal buckling of FG plates, with the Carrera unified formulation being utilized (Farrokh et al, 2021; Farrokh et al, 2022).

This paper aims to comprehensively examine advanced composite structures under varied loads, focusing specifically on a rectangular plate made of functionally graded material (FGM) subjected to mechanical and thermal stress. Using the novel CSDPT theory with five unknown variables, the study derives and solves the equations of motion through Navier's procedure. Validation occurs through comparative analyses with standard higher-order theories like PSDPT, SSDPT, and ESDPT. The study's outcomes illuminate the impact of the power index on non-dimensional displacement and stresses, addressing the thermal field's influence on deflection and stresses in the FGM plate. This examination significantly contributes to understanding the behavior of these structures under different load conditions, advancing the field of mechanical construction.

Essential formulation

Structure geometry and material gradient

Geometry of the structure of the FGM (in the context of the plate)

Figure 1 - Geometry and the coordinate system of the FGM

In the context of our research on advanced composite structures, the geometric characteristics of the rectangular plate made of functionally graded material (FGM) play a pivotal role. This plate, as depicted in Figure1, exhibits specific material properties that transition gradually from the bottom to the top surface.

Table 1 - Geometric properties of the functionally graded plate.

Characteristics Structure geometry Thickness Length Width Aspect ratio Material coordinate origin (x3)

Symbol - h a B a/b x3

Description Rectangular plate Plate thickness (x3-axis) Plate length (x1-axis) Plate width (x2-axis) Aspect ratio of the plate Middle of the plate thickness (xa = ± h / 2)

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Material gradient in the FGM plate

In the context of functionally graded material (FGM) plates, material attributes, including Young modulus (E), Poisson's ratio (v), and thermal dilation coefficient (a), are defined by applying the mixing rule (1) (Reddy, 2000) and utilizing the Power-Law function (2).

P(*3) = (Pc - Pm )V (X3 ) + Pm

V ( X3HI/2+X3/h)n

(1) (2)

Material properties can be described by the following equation:

E(X3 ) = (Ec - Em )(l/2 + X3 / hf + Em (3)

The gradient law for Young's modulus (E) (3), applies to Poisson's ratio (v) and the coefficient of thermal expansion (a) as well.

Here, x3 signifies the position along the plate's thickness, h is the total thickness, and n is a material parameter influencing the composition gradation.

The material properties (Em, vm, am) and (Ec, vc, ac) are associated with the metallic and ceramic phases, respectively.

Table 2 - Transverse shear strain functions for the FGM plate.

Theories, the author Form of function f(z)

Parabolic deformation theory of plates, (Reissner, 1945) 5z /4(1 - (4 / 3)(z / h)2)

Parabolic shear deformation theory (PSDPT), (Reddy, 1984) z(1 - (4/3)( z / h)2)

Trigonometric deformation theory of plates (SSDPT), (Touratier, 1991) (h / t) sin( -z / h)

The exponential shear deformation plate theory (ESDPT), (Karama et al, 2003) -2( z/h)2 ze ( )

Hyperbolic deformation theory of plates, (Soldatos, 1992) z cosh(1 / 2) - h sinh(z / h )

Combination functions(CSDPT), Present (e2+1)arctan(e2z/h ) 2ez K(e2+1) (e-1)2 h(e-1)2 4(e-1)2

-0.1 0.0 0.1

Shape function

0.2 0.4 0.6

Shape function dif

Figure 2 - Distribution of the functions f (z):

(a)shape function f(z),

(b) differentiation f'(z)

Figure 2 provides a comparison of how the shape function and its derivatives vary across different shear deformation theories, denoted as f(z) and its derivative f'(z).

0.2

0.0

0.8

1.0

Table 3 - Material properties - metal and ceramic materials, see (Bao & Wang, 1995;

Bouderba & Berrabah, 2022)

Property Young's modulus Poisson's ratio Thermal expansion coefficient Density

Symbol Emetal Eceramic Vmetal Vceramic O metal Qceramic P metal Pceramic

Value 70 151 0.3 0.3 23 10 7.8 7.8

Description Elastic modulus of the material in GPa Poisson's ratio of the material Linear coefficient of thermal expansion (x10-6/°C) Density of the material in g/cm3

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Theoretical formulation Formulating the displacement field

In the context of thick functionally graded material (FGM) plates, the displacement of a material point at the coordinates (xi, X2, X3) could be expressed as follows:

ôw0

u ( Xj, x2, x3 ) = u0 (x, x2 ) - x3--h ®(x3 )©Xi

dxx dwn

v(xj, x2, x3) = vq (xj, X2) - x3 - 0(x3) @x2

ôx.

(4)

w (xj, x2, x3 ) = wq (xj, x2 )

The mid surface of the structure is characterized by five unknown displacement functions, namely «0, v0, w0, and © , © .

Based on the equations describing the displacement within the field and the relationships governing strain-displacement, we deduce expressions for the strain elements derived from the displacement elements.

s

22 /12

ôun

ô2 w

dx ôx

Ôv0 ôx.

ô2 w

ô©

0 +0( x,) —^

ôx

ô©.

ôx

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d O (X3) d O (X3) 723 = ——— •© x' Yi3 =-3 •©-

dx,

dx

(5b, 5b')

where £n,f22,712,y23 and 713 are the strain and shear components, respectively.

Equations (5a) and (5b, 5b') can be expressed like this:

(6)

723 =°'( X3)y2°3,ri3 =°'( X3)yi°3,733 = 0.

The stress-strain relationships can be concisely expressed with the following equation:

£11 ° % 0 k11 kn

e22 > = - 0 S22 > + x' k22 > + ®( z) - *22

712 . 0 712 , 1 ° k1 2 , k12 ,

^11 Q11 Q12 Q13 0 0 0 *11

^22 Q12 Q22 Q23 0 0 0 S22

^33 Q13 Q23 Q33 0 0 0 £33

^12 Q66 0 0 712

^23 Q44 0 723

J13 Sym q55 _ 713,

(7)

The following equation considers the influence of thermal effects:

Q11 Q12 0

^22 > = Q22 0

r12 j Sym Q66 _

r23 ] "Q44 0 " 71

% J Sym q55 _ 17131

f £11 ^ a( x3) AT

- S12 >" - < a(x3) AT 1

V 712. 0

(8a)

(8b)

The parameters (au,a22,Tu,T23,Tn) and (e^e^y^y^yn) represent the stress and deformation components, respectively.

The mathematical formulation of the stiffness coefficients O.. is as follows:

Q11 = Q22 = Q33 =

1-v2,

Ö12 =

v E(x3)

E(x3)

1-V2 ,044 = Ö55 = Ö66 = 2(l+v)

(9a, 9b, 9c)

The temperature distribution (x1'X2'through thickness is assumed as Bouderba et al. (2013).

The following equation describes how temperature varies through the thickness of the plate:

t ( x, x2, ^ ) = T (x ' x )+~3 T ( x ' x )+® ( x )T ( x ' x )

(10)

In this study, we focused on employing the sinusoidal temperature

— 1 n distribution to conduct our analysis ®(x3) =—sin(x3-).

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Governing equations

Formulating the equilibrium governing equations involves applying the principle of virtual work, which can be articulated in the following manner in this context:

snu/ax +bn12/ax2=0 bn22 / ax2+bn12 / ax1=0

a2Mu/ax2+2 a2M12 /ax^+a2M22 / ax2 2+q = 0 (11)

a^ /axj+bs12/ax2 - g13=0

aS2^ Bx2 +aSlJ Bxi - Ö23 = 0

N, M, and S are the quantities that represent the force and moment resultants, and their definitions are as follows:

N11 N22 N12 1

M11 M22 M12 '==Jh (CT11'CT22'CT12 )' x3 ■dx3 ' (12a)

A S22 S12 , 0( xJ),

(03' Ö23 WÎ (^23 )®'( x3¥*;

(12b)

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The interval limits for simplification: -h/2 = h1, h/2 = h2,

O

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M ' = D Da < k ° > — < mt

5 _ 5ym F k1 _ sT

023 ~A44^23'Ql3 = A55^13"

N = {N11,N22,N12/ ,M = {Mn,M22,Mi2}t ,5 = {511,522,512}^ , NT ={NT ,N22,0}t ,MT ={MT ,MT2,0}t ,5T ={5U,522,0}t, £0 ^^f, ={«2^ , k1 ={kf1,k22,kj2}t,

The following can be deduced:

" A11 A12 0 " " B11 B12 0 " B 1 0 "

A = A22 0 , B = B22 0 , Ba = Ba 1 0

5ym A66 _ 5ym B66. 5ym B6

"D11 D12 0 " r Da D11 Da D12 0 r Fa Fa f12 0

D = D22 0 , Da = Da D22 0 , Fa = Fa f22 0

5ym D66 _ 5ym Da D66 _ 5ym Fa F66 _

0 = {023013}' ,Y = {A°3,aS}',Aa

a;4 °

(13a) (13b)

(13c)

(14a)

0 A£ _ (14b)

The stiffness coefficients, denoted as Ay ,By, are defined as follows:

(15a)

A11 B11 D11 Bn Da D11 „a F11

<A12 B12 D12 a B12 „a F12 » =

B66 D66 B6a6 D6 F66.

h 011 (l %4 ^ X $(X3))

(A22,B22,D22,B22,D22,F22 ) = (A11,B11,D11,B11,D11,F11 ), 4 X3)]2 ^ = 4,5)

1

v

1—v

2

3

(15b)

The stress and moment resultants on a plate element S? Nn = NliMT = MhMT = sh, due to thermal loads are defined respectively by:

NT-

ii mM

ST Sii

1

X3

®(x,)

dx3,(i = l) (16)

-B1 1 dl 1 1 w0 +(B66d22 +Blldl 1 )®x1 +(B66 +B12 )dl2®x2 =P '

(17b)

-B22d222 w0 + (B66 + B12 )d12®x1 +(B22d22 + B66d11 )0X2 =p2 ' -(B12 + 2B66 )d122u0 - B11d111u0 - (B12 + 2B66 )d112v0 - B22d222v0

+(D dm +D22d2222 )W0 +2(D2 +2D66 )dU22 W) -(Df2 +2D66 (17c)

-(Da +2Da6)d122W0 +№11 - ^44)®X1+(Fa2 +F6 M2©x2 == p4,

(17d)

The components of the generalized force vector {p} are defined as follows:

gNT _ 3N^2 _ d2MT d2MT2 _ S^ S2Sf2

P1

ONh _on22 a 0 M11 0 M22 _r, 0 S11 0 S22 f-|Q\

IX1 m p2 ^-0X2"' P3=q--8XT ~eX2r 'P4^Ix1" ' (19)

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Substituting Eq. (13a, 13b) into Eq. (11), we obtain the following equations:

(A66d22 + A11d11)u0 + (A66 + A12)d12v0 - (2B66 + B12 )d122w0 §■

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(17a)

(A66 + A12)d12u0 + (A22 d22 + A66 d11)v0 - (2B66 + B12)d112 w0 £

.c

-D11d111®x1 - (D12 + 2D66)d112®x1 - °22d222®x1 = P3' £

o

(B66 d22 + B11d11)u0 + (B66 + B12 )d12v0 - D11d111W0 £

(U

(B6a6 + BH )d12u0 +( B66d11 + B22d22)v0 - D22d222 w0 , |

In this context, {p} = {#,p2,p^p.p} is a force vector, dv, difl and dijlm are the following differential operators:

S2 S3 S4

d =-, d--, =-, d. =-, (18)

v Sxi Sxi Sxi Sxi Sxi; ijm Sxi Sxi Sx^ Sxi v '

S

di =—,(i,j,l,m=i,T),(xi =xi;xi =x2),

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Determining exact solutions for functionally graded plate structures

Rectangular plates are typically categorized based on the nature of their support systems. In this context, we are concerned with the specific boundary conditions associated with a simple support configuration.

Vq = Wq — © = Njj = Mjj — »Sjj = 0 at Xj = 0, a, x2

u0 — w — © — N22 — M22 — S22 — 0 at x2 — 0, b, xi

(20)

The displacement functions qt and T can be expanded in the

following manner, aligning them with both the boundary conditions and the governing equations. This expansion is characterized by the use of a double Fourier series.

sinO" xlsinO" X2>,(£ = 1, 2, 3)

(21)

where i = n / a, j = n / b , and ^ are constants.

Following the procedure of the Navier solution, we assume the following solution:

u0

V0

w0 > — <

©

x1

©x2 ,

U cos(" x ) sin( j x2 )

V sin(" x ) cos( j x2 ) W sin(" x ) sin( j x2 ) X cos(" x ) sin( j x2 )

Y sin(" x ) cos( j x2 )

(22)

where the arbitrary parameters u , V, w , x and y are used to determine the conditions in which the solution of equation (22) satisfies equilibrium equations (17). The following operator's equation is obtained:

{A}[Q] —

(23)

{A}={U,V,W,X,Y}t and [q] is the symmetrical matrix, in which:

Q = -(Ai2 + A j2),Q, =-i j (A + A),Q= i [BJ2 + B + 2B ) j21,

11 \ 11 66J ) 5 12 ^ V 12 66 /' 13 L 11 v 12 66 ^^ J'

Q = - (Bai2 + Blj2), Q = -i j (B" + B" ), Q = - (A i2 + A j2 ),

14 V 11 66^ / 5 15 ^ V 12 66 / 5 22 V 66 22"^ /

Q = j [B i2 + 2B i2 + B j2], Q = -i j (B" + B" ), Q = - (BaJ2 + B!, j2),

23 ^ L 12 66 22^ J> 24 ^ \ 12 66 / 5 25 \ 66 22^ / 5

(24)

4 2 2 2 2 4 V/

Q = - [Di + 2D; j + 4D i j + D j ],

-?-? L n i? J 66 J nJ J7

Q = i [D i + D j + 2D j ], Q = j [D i + 2D i + £ j ],

34 L 11 \2J 66J 35 J L 12 66 22 J?

Q = - [F'i2 + F'j2 + A' ], Q = -i j (F' + F' ), Q = - [F'i2 + F'j2 + A' ],

44 L 11 66J 44 45 J \ 12 66 /' 55 L 66 22^ 55-"

Analytical validation and numerical results

TB (0,6/2,0),

T3 (U (/) o

The components of the generalized forces vector are given by:

P = i (Ar?1 + BTt2 + BTat3), P2 = j (ATt, + BTt2 + BTat3), P3 = -q -h(i2 + j2)(BTt1 + DTt2 + D^), (25)

P4 = ih (BTA + D"rt2 + Flt3), P5 = jh (BTt + D^ + FTt3),

{AT ,BT,DT } = {h ^^(x,){1, %, x,2}, (26) f

K ,DT,FT} = J/h2 r^X^)^){1,X3,0(X3)} dx3, (27)

It is important to note % = x / h, 0(x3) = 0(%)/ h.

This section examines how materials respond to bending under thermal and mechanical loads. It aims to verify the accuracy of the theory and explore the effects of the parameters on bending, using the following non-dimensional parameters in the calculations: *

_ 102D (ab_ 1 (abh w=-4— W -z^ f 011^-^—011^^fZ

The central deflection a q0 v22y axial stress 10 q0

c (U

m

Tl2 = TTT" Tl2 (0,0,-h/3), a

The longitudinal shear stress q0 <

Transverse shear stress q0 n

D=

hiEr

The coordinate thickness x'

= x3 / h i2(i-v2 )

In this section, we validate the proposed composite shear deformation plate theory (CSDPT) using numerical results. We compare it with various standard high-order shear deformation theories (PSDPT, SSDPT, ESDPT), as well as with the first-order FSDPT and the classical plate theory CPT, as referenced in Mindlin (1951), Timoshenko & Woinowsky-Krieger (1959), Reddy (1984), Touratier (1991) and Karama et al. (2003).

Table 5 presents a comparison of dimensionless deviations between the 'present theory' and the 'standard theories,' indicating minor differences across all theories.

Figures 6 and 7 compellingly demonstrate the robust agreement between the present theory and the established standard theories.

i s

-■- CPT

—k- FSDPT -•- ESDPT SSDPT PSDPT -^J- CSDPT

0 5 10 15 20 25 30 35 40 45 50

(a/h)

Figure 3 - Dimensionless deflection ( w ) across the thickness of a square FGM plate (with n=2) under varied (a/h) ratios, subject to specific conditions (qo=100, ti=0)

0.406

0.405

0.404

0.403

0.402

0.401

0.400

Table 5 - Comparison of the volume fraction exponent (n) effects on the dimensionless displacements and stresses between the present theory and the established theories.

Deflection Theory n = 0 n = 1 n = 2 n = 5 n = 10

CPT 0.83156 1.1896 1.2976 1.3983 1.4878

FSDT (*) 0.85761 1.2252 1.3382 1.4453 1.5386

— ESDT (*) 0.85743 1.2250 1.3392 1.4482 1.5407

w SSDT (*) 0.85756 1.2251 1.3394 1.4483 1.5410

TSDT (*) 0.85760 1.2252 1.3394 1.4483 1.5410

Present 0.85761 1.2252 1.3394 1.4482 1.5410

CPT 0.50880 0.63901 0.68352 0.75682 0.83819

FSDT (*) 0.50880 0.63903 0.68355 0.75682 0.83819

— ESDT (*) 0.51180 0.64307 0.68821 0.76210 0.84369

SSDT (*) 0.51166 0.64286 0.68797 0.76182 0.84343

TSDT (*) 0.51147 0.64261 0.68769 0.76157 0.84310

Present 0.51174 0.64298 0.68814 0.76200 0.84357

CPT 0.76599 0.71701 0.69135 0.70066 0.71089

FSDT (*) 0.76599 0.71701 0.69135 0.70059 0.71093

7" ESDT (*) 0.76424 0.71574 0.68985 0.69884 0.70904

12 SSDT (*) 0.76437 0.71578 0.68993 0.69893 0.70912

TSDT (*) 0.76444 0.71582 0.69001 0.69905 0.70920

Present 0.76430 0.71570 0.68992 0.69890 0.70905

CPT / / / / /

FSDT (*) - - - - -

ESDT (*) 0.34378 0.34378 0.31986 0.29862 0.31141

SSDT (*) - - - - -

TSDT (*) 0.45704 0.45704 0.44189 0.43013 0.44157

Ti3 Present - - - - -

0.44329 0.44329 0.42779 0.41568 0.42766

0.42956 0.42957 0.41372 0.40126 0.41370

0.45200 0.45194 0.43664 0.42471 0.43641

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CP

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Figure 4 - Variation of (Tn) across the thickness of a square FGM plate (with n=2) under varied (a/h) ratios, subject to specific conditions (qo=100, ti=0)

The following figures highlight and reveal the influence of the aspect ratios on the stress distribution within functionally graded material plates. A higher a/h (e.g., a/h = 20) results in elevated stress levels, while a lower a/h (e.g., a/h = 2) leads to minimized stress levels, particularly suitable for low-stress-tolerance applications. These findings underscore the pivotal role of aspect ratios in tailoring FGM plate designs to meet diverse stress requirements.

Simultaneously, varying thermal conditions (Ti, T2, T3) significantly shapes stress distribution across all aspect ratios, highlighting the pronounced influence of the thermal.

T3 uniquely impacts the stress gradient through the plate's thickness, highlighting its distinct role in stress distribution.

Material property gradients (n values) play a vital role in shaping stress distribution.

Illustrating the relationship between a/h and n, along with the impact of the thermal field, these factors can be adjusted to meet precise design criteria.

t1=0 ;t2=10 ; t3=0 t1 = 10 ; t2=10 ; t3=0 t1=0 ; t2=0 ; t3=10 t1=0 ; t2=10 ; t3=10 t1 = 10 ;t2=10 ; t3=05 t1 = 10 ;t2=10 ; t3=10

_ 0.0 ( °11 )

Figure 5 - Effect of the thermal field on the(o11) through-the-thickness of a rectangular FGM plate with (n = 2, q0 = 100, b = 2a). a=02*h

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Figure 7 - Effect of the thermal field on the(an) through-the-thickness of a rectangular FGM plate with(n = 2, qo = 100, b = 2a). a=10*h.

t1=0 ;t2=10 ; t3=0 t1 = 10 ; t2=10 ; t3=0 t1=0 ; t2=0 ; t3=10 t1=0 ; t2=10 ; t3=10 t1 = 10 ;t2=10 ; t3=05 t1 = 10 ;t2=10 ; t3=10

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Figure 8 - Effect of the thermal field on the(an) through-the-thickness of a rectangular FGM plate with(n = 2, q0 = 100, b = 2a). a=20*h.

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Figure 9 - Effect of the thermal field on the(uu ) through-the-thickness of a rectangular FGM plate with(n = 1, q0 = 100, b = 2a). a=10*h.

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Figure10 - Effect of the thermal field on the( ) through-the-thickness of a rectangular FGM plate with(n = 6, q0 = 100, b = 2a). a=10*h.

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Conclusions

This study thoroughly examined how advanced composite structures respond to various loads - mechanical and thermal - using a new CSDPT theory that eliminates the need for shear correction factors. By comparing the results with the standard established theories, it showed strong consistency being particularly aligned, particularly pertinent when taking into account SSSS boundary conditions. The research explored how changing volume fraction exponents and side-to-thickness ratios impact the displacements and stresses of functionally graded rectangular plates under distributed loading, highlighting the significant influence of material property gradients on their response. It also emphasized how different loads affect stresses within the plates, emphasizing the intricate relationship between these loads and stress distribution. Ultimately, this theory proves to be precise and suitable for analyzing the thermo-mechanical bending response of thick functionally graded plates, contributing valuable insights for practical applications and enhancing our understanding of advanced composite materials.

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Examen de flexión de estructuras compuestas de generación avanzada con propiedades específicas expuestas a diferentes cargas

Ahmed Zitounia, autor de correspondencia, Bachir Bouderbaa, Abdelkader Dellalb, Hamza Madjid Berrabahc

a Universidad de Tissemsilt, Departamento de Ciencia y Tecnología, Laboratorio de estructuras y materiales de ingeniería mecánica, Tissemsilt, República Argelina Democrática y Popular b Universidad Tissemsilt, Departamento de ciencia y tecnología, Tissemsilt, República Argelina Democrática y Popular

c Universidad de Relizane, Departamento de Ingeniería Civil, g

Laboratorio de Estructuras y Materiales de Ingeniería Mecánica, Relizane, República Argelina Democrática y Popular

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TIPO DE ARTÍCULO: artículo científico original Resumen:

Introducción/objetivo: Este artículo presenta el examen de flexión de estructuras compuestas de generación avanzada con propiedades específicas expuestas a diferentes cargas. ^

13

Métodos: Este artículo propone e introduce una nueva teoría generalizada de la deformación cortante de cinco variables para calcular la respuesta estática de placas rectangulares funcionalmente graduadas hechas de cerámica y metal. Notablemente, nuestra teoría elimina la necesidad de un factor de corrección de corte y garantiza condiciones de tensión de corte § cero en las superficies superior e inferior. Se introducen investigaciones o numéricas para interpretar las influencias de las condiciones de carga y las J variaciones de potencia del material clasificado funcionalmente, la t proporción de módulo, la proporción de aspecto y la proporción de espesor en el comportamiento de flexión de los FGP. Estos análisis luego se comparan con los resultados disponibles en los textos. n

Resultados: Los resultados preliminares incluyen un análisis comparativo -con las teorías estándar de deformación por corte de orden superior (PSDPT, ESDPT, SSDPT), así como con las teorías de Mindlin y Kirchhoff (FSDPT y CPT).

Conclusión: Nuestra teoría contribuye junto con las teorías establecidas en | el campo, proporcionando información valiosa sobre la respuesta termomecánica estática de placas rectangulares funcionalmente graduadas. Esto abarca la influencia de los valores del exponente de la fracción de volumen en los desplazamientos y tensiones adimensionales, el impacto de las relaciones de aspecto en la deflexión y los efectos del campo térmico en la deflexión y las tensiones. Ejemplos numéricos del ^ examen de flexión de estructuras compuestas de generación avanzada i

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con propiedades específicas expuestas a diferentes cargas demuestran la precisión de la presente teoría.

Palabras claves: materiales funcionalmente graduados, flexión, teorías de deformación por corte de orden superior, termomecánica.

Исследование на изгиб усовершенствованного поколения композитных конструкций со специфическими свойствами, подверженных различным нагрузкам

Ахмед Зитуниа, корреспондент, Башир Будербаа, Абделькадер Деллал6, Хамза Маджид Беррабахв

а Университет Тиссемсилта, факультет науки и технологий, лаборатория конструкций и материалов машиностроения, г. Тиссемсилт, Алжирская Народная Демократическая Республика

б Университет Тиссемсилта, Факультет науки и технологий, г. Тиссемсилт, Алжирская Народная Демократическая Республика

в Университет Релизана, Факультет гражданского строительства, лаборатория конструкций и материалов машиностроения, г. Релизан, Алжирская Народная Демократическая Республика

РУБРИКА ГРНТИ: 55.09.43 Композиционные материалы ВИД СТАТЬИ: оригинальная научная статья

Резюме:

Введение/цель: В данной статье представлено исследование на изгиб композитных конструкций нового поколения со специфическими свойствами, подверженных различным нагрузкам.

Методы: В данной статье представлена новая обобщенная теория деформации сдвига с пятью переменными для расчета статического отклика функционально градуированных прямоугольных пластин, изготовленных из керамики и металла. Данная теория исключает необходимость использования поправочного коэффициента сдвига и обеспечивает отсутствие условий деформации сдвига как на верхней, так и на нижней поверхности пластины. Введено численное испытание при интерпретации влияния условий нагрузки и изменений прочности функционально градуированного материала, а также коэффициентов модуля, аспекта и толщины на поведение функционально градуированных пластин при изгибе. Результаты анализа были сопоставлены с результатами, доступными в литературе.

Результаты: Предварительные результаты включают сравнительный анализ со стандартными теориями сдвиговой

Абделкадер Делал6, Хамза Мауид Берабах1

а

деформации высшего порядка (PSDPT, ESDPT, SSDPT), а также теориями Миндлина и Кирхгофа (FSDPT и СРТ). Выводы: Наряду с ранее подтвержденными теориями в этой области, данная теория вносит вклад, предоставляя ценную информацию о статическом термомеханическом отклике функционально градуированных прямоугольных пластин. Он охватывает влияние значений показателя объемной доли на безразмерные смещения и напряжения, влияние коэффициентов аспекта на дефлексию, а также влияние теплового поля на дефлексию и напряжения. Численные примеры испытаний на изгиб усовершенствованного поколения композитных структур со специфическими свойствами, испытанных различными нагрузками, подтверждают точность представленной теории. -е

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Ключевые слова: функционально-сортированные материалы, 2 изгиб, теории сдвиговой деформации высшего порядка, ^ термомеханический.

Испитива^е вршено сави]а^ем напредне генераци]е композитних структура са специфичним сво]ствима изложених различитим оптерейе^има

Ахмед Зитуниа, аутор за преписку, Башир Будерба3

(Я -С

6 УЪлл-за ЬАяппг! Р^опа^аув £

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Универзитет у Тисемсилту, Одсек за науку и технологи]у, Лаборатори]а за машинске материале и конструкци]е, о

Тисемсилт, Народна Демократска Република Алжир ^

6 Универзитет у Тисемсилту, Одсек за науку и технологи]у, .1

Тисемсилт, Народна Демократска Република Алжир

5 Универзитет у Релизану, Одсек за гра^евинарство, §

Лаборатори]а за машинске материале и конструкц^е, Релизане, Народна Демократска Република Алжир

ОБЛАСТ: механика

КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад Сажетак:

Увод/цил: У раду jе представлено испитиваше сав^ашем напредне I

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генерац^е композитних структура са специфичним своjствима изложених различитим оптереПешима. Методе: Предлаже се и уводи нова генерализована теорба ш смицаша са пет вар^абли ради израчунаваша статичког одговора га" четвртастих функционално градираних керамичко-металних плоча. Теорба елиминише потребу за коришЯешем корективног фактора смицаша и обезбе^е одсуство услова за деформац^у

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нумеричко испитиваъе за тумачеъе утица]а услова оптереПеъа и варщацща снаге функционално градираног материала, као и коефици]ената модула, аспекта и деблине на понашаке функционално градираних плоча при сави]ак>у. Резултати ових анализа упоре^ени су са резултатима доступним у литератури.

Резултати: Прелиминарни резултати обухвата]у компаративну анализу са стандардним теори]ама смицаъа вишег реда (РБйРТ, ЕБйРТ, ББйРТ), као и са теори]ама Миндлина ( РБйРТ) и Кирхофа (СРТ).

Заклучак: За]едно са веЬ потвр^еним теори]ама у ово] области, представлена теори]а пружа допринос увидом у статички термомеханички одговор функционално градираних плоча. Он обухвата утица] вредности експонента запреминског удела на недимензионална помераъа и напоне, утица] коефици]ената аспекта на дефлекси]у, као и ефекте термалног пола на дефлекси]у и напоне. Нумерички примери испитиваъа вршеног сави]ак>ем напредне генераци]е композитних структура са специфичним сво]ствима изложених различитим оптереПеъима потвр^у]у тачност представлене теорбе.

Клучне речи: функционално градирани матери]али, сави]ак>е, теори'е смицаъа вишег реда, термомеханички.

Paper received on: 23.11.2023.

Manuscript corrections submitted on: 04.03.2024.

Paper accepted for publishing on: 05.03.2024.

© 2024 The Authors. Published by Vojnotehnicki glasnik / Military Technical Courier (www.vtg.mod.gov.rs, BTr.M0.ynp.cp6). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/rs/).