Scholarly article on topic 'Effect of material nonhomogeneity on the mechanical behaviors of a thick-walled functionally graded sandwich cylindrical structure'

Effect of material nonhomogeneity on the mechanical behaviors of a thick-walled functionally graded sandwich cylindrical structure Academic research paper on "Mechanical engineering"

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{Nonhomogeneity / "Functionally graded material" / Sandwich / "Cylindrical structure" / Orthotropic}

Abstract of research paper on Mechanical engineering, author of scientific article — H.M. Wang, Y.K. Wei

Abstract In this investigation, the effect of material nonhomogeneity on the mechanical behaviors of a thick-walled sandwich cylindrical structure embedded with a functionally graded (FG) interlayer is investigated. The inner of the sandwich cylindrical structure is a homogeneous and isotropic layer. The outer is a homogeneous and orthotropic layer and the middle is a transition layer. A three-parameter model is presented to model the transitional form of the FG interlayer. The laminate approach method is employed to tackle the material nonhomogeneity. The analytical solutions are obtained by employing the initial parameter method. Numerical results show that the material nonhomogeneity plays important roles on the mechanical behaviors of a sandwich cylindrical structure.

Academic research paper on topic "Effect of material nonhomogeneity on the mechanical behaviors of a thick-walled functionally graded sandwich cylindrical structure"

Results in Physics 2 (2012) 118-122

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Results in Physics

journal homepage: www.journals.elsevier.com/results-in-physics

Effect of material nonhomogeneity on the mechanical behaviors of a thick-walled functionally graded sandwich cylindrical structure

H.M. Wang *, Y.K. Wei

Department of Engineering Mechanics, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, PR China

ARTICLE INFO ABSTRACT

In this investigation, the effect of material nonhomogeneity on the mechanical behaviors of a thick-walled sandwich cylindrical structure embedded with a functionally graded (FG) interlayer is investigated. The inner of the sandwich cylindrical structure is a homogeneous and isotropic layer. The outer is a homogeneous and orthotropic layer and the middle is a transition layer. A three-parameter model is presented to model the transitional form of the FG interlayer. The laminate approach method is employed to tackle the material nonhomogeneity. The analytical solutions are obtained by employing the initial parameter method. Numerical results show that the material nonhomogeneity plays important roles on the mechanical behaviors of a sandwich cylindrical structure.

© 2012 Elsevier B.V. All rights reserved.

Article history: Received 6 July 2012 Accepted 11 September 2012 Available online 17 September 2012

Keywords: Nonhomogeneity Functionally graded material Sandwich

Cylindrical structure Orthotropic

1. Introduction

Cylindrical structures are one of the widely used configurations, such as pressure vessels, pipes, fluid reservoirs, tubes etc. Numerous investigations have been done for the cylindrical structures made of homogenous materials [1-5].

To satisfy the engineering requirement, the key issue is to improve the mechanical behaviors of the cylindrical structures. One feasible way is by choosing a proper material. But in some cases, the usage of the commaterial is hardly to match the engineering requirements. The composite materials and the multilayered structures are then designed for this special purpose. Due to their practicability and superiority, the multilayered cylindrical structures have been widely used in modern engineering and many investigations have been carried out [6-8]. For multilayered cylindrical structures, the interface is always a weak part due to the mismatch of the material properties.

With the developments in material engineering, the fabrications of the nonhomogeneous materials become a reality. The non-homogeneous materials are also called as functionally grade materials (FGMs). Practically, the use of FGMs can improve the bonding strength at the interface and reduce the interfacial effect between the bonded dissimilar materials. The studies on the structures composed of the FGMs have been increasingly interested by the engineers and scientists and many important works on cylinders, shells and plates have been reported [9-22].

* Corresponding author. Tel.: +86 571 8795 2396; fax: +86 571 8795 2570. E-mail address: wanghuiming@zju.edu.cn (H.M. Wang).

To understand and improve the performance of the newly designed engineering structures, it is necessary to make a comprehensive analysis on the mechanical behaviors. In this investigation, an analytical solution is obtained for a thick-walled sandwich cylindrical structure with a FG interlayer. Based on the obtained solution, the effect of the material nonhomogeneity on the mechanical behaviors is investigated.

2. Mechanical model

Consider an infinitely long thick-walled sandwich hollow cylinder of inner radius a and outer radius b. The inner is an isotropic tube of outer radius r1. The outer is an orthotropic tube of inner radius r2. The middle is a functionally graded interlayer. The geometry of the cross section of the sandwich hollow cylinder and the coordinate system (r, 0) is shown in Fig. 1. In the following, the quantities with the superscripts "1", "2" and "3" denote respectively, those for layer 1 (inner part - homogeneous and isotropic), layer 2 (middle part - FG) and layer 3 (outer part - homogeneous and orthotropic). For axisymmetric plane strain dynamic problem, the three-dimensional equations of the motion are simplified as

do\ dr

r(i) _ r№ @2u© rrr (i) @ ur

r P dt2 '

where i = 1, 2, 3.u^ is the radial displacement. and r«« are the radial and hoop stresses, respectively. q(l) is the mass density with

P(1) = Po,

P(3)= Pro-

The strain-displacement-stress relations are expressed as

2211-3797/$ - see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.Org/10.1016/j.rinp.2012.09.006

Orthotopic (Layer 3) FG interla: (Layer 2)

Isotropic (Layer 1)

Fig. 1. The geometry of the cross section and the coordinate system.

e(i) -tJrr -

- s(i) r(i) + s(i) r(i) — .>11 r rr + ->12rhh ,

e№ _ r(i) , s№ rC)

ehh — r — 12rrr + 22rhh, s(i) c(0

where si 1,sl2 and s2 2 are the elastic compliance coefficients which can be expressed by the engineering constants as [23]

s(i) _ s(i) _ 1 - t2 1 — J'y '

s(3) -

1 t2 ____zr

Ero Ez<

s(1) - .

to(1 + to)

.(3) _ _

Eho Ezo

s(3) -

tzrotzho \

s(2) -

t2r(r)

s(2) -.

Er(r) Ez(r)' 22 Eh(r) Ohr (r) , t zr (r)Mr)l

Ez(r) J

LEh(r) Ez(r)

where E0 and t0 are Young's modulus and Poisson's ratio of the isotropic materials (inner part). Ero, , Ezo, o«ro , tzro , are the engineering constants of cylindrically orthotropic materials (outer part). A three-parameter model is employed to model the variation forms of the material properties of the FG interlayer.

Ei (r) — aEo/!(r) -tij(r) — ato/1 (r) -p(r)— apj1(r) -

r2 - r1

ßEio/2(r) + y(ßEio - aEo)/1(r)/2(r) - ßtijof2(r)+ C(ßtjo - ato)/1(r)/2(r) (i ,j — r, h ,z) ßPr0f2(r) + C(ßPr0 - aPo)f1 (r)f2(r)•

f 2 (r) —

r - r1

r2 - r1

(r1 6 r 6 r2).

Here a, b and y are three gradient parameters which can be selected according to the engineering practices. Especially, the case for a = 1 means that the material properties between layers 1 and 2 are completely matched. And the case for b = 1 means that the material properties between layers 2 and 3 are completely matched. When choosing different y, the variation form of the material properties in the FG interlayer can be tuned.

Eq. (3) can be rewritten as

- c(i) U rr - L

11 dr du,®

,(i) ur

r(i) _ c(i) i c® ut

rhh — 12 dr + 22 r J where

c(i)_ ÜL c(0_ iL c11 W0')J c22 W('')J

c(i) _ _ in. W(i) _ s(0 s(0 _ s(0 s(0 (7\

c12 — W(0 , W — s11s22 s12s12. (/)

Particularly, for isotropic layer (inner part), we have

c(1) _ c(1) L11 — 22.

3. Solutions

For harmonic vibrations with the angular frequency m, all the fields vary harmonically. That is

{uf(r,t), r«(r,t), 4)(r,t)} = {uW(r), r«(r), exp(jmt).

where j = V^T is the imaginary unit. exp (.) stands for exponential function. For the sake of brevity, the harmonic factor exp (jmt) will be dropped in the following equations.

Substituting Eq. (6) into Eq. (1) and utilizing Eq. (9), gives

d2u® 1 du® (. 2 a2\ n + --j—+ k2 -^r u(1' = 0,

dr2 r dr with

k — mJp(t>/cfv I ^Jc22/Cf1.

For the homogeneous layers 1 and 3, ki and li (i = 1, 3) are constants. The solution of Eq. (10) involves Bessel functions of the first and second kinds. Due to the material isotropy of layer 1, we have l = 1. Then by means of the initial parameter method, the solutions of displacement and radial stress in layer 1 (a 6 r 6 r1) can be obtained as

u(1) (r) r(1)(r)

u(1)(a) ^(a)

J^r) T212)(k1r)_

'Pr1(k1a)J1(k1r)-Pj1(k1 a)Y 1(k1r)]

T11-I(k1r) — -11 D1(k1r)

t112)(k1r) — DYkr) ^J1(k1a)Y 1 (k1r)- Y1 (k1a)J1 (k1r)L

T21(k1r) — D(k71 [PY1(k1a)Pj1(k1r)- J (MP^r)]

T22(k1r) — D(k71 [J1 (k1a)PY1 (k1r ) - Y1 (k1a)Pj1 (k1r)^ A^r) —J1(k1a) PY1(k1a) - Y 1(k1 a) ^(ha) ,

PJ1(k1r — PY1(k1r) —

c111).d + c112)I

c11 dr + c12 r

J1(k1r), Y 1(k1r) ,

where J1( ) and Y1( ) are Bessel functions of the first and second kinds of order 1. Similarly, the solutions of displacement and radial stress in layer 3 (r2 6 r 6 b) can be obtained as

u(3 (r

\r<31 (r)/

T1311(k3r) T1321(k3r)

LT2311(k3r) T2321(k3r)

( u(31(r2)

T131 (Vor) =

A3 (fer )

[PY3(k3r2)Ji3 (k3r) - Pj3 ^2)) (1^)],

T132 (k3r) = Ä(k?) Jl3 (k3r2) Y13 (k3r) - Y13 (k3r2))Ji3 (k3

T 231 (k3r) =

i3(K3 1

A3(k3r)

[PY3(k3r2)Pj3(k3r) - Pj3(k3r2)Py3^3r)],

T232(k3r) Jl3 (k3r2)PY3(k3r) - Y13 (k3r2)P^r)],

A3(k3r) = jfe (k3r2)PY3 (k3r2) - Y13 (k3r2)Pj3(k3r2),

Pj3(k3r) = PY3(k3r) =

c(3) , c(3) c11 dr + c12 r

jl3 (k3r) Y

13 (k3r) •

where (■) and Yl3 (■) are Bessel functions of the first and second kinds of order i3.

It is noted that k2 and i2 of FG interlayer are functions of radial position r. For this case, Eq. (10) is very difficult to be solved. Here, the laminate approach method is employed to tackle the difficulty raised by the material nonhomogeneity. The FG interlayer is fictitiously divided into N layers with equal thickness and the radii of the fictitious interface are denoted as

Ro = r1, Rm = r1 + m(r2 - r1 ) =N (m = 1, 2,

N) •

In each sub-layer (Rm_1 6 r 6 Rm), the material properties are treated as constants and are chosen as the values at the position of the mean radius r = r2m = (Rm-1 + Rm)/2. Then the solution in each sublayer (Rm_1 6 r 6 Rm) can be derived by following the treatment same as those for homogeneous layer.

u(2m> (r) (r

TT (k2mr)

T122m) (k2mr)

(k2mr T(222m (k2mr

u(2,m> (Rm-1) ) rP,m) (Rm-1)

T121m) (k2mr) =

r [PY2 (klmRm-1 )j i2m (k2mr) - Pj2 (klmRm-1 ) Y ^ (k2mr)],

A2(k2mr)1

T122m) (fc2mr) = ^-^r j^ (k2mRm-1 ) Y^ (k2„r) - Y^ (fc2mRm-1 (k2mr )],

t(2;m 21

(k2mr) =-

A2 (k2mr) 1

r [Py2 (k2mRm-1 ) Pj2 (k2mr) -Pjl(klmRm-1 ) Py2 (k2mr) ],

A2(k2mr)1

T?2m (k2mr) = [jl2m (fc2mRm-1 ) Py2 (k2mr) - Y^ (k2mRm-1 ) Pj2 (k2mr)],

A2(k2mr) =ji2m (k2mRm-1 ) PY2^2mRm-1) -Yi2m (k2mRm-1 ) Pj2 (k2mRm-1 ) ,

Pj2(k2mr) = PY2(k2mr =

c11 (r2m) dr + c12 (r2 m) ^

cn (r2m) dr + c122 (r2m) 1

ji2m (k2mr), k2m = «

Y 12m (k2mr), 12m =

q(2) (r2m) c12 (r2^ ) '

C22 (r2m)

c(2)/r c11 (' 2m

Assuming the sandwich cylindrical structure is perfectly bonded at the interface r = r1 = R0 and r = r2 = RN, then we have

u(2,1) (r1) = u(1) (r1 ) , u(2N (r2) = u(3) (r2 ),

rf1 (r1) = r« (r1), rr2N (r2) = r(3) (r2)

(19a )

Due to the introduction of the fictitious interfaces, the continuity conditions should be supplemented. At each fictitious interface r = Rm (m = 1,2, ■■■, N - 1) , the displacement and the radial stress should be continuous and the continuity conditions can be expressed as

ir2,m+1) (Rm ) = r(2m (Rm)

u(2m+1> (Rm ) = u(2m (Rm),

(m = 1,2, •••N- 1)•

Setting r = b in Eq. (14) and utilizing Eqs. (12), (17) and (19a,b), we arrive at

u(3) (b )

rr3) (b)

H11 H21

H12 H22

u(1) (a )

(1) (a)

H11 H12 H21 H22 J

— rx(3)l rr(2)l rx(1)l

T11 (k1r1)

T12 (k1r0

T<v (k1ro T22 (k1ro

[T(2 ]=

T121m) (k2mRm)

T<21m) (k2mRm)

T(122m (k2mRm T(222m (k2mRm

T11 (k3b)

T 132 (k3b)

T2Ï(k3b) Tg(k3b)

where nij=N( ) denotes a continuous multiply symbol from N to 1. From the second equation in Eq. (20), we have

u(1)(a) = H- [ffP>(b) _H22rr1)(a)]. (22 )

Denoting Q exp (jmt) and Q, exp (jxt) as the dynamic pressures applied on the internal and external surfaces of the sandwich cylindrical structure, respectively, then the boundary conditions can be expressed as (also, the harmonic factor exp (jmt) will be dropped for the sake of brevity)

r™ (a) = Qa, of) (b ) = Qb. (23)

The substitution of Eq. (23) into Eq. (22) gives

um(a) = Qb _ H22Qa. (24 )

From Eq. (24), we learn that u(1)(a) becomes infinitely large if

H21 = 0. (25 )

Then all the fields approach to infinite. That is, Eq. (25) is the characteristic equation of resonant frequency of the thick-walled FG sandwich cylindrical structure under the radial vibration.

It should be noted here that if we set m = 0, the presented solution is the degenerated to that for the sandwich cylindrical structure subjected to the static pressures at the internal and external surfaces.

4. Numerical results

In this section, we first examine the validity of the presented solution. The results for internally pressurized functionally graded hollow cylinder are shown in Fig. 2. In this example, Young's modulus in whole cross section obeys the following relations [24]

E(r) = Ein + (Eou En b - aK

where Ein and Eou stand for Young's moduli at the internal and external surfaces, respectively. In the calculation, all the computation parameters are same as those employed in Ref. [24]. Fig. 2 shows that the present results agree well with those reported in Fig. 7(a) by Li and Peng [24]. The validity of the presented solution is then verified in this respect.

Next, the presented solution will be employed to analyze the mechanical behaviors of the thick-walled FG sandwich cylindrical structures. The inner part is chosen as aluminum: E0 = 72.2 GPa, t0 = 0.34 and p0 = 2700 kg/m3. The outer part is made of Graphite-epoxy. The engineering constants of Graphite-epoxy can be converted from Ref. [25]: Ero = 10.3Gpa, E«0 = 181.0 GPa, Ez0 = 10.3 Gpa, thr0 = 0.28, tzr0 = 0.33, tzh0 = 0.016, pr0 = 1590 kg/m3.

0.5 0.6 0.7 0.8 0.9 1.0

Fig. 2. Distributions of the hoop stress in a pressurized FG hollow cylinder.

To show the numerical results, the following non-dimensional forms are introduced as

r ____

U = U ;

y — h

Rh = ;

, r P ri

n — b; ni ^ J;

F - r2

n2 - b ;

qa — ; qb ; x0 — h\l7T; C0 — cn•

s — b ; 1 /Co

b\l Po;

In the following calculation, the geometric parameters are fixed as s = 0.6, f-i = 0.7 and n2 = 0.9. Fig. 3 shows the sensitivity of the gradient parameters a, b and y on the variation forms of c22/C0 in the FG interlayer. It shows that a wide range of adjustment of the variation forms of the material properties can be achieved by changing the gradient parameters. The convergence studies of the resonant frequency for two sets of gradient parameters are shown in Tables 1 and 2. It can be seen that the first five resonant frequencies converge rapidly with the increase of N. To guarantee the accuracy, N = 200 is employed in the following analysis. The resonant frequencies as functions of y for the first three modes are depicted in Fig. 4. We notice that the resonant frequencies for the first mode increase with the increase of y, while it is on the contrary for the second and third modes. For the first and third modes, the resonant frequencies for (a = 1.0, b = 1.0) are in the middle of those for (a = 0.8, b = 1.2) and (a = 1.2, b = 0.8) while it is not always the fact for the second mode. Fig. 5 shows the distributions of the radial displacement, radial and hoop stresses for the internally pressured thick-walled FG sandwich cylindrical structures with y = -1.0,

Table 1

Convergence study-the first five non-dimensional resonant frequencies for a = 1.0, b =1.0, y = 0.5.

N Mode

1 2 3 4 5

5 1.5604 4.8864 10.1621 15.0845 19.9271

10 1.5591 4.8753 10.1712 15.1396 19.8725

20 1.5587 4.8725 10.1730 15.1529 19.8619

50 1.5586 4.8718 10.1734 15.1565 19.8591

100 1.5586 4.8716 10.1735 15.1570 19.8587

200 1.5586 4.8716 10.1735 15.1571 19.8586

Table 2

Convergence study-the first five non-dimensional resonant frequencies for a = 0.8,

b =1.2, y = -1.0.

N Mode

1 2 3 4 5

5 1.4694 5.4598 10.1699 15.9855 20.9650

10 1.4699 5.4359 10.1648 15.9957 20.8583

20 1.4700 5.4288 10.1612 16.0017 20.8211

50 1.4701 5.4267 10.1599 16.0037 20.8097

100 1.4701 5.4264 10.1597 16.0040 20.8081

200 1.4701 5.4263 10.1596 16.0040 20.8077

(a) I-7

1.6 q 1-5 1.4 1.3

- a = 0.8, p = 1.2

----......

a= 1.0, /3 = 1.0

a = 1.2, p = 0.8 1 1 1

-1.0 -0.5 0.0 0.5 1.0

(b) 6.° 5.5 q 5.0 4.5 4.0

a= 1.2, p = 0.8

_ a = 1.0,1.0

a= 0.8, fi = 1.2

-1.0 -0.5 0.0 0.5 1.0

a = 1.2, $ = 0.8

■A / /^N. y ✓ / s / f\ ..

7\ / N. ...........

fy S ^"S*-"'*"

" ';/■*-.. .7

•* / ^ / ' / ✓ / /— •* / ....... / a = 1.0, fi = 1.0

4--7 / /

y\. / / -Y = -1.0

: ' \\ ^^ * ' -----Y = 0.0

a = 0.8, $ = 1.2 1 1 ................Y = 1.0 1

0.70 0.75 0.80 0.85 0.90

Fig. 3. Variation form of c22/C0 for different values of gradient parameters in the FGM interlayer.

a = 1.2, p = 0.8 '...............

£ = 1.0, p = 1.0 ^

Fig. 4. Resonant frequencies as functions of y (a) the first mode; (b) the second mode; (c) the third mode.

qa = -1.0, qb = 0 and X = 1.0. A discontinuous jump in hoop stress can be observed in Fig. 5c at the inner interface (f = 0.7) for a = 1.2 and at the outer interface (f = 0.9) for b = 1.2. For the case a = 1.0 and b = 1.0, the hoop stress changes continuously in whole

(a) 3.5

to 2.5

(b) 0.0 -0.2

-0.4 -0.6 -0.8 -1.0

0.6 0.7 0.8 0.9 1.0

(c) 5.°

Fig. 5. Distributions of the elastic fields for internally pressured case with y = —1.0, qa = —1.0 and X = 1.0 (a) radial displacement; (b) radial stress; (c) hoop stress.

cross section. This means that the material properties match well at two interfaces is a benefit in improving the mechanical properties of the composite structures by using the FG interlayer.

5. Conclusions

This investigation deals with the dynamic behaviors of a thick-walled sandwich cylindrical structure with a FG interlayer. By

means of the three-parameter model and the laminate approach method, the effect of the material nonhomogeneity on the mechanical behaviors is investigated. Based on the numerical results, we notice that the material nonhomogeneity has more significant effect on the hoop stress than that on the radial stress. The resonant frequencies of high-order mode are more sensitive to the material nonhomogeneity.

By employing the different values of the gradient parameters in the presented three-parameter model, the comprehensive parameter analysis can be performed. Potential applications of the presented solution can be found in designing the high performance functionally graded cylindrical structures for the special engineering requirements.

Acknowledgement

The work was supported by the Key Team of Technological Innovation of Zhejiang Province (Grant 2011R09025-05).

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