Stability of hydrostatic-pressured FGM thick rings with material nonlinearity

Applied Mathematical Modelling 45 (2017) 55–64

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Stability of hydrostatic-pressured FGM thick rings with material nonlinearity Huaiwei Huang a,∗, Yongqiang Zhang a,b, Qiang Han a a b

School of Civil Engineering and Transportation, South China University of Technology, Guangzhou, Guangdong 510640, PR China Guangzhou Salvage Bureau, Guangzhou, Guangdong 510260, PR China

a r t i c l e

i n f o

Article history: Received 24 January 2016 Revised 7 November 2016 Accepted 7 December 2016 Available online 21 December 2016 Keywords: Functionally graded materials Rings Buckling Elastoplasticity

a b s t r a c t Buckling behaviors of elastoplastic ceramic/metallic functionally graded material (FGM) rings are investigated by using the first order shear deformation theory. The hydrostaticpressured rings are assumed to be in both the plane-stress case and the plane-strain case, which lead respectively to a uniaxial and a biaxial elastoplastic stress states in prebuckling stage. A uniform strain hypothesis helps to deal with the elastoplastic stress states. By introducing in the graded material properties, the constitutive model of FGMs is formulated under the framework of J2 deformation theory. By considering the kinetic relations of von-Kárman type and employing the principle of virtual displacement, the equilibrium equations and the buckling governing equations of FGM circular rings are formulated, and the analytical solution of the anisotropic rings is obtained. Finally, the elastoplastic buckling problem is numerically solved through a semi-analytical method, which is proposed to seek the real circumferential strain of FGM rings at the buckling point and determinate the elastoplastic buckling critical hydrostatic pressure. The effects of the inhomogeneous and geometrical parameters on the buckling critical load and the position of the elastoplastic interface are discussed. Results show that, in both the plane-stress and the plane-strain cases, the elastoplastic critical loads are generally lower than their elastic counterparts due to material flow, and the plane-strain critical load is generally larger than the plane-stress one. The elastoplastic critical load does not always decrease monotonously with the increase of the inhomogeneous parameters, which is quite different from their elastic counterparts. © 2016 Elsevier Inc. All rights reserved.

1. Introductions As fundamental component or simplified model of pressure-loaded hole cylindrical structures, such as undersea oil transportation pipelines, submarines, and etc., researches in mechanical performances of circular rings have received extensive interest, especially their buckling behaviors, which are of great concern to submersible structural designers. Early researches on buckling of homogeneous rings were Timoshenko and Gere [1] and Smith and Simitses [2], which gave the analytical solutions for linear buckling of elastic thin rings under in-plane loads. Investigations on postbuckling behaviors might initiate by Flaherty et al. [3], whose work focused on the large deformation analyses with opposite sides of rings in contact, and the subsequent postbuckling theories were formulated under the inextensional or extensional assumption of the rings. Kyriakides and Babcock [4] presented elastoplastic postbuckling analyses for imperfect inextensional rings ∗

Corresponding author. E-mail address: [email protected] (H. Huang).

http://dx.doi.org/10.1016/j.apm.2016.12.007 0307-904X/© 2016 Elsevier Inc. All rights reserved.

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H. Huang et al. / Applied Mathematical Modelling 45 (2017) 55–64

and found that the inelastic nature of the material creates a distinct limit critical load, beyond which the response path is unstable. Nevertheless, for elastic circular rings, the postbuckling path is stable, as a prime result of Fu and Waas [5], where thickness effects were discussed for extensional thick ring by using an approximate first order shear deformation theory. Kim and Chaudhuri [6] employed a fully nonlinear finite-element method to study the postbuckling issue of moderately thick imperfect rings under external pressure, the transverse normal and shear strain effect were examined, and the error of von-Kárman nonlinear theory was assessed. Wu et al. [7] applied the combinations of an improved harmonic balance method and Newton’s technique to obtain the approximate analytical solutions for the nonlinear differential equations of the post-buckling response of the inextensional ring. On buckling behaviors of circular rings made of composite materials, Jones and Morgan [8] obtained analytical solutions to predict the buckling critical pressures of thin cross-ply laminated rings and cylindrical shells. Kim and Chaudhuri [9] investigated buckling behaviors of perfect thick plane-strain cross-ply rings by using finite element method. Asemi and Kiani [10] studied postbuckling behaviors of polar orthotropic linearly elastic rings in the plane-stress case, and subjected to external pressure. In their work, the Newton–Raphson iterative technique was applied to calculate the post-buckling response of the ring up to the collapse point. Recent emergence of ceramic/metallic FGMs has raised enormous research concerns about buckling behaviors of FGM structures. Kadoli and Ganesan [11] discussed thermal buckling behaviors of clamped FGM cylindrical shells. Bagherizadeh et al. [12] investigated buckling issue of FGM cylindrical shells embedded in elastic medium and subjected to combined axial and radial loads. Wu et al. [13] presented linear buckling analysis for simply-supported, multilayered FGM circular hollow cylinders under combined axial compression and external pressure. Yaghoobi and Fereidoon [14] investigated mechanical response and thermal buckling issues of FGM plates resting on elastic foundation. Shen [15] investigated postbuckling behaviors of FGM cylindrical shells and plates under thermal loads by using boundary layer theory. Woo et al. [16] focused on postbuckling behaviors of FGM plates induced by thermal loads. Tung [17] presented an analytical approach to investigate the effects of tangential edge constraints on postbuckling behaviors of FGM flat and cylindrical panels resting on elastic foundations and subjected to thermo-mechanical loads. For FGM ring structures, literature is still limited. Kerdegarbakhsh et al. [18] dealt with elastic buckling and postbuckling problems for FGM rings by using the first order shear deformation theory. However, been not taking into account the material nonlinearity, the buckling behaviors of thick FGM rings predicted by Kerdegarbakhsh may be different enormously from the real ones. Currently, little research interest has been attracted on buckling of FGM rings, especially for elastoplastic buckling. As is generally known, the submarine structures must be sufficiently stable to sustain large hydrostatic pressure. One of the most effective ways is to increase the thickness dimension of their components. However, thicker geometry tends to induce larger stress and lead to material flow. Therefore, it is essential to consider the material elastoplasticity for thick components. In this paper, elastoplastic buckling analysis of FGM circular rings is presented. The rings are assumed to be in either the plane-stress case or the plane-strain case. As an extension of our previous work [19], in which elastoplastic buckling response of FGM cylindrical shells under external pressure was investigated by using the classical shell theory, this paper deals with elastoplastic buckling problem of thick FGM rings using the first order deformation theory. The constitutive model is formulated under the framework of J2 deformation theory, and both the uniaxial and the biaxial elastoplastic stress states in prebuckling stage are analyzed based on the uniform circumferential strain hypothesis. A semi-analytical method helps to seek the real circumferential strain of FGM rings and then determinate the elastoplastic buckling load . 2. Formulations 2.1. General descriptions The FGM ring with the thickness h and the mean radius R is plotted in Fig. 1. r and θ denote the radial and the circumferential coordinate axes. z axis measures from the mid-plane along the radial direction and z = r − R. x axis is in the normal direction of the ring’s plane. 2.2. Material properties and the constitutive model The constituent distribution of FGMs considered herein follows the power exponent rule [11], given as: k

fc = (0.5 − z/h ) , fm = 1 − fc ,

(1)

where f is the volume fraction, the subscripts c, m denote the ceramic and metallic phases respectively. k is the inhomogeneous parameter governing the constituent distribution. The elastoplastic material properties of FGMs can be formulated by the TTO model, initially proposed for metallic alloy by Tamura et al. [20] and Nakamura et al. [21] It can be given in the following form:



E = Em fm (q˜ + Ec )(q˜ + Em )−1 + Ec fc



fm (q˜ + Ec )(q˜ + Em )−1 + fc

ν = νm f m + νc f c   σY = σYm fm + fc Ec (q˜ + Em )[(q˜ + Ec )Em ]−1

−1

H. Huang et al. / Applied Mathematical Modelling 45 (2017) 55–64

57

Fig. 1. Geometry and the coordinate system of FGM rings.



H = HmVm (q˜ + Ec )(q˜ + Hm )−1 + Ec fc



fm (q˜ + Ec )(q˜ + Hm )−1 + fc

−1

,

(2)

where E(z) is elastic modulus. ν (z) is Poisson’s ratio. σ Y (z) is yield limit. H(z) and Hm are the tangent modulus of FGMs and metallic constituent respectively. q˜ is the ratio of stress to strain transfer, a critical parameter to define the yield limit of metallic and ceramic composites or FGMs. The constitutive model of elastoplastic FGMs is based on J2 deformation theory, which has the general form as:

  εi j = (3/2)Es −1 σi j + K −1 − (3/2)Es −1 δi j σm ,

(3)

in which, the subscript i, j represent x, θ , r. The secant modulus in complex stress state Es = − (1 − 2ν )Es0 ]. 0 K = E/(1 − 2ν ). Es is the secant modulus in the uniaxial tension experiment. The mean stress σ m = (σ xx + σ θ θ + σ rr )/3. δ ij is unit matrix. The corresponding incremental form of Eq. (2) can be written as: 3E Es0 /[3E

  εˆi j = (3/2)Es −1 σˆ i j + K −1 − (3/2)Es −1 δi j σˆ m + (3/4)J2 −1 φ −1 Si j Sld σˆ ld ,

(4)

in which the over-fold line denotes the increment. ϕ = Et Es /(Es − Et ). The tangent modulus in complex stress state Et = 3EH/[3E − (1 − 2ν )H]. Sij is the tensor of stress deviator. J2 = Sij Sij /2. For ring structures, we have Eq. (4) reduced as:

    εˆxx = Es −1 +K −1 /3 σˆ xx + K −1 − (3/2)Es −1 σˆ θ θ /3 + (3/4)J2 −1 φ −1 (σxx − σm )σˆ ,     εˆθ θ = K −1 − (3/2)Es −1 σˆ xx /3 + Es −1 +K −1 /3 σˆ θ θ + (3/4)J2 −1 φ −1 (σθ θ − σm )σˆ , γˆrθ = 6Es −1 τˆrθ + 3J2 −1 φ −1 τrθ σˆ ,

where

(5)

 σxx2 + σθ2θ − σxx σθ θ + 3τr2θ /3, σ¯ m = (σxx + σθ θ )/3, σˆ = (σxx − σm )σˆ xx + (σθ θ − σm )σˆ θ θ + 2τrθ τˆrθ . J2 =



The equivalent stress and strain are:

 √  1 / 2 σi = 1/ 2 (σxx − σθ θ )2 + (σxx − σrr )2 + (σθ θ − σrr )2 + 6τr2θ √  1 / 2 εi = 2/3 (εxx − εθ θ )2 + (εxx − εrr )2 + (εθ θ − εrr )2 + (3/2)γr2θ .

(6)

Generally, material flow would induce material anisotropy which can be represented by the general strain-to-stress relation of anisotropic materials:

⎡

⎤ ⎡ σxx a11 ⎣σθ θ ⎦ = ⎣a21 τr θ a31

a12 a22 a32

⎤ εxx a23 ⎦⎣εθ θ ⎦. a33 γrθ a13

⎤⎡

(7)

For elastoplastic FGMs, the coefficient matrix is symmetric and aij (i, j = 1, 2, 3) are given by Eq. (5) and listed in Appendix. For the plane-stress case with σ xx = 0 and for the plane-strain states with ε xx = 0, Eq. (7) turns into:

⎡

⎤ ⎡ σxx C12 ⎣σθ θ ⎦ = ⎣C22 τr θ C32

⎤ εθ θ C23 ⎦ , γrθ C13

C33

(8)

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H. Huang et al. / Applied Mathematical Modelling 45 (2017) 55–64

where Cij = aij (i = 1, 2, 3, j = 2, 3) for plane-strain state and for plane-stress state Cij = cij , c22 = a22 − a212 /a11 , c33 = a33 , c12 = c13 = c23 = c32 = 0. 2.3. Equilibrium equations The strain-displacement relation of ring structures of von-Kárman type takes in the following form [22]:





∂V +W ∂θ

εθ θ =

r+

1 2



∂W −V ∂θ

 2 r



, γrθ =

∂W −V ∂θ



r+

∂V . ∂r

(9)

According to the first order shear deformation theory, the circumferential displacement V(θ , z) and the radial displacement W(θ , z) read as:

V = v + zϕ , W = w,

(10)

where v(θ ), w(θ ), ϕ (θ ) are respectively circumferential, radial displacements and rotation angle of the mid-plane. Generally, we have r ≈ R and z/R  1 and the rotation-angle ϕ is smaller than that aroused by w, i.e. w, θ /R, so the strain components are simplified as:







εθ θ εθ θ 0 εθ θ 1 = +z , γrθ γrθ 0 0

(11)

where [εθ θ 0 εθ θ 1 γrθ 0 ]T = [(v + w)/R + [(w − v )/R]2 /2 The internal forces N, Q and moment Mare given as:



N

M

Q

T

 =

h/2 −h/2



σθ θ

ϕ  /R

(w − v )/R + ϕ ]T .

T κτrθ dz,

z σθ θ

(12)

where the shear correction factor κ can be chosen as 5/6 or π 2 /12 according to the first order shear deformation theory [22]. Herein, the value 5/6 is used. Substituting Eqs. (8) and (11) into Eq. (12), one obtains:

⎡ ⎤

⎡

N

C220

C221

⎣M⎦ = ⎣C221 Q

C222

C320

C321

⎤ εθ θ 0 C231 ⎦⎣ εθ θ 1 ⎦, C330 κγrθ 0

C230

⎤⎡

(13)

 h/2 where Ci jl = −h/2 zl Ci j dz(i, j = 2, 3, l = 0, 1, 2). According to the virtual displacement principle, virtual potential energy δ U should vanish at the equilibrium point of system, thus we have:

δU = δUin + δUex = 0,

(14)

in which, the virtual strain energy δ Uin is given as:

δUin =



b  2π

0



h/2

−h/2

0

(σθ θ δεθ θ + κτrθ δγrθ )dzdθ dx,

(15)

where b is the thickness in the normal direction of the ring plane. By assuming that hydrostatic pressure q is always normal to the deformed ring [23], the virtual potential energy of external force δ Uex is given as:

δUex =



b  2π

0



0



  δv dθ dx.

qδ w − (q/R ) w − v

(16)

Then, the virtual potential energy is rewritten:



δU = b 02π F dθ = 0   F = Nδεθ θ 0 + Mδεθ θ 1 + Q δγrθ 0 + qδ w − (q/R ) w − v δv = 0.

(17)

Seeing that the above equations should be satisfied under arbitrary virtual displacement δ w, δ v and δφ , the equilibrium equations can be founded with the aid of part integration procedures: 

[N (w − v )] /R − (qR + N ) + Q  = 0, N  + Q + (q + N/R )(w − v ) = 0, M − RQ = 0,

(18)

and the following expressions derived from the integration procedure must hold:







RQ + N w − v

2π δ w0 = Nδv|20π = Mδϕ |20π = 0.

(19)

H. Huang et al. / Applied Mathematical Modelling 45 (2017) 55–64

59

2.4. Prebuckling stages In the prebuckling stage, we have v = ϕ = 0 and w = 0 and all the internal forces and moment should be θ -independent due to axisymmetric deformation of the rings. Accordingly, one obtains the prebuckling internal forces N¯ , Q¯ and moment M¯ , as well as the prebuckling deflection w¯ as follows:

N¯ = −qR, Q¯ = 0, M¯ = −qRC221 /C220 , w¯ = −qR2 /C220 .

(20)

By assuming the hydrostatic pressured ring is compressed uniformly in the prebuckling stage, or in other words, the circumferential strain is uniform through the thickness of the rings, the material elastoplastic interface must be circular, which is plotted in Fig. 1. The distance of this interface measuring from the middle plane of the ring is indicated by s, which can be theoretically determined, as the elastic equivalent stress σie goes up to the yield limit σ Y , i.e.:

σie = σY .

(21)

The relation between the elastic equivalent stress generated forms of Eqs. (3) and (6):

σie

and the elastic equivalent strain

εie

can be deduced from the de-

σie = 3E εie /[2(1 + ν )].

(22)

By considering Eq. (22) and the first and the third equations in Eq. (2) in Eq. (21), and assuming Poisson’s ratio to be a constant value, the position of the elastoplastic interface is solved,



1 s=h − 2



Ec + q˜ Ec − Em



3Em εie 1− 2(1 + ν )σYm

 1/k 

.

(23)

In the uniaxial stress state, τ rθ = σ xx = σ rr = 0, then we have the following relation, from which the secant modulus Es0 can be given:

σθ θ = Es0 εθ θ = σY + H (εθ θ − σY /E ).

(24)

The plane-stress prebuckling stage is similar with the uniaixal stress state, seeing that σ rr can be neglected when compared with σ θ θ , thus, the stress and strain components are expressed by ε θ θ :

σθ θ = 3K Es εθ θ /(3K + Es ), σxx = σrr = 0, εxx = εrr = −(3K − 2Es )εθ θ /[2(3K + Es )],

(25)

and the corresponding equivalent strain is:

εi = 3K εθ θ /(3K + Es ).

(26)

In the plane-strain case, we have the prebuckling stage expressed as:

σxx = 2Es (3K − 2Es )εθ∗ θ /[3(3K + 4Es )], σθ θ = 4Es (3K + Es )εθ∗ θ /[3(3K + 4Es )], σrr = 0, εxx = 0, εrr = −(3K − 2Es )εθ∗ θ /(3K + 4Es ),

(27)

and the equivalent strain is:

 εi∗ = 2εθ∗ θ 3K 2 + 2K Es + 4Es2 /3/(3K + 4Es ),

where the superscript have

(28)

∗

denotes the plane-strain parameters. To relate the plane-stress statewith the plane-strain state, we √ 2 = εi from Eqs. (26) and (28). Then considering it in Eq. (24), and noting Es0 /E ≤ 1, 3 + [(1 − 2ν )Es0 /E ] ≈ 3, and

εi∗

[(1 − 2ν )Es0 /E ]2 − 9 ≈ −9 reach to:



√  −1  ∗ Es0 + 2 3EH 3E + 2(1 − 2ν )Es0 εθ θ = (1 − H/E )σY ,

(29)

from which, the secant modulus Es0 can be solved analytically. The elastic equivalent strain in plane-stress and the plane-strain cases can be derived from Eqs. (26) and (28):



εie =

2 ( 1 + ν )ε θ θ / 3 √ 2 1 − ν + ν 2 εθ∗ θ /[3(1 − ν )]

the plane − stress case the plane − strain case

,

(30)

which is used to solve the position of the elastoplastic interface in Eq. (23). By using Eqs. (25) and (27) and their elastic degenerated form, the prebuckling internal force N is given with respect to ε θ θ or εθ∗ θ .



N=

s −h/2

σθ θ d z +



s

h/2



σθ θ d z =

ϑ εθ θ ϑ ∗ εθ∗ θ

the plane − stress case , the plane − strain case

(31)

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H. Huang et al. / Applied Mathematical Modelling 45 (2017) 55–64

where



ϑ=

s

−h/2



ϑ∗ =

s

−h/2

3K Es /(3K + Es )dz +



h/2

Edz, s

4Es (3K + Es )/[3(3K + 4Es )]dz +



h/2 s





E/ 1 − ν 2 dz.

2.5. Buckling analysis In buckling analysis, all displacements, internal forces and moment are divided into two parts, the prebuckling one and the buckling one, i.e.:

¯ + ˆ, =

(32)

in which,  represents v, w, ϕ or N, M, Q. From natural essence of buckling phenomenon, it is remarkable that the prebuckˆ,M ˆ , Qˆ should be small parameters, so the ling displacements v¯ , w¯ , ϕ¯ and the increments of the inner forces and moment N governing equations for buckling can be obtained from Eq. (18):





ˆ  − RQˆ = 0. ˆ  − vˆ ] /R − Nˆ + Qˆ  = 0, Nˆ  + κ Qˆ = 0, M [N¯ w

(33)

The incremental forms of the internal forces and moment yield from Eqs. (11) and (13) as following:

⎡ˆ⎤ N

⎡

C220

⎣Mˆ ⎦ = ⎣C221

C222

C320

C321

Qˆ

   2 ⎤ ⎤⎡ ˆ v + wˆ /R + wˆ  − vˆ /R /2 ⎢ ⎥ C231 ⎦⎣ ϕˆ  /R ⎦.    C330 κ wˆ  − vˆ /R + ϕˆ

C221

C230

(34)

Substituting Eq. (34) into Eq. (33), one obtains the governing equations with respect to displacement increments:

          ˆ + vˆ  + C221 ϕˆ = 0, C330 κ Rϕˆ + wˆ  − vˆ + C220 w       ˆ + vˆ  − C222 ϕˆ = 0. C330 κ Rϕˆ + wˆ − vˆ − C221 w



ˆ  − vˆ  − C220 w ˆ + vˆ − C221 ϕˆ  − qR w ˆ  − vˆ  = 0, C330 κ Rϕˆ  + w

Assuming the incremental displacements to be:

ˆ = w



ξw sin (nθ ), vˆ =

n



ξv cos (nθ ), ϕˆ =

n



(35)

ξϕ cos (nθ ),

(36)

n

and substituting Eq. (36) into Eq. (35) gives the following relation:



 T i j ξw ξv ξϕ = 0(i, j = 1, 2, 3), ⎡ −C220 − n2 (κ C330 − qR ) n(C220 + κ C330 − qR )   ⎣ i j = n(C220 + κ C330 ) −n2C220 − κ C330 −n(C221 − Rκ C330 ) n2C221 − Rκ C330

n(C221 − Rκ C330 )

⎤

−n2C221 + Rκ C330 ⎦.

(37)

n2C222 + R2 κ C330

To obtain the non-trivial solution for the displacement increments, the determinant of the coefficient matrix [ij ] should vanish. Then we have:

q = η/[1 + ψ /(n2 − 1 )],

(38)

2 ). κ RC330 (RC220 + C221 )/(C220C222 − C221

where η = κ C330 /R, ψ = 1 + Obviously, the minimum of q should correspond to n = 2, and the analytical critical hydrostatic pressure qcr is found:

qcr = η/(1 + ψ /3 ).

(39)

ν )(h/R )2

For the homogeneous plane-stress elastic rings, neglecting the small term 2(1 + the classic buckling critical load for hydrostatic-pressured rings [22], i.e. qcr = E h3 /(4R3 ).

can degenerate Eq. (39) into

3. Numerical results 3.1. Numerical calculating procedure In Section 2, the analytical expressions of both the prebuckling internal force and the buckling critical hydrostatic pressure have been given in Eqs. (31) and (39). The relation between the internal force and the hydrostatic pressure must hold at the buckling moment, i.e. N = qcr R. Therefore, the following task is to find, a converged result between the theoretical predictions of Eqs. (31) and (39), corresponding to the elastoplastic critical hydrostatic pressure qcr .

H. Huang et al. / Applied Mathematical Modelling 45 (2017) 55–64

61

Table 1 Verification of the present results λ of elastic FGM rings. k

0

The plane-stress case Present Kerdegarbakhsh et al. [18] Diff(%) The plane-strain case Present ABAQUS Diff(%)

0.5

2

5

10

2.989 2.98 0.28

2.504 2.53 −1.03

2.271 2.30 −1.26

2.169 2.19 −0.97

2.092 2.10 −0.39

3.242 3.244 −0.06

2.716 2.733 −0.62

2.464 2.478 −0.59

2.352 2.360 −0.33

2.269 2.270 −0.05

Fig. 2. Verification of the critical hydrostatic pressure of elastic FGM rings under the plane-strain case.

Fig. 3. Typical buckling mode of elastic FGM rings simulated by ABAQUS.

To find this result, a numerical procedure is designed as following . In the prebuckling stage, the circumferential strain of the rings, ε θ θ or εθ∗ θ , is set to be an initial value ε 0 . Then, the position of the elastoplastic interface s can be solved by Eq. (23) with Eq. (30) substituted into it, and the secant modulus Es0 is solved by Eq. (24) or Eq. (29). It should be noted that the prebuckling internal force N in Eq. (31) should consist with the buckling critical load qcr R derived from Eq. (39). Then, properly adjusting the value of ε 0 would converge the results given by Eqs. (31) and (39), and the elastoplastic buckling critical hydrostatic pressure, as well as the position of the elastoplastic interface can thus be determined. 3.2. Verifications For the plane-stress elastic FGM rings, the critical hydrostatic pressure under the plane-stress case can be verified by the work of Kerdegarbakhsh [18] as listed in Table 1, in which the non-dimensional load parameter λ = 12(qcr /Ec )(R/h)3 is introduced and the material parameters of Si3 N4 /SUS304 FGMs are used. The elastic modulus of Si3 N4 and SUS304 are 322.271 and 207.788GPa respectively. Poisson’s ratio is assumed to be constant 0.28 and the dimensional parameter R/h = 17.86. For the plane-strain elastic FGM rings, the critical hydrostatic pressures have also been verified by ABAQUS results in Table 1 with the calculating parameters same as the aforementioned. In Fig. 2, we have k = 1. It also shows that, with R/h ranging from 10 to 200, the plane-strain theoretical results are in a good agreement with those of ABAQUS, in which, a 1/4 model of the FGM elastic ring was found, as shown in Fig. 3, and the buckling mode indicates a typical buckling one, i.e. n = 2.

62

H. Huang et al. / Applied Mathematical Modelling 45 (2017) 55–64 Table 2 Dimensional effects on buckling of elastoplastic FGM rings. R/h Plane-stress

8

10

15

20

s/h qcr (MPa)

0.150 62.819 0.702 2.562

−0.265 35.340 0.769 1.564

−0.500 13.639 1.0 0 0 0.877

−0.500 5.755 1.0 0 0 0.494

s/h qcr (MPa)

0.168 69.107 0.729 2.321

−0.260 38.616 0.793 1.403

−0.500 14.469 1.0 0 0 0.749

−0.500 6.106 1.0 0 0 0.423

λcr χ ( − h/2)

Plane-strain

λcr χ ( − h/2)

Fig. 4. Comparisons of the elastoplastic buckling of FGM rings from the elastic ones.

3.3. Elastoplastic buckling analyses In the following calculation, it is assumed that the rings are made of TiB/Ti FGMs. The material properties of the ceramic and metallic constituents are obtained from Jin et al. [24] and listed as following:

Ec = 375 GPa, Em = 107 GPa, σYm = 450 MPa, Hm = 14 GPa, q˜ = 4.5 GPa. and Poisson’s ratio of FGMs is assumed to be constant 0.24. In Table 2, we choose k = 1. The results under both the plane-stress and plane-strain cases show that, elastic buckling usually occurs when R/h ≥ 15, while elastoplastic buckling occurs when R/h ≤ 10. In elastoplastic buckling states, the ratio of the equivalent stress to the yield limit χ = σ i /σ Y on the ceramic surface is always larger than 1 due to the plastic deformation of the material. In Fig. 4, we have R/h = 10. Comparisons of the elastoplastic critical hydrostatic pressures with those of the elastic one are made. The critical hydrostatic pressures basically reduce with the increasing inhomogeneous parameter k, which leads to the general conclusion that the critical hydrostatic pressure decreases with the decreasing volume fraction of ceramic constituent. In both the plane-stress and the plane-strain cases, the elastoplastic buckling curves are generally lower than their elastic counterparts due to material flow. It is worth of noting that there are flat fields when 0.05 ≤ k ≤ 0.2 and k ≥ 5, which are quite different from the smoothly decreasing elastic buckling curves. Figs. 5 and 6 respectively show the influences of the inhomogeneous parameters on the critical hydrostatic loads and the position of the elastoplastic interface. As R/h ranges from 8 to 14, the elastoplastic critical hydrostatic pressures qcr reduce monotonously with the increasing k when R/h= 10, while in R/h= 8 there observes a slight rebound when k ≥ 10. The critical hydrostatic pressures qcr in the plane-strain case are generally larger than those in the plane-stress case. Meanwhile, from Fig. 5(a) and (c), similar flat fields of the curves are found in the cases of R/h = 8, 10, 12, which are in elastoplastic buckling state characterized by − 0.5 < s/h < 0.5 from Fig. 6. For the case of R/h = 14, through there is − 0.5 < s/h < −0.2 when 0 < k < 0.05 in the plane-stress case, it is really close to an elastic buckling stage in both the plane-stress case and the plane-strain case, since the corresponding non-dimensional load parameters λcr = qcr /qecr ≈ 1 from Fig. 5(d). Herein qecr is the degenerated elastic solution of FGM rings from Eq. (39). 4. Conclusions Buckling behaviors of elastoplastic FGM circular rings under external hydrostatic pressure are investigated in this paper. Several prime conclusions are drawn from the result analyses:

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Fig. 5. Effects of the inhomogeneous parameter on elastoplastic buckling critical load.

Fig. 6. The position of the elastoplastic interface.

(i) The elastoplastic buckling would occur under the radius-to-thickness ratio of the FGM circular rings less than 15, and a lager value would lead the structure to be an elastic buckling. (ii) The elastoplastic buckling critical hydrostatic pressure of FGM circular rings is generally lower than the elastic counterpart due to material flow, and the plane-strain critical pressure is generally larger than the plane-stress one. (iii) The buckling critical hydrostatic pressure of FGM circular rings decreases with the increase of the inhomogeneous parameter or the volume fraction of metallic constituents. For both the plane-stress case and the plane-strain case,

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the elastoplastic curves of the critical hydrostatic pressure versus the inhomogeneous parameter are similar and there observes flat fields from the elastoplastic buckling curves, which are quite different from those in the elastic counterparts. Acknowledgments This work is supported by National Natural Science Foundation of China (11402093), State Key Laboratory for Strength and Vibration of Mechanical Structures (SV2016-KF-08), and Science and Technology Program of Guangzhou (201607010282). Appendix

    ρ 9C22 + C2 9C3 + C1 σxx2 − 4σxx σθ θ + 4σθ2θ ,    2  = a12 = ρ C2 −9C3 + C1 2σxx − 5σxx σθ θ + 2σθ2θ ,  2  2 = a11 + 3ρC1C2 σxx − σθ θ , a13 = a31 = a23 = a32 = 0,    2  = 0.5ρ 9C22 + 9C1C3 (σxx − σθ θ )2 + C2 18C3 + C1 5σxx − 8σxx σθ θ + 5σθ2θ ,      −1 2 2 = C2−1 9C22 + 9C1C3 σxx − 2σxx σθ θ + σθ2θ + C2 18C3 + C1 5σxx − 8σxx σθ θ + 5σθ2θ ,     −1 −1 −1 −1 = (9/4 )σi−2 Et − Es , C2 = (3/2 )Es , C3 = (2/3 )(1 − 2ν )E −1 − Es 2,  2 +σ σ 2 = σxx xx θ θ + σθ θ .

a11 = a21 a22 a33

ρ C1

σi

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