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A reﬁned theory of axisymmetric thermoporoelastic circular cylinder Di Wu b, Lianying Yu a, Yuejiao Wang a, Baosheng Zhao c, Yang Gao a, * a

Department of Applied Mechanics, College of Science, China Agricultural University, Beijing, PR China Department of Vehicle Engineering, College of Engineering, China Agricultural University, Beijing, PR China c Department of Engineering Mechanics, School of Mechanical Engineering and Automation, University of Science and Technology Liaoning, Anshan, PR China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 31 May 2014 Accepted 1 May 2015 Available online 11 May 2015

The deformations of axisymmetric transversely isotropic thermoporoelastic circular cylinders in a steady-state are analyzed and the approximate coupled ﬁelds are obtained directly without employing ad hoc stress or deformation assumptions. The expressions of the coupled ﬁelds for these materials were obtained in the light of four functions with one independent variable z at the outset. Subsequently, based on the non-homogenous boundary conditions, the reﬁned equations and approximate solutions are obtained by omitting the higher-order terms. By analyzing some coupled and uncoupled cases, the structural effect coupling phenomenon of thermoporoelastic mediums is shown. Not taking into account ﬂuid-solid or poro-thermo coupling effect, the result reduces to the corresponding solution of the thermoelastic or elastic counterpart in a straightforward way. © 2015 Elsevier Masson SAS. All rights reserved.

Keywords: Thermoporoelasticity Lur'e operator method A reﬁned theory

1. Introduction Porous mediums are one of the fastest-growing materials in the modern society. They have been widely used in many ﬁelds with poro-mechanical multi-physics coupling characteristic, such as architectural engineering, foodstuff, chemical industry and energy problem etc. Because of the inherent structural properties and great applicabilities of these materials, their characteristics and the applications in engineering have appeal to more and more scholars to study. According to the representative elementary volume and statistic mean method, because of complex structure of the porous materials, the porous system is usually considered as a continuum which is uniform distributed from the macroscopic point of view. The research on mechanical property of porous materials has made adequate progress. Starting from the basic equations of continuous medium, Biot (1941) set up three-dimensional general Biot theory of consolidation which made a foundation for the future ﬂuid-solid coupling research. In recent years, the thermoelastic consolidation problem of the porous media considering temperature effect has aroused interest among scholars. Kodashima and Kurasbige (1997)

* Corresponding author. College of Science, China Agricultural University, P.O. Box 74, Beijing 100083, PR China. E-mail address: [email protected] (Y. Gao). http://dx.doi.org/10.1016/j.euromechsol.2015.05.001 0997-7538/© 2015 Elsevier Masson SAS. All rights reserved.

analyzed the distribution of thermal stresses of a ﬂuid-saturated ron and Dureisseix (2008) develporoelastic hollow sphere. Ne oped a computational measure which is propitious to the coupled multiphysics problems for the thermoporoelastic structures. Jabbari and Dehbani (2010) presented the analytical solution for the thermoporoelasticity of thick cylinders under the radial temperature loading. The three-dimensional steady-state general solutions for isotropic (Zhao and Lu, 2014) and transversely isotropic (Li et al., 2010) thermoporoelastic media were derived, and the completeness of the general solution was proved. Hou et al. (2013a) solved the three-dimensional Green's function with a concentrated liquid or heat source in steady state by using the general solutions for transversely isotropic thermoporoelastic media. Efforts have been made to derive the exact solutions of slender and thin bodies without ad hoc assumptions. Cheng (1979) presented a method for the solution of 3D elasticity equations, and established a reﬁned theory of plates by using the Lur'e method (Lur'e, 1964). Thereafter, the reﬁned theory is extended to the transversely isotropic plates (Wang, 1990, 1991), beam (Gao et al., 2007; Gao and Shang, 2010; Lu et al., 2013) and circular cylinder (Zhao et al., 2011; Zhao and Wu, 2012). The true importance of using the reﬁned theory is in the fact that the high-dimensional theory can be transformed into low-dimensional theory to study by making use of elastic general solution and symbolic operation, and the problem can be simpliﬁed further. Moreover, the axisymmetric circular cylinders are one of the most common components in

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engineering practice, it is necessary to research on its basic theories. On account of the complexity in material properties and structure of porous medium, for instance, ﬂuid-solid coupling effects and anisotropy, there will be some difﬁculties in the utilization to linear elastic theory for thermoporoelasticity by traditional cylinder theories, so some assumptions about the deformation or the stress state are made for solving these problems, hence the results obtained from traditional cylinder theories have a certain approximation. However, making use of the reﬁned theory, the derivation process without demand of any prior assumptions, results ground on it are of high accuracy. So it can increase the likelihood of resolving complex problems, predigest the computation and enhance the calculational efﬁciency. The reﬁned theory can be as a basis for more general applications. For instance, the reﬁned theory of thermoporoelasticity can be degenerated into other well-known thermoelastic or elastic theories, various boundary value problems are solved (by considering the different boundary conditions, the different types of the solutions can be obtained), and the bending problem of an infinitely long thermoporoelastic circular cylinder with hole or crack can be researched further. Moreover, the results can be applied to practical engineering problems as well. The investigation into the reﬁned theory is extended to the study of steady-state thermoporoelastic circular cylinder. It is noted that by using the uncoupled theory, the thermoporoelastic problem to consider herein is that the porous and thermal problems can affect elasticity, but elasticity has no effect upon the porous and thermal problems as well. Moreover, the porous and thermal problems are independent of each other. A special case of nonhomogeneous boundary conditions for the circular cylinder under surface loadings, temperature loadings and pore pressure is taken into account. A reﬁned theory of thermoporoelastic circular cylinder is derived systematically from the linear thermoporoelastic theory without ad hoc assumptions, and the approximate solutions are obtained ﬁrsthand. Lastly, some illustrative examples are presented to demonstrate the application of the reﬁned theory.

Fig. 1. The axisymmetric thermoporoelastic circular cylinder.

where sjn are stress components; P and T are changes in the pore pressure and temperature, respectively. Cdf (d,f ¼ 1,2,3,4,6) are elastic constants with C66 ¼ (C11eC12)/2; a1 and a3 are Biot's effective stress coefﬁcients; b1 and b3 are thermal modules. Fluid ﬂow in porous medium is decided by Darcy law, and heat conduction obeys Fourier law. That is (Tchonkova et al., 2008; Cowin, 2004),

2. The fundamental equations and general solutions In a cylindrical coordinates system(r,q,z), a homogeneous circular cylinder occupies the region as showed in Fig. 1.

Y

¼ fðr; q; zÞjjzj l; 0 r ag

vur ; vr

εqq ¼

ur ; r

εzz ¼

vuz ; vz

qr qz

(1)

where l and a are the circular cylinder length and radius, respectively, and z-axis is the longitudinal axis which is perpendicular to the isotropic plane (r, q) of the medium. In the cylindrical coordinate system, the expressions of the strain tensor components of axisymmetric problem are

εrr ¼

εrz ¼

1 vur vuz þ ; 2 vz vr

(2)

εrq ¼ εzq ¼ 0 where u2 (2 ¼ r, z) and 3 jn (j,n ¼ r,q,z) are displacement components and strain components, respectively. Constitutive equations of a ﬂuid-saturated axisymmetric transversely isotropic thermoporoelastic circular cylinder in the respective cylindrical coordinate are (Kanj and Abousleiman, 2005):

(3)

8 9 8 9 vP > vT > > > > > = = < > < > vr vr k11 0 l11 0 hr ¼ ¼ ; 0 k33 > 0 l33 > > hz > > > ; ; : vP > : vT > vz vz

(4)

where qr and qz describe the motion of the ﬂuid volume, hr and hz are the thermal ﬂux, k11 and k33 are coefﬁcients of ﬂuid permeability, l11 and l33 are coefﬁcients of thermal conductivity. The equilibrium equations without body forces can be written as

vsrr vszr srr sqq þ þ ¼ 0; vr vz r

vszr vszz szr þ þ ¼0 vr vz r

(5)

By using uncoupled thermoporoelastic theory and assuming that the thermoporoelastic loading changes slowly over time, the rates of ﬂuid mass content and entropy vanishes simultaneously. Consequently, the pore pressure and temperature ﬁeld are timeindependent in a steady-state, and satisfy the following two equations (Hou et al., 2013a)

k11 srr ¼ C11 εrr þ C12 εqq þ C13 εzz a1 P b1 T sqq ¼ C12 εrr þ C11 εqq þ C13 εzz a1 P b1 T szz ¼ C13 εrr þ C13 εqq þ C33 εzz a3 P b3 T szr ¼ 2C44 εrz

! ! v2 1 v v2 P v2 1 v v2 T P þ k33 2 ¼ 0; l11 T þ l33 2 ¼ 0 þ þ 2 2 r vr r vr vz vz vr vr (6)

The axisymmetric general solutions for thermoporoelastic materials are given by Li et al. (2010).

D. Wu et al. / European Journal of Mechanics A/Solids 53 (2015) 187e195

ur ¼

4 X vji i¼1

vr

;

uz ¼

4 X

mi1

i¼1

vji ; si vz

P ¼ m32

v2 j3 ; s23 vz2

T ¼ m43

v2 j4 s24 vz2 (7)

where mi1, m32 and m43 are shown in the Appendix. ji (i ¼ 1, 2, 3, 4) are the harmonic functions which satisfy the following equation: 2

2

v 1 v 1 v þ þ ji ¼ 0 vr 2 r vr s2i vz2

A0 s4 B0 s2 þ D0 ¼ 0

(9)

where A0, B0 and D0 are shown in the Appendix. The Lur'e method (Lur'e, 1964) is rolled out to cylindrical coordinate, the harmonic functions ji have the following form:

ji ¼ J0 ðrvz =si Þgi ðzÞ

(10)

where vz ¼ v/vz, and gi(z) are unknown functions of z. Andrews et al. (1999) have given the expression of Bessel's function and have proved the convergence of its series. J0 and J1 denote Bessel's function of order 0 and order 1, respectively

r 2 v2 r 4 v4z J0 ðrvz =si Þ ¼ 1 2z þ þ /; 4si 64s4i

i¼1

gi0 ; si

uz ¼

4 X mi1 i¼1

si

d0i gi0 ;

(11)

P¼

m32 0 00 d g ; s23 3 3

T¼

m43 0 00 d g s24 4 4

where the differential symbol ‘‘ ' ’’ represents differentiation with respect to z, and d0i ¼ J0 ðrvz =si Þ; d1i ¼ J1 ðrvz =si Þ. Substituting equation (12) into equations (2) and (3), the stress states can be obtained: 4 X

2C66 d1i

i¼1

sqq

gi0 00 þ ui d0i gi ; rsi

" 4 X g0 ¼ 2C66 d1i i rsi i¼1

szz ¼

4 X ui i¼1

s2i

00

d0i gi ;

ui þ

szr ¼

2C66 s2i

4 X ui i¼1

si

2

L11 6 L21 6 4 0 0

L12 L22 0 0

L13 L23 L33 0

(15)

9 38 00 9 8 > g1 > L14 > qr ðzÞ > > > > > < 00 > = > < = 7 L24 7 g2 qz ðzÞ ¼ 00 0 5> <ðzÞ > > > > > > g3 > > > : ; L44 : g 00 ; wðzÞ 4

(16)

where 1 0 2C u 1 m b0 m b0 L1i ¼ 66 b d ui b d i ; L2i ¼ i b d ; L33 ¼ 32 d ; L44 ¼ 43 d4 2 3 asi vz i si i s3 s24 0 1 b d i ¼ J1 ðavz =si Þ d i ¼ J0 ðavz =si Þ; b

(17)

00

L33 g3 ¼ <ðzÞ;

!

#

00 d0i gi

;

( ) 00 L33 L44 g1 L11 L12 e1 ; U¼ L¼ ; E¼ 00 L21 L22 e2 L33 L44 g2 e1 ¼ L33 ½L44 qr ðzÞ L14 wðzÞ L13 L44 <ðzÞ; e2 ¼ L33 ½L44 qz ðzÞ L24 wðzÞ L23 L44 <ðzÞ

Taking the adjoint matrix of operator matrix L on both sides of the ﬁrst equation of equation (18), we further have:

m32 m43 b0 b0 00 L e L12 e2 d 3 d 4 אg1 ¼ 22 1 ; avz s23 s24

m32 m43 b0 b0 00 d 3 d 4 אg2 s23 s24

L21 e1 þ L11 e2 avz

2C ðu u Þ 1 b1 u1 u2 ¼ א66 2 2 1 b d1 d2 þ avz s1 s2 a2 vz

mi1 1 si

(19)

(20)

where

00

¼ C13 þ C33 mi1 si a1 mi2 b1 mi3 ¼ C44

(18)

(13)

d1i gi

C11 C13 mi1 si þ a1 mi2 þ b1 mi3 s2i

00

L44 g4 ¼ wðzÞ

where

¼

where

ui ¼

Tðr; zÞjr¼a ¼ wðzÞ

Substituting the expressions of stresses, pore pressure and temperature into equation (15), we get the following matrix equation:

LU ¼ E;

(12)

srr ¼

Pðr; zÞjr¼a

J1 ðrvz =si Þ

Substituting equation (10) into equation (7), the displacement ﬁelds, pore pressure and temperature can be obtained:

d1i

¼ <ðzÞ;

szr ðr; zÞjr¼a ¼ qz ðzÞ;

Equation (16) can be rewritten as follows

rvz r 3 v3z r 5 v5z þ þ/ ¼ 2si 16s3i 384s5i

4 X

For non-homogeneous boundary conditions, a special case of boundary conditions under surface loadings, pore pressure and temperature loading is discussed,

(8)

where the repeated lower are not summed in this paper without a pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ special annotation, s3 ¼ k11 =k33 , s4 ¼ l11 =l33 , s1 and s2 are two characteristic roots with positive real part (assumed to be distinct herein) of the following equation

ur ¼

3. Non-homogeneous boundary conditions

srr ðr; zÞjr¼a ¼ qr ðzÞ;

!

189

(14)

0 1 0 1 b d1b d2 b d b d 2 1 s2 s1

! (21)

The second and third equations of equation (18) and equation (20) are the reﬁned equations of the axisymmetric thermoporoelastic circular cylinder. These equations are of inﬁnite order which are not applicable in most cases. By using Taylor series of the Bessel's functions in equation (11) and then dropping all the terms associated with four order or the higher-order terms, we can obtain 00 00 00 00 the approximate expressions for g1, g2 , g3 and g4 as:

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D. Wu et al. / European Journal of Mechanics A/Solids 53 (2015) 187e195

s2 g1 ¼ 1 M0

("

00

s2 g2 ¼ 2 M0

u2 q r

("

00

00

g3 ¼

s23 m32

u1 qr 1þ

a2 v2z 4s23

2 C66 þ u2 s22 avz

2 C66 þ u1 s21 avz ! <;

00

g4 ¼

" # # ! ) M1 M2 a2 v2z 1 2u2 M1 M2 2 2 qz < w 1þ qz < w a vz þ 8 avz m32 m43 8s21 m32 s23 m43 s24 M3 M < 4w m32 m43 ! a2 v2z 1þ w 4s24

qz

s24 m43

# 1þ

a2 v2z

!

8s22

þ

" # ) 1 2u1 M3 M4 2 2 qz < w a v z 8 avz m32 s23 m43 s24

(22)

where M0, M1, M2, M3 and M4 are shown in the Appendix. Substitution of equation (22) into equations (12) and (13) leads to

(

2 u2 s22 u1 s21

2 a r 2 v2z u2 s22 u1 s21 M1 M3 þ M0 M M4 þ M0 ½ < 2 wþ qr avz m32 m43 8 s21 s22 ! ! #

2 1 M1 M3 M0 1 M2 M4 M0 2 2 4 4 C66 s2 C66 s1 þ u2 s2 u1 s1 qz 2 2 2 þ 2 < 2 þ 2 w m32 s21 m43 s21 s1 s2 avz s2 s3 s2 s4 " #) a2 v2z 2ðu2 u1 Þ M M3 þ M0 M M4 þ M0 qz 1 < 2 w þ avz 8 m32 s23 m43 s24 ( 2 a 2r 2 v2z k1 1 2k k k 2k2 1 M1 m11 M3 m21 M0 m31 < qr qz þ k1 qr 2 qz 3 < 4 w þ uz ¼ M0 vz m32 avz m32 m43 8 s1 s2 s1 s2 avz s1 s2 s3 #) " 1 M2 m11 M4 m21 M0 m41 a2 v2z 2k1 k3 k4 w þ þ qz < w m43 s1 s2 s4 8 avz m32 s23 m43 s24 r ur ¼ 2M0

P¼

ðu2 u1 Þqr

! a2 v2z r 2 v2z 1þ < 4s23

qz

(24)

T¼

1þ

! a2 v2z r 2 v2z w 4s24

(23)

(25)

where k1, k2, k3 and k4 are shown in the Appendix.

"

( M0 a2 r 2 C66 a2 r 2 v2z u s2 u s2 2 C66 s22 C66 s21 þ u2 s42 u1 s41 1 2 2 1 1 vz qz þ srr ¼ qr qz M0 qr þ M0 2a 8 s21 s22 s21 s22 avz ! ! #) 1 M1 M3 M0 1 M2 M4 M0 2 þ 2 < 2 þ 2 w m32 s21 m43 s21 s2 s3 s2 s4

"

( M0 a2 r 2 vz C66 3r 2 a2 v2z u s2 u s2 2 C66 s22 C66 s21 þ u2 s42 u1 s41 1 2 2 1 1 qz qr qz sqq ¼ M0 qr þ M0 2a 8 s21 s22 s21 s22 avz ! ! #) 1 M1 M3 M0 1 M2 M4 M0 þ þ < w m32 s21 m43 s21 s22 s23 s22 s24

( 2 " 2 2 2 C66 u1 s22 C66 u2 s21 þ u1 u2 s42 u1 u2 s41 a 2r 2 v2z u1 u2 s2 s1 1 2M0 szz ¼ qz þ qr qz M0 avz 8 s21 s22 s21 s22 avz ! ! #) 1 M1 u1 M3 u2 M0 u3 1 M2 u1 M4 u2 M0 u4 þ þ < w m32 m43 s21 s22 s23 s21 s22 s24

( 2 " 2 2 2 C66 u1 s22 C66 u2 s21 þ u1 u2 s42 u1 u2 s41 a r 2 v2z u1 u2 s2 s1 rvz 2M0 szr ¼ qz þ qr qz 2M0 avz 8 s21 s22 s21 s22 avz ! ! #) 1 M1 u1 M3 u2 M0 u3 1 M2 u1 M4 u2 M0 u4 þ þ < w m32 m43 s21 s22 s23 s21 s22 s24

(26)

D. Wu et al. / European Journal of Mechanics A/Solids 53 (2015) 187e195

191

Equations (23)e(26) are the approximate solutions of the axisymmetric deformation of the thermoporoelastic circular cylinder under surface loadings, pore pressure and temperature loading. It is easy to obtain the coupling ﬁelds by using the approximate solutions, once the appropriate function expressions of qr(z), qz(z), <(z) and w(z) are given. Moreover, the displacement distribution, stress states, pore pressure and temperature distribution of the thermoporoelastic circular cylinder can be shown obviously. 4. Numerical results Cdf, a1, a3, b1 and b3 can be expressed by the engineering constants as follows (Kanj and Abousleiman, 2005; Li et al., 2010; Hou et al., 2013b)

E E0 Ev02 E E0 v þ Ev02

; C12 ¼

C11 ¼ ð1 þ vÞ E0 E0 v 2Ev02 ð1 þ vÞ E0 E0 v 2Ev02

C13 ¼

EE0 v0 0

0

02

E E v 2Ev

C66 ¼ G ¼ a3 ¼ 1

; C33 ¼

E02 ð1 vÞ 0

0

02

E E v 2Ev

; C44 ¼ G0 ¼

E0 2ð1 þ v0 Þ

E C þ C12 þ C13 ; a1 ¼ 1 11 ; 2ð1 þ vÞ 3Ks

b r along the radial direction br . Fig. 3. The distribution curve of displacement u (bz ¼ 20).

radius a ¼ 10 mm. One end of the circular cylinder is ﬁxed, and the other end is free. The boundary conditions are

srr ¼ qr ðzÞ; szr ¼ qz ðzÞ; P ¼ <ðzÞ; T ¼ wðzÞ uz ¼ 0 at z ¼ l and r ¼ 0 szz ¼ 0 at z ¼ l

at r ¼ a (29)

For ease of numerical calculation and making the results got more universality and comprehensive, the following dimensionless quantities are presented

2C13 þ C33 ; b1 ¼ ðC11 þ C12 Þas1 þ C13 as3 ; 3Ks

b3 ¼ 2C13 as1 þ C33 as3 (27) where E and v are the drained Young's modulus and Poisson's ratio in the isotropic plane (rq); E0 , G0 and v0 are the drained Young's modulus, shear modulus and Poisson's ratio that perpendicular to rq plane, respectively; Ks is the bulk modulus of solid skeleton; as1 and as3 are the linear expansion coefﬁcients in the isotropic and transverse planes of the material, respectively. The material constants of thermoporoelastic medium are taken as follows (Li et al., 2010)

E ¼ 1:854 109 Pa; E0 ¼ 9:27 108 Pa; v ¼ 0:22; v0 ¼ 0:44 Ks ¼ 1:218 1012 Pa; k11 ¼ 0:1D=ðPa$sÞ; k33 ¼ 0:2D=ðPa$sÞ l11 ¼ 2:65W=ð C$mÞ; l33 ¼ 4W=ð C$mÞ as1 ¼ 6 106 1= C; as3 ¼ 1:2 105 1= C

b r ¼ r=a; b z ¼ z=a; bl ¼ l=a;

s b b p ¼ bp a1 C33 ðp ¼ 1; 3Þ;

b 2 ¼ u2 =a; b k pp ¼ kpp k33 ; u

. b ¼C C df df C33 ;

b l pp ¼ lpp l33

b ¼ P=C ; T b ¼ as T b s jn ¼ sjn C33 ; P 33 1 (30)

where T0 is reference temperature. The axisymmetric nonuniform loadings on the cylindrical surface are

qr ðzÞ ¼ qz ðzÞ ¼ <ðzÞ ¼ q0 sin ¼ w0 sin

p b lb z bl

p b lb z ; bl

wðzÞ (31)

where q0 ¼ 106Pa, w0 ¼ 1 C. Through analyzing the following coupled and uncoupled cases, the structural effect coupling phenomenon of thermoporoelastic materials are found.

(28) An axisymmetric circular cylinder is subject to distributed nonuniform surface loadings, pore pressure and temperature loading. Considering the circular cylinder length l ¼ 400 mm and

b r along the axial direction bz . Fig. 2. The distribution curve of displacement u (qz(z) ¼ <(z) ¼ w(z) ¼ 0).

Case A (A0 ): the cylinder is only subject to the nonuniform radial (axial) direction surface loading; Case B (B0 ): the cylinder is subject to the nonuniform radial (axial) direction surface loading and temperature loading;

Fig. 4. The distribution curve of stress b s rr along the radial direction br . (bz ¼ 20).

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D. Wu et al. / European Journal of Mechanics A/Solids 53 (2015) 187e195

Fig. 5. The distribution curve of stress b s zr along the radial direction br . (bz ¼ 0).

Case C (C0 ): the cylinder is subject to the nonuniform radial (axial) direction surface loading and pore pressure; Case D (D0 ): the cylinder is subject to the nonuniform radial (axial) direction surface loading, temperature loading and pore pressure; Case E: the cylinder is only subject to the nonuniform pore pressure; Case F: the cylinder is only subject to the nonuniform temperature loading; The displacement ﬁeld, stress states, pore pressure and temperature ﬁeld for the thermoporoelastic circular cylinder have been shown in Fig. 2e8. The changing rules of the displacements and stresses along the radial direction b r when the circular cylinder is subject to surface loadings, pore pressure or temperature loading are revealed. Fig. 2 illustrates the distribution of the dimensionless radial b r of the circular cylinder that is only subject to the displacement u nonuniform radial direction surface loading along the axial direction b z with different dimensionless radius b r , which shows the inhomogeneity of the boundary. The greater the dimensionless radius b r is, the bigger the peak value is. br Fig. 3 shows that the dimensionless radial displacement u (absolute value) is largest on cylindrical surface. Compared with the b r increases markcase A, the dimensionless radial displacement u edly when the temperature effect is also taken into account, and the directions of radial displacement in both cases are the same; however, the direction of radial displacement in case C is just opposite to that of the case A. As seen in Fig. 4, the largest dimensionless radial stress b s rr is on cylinder in case B, and on the neutral axis (b r ¼0) in situation C, the change of b s rr in case A is not obvious with the increase of the dimensionless radius b r . Moreover, the stress b s rr in three cases is all compression stress. Fig. 5 shows

Fig. 6. The distribution curve of stress b s zz along the radial direction br . (bz ¼ 20).

b along the radial direction br . (bz ¼ 20). Fig. 7. The distribution curve of pore pressure P

that the maximum value of the stress b s zr is near the radius b r ¼ 0.58. The stress in case C is the compression stress, but the stresses in case A and B are the tensile stress. As seen in Fig. 6, the stress b s zz is to be the minimum near the radius b r ¼ 0.7, and increases with the distance from b r ¼ 0.7 to cylindrical surface or the neutral axis along the radius. In case B0 , when b r <0.7, the stress b s zz is the tensile stress; when b r >0.7, the stress b s zz is the compression stress, which is contrary to that of the case C'. In conclusion, the directions of the displacements and stresses in case A (A0 ) are the same with those in case B (B0 ). However, there are some changes of the directions of the displacements and stresses in case C (C0 ) compared with case A (A0 ). The present analytical solutions in Case E and Case F are compared with the numerical solutions obtained by making use of Abaqus software in Figs. 7 and 8, respectively. Moreover, the distribution rules of the pore pressure and temperature ﬁeld are consistent because of the identical mathematical structures. These solutions are highly similar which shows the validity of the result obtained from the reﬁned theory (see Table 1). 4 nod linear axisymmetric elements (DCAX4) are used for the heat transfer analyses at ﬁrst. For the ﬁnite element model, the distances between the seeds in the radial and length direction are 0.002 m and 0.01 m, respectively. When the temperature ﬁelds obtained are regarded as the known conditions, the thermal stress analysis is carried out. The stress states are obtained eventually. The comparisons of stress states (b r ¼ 0:2) with numerical results in Case F are also shown in Fig. 9 and listed in Table 2. The dimensionless stresses b s zr with different order truncation in Case A0 at b z ¼ 20 are showed (see Table 3), which can be seen that the second order for calculations is enough to lead the

Fig. 8. The distribution curve of temperature Tb along the radial direction br . (bz ¼ 20).

D. Wu et al. / European Journal of Mechanics A/Solids 53 (2015) 187e195

193

Table 1 b and temperature T b with numerical results. Comparison of P b r b 106 P

b 106 T

Numerical results The reﬁned theory Relative error % Numerical results The reﬁned theory Relative error %

0

0.2

0.4

0.6

0.8

1.0

7.72045 7.72098 0.00686 5.98572 5.98603 0.00518

7.72172 7.72193 0.00272 5.98646 5.98659 0.00217

7.72469 7.72480 0.00142 5.98820 5.98827 0.00117

7.72951 7.72957 0.00078 5.99102 5.99106 0.00067

7.73623 7.73626 0.00039 5.99495 5.99497 0.00033

7.74486 7.74486 0.00000 6.00000 6.00000 0.00000

Fig. 9. The distribution curve of stress state b s rr along the axial direction bz .

solutions obtained to convergence. For the present problem, the increasing order selected can bring better accuracy indeed with higher amount of computation, but relatively makes little difference to the accuracy of the results. Fig. 10 illustrates the distribution of the dimensionless b r along the radial direction with different effective displacement u stress coefﬁcients a1 in Case E. It can be found that the dimenb r decreases with the increasing of stress sionless displacement u coefﬁcients a1, the change of the effective stress coefﬁcient a1 is with extraordinary inﬂuence on the elastic ﬁelds.

b r along the radial direction br . Fig. 10. The distribution curve of displacement u (bz ¼ 20).

ml1 ¼

n o sl ½ðC13 þ C44 Þb1 C11 b3 þ C44 b3 s2l C44 b1 þ ½ðC13 þ C44 Þb3 C33 b1 s2l

ðl ¼ 1; 2; 4Þ;

m31 ¼ m32 ¼ 0 m43 ¼

2 C11 C44 þ C13 þ 2C13 C44 C11 C33 s24 þ C33 C44 s44

5. The degenerated form

C44 b1 þ ½ðC13 þ C44 Þb3 C33 b1 s24 (32)

Some examples are taken into account for describing the application of the reﬁned theory herein. The case that a thermoporoelastic problem is degenerated to thermoelasticity will be considered ﬁrstly. Without considering the ﬂuid-solid coupling effect, let a1 ¼ a3 ¼ 0. So mi1, m32 and m43 in the Appendix are reexpressed as follows:

Supposing that the boundary conditions (15) are transformed into:

srr ðr; zÞjr¼a ¼ qr ðzÞ; ¼ 0;

szr ðr; zÞjr¼a ¼ 0;

Pðr; zÞjr¼a

Tðr; zÞjr¼a ¼ wðzÞ

(33)

Table 2 Comparison of the dimensionless stresses b s rr with numerical results. b z b s rr 108

Numerical results The reﬁned theory Relative error %

Note: Relative error is deﬁned as

5

10

15

20

25

3.82198 3.82460 0.06855

7.06285 7.06702 0.05904

9.22799 9.23339 0.05852

9.98858 - 9.99414 0.05566

9.22609 9.23339 0.07912

jThe refined theroyNumerical resultsj Numerical results

100%.

Table 3 The dimensionless stresses b s zr with different order truncation. b r b s zr

106

First order Second order Third order

0

0.2

0.4

0.6

0.8

1.0

0 0 0

1.54897 1.54754 1.54754

3.09794 3.09544 3.09544

4.64692 4.64406 4.64406

6.19589 6.19374 6.19374

7.74486 7.74486 7.74486

194

D. Wu et al. / European Journal of Mechanics A/Solids 53 (2015) 187e195

The radial displacement, radial stress, porous pressure and temperature can rewritten as

( ! r M2 M4 þ M0 a2 v2z ur ¼ ðu2 u1 Þqr 1þ w 2M0 m43 8s24 " ! #) 2 a r 2 v2z u2 s22 u1 s21 1 M2 M4 M0 qr 2 þ 2 w þ m43 s21 8 s21 s22 s2 s4 (34) ! a2 v2z r 2 v2z 1þ w 4s24

(35)

" C66 a2 r 2 v2z u2 s22 u1 s21 1 qr srr ¼ qr þ m43 8M0 s21 s22 ! # M2 M4 M0 2 þ 2 w s21 s2 s4

(36)

P ¼ 0;

T¼

Equations (34)e(36) are in the same form with the available results (Wu and Zhao, 2013). Next, considering that an axisymmetric thermoporoelastic circular cylinder is subject to distributed uniform surface loading in the radial direction. The both ends of the circular cylinder are free. The boundary conditions are

srr ¼ s0 ; szr ¼ 0 at r ¼ a Za rszz dr ¼ 0; szr ¼ 0 at z ¼ ±l 0

T ¼ 0;

P ¼ 0 in

6. Conclusions By using the general solution and the Lur'e symbolic method, the reﬁned theory of the transversely isotropic axisymmetric thermoporoelastic circular cylinder in a steady-state is derived directly without employing ad hoc assumptions concerning the deformation or the stress state. For the circular cylinder on nonhomogeneous boundary conditions, the approximate solutions are obtained which are of high accuracy. Moreover, the coupling ﬁeld distributions and the structural effect coupling phenomenon of thermoporoelastic materials are shown. For illustrating the application of the reﬁned theory, some examples are presented. The present solutions are compared with the numerical solutions. The problem of inﬂuence on the elastic ﬁelds with the changes of the effective stress coefﬁcient is studied. Moreover, the solutions got here are degenerated into the thermoelastic or elastic problem neglecting ﬂuid-solid or poro-thermo coupling effect. Results show that the theory developed in this paper is reliable, which offers a new approach to study the thermoporoelastic coupled problem and lays the foundations for its further applications. Acknowledgments

(37)

Q

where s0 is a constant stress. According to equations (23)e(26), one obtains

ðu u1 Þr k ur ¼ 2 s0 ; uz ¼ 1 s0 2M0 M0 vz

the results gained by Zhao and Wu (2012) which shows that the analysis in the reﬁned theory of this paper is credible again. Moreover, the reﬁned theory of the thermoporoelastic circular cylinder can be degenerated into other distinguished theories of elasticity, which can also be applied to the wider area based upon.

The authors are very grateful to the anonymous reviewers for their helpful suggestions. The work is supported by the National Natural Science Foundation of China (Nos. 11472299 and 11172319), Chinese Universities Scientiﬁc Fund (Nos. 2015QC050 and 2015LX001) and Program for New Century Excellent Talents in University (No. NCET-13-0552). Appendix

(38)

T ¼ 0; P ¼ 0; srr ¼ sqq ¼ s0 ; szz ¼ szr ¼ 0 It is easy to prove that equation (38) satisﬁes the boundary conditions in equation (37). A transversely isotropic thermoporoelastic circular cylinder will be reduced to a transversely isotropic elastic circular cylinder for investigating the validity of the results obtained. Under the circumstances, it really shows that a1, a3, b1 and b3 are zero. According to the equation (9),

s21 þ s22 ¼

2 2C C C11 C33 C13 C 13 44 ; s21 s22 ¼ 11 C33 C44 C33

(39)

Based on equation (14) and equation (38), the displacement ﬁeld and stress state are

C33 r

s0 ; ur ¼ 2 2 C66 C33 C11 C33 þ C13 uz ¼

C13 s0 2 v z C66 C33 C11 C33 þ C13

mi1 ¼

si A2 s4i B2 s2i þ D2 A1 s4i B1 s2i þ D1

;

m12 ¼ m22 ¼ m42 ¼ m13 ¼ m23 ¼ m33 ¼ 0; m32 ¼

m43 ¼

A0 s43 B0 s23 þ D0 l33 s23 l11 A1 s43 B1 s23 þ D1 A0 s44 B0 s24 þ D0 k33 s24 k11 A1 s44 B1 s24 þ D1

;

;

A0 ¼ C33 C44 ; 2 2C13 C44 ; B0 ¼ C11 C33 C13

(40)

T ¼ 0; P ¼ 0; srr ¼ sqq ¼ s0 ; szz ¼ szr ¼ 0 Not taking into account of poro-thermo coupling effect, the displacement ﬁeld and stress state for transversely isotropic elastic circular cylinder are obtained in a straightforward way from the above-mentioned equations. The equation (40) is consistent with

D0 ¼ C11 C44 ; A1 ¼ ðC13 þ C44 Þðk33 b3 þ l33 a3 Þ C33 ðk33 b1 þ l33 a1 Þ B1 ¼ ðC13 þ C44 Þðk11 b3 þ l11 a3 Þ C44 ðk33 b1 þ l33 a1 Þ C33 ðk11 b1 þ l11 a1 Þ

D. Wu et al. / European Journal of Mechanics A/Solids 53 (2015) 187e195

D1 ¼ C44 ðk11 b1 þ l11 a1 Þ; A2 ¼ C44 ðk33 b3 þ l33 a3 Þ B2 ¼ C11 ðk33 b3 þ l33 a3 Þ þ C44 ðk11 b3 þ l11 a3 Þ ðC13 þ C44 Þ ðk33 b1 þ l33 a1 Þ D2 ¼ C11 ðk11 b3 þ l11 a3 Þ ðC13 þ C44 Þðk11 b1 þ l11 a1 Þ

M0 ¼ C44 ðu2 u1 Þ u1 u2 s22 s21 ;

M1 ¼ C44 ðu3 u2 Þ u2 u3 s23 s22

M2 ¼ C44 ðu4 u2 Þ u2 u4 s24 s22 ;

M3 ¼ C44 ðu3 u1 Þ u1 u3 s23 s21

M4 ¼ C44 ðu4 u1 Þ u1 u4 s24 s21 k1 ¼ m11 s1 u2 m21 s2 u1 ; k2 ¼ C66 ðm11 s1 m21 s2 Þ þ s1 s2 ðm11 s2 u2 m21 s1 u1 Þ; k3 ¼ m11 s1 M1 m21 s2 M3 þ m31 s3 M0 ; k4 ¼ m11 s1 M2 m21 s2 M4 þ m41 s4 M0

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