Magneto-thermo-elastic stresses induced by a transient magnetic field in a conducting solid circular cylinder

International Journal of Solids and Structures 44 (2007) 5316–5335 www.elsevier.com/locate/ijsolstr

Magneto-thermo-elastic stresses induced by a transient magnetic field in a conducting solid circular cylinder M. Higuchi *, R. Kawamura, Y. Tanigawa Department of Mechanical Systems Engineering, Osaka Prefecture University, Gakuencho 1-1, Nakaku, Sakai, Osaka 599-8531, Japan Received 1 August 2006; received in revised form 8 December 2006; accepted 3 January 2007 Available online 10 January 2007

Abstract In the present paper, dynamic and quasi-static behaviors of magneto-thermo-elastic stresses induced by a transient magnetic field in a conducting solid circular cylinder are investigated. It is assumed that a transient magnetic field which is defined by an arbitrary function of time acts on the surface of the solid cylinder in the direction parallel to its surface. Fundamental equations of plane axisymmetrical electromagnetic, temperature, and elastic fields are formulated. Then, solutions of magnetic field, eddy current, temperature change and both dynamic solutions and quasi-static ones of stresses and deformations are analytically derived in the forms including the arbitrary function. The solutions of stresses are determined to be sums of thermal stress caused by eddy current loss and magnetic stress caused by Lorentz force. For this case that the arbitrary function is given by the smoothed ramp function with sine function, the dynamic and quasi-static behaviors of the stresses are examined by numerical calculations.  2007 Elsevier Ltd. All rights reserved. Keywords: Magneto-thermo-elasticity; Eddy current loss; Lorentz force

1. Introduction Mechanical components or structural elements which are activated in magnetic field have been increasing in recent years with the rapid progress of electromechanics. When a time-varying magnetic field acts on a conducting medium, the eddy currents are induced in the medium by electromagnetic induction, and the electric currents cause heat generation called the eddy current loss due to the Joule effect. The conducting medium is subjected to Lorentz force as well as the heat supply of eddy current loss. Thus, two kinds of stress should be generated by time-varying magnetic field: one is thermal stress caused by eddy current loss and the other is magnetic stress caused by Lorentz force. In the field of magneto-elasticity or magneto-thermo-elasticity, many studies have been conducted on an analytical treatment of an interaction between elastic, electromagnetic, and temperature fields (e.g., Kaliski and Nowacki, 1962; Kaliski and Michalec, 1963; Paria, 1967; Eringen and Maugin, 1990; Wauer, 1996 Wang *

Corresponding author. Tel.: +81 72 254 9208; fax: +81 72 254 9904. E-mail address: [email protected] (M. Higuchi).

0020-7683/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2007.01.001

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et al., 2002; Wang and Dai, 2004; Ezzat and Youssef, 2005; Wang and Dong, 2006). However, few studies have been carried out on analytical development of thermal stresses induced by time-dependent magnetic field (Moon and Chattopadhyay, 1974; Chian and Moon, 1981). Moon and Chattopadhyay (1974) have studied thermal stresses and magnetic stresses in a conducting half-space caused by an applied jump in tangential magnetic field at the boundary. Chian and Moon (1981) have extended the above work, investigating those stresses in a cylindrical conductor with a cavity caused by a pulsed magnetic field at the cavity. Pantelyat and Fe´liachi (2002) have studied mechanical behavior of metals in induction heating devices by using of finite element method. They have calculated thermo-elastic-plastic stresses induced by an alternating magnetic field, taking into account temperature dependences of material properties. In the present paper, dynamic and quasi-static behaviors of magneto-thermo-elastic stresses induced by a transient magnetic field have been investigated in a conducting solid circular cylinder which consists of nonferromagnetic metals such as copper or aluminum. Assuming that a time-dependent magnetic field which is defined as an arbitrary function of time acts on the surface of the solid cylinder in the direction parallel to its surface, we have formulated fundamental equations of plane axisymmetrical electromagnetic, temperature and elastic fields. Then, solutions of electromagnetic field, temperature change and both dynamic and quasistatic solutions of stresses and displacements have been analytically derived, respectively, in the forms including the arbitrary function. The solutions of stresses have been determined to be sums of thermal stress caused by eddy current loss and magnetic stress caused Lorentz force. Carrying out numerical calculations for the case that the arbitrary function of time is given by the smoothed ramp function with sine function, we have examined the dynamic and quasi-static behaviors of the thermal stresses and the magnetic stresses induced by a transient magnetic field in the solid cylinder. 2. Fundamental equation systems 2.1. Electromagnetic field Let us consider a conducting solid circular cylinder of radius b with cylindrical coordinate system, as shown in Fig. 1. We assume that a time-dependent axial magnetic field H0/(t), which is uniform along the h and z directions, acts on the surface of the cylinder in the direction parallel to its surface, from time t = 0. H0 is a reference magnetic field strength, and /(t) is an arbitrary function of time. If the magnetic field vector is assumed to have only the plane axisymmetric axial component, namely to be H = (0,0,Hz(r, t)), then the electric field vector is given by E = (0, Eh(r, t),0). The governing equations and the constitutive relations of electromagnetics in cylindrical coordinate are given by 1 o oBz ðrEh Þ þ ¼ 0; r or ot 

ð1Þ

oH z ¼ J h; or

ð2Þ

Fig. 1. Conditions and coordinate system of solid cylinder.

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J h ¼ rEh ;

ð3Þ

Bz ¼ lH z ;

ð4Þ

where the displacement current is disregarded in Eq. (2), and Bz is the axial component of the magnetic flux, Jh is the circumferential component of the electric current density, r and l are the electric conductivity and the magnetic permeability in the cylinder, respectively. From Eqs. (1)–(4), the fundamental equation of magnetic field is obtained as follows:   1 o oH z oH z r : ð5aÞ ¼ lr r or or ot The boundary condition and the initial condition are written as r ¼ b;

H z ðb; tÞ ¼ H 0 /ðtÞ;

ð5bÞ

t ¼ 0;

H z ðr; 0Þ ¼ 0:

ð5cÞ

The electric current density Jh(r, t) induced by time variation of magnetic field is called the eddy current, which is obtained from Eq. (2) 2.2. Temperature field The eddy current Jh generates Joule heat called the eddy current loss. The eddy current loss w(r, t) per unit time per unit volume is given by 2

½J h ðr; tÞ r: ð6Þ : We assume that the cylinder with zero initial temperature change is heated by the eddy current loss w(r, t) from time t = 0, and that the surface is insulated or subjected to surrounding medium, of which temperature is zero, with relative heat transfer coefficient h. Then, the fundamental equation of heat conduction taking into account the eddy current loss, the boundary condition and the initial condition are written as   oT 1 o oT w ¼j r ; ð7aÞ þ ot r or or Cq wðr; tÞ ¼

r ¼ b; t ¼ 0;

oT þ hT ¼ 0; or T ¼ 0;

ð7bÞ ð7cÞ

where T = T(r, t) is temperature change, and j, C and q denote the thermal conductivity, the specific heat and the mass density, respectively. If the surface is insulated, then the value of h in Eq. (7b) is zero. 2.3. Elastic field The conducting solid circular cylinder is subjected to both temperature change and Lorentz force. Lorentz force vector f is defined by 1 0 l 0 1 0 1 0 0  2 oro ½H z ðr; tÞ2 B C B B C z C f ¼ J  B ¼ @  oH @ 0 A¼@ A: 0 or A 0

lH z

ð8Þ

0

From Eq. (8), Lorentz force has only the radial component as follows: l o 2 ½H z ðr; tÞ : ð9Þ 2 or Because temperature change and Lorentz force are plane axisymmetric, we analyze stresses and deformations under the axisymmetric plane strain state. fr ðr; tÞ ¼ 

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The equation of motion in the radial direction taking into account Lorentz force is written as orrr rrr  rhh o2 u þ þ fr ¼ q 2 ; ot or r

ð10Þ

where rrr and rhh are the radial stress component and the circumferential stress component, respectively, and u is the radial displacement. Stress–displacement relations taking into account temperature change are given by ou  9 ð1mÞE m u 1þm þ  aT ;= rrr ðr; tÞ ¼ ð1þmÞð12mÞ or 1m r 1m ; ð11Þ   rhh ðr; tÞ ¼ ð1mÞE u þ m ou  1þm aT ; ð1þmÞð12mÞ r

1m or

1m

where E, m and a denote the Young’s modulus, the Poisson’s ratio and the coefficient of linear thermal expansion, respectively. The axial stress component is given by rzz ðr; tÞ ¼ mðrrr þ rhh Þ  aET :

ð12Þ

Substitution Eqs. (9) and (11) into Eq. (10) yields the displacement equation of motion:   o 1 oðruÞ 1 o2 u 1 þ m oT ð1 þ mÞð1  2mÞ l o 2 a þ ðH z Þ : ¼ 2 2þ or r or ot 1  m or ð1  mÞE 2 or CL

ð13aÞ

The cylinder is at rest prior to time t = 0 and we suppose that the surface of the cylinder is traction free (rrr = 0). Then, the mechanical boundary condition and the initial conditions are given by r ¼ b; t ¼ 0;

ou m u 1þm þ ¼ aT ; or 1  m r 1  m ou u¼ ¼ 0: ot

In Eq. (13a), CL denotes the velocity of longitudinal wave, which is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1  mÞE : CL ¼ ð1 þ mÞð1  2mÞq

ð13bÞ ð13cÞ

ð14Þ

2.4. Dimensionless quantities We define the following dimensionless quantities: 9  z ¼ H z ; s ¼ t 2 ; Jh ¼ bJ h ; r ¼ br ; H > > H0 H0 lrb > > = bf CcT rb2 w r  ¼ H 2 ; T ¼ lH 2 ;  w h ¼ bh; f r ¼ lH 2 ; 0 0 0 > > ð1mÞE > 2 hh ; r zz Þ ¼ ðrrr ;rlHhh2;rzz Þ ;  ð rrr ; r u ¼ ð1þmÞð12mÞ 2 u> ; blH 0 0

ð15Þ

2

and v1 ¼ lrj;

v2 ¼ lrbC L ;

v3 ¼

2aE : ð1  2mÞCq

ð16Þ

By use of these dimensionless quantities, Eqs. (2), (5)–(7), (9), (11)–(13) take the following form: (1) Electromagnetic field Fundamental equation system:   1 o oH z oH z r ; ¼ r or or os r ¼ 1;

H z ð1; sÞ ¼ /ðsÞ;

ð17aÞ ð17bÞ

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s ¼ 0;

H z ðr; 0Þ ¼ 0:

ð17cÞ

Eddy current: J h ðr; sÞ ¼ 

oH z ðr; sÞ ; or

ð18Þ

(2) Temperature field Eddy current loss:  ðr; sÞ ¼ ½J h ðr; sÞ2 : w Fundamental equation system:   oT o oT r ; ¼ v1 þw os or or r ¼ 1; s ¼ 0;

oT  þ hT ¼ 0; or T ðr; 0Þ ¼ 0:

ð19Þ

ð20aÞ ð20bÞ ð20cÞ

(3) Elastic field Lorentz force: 1 o ½H z ðr; sÞ2 f r ðr; sÞ ¼  2 or

ð21Þ

Stress–displacement relations: 9 u m  u rr ðr; sÞ ¼ o þ 1m  v3 T ; r > = r or  m o u u hh ðr; sÞ ¼ r þ 1m or  v3 T ; r > ; zz ðr; sÞ ¼ mðr rr þ r hh Þ  ð1  2mÞv3 T : r Fundamental equation system:   o 1 oðr uÞ 1 o2  oT o u þ ðH z Þ2 ¼ 2 2 þ v3 or r or v2 os or or r ¼ 1; s ¼ 0;

o u m  u þ ¼ v3 T ; or 1  m r o u  ¼ 0: u¼ os

ð22Þ

ð23aÞ ð23bÞ ð23cÞ

3. Solutions 3.1. Magnetic field In order to transform the inhomogeneous boundary condition (17b) into the homogeneous one, we assume that the solution H z of Eq. (17a) is given by the following expression: H z ðr; sÞ ¼ hz ðr; sÞ þ /ðsÞ: Substitution of Eq. (24) into Eq. (17) gives   1 o ohz ohz o/ðsÞ r ; þ ¼ r or os or os

ð24Þ

ð25aÞ

r ¼ 1;

hz ð1; sÞ ¼ 0;

ð25bÞ

s ¼ 0;

hz ðr; 0Þ ¼ /ð0Þ:

ð25cÞ

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By using the separation of variables technique, the solution of Eqs. (25) will be assumed in the following form: hz ðr; sÞ ¼

1 X

an ðsÞJ0 ðk nrÞ;

ð26Þ

n¼1

where an(s) are unknown functions of s, J0(Æ) is the Bessel function of the first kind of zeroth order, and kn are the positive roots of the following eigen equation: J0 ðk n Þ ¼ 0:

ð27Þ

Hence, it can be seen that the solution hz ðr; sÞ shown in Eq. (26) satisfies the homogeneous boundary conditions Eq. (25b). Substituting of Eq. (26) into Eq. (25a) with Eq. (25b), multiplying both side by rJ0 ðk mrÞ, and then integrating with respect to r from zero to one, we obtain Z Z 1 Z 1 1 1 X X dan ðsÞ 1 d/ðsÞ J0 ðk mrÞr dr; k 2n an ðsÞ J0 ðk nrÞJ0 ðk mrÞr dr ¼ J0 ðk nrÞJ0 ðk mrÞrdr þ ð28Þ  ds ds 0 0 0 n¼1 n¼1 Using the orthogonality of Bessel functions, we can derive the following equation: ( 2 Z 1 J1 ðk n Þ ðm ¼ nÞ 2 J0 ðk nrÞJ0 ðk mrÞrdr ¼ 0 0 ðm 6¼ nÞ; Substitution of Eq. (29) into Eq. (28) gives Z 1 dan ðsÞ 2 d/ðsÞ 2 þ k n an ðsÞ ¼  2 J0 ðk nrÞr dr: ds ds J1 ðk n Þ 0

ð29Þ

ð30Þ

The solutions of Eq. (30) under the initial condition (25c) are expressed as an ðsÞ ¼ 

2 ^ an ðsÞ; k n J1 ðk n Þ

where aˆn(s) are determined by the function /(s), which are given by Z s 0 2 0 d/ðs Þ ^ an ðsÞ ¼ ekn ðss Þ ds0 : ds0 0

ð31Þ

ð32Þ

From Eqs. (24), (26) and (31), the magnetic field H z is written as H z ðr; sÞ ¼ /ðsÞ  2

1 X n¼1

1 J0 ðk nrÞ^ an ðsÞ: k n J1 ðk n Þ

ð33Þ

Substituting Eq. (33) into Eq. (18), we obtain the circumferential component J h of the eddy current as follows: J h ðr; sÞ ¼ 2

1 X n¼1

1 J1 ðk nrÞ^ an ðsÞ: J1 ðk n Þ

ð34Þ

3.2. Temperature field By using the separation of variables technique, the solution of Eq. (20) will be assumed in the following form: ( 1 X b ¼ 0 ð h ¼ 0Þ; T ðr; sÞ ¼ bj ðsÞJ0 ðpjrÞ ð35Þ b ¼ 1 ð h > 0Þ; j¼b

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where bj(s) are unknown functions of s, and pj are the nonnegative roots of the following eigen equations: ) J1 ðpj Þ ¼ 0 ðpj P 0 ðj ¼ 0; 1; 2;   ÞÞ for h ¼ 0; : ð36Þ hJ0 ðpj Þ ¼ 0 ðpj > 0 ðj ¼ 1; 2; 3;   ÞÞ for h > 0; pj J1 ðpj Þ þ  Using the orthogonality of Bessel functions, we can derive the following equation: Z 1 M j ðl ¼ jÞ; J0 ðpjrÞJ0 ðplrÞr dr ¼ 0 ðl 6¼ jÞ; 0

ð37Þ

where

Mj ¼

81 > > > 22 < > > > :

J0 ðpj Þ 2 ð h2 þp2j ÞJ20 ðpj Þ 2p2j

for for

9  h ¼ 0=  h > 0;

ðj ¼ 0Þ; ðj > 0Þ:

ð38Þ

In the same manner as Section 3.1, substituting Eq. (35) into Eq. (20a) with Eq. (20b), and utilizing Eq. (37), we can derive the following equation: Z 1 dbj ðsÞ 1 2  ðr; sÞJ0 ðpjrÞr dr: þ v1 pj bj ðsÞ ¼ w ð39Þ ds Mj 0 The solutions of Eq. (39) under the initial condition (20c) are expressed as 8 R 1 R s  <2 0 0 w  ðr; s0 Þ ds0 dr; i bj ðsÞ ¼ R hR 0 0 : 1 1 s ev1 p2j ðss0 Þ w  Þ ds ð r ; s J0 ðpjrÞr dr 0 Mj 0 Here, substitution of Eq. (34) into Eq. (19) yields 1 X 1 X 1  ðr; sÞ ¼ 4 w J1 ðk mrÞJ1 ðk nrÞ^am ðsÞ^an ðsÞ: J ðk ÞJ ðk Þ m¼1 n¼1 1 m 1 n Substitution of Eq. (41) into Eq. (40) gives 8 P 1 > ^ð0Þ > ðj ¼ 0Þ; < 4 bn ðsÞ n¼1 bj ðsÞ ¼ 1 P 1 P > I 1jmn > ^ : M4j b ðsÞ ðj > 0Þ; J ðk m ÞJ ðk n Þ jmn m¼1 n¼1

1

ð40Þ

ð41Þ

ð42Þ

1

where ^ bnð0Þ ðsÞ, ^ bjmn ðsÞ are determined by the function /(s), which are given by ) Rs 2 ð0Þ ^ bn ðsÞ ¼ 0 ½^ an ðs0 Þ ds0 ; ; Rs 2 0 ^ an ðs0 Þ ds0 am ðs0 Þ^ bjmn ðsÞ ¼ 0 ev1 pj ðss Þ ^ and I1jmn are given by the following equation, and are calculated by numerical integration: Z 1 I 1jmn ¼ J1 ðk mrÞJ1 ðk nrÞJ0 ðpjrÞr dr:

ð43Þ

ð44Þ

0

Substituting Eq. (42) into Eq. (35), we obtain temperature change as follows: 1 1 1 X 1 X X X 4 I 1jmn ^ ^bjmn ðsÞ; T ðr; sÞ ¼ 4 J0 ðpjrÞ bð0Þ n ðsÞ þ M J ðk ÞJ ðk Þ j n¼1 j¼1 m¼1 n¼1 1 m 1 n in which for h > 0, the first term of right hand side in Eq. (45) is ignored.

ð45Þ

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3.3. Elastic field 3.3.1. Dynamic solutions In order to transform the inhomogeneous boundary condition Eq. (23b) into the homogeneous one, we assume that  uðr; sÞ is given by following expression:  uðr; sÞ ¼ u1 ðr; sÞ þ u2 ðr; sÞ; where u1 ðr; sÞ satisfies   o 1 oðru1 Þ ¼ 0; or r or r ¼ 1;

ou1 m u1 ¼ v3 T : þ or 1  m r

ð46Þ

ð47aÞ ð47bÞ

The solution of Eqs. (47) is expressed as u1 ¼ ð1  mÞv3 T ð1; sÞr: Substitution of Eq. (46) with Eq. (47) into Eq. (23) yields   o 1 oðru2 Þ 1 o2 u2 1 o2 u1 oT o 2 þ þ þ v3 Hz ; ¼ 2 or r or or or v2 os2 v22 os2 r ¼ 1; s ¼ 0;

ou2 m u2 þ ¼ 0; or 1  m r ou2 ou1 u2 ¼ u1 ; ¼ : os os

ð48Þ

ð49aÞ ð49bÞ ð49cÞ

By using the separation of variables technique, the solution of Eq. (49) will be assumed in the following form: u2 ðr; sÞ ¼

1 X

ci ðsÞJ1 ðgirÞ;

ð50Þ

i¼1

where ci(s) are unknown functions of s, and gi are the positive roots of the following eigen equation: gi J0 ðgi Þ 

1  2m J1 ðgi Þ ¼ 0: 1m

ð51Þ

In the same manner as Sections 3.1 and 3.2, substituting Eq. (50) with Eq. (49b) into Eq. (49a), and utilizing the orthogonality of Bessel functions, we can derive the following equation: Z 1 2 Z Z o2 ci ðsÞ 1 o u1 v22 1 oT v22 1 oH 2z 2   J J1 ðgirÞr dr; þ X c ðsÞ ¼  J ðg ðg r Þ r d r  v r Þ r d r  ð52Þ i 1 1 i 3 i i os2 N i 0 os2 N i 0 or N i 0 or where 2

Ni ¼

1 ð1  mÞ g2i  ð1  2mÞ 2 J0 ðgi Þ; 2 ð1  2mÞ2

ð53Þ

and Xi are the natural angular frequencies of ith order mode in dimensionless form, which are given by Xi ¼ v2 gi : The solutions of Eq. (52) under the initial condition Eq. (49c) are expressed as

ð54Þ

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1 ci ðsÞ ¼ Ni

Z

1



Z

s 0

0

0



u1 ðr; sÞ þ Xi sin Xi ðs  s Þu1 ðr; s Þ ds J1 ðgirÞr dr 0  Z 1 Z s v o sin Xi ðs  s0 ÞT ðr; s0 Þds0 J1 ðgirÞr dr  v3 2 gi N i 0 or 0  Z 1 Z s

2 v o sin Xi ðs  s0 Þ H z ðr; s0 Þ ds0 J1 ðgirÞr dr:  2 gi N i 0 or 0 0

ð55Þ

Substitution of Eqs. (34), (45) and (48) into Eq. (55) gives ci ðsÞ ¼ cTi ðsÞ þ cM i ðsÞ;

ð56Þ

where cTi ðsÞ and cM i ðsÞ are the terms due to temperature change and due to Lorentz force, respectively, which are given by ( 9 i P 1 h 1 1 1 P > J0 ðpj Þ P P I 1jmn T ð0Þ 4J1 ðgi Þ > T ð0Þ ^ > ci ðsÞ ¼ g2 N i v3 bn ðsÞ þ Xi^cin ðsÞ  J1 ðk m ÞJ1 ðk n Þ > Mj > i > n¼1 j¼1 m¼1 n¼1 > > > " ) =    2  g ðg Þ hJ T 0 i i ; ð57Þ  ^ bjmn ðsÞ þ p2 g2 1  gi J1 ðgi Þ Xi^cijmn ðsÞ ; > j i > > >   > > > 1 1 P 1 P P > Mð1Þ Mð2Þ 4v2 I 2imn > M 1 ; ^c ðsÞ þ ^c ðsÞ c ðsÞ ¼ J ðg Þ i

0

Ni

i

n¼1

T ð0Þ

k 2n g2i

in

m¼1 n¼1

Mð1Þ

k m k n J1 ðk m ÞJ1 ðk n Þ imn

Mð2Þ

in which ^cin ðsÞ, ^cTijmn ðsÞ, ^cii ðsÞ and ^cimn ðsÞ are defined by the function /(s), which are given by 9 Rs T ð0Þ 0 0 > ^cin ðsÞ ¼ 0 sin Xi ðs  s0 Þ^ bð0Þ > n ðs Þ ds ; > > R > s = ^jmn ðs0 Þ ds0 ; ^cTijmn ðsÞ ¼ 0 sin Xi ðs  s0 Þb ; R s Mð1Þ > ^cin ðsÞ ¼ 0 sin Xi ðs  s0 Þ/ðs0 Þ^ an ðs0 Þ ds0 ; > > > > Rs ; Mð2Þ ^cimn ðsÞ ¼ 0 sin Xi ðs  s0 Þ^ am ðs0 Þ^ an ðs0 Þ ds0 and I2imn are given by the following equation and are calculated by numerical integration: Z 1 I 2imn ¼ J0 ðk nrÞJ0 ðk mrÞJ0 ðgirÞr dr:

ð58Þ

ð59Þ

0

From Eqs. (46), (48), (50), (56), (57), we have  uT ðr; sÞ ¼ v3 ð1  mÞrT ð1; sÞ þ

1 P i¼1

M

 u ðr; sÞ ¼

1 P

9 > cTi ðsÞJ1 ðgirÞ; > = > > ;

cM rÞ i ðsÞJ1 ðgi

i¼1

;

ð60Þ

where  uT ðr; sÞ and  uM ðr; sÞ are the radial displacements due to temperature change and due to Lorentz force, respectively, and satisfy the following relation:  uðr; sÞ ¼  uT ðr; sÞ þ  uM ðr; sÞ:

ð61Þ

Substituting Eq. (61) into Eq. (22), we can obtain the dynamic solutions of the stress components as follows: h i 9 1 P T T 12m J1 ðgirÞ > rr ðr; sÞ ¼ v3 ½T ð1; sÞ  T ðr; sÞ þ ci ðsÞ gi J0 ðgirÞ  1m r ; > r > > > i¼1 > h i > = 1 P mgi 12m J1 ðgirÞ T T hh ðr; sÞ ¼ v3 ½T ð1; sÞ  T ðr; sÞ þ ci ðsÞ 1m J0 ðgirÞ þ 1m r ; r ð62Þ > i¼1 > > > 1 > P > mgi > Tzz ðr; sÞ ¼ v3 ½2mT ð1; sÞ  T ðr; sÞ þ cTi ðsÞ 1m J0 ðgirÞ; r ; i¼1

M. Higuchi et al. / International Journal of Solids and Structures 44 (2007) 5316–5335

M r; sÞ ¼ r rr ð

1 P i¼1

M r; sÞ ¼ r hh ð M r; sÞ ¼ r zz ð

1 P i¼1 1 P i¼1

h cM rÞ  12m i ðsÞ gi J0 ðgi 1m cM i ðsÞ

h

mgi J ðg rÞ 1m 0 i

J1 ðgi rÞ r

þ 12m 1m

mgi cM rÞ; i ðsÞ 1m J0 ðgi

i 9 > ; > > > > > i > = J1 ðgi rÞ ; r > > > > > > > ;

5325

ð63Þ

Tii ði ¼ r; h; zÞ and r M where r ii are thermal stresses and magnetic stresses, respectively, and satisfy the following relations: Trr þ r M rr ¼ r r rr ;

hh ¼ r Thh þ r M r hh ;

hh ¼ r Tzz þ r M r zz :

ð64Þ

From Eqs. (62) and (63), the radial and circumferential components of the thermal stress and the magnetic stress at center ðr ¼ 0Þ are expressed as 9 1 P gi Trr ð0; sÞ ¼ r Thh ð0; sÞ ¼ v3 ½T ð1; sÞ  T ð0; sÞ þ cTi ðsÞ 2ð1mÞ > r ;> = i¼1 ð65Þ 1 P > gi M M > ; M  r ð0; sÞ ¼ r ð0; sÞ ¼ c ðsÞ : rr hh i 2ð1mÞ i¼1

3.3.2. Quasi-static solutions In this section, we derive the quasi-static solutions of the displacements and stresses. Disregarding the inertia term of the right-hand side in Eq. (23a) gives the equilibrium equation:   d 1 dðr uÞ dT d 2 þ ðH z Þ : ð66Þ ¼ v3 dr r dr dr dr Solving Eq. (66) with the boundary condition (23b), we obtain the quasi-static solutions of the radial displacements due to temperature change and Lorentz force, respectively, which are given as follows:   uT ðr; sÞ ¼ v3 ½F T ðr; sÞ þ ð1  2mÞF T ð1; sÞr; ð67Þ  uM ðr; sÞ ¼ ½F M ðr; sÞ þ ð1  2mÞF M ð1; sÞr  ð1  mÞ/2 ðsÞr; where F T ðr; sÞ ¼ 1r

R

F M ðr; sÞ ¼ 1r

R

rT ðr; sÞ dr; r½H z ðr; sÞ2 dr:

) ð68Þ

Substituting Eq. (67) with the relation of Eq. (61) into Eq. (22), we obtain the quasi-static solutions of the stress components as follows: 9  1  Trr ðr; sÞ ¼ v3 12m  r F T ðr; sÞ þ F T ð1; sÞ ; r > 1m   = T 12m 1 hh ðr; sÞ ¼ v3 1m r F T ðr; sÞ þ F T ð1; sÞ  T ðr; sÞ ; ð69Þ r > ; Tzz ðr; sÞ ¼ v3 12m r ½2mF ð1; sÞ  T ð r ; sÞ; T 1m  1  9 M 12m rr ðr; sÞ ¼ 1m  r F M ðr; sÞ þ F M ð1; sÞ þ H 2z ðr; sÞ  /2 ðsÞ; > r =   2 12m 1 m 2 ð70Þ M ð r ; sÞ ¼ F ð r ; sÞ þ F ð1; sÞ þ H ð r ; sÞ  / ðsÞ; r M M hh z 1m r 1m > ; 2 M 12m m 2 zz ðr; sÞ ¼ 1m 2mF M ð1; sÞ þ 1m H z ðr; sÞ  2m/ ðsÞ: r These quasi-static thermal stresses and magnetic stresses satisfy the relations of Eq. (64). By using L’Hospital’s rule and the relations of Eq. (68), the expressions of 1r F T ðr; sÞ and 1r F T ðr; sÞ at the center ðr ¼ 0Þ in Eqs. (69) and (70) are written as )  1 F ðr; sÞr¼0 ¼ 12 T ð0; sÞ; r T  ð71Þ 1 F M ðr; sÞ ¼ 1 H 2 ð0; sÞ: r

r¼0

2

z

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From Eqs. (69)–(71), the radial and circumferential components of the thermal stress and the magnetic stress at the center ðr ¼ 0Þ are expressed as 9   1 = Trr ð0; sÞ ¼ r Thh ð0; sÞ ¼ v3 12m r F ð1; sÞ  T ð0; sÞ ; T 1m 2 h i ð72Þ 2 M M 12m 1 2 hh ð0; sÞ ¼ 1m F M ð1; sÞ  2ð12mÞ H z ð0; sÞ  / ðsÞ: ; rr ð0; sÞ ¼ r r 4. Numerical results and discussion In this paper, an arbitrary function /(s) is assumed to be the smoothed ramp function with sine function:   ( sin 2sp0 s ðs < s0 Þ; /ðsÞ ¼ ð73Þ 1 ðs P s0 Þ; T ð0Þ ^ cin ðsÞ, ^cTijmn ðsÞ, where s0 is a rise time in the nondimensional form. The functions of s: aˆn(s), ^bð0Þ n ðsÞ, bjmn ðsÞ, ^ Mð1Þ Mð2Þ ^cin ðsÞ, ^cimn ðsÞ in Eqs. (32), (43) and (58) are presented in Appendix A for this case. By use of the analytical results mentioned above, numerical calculations are carried out for aluminum, material properties of which are given by 9 l ¼ 4p  107 ½H=m; r ¼ 3:42  107 ½S=m; > > > C ¼ 2:7  103 ½J=kgK; q ¼ 0:9  103 ½kg=m3 ; j ¼ 92:6  106 ½m2 =s; = ð74Þ > m ¼ 0:33; E ¼ 70½GPa; a ¼ 24  106 ½1=K: > > ;

In addition, since the nondimensional variable v2 in Eq. (16) contains a radius b, the dimension of the radius of the solid cylinder should be set. We here set the dimension as b = 1.0 · 104[m], for reasons of the convergence of the solutions. A rise time s0 is given by s0 ¼ e 

1 ; v2

ð75Þ

where e is a parameter, and 1/v2 means the nondimensional time which stress waves created at the surface take to arrive at the center of the cylinder. Firstly, numerical results for e = 0.5 are presented in Figs. 2–7. Fig. 2 shows the time evolution of eddy current J h . The eddy current rapidly shows the peak in front of s0 at the surface ðr ¼ 1:0Þ, then it decays with time. Fig. 3 shows the time evolutions of temperature changes T for h ¼ 0:0 and 1.0 until they attain steady state. It can be seen from Fig. 3 that temperature changes take long time to attain steady state as compared with the eddy current J h . This is because the value of v1 in Eq. (16), which is the ratio of the diffusion coefficient of

0

τ0 –r = 0.5 –r = 1.0

-1 -2

ε = 0.5

– Jθ

-3 -4 -5 -6 -7 -8 0

0.2

0.4

τ

0.6

0.8

Fig. 2. Time evolution of eddy current J h , for e = 0.5.

1

M. Higuchi et al. / International Journal of Solids and Structures 44 (2007) 5316–5335

5327

– T

1.2 1

–r = 0.0 –r = 1.0

0.8

= 0.5 – h = 0.0

0.6 0.4

– h = 1.0

0.2 0 0

100

200

300

400

500

600

700

800

τ Fig. 3. Time evolutions of temperature changes T , for h ¼ 0:0 and 1.0, e = 0.5.

1.2

τ0

1 –r = 1.0

– T

0.8

– h = 0.0 – h = 1.0 ε = 0.5

0.6 0.4 –r = 0.5 0.2

–r = 0.0 0 0

0.2

0.4

τ

0.6

0.8

1

Fig. 4. Time evolution of temperature change T for short time, for h ¼ 0:0 and 1.0, e = 0.5.

temperature field j to that of magnetic field 1/lr, is very small: v1 = j/(1/lr) ffi 3.98 · 103 for aluminum. In the case of  h ¼ 1:0, temperature change converges zero, whereas in the case of h ¼ 0:0 namely the insulated case, it converges a value which is determined from Eq. (40) to be:  Z 1 Z 1  ðr; s0 Þ ds0 dr: T ¼2 w ð76Þ 0

0

However as shown in Fig. 4, which shows the time evolutions of temperature changes for short time, there is no large difference between each case, and temperature changes arise from the surface ðr ¼ 1:0Þ rapidly but still more slowly than the eddy current in Fig. 2. Numerical results of the thermal stresses are therefore shown below only for the case of  h ¼ 0:0. M Fig. 5 shows the dynamic and quasi-static behaviors of the radial and circumferential components r rr and M M hh and the axial component r zz of the magnetic stress. In Fig. 5, a new nondimensional time sE is introduced r for convenience. The nondimensional time sE is based on the longitudinal wave velocity CL, which is defined as sE ¼

s CL t: ¼ v2 b

ð77Þ

Note that the radial and circumferential stress components at the center ðr ¼ 0Þ are the same with each other as shown by Eqs. (65) and (72) It is known that magnitude of magnetic stresses should be lH 20 =2 (namely, 1 in nondimensional form with Eq. (15)) at the maximum (Moon and Chattopadhyay, 1974). The quasi-static solutions of magnetic stresses are less than 1, as shown in Fig. 5. However, it can be seen from Fig. 5 that magnitude of the dynamic magnetic stresses are larger than 1. This is because the stress wave created at

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M M M Fig. 5. Dynamic and quasi-static behaviors of the magnetic stresses r rr , r hh and r zz , for e = 0.5.

M Fig. 6. Radial variation of the radial magnetic stress r rr in the dynamic solution.

the surface due to Lorentz force propagates into the center and then it accumulates to the center as shown in M Fig. 6, which shows the radial variation of the radial magnetic stress r rr in the dynamic solution. This phenomenon is called the stress-focusing effect (e.g., Nelson, 1968; Nelson, 1969; Ho, 1976; Hata, 1994; Wang et al., 2002). If a rise time s0 equals to zero, then the ramp function become to be the step function, and the values of stresses at center will be infinite as mentioned by Ho (1976) and Hata (1994) for the thermal loading. However, a rise time will be physically nonzero. Fig. 7 shows the dynamic and quasi-static behaviors of

M. Higuchi et al. / International Journal of Solids and Structures 44 (2007) 5316–5335

5329

Trr , r Thh and r Tzz , for h ¼ 0:0, e = 0.5. Fig. 7. Dynamic and quasi-static behaviors of the thermal stresses r

Trr and r Thh and the axial component r Tzz of the thermal stress for the the radial and circumferential components r insulated case ð h ¼ 0Þ. It can be seen from Fig. 7 that although at the center ðr ¼ 0Þ the stress-focusing effects due to the rapid temperature rise in Fig. 4 are observed, the effects of thermal stresses are smaller than those of magnetic stresses in Fig. 5. Whereas at the surface ðr ¼ 1:0Þ there are no large differences between the dynamic Tzz , and the absolute values and quasi-static behaviors of both the circumferential stress rThh and the axial stress r T T hh and r zz at the surface are larger than those of the magnetic stresses r M M of the thermal stresses r hh and r zz at the surface in Fig. 5. Next, numerical results for e = 10.0 are presented in Figs. 8–10. Figs. 8–10 show the time evolutions of eddy M M M Trr , r Thh and r Tzz for h ¼ 0, respectively. current J h , the magnetic stresses r rr , r hh and r zz and the thermal stresses r It can be seen from Fig. 8 that the eddy current J h is small and varies slowly as compared with the case of τ0

0

– Jθ

-0.5

-1

–r = 0.5 –r = 1.0

ε = 10.0 -1.5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

τ Fig. 8. Time evolution of eddy current J h , for e = 10.0.

1

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M M M Fig. 9. Dynamic and quasi-static behaviors of the magnetic stresses r rr , r hh and r zz , for e = 10.0.

Trr , r Thh and r Tzz , for h ¼ 0:0, e = 10.0. Fig. 10. Dynamic and quasi-static behaviors of the thermal stresses r

M. Higuchi et al. / International Journal of Solids and Structures 44 (2007) 5316–5335

5331

e = 0.5 in Fig. 2. Therefore, the maximum absolute values of both the magnetic stresses in Fig. 9 and the thermal stresses in Fig. 10 are smaller than those for the case of e = 0.5 in Fig. 5 or Fig. 7, and there are almost no differences in magnitude between the dynamic and quasi-static solutions as shown in Figs. 9 and 10. Therefore, we conclude that if a rise time s0 of the ramp function is small (for example e = 0.5), the dynamic analysis is needed. 5. Conclusion In the present study, dynamic and quasi-static behaviors of magneto-thermo-elastic stresses induced by a transient magnetic field, which is defined by arbitrary function of time, in a conducting solid circular cylinder have been investigated. Exact solutions of electromagnetic field, temperature change, stresses and deformations have been obtained in the forms including the arbitrary function. The stresses have been determined to be sums of thermal stress and magnetic stress. Carrying out numerical calculations for the case that the arbitrary function of time is given by the smoothed ramp function with sine function, we have examined the dynamic and quasi-static behaviors of the thermal stresses and the magnetic stresses. Appendix A. The functions of nondimensional time s Mð1Þ Mð2Þ T ð0Þ The function of nondimensional time s: aˆn(s), ^ bjmn ðsÞ, ^cTijmn ðsÞ, ^cin ðsÞ, ^cimn ðsÞ and ^bnð0Þ ðsÞ, ^cin ðsÞ in Eqs. (32), (42) and (58) are given as follows: For s < s0, 9 2 > ^ an ðsÞ ¼ wn ðx sin xs þ k 2n cos xs  k 2n ekn s Þ; > > > > ðsÞ ðcÞ ð1Þ w w 2 2 2 2 2 2 2 2 m n > ^ > bjmn ðsÞ ¼ 2 fðk m þ k n Þxhj ðsÞ þ ðk m k n  x Þhj ðsÞ þ ðk m k n þ x Þhj ðsÞ > > > > ðkÞ ðckÞ ðckÞ ðskÞ ðskÞ 2 2 2 2 > þ2k m k n ½hjmn ðsÞ  fhjm ðsÞ þ hjn ðsÞg  2x½k m hjm ðsÞ þ k n hjn ðsÞg; > > > > > ðsÞ ðcÞ ð1Þ wm wn 2 2 2 2 2 2 T 2 2 = ^cijmn ðsÞ ¼ 2 fðk m þ k n Þxd ij ðsÞ þ ðk m k n  x Þd ij ðsÞ þ ðk m k n þ x Þd ij ðsÞ ðA:1Þ ðkÞ ðckÞ ðckÞ ðskÞ ðskÞ > þ2k 2m k 2n ½d ijmn ðsÞ  2fd ijm ðsÞ þ d ijn ðsÞg  2x½k 2m d ijm ðsÞ þ k 2n d ijn ðsÞg; > > > > > Mð1Þ ðsÞ ðcÞ ð1Þ ðskÞ > > ^cin ðsÞ ¼ w2n ½k 2i fi ðsÞ  xfi ðsÞ þ xfi ðsÞ  2k 2i fin ðsÞ; > > > > Mð2Þ ðsÞ ðcÞ ð1Þ wm wn 2 2 2 2 2 2 2 2 > > ^cimn ðsÞ ¼ 2 fðk m þ k n Þxfi ðsÞ þ ðk m k n  x Þfi ðsÞ þ ðk m k n þ x Þfi ðsÞ > > > ; ðckÞ ðckÞ 2 2 ðkÞ 2 ðskÞ 2 ðskÞ þ2k m k n ½fimn ðsÞ  2ffim ðsÞ þ fin ðsÞg  2x½k m fim ðsÞ þ k n fin ðsÞg; 9 n o w2n k 4n x2 4 2 ð0Þ 2 k 2n s 2k 2n s > ^ = bn ðsÞ ¼ 2 sin 2xs þ ðk n þ x Þs  k n ½cos 2xs  4e cos xs þ e þ 2 ; 2x n o ðA:2Þ 2 4 T ð0Þ ðtÞ ðcÞ ðckÞ ðkÞ ð1Þ x2 ðsÞ ; ^cin ðsÞ ¼ w2n kn2x fi ðsÞ þ ðk 4n þ x2 Þfi ðsÞ  k 2n ½fi ðsÞ  4fin ðsÞ þ finn ðsÞ þ 2fi ðsÞ ; >

where wn ¼ ðsÞ

k 4n

x ; þ x2

x¼

1 2 2 2 1 pj Þ þð2xÞ

hj ðsÞ ¼ ðv ðcÞ

hj ðsÞ ¼ ðv

1

2 2 2 1 pj Þ þð2xÞ

p ; 2s0

ðA:3Þ 2

ðv1 p2j sin 2xs  2x cos 2xs þ 2xev1 pj s Þ; 2

ð2x sin 2xs þ v1 p2j cos 2xs  v1 p2j ev1 pj s Þ; 2

ð1Þ

hj ðsÞ ¼ v 1p2 ð1  ev1 pj s Þ;

ðkÞ

hjmn ðsÞ ¼ v

1 2 2 2 1 p j ðk m þk n Þ

2

2

2

½eðkm þkn Þs  ev1 pj s ;

9 > > > > > > > > > > > > =

> > > > ðskÞ 2 1 2 > hjn ðsÞ ¼ x2 þðk2 v p2 Þ2 f½ðv1 pj  k n Þ sin xs  x cos xse þ xe g; > > > 1 j n > > > 2s 2 ðckÞ v p 2 2 1 2 k n s 2 1 j ;  ðv1 pj  k n Þe g; > hjn ðsÞ ¼ 2 2 2 2 f½x sin xs þ ðv1 p j  k n Þ cos xse 1 j

k 2n s

x þðk n v1 pj Þ

v1 p2j s

ðA:4Þ

5332

M. Higuchi et al. / International Journal of Solids and Structures 44 (2007) 5316–5335 ðsÞ

ðsÞ

1

d ij ðsÞ ¼ ðv

2 2 2 1 pj Þ þð2xÞ

ðcÞ

1

d ij ðsÞ ¼ ðv

2 2 2 1 pj Þ þð2xÞ

ð1Þ

ðcÞ

ðpÞ

½v1 p2j fi ðsÞ  2xfi ðsÞ þ 2xfij ðsÞ; ðsÞ

ðcÞ

ðpÞ

ðpÞ

d ijmn ðsÞ ¼ v

½2xfi ðsÞ þ v1 p2j fi ðsÞ  v1 p2j fij ðsÞ;

ð1Þ

ðkÞ

d ij ðsÞ ¼ v 1p2 ½fi ðsÞ  fij ðsÞ;

ðkÞ

1 2 2 2 1 p j ðk m þk n Þ

ðpÞ

½fimn ðsÞ  fij ðsÞ;

9 > > > > > > > > > > > > > > =

ðA:5Þ

> > > > > ðskÞ ðskÞ ðckÞ ðpÞ 2 1 2 > d ijn ðsÞ ¼ x2 þðk2 v p2 Þ2 ½ðv1 pj  k n Þfin ðsÞ  xfin ðsÞ þ xfij ðsÞ; > > 1 j n > > > > > ðckÞ ðskÞ ðckÞ ðpÞ 2 2 > 1 2 2 ; d ijn ðsÞ ¼ 2 2 ½xf ðsÞ þ ðv p  k Þf ðsÞ  ðv p  k Þf ðsÞ; 2 2 1 j 1 j in n in n ij 1 j

x þðk n v1 pj Þ

ðsÞ

9 > > > > > > > > > > > > > > > > > =

ðcÞ

Xi 1 fi ðsÞ ¼ X2 ð2xÞ ðsÞ ¼ X2 ð2xÞ 2 ðXi sin2xs  2x sinXi sÞ; fi 2 ðcos2xs  cos Xi sÞ; i

i

ð1Þ

fi ðsÞ ¼ X1i ð1  cosXi sÞ; ðkÞ

2

2

fimn ðsÞ ¼ X2 þðk12 þk2 Þ2 ½ðk 2m þ k 2n ÞsinXi s  Xi cos Xi s þ Xi eðk m þkn Þs ; i

m

n

> > > > > > > > 2 ðckÞ > fin ðsÞ ¼ x4 þ2x2 ðk4 X1 2 Þþðk4 þX2 Þ2 fXi ekn s ½2k 2n xsinxs þ ðk 4n þ X2i  x2 Þcosxs þ k 2n ðk 4n þ X2i þ x2 ÞsinXi s  Xi ðk 4n þ X2i  x2 ÞcosXi sg; > > > n n i i > > > > 2 > ðpÞ ðtÞ v1 pj s 1 2 1 ; f ðsÞ ¼ 2 2 4 ðv p sinXi s  Xi cosXi s þ Xi e Þ; f ðsÞ ¼ 2 ðXi s  sinXi sÞ: ðskÞ

2

fin ðsÞ ¼ x4 þ2x2 ðk4 X1 2 Þþðk4 þX2 Þ2 fXi ekn s ½ðk 4n þ X2i  x2 Þsinxs þ 2k 2n xcos xs þ xðk 4n  X2i þ x2 ÞsinXi s  2k 2n xXi cos Xi sg; n

ij

Xi þv1 pj

n

i

i

1 j

i

Xi

ðA:6Þ

For s P s0, 2

2

^ an ðsÞ

¼

wn ½xekn ðss0 Þ  k 2n xekn s ;

^ bjmn ðsÞ

¼

wm wn 2

ðs; 2Þ

fðk 2m þ k 2n Þxhj ðk; 2Þ

ðc; 2Þ

ðsÞ þ ðk 2m k 2n  x2 Þhj ðck; 2Þ

þ2k 2m k 2n ½hjmn ðsÞ  fhjm ðk2Þ þ2½x2 qjmn ðsÞ

^cTijmn ðsÞ

¼

wm wn 2



ðk3Þ xk 2m qjmn ðsÞ ðs; 2Þ

½ðk 2m þ k 2n Þxd ij ðk; 2Þ

ðck; 2Þ

ðsÞ þ hjn

ðsk; 2Þ

ðsÞg  2x½k 2m hjm

ðk3Þ xk 2n qjnm ðsÞ



þ

ðc; 2Þ

ðsÞ þ ðk 2m k 2n  x2 Þd ij ðck; 2Þ

ð1; 2Þ

ðsÞ þ ðk 2m k 2n þ x2 Þhj

ðsÞ

ðsk; 2Þ

ðsÞ þ k 2n hjn

ðsÞ

ðk1Þ k 2m k 2n qjmn ðsÞg; ð1; 2Þ

ðsÞ þ ðk 2m k 2n þ x2 Þd ij

ðck; 3Þ

ðsÞ

ðsk; 2Þ

ðsk; 2Þ

þ2k 2m k 2n fd ijmn ðsÞ  2½d ijm ðsÞ þ d ijm ðsÞg  2x½k 2m d ijm ðsÞ þ k 2n d ijn ðsÞ

> > > > > >   þ > > > > > ðc; 2Þ ð1; 2Þ ðk2Þ wn 2 ðs; 2Þ 2 ðsk; 2Þ 2 ðk1Þ > > ½k f ðsÞ  xf ðsÞ þ xf ðsÞ  2k f ðsÞ þ w ½xg ðsÞ  k g ðsÞ; > i i in n n i n in n in 2 > > > > ðs; 2Þ ðc; 2Þ ð1; 2Þ wm wn 2 2 2 2 2 2 > 2 2 > fðk þ k Þxf ðsÞ þ ðk k  x Þf ðsÞ þ ðk k þ x Þf ðsÞ > i i i m n m n m n 2 > > > > ðck; 2Þ ðck; 2Þ 2 2 ðk; 2Þ 2 ðsk; 2Þ 2 ðsk; 2Þ > > þ2k m k n ½fimn ðsÞ  2ffim ðsÞ þ fin ðsÞg  2x½k m fim ðsÞ þ k n fin ðsÞ > > > > ; ðk4Þ ðk5Þ ðk5Þ ðk3Þ 2 2 2 2 þ2½x2 gimn ðsÞ  xk m gimn ðsÞ  xk n ginm ðsÞ þ k m k n gimn ðsÞg; ðk2Þ þ2½x2 sijmn ðsÞ

Mð1Þ

^cin ðsÞ ¼ Mð2Þ

^cimn ðsÞ ¼

n 2

¼

wn 2

^cin ðsÞ ¼

w2n 2

^ bnð0Þ ðsÞ T ð0Þ

9 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > =

n

ðk3Þ xk 2m sijmn ðsÞ

ðk3Þ xk 2n sijnm ðsÞ

2

2

½ðk 4n þ x2 Þs0  k 2n þ xk2  2xekn s0   n

k 4n x2 ðs; 2Þ fi ðsÞ 2x

ðt; 2Þ

þ ðk 4n þ x2 Þfi

ðk1Þ k 2m k 2n sijmn ðsÞ;

h

x2 k 2n

2

2

2

e2kn ðss0 Þ  2xe2kn sþkn s0

ðsÞ

io 2 þ k 2n e2kn s ;

ðA:7Þ 9 > > > > > > > > > =

> ðc; 2Þ ðck; 2Þ ðk; 2Þ ð1; 2Þ > > k 2n ½fi ðsÞ  4fin ðsÞ þ finn ðsÞ þ 2fi ðsÞ > > > h io h i > > ð1Þ ðk5Þ 4 2 2 ðk3Þ 2 x2 k 2n s0 x2 ðk4Þ ; þ ðk n þ x Þs0  k n þ k2  2xe gi ðsÞ  k2 ginn ðsÞ  2xginn ðsÞ þ k n ginn ðsÞ ; > n

n

ðA:8Þ

M. Higuchi et al. / International Journal of Solids and Structures 44 (2007) 5316–5335

5333

where

2

ðsÞ

¼

2x ðv1 p2j Þ2 þð2xÞ2

ðc; 2Þ

ðsÞ

¼

 ðv

ðsÞ

¼

1 v1 p2j

hjmn ðsÞ

¼

2 2 2 1 ½eðkm þkn Þs0 ev1 pj ðss0 Þ v1 p2j ðk 2m þk 2n Þ

hj

ð1; 2Þ

hj

ðk; 2Þ

ðsk; 2Þ

ðsÞ

ðck; 2Þ

ðsÞ ¼

hjn hjn

ðk1Þ

ðk2Þ

qjmn ðsÞ ðk3Þ

qjmn ðsÞ

¼

v1 p2j

2

2

2

1 v1 p2j ðk 2m þk 2n Þ

2

 ev1 pj ðss0 Þ ;

¼

2 2 2 1 ½eðkm þkn Þsþkn s0 v1 p2j ðk 2m þk 2n Þ

 ekm s0 ev1 pj ðss0 Þ ;

ðs; 2Þ

1

ð1; 2Þ

ðsÞ

¼

1 v1 p2j

d ijmn ðsÞ

¼

ðsk; 2Þ

¼

ðck; 2Þ

ðsÞ ¼

ðv1 p2j Þ2 þð2xÞ2

ð1; 2Þ

½fi

fv1 p2j fi

ðs; 2Þ

f2xfi

ðc; 2Þ

ðsÞ  fij ðsÞ  gij ðsÞg;

ðc; 2Þ

ðsÞ  fij ðsÞ  gij ðsÞg;

ðsÞ  2x½fi

ðsÞ þ v1 p2j ½fi

ðpÞ

ðpÞ

ðp2Þ

9 > > > > > > > > > > > > > > > > > > > =

> > > > > > > > > 2 ðsk; 2Þ ðck; 2Þ ðpÞ 2 > 1 2 k n s0 ðp2Þ > fðv p  k Þ½f ðsÞ þ e g ðsÞ  x½f ðsÞ  f ðsÞg; 2 2 > 2 1 in ij in ij n j x2 þðk n v1 pj Þ > > > > > > ðsk; 2Þ ðck; 2Þ ðpÞ 2 1 k 2n s0 ðp2Þ 2 > ; fx½f ðsÞ þ e g ðsÞ þ ðv p  k Þ½f ðsÞ  f ðsÞg; 1 j in ij in ij n x2 þðk 2 v p2 Þ2 ðk; 2Þ

1

v1 p2j ðk 2m þk 2n Þ

n

ðpÞ

2

2

ðp2Þ

ðA:11Þ

½fimn ðsÞ  fij ðsÞ þ eðkm þkn Þs0 gij ðsÞ;

1 j

 eðkm þkn Þs0 gij ðsÞ;

ðk2Þ

¼

ðk4Þ 1 ½g ðsÞ v1 p2j ðk 2m þk 2n Þ ijmn

 gij ðsÞ;

¼

ðk5Þ 1 ½g ðsÞ v1 p2j ðk 2m þk 2n Þ ijmn

 ekm s0 gij ðsÞ;

ðk3Þ

ðp2Þ

ðp2Þ

ðk3Þ 1 ½g ðsÞ v1 p2j ðk 2m þk 2n Þ ijmn

sijmn ðsÞ

ðpÞ

ðsÞ  fij ðsÞ þ gij ðsÞ;

¼

sijmn ðsÞ

ðA:10Þ

> > > > > > ;

2

2

ðk1Þ

sijmn ðsÞ

9 > > > > > > =

2

2

2 2 1 ½eðkm þkn Þðss0 Þ v1 p2j ðk 2m þk 2n Þ

1 ðv1 p2j Þ2 þð2xÞ2

d ijn

2

¼

¼

d ijn ðsÞ

2

½eðkm þkn Þs  eðkm þkn Þs0 ev1 pj ðss0 Þ ;

ðsÞ

ðk; 2Þ

ðA:9Þ

2

 ev1 pj s ;

1 j

ðc; 2Þ

d ij

2

> > > > > > > > > 2s 2 ðss Þ 2 > v p v p 2 1 2 k n s0 0 1 j 1 j > ½xe þ ðv p  k Þe e ; > 2 2 2 1 2 n j > x þðk n v1 pj Þ > > > > > 2 2 2 2 v1 pj s > 1 k n s0 v1 pj ðss0 Þ 2 ; ½xe e  ðv p  k Þe ; 2 1 j n x2 þðk 2 v p2 Þ

¼

d ij

2

½ev1 pj s þ ev1 pj ðss0 Þ ;

½ev1 pj ðss0 Þ  ev1 pj s ;

ðsÞ

d ij

½ev1 pj s þ ev1 pj ðss0 Þ ;

2 2 2 1 pj Þ þð2xÞ

n

qjmn ðsÞ

ðs; 2Þ

¼

9 > > > > > > > > > > > > > > > > > > > =

2

ðs; 2Þ

hj

2

2

ðp2Þ

ðp2Þ

2

ðp2Þ

9 > > > > > = > > > > > ;

ðA:12Þ

5334

M. Higuchi et al. / International Journal of Solids and Structures 44 (2007) 5316–5335 ðs; 2Þ

2x ðsÞ ¼ X2 ð2xÞ 2 ½sin Xi s  sin Xi ðs  s0 Þ;

ðc; 2Þ

Xi ðsÞ ¼ X2 ð2xÞ 2 ½ sin Xi s0 sin Xi s  ð1 þ cos Xi s0 Þcos Xi s;

fi

9 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > =

i

fi

i

ð1; 2Þ

ðsÞ ¼ X1i ½sin Xi s0 sin Xi s  ð1  cos Xi s0 Þ cos Xi s; ðk; 2Þ fimn ðsÞ ¼ X2 þðk12 þk2 Þ2 ðk 2m þ k 2n Þ sinXi s  Xi cos Xi s fi

i

e ðsk; 2Þ

fin

m

n

o ½ðk 2m þ k 2n Þ sinXi ðs  s0 Þ  Xi cos Xi ðs  s0 Þ ;

ðk 2m þk 2n Þs0

ðsÞ ¼ x4 þ2x2 ðk4 X1 2 Þþðk4 þX2 Þ2  ½sinXi sfxðk 4n  X2i þ x2 Þ n

i

n

i

2

þekn s0 ½Xi ðk 4n þ X2i  x2 Þ sin Xi s0  k 2n ðk 4n þ X2i þ x2 Þ cos Xi s0 g n o > 2 > > þ cos Xi s 2xk 2n Xi þ ekn s0 ½Xi ðk 4n þ X2i  x2 Þcos Xi s0 þ k 2n ðk 4n þ X2i þ x2 Þsin Xi s0  ; > > > > n o >   > ðck; 2Þ 2 4 2 2 4 2 1 2 k 2n s0 2 > 2k n Xi sin Xi s0  ðk n  Xi þ x Þcos Xi s0 > fin ðsÞ ¼ x4 þ2x2 ðk4 X2 Þþðk4 þX2 Þ2  ½sin Xi s k n ðk n þ Xi þ x Þ  xe > > n n i i > > n o >  > 2  4 2 2 4 2 > >  cos Xi s Xi ðk n þ Xi  x2 Þ þ xekn s0 2k n Xi cos Xi s0 þ ðk n  Xi þ x2 Þsin Xi s0 ; > > > n > > ðp; 2Þ > > fij ðsÞ ¼ X2 þv1 2 p4 v1 p2j sin Xi s  Xi cos Xi s > > i 1 j > > o > > 2 v1 pj s0 > 2 > e ½v1 pj sin Xi ðs  s0 Þ  Xi cos Xi ðs  s0 Þ ; > > > > > ðt; 2Þ > ; fi ðsÞ ¼ 12 ½Xi s0 cos Xi ðs  s0 Þ þ sin Xi ðs  s0 Þ  sin Xi s Xi

ðk1Þ

1 gin ðsÞ ¼ X2 þk 4 fe i

k 2n s0

n

½k 2n sin Xi ðs  s0 Þ  Xi cos Xi ðs  s0 Þ þ Xi e

ðk2Þ

k 2n s

g;

k 2n ðss0 Þ

2 1 gin ðsÞ ¼ X2 þk ; 4 ½k n sin Xi ðs  s0 Þ  Xi cos Xi ðs  s0 Þ þ Xi e n i n o 2 2 2 2 ðk3Þ gimn ðsÞ ¼ X2 þðk12 þk2 Þ2 eðkm þkn Þs0 ½ðk 2m þ k 2n Þ sin Xi ðs  s0 Þ  Xi cos Xi ðs  s0 Þ þ Xi eðkm þkn Þs i

m

n

ðk4Þ

2

2

gimn ðsÞ ¼ X2 þðk12 þk2 Þ2 ½ðk 2m þ k 2n Þ sin Xi ðs  s0 Þ  Xi cos Xi ðs  s0 Þ þ Xi eðkm þkn Þðss0 Þ ; m n i n 2 o 2 2 2 ðk5Þ 1 gimn ðsÞ ¼ X2 þðk2 þk2 Þ2 ekm s0 ½ðk 2m þ k 2n Þ sin Xi ðs  s0 Þ  Xi cos Xi ðs  s0 Þ þ Xi eðkm þkn Þsþkn s0 ; i

m

n

2

ðp2Þ

gij ðsÞ ¼ X2 þv1 2 p4 ½v1 p2j sin Xi ðs  s0 Þ  Xi cos Xi ðs  s0 Þ þ Xi ev1 pj ðss0 Þ ; i

ð1Þ

j

gi ðsÞ ¼ X1i ½1  cos Xi ðs  s0 Þ:

ðA:13Þ 9 > > > > > > > > > > > > > > > > > > =

> > > > > > > > > > > > > > > > > > ; ðA:14Þ

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