含多層平面或圓柱形異質之熱彈性與黏彈性問題解析(III)

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含多層平面或圓柱形異質之熱彈性與黏彈性問題解析(3/3) 研究成果報告(完整版)

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執 行 期 間 ： 96 年 08 月 01 日至 97 年 07 月 31 日 執 行 單 位 ： 國立臺灣科技大學機械工程系

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中 華 民 國 97 年 07 月 11 日

含多層平面或圓柱形異質之熱彈性與黏彈性問題解析(3/3)

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中 華 民 國 97 年 7 月 10 日

Part 1

Thermal Stresses in a Viscoelastic Tri-Material With a Combination of a Point Heat Source and a Point Heat Sink

Abstract

A general solution for a thermoviscoelastic trimaterial combined with a point heat source and a point heat sink is presented in this work. Based on the method of analytic continuation associated with the alternation technique, the solutions to the heat conduction and thermoelastic problems for three dissimilar, sandwiched media are derived. A rapidly convergent series solution for both the temperature and stress field, expressed in terms of an explicit general term of the corresponding homogeneous potential, is obtained in an elegant form. The hereditary integral in conjunction with the Kelvin-Maxwell model is applied to simulate the thermoviscoelastic properties, while a thermorheologically simple material is considered. Based on the correspondence principle, the Laplace transformed thermoviscoelastic solution is directly determined from the corresponding thermoelastic one. The real time solution can then be solved numerically by taking the inverse Laplace transform. A typical example concerning the interfacial stresses generated from a combined arrangement of a heat source and sink are discussed in detail. The corresponding thin film problem is also discussed.

Keywords: thermoviscoelasticity; trimaterial; interfacial stresses; analytic continuation

1* Corresponding Author ([email protected])

1. Introduction

The stress analysis of a layered medium has become an important topic in recent years because of the steadily increased use of composite materials in many engineering applications. The analysis of the multilayered problem is complicated since the solutions must be forced to satisfy the continuity conditions of multiple interfaces. Consequently, the conventional approach to stress analysis for multilayered media problems requires the need to solve a system of simultaneous equations with a large number of unknown constants. For example, Iyengar and Alwar [1], as well as Chen [2], solved the semi-infinite medium composed of isotropic layers. Some more efficient methods dealing with the complicated continuity conditions have been proposed, and are mentioned as follows. Based on the transfer matrix approach, which is expressed in terms of the infinite series expansion allowing solutions with various orders of approximation to be obtained, Bufler [3] solved the elasticity problem of a multilayered medium. Based on the Fourier transform technique in association with the stiffness matrix approach, Choi and Thangjitham [4, 5] obtained solutions for multilayered anisotropic elastic media.

Choi and Earmme [6, 7] employed the alternating technique to obtain solutions for singularity problems in an isotropic and anisotropic trimaterial. Chao and Chen [8]

used analytic continuation associated with the alternating technique to solve the thermoelastic problem of an isotropic trimaterial.

For the thermoviscoelastic analysis, several authors [9, 10] derived the appropriate forms of free energy and the corresponding stress-strain relations and dissipation energy for a thermorheologically simple material from the view point of irreversible thermodynamics. Because of the complexity, most results reported in the literature are found in numerical approximation [11, 12, 13].

In this work, we consider the problem of a viscoelastic trimaterial combined with a point heat source and a point heat sink. The term, “trimaterial”, defined here, represents an infinite solid composed of three dissimilar materials bonded along two parallel interfaces. Based on the method of analytical continuation in conjunction with the alternate technique, the trimaterial solution can be derived in a series form from the corresponding homogeneous solution. A variety of problems (such as the bimaterial problem, or the thin-layer bonded to a half plane, or the finite strip of thin film, etc.), can be treated as special cases of the present study.

The general solution of the temperature field T , the total heat flux Q , the displacement derivatives and stresses for a thermoelastic medium are

1 ( ) ( )

T = 2⎡⎣θ z +θ z ⎤⎦ (1) - ( )- ( )

2

Q= k ⎡⎣θ z θ z ⎤⎦ (2)

1 2

2 (G u′+u′)= Φκ ( )z − Ω( ) (z − − Φz z) ′( ) 2z + Gβθ( )z (3)

22 i 12

σ − σ = ( )Φ z + Ω( ) (z + − Φz z) ′( )z (4) where z x= +₁ ix₂ denotes the complex coordinate, and ( )θ z is a complex

temperature function. A bar over the variable denotes the conjugate of a complex, and a prime notation indicates the derivative with respect to its argument. k and

G denote the heat conductivity and elastic shear modulus, respectively.

Furthermore, the material constants κ and β are defined as κ = −3 4ν and (1 )

β = +ν α for plane strain, and κ = −(3 ν) /(1+ν) and β α= for plane stress, where ν is Poisson’s ratio, and α is the coefficient of thermal expansion. The components of displacements and stresses can be expressed in terms of two complex stress functions, Φ( )z and Ω( )z , associated with a temperature function, ( )θ z , when the thermal effect is considered.

2. Temperature field of a trimaterial

Referring to Fig. 1, consider a dissimilar triple-layer medium with two perfectly bonded interfaces L ( x₂ = ) and 0 L (^* x₂ = ). Suppose that the regions h

D (a x₂ > ), h D (_b h x> ₂ > ), and 0 D (_c x₂ < ) are occupied by materials 0 a, b, and c, respectively.

For the first argument, we regard regions D and _a D to be composed of the _b same material b and region D of material _c c. If θ₀( )z signifies a potential for a singularity in an infinite homogeneous plane of material b, then θ_c1( )z , (analytic in D ) and _c θ₁( )z (analytic in D_a∪D_b) are introduced to satisfy the continuity conditions across L as

0 1

1

( ) ( ),

( ) ( ),

a b

c c

z z z D D

z z z D

θ θ

θ θ

+ ∈ ∪

= ⎨⎧⎩ ∈ (5)

The continuity of temperature and total heat flux across the interface, L , requires

0( )x1

θ +θ₁( )x₁ +θ₀( )x₁ +θ₁( )x₁ =θ_c1( )x₁ +θ_c1( )x₁

k [b θ₀( )x₁ +θ₁( )x₁ ]-k [_b θ₀( )x₁ +θ₁( )x₁ ]=k_cθ_c1( )x₁ -k_cθ_c1( )x₁ (6) Through standard analytic continuation arguments it follows that

1( )z

θ +θ₀( )z =θ_c1( )z , z D∈ _a ∪D_b

1( )z

θ +θ₀( )z =θ_c1( )z , z D∈ _c (7) and

kbθ₁( )z -k_bθ₀( )z =-k_cθ_c1( )z , z D∈ _a∪D_b

-k_bθ₁( )z +k_bθ₀( )z =k_cθ_c1( )z , z D∈ _c (8) Uncoupling Eqs. (7) and (8), we obtain

1( )z

θ =Λ_cbθ₀( )z , z D∈ _a∪D_b

1( )

c z

θ =Π_cbθ₀( )z , z D∈ _c (9) with

b c cb

b c

k k k k Λ = −

+ , 2 _b

cb

c b

k k k

Π = + (10)

Since this result is based on the assumption that region D is made up of material _a b, it cannot satisfy the continuity conditions at the interface L , which lies between ^* material a and b.

For the second argument, we assume regions D and _b D are made up of the _c same material b, and region D is composed of material _a a. Additional terms,

1( )

b z

θ (analytic in D_b∪D_c) and θ_a1( )z (analytic in D ) are introduced to satisfy _a the continuity conditions across the interface L such that ^*

1

0 1 1

( ) ( ),

( ) ( ) ( ),

a a

b b c

z z D

z z z z z D D

θ θ

θ θ θ

⎧ ∈

= ⎨⎩ + + ∈ ∪ (11)

Similarly, the continuity of the temperature and the total heat flux across the interface L requires *

0(x1 ih)

θ + +θ₁(x₁+ih)+θ_b1(x₁+ih)+θ₀(x₁−ih)+θ₁(x₁−ih)+θ_b1(x₁−ih)

=θ_a1(x₁+ih)+θ_a1(x₁−ih) and

k [b θ₀(x₁+ih)+θ₁(x₁+ih)+θ_b1(x₁+ih)]-k [_b θ₀(x₁−ih)+θ₁(x₁−ih)+

1( 1 )

b x ih

θ − ]

=k_aθ_a1(x₁+ih)-k_aθ_a1(x₁−ih) (12) According to analytic continuation, it leads to the results,

0(z ih)

θ + +θ₁(z ih+ )+θ_b1(z ih− )=θ_a1(z ih+ ), z D∈ _a

0(z ih)

θ − +θ₁(z ih− )+θ_b1(z ih+ )=θ_a1(z ih− ), z D∈ _b∪D_c (13) and

kbθ₀(z ih+ )+k_bθ₁(z ih+ )-k_bθ_b1(z ih− )=k_aθ_a1(z ih+ ), z D∈ _a

-k_bθ₀(z ih− )-k_bθ₁(z ih− )+k_bθ_b1(z ih+ )=-k_aθ_a1(z ih− ), z D∈ _b∪D_c (14) Uncoupling Eqs. (13) and (14), we obtain

1( )

a z

θ =Π [_ab θ₀( )z +θ₁( )z ], z D∈ _a

1( )

b z

θ =Λ [_ab θ₀(z−2 )ih +θ₁(z−2 )ih ], z D∈ _b∪D_c (15) with coefficients,

2 _b

ab

a b

k k k

Π = + , _ab ^{b a}

b a

k k k k Λ = −

+ (16) Since this result is based on the assumption that region D is made up of the same _c material b, it cannot satisfy the continuity conditions at the interface L .

For the third argument, we again assume D and _a D are made up of the same _b material b, and region D is composed of material _c c. Additional terms, θ₂( )z (analytic in D_a∪D_b) and θ_c2( )z (analytic in D ) are introduced to satisfy the _c continuity conditions across the interface L . By using a similar way to the previous approach, we can let

1 2

2

( ) ( ),

( ) ( ),

b a b

c c

z z z D D

z z z D

θ θ

θ θ

+ ∈ ∪

= ⎨⎧⎩ ∈ (17) and thereby obtain

2( )z

θ =Λ_cbθ_b1( )z , z D∈ _a ∪D_b

2( )

c z

θ =Π_cbθ_b1( )z , z D∈ _c (18) The method of analytic continuation is repeatedly performed across the two interfaces to achieve the additional terms; θ_ai( )z , ( )θ_bi z , ( )θ_ci z and θ_i( )z , for

2, 3, 4...

i= . Consequently, we find the complete solution of θ( )z as

0

( ),

( ) ( ) ( ) ( ),

( ),

a a

ba bc b

c c

z z D

z z z z z D

z z D

θ

θ θ θ θ

θ

⎧ ∈

=⎪⎨ + + ∈

⎪ ∈

⎩

(19) with

a( )z θ =

1

( )

n ai i

θ z

∑

( )

n i i

θ z

∑

θ =

1

( )

n bi i

θ z

∑

( 2 )

n i i

z ih θ

=

∑

bc( )z θ =

1

( )

n i i

θ z

∑

θ =

1

( )

n ci i

θ z

∑

( )

n bi i

θ z

∑

= Π_cbθ₀( )z + Π_cbΛ_abθ₀(z−2 )ih + Π_cbΛ_ab

1

( 2 )

n i i

z ih θ

=

∑

0

0 1

1

( ), 1

( ) [ ( 2 ) ( 2 )], 2

( 2 ), 3

cbcb

i cb ab

cb ab i

z i

z z ih z ih i

z ih i

θ

θ θ θ

θ₋

⎧ Λ =

= Λ Λ⎪⎨ − + + =

⎪ Λ Λ + ≥

⎩

(21)

For a point heat source of intensity q located at the point ₀ z , and a point heat sink _s of the same intensity located at the point z , the potential of the corresponding _k homogeneous problem is

0( ) log

2

o s

b k

q z z

z k z z

θ π

⎛ ⎞

− −

= ⎜⎝ − ⎟⎠ (22)

Note that Eq. (19) represents the solution when the singularities are located in region D . For the singularities located in other regions, the solution can also be found by b

using the same procedure.

3. Stress field of a trimaterial

Consider the stress field of a dissimilar triple-layer medium, with singularities located in the middle layer. (Refer to Fig. 1.) Similar to the previous approach, we first regard regions D and _a D as being composed of the same material _b b, and region D made of material _c c. Let

0 1

1

( ) ( ),

( ) ( ),

a b

c c

z z z D D

z z z D

Φ + Φ ∈ ∪

Φ = ⎨⎧⎩ Φ ∈

0 1

1

( ) ( ),

( ) ( ),

a b

c c

z z z D D

z z z D

Ω + Ω ∈ ∪

Ω = ⎨⎧⎩ Ω ∈ (23)

where Φ₀( )z and Ω₀( )z are the corresponding homogeneous solutions, while

1( )z

Φ , Φ_c1( )z , Ω₁( )z , and Ω_c1( )z are analytic functions. The continuity of traction and displacement across L yields

1( ) 1( ) 0( ) 1( ) 0( ) 1( )

c x c x x x x x

Φ + Ω = Φ + Φ + Ω + Ω

1 1

1 [ ( ) ( )] ( )

2 _c ^{c c} x ^c x ^{c c} x

G κ Φ − Ω +β θ =

0 1 0 1 0

1 [ ( ) ( ) ( ) ( )] [ ( ) ( ) ( )]

2 _b ^b x ^b x x x ^b x ^ba x ^bc x

G κ Φ + Φκ − Ω − Ω +β θ +θ +θ (24) By the standard analytic continuation arguments, it follows

1( ) 1( ) 0( )

c z z z

Ω = Φ + Ω , z D∈ _a ∪D_b

(25)

1( ) 0( ) 1( )

c z z z

Φ = Φ + Ω , z D∈ _c and

1( ) 2

c c

z G

−Ω = 1 _{1 0}

[ ( ) ( )] ( )

2 _b ^b z z ^{b ba} z

G κ Φ − Ω +β θ , z D∈ _a∪D_b

(26)

1( ) 2 ( )

c c

c c c

z z

G

κ ^Φ +β θ = 1 _{0 1 0}

[ ( ) ( )] [ ( ) ( )]

2 _b ^b z z ^b z ^bc z

G κ Φ − Ω +β θ +θ , z D∈ _c From Eqs. (25) and (26), we can find

1( )z

Φ =V_bcΩ₀( )z +θ_1bc( )z

1( )

c z

Ω = =(V_bc+ Ω1) ₀( )z +θ_1bc( )z

1( )z

Ω =U_bcΦ₀( )z +θ_2bc( )z

1( )

c z

Φ =(U_bc+ Φ1) ₀( )z +θ_2bc( )z where

V =bc ^{c b}

b c b

G G

G Gκ

−

+ _bc ^{c b b c}

b c c

G G

U G G

κ κ

κ

= −

+

1_bc( )z

θ = 2 _{b c b}

b c b

G G

G G

β κ

−

+ θ_bc( )z

2_bc( )z

θ = 2 ₀

{ [ ( ) ( )]- ( )}

b c

b ba c c

b c c

G G z z z

G G β θ θ β θ

κ + ⁺ (27)

and the homogeneous solutions are

0

0( ) log

( 1)

b b s

b b k

G q z z

z k z z

β π κ

⎛ − ⎞

Φ = + ⎜⎝ − ⎟⎠

0

0( ) log

( 1)

b b s s k

b b k s k

G q z z z z z z

z k z z z z z z

β π κ

⎡ ⎛ − ⎞ − − ⎤

Ω = + ⎢⎢⎣ ⎜⎝ − ⎟⎠+ − − − ⎥⎥⎦ (28)

Since the result is based on the assumption that D is made up of material _a b, it cannot satisfy the continuity condition across L . ^*

Next, we assume regions D and _b D are made up of the same material _c b, and region D is composed of material _a a. Additional terms, Φ_b1( )z and Ω_b1( )z (analytic in D_b∪D_c) and Φ_a1( )z and Ω_a1( )z (analytic in D ) are introduced to _a satisfy the continuity conditions across the interface L , so that ^*

1

1 1

( ) ( ),

( ) ( ),

a a

b b c

z z D

z z z z D D

Φ ∈

Φ = ⎨⎧⎩ Φ + Φ ∈ ∪

1

1 1

( ) ( ),

( ) ( ),

a a

b b c

z z D

z z z z D D

Ω ∈

Ω = ⎨⎧⎩ Ω + Ω ∈ ∪ (29)

The continuity of traction and displacement across L yields ^*

1( ) 1( ) 2 1( )

a x ih a x ih ih a′ x ih

Φ + + Ω − + Φ −

1(x ih) _b1(x ih) 1(x ih) _b1(x ih) 2ih 1′(x ih) 2ih ′_b1(x ih)

= Φ + + Φ + + Ω − + Ω − + Φ − + Φ −

1 1 1

1 [ ( ) ( ) 2 ( )] ( )

2 _a ^{a a} x ih ^a x ih ih ^a x ih ^{a a} x ih

G κ Φ + − Ω − − Φ′ − +β θ + =

1 1 1 1 1 1

1 [ ( ) ( ) ( ) ( ) 2 ( ) 2 ( )]

2 _b ^b x ih ^{b b} x ih x ih ^b x ih ih x ih ih ^b x ih

G κ Φ + + Φκ + − Ω − − Ω − − Φ′ − − Φ′ −

[ (0 ) ( ) ( )]

b x ih ba x ih bc x ih

β θ θ θ

+ + + + + + (30) By the standard analytic continuation arguments, it follows

1( ) 1( ) 1( ) 2 1( )

a x ih x ih b x ih ih ′b x ih

Φ + = Φ + + Ω − + Φ − , z D∈ _a

1( ) 2 1( ) 1( ) 1( ) 2 1( )

a x ih ih ′a x ih b x ih x ih ih ′ x ih

Ω − + Φ − = Φ + + Ω − + Φ − , z D∈ _b∪D_c

1

1 [ ( )] ( )

2 _a ^{a a} x ih ^{a a} x ih

G κ Φ + +β θ + =

1 1 1

1 [ ( ) ( ) 2 ( )]

2 _b ^b x ih ^b x ih ih ^b x ih

G κ Φ + − Ω − − Φ′ − +β θ_b[ (₀ x ih+ )+θ_bc(x ih+ )], z D∈ _a

1 1

1 [ ( ) 2 ( )]

2 _a ^a x ih ih ^a x ih

G −Ω − − Φ′ − =

1 1 1

1 [ ( ) ( ) 2 ( )]

2 _b ^{b b} x ih x ih ih x ih

G κ Φ + − Ω − − Φ′ − +β θ_b[ _ba(x ih+ )], z D∈ _b∪D_c (31) From Eqs. (30) and (31), we find

1( )

b z

Φ =V_ba[Ω₁(z−2 )ih +2ihΦ₁′(z−2 )]ih +θ_1ba( )z

1( )

a z

Ω =(V_ba+ Ω1)[ ₁( ) 2z − ihΦ₁′( )]z +2ihΦ′_a1( )z +θ_1ba(z−2 )ih

(32a)

1( )

b z

Ω =U_baΦ₁(z−2 )ih +2ihΦ′_a1( )z +θ_2ba(z−2 )ih

1( )

a z

Φ =(U_ba+ Φ1) ₁( )z +θ_2ba( )z where

a b

ba

b a b

G G

V G Gκ

= −

+ _ba ^{a b b a}

b a a

G G

U G G

κ κ

κ

= −

+

1_ba( )z

θ = 2

a b b ( )

ba

b a b

G G z

G G β θ κ

− +

(32b)

2_ba( )z

θ = 2 ₀

{ [ ( ) ( )] ( )}

a b

b bc a a

b a a

G G z z z

G G β θ θ β θ

κ + ^{+ −}

Since the result is based on the assumption that D is made up of material _c b, it cannot satisfy the continuity condition across L .

We again assume D and _a D are made up of the same material _b b, and region

D is composed of material c c. Additional terms, Φ₂( )z and Ω₂( )z (analytic in

a b

D ∪D ) with Φ_c2( )z and Ω_c2( )z (analytic in D ) are introduced to satisfy the _c continuity conditions across the interface L . Similar to the previous approach, we let

1 2

2

( ) ( ),

( ) ( ),

b a b

c c

z z z D D

z z z D

Φ + Φ ∈ ∪

Φ = ⎨⎧⎩ Φ ∈

(33)

1 2

2

( ) ( ),

( ) ( ),

b a b

c c

z z z D D

z z z D

Ω + Ω ∈ ∪

Ω = ⎨⎧⎩ Ω ∈

and find

2( )z

Φ =V_bcΩ₁( )z

2( )

c z

Ω = =(V_bc+ Ω1) ₁( )z

(34)

2( )z

Ω =U_bcΦ₁( )z

2( )

c z

Φ =(U_bc+ Φ1) ₁( )z

The method of analytical continuation is repeatedly performed across the two interfaces to achieve the additional terms f_ai( )z , ( )f_bi z , ( )f_ci z and f_i( )z for

2, 3, 4...

i= . Since all these procedures are similar to the previous approach, the details are suppressed here. The final results are as follows.

1

0

1 1

1

( ),

( ) ( ) ( ) ( ),

( ),

n

ai a

i

n n

bi i b

i i

n

ci c

i

z z D

z z z z z D

z z D

=

= =

=

⎧ Φ ∈

⎪⎪

Φ = Φ⎪⎨ + Φ + Φ ∈

⎪⎪

Φ ∈

⎪⎩

∑

∑ ∑

∑

(35)

1

0

1 1

1

( ),

( ) ( ) ( ) ( ),

( ),

n

ai a

i

n n

bi i b

i i

n

ci c

i

z z D

z z z z z D

z z D

=

= =

=

⎧ Ω ∈

⎪⎪

Ω = Ω⎪⎨ + Ω + Ω ∈

⎪⎪

Ω ∈

⎪⎩

∑

∑ ∑

∑

These include the following summations.

1

( )

n ai i

z

=

∑

1

( 1) ( )

n

ba i

i

U z

=

∑

1

( )

n bi i

z

=

∑

1

{ [ ( 2 ) 2 ( 2 )]}

n

ba i i

i

V z ih ih z ih

=

Ω − + Φ′ −

∑

1

( )

n i i

z

=

∑

1

( )

n bc i i

V ₋ z

=

∑

1

( )

n ci i

z

=

∑

1

( 1) ( )

n

bc i

i

U ₋ z

=

∑

1

( )

n ai i

z

=

∑

1

{( 1)[ ( ) 2 ( )] 2 ( )}

n

ba i i ai

i

V z ih z ih z

=

′ ′

+ Ω − Φ + Φ

∑

1

( )

n bi i

z

=

∑

1

{ ( 2 ) 2 ( )}

n

ba i ai

i

U z ih ih z

=

Φ − + Φ′

∑

1

( )

n i i

z

=

∑

1

( )

n

bc i i

U ₋ z

=

∑

1

( )

n ci i

z

=

∑

1

( 1) ( )

n

bc i

i

U ₋ z

=

∑

Note that the homogeneous solutions indicated in Eq. (28) are for the singularities located in region D . For the singularities located in other regions, the solution can _b also be found by using the same procedure.

4. Thermoviscoelastic formulation of a trimaterial

For a linear thermoviscoelastic material, the strains or stresses at any given time are the sum of the individual strain or stress increments through the respective time intervals during which they have been applied. By Boltzman’s superposition principle, the relationship between strains and stresses can be written in a hereditary integral [10]

( ) (0) ( ) 0 ( ) ( )

m smn n ^τsmn n d

ε τ =^ σ τ +

∫

∫

for m n, =1, 2,⋅⋅⋅ 6

Here, ξ is the dummy variable regarding the argument in question. τ =tb T( ) is designated as the reduced time, and the function b T indicates the temperature shift ( ) function which characterizes the time-dependent properties of the thermorhelogically simple material [10]. By using the Laplace transform, Eq. (37) becomes

ˆ_m( )p sˆ_mn( )p ˆ_n( )p ˆ_m( ) ( )p T pˆ

ε = σ +α (38)

where sˆ ( )_mn p = ps^ˆ_mn( )p , and αˆ_m= pα^ˆ_m( )p . Equation (38) is analogous to the thermoelastic constitutive equation. Consequently, similar to the thermoelastic problem in the previous discussion, the thermoviscoelastic field can be written as