Fundamental Solutions for Unit Point Loading

5.2 Extended Displacement Discontinuity Method and Fundamental

5.2.1 Fundamental Solutions for Unit Point Loading

In order to build the solution for general loading and crack geometry, solutions for displacement and temperature discontinuities can be constructed first as funda- mental solutions, which can then be used as Green’s functions for general loading and interface crack geometry.

Consider a penny-shaped crack with radiusalying at the interface of two bonded dissimilar materials as illustrated in Fig.5.1. The crack is located in the planexoy, and the two bonded solids are assumed to occupy the upper and lower half-space, respectively. Then the relation between the Cartesian and cylindrical coordinates can be expressed as

x¼rcos/;

y¼rsin/;

R²¼r²þz²¼x²þy²þz²: 8<

: ð5:4Þ

According to Zhao et al. [40] and Zhao and Liu [41], if the radius of the crack a approaches zero, one can obtain the fundamental solutions corresponding to a

unit, concentrated temperature discontinuity and displacement discontinuity satis- fying the governing equations of thermoelasticity and the following conditions

a!0lim Z

S

k k;u k k;v k k;w k kh

f gdS¼f1;0;0;0g; ð5:5aÞ

a!lim0

Z

S

k k;u k k;v k k;w k kh

f gdS¼f0;1;0;0g; ð5:5bÞ

a!lim0

Z

S

k k;u k k;v k k;w k kh

f gdS¼f0;0;1;0g; ð5:5cÞ

a!0lim Z

S

k k;u k k;v k k;w k kh

f gdS¼f0;0;0;1g: ð5:5dÞ

wherek k;u k k;v k k;w andk kh are the elastic displacement and temperature discon- tinuities across the crack face. In the following sections, we present the fundamental solutions for each unit point loading case, which would be readily used as weight functions for general loading cases of arbitrary crack shapes. It is worth mentioning that as the point solutions are independent of the original shape of the penny-shaped crack, these solutions can be used to build the solutions for any interface crack of arbitrary shape under various loading conditions. We will show a few examples later, where fundamental solutions have been extended and used to build analytical solutions for interface crack problems in piezoelectric, piezoelectromagnetic, and quasi-crystalline materials.

Interface z

x

o y a

 r

S⁺

S

S^-

Fig. 5.1 A penny-shaped crack with radius ofalying in the interface plane [39]

5.2 Extended Displacement Discontinuity Method and Fundamental… 175

5.2.1.1 Solution for Unit-Point Displacement Discontinuity in the Z-Direction of the Crack

In this case, the boundary condition in Eq. (5.5c) can be rewritten as:

uðn;gÞ

k k ¼0; kvðn;gÞk ¼0; kwðn;gÞk ¼dðn;gÞ; khðn;gÞk ¼0: ð5:6Þ where d(n,η) is the Dirac delta function. This is a non-torsional axisymmetric problem. Introducing the potential functions proposed by Hou et al. [42], and the relatively completed non-torsional axisymmetric general solution around thez-axis is expressed in the following form:

2lu_r ¼@w₁

@r þz@w₂

@r ; ð5:7aÞ

2lw¼@w1

@z ð34vÞw₂þz@w2

@z þ4 1ð vÞw₃; ð5:7bÞ rz¼@²w₁

@z² 2 1ð vÞ@w2

@z þz@²w₂

@z² þ2 1ð vÞ@w3

@z ; ð5:7cÞ

rzr¼@²w₁

@r@zð12vÞ@w₂

@r þz@²w₂

@r@z þ2 1ð vÞ@w₃

@r ; ð5:7dÞ 2l

Ch¼@w₃

@z ; ð5:7eÞ

hz¼ bC 2l

@²w₃

@z² ; ð5:7fÞ

whereC¼_a^{2 1v}_ð^ð_1þv^Þ_Þ.

Using the zero-order Hankel transformation technique, the potential functions can be expressed as

w₁^þ ¼ Z¹

0

nA1e^nzJ0ð Þdn;nr ð5:8aÞ

w₁ ¼ Z¹

0

nA₂e^nzJ₀ð Þdn;nr ð5:8bÞ

w₂^þ ¼ Z¹

0

nB1e^nzJ0ð Þdn;nr ð5:8cÞ

w₂ ¼ Z¹

0

nB₂e^nzJ₀ð Þdn;nr ð5:8dÞ

w₃^þ ¼ Z¹

0

nC₁e^nzJ₀ð Þdn;nr ð5:8eÞ

w₃ ¼ Z¹

0

nC2e^nzJ0ð Þdnr n; ð5:8fÞ

where the superscript“+”and“^_”denote the upper and lower domain, respectively.

When the unit displacement discontinuity inz-direction is applied on the interfacial crack, the corresponding boundary conditions are given in the cylindrical coordi- nate system as

u_r^þðr;0Þ u_rðr;0Þ ¼0; w^þðr;0Þ wðr;0Þ ¼dð Þ;r h^þðr;0Þ hðr;0Þ ¼0; 8<

:

r_z^þðr;0Þ r_zðr;0Þ ¼0; r_zr^þðr;0Þ r_zrðr;0Þ ¼0; h_z^þðr;0Þ h_zðr;0Þ ¼0: 8>

<

>: ð5:9Þ

where 0rabelongs to the crack region. After inserting Eq. (5.8) into Eq. (5.9), one can obtain

A1

l₁¼^A_l²

2;

1

2l₁½nA1ð34v₁ÞB1þ4 1ð v₁ÞC1 _2l¹

2½nA2ð34v2ÞB2þ4 1ð v2ÞC2 ¼_2p¹; ^C_l¹

1C1¼^C_l²

2C2; 8>

>>

<

>>

>:

ð5:10aÞ

nA1þ2 1ð v1ÞB12 1ð v1ÞC1 ¼nA22 1ð v2ÞB2þ2 1ð v2ÞC2; nA1þð12v1ÞB12 1ð v1ÞC1 ¼ nA2þð12v2ÞB22 1ð v2ÞC2;

C1b₁

l₁ C₁¼^C_l^2b2

2 C₂; 8<

:

ð5:10bÞ where the subscripts“1”and“2”denote the upper and lower domains, respectively.

Solving these six equations one can get

5.2 Extended Displacement Discontinuity Method and Fundamental… 177

C1¼C2¼0 ð5:11aÞ A1¼ l₁ð23v13v2þ4v1v2Þ

3l₂þl₁4l₂v1

ð Þ lð ₂3l₁þ4l₁v2Þ 2 p 1 n¼A₁1

n ð5:11bÞ A2¼ l₂ð23v13v2þ4v1v2Þ

3l₂l₁þ4l₂v₁

ð Þðl₂þ3l₁4l₁v₂Þ 2 p 1 n¼A₂1

n ð5:11cÞ B1¼ l₁l₂

2ðl13l₂þ4l₂v1Þ 2

p¼ l₁l₂

p lð ₁þk1l₂Þ ð5:11dÞ B₂¼ l₁l₂

2ðl₂þ3l₁4l₁v2Þ 2

p¼ l₁l₂

p lð ₂þl₁k2Þ ð5:11eÞ wherek_a¼34v_a,A₁,A₂,B1andB2are all constants, and the potential functions are obtained as:

w₁^þ ¼A₁1

R ð5:12aÞ

w₁ ¼A₂1

R ð5:12bÞ

w₂^þ ¼B1

z

R³ ð5:12cÞ

w₂ ¼ B2

z

R³ ð5:12dÞ

w₃^þ ¼0 ð5:12eÞ

w₃ ¼0 ð5:12fÞ

where R¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi r²þz²

p . Then the corresponding displacements and stresses are obtained, for example, the stresses of an arbitrary point in the upper domain are obtained as:

r_z^þ ¼ l₁l₂

2p lð ₁þk1l₂Þ ð1þg₂Þ 1

R³ þ3 5ð g₂Þz² R⁵30z⁴

R⁷

ð5:13aÞ

r_zr^þ ¼ l₁l₂

2p lð ₁þk1l₂Þ 3 3ð g₂Þrz

R⁵30rz³ R⁷

ð5:13bÞ

and the forms of the stresses in the Cartesian coordinates are:

r_z^þ ¼ l₁l₂

2p lð ₁þk1l₂Þ ð1þg₂Þ 1

R³ þ3 5ð g₂Þz² R⁵30z⁴

R⁷

ð5:14aÞ

r_zx^þ ¼ l₁l₂

2p lð ₁þk₁l₂Þ 3 3ð g₂Þxz

R⁵30xz³ R⁷

ð5:14bÞ

r_yz^þ ¼ l₁l₂

2p lð ₁þk1l₂Þ 3 3ð g₂Þyz

R⁵30yz³ R⁷

ð5:14cÞ

whereg₂ ¼^l_l^1þl2k1

2þl₁k2.

5.2.1.2 Unit Point Temperature Discontinuity of the Crack

The boundary condition in Eq. (5.5d) can be rewritten as uðn;gÞ

k k ¼0; kvðn;gÞk ¼0; kwðn;gÞk ¼0; khðn;gÞk ¼dðn;gÞ: ð5:15Þ It is also a non-torsional axisymmetric problem. We can use the same form of potential functions in Eq. (5.7) and adopt the same zero-order Hankel transfor- mation in Eq. (5.8). When the unit temperature discontinuity is applied on the interface crack, the corresponding boundary conditions are as follows

u_r^þðr;0Þ u_rðr;0Þ ¼0 w^þðr;0Þ wðr;0Þ ¼0 h^þðr;0Þ hðr;0Þ ¼dð Þr 8<

:

r_z^þðr;0Þ r_zðr;0Þ ¼0 r_zr^þðr;0Þ r_zrðr;0Þ ¼0 h_z^þðr;0Þ h_zðr;0Þ ¼0 8>

<

>: ð5:16Þ

thus we can obtain

A₁ l₁¼^A_l²

1 2

l1½nA1ð34v1ÞB1þ4 1ð v1ÞC1 ¼_l¹

2½nA2ð34v2ÞB2þ4 1ð v2ÞC2 _2l^C1

1C1_2l^C2

2C2¼_2p¹¹_n 8>

<

>:

ð5:17aÞ nA₁þ2 1ð v₁ÞB12 1ð v₁ÞC1 ¼nA₂2 1ð v₂ÞB2þ2 1ð v₂ÞC2

nA1þð12v1ÞB12 1ð v1ÞC1 ¼ nA2þð12v2ÞB22 1ð v2ÞC2 C1b1

l1 C1¼^C_l^2b2

2 C2

8<

:

ð5:17bÞ 5.2 Extended Displacement Discontinuity Method and Fundamental… 179

Solving these six equations, we can get

A₁¼ðv₁1ÞC2b₂l₁ðl₂þk₂l₁Þ ðv₂1ÞC1b₁l₂ðl₁þk₁l₂Þ pC1C2ðb₁þb₂Þðl₁þk1l₂Þðl₂þk2l₁Þ

2l₁ n² ¼A₁ 1

n² ð5:18aÞ A₂¼ðv11ÞC2b₂l₁ðl₂þk2l₁Þ ðv21ÞC1b₁l₂ðl₁þk1l₂Þ

pC1C2ðb₁þb₂Þðl₁þk1l₂Þðl₂þk2l₁Þ

2l₂ n² ¼A₂ 1

n² ð5:18bÞ B₁¼ 4l₁l₂ðv11Þb₂

pC1ðb₁þb₂Þðl₁þk1l₂Þ 1 n¼B₁1

n ð5:18cÞ

B2¼ 4l₁l₂ðv₂1Þb₁ pC2ðb₁þb₂Þðl₂þk2l₁Þ

1 n¼B₂1

n ð5:18dÞ

C1¼ l₁b₂ pC1ðb₁þb₂Þ

1 n¼C₁1

n ð5:18eÞ

C2¼ l₂b₁ pC2ðb₁þb₂Þ

1 n¼C₂1

n ð5:18fÞ

where A₁, A₂, B₁, B₂, C₁ and C₂ are constants, and the potential functions are determined as:

w₁^þ ¼ A₁lnðRþzÞ ð5:19aÞ w₁ ¼ A₂lnðRzÞ ð5:19bÞ

w₂^þ ¼B₁1

R ð5:19cÞ

w₂ ¼B₂1

R ð5:19dÞ

w₃^þ ¼C₁1

R ð5:19eÞ

w₃ ¼C₂1

R ð5:19fÞ

Furthermore, the corresponding displacements, stresses, temperature and heat flux caused by the unit temperature discontinuity can be obtained. For example, the stresses and heatflux of an arbitrary point in the upper domain are given as:

r_z^þ ¼ l₁l₂ 2p lð ₁þk₁l₂Þ

2½a2ð1þv2Þb₁g₂þa1ð1þv1Þb₂ b₁þb₂

z

R³12a1ð1þv1Þb₂ b₁þb₂

z³ R⁵

ð5:20aÞ

r_zr^þ ¼ l₁l₂ 2p lð ₁þk₁l₂Þ

2½a2ð1þv2Þb₁g₂þa1ð1þv1Þb₂ b₁þb₂

r

R³12a1ð1þv1Þb₂ b₁þb₂

rz² R⁵

ð5:20bÞ h_z^þ ¼ b₁b₂

2p bð ₁þb₂Þ 1 R³3z²

R⁵

ð5:20cÞ

In terms of the Cartesian coordinates, they are:

r_z^þ ¼ l₁l₂ 2p lð ₁þk1l₂Þ

2½a2ð1þv2Þb1g₂þa1ð1þv1Þb2 b₁þb₂

z

R³12a1ð1þv1Þb2

b₁þb₂ z³ R⁵

ð5:21aÞ

r_zx^þ ¼ l₁l₂ 2p lð ₁þk1l₂Þ

2½a2ð1þv2Þb1g₂þa1ð1þv1Þb2 b₁þb₂

x

R³12a1ð1þv1Þb2

b₁þb₂ xz²

R⁵

ð5:21bÞ

r_yz^þ ¼ l₁l₂ 2p lð ₁þk1l₂Þ

2½a2ð1þv2Þb1g₂þa1ð1þv1Þb2 b₁þb₂

y

R³12a1ð1þv1Þb2

b₁þb₂ yz²

R⁵

ð5:21cÞ h_z^þ ¼ b₁b₂

2p bð ₁þb₂Þ 1 R³3z²

R⁵

ð5:21dÞ

5.2.1.3 Unit-Point Displacement Discontinuity in the y-Direction of the Crack

The boundary condition in Eq. (5.5b) can be rewritten as:

uðn;gÞ

k k ¼0; kvðn;gÞk ¼dðn;gÞ; kwðn;gÞk ¼0; khðn;gÞk ¼0: ð5:22Þ Obviously this problem is no longer non-torsional axisymmetric, which requires a complete, general solution. The relatively completed, general solution around the z-axis is expressed in the following form [42]:

5.2 Extended Displacement Discontinuity Method and Fundamental… 181

2lu_r¼ @w0

r@/þ@w1

@r þz@w2

@r ; ð5:23aÞ

2lu_/¼@w0

@r þ@w1

r@/þz@w2

r@/; ð5:23bÞ 2lw¼@w1

@z ð34vÞw₂þz@w2

@z þ4 1ð vÞw₃; ð5:23cÞ r_z¼@²w₁

@z² 2 1ð vÞ@w₂

@z þz@²w₂

@z² þ2 1ð vÞ@w₃

@z ; ð5:23dÞ

rzr ¼ 1 2r

@²w₀

@/@zþ@²w₁

@r@zð12vÞ@w₂

@r þz@²w₂

@r@z þ2 1ð vÞ@w₃

@r ; ð5:23eÞ

rz/¼1 2

@²w₀

@r@zþ1 r

@²w₁

@/@zð12vÞ1 r

@w2

@/ þz r

@²w₂

@/@zþ2 1ð vÞ1 r

@w3

@/ ; ð5:23fÞ 2l

Ch¼@w3

@z ; ð5:23gÞ

h_z¼ bC 2l

@²w₃

@z² ; ð5:23hÞ

When the unit displacement discontinuity in y-direction is applied on the interfacial crack, the corresponding boundary conditions are:

u_r^þðr;0Þ u_rðr;0Þ ¼dð Þr sin/; u_/^þðr;0Þ u_/ðr;0Þ ¼dð Þr cos/; w^þðr;0Þ wðr;0Þ ¼0; h^þðr;0Þ hðr;0Þ ¼0; 8>

><

>>

:

r_z^þðr;0Þ r_zðr;0Þ ¼0; r_zr^þðr;0Þ r_zrðr;0Þ ¼0; r_z/^þðr;0Þ r_z/ðr;0Þ ¼0; h_z^þðr;0Þ h_zðr;0Þ ¼0: 8>

>>

><

>>

>>

:

ð5:24Þ

According to the boundary conditions, thefirst-order Hankel transformation is introduced, and the corresponding potential functions are:

w₁^þ ¼ Z¹

0

nA1e^nzJ1ð Þdnnr sin/; ð5:25aÞ

w₁ ¼ Z¹

0

nA2e^nzJ1ð Þdnnr sin/; ð5:25bÞ

w₂^þ ¼ Z¹

0

nB1e^nzJ1ð Þdnnr sin/; ð5:25cÞ

w₂ ¼ Z¹

0

nB₂e^nzJ₁ð Þdnnr sin/; ð5:25dÞ

w₃^þ ¼ Z¹

0

nC₁e^nzJ₁ð Þdnnr sin/; ð5:25eÞ

w₃ ¼ Z¹

0

nC2e^nzJ1ð Þdnnr sin/; ð5:25fÞ

w₀^þ ¼ Z¹

0

nD1e^nzJ1ð Þdnnr cos/; ð5:25gÞ

w₀ ¼ Z¹

0

nD2e^nzJ1ð Þdnnr cos/; ð5:25hÞ

where superscripts“+”and“^_”denote the upper and lower domains, respectively.

Substituting Eq. (5.25) into the boundary conditions in Eq. (5.24) yields:

D1A1

l₁ ^D2_l^A2

2 ¼0

D₁þA₁

2l₁ ^D2_2l^þA2

2 ¼_pn¹

1

l₁½nA1ð34v1ÞB1þ4 1ð v1ÞC1 ¼_l¹

2½nA2ð34v2ÞB2þ4 1ð v2ÞC2 ^C_l¹

1C1¼^C_l²

2C2

8>

>>

><

>>

>>

:

ð5:26aÞ nA1þ2 1ð v1ÞB12 1ð v1ÞC1¼nA22 1ð v2ÞB2þ2 1ð v2ÞC2;

nðD1þ2A1Þ 2 1ð 2v1ÞB1þ4 1ð v1ÞC1¼nðD2þ2A2Þ 2 1ð 2v2ÞB2þ4 1ð v2ÞC2; nðD1þ2A1Þ þ2 1ð 2v1ÞB14 1ð v1ÞC1¼nðD22A2Þ þ2 1ð 2v2ÞB24 1ð v2ÞC2; C1b₁

l₁ C1¼C2b₂ l₂ C2; 8>

>>

>>

<

>>

>>

>:

ð5:26bÞ where subscripts “1”and“2” denote the upper and lower domains, respectively.

Solving Eq. (5.26), one can get

5.2 Extended Displacement Discontinuity Method and Fundamental… 183

A1¼ l₂ð3þ4v1Þ þl₁ð5þ6v1þ6v28v1v2Þ l₁þl₂k₁

ð Þðl₂þl₁k₂Þ

1 pn¼A₁1

n; ð5:27aÞ A₂¼ l₁ð34v2Þ þl₂ð56v26v2þ8v1v2Þ

l₁þl₂k1

ð Þðl₂þl₁k2Þ

1 pn¼A₂1

n; ð5:27bÞ B₁¼ l₁l₂

l23l₁þ4l₁v2

ð Þ

1

p¼ l₁l₂

p lð ₂þl₁k2Þ; ð5:27cÞ B₂¼ l₁l₂

l13l₂þ4l₂v1

ð Þ

1

p¼ l₁l₂

p lð ₁þl₂k1Þ; ð5:27dÞ

C₁¼C₂¼0; ð5:27eÞ

D₁¼ l₁l₂ l₁þl₂ 1

pn¼D₁1

n; ð5:27fÞ

D2¼ l₁l₂ l₁þl₂ 1

pn¼D₂1

n; ð5:27gÞ

wherek_a¼34v_a,A₁,A₂,D₁,D₂,B₁ andB₂ are all constants, and the potential functions are obtained as:

w₀^þ ¼D₁1 r 1z

R

cos/; ð5:28aÞ

w₀ ¼D₂1 r 1þ z

R

cos/; ð5:28bÞ

w₁^þ ¼A₁1 r 1z

R

sin/; ð5:28cÞ

w₁ ¼A₂1 r 1þ z

R

sin/; ð5:28dÞ

w₂^þ ¼B1

r

R³sin/; ð5:28eÞ

w₂ ¼ B2

r

R³sin/; ð5:28fÞ

w₃^þ ¼0; ð5:28gÞ

w₃ ¼0; ð5:28hÞ

where R¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi r²þz²

p . Then, the corresponding displacements and stresses are all obtained. For example, the stresses of an arbitrary point in the upper domain are obtained in the Cartesian coordinates:

r_z^þ ¼ l₁l₂ 2p lð ₁þk₁l₂Þ

3 1ð þg₂Þyz

R⁵ 30yz³ R⁷

; ð5:29aÞ

r_zx^þ ¼ l₁l₂ 2p lð ₁þk1l₂Þ

3 1ð g₁þg₂Þxy

R⁵ 30xyz² R⁷

; ð5:29bÞ

r_yz^þ ¼ l₁l₂ 2p lð ₁þk1l₂Þ

2g₁g₂1

R³ þ3ð1g₁þg₂Þy²þð2g₂Þz²

R⁵ 30y²z² R⁷

ð5:29cÞ

5.2.1.4 Unit Point Displacement Discontinuity in the x-Direction of the Crack

Similar to Sect.5.2.1.3, as the problem is symmetric about x and y-axis, under unit point displacement discontinuity in the x-direction, the stresses of an arbitrary point in the upper domain are obtained in the Cartesian coordinates

r_z^þ ¼ l₁l₂ 2p lð ₁þk₁l₂Þ

3 1ð þg₂Þxz

R⁵ 30xz³ R⁷

; ð5:30aÞ

r_zx^þ ¼ l₁l₂ 2p lð ₁þk1l₂Þ

2g₁g₂1

R³ þ3ð1g₁þg₂Þx²þð2g₂Þz²

R⁵ 30x²z² R⁷

; ð5:30bÞ

r_yz^þ ¼ l₁l₂ 2p lð ₁þk1l₂Þ

3 1ð g₁þg₂Þxy

R⁵ 30xyz² R⁷

ð5:30cÞ

5.2.2 Boundary Integral-Differential Equations