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Corrigendum to “Solution of a crack in an electrostrictive solid” [International Journal of Solids and Structures 47 (2010) 444–453]
International Journal of Solids and Structures 48 (2011) 1082–1083
Contents lists available at ScienceDirect
International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
Corrigendum
Corrigendum to ‘‘Solution of a crack in an electrostrictive solid’’ [International Journal of Solids and Structures 47 (2010) 444–453] Cun-Fa Gao a,b,⇑, Yiu-Wing Mai b, Ning Zhang a a b
College of Aerospace Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China Centre for Advanced Materials Technology (CAMT), School of Aerospace, Mechanical and Mechatronic Engineering (J07), University of Sydney, Sydney, NSW 2006, Australia
a r t i c l e
i n f o
a b s t r a c t
Article history: Available online 5 January 2011
In our previous work [Gao, C.F., Mai, Y.W., Zhang, N., 2010. Solution of a crack in an electrostrictive solid. International Journal of Solids and Structures 47, 444–453.] the intensity factor of the total stress for an impermeable crack is directly written by using the corresponding result of a permeable crack. This is based on the fact that an impermeable crack can be considered as a special case of a permeable crack where the electric field is not zero. However, the singularity of total stresses for the impermeable crack can also be analyzed directly from the complex potentials. In this Corrigendum, the singularity of the total stresses is further studied for the impermeable crack, and the intensity factors are re-derived by using the obtained complex potentials. It is shown that for an impermeable crack, the total stresses still have an inverse square-root singularity but their intensity factor is different from that obtained by the solution of a permeable crack. Therefore, it is concluded that solutions for impermeable cracks cannot be obtained directly from those of permeable cracks, since the assumption of the electric boundary condition has not only influenced the electric fields on the crack-faces but also on the electric body force inside the material. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Electrostriction Crack Field intensity factor
For an impermeable crack, we have from Eqs. (102)–(104) of Gao et al. (2010) that:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xðzÞ ¼ iE1 z2 a2 ; 2
ð102Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 z z2 a2 ð4C1 þ C2 Þ; /ðzÞ ¼ C1 z 2 a2 : uðzÞ ¼ C2 z ð4C1 þ C2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 z2 a2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
z
ð103Þ
1 z2 a2 ; x0 ðzÞ ¼ iE2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; xðzÞ ¼ iE1 2 2 2
ð104Þ
1 ; x00 ðzÞ ¼ iE2 x0 ðzÞ ¼ iE1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
z a
z
Then, the intensity factor of the total stress is given by:
pffiffiffiffiffiffi K I ¼ rM 22 j1 pa:
ð105Þ
Below, we have re-derived the solution directly from Eqs. (102)– (104). Firstly, the total stress can be expressed by:
1 h 2 þ z/00 ðzÞ þ u0 ðzÞ:
~ 22 ðx1 Þ þ ir ~ 12 ðx1 Þs stands for the singular part of the total where ½r stresses at the crack tip. Note that:
i
z a
a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ðz2 a2 Þ z2 a2
Hence, using Eq. (108), it can be shown that ahead of the crack tip where z = jx1j > a, we obtain:
i 1 D1 2 1 1 h 0 2 j x ðzÞx0 ðzÞ þ x00 ðzÞxðzÞ ¼ j 2 ¼ j E1 : 2 2 2 2 e
ð106Þ
h i ~ 12 ðx1 Þs ¼ /0 ðzÞ þ /0 ðzÞ þ z/00 ðzÞ þ u0 ðzÞ : ~ 22 ðx1 Þ þ ir ½r
The intensity factors of the total stress can be obtained from:
Using the following identities:
pffiffiffiffiffiffiffiffiffi ~ 12 ðx1 Þs ; ~ 22 ðx1 Þ þ ir K I þ iK II ¼ lim 2pr ½r
1 z /0 ðzÞ ¼ C1 ð4C1 þ C2 Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 z2 a2 2 1 a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; /00 ðzÞ ¼ ð4C1 þ C2 Þ 2 ðz2 a2 Þ z2 a2
jx1 j > a;
ð107Þ
DOI of original article: 10.1016/j.ijsolstr.2009.10.010
⇑ Corresponding author at: College of Aerospace Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China. E-mail addresses: [email protected] (C.-F. Gao), [email protected] (Y.-W. Mai). 0020-7683/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2010.11.026
ð109Þ
Inserting Eq. (109) into (107) leads to:
r~ 22 þ ir~ 12 ¼ j x0 ðzÞx0 ðzÞ þ x00 ðzÞxðzÞ þ /0 ðzÞ þ /0 ðzÞ
r!0
ð108Þ
u0 ðzÞ ¼ C2 þ
ð4C1 þ C2 Þ 2 z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; a 2 2 2 ðz a Þ z2 a2
ð110Þ
ð111Þ
C.-F. Gao et al. / International Journal of Solids and Structures 48 (2011) 1082–1083
and for z = jx1j > a we have:
1 x1 a2 z/00 ðx1 Þ þ u0 ðx1 Þ ¼ ð4C1 þ C2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x2 a2 x2 a2 1
pffiffiffiffiffiffi K I þ iK II ¼ ð4C1 þ C2 Þ pa:
1
ð4C1 þ C2 Þ 2 x1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ C2 ¼ C2 : þ a 2 x2 a2 x2 a2 1
1083
pffiffiffi which shows that the total stresses still have the 1= r singularity at the crack tip. Substituting Eq. (116) into (107) leads to:
1
ð112Þ Thus, Eq. (110) becomes:
ð117Þ
Using Eq. (60) (from Gao et al. (2010)), we obtain:
2 1 1 2 2 4C1 þ C2 ¼ j E1 þ e1 E2 ¼ j E1 þ rM 2 2 22 j1 2
ð118Þ
Substituting Eq. (118) into (117) gives: 0
~ 12 ðx1 Þs ¼ 2Re½/ ðx1 Þ; ~ 22 ðx1 Þ þ ir ½r
jx1 j > a:
Hence, the stress intensity factor is obtained from:
1
ð114Þ Inserting Eq. (114) into (113) yields,
jx1 j > a:
ð115Þ
KI ¼
h
rM22 j1 j E1 2
rM22 j1 j E1 2
2 ipffiffiffiffiffiffi pa:
ð119Þ
Comparing Eq. (119) with (105), we find that a constant is missing in Eq. (105). This implies that K-solutions for impermeable cracks cannot be obtained directly from those of permeable cracks because the assumption of electric boundary condition not only influences the electric variables on the crack-faces but also on the electric body force inside the material. Reference
That is,
a ~ 12 ðx1 Þs ¼ ð4C1 þ C2 Þ pffiffiffiffiffiffipffiffiffi ; ~ 22 ðx1 Þ þ ir ½r 2a r
2 ipffiffiffiffiffiffi pa:
3
1 1 x1 6 7 2Re½/0 ðx1 Þ ¼ 2Re4C1 ð4C1 þ C2 Þ þ ð4C1 þ C2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5: 2 2 x2 a2
x1 ~ 12 ðx1 Þs ¼ ð4C1 þ C2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ 22 ðx1 Þ þ ir ; ½r 2 x1 a2
K I þ iK II ¼
From Eq. (111), we have:
2
h
ð113Þ
jx1 j > a
ð116Þ
Gao, C.F., Mai, Y.W., Zhang, N., 2010. Solution of a crack in an electrostrictive solid. International Journal of Solids and Structures 47, 444–453.
Contents lists available at ScienceDirect
International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
Corrigendum
Corrigendum to ‘‘Solution of a crack in an electrostrictive solid’’ [International Journal of Solids and Structures 47 (2010) 444–453] Cun-Fa Gao a,b,⇑, Yiu-Wing Mai b, Ning Zhang a a b
College of Aerospace Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China Centre for Advanced Materials Technology (CAMT), School of Aerospace, Mechanical and Mechatronic Engineering (J07), University of Sydney, Sydney, NSW 2006, Australia
a r t i c l e
i n f o
a b s t r a c t
Article history: Available online 5 January 2011
In our previous work [Gao, C.F., Mai, Y.W., Zhang, N., 2010. Solution of a crack in an electrostrictive solid. International Journal of Solids and Structures 47, 444–453.] the intensity factor of the total stress for an impermeable crack is directly written by using the corresponding result of a permeable crack. This is based on the fact that an impermeable crack can be considered as a special case of a permeable crack where the electric field is not zero. However, the singularity of total stresses for the impermeable crack can also be analyzed directly from the complex potentials. In this Corrigendum, the singularity of the total stresses is further studied for the impermeable crack, and the intensity factors are re-derived by using the obtained complex potentials. It is shown that for an impermeable crack, the total stresses still have an inverse square-root singularity but their intensity factor is different from that obtained by the solution of a permeable crack. Therefore, it is concluded that solutions for impermeable cracks cannot be obtained directly from those of permeable cracks, since the assumption of the electric boundary condition has not only influenced the electric fields on the crack-faces but also on the electric body force inside the material. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Electrostriction Crack Field intensity factor
For an impermeable crack, we have from Eqs. (102)–(104) of Gao et al. (2010) that:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xðzÞ ¼ iE1 z2 a2 ; 2
ð102Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 z z2 a2 ð4C1 þ C2 Þ; /ðzÞ ¼ C1 z 2 a2 : uðzÞ ¼ C2 z ð4C1 þ C2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 z2 a2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
z
ð103Þ
1 z2 a2 ; x0 ðzÞ ¼ iE2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; xðzÞ ¼ iE1 2 2 2
ð104Þ
1 ; x00 ðzÞ ¼ iE2 x0 ðzÞ ¼ iE1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
z a
z
Then, the intensity factor of the total stress is given by:
pffiffiffiffiffiffi K I ¼ rM 22 j1 pa:
ð105Þ
Below, we have re-derived the solution directly from Eqs. (102)– (104). Firstly, the total stress can be expressed by:
1 h 2 þ z/00 ðzÞ þ u0 ðzÞ:
~ 22 ðx1 Þ þ ir ~ 12 ðx1 Þs stands for the singular part of the total where ½r stresses at the crack tip. Note that:
i
z a
a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ðz2 a2 Þ z2 a2
Hence, using Eq. (108), it can be shown that ahead of the crack tip where z = jx1j > a, we obtain:
i 1 D1 2 1 1 h 0 2 j x ðzÞx0 ðzÞ þ x00 ðzÞxðzÞ ¼ j 2 ¼ j E1 : 2 2 2 2 e
ð106Þ
h i ~ 12 ðx1 Þs ¼ /0 ðzÞ þ /0 ðzÞ þ z/00 ðzÞ þ u0 ðzÞ : ~ 22 ðx1 Þ þ ir ½r
The intensity factors of the total stress can be obtained from:
Using the following identities:
pffiffiffiffiffiffiffiffiffi ~ 12 ðx1 Þs ; ~ 22 ðx1 Þ þ ir K I þ iK II ¼ lim 2pr ½r
1 z /0 ðzÞ ¼ C1 ð4C1 þ C2 Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 z2 a2 2 1 a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; /00 ðzÞ ¼ ð4C1 þ C2 Þ 2 ðz2 a2 Þ z2 a2
jx1 j > a;
ð107Þ
DOI of original article: 10.1016/j.ijsolstr.2009.10.010
⇑ Corresponding author at: College of Aerospace Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China. E-mail addresses: [email protected] (C.-F. Gao), [email protected] (Y.-W. Mai). 0020-7683/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2010.11.026
ð109Þ
Inserting Eq. (109) into (107) leads to:
r~ 22 þ ir~ 12 ¼ j x0 ðzÞx0 ðzÞ þ x00 ðzÞxðzÞ þ /0 ðzÞ þ /0 ðzÞ
r!0
ð108Þ
u0 ðzÞ ¼ C2 þ
ð4C1 þ C2 Þ 2 z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; a 2 2 2 ðz a Þ z2 a2
ð110Þ
ð111Þ
C.-F. Gao et al. / International Journal of Solids and Structures 48 (2011) 1082–1083
and for z = jx1j > a we have:
1 x1 a2 z/00 ðx1 Þ þ u0 ðx1 Þ ¼ ð4C1 þ C2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x2 a2 x2 a2 1
pffiffiffiffiffiffi K I þ iK II ¼ ð4C1 þ C2 Þ pa:
1
ð4C1 þ C2 Þ 2 x1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ C2 ¼ C2 : þ a 2 x2 a2 x2 a2 1
1083
pffiffiffi which shows that the total stresses still have the 1= r singularity at the crack tip. Substituting Eq. (116) into (107) leads to:
1
ð112Þ Thus, Eq. (110) becomes:
ð117Þ
Using Eq. (60) (from Gao et al. (2010)), we obtain:
2 1 1 2 2 4C1 þ C2 ¼ j E1 þ e1 E2 ¼ j E1 þ rM 2 2 22 j1 2
ð118Þ
Substituting Eq. (118) into (117) gives: 0
~ 12 ðx1 Þs ¼ 2Re½/ ðx1 Þ; ~ 22 ðx1 Þ þ ir ½r
jx1 j > a:
Hence, the stress intensity factor is obtained from:
1
ð114Þ Inserting Eq. (114) into (113) yields,
jx1 j > a:
ð115Þ
KI ¼
h
rM22 j1 j E1 2
rM22 j1 j E1 2
2 ipffiffiffiffiffiffi pa:
ð119Þ
Comparing Eq. (119) with (105), we find that a constant is missing in Eq. (105). This implies that K-solutions for impermeable cracks cannot be obtained directly from those of permeable cracks because the assumption of electric boundary condition not only influences the electric variables on the crack-faces but also on the electric body force inside the material. Reference
That is,
a ~ 12 ðx1 Þs ¼ ð4C1 þ C2 Þ pffiffiffiffiffiffipffiffiffi ; ~ 22 ðx1 Þ þ ir ½r 2a r
2 ipffiffiffiffiffiffi pa:
3
1 1 x1 6 7 2Re½/0 ðx1 Þ ¼ 2Re4C1 ð4C1 þ C2 Þ þ ð4C1 þ C2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5: 2 2 x2 a2
x1 ~ 12 ðx1 Þs ¼ ð4C1 þ C2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ 22 ðx1 Þ þ ir ; ½r 2 x1 a2
K I þ iK II ¼
From Eq. (111), we have:
2
h
ð113Þ
jx1 j > a
ð116Þ
Gao, C.F., Mai, Y.W., Zhang, N., 2010. Solution of a crack in an electrostrictive solid. International Journal of Solids and Structures 47, 444–453.