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Discussions to the paper “Closed form solution for a nonlocal elastic bar in tension” by A.A. Pisano and P. Fuschi [Int. J. Solids Struct. 40 (2003) 13–23]
International Journal of Solids and Structures 62 (2015) 272
Contents lists available at ScienceDirect
International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
Discussion
Discussions to the paper ‘‘Closed form solution for a nonlocal elastic bar in tension’’ by A.A. Pisano and P. Fuschi [Int. J. Solids Struct. 40 (2003) 13–23] L. Ming a, T. Korakianitis b, P.H. Wen c,⇑ a b c
College of Mathematics, Taiyuan University of Technology, Taiyuan, China Parks College of Engineering, Aviation and Technology, Saint Louis University, St. Louis, MO 63103, USA School of Engineering and Materials Science, Queen Mary University of London, London E1 4NS, UK
In the paper Pisano and Fuschi (2003) by Pisano and Fuschi addressed the closed form solution for a nonlocal elastic bar in tension. However, this ‘‘closed form solution’’ is not correct. As this paper was cited many times recently by different authors as benchmark, it is worth to make correction on it. For one dimensional nonlocal elasticity problem, the constitutive relation is written as
r0 E
¼ n1 eðxÞ þ
n2 2l
Z
L
0
0
ejxx j=l eðx0 Þdx
Z L n 0 0 1 2 ejxx j=l eðx0 Þdx n1 2l 0
eðxÞ ¼ e
kl ðkxlxÞ=l 1 e þ eðklLklxLþxÞ=l : 2
L
0
0
ejxx j=l dx ¼ 2 ex=l eðLxÞ=l
0
ðLþxklLÞ=l kl e eð1klÞðLxÞ=l 2ð2 klÞ 1 eð1klÞðLxÞ=l eðLxÞ=l 2 ð6Þ
r0 ¼
ð3Þ
To verify this closed form solution, following integrals should be considered
Z
0
ð2Þ
where e ¼ r0 =E. A closed form solution of the above equation was reported by Pisano and Fuschi (Pisano and Fuschi, 2003) as
0
0
ejxx j=l eðklLklx Lþx Þ=l dx ¼
0
ð1Þ
0
e
L
and kl ¼ n2 =2n1 . Substituting (2) into the right hand side of (1), it is apparent that two sides of equation in (1) are not equal. It means that the closed form solution presented in Pisano and Fuschi (2003) is not correct. In addition, when n1 is taken to be zero, (1) becomes a Fredholm integral equation of the first kind as
and also can be rewritten as
eðxÞ ¼
Z
ð4Þ
E 2l
Z
L
0
0
ejxx j=l eðx0 Þdx
ð7Þ
0
A closed form solution of the equation above is found to be
eðxÞ ¼
r0 E
½1 þ dðxÞ þ dðx LÞ
ð8Þ
where dðxÞ is the Delta function. Numerical solution with high accuracy (Li et al., 2013) also verified that the ‘‘closed form’’ solution in Pisano and Fuschi (2003) is not correct.
References
0
Z 0
L
0
0
0
ejxx j=l eðkl1Þx =l dx ¼
ð2LxklLÞ=l kl e eð1klÞx=l 2ð2 klÞ 1 eð1klÞx=l ex=l 2
⇑ Corresponding author. Tel.: +44 (0) 2078825371. E-mail address: [email protected] (P.H. Wen). http://dx.doi.org/10.1016/j.ijsolstr.2015.02.013 0020-7683/Ó 2015 Elsevier Ltd. All rights reserved.
ð5Þ
Li, M., Hon, Y.C., Korakianitis, T., Wen, P.H., 2013. Finite integration method for nonlocal elastic bar under static and dynamic loads. Eng. Anal. Boundary Elem. 37 (5), 842–849. Pisano, A.A., Fuschi, P., 2003. Closed form solution for a nonlocal elastic bar in tension. Int. J. Solids Struct. 40, 13–23.
Contents lists available at ScienceDirect
International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
Discussion
Discussions to the paper ‘‘Closed form solution for a nonlocal elastic bar in tension’’ by A.A. Pisano and P. Fuschi [Int. J. Solids Struct. 40 (2003) 13–23] L. Ming a, T. Korakianitis b, P.H. Wen c,⇑ a b c
College of Mathematics, Taiyuan University of Technology, Taiyuan, China Parks College of Engineering, Aviation and Technology, Saint Louis University, St. Louis, MO 63103, USA School of Engineering and Materials Science, Queen Mary University of London, London E1 4NS, UK
In the paper Pisano and Fuschi (2003) by Pisano and Fuschi addressed the closed form solution for a nonlocal elastic bar in tension. However, this ‘‘closed form solution’’ is not correct. As this paper was cited many times recently by different authors as benchmark, it is worth to make correction on it. For one dimensional nonlocal elasticity problem, the constitutive relation is written as
r0 E
¼ n1 eðxÞ þ
n2 2l
Z
L
0
0
ejxx j=l eðx0 Þdx
Z L n 0 0 1 2 ejxx j=l eðx0 Þdx n1 2l 0
eðxÞ ¼ e
kl ðkxlxÞ=l 1 e þ eðklLklxLþxÞ=l : 2
L
0
0
ejxx j=l dx ¼ 2 ex=l eðLxÞ=l
0
ðLþxklLÞ=l kl e eð1klÞðLxÞ=l 2ð2 klÞ 1 eð1klÞðLxÞ=l eðLxÞ=l 2 ð6Þ
r0 ¼
ð3Þ
To verify this closed form solution, following integrals should be considered
Z
0
ð2Þ
where e ¼ r0 =E. A closed form solution of the above equation was reported by Pisano and Fuschi (Pisano and Fuschi, 2003) as
0
0
ejxx j=l eðklLklx Lþx Þ=l dx ¼
0
ð1Þ
0
e
L
and kl ¼ n2 =2n1 . Substituting (2) into the right hand side of (1), it is apparent that two sides of equation in (1) are not equal. It means that the closed form solution presented in Pisano and Fuschi (2003) is not correct. In addition, when n1 is taken to be zero, (1) becomes a Fredholm integral equation of the first kind as
and also can be rewritten as
eðxÞ ¼
Z
ð4Þ
E 2l
Z
L
0
0
ejxx j=l eðx0 Þdx
ð7Þ
0
A closed form solution of the equation above is found to be
eðxÞ ¼
r0 E
½1 þ dðxÞ þ dðx LÞ
ð8Þ
where dðxÞ is the Delta function. Numerical solution with high accuracy (Li et al., 2013) also verified that the ‘‘closed form’’ solution in Pisano and Fuschi (2003) is not correct.
References
0
Z 0
L
0
0
0
ejxx j=l eðkl1Þx =l dx ¼
ð2LxklLÞ=l kl e eð1klÞx=l 2ð2 klÞ 1 eð1klÞx=l ex=l 2
⇑ Corresponding author. Tel.: +44 (0) 2078825371. E-mail address: [email protected] (P.H. Wen). http://dx.doi.org/10.1016/j.ijsolstr.2015.02.013 0020-7683/Ó 2015 Elsevier Ltd. All rights reserved.
ð5Þ
Li, M., Hon, Y.C., Korakianitis, T., Wen, P.H., 2013. Finite integration method for nonlocal elastic bar under static and dynamic loads. Eng. Anal. Boundary Elem. 37 (5), 842–849. Pisano, A.A., Fuschi, P., 2003. Closed form solution for a nonlocal elastic bar in tension. Int. J. Solids Struct. 40, 13–23.