Applied Mathematics and Mechanics

, Volume 29, Issue 1, pp 67–74

# Strain analysis of nonlocal viscoelastic Kelvin bar in tension

• Zhao Xue-chuan  (赵雪川)
• Lei Yong-jun  (雷勇军)
• Zhou Jian-ping  (周建平)
Article

## Abstract

Based on viscoelastic Kelvin model and nonlocal relationship of strain and stress, a nonlocal constitutive relationship of viscoelasticity is obtained and the strain response of a bar in tension is studied. By transforming governing equation of the strain analysis into Volterra integration form and by choosing a symmetric exponential form of kernel function and adapting Neumann series, the closed-form solution of strain field of the bar is obtained. The creep process of the bar is presented. When time approaches infinite, the strain of bar is equal to the one of nonlocal elasticity.

## Key words

nonlocal theory Kelvin viscoelasticity Neumann series bar strain analysis

O345

## 2000 Mathematics Subject Classification

74K20 70K28 74S30

## References

1. [1]
Eringen A C, Speziale C G, Kim B S. Crack-tip problem in nonlocal elasticity[J]. Journal of Mechanic Physics Solids, 1977, 25(1):339–355.
2. [2]
Eringen A C. On continuous distributions of dislocations in nonlocal elasticity[J]. International Journal of Applied Physics, 1984, 56(10):2675–2680.
3. [3]
Dai T. Advances of generalized continuum field theories in China[J]. Journal of Liaoning University (Natural Science Edition), 1999, 26(1):1–11 (in Chinese).
4. [4]
Dai T. Advances of generalized continuum field theories in China[J]. Journal of Liaoning University (Natural Science Edition). 1999, 31(4):295–301 (in Chinese).Google Scholar
5. [5]
Huang Z. New points of view on the nonlocal field theory and their application to the fracture mechanics (II)—rediscuss nonlinear constitutive equations of nonlocal thermoelastic bodies[J]. Applied Mathematics and Mechanics (English Edition), 1999, 20(7):713–720.
6. [6]
Huang Z. New points of view on the nonlocal field theory and their applications to the fracture mechanics (III)—redicuss the linear theory of nonlocal elasticity[J]. Applied Mathematics and Mechanics (English Edition), 1999, 20(11):1193–1197.
7. [7]
Zhou Z, Wang B. Investigation of a griffith crack subject to uniform tension using the nonlocal theory by a new method[J]. Applied Mathematics and Mechanics (English Edition), 1999, 20(10):1025–1032.Google Scholar
8. [8]
Zhou Z, Wang B. Investigation of the scattering of harmonic elastic waves by two collinear symmetirc cracks using the non-local theory[J]. Applied Mathematics and Mechanics (English Edition), 2001, 22(7):682–690.Google Scholar
9. [9]
Pisano A A, Fuschi P. Closed form solution for a nonlocal elastic bar in tension[J]. International Journal of Solids and Structures, 2003, 40(1):13–23.
10. [10]
Weckner O, Abeyaratne R. The effect of long-range forces on the dynamics of a bar[J]. International Journal of Solids and Structures, 2005, 53(3):705–728.
11. [11]
Lei Y J, Friswell M I, Adhikari S. A Galerkin method for distributed systems with nonlocal damping[J]. International Journal of Solids and Structures, 2006, 43(11/12):3381–3400.
12. [12]
Adhikai S, Lei Y, Friswell M I. Dynamics of non-viscously damped distributed parameter systems[ R]. AIAA-2005-1951, 2005.Google Scholar
13. [13]
Ahmadi G. Linear theory of non-local viscoelasticity[J]. International Journal of Nonlinear Mechanics, 1975, 10(6):253–258.
14. [14]
Nowinski J L. On the non-local aspects of stress in a viscoelastic medium[J]. International Journal of Nonlinear Mechanics, 1986, 21(6):439–446.
15. [15]
Zhang Y. Thermoviscoelasticcity theory[M]. Tianjin: Tianjin University Press, 2002, 4–10.Google Scholar
16. [16]
Polizzotto C. Nonlocal elasticity and related variational principles[J]. International Journal of Solids and Structures, 2001, 38(42/43):7359–7380.
17. [17]
Polizzotto C, Fuschi P, Pisano A A. A strain-difference-based nonlocal elasticity model[J]. International Journal of Solids and Structures, 2004, 41(9/10):2383–2401.

© Editorial Committee of Appl. Math. Mech. and Springer-Verlag 2008

## Authors and Affiliations

• Zhao Xue-chuan  (赵雪川)
• 1
• Lei Yong-jun  (雷勇军)
• 1
Email author
• Zhou Jian-ping  (周建平)
• 1
1. 1.College of Aerospace and Material EngineeringNational University of Defense TechnologyChangshaP. R. China