Abstract
In the paper the elasticity tensor and the relation between stress and strain of transverse isotropic material and isotropic material are deduced by tensor derivate. From the derivation the reason why there are two independent elasticity co-efficients for isotropic elastic material and five for transverse isotropic elastic material can be seen more clearly.
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Abbreviations
- U :
-
Function of strain energy
- Q :
-
Orthonormal tensor
- E :
-
Strain tensor
- E ij :
-
Components of the strain tensorE, i, j=1,2,3
- n :
-
Vector of the symmetric axis of the transverse isotropic material
- U *,E *,n * :
-
Forms ofU, E andn in another coordinate system
- J i E :
-
Main invariants of strain tensorE, i=1,2,3
- J i E,n :
-
Invariants of strain tensorE connecting with vectorn, i=4,5
- J i :
-
The abbreviated froms ofJ 1 E ,J 2 E ,J 3 E ,J /4 E, n ,J /5 E, n ,i=1,2,3,4,5
- β i :
-
Constants independent onE, n, i=1,2,3,4,5
- e i ,e m :
-
The covariant and contravariant orthonormal basis of the used coordinate system,l, m=1,2,3
- ⊗:
-
The symbol of tensor product
- I :
-
The identity tensor with the form ofe l ⊗e l
- l :
-
Fourth rank isotropic tensorI⊗I
- l (1324) :
-
The isomer of the fourth rank tensorl, with the form ofe l ⊗e m ⊗e l⊗e m
- C :
-
Elastic tensor, a fourth rank tensor
- λ, μ, α 1 ,α 2 ,α 3 :
-
Independent constants relating toβ 1 ,β 2 ,β 3 ,β 4 ,β 5
- C 11,C 12,C 13,C 33,C 44,C 55 :
-
Elastic coefficients of the transverse isotropic material
- T :
-
Stress tensor
- T ij :
-
Components of the stress tensorT, i, j=1,2,3
References
Chien Weizang, Yeh Kaiyuan.Elastic Mechanics [M]. Beijing: Science Press, 1956 (in Chinese)
Fung Y C.A First Course in Continuum Mechanics [M]. Beijing: Science Press, 1992, 211–217 (in Chinese)
Wu Jike, Wang Minzhong.Introduction to Elastic Mechanics [M]. Beijing: Peking University Press, 1981, 71–85 (in Chinese)
Hwang Kehchih, Xue Mingde, Lu Mingwan.Tensor Analysis [M]. Beijing: Tsinghua University Press, 1986, 74–78, 172–193 (in Chinese)
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Communicated by Yang Guitong
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Weiyi, C. Derivation of the general form of elasticity tensor of the transverse isotropic material by tensor derivate. Appl Math Mech 20, 309–314 (1999). https://doi.org/10.1007/BF02463857
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DOI: https://doi.org/10.1007/BF02463857