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Higher derivatives and the inverse derivative of a tensor-valued function of a tensor
Author:
Andrew N. Norris
Journal:
Quart. Appl. Math. 66 (2008), 725-741
MSC (2000):
Primary 15-XX, 15A24
Posted:
September 11, 2008
MathSciNet review:
2465142
Full-text PDF
Abstract |
References |
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Additional Information
Abstract: The  derivative of a tensor-valued function of a tensor is defined by a finite number of coefficients each with closed form expression.
- 1.
Donald
E. Carlson and Anne
Hoger, The derivative of a tensor-valued function of a tensor,
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MR 860894
(87j:53024)
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Chadwick and R.
W. Ogden, A theorem of tensor calculus and its application to
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(1971/72), 54–68. MR 0334656
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Chen and Guansuo
Dui, The derivative of isotropic tensor functions, elastic moduli
and stress rate. I. Eigenvalue formulation, Math. Mech. Solids
9 (2004), no. 5, 493–511. MR 2094142
(2005e:74001), http://dx.doi.org/10.1177/1081286504038672
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(2006), no. 2, 113–151. MR 2242836
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Itskov and Nuri
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non-symmetric tensor power series, Internat. J. Solids Structures
39 (2002), no. 24, 5963–5978. MR 2123747
(2005i:74003), http://dx.doi.org/10.1016/S0020-7683(02)00464-X
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S. Jog, Derivatives of the stretch, rotation and exponential
tensors in 𝑛-dimensional vector spaces, J. Elasticity
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O.
Kintzel and Y.
Başar, Fourth-order tensors—tensor differentiation
with applications to continuum mechanics. I. Classical tensor
analysis, ZAMM Z. Angew. Math. Mech. 86 (2006),
no. 4, 291–311. MR 2216494
(2007d:53018), http://dx.doi.org/10.1002/zamm.200410242
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R. W. Ogden, Non-linear elastic deformations, Ellis Horwood, 1984.
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L.
Rosati, A novel approach to the solution of the tensor equation
𝐴𝑋+𝑋𝐴=𝐻, Internat. J. Solids
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M. Scheidler, Time rates of generalized strain tensors. Part I: Component formulas, Mech. Materials 11 (1991), 199-210.
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Mike
Scheidler, The tensor equation
𝐴𝑋+𝑋𝐴=Φ(𝐴,𝐻), with
applications to kinematics of continua, J. Elasticity
36 (1994), no. 2, 117–153. MR 1314315
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Heng
Xiao, Otto
Timme Bruhns, and Albert
Thomas Marie Meyers, Strain rates and material spins, J.
Elasticity 52 (1998), no. 1, 1–41. MR 1673216
(99m:73021), http://dx.doi.org/10.1023/A:1007570827614
- 1.
- D. E. Carlson and A. Hoger, The derivative of a tensor-valued function of a tensor, Q. Appl. Math. 44 (1986), 409-423. MR 860894 (87j:53024)
- 2.
- P. Chadwick and R. W. Ogden, A theorem of tensor calculus and its application to isotropic elasticity, Arch. Rat. Mech. Anal. 44 (1971), no. 1, 54-68. MR 0334656 (48:12975)
- 3.
- Y-C. Chen and G-S. Dui, The derivative of isotropic tensor functions, elastic moduli and stress rate: I. Eigenvalue formulation, Math. Mech. Solids 9 (2004), no. 5, 493-511. MR 2094142 (2005e:74001)
- 4.
- W. F. Donoghue, Jr., Monotone matrix functions and analytic continuation, Springer-Verlag, New York, 1974. MR 0486556 (58:6279)
- 5.
- Guan-Suo Dui, Some basis-free formulae for the time rate and conjugate stress of logarithmic strain tensor, J. Elasticity 83 (2006), no. 2, 113-151. MR 2242836
- 6.
- R. Hill, Aspects of invariance in solid mechanics, Adv. Appl. Mech. 18 (1978), 1-75. MR 564892 (84f:73002)
- 7.
- M. Itskov and N. Aksel, A closed-form representation for the derivative of non-symmetric tensor power series, Int. J. Solids Struct. 39 (2002), no. 24, 5963-5978. MR 2123747 (2005i:74003)
- 8.
- C. S. Jog, Derivatives of the stretch, rotation and exponential tensors in n-dimensional vector spaces, J. Elasticity 82 (2006), no. 2, 175-192. MR 2230236 (2006m:74015)
- 9.
- O. Kintzel and Y. Bascedilar, Fourth-order tensors - tensor differentiation with applications to continuum mechanics. I: Classical tensor analysis, Z. Angew. Math. Mech. 86 (2006), no. 4, 291-311. MR 2216494 (2007d:53018)
- 10.
- R. W. Ogden, Non-linear elastic deformations, Ellis Horwood, 1984.
- 11.
- L. Rosati, A novel approach to the solution of the tensor equation AX+XA=H, Int. J. Solids Struct. 37 (2000), no. 25, 3457-3477. MR 1751369 (2000m:74006)
- 12.
- M. Scheidler, Time rates of generalized strain tensors. Part I: Component formulas, Mech. Materials 11 (1991), 199-210.
- 13.
- -, The tensor equation
 , with applications to kinematics of continua, J. Elasticity 36 (1994), no. 2, 117-153. MR 1314315 (96e:73019)
- 14.
- H. Xiao, O. T. Bruhns, and A. Meyers, Strain rates and material spins, J. Elasticity 52 (1998), no. 1, 1-41. MR 1673216 (99m:73021)
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Additional Information
Andrew N. Norris
Affiliation:
Mechanical and Aerospace Engineering, Rutgers University, Piscataway, New Jersey 08854-8058
Email:
norris@rutgers.edu
DOI:
http://dx.doi.org/10.1090/S0033-569X-08-01108-2
PII:
S 0033-569X(08)01108-2
Received by editor(s):
June 28, 2007
Posted:
September 11, 2008
Article copyright:
© Copyright 2008 Brown University
The copyright for this article reverts to public domain 28 years after publication.
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