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
DOI:
https://doi.org/10.1090/S0033-569X-08-01108-2
Published electronically:
September 11, 2008
MathSciNet review:
2465142
Full-text PDF Free Access
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Additional Information
Abstract: The $n^{th}$ derivative of a tensor-valued function of a tensor is defined by a finite number of coefficients each with closed form expression.
References
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References
- D. E. Carlson and A. Hoger, The derivative of a tensor-valued function of a tensor, Q. Appl. Math. 44 (1986), 409β423. MR 860894 (87j:53024)
- 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, 54β68. MR 0334656 (48:12975)
- 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, 493β511. MR 2094142 (2005e:74001)
- W. F. Donoghue, Jr., Monotone matrix functions and analytic continuation, Springer-Verlag, New York, 1974. MR 0486556 (58:6279)
- Guan-Suo Dui, Some basis-free formulae for the time rate and conjugate stress of logarithmic strain tensor, J. Elasticity 83 (2006), no. 2, 113β151. MR 2242836
- R. Hill, Aspects of invariance in solid mechanics, Adv. Appl. Mech. 18 (1978), 1β75. MR 564892 (84f:73002)
- 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, 5963β5978. MR 2123747 (2005i:74003)
- C. S. Jog, Derivatives of the stretch, rotation and exponential tensors in n-dimensional vector spaces, J. Elasticity 82 (2006), no. 2, 175β192. MR 2230236 (2006m:74015)
- 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, 291β311. MR 2216494 (2007d:53018)
- R. W. Ogden, Non-linear elastic deformations, Ellis Horwood, 1984.
- L. Rosati, A novel approach to the solution of the tensor equation AX+XA=H, Int. J. Solids Struct. 37 (2000), no. 25, 3457β3477. MR 1751369 (2000m:74006)
- M. Scheidler, Time rates of generalized strain tensors. Part I: Component formulas, Mech. Materials 11 (1991), 199β210.
- ---, The tensor equation ${AX+XA=F(A,H)}$, with applications to kinematics of continua, J. Elasticity 36 (1994), no. 2, 117β153. MR 1314315 (96e:73019)
- H. Xiao, O. T. Bruhns, and A. Meyers, Strain rates and material spins, J. Elasticity 52 (1998), no. 1, 1β41. 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
Received by editor(s):
June 28, 2007
Published electronically:
September 11, 2008
Article copyright:
© Copyright 2008
Brown University
The copyright for this article reverts to public domain 28 years after publication.
