Engineering Transactions, 54, 2, pp. 91-124, 2006
10.24423/engtrans.410.2006

Exact Representation of the Derivatives of Isotropic Tensor Functions with Respect to the Deformation Gradient F

Antonio Ercolano
Università degli studi di Cassino e del Lazio Meridionale Viale dell'Università
Italy

Expressions for derivatives of isotropic tensor functions with respect to the deformation tensor F are derived. Each derivative has the first representation in terms of eigenvectors; then, for computational conveniences, also a basis-free expression, in terms of eigen projections, is reported. Further, in the same fashion, also the time derivatives are provided. In the paper, a short review of different approaches to the problem existing in literature is presented. In order to make the exposition self-contained, some backgrounds of tensor analysis are also given.
Full Text: PDF
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).

References

Z. H. Guo, Rates of stretch tensors, J. Elasticity, 14, 263–267, 1984.

Z. H. Guo, Th. Lehmann, H.Y. Liang, and C.S. Man, Twirl tensor and tensor equation AX – XA = C, Journal of Elasticity, 27, 227–242 1992.

A. Hoger and D.E. Carlson, On the derivative of the square root of a tensor and Guo's rate theorems, J. Elasticity, 14, 329–336, 1984.

T.C.T. Ting, Determination of C112 ,C–112 and more general isotropic tensor functions of C, J. Elasticity, 15, 319–323, 1985.

T.C.T. Ting, New expression for solution for the matrix equation AT X + XA = H, J. Elasticity, 45, 61–72, 1996.

M. Scheidler, The tensor equation AX + X A = 1 (A,H), with applications to knematics of continua, J. Elasticity, 36, 117–153, 1994.

L. Rosati, A novel approach to the solutions of the tensor equation AX + X A = H, Int. J. solids struct., 37, 3457–3477, 2000.

L. Rosati, Derivatives and rates of stretch and rotation tensors, J. Elasticity, 56, 213–230, 1999.

L. Wheeler, On the derivatives of the stretch and rotation tensors with respect to the deformation gradient, J. Elasticity, 24, 129–133, 1990.

Y. Chen and L. Wheeler, Derivatives of the stretch and rotation tensors, J. Elasticity, 32, 175–182, 1993.

R. Hill, Constitutive inequalities for isotropic elastic solids under finite strain, Proc. R. Soc. London, A326, 131–147, 1970.

R. Hill, On constitutive inequalities for simple materials–I, J. Mech. Phys. Solids, 16, 229–242, 1968.

R. Hill, Aspects of invariance in solid mechanics, Advances in Applied Mechanics, 18, 1–75, 1978.

M. Scheidler, Times rates of generalized strain tensors, Part I: Component formulas, Mech. Mater., 11, 199–210, 1991.

H. Xiao, Unified explicit basis–free expression for time and conjugate stress of an arbitrary Hill's strain, Int. J. Solids Struct., 32, 3327–3340, 1995.

H. Xiao, O.T. Bruhns and A.T.M. Meyers, Strain rates and material spins, J. Elas¬ticity, 52, 1–41, 1998.

M. Itskov, Computation of the exponential and other isotropic tensor functions and their derivatives, Comput. Methods Appl. Mech. Engrg., 192, 3985–3999, 2003.

J.Lu, Exact expansions of arbitrary tensor functions F(A) and their derivatives, Int. J. Solids Struct., 41, 337–349, 2004.

C. Padovani, On the derivative of some tensor-valued functions, J. Elasticity, 58, 257¬268, 2000.

M. Silhavy, The mechanics and thermodynamics of continuos media, Springer, Berlin 1997.

G. Del Piero, Some properties of the set of fourth-order tensor with application to elasticity, J. Elasticity, 3, 245–261, 1979.

M. Itskov, On the theory of fourth–order tensors and their application in computational mechanics, Comput. Methods Appl. Mech. Engrg., 189, 419–438, 2000.

P. Halmos, Finite–dimensional vector spaces, Van Nostrand, New York 1958.

C. Truesdell and R. Toupin, The classical field theories. S. Flügge [Ed.], Handbuch der Physik, Vol. III/1, 226–858, Springer Verlag, Berlin, Gottingen, Heidelberg 1960.

C. Truesdell and W. Noll, The nonlinear field theories of mechanics, S. Flügge [Ed.], Handbuch der Physik, vol. 111/3, Springer, Berlin, 1965.

C.C. Wang and C. Truesdell, Introduction to rational elasticity, Leyden, Noordhoff 1973.

M.E. Gurtin, An introduction to continuum mechanic, Academic Press, New York 1981.

R.W. Ogden, Nonlinear elastic deformations, Ellis Horwood Chichester 1984.

T.C. Doyle and J.L. Ericksen, Nonlinear elasticity, Advances in Applied Mechanics, 4, 53–115.1956.

B.R. Setii, Generalized strain measures with applications to physical problems in Second-order effects in elasticity, Plasticity and fluid dynamics, M. Reiner and D. Abir [Eds.], Pergamon Press, Oxford, 162–172, 1964.

G. Stewart, On the early history of the singular value decomposition, SIAM Review, 35, 551–566, anonymous ftp: thales.cs.umd.edu, directory pub/reports, Dec. 1993.

J. Demmel, Applied numerical linear algebra, SIAM, Philadelphia, PA, 1997.

G. Golub and C. Van Loan, Matrix computations, Third edition, John Hopkins University Press, Baltimore, MD, 1996.

B. Parlett, The symmetric eigenvalue problem, SIAM, Philadelphia 1998.

E.P. Jiang and M. Berry, Solving total least–square problems in information retrieval, Linear Algebra and its Applications, 316, 137–156, 2000.

A. Ercolano, Uri argomento di Scienza delle Costruzioni: La Cinematica, Edizioni University di Cassino, Italy 2001.

T.P. Gialamas, D.T. Tsahalis, D.Otte, H. Van Der Auwaraer, D.A. Manolas, Substructuring technique: improvement by means of singular value decomposition (SVD), Applied Acoustics, 62, 1211–1219 2001.

M. Kobayashi, G. Dupret, Estimation of singular values of very large matrices using random sampling, Computer and Mathematics with Applications, 42, 1331–1352, 2001.

G.A. Holzapfel, Nonlinear solid mechanics, J. Wiley, England 2000.

J.D. Goddard and K. Ledniczky, On the spectral representation of stretch and rotation, J. Elasticity, 47, 255–259, 1997.

C. Miehe, Comparison of two algorithms for the computation of fourth-order isotropic tensor functions, Computers & Structures, 66, 1, 37–43 1998.




DOI: 10.24423/engtrans.410.2006