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The Pairing of Energy Momentum Tensor and Eigentransformation Rate in a Cylindrically Orthotropic Elastic Circular Tube or Bar

Published online by Cambridge University Press:  05 May 2011

Chien H. Wu*
Department of Civil and Materials Engineering, University of Illinois at Chicago, Chicago, Illinois 60607–7023, U.S.A.
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The pairing of a chemical potential and its associated concentration rate in the thermodynamic identity is well known. The existence of an experimentally determinable molar volume as a function of molar concentrations is also widely used in chemical engineering. What is perhaps less known and infrequently used is the fact that a spatially nonuniform molar volume leads to a field of geometrically incompatible eigenstrain, or eigentransformation in finite deformation. This incompatibility forces the material environment to deform, and the result is a strain energy trapped inside the material body. The change of this energy with respect to the eigentransformation is a generalized configurational stress, which, in the limit as the eigentransformation tends to the identity transformation, tends to the classical energy momentum tensor of Eshelby [1], or the so-called configurational stress. It is shown that the generalized configurational stress is an integral part of the chemical potentials that are responsible for atomic diffusion. This cycle of cause and effect is demonstrated in an axially symmetric setting where the material configuration is taken to be cylindrically orthotropic.

Symmetry can be used in many occasions to bare the simple meaning of a complex mathematical expression hiding under the disguise of tensors and index notation. We used it to elucidate the “missing” term in surface chemical potential in 1996 [2] and are now applying it to identify the chemical potential in the bulk. Both Professor Thomas C. T. Ting and I are civil engineering graduates of Tai-Da, the world-renowned National Taiwan University, but did not know each other until we joined UIC. It has been a wonderful friendship of many stimulating anisotropic discussions and numerous delicious potluck dinners. Happy birthday, Tom!

Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2003

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1Eshelby, J. D., “Energy Relations and the Energy-Momentum Tensor in Continuum Mechanics,” Inelastic Behavior of Solids, eds. Kanninen, M. F., Adler, W. F., Rosenfeld, A. R., and Jaffee, R. I., McGraw-Hill, NY, pp. 77114 (1970).Google Scholar
2Wu, C. H., “Axial Deformation as The-Simpler- The-Better Vehicle for Revealing the Missing Term in the Chemical Potential on Strained Surfaces,” The Chinese J. Mech., 12, pp. 5964 (1996).Google Scholar
3Wu, C. H., “The Role of Eshelby Stress in Composition-Generated and Stress-Assisted Diffusion,” J. Mech. Phys. Solids, 49, pp. 17711794 (2001).Google Scholar
4Sandler, S. I., Chemical and Engineering Thermodynamics, 3rd edition, John Wiley & Sons, Inc., New York (1999).Google Scholar
5Larche, F., and Cahn, J. W., “A Nonlinear Theory of Thermochemical Equilibrium of Solids under Stress,” Acta Metall., 26, pp. 5360 (1978).Google Scholar
6Ting, T. C. T., “Pressuring, Shearing, Torsion and Extension of a Circular Tube or Bar of Cylindrically Anisotropic Material,” Proc. R. Soc. Lond. A, 452, pp. 23972421(1996).Google Scholar
7Ting, T. C. T., “The Remarkable Nature of Cylindrically Orthotropic Elastic Material under Plane Strain Deformations,” Q. Jl Mech. Appl. Math., 52, pp. 387404 (1999).Google Scholar
8Ting, T. C. T., “New Solutions to Pressuring, Shearing, Torsion and Extension of a Cylindrically Anisotropic Elastic Circular Tube or Bar,” Proc. R. Soc. Lond. A, 455, pp. 35273542 (1999).Google Scholar
9Epstein, M. and Maugin, G. A., “The Energy Momentum Tensor and Material Uniformity in Finite Elasticity,” Acta Mechanica, 83, pp. 127133 (1990).Google Scholar
10Maugin, G. A., Material Inhomogeneities in Elasticity, Chapman and Hall, London (1993).Google Scholar
11Gurtin, M. E., Configurational Forces as Basic Concepts of Continuum Physics, Applied Math. Sci. 137, Springer (2000).Google Scholar
12Larche, F. and Cahn, J. W., “Thermochemical Equilibrium of Multiphase Solids under Stress,” Acta Metall., 26, pp. 15791589 (1978).Google Scholar
13Larche, F. and Cahn, J. W., “The Effect of Self-Stress on Diffusion in Solids,” Acta Metall., 30, pp. 18351845 (1982).Google Scholar
14Stephenson, G. B., “Deformation During Interdiffusion,” Acta Metall., 36, pp. 26632683 (1988).Google Scholar
15Bartholomeusz, B. J., “The Chemical Potential at the Surface of a Nonhydrostatically Stressed, Defect-Free Solid,” Phil. Magazine A, 71, pp. 489495 (1995).Google Scholar
16De Groot, S. R. and Mazur, P., Nonequilibrium Thermodynamics, Dover, New York (1984).Google Scholar
17Prigogine, I., Introduction to Thermodynamics of Irreversible Processes, 3rd ed.Wiley-Interscience, New York (1967).Google Scholar