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Eshelby Stress Tensor T: a Variety of Conservation Laws for T in Finite Deformation Anisotropic Hyperelastic Solid & Defect Mechanics, and the MLPG-Eshelby Method in Computational Finite Deformation Solid Mechanics-Part I

Z. D. Han¹, S. N. Atluri^2,3

1 Livermore Software Technology Corporation, Livermore, CA, 94551, USA
2 International Collaboratory for Fundamental Studies in Engineering and the Sciences, 4131 Engineering Gateway, University of California, Irvine, Irvine, CA, 92697, USA
3 Fellow & Eminent Scholar, Texas Institute for Advanced Study, TAMU-3474, College Station, TX 77843, USA

Computer Modeling in Engineering & Sciences 2014, 97(1), 1-34. https://doi.org/10.3970/cmes.2014.097.001

Abstract

The concept of a stress tensor [for instance, the Cauchy stress σ, Cauchy (1789-1857); the first Piola-Kirchhoff stress P, Piola (1794-1850), and Kirchhoff (1824-1889); and the second Piola-Kirchhoff stress, S] plays a central role in Newtonian continuum mechanics, through a physical approach based on the conservation laws for linear and angular momenta. The pioneering work of Noether (1882-1935), and the extraordinarily seminal work of Eshelby (1916- 1981), lead to the concept of an “energy-momentum tensor” [Eshelby (1951)]. An alternate form of the “energy-momentum tensor” was also given by Eshelby (1975) by taking the two-point deformation gradient tensor as an independent field variable; and this leads to a stress measure T (which may be named as the Eshelby Stress Tensor). The corresponding conservation laws for T in terms of the pathindependent integrals, given by Eshelby (1975), were obtained through a sequence of imagined operations to “cut the stress states” in the current configuration. These imagined operations can not conceptually be extended to nonlinear steady state or transient dynamic problems [Eshelby (1975)]. To the authors’ knowledge, these path-independent integrals for dynamic finite-deformations of inhomogeneous materials were first derived by Atluri (1982) by examining the various internal and external work quantities during finite elasto-visco-plastic dynamic deformations, to derive the energy conservation laws, in the undeformed configuration [ref. to Eq. (18) in Atluri (1982)]. The stress tensor T was derived, independently, in its path-independent integral form for computational purposes [ref. to Eq. (30) in Atluri (1982)]. The corresponding integrals were successfully applied to nonlinear dynamic fracture analysis to determine “the energy change rate”, denoted as T*. A similar analytical work for elasto-statics was reported by Hill (1986). With the use of the stress measure T for finite-deformation solid and defect mechanics, the concept of “the strength of the singularities”, labeled in this paper as the vector T*, is formulated for a defective hyperelastic anisotropic solid undergoing finite deformations, in its various path-independent integral forms.
We first derive a vector balance law for the Eshelby stress tensor T, and show that it involves a mathematically “weak-form” of a vector momentum balance law for P. In small deformation linear elasticity (where P, S and σ are all equivalent), the stress tensor σ is linear in the deformation gradient F. Even in small deformation linear elasticity, the Eshelby Stress Tensor T is quadratic in F. By considering the various weak-forms of the balance law for T itself, we derive a variety of “conservation laws” for T in Section 2. We derive four important “path-independent” integrals, T_K^∗, T_L^∗(L) , T^∗(M), T_IJ^∗(G) , in addition to many others. We show the relation of T_K^∗, T_L^∗(L) , T^∗(M) integrals to the J-, L- and M- integrals given in Knowles and Sternberg (1972). The four laws derived in this paper are, however, valid for finite-deformation anisotropic hyperelastic solid- and defect-mechanics. Some discussions related to the use of T in general computational solid mechanics of finitely deformed solids are given in Section 3. The application of the Eshelby stress tensor in computing the deformation of a one-dimensional bar is formulated in Section 4 for illustration purposes. We present two computational approaches: the Primal Meshless Local Petrov Galerkin (MLPG)-Eshelby Method, and the Mixed MLPG-Eshelby Method, as applications of the original MLPG method proposed by Atluri (1998,2004). More general applications of T directly, in computational solid mechanics of finitely deformed solids, will be reported in our forthcoming papers, for mechanical problems, in their explicitly-linearized forms, through the Primal MLPG-Eshelby and the Mixed MLPG-Eshelby Methods.

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