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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

Livermore Software Technology Corporation, Livermore, CA, 94551, USA

International Collaboratory for Fundamental Studies in Engineering and the Sciences, 4131 Engineering Gateway, University of California, Irvine, Irvine, CA, 92697, USA

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*, in addition to many others. We show the relation of

_{IJ}^{∗(G)}*T*,

_{K}^{∗}*T*,

_{L}^{∗(L)}*T*integrals to the

^{∗(M)}*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.

## Keywords

## Cite This Article

Han, Z. D., Atluri, S. N. (2014). 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.*CMES-Computer Modeling in Engineering & Sciences, 97(1)*, 1–34.