A strain space gradient plasticity theory for finite strain

Abstract

In this paper, an extension to the finite deformation regime of the infinitesimal theories of strain gradient plasticity discussed in a paper by R. Chambon, D. Caillerie and T. Matsushimas [Int. J. Solids Struct. 38 (2001) 8503–8527] is presented which extends and generalizes the previous works of R. Chambon, D. Caillerie and C. Tamagnini [C.R. Acad. Sci. 329 (Série IIb) (2001) 797–802] and C. Tamagnini, R. Chambon and D. Caillerie [C.R. Acad. Sci. 329 (Série IIb) (2001) 735–739]. Central to the proposed theory are the kinematic assumptions concerning the decomposition of the assumed measures of strain and hyperstrain into an elastic and a plastic part. Following modern treatments of finite deformation plasticity, a multiplicative decomposition of the deformation gradient is postulated, while an additive decomposition is adopted for the second deformation gradient, as in the paper by R. Chambon, D. Caillerie and C. Tamagnini [C.R. Acad. Sci. 329 (Série IIb) (2001) 797–802]. The elastic constitutive equations for Kirchhoff stress and double stress tensors are obtained by assuming the existence of a suitable free energy function in the spatial description. The requirements of (i) invariance for rigid body motions superimposed upon the intermediate configuration; and, (ii) spatial covariance, see the book by J.E. Marsden and T.J.R. Hughes [Mathematical Foundations of Elasticity, Dover Publications Inc., New York, 1994], provide the corresponding material version of the hyperelastic constitutive equations. A fully covariant formulation of the evolution equations for the plastic strain and hyperstrain tensors, as well as for the internal variables is obtained by a straightforward application of the principle of maximum dissipation, after introducing a suitable yield condition in strain space. The resulting strain-space theory of second gradient plasticity can be considered an extension of the finite deformation plasticity theory proposed by J.C. Simo [Comput. Methods Appl. Mech. Engrg. 66 (1988) 199–219] to elastoplastic media with microstructure. As an example of application, a single-mechanism, isotropic hardening second gradient model for cohesive-frictional materials is proposed in which the cohesive component of the shear strength is assumed to increase with the magnitude of elastic hyperstrains, as advocated by, e.g. N.A. Fleck and J.W. Hutchinson [Strain gradient plasticity, in: J.W. Hutchinson, T.Y. Wu (Eds.), Advances in Applied Mechanics, vol. 33 Academic Press, 1997] for metals, based on both experimental observations and micromechanical considerations. Due to the internal length scales provided by the microstructure, the model is ideally suited for the analysis of failure problems in which strain localization into shear band occurs.