Micromorphic continuum. Part I: Strain and stress tensors and their associated rates

P. Grammenoudis , Ch. Tsakmakis

Micropolar and micromorphic solids are continuum mechanics models, which take into account, in some sense, the microstructure of the considered real material. The characteristic property of such continua is that the state functions depend, besides the classical deformation of the macroscopic material body, also upon the deformation of the microcontinuum modelling the microstructure, and its gradient with respect to the space occupied by the material body. While micropolar plasticity theories, including nonlinear isotropic and nonlinear kinematic hardening, have been formulated, even for nonlinear geometry, few works are known yet about the formulation of (finite deformation) micromorphic plasticity. It is the aim of the three papers (Part I, Part II and Part III) to demonstrate how micromorphic plasticity theories may be formulated in a thermodynamically consistent way. In the present article we start by outlining the framework of the theory. Especially, we confine attention to the theory of Mindlin on continua with microstructure, which is formulated for small deformations. After precising some conceptual aspects concerning the notion of microcontinuum, we work out a finite deformation version of theory, suitable for our aims. It is examined that resulting basic field equations are the same as in the nonlinear theory of Eringen, which deals with a different definition of the microcontinuum. Furthermore, geometrical interpretations of strain and curvature tensors are elaborated. This allows to find out associated rates in a natural manner. Dual stress and double stress tensors, as well as associated rates, are then defined on the basis of the stress powers. This way, it is possible to relate strain tensors (respectively micromorphic curvature tensors) and stress tensors (respectively double stress tensors), as well as associated rates, independently of the particular constitutive properties.

Mots clés: Micromorphic plasticity; Finite deformation; Isotropic and kinematic hardening rules; Isotropic continuum damage mechanics; Finite element implementation

P. Grammenoudis, Ch. Tsakmakis. Micromorphic continuum. Part I: Strain and stress tensors and their associated rates. International Journal of Non-Linear Mechanics, 2009, 44 (9), pp.943. ⟨10.1016/j.ijnonlinmec.2009.05.005⟩. ⟨hal-00573457⟩