# Inelastic Micromorphic Polycrystals

Chapter
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 20)

The principal task here is to find the simplest yet realistic way of describing polycrystal behaviour of metals taking into account inhomogeneous strains and stresses throughout a typical RVE. In such a problem grains of diverse orientations meet at their boundaries where most dislocations are concentrated. Intergranular and intragranular plastic sliding must be accompanied by thermoelastic straining in order to preserve continuity of the body. Even without external forces, residual stresses do exist and due to discontinuous change of orientation of neighboring grain lattices it is natural to expect appearance of couple stresses. Of special interest would be to connect material constants for stress and couple stress achieving their minimal number to be calibrated from specially designed experiments. Another already mentioned issue of great importance is Eshelby’s problem: how to insert a grain larger than its available “hole” into the material of the considered body. The usual answer to this question is obtained by the so-called self-consistent methods. Again the question arises as to which part of strain do we apply such an approach. The third issue which must be analyzed is proper geometry of the considered thermo-inelastic strain history for such a polycrystalline body.

The presentation first gives a short amendment to up to this point given geometrical analysis of finite thermo-inelastic strains of polycrystalline bodies. Here issue of micro and meso-rotations is especially considered. Then conditions for homogeneous total and/or elastic and plastic strains are formulated leading to balance laws. The same analysis has been applied to materials homogenized in such a way that deformation gradient, elastic and plastic distortion are linear functions of relative position inside the RVE. Constitutive equations for stress and its moment are formulated by the homogenization method. Finally, a brief account to evolution equations following mainly [Mic02a, Mic02b] is given.

## Keywords

Constitutive Equation Slip System Representative Volume Element Couple Stress Strain Path
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