PSI - Issue 13

Milan Micunovic et al. / Procedia Structural Integrity 13 (2018) 2158–2163 Micunovic, Kudrjavceva/ Structural Integrity Procedia 00 (2018) 000 – 000

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b. Incrementality is based on almost exact experimentally observed proportionality of equivalent stress rate and equivalent plastic strain rate. It is given by a universal material constant constituting most important part of the hereditary Rabotnov's kernel. The constant holds for diverse multiaxial stress histories and a wide range of strain rates - as it has been found in [6]. c. Non-associate flow rule. The evolution equation is not necessarily of associate type, i.e. plastic stretching should not be perpendicular to loading surface. However, if it is of associate type, then for transversely isotropic materials only five material constants suffice. Experiments in JRC, Ispra, Italy [6] confirmed that for isotropic reactor AISI reactor steels with b-property just four material constants lead to very high correlation coefficient. d. Tensor function representation has been extensively used in [6-12] for its development along the line developed by Sawczuk, Murakami and Boehler in the field of inelasticity. Such an approach makes the question of induced anisotropy logical and easy. e. Thermodynamic background fits into the Vakulenko's concept of thermodynamic time [1]. However, it has been necessary to extend his concept of steady aging introducing either accelerated or decelerated aging in order to cover creep-plasticity interaction (cf. [10,11]). Most specifically the Langevin ageing function is shown to describe all three known regimes of creep behavior. f. Plastic spin issue. The controversial issue of plastic spin has been solved by the concept of fixing of orientations of intermediate reference configurations (cf. [11]). Thus, there is no need for any new material constants in the evolution equation for plastic spin apart those already appearing in the evolution equation for plastic stretching [7]. In the next sections a brief review of QRI approach as well as the J 2 approach to viscoplasticity of transversely isotropic materials, is given. The last section contains an analysis of diffuse instability in QRI and classic J 2 concept.

2. Evolution and constitutive equations 2.1. Geometric preliminaries

The total deformation tensor of a representative volume element (RVE) consists (according to Kroener’s rule) of two incompatible constituents – plastic and elastic distortion tensors. It is essential that according to [4] plastic rotation of natural state reference configuration elements is arbitrary. On the other hand, a typical RVE is composed of Nmonocrystal grains such that each  -th grain has N s slip systems, (for fcc crystals N s =12). Comparing a RVE in natural state initial and current configurations, we may write Kroener’s formula for micro -distortions. However, there is an essential difference – micro-plastic (and accordingly micro-elastic) rotations of individual grains are here constrained since arbitrary plastic meso-rotation is assumed to be unity. This has two important consequences: 1. an improved Taylor’s procedure in the numerical treatment of polycrystals and 2. abundance of new material constants in the evolution equation for plastic spin is not necessary at all.

Slip system is described by   n is the unit vector normal to the slip plane. For convenience and easier description of climb and cross slip, let us introduce a third unit vector   z normal to the considered slip plane with diads         A n z 1 and         s z A 2 . If the time rate of residual microelastic strains is negligible, the volumes of all the grains inside the considered RVE are the same, and the microspins are very small then (for details see [8]) then the plastic stretching tensor for RVE reads:   1 . T P D N             D A A (1)         s n A , where   s is the unit slip vector and