PSI - Issue 40

Artyom A. Tokarev et al. / Procedia Structural Integrity 40 (2022) 426–432 Artyom A. Tokarev, Anton Yu. Yants / Structural Integrity Procedia 00 (2022) 000 – 000

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Keywords: direct model; crystal plasticity; representative volume;

1. Introduction Since the origination of solid mechanics as a scientific discipline, full-scale experiments on loading (deformation) of macro samples of shape and size corresponding to accepted standards have been used as the main tool to establish the laws of material behavior and determine certain physical and mechanical properties of materials. In order to process the obtained values of force, moment, displacement, angle distortion and converse them into continuum characteristics (stresses, deformations), the hypothesis of a homogeneous stress – strain state of the investigated volume is usually used. However, available high-precision measurements indicate that the assumption about the homogeneity of the material behavior is not fulfilled at the meso- and microscale levels (except for the initial stage of deformation of monocrystalline samples). Based on experimental studies, mechanical engineers and physical metal scientists (Panin et al., 2006) have recently noted a significant influence of the internal (intergrain) and external boundaries of such a specimen on its behavior (both for monocrystal and for polycrystal grains) (Trusov, Shveykin, Nechaeva et al. (2012)). Therefore, the internal boundaries play an important role in forming stress concentrators and spreading plastic shear strains in the sample volume. In most cases, phenomenological models of solid mechanics are used to process full-scale experiments. Such models do not consider the multiscale nature of the ongoing sample deformation processes, as well as the influence of the internal and external boundaries of crystallites on these processes. It is known that the material microstructure and mesostructure determine its physical and mechanical properties, thus, ascertain the properties of a final product. Various models of materials are in need of carrying out of experiments and obtaining various characteristics of samples. In experiments, samples of various scales are loaded, and with the development of technologies and widespread miniaturization, experiments begin to be carried out on samples of meso- and micro-scales. This leads to the question: below what sample sizes does a significant deviation of the average response of the samples begin? Therefore, we introduce the definition of a sample representative volume as the minimum volume, when exceeding which the possible deviation of the mean response of the sample to the same load does not exceed the predetermined value. 2. Crystal plasticity model A crystal plasticity model was used to calculate the response of the material (Trusov, Shveykin, Nechaeva et al. (2012)). The crystal grain deformation is described by a sequence of plastic deformations p f (it is assumed that plastic shears do not change the orientation of the crystal lattice), rigid rotation r of an orthonormal coordinate system rigidly associated with one crystallographic direction and a crystallographic plane, and elastic distortion of the crystal lattice e f . In accordance with the proposed decomposition of motion, the spatial gradient is represented by a multiplicative decomposition (Trusov and Shveykin (2017)):

e p ,    f f f r

(1)

According to Trusov and Shveykin (2017), by neglecting the elastic distortions of the lattice ( e  f I ) and assuming that elastic distortion rates ( e  f 0 ) are not equal to zero, we obtain an additive decomposition of strain rate measure z , which is an asymmetric and indifferent (Truesdell (1965)) tensor of the 2nd rank (Trusov, Nechaeva, Shveykin (2013)):