Issue 24

Ig. S. Konovalenko et alii, Frattura ed Integrità Strutturale, 24 (2013) 75-80; DOI: 10.3221/IGF-ESIS.24.07 76 mechanical behavior of brittle porous materials from initiation of the first damages till cracks propagation and complete failure [6-9]. The calculations were carried out for a model material having mechanical properties of nanocrystalline ZrO 2 (Y 2 O 3 ) (yttria-stabilized zirconia) with the average pore size greater than the average grain size and two maxima in its pore size distribution function [2, 3]. The hierarchical model of the material with properties of the ceramics under consideration was constructed in several stages. At the first stage, the response of the ceramic material at the microscale (20÷250 μm) was simulated with explicit taking into account the porous structure of the material under various types of mechanical loading and the representative volume of this scale was determined. The result of the first stage was determination of the response function parameters of macroscale movable automata, corresponding to representative volume of the microscale. At the second stage, similar calculations were carried out at the macroscale with explicit taking into account of the porous structure of this scale. Data on the porous structure of lower scale were allowed for in the effective automata response functions defined at the first stage. At the final third stage, qualitative and quantitative verification of the developed model at the macroscale was made including comparison of simulation and experimental data. D ETERMINING THE REPRESENTATIVE VOLUME OF THE MICROSCALE AND THE MACROSCALE EFFECTIVE RESPONSE FUNCTION t the microscale of the proposed model, the representative volume is determined by analyzing the convergence of elastic and strength characteristics of porous model specimens with an increase in their dimension. Modeling of six groups of porous ceramics specimens under uniaxial compression and simple shear was performed. All the specimens in each group had the same dimension, but different pore distribution in space. Each group was consisted of six plane square specimens. Specimens under consideration had dimension (square side) of 20, 60, 100, 150, 200 and 250 μm according to the groups. It was supposed that all pores in the ceramics under investigation, as well as the model material, were equiaxed. The pore size of the model material, according to the maximum of the ceramics pore size distribution function, was equal to 3 μm [2, 3]. Diameter of the movable cellular automata, according to the average grain size, was equal to 1 μm [2, 3]. The pore structure of the specimens was specified by randomly removing individual automata and their six nearest neighbors. The total porosity for all the specimens was equal to 7% [2, 3]. Typical initial structure of one of the model specimens is shown in Fig. 1. (a) (b) (c) Figure 1 : Initial structure of a model specimen with a size of h = 60 μm and scheme of loading for shear (a) and uniaxial compression (b) ; the plot of velocity vs time for the upper automaton layer of the specimen (c) . The shear loading was simulated by setting up one and the same horizontal velocity to all automata in the upper layer with automata of the lower layer being rigidly fixed (Fig. 1,a,c). At the initial stage, the velocity of automata of the upper layer was increased by the sinusoidal law from 0 to 1 m/s and then was assumed to be constant. This scheme ensured a quasi- steady regime of loading and allowed dynamic effects to be eliminated until the first damage appeared. Duration of the loading velocity increase depended on the size of the specimen and was determined by preliminary calculations. All the samples had periodic boundary conditions in the direction of shear loading. The uniaxial compression loading was simulated by setting up one and the same velocity in vertical direction (up to 1 m/s) to all automata in the upper layer (Fig. 1,b,c). The vertical velocity of automata in the lower layer were set to zero. Displacements in horizontal direction were allowed for automata in the both lower and upper layers. The lateral surfaces of the specimen were free. The problem was solved under plane strain conditions. The response function of automata corresponded to the loading A