PSI - Issue 5

Tomasz Trzepieciński et al. / Procedia Structural Integrity 5 (2017) 562 – 568 Mojtaba Biglar et al./ Structural Integrity Procedia 00 (2017) 000 – 000

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5. Average thorium

As mentioned before, a representative volume element (RVE) that was located in the neighbourhood of the macrostructure domains was applied to show the microstructure of the materials. Thus, it is essential to show the mechanical properties such as stress and strain at different scales with different superscripts Therefore, the stress and strain on the macro-scale, which can be referred to as macro-stress or macro-strain, can be represented, using the superscript “ M ”, as σ M and ε M , while to indicate the stress and strain on the micro- scale, the superscript “m” is chosen and the micro-stress and micro-strain are denoted by σ M and ε M , respectively. One the most important points that must be considered in multi-scale modelling is making connections between different scales. The average thorium is a useful method that can be applied to achieve this important goal. Hence, the average quantity of stress and strain of the RVE can be indicated by using an overbar as m  and m  , . a) b)

Fig. 2. (a) The final model chosen from the SEM microscope and (b) RVE of the grains

It is worth noting that as long as a rate problem is regarded in computations, an overdot will be applied for stress and strain as   and   and therefore the averages of the changes in quantities of stress and strain rates are equal to the rates of average changes of stress and strain as m m      and m m      . This paper considers the properties of barium titanate piezoelectric ceramic, which is a polycrystalline brittle material, and investigates the intergranular fractures in the micro-scale of this material. The RVE that is studied here is shown in Fig. 2b, which reveals randomly oriented and distributed single anisotropic elastic grain. Due to the small ratio of the grain boundaries’ area to the grain size, the properties of the grain boundaries are ignored in this study. In other words, the grain boundaries are considered part of the grain. As mentioned before, a random orientation was assigned to each grain, which gives each grain a different property from its neighbouring grains, and thus each grain possesses different properties and is considered as solid. Interaction between grains occurs, which can lead to intergranular fracture. Considering this assumption in the computation provides an opportunity to study the initiation and propagation of cracks in polycrystalline materials. 6. Results The stress – strain curves for the elastic parts for two polycrystalline aggregates are compared in Fig. 3. The most remarkable point that can be observed in this figure is that the slopes of the curves are very close together and are equal to the experimentally determined value of c 11 = 9.54 × 10 10 , which indicates that the developed method and generated numerical codes are sufficiently accurate and can be generalized for more complicated investigations such as studies of crack initiations and propagations. Furthermore, the curve which belongs to the artificial aggregate has fewer grains and the result for this polycrystalline aggregate is more accurate than the other one from the SEM image, which has more grains. This result is close to that of Benedetti and Aliabadi (2015), who found that aggregates with fewer grains have slopes closer to the experimental one c 11 .