Abstract
Taking into account X-ray diffraction, one of the well-known methods for calculating the stress-strain of crystals is Williamson-Hall (W–H). The W-H method has three models, namely (1) Uniform deformation model (UDM); (2) Uniform stress deformation model (USDM); and (3) Uniform deformation energy density model (UDEDM). The USDM and UDEDM models are directly related to the modulus of elasticity (E). Young’s modulus is a key parameter in engineering design and materials development. Young’s modulus is considered in USDM and UDEDM models, but in all previous studies, researchers used the average values of Young’s modulus or they calculated Young’s modulus only for a sharp peak of an XRD pattern or they extracted Young’s modulus from the literature. Therefore, these values are not representative of all peaks derived from X-ray diffraction; as a result, these values are not estimated with high accuracy. Nevertheless, in the current study, the W-H method is used considering the all diffracted planes of the unit cell and super cells (2 × 2 × 2) of Hydroxyapatite (HA), and a new method with the high accuracy of the W-H method in the USDM model is presented to calculate stress (σ) and strain (ε). The accounting for the planar density of atoms is the novelty of this work. Furthermore, the ultrasonic pulse-echo test is performed for the validation of the novelty assumptions.
Highlights
We have developed a method based on the linear regression of the Young’s modulus of each plane of the crystal lattice versus planar density to obtain a reliable total Young’s modulus of materials
In this study, for the first time, we calculated the exact planar density derived from the diffracted planes of hydroxyapatite in unit cells and super cells (2 × 2 × 2)
To determine the effect of cell size on the Young’s modulus, an extensive, exact calculation of the planar density of super cells (2 × 2 × 2) of hydroxyapatite, and comparing it with the result obtained from unit cell calculations, was performed
Summary
To determine the effect of cell size on the Young’s modulus, an extensive, exact calculation of the planar density of super cells (2 × 2 × 2) of hydroxyapatite, and comparing it with the result obtained from unit cell calculations, was performed. Ratan et al calculated the stress (σ) and strain (ε) of cadmium selenide (CdSe) nanoparticles and they considered the average value of Young’s modulus from the Williamson-Hall (W-H) method in the USDM model [8]. Submitted the W-H method in the USDM model for calculating stress and strain values of hydroxyapatite and they utilized the average value of Young’s modulus [10]. In all research that has utilized the W-H method in the USDM model, the Young’s modulus values of the investigated materials are reported as an average value or selected through the literature, and reported values are not inclusive of high accuracy. The evaluated results and the extracted values of this study were in good agreement with the theoretical, experimental and literature values
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