Skip to main page content
U.S. flag

An official website of the United States government

Dot gov

The .gov means it’s official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

Https

The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

Access keys NCBI Homepage MyNCBI Homepage Main Content Main Navigation
. 2020 Oct 1;13(19):4380.
doi: 10.3390/ma13194380.

Measurement Modulus of Elasticity Related to the Atomic Density of Planes in Unit Cell of Crystal Lattices

Affiliations

Measurement Modulus of Elasticity Related to the Atomic Density of Planes in Unit Cell of Crystal Lattices

Marzieh Rabiei et al. Materials (Basel). .

Abstract

Young's modulus (E) is one of the most important parameters in the mechanical properties of solid materials. Young's modulus is proportional to the stress and strain values. There are several experimental and theoretical methods for gaining Young's modulus values, such as stress-strain curves in compression and tensile tests, electromagnetic-acoustic resonance, ultrasonic pulse echo and density functional theory (DFT) in different basis sets. Apparently, preparing specimens for measuring Young's modulus through the experimental methods is not convenient and it is time-consuming. In addition, for calculating Young's modulus values by software, presumptions of data and structures are needed. Therefore, this new method for gaining the Young's modulus values of crystalline materials is presented. Herein, the new method for calculating Young's modulus of crystalline materials is extracted by X-ray diffraction. In this study, Young's modulus values were gained through the arbitrary planes such as random (hkl) in the research. In this study, calculation of Young's modulus through the relationship between elastic compliances, geometry of the crystal lattice and the planar density of each plane is obtained by X-ray diffraction. Sodium chloride (NaCl) with crystal lattices of FCC was selected as the example. The X-ray diffraction, elastic stiffness constant and elastic compliances values have been chosen by the X'Pert software, literature and experimental measurements, respectively. The elastic stiffness constant and Young's modulus of NaCl were measured by the ultrasonic technique and, finally, the results were in good agreement with the new method of this study. The aim of the modified Williamson-Hall (W-H) method in the uniform stress deformation model (USDM) utilized in this paper is to provide a new approach of using the W-H equation, so that a least squares technique can be applied to minimize the sources of errors.

Keywords: X-ray diffraction; Young’s modulus; crystalline materials; elastic compliances; modified W–H.; planar density.

PubMed Disclaimer

Conflict of interest statement

The authors declare no conflict of interest.

Figures

Figure 1
Figure 1
X-ray diffraction of compound x.
Figure 2
Figure 2
Elaborates of stress.
Figure 3
Figure 3
Young’s modulus extracted from planes of compound x versus planar density.
Figure 4
Figure 4
Schematics of (310) expanded in three unit cells.
Figure 5
Figure 5
(a) Longitudinal compression C11 and transverse expansion C12; (b) shear modulus C44 [39].
Figure 6
Figure 6
X-ray diffraction of NaCl powder sample.
Figure 7
Figure 7
Geometry and the situation of involved atoms in diffracted planes (a) (111), (b) (200) and (c) (220).
Figure 7
Figure 7
Geometry and the situation of involved atoms in diffracted planes (a) (111), (b) (200) and (c) (220).
Figure 8
Figure 8
Young’s modulus of each plane of NaCl extracted by XRD patterns versus planar density.
Figure 9
Figure 9
Modified (William–Hall, W–H) uniform stress deformation model (USDM) plot of NaCl.

Similar articles

Cited by

References

    1. Ma Y.Z., Sobernheim D., Grazon J.R. Unconventional Oil and Gas Resources Handbook. Elsevier; Amsterdam, The Netherlands: 2016. Glossary for Unconventional Oil and Gas Resource Evaluation and Development; pp. 513–526. Chapter 19. - DOI
    1. Ravindran P., Fast L., Korzhavyi P.A., Johansson B. Density functional theory for calculation of elastic properties of orthorhombic crystals: Application to TiSi2. Appl. Phys. 1998;84:4891. doi: 10.1063/1.368733. - DOI
    1. Fine M.E., Brown L.D., Marcus H.L. Elastic constants versus melting temperature in metals. Scr. Metall. 1984;18:951–956. doi: 10.1016/0036-9748(84)90267-9. - DOI
    1. Nakamura M. Elastic constants some transition metal-disilicide-single crystals. Met. Mater. Trans. A. 1994;25:331–340. doi: 10.1007/BF02647978. - DOI
    1. Brich F. Finite Elastic Strain of Cubic Crystals. Phys. Rev. 1947;71:809. doi: 10.1103/PhysRev.71.809. - DOI

LinkOut - more resources