Estimation of Young’s Modulus in Poly-Crystalline Metals Based on the Direct Averaging Method

Abstract

Metals have elastic anisotropy depending on the crystal orientation. In single crystalline metals, Young’s modulus is small toward <100> direction but large toward <111> direction. Diffraction analysis in poly crystalline metals yields another type of Young’s modulus termed as “Diffraction Young’s modulus; E^*_hkl ”, that is obtained in the relation between lattice strain and applied stress. Generally, the elastic anisotropy of diffraction Young’s modulus is smaller than that of single crystal because of the interaction among crystal grains. This means that the values of E^*_hkl reflect the elastic deformation behavior of each crystal grain in poly crystalline metals. Therefore, the standard Young’s modulus E_s, that is of poly crystal with uniform crystal orientations, should be obtained by averaging the values of E_hkl in all crystal grains. The calculation process was named “Direct averaging method” and it is as follows: Reciprocal of E^*_hkl has a linear relationship against the orientation parameter Γ, as expressed by the orientation dependence function: 1/E^*_hkl ＝a－bΓ. Based on the orientation dependence function, the value of E^*_hkl can be calculated for any crystal orientation <hkl>. The value of E_s is finally obtained by averaging all of E^*_hkl values. In the present investigation, the values of E_s were evaluated by applying the direct averaging method on aluminum, copper, nickel, iron (bcc) and iron (fcc) and the results: 70.3 GPa, 127 GPa, 190 GPa, 209 GPa, 203 GPa are obtained on E_s respectively. These values agree with those obtained experimentally on poly crystalline metals. In addition, it was also confirmed that the Reuss equation can be applied to metals with little elastic anisotropy such an aluminum but not to metals with large elastic anisotropy such copper, austenitic stainless steel, etc. As a result, it is concluded that Young’s modulus of poly crystalline metals corresponds to the average of the diffraction Young’s modulus in all crystal grains.