Finding elastic constants for metals using molecular dynamics method

Introduction

Metals, as we know, are isotropic in nature (similar properties in various directions). But metallic single crystals show anisotropic behaviour. Generally, a material can exhibit 36 different elastic constant values.

    

   

(General relation between stress, strain and elastic constants)

The number of the independent elastic constant depends on the symmetry possessed by the symmetry of the material crystal. The metals generally are symmetric about X, Y, Z axes (Cubic). Hence they only have 3 independent elastic constants. They are C_11, C_12, C₄₄ (C₁₁= C₂₂= C₃₃, C₄₄= C₅₅= C₆₆, C₁₂= C₂₁= C₂₃= C₁₃= C₃₁= C₃₂ and remaining values are 0)

   

By computational method C₁₁, C₁₂, C₄₄ values are calculated. These are anisotropic properties of the material. To obtain isotropic properties like Young’s modulus, Shear modulus and Bulk modulus  Voigt averaging method is used :

Bulk modulus (B) = (C₁₁ + 2 C₁₂ )/ 3

Shear modulus (G) = (C₁₁ – C₁₂ + 3 C₄₄)/5

Young’s modulus (E) = (9 GB)/ (3 B + G)

Methodology

Two different molecular dynamic simulations are performed. In one simulation tensile strain is applied to fined C₁₁ and C₁₂, and in another shear strain is applied so as to fine C₄₄.

In tensile loading condition, all the strains except ε₃ (strain in the Z direction) were kept zero. So the above matrix becomes like this:

 

From the above matrix:

σ₃= C₁₁ * ε₃

C₁₁= (σ₃)/ ε₃

Also, σ₁= σ₂= C₁₂ * ε₃

C₁₂ = (σ₁)/ ε₃

In shear loading condition all the strains except ε₄ (strain in XY plane) were kept constant. So the matrix becomes like this:

 

   

From the above matrix:

σ₄= C₄₄ * ε₄

C₄₄= (σ₄)/ ε₄

_{After finding all the 3 constants, using Voigt approximation various elastic constants are found out. In my upcoming blogs, I will be showing some examples of the above mentioned calculations. Visit the blog regularly for further updates. If you have any doubts or suggestions, do mention them in the comment section.}