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In this paper, we present the procedure of generalization and implementation of the Cauchy-Born approximation to the calculation of stress at finite temperature for alloy system in which the effects of inner displacement should be incorporated. With the help of quasi-harmonic approximation, a closed form of the first Piola-Kirchhoff stress is derived as a summation of pure deformation contribution and linear term due to thermal effects. For alloy system with periodic boundary condition, a further simplified formulation of stress based on some invariance constraints is derived in reciprocal space by using Fourier transformation, in which the temperature effect can be efficiently taking account. Several numerical examples are performed for various crystalline systems to validate our generalization procedure of finite temperature Cauchy-Born (FTCB) method for alloy.

On the Cauchy-Born approximation at finite temperature for alloys