采用三维有限元方法计算了不同形状因子的γ强化相在基体为γ相的Ni基高温合金中引起的弹性应变能密度，进而建立了以形状因子为变量的弹性应变能密度表达式。通过最小化γ′强化相引起的弹性应变能和界面能之和，得到了γ′强化相的平衡形状与其特征半径之间的函数关系。本文的分析很好地解释了文献报道的Ni基高温合金中γ′强化相形状演变的实验规律，结果表明：通过三维有限元法结合强化相粒子形状近似法计算模型，可以给出复杂情况下强化相粒子引起的弹性应变能密度的表达式，并有效地应用于材料共格相变的热力学研究。
The expression for elastic energy due to coherent precipitates plays an important role in the thermodynamic calculation of phase transformations in precipitation strengthening materials for which elastic energy must be considered. However, in most cases, it was quite difficult to obtain analytic expressions for elastic strain energy in materials with anisotropic and/or inhomogeneous elasticity. The three- dimensional finite element method was a suitable straight forward technique in obtaining the expressions for elastic energy in materials with anisotropic and inhomogeneous elasticity. When the elastic energy due to coherent precipitates with different values of shape parameters were obtained by the finite element method, the approximate expression for the elastic energy can be conveniently established by means of data-fitting. As an example,the shape transitions of coherent γ′ precipitates from a sphere to a cube observed in Ni-base superalloys with γ matrix were investigated. The equilibrium shape of the γ′ precipitates was obtained by minimizing the sum of the elastic strain energy and interface energy. The calculation results are in good agreement with the theoretical and experimental data available.
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