Computational Mechanics

, Volume 63, Issue 5, pp 971–984 | Cite as

Heat capacity and thermal expansion of metal crystalline materials based on dynamic thermal vibration

  • Jieqiong ZhangEmail author
  • Junzhi Cui
  • Zihao YangEmail author
  • Yifan Yu
Original Paper


A novel approach based on dynamic thermal vibration is proposed to calculate the heat capacity and thermal expansion coefficient (TEC) for metal crystalline materials from 0 K to the melting point. The motion of metal atomic clusters is decomposed into structural deformation and thermal vibration. Then thermal vibration equations are established by the fourth-order Taylor expansion of Hamiltonian at the transient structural deformation position \({\bar{\mathbf {x}}}\). As a result, the thermal vibration frequencies dynamically change with the structural deformation positions and temperatures. A parameter \({\bar{\delta }} ({\bar{\mathbf {x}}}, T)\) is newly introduced to illustrate how the thermal vibration frequencies vary with the temperature T. Besides, the modified temperature-dependent Grüneisen parameter \({\bar{\gamma }} ({\bar{\mathbf {x}}}, T)\) is given. Finally, the formulae of heat capacity and TEC for metal crystalline materials are derived from the dynamic thermal vibration frequencies and \({\bar{\delta }} ({\bar{\mathbf {x}}}, T)\) as well as \({\bar{\gamma }} ({\bar{\mathbf {x}}}, T)\). The numerical results of heat capacity and TEC for metals Cu, Al, Au, Ag, Ni, Pd, Pt and Pb show a temperature dependence and agree well with the experimental data from 0 K to the melting point. This work suggests an efficient approach to calculate thermodynamic properties of metal materials for a wide range of temperatures, up to the melting point.


Metal crystalline materials Heat capacity Thermal expansion coefficient Dynamic thermal vibration Temperature dependence 



This research was financially supported by the National Key Research and Development Program of China (2016YFB1100602), National Natural Science Foundation of China (51739007, 11501449), the Fundamental Research Funds for the Central Universities (3102017zy043), and the China Postdoctoral Science Foundation (2018M633569).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anChina
  2. 2.LSEC, ICMSEC, Academy of Mathematics and Systems ScienceChinese Academy SciencesBeijingChina

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