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Creep and Stress Relaxation Behavior of Metal Matrix Particulate Composites Induced by Interface Diffusion with Coupled Thermomechanical Strains

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Iranian Journal of Science and Technology, Transactions of Mechanical Engineering Aims and scope Submit manuscript

Abstract

To investigate interfacial diffusion-induced creep and stress relaxation behavior of the particle-reinforced metal matrix composites, analytical solutions are obtained with coupled thermal and mechanical loads. The thermal expansion strains between the matrix and inclusion are considered as eigenstrains. Based on the Eshelby inclusion theory, the driving forces for interface diffusion and slip are obtained, respectively, and the relationship between them is elucidated. The effects of temperature-dependent elastic properties, thermal and mechanical loads on the creep rate and stress relaxation are estimated. Besides, the error caused by the scale effect is amended by fitting the finite element results with nonlinear least square method. The present results provide a straightforward guideline for designing high-quality metal matrix particulate composites.

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Acknowledgements

The authors would like to acknowledge the financial support by the Key Research and Development Plan of Shaanxi (Grant No. 2018ZDCXL-GY-03-01), the Natural Science Foundation of Shaanxi Province (Grant No. 2021JQ-466), and the Scientific Research Project of Shaanxi Education Department (Grant No. 20JK0799) and the Opening Fund of State Key Laboratory for Strength and Vibration of Mechanical Structures (Grant No. SV2021-KF-11).

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Correspondence to Yang Sun or Mabao Liu.

Appendix

Appendix

1.1 Appendix A. Explicit Formulations of Parameters in Eqs. (13) and (18)

The parameters in Eqs. (13) are listed as

$$ \left\{ \begin{gathered} k_{1} = \frac{{\alpha \left[ {\left( {3\alpha - 3 - 3f_{1} + 27f_{2} + 18\nu - 15\nu^{2} + 3\alpha f_{1} + 18\alpha f_{2} - 18\alpha \nu + 18\nu f_{1} - 72\nu f_{2} + 15\alpha \nu^{2} - 15\nu^{2} f_{1} + 45\nu^{2} f_{2} + ...} \right.} \right.}}{(\alpha - 4\nu + \alpha \nu + 2)(8\alpha - 5\nu - 10\alpha \nu + 7)} \hfill \\ \quad \frac{{\left. {15\alpha \nu^{2} f_{1} - 18\alpha \nu f_{1} - 18\alpha \nu f_{2} } \right) + \left. {\left( { - {{14E\Delta CTE} \mathord{\left/ {\vphantom {{14E\Delta CTE} \sigma }} \right. \kern-\nulldelimiterspace} \sigma } + {{10E\nu \Delta CTE} \mathord{\left/ {\vphantom {{10E\nu \Delta CTE} {\sigma + {{20E\alpha \nu \Delta CTE} \mathord{\left/ {\vphantom {{20E\alpha \nu \Delta CTE} {\sigma - {{16E\alpha \Delta CTE} \mathord{\left/ {\vphantom {{16E\alpha \Delta CTE} \sigma }} \right. \kern-\nulldelimiterspace} \sigma }}}} \right. \kern-\nulldelimiterspace} {\sigma - {{16E\alpha \Delta CTE} \mathord{\left/ {\vphantom {{16E\alpha \Delta CTE} \sigma }} \right. \kern-\nulldelimiterspace} \sigma }}}}}} \right. \kern-\nulldelimiterspace} {\sigma + {{20E\alpha \nu \Delta CTE} \mathord{\left/ {\vphantom {{20E\alpha \nu \Delta CTE} {\sigma - {{16E\alpha \Delta CTE} \mathord{\left/ {\vphantom {{16E\alpha \Delta CTE} \sigma }} \right. \kern-\nulldelimiterspace} \sigma }}}} \right. \kern-\nulldelimiterspace} {\sigma - {{16E\alpha \Delta CTE} \mathord{\left/ {\vphantom {{16E\alpha \Delta CTE} \sigma }} \right. \kern-\nulldelimiterspace} \sigma }}}}}} \right)\Delta T} \right]}}{(\alpha - 4\nu + \alpha \nu + 2)(8\alpha - 5\nu - 10\alpha \nu + 7)} \hfill \\ k_{2} = \frac{{\alpha \left[ {\left( {27 + 18\alpha - 3f_{1} - 3f_{2} - 72\nu + 45\nu^{2} + 3\alpha f_{1} + 3\alpha f_{2} - 18\alpha \nu + 18\nu f_{1} + 18\nu f_{2} - 15\nu^{2} f_{1} - 15\nu^{2} f_{2} + 15\alpha \nu^{2} f_{1} + ...} \right.} \right.}}{(\alpha - 4\nu + \alpha \nu + 2)(8\alpha - 5\nu - 10\alpha \nu + 7)} \hfill \\ \quad \frac{{\left. {15\alpha \nu^{2} f_{2} - 18\alpha \nu f_{1} - 18\alpha \nu f_{2} } \right) + \left. {\left( { - {{14E\Delta CTE} \mathord{\left/ {\vphantom {{14E\Delta CTE} \sigma }} \right. \kern-\nulldelimiterspace} \sigma } + {{10E\nu \Delta CTE} \mathord{\left/ {\vphantom {{10E\nu \Delta CTE} {\sigma + {{20E\alpha \nu \Delta CTE} \mathord{\left/ {\vphantom {{20E\alpha \nu \Delta CTE} {\sigma - {{16E\alpha \Delta CTE} \mathord{\left/ {\vphantom {{16E\alpha \Delta CTE} \sigma }} \right. \kern-\nulldelimiterspace} \sigma }}}} \right. \kern-\nulldelimiterspace} {\sigma - {{16E\alpha \Delta CTE} \mathord{\left/ {\vphantom {{16E\alpha \Delta CTE} \sigma }} \right. \kern-\nulldelimiterspace} \sigma }}}}}} \right. \kern-\nulldelimiterspace} {\sigma + {{20E\alpha \nu \Delta CTE} \mathord{\left/ {\vphantom {{20E\alpha \nu \Delta CTE} {\sigma - {{16E\alpha \Delta CTE} \mathord{\left/ {\vphantom {{16E\alpha \Delta CTE} \sigma }} \right. \kern-\nulldelimiterspace} \sigma }}}} \right. \kern-\nulldelimiterspace} {\sigma - {{16E\alpha \Delta CTE} \mathord{\left/ {\vphantom {{16E\alpha \Delta CTE} \sigma }} \right. \kern-\nulldelimiterspace} \sigma }}}}}} \right)\Delta T} \right]}}{(\alpha - 4\nu + \alpha \nu + 2)(8\alpha - 5\nu - 10\alpha \nu + 7)} \hfill \\ k_{3} = \frac{{\alpha \left[ {\left( {3\alpha - 3 + 27f_{1} - 3f_{2} + 18\nu - 15\nu^{2} + 18\alpha f_{1} + 3\alpha f_{2} - 18\alpha \nu - 72\nu f_{1} + 18\nu f_{2} + 15\alpha \nu^{2} + 45\nu^{2} f_{1} - 15\nu^{2} f_{2} + ...} \right.} \right.}}{(\alpha - 4\nu + \alpha \nu + 2)(8\alpha - 5\nu - 10\alpha \nu + 7)} \hfill \\ \quad \frac{{\left. {15\alpha \nu^{2} f_{2} - 18\alpha \nu f_{1} - 18\alpha \nu f_{2} } \right) + \left. {\left( { - {{14E\Delta CTE} \mathord{\left/ {\vphantom {{14E\Delta CTE} \sigma }} \right. \kern-\nulldelimiterspace} \sigma } + {{10E\nu \Delta CTE} \mathord{\left/ {\vphantom {{10E\nu \Delta CTE} {\sigma + {{20E\alpha \nu \Delta CTE} \mathord{\left/ {\vphantom {{20E\alpha \nu \Delta CTE} {\sigma - {{16E\alpha \Delta CTE} \mathord{\left/ {\vphantom {{16E\alpha \Delta CTE} \sigma }} \right. \kern-\nulldelimiterspace} \sigma }}}} \right. \kern-\nulldelimiterspace} {\sigma - {{16E\alpha \Delta CTE} \mathord{\left/ {\vphantom {{16E\alpha \Delta CTE} \sigma }} \right. \kern-\nulldelimiterspace} \sigma }}}}}} \right. \kern-\nulldelimiterspace} {\sigma + {{20E\alpha \nu \Delta CTE} \mathord{\left/ {\vphantom {{20E\alpha \nu \Delta CTE} {\sigma - {{16E\alpha \Delta CTE} \mathord{\left/ {\vphantom {{16E\alpha \Delta CTE} \sigma }} \right. \kern-\nulldelimiterspace} \sigma }}}} \right. \kern-\nulldelimiterspace} {\sigma - {{16E\alpha \Delta CTE} \mathord{\left/ {\vphantom {{16E\alpha \Delta CTE} \sigma }} \right. \kern-\nulldelimiterspace} \sigma }}}}}} \right)\Delta T} \right]}}{(\alpha - 4\nu + \alpha \nu + 2)(8\alpha - 5\nu - 10\alpha \nu + 7)} \hfill \\ \end{gathered} \right.. $$
(25)

where \(\alpha = {{E_{i} } \mathord{\left/ {\vphantom {{E_{i} } {E_{m} }}} \right. \kern-\nulldelimiterspace} {E_{m} }}\); Ei and Em are the modulus of inclusion and matrix, respectively; ν is the Poisson’s ratio of matrix; ΔCTE is the difference of the thermal expansion coefficient between inclusion and matrix.

The parameters in Eq. (18) are listed as follows:

$$ \left\{ \begin{gathered} \xi_{1} = \left( {0.4515\alpha + 0.0139I_{v} - 0.0023f_{0} + 0.0452\alpha I_{v} - 0.0075\alpha f_{0} - 0.0002f_{0} I_{v} + 0.0048\Delta CTE\Delta T + 0.0256f_{0}^{2} } \right) \hfill \\ \quad \times e^{{\frac{0.0045}{{ - 0.0055 - 0.0892\Delta CTE\Delta T}}}} + 0.2863 \hfill \\ \xi_{2} = \left( {0.3623\alpha + 0.0112I_{v} - 0.0045f_{0} + 0.0362\alpha I_{v} - 0.0146\alpha f_{0} - 0.0004f_{0} I_{v} - 0.0068\Delta CTE\Delta T} \right) \hfill \\ \quad \times e^{{\frac{0.0296}{{ - 0.0895 + 0.0123\Delta CTE\Delta T}}}} + 0.1686 \hfill \\ \end{gathered} \right.. $$
(26)

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Sun, Y., Li, A., Wang, X. et al. Creep and Stress Relaxation Behavior of Metal Matrix Particulate Composites Induced by Interface Diffusion with Coupled Thermomechanical Strains. Iran J Sci Technol Trans Mech Eng 46, 1121–1128 (2022). https://doi.org/10.1007/s40997-021-00483-9

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