Analytical and numerical instability analysis of functionally graded low-carbon steel

Document Type : Research Paper


1 Department of Materials Science and Engineering, School of Engineering, Shiraz University, Shiraz, Iran

2 Physics of Nanostructured Materials, Dynamics of Condensed Systems, Faculty of Physics, Vienna University, Vienna, Austria

3 Department of Materials Science and Engineering, School of Engineering, Shiraz University, Iran



The instability point of a material is one of the most important factors when choosing a material, as it could be a good representation of its formability. In this study, the instability of functionally graded materials (FGM) was investigated. An algorithm is proposed for predicting the instability of functionally graded low-carbon steel with gradient work hardening exponent (n) and strength coefficient (K). The investigated work hardening exponent and strength coefficient of the FGM vary through the cross-section as a function of radius. Numerical methods like the Simpson rule of integration were utilized to solve the equations. The mathematical and experimental results are compared, and it can be seen that the algorithm has a reliable consistency with the experimental results. The presented analysis shows that the instability of the functionally graded low-carbon steel can be predicted using the calculation of the average strain hardening exponent. The calculated average work hardening exponent (n ̅) was 0.1095 and 0.1657 for the 550 °C and 650 °C annealed samples, respectively. The instability of more complicated FGMs can be predicted with the present algorithm.


Main Subjects

[1]      B. Saleh, J. Jiang, A. Ma, D. Song, D. Yang, Effect of main parameters on the mechanical and wear behaviour of functionally graded materials by centrifugal casting: A review, Metals and Materials International. 25 (2019) 1395–1409.
[2]      R. Fathi, A. Ma, B. Saleh, Q. Xu, J. Jiang, Investigation on mechanical properties and wear performance of functionally graded AZ91-SiCp composites via centrifugal casting, Materials Today Communications. 24 (2020).
[3]      B. Saleh, J. Jiang, R. Fathi, T. Al-hababi, Q. Xu, L. Wang, D. Song, A. Ma, 30 Years of functionally graded materials: An overview of manufacturing methods, applications and future Challenges, Compos B Eng. 201 (2020).
[4]      Y. Miyamoto, W.A. Kaysser, B.H. Rabin, A. Kawasaki, R.G. Ford, Functionally Graded Materials, first ed., Springer US, New York, NY, 1999.
[5]      M. Koizumi, FGM activities in Japan, Composites Part B: Engineering. 28( 1997) 1-4.
[6]      G. Nie, Z. Zhong, Dynamic analysis of multi-directional functionally graded annular plates, Applied Mathematical Modelling. 34 (2010) 608–616.
[7]      T.P.D. Rajan, R.M. Pillai, B.C. Pai, Functionally graded Al-Al3Ni in situ intermetallic composites: Fabrication and microstructural characterization, Journal of Alloys  and Compounds. 453 (2008).
[8]      E. Efraim, Accurate formula for determination of natural frequencies of FGM plates basing on frequencies of isotropic plates, Procedia Engineering. 10 (2011) 242–247.
[9]      M. Naebe, K. Shirvanimoghaddam, Functionally graded materials: A review of fabrication and properties, Applied Materials Today. 5 (2016) 223–245.
[10]    S. Kumar Bohidar, R. Sharma, R. Mishra, Functionally graded materials: A critical review, International Journal of Research (IJR). 1 (2014).
[11]    M. Sam, R. Jojith, N. Radhika, Progression in manufacturing of functionally graded materials and impact of thermal treatment—A critical review, Journal of Manufacturing Processes. 68 (2021) 1339–1377.
[12]    R.S. Parihar, S.G. Setti, R.K. Sahu, Recent advances in the manufacturing processes of functionally graded materials: A review, Science and Engineering of Composite Materials. 25 (2018) 309–336.
[13]    C.H. Xu, G.Y. Wu, G.C. Xiao, B. Fang, Al2O3/(W,Ti)C/CaF2 multi-component graded self-lubricating ceramic cutting tool material, International Journal of Refractory Metals and Hard Materials. 45 (2014) 125–129.
[14]    P.I. Ichim, X. Hu, J.J. Bazen, W. Yi, Design optimization of a radial functionally graded dental implant, Journal of Biomedical Materials Research Part B: Applied Biomaterials. 104 (2016) 58–66.
[15]    P.S. Ghatage, V.R. Kar, P.E. Sudhagar, On the numerical modelling and analysis of multi-directional functionally graded composite structures: A review, Composite Structures. 236 (2020).
[16]    H. Qiu, L.N. Wang, T. Hanamura, S. Torizuka, Prediction of the work-hardening exponent for ultrafine-grained steels, Materials Science and Engineering A. 536 (2012) 269–272.
[17]    C.W. Sinclair, W.J. Poole, Y. Bréchet, A model for the grain size dependent work hardening of copper, Scripta Materialia. 55 (2006) 739–742.
[18]    L. Wang, B. Li, Y. Shi, G. Huang, W. Song, S. Li, Optimizing mechanical properties of gradient-structured low-carbon steel by manipulating grain size distribution, Materials Science and Engineering A. 743 (2019) 309–313.
[19]    W. F. Hosford, R. M. Caddell, Metal Forming Mechanics and Metallurgy, fourth ed., Cambridge University Press, New York, NY, 2011.
[20]    C. V. Nielsen, P.A.F. Martins, Metal Forming: Formability, Simulation, and Tool Design, Elsevier, London, 2021.