Systematic Review of Functionally Graded Pipelines and Proposal of a New Material Property Variation to Enhance Operational Stability
Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow G4 0LZ, UK Metallurgical Raw Materials Development and Investment Promotion Department, Federal Ministry of Steel Development, Abuja P.M.B 107, Nigeria
Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow G4 0LZ, UK
Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow G4 0LZ, UK
DOI: https://doi.org/10.36956/sms.v7i4.2726
Received: 12 September 2025 | Revised: 29 October 2025 | Accepted: 4 November 2025 | Published Online: 16 December 2025
Copyright © 2025 Frederick Ebili, Selda Oterkus, Erkan Oterkus. Published by Nan Yang Academy of Sciences Pte. Ltd.
This is an open access article under the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) License.
Abstract
The global discussion surrounding Functionally Graded Materials (FGMs) highlights their unique and diverse micro-material properties that result from varying two or more materials in a strategic combination profile. These combinations produce distinct physical and chemical characteristics. Changes in these characteristics may occur continuously, referred to as a gradient function, or discontinuously as a stepwise function. The changes can appear within homogeneous or heterogeneous material geometries. The variation in material properties depends on the volume fraction index function. This variation can occur in 1D, 2D, or 3D, either in the thickness or length direction within a material model. The vacuum in the review study on mechanically toughened and thermally resistant Functionally Graded (FG) pipelines prompted the current review study. This study addresses the absence of an appropriate variational function for FG cylindrical pipelines. It proposes a gradation function pattern to improve pipeline structural performance. An appraisal based on relevant FGM literature was conducted to improve the temperature differentials in traditional composite materials and stress-related issues in carbon steel pipelines. The review identifies specific FGM property variations that reduce failures that are possible in conventional materials. Reviewed articles and evaluation procedures followed the 2020 PRISMA guidelines. Literature was obtained from Scopus, Connected Papers, and other reputable sources. The study also discusses potential FG pipelines for gas and green energy transportation. The reviewed literature establishes the context for this research and addresses the gap in 3D FG model variation functions involving multiple materials.
Keywords: FGMs; Volume Fraction Index; Pipeline; Homogenous Material; Continuum; Stepwise
References
[1] Miyamoto, Y., Kaysser, W.A., Rabin, B.H., et al., Eds., 1999. Functionally Graded Materials: Design, Processing and Applications, 1st ed. Springer New York: New York, NY, USA. DOI: https://doi.org/10.1007/978-1-4615-5301-4
[2] Saleh, B., Jiang, J., Fathi, R., et al., 2020. 30 Years of functionally graded materials: An overview of manufacturing methods, Applications and Future Challenges. Composites Part B: Engineering. 201, 108376. DOI: https://doi.org/10.1016/j.compositesb.2020.108376
[3] Mohammadi, M., Rajabi, M., Ghadiri, M., 2021. Functionally graded materials (FGMs): A review of classifications, fabrication methods and their applications. Periodicals of Engineering and Natural Sciences. 15(4), 319–343. DOI: https://doi.org/10.2298/PAC2104319M
[4] Li, Y., Feng, Z., Hao, L., et al., 2020. A Review on Functionally Graded Materials and Structures via Additive Manufacturing: From Multi-Scale Design to Versatile Functional Properties. Advanced Materials Technologies. 5(6). DOI: https://doi.org/10.1002/admt.201900981
[5] Shen, M., Bever, M.B., 1972. Gradients in polymeric materials. Journal of Materials Science. 7, 741–746. DOI: https://doi.org/10.1007/BF00549902
[6] Mahamood, R.M., Akinlabi, E.T., 2018. Functionally Graded Materials. Springer: Cham, Switzerland. DOI: https://doi.org/10.1007/978-3-319-53756-6
[7] Bever, M.B., Duwez, P.E., 1972. Gradients in composite materials. Materials Science and Engineering. 10, 1–8. DOI: https://doi.org/10.1016/0025-5416(72)90059-6
[8] Hassan, A., Khalaf, H.I., Farouq, W., et al., 2022. A review of functionally graded materials including their manufacture and applications. The International Journal of Mechanical Engineering and Sciences. 7(1), 744–755. Available from: https://www.researchgate.net/publication/357649829
[9] Loknath, D., Ravindra Kumar, V.M., 2022. A review on processing and characterization of bulk functionally graded polymer materials. Materials Today: Proceedings. 56, 1192–1200. DOI: https://doi.org/10.1016/j.matpr.2021.11.152
[10] Sanjeeviprakash, K., Kannan, A.R., Shanmugam, N.S., 2023. Additive manufacturing of metal-based functionally graded materials: overview, recent advancements and challenges. Journal of the Brazilian Society of Mechanical Sciences and Engineering. 45, 241. DOI: https://doi.org/10.1007/s40430-023-04174-1
[11] Patel, Y., Karsh, P.K., 2024. A Review on Fabrication and Application of Functionally Graded Material. In Proceedings on Smart and Sustainable Developments in Engineering and Technology: (PICET 2023), Vadodara, India, 5–6 May 2023. DOI: https://doi.org/10.1063/5.0212271
[12] Teacher, M., Velu, R., 2024. Additive Manufacturing of Functionally Graded Materials: A Comprehensive Review. International Journal of Precision Engineering and Manufacturing. 25, 165–197. DOI: https://doi.org/10.1007/s12541-023-00864-x
[13] Alkunte, S., Fidan, I., Naikwadi, V., et al., 2024. Advancements and Challenges in Additively Manufactured Functionally Graded Materials: A Comprehensive Review. Journal of Manufacturing and Materials Processing. 8(1), 23. DOI: https://doi.org/10.3390/jmmp8010023
[14] Yadav, S., Liu, S., Singh, R.K., et al., 2024. A state-of-art review on functionally graded materials (FGMs) manufactured by 3D printing techniques: Advantages, existing challenges, and future scope. Journal of Manufacturing Processes. 131, 2051–2072. DOI: https://doi.org/10.1016/j.jmapro.2024.10.026
[15] Veeman, D., Subramaniyan, M.K., Browne, M.A., et al., 2025. Manufacturing and behavioral analysis of functionally graded material fabricated via wire arc additive manufacturing. Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications. 239(2), 311–318. DOI: https://doi.org/10.1177/14644207241262183
[16] Fu, G., Wang, X., Wang, B., et al., 2024. Dynamic behavior of axially functionally graded pipe conveying gas–liquid two-phase flow. Applied Ocean Research. 142, 103827. DOI: https://doi.org/10.1016/j.apor.2023.103827
[17] Zhou, J., Chang, X., Li, Y., et al., 2023. Dynamic Nonlinear Analysis of Functionally Graded Flow Pipelines with Defects Based on Different Foundation Layouts. Journal of Vibration Engineering and Technologies. 11(8), 4395–4413. DOI: https://doi.org/10.1007/s42417-022-00822-3
[18] Bever, M.B., Duwez, P.E., Tiller, W.A., 1970. On Nonstructural Applications of Composites. Materials Science and Engineering. 6(3), 149–155. DOI: https://doi.org/10.1016/0025-5416(70)90044-3
[19] Bohidar, S.K., Sharma, R., Mishra, P.R., 2014. Functionally Graded Materials: A Critical Review. International Journal of Research. 1(7). Available from: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=9a0ebaf506651ffcfe367470a1d777c6a4523af3
[20] Li, K., Fang, J., Zhan, J., et al., 2024. A critical review of biomimetic structures via laser powder bed fusion: Toward multi-functional application. Journal of Manufacturing Processes. 131, 2443–2472. DOI: https://doi.org/10.1016/j.jmapro.2024.09.087
[21] El-Galy, I.M., Saleh, B.I., Ahmed, M.H., 2019. Functionally graded materials classifications and development trends from industrial point of view. SN Applied Sciences. 1, 1378. DOI: https://doi.org/10.1007/s42452-019-1413-4
[22] Koizumi, M., Niino, M., 1995. Overview of FGM Research in Japan. MRS Bulletin. 20, 19–21.
[23] Koizumi, M., 1997. FGM activities in Japan. Composites Part B: Engineering. 28(1–2), 1–4.
[24] Busch-Vishniac, I., Busch, L., Tietjen, J.S., 2024. Women in the National Inventors Hall of Fame: The First 50 Years (Women in Engineering and Science). Springer: Cham, Switzerland.
[25] Ramteke, P.M., Kumar, E.K., Dewangan, H.C., et al., 2025. Theoretical prediction and experimental verification of thermomechanical deflection responses of geometrically nonlinear porous graded curved structure. International Journal of Mechanics and Materials in Design. 21(1), 17–42. DOI: https://doi.org/10.1007/s10999-024-09725-5
[26] Keibolahi, A., Eslami, M.R., Kiani, Y., 2025. Assessment of Axisymmetric Dynamic Snap-Through and Thermally Induced Vibrations in FGM Cylindrical Shells under Instantaneous Heating. International Journal of Structural Stability and Dynamics. 25(4). DOI: https://doi.org/10.1142/S0219455425500324
[27] Ansari, R., Talebian, A., Ershadi, M.Z., 2025. Thermo-mechanical response of axisymmetric cylindrical shells made of FGM subjected to cooling shock. Thin-Walled Structures. 212, 113145. DOI: https://doi.org/10.1016/j.tws.2025.113145
[28] Kumar, R., Lal, A., Sutaria, B.M., et al., 2024. Thermo-mechanical fracture analysis of porous functionally graded cracked plate using XFEM. Mechanics Based Design of Structures and Machines. 52(10), 7942–7961. DOI: https://doi.org/10.1080/15397734.2024.2312175
[29] Ansari, R., Ershadi, M.Z., Laskoukalayeh, H.A., et al., 2024. Thermally Induced Vibrations of Functionally Graded Shallow Spherical Shells Under Cooling Shock. AIAA Journal. 62(2), 833–841. DOI: https://doi.org/10.2514/1.J063326
[30] Rezaiee-Pajand, M., Rajabzadeh-Safaei, N., 2018. Nonlocal static analysis of a functionally graded material curved nanobeam. Mechanics of Advanced Materials and Structures. 25(7), 539–547. DOI: https://doi.org/10.1080/15376494.2017.1285463
[31] Zghal, S., Trabelsi, S., Frikha, A., et al., 2021. Thermal free vibration analysis of functionally graded plates and panels with an improved finite shell element. Journal of Thermal Stresses. 44(3), 315–341. DOI: https://doi.org/10.1080/01495739.2021.1871577
[32] Pragya, A., Ghosh, T.K., 2023. Soft Functionally Gradient Materials and Structures – Natural and Manmade: A Review. Advanced Materials. 35(49). DOI: https://doi.org/10.1002/adma.202300912
[33] Hassan, A.H.A., Kurgan, N., 2022. Modeling Functionally Graded Materials in ANSYS APDL. Research Square. DOI: https://doi.org/10.21203/rs.3.rs-1936049/v2
[34] Ghanavati, R., Naffakh-Moosavy, H., 2021. Additive manufacturing of functionally graded metallic materials: A review of experimental and numerical studies. Journal of Materials Research and Technology. 13, 1628–1664. DOI: https://doi.org/10.1016/j.jmrt.2021.05.022
[35] Sayyad, A.S., Ghugal, Y.M., 2019. Modeling and analysis of functionally graded sandwich beams: a review. Mechanics of Advanced Materials and Structures. 26(21), 1776–1795. DOI: https://doi.org/10.1080/15376494.2018.1447178
[36] Mazari, A., Attia, A., Sekkal, M., et al., 2018. Bending analysis of functionally graded thick plates with in-plane stiffness variation. Structural Engineering and Mechanics. 68(4), 409–421. DOI: https://doi.org/10.12989/sem.2018.68.4.409
[37] Su, Y., Chen, B., Tan, C., et al., 2020. Influence of composition gradient variation on the microstructure and mechanical properties of 316 L/Inconel718 functionally graded material fabricated by laser additive manufacturing. Journal of Materials Processing Technology. 283, 116702. DOI: https://doi.org/10.1016/j.jmatprotec.2020.116702
[38] Jagtap, K.R., Lal, A., Singh, B.N., 2018. Uncertainty quantification in non-linear dynamic response of functionally graded materials plate. Mechanics of Advanced Materials and Structures. 25(13), 1081–1100. DOI: https://doi.org/10.1080/15376494.2017.1329465
[39] Medeiros, M.S., Ribeiro, L.G., 2022. Micromechanical elastoplastic limit analysis of in-plane bending of Functionally Graded Pipe elbows. Thin-Walled Structures. 171, 108778. DOI: https://doi.org/10.1016/j.tws.2021.108778
[40] Reddy, J.N., 2000. Analysis of functionally graded plates. International Journal for Numerical Methods in Engineering. 47(1–3), 663–684.
[41] Pradhan, P., Sutar, M.K., Pattnaik, S., 2019. A state of the art in Functionally Graded Materials and their Analysis. Materials Today: Proceedings. 18, 3931–3936
[42] Arbind, A., Reddy, J.N., 2013. Nonlinear analysis of functionally graded microstructure-dependent beams. Composite Structures. 98, 272–281. DOI: https://doi.org/10.1016/j.compstruct.2012.10.003
[43] Reddy, J.N., 2011. Microstructure-dependent couple stress theories of functionally graded beams. Journal of the Mechanics and Physics of Solids. 59(11), 2382–2399. DOI: https://doi.org/10.1016/j.jmps.2011.06.008
[44] Shen, Y., Chen, Y., Li, L., 2016. Torsion of a functionally graded material. International Journal of Engineering Science. 109, 14–28. DOI: https://doi.org/10.1016/j.ijengsci.2016.09.003
[45] Nie, G.J., Zhong, Z., Batra, R.C., 2011. Material tailoring for functionally graded hollow cylinders and spheres. Composites Science and Technology. 71(5), 666–673. DOI: https://doi.org/10.1016/j.compscitech.2011.01.009
[46] Chandrasekaran, S., S, H., 2022. Functionally graded material and its application to marine structures. Sustainable Marine Structures. 4(1), 35–41. DOI: https://doi.org/10.36956/sms.v4i1.490
[47] Wang, Z., Soares, C.G., 2021. Upheaval thermal buckling of functionally graded subsea pipelines. Applied Ocean Research. 116, 102881. DOI: https://doi.org/10.1016/j.apor.2021.102881
[48] Singha, M.K., Prakash, T., Ganapathi, M., 2011. Finite element analysis of functionally graded plates under transverse load. Finite Elements in Analysis and Design. 47(4), 453–460. DOI: https://doi.org/10.1016/j.finel.2010.12.001
[49] Sutradhar, A., Paulino, G.H., Gray, L.J., 2002. Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method. Engineering Analysis with Boundary Elements. 26(2), 119–132. DOI: https://doi.org/10.1016/S0955-7997(01)00090-X
[50] Ali, E., Urgessa, G., 2022. Numerical Simulation of Functionally-Graded-Material Pipes Under Blast Loading. In Advances in Structural Mechanics and Applications. Springer: Cham, Switzerland. DOI: https://doi.org/10.1007/978-3-030-98335-2_11
[51] Tornabene, F., Brischetto, S., Fantuzzi, N., et al., 2016. Boundary conditions in 2D numerical and 3D exact models for cylindrical bending analysis of functionally graded structures. Shock and Vibration. 2016(1). DOI: https://doi.org/10.1155/2016/2373862
[52] Cho, J.R., 2019. A numerical evaluation of SIFs of 2-D functionally graded materials by enriched natural element method. Applied Sciences. 9(17), 3581. DOI: https://doi.org/10.3390/app9173581
[53] Zhang, S., Bu, R., Zhang, Z., et al., 2024. A systematic model for the mechanical behavior of thin-walled composite FGM pipelines subjected to strike-slip faults in geohazard area. Thin-Walled Structures. 202, 112135. DOI: https://doi.org/10.1016/j.tws.2024.112135
[54] Joshi, K.K., Kar, V.R., 2019. Bending analysis of Bi-dimensional functionally graded plate using FEA. Materials Today: Proceedings. 26(2), 1766–1770. DOI: https://doi.org/10.1016/j.matpr.2020.02.371
[55] Yas, M.H., Shakeri, M., Khanjani, M., 2011. Layer-wise finite-element analysis of a functionally graded hollow thick cylinder with a piezoelectric ring. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 225(5), 1045–1060. DOI: https://doi.org/10.1177/09544062JMES2521
[56] Tran, T.T., Nguyen, V.C., Zenkour, A.M., et al., 2025. The nonlinear vibration analysis-based an enhanced finite element procedure of multi-functionally graded sandwich plates. Thin-Walled Structures. 211, 113042. DOI: https://doi.org/10.1016/j.tws.2025.113042
[57] Asemi, K., Ashrafi, H., Shariyat, M., 2016. Three-dimensional stress and free vibration analyses of functionally graded plates with circular holes by the use of the graded finite element method. Journal of Applied Mechanics and Technical Physics. 57(4), 690–700. DOI: https://doi.org/10.1134/S0021894416040131
[58] Nguyen, N.V., Nguyen, H.X., Lee, S., et al., 2018. Geometrically nonlinear polygonal finite element analysis of functionally graded porous plates. Advances in Engineering Software. 126, 110–126. DOI: https://doi.org/10.1016/j.advengsoft.2018.11.005
[59] Zhu, L.F., Ke, L.L., Xiang, Y., et al., 2020. Free vibration and damage identification of cracked functionally graded plates. Composite Structures. 250, 112517. DOI: https://doi.org/10.1016/j.compstruct.2020.112517
[60] Hadji, L., Tounsi, A., 2021. Static deflections and stress distribution of functionally graded Sandwich plates with porosity. Smart Structures and Systems. 28(3), 343–354. DOI: https://doi.org/10.12989/sss.2021.28.3.343
[61] Choudhary, J., Patle, B.K., Ramteke, P.M., et al., 2022. Static and Dynamic Deflection Characteristics of Cracked Porous FG Panels. International Journal of Applied Mechanics. 14(7). DOI: https://doi.org/10.1142/S1758825122500764
[62] Chandrasekaran, S., Thennavan, M., 2023. Numerical analysis of coped beam with X52 Steel and FGM used in offshore topside. In Proceedings of the Thirty-third International Ocean and Polar Engineering Conference, Ottawa, ON, Canada, 19–23 June 2023; pp. 1067–1074.
[63] Page, M.J., McKenzie, J.E., Bossuyt, P.M., et al., 2021. The PRISMA 2020 statement: An updated guideline for reporting systematic reviews. BMJ. 372, 71. DOI: https://doi.org/10.1136/bmj.n71
[64] Rezaiee-Pajand, M., Masoodi, A.R., 2019. Stability Analysis of Frame Having FG Tapered Beam–Column. International Journal of Steel Structures. 19(2), 446–468. DOI: https://doi.org/10.1007/s13296-018-0133-8
[65] Xu, X.J., Meng, J.M., 2018. A model for functionally graded materials. Composites Part B: Engineering. 145, 70–80. DOI: https://doi.org/10.1016/j.compositesb.2018.03.014
[66] Wang, Q., Wu, D., Tin-Loi, F., et al., 2019. Machine learning aided stochastic structural free vibration analysis for functionally graded bar-type structures. Thin-Walled Structures. 144, 106315. DOI: https://doi.org/10.1016/j.tws.2019.106315
[67] Barbosa, J.A.T., Ferreira, A.J.M., 2010. Geometrically nonlinear analysis of functionally graded plates and shells. Mechanics of Advanced Materials and Structures. 17(1), 40–48. DOI: https://doi.org/10.1080/15376490903082870
[68] Yang, J., Shen, H.S., 2003. Non-linear analysis of functionally graded plates under transverse and in-plane loads. International Journal of Non-Linear Mechanics. 38(4), 467–482. DOI: https://doi.org/10.1016/S0020-7462(01)00070-1
[69] Shen, H.S., Noda, N., 2007. Postbuckling of pressure-loaded FGM hybrid cylindrical shells in thermal environments. Composite Structures. 77(4), 546–560. DOI: https://doi.org/10.1016/j.compstruct.2005.08.006
[70] Chen, Y.Z., 2021. A novel numerical solution for a functionally graded hollow cylinder with arbitrary elastic property along the radial direction. International Journal of Pressure Vessels and Piping. 191, 104301. DOI: https://doi.org/10.1016/j.ijpvp.2021.104301
[71] Knoppers, G.E., Gunnink, J.W., Van Den Hout, J., et al., 2004. The Reality of Functionally Graded Material Products. Available from: http://utw10945.utweb.utexas.edu/Manuscripts/2004/2004-05-Knoppers.pdf (cited 10 June 2025).
[72] Shumiya, H., Kato, K., Okubo, H., 2004. Feasibility Study on FGM (Functionally Graded Materials) Application for Gas Insulated Equipment. In Proceedings of the 17th Annual Meeting of the IEEE Lasers and Electro-Optics Society, Rio Grande, PR, USA, 7–11 November 2004.
[73] Rodrigues, T.A., Duarte, V., Miranda, R.M., et al., 2019. Current status and perspectives on wire and arc additive manufacturing (WAAM). Materials. 12(7), 1121. DOI: https://doi.org/10.3390/ma12071121
[74] Ji, S., Sun, Z., Zhang, W., et al., 2020. Microstructural evolution and high temperature resistance of functionally graded material Ti-6Al-4V/Inconel 718 coated by directed energy deposition-laser. Journal of Alloys and Compounds. 848, 156255. DOI: https://doi.org/10.1016/j.jallcom.2020.156255
[75] Attia, M.A., Eltaher, M.A., Soliman, A., et al., 2018. Thermoelastic Crack Analysis in Functionally Graded Pipelines Conveying Natural Gas by an FEM. International Journal of Applied Mechanics. 10(4). DOI: https://doi.org/10.1142/S1758825118500369
[76] Soliman, A.E., Eltaher, M.A., Attia, M.A., et al., 2018. Nonlinear transient analysis of FG pipe subjected to internal pressure and unsteady temperature in a natural gas facility. Structural Engineering and Mechanics. 66(1), 85–96. DOI: https://doi.org/10.12989/sem.2018.66.1.085
[77] Wu, K., Zhu, W.D., 2018. A new global spatial discretization method for calculating dynamic responses of two-dimensional continuous systems with application to a rectangular Kirchhoff plate. Journal of Vibration and Acoustics. 140(1). DOI: https://doi.org/10.1115/1.4037176
[78] Parida, S., Mohanty, S.C., 2018. Free Vibration Analysis of Functionally Graded Skew Plate in Thermal Environment Using Higher Order Theory. International Journal of Applied Mechanics. 10(1). DOI: https://doi.org/10.1142/S1758825118500072
[79] Ganiev, R.F., Il’gamov, M.A., Khakimov, A.G., et al., 2016. Spatial vibrations of a pipeline in a continuous medium under the action of variable internal pressure. Journal of Machinery Manufacture and Reliability. 45(6), 485–494. DOI: https://doi.org/10.3103/S105261881606008X
[80] Moussou, P., 2003. An excitation spectrum criterion for the vibration-induced fatigue of small bore pipes. Journal of Fluids and Structures. 18(2), 149–163. DOI: https://doi.org/10.1016/j.jfluidstructs.2003.07.002
[81] Liang, F., Yang, X.D., Qian, Y.J., et al., 2017. Free Vibration Analysis of Pipes Conveying Fluid Based on Linear and Nonlinear Complex Modes Approach. International Journal of Applied Mechanics. 9(8). DOI: https://doi.org/10.1142/S1758825117501125
[82] Ahmadi, H., 2019. Nonlinear primary resonance of imperfect spiral stiffened functionally graded cylindrical shells surrounded by damping and nonlinear elastic foundation. Engineering with Computers. 35(4), 1491–1505. DOI: https://doi.org/10.1007/s00366-018-0679-2
[83] Attia, M.A., El-Shafei, A.G., 2020. Investigation of multibody receding frictional indentation problems of unbonded elastic functionally graded layers. International Journal of Mechanical Sciences. 184, 105838. DOI: https://doi.org/10.1016/j.ijmecsci.2020.105838
[84] Attia, M.A., El-Shafei, A.G., 2019. Modeling and analysis of the nonlinear indentation problems of functionally graded elastic layered solids. Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology. 233(12), 1903–1920. DOI: https://doi.org/10.1177/1350650119851691
[85] Kim, S.E., Duc, N.D., Nam, V.H., et al., 2019. Nonlinear vibration and dynamic buckling of eccentrically oblique stiffened FGM plates resting on elastic foundations in thermal environment. Thin-Walled Structures. 142, 287–296. DOI: https://doi.org/10.1016/j.tws.2019.05.013
[86] Deng, J., Liu, Y., Liu, W., 2017. A Hybrid Method for Transverse Vibration of Multi-Span Functionally Graded Material Pipes Conveying Fluid with Various Volume Fraction Laws. International Journal of Applied Mechanics. 9(7). DOI: https://doi.org/10.1142/S1758825117500958
[87] Ganiev, R.F., Ilgamov, M.A., 2016. Resilient reaction of a pipeline to an internal impact pressure. Doklady Physics. 61(9), 453–456. DOI: https://doi.org/10.1134/S1028335816090044
[88] Bauomy, H.S., 2022. New controller (NPDCVF) outcome of FG cylindrical shell structure. Alexandria Engineering Journal. 61(2), 1779–1801. DOI: https://doi.org/10.1016/j.aej.2021.06.061
[89] Ding, H., Huang, L.L., Dowell, E., et al., 2019. Stress distribution and fatigue life of nonlinear vibration of an axially moving beam. Science China Technological Sciences. 62(7), 1123–1133. DOI: https://doi.org/10.1007/s11431-017-9283-4
[90] Amir, M., Talha, M., 2018. Thermoelastic Vibration of Shear Deformable Functionally Graded Curved Beams with Microstructural Defects. International Journal of Structural Stability and Dynamics. 18(11). DOI: https://doi.org/10.1142/S0219455418501353
[91] Ganiev, R.F., Il’gamov, M.A., Khakimov, A.G., et al., 2017. Spatial aperiodic vibrations of the pipelines under transient internal pressure. Journal of Machinery Manufacture and Reliability. 46(2), 87–95. DOI: https://doi.org/10.3103/S1052618817020066
[92] Lü, L., Hu, Y., Wang, X., et al., 2015. Dynamical bifurcation and synchronization of two nonlinearly coupled fluid-conveying pipes. Nonlinear Dynamics. 79(4), 2715–2734. DOI: https://doi.org/10.1007/s11071-014-1842-y
[93] Eltaher, M.A., Attia, M.A., Wagih, A., 2020. Predictive model for indentation of elasto-plastic functionally graded composites. Composites Part B: Engineering. 197, 108129. DOI: https://doi.org/10.1016/j.compositesb.2020.108129
[94] Zhang, Y.F., Liu, J.T., 2019. A widespread internal resonance phenomenon in functionally graded material plates with longitudinal speed. Scientific Reports. 9(1), 1907. DOI: https://doi.org/10.1038/s41598-018-37921-9
[95] Parida, S., Mohanty, S.C., 2018. Vibration and Stability Analysis of Functionally Graded Skew Plate Using Higher Order Shear Deformation Theory. International Journal of Applied Computational Mathematics. 4(1), 22. DOI: https://doi.org/10.1007/s40819-017-0440-3
[96] Tang, D.M., Ilgamov, M.A., Dowell, E.H., 1995. Buckling and post-buckling behavior of a pipe subjected to internal pressure. Journal of Applied Mechanics. 62(3), 595–600. DOI: https://doi.org/10.1115/1.2895987
[97] Deng, J., Liu, Y., Zhang, Z., et al., 2017. Dynamic behaviors of multi-span viscoelastic functionally graded material pipe conveying fluid. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 231(17), 3181–3192. DOI: https://doi.org/10.1177/0954406216642483
[98] Wu, K., Zhu, W.D., Fan, W., 2017. On a comparative study of an accurate spatial discretization method for one-dimensional continuous systems. Journal of Sound and Vibration. 399, 257–284. DOI: https://doi.org/10.1016/j.jsv.2017.02.027
[99] Safaei, B., Moradi-Dastjerdi, R., Qin, Z., et al., 2019. Frequency-dependent forced vibration analysis of nanocomposite sandwich plate under thermo-mechanical loads. Composites Part B: Engineering. 161, 44–54. DOI: https://doi.org/10.1016/j.compositesb.2018.10.049
[100] Il’gamov, M.A., 2017. Dynamics of a Pipeline under the Action of Internal Shock Pressure. Mechanics of Solids. 52(6), 663–674. DOI: https://doi.org/10.3103/S0025654417060061
[101] Zhang, Y., Zhang, F., 2019. Vibration and buckling of shear deformable functionally graded nanoporous metal foam nanoshells. Nanomaterials. 9(2), 271. DOI: https://doi.org/10.3390/nano9020271
[102] Attia, M.A., Mohamed, S.A., 2018. Pull-In Instability of Functionally Graded Cantilever Nanoactuators Incorporating Effects of Microstructure, Surface Energy and Intermolecular Forces. International Journal of Applied Mechanics. 10(8). DOI: https://doi.org/10.1142/S1758825118500916
[103] Wang, Y.Q., Zu, J.W., 2018. Nonlinear dynamic behavior of inhomogeneous functional plates composed of sigmoid graded metal-ceramic materials. Science China Technological Sciences. 61(11), 1654–1665. DOI: https://doi.org/10.1007/s11431-017-9167-9
[104] Attia, M.A., Mohamed, S.A., 2019. Coupling effect of surface energy and dispersion forces on nonlinear size-dependent pull-in instability of functionally graded micro-/nanoswitches. Acta Mechanica. 230(3), 1181–1216. DOI: https://doi.org/10.1007/s00707-018-2345-6
[105] Mohammadi, M., Rastgoo, A., 2020. Primary and secondary resonance analysis of FG/lipid nanoplate with considering porosity distribution based on a nonlinear elastic medium. Mechanics of Advanced Materials and Structures. 27(20), 1709–1730. DOI: https://doi.org/10.1080/15376494.2018.1525453
[106] Askari, M., Brusa, E., Delprete, C., 2021. On the vibration analysis of coupled transverse and shear piezoelectric functionally graded porous beams with higher-order theories. Journal of Strain Analysis for Engineering Design. 56(1), 29–49. DOI: https://doi.org/10.1177/0309324720922085
[107] Akbaş, Ş.D., 2019. Hygro-thermal nonlinear analysis of a functionally graded beam. Journal of Applied and Computational Mechanics. 5(2), 477–485. DOI: https://doi.org/10.22055/jacm.2018.26819.1360
[108] Parida, S., Mohanty, S.C., 2017. Thermoelastic vibration analysis of functionally graded skew plate using nonlinear finite element method. Journal of Thermal Stresses. 40(9), 1111–1133. DOI: https://doi.org/10.1080/01495739.2017.1290513
[109] Chakraverty, S., Pradhan, K.K., 2018. Flexural vibration of functionally graded thin skew plates resting on elastic foundations. International Journal of Dynamics and Control. 6(1), 97–121. DOI: https://doi.org/10.1007/s40435-017-0308-8
[110] Yulmukhametov, A.A., Shakiryanov, M.M., Utyashev, I.M., 2020. Bending vibrations of the pipeline under the influence of the internal added mass. In Proceedings of the XXVII Conference on High-Energy Processes in Condensed Matter, dedicated to the 90th anniversary of the birth of RI Soloukhin, Novosibirsk, Russia, 29 June–3 July 2020. DOI: https://doi.org/10.1063/5.0028885
[111] An, C., Su, J., 2017. Dynamic Behavior of Axially Functionally Graded Pipes Conveying Fluid. Mathematical Problems in Engineering. 2017(1). DOI: https://doi.org/10.1155/2017/6789634
[112] Amir, M., Talha, M., 2019. Nonlinear vibration characteristics of shear deformable functionally graded curved panels with porosity including temperature effects. International Journal of Pressure Vessels and Piping. 172, 28–41. DOI: https://doi.org/10.1016/j.ijpvp.2019.03.008
[113] Rong, B., Lu, K., Rui, X.T., et al., 2018. Nonlinear dynamics analysis of pipe conveying fluid by Riccati absolute nodal coordinate transfer matrix method. Nonlinear Dynamics. 92(2), 699–708. DOI: https://doi.org/10.1007/s11071-018-4084-6
[114] Ilgamov, M.A., Tang, D.M., Dowell, E.H., 1994. Flutter and Forced Response of a Cantilevered Pipe: The Influence of Internal Pressure and Nozzle Discharge. Journal of Fluids and Structures. 8, 139–156. DOI: https://doi.org/10.1006/jfls.1994.1007
[115] Shanab, R.A., Mohamed, S.A., Mohamed, N.A., et al., 2020. Comprehensive investigation of vibration of sigmoid and power law FG nanobeams based on surface elasticity and modified couple stress theories. Acta Mechanica. 231(5), 1977–2010. DOI: https://doi.org/10.1007/s00707-020-02623-9
[116] Gupta, A., Talha, M., 2015. Recent development in modeling and analysis of functionally graded materials and structures. Progress in Aerospace Sciences. 79, 1–14. DOI: https://doi.org/10.1016/j.paerosci.2015.07.001
[117] Wang, Y.Q., Zu, J.W., 2017. Vibration behaviors of functionally graded rectangular plates with porosities and moving in thermal environment. Aerospace Science and Technology. 69, 550–562. DOI: https://doi.org/10.1016/j.ast.2017.07.023
[118] Wang, Y.Q., 2018. Electro-mechanical vibration analysis of functionally graded piezoelectric porous plates in the translation state. Acta Astronautica. 143, 263–271. DOI: https://doi.org/10.1016/j.actaastro.2017.12.004
[119] Wang, Y.Q., Zu, J.W., 2017. Nonlinear steady-state responses of longitudinally traveling functionally graded material plates in contact with liquid. Composite Structures. 164, 130–144. DOI: https://doi.org/10.1016/j.compstruct.2016.12.053
[120] Wang, Y.Q., Wan, Y.H., Zhang, Y.F., 2017. Vibrations of longitudinally traveling functionally graded material plates with porosities. European Journal of Mechanics, A/Solids. 66, 55–68. DOI: https://doi.org/10.1016/j.euromechsol.2017.06.006
[121] Wang, Y.Q., Zu, J.W., 2017. Large-amplitude vibration of sigmoid functionally graded thin plates with porosities. Thin-Walled Structures. 119, 911–924. DOI: https://doi.org/10.1016/j.tws.2017.08.012
[122] Wang, Y.Q., Yang, Z., 2017. Nonlinear vibrations of moving functionally graded plates containing porosities and contacting with liquid: internal resonance. Nonlinear Dynamics. 90(2), 1461–1480. DOI: https://doi.org/10.1007/s11071-017-3739-z
[123] Wang, Y.Q., Zu, J.W., 2017. Nonlinear dynamic thermoelastic response of rectangular FGM plates with longitudinal velocity. Composites Part B: Engineering. 117, 74–88. DOI: https://doi.org/10.1016/j.compositesb.2017.02.037
[124] Wang, Y.Q., Zu, J.W., 2017. Porosity-dependent nonlinear forced vibration analysis of functionally graded piezoelectric smart material plates. Smart Materials and Structures. 26(10). DOI: https://doi.org/10.1088/1361-665X/aa8429
[125] Wang, Y.Q., Zu, J.W., 2018. Nonlinear Dynamics of a Translational FGM Plate with Strong Mode Interaction. International Journal of Structural Stability and Dynamics. 18(3). DOI: https://doi.org/10.1142/S0219455418500311
[126] Yao, L., Ramesh, A., Xiao, Z., et al., 2023. Multimetal Research in Powder Bed Fusion: A Review. Materials. 16(12), 4287. DOI: https://doi.org/10.3390/ma16124287
[127] Pasha, A., B.m, R., 2022. Functionally graded materials (FGM) fabrication and its potential challenges & applications. Materials Today: Proceedings. 52(3), 413–418. DOI: https://doi.org/10.1016/j.matpr.2021.09.077
[128] Yadav, A., Srivastava, M., Jain, P.K., et al., 2025. Development of ferritic–austenitic functionally graded structure via twin wire arc additive manufacturing route. Progress in Additive Manufacturing. 10(5), 3489–3500. DOI: https://doi.org/10.1007/s40964-024-00823-6
[129] Jin, X., Liu, J., Fan, W., et al., 2025. Research and Applications of Additive Manufacturing in Oil and Gas Extraction and Gathering Engineering. Materials. 18(14), 3353. DOI: https://doi.org/10.3390/ma18143353
[130] Cao, J., 2022. Research on Dynamics Behaviors of Two Coaxial Circular Cylindrical Thin Shells Constructed with Functionally Graded Materials. Journal of Physics: Conference Series. 2230, 012010. DOI: https://doi.org/10.1088/1742-6596/2230/1/012010
[131] Jing, J., Mao, X., Ding, H., et al., 2024. Parametric resonance of axially functionally graded pipes conveying pulsating fluid. Applied Mathematics and Mechanics. 45(2), 239–260. DOI: https://doi.org/10.1007/s10483-024-3083-6
[132] Tuo, Y.-H., Fu, G.-M., Sun, B.-J., et al., 2022. Stability of axially functionally graded pipe conveying fluid: Generalized integral transform solution. Applied Ocean Research. 125, 103218. DOI: https://doi.org/10.1016/j.apor.2022.103218
[133] Zhang, B., Long, Z., Liu, G., et al., 2025. Investigating the distribution of heat transfer in a thick-walled functionally graded cylindrical shell under heat flux. Journal of Engineering and Applied Science. 72(1). DOI: https://doi.org/10.1186/s44147-024-00571-y
[134] Zeng, B., Yang, J., Ni, Z., et al., 2024. Improved pyroelectric performances of functionally graded graphene nanoplatelet reinforced polyvinylidene fluoride composites: Experiment and modelling. Composites Part A: Applied Science and Manufacturing. 176, 107883. DOI: https://doi.org/10.1016/j.compositesa.2023.107883
[135] Zhang, X., Zhou, J., Shen, X., 2025. Longitudinal thermoelastic guided waves in functionally graded hollow cylinders with Green-Naghdi thermoelastic theory. Smart Materials and Structures. 34(1). DOI: https://doi.org/10.1088/1361-665X/ad9800
[136] Li, S.P., Zuo, G.Q., Zhang, C.L., et al., 2025. Recent progress in thermal structures: Materials, structures, and analyses. Composite Structures. 359, 119037. DOI: https://doi.org/10.1016/j.compstruct.2025.119037
[137] Zemani, K., May, A., Khatir, S., et al., 2024. Numerical analysis of an experimental ballistic test of Al/SiC functionally graded materials. Composite Structures. 333, 117909. DOI: https://doi.org/10.1016/j.compstruct.2024.117909
[138] Turan, M., Uzun Yaylacı, E., Yaylacı, M., 2023. Free vibration and buckling of functionally graded porous beams using analytical, finite element, and artificial neural network methods. Archive of Applied Mechanics. 93(4), 1351–1372. DOI: https://doi.org/10.1007/s00419-022-02332-w
[139] Ebili, F., Oterkus, S., Oterkus, E., 2025. A Detailed Structural Review of Onshore and Offshore Pipelines Containing Defects. Sustainable Marine Structures. 7(3), 177–208. DOI: https://doi.org/10.36956/sms.v7i3.2320
[140] Yildirim, S., 2020. Hydrogen elasticity solution of functionally-graded spheres, cylinders and disks. International Journal of Hydrogen Energy. 45(41), 22094–22101. DOI: https://doi.org/10.1016/j.ijhydene.2020.05.272
[141] Neff, H., 1992. Ceramics and Evolution. Archaeological Method and Theory. 4, 141–193. Available from: https://www.jstor.org/stable/20170223?seq=1&cid=pdf-
[142] Yan, L., Chen, Y., Liou, F., 2020. Additive manufacturing of functionally graded metallic materials using laser metal deposition. Additive Manufacturing. 31, 100901. DOI: https://doi.org/10.1016/j.addma.2019.100901
[143] Vijayakumar, K., Mayuram, M.M., Krishnamurthy, R., 2003. Investigations into Erosion Wear Performance of Functionally Graded Chromium Oxide - Alumina Titania Ceramic Composite Deposits. Materials Science Forum. 437–438, 189–192. DOI: https://doi.org/10.4028/www.scientific.net/msf.437-438.189
[144] Boggarapu, V., Gujjala, R., Ojha, S., et al., 2021. State of the art in functionally graded materials. Composite Structures. 262, 113596. DOI: https://doi.org/10.1016/j.compstruct.2021.113596
[145] Xin, L., Xu, J., Li, Z., et al., 2023. A Mori-Tanaka method based theoretical approximation for functionally graded thick wall tube under combined thermal and mechanical loads. Journal of Thermal Stresses. 46(3), 229–250. DOI: https://doi.org/10.1080/01495739.2022.2155743
[146] Du, B.X., Ran, Z.Y., Li, J., et al., 2019. Novel insulator with interfacial σ-FGM for DC compact gaseous insulated pipeline. IEEE Transactions on Dielectrics and Electrical Insulation. 26(3), 818–825. DOI: https://doi.org/10.1109/TDEI.2019.8726029
[147] Khodabakhsh, R., Saidi, A.R., Bahaadini, R., 2020. An analytical solution for nonlinear vibration and post-buckling of functionally graded pipes conveying fluid considering the rotary inertia and shear deformation effects. Applied Ocean Research. 101, 102277. DOI: https://doi.org/10.1016/j.apor.2020.102277
[148] Gunes, R., Apalak, M.K., Yildirim, M., 2007. The free vibration analysis and optimal design of an adhesively bonded functionally graded single lap joint. International Journal of Mechanical Sciences. 49(4), 479–499. DOI: https://doi.org/10.1016/j.ijmecsci.2006.09.010
[149] Walters, M.C., Paulino, G.H., Dodds Jr., R.H., 2004. Stress-intensity factors for surface cracks in functionally graded materials under mode-I thermomechanical loading. International Journal of Solids and Structures. 41(3–4), 1081–1118. DOI: https://doi.org/10.1016/j.ijsolstr.2003.09.050
[150] Cho, J.R., Ha, D.Y., 2001. Thermo-elastoplastic characteristics of heat-resisting functionally graded composite structures. Structural Engineering and Mechanics. 11(1), 49–70. DOI: https://doi.org/10.12989/sem.2001.11.1.049
[151] Chi, S.H., Chung, Y.L., 2006. Mechanical behavior of functionally graded material plates under transverse load-Part II: Numerical results. International Journal of Solids and Structures. 43(13), 3675–3691. DOI: https://doi.org/10.1016/j.ijsolstr.2005.04.010
[152] Akshaya, S.L., Prakash, A., Raj, J.B., 2021. Applications of Functionally Graded Materials in Structural Engineering—A Review. In: Dasgupta, K., Sudheesh, T.K., Praseeda, K.I., et al. (eds.). Proceedings of SECON 2020. Springer: Cham ,Switzerland. pp. 553–566. DOI: https://doi.org/10.1007/978-3-030-55115-5_51
[153] Amos, S.O. Numerical Analysis and Material Selection of Functionally Graded Pipes Based on Metals and Ceramics for Deep Offshore Oil and Gas Operations in Gulf of Guinea [PhD thesis]. University of Strathclyde: Glasgow, UK. DOI: https://doi.org/10.48730/wyza-jq20
[154] Mao, X.Y., Jing, J., Ding, H., et al., 2023. Dynamics of axially functionally graded pipes conveying fluid. Nonlinear Dynamics. 111(12), 11023–11044. DOI: https://doi.org/10.1007/s11071-023-08470-2
[155] Bhardwaj, Y.K., Bansal, V., Mahur, B.P., 2016. Analysis of Functionally Graded Cylinder Subjected to Internal Pressure. In Proceedings of the 4th International Conference on Science, Technology and Management (ICSTM-16), New Delhi, India, 16 May 2016.
[156] Abdali, N.K., Madeh, A.R., 2021. Structural analysis of functionally graded material using sigmioadal and power law. Diagnostyka. 22(4), 59–65. DOI: https://doi.org/10.29354/diag/144171
[157] Sidhoum, I.A., Boutchicha, D., Benyoucef, S., et al., 2017. An original HSDT for free vibration analysis of functionally graded plates. Steel and Composite Structures. 25(6), 735–745. DOI: https://doi.org/10.12989/scs.2017.25.6.735
[158] Wang, Z.M., Liu, Y.Z., 2016. Transverse vibration of pipe conveying fluid made of functionally graded materials using a symplectic method. Nuclear Engineering and Design. 298, 149–159. DOI: https://doi.org/10.1016/j.nucengdes.2015.12.007
[159] Fan, J., Chang, X., Chen, B., et al., 2024. Stability optimization of spinning FGM pipes conveying fluid via intermediate elastic supports. Ocean Engineering. 292, 116368. DOI: https://doi.org/10.1016/j.oceaneng.2023.116368
[160] Akbaş, Ş.D., 2019. Forced vibration analysis of functionally graded sandwich deep beams. Coupled Systems Mechanics. 8(3), 259–271. DOI: https://doi.org/10.12989/csm.2019.8.3.259
[161] Li, H.C., Ke, L.L., Yang, J., et al., 2020. Free vibration of variable thickness FGM beam submerged in fluid. Composite Structures. 233, 111582. DOI: https://doi.org/10.1016/j.compstruct.2019.111582
[162] Chang, X., Zhou, J., 2022. Static and dynamic characteristics of post-buckling of porous functionally graded pipes under thermal shock. Composite Structures. 288, 115373. DOI: https://doi.org/10.1016/j.compstruct.2022.115373
[163] Li, C., Shen, H.S., Wang, H., et al., 2020. Large amplitude vibration of sandwich plates with functionally graded auxetic 3D lattice core. International Journal of Mechanical Sciences. 174, 105472. DOI: https://doi.org/10.1016/j.ijmecsci.2020.105472
[164] Shen, H.S., Wang, Z.X., 2012. Assessment of Voigt and Mori-Tanaka models for vibration analysis of functionally graded plates. Composite Structures. 94(7), 2197–2208. DOI: https://doi.org/10.1016/j.compstruct.2012.02.018
[165] Xin, L., Yang, S., Zhou, D., et al., 2016. An approximate analytical solution based on the Mori-Tanaka method for functionally graded thick-walled tube subjected to internal pressure. Composite Structures. 135, 74–82. DOI: https://doi.org/10.1016/j.compstruct.2015.08.104
[166] Kiani, Y., Eslami, M.R., 2015. Thermal Postbuckling of Imperfect Circular Functionally Graded Material Plates: Examination of Voigt, Mori-Tanaka, and Self-Consistent Schemes. Journal of Pressure Vessel Technology, Transactions of the ASME. 137(2). DOI: https://doi.org/10.1115/1.4026993
[167] Najibi, A., Kianifar, M., Ghazifard, P., 2023. On the natural frequency investigation of the thick hollow cylinder with 2D-FGM Mori-Tanaka scheme. Ships and Offshore Structures. 18(12), 1638–1649. DOI: https://doi.org/10.1080/17445302.2022.2133881
[168] Foroutan, K., Torabi, F., Patel, A.P., 2025. Nonlinear Vibration and Post-Buckling Behaviors of Metal and FGM Pipes Transporting Heavy Crude Oil. Applied Sciences. 15(15), 8515. DOI: https://doi.org/10.3390/app15158515