Effects of Periodic Loading on Longitudinal Fracture in ‎Viscoelastic Functionally Graded Beam Structures‎

Document Type : Research Paper


Department of Technical Mechanics, University of Architecture, Civil Engineering and Geodesy, 1 Chr. Smirnensky blvd., Sofia, 1046, Bulgaria‎


This paper analyzes the effects of periodic external loading on the longitudinal fracture of a beam structure made of linear viscoelastic material. A longitudinal crack splits the cracked part of the beam into left-hand and right-hand crack arms. The beam is under a bending moment applied at the free end of the right-hand crack arm. The bending moment varies periodically with time. The material of the beam is continuously inhomogeneous (functionally graded) along the thickness. Thus, the modulus of elasticity and the coefficient of viscosity of the material vary continuously in transversal direction of the beam. The balance of the energy in the beam under periodic bending moment is analyzed in order to derive the strain energy release rate for the longitudinal crack. The solution is verified by deriving the strain energy release rate by analyzing of the compliance of the beam subjected to periodic bending moment. An investigation is carried-out by applying the solutions obtained in order to evaluate the effects of the parameters of the periodic loading on the strain energy release rate.


Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Markworth, A.J., Ramesh, K.S., Parks, Jr.W.P., Review: modeling studies applied to functionally graded materials, J. Mater. Sci., 30(3), 1995, 2183-2193.  
[2] Butcher, R.J., Rousseau, C.E., Tippur, H.V., A functionally graded particulate composite: Measurements and Failure Analysis, Acta. Mater., 47(2), 1999, 259-268.
[3] Hirai, T., Chen, L., Recent and prospective development of functionally graded materials in Japan, Mater Sci. Forum, 308-311(4), 1999, 509-514.
[4] Miyamoto, Y., Kaysser, W.A., Rabin, B.H., Kawasaki, A., Ford, R.G., Functionally Graded Materials: Design, Processing and Applications, Kluwer Academic Publishers, Dordrecht/London/Boston, 1999.
[5] Han, X., Xu, Y.G., Lam, K.Y., Material characterization of functionally graded material by means of elastic waves and a progressive-learning neural network, Compos. Sci. Technol., 61(10), 2001, 1401-1411.
[6] Gasik, M.M., Functionally graded materials: bulk processing techniques, Int. J. Mater. Prod. Technol., 39(1-2), 2010, 20-29.
[7] Nemat-Allal, M.M., Ata, M.H., Bayoumi, M.R., Khair-Eldeen, W., Powder metallurgical fabrication and microstructural investigations of Aluminum/Steel functionally graded material, Mater. Sci. Appl., 2(5), 2011, 1708-1718.
[8] Shrikantha Rao, S., Gangadharan, K.V., Functionally graded composite materials: an overview, Proc. Mater. Sci., 5(1), 2014, 1291-1299.
[9] Wu, X.L., Jiang, P., Chen, L., Zhang, J.F., Yuan, F.P., Zhu, Y.T., Synergetic strengthening by gradient structure, Mater. Res. Lett., 2(1), 2014, 185–191.  
[10] Arefi, M., Rahimi, G.H., Nonlinear analysis of a functionally graded beam with variable thickness, Sci. Res. Essays, 8(6), 2013, 256-264.
[11] Arefi, M., Nonlinear analysis of a functionally graded beam resting on the elastic nonlinear foundation, J. Theor. Appl. Mech., 44(2), 2014, 71-82.
[12] Arefi, M., Elastic solution of a curved beam made of functionally graded materials with different cross section, Steel Compos. Struct., 18(3), 2015, 659-672.  
[13] Saiyathibrahim, A., Subramaniyan, R. Dhanapl, P., Centrefugally cast functionally graded materials – review, In: International Conference on Systems, Science, Control, Communications, Engineering and Technology, 2016.
[14] Mahamood, R.M., Akinlabi, E.T., Functionally Graded Materials, Springer, 2017. 
[15] Dolgov, N.A., Effect of the elastic modulus of a coating on the serviceability of the substrate-coating system, Strength Mater., 37(2), 2002, 422-431.
[16] Dolgov, N.A., Determination of Stresses in a Two-Layer Coating, Strength Mater., 37(2), 2005, 422-431.
[17] Dolgov, N.A., Analytical Methods to Determine the Stress State in the Substrate–Coating System Under Mechanical Loads, Strength Mater., 48(1), 2016, 658-667.
[18] Rizov, V.I., Influence of the viscoelastic material behaviour on the delamination in multilayered beam, Procedia Struct. Integr., 25, 2020, 88-100.
[19] Rizov, V.I., Longitudinal fracture analysis of continuously inhomogeneous beam in torsion with stress relaxation, Procedia Struct. Integr., 28, 2020, 1212-122.
[20] Rizov, V.I., Continuously inhomogeneous beam structure with creep: a longitudinal crack study, Struct. Monit. Maint., 8(1), 2021, 111-124.