PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 125–133 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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the strain energy release rate and time in Fig. 3 are expressed in non-dimensional form by using formulae   1 2 / G G G R C N  and 1 1 / C C N t tG   , respectively. The increase of the strain energy release rate with time is caused by increase of the angle of twist,  . One can observe the evolution of the strain energy release rate with increase of parameter, 1 f , in Fig. 4. The curves in Fig. 4 indicate that the increase of 1 f leads to reduction of the strain energy release rate. A solution of the strain energy release rate is derived also assuming that the beam is statically determinate structure (the right-hand end of the beam is clamped, while the left-hand end is free). In this case, the torsion moments, 1 3 L L T and 3 4 L L T , are found as 0 1 3  L L T , (31) T T L L  3 4 . (32) The strain energy release rate in the statically determined beam is shown also in Fig. 4. It can be observed that the static indeterminacy reduces the strain energy release rate (Fig. 4). The evolution of the strain energy release rate with increase of parameter, 1 p , is illustrated in Fig. 5 at three 1 2 / R R ratio. The curves in Fig. 5 show that increase of 1 p and 1 2 / R R causes decrease of the strain energy release rate. 4. Conclusions A non-linear viscoelastic statically undetermined beam structure with a lengthwise crack is considered. A viscoelastic mechanical model consisting of non-linear springs and non-linear dashpots arranged in parallel is used for treating of the time-dependent behaviour of the beam in this study. The beam has a circular cross-section and is subjected to pure torsion so as the angle of twist of the cross-section in which the torsion moment is applied increases linearly with time. The material of the beam is continuously inhomogeneous in radial direction. An approach for resolving the static indeterminacy is presented. The strain energy release rate is derived by considering the complementary strain energy in the beam. The balance of the energy is analyzed to verify the solution of the strain energy release rate. In order to evaluate the effect of static indeterminacy, the strain energy release rate in the statically undetermined beam is compared with that in statically determined beam. It is found that the static indeterminacy reduces the strain energy release rate. The analysis indicates also that increase of parameters, 1 f and 1 p , and 1 2 / R R ratio reduces the strain energy release rate. References Chatzigeorgiou, G., Charalambakis, G., 2005. Instability Analysis of Non-Homogeneous Materials Under Biaxial Loading, Int. J. Plast. 21, 0749 6419. Chen, Y., Lin, X., 2008. Elastic analysis for thick cylinders and spherical pressure vessels made of functionally graded materials. Computational Materials Science 44, 581-581. Chikh, A., 2019. Investigations in static response and free vibration of a functionally graded beam resting on elastic foundations. Frattura ed Integrità Strutturale 14, 115-126. Ganapathi, M., 2007, Dynamic stability characteristics of functionally graded materials shallow spherical shells, Composite Structures 79, 338 343. Han, X., Liu, G.R., Lam, K.Y., 2001. Transient waves in plates of functionally graded materials, International Journal for Numerical Methods in Engineering 52, 851-865. Hao, Y.X, Chen, L.H., Zhang, W., Lei, J. G., 2002. Nonlinear oscillations and chaos of functionally graded materials plate. Journal of Sound and vibration 312, 862-892. Kou, X.Y., Parks, G.T., Tan, S.T., 2012. Optimal design of functionally graded materials, using a procedural model and particle swarm optimization, Computer Aided Design 44, 300-310. Kyung-Su Na, Ji-Hwan Kim, 2004. Three-dimensional thermal buckling analysis of functionally graded materials, Composites Part B: Engineering 35, 429-437. Mahamood, R.M., Akinlabi, E.T., 2017. Functionally Graded Materials. Springer.