Abstract
Transient Galerkin finite volume method (GFVM) is developed to solve time-dependent problems and analysis of the dynamic stress intensity factors (DSIFs) for cracked problem. An interesting feature of the developed method is its matrix free operations; therefore, it obviously reduces the computation workloads for dynamic cases with small time marching. The two-point displacement extrapolation method is used for calculating the stress intensity factors (SIFs). To show the ability of this method, the structural problem, such as a beam under dynamic load, is considered as the first case study. The computed transient deflections are used for evaluating the accuracy of the GFVM in comparison with the results of the explicit finite element method (explicit-FEM) and meshless method solvers. A comparison of the CPU time consumption of the GFVM and explicit-FEM solvers shows that the GFVM entails lesser time consumption than the explicit-FEM, without reducing the accuracy of the results. In the second case study, the SIFs are computed for plate with inner crack under constant loading. For the third and fourth case study, the ability of the proposed GFVM solver to cope with DSIFs for a plate with an edge crack and L-shape plate with an inclined crack under dynamic load were tested. The comparison indicates that the GFVM not only provide compatible accuracy close to other common numerical solvers, also offers considerable CPU-time consumption, in comparison with the methods that requires matrix manipulations.
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Aghajanian, S., Baghi, H., Amini, F., & Samani, M. (2014). Optimal control of steel structures by improved particle swarm. International Journal of Steel Structures, 14, 223–230. https://doi.org/10.1007/s13296-014-2003-3.
Amini, F., & Zabihi-Samani, M. (2014). A wavelet-based adaptive pole assignment method for structural control. Computer-Aided Civil and Infrastructure Engineering, 29, 464–477. https://doi.org/10.1111/mice.12072.
Anderson, T. L. (2004). Fracture mechanics: Fundamentals and applications (3rd ed.). Boca Raton: CRC Press.
Bijelonja, I., Demirdžić, I., & Muzaferija, S. (2006). A finite volume method for incompressible linear elasticity. Computer Methods in Applied Mechanics and Engineering, 195, 6378–6390. https://doi.org/10.1016/j.cma.2006.01.005.
Chan, S. K., Tuba, I. S., & Wilson, W. K. (1970). On the finite element method in linear fracture mechanics. Engineering Fracture Mechanics, 2, 1–17. https://doi.org/10.1016/0013-7944(70)90026-3.
Fedelinski, P., Aliabadi, M. H., & Rooke, D. P. (1996). The Laplace transform DBEM for mixed-mode dynamic crack analysis. Computers & Structures, 59, 1021–1031. https://doi.org/10.1016/0045-7949(95)00347-9.
Fryer, Y. D., Bailey, C., Cross, M., & Lai, C. H. (1991). A control volume procedure for solving the elastic stress–strain equations on an unstructured mesh. Applied Mathematical Modelling, 15, 639–645. https://doi.org/10.1016/S0307-904X(09)81010-X.
Ghanooni-Bagha, M., Shayanfar, M., Reza-Zadeh, O., & Zabihi-Samani, M. (2017). The effect of materials on the reliability of reinforced concrete beams in normal and intense corrosions. Eksploatacja i Niezawodnosc – Maintenance and Reliability, 19, 393–402. https://doi.org/10.17531/ein.2017.3.10.
Gu, Y. T., & Liu, G. R. (2001). A meshless local Petrov–Galerkin (MLPG) method for free and forced vibration analyses for solids. Computational Mechanics, 27, 188–198. https://doi.org/10.1007/s004660100237.
Irwin, G. R. (1957). Analysis of stresses and strains near the end of a crack traversing a plate. Journal of Applied Mechanics, 24, 361–364.
Naeiji, A., Raji, F., & Zisis, I. (2017). Wind loads on residential scale rooftop photovoltaic panels. Journal of Wind Engineering and Industrial Aerodynamics, 168, 228–246. https://doi.org/10.1016/j.jweia.2017.06.006.
Nayroles, B., Touzot, G., & Villon, P. (1992). Generalizing the finite element method: Diffuse approximation and diffuse elements. Computational Mechanics, 10, 307–318. https://doi.org/10.1007/BF00364252.
Rice, J. R. (1968). A path independent integral and the approximate analysis of strain concentration by notches and cracks. Journal of Applied Mechanics, 35, 379–386. https://doi.org/10.1115/1.3601206.
Rybicki, E. F., & Kanninen, M. F. (1977). A finite element calculation of stress intensity factors by a modified crack closure integral. Engineering Fracture Mechanics, 9, 931–938. https://doi.org/10.1016/0013-7944(77)90013-3.
Sabbagh-Yazdi, S. R., Mastorakis, N. E., & Esmaili, M. (2008). Explicit 2D matrix free Galerkin finite volume solution of plane strain structural problems on triangular meshes. International Journal of Mathematics and Computers in Simulation, 2, 1–8.
Sabbagh-Yazdi, S. R., & Ali-Mohammadi, S. (2011). Performance evaluation of iterative GFVM on coarse unstructured triangular meshes and comparison with matrix manipulation based solution methods. Scientia Iranica, 18, 131–138. https://doi.org/10.1016/j.scient.2011.03.023.
Sabbagh-Yazdi, S. R., Ali-Mohammadi, S., & Pipelzadeh, M. K. (2012). Unstructured finite volume method for matrix free explicit solution of stress–strain fields in two dimensional problems with curved boundaries in equilibrium condition. Applied Mathematical Modelling, 36, 2224–2236. https://doi.org/10.1016/j.apm.2011.08.001.
Sabbagh-Yazdi, S. R., Farhoud, A., & Asil Gharebaghi, S. (2018). Simulation of 2D linear crack growth under constant load using GFVM and two-point displacement extrapolation method. Applied Mathematical Modelling, 61, 650–667. https://doi.org/10.1016/j.apm.2018.05.022.
Slone, A. K., Bailey, C., & Cross, M. (2003). Dynamic solid mechanics using finite volume methods. Applied Mathematical Modelling, 27, 69–87. https://doi.org/10.1016/S0307-904X(02)00060-4.
Suliman, R., Oxtoby, O. F., Malan, A. G., & Kok, S. (2014). An enhanced finite volume method to model 2D linear elastic structures. Applied Mathematical Modelling, 38, 2265–2279. https://doi.org/10.1016/j.apm.2013.10.028.
Timoshenko, S. P., & Goodier, J. N. (1970). Theory of elasticity (3rd ed.). New York: McGraw-Hill.
Tran, V.-X., Geniaut, S., Galenne, E., & Nistor, I. (2013). A modal analysis for computation of stress intensity factors under dynamic loading conditions at low frequency using extended finite element method. Engineering Fracture Mechanics, 98, 122–136. https://doi.org/10.1016/j.engfracmech.2012.12.005.
Wang, J., Liew, K. M., Tan, M. J., & Rajendran, S. (2002). Analysis of rectangular laminated composite plates via FSDT meshless method. International Journal of Mechanical Sciences, 44, 1275–1293. https://doi.org/10.1016/S0020-7403(02)00057-7.
Wheel, M. A. (1996). A geometrically versatile finite volume formulation for plane elastostatic stress analysis. The Journal of Strain Analysis for Engineering Design, 31, 111–116. https://doi.org/10.1243/03093247V312111.
Xuan, Z. C., Khoo, B. C., & Li, Z. R. (2006). Computing bounds to mixed-mode stress intensity factors in elasticity. Archive of Applied Mechanics, 75, 193–209. https://doi.org/10.1007/s00419-005-0388-3.
Zabihi-Samani, M., & Ghanooni-Bagha, M. (2018). Optimal semi-active structural control with a wavelet-based cuckoo-search fuzzy logic controller. Iranian Journal of Science and Technology, Transactions of Civil Engineering. https://doi.org/10.1007/s40996-018-0206-0.
Zisis, I., Raji, F., & Candelario Jose, D. (2017). Large-scale wind tunnel tests of canopies attached to low-rise buildings. Journal of Architectural Engineering, 23, B4016005. https://doi.org/10.1061/(ASCE)AE.1943-5568.0000235.
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Sabbagh-Yazdi, S.R., Farhoud, A. & Zabihi-Samani, M. Transient Galerkin finite volume solution of dynamic stress intensity factors. Asian J Civ Eng 20, 371–381 (2019). https://doi.org/10.1007/s42107-018-00111-z
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DOI: https://doi.org/10.1007/s42107-018-00111-z