Bézier base extended isogeometric numerical method for thermo elastic-plastic analysis of crack propagation in cracked plate under welding residual stress and thermal load
Abstract
A new procedure in the field of Bézier base extended isogeometric method (XIGA) has been introduced to analyze the effect of welding residual stress and thermal load on crack propagation rate and fatigue life. This new procedure is based on the constitutive thermoelastic plastic equation. The main parts of this procedure are using the B´ezier base XIGA method to calculate the redistribution of welding residual stress due to crack growth and to compute the value of stress intensity factor (SIF) in the welding residual stress field. For this purpose, the grid points of Bézier elements (with C0-continuity) around the crack line and the crack tip are identified by the level set representation. Then, discontinuous enrichment functions are added to the isogeometric analysis approximation. Thus, this method does not require the re-meshing process. The results show that there is a good agreement between the results of proposed numerical method and the Hole-Drilling Strain-Gage method. The interaction integral method has been used to extract SIF. The effects of welding residual stress and thermal load on the SIF are considered using the superposition method. Also, the Walker equation has been modified to calculate the fatigue life caused by thermal loading and welding residual stress. The results display a good agreement between the proposed method and the finite element method. Due to the advantages of the Bézier based XIGA method, which eliminates parametric space and allows the precise addition of enrichment functions to the basis functions of cracked elements (crack line or crack tip), the obtained results are highly accurate that shows this method is effective for analyzing discontinuous problems.
Keyword : isogeometric method, Bézier extraction operator, Welding residual stress, stress intensity factor, fatigue life
How to Cite
Shoheib, M. M., Shahrooi, S., Shishehsaz, M., & Hamzehei , M. (2022). Bézier base extended isogeometric numerical method for thermo elastic-plastic analysis of crack propagation in cracked plate under welding residual stress and thermal load. Mathematical Modelling and Analysis, 27(4), 629–651. https://doi.org/10.3846/mma.2022.15503
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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D. Bremberg and J. Faleskog. A numerical procedure for interaction integrals developed for curved cracks of general shape in 3D. International Journal of Solids and Structures, 62:144–157, 2015. https://doi.org/10.1016/j.ijsolstr.2015.02.022
B. Brickstad and B.L. Josefson. A parametric study of residual stresses in multipass butt-welded stainless steel pipes. International Journal of pressure Vessels and piping, 75(1):11–25,1998. https://doi.org/10.1016/S0308-0161(97)00117-8
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A. de Klerk, A.G. Visser and A.A. Groenwold. Lower and upper bound estimation of isotropic and orthotropic fracture mechanics problems using elements with rotational degrees of freedom. Communications in Numerical Methods in Engineering, 24(5):335–353, 2008. https://doi.org/10.1002/cnm.973
H. Dehestani, Y. Ordokhani and M. Razzaghi. Numerical solution of variableorder time fractional weakly singular partial integro-differential equations with error estimation. Mathematical Modelling and Analysis, 25(4):680–701, 2020. https://doi.org/10.3846/mma.2020.11692
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G. Glinka. Effect of residual stresses on fatigue crack growth in steel weldments under constant and variable amplitude loads. In Fracture Mechanics: Proceedings of the Eleventh National Symposium on Fracture Mechanics: Part I. ASTM International, 1979. https://doi.org/10.1520/STP34914S
J. Huang, N. Nguyen-Thanh, W. Li and K. Zhou. An adaptive isogeometric-meshfree coupling approach for the limit analysis of cracked structures. Theoretical and Applied Fracture Mechanics, 106:102426, 2020. https://doi.org/10.1016/j.tafmec.2019.102426
X. Huang, Y. Liu and X. Huang. Analytical characterizations of crack tip plastic zone size for central-cracked unstiffened and stiffened plates under biaxial loading. Engineering Fracture Mechanics, 206:1–20, 2019. https://doi.org/10.1016/j.engfracmech.2018.11.047
T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer methods in applied mechanics and engineering, 194(39):4135–4195, 2005. https://doi.org/10.1016/j.cma.2004.10.008
A. Jameel and G.A. Harmain. A coupled FE-IGA technique for modeling fatigue crack growth in engineering materials. Mechanics of Advanced Materials and Structures, 26(21):1764–1775, 2019. https://doi.org/10.1080/15376494.2018.1446571
A. Jameel and G.A. Harmain. Extended iso-geometric analysis for modeling three-dimensional cracks. Mechanics of Advanced Materials and Structures, 26(11):915–923, 2019. https://doi.org/10.1080/15376494.2018.1430275
A. Jameel and G.A. Harmain. Fatigue crack growth analysis of cracked specimens by the coupled finite element-element free Galerkin method. Mechanics of Advanced Materials and Structures, 26(16):1343–1356, 2019. https://doi.org/10.1080/15376494.2018.1432800
A. Jameel and G.A. Harmain. Large deformation in bi-material components by XIGA and coupled FE-IGA techniques. Mechanics of Advanced Materials and Structures, 29(6):850–872, 2022. https://doi.org/10.1080/15376494.2020.1799120
B. Karaman. On the numerical simulation of time-space fractional coupled nonlinear schro¨dinger equations utilizing wendland’s compactly supported function collocation method. Mathematical Modelling and Analysis, 26(1):94–115, 2021. https://doi.org/10.3846/mma.2021.12262
M. Kern, A. Taakili and M.M. Zarrouk. Preconditioned iterative method for reactive transport with sorption in porous media. Mathematical Modelling and Analysis, 25(4):546–568, 2020. https://doi.org/10.3846/mma.2020.10626
W. Li, N. Nguyen-Thanh, J. Huang and K. Zhou. Adaptive analysis of crack propagation in thin-shell structures via an isogeometric-meshfree moving leastsquares approach. Computer Methods in Applied Mechanics and Engineering, 358:112613, 2020. https://doi.org/10.1016/j.cma.2019.112613
A.M. Malik, E.M. Qureshi, N.U. Dar and I. Khan. Analysis of circumferentially arc welded thin-walled cylinders to investigate the residual stress fields. Thin-Walled Structures, 46(12):1391–1401, 2008. https://doi.org/10.1016/j.tws.2008.03.011
A. Moarrefzadeh, S. Shahrooi and M.J. Azizpour. Predicting fatigue crack propagation in residual stress field due to welding by meshless local PetrovGalerkin method. Journal of Manufacturing Processes, 45:379–391, 2019. https://doi.org/10.1016/j.jmapro.2019.07.019
S. Murugan, S.K. Rai, P.V. Kumar, T. Jayakumar, B. Raj and M.S.C Bose. Temperature distribution and residual stresses due to multipass welding in type 304 stainless steel and low carbon steel weld pads. International Journal of Pressure vessels and piping, 78(4):307–317, 2001. https://doi.org/10.1016/S0308-0161(01)00047-3
L.B. Nguyen, C.H. Thai, A.M. Zenkour and H. Nguyen-Xuan. An isogeometric B´ezier finite element method for vibration analysis of functionally graded piezoelectric material porous plates. International Journal of Mechanical Sciences, 157-158:165–183, 2019. https://doi.org/10.1016/j.ijmecsci.2019.04.017
N. Nguyen-Thanh, J. Huang and K. Zhou. An isogeometric-meshfree coupling approach for analysis of cracks. International Journal for Numerical Methods in Engineering, 113(10):1630–1651, 2018. https://doi.org/10.1002/nme.5713
N. Nguyen-Thanh, N. Valizadeh, M.N. Nguyen, H. Nguyen-Xuan, X. Zhuang, P. Areias, G. Zi, Y. Bazilevs, L. De Lorenzis and T. Rabczuk. An extended isogeometric thin shell analysis based on Kirchhoff–Love theory. Computer Methods in Applied Mechanics and Engineering, 284:265–291, 2015. https://doi.org/10.1016/j.cma.2014.08.025
L. Piegl and W. Tiller. The NURBS book. Springer Science & Business Media, 2012. https://doi.org/10.1007/978-3-642-97385-7
M. Ratas, A. Salupere and J. Majak. Solving nonlinear PDEs using the higher order Haar wavelet method on nonuniform and adaptive grids. Mathematical Modelling and Analysis, 26(1):147–169, 2021. https://doi.org/10.3846/mma.2021.12920
A.H.S. Shayegan, A. Zakeri and S.M. Hosseini. A numerical method for solving two-dimensional nonlinear parabolic problems based on a preconditioning operator. Mathematical Modelling and Analysis, 25(4):531–545, 2020. https://doi.org/10.3846/mma.2020.4310
J. Shi, D. Chopp, J. Lua, N. Sukumar and T. Belytschko. Abaqus implementation of extended finite element method using a level set representation for three-dimensional fatigue crack growth and life predictions. Engineering Fracture Mechanics, 77(14):2840–2863, 2010. https://doi.org/10.1016/j.engfracmech.2010.06.009
A.K. Singh, A. Jameel and G.A. Harmain. Investigations on crack tip plastic zones by the extended iso-geometric analysis. Materials Today: Proceedings, 5(9):19284–19293, 2018. https://doi.org/10.1016/j.matpr.2018.06.287 Materials Processing and characterization, 16th – 18th March 2018
J. Sladek, P. Stanak, Z.D. Han, V. Sladek and S.N. Atluri. Applications of the MLPG method in engineering & sciences: a review. CMES: Computer Modeling in Engineering & Sciences, 92(5):423–475, 2013.
M. Spaniel, J. Jurenka and J. Kuˇzelka. Verification of FE model of fatigue crackˇ propagation under mixed mode conditions. Meccanica, 44(2):189–195, 2009. https://doi.org/10.1007/s11012-008-9164-0
G.F. Sun, Z.D. Wang, Y. Lu, R. Zhou, Z.H. Ni, X. Gu and Z.G. Wang. Numerical and experimental investigation of thermal field and residual stress in laser-mig hybrid welded NV E690 steel plates. Journal of Manufacturing Processes, 34:106–120, 2018. https://doi.org/10.1016/j.jmapro.2018.05.023
A. Sutradhar and G.H. Paulino. Symmetric Galerkin boundary element computation of T-stress and stress intensity factors for mixed-mode cracks by the interaction integral method. Engineering Analysis with Boundary Elements, 28(11):1335–1350, 2004. https://doi.org/10.1016/j.enganabound.2004.02.009
L.V. Tran, A.J.M. Ferreira and H. Nguyen-Xuan. Isogeometric analysis of functionally graded plates using higher-order shear deformation theory. Composites Part B: Engineering, 51:368–383, 2013. https://doi.org/10.1016/j.compositesb.2013.02.045
R. Vaghefi, M.R. Hematiyan, A. Nayebi and A. Khosravifard. A parametric study of the MLPG method for thermo-mechanical solidification analysis. Engineering Analysis with Boundary Elements, 89:10–24, 2018. https://doi.org/10.1016/j.enganabound.2018.01.006
W. Wen, S. Luo, S. Duan, J. Liang and D. Fang. Improved quadratic isogeometric element simulation of one-dimensional elastic wave propagation with central difference method. Applied Mathematics and Mechanics, 39(5):703–716, 2018. https://doi.org/10.1007/s10483-018-2330-6
B. Yahiaoui and P. Petrequin. Etudé de la propagation de fissures par fatigue dans des aciers inoxydables austénitiques à bas carbone du type 304l et 316l. Revue de Physique Appliquée, 9(4):683–690, 1974. https://doi.org/10.1051/rphysap:0197400904068300
S. Yin, T. Yu, T.Q. Bui, X. Zheng and S. Gu. Static and dynamic fracture analysis in elastic solids using a multiscale extended isogeometric analysis. Engineering Fracture Mechanics, 207:109–130, 2019. https://doi.org/10.1016/j.engfracmech.2018.12.024
H. Yuan, W.J. Liu and Y.J. Xie. Mode-I stress intensity factors for cracked special-shaped shells under bending. Engineering Fracture Mechanics, 207:131– 148, 2019. https://doi.org/10.1016/j.engfracmech.2018.12.026
M. Zubairuddin, S.K. Albert, M. Vasudevan, S. Mahadevan, V. Chaudhari and V.K. Suri. Numerical simulation of multi-pass GTA welding of grade 91 steel. Journal of Manufacturing Processes, 27:87–97, 2017. https://doi.org/10.1016/j.jmapro.2017.04.031
D.J. Benson, Y. Bazilevs, M.-C. Hsu and T.J.R. Hughes. A large deformation, rotation-free, isogeometric shell. Computer Methods in Applied Mechanics and Engineering, 200(13):1367–1378, 2011. https://doi.org/10.1016/j.cma.2010.12.003
G. Bhardwaj, I.V. Singh and B.K. Mishra. Fatigue crack growth in functionally graded material using homogenized XIGA. Composite Structures, 134:269–284, 2015. https://doi.org/10.1016/j.compstruct.2015.08.065
D. Bremberg and J. Faleskog. A numerical procedure for interaction integrals developed for curved cracks of general shape in 3D. International Journal of Solids and Structures, 62:144–157, 2015. https://doi.org/10.1016/j.ijsolstr.2015.02.022
B. Brickstad and B.L. Josefson. A parametric study of residual stresses in multipass butt-welded stainless steel pipes. International Journal of pressure Vessels and piping, 75(1):11–25,1998. https://doi.org/10.1016/S0308-0161(97)00117-8
R. Cimrman, M. Novák, R. Kolman, M. Tuma, J. Plešek and J. Vackář. Convergence study of isogeometric analysis based on bézier extraction in electronic structure calculations. Applied Mathematics and Computation, 319:138–152, 2018. https://doi.org/10.1016/j.amc.2017.02.023
A. de Klerk, A.G. Visser and A.A. Groenwold. Lower and upper bound estimation of isotropic and orthotropic fracture mechanics problems using elements with rotational degrees of freedom. Communications in Numerical Methods in Engineering, 24(5):335–353, 2008. https://doi.org/10.1002/cnm.973
H. Dehestani, Y. Ordokhani and M. Razzaghi. Numerical solution of variableorder time fractional weakly singular partial integro-differential equations with error estimation. Mathematical Modelling and Analysis, 25(4):680–701, 2020. https://doi.org/10.3846/mma.2020.11692
D. Deng and H. Murakawa. Numerical simulation of temperature field and residual stress in multi-pass welds in stainless steel pipe and comparison with experimental measurements. Computational materials science, 37(3):269–277, 2006. https://doi.org/10.1016/j.commatsci.2005.07.007
G. Glinka. Effect of residual stresses on fatigue crack growth in steel weldments under constant and variable amplitude loads. In Fracture Mechanics: Proceedings of the Eleventh National Symposium on Fracture Mechanics: Part I. ASTM International, 1979. https://doi.org/10.1520/STP34914S
J. Huang, N. Nguyen-Thanh, W. Li and K. Zhou. An adaptive isogeometric-meshfree coupling approach for the limit analysis of cracked structures. Theoretical and Applied Fracture Mechanics, 106:102426, 2020. https://doi.org/10.1016/j.tafmec.2019.102426
X. Huang, Y. Liu and X. Huang. Analytical characterizations of crack tip plastic zone size for central-cracked unstiffened and stiffened plates under biaxial loading. Engineering Fracture Mechanics, 206:1–20, 2019. https://doi.org/10.1016/j.engfracmech.2018.11.047
T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer methods in applied mechanics and engineering, 194(39):4135–4195, 2005. https://doi.org/10.1016/j.cma.2004.10.008
A. Jameel and G.A. Harmain. A coupled FE-IGA technique for modeling fatigue crack growth in engineering materials. Mechanics of Advanced Materials and Structures, 26(21):1764–1775, 2019. https://doi.org/10.1080/15376494.2018.1446571
A. Jameel and G.A. Harmain. Extended iso-geometric analysis for modeling three-dimensional cracks. Mechanics of Advanced Materials and Structures, 26(11):915–923, 2019. https://doi.org/10.1080/15376494.2018.1430275
A. Jameel and G.A. Harmain. Fatigue crack growth analysis of cracked specimens by the coupled finite element-element free Galerkin method. Mechanics of Advanced Materials and Structures, 26(16):1343–1356, 2019. https://doi.org/10.1080/15376494.2018.1432800
A. Jameel and G.A. Harmain. Large deformation in bi-material components by XIGA and coupled FE-IGA techniques. Mechanics of Advanced Materials and Structures, 29(6):850–872, 2022. https://doi.org/10.1080/15376494.2020.1799120
B. Karaman. On the numerical simulation of time-space fractional coupled nonlinear schro¨dinger equations utilizing wendland’s compactly supported function collocation method. Mathematical Modelling and Analysis, 26(1):94–115, 2021. https://doi.org/10.3846/mma.2021.12262
M. Kern, A. Taakili and M.M. Zarrouk. Preconditioned iterative method for reactive transport with sorption in porous media. Mathematical Modelling and Analysis, 25(4):546–568, 2020. https://doi.org/10.3846/mma.2020.10626
W. Li, N. Nguyen-Thanh, J. Huang and K. Zhou. Adaptive analysis of crack propagation in thin-shell structures via an isogeometric-meshfree moving leastsquares approach. Computer Methods in Applied Mechanics and Engineering, 358:112613, 2020. https://doi.org/10.1016/j.cma.2019.112613
A.M. Malik, E.M. Qureshi, N.U. Dar and I. Khan. Analysis of circumferentially arc welded thin-walled cylinders to investigate the residual stress fields. Thin-Walled Structures, 46(12):1391–1401, 2008. https://doi.org/10.1016/j.tws.2008.03.011
A. Moarrefzadeh, S. Shahrooi and M.J. Azizpour. Predicting fatigue crack propagation in residual stress field due to welding by meshless local PetrovGalerkin method. Journal of Manufacturing Processes, 45:379–391, 2019. https://doi.org/10.1016/j.jmapro.2019.07.019
S. Murugan, S.K. Rai, P.V. Kumar, T. Jayakumar, B. Raj and M.S.C Bose. Temperature distribution and residual stresses due to multipass welding in type 304 stainless steel and low carbon steel weld pads. International Journal of Pressure vessels and piping, 78(4):307–317, 2001. https://doi.org/10.1016/S0308-0161(01)00047-3
L.B. Nguyen, C.H. Thai, A.M. Zenkour and H. Nguyen-Xuan. An isogeometric B´ezier finite element method for vibration analysis of functionally graded piezoelectric material porous plates. International Journal of Mechanical Sciences, 157-158:165–183, 2019. https://doi.org/10.1016/j.ijmecsci.2019.04.017
N. Nguyen-Thanh, J. Huang and K. Zhou. An isogeometric-meshfree coupling approach for analysis of cracks. International Journal for Numerical Methods in Engineering, 113(10):1630–1651, 2018. https://doi.org/10.1002/nme.5713
N. Nguyen-Thanh, N. Valizadeh, M.N. Nguyen, H. Nguyen-Xuan, X. Zhuang, P. Areias, G. Zi, Y. Bazilevs, L. De Lorenzis and T. Rabczuk. An extended isogeometric thin shell analysis based on Kirchhoff–Love theory. Computer Methods in Applied Mechanics and Engineering, 284:265–291, 2015. https://doi.org/10.1016/j.cma.2014.08.025
L. Piegl and W. Tiller. The NURBS book. Springer Science & Business Media, 2012. https://doi.org/10.1007/978-3-642-97385-7
M. Ratas, A. Salupere and J. Majak. Solving nonlinear PDEs using the higher order Haar wavelet method on nonuniform and adaptive grids. Mathematical Modelling and Analysis, 26(1):147–169, 2021. https://doi.org/10.3846/mma.2021.12920
A.H.S. Shayegan, A. Zakeri and S.M. Hosseini. A numerical method for solving two-dimensional nonlinear parabolic problems based on a preconditioning operator. Mathematical Modelling and Analysis, 25(4):531–545, 2020. https://doi.org/10.3846/mma.2020.4310
J. Shi, D. Chopp, J. Lua, N. Sukumar and T. Belytschko. Abaqus implementation of extended finite element method using a level set representation for three-dimensional fatigue crack growth and life predictions. Engineering Fracture Mechanics, 77(14):2840–2863, 2010. https://doi.org/10.1016/j.engfracmech.2010.06.009
A.K. Singh, A. Jameel and G.A. Harmain. Investigations on crack tip plastic zones by the extended iso-geometric analysis. Materials Today: Proceedings, 5(9):19284–19293, 2018. https://doi.org/10.1016/j.matpr.2018.06.287 Materials Processing and characterization, 16th – 18th March 2018
J. Sladek, P. Stanak, Z.D. Han, V. Sladek and S.N. Atluri. Applications of the MLPG method in engineering & sciences: a review. CMES: Computer Modeling in Engineering & Sciences, 92(5):423–475, 2013.
M. Spaniel, J. Jurenka and J. Kuˇzelka. Verification of FE model of fatigue crackˇ propagation under mixed mode conditions. Meccanica, 44(2):189–195, 2009. https://doi.org/10.1007/s11012-008-9164-0
G.F. Sun, Z.D. Wang, Y. Lu, R. Zhou, Z.H. Ni, X. Gu and Z.G. Wang. Numerical and experimental investigation of thermal field and residual stress in laser-mig hybrid welded NV E690 steel plates. Journal of Manufacturing Processes, 34:106–120, 2018. https://doi.org/10.1016/j.jmapro.2018.05.023
A. Sutradhar and G.H. Paulino. Symmetric Galerkin boundary element computation of T-stress and stress intensity factors for mixed-mode cracks by the interaction integral method. Engineering Analysis with Boundary Elements, 28(11):1335–1350, 2004. https://doi.org/10.1016/j.enganabound.2004.02.009
L.V. Tran, A.J.M. Ferreira and H. Nguyen-Xuan. Isogeometric analysis of functionally graded plates using higher-order shear deformation theory. Composites Part B: Engineering, 51:368–383, 2013. https://doi.org/10.1016/j.compositesb.2013.02.045
R. Vaghefi, M.R. Hematiyan, A. Nayebi and A. Khosravifard. A parametric study of the MLPG method for thermo-mechanical solidification analysis. Engineering Analysis with Boundary Elements, 89:10–24, 2018. https://doi.org/10.1016/j.enganabound.2018.01.006
W. Wen, S. Luo, S. Duan, J. Liang and D. Fang. Improved quadratic isogeometric element simulation of one-dimensional elastic wave propagation with central difference method. Applied Mathematics and Mechanics, 39(5):703–716, 2018. https://doi.org/10.1007/s10483-018-2330-6
B. Yahiaoui and P. Petrequin. Etudé de la propagation de fissures par fatigue dans des aciers inoxydables austénitiques à bas carbone du type 304l et 316l. Revue de Physique Appliquée, 9(4):683–690, 1974. https://doi.org/10.1051/rphysap:0197400904068300
S. Yin, T. Yu, T.Q. Bui, X. Zheng and S. Gu. Static and dynamic fracture analysis in elastic solids using a multiscale extended isogeometric analysis. Engineering Fracture Mechanics, 207:109–130, 2019. https://doi.org/10.1016/j.engfracmech.2018.12.024
H. Yuan, W.J. Liu and Y.J. Xie. Mode-I stress intensity factors for cracked special-shaped shells under bending. Engineering Fracture Mechanics, 207:131– 148, 2019. https://doi.org/10.1016/j.engfracmech.2018.12.026
M. Zubairuddin, S.K. Albert, M. Vasudevan, S. Mahadevan, V. Chaudhari and V.K. Suri. Numerical simulation of multi-pass GTA welding of grade 91 steel. Journal of Manufacturing Processes, 27:87–97, 2017. https://doi.org/10.1016/j.jmapro.2017.04.031