Document Type : Review Articles

Authors

1 Department of Mechanical Engineering, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran

2 Department of Mechanical Engineering, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran

10.30495/admt.2021.1927025.1271

Abstract

This study investigates the fatigue life of a cracked plate subjected to cyclic load under linear elastic fracture mechanics, using a numerical method of extended isogeometric analysis (XIGA) with a K-refinement approach. XIGA is applied to simulate discontinuity problems without meshing and without the necessity for element boundaries to be aligned to crack faces. In this method, the crack faces are simulated by discontinuous Heaviside functions, whereas the singularity in the stress field at the crack tip is simulated by crack tip enrichment functions. The stress intensity factors for the cracks are numerically calculated using the interaction integral method. Paris law of fatigue crack growth is utilized for predicting the fatigue life of a cracked plate. In the standard finite element analysis, there is no refinement method similar to k-refinement. The effect of the k-refinement on the accuracy of the values stress intensity factor and fatigue life is investigated. To achieve this, the order of Non-uniform rational B-Splines (NURBS) basic function is considered as linear, quadratic, and cubic. It is observed that as NURBS orders are increased in k-refinement, results are improved, and the error is lower compared with the analytical solution. The results show that values of stress intensity factor and fatigue life obtained using XIGA are more accurate compared to those obtained by the finite element method. In addition, and they are closer to the results of the analytical solution, and the XIGA method is more efficient.

Keywords

  • Nguyen-Xuan, H., Liu, G., Bordas, S., Natarajan, S., and Rabczuk, T., An Adaptive Singular ES-FEM for Mechanics Problems with Singular Field of Arbitrary Order, Computer Methods in Applied Mechanics and Engineering, Vol. 253, 2013, pp. 252-273.
  • Yan, X., A Boundary Element Modeling of Fatigue Crack Growth in A Plane Elastic Plate, Mechanics Research Communications, Vol. 33, No. 4, 2006, pp. 470-481.
  • Chen, J. W., Zhou, X. P., The Enhanced Extended Finite Element Method for the Propagation of Complex Branched Cracks, Engineering Analysis with Boundary Elements, Vol. 104, 2019, pp. 46-62.
  • Xin, H., Veljkovic, M., Fatigue crack Initiation Prediction Using Phantom Nodes-Based Extended Finite Element Method for S355 and S690 Steel Grades, Engineering Fracture Mechanics, Vol. 214, 2019, pp. 164-176.
  • Hou, J., Zuo, H., Li, Q., Jiang, R., and Zhao, L., Mintegral Analysis For Cracks in a Viscoplastic Material with Extended Finite Element Method, Engineering Fracture Mechanics, Vol. 200, 2018, pp. 294-311.
  • Apprich, C., Höllig, K., Hörner, J., and Reif, U., Collocation with WEB–Splines, Advances in Computational Mathematics, Vol. 42, No. 4, 2016, pp. 823-842.
  • Muthu, N., Maiti, S., Yan, W., and Falzon, B., Modelling Interacting Cracks Through a Level Set Using the Element-Free Galerkin Method, International Journal of Mechanical Sciences, Vol. 134, 2017, pp. 203-215.
  • Hughes, T. J., Cottrell, J. A., and Bazilevs, Y., Isogeometric Analysis: CAD, Finite Elements, Nurbs, Exact Geometry and Mesh Refinement, Computer Methods in Applied Mechanics and Engineering, Vol. 194, No. 39, 2005, pp. 4135-4195.
  • Belytschko, T., Black, T., Elastic Crack Growth in Finite Elements with Minimal Remeshing, International Journal for Numerical Methods in Engineering, Vol. 45, No. 5, 1999, pp. 601-620.
  • Moës, N., Dolbow, J., and Belytschko, T., A Finite Element Method for Crack Growth Without Remeshing, International Journal for Numerical Methods in Engineering, Vol. 46, No. 1, 1999, pp. 131-150.
  • Baietto, M. C., Pierres, E., Gravouil, A., Berthel, B., Fouvry, S., and Trolle, B., Fretting Fatigue Crack Growth Simulation Based on a Combined Experimental and XFEM Strategy, International Journal of Fatigue, Vol. 47, 2013, pp. 31-43.
  • Stolarska, M., Chopp, D. L., Moës, N., and Belytschko, T., Modelling Crack Growth by Level Sets in the Extended Finite Element Method, International Journal for Numerical Methods in Engineering, Vol. 51, No. 8, 2001, pp. 943-960.
  • Belytschko, T., Chen, H., Singular Enrichment Finite Element Method for Elastodynamic Crack Propagation, International Journal of Computational Methods, Vol. 1, No. 01, 2004, pp. 1-15.
  • Ghorashi, S. S., Valizadeh, N., Mohammadi, S., and Rabczuk, T., T-spline Based XIGA for Fracture Analysis of Orthotropic Media, Computers & Structures, Vol. 147, 2015, pp. 138-146.
  • Gu, J., Yu, T., Nguyen, T. T., Yang, Y., and Bui, T. Q., Fracture Modeling with the Adaptive Xiga Based on Locally Refined B-Splines, Computer Methods in Applied Mechanics and Engineering, Vol. 354, 2019, pp. 527-567.
  • Peng, X., Atroshchenko, E., Kerfriden, P., and Bordas, S., Isogeometric Boundary Element Methods for Three Dimensional Static Fracture and Fatigue Crack Growth, Computer Methods in Applied Mechanics and Engineering, Vol. 316, 2017, pp. 151-185.
  • Kumar, S., Singh, I., and Mishra, B., A Coupled Finite Element and Element-Free Galerkin Approach for The Simulation of Stable Crack Growth in Ductile Materials, Theoretical and Applied Fracture Mechanics, Vol. 70, 2014, pp. 49-58.
  • Shedbale, A., Singh, I., and Mishra, B., A Coupled Fe–EFG Approach for Modelling Crack Growth in Ductile Materials, Fatigue & Fracture of Engineering Materials & Structures, Vol. 39, No. 10, 2016, pp. 1204-1225.
  • Shedbale, A., Singh, I., Mishra, B., and Sharma, K., Ductile Failure Modeling and Simulations Using Coupled FE–EFG Approach, International Journal of Fracture, Vol. 203, No. 1-2, 2017, pp. 183-209.
  • Shedbale, A., Singh, I., Mishra, B., and Sharma, K., Evaluation of Mechanical Properties Using Spherical Ball Indentation and Coupled Finite Element–Element-Free Galerkin Approach, Mechanics of Advanced Materials and Structures, Vol. 23, No. 7, 2016, pp. 832-843.
  • Singh, I., Mishra, B., Bhattacharya, S., and Patil, R., The Numerical Simulation of Fatigue Crack Growth Using Extended Finite Element Method, International Journal of Fatigue, Vol. 36, No. 1, 2012, pp. 109-119.
  • Kumar, S., Shedbale, A., Singh, I., and Mishra, B., Elasto-Plastic Fatigue Crack Growth Analysis of Plane Problems in The Presence of Flaws Using XFEM, Frontiers of Structural and Civil Engineering, Vol. 9, No. 4, 2015, pp. 420-440.
  • Menouillard, T., Belytschko, T., Dynamic Fracture with Meshfree Enriched XFEM, Acta Mechanica, Vol. 213, No. 1-2, 2010, pp. 53-69.
  • Tanaka, S., Suzuki, H., Sadamoto, S., Imachi, M., and Bui, T. Q., Analysis of Cracked Shear Deformable Plates by an Effective Meshfree Plate Formulation, Engineering Fracture Mechanics, Vol. 144, 2015, pp. 142-157.
  • Tanaka, S., Suzuki, H., Sadamoto, S., Okazawa, S., Yu, T., and Bui, T., Accurate Evaluation of Mixed-Mode Intensity Factors of Cracked Shear-Deformable Plates by an Enriched Meshfree Galerkin Formulation, Archive of Applied Mechanics, Vol. 87, No. 2, 2017, pp. 279-298.
  • Tran, L. V., Ferreira, A., and Nguyen-Xuan, H., Isogeometric Analysis of Functionally Graded Plates Using Higher-Order Shear Deformation Theory, Composites Part B: Engineering, Vol. 51, 2013, pp. 368-383.
  • Yuan, H., Liu, W., and Xie, Y., Mode-I Stress Intensity Factors for Cracked Special-Shaped Shells Under Bending, Engineering Fracture Mechanics, Vol. 207, 2019, pp. 131-148.
  • Nguyen, B., Tran, H., Anitescu, C., Zhuang, X., and Rabczuk, T., An Isogeometric Symmetric Galerkin Boundary Element Method for Two-Dimensional Crack Problems, Computer Methods in Applied Mechanics and Engineering, Vol. 306, 2016, pp. 252-275.
  • Verhoosel, C. V., Scott, M. A., De Borst, R., and Hughes, T. J., An Isogeometric Approach to Cohesive Zone Modeling, International Journal for Numerical Methods in Engineering, Vol. 87, No. 1‐5, 2011, pp. 336-360.
  • Verhoosel, C. V., Scott, M. A., Hughes, T. J., and De Borst, R., An Isogeometric Analysis Approach to Gradient Damage Models, International Journal for Numerical Methods in Engineering, Vol. 86, No. 1, 2011, pp. 115-134.
  • Hao, P., Yuan, X., Liu, H., Wang, B., Liu, C., Yang, D., and Zhan, S., Isogeometric Buckling Analysis of Composite Variable-Stiffness Panels, Composite Structures, Vol. 165, 2017, pp. 192-208.
  • Tran, L. V., Ly, H. A., Lee, J., Wahab, M. A., and Nguyen-Xuan, H., Vibration Analysis of Cracked Fgm Plates Using Higher-Order Shear Deformation Theory and Extended Isogeometric Approach, International Journal of Mechanical Sciences, Vol. 96, 2015, pp. 65-78.
  • Bhardwaj, G., Singh, I., and Mishra, B., Numerical Simulation of Plane Crack Problems Using Extended Isogeometric Analysis, Procedia Engineering, Vol. 64, 2013, pp. 661-670.
  • Bhardwaj, G., Singh, I., Mishra, B., and Kumar, V., Numerical Simulations of Cracked Plate Using Xiga Under Different Loads and Boundary Conditions, Mechanics of Advanced Materials and Structures, Vol. 23, No. 6, 2016, pp. 704-714.
  • Nguyen-Thanh, N., Valizadeh, N., Nguyen, M., Nguyen-Xuan, H., Zhuang, X., Areias, P., Zi, G., Bazilevs, Y., De Lorenzis, L., and Rabczuk, T., An Extended Isogeometric Thin Shell Analysis Based on Kirchhoff–Love Theory, Computer Methods in Applied Mechanics and Engineering, Vol. 284, 2015, pp. 265-291.
  • Singh, A. K., Jameel, A., and Harmain, G., Investigations on Crack Tip Plastic Zones by the Extended Iso-Geometric Analysis, Materials Today: Proceedings, Vol. 5, No. 9, 2018, pp. 19284-19293.
  • Roh, H. Y., Cho, M., The Application of Geometrically Exact Shell Elements to B-Spline Surfaces, Computer Methods in Applied Mechanics and Engineering, Vol. 193, No. 23-26, 2004, pp. 2261-2299.
  • Piegl, L., Tiller, W., The NURBS Book, Second Edition, Springer Science & Business Media, Germany, 2012, 3642592236.
  • Cottrell, J., Hughes, T., and Reali, A., Studies of Refinement and Continuity in Isogeometric Structural Analysis, Computer Methods in Applied Mechanics and Engineering, Vol. 196, No. 41-44, 2007, pp. 4160-4183.
  • Cottrell, J. A., Hughes, T. J., and Bazilevs, Y., Isogeometric Analysis Toward Integration of CAD and FEA, John Wiley & Sons, United Kingdom, 2009, 0470749091.
  • Sutradhar, A., Paulino, G. H., 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, Vol. 28, No. 11, 2004, pp. 1335-1350.
  • Mohammadi, S., Extended Finite Element Method for Fracture Analysis of Structures, John Wiley & Sons, Oxford OX4 2DQ, United Kingdom, 2008, 0470697997.
  • Williams, M., Gn the Stress Distribution at the Base of a Stationary Crack. I. Appl. 1957, Mech.
  • Guo, F., Guo, L., Huang, K., Bai, X., Zhong, S., and Yu, H., An Interaction Energy Integral Method for T-Stress Evaluation in Nonhomogeneous Materials Under Thermal Loading, Mechanics of Materials, Vol. 83, 2015, pp. 30-39.
  • Bremberg, D., Faleskog, J., A Numerical Procedure for Interaction Integrals Developed for Curved Cracks of General Shape in 3-D, International Journal of Solids and Structures, Vol. 62, 2015, pp. 144-157.
  • Yoon, M., Cho, S., Isogeometric Shape Design Sensitivity Analysis of Elasticity Problems Using Boundary Integral Equations, Engineering Analysis with Boundary Elements, Vol. 66, 2016, pp. 119-128.