NURBS-Based Isogeometric Analysis Method Application to Mixed-Mode Computational Fracture Mechanics

Document Type: Research Paper

Author

Department of Civil Engineering, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran

Abstract

An interaction integral method for evaluating mixed-mode stress intensity factors (SIFs) for two dimensional crack problems using NURBS-based isogeometric analysis method is investigated. The interaction integral method is based on the path independent J-integral. By introducing a known auxiliary field solution, the mixed-mode SIFs are calculated simultaneously. Among features of B-spline basis functions, the possibility of enhancing a B-spline basis with discontinuities by means of knot insertion makes isogeometric analysis method a suitable candidate for modelling discrete cracks. Moreover, the repetition of two different control points between two patches can create a discontinuity and also demonstrates a singularity in the stiffness matrix. In the case of a pre-defined interface, non-uniform rational B-splines are used to obtain an efficient discretization. Various numerical simulations for edge and center cracks demonstrate the suitability of the isogeometric analysis approach to fracture mechanics.

Keywords

Main Subjects

[1] Babuska, I., Melenk, J., The Partition of unity method, International Journal for Numerical Methods in Engineering, 40 (1997) 727–758.

[2] Babuška, I., Zhang, Z., The partition of unity method for the elastically supported beam, Computer Methods in Applied Mechanics and Engineering, 152(1–2) (1998) 1-18.

[3] Bandyopadhyay, S.N., Deysarker, H.K., Stress intensity factor for a crack emanating from the root of a semi-circular edge notch in a tension plate, Engineering Fracture Mechanics, 14(2) (1981) 373-384.

[4] Bazilevs, Y. et al., Isogeometric analysis using T-splines, Computer Methods in Applied Mechanics and Engineering, 199(5–8) (2010) 229-263.

[5] Bazilevs, Y., Hsu, M.C., Scott, M.A., Isogeometric fluid–structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines, Computer Methods in Applied Mechanics and Engineering, 249–252 (2012) 28-41.

[6] Belytschko, T., Black, T., Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering, 45(5) (1999) 601-620.

[7] Bhardwaj, G., Singh, I.V., Mishra, B.K., Bui, T.Q., Numerical simulation of functionally graded cracked plates using NURBS based XIGA under different loads and boundary conditions, Composite Structures, 126 (2015) 347-359.

[8] Bui, T.Q., Extended isogeometric dynamic and static fracture analysis for cracks in piezoelectric materials using NURBS, Computer Methods in Applied Mechanics and Engineering, 295 (2015) 470-509.

[9] Chen, T., Xiao, Z.-G., Zhao, X.-L., Gu, X.-L., A boundary element analysis of fatigue crack growth for welded connections under bending, Engineering Fracture Mechanics, 98 (2013) 44-51.

[10] Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, 2009.

[11] Cottrell, J.A., Hughes, T.J.R., Reali, A., Studies of refinement and continuity in isogeometric structural analysis, Computer Methods in Applied Mechanics and Engineering, 196(41–44) (2007) 4160-4183.

[12] Cottrell, J.A., Reali, A., Bazilevs, Y., Hughes, T.J.R., Isogeometric analysis of structural vibrations, Computer Methods in Applied Mechanics and Engineering, 195(41–43) (2006) 5257-5296.

[13] COX, M.G., The Numerical Evaluation of B-Splines, IMA Journal of Applied Mathematics, 10(2) (1972) 134-149.

[14] Daxini, S.D., Prajapati, J.M., A Review on Recent Contribution of Meshfree Methods to Structure and Fracture Mechanics Applications, The Scientific World Journal, 2014 (2014) ID 247172.

[15] de Boor, C., On calculating with B-splines, Journal of Approximation Theory, 6(1) (1972) 50-62.

[16] de Klerk, A., Visser, A.G., Groenwold, A.A., 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) (2008) 335-353.

[17] De Luycker, E., Benson, D.J., Belytschko, T., Bazilevs, Y., Hsu, M.C., X-FEM in isogeometric analysis for linear fracture mechanics, International Journal for Numerical Methods in Engineering, 87(6) (2011) 541-565.

[18] Evans, J.A., Bazilevs, Y., Babuška, I., Hughes, T.J.R., n-Widths, sup–infs, and optimality ratios for the k-version of the isogeometric finite element method, Computer Methods in Applied Mechanics and Engineering, 198(21–26) (2009) 1726-1741.

[19] Gdoutos, E.E., Fracture Mechanics: An Introduction, Springer, 2005.

[20] Ghorashi, S.S., Valizadeh, N., Mohammadi, S., Extended isogeometric analysis for simulation of stationary and propagating cracks, International Journal for Numerical Methods in Engineering, 89(9) (2012) 1069-1101.

[21] Gómez, H., Calo, V.M., Bazilevs, Y., Hughes, T.J.R., Isogeometric analysis of the Cahn–Hilliard phase-field model, Computer Methods in Applied Mechanics and Engineering, 197(49–50) (2008) 4333-4352.

[22] Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194(39–41) (2005) 4135-4195.

[23] Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y., Isogeometric Analysis Toward integration of CAD and FEM, John Wiley & Sons, Inc., 2009.

[24] Joseph, R.P., Purbolaksono, J., Liew, H.L., Ramesh, S., Hamdi, M., Stress intensity factors of a corner crack emanating from a pinhole of a solid cylinder, Engineering Fracture Mechanics, 128 (2014) 1-7.

[25] Kim, H.-K., Lee, Y.-H., Decoupling of Generalized Mode I and II Stress Intensity Factors in the Complete Contact Problem of Elastically Dissimilar Materials, Procedia Materials Science, 3 (2014) 245-250.

[26] Kim, J.-H., Paulino, G.H., T-stress, mixed-mode stress intensity factors, and crack initiation angles in functionally graded materials: a unified approach using the interaction integral method, Computer Methods in Applied Mechanics and Engineering, 192(11–12) (2003) 1463-1494.

[27] Likeb, A., Gubeljak, N., Matvienko, Y., Stress Intensity Factor and Limit Load Solutions for New Pipe-ring Specimen with Axial Cracks, Procedia Materials Science, 3 (2014) 1941-1946.

[28] Mohammadi, S., Extended Finite Element Method: for Fracture Analysis of Structures, John Wiley & Sons, Inc., 2008.

[29] Oliver, J., Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Part 2: numerical simulation, International Journal for Numerical Methods in Engineering, 39(21) (1996) 3601-3623.

[30] Piegl, L., Tiller, W., The NURBS Book, U.S. Government Printing Office, 1997.

[31] Piegl, L.A., Tiller, W., The Nurbs Book, Springer-Verlag GmbH, 1997.

[32] Rice, J.R., A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks, Journal of Applied Mechanics, 35(2) (1968) 379-386.

[33] Rogers, D.F., An introduction to NURBS: with historical perspective, Morgan Kaufmann Publishers, 2001.

[34] Rots, J., Smeared and discrete representations of localized fracture, International Journal of Fracture, 51(1) (1991) 45-59.

[35] Schellekens, J.C.J., De Borst, R., On the numerical integration of interface elements, International Journal for Numerical Methods in Engineering, 36(1) (1993) 43-66.

[36] Scott, M.A., Li, X., Sederberg, T.W., Hughes, T.J.R., Local refinement of analysis-suitable T-splines, Computer Methods in Applied Mechanics and Engineering, 213–216 (2012) 206-222.

[37] Simo, J.C., Oliver, J., Armero, F., An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids, Computational Mechanics, 12(5) (1993) 277-296.

[38] Smith, D.J., Ayatollahi, M.R., Pavier, M.J., The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading, Fatigue and Fracture of Engineering Materials and Structures, 24(2) (2001) 137-150.

[39] 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, 28(11) (2004) 1335-1350.

[40] Tur, M., Giner, E., Fuenmayor, F.J., A contour integral method to compute the generalized stress intensity factor in complete contacts under sliding conditions, Tribology International, 39(10) (2006) 1074-1083.

[41] Valizadeh, N., Bui, T.Q., Vu, T.V., Minh, H.C., Thai, T., Nguyen, M.N., Isogeometric simulation for buckling, free and forced vibration of orthotropic plates, International Journal of Applied Mechanics, 5(2) (2013) 1350017.

[42] Williams, M.L., On the stress distribution at the base of a stationary crack, Journal of Applied Mechanics, 24(1) (1957) 109–114.

[43] Yin, S., Yu, T., Bui, T. Q., Liu, P., Hirose, S., Buckling and vibration extended isogeometric analysis of imperfect graded Reissner-Mindlin plates with internal defects using NURBS and level sets, Computers & Structures, 177 (2016) 23-38.

[44] Zhuang, Z., Liu, Z., Cheng, B., Liao, J., Fundamental Linear Elastic Fracture Mechanics, In: Zhuang Z, Liu Z, Cheng B, Liao J (eds) Extended Finite Element Method, Academic Press, Oxford, 2014.