Comparison between Isogeometric Analysis and Extended Finite Element Methods in Discontinuities Modeling and Stress Intensity Factor Calculation
Authors
Abstract
This study shows how to create different types of crack and discontinuities by using isogeometric analysis approach (IGA) and extended finite element method (XFEM). In this contribution, two unique features of isogeometric analysis approach are utilized to create discontinuous zones. Discontinuities consist of crack and cohesive zone. In isogeometric analysis method NURBS is used to approximate both geometry and primary variable. NURBS can create quadratic shapes exactly. Also, stress intensity factors are calculated in the vicinity of the crack tips for two dimensional problems and are compared with corresponding analytical and numerical counterparts. Extended finite element method is the other numerical method which is used in this work. The enrichment procedure is utilized in extended finite element method to create discontinuities. The well-known path independent J-integral approach is used in order to calculate the stress intensity factors. Also, in mixed mode situation, the interaction integral (M-integral) is considered to calculate the stress intensity factors. Results show that isogeometric analysis method has desirable accuracy as it uses lower degree of freedoms and consequently lower computational efforts than extended finite element method. In addition, creating the internal cohesive zone as one of the most important issues in computational fracture mechanics is feasible due to the special features of isogeometric analysis. The present study demonstrates the capability of isogeometric analysis parametric space to control the inter-element continuity and create the cohesive zone.
Keywords
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3. Joseph, R. P., Purbolaksono, J., Liew, H. L., Ramesh, S., and Hamdi, M., “Stress Intensity Factors of a Corner Crack Emanating from a Pinhole of a Solid Cylinder”, Engineering Fracture Mechanics, Vol. 128, pp. 1-7, 2014.
4. Ghorashi, S. S., Valizadeh, N., and Mohammadi, S., “Extended Isogeometric Analysis for Simulation of Stationary and Propagating Cracks”, International Journal for Numerical Methods in Engineering, Vol. 89, pp. 1069-1101, 2012.
5. De Luycker, E., Benson, D. J., Belytschko, T., Bazilevs, Y., and Hsu, M. C., “X-FEM in Isogeometric Analysis for Linear Fracture Mechanics”, International Journal for Numerical Methods in Engineering, Vol. 87, pp. 541-565, 2011.
6. Mohammadi, S., “Extended Finite Element Method: for Fracture Analysis of Structures”, Wiley, 2008.
7. Dugdale, D. S., “Yielding of Steel Sheets Containing Slits”, Journal of the Mechanics and Physics of Solids, Vol. 8, pp. 100-104, 1960.
8. Barenblatt, G. I., “The Mathematical Theory of Equilibrium Cracks in Brittle Fracture”, Advances in Applied Mechanics, pp. 55-129, 1962.
9. Griffith, A. A., “The Phenomena of Rupture and Flow in Solids”, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, Vol. 221, pp. 163-198, 1921.
10. Schellekens, J. C. J., and De Borst, R., “On the Numerical Integration of Interface Elements”, International Journal for Numerical Methods in Engineering, Vol. 36, pp. 43-66, 1993.
11. Rots, J., “Smeared and Discrete Representations of Localized Fracture”, International Journal of Fracture, Vol. 51, pp. 45-59, 1991.
12. Jirásek, M., and Zimmermann, T., “Embedded Crack Model: I. Basic Formulation”, International Journal for Numerical Methods in Engineering, Vol. 50, pp. 1269-1290, 2001.
13. Oliver, J., “Modelling Strong Discontinuities in Solid Mechanics Via Strain Softening Constitutive Equations. Part 2: Numerical Simulation”, International Journal for Numerical Methods in Engineering, Vol. 39, pp. 3601-3623, 1996.
14. Simo, J. C., Oliver, J., and Armero, F., “An Analysis of Strong Discontinuities Induced by Strain-Softening in Rate-Independent Inelastic Solids”, Computational Mechanics, Vol. 12, pp. 277-296, 1993.
15. Babuška, I., and Zhang, Z., “The Partition of Unity Method for the Elastically Supported Beam”, Computer Methods in Applied Mechanics and Engineering, Vol. 152, pp. 1-18, 1998.
16. Melenk, J. M., and Babuška, I., “The Partition of Unity Finite Element Method: Basic Theory and Applications”, Computer Methods in Applied Mechanics and Engineering, Vol. 139, pp. 289-314, 1996.
17. Hughes, T. J. R., 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, pp. 4135-4195, 2005.
18. Bazilevs, Y., Beirão Da Veiga, L., Cottrell, J. A., Hughes, T. J. R., and Sangalli, G., “Isogeometric Analysis: Approximation, Stability and Error Estimates for H-Refined Meshes”, Mathematical Models and Methods in Applied Sciences, Vol. 16, pp. 1031-1090, 2006.
19. Hughes, T. J. R., Reali, A., and Sangalli, G., “Efficient Quadrature for NURBS-Based Isogeometric Analysis”, Computer Methods in Applied Mechanics and Engineering, Vol. 199, pp. 301-313, 2010.
20. Wang, D., and Xuan, J., “An Improved NURBS-Based Isogeometric Analysis with Enhanced Treatment of Essential Boundary Conditions”, Computer Methods in Applied Mechanics and Engineering, Vol. 199, pp. 2425-2436, 2010.
21. Mysore, K., Morgan, O. T., and Subbarayan, G., “NURBS Representational Strategies for Tracking Moving Boundaries and Topological Changes During Phase Evolution”, Computer Methods in Applied Mechanics and Engineering, Vol. 200, pp. 2594-2610, 2011.
22. Evans, J. A., Bazilevs, Y., Babuška, I., and 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, Vol. 198, pp. 1726-1741, 2009.
23. Cottrell, J. A., Hughes, T. J. R., and Reali, A., “Studies of Refinement and Continuity in Isogeometric Structural Analysis”, Computer Methods in Applied Mechanics and Engineering, Vol. 196, pp. 4160-4183, 2007.
24. Cottrell, J. A., Reali, A., Bazilevs, Y., and Hughes, T. J. R., “Isogeometric Analysis of Structural Vibrations”, Computer Methods in Applied Mechanics and Engineering, Vol. 195, pp. 5257-5296, 2006.
25. De Boor, C., “On Calculating with B-Splines”, Journal of Approximation Theory, Vol. 6, pp. 50-62, 1972.
26. Cox, M. G., “The Numerical Evaluation of B-Splines”, IMA Journal of Applied Mathematics, Vol. 10, pp. 134-149, 1972.
27. Piegl, L. A., and Tiller, W., “The Nurbs Book”, Springer-Verlag GmbH, 1997.
28. Cottrell, J. A., Hughes, T. J. R., and Bazilevs, Y., “Isogeometric Analysis: Toward Integration of CAD and FEA”, Wiley Publishing, 2009.
29. Hassani, B., Tavakkoli, S. M., and Moghadam, N. Z., “Application of Isogeometric Analysis in Structural Shape Optimization”, Scientia Iranica, Vol. 18, pp. 846-852, 2011.
30. Gdoutos, E. E., “Fracture Mechanics: An Introduction”, Springer, 2005.
31. Williams, M. L., “On the Stress Distribution at the Base of a Stationary Crack”, ASME Journal of Applied Mechanics, Vol. 24, pp. 109-114, 1957.
32. Yau, J. F., Wang, S. S., and Corten, H. T., “A Mixed-Mode Crack Analysis of Isotropic Solids Using Conservation Laws of Elasticity”, Journal of Applied Mechanics-Transactions of the ASME, Vol. 47, pp., 1980.
33. Belytschko, T., and Black, T., “Elastic Crack Growth in Finite Elements with Minimal Remeshing”, International Journal for Numerical Methods in Engineering, Vol. 45, pp. 601-620, 1999.
34. Moës, N., and Belytschko, T., “Extended Finite Element Method for Cohesive Crack Growth”, Engineering Fracture Mechanics, Vol. 69, pp. 813-833, 2002.
35. Belytschko, T., Moës, N., Usui, S., and Parimi, C., “Arbitrary Discontinuities in Finite Elements”, International Journal for Numerical Methods in Engineering, Vol. 50, pp. 993-1013, 2001.
36. Dolbow, J., Moës, N., and Belytschko, T., “Discontinuous Enrichment in Finite Elements with a Partition of Unity Method”, Finite Elements in Analysis and Design, Vol. 36, pp. 235-260, 2000.
37. 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, pp. 131-150, 1999.
38. Belytschko, T., and Fleming, M., “Smoothing, Enrichment and Contact in the Element-free Galerkin Method”, Computers & Structures, Vol. 71, pp. 173-195, 1999.
39. Fleming, M., Chu, Y. A., Moran, B., and Belytschko, T., “Enriched Element-Free Galerkin Methods for Crack Tip Fields”, International Journal for Numerical Methods in Engineering, Vol. 40, pp. 1483-1504, 1997.
40. Sethian, J. A., “A Fast Marching Level Set Method for Monotonically Advancing Fronts”, Proceedings of the National Academy of Sciences, Vol. 93, pp. 1591-1595, 1996.
41. Adalsteinsson, D., and Sethian, J. A., “A Fast Level Set Method for Propagating Interfaces”, Journal of Computational Physics, Vol. 118, pp. 269-277, 1995.
42. Osher, S., and Sethian, J. A., “Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations”, Journal of Computational Physics, Vol. 79, pp. 12-49, 1988.
43. Sukumar, N., Chopp, D. L., Moës, N., and Belytschko, T., “Modeling Holes and Inclusions by Level Sets in the Extended Finite-Element Method”, Computer Methods in Applied Mechanics and Engineering, Vol. 190, pp. 6183-6200, 2001.
44. Jiang, H., “Cohesive Zone Model for Carbon Nanotube Adhesive Simulation and Fracture/Fatigue Crack Growth”, Ph.D. Thesis, University of Akron, 2010.
45. Verhoosel, C., Scott, M. A., Borst, R. D., and Hughes, T. J. R., “An Isogeometric Approach To Cohesive Zone Modeling”, The Institute for Computational Engineering and Sciences, The University of Texas at Austin Austin, Texas 78712, 2010.
2. Torres, Y., Tarrago, J. M., Coureaux, D., Tarrés, E., Roebuck, B., Chan, P., James, M., Liang, B., Tillman, M., Viswandham, R. K., Mingrad, K. P., Mestra, A., and Lianes, L., “Fracture and Fatigue of Rock Bit Cemented Carbides: Mechanics and Mechanisms of Crack Growth Resistance under Monotonic and Cyclic Loading”, International Journal of Refractory Metals and Hard Materials, Vol. 45, pp. 179-188, 2014.
3. Joseph, R. P., Purbolaksono, J., Liew, H. L., Ramesh, S., and Hamdi, M., “Stress Intensity Factors of a Corner Crack Emanating from a Pinhole of a Solid Cylinder”, Engineering Fracture Mechanics, Vol. 128, pp. 1-7, 2014.
4. Ghorashi, S. S., Valizadeh, N., and Mohammadi, S., “Extended Isogeometric Analysis for Simulation of Stationary and Propagating Cracks”, International Journal for Numerical Methods in Engineering, Vol. 89, pp. 1069-1101, 2012.
5. De Luycker, E., Benson, D. J., Belytschko, T., Bazilevs, Y., and Hsu, M. C., “X-FEM in Isogeometric Analysis for Linear Fracture Mechanics”, International Journal for Numerical Methods in Engineering, Vol. 87, pp. 541-565, 2011.
6. Mohammadi, S., “Extended Finite Element Method: for Fracture Analysis of Structures”, Wiley, 2008.
7. Dugdale, D. S., “Yielding of Steel Sheets Containing Slits”, Journal of the Mechanics and Physics of Solids, Vol. 8, pp. 100-104, 1960.
8. Barenblatt, G. I., “The Mathematical Theory of Equilibrium Cracks in Brittle Fracture”, Advances in Applied Mechanics, pp. 55-129, 1962.
9. Griffith, A. A., “The Phenomena of Rupture and Flow in Solids”, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, Vol. 221, pp. 163-198, 1921.
10. Schellekens, J. C. J., and De Borst, R., “On the Numerical Integration of Interface Elements”, International Journal for Numerical Methods in Engineering, Vol. 36, pp. 43-66, 1993.
11. Rots, J., “Smeared and Discrete Representations of Localized Fracture”, International Journal of Fracture, Vol. 51, pp. 45-59, 1991.
12. Jirásek, M., and Zimmermann, T., “Embedded Crack Model: I. Basic Formulation”, International Journal for Numerical Methods in Engineering, Vol. 50, pp. 1269-1290, 2001.
13. Oliver, J., “Modelling Strong Discontinuities in Solid Mechanics Via Strain Softening Constitutive Equations. Part 2: Numerical Simulation”, International Journal for Numerical Methods in Engineering, Vol. 39, pp. 3601-3623, 1996.
14. Simo, J. C., Oliver, J., and Armero, F., “An Analysis of Strong Discontinuities Induced by Strain-Softening in Rate-Independent Inelastic Solids”, Computational Mechanics, Vol. 12, pp. 277-296, 1993.
15. Babuška, I., and Zhang, Z., “The Partition of Unity Method for the Elastically Supported Beam”, Computer Methods in Applied Mechanics and Engineering, Vol. 152, pp. 1-18, 1998.
16. Melenk, J. M., and Babuška, I., “The Partition of Unity Finite Element Method: Basic Theory and Applications”, Computer Methods in Applied Mechanics and Engineering, Vol. 139, pp. 289-314, 1996.
17. Hughes, T. J. R., 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, pp. 4135-4195, 2005.
18. Bazilevs, Y., Beirão Da Veiga, L., Cottrell, J. A., Hughes, T. J. R., and Sangalli, G., “Isogeometric Analysis: Approximation, Stability and Error Estimates for H-Refined Meshes”, Mathematical Models and Methods in Applied Sciences, Vol. 16, pp. 1031-1090, 2006.
19. Hughes, T. J. R., Reali, A., and Sangalli, G., “Efficient Quadrature for NURBS-Based Isogeometric Analysis”, Computer Methods in Applied Mechanics and Engineering, Vol. 199, pp. 301-313, 2010.
20. Wang, D., and Xuan, J., “An Improved NURBS-Based Isogeometric Analysis with Enhanced Treatment of Essential Boundary Conditions”, Computer Methods in Applied Mechanics and Engineering, Vol. 199, pp. 2425-2436, 2010.
21. Mysore, K., Morgan, O. T., and Subbarayan, G., “NURBS Representational Strategies for Tracking Moving Boundaries and Topological Changes During Phase Evolution”, Computer Methods in Applied Mechanics and Engineering, Vol. 200, pp. 2594-2610, 2011.
22. Evans, J. A., Bazilevs, Y., Babuška, I., and 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, Vol. 198, pp. 1726-1741, 2009.
23. Cottrell, J. A., Hughes, T. J. R., and Reali, A., “Studies of Refinement and Continuity in Isogeometric Structural Analysis”, Computer Methods in Applied Mechanics and Engineering, Vol. 196, pp. 4160-4183, 2007.
24. Cottrell, J. A., Reali, A., Bazilevs, Y., and Hughes, T. J. R., “Isogeometric Analysis of Structural Vibrations”, Computer Methods in Applied Mechanics and Engineering, Vol. 195, pp. 5257-5296, 2006.
25. De Boor, C., “On Calculating with B-Splines”, Journal of Approximation Theory, Vol. 6, pp. 50-62, 1972.
26. Cox, M. G., “The Numerical Evaluation of B-Splines”, IMA Journal of Applied Mathematics, Vol. 10, pp. 134-149, 1972.
27. Piegl, L. A., and Tiller, W., “The Nurbs Book”, Springer-Verlag GmbH, 1997.
28. Cottrell, J. A., Hughes, T. J. R., and Bazilevs, Y., “Isogeometric Analysis: Toward Integration of CAD and FEA”, Wiley Publishing, 2009.
29. Hassani, B., Tavakkoli, S. M., and Moghadam, N. Z., “Application of Isogeometric Analysis in Structural Shape Optimization”, Scientia Iranica, Vol. 18, pp. 846-852, 2011.
30. Gdoutos, E. E., “Fracture Mechanics: An Introduction”, Springer, 2005.
31. Williams, M. L., “On the Stress Distribution at the Base of a Stationary Crack”, ASME Journal of Applied Mechanics, Vol. 24, pp. 109-114, 1957.
32. Yau, J. F., Wang, S. S., and Corten, H. T., “A Mixed-Mode Crack Analysis of Isotropic Solids Using Conservation Laws of Elasticity”, Journal of Applied Mechanics-Transactions of the ASME, Vol. 47, pp., 1980.
33. Belytschko, T., and Black, T., “Elastic Crack Growth in Finite Elements with Minimal Remeshing”, International Journal for Numerical Methods in Engineering, Vol. 45, pp. 601-620, 1999.
34. Moës, N., and Belytschko, T., “Extended Finite Element Method for Cohesive Crack Growth”, Engineering Fracture Mechanics, Vol. 69, pp. 813-833, 2002.
35. Belytschko, T., Moës, N., Usui, S., and Parimi, C., “Arbitrary Discontinuities in Finite Elements”, International Journal for Numerical Methods in Engineering, Vol. 50, pp. 993-1013, 2001.
36. Dolbow, J., Moës, N., and Belytschko, T., “Discontinuous Enrichment in Finite Elements with a Partition of Unity Method”, Finite Elements in Analysis and Design, Vol. 36, pp. 235-260, 2000.
37. 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, pp. 131-150, 1999.
38. Belytschko, T., and Fleming, M., “Smoothing, Enrichment and Contact in the Element-free Galerkin Method”, Computers & Structures, Vol. 71, pp. 173-195, 1999.
39. Fleming, M., Chu, Y. A., Moran, B., and Belytschko, T., “Enriched Element-Free Galerkin Methods for Crack Tip Fields”, International Journal for Numerical Methods in Engineering, Vol. 40, pp. 1483-1504, 1997.
40. Sethian, J. A., “A Fast Marching Level Set Method for Monotonically Advancing Fronts”, Proceedings of the National Academy of Sciences, Vol. 93, pp. 1591-1595, 1996.
41. Adalsteinsson, D., and Sethian, J. A., “A Fast Level Set Method for Propagating Interfaces”, Journal of Computational Physics, Vol. 118, pp. 269-277, 1995.
42. Osher, S., and Sethian, J. A., “Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations”, Journal of Computational Physics, Vol. 79, pp. 12-49, 1988.
43. Sukumar, N., Chopp, D. L., Moës, N., and Belytschko, T., “Modeling Holes and Inclusions by Level Sets in the Extended Finite-Element Method”, Computer Methods in Applied Mechanics and Engineering, Vol. 190, pp. 6183-6200, 2001.
44. Jiang, H., “Cohesive Zone Model for Carbon Nanotube Adhesive Simulation and Fracture/Fatigue Crack Growth”, Ph.D. Thesis, University of Akron, 2010.
45. Verhoosel, C., Scott, M. A., Borst, R. D., and Hughes, T. J. R., “An Isogeometric Approach To Cohesive Zone Modeling”, The Institute for Computational Engineering and Sciences, The University of Texas at Austin Austin, Texas 78712, 2010.
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