Hybrid IG-FE method applied to cohesive fracture/contact in particle-filled elastomeric composites

  • Saeed Maleki Jebeli
  • Mahmoud Musavi Mashhadi
  • Mostafa BaghaniEmail author


In this paper, employing a new numerical framework, a 2D investigation is conducted on the effect of fiber-matrix contact/debonding on the stress–strain response of fiber-reinforced elastomeric composites. The analysis is carried out in the framework of large deformation/sliding. The main novelty of this work resides in using hybrid IsoGeometric-Finite Element (IG-FE) technique to discretize the domain where interfaces together with a band around them are represented by NURBS patches. These patches are coupled to the surrounding Lagrangian domain thanks to the transition elements. Interfaces’ discontinuity is readily induced through knot insertion capability within IGA. Moreover, contact constraints together with cohesive behavior are incorporated in the numerical model employing a unified mortar framework with appropriate definitions of potentials. Two numerical examples are considered within the proposed context and results are discussed in terms of stress–strain curves and stress contours. Due to proven higher accuracy of IGA in modeling contact/cohesive interfaces under large deformation and efficient use of NURBS around discontinuities, the proposed numerical methodology seems to be an appropriate (yet efficient) candidate to model such kind of highly nonlinear problems.


Fiber-matrix contact/debonding Hybrid IG-FE discretization Knot insertion Isogeometric analysis 



  1. Al-Anany, Y.M., Tait, M.J.: Experimental assessment of utilizing fiber reinforced elastomeric isolators as bearings for bridge applications. Compos. B Eng. 114, 373–385 (2017)CrossRefGoogle Scholar
  2. Alberdi, R., Zhang, G., Khandelwal, K.: A framework for implementation of RVE-based multiscale models in computational homogenization using isogeometric analysis. Int. J. Numer. Meth. Eng. 114(9), 1018–1051 (2018). MathSciNetCrossRefGoogle Scholar
  3. Bathe, K.-J.: Finite Element Procedures. Klaus-Jurgen Bathe, Watertown (2006)zbMATHGoogle Scholar
  4. Bathe, K.J., Saunders, H.: Finite element procedures in engineering analysis. J. Pressure Vessel Technol. Trans. ASME 106(4), 421–422 (1984)CrossRefGoogle Scholar
  5. Bazilevs, Y., Calo, V., Hughes, T., Zhang, Y.: Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput. Mech. 43(1), 3–37 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. Bazilevs, Y., Gohean, J., Hughes, T., Moser, R., Zhang, Y.: Patient-specific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device. Comput. Methods Appl. Mech. Eng. 198(45), 3534–3550 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. Belgaid, H., Bouazzouni, A.: Vibration analysis of mechanical structures with a new formulation of the isogeometric collocation method. Eur. J. Mech. A. Solids 68, 88–103 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  8. Bonet, J., Wood, R.D.: Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  9. Catapano, A., Jumel, J.: A numerical approach for determining the effective elastic symmetries of particulate-polymer composites. Compos. Part B Eng. 78, 227–243 (2015)CrossRefGoogle Scholar
  10. Cho, J.R., Kim, K.W., Jeon, D.H., Yoo, W.S.: Transient dynamic response analysis of 3-D patterned tire rolling over cleat. Eur. J. Mech. A. Solids 24(3), 519–531 (2005)zbMATHCrossRefGoogle Scholar
  11. Corbett, C.J., Sauer, R.A.: NURBS-enriched contact finite elements. Comput. Methods Appl. Mech. Eng. 275, 55–75 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  12. Corbett, C.J., Sauer, R.A.: Three-dimensional isogeometrically enriched finite elements for frictional contact and mixed-mode debonding. Comput. Methods Appl. Mech. Eng. 284, 781–806 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  13. Cottrell, J., Reali, A., Bazilevs, Y., Hughes, T.: Isogeometric analysis of structural vibrations. Comput. Methods Appl. Mech. Eng. 195(41), 5257–5296 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  14. Cottrell, J.A., Hughes, T.J., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, New York (2009)zbMATHCrossRefGoogle Scholar
  15. De Falco, C., Reali, A., Vázquez, R.: GeoPDEs: a research tool for isogeometric analysis of PDEs. Adv. Eng. Softw. 42(12), 1020–1034 (2011)zbMATHCrossRefGoogle Scholar
  16. De Lorenzis, L., Temizer, I., Wriggers, P., Zavarise, G.: A large deformation frictional contact formulation using NURBS-based isogeometric analysis. Int. J. Numer. Methods Eng. 87(13), 1278–1300 (2011)MathSciNetzbMATHGoogle Scholar
  17. De Lorenzis, L., Wriggers, P., Zavarise, G.: A mortar formulation for 3D large deformation contact using NURBS-based isogeometric analysis and the augmented Lagrangian method. Comput. Mech. 49(1), 1–20 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  18. Dimitri, R., De Lorenzis, L., Wriggers, P., Zavarise, G.: NURBS-and T-spline-based isogeometric cohesive zone modeling of interface debonding. Comput. Mech. 54(2), 369–388 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  19. Dimitri, R., Trullo, M., De Lorenzis, L., Zavarise, G.: Coupled cohesive zone models for mixed-mode fracture: a comparative study. Eng. Fract. Mech. 148, 145–179 (2015)CrossRefGoogle Scholar
  20. Dimitri, R., Zavarise, G.: Isogeometric treatment of frictional contact and mixed mode debonding problems. Comput. Mech. 60(2), 315–332 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  21. Elices, M., Guinea, G.V., Gómez, J., Planas, J.: The cohesive zone model: advantages, limitations and challenges. Eng. Fract. Mech. 69(2), 137–163 (2002). CrossRefGoogle Scholar
  22. Fischer, K.A., Wriggers, P.: Frictionless 2D Contact formulations for finite deformations based on the mortar method. Comput. Mech. 36(3), 226–244 (2005). zbMATHCrossRefGoogle Scholar
  23. Ge, J., Guo, B., Yang, G., Sun, Q., Lu, J.: Blending isogeometric and Lagrangian elements in three-dimensional analysis. Finite Elem. Anal. Des. 112, 50–63 (2016)MathSciNetCrossRefGoogle Scholar
  24. Gilormini, P., Toulemonde, P.-A., Diani, J., Gardere, A.: Stress-strain response and volume change of a highly filled rubbery composite: experimental measurements and numerical simulations. Mech. Mater. 111, 57–65 (2017)CrossRefGoogle Scholar
  25. Hassani, B.: Isogeometric shape optimization of three dimensional problems. In: 8th World Congress on Structural and Multidisciplinary Optimization (2009)Google Scholar
  26. Hassani, B., Tavakkoli, S., Moghadam, N.: Application of isogeometric analysis in structural shape optimization. Scientia Iranica 18(4), 846–852 (2011)CrossRefGoogle Scholar
  27. Hughes, T.J., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39), 4135–4195 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  28. Jiménez, F.L.: Modeling of soft composites under three-dimensional loading. Compos. B Eng. 59, 173–180 (2014)CrossRefGoogle Scholar
  29. Kapoor, H., Kapania, R.: Geometrically nonlinear NURBS isogeometric finite element analysis of laminated composite plates. Compos. Struct. 94(12), 3434–3447 (2012)CrossRefGoogle Scholar
  30. Kiendl, J., Bazilevs, Y., Hsu, M.-C., Wüchner, R., Bletzinger, K.-U.: The bending strip method for isogeometric analysis of Kirchhoff–Love shell structures comprised of multiple patches. Comput. Methods Appl. Mech. Eng. 199(37), 2403–2416 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  31. Kostas, K., Ginnis, A., Politis, C., Kaklis, P.: Ship-hull shape optimization with a T-spline based BEM–isogeometric solver. Comput. Methods Appl. Mech. Eng. 284, 611–622 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  32. Laursen, T.A.: Computational Contact and Impact Mechanics: Fundamentals of Modelling Interfacial Phenomena in Nonlinear Finite Element Analysis. Springer, Berlin (2002)zbMATHGoogle Scholar
  33. López Jiménez, F., Pellegrino, S.: Constitutive modeling of fiber composites with a soft hyperelastic matrix. Int. J. Solids Struct. 49(3), 635–647 (2012)CrossRefGoogle Scholar
  34. Maleki-Jebeli, S., Mosavi-Mashhadi, M., Baghani, M.: A large deformation hybrid isogeometric-finite element method applied to cohesive interface contact/debonding. Comput. Methods Appl. Mech. Eng. 330(Supplement C), 395–414 (2018)MathSciNetCrossRefGoogle Scholar
  35. Matouš, K., Geubelle, P.H.: Finite element formulation for modeling particle debonding in reinforced elastomers subjected to finite deformations. Comput. Methods Appl. Mech. Eng. 196(1), 620–633 (2006a)zbMATHCrossRefGoogle Scholar
  36. Matouš, K., Geubelle, P.H.: Multiscale modelling of particle debonding in reinforced elastomers subjected to finite deformations. Int. J. Numer. Meth. Eng. 65(2), 190–223 (2006b)MathSciNetzbMATHCrossRefGoogle Scholar
  37. Matzen, M., Cichosz, T., Bischoff, M.: A point to segment contact formulation for isogeometric, NURBS based finite elements. Comput. Methods Appl. Mech. Eng. 255, 27–39 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  38. Moraleda, J., Segurado, J., Llorca, J.: Effect of interface fracture on the tensile deformation of fiber-reinforced elastomers. Int. J. Solids Struct. 46(25), 4287–4297 (2009)zbMATHCrossRefGoogle Scholar
  39. Nguyen, V.P., Kerfriden, P., Bordas, S.: Isogeometric cohesive elements for two and three dimensional composite delamination analysis. arXiv preprint arXiv:1305.2738 (2013)
  40. Nguyen, V.P., Kerfriden, P., Bordas, S.P.A.: Two- and three-dimensional isogeometric cohesive elements for composite delamination analysis. Compos. B Eng. 60, 193–212 (2014a)CrossRefGoogle Scholar
  41. Nguyen, V.P., Kerfriden, P., Brino, M., Bordas, S.P., Bonisoli, E.: Nitsche’s method for two and three dimensional NURBS patch coupling. Comput. Mech. 53(6), 1163–1182 (2014b)MathSciNetzbMATHCrossRefGoogle Scholar
  42. Piegl, L., Tiller, W.: The NURBS Book. Monographs in Visual Communication. Springer, New York (1997)zbMATHGoogle Scholar
  43. Ping, X.-C., Chen, M.-C.: Effective elastic properties of solids with irregularly shaped inclusions. Int. J. Mech. Mater. Des. 5(3), 231 (2009)MathSciNetCrossRefGoogle Scholar
  44. Ponnamma, D., Sadasivuni, K.K., Grohens, Y., Guo, Q., Thomas, S.: Carbon nanotube based elastomer composites—an approach towards multifunctional materials. J. Mater. Chem. C 2(40), 8446–8485 (2014)CrossRefGoogle Scholar
  45. Rasool, R., Corbett, C.J., Sauer, R.A.: A strategy to interface isogeometric analysis with Lagrangian finite elements—application to incompressible flow problems. Comput. Fluids 127, 182–193 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  46. Reali, A., Gomez, H.: An isogeometric collocation approach for Bernoulli–Euler beams and Kirchhoff plates. Comput. Methods Appl. Mech. Eng. 284, 623–636 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  47. Spring, D.W., Paulino, G.H.: Computational homogenization of the debonding of particle reinforced composites: the role of interphases in interfaces. Comput. Mater. Sci. 109, 209–224 (2015)CrossRefGoogle Scholar
  48. Taylor, R., Simo, J., Zienkiewicz, O., Chan, A.: The patch test—a condition for assessing FEM convergence. Int. J. Numer. Methods Eng. 22(1), 39–62 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  49. Thai, C.H., Ferreira, A.J.M., Bordas, S.P.A., Rabczuk, T., Nguyen-Xuan, H.: Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory. Eur. J. Mech. A Solids 43, 89–108 (2014)zbMATHCrossRefGoogle Scholar
  50. Toulemonde, P.-A., Diani, J., Gilormini, P., Desgardin, N.: On the account of a cohesive interface for modeling the behavior until break of highly filled elastomers. Mech. Mater. 93, 124–133 (2016a)CrossRefGoogle Scholar
  51. Toulemonde, P.A., Diani, J., Gilormini, P., Lacroix, G., Desgardin, N.: Roles of the interphase stiffness and percolation on the behavior of solid propellants. Propellants Explos. Pyrotech. 41(6), 978–986 (2016b)CrossRefGoogle Scholar
  52. Trofimov, A., Drach, B., Sevostianov, I.: Effective elastic properties of composites with particles of polyhedral shapes. Int. J. Solids Struct. 120, 157–170 (2017)CrossRefGoogle Scholar
  53. Tur, M., Fuenmayor, F., Wriggers, P.: A mortar-based frictional contact formulation for large deformations using Lagrange multipliers. Comput. Methods Appl. Mech. Eng. 198(37), 2860–2873 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  54. Verhoosel, C.V., Scott, M.A., de Borst, R., Hughes, T.J.: An isogeometric approach to cohesive zone modeling. Int. J. Numer. Methods Eng. 87(1–5), 336–360 (2011)zbMATHCrossRefGoogle Scholar
  55. Wall, W.A., Frenzel, M.A., Cyron, C.: Isogeometric structural shape optimization. Comput. Methods Appl. Mech. Eng. 197(33), 2976–2988 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  56. Wang, D., Zhang, H.: A consistently coupled isogeometric–meshfree method. Comput. Methods Appl. Mech. Eng. 268, 843–870 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  57. Wriggers, P.: Computational Contact Mechanics. Springer, Berlin (2006)zbMATHCrossRefGoogle Scholar
  58. Xu, L., Fan, H., Sze, K., Li, C.: Elastic property prediction by finite element analysis with random distribution of materials for heterogeneous solids. Int. J. Mech. Mater. Des. 3(4), 319–327 (2006)CrossRefGoogle Scholar
  59. Yu, T., Lai, W., Bui, T.Q.: Three-dimensional elastoplastic solids simulation by an effective IGA based on Bézier extraction of NURBS. Int. J. Mech. Mater. Des. 15(1), 175–197 (2018)CrossRefGoogle Scholar
  60. Zhang, H., Wang, D.: An isogeometric enriched quasi-convex meshfree formulation with application to material interface modeling. Eng Anal Bound Elem 60, 37–50 (2015)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mechanical EngineeringUniversity of TehranTehranIran

Personalised recommendations