Computational Methods for Debonding in Composites

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278 M. Kästner et al. Table 13.4 Comparison of engineering constants computed from numerical and experimental effective stiffness properties ¯Cijkl with results from standard tensile tests Property Ē11[GPa] ¯G12[GPa] ¯ν12[-] Homogenization 14.37 1.50 0.11 Ultrasonic test 13.45 a – b 0.38 a Tensile test 13.05 1.11 0.09 a Results for assumed stiffness property value ¯C1122 = 4.0 GPa b Arbitrary as ¯G12 = ¯C1212 13.3.3 Experimental Verification The comparison of numerically determined effective stiffness properties to those measured by ultrasonic tests in Table 13.3 demonstrates a very good agreement of the tensile stiffness ¯C1111 and ¯C2222. On the other hand a quite significant difference is observed for the transverse and shear properties which are subject to further investigation on the influence of scatter in the input data as well as the geometric assumptions made. In addition, the effective stiffness properties obtained from homogenization and ultrasonic testing have been converted into the well-known engineering constants. In order to be able to invert the stiffness matrix a reasonable estimate for ¯C1122 in Table 13.3 has been made. The properties characterizing the in-plane material behaviour are given in Table 13.4. They are compared to results from tensile testing. This comparison reveals a general correlation between experimental and numerical results. On the other hand a fairly large deviation of the two experimental testing methods can be observed. 13.4 Conclusion In this work an efficient modelling strategy for textile-reinforced composites has been proposed. A computational homogenization technique based on the equivalence of strain energy in a locally heterogeneous material and an effective homogeneous medium is applied. In addtion an X-FEM approach to handle the complexity of textile reinforcement is developed, resulting in an algorithm for automated modelling of RVE based on a 3D geometric model. As it was demonstrated in the previous section, the main advantage of simulation over experimental tests is that all stiffness properties required for a structural analysis can be quantified more efficiently. In contrast to procedures that identify parameters of an assumed constitutive relation from experiments the material behaviour is predicted using geometrical and physical characteristics of two different length scales.
13 Effective Stiffness Properties of Textile-Reinforced Composites 279 In order to improve the flexibility of the approach it has to be extended to more complex weft-knitted fabrics. Furthermore, the combination of X-FEM and nonlinear material behaviour will be adressed in the future. Acknowledgements Financial support for this research project from the German Research Fundation (DFG) in the framework of the Collaborative Research Centre (SFB) 639 is gratefully acknowledged. Additionally, the authors thank the Institute of Lightweight Construction and Polymer Technology (ILK) at TU Dresden for providing experimental data obtained from ultrasonic and tensile tests. References 1. Belytschko T, Moës N, Usui S, Parimi C (2001) Arbitrary discontinuities in finite elements. Int J Numer Meth Eng 50:993–1013 2. Daux C, Moës N, Dolbow J et al (2000) Arbitrary branched and intersecting cracks with the extended finite element method. Int J Numer Meth Eng 48:1741–1760 3. Fagerström M, Larsson R (2006) Theory and numerics for finite deformation fracture modelling using strong discontinuities. Int J Numer Meth Eng 66:911–948 4. Gravouil A, Moës N, Belytschko T (2002) Non-planar 3D crack growth by the extended finite element and level sets – Part I: mechanical model. Int J Numer Meth Eng 53:2549–2568 5. Gravouil A, Moës N, Belytschko T (2002) Non-planar 3D crack growth by the extended finite element and level sets – Part II: level set update. Int J Numer Meth Eng 53:2569–2586 6. Haasemann G, Kästner M, Ulbricht V (2006) Multi-Scale modelling and simulation of textilereinforced materials. CMC 3:131–146 7. Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11:357–372 8. Hufenbach W, Böhm R, Langkamp A, Kroll L, Ritschel T (2006) Ultrasonic evaluation of anisotropic damage in textile multi-axial reinforced thermoplastic composites made from hybrid yarns. Mech Compos Mater 42:221–234 9. Kästner M, Ulbricht V (2006) Homogenization of fibre-reinforced composites using X-FEM. Proc Appl Math Mech 6:489–490 10. Lukkassen D, Persson LE, Wall P (1995) Some engineering and mathematical aspects on the homogenization method. Compos Eng 5:519–531 11. Melenk JM, Babuˇska I (1996) The partition of unity finite element method: basic theory and applications. Research Report 12. Moës N, Cloirec M, Cartraud P, Remacle J (2003) A computational approach to handle complex microstructure geometries. Comput Methods Appl Mech Eng 192:3163–3177 13. Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Meth Eng 46:131–150 14. Sanchez-Palencia E, Zaoui E (1987) Homogenization techniques for composite media. Lecture Notes in Physics 272. Springer, Berlin 15. Stazi L, Budyn E, Chessa J, Belytschko T (2003) An extended finite element method with higher-order elements for curved cracks. Comput Mech 31:38–48 16. Sukumar N, Chopp D, Moës N, Belytschko T (2001) Modeling holes and inclusions by level sets in the extended finite element method. Comput Methods Appl Mech Eng 190:6183–6200 17. Sukumar N, Huang ZY, Prévost J, Suo Z (2004) Partition of unity enrichment for bimaterial interface cracks. Int J Numer Meth Eng 59:1075–1102 18. Sukumar N, Moës N, Moran B, Belytschko T (2000) Extended finite element method for three-dimensional crack modelling. Int J Numer Meth Eng 48:1549–1570 19. Zi G, Belytschko T (2003) New crack-tip elements for XFEM and applications to cohesive cracks. Int J Numer Meth Eng 57:2221–2240
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Mechanical Response of Composites
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Pedro P. Camanho • C.G. Dávila S
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Preface The methodology for designi
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Acknowledgements The Editors of thi
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x Contents 3 Practical Challenges i
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xii Contents 8.2.4 Modelling Effect
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xiv Contents 13.1.2 EffectiveProper
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xvi List of Contributors Clara Schu
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Chapter 1 Computational Methods for
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1 Computational Methods for Debondi
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Chapter 4 Analytical and Numerical
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4 Analytical and Numerical Investig
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10 Elastoplastic Modeling of Multi-
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Computational Methods for Debonding in Composites

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