Computational Methods for Debonding in Composites

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240 J.A. Oliveira et al. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Fig. 11.15 Y-periodicity of the characteristic displacements – shear mode χχχ 23 using alternative RUC: (a–b) reference RUC, (c–d) C1,(e–f) C2,(g–h) C3and(i–j) C4 scales is not numerically relevant. In fact, ε does not even appear in the explicit formulation of the method for a first order approximation (i.e. the conventional AEH method), as considered in this work. Tiles of four RUC are shown in Fig. 11.15 in order to illustrate the fact the correctors are indeed representative of the microscale heterogeneities. It is now easier to see that the correctors exactly represent the same deformed geometry of the overall composite material, even if obtained from different representative volumes. Note that the displacement values shown in Figs. 11.15f, h seem to represent a different deformed geometry when compared to the other results. Nevertheless, as seen in Figs. 11.15e, g, it is exactly the same. The differences in terms of displacement are derived from the fact that the fixed points used to avoid rigid motion are located at different places within the material microstructure. 11.6 Final Remarks The main conclusion of this work is the validation of the use of the AEH approach, both in terms of global finite element simulation applications and prediction of effective material properties for periodic microstructure composite materials. The numerical simulation results show that the homogenised elasticity matrices D h of the fibre reinforced composite material define a particular case of orthotropic material: the tetragonal structure material. The convergence study showed the simultaneous convergence of all D h matrix components, as well as of the Frobenius norm, for structured and unstructured/nonperiodical meshes. Moreover, in the AEH the convergence of the results depends not only on the number of degrees of freedom but also on the number of finite elements.
11 Prediction of Mechanical Properties of Composite Materials by AEH 241 The AEH proved to be accurate enough to represent the behaviour of a given periodic microstructure composite material as long as the chosen volume element is representative of the microscale material distribution. Additionally, the fact that the homogenised elastic properties converge to the same values shows that the AEH procedure leads to correctors χχχ that are representative of the same heterogeneities. The obtained numerical results indicate that the tools that were developed allow the use of non-periodical meshes within a finite element homogenisation approach for periodic microstructures. The results of the AEH method are good approximations for the experimental results. In comparison to the rest of the prediction methods used, the AEH gives the best global approximations. References 1. Banks-Sills L, Leiderman V, Fang D (1997) On the effect of particle shape and orientation on elastic properties of metal matrix composites. Compos Part B 28:465–481 2. Barrett R, Berry M, Chan TF et al. (1994) Templates for the solution of linear systems: Building blocks for iterative methods. SIAM, Philadelphia. PA 3. Böhm HJ (1998) A short introduction to basic aspects of continuum micromechanics, CDL- FMD-Report, TU Wien, Wien 4. Chamis CC (1984) Simplified Composite Micromechanical Equations for Hygral, Thermal, and Mechanical Properties. Sampe Quartely 15:14–23 5. Chung PW (1999) Computational methods for multi-scale/multi-physics problems in heterogenous/composite structures. Ph.D. thesis, University of Minesota, Minesota 6. Chung PW, Tamma KK, Namburu RR (2001) Asymptotic expansion homogenization for heterogeneous media: Computational issues and applications. Compos Part A 32:1291–1301 7. Cioranescu D, Donato P (1999) An introduction to homogenization. Oxford University Press, Oxford 8. Felippa C (2004) Introduction to finite element methods. University of Colorado, Boulder, CO 9. Golub GH, O’Leary DP (1989) Some history of the conjugate gradient and lanczos algorithms: 1948–1976. SIAM Rev 31:50–102 10. Guedes JM, Kikuchi N (1990) Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Comput Methods Appl Mech Eng 83:143–198 11. Hestenes MR, Stiefel EL (1952) J Res Natl Bur Stand 49:409–436 12. Kenaga D, Doyle JF, Sun CT (1987) The characterization of boron/aluminum composite in the nonlinear range as an orthotropic elastic-plastic material. J Compos Mater 21:516–531 13. Oliveira JA (2006) Micromechanical modelling of the behaviour of aluminium matrix composites. M.Sc. thesis (in portuguese), Aveiro University, Aveiro 14. Oliveira JA, Pinho-da-Cruz J, Andrade-Campos A, Teixeira-Dias F (2004) On the modelling of representative unit-cell geometries with GiD. In: 2nd Conference on Advances and Applications of GiD. SEMNI, Barcelona 15. Pinho-da-Cruz J (2007) Thermomechanical characterisation of multiphasic materials using homogenisation procedures. Ph.D. thesis (in portuguese), Aveiro University, Aveiro 16. Rizzo AR (1991) Estimating errors in FE analyses. Mech Eng 113:61–63 17. Segurado J, Llorca J (2002) A numerical approximation to the elastic properties of spherereinforced composites. J Mech Phys Solids 50:2107–2121 18. Sun CT, Chen JL (1991) A micromechanical model for plastic behavior of fibrous composites. Compos Sci Technol 40:115–129 19. Sun CT, Vaidya RS (1996) Prediction of composite properties from a representative volume element. Compos Sci Technol 56:171–179
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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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28 R. Rolfes et al. functions by me
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30 R. Rolfes et al. Microscale fibe
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32 R. Rolfes et al. symmetric, whic
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34 R. Rolfes et al. Fig. 2.5 Discre
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36 R. Rolfes et al. Unit cell (a) S
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38 R. Rolfes et al. stress state. T
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40 R. Rolfes et al. biaxial compres
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42 R. Rolfes et al. Damage Evolutio
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44 R. Rolfes et al. Fig. 2.14 Radia
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46 R. Rolfes et al. Table 2.2 Elast
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48 R. Rolfes et al. A dev is the de
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50 R. Rolfes et al. uniaxial compre
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52 R. Rolfes et al. Stress in MPa -
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54 R. Rolfes et al. 2.4.2 Results o
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56 R. Rolfes et al. 12. Hughes TJR
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58 B.N. Cox et al. inform models; a
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60 B.N. Cox et al. 3.2 The Structur
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62 B.N. Cox et al. be successfully
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64 B.N. Cox et al. high fluxes of c
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66 B.N. Cox et al. provides poor in
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68 B.N. Cox et al. For example, a c
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70 B.N. Cox et al. failures in comp
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72 B.N. Cox et al. mixed-mode crack
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74 B.N. Cox et al. 24. Elices M, Gu
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Chapter 4 Analytical and Numerical
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4 Analytical and Numerical Investig
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Chapter 6 Study of Delamination in
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6 Study of Delamination with in Com
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8 A Numerical Material Model for Pr
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180 N. Zobeiry et al. There are sev
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182 N. Zobeiry et al. for construct
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184 N. Zobeiry et al. In compressio
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186 N. Zobeiry et al. c = 38.7 mm h
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