COMATCOMP 09 : Donostia - San Sebastian, 7, 8 y 9 de octubre de 2009The growth of interface cracks corresponding to tension dominated inter-fibre failure
under bi-directional loads is studied. The Boundary Element Method is employed for the
analysis of the fracture process and the results evaluated by means of Interfacial Fracture
Mechanics. The conclusions of the numerical analysis contribute to clarify the failure
onset.Ministerio de Educación y Ciencia TRA2006 08077Junta de Andalucía TEP-02045Junta de Andalucía TEP- 0405

Correa Montoto, Elena

Mantic, Vladislav

París Carballo, Federico

idUS. Depósito de Investigación Universidad de Sevilla

T8 COMATCOMP 09 915  Tension dominated inter-fibre failure under bi-directional loads. Micromechanical approach  E. Correa, F. París, V. Mantič Grupo de Elasticidad y Resistencia de Materiales. Mecánica de Medios Continuos, Teoría de Estructuras e Ingeniería del Terreno. Escuela Superior de Ingenieros, Universidad de Sevilla, España   ABSTRACT The growth of interface cracks corresponding to tension dominated inter-fibre failure under bi-directional loads is studied. The Boundary Element Method is employed for the analysis of the fracture process and the results evaluated by means of Interfacial Fracture Mechanics. The conclusions of the numerical analysis contribute to clarify the failure onset.   1. INTRODUCTION  The inter-fibre failure under uniaxial tension has already been the objective of several studies by the authors, París et al. (2007). These studies allowed the first stages of the mechanism of damage at micromechanical level to be identified, showing that it starts with the appearance of small debonds at the fibre-matrix interfaces. The initial defects grow unstably and symmetrically along the interfaces (interface cracks). This period ends when these cracks have reached a certain length at the interface coinciding with the appearance of a physically relevant contact zone at the tip. From that moment on the growth of the interface cracks becomes stable, which favours the occurrence of the kinking towards the matrix. The coalescence between different cracks in the matrix leads to the final macro-failure. Many of the existing proposals for the prediction of the inter-fibre failure at lamina level are based on the hypothesis that the failure taking place at a plane is governed by the components of the stress vector associated to that plane. In the present work this assumption is revised for the tension dominated inter-fibre failure by means of a single-fibre Boundary Element model, Fig. 1, using the same bi-material system employed in París et al. (2007). An analysis of the influence of an out of failure plane stress component (tension or compression) on the generation of the damage dominated by a transverse tension is carried out. Several aspects of this problem have already been analysed by the authors in París et al. (2003). Interfacial Fracture Mechanics has been used to analyse the numerical results.  2. ORIGIN OF FAILURE  The presence of an external secondary load, 33σ , acting at the same time as 22σ  could alter the origin of the failure and, thus, the development of the interface crack. The initiation of the failure at the interface has been considered in this work to be controlled by the rrσ  distribution, Goodier (1933), at the undamaged interface, as was already done in París et al. (2007). Then it is fundamental to analyse the influence that the different levels of  33σ  have on rrσ  of the undamaged interface. Three different values T8 COMATCOMP 09 916  of n  coefficient have been considered: 0, 0.5 and 1. Based on this the notation employed to distinguish between the different bi-directional cases follows the scheme: T-nC and T-nT (T=tension, C=compression). 32FibreMatrixaσ0σ0θd± nσ0± nσ0 Fig. 1. Single fibre BEM model.  Curves presented in Fig. 2a (T-nC case) show that the presence of 33σ  does not qualitatively alter the distribution of rrσ  at the interface. Quantitatively, and with reference to the T-0 case, compressive rrσ  significantly increases as 33σ  does, whereas maximum tensile rrσ  increases only slightly. Referring to the T-nT case, Fig. 2b, the rrσ  level at the interface is maintained with reference to the T-0 case though presenting some qualitative alterations, consisting in an increase of the rrσ  level in those interfacial points that were less stressed in the unidirectional case. Thus the tendency generated is to level the rrσ  state between all interfacial points, as is shown in the limit case T-T.   -2-1.5-1-0.500.511.520 45 90 135 180α (º)SRR (T-0)SRR (T-0.5C)SS (T-C)σ rr -0)σ rr -0.5C)σ rr (T-C)UndamagedinterfaceFibreMatrixασ0σ0nσ0 Fig. 2a. rrσ distribution. (T-nC case).  UndamagedinterfaceFibreMatrixασ0σ0nσ0-2-1.5-1-0.500.511.520 45 90 135 180α (º)SRR (T-0)SRR (T-0.5T)SRR (T-T)σ rr -0)σ rr -0.5T)σ rr -T) Fig. 2b. rrσ distribution. (T-nT case).   Based on the former evidence, an initial debond at the interface of 10º length centred in axis 2 (position at which rrσ  is maximum) will be assumed for the study of the growth of the interface crack, both for the T-nC case and the T-nT case.      T8 COMATCOMP 09 917  3. INTERFACIAL GROWTH  The evolution of the first debond at the interface is studied by means of the BEM model (París and Cañas (1997), Graciani (2006)), shown in Fig. 1a and its growth evaluated in terms of the Energy Release Rate, G  (Irwin (1957)).  3.1 Tension-compression biaxial case The results obtained show that the presence of 033 <σ  does not qualitatively alter the evolution of G  versus dθ , though it is found that its level increases as 033 <σ  does, which means that the load level required for crack propagation is lower.  The prediction of growth of the interface crack is made by comparing G  with its corresponding critical value, cG , Hutchinson and Suo (1992).G - cG  comparisons for the cases T-0 and T-C (taken as representative of all T-nC cases) are plotted in Fig. 3. A value of 0.2 has been chosen for λ  and cG1  has been taken as the value that makes G  and cG  coincide for the first debonding angle, º10=dθ in this case. This criterion for the election of cG1  can be implemented once a scaled representation of the G  curves, that makes all curves to coincide at º10=dθ , has been considered. The results shown in Fig. 3 predict an unstable growth of the interface crack that reaches lower debonding angles as the presence of 033 <σ increases, though the value of 0σ  needed for the initiation of the crack growth is lower as the presence of 33σ  increases, which means that the presence of a compression superimposed on the tension nominally responsible for the failure accelerates the failure.  0.000.100.200.300.400.500.600.700.800.901.000 20 40 60 80 100 120 140Debonding angle, θ d  (º)GT-C/G 0 (σ =0.898σ 0)GT-0/G 0  (σ )G cT-0/G 0 (λ =0.2)G cT-C/G 0 (λ =0.2) Fig. 3. G-Gc  comparison. (T-nC case). 0.000.100.200.300.400.500.600.700.800.901.000 20 40 60 80 100 120 140Debonding angle, θ d  (º)T-0T-0.5TT-TGc T-0Gc T-0.5TGc T-TG /G 0    T-0G /G 0    T-0.5T (σ /σ 0=1.059)G /G 0    T-T (σ /σ 0=1.117)G c(λ =0.2)/G 0  T-0 G c(λ =0.2)/G 0  T-0.5TG c(λ =0.2)/G 0  T-T Fig. 4. G-Gc comparison. (T-nT case).  3.2 Tension-tension biaxial case The results obtained for this case show that the presence of 033 >σ does not significantly alters the G  level obtained in the T-0 case, though a translation of the position of the maxima has been detected. It can also be deduced that the propagation of the initial debond requires a slightly higher level of the external load as n increases.G  and cG  curves associated to T-nT cases corresponding to n=0,0.5 and 1 are represented together in Fig. 4, having used the same scaled representation as in the T-nC case (Fig. 3). The results obtained predict an unstable growth of the interface crack that extends T8 COMATCOMP 09 918  towards larger debonding angles as the presence of 033 >σ  increases, though the amount of load required for the initiation of growth is also greater as this presence increases. For the T-T limit case this tendency could lead to a particularly large extension of the crack at the interface that, in conjunction with the special morphology of the crack detected in this case (consisting in the opening of the previously developed contact zone near the tip), would impede the kinking of the crack towards the matrix.  3. CONCLUSIONS  The first stages of the development of the tension dominated inter-fibre failure under bidirectional loads have been studied by means of a single fibre BEM model. A secondary external load has been considered to act simultaneously with the tension nominally responsible for the failure, and both cases (tension and compression) have been analysed. The results obtained show that the presence of a secondary load could alter several aspects of the stages already detected for the tension uniaxial case leading then to the conclusion that the presence of an out plane stress component could affect the development of the failure.  ACKNOWLEDGEMENTS  This work was supported by the Spanish Ministry of Education and Science (Project TRA2006 08077) and Junta de Andalucía (Projects of Excellence TEP-02045 and TEP-04051).  REFERENCES  GOODIER, J.N. (1933). Concentration of stress around spherical and cylindrical inclusions and flaws. Applied Mechanics,55(7), 39-44. GRACIANI, E. (2006). Formulación e implementación del método de los elementos de contorno para problemas axisimétricos con contacto. Aplicación a la caracterización de la interfase fibra matriz en materiales compuestos, PhD Thesis. University of Seville. HUTCHINSON, J.W., SUO, Z. (1992). Mixed mode cracking in layered materials, Advanced Applied Mechanics, 29, 63-191. IRWIN, G.R. (1957). Analysis of stresses and strain near the end of a crack transversing a plate. Journal of Applied Mechanics, 24, 361-364. PARÍS, F., CAÑAS, J. (1997). Boundary Element Method. Fundamentals and Applications. Oxford, OUP. PARÍS, F., CORREA, CAÑAS, J. (2003). Micromechanical view of failure of the matrix in fibrous composite materials. Composites Science and Technology, 63, 1041-1052. PARÍS, F., CORREA, E. MANTIČ, V (2007). Kinking of transverse interface cracks between fibre and matrix. Journal of Applied Mechanics, 74(4), 703-716. 

Tension dominated inter-fibre failure under bi-directional loads. Micromechanical approach

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Tension dominated inter-fibre failure under bi-directional loads. Micromechanical approach

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