Finite element analysis of longitudinal debonding between fibre and matrix interface

Transcription

1 Indian Journal of Engineering & Materials Sciences Vol. 11, February 2004, pp Finite element analysis of longitudinal debonding between fibre and matrix interface K Aslantaş & S Taşgetiren Department of Mechanical Education, University of Afyon Kocatepe, Afyon, Turkey Received 4 February 2003; accepted 15 December 2003 In this paper, a numerical model developed for the analysis of a cylindrical element of matrix containing a single fibre is presented. A ring-shaped crack is assumed at the interface of fibre and matrix. Both layers in the model are bonded perfectly with the exception of the crack faces. Contact elements, which have bonded feature, are used between fibre and matrix. Displacement correlation method is used to calculate opening-mode and sliding-mode stress intensity factors. These results obtained from the analysis help to understand the debonding phenomenon between fibre and matrix interface. Adhesion between fibre and matrix is controlled by properties of the interface. Generally high degree of adhesion is desirable to provide efficient transfer of load between fibre and matrix. The strength of the adhesion controls properties of composite in a direction transverse to fibres as well as shear properties. Continuous fibre reinforced metal matrix composites are attractive as a material for high-performance components in aeroplane jet engines, for example, for compressor disks or blades 1-3. During service of highly loaded components, fatigue cracks can be initiated. After initiation, these fatigue cracks can grow until final failure occurs. Gdoutos et al. 4 studied singular stress field in the neighborhood of the periphery of an annular crack. The case of fibre debonding originating from the annular crack was also considered. In the study, K I and K II stress intensity factors and energy release rates were calculated. Liu and Kagawa 5 have derived the energy release rate for an interfacial debonding of a crack in a ceramic-matrix composite by the use of the Lame solution for an axisymmetric cylindrical fibre/matrix model. Aslantas and Taşgetiren 6 studied a numerical solution is carried out for the problem of interface crack. Variations in the stress intensity factors K I and K II, with load position are obtained for various cases such as different combinations of material of coating layer and substrate, changes in the coefficient of friction on the surface. Fatigue crack initiation in SiC fibre (SCS-6) reinforced titanium has been analysed on the basis of a finite element model by Xia et al. 7 Their results showed that the formation of matrix crack largely depends on the applied stress and reaction layer thickness. Bjerken and Persson 8 presented a method for obtaining the complex stress intensity factor (or alternatively the corresponding energy release rate and mode mixity) for an interface crack in a bimaterial using a minimum number of computations. Aslantas 9 developed a numerical model for the analysis of a cylindrical element of matrix containing a single fibre. A ring-shaped crack is assumed at interface of fibre and matrix. These results obtained from the analysis help to understand the debonding phenomenon between fibre and matrix interface. Effects of the mechanical properties of fibre and matrix on direction of crack propagation are also discussed. In general, stress intensity factors at the crack tip is calculated based on energy release rate or J integral 3-5. In this study, a numerical analysis has been carried out for analysis of a cylindrical element of matrix with a single fibre having a ring-shaped crack at the interface. Special contact elements, which have bonded feature, are used between fibre and matrix. Displacement correlation method is used to calculate K 1 and K II. Numerical Analysis Model description Fig. 1 shows a micromechanical model used in this study. A ring-shaped crack is assumed at the interface between fibre and matrix. Both layers in the model are bonded perfectly with the exception of the crack faces. The sliding behaviour at the interface is characterised by Coulomb s friction law in which the coeffi-

2 44 INDIAN J. ENG. MATER. SCI., FEBRUARY 2004 Fig. 1 (a) The composite cylinder model, (b) cross-section of continuous fibre reinforced composite, and (c) axi-symmetric model of the composite with crack Fig. 2 (a) Finite element mesh and boundary conditions, and (b) quarter point elements at the interface crack tip cient of friction is assumed to be zero. Cylindrical model showing a fibre and matrix is given in Fig. 1. Finite element model Numerical modelling is done by the ANSYS 6.1 finite element program. Fig. 2 shows the configuration of a crack at the interface between fibre and matrix. Eight nodes isoparametric finite element elements are used for the model of the solution domain except those contacting the crack tip. These elements are six nodes triangular quarter point elements. The element and node numbers are 4512 and 6680 respectively. The elements near the crack are taken as small as possible in order to simulate the stress distribution and deformation near the crack more accurately, as shown in Fig. 2b. TARGE169 and CONTA172 contact elements are used to represent various axisymmetric contacting surfaces between fibre and matrix. The target and associated contact surfaces are identified by a shared real constant set. This real constant set includes all real constants for both the target and contact elements. For the surfaces between fibre and

3 ASLANTAS & TASGETIREN: ANALYSIS OF LONGITUDINAL DEBONDING BETWEEN FIBRE & MATRIX INTERFACE 45 matrix, except the crack surfaces, the property of bonded contact elements is used. Calculation of stress intensity factors The mixed-mode crack propagation scheme can be implemented by Finite Element or Boundary Element Analysis. The advantage of such a numerical method is that it calculates the stress intensity factors more accurately in terms of near crack tip nodal displacements 10. The method is called displacement correlation method. The definitions are given in Fig. 3 for the application of the method. After finite element or boundary element solutions for cracked structure is obtained, nodal displacement values of nodes a, b, c, d and e (Fig. 3a) are determined. Opening mode K I and shear mode K II are defined as 11, ( 1 2π e d a KI = D v + 4v 3v L ) c b a D 2 v + 4v 3v (1) Fig. 3 (a) The interface crack between fibre and matrix, and (b) nodes at the crack surfaces Fig. 4 Variation of non-dimensional τ xy shear stress at the fibre matrix interface for different E f /E m values

4 46 INDIAN J. ENG. MATER. SCI., FEBRUARY 2004 ( 1 2π e d a KII = D u + 4u 3u L c b a D 2 u + 4u 3u ) (2) where L is distance between nodes of a-c or a-e. v i is the displacement along y at nodes a, b, c, d, e and u i is displacement along x at nodes a, b, c, d, e. D 1 and D 2 can be expressed as: (1+γ)λ D=. G πε -πε cosh(πε) κ1e +γe (1+γ)λ D=. G πε πε cosh(πε) κγe 2 +e 1 ε= lnγ 2π 1/2 G+κ 1 1G2 1 2 γ= λ 0= +ε G+κ 2 2 G 1 4 (3) (4) (5) where κ is defined for axi-symmetric problems as 3-4ν i. ε is the bimaterial constant, and G i, ν i are shear modulus and Poisson s ratios for respective material. When the mode I and II stress intensity factors are known, the predicted crack propagation angle can be estimated under mixed mode loading. The most widely accepted method is the maximum principal stress theory 12. According to this theory, the crack propagates in a direction perpendicular to maximum tangential stress. In the present study, the angle of crack growth is obtained from the maximum tangential stress theory K I K I θ = 2tan ± KII K II Results and Discussion (7) Stress distribution at the interface In Fig. 4, τ xy shear stress at the fibre matrix interface is given for a/r f =0.5 and different E f /E m values. Von-Misses stress distribution at the interface for a/r f =0.5 and various E f /E m values is given in Fig. 5. Stress values are normalized with σ a, applied stress on fibre. It can be seen from the Fig. 4 max. τ xy shear stresses at the fibre matrix interface occur at y/l= In addition max. von-misses stresses at the interface occur at y/l=0.05 for E f /E m = 1. But max. von-misses stresses occur at y/l=0 for E f /E m > 1 (Fig. 5). Fig. 6 displays a profile of deformed mesh for three different E f /E m ratios. It is observed from Fig. 6 that elastic deformation at the crack faces decreases as the E f /E m ratio increases. So the stress intensity factors will decrease as node displacement values at the crack faces decreases. Variation in stress intensity factors For the evaluation of stress intensity factors, K I and K II at the interfacial crack tip, the relative displacements, Δy and Δx, of five points on upper and lower crack faces are determined by using the ANSYS 6.1 finite element software. Fig. 7 shows the variation of Fig. 5 Variation of non-dimensional σ von-misses stress at the fibre matrix interface for different E f /E m values

5 ASLANTAS & TASGETIREN: ANALYSIS OF LONGITUDINAL DEBONDING BETWEEN FIBRE & MATRIX INTERFACE 47 Fig. 6 Deformed mesh of the composite material for a/r f =0.3 (displacement multiplier x20) Fig. 7 Non-dimensional K I stress intensity factor versus interface crack length for different E f /E m values Fig. 8 Non-dimensional K II stress intensity factor versus interface crack length for different E f /E m values.

6 48 INDIAN J. ENG. MATER. SCI., FEBRUARY 2004 the normalized K I stress intensity factor against the normalized interface crack. Fig. 8 also shows the variation of the K II. Both K I and K II curves are obtained for four different E f /E m values. Opening mode K I takes positive values for all E f /E m values and the applied stress on fibre exerts to open the crack faces. But, note that for all values of E f /E m, K I decreases as the interface crack length increases. In addition to this, K I decreases as the value of E f /E m increases (Fig. 7). K II affects the trajectory of the path of propagation of the crack. Erdogan and Sih 12 have shown that a crack continues to advance in its own plane when it is subjected to pure mode I. The presence of positive K II deflects the crack in -θ direction, and negative K II causes the crack to deviate in +θ direction according to Fig. 3. It can be seen from Figs 7 and 8 that absolute K II values are much higher than K I values. Furthermore, sliding mode K II stress intensity factor is affected more than opening mode K I stress intensity factor as the E f /E m value increases. These results indicate that deformation fields in the vicinity of the crack tip are dominated by shear mode. Conclusions The present study has provided a linear elastic fracture mechanics analysis of interface crack problem in continuous fibre reinforced metal matrix composites. The finite element mesh is formed for cylindrical element of matrix with a single fibre. A ring-shaped crack is assumed between fibre and matrix. Displacement correlation method is used to calculate K 1 and K 2 stress intensity factors at the crack tip. The analysis results show that absolute K II values are much higher than K I values. Sliding mode K II stress intensity factor is affected more than opening mode K I stress intensity factor as the E f /E m value increases. The deformation fields in the vicinity of the crack tip are dominated by shear mode. Debonding between fibre and matrix most probably occurs due to shear mode under axial loading. References 1 Masud A, Zhang Z & Botsis J, Composites Pt B:Eng, 29(5) (1998) Peters P W M, Xia Z, Hemptenmacher J & Assler H, Composites Pt A: Appl Sci Manufact, 32(3-4) (2001) Rauchs G, Thomason P F & Withers P J, Comput Mater Sci, 25(1-2) (2002) Gdoutos E E, Giannakopoulou A & Zacharopoulos D A, Int J Fract, 98 (1999) LiuY F & Kagawa Y, Composit Sci Technol, 60(2) (2000) Aslantaş K & Taşgetiren S, Mater Design, 43(2002) Xia Z H, Peters P W M & Dudek H J, Composites Pt A: Appl Sci Manufact, 31(10) (2000) Bjerken C & Persson C, Eng Fractllre Mech, 68(2) (2001) Aslantaş K, Int J Solid Struct, 40 (2003) Chan S K, Tuba I S & Wilson W K, Eng Fractllre Mech, 2 (1970) Tan C L, Gao Y L, Eng Fract Mech, 36 (1990) Erdogan F & Sih G C, J Basic Eng, 85 (1963)