J. Adhesion Sci. Technol., Vol. 21, No 8, PP. 725-734(2007) Alsoavailableonline-www.brill.nl/jast Finite element analysis of load transfer at a fibre-matrix interface during pull-out loading MOHAMED KHARRAT 1, 2,* MAHER DAMMAK I ,2 and MOHAMED TRABELSI2 ent of Technology, Institut Preparatoire aux Etudes d'ingenieurs de sfax, Rte Menzel Chaker km 0.5, B.P. 805, 3018 Sfar, Tunisia 2 Laboratoire des Systemes Electromecaniques, Ecole Nationale d'Ingenieurs de sfax, Tunisia Received in final form 27 February 2007 Abstract-Load transfer ability of the fibre-matrix interface is well known to mainly control the mechanical behaviour of fibre-reinforced materials. This load transfer phenomenon is of great importance in dentistry when a post is used for fixing a ceramic crown on the tooth. The pull-out test has been well accepted as the most important micromechanical test for evaluating the interaction properties between the fibre and matrix. In this study, a finite element model is developed to analyse the pull-out process of a steel fibre from an epoxy matrix. Based on the pull-out force-displacement curves, developed in our previous experimental work, specific load transfer laws at the fibre-matrix interface have been proposed for each stage of the pull-out process, i.e., before and after fibre- matrix debonding. Predicted initial extraction forces for different implantation lengths were fitted alues and an initial interference fit of 4 um was determined. An interfacial shear trength of 21 MPa was then determined by fitting the predicted debonding forces for different implantation lengths to the experimental values. According to the load transfer laws considered analysis of the interfacial shear stress indicates that fibre-matrix debonding initiates simultaneously at both the lower and upper extremities of the interface. Keywords: Fibre; matrix; interface; friction; pull-out; debonding; finite element. 1 INTRODUCTION It is well known that the mechanical behaviour of fibre-reinforced materials mainly controlled by the load transfer ability of the fibre-matrix interface. Us of these materials in mechanical structures complicates the design problem for fatigue life, which relies on the interaction between the fibre and matrix. In order To whom correspondence should be addressed. Tel. (216)7424-1403, ext. 154; Fax: (216)7424- 6347; e-mail: Mohamed Kharrat @ipesrmutn
J. Adhesion Sci. Technol., Vol. 21, No. 8, pp. 725–734 (2007) VSP 2007. Also available online - www.brill.nl/jast Finite element analysis of load transfer at a fibre–matrix interface during pull-out loading MOHAMED KHARRAT 1,2,∗, MAHER DAMMAK 1,2 and MOHAMED TRABELSI 2 1 Department of Technology, Institut Préparatoire aux Etudes d’Ingénieurs de Sfax, Rte Menzel Chaker km 0.5, B.P. 805, 3018 Sfax, Tunisia 2 Laboratoire des Systèmes Electromécaniques, Ecole Nationale d’Ingénieurs de Sfax, Tunisia Received in final form 27 February 2007 Abstract—Load transfer ability of the fibre–matrix interface is well known to mainly control the mechanical behaviour of fibre-reinforced materials. This load transfer phenomenon is of great importance in dentistry when a post is used for fixing a ceramic crown on the tooth. The pull-out test has been well accepted as the most important micromechanical test for evaluating the interaction properties between the fibre and matrix. In this study, a finite element model is developed to analyse the pull-out process of a steel fibre from an epoxy matrix. Based on the pull-out force–displacement curves, developed in our previous experimental work, specific load transfer laws at the fibre–matrix interface have been proposed for each stage of the pull-out process, i.e., before and after fibre– matrix debonding. Predicted initial extraction forces for different implantation lengths were fitted to experimental values and an initial interference fit of 4 µm was determined. An interfacial shear strength of 21 MPa was then determined by fitting the predicted debonding forces for different implantation lengths to the experimental values. According to the load transfer laws considered, analysis of the interfacial shear stress indicates that fibre–matrix debonding initiates simultaneously at both the lower and upper extremities of the interface. Keywords: Fibre; matrix; interface; friction; pull-out; debonding; finite element. 1. INTRODUCTION It is well known that the mechanical behaviour of fibre-reinforced materials is mainly controlled by the load transfer ability of the fibre–matrix interface. Use of these materials in mechanical structures complicates the design problem for fatigue life, which relies on the interaction between the fibre and matrix. In order ∗To whom correspondence should be addressed. Tel.: (216) 7424-1403, ext. 154; Fax: (216) 7424- 6347; e-mail: Mohamed.Kharrat@ipeis.rnu.tn
M. Kharrat et al Crown Dental po Cement Aveolar bon Figure 1. Schematic drawing of an endodontically-treated tooth with a dental post. to determine this interaction several micromechanical tests which. in general use composite model specimens containing a single fibre, have been developed [1-4]. Based on how the external load is applied to the composite specimen, micromechanical tests can be divided into two groups. The first group in which the external load is directly applied to the fibre includes the micro-indentation test, the push-out test and the pull-out test. In the second group the external load is applied to the matrix and includes the fragmentation test and the broutman test. there has been extensive discussion in the literature about the choice of an appropriate micromechanical test for interface characterization [4 ]. The pull-out test has been well accepted as the most suitable micromechanical test for evaluating the bond quality at the fibre-matrix interface [4-7]. In addition to its relative simplicity of sample preparation and measurement, this test is expected to provide realistic investigation of interfacial adhesion for composites with both ductile and brittle matrices In prosthetic dentistry, a ceramic crown is fixed by placing a dental post within the tooth root using cement. The visible part of the post is then bonded to the crown with a resin core(Fig. 1). When forces are applied to the crown, they are transferred to the dentin through the core and the post. Stress concentration at the end of the post often initiates root fracture [8]. This phenomenon depends on the post-core and post-tooth root interaction. The pull-out test can also be used to evaluate the retentive strength of the post to the tooth root and to the core foundation [9] Experimental pull-out force-displacement curves are used to determine the intrin sic properties of the fibre-matrix interface. For this purpose, several analytical mod- els based on elastic (shear-lag) and frictional load transfer considerations between
726 M. Kharrat et al. Figure 1. Schematic drawing of an endodontically-treated tooth with a dental post. to determine this interaction, several micromechanical tests, which, in general, use composite model specimens containing a single fibre, have been developed [1–4]. Based on how the external load is applied to the composite specimen, micromechanical tests can be divided into two groups. The first group in which the external load is directly applied to the fibre includes the micro-indentation test, the push-out test and the pull-out test. In the second group the external load is applied to the matrix and includes the fragmentation test and the Broutman test. There has been extensive discussion in the literature about the choice of an appropriate micromechanical test for interface characterization [4]. The pull-out test has been well accepted as the most suitable micromechanical test for evaluating the bond quality at the fibre–matrix interface [4–7]. In addition to its relative simplicity of sample preparation and measurement, this test is expected to provide realistic investigation of interfacial adhesion for composites with both ductile and brittle matrices. In prosthetic dentistry, a ceramic crown is fixed by placing a dental post within the tooth root using cement. The visible part of the post is then bonded to the crown with a resin core (Fig. 1). When forces are applied to the crown, they are transferred to the dentin through the core and the post. Stress concentration at the end of the post often initiates root fracture [8]. This phenomenon depends on the post-core and post-tooth root interaction. The pull-out test can also be used to evaluate the retentive strength of the post to the tooth root and to the core foundation [9]. Experimental pull-out force–displacement curves are used to determine the intrinsic properties of the fibre–matrix interface. For this purpose, several analytical models based on elastic (shear-lag) and frictional load transfer considerations between
Finite element analysis of load transfer during a pull-out process the fibre and matrix have been developed [5, 6, 10]. For the same purpose, finite el- ement models have also been developed [ll, 12]. To analyse the effect of specimen size on the interfacial shear and normal stresses, Yang et al. [ll] developed a 2-D finite element model using ANSYS program. For simplicity, the effects of thermal residual stress and friction between crack faces have been ignored by these authors They concluded that both normal and shear interfacial stresses concentrations exist near the fibre ends whose values are affected by the length of the fibre embed- ded in the matrix and the thickness of the matrix around the fibre. To analyse the crack bridging ability in laminated composites with through-thickness reinforce- nent (TTR), Meo et al. [12] developed a finite element model. A specific contact frictional law has been used to describe the several phases of the pull-out process In their work, Meo et al. carried out a parametric study to investigate the effect of the contact frictional law parameters on the simulated pull-out force-displacement curve Experimental characterization of the fibre-matrix interface for a steel/epoxy composite system has been developed using the pull-out test in our previous work [10]. The objective of this study was to develop a finite element model for the analysis of the experimental results. A specific load transfer law at the fibre-matrix interface has been proposed for each stage of the pull-out process, i.e., before and after fibre-matrix debonding. Using this finite element model. the interfacial shear trength has been determined and the interfacial stresses have been analysed 2. EXPERIMENTAL The composite model specimen used in the pull-out experiment [10] was made from stainless steel fibre embedded in a cylindrical epoxy matrix with different implantation lengths(the length of the fibre embedded in the matrix). To carry out the pull-out experiments, an apparatus mounted on a standard traction-compression machine was designed [10]. The top surface of the matrix cylinder was not permitted to move with respect to the pull-out test apparatus(Fig. 2). A monotonic displacement with a constant speed of 5 mm/min was applied to the end of the steel fibre. Typical pull-out force versus displacement curve for 10 mm implantation length is shown in Fig 3. The features of this curve are outlined as follows [ 5] In the initial quasi-linear region, the fibre extends as the force rises. When the maximum force Fa(debonding force)is reached the fibre debonds from the matrix along the full embedded length. This is followed by a sudden drop of the force to initial Fi (initial extraction force)required for pulling out the debonded fibre fron the matrix. After that the force continues to decrease while the extracted length increases until the whole fibre is extracted from the matrix FINITE ELEMENT MODELLING A 2-D axisymmetric finite element was used in the pull-out simulation, where a
Finite element analysis of load transfer during a pull-out process 727 the fibre and matrix have been developed [5, 6, 10]. For the same purpose, finite element models have also been developed [11, 12]. To analyse the effect of specimen size on the interfacial shear and normal stresses, Yang et al. [11] developed a 2-D finite element model using ANSYS program. For simplicity, the effects of thermal residual stress and friction between crack faces have been ignored by these authors. They concluded that both normal and shear interfacial stresses concentrations exist near the fibre ends whose values are affected by the length of the fibre embedded in the matrix and the thickness of the matrix around the fibre. To analyse the crack bridging ability in laminated composites with through-thickness reinforcement (TTR), Meo et al. [12] developed a finite element model. A specific contact frictional law has been used to describe the several phases of the pull-out process. In their work, Meo et al. carried out a parametric study to investigate the effect of the contact frictional law parameters on the simulated pull-out force–displacement curve. Experimental characterization of the fibre–matrix interface for a steel/epoxy composite system has been developed using the pull-out test in our previous work [10]. The objective of this study was to develop a finite element model for the analysis of the experimental results. A specific load transfer law at the fibre–matrix interface has been proposed for each stage of the pull-out process, i.e., before and after fibre–matrix debonding. Using this finite element model, the interfacial shear strength has been determined and the interfacial stresses have been analysed. 2. EXPERIMENTAL The composite model specimen used in the pull-out experiment [10] was made from stainless steel fibre embedded in a cylindrical epoxy matrix with different implantation lengths (the length of the fibre embedded in the matrix). To carry out the pull-out experiments, an apparatus mounted on a standard traction-compression machine was designed [10]. The top surface of the matrix cylinder was not permitted to move with respect to the pull-out test apparatus (Fig. 2). A monotonic displacement with a constant speed of 5 mm/min was applied to the end of the steel fibre. Typical pull-out force versus displacement curve for 10 mm implantation length is shown in Fig. 3. The features of this curve are outlined as follows [5]: In the initial quasi-linear region, the fibre extends as the force rises. When the maximum force Fd (debonding force) is reached the fibre debonds from the matrix along the full embedded length. This is followed by a sudden drop of the force to the initial Fi (initial extraction force) required for pulling out the debonded fibre from the matrix. After that, the force continues to decrease while the extracted length increases until the whole fibre is extracted from the matrix. 3. FINITE ELEMENT MODELLING A 2-D axisymmetric finite element was used in the pull-out simulation, where a
M. Kharrat et al Pull-out force Steel fibre 囫H Epoxy matrix Figure 2. Schematic representation of the experimental pull-out test configuration. Displacement(mm) Figure 3. Experimental pull-out force-displacement curve for 10 mm implantation length [10]. Fd debonding force; Fi. initial extraction force. steel fibre of 1. 2 mm in diameter was embedded in an epoxy cylinder of 30 mm outside diameter and 20 mm length. The implantation length L of the fibre in the matrix was varied in the range of 4-12 mm, while the non-embedded length of fibre( the surface of the matrix cylinder and the point of loading)was kept at 28 mm. As we do not expect large deformations in the fibre or the matrix
728 M. Kharrat et al. Figure 2. Schematic representation of the experimental pull-out test configuration. Figure 3. Experimental pull-out force–displacement curve for 10 mm implantation length [10]. Fd, debonding force; Fi, initial extraction force. steel fibre of 1.2 mm in diameter was embedded in an epoxy cylinder of 30 mm outside diameter and 20 mm length. The implantation length L of the fibre in the matrix was varied in the range of 4–12 mm, while the non-embedded length of fibre (between the surface of the matrix cylinder and the point of loading) was kept at 28 mm. As we do not expect large deformations in the fibre or the matrix
Finite element analysis of load transfer during a pull-out process Properties of steel fibre and epoxy matrix used Elastic modulus(MPa) Poissons coefficient Epoxy matrix 4080±180 Araldite LY 556 with Aradur 917 and DY070(Huntsman) Steel fibre X2CrNiMo17-12(316L) =0 Figure 4. Axisymmetric finite element mesh for pull-out loading. I/1, radial direction: 2, axial direction; F, pull-out force. during pull-out loading, both fibre and matrix are assumed to be linear elastic. Their material properties are given in Table 1 For each implantation length, an axisymmetric nonlinear finite element model was developed using the ABAQUS program(ABAQUS 6.4, 2003). Small displacement conditions were considered for this model. Therefore, we chose to limit the analysis of the pull-out process to the beginning of the extraction process(F= Fi). The model subdivision used for calculations is shown in Fig 4. The model consists of 760 8-node elements and 84 6-node elements. a total of 2747 nodes were used in his model
Finite element analysis of load transfer during a pull-out process 729 Table 1. Properties of steel fibre and epoxy matrix used Elastic modulus (MPa) Poisson’s coefficient Epoxy matrix 4080 ± 180 0.3 Araldite LY 556 with Aradur 917 and DY070 (Huntsman) Steel fibre 200 000 0.22 X2CrNiMo17-12 (316L) Figure 4. Axisymmetric finite element mesh for pull-out loading. u1, radial direction; u2, axial direction; F, pull-out force. during pull-out loading, both fibre and matrix are assumed to be linear elastic. Their material properties are given in Table 1. For each implantation length, an axisymmetric nonlinear finite element model was developed using the ABAQUS program (ABAQUS 6.4, 2003). Small displacement conditions were considered for this model. Therefore, we chose to limit the analysis of the pull-out process to the beginning of the extraction process (F = Fi). The model subdivision used for calculations is shown in Fig. 4. The model consists of 760 8-node elements and 84 6-node elements. A total of 2747 nodes were used in this model
730 M. Kharrat et al 1=7 12=O11 Interfacial shear strain, y12 Figure 5. Interfacial shear stress versus interfacial shear strain. (a) Linear elastic shear stress with maximum value td.(b) Coulomb frictional shear stress with maximum value ual interfacial coefficient of friction; all, interfacial radial stress. On the basis of the analysis of experimental pull-out curves, two load transfer laws at the fibre-matrix interface have been proposed. Before the debonding force is reached, an elastic load transfer is assumed to take place at the fibre-matrix interface. For each contact element, a linear elastic shear stress t1 was introduced with a maximum shear stress td(Fig. 5, curve a). After debonding, a Coulomb frictional shear stress [12 was introduced for each contact element with a maximum value Hon(Fig. 5, curve b), where u is the interfacial coefficient of friction and oil is the interfacial radial stress. For this Coulomb law, u=0.3 was determined from the sliding friction experiment using the epoxy matrix sample in contact with steel fibres [10]. In addition the initial interference fit Ar(difference between the fibre radius and the inner radius of the matrix cylinder) which generates radial stress at the fibre-matrix interface has been considered. This interference fit, which generates compressive residual stress, was induced by resin shrinkage during cure process, as well as by the difference of thermal expansion coefficients of the fibre and the matrix 4. RESULTS AND DISCUSSION 4.1. Finite element analysis of the initial extraction force Finite element values of the initial extraction force Fi for different implantation lengths have been fitted to the experimental values [10] using the interference fit Ar at the fibre-matrix interface as an adjustable parameter. Finite element and experimental results(Fi L plots) are reported in Fig. 6, as well as the prediction of our previous analytical model [10]. It can be seen from this figure that the finite element analysis is satisfactory in describing both the experimental and analytical results. The resultant value for the interference fit Ar=4 um is found. This value will be used for the analysis of the debonding force using the finite element method
730 M. Kharrat et al. Figure 5. Interfacial shear stress versus interfacial shear strain. (a) Linear elastic shear stress with maximum value τd. (b) Coulomb frictional shear stress with maximum value µσ11. µ, interfacial coefficient of friction; σ11, interfacial radial stress. On the basis of the analysis of experimental pull-out curves, two load transfer laws at the fibre–matrix interface have been proposed. Before the debonding force is reached, an elastic load transfer is assumed to take place at the fibre–matrix interface. For each contact element, a linear elastic shear stress τ12 was introduced with a maximum shear stress τd (Fig. 5, curve a). After debonding, a Coulomb frictional shear stress τ12 was introduced for each contact element with a maximum value µσ11 (Fig. 5, curve b), where µ is the interfacial coefficient of friction and σ11 is the interfacial radial stress. For this Coulomb law, µ = 0.3 was determined from the sliding friction experiment using the epoxy matrix sample in contact with steel fibres [10]. In addition, the initial interference fit r (difference between the fibre radius and the inner radius of the matrix cylinder) which generates radial stress at the fibre–matrix interface has been considered. This interference fit, which generates compressive residual stress, was induced by resin shrinkage during cure process, as well as by the difference of thermal expansion coefficients of the fibre and the matrix. 4. RESULTS AND DISCUSSION 4.1. Finite element analysis of the initial extraction force Finite element values of the initial extraction force Fi for different implantation lengths have been fitted to the experimental values [10] using the interference fit r at the fibre–matrix interface as an adjustable parameter. Finite element and experimental results (Fi–L plots) are reported in Fig. 6, as well as the prediction of our previous analytical model [10]. It can be seen from this figure that the finite element analysis is satisfactory in describing both the experimental and analytical results. The resultant value for the interference fit r = 4 µm is found. This value will be used for the analysis of the debonding force using the finite element method.
Finite element analysis of load transfer during a pull-out process Figure 6. Initial extraction force versus implantation length. Experimental; ( element model; - analytical model [10] 1000 800 忑600 400 200 Implantation length(mm) Figure 7. Debonding force versus implantation length (O) Experimental; ( finite element 4.2. Finite element analysis of the debonding force For the debonding force Fd, the finite element predicted values have been fitted to the measured values for different implantation lengths using the maximum shear stress td as an adjustable parameter. Finite element and experimental results ( Fa-L plots) are reported in Fig. 7. The predicted results for the Fa-L evolution from our previous analytical model [10] are also plotted in Fig. 7. It can be seen that the finite element simulation is in good agreement with both the experimental and analytical results for implantation lengths lower than 10 mm. For higher mplantation lengths, the finite element model seems to overestimate. The resultant
Finite element analysis of load transfer during a pull-out process 731 Figure 6. Initial extraction force versus implantation length. ( ) Experimental; ( ) finite element model; ( ) analytical model [10]. Figure 7. Debonding force versus implantation length. ( ) Experimental; ( ) finite element model; ( ) analytical model [10]. 4.2. Finite element analysis of the debonding force For the debonding force Fd, the finite element predicted values have been fitted to the measured values for different implantation lengths using the maximum shear stress τd as an adjustable parameter. Finite element and experimental results (Fd–L plots) are reported in Fig. 7. The predicted results for the Fd–L evolution from our previous analytical model [10] are also plotted in Fig. 7. It can be seen that the finite element simulation is in good agreement with both the experimental and analytical results for implantation lengths lower than 10 mm. For higher implantation lengths, the finite element model seems to overestimate. The resultant
M. Kharrat et al Figure 8. Interfacial radial stress all versus interface position u2 for 10 mm implantation length. )F=0N;( )F=Fd=745N value for the interfacial shear strength(maximum shear stress), obtained by finite element analysis, is ta= 21 MPa 4.3. Interfacial stresses analysis In order to understand the debonding process at the fibre-matrix interface, interfa- cial stresses have been analysed by applying increasing pull-out force. The profiles of the radial stress oll along the fibre-matrix interface are plotted in Fig. 8 for 10 mm implantation length condition. The initial interference fit generates a quasi- uniform compressive radial stress along the fibre-matrix interface with a mean value of-20 MPa. This radial stress is affected by the applied pull-out force. When the debonding force is reached, the compressive radial stress attains a value of-40 MPa at the lower extremity of the fibre-matrix interface(u2 = 0; see Fig. 4) and falls to zero at the upper extremity(u2=L; see Fig 4). In Fig 9 are plotted the shear stress [12 profiles along the fibre-matrix interface for different values of the pull-out force. According to the maximum shear stress criterion introduced for the analysis of the load transfer at the fibre-matrix interface before debonding (fig. 5, curve a), it appears from Fig. 9 that the debonding initiates at both the lower and upper extremi- ties of the interface simultaneously. The debonding progresses toward the middle of the interface length from these two nucleation sites as the pull-out force increases When the pull-out force reaches Fd, a complete debonding occurs at the fibre-matrix interface. This result illustrates the limitation of the simplified shear-lag approach according to which debonding initiates at the upper extremity of the fibre-matrix interface [4
732 M. Kharrat et al. Figure 8. Interfacial radial stress σ11 versus interface position u2 for 10 mm implantation length. ( ) F = 0 N; ( ) F = Fd = 745 N. value for the interfacial shear strength (maximum shear stress), obtained by finite element analysis, is τd = 21 MPa. 4.3. Interfacial stresses analysis In order to understand the debonding process at the fibre–matrix interface, interfacial stresses have been analysed by applying increasing pull-out force. The profiles of the radial stress σ11 along the fibre–matrix interface are plotted in Fig. 8 for 10 mm implantation length condition. The initial interference fit generates a quasiuniform compressive radial stress along the fibre–matrix interface with a mean value of −20 MPa. This radial stress is affected by the applied pull-out force. When the debonding force is reached, the compressive radial stress attains a value of −40 MPa at the lower extremity of the fibre–matrix interface (u2 = 0; see Fig. 4) and falls to zero at the upper extremity (u2 = L; see Fig. 4). In Fig. 9 are plotted the shearstress τ12 profiles along the fibre–matrix interface for different values of the pull-out force. According to the maximum shear stress criterion introduced for the analysis of the load transfer at the fibre–matrix interface before debonding (Fig. 5, curve a), it appears from Fig. 9 that the debonding initiates at both the lower and upper extremities of the interface simultaneously. The debonding progresses toward the middle of the interface length from these two nucleation sites as the pull-out force increases. When the pull-out force reaches Fd, a complete debonding occurs at the fibre–matrix interface. This result illustrates the limitation of the simplified shear-lag approach according to which debonding initiates at the upper extremity of the fibre–matrix interface [4]
Finite element analysis of load transfer during a pull-out process 10 ee-eHes-s-B 6 10 Interface position(mm) Ire 9. Interfacial shear stress [12 versus interface position u2 for 10 mm implantation length. )F=0N;( )F=520N;(口)F=720N;( )F=Fd=745N 5 CONCLUSIONS An axisymmetric non-linear finite element model was developed using the ABAQUS program for the analysis of the pull-out behaviour of a stainless steel fi- bre/epoxy matrix composite. After debonding was experienced, Coulomb frictional shear stress was introduced for each contact element. An initial interference fit of 4 um was found by fitting the predicted FiL plots to experimental results. Before the debonding force was reached a linear elastic shear stress was introduced for each contact element with a maximum shear stress td. An interfacial shear strength of 21 MPa was determined by fitting the predicted Fa-L plots to experimental re- sults. Predicted interfacial stresses have been analysed by applying increasing pull- out force. The compressive radial stress, which has an initial quasi-uniform profile increases at the lower extremity of the fibre-matrix interface and decreases at the upper extremity as the pull-out force increases. Analysis of the interfacial shear stress indicates that the fibre-matrix debonding initiates at both the lower and upper extremities of the interface simultaneously. This finding illustrates the limitation of the simplified shear-lag approach according to which the maximum shear stress is cated at the upper extremity of the fibre-matrix interface REFERENCES 1. M. Kharrat, A. Chateauminois, L. Carpentier and P. Kapsa, Composites Part A 28A, 39(1997) 2. J. Li, H. Ma and Y huang, Mater: Chem. Phys. 89, 367(2005) 3. S Guo, K Honda and Y Kagawa, Composites Sci. Technol. 65, 1808(2005) 4. S. Zhandarov and E Mader, Composites Sci. Technol. 65, 149(2005) 5. A. Takaku and R. G. C. Arridge, J. Phys. D: Appl. Phys. 6, 2038(1973 6. S. Y Fu, C. Y. Yue, X Hu and Y. w Mai, Composites Sci. Technol. 60, 569(2000) 7. S. Zhandarov, Y. Gorbatkina and E. Mader, Composites Sci. Technol. 66, 2610(2006)
Finite element analysis of load transfer during a pull-out process 733 Figure 9. Interfacial shear stress τ12 versus interface position u2 for 10 mm implantation length. ( ) F = 0 N; ( ) F = 520 N; ( ) F = 720 N; ( ) F = Fd = 745 N. 5. CONCLUSIONS An axisymmetric non-linear finite element model was developed using the ABAQUS program for the analysis of the pull-out behaviour of a stainless steel fi- bre/epoxy matrix composite. After debonding was experienced, Coulomb frictional shear stress was introduced for each contact element. An initial interference fit of 4 µm was found by fitting the predicted Fi–L plots to experimental results. Before the debonding force was reached, a linear elastic shear stress was introduced for each contact element with a maximum shear stress τd. An interfacial shear strength of 21 MPa was determined by fitting the predicted Fd–L plots to experimental results. Predicted interfacial stresses have been analysed by applying increasing pullout force. The compressive radial stress, which has an initial quasi-uniform profile, increases at the lower extremity of the fibre–matrix interface and decreases at the upper extremity as the pull-out force increases. Analysis of the interfacial shear stress indicates that the fibre–matrix debonding initiates at both the lower and upper extremities of the interface simultaneously. This finding illustrates the limitation of the simplified shear-lag approach according to which the maximum shear stress is located at the upper extremity of the fibre–matrix interface. REFERENCES 1. M. Kharrat, A. Chateauminois, L. Carpentier and P. Kapsa, Composites Part A 28A, 39 (1997). 2. J. Li, H. Ma and Y. Huang, Mater. Chem. Phys. 89, 367 (2005). 3. S. Guo, K. Honda and Y. Kagawa, Composites Sci. Technol. 65, 1808 (2005). 4. S. Zhandarov and E. Mäder, Composites Sci. Technol. 65, 149 (2005). 5. A. Takaku and R. G. C. Arridge, J. Phys. D: Appl. Phys. 6, 2038 (1973). 6. S. Y. Fu, C. Y. Yue, X. Hu and Y. W. Mai, Composites Sci. Technol. 60, 569 (2000). 7. S. Zhandarov, Y. Gorbatkina and E. Mäder, Composites Sci. Technol. 66, 2610 (2006)
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734 M. Kharrat et al. 8. K. Fujihara, K. Teo, R. Gopal, P. L. Loh, V. K. Ganesh, S. Ramakrishna, K. W. C. Foong and C. L. Chew, Composites Sci. Technol. 64, 775 (2004). 9. F. Al-harbi and D. Nathanson, J. Prosthet. Dent. 90, 547 (2003). 10. M. Kharrat, M. Dammak and A. Charfi, J. Mater. Sci. Technol. 22, 552 (2006). 11. Q. S. Yang, Q. H. Qin and X. R. Peng, Composite Struct. 61, 193 (2003). 12. M. Meo, F. Achard and M. Grassi, Composite Struct. 71, 383 (2005)