COMPOSITES SCIENCE AND TECHNOLOGY ELSEVIER Composites Science and Technology 59(1999)1871-1879 The role of interfacial debonding in increasing the strength and reliability of unidirectional fibrous composites Koichi goda Department of Mechanical Engineering, Yamaguchi University, Tokiwadai, Ube 755, Japan Received 30 December 1996: received in revised form I July 1998: accepted 22 February 1999 Abstract A Monte-Carlo simulation technique based on a finite-element method has been developed in order to clarify the effect of inte facial shear strength on the tensile strength and reliability of fibrous composites. In the simulation a boron/epoxy monolayer omposite was modelled, and five hundred simulations were carried out for various interfacial shear strengths. The interfacial shear strength value which raised the average strength of the composite corresponded approximately to the value which reduced the coefficient of variation. This implies the existence of an optimum value of interfacial shear strength which can increase the strength d reliability. The simulated strength and reliability were closely related to the degree and type of damage around a fiber break That is to say, large-scale debonding caused by a weak interfacial bond and matrix cracking caused by a strong bond reduced the number of fiber breaks accumulated up to the maximum stress, and decreased the strength and reliability. On the other hand, small- scale debonding promoted comparatively the cumulative effect of fiber breaks and played a key role in increasing the composite trength and reliability. c 1999 Elsevier Science Ltd. All rights reserved Keywords: Composite materials: Strength and reliability; Interfacial debonding: Monte-Carlo simulation; Finite element method 1. Introduction large-scale debonding and matrix cracking are major factors to reduce the strengths of both polymer-matrix Advanced composites reinforced with inorganic fibers, [5,6] and metal-matrix composites [8]. However, there such as carbon and boron, are expected for applications are only a few reports which attempt analytical approa- as structural materials requiring high reliability and dur- ches to explain the above phenomena. For example, Shih ability. Since composites are in general composed of dif- and Ebert [9] reported ect of interfacial shear ferent constituents, there exist several factors which can strength on the axial strength by incorporating the Pig influence the strength and lifetime by comparison with got model [10] into the Rosen model [11]. Their results monolithic materials. In addition, mechanical properties show that the axial strength monotonically increased of composites are often discussed from the viewpoint of with an increase in interfacial shear strength. However reliability engineering. In particular, the tensile strength, in their analysis the effect of matrix cracking is neglected one of the most fundamental mechanical properties, of The present study simulates the above phenomena to such composites has been theoretically evaluated as a clarify the effect of interfacial shear strength on the tensile statistical quantity caused by a statistical variation of strength and reliability of fibrous composites by using a fiber tensile strengths(e.g. Refs. [1-4D) Monte-Carlo simulation technique based on a finite-ele- bond properties and mechanical properties of the matrix debonding in increasing the strength and reau facial It is well-known that. on the other hand interfacial ment method and discusses the role of interfacial can also significantly influence the tensile strengths of composites [5,6]. That is, a low interfacial bond pro- motes large-scale debonding and reduces the load-car- 2. Analysis rying capacity of the broken fibers. Furthermore, a high interfacial bond tends to extend the crack transversely 2. 1. Finite element model and mesh into the matrix at fiber breaks and results in increasing stress concentrations around these breaks The same Microdamage following fiber breaks in a fiber-rein- phenomenon occurs when a matrix is brittle [7]. Such forced polymer-matrix composite is as follows [12] 0266-3538/99/S- see front matter C 1999 Elsevier Science Ltd. All rights reserved. PlI:S0266-3538(99)00046-9
The role of interfacial debonding in increasing the strength and reliability of unidirectional ®brous composites Koichi Goda Department of Mechanical Engineering, Yamaguchi University, Tokiwadai, Ube 755, Japan Received 30 December 1996; received in revised form 1 July 1998; accepted 22 February 1999 Abstract A Monte-Carlo simulation technique based on a ®nite-element method has been developed in order to clarify the eect of interfacial shear strength on the tensile strength and reliability of ®brous composites. In the simulation a boron/epoxy monolayer composite was modelled, and ®ve hundred simulations were carried out for various interfacial shear strengths. The interfacial shear strength value which raised the average strength of the composite corresponded approximately to the value which reduced the coecient of variation. This implies the existence of an optimum value of interfacial shear strength which can increase the strength and reliability. The simulated strength and reliability were closely related to the degree and type of damage around a ®ber break. That is to say, large-scale debonding caused by a weak interfacial bond and matrix cracking caused by a strong bond reduced the number of ®ber breaks accumulated up to the maximum stress, and decreased the strength and reliability. On the other hand, smallscale debonding promoted comparatively the cumulative eect of ®ber breaks and played a key role in increasing the composite strength and reliability. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Composite materials; Strength and reliability; Interfacial debonding; Monte-Carlo simulation; Finite element method 1. Introduction Advanced composites reinforced with inorganic ®bers, such as carbon and boron, are expected for applications as structural materials requiring high reliability and durability. Since composites are in general composed of different constituents, there exist several factors which can in¯uence the strength and lifetime by comparison with monolithic materials. In addition, mechanical properties of composites are often discussed from the viewpoint of reliability engineering. In particular, the tensile strength, one of the most fundamental mechanical properties, of such composites has been theoretically evaluated as a statistical quantity caused by a statistical variation of ®ber tensile strengths (e.g. Refs. [1±4]). It is well-known that, on the other hand, interfacial bond properties and mechanical properties of the matrix can also signi®cantly in¯uence the tensile strengths of composites [5,6]. That is, a low interfacial bond promotes large-scale debonding and reduces the load-carrying capacity of the broken ®bers. Furthermore, a high interfacial bond tends to extend the crack transversely into the matrix at ®ber breaks and results in increasing stress concentrations around these breaks. The same phenomenon occurs when a matrix is brittle [7]. Such large-scale debonding and matrix cracking are major factors to reduce the strengths of both polymer-matrix [5,6] and metal-matrix composites [8]. However, there are only a few reports which attempt analytical approaches to explain the above phenomena. For example, Shih and Ebert [9] reported the eect of interfacial shear strength on the axial strength by incorporating the Piggot model [10] into the Rosen model [11]. Their results show that the axial strength monotonically increased with an increase in interfacial shear strength. However, in their analysis the eect of matrix cracking is neglected. The present study simulates the above phenomena to clarify the eect of interfacial shear strength on the tensile strength and reliability of ®brous composites by using a Monte-Carlo simulation technique based on a ®nite-element method, and discusses the role of interfacial debonding in increasing the strength and reliability. 2. Analysis 2.1. Finite element model and mesh Microdamage following ®ber breaks in a ®ber-reinforced polymer-matrix composite is as follows [12]: Composites Science and Technology 59 (1999) 1871±1879 0266-3538/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(99)00046-9
1872 K Goda/Composites Science and Technology 59(1999)1871-1879 (a) If the interface is weak, a shear stress concentra- bond layer and the shear modulus, similar to the for- tion parallel to the fiber/matrix interface often mulation taken for a 2-node line element. A global causes interfacial shear debonding along the stiffness matrix is constituted from the three-element fiber -axis stiffness matrices. and therefore the whole structural (b)However, if the interface has a strong bond, a analysis can be carried out following an ordinary finite rack initiates at the fiber break and extends into element procedure. In this study a relatively brittle the matrix perpendicular to the fiber axis. material such as epoxy is used as a matrix, so that the (c) If the matrix consists of a ductile material, it yields effect of (3)was not taken into account. Thus, it is and the yield zone spreads along the broken fiber assumed that the matrix and interface elements as well as the fiber element behave as a linear elastic body, The shear-lag model [ 13] is widely used in estimating respectively, and are statically fractured when the local axial fiber stress distributions around fiber break points stress satisfies a fracture criterion. Namely, the Youngs in a composite, simulating its axial fracture process and modulus of a fiber element is changed to zero if the so on. However, the effect of (2)is not contained in the normal stress of the fiber element achieves its tensile shear-lag model. Therefore, in the present study a finite strength. The shear modulus of an interface element is element method is applied for modeling interfacial changed to zero if the shear stress of the matrix element debonding and matrix cracking. The present finite-ele- achieves the so-called interfacial shear strength. For a ment model is based on the model of a monolayer matrix element, the Von Mises criterion is applied, in composite suggested by Mandel et al. [14]. Fig. I shows which the elastic modulus of this element is changed to the model and mesh, in which a 2-node line element zero if the equivalent stress of this element achieves its representing a fiber element is incorporated into the tensile strength. In the remainder of this article, we call nodes along the y axis of a 4-node isoparametric ele- their fractures "damages, and individually we call ment based on a plane stress condition. This plane ele- them fiber break, interfacial debonding and matrix ment represents a matrix element and takes into account cracking, respectively a multi-axial stress state of tensile and shear stresses The composite model used in this study is a boron around a fiber break epoxy monolayer, and 10 fibers are placed in the finite Furthermore, a shear spring element representing an element mesh, as shown in Fig. 1. Prior to the present interfacial bond (referred to as"interface element") simulation, the effect of the division number per fiber connects the fiber and matrix elements. Deformation was preliminarily investigated in the cases of 10, 20 and resistance of the interface element is determined by the 30 elements per fiber. The calculation results of 20 and spring constant and the relative displacement of the 30 elements showed almost the same stress distributions fiber and matrix elements. The stiffness matrix of a around a broken fiber. around which the most drastic shear spring element is determined by the size of the change in stress occurs. Therefore 20 elements per fiber were selected for the actual simulation. according to this meshing, the number of nodes is 462, and the numbers ▲▲▲▲▲▲▲▲▲▲▲▲ Fiber elemet of fiber. matrix and interface elements are 200. 190 and Interface 220, respectively Matrix element 2. 2. Simulation Occurrences of fiber breaks, matrix cracking and interfacial debonding would cause complicated stress distributions throughout method for estimating reasonably what type of damage occurs in each element, should be incorporated within the simulation procedure. In order to achieve such an estimation, an Imin method [15]is employed in this study, which was originally used in searching for yielding △22 in a metal with an elasto-plastic finite element method. According to this method. a ratio of the dif- ference between strength and stress to the stress incre- ment is calculated by each element, and the element giving the minimum ratio causes one of the damages, i.e Fig. 1. Finite-element model and mesh. Reprinted with permission the fiber break, the matrix cracking and the interfacial from Trans JSME 1997; 63A: 445-452. C 1999 The Japan Society of debonding. The following is the present simulation Mechanical Engineers [16- procedure:
(a) If the interface is weak, a shear stress concentration parallel to the ®ber/matrix interface often causes interfacial shear debonding along the ®ber-axis. (b) However, if the interface has a strong bond, a crack initiates at the ®ber break and extends into the matrix perpendicular to the ®ber axis. (c) If the matrix consists of a ductile material, it yields and the yield zone spreads along the broken ®ber. The shear-lag model [13] is widely used in estimating axial ®ber stress distributions around ®ber break points in a composite, simulating its axial fracture process and so on. However, the eect of (2) is not contained in the shear-lag model. Therefore, in the present study a ®nite element method is applied for modeling interfacial debonding and matrix cracking. The present ®nite-element model is based on the model of a monolayer composite suggested by Mandel et al. [14]. Fig. 1 shows the model and mesh, in which a 2-node line element representing a ®ber element is incorporated into the nodes along the y axis of a 4-node isoparametric element based on a plane stress condition. This plane element represents a matrix element and takes into account a multi-axial stress state of tensile and shear stresses around a ®ber break. Furthermore, a shear spring element representing an interfacial bond (referred to as ``interface element'') connects the ®ber and matrix elements. Deformation resistance of the interface element is determined by the spring constant and the relative displacement of the ®ber and matrix elements. The stiness matrix of a shear spring element is determined by the size of the bond layer and the shear modulus, similar to the formulation taken for a 2-node line element. A global stiness matrix is constituted from the three-element stiness matrices, and therefore the whole structural analysis can be carried out following an ordinary ®nite element procedure. In this study a relatively brittle material such as epoxy is used as a matrix, so that the eect of (3) was not taken into account. Thus, it is assumed that the matrix and interface elements as well as the ®ber element behave as a linear elastic body, respectively, and are statically fractured when the local stress satis®es a fracture criterion. Namely, the Young's modulus of a ®ber element is changed to zero if the normal stress of the ®ber element achieves its tensile strength. The shear modulus of an interface element is changed to zero if the shear stress of the matrix element achieves the so-called interfacial shear strength. For a matrix element, the Von Mises criterion is applied, in which the elastic modulus of this element is changed to zero if the equivalent stress of this element achieves its tensile strength. In the remainder of this article, we call their fractures ``damages'', and individually we call them ®ber break, interfacial debonding and matrix cracking, respectively. The composite model used in this study is a boron/ epoxy monolayer, and 10 ®bers are placed in the ®nite element mesh, as shown in Fig. 1. Prior to the present simulation, the eect of the division number per ®ber was preliminarily investigated in the cases of 10, 20 and 30 elements per ®ber. The calculation results of 20 and 30 elements showed almost the same stress distributions around a broken ®ber, around which the most drastic change in stress occurs. Therefore 20 elements per ®ber were selected for the actual simulation. According to this meshing, the number of nodes is 462, and the numbers of ®ber, matrix and interface elements are 200, 190 and 220, respectively. 2.2. Simulation procedure Occurrences of ®ber breaks, matrix cracking and interfacial debonding would cause complicated stress distributions throughout a composite. Therefore, a method for estimating reasonably what type of damage occurs in each element, should be incorporated within the simulation procedure. In order to achieve such an estimation, an rmin method [15] is employed in this study, which was originally used in searching for yielding regions in a metal with an elasto±plastic ®nite element method. According to this method, a ratio of the difference between strength and stress to the stress increment is calculated by each element, and the element giving the minimum ratio causes one of the damages, i.e. the ®ber break, the matrix cracking and the interfacial debonding. The following is the present simulation procedure: Fig. 1. Finite-element model and mesh. Reprinted with permission from Trans JSME 1997;63A:445±452. # 1999 The Japan Society of Mechanical Engineers [16]. 1872 K. Goda / Composites Science and Technology 59 (1999) 1871±1879
K Goda/Composites Science and Technology 59(1999)1871-1879 (a)A strength of a fiber element obeys the following (c)Next, a possibility of the damage occurrence is 2-parameter Weibull distribution determined by the Imin method. The outcome will change the boundary condition, as shown in Fo) ig. 2 and as follows: (i) If I'min l, all the dis- placements calculated in this stage are modified act displacements by multiplying Imin where m and oo: the Weibull shape and scale with the displacement increment in each element parameters, L: an arbitrary fiber length, and in [see Fig. 2(a). Then the [D] matrix components this study is equivalent to the fiber element of the element giving the rmin are changed to zero length, Lo: a standard gage length at which the If this element is a fiber element or a matrix ele- Weibull parameters are estimated. By substitut- ment, then in the next stage the load Pi acting in ing a uniform random number into the inverse this element is released through its nodes along function of Eq (1), a random Weibull strength the y axis under the boundary condition of load assigned to a fiber element can be generated increment and the fixed condition at the fiber and (b) The unknown nodal displacements, Aui, are com- matrix ends [see Fig. 2(b)]. If r< l is still satisfied outed under the boundary condition of displace- the next stage is consumed to release the load hent increment AU at the fiber and matrix ends acting in the new damaged element, together with as shown in Fig. 1. The computation is carried the residual load of the present stage. This pro out incrementally, but the increment width is not cedure is repeated until Imin >1.(ii)If Imin >1 fixed. In this study an arbitrary increment large then damage does not occur. The boundary con nough to damage almost all the elements was dition of displacement increment is applied again given to the ends even at the first calculation at the fiber and matrix ends, as described in(b) stage. The stresses and the stress components (d)As the damage accumulates in a composite, the acting in the elements are calculated from the support force along the y axis decreases largely at computed displacements. Then, the ratios rare a certain strain level. It was assumed that when calculated for all the elements(see the Appendix) such behavior occurs, or when the composite stress reaches a stress level less than 80% of the AU maximum stress, the composite fracture criterion [min x△U is satisfied e)Following the above procedure, 500 simulations were carried out under different sets of random numbers. Finally the average and coefficient of variation in the simulated strengths were calculated 2.3. Material constants The present study simulates the tensile strength and reliability of a boron/epoxy monolayer composite con sisting of 10 fibers, as mentioned above. Table l shows (a)Method 1 Table i Material constants used in the present simulation. permission from Trans JSME 1997: 63A: 445-452. C 1999 The Japan Society of Mechanical Engineers [16] Young's modulus of fiber 397.9GPa Diameter of fiber 0.142mm Fiber element length Weibull shape parameter of fiber strength Weibull scale parameter of fiber strength at 6 mm Youngs modulus of matrix P Poisson,s ratio of matri Tensile strength of matrix 45.57MPa Thickness of matrix element Shear modulus of interface element 1.186GPa Thickness of interface element 0.142 Width of interface element Distance between fibers 0.259mm Fig. 2. Calculation methods for damage process simulation
(a) A strength of a ®ber element obeys the following 2-parameter Weibull distribution: F 1 ÿ exp ÿ L L0 0 m 1 where m and 0: the Weibull shape and scale parameters, L: an arbitrary ®ber length, and in this study is equivalent to the ®ber element length, L0: a standard gage length at which the Weibull parameters are estimated. By substituting a uniform random number into the inverse function of Eq. (1), a random Weibull strength assigned to a ®ber element can be generated. (b) The unknown nodal displacements, ui, are computed under the boundary condition of displacement increment U at the ®ber and matrix ends, as shown in Fig. 1. The computation is carried out incrementally, but the increment width is not ®xed. In this study an arbitrary increment large enough to damage almost all the elements was given to the ends even at the ®rst calculation stage. The stresses and the stress components acting in the elements are calculated from the computed displacements. Then, the ratios r are calculated for all the elements (see the Appendix). (c) Next, a possibility of the damage occurrence is determined by the rmin method. The outcome will change the boundary condition, as shown in Fig. 2 and as follows: (i) If rmin 1, all the displacements calculated in this stage are modi®ed to the exact displacements by multiplying rmin with the displacement increment in each element [see Fig. 2(a)]. Then the [D] matrix components of the element giving the rmin are changed to zero. If this element is a ®ber element or a matrix element, then in the next stage the load Pi acting in this element is released through its nodes along the y axis under the boundary condition of load increment and the ®xed condition at the ®ber and matrix ends [see Fig. 2(b)]. If r41 is still satis®ed, the next stage is consumed to release the load acting in the new damaged element, together with the residual load of the present stage. This procedure is repeated until rmin > 1. (ii) If rmin > 1, then damage does not occur. The boundary condition of displacement increment is applied again at the ®ber and matrix ends, as described in (b). (d) As the damage accumulates in a composite, the support force along the y axis decreases largely at a certain strain level. It was assumed that when such behavior occurs, or when the composite stress reaches a stress level less than 80% of the maximum stress, the composite fracture criterion is satis®ed. (e) Following the above procedure, 500 simulations were carried out under dierent sets of random numbers. Finally the average and coecient of variation in the simulated strengths were calculated. 2.3. Material constants The present study simulates the tensile strength and reliability of a boron/epoxy monolayer composite consisting of 10 ®bers, as mentioned above. Table 1 shows Fig. 2. Calculation methods for damage process simulation. Table 1 Material constants used in the present simulation. Reprinted with permission from Trans JSME 1997;63A:445±452. # 1999 The Japan Society of Mechanical Engineers [16] Young's modulus of ®ber 397.9 GPa Diameter of ®ber 0.142 mm Fiber element length 0.3 mm Weibull shape parameter of ®ber strength 7.16 Weibull scale parameter of ®ber strength at 6 mm 3.665 GPa Young's modulus of matrix 3.296 GPa Poisson's ratio of matrix 0.39 Tensile strength of matrix 45.57 MPa Thickness of matrix element 0.371 mm Shear modulus of interface element 1.186 GPa Thickness of interface element 0.142 mm Width of interface element 1.42 mm Distance between ®bers 0.259 mm K. Goda / Composites Science and Technology 59 (1999) 1871±1879 1873
1874 K Goda/Composites Science and Technology 59(1999)1871-1879 the material constants used in the simulation. Some of These results imply that the more cumulative fiber break the material constants, i.e. the Weibull parameters of pattern increases the composite strength. Such stress/ the fiber strength, the Youngs modulus and tensile strain behavior and strengths are closely related with the strength of the matrix, were determined in experiment degrees of matrix and interfacial damages following [16], in which boron fibers with a diameter of 142 um fiber breaks. In the next session the relation between produced by AVCO and an epoxy resin (Araldite the damage type and fiber stress distribution around a CY230/hardner HY2967) supplied by Ciba-Geigy Co. broken fiber is described were used as the test materials, a shear modulus of the interface element is supposed to be equal to that of the 2400 matrix, because the interface layer does not exist as an 日v=117MPa appreciable thickness and is used as a model to express 2300 -O t, =20. 4 MPa nterfacial debonding. It is also assumed that the width of 4 =35.0 MPa the interface element corresponds to the width of pro- 32200& jection of the fiber, i.e. the fiber diameter. For simplicity the thickness of the interface element was set to be a 1/100 of the fiber diameter. According to the material constants shown in table l. the fiber volume fraction of 态2000 the composite is relatively low, approximately 0.1 1900 However, the distance between fibers used here gives the fiber volume fraction of 0.53 if the fibers are distributed Fiber element order next to broken element in hexagonal array (a) Stress concentration on fiber elements perpendicular to fiber axis 3. Results a1500 一t,=117MPa 3. 1. Stress/ strain curve 1000v=350MPa Fig. 3 shows typical stress/strain diagrams of the simulation results. In the computation, interfacial shear strengths, t, of 11.7, 20.4 and 35.0 MPa were used 500 under the same set of random fiber strengths. The stress levels at the first fiber break are therefore all the same but the behavior following the break is completely dif- ferent. Fig. 3(a) shows a similar fracture process to that of a bundle consisting of small number of fibers. That is Distance from broken point along fiber axis mm the first fiber break indicates the maximum stress and (b)Stress recovery of broken fiber element along fiber axis is followed by the other individual fiber breaks. In Fig.3(b)the level of the second peak is higher than the Fig. 4. Fiber stress distributions around a broken fiber element the fiber axis. shows that the stress level drops rapidly after the first Reprinted with permission from Trans JSME 1997: 63A: 445-452 peak, though recovering slightly around 0.8% strain. C 1999 The Japan Society of Mechanical Engineers [16] 400 400 d300 d300 的200 0204060.81.0 002040.60.81.0 0.20.40.60.81.0 Strain Fig 3. Simulated stress/strain diagrams of a boron/epoxy composite (a)tr=11.7 MPa(b)tr=20.4 MPa(c)I,=350 MPa. Reprinted with pe mission from Trans JSME 1997: 63A: 445-452.@ 1999 The Japan Society of Mechanical Engineers [16]
the material constants used in the simulation. Some of the material constants, i.e. the Weibull parameters of the ®ber strength, the Young's modulus and tensile strength of the matrix, were determined in experiment [16], in which boron ®bers with a diameter of 142 mm produced by AVCO and an epoxy resin (Araldite CY230/hardner HY2967) supplied by Ciba-Geigy Co. were used as the test materials. A shear modulus of the interface element is supposed to be equal to that of the matrix, because the interface layer does not exist as an appreciable thickness and is used as a model to express interfacial debonding. It is also assumed that the width of the interface element corresponds to the width of projection of the ®ber, i.e. the ®ber diameter. For simplicity the thickness of the interface element was set to be a 1/100 of the ®ber diameter. According to the material constants shown in Table 1, the ®ber volume fraction of the composite is relatively low, approximately 0.1. However, the distance between ®bers used here gives the ®ber volume fraction of 0.53 if the ®bers are distributed in hexagonal array. 3. Results 3.1. Stress/strain curve Fig. 3 shows typical stress/strain diagrams of the simulation results. In the computation, interfacial shear strengths, I, of 11.7, 20.4 and 35.0 MPa were used under the same set of random ®ber strengths. The stress levels at the ®rst ®ber break are therefore all the same, but the behavior following the break is completely different. Fig. 3(a) shows a similar fracture process to that of a bundle consisting of small number of ®bers. That is, the ®rst ®ber break indicates the maximum stress and is followed by the other individual ®ber breaks. In Fig. 3(b) the level of the second peak is higher than the ®rst level and indicates the maximum stress. Fig. 3(c) shows that the stress level drops rapidly after the ®rst peak, though recovering slightly around 0.8% strain. These results imply that the more cumulative ®ber break pattern increases the composite strength. Such stress/ strain behavior and strengths are closely related with the degrees of matrix and interfacial damages following ®ber breaks. In the next session the relation between the damage type and ®ber stress distribution around a broken ®ber is described. Fig. 3. Simulated stress/strain diagrams of a boron/epoxy composite. (a) I 11:7MPa (b) I 20:4MPa (c) I 35:0 MPa. Reprinted with permission from Trans JSME 1997;63A:445±452. # 1999 The Japan Society of Mechanical Engineers [16]. Fig. 4. Fiber stress distributions around a broken ®ber element: (a) stress concentration on ®ber elements perpendicular to the ®ber axis; (b) stress recovery of broken ®ber element along the ®ber axis. Reprinted with permission from Trans JSME 1997;63A:445±452. # 1999 The Japan Society of Mechanical Engineers [16]. 1874 K. Goda / Composites Science and Technology 59 (1999) 1871±1879
K Goda/Composites Science and Technology 59(1999)1871-1879 1875 3. 2. Fiber stress distribution around a broken fiber fibers and a high load-carring capacity for broken fibers And the appropriate bond strength can possibly Fig. 4 shows the fiber stress distributions around a increase the composite strength, as shown in the pre- broken fiber element simulated for t=11.7, 20.4 and vious section. Such a qualitative relation between com- 35.0 MPa In the figure the fiber element, fifth from the posite strength, interfacial strength and damage is also eft-hand and tenth from the fiber end, was broken verified in the experiment in which the effect of inter- intentionally at the fiber stress of 1960 MPa. Fig. 5 facial bond on the tensile strength of a boron/epoxy shows the damage states of the and interface composite is investigated [5] obtained in the above simulations 4(a)the stress distributions on the fiber elements adjacent to the bro- 3.3. Efect of interfacial shear strength on strength and ken element are shown, in which the largest stress acts reliability on the nearest fiber element. The degree of the stress concentration depends largely on the interfacial shear Fig. 6 shows the effect of the interfacial shear strength strengths. That is to say, the strongest bond, i.e. on the average and coefficient of variation in simulated [I= 35OMPa, propagates matrix cracks into the sur- strengths. Closed symbols for t/= OMPa in the figure rounding matrix elements, as shown in Fig. 5(c), and indicates the results of 5000 bundle simulations. The gives the largest stress concentration. On the other results show that the average strength gradually hand, the lowest interfacial shear strength, i.e. increases with increasing interfacial shear strength, but 11.7MPa, promotes large-scale debonding, as decreases over the peak at t= 20.4MPa. The coefficient shown in Fig. 5(a), and gives the lowest stress con- of variation decreases up to t=233MPa and then centration, as shown in Fig. 4(a). Fig. 4(b) shows the increases abruptly. It is predicted from both of the stress distributions of fiber element along the broken behaviors that there is an optimum interfacial shear fiber. Since in t/= 35.0MPa the load-carrying capacity strength which raises further the strength and reliability of the broken fiber is reduced, particularly in the region between t/= 20.4 and 23.3 MPa. It is also prec where the matrix cracks occur, the stress recovery is that the damage type giving the highest strength and delayed. The weakest bond, i.e. t/= 11.7MPa, yield he lowest coefficient of variation is small-scale he poorest load-carrying capacity for the broken fiber, debonding due to large-scale debonding. The intermediate value for The shear strength of epoxy is estimated to be 26.3 bond strength, i.e. TI=20.4MPa, yields small-scale MPa according to von Mises'criterion. Fig. 6 also debonding and brings the highest load-carying capacity. shows that an optimal interfacial shear strength descri- Figs 4 and 5 imply that there is an appropriate inter- bed in the above would be slightly less than 26.3 MPa facial shear strength which can generate a state with a Incidentally, the averages and coefficients of variation relatively small stress concentration around broken result in showing almost the same values for levels of (a)t= 11.7 MPa (b)t=20.4 MPa (c)t1=35.0MPa x Fiber break Interfacial debonding Matrix cracking Fig. 5. Damages of matrix and interface around a broken fiber element: (a)t=11.7MPa(b)I= 20.4MPa(c)I=35.0MPa
3.2. Fiber stress distribution around a broken ®ber Fig. 4 shows the ®ber stress distributions around a broken ®ber element simulated for I 11:7, 20.4 and 35.0 MPa. In the ®gure the ®ber element, ®fth from the left-hand and tenth from the ®ber end, was broken intentionally at the ®ber stress of 1960 MPa. Fig. 5 shows the damage states of the matrix and interface obtained in the above simulations. In Fig. 4(a) the stress distributions on the ®ber elements adjacent to the broken element are shown, in which the largest stress acts on the nearest ®ber element. The degree of the stress concentration depends largely on the interfacial shear strengths. That is to say, the strongest bond, i.e. I 35:0MPa, propagates matrix cracks into the surrounding matrix elements, as shown in Fig. 5(c), and gives the largest stress concentration. On the other hand, the lowest interfacial shear strength, i.e. I 11:7MPa, promotes large-scale debonding, as shown in Fig. 5(a), and gives the lowest stress concentration, as shown in Fig. 4(a). Fig. 4(b) shows the stress distributions of ®ber element along the broken ®ber. Since in I 35:0MPa the load-carrying capacity of the broken ®ber is reduced, particularly in the region where the matrix cracks occur, the stress recovery is delayed. The weakest bond, i.e. I 11:7MPa, yields the poorest load-carrying capacity for the broken ®ber, due to large-scale debonding. The intermediate value for bond strength, i.e. I 20:4MPa, yields small-scale debonding and brings the highest load-carying capacity. Figs. 4 and 5 imply that there is an appropriate interfacial shear strength which can generate a state with a relatively small stress concentration around broken ®bers and a high load-carring capacity for broken ®bers. And the appropriate bond strength can possibly increase the composite strength, as shown in the previous section. Such a qualitative relation between composite strength, interfacial strength and damage is also veri®ed in the experiment in which the eect of interfacial bond on the tensile strength of a boron/epoxy composite is investigated [5]. 3.3. Eect of interfacial shear strength on strength and reliability Fig. 6 shows the eect of the interfacial shear strength on the average and coecient of variation in simulated strengths. Closed symbols for I 0MPa in the ®gure indicates the results of 5000 bundle simulations. The results show that the average strength gradually increases with increasing interfacial shear strength, but decreases over the peak at I 20:4MPa. The coecient of variation decreases up to I 23:3MPa and then increases abruptly. It is predicted from both of the behaviors that there is an optimum interfacial shear strength which raises further the strength and reliability between I 20:4 and 23.3 MPa. It is also predicted that the damage type giving the highest strength and the lowest coecient of variation is small-scale debonding. The shear strength of epoxy is estimated to be 26.3 MPa according to von Mises' criterion. Fig. 6 also shows that an optimal interfacial shear strength described in the above would be slightly less than 26.3 MPa. Incidentally, the averages and coecients of variation result in showing almost the same values for levels of Fig. 5. Damages of matrix and interface around a broken ®ber element: (a) I 11:7MPa (b) I 20:4MPa (c) I 35:0MPa. K. Goda / Composites Science and Technology 59 (1999) 1871±1879 1875