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 r1.(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
1876 K Goda/Composites Science and Technology 59(1999)1871-1879 300 12 0.7 a290○ 11 09 0.6/-A2-break ○3- break or more 0.5 280 △105 04 270 0.3 260 8 -O- Average strength 0.2 △ Coefficient of variation 250 0.1 01020304050 Interfacial shear strength MPa 0.0 01020304050 Fig. 6. Effect of interfacial shear strength on the strength and reliability of boron/epoxy composites. Reprinted with permission Interfacial shear strength MPa from Trans JSME 1997: 63A: 445-452. c 1999 The Japan Society of Fig. 7. Ratio of the number of j-break occurrence to the total num ber [/=297MPa and more. In these levels many of fiber mode, shows the minimum value at t/=175MPa, and breaks are followed by matrix cracking, not interfacial increases up to around 0.5 with an increase in interfacial debonding, because ti is beyond the shear strength of the shear strength. On the other hand, the ratio of 3-break matrix. Consequently the effect of interfacial debonding or more cumulative failure mode behaves to be con- trary to the case of l-break. Namely, this ratio gives the peak at tI=17.5MPa, decreases to around 0. 15, and then shows almost a constant. In the case of 2-break 4. Discussion intermediate failure mode, the ratio shows approxi- ightly higher level 4.1. Concerns of interfacial shear debonding with respect at t/=11.7MPa to 23.3 MPa. Such a correspondence to to strength and reliability Fig. 6 tells us that composites with large-scale debond ing and matrix cracking yield a smaller i-break than a It was shown that the small and large interfacial shear composite with small-scale debonding. If failure of the strengths cause large-scale debonding and matrix weakest fiber in a composite immediately leads to com- cracking, respectively, and they both reduced the com- posite fracture without accumulation of fiber breaks, a posite's average strength On the other hand, the inter- variability in composite strength would reflect that in mediate values of interfacial shear strengths resulted fiber strength. Because the scatter of the weakest fiber in increasing the average strength. Why do the inter- strength is the same as that of the whole fiber strength. mediate values raising the average strength also produce Thus the scatter is large, corresponding to an upper the low coefficients of variation? The author considers bound. A less cumulative failure mode will approach to that the strength and reliability of fibrous composites is this degree of scatter. Therefore both small and large closely related with a damage quantity accumulated up interfacial shear strengths induce significant scatter in to the maximum stress, the number of broken fiber ele- composite strength On the other hand, more cumula ments. For example, in Fig. 3(a(c) these numbers are tive fiber breaks further increases the stress level, so M Fig. 7 shows the ratio of the number of i-break width toward the upper side. The shrinkage might be 2 and 1, respectively(referred to as 'i-break) that the survival fiber distribution would shrink in occurrence to the total number of simulations versus the interfacial shear strength. In the present simulation most of composites indicated 1-, 2-or 3-break, but four a composite consists of N fibers and the fiber st or more breaks was not a major figure (in any lows a 2-parameter Weibull distribution, the cumulative distribution function for the weakest fiber strength. i.e. for the first order statistic approximately I to 3% The transition of the ratio is given as follows: A()=1-( -Fo)]=1-exp(-N(a is comparable to the pattern shown in Fig. 6. That is This form is also a 2-parameter Weibull distribution, and this shape to say, the ratio of l-break, non-cumulative failure parameter, m, agrees with that of its population F(a)
I 29:7MPa and more. In these levels many of ®ber breaks are followed by matrix cracking, not interfacial debonding, because I is beyond the shear strength of the matrix. Consequently the eect of interfacial debonding disappears. 4. Discussion 4.1. Concerns of interfacial shear debonding with respect to strength and reliability It was shown that the small and large interfacial shear strengths cause large-scale debonding and matrix cracking, respectively, and they both reduced the composite's average strength. On the other hand, the intermediate values of interfacial shear strengths resulted in increasing the average strength. Why do the intermediate values raising the average strength also produce the low coecients of variation? The author considers that the strength and reliability of ®brous composites is closely related with a damage quantity accumulated up to the maximum stress, the number of broken ®ber elements. For example, in Fig. 3(a)±(c) these numbers are 1, 2 and 1, respectively (referred to as `i-break'). Fig. 7 shows the ratio of the number of i-break occurrence to the total number of simulations versus the interfacial shear strength. In the present simulation most of composites indicated 1-, 2- or 3-break, but four or more breaks was not a major ®gure (in any case approximately 1 to 3%.). The transition of the ratio is comparable to the pattern shown in Fig. 6. That is to say, the ratio of 1-break, non-cumulative failure mode, shows the minimum value at I 17:5MPa, and increases up to around 0.5 with an increase in interfacial shear strength. On the other hand, the ratio of 3-break or more, cumulative failure mode, behaves to be contrary to the case of 1-break. Namely, this ratio gives the peak at I 17:5MPa, decreases to around 0.15, and then shows almost a constant. In the case of 2-break, intermediate failure mode, the ratio shows approximately a constant, though it keeps a slightly higher level at I 11:7MPa to 23.3 MPa. Such a correspondence to Fig. 6 tells us that composites with large-scale debonding and matrix cracking yield a smaller i-break than a composite with small-scale debonding. If failure of the weakest ®ber in a composite immediately leads to composite fracture without accumulation of ®ber breaks, a variability in composite strength would re¯ect that in ®ber strength. Because the scatter of the weakest ®ber strength is the same as that of the whole ®ber strength.1 Thus the scatter is large, corresponding to an upper bound. A less cumulative failure mode will approach to this degree of scatter. Therefore both small and large interfacial shear strengths induce signi®cant scatter in composite strength. On the other hand, more cumulative ®ber breaks further increases the stress level, so that the survival ®ber distribution would shrink in width toward the upper side. The shrinkage might be Fig. 6. Eect of interfacial shear strength on the strength and reliability of boron/epoxy composites. Reprinted with permission from Trans JSME 1997;63A:445±452. # 1999 The Japan Society of Mechanical Engineers [16]. Fig. 7. Ratio of the number of i-break occurrence to the total number of simulations vs. interfacial shear strength. 1 When a composite consists of N ®bers and the ®ber strength follows a 2-parameter Weibull distribution, the cumulative distribution function for the weakest ®ber strength, i.e. for the ®rst order statistic, is given as follows: F 1 1 ÿ f g 1 ÿ F N 1 ÿ exp ÿN 0 n o m . This form is also a 2-parameter Weibull distribution, and this shape parameter, m, agrees with that of its population F . 1876 K. Goda / Composites Science and Technology 59 (1999) 1871±1879
K Goda/Composites Science and Technology 59(1999)1871-1879 concerned with decreasing a scatter in composite strength. Such a situation accompanied with the more ErA cumulative fiber breaks is generated by small-scale debondings,and therefore contribute to improving the Ea, d+1-2u; -li-1)=0 strength and reliability of composites. Changes in the strength and reliability can also be explained using a stochastic model, in which the failure EyA/ (uN-lN-1)=0 probability of a fibrous composite can be estimated As described in the previous chapter, the small inter- facial shear strength results in a poor load-bearing where ui(i=2, 3. N-1): displacement of fiber capacity. a decrease in the capacity corresponds to an for x axis, E: elastic modulus of the fiber,A:cross- increase in the ineffective length [1-4], i.e. the stress sectional area of the fiber, Gm: shear modulus of the region unrecovered from its broken point. It is reported matrix, h: thickness of the monolayer, d: distance that a decrease in ineffective length of broken fibers between the fibers. The finite difference approximation brings an increase in scatter in composite strength, as for the differential terms of Eq (2)enables the problem well as a decrease in strength, because the number of to solve it as linear equations. By giving appropriate conceptual bundle elements consisting of ineffective boundary conditions to some of the finite difference length is reduced [17]. Therefore, large-scale debonding terms, one can simulate the composite fracture process which leads to an increase in ineffective length, reduces In this study a fiber element corresponds to the part the strength and reliability of composites. Incidentally, between two axial finite difference nodes, and a matrix an increase in the local stress concentrations around element does to the part between two transverse nodes broken fibers also reduces the strength and reliability The simulation procedure of this model follows the Pitt and Phoenix [18] developed a stochastic model previously proposed technique [19]. In order to compare under a tapered load sharing rule in a planar composite, with the present finite element simulation, an interfacial in which the four nearest fibers around broken fibers condition without frictional stress was assumed for the support the lost loads. They compared the stochastic finite difference shear-lag simulation. Fig. 8 shows the behavior with the conventional model with a local load results of five hundreds of the shear-lag simulation. The sharing rule, in which only the nearest two fibers support average strength shows almost a constant at the rela the lost loads. Accordingly in this rule stress concentra- tively low interfacial shear strengths and begins to ion acts in the fibers more locally and intensively. The increase monotonically around ti= 30MPa with further results showed that the local load sharing rule reduced increase in the interfacial shear strength. Also, the coef- composite strength and reliability, i.e. decreased the ficient of variation begins to decrease around the same strength and increased the scatter in strength. It is pre- value, but never increases. In the shear-lag simulati dicted from the fact that matrix cracking following broken fibers, which leads to a further increase in stress 300 2 concentration, also reduces the strength and reliability Average strength of composites. On the other hand, small-scale debonding produces a short ineffective length and a relaxed stress 290 11 concentration such as a tapered load sharing rule, and consequently leads to a high strength and reliability 280 00-0OO 105 4. 2. Comparison with shear -lag model a shear-lag model is often used in estimating stress 70 distributions around broken fibers in a composite and composite strength properties, as described earlier. In this section the effect of interfacial shear strength on the 260 8 tensile strength of a boron/epoxy composite monolayer is determined with aid of a monte- Carlo simulation -A- Coefficient of variation based on a shear-lag model, and compared with the 250 finite element simulation results. This model assumes 01020304050 that only the fibers in the composite sustain tensile Interfacial shear strength MPa load, and the matrix deforms only in shear and trans- Fig. 8. Effect of interfacial shear strength on the strength and relia- mits tensile load to the fibers. The force equilibrium bility of boron/epoxy composites by shear-lag model simulation quations for a composite monolayer with N fibers are Reprinted with on from Trans JSME 1997: 63A: 445-452 given as simultaneous differential equations as follows: o 1999 The Japan Society of Mechanical Engineers [161
concerned with decreasing a scatter in composite strength. Such a situation accompanied with the more cumulative ®ber breaks is generated by small-scale debondings, and therefore contribute to improving the strength and reliability of composites. Changes in the strength and reliability can also be explained using a stochastic model, in which the failure probability of a ®brous composite can be estimated. As described in the previous chapter, the small interfacial shear strength results in a poor load-bearing capacity. A decrease in the capacity corresponds to an increase in the ineective length [1±4], i.e. the stress region unrecovered from its broken point. It is reported that a decrease in ineective length of broken ®bers brings an increase in scatter in composite strength, as well as a decrease in strength, because the number of conceptual bundle elements consisting of ineective length is reduced [17]. Therefore, large-scale debonding which leads to an increase in ineective length, reduces the strength and reliability of composites. Incidentally, an increase in the local stress concentrations around broken ®bers also reduces the strength and reliability. Pitt and Phoenix [18] developed a stochastic model under a tapered load sharing rule in a planar composite, in which the four nearest ®bers around broken ®bers support the lost loads. They compared the stochastic behavior with the conventional model with a local load sharing rule, in which only the nearest two ®bers support the lost loads. Accordingly in this rule stress concentration acts in the ®bers more locally and intensively. The results showed that the local load sharing rule reduced composite strength and reliability, i.e. decreased the strength and increased the scatter in strength. It is predicted from the fact that matrix cracking following broken ®bers, which leads to a further increase in stress concentration, also reduces the strength and reliability of composites. On the other hand, small-scale debonding produces a short ineective length and a relaxed stress concentration such as a tapered load sharing rule, and consequently leads to a high strength and reliability. 4.2. Comparison with shear-lag model A shear-lag model is often used in estimating stress distributions around broken ®bers in a composite and composite strength properties, as described earlier. In this section the eect of interfacial shear strength on the tensile strength of a boron/epoxy composite monolayer is determined with aid of a Monte-Carlo simulation based on a shear-lag model, and compared with the ®nite element simulation results. This model assumes that only the ®bers in the composite sustain tensile load, and the matrix deforms only in shear and transmits tensile load to the ®bers. The force equilibrium equations for a composite monolayer with N ®bers are given as simultaneous dierential equations as follows: EfAf d2u1 dx2 Gmh d u2 ÿ u1 0 EfAf d2ui dx2 Gmh d ui1 ÿ 2ui ÿ uiÿ1 0 EfAf d2uN dx2 ÿ Gmh d uN ÿ uNÿ1 0 2 where ui i 2; 3; ...... ; N ÿ 1: displacement of ®ber for x axis, Ef: elastic modulus of the ®ber, Af: crosssectional area of the ®ber, Gm: shear modulus of the matrix, h: thickness of the monolayer, d: distance between the ®bers. The ®nite dierence approximation for the dierential terms of Eq. (2) enables the problem to solve it as linear equations. By giving appropriate boundary conditions to some of the ®nite dierence terms, one can simulate the composite fracture process. In this study a ®ber element corresponds to the part between two axial ®nite dierence nodes, and a matrix element does to the part between two transverse nodes. The simulation procedure of this model follows the previously proposed technique [19]. In order to compare with the present ®nite element simulation, an interfacial condition without frictional stress was assumed for the ®nite dierence shear-lag simulation. Fig. 8 shows the results of ®ve hundreds of the shear-lag simulation. The average strength shows almost a constant at the relatively low interfacial shear strengths and begins to increase monotonically around I 30MPa with further increase in the interfacial shear strength. Also, the coef- ®cient of variation begins to decrease around the same value, but never increases. In the shear-lag simulation Fig. 8. Eect of interfacial shear strength on the strength and reliability of boron/epoxy composites by shear-lag model simulation. Reprinted with permission from Trans JSME 1997;63A:445±452. # 1999 The Japan Society of Mechanical Engineers [16]. K. Goda / Composites Science and Technology 59 (1999) 1871±1879 1877
1878 K Goda/Composites Science and Technology 59(1999)1871-1879 also small-scale debondings govern the damage type in a Appendix composite with an increase in the interfacial shear strength. In this point the behavior is the same as the An Imin method was developed in the field of elaste present finite-element simulation. However, since a plastic finite element analysis by Yamada et al. 12 shear-lag model restricts its matrix deformation only in which has been used in searching for yielding regions in shear, the effect of stress concentrations caused by the a metal. In this study this method is used for what type matrix cracking cannot be simulated. This explains why of element takes precedence for the damage and where the strength never decreases again. Therefore it is con- the damage occurs. When in the n-stage computation an cluded that the shear-lag model is unsuitable forevaluation arbitrary displacement increment is applied at the fiber of composites in the case that matrix cracking is a and matrix ends as a boundary condition, the stresses of dominant failure mechanism the fiber. matrix and interface element are as follows (af)n=(o)n-1+(△a/)n (A1) For the purpose of uncovering the effect of interfacial shear strength on the tensile strength and reliability of Matrix fibrous composites a Monte-Carlo simulation technique based on a finite-element method has been develope (ax)n=(0x)-1+(△ax)n,(ay)n=(0y)-1+(△ay)n In the simulation a boron/epoxy monolayer composite (Txr)n=(txv)mn-I+(Atxy) with 10 fibers was modelled. and 500 simulations were (A2) arried out using various interfacial shear strengths. The main results of this work were as follow (a)The interfacial shear strength value which (tin=(tim-1+(Ati), (A3) increased the average strength of the composites corresponded to the value which decreased their Note that the fiber and interface elements are one dimen coefficient of variation. This implied an existence sional element. In order to equalize these element stresses of an optimum value of interfacial shear strength with the fiber and matrix strengths, the following equa which can increase the strength and reliability. tions should be satisfied This value was estimated to be slightly less than the matrix shear strength (an-1+r(△a)n=0F,()n-1+r(△t)n=T (A4) b) The simulated strength and reliability was closely related with the degree of damage and its where oF is the assigned Weibull fiber strength, and t/ is following a fiber break. That is to say, large-scale the interface shear strength. r is the ratio equalizing the debonding and matrix cracking reduced the num- left side with the right side. In other words, r is an index ber of fiber breaks accumulated up to the max- which judges the degree of damage. On the other hand imum achieved stress, and decreased the strength r of the matrix element is determined from the equiva bility. On the other hand, small-scale lent stress as follows: debonding promoted comparatively the cumula tive effect of fiber breaks and play a key role in ()n1+r(△om}2-{obn-1+r(△an} increasing composite strength and reliability (c) Since it was unable to simulate matrix cracking, a {(a)n-1+n(△a)n}+{(ay)n-1+r△am(A5) Monte-Carlo simulation based on a shear-lag {(xy)n-1+(△txy)}=a model is unsuitable for evaluation for composites with a rigid interfacial shear bond where om is the matrix strength. r is calculated from Eqs.(A4)and(A5)by each element as follows: Acknowledgements Fiber Part of this work was carried out at the Department r=oF-(an-I (A6) of Theoretical and Applied Mechanics of Cornell Uni- versity, when the author visited there in the summer of Matrix 1996. The author is very grateful to Professor S. Leigh Phoenix and Ms Irene J. Beyerlein of the Department -b+vb2-4ac (A7) who gave him helpful comments for this work
also small-scale debondings govern the damage type in a composite with an increase in the interfacial shear strength. In this point the behavior is the same as the present ®nite-element simulation. However, since a shear-lag model restricts its matrix deformation only in shear, the eect of stress concentrations caused by the matrix cracking cannot be simulated. This explains why the strength never decreases again. Therefore it is concluded that the shear-lag model is unsuitable for evaluation of composites in the case that matrix cracking is a dominant failure mechanism. 5. Conclusion For the purpose of uncovering the eect of interfacial shear strength on the tensile strength and reliability of ®brous composites a Monte-Carlo simulation technique based on a ®nite-element method has been developed. In the simulation a boron/epoxy monolayer composite with 10 ®bers was modelled, and 500 simulations were carried out using various interfacial shear strengths. The main results of this work were as follows: (a) The interfacial shear strength value which increased the average strength of the composites corresponded to the value which decreased their coecient of variation. This implied an existence of an optimum value of interfacial shear strength which can increase the strength and reliability. This value was estimated to be slightly less than the matrix shear strength. (b) The simulated strength and reliability was closely related with the degree of damage and its type following a ®ber break. That is to say, large-scale debonding and matrix cracking reduced the number of ®ber breaks accumulated up to the maximum achieved stress, and decreased the strength and reliability. On the other hand, small-scale debonding promoted comparatively the cumulative eect of ®ber breaks and play a key role in increasing composite strength and reliability. (c) Since it was unable to simulate matrix cracking, a Monte-Carlo simulation based on a shear-lag model is unsuitable for evaluation for composites with a rigid interfacial shear bond. Acknowledgements Part of this work was carried out at the Department of Theoretical and Applied Mechanics of Cornell University, when the author visited there in the summer of 1996. The author is very grateful to Professor S. Leigh Phoenix and Ms. Irene J. Beyerlein of the Department who gave him helpful comments for this work. Appendix An rmin method was developed in the ®eld of elastoplastic ®nite element analysis by Yamada et al. [12] which has been used in searching for yielding regions in a metal. In this study this method is used for what type of element takes precedence for the damage and where the damage occurs. When in the n-stage computation an arbitrary displacement increment is applied at the ®ber and matrix ends as a boundary condition, the stresses of the ®ber, matrix and interface element are as follows: Fiber: fn fnÿ1 fn A1 Matrix: xn xnÿ1 xn; yn ynÿ1 yn; xyn xynÿ1 xyn A2 Interface: in inÿ1 in A3 Note that the ®ber and interface elements are one dimensional element. In order to equalize these element stresses with the ®ber and matrix strengths, the following equations should be satis®ed: fnÿ1 r fn F; inÿ1 r in I A4 where F is the assigned Weibull ®ber strength, and I is the interface shear strength. r is the ratio equalizing the left side with the right side. In other words, r is an index which judges the degree of damage. On the other hand, r of the matrix element is determined from the equivalent stress as follows: xnÿ1 r xn 2 ÿ xnÿ1 r xn ynÿ1 r yn ynÿ1 r yn 2 ÿ 3 xynÿ1 r xyn 2 m A5 where m is the matrix strength. r is calculated from Eqs. (A4) and (A5) by each element as follows: Fiber: r F ÿ fnÿ1 fn A6 Matrix: r ÿb b2 ÿ 4ac p 2a A7 1878 K. Goda / Composites Science and Technology 59 (1999) 1871±1879
K Goda/Composites Science and Technology 59(1999)1871-1879 where 2 Harlow DO model for the strength of fibrous materials I: Analysis and con- a={(△ox)n}+{(△o)}-(△ax)(△yhn+3(△rxyn tures. Il: A numerical study of convergence. J Comp Mater 1978:12:195-214,31434. b=2{(a)n-1(△xn+(o)h-(△a)n} 3 Batdorf SB. Tensile strength of unidirectionally reinforced com- {(ax)n-1(△a)n+(yh1(△am posites I. J Reinf Plas Comp 1982: 1: 153-62. 4 Subramanian S, Reifsnider KL, Stinchcomb ww. Tensile strength c={o)=1}2+{(o)12-()-1(a)h1+3(xh-1}2 of unidirectional composites: the role of efficiency and strength of fiber-matrix interface. J Comp Tech Res 1995: 17: 289-300 5 Gatti A, Mullin JV, Berry JM. The role of bond strength in the fracture of advanced filament reinforced composites. ASTM STP 1969:460:573-82. Interface [6 Mullin JV. Influence of fiber property variation on composite failure mechanisms. ASTM STP 1973: 521: 349-66 (A8) Reliability evaluation on mechanical characteristics of CFRP Proc 6th Int Conf Mechanical Behavior of Materials 1991: 1: 677- Only the element giving the minimum ratio of all the [8] Ochiai S, Osamura K. Influences of matrix ductility, interfacial elements suffers the damage. i.e. the fiber break. the bonding strength, and fiber volume fraction on tensile strength matrix cracking or the interfacial debonding. Finally the unidirectional metal matrix composite. Metall Trans A exact stresses are given as follows: 1990;21A:971-7 9 Shih GC, Ebert [J. Theoretical modelling of the effect of the nterfacial shear strength on the longitudinal tensile strength of Fiber unidirectional composites. J Comp Mater 1987: 21: 207-24 [10 Piggott ad bearing fiber composites. Pergamon Press, (o)n=(o)n-1+rmin(△n 80.p.83-99 [1 Rosen Bw. Tensile failure of fibrous composites. AIAA J 1964:2:1985-9 Matrix. [2] Hull D. An introduction to composite materials. Cambridge (ax)n=(axn-1+rmin(△ax)n,(on=(ay)n-1+rmin [3 Hedgepeth JM. Stress concentations in filamentary structures. (△a)n,(x)n=(τxy)h-1=rmin(△rxy) NASA Tec Note 1961: D-882: 1-30 [14 Mandel JA, Pack SC, Tarazi S. Micromechanical studies of (A10) crack growth in fiber reinforced materials. Eng Fract Mech 1982;16:741-54 Interface [5 Yamada Y, Yoshimura N, Sakurai T. Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by (th)n=(tb)mn-I+Imin(Atb) the finite element method. Int Mech Sci 1968: 10: 343-54. [16 Goda K. Role of interfacial shear debonding involving the By multiplying Imin with its increment, the exact strain ength and reliability of fiber-reinforced composites. Trans Jpn Soc Mech Eng 1997: 63A: 445-52(in Japanese). and displacement are also calculated. This procedure is [I7] Goda K, Fukunaga H. Consideration of the reliability of tensile also applicable in the same way at the load release pro- trength at elevated temperature of unidirectional metal matrix cess, following a fiber element break. omposites. Comp Sci Tech 1989: 35: 181-93 [18 Pitt RE, Phoenix SL. Probability distribution for the strength of omposite materials IV: localized load-sharing with tapering. Int References J Fract1983;22:243-76 [19 Goda K, Phoenix SL. Reliability approach to the tensile strength [1 Zweben C. Tensile failure of fiber composites. AIAA J of unidirectional CFRP composites by Monte-Carlo simulation n a shear-lag model. Comp Sci Tech 1994: 50: 457-68
where a xn 2 yn 2 ÿ xn yn 3 xyn 2 b 2 xnÿ1 xn ynÿ1 yn ÿ xnÿ1 yn ynÿ1 xn 6 xynÿ1 yn c xnÿ1 2 ynÿ1 2 ÿ xnÿ1 ynÿ1 3 xynÿ1 2 ÿ 2 m Interface: r I ÿ inÿ1 in A8 Only the element giving the minimum ratio of all the elements suers the damage, i.e. the ®ber break, the matrix cracking or the interfacial debonding. Finally the exact stresses are given as follows: Fiber: fn fnÿ1 rmin fn A9 Matrix: xn xnÿ1 rmin xn; yn ynÿ1 rmin yn; xyn xynÿ1 rmin xyn A10 Interface: bn bnÿ1 rmin bn A11 By multiplying rmin with its increment, the exact strain and displacement are also calculated. This procedure is also applicable in the same way at the load release process, following a ®ber element break. References [1] Zweben C. Tensile failure of ®ber composites. AIAA J 1968;6:2325±31. [2] Harlow DG, Phoenix SL. The chain-of-bundles probability model for the strength of ®brous materials I: Analysis and conjectures. II: A numerical study of convergence. J Comp Mater 1978;12:195±214, 314±34. [3] Batdorf SB. Tensile strength of unidirectionally reinforced composites I. J Reinf Plas Comp 1982;1:153±62. [4] Subramanian S, Reifsnider KL, Stinchcomb WW. Tensile strength of unidirectional composites: the role of eciency and strength of ®ber-matrix interface. J Comp Tech Res 1995;17:289±300. [5] Gatti A, Mullin JV, Berry JM. The role of bond strength in the fracture of advanced ®lament reinforced composites. ASTM STP 1969;460:573±82. [6] Mullin JV. In¯uence of ®ber property variation on composite failure mechanisms. ASTM STP 1973;521:349±66. [7] Maekawa Z, Hamada H, Yokoyama A, Lee K, Ishibashi S. Reliability evaluation on mechanical characteristics of CFRP. Proc 6th Int Conf Mechanical Behavior of Materials 1991;1:677± 82. [8] Ochiai S, Osamura K. In¯uences of matrix ductility, interfacial bonding strength, and ®ber volume fraction on tensile strength of unidirectional metal matrix composite. Metall Trans A 1990;21A:971±7. [9] Shih GC, Ebert LJ. Theoretical modelling of the eect of the interfacial shear strength on the longitudinal tensile strength of unidirectional composites. J Comp Mater 1987;21:207±24. [10] Piggott MR. Load bearing ®ber composites. Pergamon Press, 1980. p. 83±99. [11] Rosen BW. Tensile failure of ®brous composites. AIAA J 1964;2:1985±91. [12] Hull D. An introduction to composite materials. Cambridge, 1981. [13] Hedgepeth JM. Stress concentations in ®lamentary structures. NASA Tec Note 1961;D-882:1±30. [14] Mandel JA, Pack SC, Tarazi S. Micromechanical studies of crack growth in ®ber reinforced materials. Eng Fract Mech 1982;16:741±54. [15] Yamada Y, Yoshimura N, Sakurai T. Plastic stress±strain matrix and its application for the solution of elastic±plastic problems by the ®nite element method. Int Mech Sci 1968;10:343±54. [16] Goda K. Role of interfacial shear debonding involving the strength and reliability of ®ber-reinforced composites. Trans Jpn Soc Mech Eng 1997;63A:445±52 (in Japanese). [17] Goda K, Fukunaga H. Consideration of the reliability of tensile strength at elevated temperature of unidirectional metal matrix composites. Comp Sci Tech 1989;35:181±93. [18] Pitt RE, Phoenix SL. Probability distribution for the strength of composite materials IV: localized load-sharing with tapering. Int J Fract 1983;22:243±76. [19] Goda K, Phoenix SL. Reliability approach to the tensile strength of unidirectional CFRP composites by Monte-Carlo simulation in a shear-lag model. Comp Sci Tech 1994;50:457±68. K. Goda / Composites Science and Technology 59 (1999) 1871±1879 1879