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brittle matrix composite with weak interface
Composites Science and Technology 63 (2003) 1027–1040 www.elsevier.com/locate/compscitech
Modeling of residual stress-induced stress–strain behavior of unidirectional brittle fiber/brittle matrix composite with weak interface S. Ochiaia,*, H. Tanakab, S. Kimurab, M. Tanakab, M. Hojob, K. Okudaa a
Division of Research Initiatives, International Innovation Center, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan b Graduate School of Engineering, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan Received 17 April 2002; received in revised form 9 December 2002; accepted 14 December 2002
Abstract To describe the residual stress-induced interfacial debonding and stress–strain behavior of unidirectional brittle fiber/brittle matrix composite with weak interface, a simulation method, based on the modified shear lag analysis which can take residual stresses into account and Monte Carlo procedure, was presented, and was applied to a model composite similar to polymer-derived SiC/ceramic glass system. Main results are summarized as follows. (1) It was shown concretely that the broken component-induced debonding is encouraged and discouraged when the corresponding component has tensile and compressive residual stresses, respectively. (2) The feature of the damage- (breakage of fiber, breakage of matrix and interfacial debonding)-accumulation process and its influence on the stress–strain curve could be realized in the computer; the damages are accumulated intermittently, resulting in a serrated stress–strain curve of composites. In such a damage accumulation process, the existence of residual stresses change the order and location of occurrence of damages and therefore the stress–strain curve and strength of the composite. (3) The dependence of the composite strength on the matrix strength in the presence of residual stresses was clarified; the composite strength increases significantly with increasing matrix strength when the fiber and matrix have compressive and tensile residual stresses along the fiber axis, respectively, but only marginally when they have reversed residual stresses. # 2003 Elsevier Science Ltd. All rights reserved. Keywords: A. Ceramic-matrix composites; B. Debonding; B. Mechanical properties; C. Computational simulation; C. Residual stress
1. Introduction It is known that if the interface is strong in unidirectional brittle fiber/brittle matrix composites whose components have low fracture toughness, the crack arrest-capacity is low and therefore high toughness cannot be achieved. Thus the interface in such composites should be made weak to promote interfacial debonding [1,2]. The debonding and overall fracture processes of such weakly bonded composites are strongly affected by the residual stresses retained in the composites and by the strengths of fiber and matrix [1–4]. The aim of the present work is to develop a simulation method and to * Corresponding author. Tel.: +81-75-753-4834; fax: +81-75-7534841. E-mail address: [email protected] or [email protected] (S. Ochiai).
understand the influences of the residual stresses and component-strengths on the damage accumulation process and stress–strain behavior of brittle fiber/brittle matrix composites with weak interface. In unidirectional fiber-composites, failures such as breakage of fiber and matrix, and interfacial debonding arise at many places, and they interact mechanically on each other. Such mechanical interactions determine the species and location of the next damage, one after another. Thus the damage map is changed and therefore the mechanical interaction among damages is also changed with progressing fracture. As a result of the consecutive change of the damage map-dependent interaction, overall response of composites such as stress–strain curve, strength and fracture morphology is determined. Thus, to describe the influence of residual stresses and component-strengths on overall mechanical response of
0266-3538/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0266-3538(03)00015-0
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multi-filamentary weakly bonded composites, clarification of the mechanical interactions and change of the damage map with increasing applied strain is required. However, it is extremely difficult to obtain a rigid solution for the interactions among many spatially distributed damages (so-called multi-body problem). In addition, the damage map is changed at every occurrence of new damage, and therefore the new interaction shall be calculated for the new damage map one after another. Thus, simplified and approximated methods, which make it possible to solve the change of damage map and the resultant interaction consecutively, shall be developed as a first approximation. The shear lag analysis [4–11] has been studied as one of the tools to obtain the approximate solution for the problems stated above. The ordinary shear lag analysis has, however, been developed using the approximation that only fibers carry the applied stress and the matrix acts only as a stress-transfer medium. Due to this approximation, it has been applied only to the composites containing soft matrices such as polymer- and low yield stress metal-matrix ones but not to ceramic matrix ones [5–10]. Also, due to this approximation, the residual stresses could not be taken into account. Recently, the present authors [11] have proposed a modified method to overcome these disadvantages, with which the general situation that both fiber and matrix carry applied stress and both acts as stress transfer media in the presence of residual stresses can be described. The strength of components (fiber, matrix) is not unique but scattered. Namely, the strength-value of the components is different from position to position. The spatial distribution of strength-value affects on the spatial distribution of damages (damage map). Thus, the fracture process of composites shall be analyzed by taking the influence of the spatial distribution of the strength-value on the damage process and the mechanical interaction among the damages mentioned above. In the present study, the modified shear lag analysis mentioned above [11], which can describe the mechanical interactions among damages in the presence of residual stresses, was combined with the Monte Carlo technique which can give the spatial distribution of strength of the components [6,12–14]. With such a combined method, it was attempted to simulate the influences of residual stresses and component-strengths on stress–strain behavior of unidirectional brittle fiber/ brittle matrix composites with weak interface. The present work has the following features. (1) Until now, the shear lag+Monte Carlo simulation cannot deal with the influence of residual stresses on the damage process and strength of the composite. Such a problem is overcome. (2) The species (fiber breakage, matrix breakage and interfacial debonding), location and order of occurrence of the damages can be traced in a concrete manner and therefore the change of the
damage map with increasing applied strain can be expressed visually, as will be shown in Figs. 2, 4, 6, 7 and 8. In this way, the present approach can provide concrete and visual solutions for the damage process. (3) The influence of residual stresses becomes different, depending on the matrix strength. This work is the first one to study systematically the sensitivity of the composite strength to the matrix strength under different residual stress states. The result (shown later in Section 3.2) can be used as a guide for selection of matrix material for fabrication of composites.
2. Modeling and simulation method 2.1. Modeling and equations A two-dimensional model composite with the width W, length L and thickness h, composed of continuous fiber and matrix alternatively was employed, as shown in Fig. 1. The Young’s modulus, shear modulus,
Fig. 1. Schematic representation of the two-dimensional model composite for simulation. The fiber and matrix were regarded to be composed of short elements, whose locations were numbered as shown in the upper part. The (i,j)-component-element, (i,j)-interface-element and the displacement ui,j were defined as indicated in the lower part.
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Fig. 2. Difference in progress of debonding among the cases of f=m, f <m and f >m under the existence of one precut fiber-element and one precut matrix-element. The white and colored components correspond to the matrix and fiber, respectively. The thick transverse and longitudinal lines indicate the locations of the precut component-elements and debonded interface-elements, respectively. The numbers 1,2,. . . show the order of the debonding of the interface-element.
Fig. 3. Variations of number of interface-elements debonded from the cut-ends of the fiber (Nd,f) and matrix (Nd,m) with increasing applied strain "c under the existence of one precut matrix-element and one precut fiber-element for f=m, f < m and f >m. The corresponding variations of the damage map are shown in Fig. 2.
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Fig. 4. Stress ( c)–strain ("c) curves for f=m, f <m and f >m under the existence of many precut component-elements, together with the variation of damage map due to the progress of debonding with increasing strain. The arrows in the stress–strain curves indicate the strains corresponding to the damage maps shown in the left side.
coefficient of thermal expansion, cross-sectional area and diameter (=width in the present two-dimensional model) of fiber were noted as Ef, Gf, f, Af and df, and those of matrix as Em, Gm, m, Am and dm, respectively. The residual strains of fiber ("f,r) and matrix ("m,r) along the fiber axis were, to a first approximation, calculated by applying the rule of mixtures for the axial coefficient of thermal expansion [15] under the assumption that the composite has no residual stress at a high temperature for fabrication T1 and is cooled down to T2 (room temperature);
Fig. 5. Variation of number of debonded interface-element, Nd, with increasing applied strain "c under the existence of many precut component-element. The corresponding variations of the damage map are shown in Fig. 4.
"f;r ¼ ðm f ÞEm Vm T=ðEf Vf þ Em Vm Þ
ð1Þ
"m;r ¼ ðf E ÞEf Vf T=ðEf Vf þ Em Vm Þ
ð2Þ
where T is the difference in temperature (T2T1), and Vf and Vm are volume fractions of fiber and matrix, respectively. As shown in Fig. 1, the distance x was taken from the top end of the composite. The matrix component at the left side was numbered as 1, the neighboring component as 2 (fiber), and the next one as 3 (matrix), 4 (fiber),
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Fig. 6. Stress–strain curve of the composite and damage maps at the strains indicated by the arrows in the curve for low matrix strength 0,m=300 MPa. This example monitors the influence of residual stresses on behavior of composite for the case where the matrix tends to be broken prior to fiber commonly for f=m, f < m and f >m. Nf, Nm and Ni refer to the number of broken fiber-, matrix- and interface-elements normalized with respect to the values at fracture of composite.
. . .i. . . to N (matrix) to the right side. The Young’s modulus, shear modulus, width, cross-sectional area and residual strains of the ‘‘i’’ component were given by Ei, Gi, i, Ai (=hdi) and "i,r, where i=odd and even refer to the matrix and fiber, respectively. Each component (i=1–N) with a gage length, L, was composed of k+1 short component-elements with a length x. The position at x=0 was numbered as 0 and then 1,2,3,. . .j. . .k+1 downward, in step of x. L is equal to (k+1) x. j=0 and j=k+1 correspond to x=0 and L, respectively. The ‘‘i’’ component from x=(j1)x to jx was named the (i,j)-component-element (i=1–N, j=1–k+1), and the interface from x=(j1/2) x to (j+1/2) x between ‘‘i’’ and ‘‘i+100 components the (i,j)-interfaceelement (i=1–N1, j=1–k), as shown in the lower part of Fig. 1. The displacement of the (i,j)-component-element at x=jx is noted as ui,j for i=1–N and j=0–k+1. Three following parameters were introduced for mathematical convenience in calculation; interfacial parameter, i,j, which is equal to 1 (unity) when the (i,j)interface is bonded and is 0 (zero) when debonded;
shear-friction-parameter, i,j, for the debonded interface, which is 1 and 1 when ui+1,jui,j > 0 and < 0, respectively; and component parameter, i,j, which is 1 and 0 when the (i,j)-component-element is not broken and broken, respectively. The damage map could be drawn with ease from the spatial distribution of the debonded interface-element with i,j=0 and the broken component-ones with i,j=0. The tensile stress of the (i,j)-component i,j and the shear stress at the (i,j) interface i,j can be calculated by i;j ¼ i;j Ei ui;j ui;j1 =x þ "i;r ð3Þ
i;j ¼ i;j 2Gi Giþ1 =ðdi Giþ1 þ diþ1 Gi Þ uiþ1;j ui;j þ i:j 1 i;j f
ð4Þ
where f is the frictional shear stress at the debonded interface. From the stress balance based on the stress transfer through the interfacial shear stress, the following equation has been derived [11], from which ui,jvalues (i=1–N, j=1–k) can be calculated.
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Fig. 7. Stress–strain curve of the composite and damage maps at the strains indicated by the arrows in the curve for 0,m=944 MPa, corresponding to the case where the distribution of the failure strain of matrix is the same as that of fiber under no residual stresses. This example monitors the influence of residual stresses on behavior of composite for the case where the fiber and matrix are broken comparably (f=m), the matrix tends to be broken prior to the fiber (f <m) and the fiber tends to be broken prior to the matrix (f <m).
B1ði; jÞui;j1 þ B2ði; jÞui1;j þ B3ði; jÞui;j þ B4ði; jÞui;jþ1 þ B5ði; jÞuiþ1;j ¼ B6ði; jÞ
ð5Þ
B1ði; jÞ ¼ 4 i;j = 2 þ i;j þ i;jþ1 x2
The displacements at x=0 and L were given by zero and uc (=L"c), respectively, where "c is the overall strain of the composite. Under this condition, i,j, i,j and i,j and then B1(i,j) to B6(i,j) were determined at each occurrence of new damage and the new ui,j-values were calculated from Eq. (5).
B2ði; jÞ ¼ 2i1;j Gi1 Gi =ðdi1 Gi þ di Gi1 Þ =ðAi Ei Þ
2.2. Procedure for simulation of stress–strain behavior
B3ði; jÞ ¼ 4 i;j þ i;j1 = 2 þ i;j þ i;jþ1 x2
In the present study dealing with weakly bonded composites, the maximum stress criterion was used for the breakage of fiber and matrix and interfacial debonding; namely it was assumed that the breakage of fiber and matrix elements occurs when the exerted tensile stress i,j reaches the strength Si,j and the interfacial debonding occurs when the exerted shear stress i,j exceeds the shear strength c. The following simulations were carried out in this work. Simulation (I): Debonding behavior in the presence of precut components to monitor the fundamental debonding process. Simulation (II): Stress–strain behavior of the composite to monitor the cumulative breakage of fiber and matrix and interfacial debonding.
2i;j Gi Giþ1 =ðdi Giþ1 þ di Gi Þ h=ðAi Ei Þ 2i1;j Gi1 Gi =ðdi Gi1 þ di1 Gi Þ h=ðAi Ei Þ B4ði; jÞ ¼ 4 i;jþ1 = 2 þ i;j þ i;jþ1 x2 B5ði; jÞ ¼ 2i;j Gi Giþ1 =ðdi Giþ1 þ diþ1 Gi Þ h=ðAi Ei Þ B6ði; jÞ ¼ i;j 1 i;j i1;j 1 i1;j f h=ðAi Ei Þ 4"i;r i;j þ i;jþ1 i;jþ1 i;j = ð2þ i;j þ i;jþ1 x
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Fig. 8. Stress–strain curve of the composite and damage maps at the strains indicated by the arrows in the curve for high matrix strength 0,m=1500 MPa. This example monitors the influence of residual stresses on behavior of composite for the case where the matrix tends to be broken prior to the fiber commonly for f=m, f <m and f >m.
Simulation (I) was carried out by modifying Simulation (II) on the points; (a) some component-elements were precut in advance and (b) the uncut componentelements was treated to be never broken under the externally applied stress. Simulation (II) was carried out in the following procedure. 1. The strength distributions of fiber and matrix were assumed to obey the Weibull distribution function [16]. The strength of each component Si,j was determined by generating a random value based on the Monte Carlo procedure. A fixed shear strength value ( c) was given for all interface-elements for simplicity. 2. The displacements of the fiber and matrix at the ends of the specimen (j=0 and k+1 in Fig. 1) were constricted to be the same for i=1–N. 3. Whether new damage occurs or not at a given strain, and also whether more damages occur followed by the newly formed damage at the same strain or not, were judged in the following procedure. (i) Under the corresponding damage map, exerted tensile stress i,j for all component (fiber, matrix)-elements were calculated, and the
component-element having the maximum i,j/ Si,j-value, say (m,n)-component, was identified. Also, the interface-element with the maximum i,j/ c-value, say (m0 ,n0 )-interface, was identified. (ii) If m,n/Sm,n < 1 and m0 ,n0 / c < 1, it was judged that no breakage of component and no interfacial debonding occur. (iii)If m,n/Sm,n > =1 and m0 ,n0 / c < 1, (m,n)component-element was judged to be broken. If m,n/Si,j < 1 and m0 ,n0 / c > =1, (m0 ,n0 )interface-element was judged to be debonded. If m,n/Sm,n > =1 and m0 ,n0 / c > =1, (m,n)component-element was judged to be broken when m,n/Sm,n > m0 ,n0 / c, while (m0 ,n0 )-interface-element to be debonded when m,n/ Sm,n < m0 ,n0 / c. In this way, which damage among the breakage of fiber, breakage of matrix and interfacial debonding occurs and where it occurs were identified. (iv)At a given strain, once a new damage occurs, the damage map is changed and therefore the exerted tensile stress of each component-element and shear stress of each interface-element also change from those before the
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occurrence of the new damage. Thus two cases are possible following the occurrence of the new damage at the same strain: in one case, no more damage occurs, and in another case, more damages occur following the preceding one. Then the above steps (i), (ii) and (iii) were repeated and the next damage was identified. Such a procedure was repeated until no more damage occurs at a given strain. 4. When no or no more damage occurs at a given strain, the applied strain was raised in a step of 0.01 0.025%, and the procedure (3) was repeated at each strain until overall fracture of the composite. 5. The overall fracture was said to have occurred when the broken components and debonded interfaces are connected over the entire specimen width.
2.3. Input values 2.3.1. Properties of fiber and matrix Due to the reasons mentioned below, the following values were employed in the present simulation; df=12 mm, Vf (volume fraction of fiber)=0.5, x=24 mm, Ef=180 GPa, Em=100 GPa, Gf=72 GPa, Gm=40 GPa, f=10 MPa, c=100 MPa and T=1000 K. Concerning the values of f and m, following three cases were assumed to give different residual stress-state. (i) f=m (no residual stress exists). (ii) f < m (mf=4106/K; corresponding to compressive and tensile residual stresses in fiber and matrix along the fiber axis, respectively. In practice, the values of f=1106/K and m=5106/K were used for convenience. As the residual stresses are determined only by the difference (mf) [Eqs. (1) and (2)], the simulation result is the same if we use other pair values of f and m as long as the relation of mf=4106/K is kept (f=3106/K and m=7106/K, for instance.) (iii) f > m(fm=4106/K; corresponding to tensile and compressive residual stresses in fiber and matrix along the fiber axis, respectively. In practice, the values of f=5106/K and m=1106/K were used for convenience due to the same reason stated above). The employed material properties mentioned above were set due to the following reasons by imaging the polymer-derived SiC fiber-reinforced ceramic glass matrix composite. (a) The diameter df of the polymerderived SiC fiber such as Nicalon and Tyranno has been
reported to be 11–14 mm [17,18], from which df was set to be 12 mm. (b) The Young’s modulus Ef of them has been reported to be around 180–220 GPa [17,18], from which the value of F-type Tyrrano fiber (180 GPa [18]) was employed as the Ef-value. (c) The Young’s modulus of the BaO–MgO–Al2O3–SiO2 ceramic glass is around 80 GPa [19] and that of Al2O3–SiO2–Y2O3 around 115 GPa [20], from which Em was set to be 100 GPa. (d) The coefficient of thermal expansion of the Nicalon and Tyrrano fibers has been reported to be around 3106/ K [17,18] but that of ceramic glasses is in the wide range (for instance, 0.53, 2.1, 3.0 and 6.5106/K for silica glass (7930) [21], BaO–MgO–Al2O3–SiO2 [18], MgO– Al2O3–SiO2 [18] and BaO–Al2O3–SiO2 [18] glasses, respectively. Thus, the difference in the coefficient of thermal expansion (mf) can be positive and negative, depending on the choice of the matrix material for fabrication of the SiC fiber/ceramic glass matrix composites. As the residual strains of fiber ("f, r) and matrix ("m, r) are depending on (mf), three model cases were assumed; mf=0, 4106/K and +4106/K. (e) As the difference between the fabrication (softening) and room temperatures is around 1000 K [18], T was set to be 1000 K. (f) The volume fraction of the fiber Vf was taken to be 0.5 as an example. (g) The interfacial frictional shear stress f was taken to be similar to the reported experimental value (8–12 MPa) for Nicalon/ CaO–Al2O3–SiO2.composite [22]. (h) The interfacial shear strength c was assumed to be 100 MPa. With this value, the feature of fracture behavior of weakly bonded composites could be traced as shown in the results. 2.3.2. Size of sample (k,N) For Simulation (I) to monitor the fundamental debonding process in the presence of precut components, mini-sized model composite with N=9 and k=12 was employed for the following reasons: (i) the fundamental debonding process is not so much affected by the size of sample; (ii) mini-composite is convenient for visual description of the location and order of occurrence of debonding in the damage map. For Simulation (II) to monitor the stress–strain behavior and strength of the composite, in which fractures of fiber, matrix and interface occur concurrently, the size of the model composite was made large; N=41 and k=34. 2.3.3. Strength distribution of component-elements For Simulation (II), the Weibull’s shape parameters for the strength distribution of fiber (mf) and matrix (mm) were assumed to be 5, corresponding to the coefficient of variation 0.229. The scale parameters for fiber ( 0,f) was assumed to be 1700 MPa, corresponding to the average strength 1560 357 MPa. These values are nearly the same as those inferred from the insitu observation of fracture behavior and measured tensile
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strength of the SiC(Tyranno)/BaO–MgO–Al2O3–SiO2 composite [19]. As the fracture behavior and strength of composites are dependent on strength level of matrix, the scale parameter for matrix ( 0,m) was varied from 300 to 2300 MPa (corresponding to the average strengths 276 63 MPa to 2120 485 MPa). It is noted here that the strength-value of each component-element was given by the same series of random values in all specimens. Thus the spatial distribution of the component-strength was common in all cases of f=m, f < m and f > m for a given 0,m-value. By doing so, the influence of residual stresses on fracture behavior could be evaluated.
3. Results and discussion 3.1. Influence of residual stresses on progress of debonding in the presence of precut components 3.1.1. Progress of debonding in the presence of one precut matrix-element and one precut fiber-element Fig. 2 shows the progress of debonding with increasing applied strain "c in the presence of one precut matrix-element and fiber-element each. The thick transverse lines show the precut component-elements. An example of the morphology before occurrence of debonding is found in the damage map at "c=0.14% for f=m. The thick longitudinal lines show the debonded interface-elements. The numbers 1,2. . . show the order of the debonding of interface-element. Fig. 3 shows the variation of number of debonded interface-element arising from the cut-ends of fiber (Nd,f) and matrix (Nd,m) with increasing "c. The following features were found. (a) When no residual stress exists (f=m) and fiber volume fraction is high as in the present examples (0.5), the debonding induced from the cut-ends of fiber (hereafter noted as cut fiber-induced debonding) occurs prior to the cut matrixinduced one. Also, the former progresses in a narrower strain range ("c=0.30–0.53%) than the latter ("c=0.48–0.78%). (b) When the residual stresses along the fiber axis are tensile and compressive in the matrix and fiber, respectively (f < m), the cut matrix-induced debonding is encouraged but the cut fiberinduced one is discouraged in comparison with those without residual stress (f=m). On the other hand, when the residual stresses are reverse (f > m), the cut fiber-induced debonding is encouraged but the cut matrix-induced one is discouraged. These features can be summarized as follows; cut component-induced debonding is discouraged when the corresponding component
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has compressive residual stress but is encouraged when the component has tensile residual stress. Such a feature is explained as follows. If the residual stress of the cut-component is tensile, the direction of residual stress-induced shear stress (say, + r) is same that of the applied strain-induced shear stress (say, + a). If the residual stress of the cut-component is compressive, the direction of residual stress-induced shear stress (say, r) is reverse to that of the applied strain-induced shear stress (+ a). a increases with increasing applied strain, and the interfacial debonding occurs when ‘‘ r+ a‘‘ and ‘‘ r+ a‘‘ reaches the shear strength, c, for tensile and compressive residual stresses, respectively. Thus, higher applied strain is needed for the shear stress to reach c in the case of compressive residual stress but lower strain is enough in the case of tensile one. Consequently, the interfacial debonding is discouraged and encouraged by compressive and tensile residual stresses. (c) Debonding grows to some extent and stops at a given strain, but then grows again after increment of applied strain, as known from the stepwise variation of Nd,f and Nd,m in Fig. 3. Namely debonding progresses intermittently. This feature will be discussed below in detail.
3.1.2. Progress of debonding in the presence of many precut components Fig. 4 shows the stress ( c)-strain ("c) curves for f=m, f < m and f > m in the presence of many precut components, together with the variation of damage map. The damage maps shown in the right sides correspond to the strains indicated by the arrows in the stress–strain curves in the left side. The variation of the total number of debonded interface-elements, Nd, is shown in Fig. 5. Following features are read from Figs. 4 and 5. (a) In the example without residual stresses (f=m) shown in the upper part in Fig. 4, the feature of the progress of debonding is read as follows. Debonding occurs first at "c=0.22%. Due to the progress of debonding, the stress-carrying capacity of the present mini-composite is reduced. The reduction of stress at "c=0.22% in the stress–strain curve corresponds to such an interfacial debonding-induced loss of stress carrying capacity. Beyond "c =0.22%, the composite stress increases again when applied strain is raised. In this example, until "c reaches 0.24%, no new debonding occurs and the composite stress increases monotonically. When "c reaches
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0.24%, debonding occurs at many interface-elements as shown in the damage map, resulting in loss of stress carrying capacity. After the stoppage of debonding at this strain, the composite stress increases again with increasing strain. As shown in this example, the overall debonding progresses intermittently with repetition of growth and stoppage, resulting in the serrated stress–strain curve. (b) The cut matrix-induced debonding is encouraged for f < m and also the cut fiber-induced one is encouraged for f > m. As a result, the strain at which debonding starts (0.02% for f < m and 0.06% for f > m) becomes lower than that (0.22%) for f=m. Also the number of debonded interface-elements for f < m and f > m increases faster than that for f=m with increasing strain. (c) Comparing the variation of Nd,f+Nd,m in Fig. 3 with that of Nd in Fig. 5, the debonding is encouraged in the presence of many precut components. The reason for this can be attributed to the enhanced mechanical interaction, raising the interfacial shear stress [11]. (d) The debonding progresses differently among the cases of f=m, f < m and f > m, resulting in different damage-map and therefore the stressstrain curve. As a result, the stress of composite at given strain is quite different. For instance, at "c=0.2%, the stresses of the composites for f=m, f < m and f > m are 246, 148 and 141 MPa, respectively.
3.2. Stress–strain behavior of composite in which breakage of components and interfacial debonding occur concurrently The results mentioned above showed the features of interfacial debonding affected by residual stresses in the presence of precut components. In practical composites, both breakages of components and interfacial debonding occur concurrently. In this part, the spatial distribution of strength of components was given by the Monte Carlo procedure and the influence of residual stresses on stress–strain behavior and strength of composites were studied. The influences of residual stresses on the stress–strain behavior of composites are dependent on the sequence of breakage of components since the debonding is caused from the broken ends of the components. For instance, if the fracture strain of the matrix is lower than that of fiber, matrix tends to be broken prior to fiber, and the broken matrix-induced debonding tends to occur in advance of broken fiber-induced debonding. Thus the damage map is governed by breakage of
matrix. If the fracture strain of matrix is higher than that of fiber, the reverse situation arises. In the present simulation, the value of 0,m was varied from 300 to 2300 MPa under a fixed value of 0,f=1700 MPa, as stated in Section 2.3.3. The average strengths of fiber and matrix are given by 0,f (1+1/mf) and 0,m (1+1/ mm), respectively, where is the gamma function, being 0.918 for mf=mm=5. Noting the residual stress of fiber as f,r (=(mf)EmEfVmT/(EfVf+EmVm)) and that of matrix as m,r (=(fm)EfEmVfT/(EfVf+EmVm)), the matrix tends to be broken prior to fiber during tensile loading when the value of 0,m satisfies 0;m Gð1þ1=mm Þm;r =Em < 0;f Gð1þ1=mf Þf;r =Ef ð6Þ Substituting r,f= r,m=0 for f=m, r,f=260 MPa and r,m=260 MPa for f < m, and r,f=260 MPa and r,m=260 MPa for f > m into Eq. (6), we have the condition for the matrix to be broken in advance of fiber as 0,m < 944, 1380 and 500 MPa for f=m, f < m and f > m, respectively. 3.2.1. Influence of residual stresses on stress–strain curve and damage map of composites for low, intermediate and high matrix strengths The simulated stress–strain curves of the composite and damage maps at the strains indicated in the curves for 0,m=300 (matrix tends to be broken prior to fiber commonly for f=m, f < m and f > m), 944 (fiber and matrix tend to be broken simultaneously for f=m, matrix tends to be broken prior to fiber for f < m, and fiber tends to be broken prior to matrix for f > m), and 1500 MPa (fiber tends to be broken prior to matrix commonly for f=m, f < m and f > m), are shown in Figs. 6–8, respectively. The strains ("c) at which the damage maps are shown in the right side are indicated by the arrows in the stress–strain curves in the left side. "c="cu corresponds to the strain at ultimate load. As the numbers of elements of broken fiber (Nf,e), matrix (Nm,e) and interface (Ni,E) as a function of applied strain were quite different to each other and could not be clearly shown on the same scale, they were normalized with respect to the final values Nf,f, Nm,f and Ni,f at the fracture of composite, respectively. Such normalized values of Nf=Nf,e/Nf,f, Nm=Nm,e/Nm,f and Ni=Ni,e/Ni,f are presented in Figs. 6–8. It is noted that the stress–strain curves for all samples show serration due to the breakage of the components and interfacial debonding. From the variations of stress–strain curve, damage map and Nf, Nm and Ni as a function of "c, following features are read for low ( 0,m=300 MPa), intermediate (944 MPa) and high (1500 MPa) matrix strengths.
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3.2.1.1. s0,m=300 MPa (Fig. 6). While the matrix is broken prior to fiber in all cases of f=m, f < m and f > m when 0,m=300 MPa, the strains of occurrence of matrix breakage and interfacial debonding and therefore the composite strength are different among the cases. In the case of f=m, many matrix-elements are broken but interface-elements are not debonded at "c=0.35%. In the case of f < m, at the same strain of "c=0.35%, more matrix-elements are broken due to the tensile residual stress in matrix and also many interfaceelements are debonded due to the matrix breakageinduced debonding enhanced by the residual stresses. On the contrary, in the case of f > m, no matrix component-element is broken at the same strain due to the suppression of matrix breakage by the compressive residual stress in matrix. Therefore no matrix breakageinduced debonding occurs. At "c=0.55%, in the cases of f=m and f < m, the number of debonded interface-element increase in comparison with that at "c=0.35%, while still no fiber breakage occurs. The number of broken matrix-element increases only slightly since the growth of debonding reduces the efficiency of stress transfer to the broken matrix. In the case of f > m, matrix breakage, which has been suppressed at "c=0.35% due to the compressive residual stress of matrix, occurs at "c=0.55%. However, the number of broken matrix-elements is still small in comparison with that in the cases of f=m and f < m. On the contrary, due to the enhancement of fiber breakage by the tensile residual stress in fiber, the weaker fibers are broken. The fiber-breakage-induced interfacial debonding is also enhanced by the tensile residual stress. The damage maps at ultimate load are shown in the right side in Fig. 6. The strains "cu at ultimate load (0.65, 0.8 and 0.6% in the cases of f=m, f < m and f > m, respectively) are different due to the different residual stresses. In the case of f=m and f < m, at ultimate load, almost all fiber-elements carry applied stress but most matrix-elements have lost the stress carrying capacity due to the breakage, followed by debonding. However, comparing the damage map for f=m with that for f < m in detail, the matrix is broken more and interface is debonded more in the latter. Thus the strength of composite for f=m is slightly higher than that for f < m. In the case of f > m, the matrix-breakage-induced debonding is still suppressed, but the fiber breakage and fiber-breakage-induced debonding are enhanced. However, still many fibers and matrix can carry applied stress, resulting in higher strength than the strengths in other cases.
modulus is longer than that of the matrix with low Young’s modulus, as shown in the damage maps. Therefore, the matrix-elements apart from the brokenelement in the longitudinal direction carry the applied stress, while the stress carrying capacity of broken fiber is lost for long longitudinal distance. Thus the load at maximum is carried by the surviving fiber, surviving matrix and also a part of broken matrix apart from the broken-ends in longitudinal direction. In the case of f < m, the matrix and fiber breakage’s are encouraged and discouraged, respectively. The matrix breakage-induced debonding occurs dominantly, resulting in loss of stress carrying capacity of broken matrix for long longitudinal distance. At ultimate point, the load is supported mainly by fiber and surviving matrix. In the case of f > m, the matrix and fiber breakage’s are discouraged and encouraged, respectively. The fiber breakage-induced debonding is dominant, resulting in loss of stress carrying capacity of broken fiber. Thus the surviving fiber and surviving matrix carry the stress at ultimate load. In this case, the overall stress carrying capacity is lower than that in the case of f=m due to the breakage of fiber, followed by debonding for long distance, enhanced by the tensile residual stress in fiber. Within the present work, the stress carrying capacity of the composite in the case of f=m was higher than that in the case of f < m due to the encouraged matrix breakage and matrix breakage-induced debonding in the latter case. The stress carrying capacity in the case of f=m is also higher than that in the case of f > m due to the encouraged fiber-breakage and fiber-breakageinduced debonding in the latter case. As a result, the composite strength in the case of f=m is higher than that in the cases of f > m and f < m.
3.2.1.2. s0,m=944 MPa (Fig. 7). When 0,m=944 MPa, matrix and fiber are broken simultaneously in the case of f=m. Under this situation, the debonded length from the broken-ends of fiber with high Young’s
3.2.2. Sensitivity of composite strength to matrix strength for af=am, af < am and af > am Fig. 9 shows the influence of residual stresses on composite strength. While there is a tendency that the
3.2.1.3. s0,m=1500 MPa (Fig. 8). When 0,m=1500 MPa, the fiber breakage tends to occur prior to matrix. While the fiber is broken prior to matrix in all cases of f=m, f < m and f > m, following differences are found from the comparison of the damage map at "c=0.60, 75 and 0.90% among the cases. In the cases of f < m and f > m, the fiber breakage and fiber breakage-induced debonding, which act to reduce the stress carrying capacity of composite, are discouraged and encouraged due to the compressive and tensile residual stresses in fiber, respectively, in comparison with those in the case of f=m. As a result, the stress carrying capacity and strength of the composite in the cases of f < m and f > m become higher and lower than those without residual stresses (f=m), respectively.
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both by surviving fibers and surviving matrix, the composite strength is higher than that for f < m.
Fig. 9. Influence of residual stresses on variation of composite strength c as a function of Weibull’s scales parameter 0,m of matrix strength.
composite strength increases with increasing matrix strength for all cases of f=m, f < m and f > m, the quantitative dependence of composite strength on matrix strength is different. In the cases of f=m, f < m and f > m, the composite strength varies from 650 to 1050 MPa, from 600 to 1200 MPa and from 800 to 900 MPa, respectively, with increasing matrix strength. This means that the sensitivity of composite strength to matrix strength is high, low and intermediate for f < m, f=m and f > m, respectively. The reason for this can be explained as follows. 3.2.2.1. Low s0,m region. In the case of f < m, the matrix and fiber have tensile and compressive residual stresses along the fiber axis, respectively. The tensile residual stress in the matrix encourages the breakage of matrix and thus the matrix breakage-induced debonding when 0,m is low. Thus the stress carrying capacity of matrix is almost lost in the early stage, as known from the variation of damage maps for f < m in Fig. 6. In such a case, as only surviving fibers can carry the applied stress to a first approximation, the composite strength is low. On the other hand, in the case of f > m, the matrix and fiber have compressive and tensile residual stresses along fiber axis, respectively. The compressive residual stress in the matrix discourages the fracture of matrix and thus the matrix breakage-induced debonding. Therefore the damage is not accumulated up to high strain as known from the damage maps for f > m in Fig. 6. It is emphasized that the matrix can carry applied stress even if it is broken, since debonding is suppressed. In such case, as the applied stress is carried
3.2.2.2. High s0,m region. When 0,m is high, the fiber tends to be broken prior to the matrix and therefore the matrix can carry the applied stress at ultimate load. In the case of f < m, the compressive residual stress in the fiber suppresses the breakage of fiber and also discourages the fiber breakage-induced growth of debonding. Thus, the damage is not accumulated up to high strain, as known from the variation of damage maps for f < m in Fig. 8. On the contrary, in the case of f > m, the tensile residual stress in the fiber encourages the breakage of fiber and fiber breakage-induced debonding. Thus the damage is accumulated at low strain, as known from the variation of damage maps for f > m in Fig. 8. Due to such a difference in the damage accumulation process, the composite strength for f < m is higher than that for f > m. As shown above, in all cases of f=m, f < m and f < m, matrix tends to be broken prior to fiber when 0,m is low, but fiber tends to be broken prior to matrix when 0,m is high. As the strength of weakly bonded composite is determined by the damage accumulation process up to the ultimate load, premature breakage of matrix, followed by matrix breakage-induced debonding for low 0,m, and premature fracture of fiber, followed by fiber breakage-induced debonding for high s0,m, play dominant role to reduce the composite strength. Thus, the composite strength for f < m is lower than that for f > m when 0,m is low but it becomes higher when s0,m is high. As a result, the composite strength for f < m and f > m increases significantly and marginally with increasing matrix strength. In this way, the feature that the sensitivity of composite strength to matrix strength is high when f < m but low when f > m could be accounted for. The present result indicates that (1) if the materials with high fracture strain are available as the matrix, the high strength composite can be realized, especially when the material with higher coefficient of thermal expansion (CTE) than fiber is employed, (2) as long as the materials with low fracture strain are available, the employment of the material with lower CTE than fiber leads to higher composite strength and (3) when the materials with the fracture strain comparable to fiber are available, the employment of the material with the similar CTE to fiber leads to higher composite strength. These indications are useful as the guide for selection of matrix material for fabrication of composites.
4. Conclusions The influences of residual stresses on interfacial debonding and stress–strain behavior of unidirectional
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continuous fiber composites with weak interface were studied using the proposed simulation method. Main results are summarized as follows. 1. Influences of broken component and residual stresses along the fiber axis on interfacial debonding. (a) When the residual stresses along the fiber axis are compressive and tensile for the fiber and matrix, respectively, the matrix breakageinduced debonding is encouraged but the fiber breakage-induced debonding is discouraged. (b) On the other hand, when the residual stresses are tensile and compressive for the fiber and matrix, respectively, the fiber breakageinduced debonding is encouraged but the matrix breakage-induced debonding is discouraged. (c) The features (a) and (b) can be unified; broken component-induced debonding is encouraged and discouraged when the corresponding component has tensile and compressive residual stresses, respectively. 2. Damage process (a) In both composites with and without residual stresses, the damages (breakage of fiber, breakage of matrix and interfacial debonding) arise with repetition of growth and stoppage. (b) At the strains at which damages arise, the stress-carrying capacity of composite is reduced and the composite stress drops in the stress– strain curve. Thus, the damages are commonly accumulated intermittently, resulting in a serrated stress-strain curve of composite. (c) In such a damage accumulation process, the existence of residual stresses changes the order and location of occurrence of damages and therefore the stress–strain curve and strength. 3. Influences of residual stresses and matrix strength on composite strength (a) When the matrix is weak, the strength of the composite with compressive and tensile residual stresses for the fiber and matrix, respectively, is lower than that of the composite with the reversed residual stresses. The reason for this is attributed to the more loss of the stress carrying capacity of the former composite due to the encouraged matrix breakage and matrix breakage-induced debonding. (b) When the matrix is strong, the strength of the composite with compressive and tensile resi-
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dual stresses for the fiber and matrix, respectively, is higher than that of the composite with the reversed residual stresses, due to the encouraged fiber breakage and fiber breakage-induced debonding in the latter one. (c) As a result, the strength of composite increases significantly with increasing matrix strength when the fiber and matrix have compressive and tensile residual stresses along the fiber axis, respectively, but only marginally when they have reversed residual stresses.
Acknowledgements This work was supported by the New Energy and Industrial Technology Development Organization (NEDO) of Japan.
References [1] Evans AG, Zok FW. The physics and mechanics of fibre-reinforced brittle matrix composites. J Mater Sci 1994;29:3857–96. [2] Evans AG. The mechanical properties of reinforced ceramic, metal and intermetallic matrix composites. Mater Sci Eng 1989; A107:227–41. [3] Ochiai S, Fujita T, Tanaka M, Hojo M, Tanaka R, Miyamura K, et al. Influences of distribution of fiber strength and residual stresses on tensile behavior of SiC fiber-reinforced TiAl matrix composites. J Japan Inst Metals 1999;63:1567–75. [4] Ochiai S, Tanaka M, Hojo M. Effects of residual stress and interfacial frictional shear stress on interfacial debonding at notch-tip in two-dimensional unidirectional composite. Comp Interfaces 1998;5:437–60. [5] Hedgepeth JM. Stress concentrations in filamentary structures [NASA TN D-882]. Washington; 1961. [6] Oh KP. A Monte Carlo study of the strength of unidirectional fiber-reinforced composites. J Comp Mater 1979;13:311–28. [7] Goree JG, Gross RS. Analysis of a unidirectional composite containing broken fibers and matrix damage. Eng Fract Mech 1980;13:563–78. [8] Dharani LR, Jones WF, Goree JG. Mathematical modeling of damage in unidirectional composites. Eng Fract Mech 1983;17: 555–73. [9] Narin JA. Fracture mechanics of unidirectional composites using shear-lag model I: theory. J Comp Mater 1988;22:561–88. [10] Ochiai S, Tokinori K, Osamura K, Nakatani E, Yamatsuta Y. Stress concentration at a notch-tip in unidirectional metal matrix composites. Metall Trans 1991;22A:2085–95. [11] Ochiai S, Okumura I, Tanaka M, Hojo M, Inoue T. Influences of residual stresses, frictional shear stress at debonded interface and interactions among broken components on interfacial debonding in unidirectional multifilamentary composites. Comp Interfaces 1998;5:363–81. [12] Ochiai S, Hojo M. Application of Monte Carlo simulation to mesomechanics of fiber-reinforced composite. Materia Japan 1994;33:1397–406. [13] 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 Technol 1994;50:457–68.
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[14] Ochiai S, Sawada T, Hojo M. Application of Monte Carlo simulation to tensile behavior of FRM. J Sci Eng Compos Mater 1994;56:63–76. [15] Hale DK. Review: the physical properties of composite materials. J Mater Sci 1976;11:2105–41. [16] Weibull W. Statistical distribution function of wide applicability. J Appl Mech 1951;18:293–7. [17] Ichikawa H. Present status and future trend on development and application of continuous SiC fiber. Ceramic Trans, American Ceramic Society 2002;144:153–64. [18] Tamura M, Miyamoto N, Sato M, Sakamoto O, Yamamura T. Preparation and property of Si–Ti–C–O fibre/glass-ceramics composites. In: Proceedings of 1993 Powder Metallurgy World
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Congress. Japan Society of Powder and Powder Metallurgy; 1993. p. 1350–3. Tanaka M, Okumura I, Ochiai S, Hojo M, Yamamura T, Sato M, et al. In-situ Observation and simulation of tensile behavior of unidirectional Si-Ti-C-O/BMAS composite materials. J Soc Mater Sci, Japan 2000;49(5):502–9. Yano T. Special glasses, handbook of ceramics, 2nd ed. Japan Ceramic Society; 2002. p. 360–5. Prewo KM, Brennan JJ. Silicon carbide yarn reinforced glass matrix composites. J Mater Sci 1982;17:1201–6. Karandikar PG, Chou T-W, Parvizi-Majidi A, Takeda N, Kishi T. Silicon carbide (NicalonTM) fiber-reinforced borosilicate glass composites: mechanical properties. J Mater Sci 1997;32:6459–69.
Modeling of residual stress-induced stress–strain behavior of unidirectional brittle fiber/brittle matrix composite with weak interface S. Ochiaia,*, H. Tanakab, S. Kimurab, M. Tanakab, M. Hojob, K. Okudaa a
Division of Research Initiatives, International Innovation Center, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan b Graduate School of Engineering, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan Received 17 April 2002; received in revised form 9 December 2002; accepted 14 December 2002
Abstract To describe the residual stress-induced interfacial debonding and stress–strain behavior of unidirectional brittle fiber/brittle matrix composite with weak interface, a simulation method, based on the modified shear lag analysis which can take residual stresses into account and Monte Carlo procedure, was presented, and was applied to a model composite similar to polymer-derived SiC/ceramic glass system. Main results are summarized as follows. (1) It was shown concretely that the broken component-induced debonding is encouraged and discouraged when the corresponding component has tensile and compressive residual stresses, respectively. (2) The feature of the damage- (breakage of fiber, breakage of matrix and interfacial debonding)-accumulation process and its influence on the stress–strain curve could be realized in the computer; the damages are accumulated intermittently, resulting in a serrated stress–strain curve of composites. In such a damage accumulation process, the existence of residual stresses change the order and location of occurrence of damages and therefore the stress–strain curve and strength of the composite. (3) The dependence of the composite strength on the matrix strength in the presence of residual stresses was clarified; the composite strength increases significantly with increasing matrix strength when the fiber and matrix have compressive and tensile residual stresses along the fiber axis, respectively, but only marginally when they have reversed residual stresses. # 2003 Elsevier Science Ltd. All rights reserved. Keywords: A. Ceramic-matrix composites; B. Debonding; B. Mechanical properties; C. Computational simulation; C. Residual stress
1. Introduction It is known that if the interface is strong in unidirectional brittle fiber/brittle matrix composites whose components have low fracture toughness, the crack arrest-capacity is low and therefore high toughness cannot be achieved. Thus the interface in such composites should be made weak to promote interfacial debonding [1,2]. The debonding and overall fracture processes of such weakly bonded composites are strongly affected by the residual stresses retained in the composites and by the strengths of fiber and matrix [1–4]. The aim of the present work is to develop a simulation method and to * Corresponding author. Tel.: +81-75-753-4834; fax: +81-75-7534841. E-mail address: [email protected] or [email protected] (S. Ochiai).
understand the influences of the residual stresses and component-strengths on the damage accumulation process and stress–strain behavior of brittle fiber/brittle matrix composites with weak interface. In unidirectional fiber-composites, failures such as breakage of fiber and matrix, and interfacial debonding arise at many places, and they interact mechanically on each other. Such mechanical interactions determine the species and location of the next damage, one after another. Thus the damage map is changed and therefore the mechanical interaction among damages is also changed with progressing fracture. As a result of the consecutive change of the damage map-dependent interaction, overall response of composites such as stress–strain curve, strength and fracture morphology is determined. Thus, to describe the influence of residual stresses and component-strengths on overall mechanical response of
0266-3538/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0266-3538(03)00015-0
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multi-filamentary weakly bonded composites, clarification of the mechanical interactions and change of the damage map with increasing applied strain is required. However, it is extremely difficult to obtain a rigid solution for the interactions among many spatially distributed damages (so-called multi-body problem). In addition, the damage map is changed at every occurrence of new damage, and therefore the new interaction shall be calculated for the new damage map one after another. Thus, simplified and approximated methods, which make it possible to solve the change of damage map and the resultant interaction consecutively, shall be developed as a first approximation. The shear lag analysis [4–11] has been studied as one of the tools to obtain the approximate solution for the problems stated above. The ordinary shear lag analysis has, however, been developed using the approximation that only fibers carry the applied stress and the matrix acts only as a stress-transfer medium. Due to this approximation, it has been applied only to the composites containing soft matrices such as polymer- and low yield stress metal-matrix ones but not to ceramic matrix ones [5–10]. Also, due to this approximation, the residual stresses could not be taken into account. Recently, the present authors [11] have proposed a modified method to overcome these disadvantages, with which the general situation that both fiber and matrix carry applied stress and both acts as stress transfer media in the presence of residual stresses can be described. The strength of components (fiber, matrix) is not unique but scattered. Namely, the strength-value of the components is different from position to position. The spatial distribution of strength-value affects on the spatial distribution of damages (damage map). Thus, the fracture process of composites shall be analyzed by taking the influence of the spatial distribution of the strength-value on the damage process and the mechanical interaction among the damages mentioned above. In the present study, the modified shear lag analysis mentioned above [11], which can describe the mechanical interactions among damages in the presence of residual stresses, was combined with the Monte Carlo technique which can give the spatial distribution of strength of the components [6,12–14]. With such a combined method, it was attempted to simulate the influences of residual stresses and component-strengths on stress–strain behavior of unidirectional brittle fiber/ brittle matrix composites with weak interface. The present work has the following features. (1) Until now, the shear lag+Monte Carlo simulation cannot deal with the influence of residual stresses on the damage process and strength of the composite. Such a problem is overcome. (2) The species (fiber breakage, matrix breakage and interfacial debonding), location and order of occurrence of the damages can be traced in a concrete manner and therefore the change of the
damage map with increasing applied strain can be expressed visually, as will be shown in Figs. 2, 4, 6, 7 and 8. In this way, the present approach can provide concrete and visual solutions for the damage process. (3) The influence of residual stresses becomes different, depending on the matrix strength. This work is the first one to study systematically the sensitivity of the composite strength to the matrix strength under different residual stress states. The result (shown later in Section 3.2) can be used as a guide for selection of matrix material for fabrication of composites.
2. Modeling and simulation method 2.1. Modeling and equations A two-dimensional model composite with the width W, length L and thickness h, composed of continuous fiber and matrix alternatively was employed, as shown in Fig. 1. The Young’s modulus, shear modulus,
Fig. 1. Schematic representation of the two-dimensional model composite for simulation. The fiber and matrix were regarded to be composed of short elements, whose locations were numbered as shown in the upper part. The (i,j)-component-element, (i,j)-interface-element and the displacement ui,j were defined as indicated in the lower part.
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Fig. 2. Difference in progress of debonding among the cases of f=m, f <m and f >m under the existence of one precut fiber-element and one precut matrix-element. The white and colored components correspond to the matrix and fiber, respectively. The thick transverse and longitudinal lines indicate the locations of the precut component-elements and debonded interface-elements, respectively. The numbers 1,2,. . . show the order of the debonding of the interface-element.
Fig. 3. Variations of number of interface-elements debonded from the cut-ends of the fiber (Nd,f) and matrix (Nd,m) with increasing applied strain "c under the existence of one precut matrix-element and one precut fiber-element for f=m, f < m and f >m. The corresponding variations of the damage map are shown in Fig. 2.
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Fig. 4. Stress ( c)–strain ("c) curves for f=m, f <m and f >m under the existence of many precut component-elements, together with the variation of damage map due to the progress of debonding with increasing strain. The arrows in the stress–strain curves indicate the strains corresponding to the damage maps shown in the left side.
coefficient of thermal expansion, cross-sectional area and diameter (=width in the present two-dimensional model) of fiber were noted as Ef, Gf, f, Af and df, and those of matrix as Em, Gm, m, Am and dm, respectively. The residual strains of fiber ("f,r) and matrix ("m,r) along the fiber axis were, to a first approximation, calculated by applying the rule of mixtures for the axial coefficient of thermal expansion [15] under the assumption that the composite has no residual stress at a high temperature for fabrication T1 and is cooled down to T2 (room temperature);
Fig. 5. Variation of number of debonded interface-element, Nd, with increasing applied strain "c under the existence of many precut component-element. The corresponding variations of the damage map are shown in Fig. 4.
"f;r ¼ ðm f ÞEm Vm T=ðEf Vf þ Em Vm Þ
ð1Þ
"m;r ¼ ðf E ÞEf Vf T=ðEf Vf þ Em Vm Þ
ð2Þ
where T is the difference in temperature (T2T1), and Vf and Vm are volume fractions of fiber and matrix, respectively. As shown in Fig. 1, the distance x was taken from the top end of the composite. The matrix component at the left side was numbered as 1, the neighboring component as 2 (fiber), and the next one as 3 (matrix), 4 (fiber),
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Fig. 6. Stress–strain curve of the composite and damage maps at the strains indicated by the arrows in the curve for low matrix strength 0,m=300 MPa. This example monitors the influence of residual stresses on behavior of composite for the case where the matrix tends to be broken prior to fiber commonly for f=m, f < m and f >m. Nf, Nm and Ni refer to the number of broken fiber-, matrix- and interface-elements normalized with respect to the values at fracture of composite.
. . .i. . . to N (matrix) to the right side. The Young’s modulus, shear modulus, width, cross-sectional area and residual strains of the ‘‘i’’ component were given by Ei, Gi, i, Ai (=hdi) and "i,r, where i=odd and even refer to the matrix and fiber, respectively. Each component (i=1–N) with a gage length, L, was composed of k+1 short component-elements with a length x. The position at x=0 was numbered as 0 and then 1,2,3,. . .j. . .k+1 downward, in step of x. L is equal to (k+1) x. j=0 and j=k+1 correspond to x=0 and L, respectively. The ‘‘i’’ component from x=(j1)x to jx was named the (i,j)-component-element (i=1–N, j=1–k+1), and the interface from x=(j1/2) x to (j+1/2) x between ‘‘i’’ and ‘‘i+100 components the (i,j)-interfaceelement (i=1–N1, j=1–k), as shown in the lower part of Fig. 1. The displacement of the (i,j)-component-element at x=jx is noted as ui,j for i=1–N and j=0–k+1. Three following parameters were introduced for mathematical convenience in calculation; interfacial parameter, i,j, which is equal to 1 (unity) when the (i,j)interface is bonded and is 0 (zero) when debonded;
shear-friction-parameter, i,j, for the debonded interface, which is 1 and 1 when ui+1,jui,j > 0 and < 0, respectively; and component parameter, i,j, which is 1 and 0 when the (i,j)-component-element is not broken and broken, respectively. The damage map could be drawn with ease from the spatial distribution of the debonded interface-element with i,j=0 and the broken component-ones with i,j=0. The tensile stress of the (i,j)-component i,j and the shear stress at the (i,j) interface i,j can be calculated by i;j ¼ i;j Ei ui;j ui;j1 =x þ "i;r ð3Þ
i;j ¼ i;j 2Gi Giþ1 =ðdi Giþ1 þ diþ1 Gi Þ uiþ1;j ui;j þ i:j 1 i;j f
ð4Þ
where f is the frictional shear stress at the debonded interface. From the stress balance based on the stress transfer through the interfacial shear stress, the following equation has been derived [11], from which ui,jvalues (i=1–N, j=1–k) can be calculated.
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Fig. 7. Stress–strain curve of the composite and damage maps at the strains indicated by the arrows in the curve for 0,m=944 MPa, corresponding to the case where the distribution of the failure strain of matrix is the same as that of fiber under no residual stresses. This example monitors the influence of residual stresses on behavior of composite for the case where the fiber and matrix are broken comparably (f=m), the matrix tends to be broken prior to the fiber (f <m) and the fiber tends to be broken prior to the matrix (f <m).
B1ði; jÞui;j1 þ B2ði; jÞui1;j þ B3ði; jÞui;j þ B4ði; jÞui;jþ1 þ B5ði; jÞuiþ1;j ¼ B6ði; jÞ
ð5Þ
B1ði; jÞ ¼ 4 i;j = 2 þ i;j þ i;jþ1 x2
The displacements at x=0 and L were given by zero and uc (=L"c), respectively, where "c is the overall strain of the composite. Under this condition, i,j, i,j and i,j and then B1(i,j) to B6(i,j) were determined at each occurrence of new damage and the new ui,j-values were calculated from Eq. (5).
B2ði; jÞ ¼ 2i1;j Gi1 Gi =ðdi1 Gi þ di Gi1 Þ =ðAi Ei Þ
2.2. Procedure for simulation of stress–strain behavior
B3ði; jÞ ¼ 4 i;j þ i;j1 = 2 þ i;j þ i;jþ1 x2
In the present study dealing with weakly bonded composites, the maximum stress criterion was used for the breakage of fiber and matrix and interfacial debonding; namely it was assumed that the breakage of fiber and matrix elements occurs when the exerted tensile stress i,j reaches the strength Si,j and the interfacial debonding occurs when the exerted shear stress i,j exceeds the shear strength c. The following simulations were carried out in this work. Simulation (I): Debonding behavior in the presence of precut components to monitor the fundamental debonding process. Simulation (II): Stress–strain behavior of the composite to monitor the cumulative breakage of fiber and matrix and interfacial debonding.
2i;j Gi Giþ1 =ðdi Giþ1 þ di Gi Þ h=ðAi Ei Þ 2i1;j Gi1 Gi =ðdi Gi1 þ di1 Gi Þ h=ðAi Ei Þ B4ði; jÞ ¼ 4 i;jþ1 = 2 þ i;j þ i;jþ1 x2 B5ði; jÞ ¼ 2i;j Gi Giþ1 =ðdi Giþ1 þ diþ1 Gi Þ h=ðAi Ei Þ B6ði; jÞ ¼ i;j 1 i;j i1;j 1 i1;j f h=ðAi Ei Þ 4"i;r i;j þ i;jþ1 i;jþ1 i;j = ð2þ i;j þ i;jþ1 x
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Fig. 8. Stress–strain curve of the composite and damage maps at the strains indicated by the arrows in the curve for high matrix strength 0,m=1500 MPa. This example monitors the influence of residual stresses on behavior of composite for the case where the matrix tends to be broken prior to the fiber commonly for f=m, f <m and f >m.
Simulation (I) was carried out by modifying Simulation (II) on the points; (a) some component-elements were precut in advance and (b) the uncut componentelements was treated to be never broken under the externally applied stress. Simulation (II) was carried out in the following procedure. 1. The strength distributions of fiber and matrix were assumed to obey the Weibull distribution function [16]. The strength of each component Si,j was determined by generating a random value based on the Monte Carlo procedure. A fixed shear strength value ( c) was given for all interface-elements for simplicity. 2. The displacements of the fiber and matrix at the ends of the specimen (j=0 and k+1 in Fig. 1) were constricted to be the same for i=1–N. 3. Whether new damage occurs or not at a given strain, and also whether more damages occur followed by the newly formed damage at the same strain or not, were judged in the following procedure. (i) Under the corresponding damage map, exerted tensile stress i,j for all component (fiber, matrix)-elements were calculated, and the
component-element having the maximum i,j/ Si,j-value, say (m,n)-component, was identified. Also, the interface-element with the maximum i,j/ c-value, say (m0 ,n0 )-interface, was identified. (ii) If m,n/Sm,n < 1 and m0 ,n0 / c < 1, it was judged that no breakage of component and no interfacial debonding occur. (iii)If m,n/Sm,n > =1 and m0 ,n0 / c < 1, (m,n)component-element was judged to be broken. If m,n/Si,j < 1 and m0 ,n0 / c > =1, (m0 ,n0 )interface-element was judged to be debonded. If m,n/Sm,n > =1 and m0 ,n0 / c > =1, (m,n)component-element was judged to be broken when m,n/Sm,n > m0 ,n0 / c, while (m0 ,n0 )-interface-element to be debonded when m,n/ Sm,n < m0 ,n0 / c. In this way, which damage among the breakage of fiber, breakage of matrix and interfacial debonding occurs and where it occurs were identified. (iv)At a given strain, once a new damage occurs, the damage map is changed and therefore the exerted tensile stress of each component-element and shear stress of each interface-element also change from those before the
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occurrence of the new damage. Thus two cases are possible following the occurrence of the new damage at the same strain: in one case, no more damage occurs, and in another case, more damages occur following the preceding one. Then the above steps (i), (ii) and (iii) were repeated and the next damage was identified. Such a procedure was repeated until no more damage occurs at a given strain. 4. When no or no more damage occurs at a given strain, the applied strain was raised in a step of 0.01 0.025%, and the procedure (3) was repeated at each strain until overall fracture of the composite. 5. The overall fracture was said to have occurred when the broken components and debonded interfaces are connected over the entire specimen width.
2.3. Input values 2.3.1. Properties of fiber and matrix Due to the reasons mentioned below, the following values were employed in the present simulation; df=12 mm, Vf (volume fraction of fiber)=0.5, x=24 mm, Ef=180 GPa, Em=100 GPa, Gf=72 GPa, Gm=40 GPa, f=10 MPa, c=100 MPa and T=1000 K. Concerning the values of f and m, following three cases were assumed to give different residual stress-state. (i) f=m (no residual stress exists). (ii) f < m (mf=4106/K; corresponding to compressive and tensile residual stresses in fiber and matrix along the fiber axis, respectively. In practice, the values of f=1106/K and m=5106/K were used for convenience. As the residual stresses are determined only by the difference (mf) [Eqs. (1) and (2)], the simulation result is the same if we use other pair values of f and m as long as the relation of mf=4106/K is kept (f=3106/K and m=7106/K, for instance.) (iii) f > m(fm=4106/K; corresponding to tensile and compressive residual stresses in fiber and matrix along the fiber axis, respectively. In practice, the values of f=5106/K and m=1106/K were used for convenience due to the same reason stated above). The employed material properties mentioned above were set due to the following reasons by imaging the polymer-derived SiC fiber-reinforced ceramic glass matrix composite. (a) The diameter df of the polymerderived SiC fiber such as Nicalon and Tyranno has been
reported to be 11–14 mm [17,18], from which df was set to be 12 mm. (b) The Young’s modulus Ef of them has been reported to be around 180–220 GPa [17,18], from which the value of F-type Tyrrano fiber (180 GPa [18]) was employed as the Ef-value. (c) The Young’s modulus of the BaO–MgO–Al2O3–SiO2 ceramic glass is around 80 GPa [19] and that of Al2O3–SiO2–Y2O3 around 115 GPa [20], from which Em was set to be 100 GPa. (d) The coefficient of thermal expansion of the Nicalon and Tyrrano fibers has been reported to be around 3106/ K [17,18] but that of ceramic glasses is in the wide range (for instance, 0.53, 2.1, 3.0 and 6.5106/K for silica glass (7930) [21], BaO–MgO–Al2O3–SiO2 [18], MgO– Al2O3–SiO2 [18] and BaO–Al2O3–SiO2 [18] glasses, respectively. Thus, the difference in the coefficient of thermal expansion (mf) can be positive and negative, depending on the choice of the matrix material for fabrication of the SiC fiber/ceramic glass matrix composites. As the residual strains of fiber ("f, r) and matrix ("m, r) are depending on (mf), three model cases were assumed; mf=0, 4106/K and +4106/K. (e) As the difference between the fabrication (softening) and room temperatures is around 1000 K [18], T was set to be 1000 K. (f) The volume fraction of the fiber Vf was taken to be 0.5 as an example. (g) The interfacial frictional shear stress f was taken to be similar to the reported experimental value (8–12 MPa) for Nicalon/ CaO–Al2O3–SiO2.composite [22]. (h) The interfacial shear strength c was assumed to be 100 MPa. With this value, the feature of fracture behavior of weakly bonded composites could be traced as shown in the results. 2.3.2. Size of sample (k,N) For Simulation (I) to monitor the fundamental debonding process in the presence of precut components, mini-sized model composite with N=9 and k=12 was employed for the following reasons: (i) the fundamental debonding process is not so much affected by the size of sample; (ii) mini-composite is convenient for visual description of the location and order of occurrence of debonding in the damage map. For Simulation (II) to monitor the stress–strain behavior and strength of the composite, in which fractures of fiber, matrix and interface occur concurrently, the size of the model composite was made large; N=41 and k=34. 2.3.3. Strength distribution of component-elements For Simulation (II), the Weibull’s shape parameters for the strength distribution of fiber (mf) and matrix (mm) were assumed to be 5, corresponding to the coefficient of variation 0.229. The scale parameters for fiber ( 0,f) was assumed to be 1700 MPa, corresponding to the average strength 1560 357 MPa. These values are nearly the same as those inferred from the insitu observation of fracture behavior and measured tensile
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strength of the SiC(Tyranno)/BaO–MgO–Al2O3–SiO2 composite [19]. As the fracture behavior and strength of composites are dependent on strength level of matrix, the scale parameter for matrix ( 0,m) was varied from 300 to 2300 MPa (corresponding to the average strengths 276 63 MPa to 2120 485 MPa). It is noted here that the strength-value of each component-element was given by the same series of random values in all specimens. Thus the spatial distribution of the component-strength was common in all cases of f=m, f < m and f > m for a given 0,m-value. By doing so, the influence of residual stresses on fracture behavior could be evaluated.
3. Results and discussion 3.1. Influence of residual stresses on progress of debonding in the presence of precut components 3.1.1. Progress of debonding in the presence of one precut matrix-element and one precut fiber-element Fig. 2 shows the progress of debonding with increasing applied strain "c in the presence of one precut matrix-element and fiber-element each. The thick transverse lines show the precut component-elements. An example of the morphology before occurrence of debonding is found in the damage map at "c=0.14% for f=m. The thick longitudinal lines show the debonded interface-elements. The numbers 1,2. . . show the order of the debonding of interface-element. Fig. 3 shows the variation of number of debonded interface-element arising from the cut-ends of fiber (Nd,f) and matrix (Nd,m) with increasing "c. The following features were found. (a) When no residual stress exists (f=m) and fiber volume fraction is high as in the present examples (0.5), the debonding induced from the cut-ends of fiber (hereafter noted as cut fiber-induced debonding) occurs prior to the cut matrixinduced one. Also, the former progresses in a narrower strain range ("c=0.30–0.53%) than the latter ("c=0.48–0.78%). (b) When the residual stresses along the fiber axis are tensile and compressive in the matrix and fiber, respectively (f < m), the cut matrix-induced debonding is encouraged but the cut fiberinduced one is discouraged in comparison with those without residual stress (f=m). On the other hand, when the residual stresses are reverse (f > m), the cut fiber-induced debonding is encouraged but the cut matrix-induced one is discouraged. These features can be summarized as follows; cut component-induced debonding is discouraged when the corresponding component
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has compressive residual stress but is encouraged when the component has tensile residual stress. Such a feature is explained as follows. If the residual stress of the cut-component is tensile, the direction of residual stress-induced shear stress (say, + r) is same that of the applied strain-induced shear stress (say, + a). If the residual stress of the cut-component is compressive, the direction of residual stress-induced shear stress (say, r) is reverse to that of the applied strain-induced shear stress (+ a). a increases with increasing applied strain, and the interfacial debonding occurs when ‘‘ r+ a‘‘ and ‘‘ r+ a‘‘ reaches the shear strength, c, for tensile and compressive residual stresses, respectively. Thus, higher applied strain is needed for the shear stress to reach c in the case of compressive residual stress but lower strain is enough in the case of tensile one. Consequently, the interfacial debonding is discouraged and encouraged by compressive and tensile residual stresses. (c) Debonding grows to some extent and stops at a given strain, but then grows again after increment of applied strain, as known from the stepwise variation of Nd,f and Nd,m in Fig. 3. Namely debonding progresses intermittently. This feature will be discussed below in detail.
3.1.2. Progress of debonding in the presence of many precut components Fig. 4 shows the stress ( c)-strain ("c) curves for f=m, f < m and f > m in the presence of many precut components, together with the variation of damage map. The damage maps shown in the right sides correspond to the strains indicated by the arrows in the stress–strain curves in the left side. The variation of the total number of debonded interface-elements, Nd, is shown in Fig. 5. Following features are read from Figs. 4 and 5. (a) In the example without residual stresses (f=m) shown in the upper part in Fig. 4, the feature of the progress of debonding is read as follows. Debonding occurs first at "c=0.22%. Due to the progress of debonding, the stress-carrying capacity of the present mini-composite is reduced. The reduction of stress at "c=0.22% in the stress–strain curve corresponds to such an interfacial debonding-induced loss of stress carrying capacity. Beyond "c =0.22%, the composite stress increases again when applied strain is raised. In this example, until "c reaches 0.24%, no new debonding occurs and the composite stress increases monotonically. When "c reaches
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0.24%, debonding occurs at many interface-elements as shown in the damage map, resulting in loss of stress carrying capacity. After the stoppage of debonding at this strain, the composite stress increases again with increasing strain. As shown in this example, the overall debonding progresses intermittently with repetition of growth and stoppage, resulting in the serrated stress–strain curve. (b) The cut matrix-induced debonding is encouraged for f < m and also the cut fiber-induced one is encouraged for f > m. As a result, the strain at which debonding starts (0.02% for f < m and 0.06% for f > m) becomes lower than that (0.22%) for f=m. Also the number of debonded interface-elements for f < m and f > m increases faster than that for f=m with increasing strain. (c) Comparing the variation of Nd,f+Nd,m in Fig. 3 with that of Nd in Fig. 5, the debonding is encouraged in the presence of many precut components. The reason for this can be attributed to the enhanced mechanical interaction, raising the interfacial shear stress [11]. (d) The debonding progresses differently among the cases of f=m, f < m and f > m, resulting in different damage-map and therefore the stressstrain curve. As a result, the stress of composite at given strain is quite different. For instance, at "c=0.2%, the stresses of the composites for f=m, f < m and f > m are 246, 148 and 141 MPa, respectively.
3.2. Stress–strain behavior of composite in which breakage of components and interfacial debonding occur concurrently The results mentioned above showed the features of interfacial debonding affected by residual stresses in the presence of precut components. In practical composites, both breakages of components and interfacial debonding occur concurrently. In this part, the spatial distribution of strength of components was given by the Monte Carlo procedure and the influence of residual stresses on stress–strain behavior and strength of composites were studied. The influences of residual stresses on the stress–strain behavior of composites are dependent on the sequence of breakage of components since the debonding is caused from the broken ends of the components. For instance, if the fracture strain of the matrix is lower than that of fiber, matrix tends to be broken prior to fiber, and the broken matrix-induced debonding tends to occur in advance of broken fiber-induced debonding. Thus the damage map is governed by breakage of
matrix. If the fracture strain of matrix is higher than that of fiber, the reverse situation arises. In the present simulation, the value of 0,m was varied from 300 to 2300 MPa under a fixed value of 0,f=1700 MPa, as stated in Section 2.3.3. The average strengths of fiber and matrix are given by 0,f (1+1/mf) and 0,m (1+1/ mm), respectively, where is the gamma function, being 0.918 for mf=mm=5. Noting the residual stress of fiber as f,r (=(mf)EmEfVmT/(EfVf+EmVm)) and that of matrix as m,r (=(fm)EfEmVfT/(EfVf+EmVm)), the matrix tends to be broken prior to fiber during tensile loading when the value of 0,m satisfies 0;m Gð1þ1=mm Þm;r =Em < 0;f Gð1þ1=mf Þf;r =Ef ð6Þ Substituting r,f= r,m=0 for f=m, r,f=260 MPa and r,m=260 MPa for f < m, and r,f=260 MPa and r,m=260 MPa for f > m into Eq. (6), we have the condition for the matrix to be broken in advance of fiber as 0,m < 944, 1380 and 500 MPa for f=m, f < m and f > m, respectively. 3.2.1. Influence of residual stresses on stress–strain curve and damage map of composites for low, intermediate and high matrix strengths The simulated stress–strain curves of the composite and damage maps at the strains indicated in the curves for 0,m=300 (matrix tends to be broken prior to fiber commonly for f=m, f < m and f > m), 944 (fiber and matrix tend to be broken simultaneously for f=m, matrix tends to be broken prior to fiber for f < m, and fiber tends to be broken prior to matrix for f > m), and 1500 MPa (fiber tends to be broken prior to matrix commonly for f=m, f < m and f > m), are shown in Figs. 6–8, respectively. The strains ("c) at which the damage maps are shown in the right side are indicated by the arrows in the stress–strain curves in the left side. "c="cu corresponds to the strain at ultimate load. As the numbers of elements of broken fiber (Nf,e), matrix (Nm,e) and interface (Ni,E) as a function of applied strain were quite different to each other and could not be clearly shown on the same scale, they were normalized with respect to the final values Nf,f, Nm,f and Ni,f at the fracture of composite, respectively. Such normalized values of Nf=Nf,e/Nf,f, Nm=Nm,e/Nm,f and Ni=Ni,e/Ni,f are presented in Figs. 6–8. It is noted that the stress–strain curves for all samples show serration due to the breakage of the components and interfacial debonding. From the variations of stress–strain curve, damage map and Nf, Nm and Ni as a function of "c, following features are read for low ( 0,m=300 MPa), intermediate (944 MPa) and high (1500 MPa) matrix strengths.
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3.2.1.1. s0,m=300 MPa (Fig. 6). While the matrix is broken prior to fiber in all cases of f=m, f < m and f > m when 0,m=300 MPa, the strains of occurrence of matrix breakage and interfacial debonding and therefore the composite strength are different among the cases. In the case of f=m, many matrix-elements are broken but interface-elements are not debonded at "c=0.35%. In the case of f < m, at the same strain of "c=0.35%, more matrix-elements are broken due to the tensile residual stress in matrix and also many interfaceelements are debonded due to the matrix breakageinduced debonding enhanced by the residual stresses. On the contrary, in the case of f > m, no matrix component-element is broken at the same strain due to the suppression of matrix breakage by the compressive residual stress in matrix. Therefore no matrix breakageinduced debonding occurs. At "c=0.55%, in the cases of f=m and f < m, the number of debonded interface-element increase in comparison with that at "c=0.35%, while still no fiber breakage occurs. The number of broken matrix-element increases only slightly since the growth of debonding reduces the efficiency of stress transfer to the broken matrix. In the case of f > m, matrix breakage, which has been suppressed at "c=0.35% due to the compressive residual stress of matrix, occurs at "c=0.55%. However, the number of broken matrix-elements is still small in comparison with that in the cases of f=m and f < m. On the contrary, due to the enhancement of fiber breakage by the tensile residual stress in fiber, the weaker fibers are broken. The fiber-breakage-induced interfacial debonding is also enhanced by the tensile residual stress. The damage maps at ultimate load are shown in the right side in Fig. 6. The strains "cu at ultimate load (0.65, 0.8 and 0.6% in the cases of f=m, f < m and f > m, respectively) are different due to the different residual stresses. In the case of f=m and f < m, at ultimate load, almost all fiber-elements carry applied stress but most matrix-elements have lost the stress carrying capacity due to the breakage, followed by debonding. However, comparing the damage map for f=m with that for f < m in detail, the matrix is broken more and interface is debonded more in the latter. Thus the strength of composite for f=m is slightly higher than that for f < m. In the case of f > m, the matrix-breakage-induced debonding is still suppressed, but the fiber breakage and fiber-breakage-induced debonding are enhanced. However, still many fibers and matrix can carry applied stress, resulting in higher strength than the strengths in other cases.
modulus is longer than that of the matrix with low Young’s modulus, as shown in the damage maps. Therefore, the matrix-elements apart from the brokenelement in the longitudinal direction carry the applied stress, while the stress carrying capacity of broken fiber is lost for long longitudinal distance. Thus the load at maximum is carried by the surviving fiber, surviving matrix and also a part of broken matrix apart from the broken-ends in longitudinal direction. In the case of f < m, the matrix and fiber breakage’s are encouraged and discouraged, respectively. The matrix breakage-induced debonding occurs dominantly, resulting in loss of stress carrying capacity of broken matrix for long longitudinal distance. At ultimate point, the load is supported mainly by fiber and surviving matrix. In the case of f > m, the matrix and fiber breakage’s are discouraged and encouraged, respectively. The fiber breakage-induced debonding is dominant, resulting in loss of stress carrying capacity of broken fiber. Thus the surviving fiber and surviving matrix carry the stress at ultimate load. In this case, the overall stress carrying capacity is lower than that in the case of f=m due to the breakage of fiber, followed by debonding for long distance, enhanced by the tensile residual stress in fiber. Within the present work, the stress carrying capacity of the composite in the case of f=m was higher than that in the case of f < m due to the encouraged matrix breakage and matrix breakage-induced debonding in the latter case. The stress carrying capacity in the case of f=m is also higher than that in the case of f > m due to the encouraged fiber-breakage and fiber-breakageinduced debonding in the latter case. As a result, the composite strength in the case of f=m is higher than that in the cases of f > m and f < m.
3.2.1.2. s0,m=944 MPa (Fig. 7). When 0,m=944 MPa, matrix and fiber are broken simultaneously in the case of f=m. Under this situation, the debonded length from the broken-ends of fiber with high Young’s
3.2.2. Sensitivity of composite strength to matrix strength for af=am, af < am and af > am Fig. 9 shows the influence of residual stresses on composite strength. While there is a tendency that the
3.2.1.3. s0,m=1500 MPa (Fig. 8). When 0,m=1500 MPa, the fiber breakage tends to occur prior to matrix. While the fiber is broken prior to matrix in all cases of f=m, f < m and f > m, following differences are found from the comparison of the damage map at "c=0.60, 75 and 0.90% among the cases. In the cases of f < m and f > m, the fiber breakage and fiber breakage-induced debonding, which act to reduce the stress carrying capacity of composite, are discouraged and encouraged due to the compressive and tensile residual stresses in fiber, respectively, in comparison with those in the case of f=m. As a result, the stress carrying capacity and strength of the composite in the cases of f < m and f > m become higher and lower than those without residual stresses (f=m), respectively.
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both by surviving fibers and surviving matrix, the composite strength is higher than that for f < m.
Fig. 9. Influence of residual stresses on variation of composite strength c as a function of Weibull’s scales parameter 0,m of matrix strength.
composite strength increases with increasing matrix strength for all cases of f=m, f < m and f > m, the quantitative dependence of composite strength on matrix strength is different. In the cases of f=m, f < m and f > m, the composite strength varies from 650 to 1050 MPa, from 600 to 1200 MPa and from 800 to 900 MPa, respectively, with increasing matrix strength. This means that the sensitivity of composite strength to matrix strength is high, low and intermediate for f < m, f=m and f > m, respectively. The reason for this can be explained as follows. 3.2.2.1. Low s0,m region. In the case of f < m, the matrix and fiber have tensile and compressive residual stresses along the fiber axis, respectively. The tensile residual stress in the matrix encourages the breakage of matrix and thus the matrix breakage-induced debonding when 0,m is low. Thus the stress carrying capacity of matrix is almost lost in the early stage, as known from the variation of damage maps for f < m in Fig. 6. In such a case, as only surviving fibers can carry the applied stress to a first approximation, the composite strength is low. On the other hand, in the case of f > m, the matrix and fiber have compressive and tensile residual stresses along fiber axis, respectively. The compressive residual stress in the matrix discourages the fracture of matrix and thus the matrix breakage-induced debonding. Therefore the damage is not accumulated up to high strain as known from the damage maps for f > m in Fig. 6. It is emphasized that the matrix can carry applied stress even if it is broken, since debonding is suppressed. In such case, as the applied stress is carried
3.2.2.2. High s0,m region. When 0,m is high, the fiber tends to be broken prior to the matrix and therefore the matrix can carry the applied stress at ultimate load. In the case of f < m, the compressive residual stress in the fiber suppresses the breakage of fiber and also discourages the fiber breakage-induced growth of debonding. Thus, the damage is not accumulated up to high strain, as known from the variation of damage maps for f < m in Fig. 8. On the contrary, in the case of f > m, the tensile residual stress in the fiber encourages the breakage of fiber and fiber breakage-induced debonding. Thus the damage is accumulated at low strain, as known from the variation of damage maps for f > m in Fig. 8. Due to such a difference in the damage accumulation process, the composite strength for f < m is higher than that for f > m. As shown above, in all cases of f=m, f < m and f < m, matrix tends to be broken prior to fiber when 0,m is low, but fiber tends to be broken prior to matrix when 0,m is high. As the strength of weakly bonded composite is determined by the damage accumulation process up to the ultimate load, premature breakage of matrix, followed by matrix breakage-induced debonding for low 0,m, and premature fracture of fiber, followed by fiber breakage-induced debonding for high s0,m, play dominant role to reduce the composite strength. Thus, the composite strength for f < m is lower than that for f > m when 0,m is low but it becomes higher when s0,m is high. As a result, the composite strength for f < m and f > m increases significantly and marginally with increasing matrix strength. In this way, the feature that the sensitivity of composite strength to matrix strength is high when f < m but low when f > m could be accounted for. The present result indicates that (1) if the materials with high fracture strain are available as the matrix, the high strength composite can be realized, especially when the material with higher coefficient of thermal expansion (CTE) than fiber is employed, (2) as long as the materials with low fracture strain are available, the employment of the material with lower CTE than fiber leads to higher composite strength and (3) when the materials with the fracture strain comparable to fiber are available, the employment of the material with the similar CTE to fiber leads to higher composite strength. These indications are useful as the guide for selection of matrix material for fabrication of composites.
4. Conclusions The influences of residual stresses on interfacial debonding and stress–strain behavior of unidirectional
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continuous fiber composites with weak interface were studied using the proposed simulation method. Main results are summarized as follows. 1. Influences of broken component and residual stresses along the fiber axis on interfacial debonding. (a) When the residual stresses along the fiber axis are compressive and tensile for the fiber and matrix, respectively, the matrix breakageinduced debonding is encouraged but the fiber breakage-induced debonding is discouraged. (b) On the other hand, when the residual stresses are tensile and compressive for the fiber and matrix, respectively, the fiber breakageinduced debonding is encouraged but the matrix breakage-induced debonding is discouraged. (c) The features (a) and (b) can be unified; broken component-induced debonding is encouraged and discouraged when the corresponding component has tensile and compressive residual stresses, respectively. 2. Damage process (a) In both composites with and without residual stresses, the damages (breakage of fiber, breakage of matrix and interfacial debonding) arise with repetition of growth and stoppage. (b) At the strains at which damages arise, the stress-carrying capacity of composite is reduced and the composite stress drops in the stress– strain curve. Thus, the damages are commonly accumulated intermittently, resulting in a serrated stress-strain curve of composite. (c) In such a damage accumulation process, the existence of residual stresses changes the order and location of occurrence of damages and therefore the stress–strain curve and strength. 3. Influences of residual stresses and matrix strength on composite strength (a) When the matrix is weak, the strength of the composite with compressive and tensile residual stresses for the fiber and matrix, respectively, is lower than that of the composite with the reversed residual stresses. The reason for this is attributed to the more loss of the stress carrying capacity of the former composite due to the encouraged matrix breakage and matrix breakage-induced debonding. (b) When the matrix is strong, the strength of the composite with compressive and tensile resi-
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dual stresses for the fiber and matrix, respectively, is higher than that of the composite with the reversed residual stresses, due to the encouraged fiber breakage and fiber breakage-induced debonding in the latter one. (c) As a result, the strength of composite increases significantly with increasing matrix strength when the fiber and matrix have compressive and tensile residual stresses along the fiber axis, respectively, but only marginally when they have reversed residual stresses.
Acknowledgements This work was supported by the New Energy and Industrial Technology Development Organization (NEDO) of Japan.
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