MATERIALS HIENGE& ENGIEERING ELSEVIER Materials Science and Engineering A 489(2008)120-126 www.elsevier.com/locate/msea Dynamic compressive mechanical properties and a new constitutive model of 2D-C/Sic composites Liu Mingshuang, Li Yulong,", Xu Fei, Xu Zejian, Cheng Laifei a School of Aeronautics, Northwestem Polytechnical University, Xi an, 710072, China b School of Materials Science and Engineering. Northwestern Polytechnical University, Xi'an,710072, China Received 15 June 2007; received in revised form 2 December 2007; accepted 3 January 2008 Abstract The layer-directional compressive properties of 2D-C/SiC composites were investigated at strain rates ranging from 10- to 2. x 103s-I.The quasi-static experiments were performed using the electronic universal testing machine, and the dynamic experiments were conducted by the split Hopkinson pressure bar system. The results show that the dynamic compressive stress-strain curves are non-linear. The failure strength and the elasticity modulus vary linearly to the logarithm of the strain rate, and the failure strain reduces with an ing strain rate. Scatter of dynamic compressive failure strength obeys Weibull distribution and the Weibull parameter m is 5.27. The damage angle of dynamic compression is larger than that of static loading. Observed on SEM, the ruptured surface is smooth at high loading rate and more cracked fibers appear in the specimen than at lower strain rate. Based on the experimental results, a new constitutive model is proposed in this paper. C 2008 Elsevier B. V. All rights reserved. Keywords: Ceramic matrix composites( CMCs); Weibull distribution; Dynamical constitutive model; Damage angle 1. Introduction the non-linear mechanical behavior of 2D-C/SiC composites Two damage modes were emphasized that transverse microc- Carbon-fiber-reinforced Sic-matrix( C/SiC)composites fab- racks were characterized by a deterministic accumulation and ricated by the chemical vapor infiltration process(CVi)have a random development of longitudinal micro cracking, 1.e been developed and are widely used in high technology struc- fiber/matrix and bundle/matrix debonding. Baste [3]summa- tural applications, especially in aerospace field, in which the rized a methodology for the formulation and identification of ratios of stiffness/weight and strength/weight, and high temper- the constitutive laws of ceramic-matrix composites. It relied ature prop re of great int on an anisotropic damage evaluation Sarva and Nemat-Nasser Many investigations have been conducted on 2D-C/SiC com- [4] investigated the effect of the strain rate on the compres- posites. Camus et al. [1] investigated the mechanical responses sive strength of Sic under uniaxial loading. They found a of 2D-C/SiC composites subjected to uniaxial tensile and com- marked increase in the compressive strength at strain rates pressive loadings. An extended non-linear stress-strain response greater than 10-s. Futakawa et al. [5] used a tensile split and a multi-stage development of damage involving trans- Hopkinson pressure bar to investigate the Sicr/Sicm compos- verse matrix microcracking, bundle/matrix and inter-bundle ites From the results, the interface friction stress was evaluated debonding as well as thermal residual stress release were evi- by the fiber pullout length, which was measured through micro- denced In compression, after an initial stage involving closure scopic observations of fractured specimens. And it was larg of the thermal microcracks generated from processing, the in dynamic loading than that in static loading. Weeks and opposites displayed a linear-elastic behavior until fai Sun [6] investigated the rate-dependent behavior of AS4/PEEK Bouazzaoui et al. [2] used an ultrasonic method to investi (APC-2)thermoplastic composites over a wide strain rate (10-3to 102s-)and introduced two Corresponding author at: P.O. Box 118, Northwestern Polytechnical Uni- Although a wealth of literature is reported on static mechan- versity, Xi'an 710072, PR China. Tel +862988494859: fax:+862988494859. ical properties, the dynamic compressive behavior of C/Sic E-mailaddress:liyulong@nwpu.edu.cn(L.Yulong) composites has not yet been reported to our knowledge. In this 0921-5093 ter 2008 Elsevier B v. All rights reserved
Materials Science and Engineering A 489 (2008) 120–126 Dynamic compressive mechanical properties and a new constitutive model of 2D-C/SiC composites Liu Mingshuang a, Li Yulong a,∗, Xu Fei a, Xu Zejian a, Cheng Laifei b a School of Aeronautics, Northwestern Polytechnical University, Xi’an, 710072, China b School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an, 710072, China Received 15 June 2007; received in revised form 2 December 2007; accepted 3 January 2008 Abstract The layer-directional compressive properties of 2D-C/SiC composites were investigated at strain rates ranging from 10−4 to 2.8 × 103 s−1. The quasi-static experiments were performed using the electronic universal testing machine, and the dynamic experiments were conducted by the split Hopkinson pressure bar system. The results show that the dynamic compressive stress–strain curves are non-linear. The failure strength and the elasticity modulus vary linearly to the logarithm of the strain rate, and the failure strain reduces with an increasing strain rate. Scatter of dynamic compressive failure strength obeys Weibull distribution and the Weibull parameter m is 5.27. The damage angle of dynamic compression is larger than that of static loading. Observed on SEM, the ruptured surface is smooth at high loading rate and more cracked fibers appear in the specimens than at lower strain rate. Based on the experimental results, a new constitutive model is proposed in this paper. © 2008 Elsevier B.V. All rights reserved. Keywords: Ceramic matrix composites (CMCs); Weibull distribution; Dynamical constitutive model; Damage angle 1. Introduction Carbon-fiber-reinforced SiC-matrix (C/SiC) composites fabricated by the chemical vapor infiltration process (CVI) have been developed and are widely used in high technology structural applications, especially in aerospace field, in which the ratios of stiffness/weight and strength/weight, and high temperature properties are of great interest. Many investigations have been conducted on 2D-C/SiC composites. Camus et al. [1] investigated the mechanical responses of 2D-C/SiC composites subjected to uniaxial tensile and compressive loadings. An extended non-linear stress–strain response and a multi-stage development of damage involving transverse matrix microcracking, bundle/matrix and inter-bundle debonding as well as thermal residual stress release were evidenced. In compression, after an initial stage involving closure of the thermal microcracks generated from processing, the composites displayed a linear-elastic behavior until failure. Bouazzaoui et al. [2] used an ultrasonic method to investigate ∗ Corresponding author at: P.O. Box 118, Northwestern Polytechnical University, Xi’an 710072, PR China. Tel.: +86 2988494859; fax: +86 2988494859. E-mail address: liyulong@nwpu.edu.cn (L. Yulong). the non-linear mechanical behavior of 2D-C/SiC composites. Two damage modes were emphasized that transverse microcracks were characterized by a deterministic accumulation and a random development of longitudinal micro cracking, i.e. fiber/matrix and bundle/matrix debonding. Baste [3] summarized a methodology for the formulation and identification of the constitutive laws of ceramic–matrix composites. It relied on an anisotropic damage evaluation. Sarva and Nemat-Nasser [4] investigated the effect of the strain rate on the compressive strength of SiC under uniaxial loading. They found a marked increase in the compressive strength at strain rates greater than 102 s−1. Futakawa et al. [5] used a tensile split Hopkinson pressure bar to investigate the SiCf/SiCm composites. From the results, the interface friction stress was evaluated by the fiber pullout length, which was measured through microscopic observations of fractured specimens. And it was larger in dynamic loading than that in static loading. Weeks and Sun [6] investigated the rate-dependent behavior of AS4/PEEK (APC-2) thermoplastic composites over a wide strain rate range (10−5 to 102 s−1) and introduced two rate-dependent models. Although a wealth of literature is reported on static mechanical properties, the dynamic compressive behavior of C/SiC composites has not yet been reported to our knowledge. In this 0921-5093/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2008.01.014
L Mingshuang et al. Materials Science and Engineering A 489(2008)120-126 121 Socket Projectile Laser elocity meter Endergonic bar nput bar _口■■ High dynamic ve collection syste Fig. 1. The schematic illustration of the split Hopkinson pressure bar systems. paper, the layer-directional dynamic compressive behavior of a 2D-C/SiC composites was investigated, and a new constitutive equation including the rate-dependent and damage-softening effects was proposed, which agreed very well with the exper- As imental results 2. Material and experimental techniques es 2. Material where ER and et are the transmitted and reflected strain pulses, respectively; Co = VE/po denotes longitudinal elastic wave The 2D-C/SiC composite materials are supplied by the State velocity in the Hopkinson bars; E Young's modulus of the Hop- Key Laboratory of Solidification Processing in Northwestern kinson bars, po the density of the Hopkinson bars, Is and As Polytechnical University, People's Republic of China. PAN- the length and cross-sectional area of the specimen, A the cross- based carbon fiber clothes were laid up at T300-1K. And then sectional area of the Hopkinson bars, respectively the chemical vapor infiltration process(CVn) was employed Fracture surfaces of the broken specimens were observed to deposit a thin pyrolytic carbon layer and Sic-matrix. The sing a scanning electron microscope(SEM) dimension of the 2D-C/Sic composite specimens is about Φ5mm×4.3mm. 3. Experimental results and discussion 3.. Stress-strain curves 2.2. Experimental techniques The 2D-C/SiC composites were compressed under uniaxial The quasi-static compressive strengths of 2D-C/SiC com- compressive loading at different strain rates. The compressive posites were determined by using a universal test machine. strength increases from 360 MPa at the strain rate of 10-4s" Static tests were performed at strain rates of 10-4 and up to 430 MPa at the strain rate of 2800s-, which is 19.4% 10-2s-. The stiffness of the test machine was calibrated higher compared with quasi-static result. All the compressive and then taken into account in the displacement measure- stress-strain curves are non-linear. Fig. 2 shows the variation ment of failure stress with the logarithm of strain rate. It is clear that The dynamic compressive experiments were performed by the compressive failure strength of the 2D-C/SiC composites the split Hopkinson pressure bar(SHPB)apparatus(as shown increases with an increasing strain rate. As previously observed Fig. 1). Different strain rates were obtained by changing the in the tests of carbon/epoxy matrix composites [7]and C/C com- striker's length and the gas pressure. Based on the theory of posites[8], the elastic modulus for the 2D-C/SiC composites also one-dimensional elastic wave propagation, the average strain increases with loading rate in the present study. But the intrinsic ES, strain rate Es and stress os in the specimen can be evaluated mechanism is not clear
L. Mingshuang et al. / Materials Science and Engineering A 489 (2008) 120–126 121 Fig. 1. The schematic illustration of the split Hopkinson pressure bar systems. paper, the layer-directional dynamic compressive behavior of 2D-C/SiC composites was investigated, and a new constitutive equation including the rate-dependent and damage-softening effects was proposed, which agreed very well with the experimental results. 2. Material and experimental techniques 2.1. Material The 2D-C/SiC composite materials are supplied by the State Key Laboratory of Solidification Processing in Northwestern Polytechnical University, People’s Republic of China. PANbased carbon fiber clothes were laid up at T300-1 K. And then the chemical vapor infiltration process (CVI) was employed to deposit a thin pyrolytic carbon layer and SiC-matrix. The dimension of the 2D-C/SiC composite specimens is about 5 mm × 4.3 mm. 2.2. Experimental techniques The quasi-static compressive strengths of 2D-C/SiC composites were determined by using a universal test machine. Static tests were performed at strain rates of 10−4 and 10−2 s−1. The stiffness of the test machine was calibrated and then taken into account in the displacement measurement. The dynamic compressive experiments were performed by the split Hopkinson pressure bar (SHPB) apparatus (as shown in Fig. 1). Different strain rates were obtained by changing the striker’s length and the gas pressure. Based on the theory of one-dimensional elastic wave propagation, the average strain εS, strain rate ε˙S and stress σS in the specimen can be evaluated as: ⎧ ⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩ σS = E A AS εT εS = −2C0 lS t 0 εR dτ ε˙S = −2C0 lS εR (1) where εR and εT are the transmitted and reflected strain pulses, respectively; C0 = √E/ρ0 denotes longitudinal elastic wave velocity in the Hopkinson bars; E Young’s modulus of the Hopkinson bars, ρ0 the density of the Hopkinson bars, lS and AS the length and cross-sectional area of the specimen, A the crosssectional area of the Hopkinson bars, respectively. Fracture surfaces of the broken specimens were observed using a scanning electron microscope (SEM). 3. Experimental results and discussion 3.1. Stress–strain curves The 2D-C/SiC composites were compressed under uniaxial compressive loading at different strain rates. The compressive strength increases from 360 MPa at the strain rate of 10−4 s−1 up to 430 MPa at the strain rate of 2800 s−1, which is 19.4% higher compared with quasi-static result. All the compressive stress–strain curves are non-linear. Fig. 2 shows the variation of failure stress with the logarithm of strain rate. It is clear that the compressive failure strength of the 2D-C/SiC composites increases with an increasing strain rate. As previously observed in the tests of carbon/epoxy matrix composites[7] and C/C composites[8], the elastic modulus for the 2D-C/SiC composites also increases with loading rate in the present study. But the intrinsic mechanism is not clear.
L Mingshuang et al. / Materials Science and Engineering A 489(2008)120-126 250 三 2D-C/Sic Fig 4. The typical true stress vs time curve of the 2D-C/SiC composites under Fig. 2. The failure strength vs. logarithm strain rate curve of 2D-C/SiC com- dynamic compression. that the stress state In the deons of the suss yokgjque is the logarithm of strain rate. Therefore, it can be defined in the As shown in Fig. 5, the elastic modulus varies linearly with following equation neous Clearly some time is needed for the specimen to achieve a homogeneous stress state after first loading. According to Ravichandran and Subhash [9], the homogeneity factor a(t)is Ed= es A ln defined where Ed and es are the elastic modulus at different strain rate a(D)、|o1(t)-02( (2) and referenced elastic modulus, respectively. e and Eo are the a1(t)+a2(1)/2 strain rate and referenced strain rate, respectively. A is a param- where o(t)and o(t) are the mean axial stresses on the inter- eter to be ascertained faces between the specimen and the incident bar as well as the specimen and the transmission bar, respectively. From Fig 3. it 3.2. The Weibull distribution of the dynamic failure strength is clear that the homogeneity factor falls down to 5% when the loading time is 9 us. The stress vs time curve of this specimen The dynamic failure strengths of the 2D-C/SiC compos ites are also scattered at the strain rate of 2800s(as shown 9 us). the stress is about 60 MPa and the strain is about 0. 01. in Fig. 6). The average strength and mean squared error are It accords with the investigation of Sharpe and Hoge [10] 427.5 and 56.7 MPa, respectively. As well be known, the The elastic modulus of the 2D-C/SiC composites under strength of brittle materials tallies with Weibull distribution. For dynamic loading can be calculated from the experimental data. fiber-reinforced ceramic matrix composites, the three basic con- 642 10 8642 Elastic modulus -a % Time /us Fig. 5. The elastic modular vs logarithm strain rate curve of 2D-C/SiC com- Fig3. The typical curve of a-i in the 2D-C/SiC composite sample. posites
122 L. Mingshuang et al. / Materials Science and Engineering A 489 (2008) 120–126 Fig. 2. The failure strength vs. logarithm strain rate curve of 2D-C/SiC composites. One of the basic assumptions of the SHPB technique is that the stress state in the deforming specimen is homogeneous. Clearly some time is needed for the specimen to achieve a homogeneous stress state after first loading. According to Ravichandran and Subhash [9], the homogeneity factor α(t) is defined as: α(t) = |σ1(t) − σ2(t)| [σ1(t) + σ2(t)]/2 (2) where σ1(t) and σ2(t) are the mean axial stresses on the interfaces between the specimen and the incident bar as well as the specimen and the transmission bar, respectively. From Fig. 3, it is clear that the homogeneity factor falls down to 5% when the loading time is 9 s. The stress vs. time curve of this specimen is shown in Fig. 4. When the stress state reaches homogeneous (t = 9s), the stress is about 60 MPa and the strain is about 0.01. It accords with the investigation of Sharpe and Hoge [10]. The elastic modulus of the 2D-C/SiC composites under dynamic loading can be calculated from the experimental data. Fig. 3. The typical curve of α–t in the 2D-C/SiC composite sample. Fig. 4. The typical true stress vs. time curve of the 2D-C/SiC composites under dynamic compression. As shown in Fig. 5, the elastic modulus varies linearly with the logarithm of strain rate. Therefore, it can be defined in the following equation: Ed = ES + A ln ε˙ ε˙0 (3) where Ed and ES are the elastic modulus at different strain rate and referenced elastic modulus, respectively. ε˙ and ε˙0 are the strain rate and referenced strain rate, respectively. A is a parameter to be ascertained. 3.2. The Weibull distribution of the dynamic failure strength The dynamic failure strengths of the 2D-C/SiC composites are also scattered at the strain rate of 2800 s−1 (as shown in Fig. 6). The average strength and mean squared error are 427.5 and 56.7 MPa, respectively. As well be known, the strength of brittle materials tallies with Weibull distribution. For fiber-reinforced ceramic matrix composites, the three basic conFig. 5. The elastic modular vs. logarithm strain rate curve of 2D-C/SiC composites.
L Mingshuang et al. /Materials Science and Engineering A 489(2008)120-126 +l# sampl -3# sample 47# sample Fig. 8. The ruptured scratch of the 2D-C/SiC composites under static and 20 10 33. fracture observation 000010020030.040.05006007008 The specimens fail by shearing under static loading while True strain they are shattered to more fragments under dynamic conditions which indicated a multiple fracture mode. Therefore, a pad was ag o. in true suess yvs. stran curves or te -usl composites win the used to limit the compressive strain to about 6%(as shown in Fig. 1). The pictures of the ruptured 2D-C/SiC specimens under stituents(fiber, interphase, and matrix) are essentially brittle and static and dynamic compression are shown in Fig. 8. It can the cracking involves defect-induced random failures. There- be seen that shear fracture angles of the failed specimens are fore, statistical distribution of the strengths for the 2D- C/Sic approximately 500 at the strain rate of 10-4s-, and 550 at the composites can be described by Weibull equation, strain rate of 850s-, respectively. Garland et al. [13]investi gated the development of compressive damage zones in fibrous (4) composites, and the results showed that the magnitude of frac- interface would result in a large shear fracture angle, and weak where F(o)is a probability function; o is the applied stress m, interface induces a small fracture angle. Therefore, it seems that the Weibull modulus, is the most important parameter that char- the increase of fracture angle at the strain rate of 850s-could be acterizes the scatter of the material. The larger the value of m, contributed to the strain rate effect of interface strength, which the better the scatter of the material oo is the scale parameter or has been identified by the experiments made by Bi et al. [14] characteristic strength and the maximum density of the Weibull The observation that larger fracture angle corresponds to ahigher distribution is located at the position of o=oo. It could be found from Fig. 7 that the dynamIc compressive failure strength of 2D. compressive strength in the present study is also consistent with C/SiC composites also obeys Weibull distribution. The slope of the work of Narayanan and Schadler [15], who found a simi- the solid line, corresponding to the Weibull modulus m, and the lar relationship between the fracture angle and the compressive strength. Moreover, it can be clearly seen in Fig. that more shear haracteristic strength oo are 5.27 and 454.6 MPa, respectively. bands exist in the failed composite specimen tested under the The result is consistent with those of 3 D-C/SiC [11]and SiC/Sic strain rate of 850s-l. Obviously, the multiple shear bands could composites [12] improve the toughness of the composites through the forma tion of new failure surfaces under dynamic compressive loading Hence the fracture angle increased at a higher loading rate 1.0 Fig 9 shows the SEM micrographs of the 2D-C/SiC com- posites under static and dynamic compression, which were taken parallel with the loading direction. Compared with the clear rup- tured surface formed at the strain rate of 10-4s-I. the rough ruptured surface at the strain rate of 850s- indicates that more small cracked fiber and matrix fragments generates with the increase of strain rate, which further identified the multiple frac ture characteristic for the 2D-C/SiC composites under higher 2D.C/SiC 3.4. Constitutive model of 2D-C/SiC composites with a strain rate-dependent In(dynamic failure strength) A new continuum damage-mechanics-based constituti model for composite materials was implemented by Nandlall Fig. 7. The Weibull distribution of the 2D-C/SiC composites. et al. [16] in the ballistic response simulation of grP plates
L. Mingshuang et al. / Materials Science and Engineering A 489 (2008) 120–126 123 Fig. 6. The true stress vs. strain curves of the 2D-C/SiC composites with the same strain rate under dynamic compression. stituents (fiber, interphase, and matrix) are essentially brittle and the cracking involves defect-induced random failures. Therefore, statistical distribution of the strengths for the 2D-C/SiC composites can be described by Weibull equation, F(σ) = 1 − exp − σ σ0 m (4) where F(σ) is a probability function; σ is the applied stress; m, the Weibull modulus, is the most important parameter that characterizes the scatter of the material. The larger the value of m, the better the scatter of the material. σ0 is the scale parameter or characteristic strength and the maximum density of the Weibull distribution is located at the position of σ = σ0. It could be found from Fig. 7 that the dynamic compressive failure strength of 2DC/SiC composites also obeys Weibull distribution. The slope of the solid line, corresponding to the Weibull modulus m, and the characteristic strength σ0 are 5.27 and 454.6 MPa, respectively. The result is consistent with those of 3D-C/SiC[11] and SiC/SiC composites [12]. Fig. 7. The Weibull distribution of the 2D-C/SiC composites. Fig. 8. The ruptured scratch of the 2D-C/SiC composites under static and dynamic compression. 3.3. Fracture observation The specimens fail by shearing under static loading while they are shattered to more fragments under dynamic conditions which indicated a multiple fracture mode. Therefore, a pad was used to limit the compressive strain to about 6% (as shown in Fig. 1). The pictures of the ruptured 2D-C/SiC specimens under static and dynamic compression are shown in Fig. 8. It can be seen that shear fracture angles of the failed specimens are approximately 50◦ at the strain rate of 10−4 s−1, and 55◦ at the strain rate of 850 s−1, respectively. Garland et al. [13] investigated the development of compressive damage zones in fibrous composites, and the results showed that the magnitude of fracture angles depends on the bonding strength of interface. Strong interface would result in a large shear fracture angle, and weak interface induces a small fracture angle. Therefore, it seems that the increase of fracture angle at the strain rate of 850 s−1 could be contributed to the strain rate effect of interface strength, which has been identified by the experiments made by Bi et al. [14]. The observation that larger fracture angle corresponds to a higher compressive strength in the present study is also consistent with the work of Narayanan and Schadler [15], who found a similar relationship between the fracture angle and the compressive strength. Moreover, it can be clearly seen in Fig. 8 that more shear bands exist in the failed composite specimen tested under the strain rate of 850 s−1. Obviously, the multiple shear bands could improve the toughness of the composites through the formation of new failure surfaces under dynamic compressive loading. Hence the fracture angle increased at a higher loading rate. Fig. 9 shows the SEM micrographs of the 2D-C/SiC composites under static and dynamic compression, which were taken parallel with the loading direction. Compared with the clear ruptured surface formed at the strain rate of 10−4 s−1, the rough ruptured surface at the strain rate of 850 s−1 indicates that more small cracked fiber and matrix fragments generates with the increase of strain rate, which further identified the multiple fracture characteristic for the 2D-C/SiC composites under higher strain rate. 3.4. Constitutive model of 2D-C/SiC composites with a strain rate-dependent A new continuum damage-mechanics-based constitutive model for composite materials was implemented by Nandlall et al. [16] in the ballistic response simulation of GRP plates
L Mingshuang et aL. Materials Science and Engineering A 489(2008)120-126 150y:2 Fig 9. SEM micrographs of the 2D-C/SiC composite ruptured surface under static and dynamic compression. which shown as: Xu et al. [17] presented the parameter n was relative to strain rate. Therefore, a new constitutive model was proposed E(l-D)E J=EE(I-D) where D is the damage parameter, assumed to follow a Weibull distribution where E, D, e and eo have the same meaning as before, while q is a rate-dependent parameter. For the 2D-C/SiC composites, the elastic modulus is linear to the logarithm of strain rate, which is defined as Eq. (3). Thus, where e.s and y are the elastic modulus. failure strain and Eq. (8)can be expressed as strength of the material associated with the damage mod respectively. e is the base of the natural logarithm and the param =E(-D\(a eter n defines the shape of the damage growth curve. However, the meaning of n is not illuminated For 2D-C/SiC composites D=1-exp (-m(y) in the paper, n can be defined as ed= es A ln n=a1+ a2 exp +a2 exp where aj=2.59, a2=13.93, a3=4. 15, as shown in Fig 10 The parameters of the constitutive model are shown Table 1 Fig. 1l compares the constitutive model with experimental data of 2D-C/SiC composites at different strain rate. It is shown 16A hat, the constitutive model agrees with the experimental data very well 3.5. Discussion For 2D-C/SiC composites, most of cracks initiate in the stress concentration center and the pore largely exists in the com- Fitted curve posites. Many microcracks exist due to the different thermal expansion coefficients of the base phase and reinforced pha These cracks will spread and induce brittle failure. Under 8 10 12 14 16 18 he parameters of the 2D-C/SiC composite dynamical constitutive mode Es(GPa) 7.6 360 0.0001 0.31 Fig. 10. The curve of the n vs logarithm strain rate
124 L. Mingshuang et al. / Materials Science and Engineering A 489 (2008) 120–126 Fig. 9. SEM micrographs of the 2D-C/SiC composite ruptured surface under static and dynamic compression. which was shown as: σ = E(1 − D)ε (5) where D is the damage parameter, assumed to follow a Weibull distribution: D = 1 − exp − 1 neEε Y n (6) where E, ε and Y are the elastic modulus, failure strain and strength of the material associated with the damage model, respectively. e is the base of the natural logarithm and the parameter n defines the shape of the damage growth curve. However, the meaning of n is not illuminated. For 2D-C/SiC composites in the paper, n can be defined as: n = a1 + a2 exp −ln(ε/˙ ε˙0) a3 (7) where a1 = 2.59, a2 = 13.93, a3 = 4.15, as shown in Fig. 10. Fig. 10. The curve of the n vs. logarithm strain rate. Xu et al. [17] presented the parameter n was relative to strain rate. Therefore, a new constitutive model was proposed: σ = Eε(1 − D) ε˙ ε˙0 q (8) where E, D, ε˙ and ε˙0 have the same meaning as before, while q is a rate-dependent parameter. For the 2D-C/SiC composites, the elastic modulus is linear to the logarithm of strain rate, which is defined as Eq. (3). Thus, Eq. (8) can be expressed as: ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ σ = Edε(1 − D) ε˙ ε˙0 q D = 1 − exp − 1 neEdε Y n Ed = ES + A ln ε˙ ε˙0 n = a1 + a2 exp −ln(ε/˙ ε˙0) a3 (9) The parameters of the constitutive model are shown in Table 1. Fig. 11 compares the constitutive model with experimental data of 2D-C/SiC composites at different strain rate. It is shown that, the constitutive model agrees with the experimental data very well. 3.5. Discussion For 2D-C/SiC composites, most of cracks initiate in the stress concentration center and the pore largely exists in the composites. Many microcracks exist due to the different thermal expansion coefficients of the base phase and reinforced phase. These cracks will spread and induce brittle failure. Under high Table 1 The parameters of the 2D-C/SiC composite dynamical constitutive model Es (GPa) Y (MPa) ε˙0 (s−1) q A 7.6 360 0.0001 0.01 0.31
L Mingshuang et al. Materials Science and Engineering A 489(2008)120-126 4 ∽∽2历 sxs 50 perimental data(10s) 00010020.03004005006 Tr le stral True strain Os n=3 100 experimental data(85(s) ntal data(2800s) tutive moedl 00B0040050.06 00100200B0040.050.06007008 True strain True strain Fig. 11. The comparative charts of the constitutive model vs. experimental data of 2D-C/SiC composites at different strain rates. strain rate, high stress is needed to make the cracks spread, as (4) Based on the experimental results of 2D-C/SiC composites, the time of stress reaction is very short and the accumulation a new constitutive model is proposed and it agrees with the ergy is not enough to make the composite materials failure exPerimen tal data very well. As a result, the dynamic compressive strength is larger than the static one Acknowledgements The correlative exponents of Eqs. (3)and (7) for the elastic modulus E and parameter n are 0.968 and 0.9816, respectively, This project was supported by the National Natural Science which fit the results very well. Based on the deformation mecha- Foundation of China(No 90405016), China Aviation Science nism of the 2D-C/SiC composites, the rate-dependent non-linear Foundation(No 2006ZF53060) and the 111 project(B07050 feature should result from the inner damages and their evolution. to the Northwestern Polytechnical University in Xi'an Therefore it is rational to consider the factor of damage in the constitutive model References 4. Conclusions [1] G. Camus, L Guillaumat, S. Baste, Compos. Sci. Technol. 56(1996) 1363-1372 (1)The mechanical properties of the 2D-C/SiC composites [2] R.E.L. Bouazzaoui, S Baste, G. Camus, Compos. Sci. Technol. 56(1996) are rate-dependent. The failure strength and elastic mod 1373-1382 ulus increase and failure strain decreases with an increasing [3]S Baste, Compos. Sci. Technol. 61(2001)2285-2297 strain rate [4 S Sarva, S Nemat-Nasser, Mater Sci Eng. A317(2001)140-144. (2) The specimens of 2D-C/SiC composites fail by shearing M. Futakawa, Y. Tannabe, T. Wakui, et al., Int J. Impact Eng. 25(2001) under compression, and a larger shear fracture angle, 55 [6] C.A. Weeks, C.T. Sun, Compos. Sci. Technol. 58(1998)603-611 are observed at the higher strain rate of 850s-, comparing [71 M.V. Hosur, M. Adya, U.K. Vaidya, A. Mayer, S. Jeelani, Compos.Struct. with 500 at 10-4s- More cracked fibers are also observed 59(2003)507-523. at the fracture surface under higher strain rates. [8] Q.. Yuan, Y.L. Li, HJ. Li, S.P. Li, L.J. Guo, J. Inorg. Mater. 22(2007) 311-314 (in Chinese) (3)The elastic modulus of the 2D-C/SiC composites varies (9) G Ravichandran, G. Subhash, J Am. Ceram Soc. 77(1994)263-267 linearly to the logarithm of strain rates. [10] W N.J. Sharpe, K.G. Hoge, Exp Mech. 12(1972)570
L. Mingshuang et al. / Materials Science and Engineering A 489 (2008) 120–126 125 Fig. 11. The comparative charts of the constitutive model vs. experimental data of 2D-C/SiC composites at different strain rates. strain rate, high stress is needed to make the cracks spread, as the time of stress reaction is very short and the accumulation energy is not enough to make the composite materials failure. As a result, the dynamic compressive strength is larger than the static one. The correlative exponents of Eqs. (3) and (7) for the elastic modulus E and parameter n are 0.968 and 0.9816, respectively, which fit the results very well. Based on the deformation mechanism of the 2D-C/SiC composites, the rate-dependent non-linear feature should result from the inner damages and their evolution. Therefore, it is rational to consider the factor of damage in the constitutive model. 4. Conclusions (1) The mechanical properties of the 2D-C/SiC composites are rate-dependent. The failure strength and elastic modulus increase and failure strain decreases with an increasing strain rate. (2) The specimens of 2D-C/SiC composites fail by shearing under compression, and a larger shear fracture angle, 55◦ are observed at the higher strain rate of 850 s−1, comparing with 50◦ at 10−4 s−1. More cracked fibers are also observed at the fracture surface under higher strain rates. (3) The elastic modulus of the 2D-C/SiC composites varies linearly to the logarithm of strain rates. (4) Based on the experimental results of 2D-C/SiC composites, a new constitutive model is proposed and it agrees with the experimental data very well. Acknowledgements This project was supported by the National Natural Science Foundation of China (No. 90405016), China Aviation Science Foundation (No. 2006ZF53060) and the 111 project (B07050) to the Northwestern Polytechnical University in Xi’an. References [1] G. Camus, L. Guillaumat, S. Baste, Compos. Sci. Technol. 56 (1996) 1363–1372. [2] R.E.I. Bouazzaoui, S. Baste, G. Camus, Compos. Sci. Technol. 56 (1996) 1373–1382. [3] S. Baste, Compos. Sci. Technol. 61 (2001) 2285–2297. [4] S. Sarva, S. Nemat-Nasser, Mater. Sci. Eng. A317 (2001) 140–144. [5] M. Futakawa, Y. Tannabe, T. Wakui, et al., Int. J. Impact Eng. 25 (2001) 29–40. [6] C.A. Weeks, C.T. Sun, Compos. Sci. Technol. 58 (1998) 603–611. [7] M.V. Hosur, M. Adya, U.K. Vaidya, A. Mayer, S. Jeelani, Compos. Struct. 59 (2003) 507–523. [8] Q.L. Yuan, Y.L. Li, H.J. Li, S.P. Li, L.J. Guo, J. Inorg. Mater. 22 (2007) 311–314 (in Chinese). [9] G. Ravichandran, G. Subhash, J. Am. Ceram. Soc. 77 (1994) 263–267. [10] W.N.J. Sharpe, K.G. Hoge, Exp. Mech. 12 (1972) 570
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