H Mei/Composites Science and Technology 68(2008)3285-3292 3289 Table 2 Parameters and values used in calculation 70 ng s modulus of matrix Volume fraction of matrix 亏65 fraction of c fber 60 160 MPa (axial) C fber radit Coefficient related to weave b 200 MPa Interface sliding stress for C/SiC MPa 100 150 Processing temperature Youngs modulus of Sic fiber ￥ Ptvaican unloading -reloading cues. calcuated om tne reloading curves cTE of sic fil SiC fber radius Interface sliding stress for SiC/SiC MPa loading curves of several unloading-reloading loops. It is clear that the tangent modulus of the composite at each loading cycle de- CTE, coefficient of thermal expansion. creases with increasing reloading stress up to the final distinctive steady state (i.e. SSM), which indicating that the top linear behav or with no sliding indeed exists for the present composite system Both initial modulus and final SSM of the composite at each hyster- esis loop decrease as the peak applied stress increases such as from 160 160 MPa. 180 MPa to 200 MPa. E=iEV, 3.3. Experimental measurement of TRs According to Steen and Camus [8, 14, the axial residual stress state at a given temperature and for a given composite specimen can be determined directly from the coordinates of that commor intersection point by extrapolation of the compliance slopes of the top linear portion at each reloading-unloading loop As shown in Fig. 7, these compliance slopes happen to meet at a single point O(Er, or)localized in the compression domain. Specifically, if the -0.30.2-0.10.00.10.20.30.40.50.60.7 matrix of the composite is in residual compression, the intersection of the regression lines of the reloading-unloading loops lies in the positive stress-strain quadrant(I quadrant ) and if the matrix of Fig 10. Typical step-loading curves of 2D C/Sic showing the intersection point of gative stress-strain quadrant(ll quadrant). In the present C/ stress-strain quadrant ons the composite is in residual tension, the intersection lies in the tive unloading-reloading loops in the negative SiC, below the processing temperature of 1000C the Sic matrix is in residual tensile stress whereas the carbon fiber reinforcement ite specimen at the last hysteresis loop(see Table 1).This so-called is in residual compressive stress since the SiC matrix normally has 'EVe line was also plotted in the Fig. 7. Above this line all the com- greater CTE than the axial carbon fiber(see Table 2). As a result, liance slopes can intersect at the origin point O and the envelop the C/sic materials have a pre-cracked as-received condition due to tensile curve of the composite(defined as tensile mastercurve by to the residual tensile stress state in the brittle SiC matrix once Steen)exhibits large nonlinear phenomena associated with time- cooled down from the processing temperature and the speci dependent matrix cracking. The intersection of the tensile master- intersection for indication of the axial residual stress state local- curve with this line indicates that a saturated matrix cracking state The lower-right solid straight line through the origin o in illustrated in Fig. 7) and the longitudinal 90 fibers in the load direc final failure with the minimum modulus cracks. High modulus fibers and bundles now bore the global load- Ep=ErVE (4) in a completely cracked matrix, leading to macroscopic"stiffen- ing"phenomenon and linear stress-strain relationship of the fibers where Ef and Vf denote the modulus and volume fraction of the with"some slope recovery". The tensile mastercurve with"slo intact fibers oriented in the direction of loading. The parameter i recovery"continuously climbs up along with Emin line until the is defined as a modified coefficient related to fiber woven architec fibres failure occurs leading to a decrease in the value of Vr and then ture of the composite perform, and here it is equal to 0.5 because final rupture of the composite is coming. The"stiffening"phenom- only one half of the fibers(longitudinal fibers) in 2D [0 /90]C/ enon and linear stress-strain response of the composite prior to Sic has a contribution to the modulus of the whole composite in rupture caused by the solely load bearing fibers were also observed the loading direction. Using the data listed in Table 2 and Eq (4). by the other researchers [15-17]. Fig 2a in a previously reported the minimum modulus of the composite approximates to 46 GPa, work by Mei et al. [18 indicated this behavior of the 2D C/Sic which is slightly lower than the SSM Ep of 52.65 GPa of the compos- composite In the present study, however, this phenomenon couldloading curves of several unloading–reloading loops. It is clear that the tangent modulus of the composite at each loading cycle de￾creases with increasing reloading stress up to the final distinctive steady state (i.e., SSM), which indicating that the top linear behav￾ior with no sliding indeed exists for the present composite system. Both initial modulus and final SSM of the composite at each hyster￾esis loop decrease as the peak applied stress increases such as from 160 MPa, 180 MPa to 200 MPa. 3.3. Experimental measurement of TRS According to Steen and Camus [8,14], the axial residual stress state at a given temperature and for a given composite specimen can be determined directly from the coordinates of that common intersection point by extrapolation of the compliance slopes of the top linear portion at each reloading-unloading loop. As shown in Fig. 7, these compliance slopes happen to meet at a single point O0 (er,rr) localized in the compression domain. Specifically, if the matrix of the composite is in residual compression, the intersection of the regression lines of the reloading–unloading loops lies in the positive stress–strain quadrant (I quadrant); and if the matrix of the composite is in residual tension, the intersection lies in the negative stress–strain quadrant (III quadrant). In the present C/ SiC, below the processing temperature of 1000 C the SiC matrix is in residual tensile stress whereas the carbon fiber reinforcement is in residual compressive stress since the SiC matrix normally has a greater CTE than the axial carbon fiber (see Table 2). As a result, the C/SiC materials have a pre-cracked as-received condition due to the residual tensile stress state in the brittle SiC matrix once cooled down from the processing temperature and the special intersection for indication of the axial residual stress state local￾ized in the III quadrant in Fig. 10. The lower-right solid straight line through the origin O0 in Fig. 10 represents an elastic behavior of the composite before the final failure with the minimum modulus Emin p ¼ kEfVf ð4Þ where Ef and Vf denote the modulus and volume fraction of the intact fibers oriented in the direction of loading. The parameter k is defined as a modified coefficient related to fiber woven architec￾ture of the composite perform, and here it is equal to 0.5 because only one half of the fibers (longitudinal fibers) in 2D [0/90] C/ SiC has a contribution to the modulus of the whole composite in the loading direction. Using the data listed in Table 2 and Eq. (4), the minimum modulus of the composite approximates to 46 GPa, which is slightly lower than the SSM Ep of 52.65 GPa of the compos￾ite specimen at the last hysteresis loop (see Table 1). This so-called ‘EfVf line’ was also plotted in the Fig. 7. Above this line all the com￾pliance slopes can intersect at the origin point O0 and the envelope to tensile curve of the composite (defined as tensile mastercurve by Steen) exhibits large nonlinear phenomena associated with time￾dependent matrix cracking. The intersection of the tensile master￾curve with this line indicates that a saturated matrix cracking state is reached (this stress is called matrix crack saturation stress rs as illustrated in Fig. 7) and the longitudinal 90 fibers in the load direc￾tion were completely bridged by the transverse saturated matrix cracks. High modulus fibers and bundles now bore the global load￾ing in a completely cracked matrix, leading to macroscopic ‘‘stiffen￾ing” phenomenon and linear stress–strain relationship of the fibers with ‘‘some slope recovery”. The tensile mastercurve with ‘‘slope recovery” continuously climbs up along with Emin p line until the fibres failure occurs leading to a decrease in the value of Vf and then final rupture of the composite is coming. The ‘‘stiffening” phenom￾enon and linear stress-strain response of the composite prior to rupture caused by the solely load bearing fibers were also observed by the other researchers [15–17]. Fig. 2a in a previously reported work by Mei et al. [18] indicated this behavior of the 2D C/SiC composite. In the present study, however, this phenomenon could Table 2 Parameters and values used in calculation Parameter Symbol Value Units Porosity of composite P 13 % Young’s modulus of matrix Em 350 GPa Volume fraction of matrix Vm 60 % CTE* of SiC matrix am 4.6 106 /K Young’s modulus of C fiber Ef 230 GPa Volume fraction of C fiber Vf 40 % CTE of C fiber af 0 (axial) 106 /K 10 (radial) C fiber radius R 3.5 lm Coefficient related to weave k 0.5 Crack spacing before failure d 150–160 lm Interface sliding stress for C/SiC s 6 MPa Room temperature To 298 K Processing temperature Tp 1273 K Matrix fracture energy Cm 6 J/m2 Young’s modulus of SiC fiber Ef 200 GPa Volume fraction of SiC fiber Vf 40 % CTE of SiC fiber af 4.6 106 /K SiC fiber radius R 7 lm Interface sliding stress for SiC/SiC s 20 MPa * CTE, coefficient of thermal expansion. Fig. 10. Typical step-loading curves of 2D C/SiC showing the intersection point of the regression lines of consecutive unloading–reloading loops in the negative stress–strain quadrant. Fig. 9. Typical tangent modulus change curves calculated from the reloading curves of several typical unloading–reloading cycles. H. Mei / Composites Science and Technology 68 (2008) 3285–3292 3289