292 not be seen because the tested composite sample ruptured at Ep- opening displacement)model. The axial residual stress in matrix 52.65 GPa before e= 46 GPa can be analytically expressed by 10 The intersection coordinates(Er, Or), as shown in Fig. 7, can be simply calculated by using arbitrary two pairs of ogous trian- tEVA+正E (10) gles,eg,△ORG~△FHG,as and the misfit stress, oT, can be extracted from the permanent strain E 11(ie the inelastic strain ai defined in Fig. 7) Thus ing to Eq(3), the TRS Or can be linearly expressed as a gT (Ep-+2 function of TR strain in each reloading-unloading loop I -) or=Eper+b where the SSM Ep is given by where R is the fiber radius, E and Em are the Young 's modulus of the composite and matrix, E is the modulus of the composite contain- ing(see Fig. 7). Vm is the tion of the matrix and t is interface sliding stress, and d is the and the intercept b is given by matrix crack spacing. The coefficients a, bi(appear in Eq. (11)were defined by Hutchinson and jensen [19] b=Epbr=EpE”-Gp (8) Fig 11 is the theoretical predictions of the permanent strain Ej as function of the peak applied stress op with three different level of and the known parameters such as the total strain g'and the peak ap- misfit stress o of 70, 100 and 150 MPa by using the Eq (11)and the plied stress ap. Table 1 gives the values of all the parameters for the data listed in Tables 1 and 2. The detailed analysis and calculation I hysteresis loops of the tested 2D C/SiC composite specimen. It procedure related to the misfit stress of the presently tested 2D C/ should be noted that the line determined by the Eq (6)just denotes SiC composite can be found in another paper by Wang et al. [201 the fitting line of the top linear portion of each hysteresis loop, e.g. It is clear from Fig. 11 that the theoretical prediction of the perm line O,Fin Fig. 7. Fig. 10 plotted all the 11 lines(dashed line)together nent strain ei perfectly matches the experimental results when o is with the periodic loading/unloading stress-strain curves. Evidently, equal to 100 MPa. Through the data in Table 2 and Eq (10), analy these compliance slopes are found to nearly intersect in a single ically calculated value of the trs of the tested 2D C/Sic materials oint o(Er, or), which can be termed"thermal-residual-stress-free containing matrix cracks and interface sliding is 136.71 MPa, which origin( 14. It can be easily calculated out by solving the coordinates of all the 55 intersections of the 11 lines(C1= 55)as E 3100 MPa 后012 0=150 MPa According to the data in Table 1, the tRstress and strain could be easy to be obtained by using Matlab" programming: Er=-012+0.04% 13084+34.53 MPa. all the calculational results of the 55 intersections of the 11 lines are listed in table 3 3.4. Analytical calculation of trs considering matrix cracking and interfacial sliding Evans and his colleagues [10-11 developed analytical formulas Peak maximum stress(MPa) for the calculation of TRS of LdE(large debond energy) materials considering matrix cracking and interface sliding, mainly based Fig. 11. Accumulation of residual strains with applied stress. Misfit stress can be on the analysis of interface inelastic deformation and CoD(crack obtained by fitting them with difterent values on fixed >h Table 3 Calculational coordinates of all the 55 intersections of the 11 lines 020-173.66 167.85 0.09-108.38 0.12 -1323 L3-0.0 -019-16983L25-0.13-135.56L39-0.10-116 45 -014-14277128010-12001n-012-12937159-00-10183210-013-13803 010-118.54L10-0.10-117.6 0.18 178.22 11-12739L210-0.11-12386 0.10-10915L5-0.12-12383L310-0.13-14219 12-12950L2 0.13-13421L47-0.07 03-5889Ls,11-0.21-207.77 -013-13886L35-0.14-139.82L49-0.09-10597 0.09103.16L9n-0.29-281.29 0.19-16758L 0.12 0-0.09-108.60L1 0.14-129.3 Thermal residual strain and stress(AVE* STDEV) 2513084±345Ma Ly denotes the intersection of lines i andjnot be seen because the tested composite sample ruptured at Ep = 52.65 GPa before Emin p = 46 GPa. The intersection coordinates (er,rr), as shown in Fig. 7, can be simply calculated by using arbitrary two pairs of analogous trian￾gles, e.g., DO’RG DFHG, as rr rp ¼ er þ eT es þ ee ð5Þ Thus, according to Eq. (3), the TRS rr can be linearly expressed as a function of TR strain in each reloading–unloading loop n: rr ¼ Eper þ b ð6Þ where the SSM Ep is given by Ep ¼ rp es þ ee ð7Þ and the intercept b is given by b ¼ EpeT ¼ Epe  rp ð8Þ The SSM Ep and the intercept b are now obtained by the linear fitting and the known parameters such as the total strain e* and the peak ap￾plied stress rp. Table 1 gives the values of all the parameters for the 11 hysteresis loops of the tested 2D C/SiC composite specimen. It should be noted that the line determined by the Eq. (6) just denotes the fitting line of the top linear portion of each hysteresis loop, e.g. line O’F in Fig. 7. Fig. 10 plotted all the 11 lines (dashed line) together with the periodic loading/unloading stress–strain curves. Evidently, these compliance slopes are found to nearly intersect in a single point O0 (er,rr), which can be termed ‘‘thermal-residual-stress-free” origin [14]. It can be easily calculated out by solving the coordinates of all the 55 intersections of the 11 lines (C2 11 ¼ 55) as, rr ¼ E1 p . . . E11 p 2 6 6 6 4 3 7 7 7 5 er þ b1 . . . b11 2 6 6 4 3 7 7 5 ð9Þ According to the data in Table 1, the TR stress and strain could be easy to be obtained by using Matlab programming: er = 0.12 ± 0.04%, and rr = 130.84 ± 34.53 MPa. All the calculational results of the 55 intersections of the 11 lines are listed in Table 3. 3.4. Analytical calculation of TRS considering matrix cracking and interfacial sliding Evans and his colleagues [10–11] developed analytical formulas for the calculation of TRS of LDE (large debond energy) materials considering matrix cracking and interface sliding, mainly based on the analysis of interface inelastic deformation and COD (crack opening displacement) model. The axial residual stress in matrix can be analytically expressed by [10] rm r ¼ Em EmVm þ kEfVf rT ð10Þ and the misfit stress, rT , can be extracted from the permanent strain ei [11] (i.e., the inelastic strain ei defined in Fig. 7) rT ¼ ð 1 Ep  1 E Þ 2 8 b2ð1a1VfÞ 2R 4dsEmV2 f ð 1 Ep  1 EÞ  Erp 1 Ep  ei rp ð11Þ where R is the fiber radius, E and Em are the Young’s modulus of the composite and matrix, E* is the modulus of the composite contain￾ing matrix cracks upon unloading (see Fig. 7). Vm is the volume frac￾tion of the matrix and s is interface sliding stress, and d is the matrix crack spacing. The coefficients ai, bi (appear in Eq. (11)) were defined by Hutchinson and Jensen [19]. Fig. 11 is the theoretical predictions of the permanent strain ei as a function of the peak applied stress rp with three different level of misfit stress rT of 70, 100 and 150 MPa by using the Eq. (11) and the data listed in Tables 1 and 2. The detailed analysis and calculation procedure related to the misfit stress of the presently tested 2D C/ SiC composite can be found in another paper by Wang et al. [20]. It is clear from Fig. 11 that the theoretical prediction of the perma￾nent strain ei perfectly matches the experimental results when rT is equal to 100 MPa. Through the data in Table 2 and Eq. (10), analyt￾ically calculated value of the TRS of the tested 2D C/SiC materials containing matrix cracks and interface sliding is 136.71 MPa, which Table 3 Calculational coordinates of all the 55 intersections of the 11 lines Li,j * er rr Li,j er rr Li,j er rr Li,j er rr Li,j er rr L1,2 0.20 173.66 L2,4 0.19 167.85 L3,7 0.09 108.38 L4,11 0.12 122.50 L6,11 0.13 132.39 L1,3 0.20 170.71 L2,5 0.15 148.00 L3,8 0.09 111.55 L5,6 0.08 98.52 L7,8 0.12 130.78 L1,4 0.19 169.83 L2,6 0.13 135.56 L3,9 0.10 116.15 L5,7 0.05 77.45 L7,9 0.13 140.79 L1,5 0.16 154.10 L2,7 0.10 118.54 L3,10 0.10 117.62 L5,8 0.07 90.60 L7,10 0.13 138.03 L1,6 0.14 142.77 L2,8 0.10 120.01 L3,11 0.12 129.37 L5,9 0.09 101.83 L7,11 0.18 178.22 L1,7 0.11 127.00 L2,9 0.11 123.15 L4,5 0.11 118.10 L5,10 0.09 106.11 L8,9 0.14 151.78 L1,8 0.11 127.39 L2,10 0.11 123.86 L4,6 0.10 109.15 L5,11 0.12 123.83 L8,10 0.13 142.19 L1,9 0.12 129.50 L2,11 0.13 134.21 L4,7 0.07 91.98 L6,7 0.03 58.89 L8,11 0.21 207.77 L1,10 0.12 129.66 L3,4 0.19 168.10 L4,8 0.08 98.72 L6,8 0.06 86.28 L9,10 0.12 131.70 L1,11 0.13 138.86 L3,5 0.14 139.82 L4,9 0.09 105.97 L6,9 0.09 103.16 L9,11 0.29 281.29 L2,3 0.19 167.58 L3,6 0.12 126.50 L4,10 0.10 108.83 L6,10 0.09 108.60 L10,11 0.14 129.31 Thermal residual strain and stress (AVE ± STDEV) er ¼ 0:12  0:04% rr ¼ 130:84  34:53 MPa * Li,j denotes the intersection of lines i and j. Fig. 11. Accumulation of residual strains with applied stress. Misfit stress can be obtained by fitting them with different values on fixed Ri. 3290 H. Mei / Composites Science and Technology 68 (2008) 3285–3292