84 2 Brittle and Ductile Fracture spaced by I =30,60,90,120um,were considered.Moreover,geometrically identical particles of negligibly small moduli (holes)were also studied for comparative reasons.Note that the range d/l E(0.1,3)corresponds to the particle volume fraction fpE(0.04,25)%.Several thousands of possible po- sitions of the crack tip were analyzed within an investigated area in between a pair of spherical particles,according to the scheme in Figure 2.6.This area was long enough to incorporate the influence of further neighbouring pairs of particles(behind and ahead of the investigated area,dashed lined)which were not explicitly considered in the analysis. (a) (b) Figure 2.6 The scheme related to the model of particle-induced crack tip shielding: (a)position of the investigated region in the testing sample,and (b)detail of the region (black rectangle)and circular particles This enabled us to generalize the results to a periodic square network of particles by multiplication of normalized effective SIFs in the points which lie within both the left-hand and the right-hand parts of the investigated region and are associated owing to the translation periodicity. The effective SIF(kef,r=√号.，+kir)was used to assess the effective crack driving force.Averaged values of as functions of the ratio d/l for all analyzed types of particles are displayed in Figure 2.7.One can see that the rigid inclusions start to produce some shielding after reaching the critical value (d/T)c =0.2 (or fpe =0.5%).Then the normalized effective k-factor rather slowly drops to the value of 0.9 that corresponds to (d/l)=3 (or fp=25%).Practically the same decrease was identified in the case of holes. Here,however,the drop was shifted to a higher critical value (d/l)c=1 (or fpe=6.5%).Despite this rather slight difference,the shielding effect of both rigid particles and holes appeared to be similar. In order to asses the RIS,the pyramidal model was applied in the first approximation by using the roughness characteristics from Table 2.1.The dependence of the relative fracture toughness Kre(X%)/KIe(0%)(where KIe(0%)=KIci =KIcm),calculated using Equations 2.16,2.17 and 2.18, on the volume fraction of Al2O3 platelets is plotted by the dashed line in
84 2 Brittle and Ductile Fracture spaced by l = 30, 60, 90, 120 μm, were considered. Moreover, geometrically identical particles of negligibly small moduli (holes) were also studied for comparative reasons. Note that the range d/l ∈ (0.1, 3) corresponds to the particle volume fraction fp ∈ (0.04, 25)%. Several thousands of possible positions of the crack tip were analyzed within an investigated area in between a pair of spherical particles, according to the scheme in Figure 2.6. This area was long enough to incorporate the influence of further neighbouring pairs of particles (behind and ahead of the investigated area, dashed lined) which were not explicitly considered in the analysis. x x y y (a) (b) Figure 2.6 The scheme related to the model of particle-induced crack tip shielding: (a) position of the investigated region in the testing sample, and (b) detail of the region (black rectangle) and circular particles This enabled us to generalize the results to a periodic square network of particles by multiplication of normalized effective SIFs in the points which lie within both the left-hand and the right-hand parts of the investigated region and are associated owing to the translation periodicity. The effective SIF (keff ,r = k2 I,r + k2 II,r) was used to assess the effective crack driving force. Averaged values of ¯ keff as functions of the ratio d/l for all analyzed types of particles are displayed in Figure 2.7. One can see that the rigid inclusions start to produce some shielding after reaching the critical value (d/l)c = 0.2 (or fpc = 0.5%). Then the normalized effective k-factor rather slowly drops to the value of 0.9 that corresponds to (d/l) = 3 (or fp = 25%). Practically the same decrease was identified in the case of holes. Here, however, the drop was shifted to a higher critical value (d/l)c = 1 (or fpc = 6.5%). Despite this rather slight difference, the shielding effect of both rigid particles and holes appeared to be similar. In order to asses the RIS, the pyramidal model was applied in the first approximation by using the roughness characteristics from Table 2.1. The dependence of the relative fracture toughness KIc(X%)/KIc(0%) (where KIc(0%) = KIci = KIcm), calculated using Equations 2.16, 2.17 and 2.18, on the volume fraction of Al2O3 platelets is plotted by the dashed line in
2.1 Brittle Fracture 85 1.0 0 …holes 0.9 ★ rigid particles 0.1 10 dil Figure 2.7 Averaged values of the effective stress intensity factor as a function of the particle size/spacing ratio for all analyzed types of particles Figure 2.8 along with the experimental data.As expected,the maximal pre- dicted relative increase of 40%in the fracture toughness cannot fully explain the real improvement of 120%that was achieved by the 30%volume fraction of platelets. 2.4 0 experiment 22 ◇ RIS RIS+E 2.0 女 RIS+E+PS ☆ 1.8 T 1.6 1.4 a 1.2 1.0F 0 10 15 20 25 30 Volume fraction of AlO platelets V,[%] Figure 2.8 Theoretical curves of the relative fracture toughness as functions of the percentage of Al2Os particles in comparison with experimental data.The full line shows the theoretical prediction including all considered corrections
2.1 Brittle Fracture 85 Figure 2.7 Averaged values of the effective stress intensity factor as a function of the particle size/spacing ratio for all analyzed types of particles Figure 2.8 along with the experimental data. As expected, the maximal predicted relative increase of 40% in the fracture toughness cannot fully explain the real improvement of 120% that was achieved by the 30% volume fraction of platelets. Figure 2.8 Theoretical curves of the relative fracture toughness as functions of the percentage of Al2O3 particles in comparison with experimental data. The full line shows the theoretical prediction including all considered corrections
86 2 Brittle and Ductile Fracture Indeed,one must also consider the decrease in the crack driving force related to both the increase in Young's modulus and the shielding induced by platelets 177.The increase in fracture toughness due to the increase in Young's modulus can be calculated from E Ge KicnVEm GIem (2.19) where E is Young's modulus of the composite and Em =E(0%)is Young's modulus of the matrix,the values of which are given in Table 2.2 [176].The improved prediction (RIS+E)including this effect is shown by the dotted line in Figure 2.8. Because the difference in shielding produced by rigid particles and holes was found to be negligible,one can use the result plotted in Figure 2.7 as a further correction of the theoretical curve.By considering the relevant volume fractions of experimental samples (the value kefr=0.9 was used for X= 30%),the final theoretical prediction(RIS+E+PS)is shown by the full line in Figure 2.8. Table 2.2 Young's moduli of borosilicate glass matrix composite containing different volume fractions of Al2Os platelets Platelets content vol.%]E [GPa] 0 63 6 65 10 70 15 79 30 102 One can see that the agreement between theory and experiment is reason- able. A somewhat more complex and exact model was proposed by Kotoul et al.[184].This model considered several additional toughening mechanisms, such as compressive residual stresses or crack front trapping at platelets that may be effective in these composites.Moreover,it could explain the experi- mental fact that the fracture roughness ceased to increase from about 15vol% of Al2O3.The model follows from the theory of particle-induced crack de- flection that was developed by Faber and Evans.However,some errors in the expression for the strain energy release rate,appearing in their original paper 169,had to be corrected.After relevant modification,the following equation for the normalized effective crack driving force was obtained:
86 2 Brittle and Ductile Fracture Indeed, one must also consider the decrease in the crack driving force related to both the increase in Young’s modulus and the shielding induced by platelets [177]. The increase in fracture toughness due to the increase in Young’s modulus can be calculated from KIc KIcm = E Em Gc GIcm , (2.19) where E is Young’s modulus of the composite and Em = E(0%) is Young’s modulus of the matrix, the values of which are given in Table 2.2 [176]. The improved prediction (RIS+E) including this effect is shown by the dotted line in Figure 2.8. Because the difference in shielding produced by rigid particles and holes was found to be negligible, one can use the result plotted in Figure 2.7 as a further correction of the theoretical curve. By considering the relevant volume fractions of experimental samples (the value k¯eff ,r = 0.9 was used for X = 30%), the final theoretical prediction (RIS+E+PS) is shown by the full line in Figure 2.8. Table 2.2 Young’s moduli of borosilicate glass matrix composite containing different volume fractions of Al2O3 platelets Platelets content [vol.%] E [GPa] 0 63 5 65 10 70 15 79 30 102 One can see that the agreement between theory and experiment is reasonable. A somewhat more complex and exact model was proposed by Kotoul et al. [184]. This model considered several additional toughening mechanisms, such as compressive residual stresses or crack front trapping at platelets that may be effective in these composites. Moreover, it could explain the experimental fact that the fracture roughness ceased to increase from about 15vol% of Al2O3. The model follows from the theory of particle-induced crack de- flection that was developed by Faber and Evans. However, some errors in the expression for the strain energy release rate, appearing in their original paper [169], had to be corrected. After relevant modification, the following equation for the normalized effective crack driving force was obtained:
2.1 Brittle Fracture 87 geff,r cos2 2 2sim2中+cos25)cos4b+ +c0s2 sin2 cos4 2 (2.20) cos2 1-w 2v-cos2 Here is a tilt angle and is a twist angle of crack front elements induced by their interaction with platelets.Note that Equation 2.20 possesses the required limiting properties,i.e., lim。gef,r→0， 中子T/2 卿9er→cos号 中→0 which was not the case in the originally derived expressions in 169.De- tails concerning the calculation of the averaged crack driving force geff,r can be found elsewhere [184].This solution also involves the contribution of the change in Young's modulus according to Equation 2.19.The theoretical pre- diction was in very good agreement with experimental data.This result re- vealed that,most probably,the contributions of residual stresses as well as crack trapping could be negligible.Indeed,as shown in [184],the presence of high residual radial tensile stresses along the platelet circumference leads to crack front propagation around the particle to relieve these stresses(no crack trapping).Simultaneously,however,the segment of the crack front propa- gating through the matrix is shortened and the corresponding twist angle decreases which results in flattening of the crack front in the matrix.This raises the energy release rate ger and makes it easier for the crack prop- agation in the tangential compressive stress field within the matrix.As a result,the net toughening increment given by gef,r remains unchanged and the residual stress contribution also does not take any effect. Taking the above-mentioned considerations into account,the peculiar oc- currence of a plateau in the plot of fracture surface roughness as a function of platelet volume fraction (Figure 2.3)can also be elucidated.There are two contributions to the surface roughness related to(1)crack propagation around the platelets and(2)crack propagation within the matrix.The for- mer grows with increasing platelet concentration.The latter decreases with increasing platelets concentration because the fracture surface in the matrix flattens.Moreover,due to clustering of platelets,their vicinity becomes less effective at deflecting cracks and,as a result,the corresponding contribution to the surface roughness further decreases.Beyond about 15%volume frac- tion of platelets,the positive and the negative contributions to the surface roughness mutually compensate and the increase in surface roughness stops. In summary,one can say that the models based on coupled shielding effects are able to elucidate quantitatively the increase in fracture toughness caused by particle reinforcement of glass-based ceramics
2.1 Brittle Fracture 87 geff ,r = cos2 λ 2 2ν sin2 φ + cos2 λ 2 2 cos4 φ+ + cos2 φ sin2 λ 2 cos4 λ 2 + + cos2 λ 2 sin2 φ cos2 φ 1 − ν 2ν − cos2 λ 2 2 . (2.20) Here Θ is a tilt angle and Φ is a twist angle of crack front elements induced by their interaction with platelets. Note that Equation 2.20 possesses the required limiting properties, i.e., lim φ→π/2 geff ,r → 0, lim φ→0 geff ,r → cos4 θ 2 which was not the case in the originally derived expressions in [169]. Details concerning the calculation of the averaged crack driving force ¯geff ,r can be found elsewhere [184]. This solution also involves the contribution of the change in Young’s modulus according to Equation 2.19. The theoretical prediction was in very good agreement with experimental data. This result revealed that, most probably, the contributions of residual stresses as well as crack trapping could be negligible. Indeed, as shown in [184], the presence of high residual radial tensile stresses along the platelet circumference leads to crack front propagation around the particle to relieve these stresses (no crack trapping). Simultaneously, however, the segment of the crack front propagating through the matrix is shortened and the corresponding twist angle decreases which results in flattening of the crack front in the matrix. This raises the energy release rate ¯geff ,r and makes it easier for the crack propagation in the tangential compressive stress field within the matrix. As a result, the net toughening increment given by ¯geff ,r remains unchanged and the residual stress contribution also does not take any effect. Taking the above-mentioned considerations into account, the peculiar occurrence of a plateau in the plot of fracture surface roughness as a function of platelet volume fraction (Figure 2.3) can also be elucidated. There are two contributions to the surface roughness related to (1) crack propagation around the platelets and (2) crack propagation within the matrix. The former grows with increasing platelet concentration. The latter decreases with increasing platelets concentration because the fracture surface in the matrix flattens. Moreover, due to clustering of platelets, their vicinity becomes less effective at deflecting cracks and, as a result, the corresponding contribution to the surface roughness further decreases. Beyond about 15% volume fraction of platelets, the positive and the negative contributions to the surface roughness mutually compensate and the increase in surface roughness stops. In summary, one can say that the models based on coupled shielding effects are able to elucidate quantitatively the increase in fracture toughness caused by particle reinforcement of glass-based ceramics