2.1 Brittle Fracture 79 The first step can be achieved by the 3D reconstruction of fracture mor- phology.The second problem can be solved,for example,by using the soft- ware package FRANC3D based on the boundary element method [172].The third step is solvable by standard mathematics.A nearly exact numerical so- lution by means of the FRANC3D code is,however,usually extremely time consuming.Therefore,a simple pyramidal model of the crack front was pro- posed for approximate analytical estimations [168,173.This model is based on a pyramid-like periodic approximation of the tortuous crack front,each element of which is characterized by the twist angle and the highest tilt angle em towards the macroscopic crack plane;see Figure 2.2. ⊙ Figure 2.2 A periodic element of the pyramidal model of tortuous crack front The profile roughness RL (measured along the crack front)and the pe- riodicity Apt(App)measured parallel (perpendicular)to the crack front are associated with the angles p and em by the following simple equations: 入pp tan日m=入pl tan重，Rz=cos-l重. (2.16) The characteristic periodicities Apt and App can be determined either by the Fourier analysis of roughness profiles measured at appropriate locations on the fracture surface,or simply identified with a characteristic microstructural periodicity,e.g.,with the mean grain size.The effective stress intensity factor kef.r(normalized to the remote KI factor)at each point of the pyramidal front can be calculated by using Equation 2.7 with the following approximate analytical expressions for local stress intensity factors: k1r cos S 重+cos 29 k2r sin cos2 2 (2.17) k3r cos 2 sinΦcosΦ 2w-ca( The results calculated according to Equation 2.17 are sufficiently accurate provided that pp2a,where a is the pre-crack length.The global effective factor er,averaged for the periodic crack front geometry composed of identical pyramidal elements,can then be computed as
2.1 Brittle Fracture 79 The first step can be achieved by the 3D reconstruction of fracture morphology. The second problem can be solved, for example, by using the software package FRANC3D based on the boundary element method [172]. The third step is solvable by standard mathematics. A nearly exact numerical solution by means of the FRANC3D code is, however, usually extremely time consuming. Therefore, a simple pyramidal model of the crack front was proposed for approximate analytical estimations [168, 173]. This model is based on a pyramid-like periodic approximation of the tortuous crack front, each element of which is characterized by the twist angle Φ and the highest tilt angle Θm towards the macroscopic crack plane; see Figure 2.2. m a pp/2 pl Figure 2.2 A periodic element of the pyramidal model of tortuous crack front The profile roughness RL (measured along the crack front) and the periodicity λpl (λpp) measured parallel (perpendicular) to the crack front are associated with the angles Φ and Θm by the following simple equations: λpp tan Θm = λpl tan Φ, RL = cos−1 Φ. (2.16) The characteristic periodicities λpl and λpp can be determined either by the Fourier analysis of roughness profiles measured at appropriate locations on the fracture surface, or simply identified with a characteristic microstructural periodicity, e.g., with the mean grain size. The effective stress intensity factor keff ,r (normalized to the remote KI factor) at each point of the pyramidal front can be calculated by using Equation 2.7 with the following approximate analytical expressions for local stress intensity factors: k1r = cos Θ 2 2ν sin2 Φ + cos2 Θ 2 cos2 Φ , k2r = sin Θ 2 cos2 Θ 2 , k3r = cos Θ 2 sin ΦcosΦ 2ν − cos2 Θ 2 . (2.17) The results calculated according to Equation 2.17 are sufficiently accurate provided that λpp 2a, where a is the pre-crack length. The global effective factor ¯ keff ,r, averaged for the periodic crack front geometry composed of identical pyramidal elements, can then be computed as
80 2 Brittle and Ductile Fracture T-2 不ef,r=28n2R+T-） ++) (2.18) -m Comparison of results obtained by means of the pyramidal model and the FRANC3D code revealed that,in the whole range of both the surface rough- ness and the roughness periodicity typical for real intergranular surfaces,the difference lies within the 10%of error band 168.Although the pyramidal model yields very promising results predominantly in the case of intercrys- talline fracture (see Section 2.2.2),it can also be quite successfully applied to other brittle fracture modes,as shown in the next section. 2.1.3 Fracture Toughness of Particle Reinforced Glass Composite Traditional ceramic materials such as glass or porcelain possess amorphous microstructures.An absence of crystallographically conditioned dislocations makes these materials extremely brittle.However,the very low intrinsic frac- ture toughness of glass in the range KIciE(0.5,1)MPam1/2 may be im- proved,for example,by reinforcing with second constituents with high mod- ulus,high strength and/or high ductility in the form of fibres,whiskers, platelets or particulates embedded into the matrix [174,175].A success- ful example of ceramic platelet reinforcement of glass is the borosilicate glass/Al2O3 platelet composite that was first introduced by Boccaccini et al.[176.Based on this system,environmentally friendly and cost-effective materials can be produced as alumina platelets for the building industry or as abrasives for the polishing industry.The enhancement in fracture toughness can be ascribed here to four concurrent phenomena [176-180]:the Young's modulus increment resulting from the platelets addition (the intrinsic com- ponent),the presence of a compressive residual stress in the glass matrix, the crack tip shielding produced by platelets and the crack deflection mech- anism (extrinsic components).The shielding effect is a result of local mixed- mode I+II+III induced by rigid particles surrounding the crack tip.The crack deflection is forced particularly by a necessity to bypass rigid parti- cles when searching the direction of the highest crack driving force (com- pare Section 2.2.1).This leads to a zig-zag crack propagation in between the platelets (crack tortuosity)and a reduction of the crack driving force in comparison to that of the straight crack.This must be associated with an enhanced microroughness of fracture surfaces.A direct correlation between the roughness of the fracture surface and the fracture toughness of dispersion reinforced ceramic and glass composites has been suggested and experimen-
80 2 Brittle and Ductile Fracture ¯ keff ,r = π − 2 2Θm(2RL + π − 4) Θm −Θm k2 1r + k2 2r + k2 3r 1 − ν dΘ. (2.18) Comparison of results obtained by means of the pyramidal model and the FRANC3D code revealed that, in the whole range of both the surface roughness and the roughness periodicity typical for real intergranular surfaces, the difference lies within the 10% of error band [168]. Although the pyramidal model yields very promising results predominantly in the case of intercrystalline fracture (see Section 2.2.2), it can also be quite successfully applied to other brittle fracture modes, as shown in the next section. 2.1.3 Fracture Toughness of Particle Reinforced Glass Composite Traditional ceramic materials such as glass or porcelain possess amorphous microstructures. An absence of crystallographically conditioned dislocations makes these materials extremely brittle. However, the very low intrinsic fracture toughness of glass in the range KIci ∈ (0.5, 1)MPa m1/2 may be improved, for example, by reinforcing with second constituents with high modulus, high strength and/or high ductility in the form of fibres, whiskers, platelets or particulates embedded into the matrix [174, 175]. A successful example of ceramic platelet reinforcement of glass is the borosilicate glass/Al2O3 platelet composite that was first introduced by Boccaccini et al. [176]. Based on this system, environmentally friendly and cost-effective materials can be produced as alumina platelets for the building industry or as abrasives for the polishing industry. The enhancement in fracture toughness can be ascribed here to four concurrent phenomena [176–180]: the Young’s modulus increment resulting from the platelets addition (the intrinsic component), the presence of a compressive residual stress in the glass matrix, the crack tip shielding produced by platelets and the crack deflection mechanism (extrinsic components). The shielding effect is a result of local mixedmode I+II+III induced by rigid particles surrounding the crack tip. The crack deflection is forced particularly by a necessity to bypass rigid particles when searching the direction of the highest crack driving force (compare Section 2.2.1). This leads to a zig-zag crack propagation in between the platelets (crack tortuosity) and a reduction of the crack driving force in comparison to that of the straight crack. This must be associated with an enhanced microroughness of fracture surfaces. A direct correlation between the roughness of the fracture surface and the fracture toughness of dispersion reinforced ceramic and glass composites has been suggested and experimen-
2.1 Brittle Fracture 81 tally proved [180-182].Because these systems provide an excellent possibility to verify theoretical GIS models,quantitative assessments of all the above- mentioned intrinsic and extrinsic effects have been performed 183,184.More- over,extended experimental analysis of fracture toughness,fracture surface roughness and microstructure was performed on samples made of borosilicate glass containing different volume fractions of alumina platelets. 2.1.3.1 Experimental Procedure and Results The experimental glass matrix composite was fabricated via powder technol- ogy and hot pressing.Alumina platelets of a hexagonal shape,with major axes between 5-25um and axial ratio of 0.2,were used.The commercially available borosilicate glass was selected for the composite matrix.The mi- crostructure of specimens containing 0,5,10,15 and 30 vol.of platelets [176 consisted of a dense glass matrix with a more or less homogeneous distribu- tion of platelets.A strong bond between the matrix and the platelets was confirmed by transmission electron microscopy [185.Upon cooling from the processing temperature,the thermal expansion mismatch between matrix and reinforcement induces tangential compressive and radial tensile residual stress in the matrix around the particles.Fracture toughness values were obtained using test pieces of a standard cross-section(3 x 4 mm2)with the chevron notch machined by an ultra thin diamond blade.A Zwick/Roell electromechanical testing machine was utilized for the three-point bending test with a span of 20mm.Scanning electron microscopy (SEM)was used for the fractographic analyses of fracture surfaces.Roughness parameters were measured by the optical profilometer MicroProf FRT based on a chro- matic aberration of its lens.The device works with vertical resolution of 3 nm and lateral resolution of about 1 um.A three-dimensional reconstruction of surface topography was performed by means of the software Mark III.The surface roughness was quantified by the average area roughness,RA,defined on the basis of the ISO 4278 norm as the arithmetic mean of the deviations of the roughness profile from the central line.The profile roughness RL,de- fined in a standard manner as the true profile length divided by its projected length,was also determined.The profiles obtained from 3D fracture surface morphology quantification were subjected to Fourier analysis in order to de- termine the characteristic periodicities App and Apt.The measured values for all specimens are displayed in Table 2.1. Note that values of App are an order of magnitude lower than the double- length of the pre-crack (2a 4mm)which ensures a reasonable validity of the pyramidal model. Dependencies of both the relative area roughness Rr RA(X%)/RA(0%) and the average fracture toughness on different alumina platelet volume con- tents (X%)are shown in Figure 2.3.It is seen that both curves increase linearly with increasing content of alumina platelets in the matrix approx-
2.1 Brittle Fracture 81 tally proved [180–182]. Because these systems provide an excellent possibility to verify theoretical GIS models, quantitative assessments of all the abovementioned intrinsic and extrinsic effects have been performed [183,184]. Moreover, extended experimental analysis of fracture toughness, fracture surface roughness and microstructure was performed on samples made of borosilicate glass containing different volume fractions of alumina platelets. 2.1.3.1 Experimental Procedure and Results The experimental glass matrix composite was fabricated via powder technology and hot pressing. Alumina platelets of a hexagonal shape, with major axes between 5 − 25 μm and axial ratio of 0.2, were used. The commercially available borosilicate glass was selected for the composite matrix. The microstructure of specimens containing 0, 5, 10, 15 and 30 vol.% of platelets [176] consisted of a dense glass matrix with a more or less homogeneous distribution of platelets. A strong bond between the matrix and the platelets was confirmed by transmission electron microscopy [185]. Upon cooling from the processing temperature, the thermal expansion mismatch between matrix and reinforcement induces tangential compressive and radial tensile residual stress in the matrix around the particles. Fracture toughness values were obtained using test pieces of a standard cross-section (3 × 4 mm2) with the chevron notch machined by an ultra thin diamond blade. A Zwick/Roell electromechanical testing machine was utilized for the three-point bending test with a span of 20 mm. Scanning electron microscopy (SEM) was used for the fractographic analyses of fracture surfaces. Roughness parameters were measured by the optical profilometer MicroProf FRT based on a chromatic aberration of its lens. The device works with vertical resolution of 3 nm and lateral resolution of about 1 μm. A three-dimensional reconstruction of surface topography was performed by means of the software Mark III. The surface roughness was quantified by the average area roughness, RA, defined on the basis of the ISO 4278 norm as the arithmetic mean of the deviations of the roughness profile from the central line. The profile roughness RL, de- fined in a standard manner as the true profile length divided by its projected length, was also determined. The profiles obtained from 3D fracture surface morphology quantification were subjected to Fourier analysis in order to determine the characteristic periodicities λpp and λpl. The measured values for all specimens are displayed in Table 2.1. Note that values of λpp are an order of magnitude lower than the doublelength of the pre-crack (2a = 4 mm) which ensures a reasonable validity of the pyramidal model. Dependencies of both the relative area roughness Rr = RA(X%)/RA(0%) and the average fracture toughness on different alumina platelet volume contents (X%) are shown in Figure 2.3. It is seen that both curves increase linearly with increasing content of alumina platelets in the matrix approx-
82 2 Brittle and Ductile Fracture Table 2.1 Characteristics of the pyramidal model related to tortuous cracks in mea- sured specimens Al203 (vol.%) RL 入ppm] 入plum 日m keff.r 0 1.011 373 114 0.0455 0.983 5 1.053 412 171 0.3178 0.924 10 1.199 102 32 0.2040 0.763 15 1.115 341 170 0.2410 0.719 30 1.229 102 128 0.7311 0.714 5 2.0 工 4 1.6 3 1.2 sseu 2 g.. 10.8 RRAS 0.4 0 0.0 0 510 .152025 30 Volume fraction of Al O,platelets V,[%] Figure 2.3 Dependence of the relative surface roughness and the fracture toughness on the volume content of alumina platelets in the glass matrix.Reprinted with permission from Elsevier B.V.(see page 265) imately up to X =15%.The increase in roughness is,unlike that of the fracture toughness,effectively stopped at higher platelet contents.This also means that other mechanism(s)should be acting to counteract the loss of effectiveness of crack deflection here.Typical examples of reconstructed frac- ture surfaces obtained from the profilometric measurement for both 0 and 30 vol%of alumina platelets are depicted in Figure 2.4.It is evident that the fracture surface roughness was significantly increased when alumina platelets were incorporated into the borosilicate glass matrix. At the highest volume fraction of alumina platelets (30 vol%),however, platelet clusters are already observed as shown in Figure 2.5.It seems to be plausible that the crack front interacts with the whole cluster rather than with all its individual platelets.Thus,some particles inside clusters do not directly contribute to the crack front deflection (the surface roughness)
82 2 Brittle and Ductile Fracture Table 2.1 Characteristics of the pyramidal model related to tortuous cracks in measured specimens Al2O3 (vol.%) RL λpp [μm] λpl [μm] Θm k¯eff ,r 0 1.011 373 114 0.0455 0.983 5 1.053 412 171 0.3178 0.924 10 1.199 102 32 0.2040 0.763 15 1.115 341 170 0.2410 0.719 30 1.229 102 128 0.7311 0.714 Figure 2.3 Dependence of the relative surface roughness and the fracture toughness on the volume content of alumina platelets in the glass matrix. Reprinted with permission from Elsevier B.V. (see page 265) imately up to X = 15%. The increase in roughness is, unlike that of the fracture toughness, effectively stopped at higher platelet contents. This also means that other mechanism(s) should be acting to counteract the loss of effectiveness of crack deflection here. Typical examples of reconstructed fracture surfaces obtained from the profilometric measurement for both 0 and 30 vol% of alumina platelets are depicted in Figure 2.4. It is evident that the fracture surface roughness was significantly increased when alumina platelets were incorporated into the borosilicate glass matrix. At the highest volume fraction of alumina platelets (30 vol%), however, platelet clusters are already observed as shown in Figure 2.5. It seems to be plausible that the crack front interacts with the whole cluster rather than with all its individual platelets. Thus, some particles inside clusters do not directly contribute to the crack front deflection (the surface roughness)
2.1 Brittle Fracture 83 .m (a) (b) Figure 2.4 Three-dimensional reconstructed fracture surfaces for 0 and 30 vol%of alumina platelets.Reprinted with permission from Elsevier B.V.(see page 265) 4um Figure 2.5 Clusters of platelets in the sample with 30%reinforcement volume. Reprinted with permission from Elsevier B.V.(see page 265) 2.1.3.2 Theoretical Assessment of the Shielding Effect Besides the roughness-induced shielding(RIS),the crack tip shielding caused by surrounding rigid particles has to also be considered.This effect can be approximately assessed according to results reported in [178,183].In these works,the shielding effect produced by rigid circular particles was analyzed in the frame of the 2D ANSYS model based on the finite element method. The presence of such inclusions generally induces the mixed-mode I+II at the tip of the straight crack. The rigid particles possessed 20 times higher Young's modulus than the matrix.Particles of different sizes (diameter d =6,12,30,60,120,240um)
2.1 Brittle Fracture 83 (a) (b) Figure 2.4 Three-dimensional reconstructed fracture surfaces for 0 and 30 vol% of alumina platelets. Reprinted with permission from Elsevier B.V. (see page 265) Figure 2.5 Clusters of platelets in the sample with 30% reinforcement volume. Reprinted with permission from Elsevier B.V. (see page 265) 2.1.3.2 Theoretical Assessment of the Shielding Effect Besides the roughness-induced shielding (RIS), the crack tip shielding caused by surrounding rigid particles has to also be considered. This effect can be approximately assessed according to results reported in [178, 183]. In these works, the shielding effect produced by rigid circular particles was analyzed in the frame of the 2D ANSYS model based on the finite element method. The presence of such inclusions generally induces the mixed-mode I+II at the tip of the straight crack. The rigid particles possessed 20 times higher Young’s modulus than the matrix. Particles of different sizes (diameter d = 6, 12, 30, 60, 120, 240 μm)