Chapter 1:Account for Random Microstructure 11 Fig.1.3.Variations in local volume fraction of fibres(see Fig.1.1) Obviously,the size of Vo will affect the level of the local volume fraction; but even for the same /o it depends on x.All the moments of this variable are studied for various systems in [51]. Two schemes are used in [52]to estimate variability of the volume fraction in carbon-epoxy T300/914 unidirectional composite.For macro- scopic specimens with a cross-section 10 x 1 mm,a direct measurement of the Young's modulus of the composite is employed to calculate Vr using a linear rule of mixtures.An image analysis of fields 0.1 x 0.1 mm provides the data for direct estimation of the volume fraction of fibres (and its variation).In both cases,the distributions peak at (or close to)the nominal volume fraction V=0.6.The increase in the window dimension results in the decrease in the scatter;still,even for a macroscopic specimens of the first method the measured interval of Vr was from 0.5 to 0.68. Another analysis is performed for a micrograph of the ply's cross- sectional area of a carbon/epoxy composite,containing 603 fibres with diameter d=10μm[48];the size of the window is345×250μum(its part is shown in Fig.1.1).With the increase in the window size,the distribution of local magnitudes of Vr changes its shape and bounds-maximal (V) and minimal(Vmi).The respective evolution of these bounds is shown in Fig.1.4.For sufficiently small window size,these two bonds demonstrate mono-phase asymptotes:Vmax1 and Vmin0.With the increase in the window size,both bounds should converge to the average value yax→可and Vmin→可. (1.20)
11 Fig. 1.3. Variations in local volume fraction of fibres (see Fig. 1.1) Obviously, the size of V0 will affect the level of the local volume fraction; but even for the same V0 it depends on x. All the moments of this variable are studied for various systems in [51]. Two schemes are used in [52] to estimate variability of the volume fraction in carbon–epoxy T300/914 unidirectional composite. For macroscopic specimens with a cross-section 10 × 1 mm, a direct measurement of the Young’s modulus of the composite is employed to calculate Vf using a linear rule of mixtures. An image analysis of fields 0.1 × 0.1 mm provides the data for direct estimation of the volume fraction of fibres (and its variation). In both cases, the distributions peak at (or close to) the nominal volume fraction Vf = 0.6. The increase in the window dimension results in the decrease in the scatter; still, even for a macroscopic specimens of the first method the measured interval of Vf was from 0.5 to 0.68. Another analysis is performed for a micrograph of the ply’s crosssectional area of a carbon/epoxy composite, containing 603 fibres with diameter d f = 10 µm [48]; the size of the window is 345 × 250 µm (its part is shown in Fig. 1.1). With the increase in the window size, the distribution of local magnitudes of Vf changes its shape and bounds – maximal max f ( ) V and minimal min f ( ) V . The respective evolution of these bounds is shown in Fig. 1.4. For sufficiently small window size, these two bonds demonstrate mono-phase asymptotes: max f V →1 and min f V → 0 . With the increase in the window size, both bounds should converge to the average value max min f f ff VV VV → → and . (1.20) Chapter 1: Account for Random Microstructure
12 V.V.Silberschmidt Though this trend is distinct in Fig.1.4,the full convergence of the bounds is not reached even at the length scale of 115 um. The spatial variation in the volume fraction of fibres causes con- siderable variations in the local values of stiffness:For the window size 30 um,the axial and shear moduli demonstrate the scatter of more than 100%and the transverse module more than 40%. An important parameter of local variations of the volume fraction of fibres Vr,treated as a random variable,can be linked to the standard deviation of its local magnitude.Such a parameter,named coarseness C,is introduced in [15,36]as a the standard deviation of the volume fraction of filaments normalised by its mean value (1.21) 09 ◆-max o—min average 0.1 0 山 0 20 40 60 80 100 120 window size,um Fig.1.4.Evolution of bounds for local volume fraction with window size The change of coarseness C with the window size,calculated for the arrangement of 603 fibres that was treated before(see Fig.1.4),is pre- sented in Fig.1.5.Obviously,C=0 for an infinite area
Though this trend is distinct in Fig. 1.4, the full convergence of the bounds is not reached even at the length scale of 115 µm. The spatial variation in the volume fraction of fibres causes considerable variations in the local values of stiffness: For the window size 30 µm, the axial and shear moduli demonstrate the scatter of more than 100% and the transverse module more than 40%. An important parameter of local variations of the volume fraction of fibres Vf, treated as a random variable, can be linked to the standard deviation of its local magnitude. Such a parameter, named coarseness C, is introduced in [15, 36] as a the standard deviation of the volume fraction of filaments normalised by its mean value Vf 2 2 f f f 1 C VV . V = 〈 〉− (1.21) Fig. 1.4. Evolution of bounds for local volume fraction with window size The change of coarseness C with the window size, calculated for the arrangement of 603 fibres that was treated before (see Fig. 1.4), is presented in Fig. 1.5. Obviously, C = 0 for an infinite area. 12 V.V. Silberschmidt