J. Shi, C. Kumar/ Materials Science and Engineering 4250(1998)194-208 3. 1. The interface waviness effect (interface roughnes the quarter of the wavelength which is required for the of 0.1 um) sliding of the fibre peak over the matrix peak to occur. nstead. the first the loading end on fibr results for straight, and matrix stick together, transmitting a large portion three. ten and 30 wave in models with a fixed of the external load from the fibre to the matrix interface roughness are with the emphasis Admittedly, the high pressure experienced there is most placed on the wavelength of surface roughness using a likely to reduce the peak and thus make way for easier fixed surface roughness of 0. 1 um. In the next section liding over. This is not modelled in the present study the discussion will be focused on the amplitude of the Nevertheless, it does suggest that upon loading the surface roughness, using the 30 wave model only. In reduction of asperity may occur at the matrix crack both cases the focus is on the global response and end, and thus relieve the high local high stresses. With interfacial stresses. while in the last sub-section stresses further loading, the reduction process propagates away within the unit cell are discussed from the matrix crack end The stress-strain curves (Fig. 3) show a marked As shown in Fig. 5(a-d), the interface sliding is difference between the smooth and the non-smooth highly non-uniform when the composite is stressed, cases. Firstly, the distinctive straight lines, which corre- hich makes the amount of radial displacement non- spond to the separation of the fibre from the matrix at niform as well. So the assumption of uniform sliding high stresses, have disappeared for the fibre with a may not be valid. rough surface, indicating that the two did not separate Secondly, the hysteresis of the rough fibre shows a Sorensen [8] has proposed that the surface roughness smaller loop width when the number of waves is larger, may account for the positive transverse strain after ading to a lower energy dissipation. With a larger number of ves. the stiffnes of the matrix cracking and interface debonding. Basically, as comes larger. Both are caused by larger interface pres- the peak of surface roughness on the fibre side climbs sure and friction, which makes sliding more difficult from the trough to the peak of the surface roughness on and hence more stress is transferred to the matrix the matrix side(Fig. 4), the matrix is displaced trans- The interface pressure (Fig. Sa and Fig. 5b)and versely by an amount twice the magnitude of the rough hear force(Fig. 5c and Fig. 5d) also demonstrate a ess,resulting in net positive transverse displacement dramatic change, the non-smooth interface exhibits an after matrix contraction due to Poisson effect(see Fig. oscillatory response reflecting the modulation of the interface. Clearly the waviness of the interface and the However, this did not happen in the simulation. In number of waves generate high peak stresses in both fact, the axial relative displacement is far smaller than the fibre and the matrix, and hence have serious impli- cations on the strength of the composite. At the end of the first step, the peak interface pressure distribution (Fig. 5a) is fairly uniform as the relative displacement direction rather than in the longitudinal direction. Be- cause of the uneven surfaces of the fibre and the matrix the amount of relative radial contraction is the largest in the fibre surface troughs (or at the matrix surface peaks). At these positions the fibre contracts the least whereas the matrix shrinks the most. The opposite is rue for those positions at the fibre surface peaks and the matrix surface troughs. In fact, fibre and matrix have separated at these places(Fig. Sc)and the contact pressure is thus zero(Fig. Sa and Fig. 5b). This cer- tainly depends on the degree of asperity The increase in interface pressure with the number of waves' is a direct consequence of wavelength reduction With a fixed longitudinal relative sliding(controlled by the thermal mismatch), the smaller the roughness wave- length, the more radial relative displacement (Fig. 6) and hence the higher the interface pressure. This is Relative sliding and shear stress: (a) initial configuration;(b) confirmed by the slightly higher interface pressure at due to mismatch of thermal expansion coefficient and the the matrix crack (distance=0 in Fig. 5a), where the mechanical loading: (c) slide over longitudinal displacement due to thermal contraction isJ. Shi, C. Kumar / Materials Science and Engineering A250 (1998) 194–208 197 3.1. The interface wa6iness effect (interface roughness of 0.1 mm) In this section, finite element results for straight, three, ten and 30 wave interface models with a fixed interface roughness are discussed with the emphasis placed on the wavelength of surface roughness using a fixed surface roughness of 0.1 mm. In the next section the discussion will be focused on the amplitude of the surface roughness, using the 30 wave model only. In both cases the focus is on the global response and interfacial stresses, while in the last sub-section stresses within the unit cell are discussed. The stress–strain curves (Fig. 3) show a marked difference between the smooth and the non-smooth cases. Firstly, the distinctive straight lines, which corre￾spond to the separation of the fibre from the matrix at high stresses, have disappeared for the fibre with a rough surface, indicating that the two did not separate. Sørensen [8] has proposed that the surface roughness may account for the positive transverse strain after matrix cracking and interface debonding. Basically, as the peak of surface roughness on the fibre side climbs from the trough to the peak of the surface roughness on the matrix side (Fig. 4), the matrix is displaced trans￾versely by an amount twice the magnitude of the rough￾ness, resulting in net positive transverse displacement after matrix contraction due to Poisson effect (see Fig. 4c). However, this did not happen in the simulation. In fact, the axial relative displacement is far smaller than the quarter of the wavelength which is required for the sliding of the fibre peak over the matrix peak to occur. Instead, the first two waves at the loading end on fibre and matrix stick together, transmitting a large portion of the external load from the fibre to the matrix. Admittedly, the high pressure experienced there is most likely to reduce the peak and thus make way for easier sliding over. This is not modelled in the present study. Nevertheless, it does suggest that upon loading the reduction of asperity may occur at the matrix crack end, and thus relieve the high local high stresses. With further loading, the reduction process propagates away from the matrix crack end. As shown in Fig. 5(a–d), the interface sliding is highly non-uniform when the composite is stressed, which makes the amount of radial displacement non￾uniform as well. So the assumption of uniform sliding may not be valid. Secondly, the hysteresis of the rough fibre shows a smaller loop width when the number of waves is larger, leading to a lower energy dissipation. With a larger number of waves, the stiffness of the composite be￾comes larger. Both are caused by larger interface pres￾sure and friction, which makes sliding more difficult and hence more stress is transferred to the matrix. The interface pressure (Fig. 5a and Fig. 5b) and shear force (Fig. 5c and Fig. 5d) also demonstrate a dramatic change, the non-smooth interface exhibits an oscillatory response reflecting the modulation of the interface. Clearly the waviness of the interface and the number of waves generate high peak stresses in both the fibre and the matrix, and hence have serious impli￾cations on the strength of the composite. At the end of the first step, the peak interface pressure distribution (Fig. 5a) is fairly uniform as the relative displacement between the fibre and the matrix is mainly in the radial direction rather than in the longitudinal direction. Be￾cause of the uneven surfaces of the fibre and the matrix, the amount of relative radial contraction is the largest in the fibre surface troughs (or at the matrix surface peaks). At these positions the fibre contracts the least, whereas the matrix shrinks the most. The opposite is true for those positions at the fibre surface peaks and the matrix surface troughs. In fact, fibre and matrix have separated at these places (Fig. 5c) and the contact pressure is thus zero (Fig. 5a and Fig. 5b). This cer￾tainly depends on the degree of asperity. The increase in interface pressure with the number of ‘waves’ is a direct consequence of wavelength reduction. With a fixed longitudinal relative sliding (controlled by the thermal mismatch), the smaller the roughness wave￾length, the more radial relative displacement (Fig. 6) and hence the higher the interface pressure. This is confirmed by the slightly higher interface pressure at the matrix crack (distance=0 in Fig. 5a), where the longitudinal displacement due to thermal contraction is Fig. 4. Relative sliding and shear stress: (a) initial configuration; (b) sliding due to mismatch of thermal expansion coefficient and the mechanical loading; (c) slide over