MIATERIALS MENE& ENGEERNG ELSEVIER Materials Science and Engineering A250(1998)194-208 The effect of fibre surface roughness on the mechanical behaviour of ceramic matrix composites J. Shi*.c. Kumar methods Group, Aero and Technology Products, Mechanical Engineering Centre, GEC-ALSTHOM, Cambridge Road, Whetstone, Leicester LE8 6LH. UK Abstract The interface between the matrix and the fibre plays an important role in controlling the strength and toughness of ceramic matrix composites. It has been found experimentally that depending on manufacturing process, the interface may show substantial surface roughness, which has been modelled analytically with certain degree of success. The analytical models, however, do not take into account the interface geometry. Instead, only the magnitude of the surface undulation is included. In this paper, a direct simulation of fibre-matrix interface roughness by the finite element method is performed on an axisymmetric unit cell with a fully debonded interface. The simulation is employed to account for the three dimensional stress state, surface roughness and interface friction, which are normally simplified or idealised in theoretical studies. The model gives the highly non-uniform interface shear and pressure, which have direct implications on the interface damage and composite behaviour. Under the approximation made in the model, the positive transverse strains does not show up in the simulation despite the fact that two different surface roughness are used. o 1998 Elsevier Science S.A. All rights reserved Keywords: Interface roughness; Simulation 1. Introduction experimental characterisation, macro-mechanics and micro-mechanics analysis. Although it is believed that a For aeroengines to achieve a thrust to weight ratio of comprehensive understanding of CMCs is only possible 15: I by the turn of this century, materials of high by combining all these approaches, the micro-mechan- specific stiffness and strength, which also retain these cs is important in interpreting experimental findings, properties in a highly corrosive, erosive and oxidiser building macro-mechanics models and guiding material operating environment, are required. Such materials are designs. Consequently a large number of micro-me- also needed to increase engine thermal efficiency and to chanics models have been put forward to appreciate the decrease CO and NOx emissions in view of the tighter fundamental deformation and failure mechanisms of nd tighter regulations on environmental impact. Ce CMCs. Some of these studies have been concentrated ramic matrix composites(CMCs) have potential for on the damage and energy dissipation mechanisms, almost all of these stringent requirements and thus hich have direct impact on strength and toughness stand out as strong candidates for future engine compo- The interface between fibre and matrix plays an nents. In consequence, CMCs are widely recognised as important role in controlling the toughness, ultimate key materials for the next generation of gas turbine tensile strength and fatigue life of CMCs. In conse- engines. quence, there has been extensive study on various prop. Before large scale application of these relatively new erties of the interface and their effect on composite materials is possible, a clear understanding of thei performance. Experimentally, a number of tests have thermal and mechanical properties is essential. This has been devised to determine the interface shear strength been approached from a number of angles in the past: and slide stress [l]. Among these, the single fibre push in/push-out test seems to be the most popular. Various Corresponding author. Tel +44 116 2750750 fax: +44 116 theoretical models exist in order to model the tests and 750768 to interpret the results e.g. Ref [2]. One interesting 0921-5093/98/S1900c 1998 Elsevier Science S.A. All rights reserved PIs0921-5093(98)00592-9
Materials Science and Engineering A250 (1998) 194–208 The effect of fibre surface roughness on the mechanical behaviour of ceramic matrix composites J. Shi *, C. Kumar Methods Group, Aero and Technology Products, Mechanical Engineering Centre, GEC-ALSTHOM, Cambridge Road, Whetstone, Leicester LE8 6LH, UK Abstract The interface between the matrix and the fibre plays an important role in controlling the strength and toughness of ceramic matrix composites. It has been found experimentally that depending on manufacturing process, the interface may show substantial surface roughness, which has been modelled analytically with certain degree of success. The analytical models, however, do not take into account the interface geometry. Instead, only the magnitude of the surface undulation is included. In this paper, a direct simulation of fibre–matrix interface roughness by the finite element method is performed on an axisymmetric unit cell with a fully debonded interface. The simulation is employed to account for the three dimensional stress state, surface roughness and interface friction, which are normally simplified or idealised in theoretical studies. The model gives the highly non-uniform interface shear and pressure, which have direct implications on the interface damage and composite behaviour. Under the approximation made in the model, the positive transverse strains does not show up in the simulation despite the fact that two different surface roughness are used. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Interface roughness; Simulation 1. Introduction For aeroengines to achieve a thrust to weight ratio of 15:1 by the turn of this century, materials of high specific stiffness and strength, which also retain these properties in a highly corrosive, erosive and oxidising operating environment, are required. Such materials are also needed to increase engine thermal efficiency and to decrease CO and NOX emissions in view of the tighter and tighter regulations on environmental impact. Ceramic matrix composites (CMCs) have potential for almost all of these stringent requirements and thus stand out as strong candidates for future engine components. In consequence, CMCs are widely recognised as key materials for the next generation of gas turbine engines. Before large scale application of these relatively new materials is possible, a clear understanding of their thermal and mechanical properties is essential. This has been approached from a number of angles in the past: experimental characterisation, macro-mechanics and micro-mechanics analysis. Although it is believed that a comprehensive understanding of CMCs is only possible by combining all these approaches, the micro-mechanics is important in interpreting experimental findings, building macro-mechanics models and guiding material designs. Consequently a large number of micro-mechanics models have been put forward to appreciate the fundamental deformation and failure mechanisms of CMCs. Some of these studies have been concentrated on the damage and energy dissipation mechanisms, which have direct impact on strength and toughness. The interface between fibre and matrix plays an important role in controlling the toughness, ultimate tensile strength and fatigue life of CMCs. In consequence, there has been extensive study on various properties of the interface and their effect on composite performance. Experimentally, a number of tests have been devised to determine the interface shear strength and slide stress [1]. Among these, the single fibre pushin/push-out test seems to be the most popular. Various theoretical models exist in order to model the tests and to interpret the results e.g. Ref [2]. One interesting * Corresponding author. Tel.: +44 116 2750750; fax: +44 116 2750768 0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S09 21- 5093(98)0059 2 - 9
J Shi, C. Kumar/ Materials Science and Engineering 4250(1998)194-208 phenomenon observed in these tests is the load drop in P=157E03N/山m re-seating due to the surface roughness [3, 4 To account for the 3d stress field and to include the frictional effect. finite element models have been devel- ped (see e.g. [5-7D). In particular, a unit cell corre 0.00015m sponding to a unidirectional CMC, with a fully bridged matrix crack and complete matrix-fibre debonding was studied in great detail by Sorensen [5]. A deep FibreMatrix insight was gained into the load transfer between the Fibre E=0.20N/um2 fibre and the matrix. The finite element study [5] was =0.35 based on a smooth fibre-matrix interface. whereas the a=3.0E06/C interface roughness was investigated analytically in [8]. 1267r and it was shown that interface roughness affects the Matrix E=0.098 N/um overall response of CMCs. The analytical solution pro- v=0.3 a=50E06C vides useful qualitative information of the surface roughness effects. However, a quantitative knowledge of these effects is clearly desirable In these investigations only the surface roughness amplitude at most is considered, while surface shape is ignored. Furthermore uniform sliding of the interface has been assumed in analytical models to simplify the analysis. In this paper, the surface roughness is approx 5gm516985μm imated by sinusoidal waves and accounted for explicitly Fig. l. Unit cell model of the composite with fully cracked matrix and by direct finite element modelling. (With the limit of interface memory size and speed for todays computers, unlikely to simulate a true or measured interface com pletely and successfully. The global response, interfa- unit cell is worked out from the volume fraction ratio cial shear and pressure as well as local stress states, of 0.35. The length of the model is 80 um, which hich are not available from an analytical solution, will be examined in detail probably agrees well with the saturated matrix crack spacing. Only fully cracked matrix and interface are modelled and the fibre is assumed to remain intact This suggests that the composite behaviour beyond 2. The unit cell model and its finite element proportional limit, but below ultimate tensile strength is representation studied The general finite element analysis package In this paper, a unit cell(Fig. 1)similar to that of ABAQUS was chosen for its contact modelling Sorensen [5] was used, the only difference being that the capability. Four-noded axisymmetric element was used interface consists of sine waves. to simulate the surface to model the unit cell, as the eight node element gave roughness. The unit cell is made of two concentric strong stress oscillation and hence was not used. The cylinders: the inner one corresponds to the fibre, while he outer one corresponds to the matrix. No attempt has been made to model the interface layer, which is so thin that it does not make any contribution to the composite stiffness, particularly when the interface fully debonded In the present study, the interface is made of three, ten and 30 sine waves the magnitude of which is 0. 1 or 0.01 um. In reality, the surface roughness depends on the manufacturing process and can vary a great deal [3] Further. the in situ interface morphology compared with as fabricated fibre. The surface roughness causes uneven pressure and friction istribution which leads to differential wear and reduc Fig. 2. Finite element model of the unit Tor different types of tion of asperity along the interface. But neither of these interfaces: (a)straight interface: (b)three sine waves: (c) ten sine complications are considered
J. Shi, C. Kumar / Materials Science and Engineering A250 (1998) 194–208 195 phenomenon observed in these tests is the load drop in re-seating due to the surface roughness [3,4]. To account for the 3D stress field and to include the frictional effect, finite element models have been developed (see e.g. [5–7]). In particular, a unit cell corresponding to a unidirectional CMC, with a fully bridged matrix crack and complete matrix–fibre debonding, was studied in great detail by Sørensen [5]. A deep insight was gained into the load transfer between the fibre and the matrix. The finite element study [5] was based on a smooth fibre–matrix interface, whereas the interface roughness was investigated analytically in [8], and it was shown that interface roughness affects the overall response of CMCs. The analytical solution provides useful qualitative information of the surface roughness effects. However, a quantitative knowledge of these effects is clearly desirable. In these investigations only the surface roughness amplitude at most is considered, while surface shape is ignored. Furthermore uniform sliding of the interface has been assumed in analytical models to simplify the analysis. In this paper, the surface roughness is approximated by sinusoidal waves and accounted for explicitly by direct finite element modelling. (With the limit of memory size and speed for today’s computers, it is unlikely to simulate a true or measured interface completely and successfully.) The global response, interfacial shear and pressure as well as local stress states, which are not available from an analytical solution, will be examined in detail. 2. The unit cell model and its finite element representation In this paper, a unit cell (Fig. 1) similar to that of Sørensen [5] was used, the only difference being that the interface consists of sine waves, to simulate the surface roughness. The unit cell is made of two concentric cylinders: the inner one corresponds to the fibre, while the outer one corresponds to the matrix. No attempt has been made to model the interface layer, which is so thin that it does not make any contribution to the composite stiffness, particularly when the interface is fully debonded. In the present study, the interface is made of three, ten and 30 sine waves, the magnitude of which is 0.1 or 0.01 mm. In reality, the surface roughness depends on the manufacturing process and can vary a great deal [3]. Further, the in situ interface may have a different morphology compared with as fabricated fibre. The surface roughness causes uneven pressure and friction distribution, which leads to differential wear and reduction of asperity along the interface. But neither of these complications are considered. Fig. 1. Unit cell model of the composite with fully cracked matrix and interface. The fibre diameter is 15 mm, while the diameter of the unit cell is worked out from the volume fraction ratio of 0.35. The length of the model is 80 mm, which probably agrees well with the saturated matrix crack spacing. Only fully cracked matrix and interface are modelled and the fibre is assumed to remain intact. This suggests that the composite behaviour beyond proportional limit, but below ultimate tensile strength is studied. The general finite element stress analysis package ABAQUS was chosen for its good contact modelling capability. Four-noded axisymmetric element was used to model the unit cell, as the eight node element gave strong stress oscillation and hence was not used. The Fig. 2. Finite element model of the unit cell for different types of interfaces: (a) straight interface; (b) three sine waves; (c) ten sine waves; (d) 30 sine waves
J. Shi, C. Kumar/ Materials Science and Engineering 4250(1998)194-208 Table I Details of finite element unit cell models Smooth interface Three wave interface Ten wave interface 30 wave interface ber of elements 1382 4142 ber of nodes 1524 30wave long -O owave rad ◇…10 wave rad 。-3 wave rad 0.8 Strain(%) Fig. 3. Effect of interface roughness(0.I um) on stress-strain curves. number of elements in the model is decided by the The fibre is constrained radially at the left hand number of points required to represent a sine wave vertical side and axially at the bottom side, whereas the ccurately. To save computer time and memory matrix is constrained at the bottom axially but at the each full sine wave has the minimum number of nine right hand vertical side it is constrained in such a way nodes. Consequently there are four finite element mod- that it remains vertical sponding to the straight interface, three, ten and Exactly the same thermal and mechanical loads and 30 wave interface models(see Fig. 2). The sizes of these models are indicated in Table 1. The three wave and boundary conditions were applied as in Ref [5].After an initial thermal step, representing the cooling down from large roughness model represents fibres of long wave- manufacturing temperature, the unit cell is loaded in the length and high undulations, while the 30 wave and fibre direction. It is then unloaded and loaded again. At small undulation model mimics fibres of short wave- the top surface of the fibre, pressure is applied.The length and small undulations, as reported in Ref. [3] To avoid the confusion caused by round-off errors in magnitude (1.57 X 10-3 N um-2) is equivalent to a deciding overclosure, an artificial gap of 0.00015 um composite remote stress of 550 MPa. The temperature between the fibre and the matrix is introduced in the drops from I000° to room temperature(20°Cto finite element model. The artificial gap is so small that simulate the thermal residual stress induced in manu- it does not affect the results. A surface to surface finite facturing. The higher thermal expansion coefficient sliding option in ABAQUS is invoked, and friction is of the matrix leads to a clamping pressure at the described by the classical Coulomb law interface
196 J. Shi, C. Kumar / Materials Science and Engineering A250 (1998) 194–208 Table 1 Details of finite element unit cell models Smooth interface Three wave interface Ten wave interface 30 wave interface Number of elements 1382 462 462 4142 Number of nodes 524 524 1524 4526 Fig. 3. Effect of interface roughness (0.1 mm) on stress–strain curves. number of elements in the model is decided by the number of points required to represent a sine wave accurately. To save computer time and memory size, each full sine wave has the minimum number of nine nodes. Consequently there are four finite element models, corresponding to the straight interface, three, ten and 30 wave interface models (see Fig. 2). The sizes of these models are indicated in Table 1. The three wave and large roughness model represents fibres of long wavelength and high undulations, while the 30 wave and small undulation model mimics fibres of short wavelength and small undulations, as reported in Ref. [3]. To avoid the confusion caused by round-off errors in deciding overclosure, an artificial gap of 0.00015 mm between the fibre and the matrix is introduced in the finite element model. The artificial gap is so small that it does not affect the results. A surface to surface finite sliding option in ABAQUS is invoked, and friction is described by the classical Coulomb law. The fibre is constrained radially at the left hand vertical side and axially at the bottom side, whereas the matrix is constrained at the bottom axially but at the right hand vertical side it is constrained in such a way that it remains vertical. Exactly the same thermal and mechanical loads and boundary conditions were applied as in Ref. [5]. After an initial thermal step, representing the cooling down from manufacturing temperature, the unit cell is loaded in the fibre direction. It is then unloaded and loaded again. At the top surface of the fibre, pressure is applied. The magnitude (1.57×10−3 N mm−2 ) is equivalent to a composite remote stress of 550 MPa. The temperature drops from 1000°C to room temperature (20°C) to simulate the thermal residual stress induced in manufacturing. The higher thermal expansion coefficient of the matrix leads to a clamping pressure at the interface
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 is
J. 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 correspond 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 transversely by an amount twice the magnitude of the roughness, 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 nonuniform 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 becomes larger. Both are caused by larger interface pressure 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 implications 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. Because 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 certainly 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 wavelength, 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
J. Shi, C. Kumar/ Materials Science and Engineering 4250(1998)194-208 3.50E+02 -10 wave - -----3 wave traight 3.00E+02 2.50E+0 8200E+02 1.50E+02 H群 500E+01 0.00E+00 (a) Distance along contact (um) 1.60E+03 1.40E+03 30 wave 8800E+02 6.00E+02 2.00E+02 00600 Distance along contact (um) Fig. 5.(a) Interface pressure at the end of the first step(interface roughness=0.I um).(b)Interface shear at the end of the first step(interface roughness=0.1 um).(c)Interface pressure at the end of the second step(interface roughness =0. 1 um).(d) Interface shear at the end of the second step (interface roughness=0. 1 um)
198 J. Shi, C. Kumar / Materials Science and Engineering A250 (1998) 194–208 Fig. 5. (a) Interface pressure at the end of the first step (interface roughness=0.1 mm). (b)Interface shear at the end of the first step (interface roughness=0.1 mm). (c)Interface pressure at the end of the second step (interface roughness=0.1 mm). (d) Interface shear at the end of the second step (interface roughness=0.1 mm)
J Shi, C. Kumar/ Materials Science and Engineering 4250(1998)194-208 1.10E+02 100E+02 900E+01 8.00E+01 -10 wave 7.00E+01 E600E+01 straight 500E+01 300E+01 200E+01 100E+01 0.00E+00 时人A Distance along contact (um) 3.50E+02 10 ave 300E+02 straight 2.50E+02 6200E+02 150E+02 100E+02 4.00E05 Distance along contact (um)
J. Shi, C. Kumar / Materials Science and Engineering A250 (1998) 194–208 199 Fig. 5. (Continued)
J Shi, C. Kumar/ Materials Science and Engineering 4250(1998)194-208 which is confirmed by the interface shear). For the first step, the contact shear stress(Fig. 5b)equals to the interface pressure(Fig. 5a) multiplied by the fric- tion coefficient (0.3), and hence sliding has occurred only for one quarter of the interface. For the second step, a larger portion of the interface has slid The interface pressure at the end of step I for a model is given in Fig. 7. The analytical solution used was obtained by Evans et al [9]. In the solution an empirical constant B is used, to account for the partial contact due to surface roughness, but he actual magnitude is not given. In Fig. 7, it can be seen that the analytical solution with B=l is much higher than the finite element solution. In this partic g rface with a short ngth (a) has a larger relative ular case, B should be s0. 4 to have agreement be acement than that with a long wavelength(b) tween the two solutions. However, in general, the interface pressure distribution is highly localised as can be seen in Fig 5a and Fig 5c. In consequence, B the largest, whereas at the other end (distance= 80 in should vary along the interface and it is not longer a Fig. 5a), the smallest displacement is expected and constant thus a lower pressure. When the fibre (or composite since the matrix is 3. 2. The effect of the surface roughness magnitude(30 cracked)is loaded up to 550 MPa, the peak interface wave interface) pressure(Fig. 5c) becomes highly non-uniform. The stress applied on the fibre at the matrix crack end To investigate the effect of the surface roughness makes the fibre slide more over the matrix than ther- magnitude, a second finite element model was consid mal contraction in the first step. In consequence the ered, with a surface roughness magnitude of 0.01 um, interface pressure is much greater. Since the current as opposed to 0. 1 um, and the number of waves is model does not allow interface wear and a large in- fixed at 30. The stress-strain curves for the two terface roughness is used, the pressure becomes exces- rough interface models are plotted in Fig. 8 together ively high (A 1.6 GPa, the strength of the fibre) with the smooth interface model. Clearly, the larger Nevertheless, the highest interface pressure is expected roughness model leads to the stiffest response, which to be at the matrix crack end, and as a result the is not surprising as the larger roughness makes it wear damage is more sever there (assuming that a more difficult for the fibre to slide over the matrix sliding rather than sticking contact condition applies, As a result, a complete load transfer from the fibre to Analyt Interface shape 626 Top→> Fig. 7. Analytical and finite element interface pressure at the end of the first step(0. I um) for a 30 wave interface model
200 J. Shi, C. Kumar / Materials Science and Engineering A250 (1998) 194–208 Fig. 6. Interface with a short wavelength (a) has a larger relative radial displacement than that with a long wavelength (b). which is confirmed by the interface shear). For the first step, the contact shear stress (Fig. 5b) equals to the interface pressure (Fig. 5a) multiplied by the friction coefficient (0.3), and hence sliding has occurred only for one quarter of the interface. For the second step, a larger portion of the interface has slid. The interface pressure at the end of step 1 for a ten wave model is given in Fig. 7. The analytical solution used was obtained by Evans et al [9]. In the solution an empirical constant b is used, to account for the partial contact due to surface roughness, but the actual magnitude is not given. In Fig. 7, it can be seen that the analytical solution with b=1 is much higher than the finite element solution. In this particular case, b should be :0.4 to have agreement between the two solutions. However, in general, the interface pressure distribution is highly localised as can be seen in Fig. 5a and Fig. 5c. In consequence, b should vary along the interface and it is not longer a constant. 3.2. The effect of the surface roughness magnitude (30 wa6e interface) To investigate the effect of the surface roughness magnitude, a second finite element model was considered, with a surface roughness magnitude of 0.01 mm, as opposed to 0.1 mm, and the number of waves is fixed at 30. The stress–strain curves for the two rough interface models are plotted in Fig. 8 together with the smooth interface model. Clearly, the larger roughness model leads to the stiffest response, which is not surprising as the larger roughness makes it more difficult for the fibre to slide over the matrix. As a result, a complete load transfer from the fibre to the largest, whereas at the other end (distance=80 in Fig. 5a), the smallest displacement is expected and thus a lower pressure. When the fibre (or composite since the matrix is cracked) is loaded up to 550 MPa, the peak interface pressure (Fig. 5c) becomes highly non-uniform. The stress applied on the fibre at the matrix crack end makes the fibre slide more over the matrix than thermal contraction in the first step. In consequence the interface pressure is much greater. Since the current model does not allow interface wear and a large interface roughness is used, the pressure becomes excessively high (:1.6 GPa, the strength of the fibre) Nevertheless, the highest interface pressure is expected to be at the matrix crack end, and as a result the wear damage is more sever there (assuming that a sliding rather than sticking contact condition applies, Fig. 7. Analytical and finite element interface pressure at the end of the first step (0.1 mm) for a 30 wave interface model
J Shi, C. Kumar/ Materials Science and Engineering 4250(1998)194-208 U2 rough interface(0.1) --U2 smaller interface(0.01) -x-U2 straight interface -*-U1 rough interface(0. 1) HU1 smaller interface (0.01) ▲U1 straight interface strain(%) Fig 8. The effect of surface roughness magnitude on stress-strain curve the matrix is possible and more material is thus due to the large pulling force applied on the fibre stressed. It can be envisaged that as the surface which made a larger part of the fibre to slide. In the oughness approaches zero, the rough interface results middle part of the fibre the smaller roughness model should approach those of the smooth interface model. gives a higher pressure, whereas for the bottom end moreover, when the surface roughness is reduced, it the two models give the same interface pressure should be easier for the matrix and the fibre to slide When the fibre is unloaded in the third step, much relative to each other. However, any surface rough- of the interface pressure created in the second step is ness significantly smaller than 0.01 um seems to be 'locked-in, even though the external loading becomes not meaningful in practice [3]. Hence no further simu- zero. This is typical for a frictional contact problem lation is performed as illustrated in Fig. 10. The reverse sliding during The interface pressure and shear for the 30 wave unloading is smaller than the forward sliding during model at the end of steps 1, 2 and 3, are given in loading, which has actually resulted in the permanent Fig 9. At the end of the first thermal step, the inter- strain and the hystersis in the stress-strain curve. Be- face pressure is higher for the smaller roughness cause of the smaller reverse sliding and the surface model, as it has a smaller slope. In consequence, the roughness, the matrix remains displaced radially after resistance to sliding resulting from the pressure unloading, and hence the large interface pressure. The smaller. In other words, a larger normal pressure can interface shear plot(Fig. 9f) shows that while the top be tolerated. Despite the fairly uniform pressure dis- 25% of the fibre tends to slide back into the matrix, tortion, the shear stress is concentrated at the matrix the remaining 75% still tries to slide in the opposite crack end. In both cases, about 20% of fibre(matrix) direction. This suggests that the majority of the slid has reached sliding stage, while the rest remains ing damage occurs in this small zone. As the axial ticking state. In the latter part, the shear stresses in loading is zero, it can be seen that the shear stresses both the fibre and the matrix are nearly the same (area under the curve)in the two parts balance each At the end of the second step, in which a mechani- other. Again, the larger roughness model predicts cal load is applied on the fibre to pull it through a higher interface pressure. As shown in [10] the matrix, the interface pressure is higher for the bigger interface shear strongly influences the roughness model at the matrix end. This is mainly timate tensile strength. It can be envisaged that
J. Shi, C. Kumar / Materials Science and Engineering A250 (1998) 194–208 201 Fig. 8. The effect of surface roughness magnitude on stress–strain curve. the matrix is possible and more material is thus stressed. It can be envisaged that as the surface roughness approaches zero, the rough interface results should approach those of the smooth interface model. Moreover, when the surface roughness is reduced, it should be easier for the matrix and the fibre to slide relative to each other. However, any surface roughness significantly smaller than 0.01 mm seems to be not meaningful in practice [3]. Hence no further simulation is performed. The interface pressure and shear for the 30 wave model at the end of steps 1, 2 and 3, are given in Fig. 9. At the end of the first thermal step, the interface pressure is higher for the smaller roughness model, as it has a smaller slope. In consequence, the resistance to sliding resulting from the pressure is smaller. In other words, a larger normal pressure can be tolerated. Despite the fairly uniform pressure distortion, the shear stress is concentrated at the matrix crack end. In both cases, about 20% of fibre (matrix) has reached sliding stage, while the rest remains in sticking state. In the latter part, the shear stresses in both the fibre and the matrix are nearly the same. At the end of the second step, in which a mechanical load is applied on the fibre to pull it through the matrix, the interface pressure is higher for the bigger roughness model at the matrix end. This is mainly due to the large pulling force applied on the fibre which made a larger part of the fibre to slide. In the middle part of the fibre the smaller roughness model gives a higher pressure, whereas for the bottom end the two models give the same interface pressure. When the fibre is unloaded in the third step, much of the interface pressure created in the second step is ‘locked-in’, even though the external loading becomes zero. This is typical for a frictional contact problem as illustrated in Fig. 10. The reverse sliding during unloading is smaller than the forward sliding during loading, which has actually resulted in the permanent strain and the hystersis in the stress–strain curve. Because of the smaller reverse sliding and the surface roughness, the matrix remains displaced radially after unloading, and hence the large interface pressure. The interface shear plot (Fig. 9f) shows that while the top 25% of the fibre tends to slide back into the matrix, the remaining 75% still tries to slide in the opposite direction. This suggests that the majority of the sliding damage occurs in this small zone. As the axial loading is zero, it can be seen that the shear stresses (area under the curve) in the two parts balance each other. Again, the larger roughness model predicts a higher interface pressure. As shown in [10] the interface shear strongly influences the ultimate tensile strength. It can be envisaged that
J Shi, C. Kumar/ Materials Science and Engineering 4250(1998)194-208 Interface roughness=0.lum rface roughness=0.01 300 8200 Distance along contact (um) Interface roughness =0. 1um UIItUH IAA AA人人人人 Distance along conta Fig 9.(a)The effect of surface roughness magnitude on interface pressure(step I).(b) The effect of surface roughness magnitude on interface hear(step 1).(c) The effect of surface roughness magnitude on interface pressure(step 2).(d) The effect of surface roughness magnitude on interface shear(step 2).(e) The effect of surface roughness magnitude on interface pressure(step 3).(f) The effect of surface roughness magnitude
202 J. Shi, C. Kumar / Materials Science and Engineering A250 (1998) 194–208 Fig. 9. (a) The effect of surface roughness magnitude on interface pressure (step 1). (b) The effect of surface roughness magnitude on interface shear (step 1). (c) The effect of surface roughness magnitude on interface pressure (step 2). (d) The effect of surface roughness magnitude on interface shear (step 2). (e) The effect of surface roughness magnitude on interface pressure (step 3). (f) The effect of surface roughness magnitude on interface shear (step 3)
J Shi, C. Kumar/ Materials Science and Engineering 4250(1998)194-208 Interface roughness=0. lum Interface roughness=0.01um e3%5 连AAAA 50 Distance along contact (um) -- Interface roughness =0. lum Interface roughness=0. olum Distance along contact (um) for CMCs whose debonded interface exhibits a large 3.3. Stress distribution within the unit cell roughness it would give a high strength. However, the failure strain would be smaller due to a large So far only stresses along the interface have been stiffness discussed. The radial and axial direct stresses plus the
J. Shi, C. Kumar / Materials Science and Engineering A250 (1998) 194–208 203 Fig. 9. (Continued) for CMCs whose debonded interface exhibits a large roughness it would give a high strength. However, the failure strain would be smaller due to a large stiffness. 3.3. Stress distribution within the unit cell So far only stresses along the interface have been discussed. The radial and axial direct stresses plus the