1050 rensen r talre pull out. This is incorrect, since matrix cracking, the matrix before matrix cracking are proportional ibre/matrix debonding, fibre sliding and fibre fail- to the axial strain of the composite. This approach ure continue to take place during this part of the also makes it easier to compare behaviour of bending response, as the failure locus moves unidirectional and cross-ply composites. For the across the depth of the specimen towards the com- SiC/CAS II composite shown in Fig. 3, U was pression side measured to be 3. 1 MJ/m, which is almost 40 Uniaxial tension experiments, on the other times that of the pure matrix material. However, hand,allow an unambiguous interpretation, since due to loading rate cffcct h the damage mechanisms initiate and develop at stress-strain behaviour, the value for SIC/CAS II identifiable stress or strain levels, 5-19 and take can vary from 1.7 MJ/m(at 0.01 MPa/s)to 5 4 place uniformly over the entire volume of the gage MJ/m3(500 MPa/s) section of the specimen The non-linear stress-strain curve reflects the Unfortunately, tensile tests are difficult to various stages of damage( see ref. 19 for a detailed perform, since an accurate alignment is needed to discussion). At low strain the material response is prevent bending effects. 4 Also, although tensile linearly elastic and reversible( Stage I). At highe tests are performed with much care, it is not strain matrix cracks initiate and evolve into fairly unusual that failure takes place at tabs so the true regularly spaced multiple matrix cracks( Stage in) strength of the composite cannot always be mea- This takes place over a certain strain range, rather sured. However, if there is no bending in the test than at a specific strain value. Fibre/mat section, then the relationship between stress, strain debonding also takes place during Stage Il, and and damage mechanisms can readily be identified. facilitates frictional sliding along the fibre/matrix interface. These damage processes cause the non 2.4 Experimental characterization of damage in lincar part of the strcss-strain curve. The corre- uniaxial tension sponding changes in the overall unloading Fi ental stress-strain curve modul of a hot pressed unidirectional Sic-fibre-rein- be modelled by a continuum damage model forced calcium alumino silicate(denoted CAS II Increasing the applied stress further leads to a se from Corning Inc, NY, USA)glass-ceramic ond linear response( Stage III), where the stiffness matrix composite, tested in uniaxial tension along is mainly due to the fibres. However, significant fibres at a strain rate of 4 x 104 min hysteresis appears during unloading due to effects Throughout this paper we will refer the evolution from interfacial friction. For some specimens a of damage to the axial strain of the composite small amount of distributed fibre fracture( Stage rather than the composite stress, since the axial IV)may take place just prior to failure. Final fail stress component in the fibres resulting from the ure occurs by localized fibre fracture. Recalling external load and the axial stress component in the preceding discussion, the area under the stress-strain curve represents material has absorbed per unit volume of the composite up to failure. The energy quantity mea sured by U is the energy that has been absorbed Matrix by distributed mechanisms. Mechanisms operating (MPa) Initiation during the localized fracture, such as fibre fracture and pull out do not contribute to this quantity 3 Model of Energy Uptake 3.1 Basic consideratior 200 In the fol III IV nechanisms of a unidirectional ceramic fibre rein- 100 forced ceramic composite is estimated, based on the damage mechanisms and 000.250507510 above. The tensile stress-strain curve represents re quasI-static Fig. 3. The non-inear stress-strain curve of SiCCAS I mea- ing in equilibrium with the stresses in the material, ores (strain rate and therefore, according to the first law of thermo- 0/ min). applied stress as function of strain. The stages of dynamics, the work of the external force must be damage evolution are indicated(after ref. 19) equal to the change of the energy in the specimen1050 B. F. Swensen, R. Talreja pull out. This is incorrect, since matrix cracking, fibre/matrix debonding, fibre sliding and fibre fail￾ure continue to take place during this part of the bending response, as the failure locus moves across the depth of the specimen towards the com￾pression side. Uniaxial tension experiments, on the other hand, allow an unambiguous interpretation, since the damage mechanisms initiate and develop at identifiable stress or strain levels,‘5-‘9 and take place uniformly over the entire volume of the gage section of the specimen. Unfortunately, tensile tests are difficult to perform, since an accurate alignment is needed to prevent bending effects. 24 Also, although tensile tests are performed with much care, it is not unusual that failure takes place at tabs,19 so the true strength of the composite cannot always be mea￾sured. However, if there is no bending in the test section, then the relationship between stress, strain and damage mechanisms can readily be identified. 2.4 Experimental characterization of damage in uniaxial tension Figure 3 shows an experimental stress-strain curve of a hot pressed unidirectional SiC-fibre-rein￾forced calcium alumino silicate (denoted CAS II from Corning Inc., NY, USA) glass-ceramic matrix composite, tested in uniaxial tension along fibres19 at a strain rate of 4 X lOA min. Throughout this paper we will refer the evolution of damage to the axial strain of the composite rather than the composite stress, since the axial stress component in the fibres resulting from the external load and the axial stress component in Fibre Failure 0.0 0.25 0.5 0.75 1.0 (%I Fig. 3. The non-linear stress-strain curve of SiCKAS II mea￾sured in uniaxial tension along fibres (strain rate of 4 X lO?min). Applied stress as function of strain. The stages of damage evolution are indicated (after ref. 19). the matrix before matrix cracking are proportional to the axial strain of the composite. This approach also makes it easier to compare behaviour of unidirectional and cross-ply composites. For the SiCKAS II composite shown in Fig. 3, U was measured to be 3.1 MJ/m3, which is almost 40 times that of the pure matrix material. However, due to loading rate effects on the monotonic stress-strain behaviour, 25 the value for SiCKAS II can vary from 1.7 MJ/m3 (at 0.01 MPa/s) to 5.4 MJ/m3 (500 MPa/s). The non-linear stress-strain curve reflects the various stages of damage (see ref. 19 for a detailed discussion). At low strain the material response is linearly elastic and reversible (Stage I). At higher strain matrix cracks initiate and evolve into fairly regularly spaced multiple matrix cracks (Stage II). This takes place over a certain strain range, rather than at a specific strain value. Fibre/matrix debonding also takes place during Stage II, and facilitates frictional sliding along the fibre/matrix interface. These damage processes cause the non￾linear part of the stress-strain curve. The corre￾sponding changes in the overall unloading modulus and Poisson’s ratio in Stage II can be modelled by a continuum damage model.* Increasing the applied stress further leads to a sec￾ond linear response (Stage III), where the stiffness is mainly due to the fibres. However, significant hysteresis appears during unloading due to effects from interfacial friction. For some specimens a small amount of distributed fibre fracture (Stage IV) may take place just prior to failure. Final fail￾ure occurs by localized fibre fracture. Recalling the preceding discussion, the area under the stress-strain curve represents the energy that the material has absorbed per unit volume of the composite up to failure. The energy quantity mea￾sured by ti is the energy that has been absorbed by distributed mechanisms. Mechanisms operating during the localized fracture, such as fibre fracture and pull out do not contribute to this quantity. 3 Model of Energy Uptake 3.1 Basic considerations In the following the energy uptake by distributed mechanisms of a unidirectional ceramic fibre rein￾forced ceramic composite is estimated, based on the damage mechanisms and scenario identified above. The tensile stress-strain curve represents the response of the material to a quasi-static load￾ing in equilibrium with the stresses in the material, and therefore, according to the first law of thermo￾dynamics, the work of the external force must be equal to the change of the energy in the specimen