1048 B. F. Sorensen, R. talreja Most theoretical works on continuous fibre reinforced ceramics focus on two topics: progression P pation due to fibre pull out. 4/ Multiple matri cracking is of importance since it leads to non linear constitutive behaviour that must be taken into account when these materials are used. see e.g. ref. 8. It is equally important to point out that in general there is no correlation between the ER proportional limit, i.e. the stress level at which u=o de multiple matrix cracking causes non-linearity in the stress-strain curve and the fatigue limit. The energy dissipation due to pull out is important for assessment of the final fracture behaviour. A central part of all these models is the role of the interface(see ref. 10 for a recent review ) The energy uptake by distributed mechanisms (i.e. prior to localization) has not been addressed sufficiently in the previous works. experimentally a remarkably non-linear stress-strain response has been found for several continuous fibre-reinforced ceramics composites.-4 However, the details of -di =g,b dc the associated evolution of damage have been studied only recentl on the basis of ecent findings, this paper develops a model p Quasi defined here as toughness, i.e. the total energy uptake until failure in volume-distributed mechan isms. This energy comprises both the recoverable and non-recoverable parts. It is of major impor- tance, because this is the maximum energy that can be absorbed per unit volume without causing fracture of a component, i.e.(the component will be damaged but will retain its integrity such that it can still continue to carry loads 2 Measurements of energy for material Fig. 1. Energy measures for materials:(A)the toughness, U characterization B) the critical energy release rate, G, and (C)the work of fracture, wor 2.1 Energy quantities Firstly, for clarity, we start off reviewing some where o and e are stress and strain, respectively energy concepts and quantities that are often used and e, is the failure strain (onset of localization to characterize the fracture behaviour of materials. It is the area under the stress-strain curve, and For simplicity the considerations are limited to the has the sI unit J/m case where a specimen is loaded by a single tension The energy release rate is defined in fracture force.Figure I illustrates three commonly used mechanics by the decrease of the potential energy quantities: the toughness, the critical energy per new crack area during a quasi-static crack release rate(fracture toughness) and the work of Increment fracture 1 dl The toughness20 is the ability of a material to uptake energy (per unit volume) prior to failure, e. the strain energy density to failure 20 where I is the potential energy, and c and B are the length and the width of the crack, respectively The critical value of G is equivalent to the energy U=de)dE, absorbed per unit new crack area, during an incre- mental(stable) crack growth. The SI unit for G is1048 B. F. Smensen, R. Talreja Most theoretical works on continuous fibre reinforced ceramics focus on two topics: progression of multiple matrix crackinglm3 and energy dissi￾pation due to fibre pull out.47 Multiple matrix cracking is of importance since it leads to non￾linear constitutive behaviour that must be taken into account when these materials are used, see e.g. ref. 8. It is equally important to point out that in general there is no correlation between the ‘proportional limit’, i.e. the stress level at which multiple matrix cracking causes non-linearity in the stress-strain curve and the fatigue limit.’ The energy dissipation due to pull out is important for assessment of the final fracture behaviour. A central part of all these models is the role of the interface (see ref. 10 for a recent review). The energy uptake by distributed mechanisms (i.e. prior to localization) has not been addressed sufficiently in the previous works. Experimentally, a remarkably non-linear stress-strain response has been found for several continuous fibre-reinforced ceramics composites. 9-14 However, the details of the associated evolution of damage have been studied only recently. 15-19 On the basis of these recent findings, this paper develops a model for the energy uptake by distributed mechanisms, defined here as toughness, i.e. the total energy uptake until failure in volume-distributed mechan￾isms. This energy comprises both the recoverable and non-recoverable parts. It is of major impor￾tance, because this is the maximum energy that can be absorbed per unit volume without causing fracture of a component, i.e. (the component will be damaged but will retain its integrity such that it can still continue to carry loads). 2 Measurements of energy for material characterization 2.1 Energy quantities Firstly, for clarity, we start off reviewing some energy concepts and quantities that are often used to characterize the fracture behaviour of materials. For simplicity the considerations are limited to the case where a specimen is loaded by a single tension force. Figure 1 illustrates three commonly used quantities: the toughness, the critical energy release rate (fracture toughness) and the work of fracture. The toughness20 is the ability of a material to uptake energy (per unit volume) prior to failure, i.e. the strain energy density to failure2’; C” U = s O(E) de, (1) 0 4-I = GIG B de (4 (b) YWOF =s E dA OA P A tA f 1 1 Fig. 1. Energy measures for materials: (A) the toughness, U, (B) the critical energy release rate, G, and (C) the work of fracture, -ywOF. where (T and E are stress and strain, respectively, and l U is the failure strain (onset of localization). It is the area under the stress-strain curve, and has the SI unit J/m3. The energy release rate is defined in fracture mechanics by the decrease of the potential energy per new crack area during a quasi-static crack increment21 G=-L!!Z B dc ’ where 17 is the potential energy, and c and B are the length and the width of the crack, respectively. The critical value of G is equivalent to the energy absorbed per unit new crack area, during an incre￾mental (stable) crack growth. The SI unit for G is