13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- 2 2 2 1 1 2 5 3 ln( ) 3 3 (2 3 ) 2 ( 1) m rr c rr T f d T f hT f ρ σ σ ε Σ − − = − + − + Δ ∫ . (40) where the notation ( ) rr a c r σ = and the boundary condition (34) at e r r = have been used. Note that no boundary condition is available at i r r = . The physical meaning of (40) is the following: Whenever there exists a constant c such that (40) is fulfilled (with 1 ε=+ or -1), then mΣ is an admissible loading for the value of ρ at stake. We seek the highest possible value 0 m +Σ > of mΣ (isotropic strength in traction) and the lowest one, denoted by 0 m −Σ < (isotropic strength on compression). For the simplification of the following discussion, the denominator in the integral of (40) is denoted byDε: 2 2 (2 3 ) 2 ( 1) rr D T f hT f ε σ ε = − + − + Δ. (41) In order for this integral to be defined, two mathematical conditions are to be met, namely 0 Δ≥ and 0 Dε ≠ . This remark leads to introduce the solutions to the equations of 0 Δ= and of 0 Dε = . First, let 1m ±Σ denote the solutions to 0 Δ= , which read: ( ) ( )( ) ( ) ( )( ) 2 1 2 2 1 2 2 3 2 6 3 2 5 2 3 (1 ) (3 2 )(2 3 ) 2 3 2 6 3 2 5 2 3 (1 ) (3 2 )(2 3 ) m m T f f f f T f hT f T f T f f f f T f hT f T f ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ + ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ − ⎣ ⎦ + − + − + − Σ = + − + + + − + − Σ = + − . (42) Secondly, let 2m +Σ (resp. 2m −Σ ) denote the solution to 0 D+ = (resp. 0 D− = ): 2 2 2 6 ( 1) 3 2 m T f f Th f T ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ + − Σ = − . (43) 2 2 2 6 ( 1) 3 2 m T f f Th f T ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ + ⎝ ⎠ − − Σ = − . (44) After some reasoning, the static solution can be finally determined by numerical integration. Then the static solution is compared to the cinematic solution in the following subsection. 6.2. Comparison between static and cinematic solutions The comparison between the static solution and the cinematic solution as functions of the rigid core volume fraction ρ (f=0.25 and T=0.525) has been performed. It is found that the two solutions can hardly be differentiated. it can be concluded that they can be regarded as the exact strength of the composite material, within the rigid core model. . 6.3. Analytical expressions of the strength under isotropic loading Due to the complexity of the integrals in (40), these equations can hardly be solved analytically.
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