13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- In [15] we have proposed a model incorporating mechanical and particle size effects and demonstrated that it provides excellent predictions for failure probability profiles. In this work we report on application of the model to predict cleavage fracture toughness in DBT regime. We focus on the first two possibilities mentioned above: to improve individual probability of failure and to account directly for the real particle size distribution rather than approximating the tail. The validity of the weakest-link assumption for the calculation of global probability is maintained. The model is applied to specific RPV steel, Euro Material A, for which fracture toughness data is available in [21] and particle size distribution, obtained via metallographic analyses, is given in [22]. 2. Theory and model LA methods share a common philosophy based on two distinct components. Firstly, the local mechanical fields provide a local or ‘individual’ probability of failure when linked to the size distribution of the micro-crack initiators. The individual probability of failure at location i is ( ) p f r dr p c i r c i f i ∫ ∞ = , , , , (2) where f(r) is the probability density of initiators’ sizes, pc,i is the probability of micro-crack nucleation, and rc,i is a critical micro-crack size at location i. Note, that pc f(r) is the probability density of nucleated micro-crack sizes. Existing LAs can be recast into Eq. (2) albeit with different definitions of pc and rc. For example in [2-4] pc = 0 or 1 for zero and non-zero plastic strains. In [11, 12] pc scales with the equivalent plastic strain, while in [14] pc scales with the equivalent plastic strain and exponent of the stress triaxiality. In [13] pc is a more complex function of stress and plastic strain increments. In all cases rc is defined via Griffith or plasticity-modified Griffith criterion. Common feature of [2-4, 11-14] is that pc is independent of particle size. Secondly, it is assumed that the individual failure events are independent. This allows the weakest-link argument to be invoked for calculating the global failure probability, so that ( ) ( ) ∏ = = − − N i f i f p P V 1 , 1 1 , (3) where N is the number of possible weakest-links, i.e. active micro-cracks, in volume V. In practice, LAs are applied to FE solutions of cracked components, where the mechanical fields are constant within an integration point volume, Vi. The failure probability of such a volume is thus ( ) ( ) ( ) i i i V f i N f i i f p p P V ρ , , 1 1 1 1 = − − = − − , (4) where Ni = ρi Vi is the number of micro-cracks in Vi, and ρi is the density of the micro-cracks. Strictly speaking ρi = ρ pc,i, where ρ is the density of initiating particles in the material. The component probability of failure is then calculated by repeated application of (3) to get ( ) ( ) ∏ = = − − IP i V f i f i i p P V 1 , 1 1 ρ , (5) where the product is taken over all integration points. When pc is independent of particle size, Eq. (2) can have a closed-form solution in terms of critical micro-crack sizes, rc. With a power law approximation for particle sizes and assuming small individual failure probabilities, Eq. (5) yields the Weibull-stress function in Eq. (1). We use Eq. (5) directly for global failure probability, because the experimental particle size distribution, Section 2.1, and our expression for individual failure probability, Section 2.3, do not allow for closed form solution of Eq. (2).
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