13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- different impact angles and velocities, and compare the present simulation results with that using other computational and theoretical models. 2. Modeling 2.1. Material model 2.1.1 General property The erosion process has been studied widely using numerical approaches such as finite element method. One key point in simulation is the choice of material model considering strain, strain rate and temperature. To study erosion process, a model must have three important components: elasticity, plasticity, damage initiation and damage growth. In this study, the substrate material is Ti-6Al-4V, and elastic response of the material is assumed to be linear and defined by elastic modulus and Poisson’s ratio. Thermal response is ignored because of transient process. 2.1.2. Plasticity model The Johnson-Cook visco-plastic model (J-C) is used in this study [10, 11]. In this model, flow stress σഥ depends on equivalent plastic strain (εത), equivalent plastic strain rate (εതሶ ), and temperature. The model can be expressed as follows: σഥൌሺABεത୬ሻቀ1Clnቀ கത கത ሶ ሶబቁቁሺ1െT∗୫ሻ, (1) where ܣ, ܤ, ܥ and ݉ are material constants, ݊ is strain hardening exponent, εതሶ εതሶ ⁄ is the normalized equivalent plastic strain rate (typically normalized by a strain rate of 1.0 s-1), and T∗ is the homologous temperature which is defined as: T∗ ൌ ି ౨ ౣି ౨, (2) where T is the current temperature, T୰ is the reference temperature, T୫ is the melting temperature of material. The model assumes that strength is isotropic. 2.1.3. Failure model Johnson-Cook failure model is applied for the ductile failure criterion [10, 11], in which the equivalent plastic strain at the onset of damage, ̅ߝ , is assumed to be a function of stress triaxiality (ߟ ), strain rate (εሶ ∗) and temperature. Johnson-Cook damage model is expressed in term of the failure strain as follows: ̅ߝ ൌ ሾ݀ ଵ ݀ ଶ݁ ݔ ሺെ݀ ଷ ߟ ሻሿሺ1݀ ସ ln ߝ ሶ ∗ሻሺ1݀ ହܶ ∗ሻ, ߟ ൌ (3) where ݀ ଵ െ݀ ହ are material constants, is the pressure stress (positive in tension), q is the von-Mises stress, and ܶ ∗ is the homologous temperature. In the explicit finite element method, the overall damage variable ܦ captures the combined effects of all active damage mechanisms, and is computed in terms of the individual damage variables. The damage parameter ܦ is defined as: ܦ ൌ∑௱ఌ ఌതವ (4) In each finite element, ሺ∑Δεሻ is calculated, and the damage parameter ܦ for element is subsequently calculated during each time step. When the damage parameter ܦ reaches the value of
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