Numerical methods for accurate and efficient ductile fracture predictions

Numerical methods for accurate and efficient ductile fracture predictions

An uncoupled fracture model for Advanced High Strength Steel sheets

We make use of a rate-independent simplification of the plasticity model proposed in Chapter 4. The onset of ductile farcture will be modeled independently using the so-called Modified Mohr-Coulomb (MMC) model (Bai and Wierzbicki, 2010, [7]). The predictive capabilities of the constitutive and fracture models described thereafter are not investigated here. The reader is referred to Mohr et al. (2010, [122]) and Dunand and Mohr (2011, [49]) for a critical evaluation.

Plasticity model

For the present sheet material, nearly the same stress-strain curve is measured for different specimen orientations even though the r-values are direction dependent. As 90 Chapter 5. Numerical methods for accurate and efficient ductile fracture predictions detailed in Mohr et al. (2010, [122]), we make use of a planar isotropic quadratic yield function, ( ) ̅ ̅ √( ) (5-1) in conjunction with a non-associated flow rule (5-2) denotes the plastic multiplier. The anisotropic quadratic flow potential reads √( ) (5-3) P and G are symmetric positive-semidefinite matrices, with ̅ and if and only if is a hydrostatic stress state. The values for the non-zero components of P and G are given in Table 5-1. denotes the Cauchy stress vector in material coordinates, [ ] (5-4) The components , and represent the true normal stress in the rolling, transverse and out-of-plane directions; denotes the corresponding in-plane shear stress, while and represent the corresponding out-of-plane shear stresses. Isotropic strain hardening is described as ( ̅ ) ̅ (5-5) where ( ̅ ) defines the strain hardening modulus. The strain hardening response of the material is modeled by a saturation law.

Fracture model

The original Mohr-Coulomb failure criterion (Mohr, 1900, [127]) is formulated in the stress space and assumes that failure occurs when the shear and normal stresses on any plane of normal vector n verify the condition.

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Influence of Finite

Element modeling on ductile fracture predictions with the friction coefficient and the cohesion . Bai and Wierzbicki (2010, [7]) transformed Eq. (5-7) into the space of stress triaxiality, Lode angle and equivalent plastic strain to fracture assuming proportional monotonic loading, a pressure and Lode angle dependent isotropic plasticity model, and isotropic strain hardening according to the power law. The resulting explicit expression for the fracture strain reads . The exponent n describes the strain hardening of the material. The coefficient is related to in Eq. (5-7), while and characterize the dependence of the underlying plasticity model on the third stress invariant. controls the amount of Lode angle dependence of the fracture locus and ( ̅) controls the asymmetry of the fracture locus with respect to the plane ̅ . Despite the discontinuity of ( ̅), the fracture strain ̂( ̅) is a continuous function of the stress invariants and ̅. To apply the MMC fracture model for non-proportional loadings, Bai and Wierzbicki (2010, [7]) make use of Eq. (5-8) as reference strain in Eq. (2-19). 5.3 Influence of Finite Element modeling on ductile fracture predictions The influence of the type of elements (brick vs shell) and their characteristic dimensions are investigated on numerical simulations of a tensile experiment on a flat specimen with circular notches of radius 6.67mm, as sketched in Fig. 4-1c. Parameters for the constitutive model described in Section 5.2 have been calibrated on an extensive  set of multi-axial fracture experiments carried out on TRIP780 steel (Mohr et al., 2010, [122]; Dunand and Mohr, 2011, [49]). Note that this material comes from a different production batch than the one used in Chapters 3, 4 and 6. Materials parameters are thus slightly different. 

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