PSI - Issue 13

Andrzej Neimitz et al. / Structural Integrity Procedia 00 (2018) 000 – 000

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Andrzej Neimitz et al. / Procedia Structural Integrity 13 (2018) 862–867

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4) For the PR specimen, the power approximation leads to favourable results over the entire range of test temperatures. The calibration of the constitutive equations is performed according to the procedure proposed by Bai-Wierzbicki [2008]. They proposed the yield function in the form: = ̅( ̅̅̅̅)[1 − ƞ (ƞ − ƞ 0 )][ + ( − )( − +1 ) + ) 1 )] (1) where  0 is a reference value of the triaxiality coefficient; and  0 =1/3 for the uniaxial tensile test. The  function represents a curve drawn at the deviatoric surface between the contours defined by the Huber – von Mises and Tresca criteria in the principal stress space. The  function satisfies the inequality 0  1, and  =0 for plane stress or pure shear, and  =1 for axial symmetry. Bai and Wierzbicki postulated that the  function takes the following form: = cos⁡( /6) 1 − cos⁡( /6) [ cos( 1 − /6) − 1] = 6.464[sec ( − 6 ) − 1] (2) In Eq. 1, the quantity ax c  is defined as follows: = ⁡ ̅ <0 ⁡ ̅ ≥0 (3) Eq. 1 contains four parameters to be determined: , ⁡ , ⁡ ⁡ and m . The term containing the m parameter is added to make the yield surface smooth and differentiable with respect to the Lode angle  in the neighbourhood of  =1. These parameters must be determined experimentally. Wierzbicki et al. (2010) and Algarni et al. (2015) proposed also other yield functions. Probably, authors' intension was to formulate a universal function, independent of the specimen geometry. Present authors' efforts to find a unique set of parameters in Eq.1 to calibrate constitutive equation leading to the numerical and experimental curves convergence were not satisfactory. The curves were close to each other but the convergence was not as good as after individual calibration performed for each geometry separate. Perfect convergence of the experimental and numerical curves (see Fig. 2) could be reached after certain modifications introduced to Bai-Wierzbicki procedure. Both η and L parameters change over the critical plane and over time during the loading process, the average values of these quantities over the critical plane are introduced into Eq. 1. The η function change according to Eq. 4 (the linear approximation). ƞ = ƞ − ( ƞ − ƞ _ _ ) _ (4) where index i denotes the initial state, index f denotes the final state, ɛ pl_avr_final is the average value of the effective plastic strains at the critical plane before the failure ɛ pl_avr and is the actual average effective plastic strain in the critical plane. A similar formula was used for the Lode parameter. The average values over the critical plane are assumed since the force – elongation curve represents the average response of the specimen to the external loading. In addition, the coefficient ɳ , in Eq.1 is adjusted when necessary. At the final stage of loading, the voids within loaded specimens rapidly grow and coalesce. The force – elongation curve may drop sharply. Eq. 5 describes this phenomenon. ƞ = [1 + ( 0 )( − _0 ] (5) where ε pl_o denotes the presumed value of the effective plastic strain at the onset of rapid void growth. H(ε pl_o ) is the Heaviside function. Coefficient α is a constant, typically smaller than 0.1, that should be determined experimentally by curve fitting. In addition, the power exponent  should be determined experimentally. Exemplary curves before and after the calibration processes are shown in Fig. 4. The correction term shown by Eq. 5 replaces the Gurson – Needlemen – Tvergaard model to some degree. This correction is not as precisely defined but can be more easily calibrated.