PSI - Issue 13

Florian Fehringer et al. / Procedia Structural Integrity 13 (2018) 932–938 Fehringer, F., Schuler, X., Seidenfuß, M. / Structural Integrity Procedia 00 (2018) 000 – 000

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3

The main part of the newly proposed strain based concept is the so-called limit strain curve. The limit strain curve defines a tolerable plastic strain value depending on different influence factors i.e. the stress triaxiality and the Lode angle. To determine this limit strain curve, various simulations are carried out using the extended Rousselier model. In this manner, the failure behavior (crack initiation spot, crack initiation load, crack initiation time, crack growth) can be derived for each specimen and loading condition. Subsequently, the simulations are repeated using an elastic-plastic material model. These simulation results are needed to determine the plastic strain, the stress triaxiality and the Lode angle for the crack initiation location at crack initiation load, as these values cannot be obtained accurately from the damage mechanics simulations. The limit strain curve can be derived frommany different simulations. Finally, a safety factor needs to be applied to the strain values to consider additional effects on limit strain (i. e. strain rate or temperature), or possible scattering of the material. Later in industrial application, only a small number of specimen will be needed for the calibration of the material dependent model parameters and a limit strain curve can be generated numerically. To evaluate a component under a certain load case, a standard elastic-plastic finite element calculation will be sufficient. For every part of the component, depending on the occurring stress triaxiality and Lode angle a tolerable plastic strain can be derived from the limit strain curve.

3. Damage mechanics approach

Ductile failure under tensile load and its numerical description can be divided in three different stages.  void initiation: This process is described by the equivalent stress value exceeding the material yield strength. As a result, the void volume fraction is set to an initial void volume fraction f 0 (Seidenfuß (2012)).  void growth: For high stress triaxialities void growth can be described by the standard Rousselier model (eq. (1)).

     y 

  

v 



(1)

D f exp

0



k 





  

m





f

f

1

1

k 





 coalescence of voids: To describe the coalescence of voids, the element stiffness is strongly decreased when the void volume fraction f reaches a critical value f c (Seidenfuß (2012)). By using the von Mises equivalent stress in eq. (1), the influence of the Lode angle on the flow behavior of the material is not considered. So, when simulating torsional tests, the model behavior is too rigid compared to the experimental results (see Fig. 2 (a)). Furthermore, the Rousselier model is not able to predict failure under shear stress conditions (see Fig. 2 (b)). Lastly, the standard Rousselier model contains no kinematic hardening terms. Therefore, an accurate simulation of cyclic behavior is not possible (see Fig. 2 (c)). To overcome these disadvantages, the Rousselier model is extended in three stages. 1) To take a Lode angle dependent flow behavior into account, Hosford’s generalized isotropic yield criterion (Hosford (1972)) is used (see eq. (2)).       n / n n n hosf v 1 3 1 3 2 2 1 2                with 3 2 1      and    n 1 (2) Fig. 3 (a) shows the influence of n on the yield surface for a 2-dimensional stress state, with n = 2 being the von Mises yield surface and n = 1 being the Tresca yield surface. According to literature, n = 1.6 is a good value for body centered cubic (bcc) metals (Hosford (1972)). 2) To predict failure, not only for high stress triaxialities, but also under shear stress conditions, the Rousselier model is enhanced by a term developed by Nahshon and Hutchinson (2008). This term is added to the evolution equation of the void volume fraction (see eq. (3)).    Hutchinson and Nahshon q p k f f f            1 (3)        