PSI - Issue 42

Pauline Herr et al. / Procedia Structural Integrity 42 (2022) 498–505 P. Herr et al. / Structural Integrity Procedia 00 (2019) 000–000

501

4

cover the whole range of load cases, which is why non-linear yield criteria or piecewise linear yield conditions are required to model more complex load scenarios, e.g., Suwanpakpraek et al. (2020). In this work, we focus on the region of a negative hydrostatic pressure, which is dominant in the mode I FFT simulations owing to the uniaxial strain state, see Section 2.2. Therefore, the slope of the linear Drucker-Prager was set rather high. Furthermore, a non-associative, von Mises type plastic flow potential was used. The implementation of the model was taken from Lee et al. (2019) and the parameters were adapted to match the form of Equation 2. The linear evolution equation of the ductile, nonlocal gradient damage model for the fracture behavior of the polymer is given by D =    0 , for ¯ κ nl < ¯ κ 0 nl ¯ κ nl − ¯ κ 0 nl ¯ κ f nl − ¯ κ 0 nl , for ¯ κ 0 nl ≤ ¯ κ nl < ¯ κ f nl 1 , for ¯ κ nl ≥ ¯ κ f nl (3) with the nonlocal equivalent plastic strain ¯ κ nl . The nonlocal regularization is done with the Helmoltz type equation

2 ∆ ¯ κ

¯ κ nl − l

nl = ¯ κ l ,

(4)

where the local equivalent plastic strain acts as source term. l corresponds to an intrinsic characteristic length that represents a measure for the size of the damage localization zone. The e ff ective stress from the elastic plastic model is scaled by the factor (1 − D ). The gradient damage model was implemented in the FFT-based homogenization framework according to Magri et al. (2021); we refer to Bo¨deker et al. (2022) for more details. All parameters for the material models are summarized in Table 1.

Table 1. Material parameters used

f nl ( - )

d 0 ( MPa )

¯ κ 0

l 2 ( mm 2 )

β ( ◦ )

Youngs Modulus ( MPa )

Poisson’s ratio ( - )

H ( MPa )

nl ( - )

¯ κ

Polymer

2000

0.4

40

45

100

0.07

0.5

0.0015625

Glass beads Sti ff layers

68900 70000

0.21 0.33

- -

- -

- -

- -

- -

- -

2.2. Data generation

In the next step, the FFT-based homogenization scheme for cohesive zones (see Bo¨deker et al. (2022)) originated from Matousˇ et al. (2008) was used to collect the required data for FFNN training. In FFT-based homogenization, the volumetric average strain is usually prescribed to establish the transition from macroscale to microscale. Following Matousˇ et al. (2008), the average strain in the adhesive layer under a mode I loading can be computed according to

E V

m = (0 0 u / t 0 0 0) T

(5)

from mode I separation u and cohesive zone thickness t , whereby Voigt notation is used and orientation of the cohesive zone is the z direction, without loss of generality. The micro-to-macro transition is then given by t M = σ m x · n , (6)