PSI - Issue 42

Felix Bödeker et al. / Procedia Structural Integrity 42 (2022) 490–497

493

4

F. Bo¨deker et al. / Structural Integrity Procedia 00 (2019) 000–000

Algorithm 1 Barzilai-Borwein scheme 1: Initialize: ε 1 = E , i = 0 and s 1 = 2 α + + α − 2: Iterate: i = i + 1 until convergence: 3: compute σ i from ε i and state variables

constitutive law

V

if i > 1 then s i = 1 − Γ : FFT σ i , else ε i + 1 = ε i − s i FFT − 1 d ε i =   0 ,

− 1

σ i : d ε i − 1 ) V d ε i − 1 : d ε i − 1 for ξ = 0

s i − 1

update step size

4:

5: d ε i

apply Green operator

6: 7:

update strain field

via the stress tensor σ x V = σ

Macro follows from the energetic consistency, the Hill-Mandel condition (Matousˇ et

al. (2017)). The scale transition relations are also summarized in Figure 1a.

2.1. Damage model

A non-local, ductile, implicit gradient damage model is used to model the fracture behavior at the microscale in this work, which is described by

2 ∆ p

p nl − l

nl = p l ,

(3)

neglecting the spatial dependencies again for notational clarity, cf. Magri et al. (2021). The non-local equivalent plastic strain p nl is thereby regularized by a Helmholtz-type equation with the local equivalent plastic strain p l as source term. The damage field is distributed within a certain localization region, whose size is related to the characteristic internal length l . Furthermore, the evolution equation for the scalar damage variable D is given by D =    0 , for p nl < p 0 nl p nl − p 0 nl p f nl − p 0 nl , for p 0 nl ≤ p nl < p f nl 1 , for p nl ≥ p f nl (4) with initiation parameter p 0 nl and failure parameter p f nl . In addition, the stress is computed as

σ = (1 − D ) ¯ σ,

(5)

where ¯ σ is the undamaged, e ff ective stress from standard von Mises plasticity. The damage model is valid in the damaging phase only; however, in FFT-based methods we need an equation that is valid in the whole RVE owing to the Fourier transform. Following Magri et al. (2021) the Helmholtz-type equation 3 is modified to p nl − div l 2 x grad p nl = p l (6)