Table 2 Stress-updating procedure –... | Emerald Publishing

Table 2

Stress-updating procedure – small strain

(a) Initialise
$k = 0$ ⁠, $α_{d_{n + 1}}^{(0)} = α_{d_{n}}$ ⁠, $σ_{n + 1}^{(0)} = σ_{n + 1}^{trial}$
(b) Check the yield condition and evaluate residuals
$\begin{array}{l} f_{n + 1}^{(k)} = f (σ_{n + 1}^{(k)}, α_{d n + 1}^{(k)}) \\ r_{n + 1}^{(k)} = {\begin{array}{c} σ_{n + 1}^{(k)} - (1 - α_{d n + 1}) σ_{n + 1}^{trial} \\ α_{d n + 1}^{(k)} - α_{d n} \end{array}} + Δ t {\begin{array}{c} \dot{γ} (1 - α_{d n + 1}^{(k)}) D^{e} \partial_{σ} f_{n + 1}^{(k)} \\ - {\dot{α}}_{d} \end{array}} \\ \begin{array}{c} If ‖ r_{n + 1}^{(k)} ‖ < ϵ_{r} and f_{n + 1}^{(k)} < ϵ_{f}, then \\ set {(\cdot)}_{n + 1} = {(\cdot)}_{n + 1}^{(k)} and exit \end{array} \\ Else \end{array}$
(c) Compute the algorithmic modulus
$A_{n + 1}^{(k)} = [\begin{matrix} A_{n + 1}^{σ σ} & A_{n + 1}^{σ α_{d}} \\ A_{n + 1}^{α_{d} σ} & A_{n + 1}^{α_{d} α_{d}} \end{matrix}]$
where
$\begin{array}{l} A_{n + 1}^{σ σ} = I + [Δ γ \partial_{σ σ}^{2} f_{n + 1}^{(k)} + \partial_{σ} f_{n + 1}^{(k)} {\partial_{σ} Δ γ_{n + 1}^{(k)}}^{T}] (1 - α_{d n + 1}^{(k)}) D^{e} \\ A_{n + 1}^{σ α_{d}} = [Δ γ \partial_{σ α_{d}}^{2} f_{n + 1}^{(k)} + \partial_{σ} f_{n + 1}^{(k)} \partial_{α_{d}} Δ γ_{n + 1}^{(k)}] (1 - α_{d n + 1}^{(k)}) D^{e} + σ_{n + 1}^{trial} - Δ γ_{n + 1}^{(k)} \partial_{σ} f_{n + 1}^{(k)} \\ A_{n + 1}^{α_{d} σ} = {\partial_{σ} Δ α_{d n + 1}^{(k)}}^{T} \\ A_{n + 1}^{α_{d} α_{d}} = 1 - \partial_{α_{d}} Δ α_{d n + 1}^{(k)} \end{array}$
(d) Evaluate increment of stresses and internal variable
${\begin{matrix} Δ σ_{n + 1}^{(k + 1)} \\ Δ α_{d}_{n + 1}^{(k + 1)} \end{matrix}} = - {(A_{n + 1}^{(k)})}^{- 1} : r_{n + 1}^{(k)}$
(e) Update stresses and internal variable
$\begin{array}{l} {\begin{matrix} σ_{n + 1}^{(k + 1)} \\ α_{d n + 1}^{(k + 1)} \end{matrix}} = {\begin{matrix} σ_{n + 1}^{(k)} \\ α_{d n + 1}^{(k)} \end{matrix}} + {\begin{matrix} {Δ σ}_{n + 1}^{(k + 1)} \\ Δ α_{d}_{n + 1}^{(k + 1)} \end{matrix}} \\ Set k = k + 1 and go to (b) \end{array}$