Crack phase-field model equipped with plastic driving force and degrading fracture toughness for ductile fracture simulation

Abstract

This study presents a novel phase-field model for ductile fracture by the introduction of both the plastic driving force and the degrading fracture toughness into crack phase-field computations based on the phenomenological justification for ductile fracture in elastoplastic materials. Assuming that the constitutive work density consists of elastic, pseudo-plastic and crack components, we derive the governing equations from local and global optimization problems within the continuum thermodynamics framework. In addition to the elastic strain energy, the plastic strain energy also works as a driving force to sustain damage evolution. Additionally, we introduce a degrading fracture toughness to reflect the evolution of micro-defects and their coalescences into each other that are caused by accumulated plastic deformation. Equipped with these ingredients, the proposed model realizes the reduction of both stiffness and fracture toughness to simulate the failure phenomena of elastoplastic materials. Several numerical examples are presented to demonstrate the capability of the proposed model in reproducing some typical ductile fracture behaviors. The findings and perspectives are subsequently summarized.

Appendices

A Return mapping algorithm for internal variables

We show a pseudo-code for the local return mapping algorithm in Fig. 21. During this process, the plastic multiplier $\lambda ^\mathrm {p}$ and flow tensor $\varvec{{\mathfrak {n}}}$ are considered as internal variables to be determined. Also, to ensure incompressibility of the plastic deformation, an exponential mapping is utilized.

Fig. 21

Return mapping algorithm

The residuals of two local differential equations yield

$$\begin{aligned}&{\mathsf {r}}^{\lambda ^\mathrm {p}}_{k}= -\left( ||{\varvec{\tau }_0}_{\mathrm {dev},k}||-\sqrt{\dfrac{2}{3}}r^\mathrm {p}_{0,k}-\dfrac{2{\eta }_{\mathrm {p}}}{3g\left( d\right) }\lambda ^\mathrm {p}_{k}\right) \end{aligned}$$

(A.1)

$$\begin{aligned}&{\mathsf {r}}^{\varvec{{\mathfrak {n}}}_{k}}=-\left( \dfrac{{\varvec{\tau }_0}_{\mathrm {dev},k}}{||{\varvec{\tau }_0}_{\mathrm {dev},k}||}-\varvec{{\mathfrak {n}}}_{k}\right) , \end{aligned}$$

(A.2)

where the index k represents the iteration of the local Newton-Raphson scheme. By numerical linearization, its tangent matrix yields

$$\begin{aligned} \varvec{{\mathsf {k}}}=\left[ \begin{array}{cc} {\mathsf {k}}_{\lambda ^\mathrm {p}\lambda ^\mathrm {p}} &{}{\mathsf {k}}_{\lambda ^\mathrm {p}\varvec{{\mathfrak {n}}}} \\ {\mathsf {k}}_{\varvec{{\mathfrak {n}}}\lambda ^\mathrm {p}} &{} {\mathsf {k}}_{\varvec{{\mathfrak {n}}}\varvec{{\mathfrak {n}}}} \end{array} \right] \end{aligned}$$

(A.3)

with diagonal components

$$\begin{aligned} \begin{aligned}&{\mathsf {k}}_{\lambda ^\mathrm {p}\lambda ^\mathrm {p}}= \dfrac{\partial ||{\varvec{\tau }_0}_{\mathrm {dev},k}||}{\partial \varvec{b}^\mathrm {e}}: \left\{ \dfrac{\partial \exp \left( -2{\varDelta }t_{n+1}\lambda ^\mathrm {p}_k\varvec{{\mathfrak {n}}}_k\right) \cdot \varvec{b}^{\mathrm {e,tr}}_{n+1}}{\partial \lambda ^\mathrm {p}} \right\} \\&\quad -\sqrt{\dfrac{2}{3}}\dfrac{\partial r^\mathrm {p}_{0,k}}{\partial \lambda ^\mathrm {p}}-\dfrac{2{\eta }_{\mathrm {p}}}{3g\left( d\right) }\\&\text {with}\quad \dfrac{\partial \exp \left( -2{\varDelta }t_{n+1}\lambda ^\mathrm {p}\varvec{{\mathfrak {n}}}\right) \cdot \varvec{b}^{\mathrm {e,tr}}_{n+1}}{\partial \lambda ^\mathrm {p}}\\&\quad = \left\{ \text {D}\exp \left( -2{\varDelta }t_{n+1}\lambda ^\mathrm {p}\varvec{{\mathfrak {n}}}\right) : \left( -2{\varDelta }t_{n+1}\varvec{{\mathfrak {n}}}\right) \right\} \cdot \varvec{b}^{\mathrm {e,tr}}_{n+1} \end{aligned} \end{aligned}$$

(A.4)

and

$$\begin{aligned} \begin{aligned}&{\mathsf {k}}_{\varvec{{\mathfrak {n}}}\varvec{{\mathfrak {n}}}}= \dfrac{1}{||{\varvec{\tau }_0}_{\mathrm {dev},k}||}\dfrac{\partial {\varvec{\tau }_0}_{\mathrm {dev},k}}{\partial \varvec{{\mathfrak {n}}}}- \dfrac{{\varvec{\tau }_0}_{\mathrm {dev},k}}{||{\varvec{\tau }_0}_{\mathrm {dev},k}||^2}\otimes \dfrac{\partial ||{\varvec{\tau }_0}_{\mathrm {dev},k}||}{\partial \varvec{{\mathfrak {n}}}} -\underline{\varvec{1}}_{sym.}\\&\quad = \left( \dfrac{1}{||{\varvec{\tau }_0}_{\mathrm {dev},k}||}\dfrac{\partial {\varvec{\tau }_0}_{\mathrm {dev},k}}{\partial \varvec{b}^\mathrm {e}}- \dfrac{{\varvec{\tau }_0}_{\mathrm {dev},k}}{||{\varvec{\tau }_0}_{\mathrm {dev},k}||^2}\otimes \dfrac{\partial ||{\varvec{\tau }_0}_{\mathrm {dev},k}||}{\partial \varvec{b}^\mathrm {e}}\right) \\&\qquad :\dfrac{\partial \exp \left( -2{\varDelta }t_{n+1}\lambda ^\mathrm {p}_k\varvec{{\mathfrak {n}}}_k\right) \cdot \varvec{b}^{\mathrm {e,tr}}_{n+1}}{\partial \varvec{{\mathfrak {n}}}} -\underline{\varvec{1}}_{sym.}\\&\text {with}\quad \dfrac{\partial \exp \left( -2{\varDelta }t_{n+1}\lambda ^\mathrm {p}\varvec{{\mathfrak {n}}}\right) \cdot \varvec{b}^{\mathrm {e,tr}}_{n+1}}{\partial \varvec{{\mathfrak {n}}}}\\&\quad = \underbrace{ \left\{ \text {D}\exp \left( -2{\varDelta }t_{n+1}\lambda ^\mathrm {p}\varvec{{\mathfrak {n}}}\right) : \left( -2{\varDelta }t_{n+1}\lambda ^\mathrm {p}\underline{\varvec{1}}_{sym.}\right) \right\} }_{iakl} *\underbrace{\varvec{b}^{\mathrm {e,tr}}_{n+1}}_{aj} \end{aligned} \end{aligned}$$

(A.5)

and coupling components

$$\begin{aligned} {\mathsf {k}}_{\lambda ^\mathrm {p}\varvec{{\mathfrak {n}}}}&= \dfrac{\partial ||{\varvec{\tau }_0}_{\mathrm {dev},k}||}{\partial \varvec{b}^\mathrm {e}}: \dfrac{\partial \exp \left( -2{\varDelta }t_{n+1}\lambda ^\mathrm {p}_k\varvec{{\mathfrak {n}}}_k\right) \cdot \varvec{b}^{\mathrm {e,tr}}_{n+1}}{\partial \varvec{{\mathfrak {n}}}} \end{aligned}$$

(A.6)

and

$$\begin{aligned} {\mathsf {k}}_{\varvec{{\mathfrak {n}}}\lambda ^\mathrm {p}}&= \dfrac{1}{||{\varvec{\tau }_0}_{\mathrm {dev},k}||}\dfrac{\partial {\varvec{\tau }_0}_{\mathrm {dev},k}}{\partial \lambda ^\mathrm {p}}- \dfrac{{\varvec{\tau }_0}_{\mathrm {dev},k}}{||{\varvec{\tau }_0}_{\mathrm {dev},k}||^2}\dfrac{\partial ||{\varvec{\tau }_0}_{\mathrm {dev},k}||}{\partial \lambda ^\mathrm {p}}\nonumber \\&\quad = \left( \dfrac{1}{||{\varvec{\tau }_0}_{\mathrm {dev},k}||}\dfrac{\partial {\varvec{\tau }_0}_{\mathrm {dev},k}}{\partial \varvec{b}^\mathrm {e}}- \dfrac{{\varvec{\tau }_0}_{\mathrm {dev},k}}{||{\varvec{\tau }_0}_{\mathrm {dev},k}||^2}\otimes \dfrac{\partial ||{\varvec{\tau }_0}_{\mathrm {dev},k}||}{\partial \varvec{b}^\mathrm {e}}\right) \nonumber \\&\qquad : \left\{ \dfrac{\partial \exp \left( -2{\varDelta }t_{n+1}\lambda ^\mathrm {p}_k\varvec{{\mathfrak {n}}}_k\right) }{\partial \lambda ^\mathrm {p}}\cdot \varvec{b}^{\mathrm {e,tr}}_{n+1}\right\} . \end{aligned}$$

(A.7)

Here, $\text {D}\exp$ denotes the derivative of the second order exponential tensor and $\underline{\varvec{1}}_{sym.}$ is the fourth order symmetric identity tensor. Also, to show formulations explicitly in tensor notations, we define an operator $*$ that denotes an inner product of the second basis of a fourth order tensor ${\mathbb {X}}$ and the first basis of a second order tensor $\varvec{Y}$, which is shown in index notation as $X_{iajk}Y_{aj}$.

B Components accounting for global tangent matrix

The component in Eq. (67) is written in an explicit form as

$$\begin{aligned} \dfrac{\partial \varvec{\tau }}{\partial \varvec{F}}&= {\left\{ \begin{array}{ll} g\left( d \right) \dfrac{\partial {\varvec{\tau }_{0}}_{\mathrm {vol}}}{\partial \varvec{F}}+ g\left( d \right) \dfrac{\partial {\varvec{\tau }_{0}}_{\mathrm {dev}}}{\partial \varvec{F}} \quad \text {for}\quad J^\mathrm {e}\ge 1 \\ \quad \dfrac{\partial {\varvec{\tau }_{0}}_{\mathrm {vol}}}{\partial \varvec{F}}+ g\left( d \right) \dfrac{\partial {\varvec{\tau }_{0}}_{\mathrm {dev}}}{\partial \varvec{F}} \quad \text {for}\quad J^\mathrm {e}< 1 \end{array}\right. } \end{aligned}$$

(B.1)

with

$$\begin{aligned} \begin{aligned} \dfrac{\partial \varvec{\tau }_0}{\partial \varvec{F}}&= \left\{ \dfrac{\partial {\varvec{\tau }_{0}}_{\mathrm {vol}}}{\partial J^\mathrm {e}}\otimes \dfrac{\partial J^\mathrm {e}}{\partial \varvec{F}^\mathrm {e}}+ \dfrac{\partial {\varvec{\tau }_{0}}_{\mathrm {dev}}}{\partial \bar{\varvec{b}}^\mathrm {e}}:\dfrac{\partial \bar{\varvec{b}}^\mathrm {e}}{\partial \varvec{F}^\mathrm {e}} \right\} :\dfrac{\partial \varvec{F}^\mathrm {e}}{\partial \varvec{F}}\\&= \bigg \{ \kappa {J^{\mathrm {e}}}^2\varvec{1}\otimes \varvec{F}^{\mathrm {e}-\mathrm {T}} +\mu {J^\mathrm {e}}^{-2/3}\\&\quad \text {dev}\bigg [\varvec{1}\ \overline{\otimes }\ \varvec{F}^\mathrm {e}+\varvec{F}^\mathrm {e}\ \underline{\otimes }\ \varvec{1} -\dfrac{2}{3}\varvec{b}^\mathrm {e}\otimes \varvec{F}^{\mathrm {e}-\mathrm {T}}\bigg ] \bigg \}:\dfrac{\partial \varvec{F}^\mathrm {e}}{\partial \varvec{F}} . \end{aligned} \end{aligned}$$

(B.2)

Using the total differential, the last term in Eq. (B.2) yields

$$\begin{aligned} \begin{aligned} {\dfrac{\partial \varvec{F}^\mathrm {e}}{\partial \varvec{F}}}= \left. \dfrac{\partial \varvec{F}^\mathrm {e}}{\partial \varvec{F}}\right| _{\lambda ^\mathrm {p},\varvec{{\mathfrak {n}}}=\text {const.}}+ \left. \dfrac{\partial \varvec{F}^\mathrm {e}}{\partial \lambda ^\mathrm {p}}\right| _{\varvec{{\mathfrak {n}}},\varvec{F}=\text {const.}}\otimes \dfrac{\partial \lambda ^\mathrm {p}}{\partial \varvec{F}} \\ +\left. \dfrac{\partial \varvec{F}^\mathrm {e}}{\partial \varvec{{\mathfrak {n}}}}\right| _{\varvec{F},\lambda ^\mathrm {p}=\text {const.}}: \dfrac{\partial \varvec{{\mathfrak {n}}}}{\partial \varvec{F}} \end{aligned} \end{aligned}$$

(B.3)

with

$$\begin{aligned}&\left. \dfrac{\partial \varvec{F}^\mathrm {e}}{\partial \varvec{F}}\right| _{\lambda ^\mathrm {p},\varvec{{\mathfrak {n}}}=\text {const.}}= \exp \left( -{\varDelta }t_{n+1}\lambda ^\mathrm {p}\varvec{{\mathfrak {n}}}\right) \ \overline{\otimes }\ \varvec{F}^{\mathrm {p}-\mathrm {T}}_{n} , \end{aligned}$$

(B.4)

$$\begin{aligned}&\left. \dfrac{\partial \varvec{F}^\mathrm {e}}{\partial \lambda ^\mathrm {p}}\right| _{\varvec{{\mathfrak {n}}},\varvec{F}=\text {const.}}= \left\{ \text {D}\exp \left( -{\varDelta }t_{n+1}\lambda ^\mathrm {p}\varvec{{\mathfrak {n}}}\right) : \left( -{\varDelta }t_{n+1}\varvec{{\mathfrak {n}}}\right) \right\} \cdot \varvec{F}^{\mathrm {e,tr}} , \end{aligned}$$

(B.5)

$$\begin{aligned}&\left. \dfrac{\partial \varvec{F}^\mathrm {e}}{\partial \varvec{{\mathfrak {n}}}}\right| _{\varvec{F},\lambda ^\mathrm {p}=\text {const.}}= \underbrace{ \left\{ \text {D}\exp \left( -{\varDelta }t_{n+1}\lambda ^\mathrm {p}\varvec{{\mathfrak {n}}}\right) : \left( -{\varDelta }t_{n+1}\lambda ^\mathrm {p}\underline{\varvec{1}}_{sym.}\right) \right\} }_{iakl} * \underbrace{\varvec{F}^{\mathrm {e,tr}}}_{aj}. \end{aligned}$$

(B.6)

Then, $\dfrac{\partial \lambda ^\mathrm {p}}{\partial \varvec{F}}$ and $\dfrac{\partial \varvec{{\mathfrak {n}}}}{\partial \varvec{F}}$ are obtained from the total differential of two local residuals

$$\begin{aligned}&\text {d}_{\varvec{F}}{\mathsf {r}}^{\lambda ^\mathrm {p}}= \text {d}_{\varvec{F}}\left[ -||{\varvec{\tau }_0}_{\mathrm {dev},n+1}||+\sqrt{\dfrac{2}{3}}r^\mathrm {p}_{0,n+1}+\dfrac{2{\eta }_{\mathrm {p}}}{3g\left( d\right) }\lambda ^\mathrm {p}_{n+1}\right] =0 \nonumber \\&\quad \rightarrow \dfrac{\partial {\mathsf {r}}^{\lambda ^\mathrm {p}}}{\partial \varvec{F}}+ \dfrac{\partial {\mathsf {r}}^{\lambda ^\mathrm {p}}}{\partial \lambda ^\mathrm {p}}\dfrac{\partial \lambda ^\mathrm {p}}{\partial \varvec{F}}+ \dfrac{\partial {\mathsf {r}}^{\lambda ^\mathrm {p}}}{\partial \varvec{{\mathfrak {n}}}}:\dfrac{\partial \varvec{{\mathfrak {n}}}}{\partial \varvec{F}} \nonumber \\&\quad =-\dfrac{\partial ||{\varvec{\tau }_0}_{\mathrm {dev}}||}{\partial \varvec{F}} -{\mathsf {k}}_{\lambda ^\mathrm {p}\lambda ^\mathrm {p}}\dfrac{\partial \lambda ^\mathrm {p}}{\partial \varvec{F}} -{\mathsf {k}}_{\lambda ^\mathrm {p}\varvec{{\mathfrak {n}}}}:\dfrac{\partial \varvec{{\mathfrak {n}}}}{\partial \varvec{F}} =\varvec{0} \end{aligned}$$

(B.7)

and

$$\begin{aligned} \begin{aligned}&\text {d}_{\varvec{F}}{\mathsf {r}}^{\varvec{{\mathfrak {n}}}}= \text {d}_{\varvec{F}}\left[ -\dfrac{{\varvec{\tau }_0}_{\mathrm {dev},n+1}}{||{\varvec{\tau }_0}_{\mathrm {dev},n+1}||}+\varvec{{\mathfrak {n}}}\right] =0\\&\quad \rightarrow \dfrac{\partial {\mathsf {r}}^{\varvec{{\mathfrak {n}}}}}{\partial \varvec{F}}+ \dfrac{\partial {\mathsf {r}}^{\varvec{{\mathfrak {n}}}}}{\partial \lambda ^\mathrm {p}}\otimes \dfrac{\partial \lambda ^\mathrm {p}}{\partial \varvec{F}}+ \dfrac{\partial {\mathsf {r}}^{\varvec{{\mathfrak {n}}}}}{\partial \varvec{{\mathfrak {n}}}}:\dfrac{\partial \varvec{{\mathfrak {n}}}}{\partial \varvec{F}}\\&\qquad =-\dfrac{\partial }{\partial \varvec{F}}\dfrac{{\varvec{\tau }_0}_{\mathrm {dev}}}{||{\varvec{\tau }_0}_{\mathrm {dev}}||} -{\mathsf {k}}_{\varvec{{\mathfrak {n}}}\lambda ^\mathrm {p}}\otimes \dfrac{\partial \lambda ^\mathrm {p}}{\partial \varvec{F}} -{\mathsf {k}}_{\varvec{{\mathfrak {n}}}\varvec{{\mathfrak {n}}}}:\dfrac{\partial \varvec{{\mathfrak {n}}}}{\partial \varvec{F}} =\underline{\varvec{0}} . \end{aligned} \end{aligned}$$

(B.8)

Taking into account the symmetry of flow tensor, an equation in Vogit notation for Eqs. (B.7) and (B.8) is obtained

$$\begin{aligned}&\underbrace{\left[ \begin{array}{ccccccc} {\mathsf {k}}_{\lambda ^\mathrm {p}\lambda ^\mathrm {p}} &{}{\mathsf {k}}_{\lambda ^\mathrm {p}{\mathfrak {n}}_{11}} &{}{\mathsf {k}}_{\lambda ^\mathrm {p}{\mathfrak {n}}_{22}} &{}{\mathsf {k}}_{\lambda ^\mathrm {p}{\mathfrak {n}}_{33}} &{}2{\mathsf {k}}_{\lambda ^\mathrm {p}{\mathfrak {n}}_{23}} &{}2{\mathsf {k}}_{\lambda ^\mathrm {p}{\mathfrak {n}}_{13}} &{}2{\mathsf {k}}_{\lambda ^\mathrm {p}{\mathfrak {n}}_{12}} \\ {\mathsf {k}}_{{\mathfrak {n}}_{11}\lambda ^\mathrm {p}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{11}{\mathfrak {n}}_{11}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{11}{\mathfrak {n}}_{22}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{11}{\mathfrak {n}}_{33}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{11}{\mathfrak {n}}_{23}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{11}{\mathfrak {n}}_{13}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{11}{\mathfrak {n}}_{12}} \\ {\mathsf {k}}_{{\mathfrak {n}}_{22}\lambda ^\mathrm {p}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{22}{\mathfrak {n}}_{11}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{22}{\mathfrak {n}}_{22}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{22}{\mathfrak {n}}_{33}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{22}{\mathfrak {n}}_{23}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{22}{\mathfrak {n}}_{13}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{22}{\mathfrak {n}}_{12}} \\ {\mathsf {k}}_{{\mathfrak {n}}_{33}\lambda ^\mathrm {p}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{33}{\mathfrak {n}}_{11}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{33}{\mathfrak {n}}_{22}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{33}{\mathfrak {n}}_{33}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{33}{\mathfrak {n}}_{23}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{33}{\mathfrak {n}}_{13}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{33}{\mathfrak {n}}_{12}} \\ 2{\mathsf {k}}_{{\mathfrak {n}}_{23}\lambda ^\mathrm {p}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{23}{\mathfrak {n}}_{11}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{23}{\mathfrak {n}}_{22}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{23}{\mathfrak {n}}_{33}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{23}{\mathfrak {n}}_{23}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{23}{\mathfrak {n}}_{13}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{23}{\mathfrak {n}}_{12}} \\ 2{\mathsf {k}}_{{\mathfrak {n}}_{13}\lambda ^\mathrm {p}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{13}{\mathfrak {n}}_{11}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{13}{\mathfrak {n}}_{22}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{13}{\mathfrak {n}}_{33}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{13}{\mathfrak {n}}_{23}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{13}{\mathfrak {n}}_{13}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{13}{\mathfrak {n}}_{12}} \\ 2{\mathsf {k}}_{{\mathfrak {n}}_{12}\lambda ^\mathrm {p}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{12}{\mathfrak {n}}_{11}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{12}{\mathfrak {n}}_{22}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{12}{\mathfrak {n}}_{33}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{12}{\mathfrak {n}}_{23}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{12}{\mathfrak {n}}_{13}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{12}{\mathfrak {n}}_{12}} \end{array} \right] }_{\varvec{{\mathsf {k}}}}&\quad \left[ \begin{array}{c} \dfrac{\partial \lambda ^\mathrm {p}}{\partial \varvec{F}}\\ \dfrac{\partial {\mathfrak {n}}_{11}}{\partial \varvec{F}}\\ \dfrac{\partial {\mathfrak {n}}_{22}}{\partial \varvec{F}}\\ \dfrac{\partial {\mathfrak {n}}_{33}}{\partial \varvec{F}}\\ \dfrac{\partial {\mathfrak {n}}_{23}}{\partial \varvec{F}}\\ \dfrac{\partial {\mathfrak {n}}_{13}}{\partial \varvec{F}}\\ \dfrac{\partial {\mathfrak {n}}_{12}}{\partial \varvec{F}} \end{array} \right] \nonumber \\&= -\left[ \begin{array}{c} \dfrac{\partial ||{\varvec{\tau }_0}_{\mathrm {dev}}||}{\partial \varvec{F}}\\ \dfrac{\partial }{\partial \varvec{F}}\dfrac{{{\tau }_0}_{\mathrm {dev},11}}{||{\varvec{\tau }_0}_{\mathrm {dev}}||}\\ \dfrac{\partial }{\partial \varvec{F}}\dfrac{{{\tau }_0}_{\mathrm {dev},22}}{||{\varvec{\tau }_0}_{\mathrm {dev}}||}\\ \dfrac{\partial }{\partial \varvec{F}}\dfrac{{{\tau }_0}_{\mathrm {dev},33}}{||{\varvec{\tau }_0}_{\mathrm {dev}}||}\\ 2\dfrac{\partial }{\partial \varvec{F}}\dfrac{{{\tau }_0}_{\mathrm {dev},23}}{||{\varvec{\tau }_0}_{\mathrm {dev}}||}\\ 2\dfrac{\partial }{\partial \varvec{F}}\dfrac{{{\tau }_0}_{\mathrm {dev},13}}{||{\varvec{\tau }_0}_{\mathrm {dev}}||}\\ 2\dfrac{\partial }{\partial \varvec{F}}\dfrac{{{\tau }_0}_{\mathrm {dev},12}}{||{\varvec{\tau }_0}_{\mathrm {dev}}||} \end{array} \right] \end{aligned}$$

(B.9)

with

$$\begin{aligned} \begin{aligned} \dfrac{\partial ||{\varvec{\tau }_0}_{\mathrm {dev}}||}{\partial \varvec{F}_{n+1}}&= \dfrac{\partial ||{\varvec{\tau }_0}_{\mathrm {dev}}||}{\partial \varvec{F}^{e}_{n+1}}: \exp \left( -{\varDelta }t_{n+1}\lambda ^\mathrm {p}\varvec{{\mathfrak {n}}}\right) \ \overline{\otimes }\ \varvec{F}^{\mathrm {p}-\mathrm {T}}_{n} ,\\ \dfrac{\partial }{\partial \varvec{F}_{n+1}}\dfrac{{\varvec{\tau }_0}_{\mathrm {dev}}}{||{\varvec{\tau }_0}_{\mathrm {dev}}||}&= \left( \dfrac{1}{||{\varvec{\tau }_0}_{\mathrm {dev}}||}\dfrac{\partial {\varvec{\tau }_0}_{\mathrm {dev}}}{\partial \varvec{F}^\mathrm {e}_{n+1}}- \dfrac{{\varvec{\tau }_0}_{\mathrm {dev}}}{||{\varvec{\tau }_0}_{\mathrm {dev}}||^2}\otimes \dfrac{\partial ||{\varvec{\tau }_0}_{\mathrm {dev}}||}{\partial \varvec{F}^\mathrm {e}_{n+1}}\right) \\&\quad :\exp \left( -{\varDelta }t_{n+1}\lambda ^\mathrm {p}\varvec{{\mathfrak {n}}}\right) \ \overline{\otimes }\ \varvec{F}^{\mathrm {p}-\mathrm {T}}_{n} . \end{aligned} \end{aligned}$$

(B.10)

In a same manner, ${\dfrac{\partial \varvec{F}^\mathrm {e}}{\partial \alpha }}$ yields

$$\begin{aligned} {\dfrac{\partial \varvec{F}^\mathrm {e}}{\partial \alpha }}= \left. \dfrac{\partial \varvec{F}^\mathrm {e}}{\partial \lambda ^\mathrm {p}}\right| _{\varvec{{\mathfrak {n}}}=\text {const.}} \dfrac{\partial \lambda ^\mathrm {p}}{\partial \alpha }+ \left. \dfrac{\partial \varvec{F}^\mathrm {e}}{\partial \varvec{{\mathfrak {n}}}}\right| _{\lambda ^\mathrm {p}=\text {const.}}: \dfrac{\partial \varvec{{\mathfrak {n}}}}{\partial \alpha } \end{aligned}$$

(B.11)

with

$$\begin{aligned} \begin{aligned}&\text {d}_{\alpha }{\mathsf {r}}^{\lambda ^\mathrm {p}}= \text {d}_{\alpha }\left[ -||{\varvec{\tau }_0}_{\mathrm {dev},n+1}||+\sqrt{\dfrac{2}{3}}r^\mathrm {p}_{0,n+1}+\dfrac{2{\eta }_{\mathrm {p}}}{3g\left( d\right) }\lambda ^\mathrm {p}_{n+1}\right] =0 \\&\qquad \rightarrow \dfrac{\partial {\mathsf {r}}^{\lambda ^\mathrm {p}}}{\partial \alpha }+ \dfrac{\partial {\mathsf {r}}^{\lambda ^\mathrm {p}}}{\partial \lambda ^\mathrm {p}}\dfrac{\partial \lambda ^\mathrm {p}}{\partial \alpha }+ \dfrac{\partial {\mathsf {r}}^{\lambda ^\mathrm {p}}}{\partial \varvec{{\mathfrak {n}}}}:\dfrac{\partial \varvec{{\mathfrak {n}}}}{\partial \alpha }\\&\quad =\sqrt{\dfrac{2}{3}}\dfrac{\partial r^\mathrm {p}_{0}}{\partial \alpha } -{\mathsf {k}}_{\lambda ^\mathrm {p}\lambda ^\mathrm {p}}\dfrac{\partial \lambda ^\mathrm {p}}{\partial \alpha } -{\mathsf {k}}_{\lambda ^\mathrm {p}\varvec{{\mathfrak {n}}}}:\dfrac{\partial \varvec{{\mathfrak {n}}}}{\partial \alpha } =0 \end{aligned} \end{aligned}$$

(B.12)

and

$$\begin{aligned} \begin{aligned}&\text {d}_{\alpha }{\mathsf {r}}^{\varvec{{\mathfrak {n}}}}= \text {d}_{\alpha }\left[ -\dfrac{{\varvec{\tau }_0}_{\mathrm {dev},n+1}}{||{\varvec{\tau }_0}_{\mathrm {dev},n+1}||}+\varvec{{\mathfrak {n}}}\right] =0\\&\quad \rightarrow \dfrac{\partial {\mathsf {r}}^{\varvec{{\mathfrak {n}}}}}{\partial \lambda ^\mathrm {p}}\dfrac{\partial \lambda ^\mathrm {p}}{\partial \alpha }+ \dfrac{\partial {\mathsf {r}}^{\varvec{{\mathfrak {n}}}}}{\partial \varvec{{\mathfrak {n}}}}:\dfrac{\partial \varvec{{\mathfrak {n}}}}{\partial \alpha } =-{\mathsf {k}}_{\varvec{{\mathfrak {n}}}\lambda ^\mathrm {p}}\dfrac{\partial \lambda ^\mathrm {p}}{\partial \alpha } -{\mathsf {k}}_{\varvec{{\mathfrak {n}}}\varvec{{\mathfrak {n}}}}:\dfrac{\partial \varvec{{\mathfrak {n}}}}{\partial \alpha } =\varvec{0} , \end{aligned} \end{aligned}$$

(B.13)

Taking into account of the symmetry of flow tensor, an equation in Vogit notation for Eqs. (B.12) and (B.13) is obtained

$$\begin{aligned} \begin{aligned}&\underbrace{\left[ \begin{array}{ccccccc} {\mathsf {k}}_{\lambda ^\mathrm {p}\lambda ^\mathrm {p}} &{}{\mathsf {k}}_{\lambda ^\mathrm {p}{\mathfrak {n}}_{11}} &{}{\mathsf {k}}_{\lambda ^\mathrm {p}{\mathfrak {n}}_{22}} &{}{\mathsf {k}}_{\lambda ^\mathrm {p}{\mathfrak {n}}_{33}} &{}2{\mathsf {k}}_{\lambda ^\mathrm {p}{\mathfrak {n}}_{23}} &{}2{\mathsf {k}}_{\lambda ^\mathrm {p}{\mathfrak {n}}_{13}} &{}2{\mathsf {k}}_{\lambda ^\mathrm {p}{\mathfrak {n}}_{12}} \\ {\mathsf {k}}_{{\mathfrak {n}}_{11}\lambda ^\mathrm {p}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{11}{\mathfrak {n}}_{11}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{11}{\mathfrak {n}}_{22}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{11}{\mathfrak {n}}_{33}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{11}{\mathfrak {n}}_{23}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{11}{\mathfrak {n}}_{13}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{11}{\mathfrak {n}}_{12}} \\ {\mathsf {k}}_{{\mathfrak {n}}_{22}\lambda ^\mathrm {p}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{22}{\mathfrak {n}}_{11}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{22}{\mathfrak {n}}_{22}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{22}{\mathfrak {n}}_{33}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{22}{\mathfrak {n}}_{23}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{22}{\mathfrak {n}}_{13}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{22}{\mathfrak {n}}_{12}} \\ {\mathsf {k}}_{{\mathfrak {n}}_{33}\lambda ^\mathrm {p}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{33}{\mathfrak {n}}_{11}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{33}{\mathfrak {n}}_{22}} &{}{\mathsf {k}}_{{\mathfrak {n}}_{33}{\mathfrak {n}}_{33}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{33}{\mathfrak {n}}_{23}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{33}{\mathfrak {n}}_{13}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{33}{\mathfrak {n}}_{12}} \\ 2{\mathsf {k}}_{{\mathfrak {n}}_{23}\lambda ^\mathrm {p}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{23}{\mathfrak {n}}_{11}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{23}{\mathfrak {n}}_{22}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{23}{\mathfrak {n}}_{33}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{23}{\mathfrak {n}}_{23}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{23}{\mathfrak {n}}_{13}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{23}{\mathfrak {n}}_{12}} \\ 2{\mathsf {k}}_{{\mathfrak {n}}_{13}\lambda ^\mathrm {p}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{13}{\mathfrak {n}}_{11}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{13}{\mathfrak {n}}_{22}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{13}{\mathfrak {n}}_{33}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{13}{\mathfrak {n}}_{23}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{13}{\mathfrak {n}}_{13}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{13}{\mathfrak {n}}_{12}} \\ 2{\mathsf {k}}_{{\mathfrak {n}}_{12}\lambda ^\mathrm {p}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{12}{\mathfrak {n}}_{11}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{12}{\mathfrak {n}}_{22}} &{}2{\mathsf {k}}_{{\mathfrak {n}}_{12}{\mathfrak {n}}_{33}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{12}{\mathfrak {n}}_{23}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{12}{\mathfrak {n}}_{13}} &{}4{\mathsf {k}}_{{\mathfrak {n}}_{12}{\mathfrak {n}}_{12}} \end{array} \right] }_{\varvec{{\mathsf {k}}}}&\quad \left[ \begin{array}{c} \dfrac{\partial \lambda ^\mathrm {p}}{\partial \alpha }\\ \dfrac{\partial {\mathfrak {n}}_{11}}{\partial \alpha }\\ \dfrac{\partial {\mathfrak {n}}_{22}}{\partial \alpha }\\ \dfrac{\partial {\mathfrak {n}}_{33}}{\partial \alpha }\\ \dfrac{\partial {\mathfrak {n}}_{23}}{\partial \alpha }\\ \dfrac{\partial {\mathfrak {n}}_{13}}{\partial \alpha }\\ \dfrac{\partial {\mathfrak {n}}_{12}}{\partial \alpha } \end{array} \right] \\&\quad = \left[ \begin{array}{c} \sqrt{\dfrac{2}{3}}\dfrac{\partial r^\mathrm {p}_{0}}{\partial \alpha }\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0 \end{array} \right] \end{aligned} \end{aligned}$$

(B.14)

with

$$\begin{aligned} \begin{aligned} \dfrac{\partial r^\mathrm {p}_{0}}{\partial \alpha }&= \dfrac{\partial }{\partial \alpha }\left\{ {\hat{y}}\left( {\bar{\alpha }}\right) +p_\mathrm {p}\left( {\bar{\alpha }}-\alpha \right) \right\} =-p_\mathrm {p}. \end{aligned} \end{aligned}$$

(B.15)

In addition, $\dfrac{\partial {\bar{\alpha }}}{\partial \alpha }$ and $\dfrac{\partial {\bar{\alpha }}}{\partial \varvec{F}}$ are obtained as

$$\begin{aligned} \begin{aligned} \dfrac{\partial {\bar{\alpha }}}{\partial \alpha }= \dfrac{\partial }{\partial \alpha }\left( {\bar{\alpha }}_n+\sqrt{\dfrac{2}{3}}\lambda ^\mathrm {p}{\varDelta }t_{n+1}\right) = \sqrt{\dfrac{2}{3}}\dfrac{\partial \lambda ^\mathrm {p}}{\partial \alpha }{\varDelta }t_{n+1} \end{aligned} \end{aligned}$$

(B.16)

and

$$\begin{aligned} \begin{aligned} \dfrac{\partial {\bar{\alpha }}}{\partial \varvec{F}}= \dfrac{\partial }{\partial \varvec{F}}\left( {\bar{\alpha }}_n+\sqrt{\dfrac{2}{3}}\lambda ^\mathrm {p}{\varDelta }t_{n+1}\right) = \sqrt{\dfrac{2}{3}}\dfrac{\partial \lambda ^\mathrm {p}}{\partial \varvec{F}}{\varDelta }t_{n+1} \end{aligned} . \end{aligned}$$

(B.17)

C An issue about crack phase-field modeling

Generally, once the material experiences plastic deformation, it does not show elastic behavior again except unloadings. Nevertheless, due to the setup of damage modeling incorporated into the elastoplastic constitutive model such as [42, 50, 51], the material shows elastic behavior again when the damage accumulates significantly. We explain this elastic-plastic-elastic transition using the same setup as in Sect. 2. We assume the material is under the plastic state and damaged by external loadings. Then, the damage variable without the plastic driving force at the end of the loading step n yields

$$\begin{aligned} \begin{aligned}&d_n=\dfrac{\hat{{\varPsi }}^{\mathrm {e}}_n}{G_\mathrm {c}/2l_\mathrm {f}+\hat{{\varPsi }}^{\mathrm {e}}_n}\\&\quad \text {with}\quad \hat{{\varPsi }}^{\mathrm {e}}_n=\dfrac{1}{2}E {\varepsilon ^\mathrm {e}_n}^2 ,\quad \varepsilon ^\mathrm {e}_n =\varepsilon _n - \varepsilon ^\mathrm {p}_n \quad \text {and}\quad \varepsilon _n =\dfrac{u_n}{L+u_n} , \end{aligned} \end{aligned}$$

(C.1)

where L and $u_n$ denote the structure’s length and the total displacement by the loading step n. When the next loading is introduced at the loading step $n+1$, we start from checking the deformation state by using the yield function ${\hat{\varPhi }}^\mathrm {p}_{n+1,k=1}$ at the first global Newton-Raphson iteration $k=1$. By using the variables obtained in the previous loading step n, the yield function is written as

$$\begin{aligned} \begin{aligned}&{\hat{\varPhi }}^\mathrm {p}_{n+1,k=1}=\left( 1-d_n\right) ^2\sigma _{n+1,k=1} - \left( \sigma _0 + h \varepsilon ^\mathrm {p}_n\right) \\&\quad \text {with}\quad \sigma _{n+1,k=1}=E\left( \varepsilon _{n+1} - \varepsilon ^\mathrm {p}_n\right) \quad \text {and}\quad \varepsilon _{n+1}\\&\quad =\dfrac{u_n+\varDelta u_{n+1}}{L+u_n+\varDelta u_{n+1}}, \end{aligned} \end{aligned}$$

(C.2)

where $\varDelta u_{n+1}$ is the new displacement increment. Here, Eq. (C.2) may be lower than zero due to the updated damage variable $d_n$, which represents that the material shows elastic behavior despite new loading. By conducting some algebra, we obtain the explicit displacement increment

$$\begin{aligned} \begin{aligned} \varDelta u_{n+1}\ge \dfrac{C_{n+1}\left( L+u_n\right) -u_n}{C_{n+1}-1} \quad \text {with}\quad C_{n+1}=\dfrac{\left( \sigma _0 + h \varepsilon ^\mathrm {p}_n\right) }{\left( 1-d_n\right) ^2 E}+ \varepsilon ^\mathrm {p}_n \end{aligned} , \end{aligned}$$

(C.3)

which leads the material into the elastic state again. Thus, we may infer that this setup is not absolutely appropriate to explain ductile fracture in elastoplastic materials. On the other hand, we also realize that this unrealistic transition of deformation state contributes to computational stability. As widely known, the violent plastic deformation due to the severely damaged mesh’s softening usually causes unstable tendencies, especially in the local return mapping process. This problem may be avoided by the unreal state transition since the severely damaged mesh does not show the plastic behavior.

However, the proposed model introduces the plastic deformation, which results in the inevitable of the violent plastic deformation in the severely damaged region. Therefore, we set a threshold around $d\approx 0.9\sim 0.95$ to terminate the plastic deformation to maintain computational stability.