Phase-field modeling of ductile fracture

Abstract

Phase-field modeling of brittle fracture in elastic solids is a well-established framework that overcomes the limitations of the classical Griffith theory in the prediction of crack nucleation and in the identification of complicated crack paths including branching and merging. We propose a novel phase-field model for ductile fracture of elasto-plastic solids in the quasi-static kinematically linear regime. The formulation is shown to capture the entire range of behavior of a ductile material exhibiting $J_2$-plasticity, encompassing plasticization, crack initiation, propagation and failure. Several examples demonstrate the ability of the model to reproduce some important phenomenological features of ductile fracture as reported in the experimental literature.

Appendix

1.1 Perturbation analysis for the $s$ and $p$ mutual interaction

One of the most important aspects in our phase-field model of elasto-plastic ductile fracture is the coupling function $g(s,p):=s^{2p}$. Introduction of the exponent $p$, which accounts for the accumulation and localization of plastic strains, into the classic brittle-case function $g(s)=s^{2}$ allows us to let $s$ explicitly depend also on the plastic energy density $\Psi _\mathrm{p}$, and not only on the elastic one, $\Psi _\mathrm{e}$. This can be formally seen from the corresponding evolution equation for $s$ (25). More importantly, we show below that the evolution of $s$ in this case will mainly be governed by $p\sim \Psi _\mathrm{p}$ rather than $\Psi _\mathrm{e}$, thus delaying the brittle-like fracture development and enabling the desired plastic-driven fracture mechanism.

Without loss of generality, we restrict ourselves to the pure tensile loading situation so that no split of the elastic energy density function $\Psi _\mathrm{e}$ is required and the evolution equation for $s$ is given by

$$\begin{aligned} -4\ell ^2\Delta s+s+\frac{4\ell }{G_c}\Psi _\mathrm{e}(\varvec{\varepsilon })p s^{2p-1}=1. \end{aligned}$$

(31)

Equation (31) is a non-linear partial differential equation whose explicit analytical solution for $s$ is out of reach: one can only conclude that with $p\equiv 0$ (zero plastic strains), a trivial solution is $s=1$ (no phase-field development occurs). With a suitable rescaling of variables (i.e. non-dimensionalization), one can obtain a dimensionless version of (31) containing a small (perturbation) parameter. An approximate analytical solution of such perturbed non-linear equation can then be constructed explicitly, establishing a relation between $s$ and $p$.

Let $L$ be the maximal macroscopic dimension of the considered body. We introduce the following non-dimensional quantities:

$$\begin{aligned} (\bar{x},\bar{y}):=\left( \frac{x}{L},\frac{y}{L}\right) , \quad \bar{\varvec{u}}:=\frac{\varvec{u}}{A}, \quad \bar{\mathbb C}:=\frac{\mathbb C}{E}, \quad \bar{\ell }:=\frac{\ell }{\frac{1}{2}L}, \end{aligned}$$

(32)

where $A>0$ is a constant whose particular choice will be motivated below. The scaled strain tensor $\bar{\varvec{\varepsilon }}=\frac{1}{2}(\bar{\nabla }\bar{\varvec{u}}+\bar{\nabla }\bar{\varvec{u}}^T)$, with $\bar{\nabla }$ denoting the gradient w.r.t. the non-dimensional variables $(\bar{x},\bar{y})$, relates to the actual strain tensor $\varvec{\varepsilon }$ as

$$\begin{aligned} \varvec{\varepsilon }(\varvec{u})=\frac{A}{L}\bar{\varvec{\varepsilon }}(\bar{\varvec{u}}), \end{aligned}$$

(33)

yielding for $\Psi _\mathrm{e}$ the relation

$$\begin{aligned} \Psi _\mathrm{e}(\varvec{\varepsilon }) =\frac{1}{2}\mathbb {C}\varvec{\varepsilon }^2 =\frac{1}{2} E \bar{\mathbb {C}} \left( \frac{A}{L} \right) ^2 \bar{\varvec{\varepsilon }}^2 =E\left( \frac{A}{L}\right) ^2 \bar{\Psi }_\mathrm{e}(\bar{\varvec{\varepsilon }}). \end{aligned}$$

(34)

Also, we set $\bar{s}:=s$ so that $\Delta s=\frac{1}{L^2}\bar{\Delta }\bar{s}$, with $\bar{\Delta }$ standing for the Laplace operator w.r.t. $(\bar{x},\bar{y})$. Finally, a non-dimensional plastic strain variable is defined as $\overline{\varvec{\varepsilon }^\mathrm{p}}:=\varvec{\varepsilon }^\mathrm{p}$ leading to the corresponding definition of $\bar{p}$:

$$\begin{aligned} \bar{p}:=\frac{ \overline{\varepsilon ^\mathrm{p}_\mathrm{eq}} }{\varepsilon ^\mathrm{p}_\mathrm{eq,crit}}, \quad \overline{\varepsilon ^\mathrm{p}_\mathrm{eq}}(t):= \sqrt{\frac{2}{3}}\int _{0}^t \sqrt{ \dot{\overline{\varvec{\varepsilon }^\mathrm{p}}}:\dot{\overline{\varvec{\varepsilon }^\mathrm{p}}} } \;d\tau . \end{aligned}$$

(35)

Note that the denominator remains the same as in the definition of $p$.

Using (32)–(35) we turn (31) into

$$\begin{aligned} -\bar{\ell }^2\bar{\Delta }\bar{s}+\bar{s} +\frac{2\bar{\ell } L}{G_c} E \left( \frac{A}{L} \right) ^2 \bar{\Psi }_\mathrm{e}(\bar{\varvec{\varepsilon }}) \bar{p} \bar{s}^{2\bar{p}-1}=1. \end{aligned}$$

In the above, the scaling constant $A$ can be chosen to provide $\frac{2L}{G_c} E \left( \frac{A}{L} \right) ^2=1$, that is, $A:=\sqrt{\frac{G_c L}{2E}}$, and the dimensionless counterpart of (31) we end up with reads as

$$\begin{aligned} -\bar{\ell }^2\bar{\Delta }\bar{s}+\bar{s} +\bar{\ell } \bar{\Psi }_\mathrm{e}(\bar{\varvec{\varepsilon }}) \bar{p} \bar{s}^{2\bar{p}-1}=1. \end{aligned}$$

(36)

In what follows, we set $\gamma :=\bar{\ell }$ and also drop the bars over all remaining variables in (36). This yields

$$\begin{aligned} -\gamma ^2\Delta s+s+\gamma \Psi _\mathrm{e}(\varvec{\varepsilon }) p s^{2p-1}=1, \end{aligned}$$

(37)

where, by definition, $\gamma$ is a small (perturbation) parameter, that is, $0<\gamma \ll 1$.

According to the perturbation analysis framework [59], an asymptotic expansion of $s$ is taken in a form of a power series in $\gamma$:

$$\begin{aligned} s=s_0+\gamma s_1+\gamma ^2 s_2+\cdots , \end{aligned}$$

(38)

what yields for $\Delta s$ and $s^{2p-1}$ the expansions

$$\begin{aligned} \Delta s=\Delta s_0+\gamma \Delta s_1+\gamma ^2 \Delta s_2+\cdots , \end{aligned}$$

(39)

and

$$\begin{aligned} s^{2p-1}&= s_0^{2p-1}+\gamma (2p-1)s_0^{2p-2}s_1 \nonumber \\&\quad \ + \gamma ^2(2p-1)s_0^{2p-3}\left( s_0s_2+s_1^2(p-1)\right) +\cdots , \end{aligned}$$

(40)

respectively. In (39) and (40), only the terms up to order $O(\gamma ^2)$ have been retained, consistently with the assumed expansion (38). We now substitute (38)–(40) into the governing equation (37) and collect the coefficients of like powers of $\gamma$ to obtain:

$$\begin{aligned}&s_0+\gamma \left( s_1+p\Psi _\mathrm{e}s_0^{2p-1} \right) \nonumber \\&\quad + \gamma ^2 \left( -\Delta s_0+s_2+p(2p-1)\Psi _\mathrm{e}s_0^{2p-2}s_1 \right) +\cdots =1. \end{aligned}$$

(41)

Equating the coefficients of each power of $\gamma$ to zero we ’extract’ the equations for $s_0,s_1$ and $s_2$. The corresponding solutions are

$$\begin{aligned} s_0=1, \quad s_1=-p\Psi _\mathrm{e}, \quad s_2=p^2(2p-1)\Psi _\mathrm{e}^2, \end{aligned}$$

(42)

and, due to (38), we finally obtain

$$\begin{aligned} s = 1-\gamma p\Psi _\mathrm{e} + \gamma ^2 p^2(2p-1)\Psi _\mathrm{e}^2+\cdots , \end{aligned}$$

(43)

to be treated as an approximate analytical solution of equation (37). Note that with $\gamma \rightarrow 0$, equation (37) reduces to $s=1$. Sending $\gamma$ to zero also in (43), we recover the same result, i.e. $s=1$, as expected. Recall also that setting $p=0$ in (37), the trivial solution of the reduced equation is $s=1$. Similar result is obtained while substituting $p=0$ in (43).

From (43) the influence of $p$ on the phase-field $s$ can be grasped. In the linear elastic regime (no plastic strains, $p=0$) and at the beginning of the plastic regime (plastic strains are still negligibly small, $0<p\ll 1$), the evolution of $s$ is minor even with a (possibly large) non-zero contribution of $\Psi _\mathrm{e}$. In other words, the presence of $p$ in $g(s,p)=s^{2p}$ delays the brittle-like fracture formation. This is in a contrast to the brittle phase-field formulation, when the use of $g(s)=s^2$ results in $s$ depending solely on $\Psi _\mathrm{e}$ (expansion (43) with $p\equiv 1$) meaning that the crack development occurs independently of the evolution of the plastic strains.

1.2 Proof of thermodynamic consistency

In this Appendix, the governing equations of the proposed model are rewritten in the framework of the principle of virtual power as developed by Gurtin, see [60, 61], and their thermodynamic consistency is investigated.

1.2.1 Kinematics

The kinematics is based on the decomposition of the displacement gradient into elastic and plastic components $\nabla \mathbf {u}=\mathbf {H}^\mathrm{e}+\mathbf {H}^\mathrm{p}$. Correspondingly, we define the elastic and plastic strain tensors as

$$\begin{aligned} {\varvec{\varepsilon }}^\mathrm{e}=\frac{1}{2}\left( \mathbf {H}^\mathrm{e}+\mathbf {H}^{\mathrm{e}T}\right) \quad {\varvec{\varepsilon }}^\mathrm{p}=\frac{1}{2}\left( \mathbf {H}^\mathrm{p}+\mathbf {H}^{\mathrm{p}T}\right) \end{aligned}$$

(44)

so that the total strain tensor is given by ${\varvec{\varepsilon }}={\varvec{\varepsilon }}^\mathrm{e}+{\varvec{\varepsilon }}^\mathrm{p}$. Finally, we assume that $\mathbf {H}^\mathrm{p}$ is purely deviatoric, i.e. $\mathrm{tr}\,\mathbf {H}^\mathrm{p}=\mathrm{tr}\,{\varvec{\varepsilon }}^\mathrm{p}=0$.

1.2.2 Balance equations

Let us introduce the internal power in the form

$$\begin{aligned} \mathcal {I}\left( P\right)= & {} \int _{P}{{\varvec{\sigma }}}^\mathrm{e}:\dot{\mathbf {H}}^\mathrm{e}\;dv+\int _{P}{\varvec{\sigma }}^\mathrm{p}:\dot{\mathbf {H}}^\mathrm{p}\;dv+\int _{P}{\varvec{\xi }}\cdot \nabla \dot{s}\;dv\nonumber \\&+ \int _{P}\zeta \cdot \dot{s}\;dv \end{aligned}$$

(45)

where ${{\varvec{\sigma }}}^\mathrm{e}$ and ${{\varvec{\sigma }}}^\mathrm{p}$ are respectively an elastic stress, power-conjugate to $\dot{\mathbf {H}}^\mathrm{e}$, and a plastic stress, power-conjugate to $\dot{\mathbf {H}}^\mathrm{p}$. Since $\mathbf {H}^\mathrm{p}$ is deviatoric, we may assume without loss of generality that ${{\varvec{\sigma }}}^\mathrm{p}$ is deviatoric, i.e. that $\mathrm{tr}{\varvec{\sigma }}^\mathrm{p}=0$. Moreover, ${\varvec{\xi }}$ is the microscopic stress power-conjugate to $\nabla \dot{s}$, and $\zeta$ is the microscopic internal body force power-conjugate to $\dot{s}$, where $s$ is the crack field parameter, $0\le s\le 1$. The integration domain is any subregion $P$ of the body under consideration. The external power is given by

$$\begin{aligned} \mathcal {W}\left( P\right)= & {} \int _{\partial P}\varvec{t}\left( \varvec{n}\right) \cdot \dot{\varvec{u}}\;da+\int _{P}\varvec{b}\cdot \dot{\varvec{u}}\;dv\nonumber \\&+\int _{\partial P}\chi \left( \varvec{n}\right) \dot{s}\;da+\int _{P}\gamma \dot{s}\;dv \end{aligned}$$

(46)

where $\varvec{t}$ is the traction vector on the elementary area $da$ of the surface of $P,\, \partial P$, with outward unit normal $\varvec{n},\, \varvec{b}$ is the body force, $\chi$ and $\gamma$ are respectively the microscopic external traction and the microscopic external body force, both power-conjugate to $\dot{s}$. Considering a generalized virtual velocity

$$\begin{aligned} \mathcal {V}=\left( \tilde{\varvec{u}},\tilde{\mathbf {H}}^\mathrm{e},\tilde{\mathbf {H}}^\mathrm{p},\tilde{s}\right) \end{aligned}$$

(47)

satisfying the kinematical constraints above, the principle of virtual power reads

$$\begin{aligned}&\int _{P}{{\varvec{\sigma }}}^\mathrm{e}:\tilde{\mathbf {H}}^\mathrm{e}\;dv\!+\!\int _{P}{{\varvec{\sigma }}}^\mathrm{p}:\tilde{\mathbf {H}}^\mathrm{p}\;dv\!+\!\int _{P}{\varvec{\xi }}\cdot \nabla \tilde{s}\;dv\!+\!\int _{P}\zeta \cdot \tilde{s}\;dv\nonumber \\&=\int _{\partial P}\varvec{t}\left( \varvec{n}\right) \cdot \varvec{\tilde{u}}\;da\!+\!\int _{P}\varvec{b}\cdot \varvec{\tilde{u}}\;dv\!+\!\int _{\partial P}\chi \left( \varvec{n}\right) \tilde{s}\;da\!+\!\int _{P}\gamma \tilde{s}\;dv\nonumber \\ \end{aligned}$$

(48)

for any subregion $P$ of the body and any $\mathcal {V}$. Frame invariance implies that ${\varvec{\sigma }}^\mathrm{e}$ is symmetric, therefore ${\varvec{\sigma }}^\mathrm{e}:\tilde{\mathbf {H}}^\mathrm{e}={\varvec{\sigma }}^\mathrm{e}:\tilde{{{\varvec{\varepsilon }}}}^\mathrm{e}$. Applying Eq. (48) with $\mathcal {V}=\left( \tilde{\varvec{u}},\nabla \tilde{\varvec{u}}, \mathbf {0},0\right)$ leads to the macroscopic force balance $\mathrm{div}\varvec{\sigma }+\varvec{b}=\mathbf {0}$ and to the expression of the macroscopic traction $\varvec{t}\left( \varvec{n}\right) ={\varvec{\sigma }}\cdot \varvec{n}$, where we have introduced ${\varvec{\sigma }}:={\varvec{\sigma }}^\mathrm{e}$ as the (standard) Cauchy stress. Considering next a virtual velocity $\mathcal {V}=\left( \mathbf {0},-\tilde{\mathbf {H}}^\mathrm{p},{\tilde{\mathbf {H}}}^\mathrm{p},0\right)$ delivers the plastic microscopic force balance [61], ${\varvec{\sigma }}_{dev}={\varvec{\sigma }}^\mathrm{p}$, where ${\varvec{\sigma }}_{dev}$ is the deviatoric component of the stress tensor. Note that thus also ${\varvec{\sigma }}^\mathrm{p}$ is symmetric, therefore ${\varvec{\sigma }}^\mathrm{p}:\tilde{\mathbf {H}}^\mathrm{p}={\varvec{\sigma }}^\mathrm{p}:\tilde{{{\varvec{\varepsilon }}}}^\mathrm{p}$. Finally, insertion in Eq. (48) of a virtual velocity $\mathcal {V}=\left( \mathbf {0},\mathbf {0},\mathbf {0},\tilde{s}\right)$ yields

$$\begin{aligned} \int _{P}{\varvec{\xi }}\cdot \nabla \tilde{s}\;dv+\int _{P}\zeta \cdot \tilde{s}\;dv=\int _{\partial P}\chi \left( \varvec{n}\right) \tilde{s}\;da+\int _{P}\gamma \tilde{s}\;dv. \end{aligned}$$

(49)

Through the divergence theorem, Eq. (49) leads to the phase-field microscopic force balance

$$\begin{aligned} \mathrm{div}{\varvec{\xi }}-\zeta +\gamma =0 \end{aligned}$$

(50)

and to the expression of the phase-field microscopic traction $\chi \left( \varvec{n}\right) ={{\varvec{\xi }}}\cdot \varvec{n}$.

As a consequence of the above, and if we further introduce the codirectionality constraint valid for $J_{2}$ plasticity, $\frac{\dot{{\varvec{\varepsilon }}}^\mathrm{p}}{\left\| \dot{{\varvec{\varepsilon }}}^\mathrm{p}\right\| }=\varvec{n}^\mathrm{p}$ with $\varvec{n}^\mathrm{p}=\frac{{\varvec{\sigma }}_{dev}}{\left\| {\varvec{\sigma }}_{dev}\right\| }$, the power balance takes the final form

$$\begin{aligned}&\int _{P}{{\varvec{\sigma }}}:\dot{{\varvec{\varepsilon }}}^\mathrm{e}\;dv+\int _{P}\tau ^\mathrm{p}\dot{e}^\mathrm{p}\;dv+\int _{P}{\varvec{\xi }}\cdot \nabla \dot{s}\;dv+\int _{P}\zeta \cdot \dot{s}\;dv\\&= \int _{\partial P}\varvec{t}\left( \varvec{n}\right) \cdot \dot{\varvec{u}}\;da+\int _{P}\varvec{b}\cdot \dot{\varvec{u}}\;dv\!+\!\int _{\partial P}\chi \left( \varvec{n}\right) \dot{s}\;da\!+\!\int _{P}\!\gamma \dot{s}\;dv, \end{aligned}$$

where

$$\begin{aligned} \tau ^\mathrm{p}:=\left\| {\varvec{\sigma }}_{dev}\right\| , \quad \dot{e}^\mathrm{p}:=\left\| \dot{{\varvec{\varepsilon }}}^\mathrm{p}\right\| \ge 0, \quad e^\mathrm{p}\left( t\right) =\int _{0}^{t}\left\| \dot{{\varvec{\varepsilon }}}^\mathrm{p}\right\| \;d\tau \end{aligned}$$

(51)

1.2.3 Dissipation inequality and constitutive laws

Based on the formulation in Sect. 1, the dissipation inequality can be written (in local form) as

$$\begin{aligned} \dot{E}_\ell -{\varvec{\sigma }}:\dot{{\varvec{\varepsilon }}}^\mathrm{e}-\tau ^\mathrm{p}\dot{e}^\mathrm{p}-{\varvec{\xi }}\cdot \nabla \dot{s}-\zeta \cdot \dot{s}=-D\le 0 \end{aligned}$$

(52)

with $E_\ell$ as the free energy and $D$ as the dissipation rate, both per unit volume. Let us consider the following form of the free energy

$$\begin{aligned} E_\ell =E_\ell \left( {\varvec{\varepsilon }}^\mathrm{e},e^\mathrm{p},\alpha ,s,\nabla s\right) . \end{aligned}$$

(53)

This postulated form contains the “standard” dependencies on the elastic strain ${\varvec{\varepsilon }}^\mathrm{e}$ and on the hardening variable $\alpha$, plus further dependencies on the phase field and on its gradient, as well as on $e^\mathrm{p}$. The latter is a scalar measure of plastic strain accumulation defined in Eq. (51). The dependency of the free energy on $e^\mathrm{p}$ is needed to realize a coupling between the evolution of the phase-field parameter and the accumulation of plastic strains, and is an essential ingredient of the proposed model. Substitution of Eq. (53) in the dissipation inequality (52) leads to

$$\begin{aligned}&\left( \frac{\partial E_\ell }{\partial {\varvec{\varepsilon }}^\mathrm{e}}-{\varvec{\sigma }}\right) :\dot{{\varvec{\varepsilon }}}^\mathrm{e}+\left( \frac{\partial E_\ell }{\partial e^\mathrm{p}}-\tau ^\mathrm{p}\right) \dot{e}^\mathrm{p}+\frac{\partial E_\ell }{\partial \alpha }\dot{\alpha }\nonumber \\&+\left( \frac{\partial E_\ell }{\partial s}-\zeta \right) \dot{s}+\left( \frac{\partial E_\ell }{\partial \nabla s}-{\varvec{\xi }}\right) \cdot \nabla \dot{s}=-D\le 0. \end{aligned}$$

(54)

Well-known arguments lead immediately to the elastic stress-strain relationship

$$\begin{aligned} {\varvec{\sigma }}=\frac{\partial E_\ell }{\partial {\varvec{\varepsilon }}^{e}}. \end{aligned}$$

(55)

Additional consequences can be driven by applying the inequality individually to the group of terms related to the phase field

$$\begin{aligned} \left( \frac{\partial E_\ell }{\partial s}-\zeta \right) \dot{s}+\left( \frac{\partial E_\ell }{\partial \nabla s}-{\varvec{\xi }}\right) \cdot \nabla \dot{s}\le 0. \end{aligned}$$

(56)

From (56) follow the phase-field microscopic constitutive equations

$$\begin{aligned} {\varvec{\xi }}=\frac{\partial E_\ell }{\partial \nabla s}, \qquad \zeta =\frac{\partial E_\ell }{\partial s}. \end{aligned}$$

(57)

Substituting eqs. (57) into (50) leads to the phase-field evolution equation

$$\begin{aligned} \mathrm{div}\left( \frac{\partial E_\ell }{\partial \nabla s}\right) -\frac{\partial E_\ell }{\partial s}=0, \end{aligned}$$

(58)

where we have further assumed $\gamma =0$.

As a result of Eq. (55) and inequality (56), and introducing the thermodynamic force power-conjugate to $\dot{\alpha }$,

$$\begin{aligned} t_{\alpha }:=-\frac{\partial E_\ell }{\partial \alpha }, \end{aligned}$$

(59)

the following reduced dissipation inequality is obtained from (54)

$$\begin{aligned} D:=\tau ^\mathrm{p}\dot{e}^\mathrm{p}+t_{\alpha }\dot{\alpha }-\frac{\partial E_\ell }{\partial e^\mathrm{p}}\dot{e}^\mathrm{p}\ge 0. \end{aligned}$$

(60)

Note that, being $\dot{e}^\mathrm{p}\ge 0$, inequality (60) can be further reduced to its “classical” version

$$\begin{aligned} \tau ^\mathrm{p}\dot{e}^\mathrm{p}+t_{\alpha }\dot{\alpha }\ge 0, \end{aligned}$$

(61)

(whose satisfaction is ensured by the “classical” flow rule and hardening evolution equation of $J_{2}$-plasticity used in this work) provided that the following inequality holds

$$\begin{aligned} \frac{\partial E_\ell }{\partial e^\mathrm{p}}\le 0. \end{aligned}$$

(62)

From the specific choice of the free energy in Eq. (16) follows $\frac{\partial E_\ell }{\partial e^\mathrm{p}}=\frac{\partial g}{\partial e^\mathrm{p}}\Psi _\mathrm{e}^{+}({\varvec{\varepsilon }}^\mathrm{e})$ and therefore inequality (62) holds [and the reduced dissipation inequality takes the form (61)] provided that

$$\begin{aligned} \frac{\partial g}{\partial e^\mathrm{p}}\le 0. \end{aligned}$$

(63)

A possible specific choice of degradation function $g$ complying with (63) is the following

$$\begin{aligned} g\left( s,e^\mathrm{p}\right) =s^{2e^\mathrm{p}/e_\mathrm{crit}^\mathrm{p}}+\eta \end{aligned}$$

(64)

with coincides with the one in Eq. (17) by taking

$$\begin{aligned} e_\mathrm{crit}^\mathrm{p}=\sqrt{\frac{3}{2}}\varepsilon _\mathrm{eq,crit}^\mathrm{p} \end{aligned}$$

(65)

This proves the thermodynamic consistency of the proposed model.