Consider the continua
\(\Omega \in {\mathbb {R}}^D\) in
D-dimensional space, depicted in Fig.
1, with its boundary represented as
\(\Gamma \). The boundary
\(\Gamma \) is subjected to Neumann boundary conditions on
\(\Gamma _t\) and Dirichlet boundary conditions on
\(\Gamma _u\), where
\(\Gamma _u \cup \Gamma _t = \Gamma \) and
\(\Gamma _u \cap \Gamma _t = \varnothing \). The set of discontinuities in the domain is represented by a discrete surface
\(\Gamma _d\). The crack’s intersection with the boundary is considered a Neumann-type boundary condition.
According to the phase-field formulation, the fracture’s discrete surface
\(\Gamma _d\) is approximated implicitly as a continuous function over a width
l using the Allen–Cahn fracture surface density function
\(\gamma (d)\) as
$$\begin{aligned} \gamma (d) = \frac{1}{c_0 l} \left( w(d) + l^2 \Vert \nabla d \Vert ^2 \right) , \; \text {with} \; c_0 = 4\int _0^1 \sqrt{w(l)} dl \nonumber \\ \end{aligned}$$
(1)
where
d is the phase-field variable, with
\(d=0\) presenting the intact part of the domain while
\(d=1\) expressing a point on
\(\Gamma _d\).
w(
d) is the transition function, also known as the dissipation function, defined for cohesive cracks as
\(w(d)=d\) [
85,
86], hence
\(c_0=\frac{8}{3}\). Accordingly, a surface integral
\(\int ds\) is approximated using a volume integral as
\(\int ds \approx \int \gamma (d)~dv\).
Assuming small deformation kinematics, and given the displacement field
\(\varvec{u}\), the strain field is expressed by
\(\varvec{\varepsilon }=(\nabla \varvec{u} + \nabla \varvec{u}^T)/2\), and the crack surface density function
\(\gamma (d)\), the total energy of a fractured continua, occupying the domain
\(\Omega \) and bounded by the boundary
\(\Gamma \), shown in Fig.
1, is expressed as
$$\begin{aligned} \Psi = -\Psi ^{external} + \Psi ^{internal} + \Psi ^{fracture}, \end{aligned}$$
(2)
where
\(\Psi ^{external}\) is the work done by the external traction stress
\(\varvec{\tau }\) and body force
\(\varvec{b}\), and expressed as
$$\begin{aligned} \Psi ^{external}&= \int _{\Gamma _t} \varvec{u}\cdot \varvec{\tau }~ds + \int _{\Omega } \varvec{u}\cdot \varvec{b}~dv. \end{aligned}$$
(3)
The fracture energy, i.e.,
\(\Psi ^{fracture}\), is the energy dissipated from the system to create a fracture surface
\(\Gamma _d\). Given the energy release rate
\(\mathcal {G}_c\) (per unit fracture length),
\(\Psi ^{fracture}\) is expressed as
$$\begin{aligned} \Psi ^{fracture}&= \int _{\Gamma _d} \mathcal {G}_c ~ds \approx \int _{\Omega } \mathcal {G}_c \gamma (d)~dv. \end{aligned}$$
(4)
The stored internal energy of the system
\(\Psi ^{internal}\) consists of the elastic stored energy in the intact part of the domain and stored energy in the damaged part of the domain, expressed as
$$\begin{aligned} \Psi ^{internal}&= \int _{\Omega } \psi (\varvec{\varepsilon }, d)~dv. \end{aligned}$$
(5)
The internal energy density function
\(\psi (\varvec{\varepsilon }, d)\) is defined as
\(\psi (\varvec{\varepsilon },d) = \tfrac{1}{2} \varvec{\sigma }: \varvec{\varepsilon }\), which consists of both inactive and damaged counterparts. For the intact part of continuum, i.e., where
\(d=0\), the Cauchy stress tensor
\({\varvec{\sigma }}(\varvec{\varepsilon }, d=0)\) is expressed using Hook’s law as
$$\begin{aligned} {\varvec{\sigma }}(\varvec{\varepsilon }, d=0) = \left( \kappa -\frac{2}{3}\mu \right) \varepsilon _v \varvec{1} + 2\mu \varvec{\varepsilon }, \end{aligned}$$
(6)
where
\(\kappa \) and
\(\mu \) are bulk and shear moduli of the intact material, respectively, and
\(\varepsilon _v\) is the volumetric strain, expressed as
\(\varepsilon _v = \text {tr}(\varvec{\varepsilon })\). For the parts of the domain where
\(d > 0\), the Cauchy stress tensor is decomposed into inactive part
\(\varvec{\sigma }^I\) and active part
\(\varvec{\sigma }^A\) as
$$\begin{aligned} \begin{aligned} \varvec{\sigma }(\varvec{\varepsilon }, d)&= \varvec{\sigma }^I(\varvec{\varepsilon }) + \varvec{\sigma }^A(\varvec{\varepsilon },d) \\&= \varvec{\sigma }^I(\varvec{\varepsilon }) + g(d) {\hat{\varvec{\sigma }}}(\varvec{\varepsilon }) + (1-g(d)){\tilde{\varvec{\sigma }}}(\varvec{\varepsilon }). \end{aligned} \end{aligned}$$
(7)
The active part of the stress tensor undergoes the damage process, and
g(
d) is a degradation function that expresses the stress transition from bulk (
\({\hat{\varvec{\sigma }}}\)) to fracture (
\({\tilde{\varvec{\sigma }}}\)). We will discuss these in more details in the next sections.
Therefore, there are two solution variables associated with the phase-field formulation, the standard displacement field
\(\varvec{u}\) and the additional phase-field variable
d. Taking the variation of
\(\Psi \) with respect to
\(\varvec{u}\) and
d, and following the standard
weak to strong form steps of the FEM [
87,
88] and phase-field [
73,
82,
89], we can arrive at the following governing relations:
$$\begin{aligned}&\nabla \cdot \varvec{\sigma }(\varvec{\varepsilon },d) + \varvec{b} = 0, \end{aligned}$$
(8)
$$\begin{aligned}&\frac{3\mathcal {G}_c}{8l}\left( 2l^2\nabla ^2 d - 1\right) - g'(d) \mathcal {H}(\varvec{\varepsilon }) = 0. \end{aligned}$$
(9)
The irreversibility of the fracture process is guaranteed with the local history field of maximum stored shear energy
\(\mathcal {H}^+(\varvec{\varepsilon })\) that allows us to solve the constrained minimization of Eq. (
9) in a straightforward way [
67] and avoids unphysical self-healing.
\(\mathcal {H}^+(\varvec{\varepsilon })\) is defined as follows:
$$\begin{aligned} \mathcal {H}^+(\varvec{\varepsilon }) = \max _{s\in [0,t]}\left( \mathcal {H}(\varvec{\varepsilon }) \right) , \end{aligned}$$
(10)
where
t is time. Equation (
9) is then rewritten as follows:
$$\begin{aligned} \frac{3\mathcal {G}_c}{8l}\left( 2l^2\nabla ^2 d - 1\right) - g'(d) \mathcal {H}^+(\varvec{\varepsilon }) = 0. \end{aligned}$$
(11)
Since
\(\dot{\mathcal {H}}\ge 0\), non-negative
\({\dot{d}}\) is guaranteed and, consequently, the irreversibility of the fracture growth. We define
\(\mathcal {H}(\varvec{\varepsilon })\) after describing the stress decomposition approach.