In this section the material model, combining isotropic elasticity with isotropic damage, is derived. For the gradient-enhanced model a Helmholtz free energy is proposed as
$$\begin{aligned} \psi = \psi ^\mathrm {vol}(\varvec{F},d)+\psi ^\mathrm {iso}(\varvec{F},d)+\psi ^\mathrm {grd}(\nabla _{\!X}d), \end{aligned}$$
(13)
where
d is the scalar variable associated to isotropic damage. The elastic energy is additively split into a volumetric
\(\psi ^\mathrm {vol}\) and an isochoric
\(\psi ^\mathrm {iso}\) contribution. Gradient enhancement is captured in the term
\(\psi ^\mathrm {grd}\). The damage variable
d deteriorates the elastic properties of the material model via so-called damage functions
\(f^\bullet \), i.e.
$$\begin{aligned} \psi ^\mathrm {vol}+\psi ^\mathrm {iso} = f^\mathrm {vol}(d)\bar{\psi }^\mathrm {vol}(\varvec{F})+f^\mathrm {iso}(d)\bar{\psi }^\mathrm {iso}(\varvec{F}). \end{aligned}$$
(14)
The damage functions
\(f^\bullet \) are a generalisation of the well-known
\([1-d]\)-approach commonly found in literature. They map the damage variable
\(d \in [0,\infty [\) to the interval ]0, 1], i.e.
$$\begin{aligned} f^\bullet : \left\{ \mathbb {R}^+_0\rightarrow ]0,1]\mid f^\bullet (0)=1, \lim \limits _{d\rightarrow \infty }f^\bullet (d)=0\right\} , \end{aligned}$$
(15)
such that no additional constraints to limit the value of
d are necessary. Following the framework of generalised standard dissipative materials together with the framework of the introduction of a non-locality residual, the Clausius–Duhem inequality reads
$$\begin{aligned} \mathcal {D}&= \varvec{P}:\dot{\varvec{F}} - \dot{\psi }+\mathcal {P} \nonumber \\&= [\varvec{P}-\partial _F\psi ]:\dot{\varvec{F}}-\partial _d\psi \,\dot{d}-\partial _{\nabla _X d}\psi \,\dot{\overline{\nabla _{\!X}d}}+\mathcal {P}\ge 0, \end{aligned}$$
(16)
with the Piola stresses
\(\varvec{P}\). Thereby,
\(\mathcal {P}\) denotes the non-locality residual, see [
15,
22,
28,
34]. Moreover, the notation
\(\dot{\bullet }\) represents the material time derivative of the quantity
\(\bullet \). The non-locality residual accounts for the energy exchange between the particles in the damaged area
\(\mathcal {B}_d\) and has to satisfy the insulation condition
$$\begin{aligned} \int _{\mathcal {B}_d} \mathcal {P}\,\,\mathrm {d}V= 0, \end{aligned}$$
(17)
which states that no energy is exchanged between particles inside and outside the damaged area. In the elastic domain
\(\mathcal B_e := \mathcal B \backslash \mathcal B_d\) the non-locality residual has to be pointwise zero, i.e.
$$\begin{aligned} \mathcal P = 0 \qquad \text {in}\quad \mathcal B_e. \end{aligned}$$
(18)
With the definition of the stresses and driving forces
$$\begin{aligned} \varvec{P} := \frac{\partial \psi }{\partial \varvec{F}},\quad Y := -\frac{\partial \psi }{\partial d},\quad \varvec{Y} := -\frac{\partial \psi }{\partial \nabla _{\!X}d}, \end{aligned}$$
(19)
the dissipation inequality (
16) reduces to
$$\begin{aligned} \mathcal {D}^\mathrm {red} = Y\dot{d} + \varvec{Y}\cdot \dot{\overline{\nabla _{\!X}d}} + \mathcal {P} \ge 0. \end{aligned}$$
(20)
Since
d and
\(\nabla _{\!X}d\) cannot evolve independently from each other, the reduced dissipation inequality is assumed to take the bilinear form
$$\begin{aligned} \mathcal {D}^\mathrm {red} = \bar{Y}\dot{d} \ge 0. \end{aligned}$$
(21)
Comparing both forms of the reduced dissipation, (
20) and (
21), yields an expression for the non-locality residual in the form of
$$\begin{aligned} \mathcal {P} = \bar{Y}\dot{d} - Y\dot{d} - \varvec{Y}\cdot \dot{\overline{\nabla _{\!X}d}}. \end{aligned}$$
(22)
Combining the insulation condition (
17) with the vanishing non-locality residual in the elastic domain (
18) renders the integral of
\(\mathcal P\) over the whole domain
\(\mathcal B\) to vanish. Additionally applying integration by parts and the Gauss theorem leads to
$$\begin{aligned} \begin{aligned} \int _{\mathcal {B}}\mathcal {P}\,\,\mathrm {d}V&= \int _{\mathcal {B}}[\bar{Y} - Y]\dot{d}\,\,\mathrm {d}V- \int _{\mathcal {B}}\varvec{Y}\cdot \dot{\overline{\nabla _{\!X}d}}\,\,\mathrm {d}V\\&=\int _{\mathcal {B}}[\bar{Y}+\nabla _X\cdot \varvec{Y}-Y]\dot{d}\,\,\mathrm {d}V\\&\quad - \int _{\partial \mathcal {B}}[\varvec{Y}\cdot \varvec{N}]\dot{d}\,\,\mathrm {d}A= 0, \end{aligned} \end{aligned}$$
(23)
with
\(\varvec{N}\) being the referential outward unit normal vector. Since (
23) has to be fulfilled for arbitrary damage rates
\(\dot{d}\) one obtains the following conditions
$$\begin{aligned}&\bar{Y} = Y - \nabla _X\cdot \varvec{Y} \quad \text {in} \quad \mathcal {B}, \end{aligned}$$
(24)
$$\begin{aligned}&\varvec{Y}\cdot \varvec{N} = 0\qquad \qquad \, \text {on}\quad \partial \mathcal {B}. \end{aligned}$$
(25)
The quantity
\(\bar{Y}\) can be interpreted as a non-local driving force. Consequently, damage potential
\(\phi ^d\)—governing onset and evolution of the damage variable
d in this associated framework—is formulated in terms of
\(\bar{Y}\) such that the elastic domain is introduced as
$$\begin{aligned} \mathbb E := \{ \bar{Y}\,|\,\phi ^d(\bar{Y}) \le 0 \}. \end{aligned}$$
(26)
Homogeneous Neumann boundary conditions are applied to the total boundary and no Dirichlet boundary conditions are considered, see (
25). The constrained minimisation problem ensuing from the postulate of maximum dissipation is solved with the Lagrange functional, i.e.
$$\begin{aligned} \mathcal {L}=-\mathcal {D}^\mathrm {red}+\lambda \phi ^d=-\bar{Y}\dot{d}+\lambda \phi ^d\rightarrow \text {stat.}, \end{aligned}$$
(27)
with Lagrange parameter
\(\lambda \ge 0\). This leads to the Karush–Kuhn–Tucker (KKT) conditions. The evolution equation for the damage variable is obtained as
$$\begin{aligned} \dot{d} = \lambda \,\frac{\partial \phi ^d}{\partial \bar{Y}} \end{aligned}$$
(28)
and the loading/unloading conditions are
$$\begin{aligned} \phi ^d\le 0,\quad \lambda \ge 0,\quad \lambda \phi ^d=0. \end{aligned}$$
(29)
In this work, the elastic free Helmholtz energy is chosen to be of Neo-Hookean type, whereas the gradient part is a simple quadratic form, i.e.
$$\begin{aligned} \begin{aligned} \bar{\psi }^\mathrm {vol}&= \frac{1}{2}\,K\,\left[ \frac{1}{2}\,[J^2-1]-\ln (J)\right] ,\\ \bar{\psi }^\mathrm {iso}&= \frac{1}{2}\,\mu \,\left[ \text {tr}(\varvec{C}^\mathrm {iso}) - 3\right] ,\\ \psi ^\mathrm {grd}&= \frac{1}{2}\,c_d\,\Vert \nabla _{\!X}d\Vert ^2, \end{aligned} \end{aligned}$$
(30)
where
K is the bulk modulus,
\(\mu \) the shear modulus,
\(c_d\) shall be denoted as regularisation parameter and
\(\Vert \bullet \Vert = \sqrt{\bullet \cdot \bullet }\). Moreover,
\(\varvec{C}^\mathrm {iso}=J^{-\frac{2}{3}}\varvec{C}\) is the isochoric part of the right Cauchy-Green tensor
\(\varvec{C}\). The conditions for the damage function
\(f^\bullet \) are fulfilled by the exponential function. We choose
$$\begin{aligned} f^\bullet (d) := \exp (-\eta _\bullet \,d), \end{aligned}$$
(31)
where
\(\eta _\mathrm {vol}\) and
\(\eta _\mathrm {iso}\) are material parameters controlling the speed of deterioration of the elastic properties. The stresses and driving forces are then given by
$$\begin{aligned} \begin{aligned}&\varvec{P}^\mathrm {vol} = f^\mathrm {vol}(d)\,\left[ \frac{1}{2}\,K\,[J^2-1]\,\varvec{F}^{-\mathrm {t}}\right] , \\&\varvec{P}^\mathrm {iso} = f^\mathrm {iso}(d)\,\left[ J^{-\frac{2}{3}}\,\mu \,\left[ \varvec{F} - \frac{\text {tr}(\varvec{C})}{3}\,\varvec{F}^{-\mathrm {t}}\right] \right] ,\\&\varvec{P} = \varvec{P}^\mathrm {vol} + \varvec{P}^\mathrm {iso}, \\&Y = \eta _\mathrm {vol}\,f^\mathrm {vol}(d)\,\bar{\psi }^\mathrm {vol}(\varvec{F}) + \eta _\mathrm {iso}\,f^\mathrm {iso}(d)\,\bar{\psi }^\mathrm {iso}(\varvec{F}),\\&\varvec{Y} = -\,c_d\,\nabla _{\!X}d. \end{aligned} \end{aligned}$$
(32)
Finally, damage potential
\(\phi ^d\) is chosen as
$$\begin{aligned} \phi ^d = \bar{Y} - y_0, \end{aligned}$$
(33)
so that damage driving force
\(\bar{Y}\) is compared to a constant threshold value
\(y_0\). This results in the evolution equation
$$\begin{aligned} \dot{d} = \lambda , \end{aligned}$$
(34)
cf. (
28). The quantity
d is introduced as a field variable in the algorithmic setting and the Lagrange multiplier
\(\lambda \) is solved for by the implicit Backward–Euler scheme, which leads to the discrete update at time step
\(n+1\)$$\begin{aligned} \lambda _{n+1} = \frac{d_{n+1} - d_n}{\varDelta t}, \end{aligned}$$
(35)
with the incremental time step
\(\varDelta t = t_{n+1} - t_n\). This means that the loading/unloading conditions cannot be fulfilled by an evolution of the damage variable
d at integration point level within the finite element formulation discussed as this work proceeds—instead, the loading/unloading conditions need to be solved globally.