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}{\rho }}. \end{aligned}$$

(13)

The process zone resulting from the assumed displacement field is illustrated in Fig. 2a; the displacement profile tends to a jump in the back of \(\varGamma \), where a strong discontinuity, or a macro-crack, is expected, and to a regularised profile in the frontal portion of \(\varGamma \), denoting the emergence of a cohesive zone.

The variation of \(\varvec{u}_\rho \) is

$$\begin{aligned} D \varvec{u}_\rho = \nabla \varvec{v}+ {H}_{\rho }\nabla \varvec{\jmath }+ {\delta }_{\rho }(\varvec{n}\otimes _s \varvec{\jmath }), \end{aligned}$$

(14)

where the term \({\delta }_{\rho }=\Vert \nabla {H}_{\rho }\Vert \) denotes the norm of the derivative of \(H_\rho \). \({\delta }_{\rho }\) is localized within a crack band whose sketch 54 crack - Activators Patch corresponds to the support of the regularization. Since we consider linear elasticity, in the previous equation, we restrict the approximate gradients to their symmetric part. For this reason, \(\varvec{n}\otimes _s \varvec{\jmath }\) denotes the symmetric part of tensor \(\varvec{n}\otimes \varvec{\jmath }\).

The regularized displacement jump (a) and the crack-density (b) evolve along the fracture process zone depending on the advancement of the damage localization process and on the value of the regularization length \(\rho \)

Full size image

Function \({\delta }_{\rho }\) can be recognized as the crack field density and plays a role analogous to the crack density of phase-field models [49]. Hence, it will also be referred to as the crack density function in the following sections. A pictorial view of \({\delta }_{\rho }\) for variable \(\rho \) is shown in Fig. 2b.

Width of the regularized damage band The crack-density \({\delta }_{\rho }\) has a support whose width is expected to shrink as the damaging process advances [16]. This effect is reached by modulating the regularization length according to the damage level as follows.

We define two regularization lengths, \(\rho _m\) and \(\rho _M\), which represent the minumum and the maximum value that \(\rho \) can take, respectively. The value of \(\rho \) is assumed being a function decreasing with the value of the damage \(d_\rho \) according to the following law:

$$\begin{aligned} \rho = {\left\{ \begin{array}{ll} \rho _M &{} \;d_\rho = d_m,\\ \alpha d_\rho + \beta &{} d_m \le d \le d_M,\\ \rho _m &{} \;d_\rho \ge d_M, \end{array}\right. } \end{aligned}$$

(15)

where

$$\begin{aligned} \alpha = \dfrac{\rho _m -\rho _M}{d_M - d_m},\qquad \beta = \dfrac{d_M \rho _M-d_m\rho _m}{d_M-d_m}. \end{aligned}$$

(16)

Remarkably, law (15) describes a three-stage evolution of \(\rho \), starting from a maximal value \(\rho _M\), linearly decreasing between \(d_m\) and \(d_M\), with \(d_m \le d_M\), up to assuming the minimum value \(\rho _m\) for \(d > d_M\).

Being the support of \({\delta }_{\rho }\) noncompact, it is necessary to introduce a truncation length beyond which \({\delta }_{\rho }\) is not evaluated. The approximation issues related to the choice of the truncation length were investigated in [20]. In particular, it was found that a truncated support length of \(40\rho \) provides a satisfying compromise between computational burden and accuracy, at least for the present \(H_\rho \). In the following developments, the lengths \(\ell _{\rho ,M}\) and \(\ell _{\rho ,m}\) indicate the width of the truncated supports of \({\delta }_{\rho }\) for \(\rho =\rho _M\) and \(\rho =\rho _m\), respectively.

Constitutive equations

We express \(D \varvec{u}_\rho \) (14) as the sum of a bounded bulk term \( \varvec{\varepsilon }\) and a localized terms \( \varvec{\varepsilon }_\rho \) defined as:

$$\begin{aligned}&\varvec{\varepsilon }= \nabla \varvec{v}+H_\rho \nabla \varvec{\jmath }, \end{aligned}$$

(17a)

$$\begin{aligned}&\varvec{\varepsilon }_\rho = {\delta }_{\rho }(\varvec{n}\otimes _s \varvec{\jmath }), \end{aligned}$$

(17b)

in the following way:

$$\begin{aligned} D \varvec{u}_\rho = \varvec{\varepsilon }+ \varvec{\varepsilon }_\rho. \end{aligned}$$

(18)

Let the material associated with the bulk and the zone around the crack display an isotropic elastodamaging behavior. The set of the state variables includes the strain fields \(\varepsilon \) and \(\varepsilon _\rho \), and the scalar damage variables d and \(d_\rho \), that take any value in the range [0, 1] from sound to completely damaged materials.

The idea we pursue sketch 54 crack - Activators Patch is that the expression of the free energy functional should be reminiscent of the structure of regularised free discontinuity energy (6), in particular, it should contain a bulk contribution and a distinct \(\rho -\)regularized energy term that tends to a surface energy for vanishing \(\rho \). We previously referred to this latter term to as a cohesive-like term [12]. In particular, we define the following energy density:

$$\begin{aligned} \psi (\varvec{\varepsilon }, \varvec{\varepsilon }_\rhod, d_\rho ) = \hat{\psi }(\varvec{\varepsilon },d) + \psi _{c}(\varvec{\varepsilon }_\rho ,d_\rho ). \end{aligned}$$

(19)

At the r.h.s of Eq. (19), the first term is written as

$$\begin{aligned} \hat{\psi }(\varvec{\varepsilon },d) = \frac{1}{2} \varvec{\varepsilon }\cdot (1-d) \varvec{C}\,\varvec{\varepsilon }. \end{aligned}$$

(20)

It denotes the energy associated with the standard elastodamaging energy of a bulk whose elasticity matrix is \(\varvec{C}\). The second term related to the cohesive-like contribution is set as

$$\begin{aligned} \begin{aligned}&\psi _{c}(\varvec{\varepsilon }_\rho ,d_\rho ) = \frac{1}{2} {\delta }_{\rho }(1-d_\rho ) (\varvec{n} \otimes _s \varvec{\jmath })\cdot \bar{\varvec{C}} (\varvec{n} \otimes _s \varvec{\jmath }), \end{aligned} \end{aligned}$$

(21)

where \(\bar{\varvec{C}}\) indicates \(\varvec{C}/t\), t being a unit length to be introduced for dimensional consistency [14]. Finally, the following regularized energy functional is defined:

$$\begin{aligned} \begin{aligned}&\mathcal {E}_\rho (\varvec{\varepsilon },\varvec{\varepsilon }_\rho ,d,d_\rho ) = \\&\int _\varOmega \hat{\psi }(\varvec{\varepsilon },d) dV + \int _\varOmega {\psi }_c(\varvec{\varepsilon }_\rho ,d_\rho ) dV. \end{aligned} \end{aligned}$$

(22)

Remark on the fracture energy recovery The model is naturally endowed with a crack density function \({\delta }_{\rho }\) analogous to the crack density potential of phase-field models [60]. It is \({\delta }_{\rho }\) that makes it possible to spread the cracking process over a zone whose width shrinks for increasing damage through a smooth continuous-discontinuous transition.

While in phase-field models the fracture energy density \(g_c\) is a model parameter, here, we recover the total surface energy \(G_c\) in the limit. More precisely, in phase-field models, the global constitutive dissipation functional for a rate-independent fracture process is defined through an expression of the type [60]:

$$\begin{aligned} \mathcal {D} (d,\dot{d}) = \int _\varOmega g_c f(d,\nabla d) dV \end{aligned}$$

(23)

contains \(g_c\) a parameter related to the fracture energy density.

In the present case, it has been proved that [19] \(\int _\varOmega 2\psi _{c} dV\) converges to the surface work carried out by the traction \(\varvec{t}\) across \(\varGamma \) to open a discontinuity \(\llbracket \varvec{u} \rrbracket \), namely to the critical cohesive fracture energy \(\mathcal {G}_c\) as follows:

$$\begin{aligned} \lim _{\rho \rightarrow 0} \int _\varOmega 2 \psi _{c} dV = \int _{\varGamma } \varvec{t}\cdot \llbracket \varvec{u} \rrbracket dS = \mathcal {G}_c. \end{aligned}$$

(24)

Therefore, the two-terms structure of function \(\mathcal {E}_\rho \) (22) mirrors the form of the regularized free-energy functionals for phase-field models based on Ambrosio and Tortorelli functional (6).

Constitutive laws

The first thermodynamics principle

$$\begin{aligned} \varvec{\sigma }\cdot \dot{\varvec{\varepsilon }} + \varvec{\sigma }_\rho \cdot \dot{\varvec{\varepsilon }}_\rho - \dot{\psi } \ge 0 \end{aligned}$$

(25)

makes it possible to deduce the constitutive equations

$$\begin{aligned}&\varvec{\sigma }= \frac{\partial \hat{\psi }}{\partial \varvec{\varepsilon }} = (1-d) {\varvec{C}} (\nabla \varvec{v}+ {H}_{\rho }\nabla \varvec{\jmath }), \end{aligned}$$

(26a)

$$\begin{aligned}&\varvec{\sigma }_\rho = \frac{\partial \psi _c}{\partial \varvec{\varepsilon }_\rho } = (1-d_\rho ) \bar{\varvec{C}} (\varvec{n}\otimes _s \varvec{\jmath }). \end{aligned}$$

(26b)

Here, \(\varvec{\sigma }\) and \(\varvec{\sigma }_\rho \) are the stress fields that are work-conjugated to \(\varvec{\varepsilon }\) and \(\varvec{\varepsilon }_\rho \), respectively.

Damage evolution

First, we define an appropriate effective stress \(\tilde{\varvec{\sigma }}\) for the evolution of d and associate another one with the evolution \(d_\rho \) as

$$\begin{aligned} \tilde{\varvec{\sigma }} = \varvec{C}\nabla \varvec{v}, \end{aligned}$$

(27)

and

$$\begin{aligned} \tilde{\varvec{\sigma }}_\rho = w_\rho (s(\varvec{x})) \bar{\varvec{C}}(\varvec{n}\otimes _s \varvec{\jmath }), \end{aligned}$$

(28)

respectively. In Eq. (28), \(w_\rho \) plays the role of a weight function of the signed distance \(s(\varvec{x})\) and is cast as

$$\begin{aligned} w_\rho (s(\varvec{x})) sketch 54 crack - Activators Patch {\left\{ \begin{array}{ll}1 &{} s(\varvec{x})\le \ell _{\rho ,{m}}/2,\\ {\delta }_{\rho }&{} s(\varvec{x}) > \ell _{\rho ,{m}}/2, \end{array}\right. } \end{aligned}$$

(29)

where \(\ell _{\rho ,m}\) denotes the truncated support of \(\delta _{\rho ,m}\). The adoption of \(w_\rho \) makes it possible to carry out the transition from a thick to a thin process zone, and overcomes a previous limitation [16]. A plot of \(w_\rho \) for variable values of \(\rho \) is shown in Fig. 3b.

We assume an isotropic damage Rankine model and define the following damage activation functions

$$\begin{aligned} g = {\tau } - \kappa ,\qquad g_\rho = {\tau }_\rho - \kappa\end{aligned}$$

(30)

\(\tau \) and \(\tau _\rho \) being equivalent stress scalars defined as:

$$\begin{aligned} {\tau }= \langle \text {max}(\text {eig}(\tilde{\varvec{\sigma }})\rangle\qquad {\tau }_\rho = \langle \text {max}(\text {eig}(\tilde{\varvec{\sigma }}_\rho )\rangle\end{aligned}$$

(31)

where the Macaulay brackets \(\langle \cdot \rangle \) are such that \(\langle x \rangle \) is the ramp function.

Plot of the crack density function \({\delta }_{\rho }\) (a) and the weight function \(w_\rho \) (b) for variable \(\rho \); the continuous line has been obtained with \(\rho _M\), the dashed line refers to \(\rho _m< \rho < \rho _M\), and the dotted line to \(\rho = \rho _m\)

Full size image

The evolution of the damage variable d and \(d_\rho \) stems from the loading–unloading conditions

$$\begin{aligned} \begin{aligned}&g\le 0, \qquad \dot{d}{g} = 0,\qquad \dot{d} \ge 0,\\&\dot{d} > 0 \qquad \text {if}\;\;\;g = 0, \end{aligned} \end{aligned}$$

(32)

and

$$\begin{aligned} \begin{aligned}&g_\rho \le 0, \qquad \dot{d}_\rho {g}_\rho = 0,\qquad \dot{d}_\rho \ge 0,\\&\dot{d} _\rho > 0 \qquad \text {if}\;\;\;g _\rho = 0, \end{aligned} \end{aligned}$$

(33)

respectively. The damage variables increase monotonically with the equivalent stress, and thus, with the current damage threshold \(\kappa \) and \(\kappa _\rho \). In particular, given the values computed at the previous instant \(\hat{d}\) and \(\hat{d}_{\rho }\), the damage variables evolve as non-decreasing functions of the associated damage thresholds through the relationships [11]

$$\begin{aligned} d =\max \{\hat{d},G(\kappa )\}, \;\; d_\rho = \max \{\hat{d}_{\rho }, G(\kappa _\rho )\}. \end{aligned}$$

(34)

We adopt the following exponential law [28]

$$\begin{aligned} G(\tilde{\kappa }) = 1- \frac{\tilde{\kappa }_0}{\tilde{\kappa }} e^{h(\tilde{\kappa })},\; h(\tilde{\kappa })= -2H \dfrac{\tilde{\kappa }-\tilde{\kappa }_0}{\tilde{\kappa }_0}, \end{aligned}$$

(35)

where H is a hardening modulus, \(\tilde{\kappa }\) is the current threshold and \(\tilde{\kappa }_0\) is an initial threshold that is a material parameter.

Because the laws of damage evolution are non-associative, their deduction from an energy-based variational principle is not as straightforward as in the case of associative damage. For this purpose, we deduce the rate form of the variational principle following the strain-driven damage approach described in [11]. Basically, the thresholds \(\kappa \) and \(\kappa _\rho \) turn out being non decreasing functions of the strain field, and, therefore, the damage variables too can be regarded as non decreasing functions of the current strain of the type

$$\begin{aligned} d = \tilde{G}(\varvec{\varepsilon }), \qquad d_\rho = \tilde{G}(\varvec{\varepsilon }_\rho ). \end{aligned}$$

(36)

The strain-driven damage assumption will be exploited in the forthcoming section.

Variational formulation

We compute the rate form of the energy functional as

$$\begin{aligned} \begin{aligned}&\dot{\mathcal {E}}_\rho (\dot{\varvec{\varepsilon }},\dot{\varvec{\varepsilon }}_\rho ,\dot{d},\dot{d_\rho }) = \\&\int _{\varOmega } \bigl (\varvec{\sigma }\cdot \dot{\varvec{\varepsilon }} + \varvec{\sigma }_\rho \cdot \dot{\varvec{\varepsilon }} _\rho - Y\dot{d} -Y_\rho \dot{d}_\rho \bigr ) dV, \end{aligned} \end{aligned}$$

(37)

where the constitutive laws (26) have been replaced.

For the present strain-driven damage process, after assuming the dissipation potential [11]

$$\begin{aligned} \begin{aligned}&\mathcal {D} (\dot{\varvec{\varepsilon }}, \dot{\varvec{\varepsilon }}_\rho ) = \\&\int _{\varOmega } \biggl [\bigl (\frac{1}{2} \frac{\partial d}{\partial \varvec{\varepsilon }} \otimes \varvec{C}\varvec{\varepsilon }\bigr ) \dot{\varvec{\varepsilon }}+ \bigl (\frac{1}{2} \frac{\partial d_\rho }{\partial \varvec{\varepsilon }_\rho } \otimes \varvec{C}\varvec{\varepsilon }_\rho \bigr ) \dot{\varvec{\varepsilon }}_\rho \biggr ] dV, \end{aligned} \end{aligned}$$

(38)

the rate form of the total potential is cast as:

$$\begin{aligned} \begin{aligned}&\mathcal {P}(\dot{\varvec{v}},\dot{\varvec{\varepsilon }},\dot{\varvec{\varepsilon }}_\rho ,\dot{d},\dot{d_\rho })=\\&\dot{\mathcal {E}}_\rho (\dot{\varvec{\varepsilon }},\dot{\varvec{\varepsilon }}_\rho\dot{d},\dot{d_\rho }) + \mathcal {D} (\dot{\varvec{\varepsilon }}, \dot{\varvec{\varepsilon }}_\rho ) - \mathcal {P}_{ext}(\dot{\varvec{v}}), \end{aligned} \end{aligned}$$

(39)

where \(\mathcal {P}_{ext}(\dot{\varvec{v}})\) denotes the external work contributed by the surface forces applied on the boundary \(\partial \varOmega _p\). Finally, the following variational principle governs the problem at hand:

Find \(\{ \dot{v},\dot{\varvec{\varepsilon }}, \dot{\varvec{\varepsilon }}_\rho\dot{d},\dot{d_\rho }\}\)that make functional\(\mathcal {P}(\dot{\varvec{v}},\dot{\varvec{\varepsilon }},\dot{\varvec{\varepsilon }}_\rho\dot{d}, \dot{d}_\rho )\)stationary, given the assumptions\(d = \tilde{G}(\varvec{\varepsilon })\), \(d_\rho = \tilde{G}(\varvec{\varepsilon }_\rho )\)and the boundary condition\(\varvec{\sigma }\cdot \varvec{n}= \bar{\varvec{p}}\)on\(\partial \varOmega _p {\setminus } \varGamma \).

We consider the first variation of functional \(\mathcal {P}\) with respect to any admissible variation of \(\varvec{\varepsilon }\), \(\varvec{\varepsilon }_\rho \) and \(\varvec{v}\) such that \(\delta \dot{\varvec{v}}\) satisfies homogeneous boundary conditions on \(\partial \varOmega _u\). The condition that the first variation vanishes for any admissible variation of the field variables is equivalent to impose that

$$\begin{aligned} \int _\varOmega \varvec{\sigma }\cdot \delta \dot{\varvec{\varepsilon }} dV + \int _\varOmega \varvec{\sigma }_\rho \cdot \delta \dot{\varvec{\varepsilon }}_\rho dV - \int _{\partial \varOmega _p {\setminus } \varGamma } \bar{\varvec{p}} \cdot \delta \dot{\varvec{v}} dS = 0\nonumber \\ Norton Internet Security Free Activate (40)

for abitrary \(\delta \dot{\varvec{\varepsilon }}\), \(\delta \dot{\varvec{\varepsilon }}_\rho \) and \(\delta \dot{v}\).

After replacing the strain expressions into the rate form of the variational principle (40), we write the first variation of \(\mathcal {P}\) with respect to \(\delta \dot{\varvec{v}}\) and \(\delta \dot{\varvec{\jmath }}\) as:

$$\begin{aligned} \begin{aligned}&\int _\varOmega \varvec{\sigma }\cdot (\nabla _s \delta \dot{\varvec{v}} + H_\rho \nabla _s \delta \dot{\varvec{\jmath }}) dV + \\&\int _\varOmega \varvec{\sigma }_\rho \cdot {\delta }_{\rho }(\varvec{n}\otimes _s \delta \dot{\varvec{\jmath }}) dV - \int _{\partial \varOmega _p {\setminus } \varGamma } \varvec{\bar{p}} \cdot \delta \dot{\varvec{v}} dS = 0, \end{aligned} \end{aligned}$$

(41)

for any admissible variation \(\delta \dot{\varvec{v}}\) and \(\delta \dot{\varvec{\jmath }}\). To preserve the convergence of the regularized formulation to the cohesive-like formulation for vanishing regularization [12, 19], the stress \(\varvec{\sigma }\) is mechanically decoupled from the Dirac-like term containing \({\delta }_{\rho }\), while the stress \(\varvec{\sigma }_\rho \) is not work-conjugated with \(\varvec{\varepsilon }\), the standard part of the strain. Consequently, the following Euler-Lagrange equations are obtained [12]

$$\begin{aligned}&\int _\varOmega \text {div}\varvec{\sigma }\cdot \delta \dot{\varvec{v}} dV = 0, \end{aligned}$$

(42a)

$$\begin{aligned}&\int _\varOmega \bigl ( \varvec{\sigma }\cdot \nabla _s\delta \dot{\varvec{\jmath }}+ \varvec{\sigma }_\rho \cdot {\delta }_{\rho }(\varvec{n}\otimes _s \delta \dot{\varvec{\jmath }}) \bigr ) dV = 0, \end{aligned}$$

(42b)

$$\begin{aligned}&\int _{\partial \varOmega _p} (\varvec{\sigma }\varvec{n}- \bar{\varvec{p}})\cdot \delta \dot{\varvec{v}} dV = 0, \end{aligned}$$

(42c)

for any admissible virtual variation \(\delta \dot{\varvec{v}}\) and \(\delta \dot{\varvec{\jmath }}\).

Space discretization

We adopt the extended finite element method [8, 61], and approximate the displacement in the enriched elements by means of the shifted basis approach as [80]:

$$\begin{aligned} \begin{aligned} \varvec{u}_h (\varvec{x})&= \sum _{I \in \mathcal {N}} N_I(\varvec{x}) \varvec{v}_I \\&\quad +\sum _{I\in \mathcal {N}_{enr}} N_I(\varvec{x})\bigl ( H_{\rho }(s(\varvec{x}))-H_{\rho ,I} \bigr ) \varvec{\jmath }_I, \end{aligned} \end{aligned}$$

(43)

where \(\mathcal {N}_{enr}\) denotes the number of enriched nodes, \(H_{\rho ,I} = {H}_{\rho }(s({\varvec{x}}_I))\), and \(\varvec{v}_I\) and \(\varvec{\jmath }_I\) are the nodal vector variables of the standard part of the displacement and the enrichment, respectively. For the sake of simplicity, we have here chosen the same shape functions N for both the standard degrees of freedom in \(\mathcal {N}\) and the enriched degrees of freedom associated with \(\mathcal {N}_{enr}\). In general, any partition of unity could be used as enrichment function [5]. The final picture emerging from the approximation (43) displays a set of enriched elements that are crossed by the crack line, and another set of enriched elements whose distance from the crack line is smaller or equal to the semi-diameter \(\ell _\rho /2\) but are not crossed by the crack line. The remaining elements are not enriched and are governed by a standard finite element approximation.

Let \(\varvec{V}\) and \(\varvec{J}\) be the vectors collecting the nodal degrees of freedom, and \(\varvec{B}\) and \(\bar{\varvec{N}}\) denote the compatibility matrices. Then the discrete form of the strain and stress fields can be written in the compact form

$$\begin{aligned}&\varvec{\varepsilon }_h (\varvec{x}) = \varvec{B}(\varvec{x}) \varvec{V}+ \tilde{{H}_{\rho }}(s(\varvec{x}))\varvec{B}(\varvec{x}) \varvec{J},\end{aligned}$$

(44a)

$$\begin{aligned}&\varvec{\varepsilon }_{\rho ,h} (\varvec{x}) = {\delta }_{\rho }(\varvec{x})\tilde{\varvec{N}}(\varvec{x}) \varvec{J}, \end{aligned}$$

(44b)

and

$$\begin{aligned}&\varvec{\sigma }_h(\varvec{x}) = \varvec{B}(\varvec{x})\varvec{C}_d \bigl (\varvec{V}+ \tilde{{H}_{\rho }}(s(\varvec{x}))\varvec{J}\bigr ),\end{aligned}$$

(45a)

$$\begin{aligned}&\varvec{\sigma }_{\rho ,h}(\varvec{x}) = \varvec{C}_\rho \bar{\varvec{N}}(\varvec{x}) \varvec{J}, \end{aligned}$$

(45b)

where the positions

$$\begin{aligned} \varvec{C}_d = (1-d)\varvec{C},\qquad \varvec{C}_\rho =(1-d_\rho )\bar{\varvec{C}} \end{aligned}$$

(46)

have been set for the sake of conciseness of notation. Finally, the solving equations can be obtained after replacement of the previous expressions in the Euler-Lagrange equations (42) [12].

Time discrete formulation

Let the loading history from instant \(t^0\) to instant \(t^N\) be subdivided into N non-overlapping intervals, \([t^0, t^N] = \bigcup _{n=1, N} [t^{n-1}, t^{n}]\). Given \(\varvec{\varepsilon }^{n-1}\) and \(\varvec{\varepsilon }_{\rho }^{n-1}\) at instant \(t^{n-1}\), so that \(d^{n-1}\) and \(d_\rho ^{n-1}\) are known, after a new load step during the interval \([t^{n-1}, t^n]\), we compute \(\varvec{V}^{n}\) and \(\varvec{J}^{n}\) and update the values of the damage, \(d^{n}\) and \(d_\rho ^{n}\), following the classic Backward Euler integration scheme [73]. In particular, at each Gauß point, the loading functions at the current instant \(t^n\) read

$$\begin{aligned} g^{n}= \tau (\varvec{\varepsilon }^{n})- \kappa ^{n}, \qquad g_\rho ^{n}= \tau _\rho (\varvec{\varepsilon }_\rho ^{n})- \kappa _\rho ^{n}, \end{aligned}$$

(47)

where thresholds \(\kappa ^{n}\) and \(\kappa _\rho ^{n} \) are the maximum values that equivalent stress scalars have ever reached during the loading history up to the current instant \(t^{n}\) as follows

$$\begin{aligned} \kappa ^{n}(\varvec{\varepsilon }^{n}) = \sup _{i \in [0, {n}]}\{\tau (\varvec{\varepsilon }^{i})\}, \;\kappa _\rho ^{n}(\varvec{\varepsilon }_\rho ^{n}) = \sup _{i \in [0, {n}]} \{\tau _\rho (\varvec{\varepsilon }_\rho ^{i})\}. \end{aligned}$$

sketch 54 crack - Activators Patch (48)

Since \(\kappa \) and \(\kappa _\rho \) are non-decreasing functions of the strain fields, the rate form of the loading unloading conditions can be integrated over the time.

We formulate the following damage evolution laws

$$\begin{aligned} d^{n} = \left\{ \begin{array}{ll} d^{n-1} &{} \text {if} \;\kappa ^{n} < \kappa ^{n-1}, \\ G(\kappa ^{n}) &{} \text {if} \;\kappa ^{n} \ge \kappa ^{n-1} \; \text {and} \; G(\kappa ^{n}) \le d_m, \\ d_m &{} \text {if}\; G(\kappa ^{n}) > d_m, \end{array}\right. \end{aligned}$$

(49)

and

$$\begin{aligned} d_\rho ^{n} = \left\{ \begin{array}{ll} d_m &{} \text {if}\; G(\kappa _\rho ^{n})< d_m,\\ d_\rho ^{n-1} &{} \text {if}\;\kappa _\rho ^{n} < \kappa _\rho ^{n-1}, \\ G(\kappa _\rho ^{n}) &{} \text {if}\;\kappa ^{n} \ge \kappa _\rho ^{n-1}. \\ \end{array}\right. \end{aligned}$$

(50)

Whenever \(d = d_m\), the transition from the continuous to the discontinuous setting becomes possible, though not automatic as it will be clarified in the next sections.

Finally, the space-time discrete form of the work principle is cast as [12]:

$$\begin{aligned}&\int _\varOmega \varvec{C}_d^n \varvec{B}( \varvec{V}^n + H_\rho \varvec{J}^n) \cdot \varvec{B}( \tilde{\varvec{V}}^n + H_\rho \tilde{\varvec{J}}^n) dV + \nonumber \\&+ \int _\varOmega \delta _\rho \bar{\varvec{N}} \varvec{J}^n \cdot \varvec{C}_\rho ^n \bar{\varvec{N}} \tilde{\varvec{J}}^n dV = \int _{\partial \varOmega _p {\setminus } \varGamma } \varvec{F}^n \cdot \varvec{N}\tilde{\varvec{V}}^n dS, \end{aligned}$$

(51)

where \(\tilde{\varvec{V}}^n\) and \(\tilde{\varvec{J}}^n\) are arbitrary and the damage variables are computed by means of the loading-unloading conditions (50). In virtue of the arbitrariness of \(\tilde{\varvec{V}}^n\) and \(\tilde{\varvec{J}}^n\), a solving system of equations is obtained and subsequently solved through a Newton-Raphson procedure, the loading increments being applied via an arc-length algorithm with indirect control of purposely selected monotonically increasing degrees of freedom.