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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 }$$. Function $${\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$$ is the ramp function. The evolution of the damage variable d and $$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.