Abstract
We consider a class of separately convex phase field energies employed in fracture mechanics, featuring non-interpenetration and a general softening behavior. We analyze the time-discrete evolutions generated by a staggered minimization scheme, where fracture irreversibility is modeled by a monotonicity constraint on the phase field variable. After recasting the staggered scheme by means of gradient flows, we characterize the time-continuous limits of the discrete solutions in terms of balanced viscosity evolutions, parametrized by their arc-length with respect to the \(L^2\)-norm (for the phase field) and the \(H^1\)-norm (for the displacement field). By a careful study of the energy balance we deduce that time-continuous evolutions may still exhibit an alternate behavior in discontinuity times.
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References
Almi, S., Belz, S.: Consistent finite-dimensional approximation of phase-field models of fracture. Ann. Mat. Pura Appl. (4)198(4), 1191–1225, 2019
Almi, S., Belz, S., Negri, M.: Convergence of discrete and continuous unilateral flows for Ambrosio–Tortorelli energies and application to mechanics. ESAIM Math. Model. Numer. Anal. 53(2), 659–699, 2019
Ambati, M., Gerasimov, T., De Lorenzis, L.: A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput. Mech. 55(2), 383–405, 2015
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel 2005
Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via \(\Gamma \)-convergence. Commun. Pure Appl. Math. 43(8), 999–1036, 1990
Amor, H., Marigo, J.J., Maurini, C.: Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J. Mech. Phys. Solids57, 1209–1229, 2009
Babadjian, J.F., Millot, V.: Unilateral gradient flow of the Ambrosio–Tortorelli functional by minimizing movements. Ann. Inst. H. Poincaré Anal. Non Linéaire31(4), 779–822, 2014
Balder, E.J.: An extension of Prohorov’s theorem for transition probabilities with applications to infinite-dimensional lower closure problems. Rend. Circ. Mat. Palermo (2)34(3), 427–447, 1985
Bourdin, B.: Numerical implementation of the variational formulation for quasi-static brittle fracture. Interfaces Free Bound. 9(3), 411–430, 2007
Bourdin, B., Francfort, G.A., Marigo, J.J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids48(4), 797–826, 2000
Brézis, H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50)North-Holland Publishing Co., Amsterdam 1973
Burke, S., Ortner, C., Süli, E.: An adaptive finite element approximation of a variational model of brittle fracture. SIAM J. Numer. Anal. 48(3), 980–1012, 2010
Chambolle, A.: An approximation result for special functions with bounded deformation. J. Math. Pures Appl. (9)83(7), 929–954, 2004
Chambolle, A., Conti, S., Francfort, G.A.: Approximation of a brittle fracture energy with a constraint of non-interpenetration. Arch. Ration. Mech. Anal. 228(3), 867–889, 2018
Chambolle, A., Crismale, V.: A density result in \({GSBD}^p\) with applications to the approximation of brittle fracture energies. Arch. Ration. Mech. Anal. 232(3), 1329–1378, 2019
Comi, C., Perego, U.: Fracture energy based bi-dissipative damage model for concrete. Int. J. Solids Struct. 38(36), 6427–6454, 2001
Dal Maso, G.: Generalised functions of bounded deformation. J. Eur. Math. Soc. (JEMS)15(5), 1943–1997, 2013
Dal Maso, G., Francfort, G.A., Toader, R.: Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176(2), 165–225, 2005
Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: \(L^p\)Spaces. Springer Monographs in MathematicsSpringer, New York 2007
Giacomini, A.: Ambrosio–Tortorelli approximation of quasi-static evolution of brittle fractures. Calc. Var. Partial Differ. Equ. 22(2), 129–172, 2005
Herzog, R., Meyer, C., Wachsmuth, G.: Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions. J. Math. Anal. Appl. 382(2), 802–813, 2011
Iurlano, F.: A density result for GSBD and its application to the approximation of brittle fracture energies. Calc. Var. Partial Differ. Equ. 51(1–2), 315–342, 2014
Karma, A., Kessler, D.A., Levine, H.: Phase-field model of mode III dynamic fracture. Phys. Rev. Lett. 87, 045–501, 2001
Knees, D., Negri, M.: Convergence of alternate minimization schemes for phase field fracture and damage. Math. Models Methods Appl. Sci. 27(9), 1743–1794, 2017
Knees, D., Rossi, R., Zanini, C.: A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23(4), 565–616, 2013
Knees, D., Rossi, R., Zanini, C.: A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains. Nonlinear Anal. Real World Appl. 24, 126–162, 2015
Knees, D., Rossi, R., Zanini, C.: Balanced viscosity solutions to a rate-independent system for damage. Eur. J. Appl. Math. 30(1), 117–175, 2019
Łojasiewicz, S.: Sur la géométrie semi- et sous-analytique. Ann. Inst. Fourier (Grenoble)43(5), 1575–1595, 1993
Mielke, A.: Evolution of rate-independent systems. In: Dafermos, C., Feireisl, E. (eds.) Evolutionary Equations. Handbook of Differential Equations, vol. II, pp. 461–559. Elsevier, Amsterdam 2005
Mielke, A., Rossi, R., Savaré, G.: Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems. J. Eur. Math. Soc. (JEMS)18(9), 2107–2165, 2016
Mielke, A., Roubíček, T.: Rate-Independent Systems: Theory and Application. Applied Mathematical Sciences, vol. 193. Springer, New York 2015
Negri, M.: A unilateral \(L^2\)-gradient flow and its quasi-static limit in phase-field fracture by alternate minimization. Adv. Calc. Var. 12(1), 1–29, 2019
Negri, M., Kimura, M.: Weak solutions for gradient flows under monotonicity contraints. arxiv:1908.10111
Thomas, M.: Quasistatic damage evolution with spatial BV-regularization. Discrete Contin. Dyn. Syst. Ser. S6(1), 235–255, 2013
Wu, J.Y.: A unified phase-field theory for the mechanics of damage and quasi-brittle failure. J. Mech. Phys. Solids103, 72–99, 2017
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Communicated by M. Ortiz
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The work of S.A. was supported by the SFB TRR109. M.N. is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Instituto Nazionale di Alta Matematica (INdAM).
Appendices
Comparing Different Parametrizations
In this appendix we will compare, qualitatively, the evolutions of Theorem 1 with those of [24, Theorem 4.2], or, more precisely, we will compare the evolutions obtained here, employing \(H^1\)-norm for u and \(L^2\)-norm for z, with those obtained employing energy norms. As we will see, the evolutions will be qualitatively the same (up to subsequences) even if these norms are not equivalent.
We need to consider the setting of [24], otherwise energy norms would not be defined. Let \(\mathcal {J} :[0,T] \times H^1_0 (\varOmega , \mathbb {R}^2) \times H^1 (\varOmega ; [0,1]) \rightarrow [0,+\infty )\) given by
where we assume that the boundary datum g belongs to \(C^{1,1}([0,T]; W^{1,p}(\varOmega ;\mathbb {R}^{2}))\), \(p>2\). Note that this energy is separately quadratic, thus it is natural, and technically convenient, to introduce a couple of energy (instrinsic) norms
which correspond, respectively, to the quadratic part of the energies \(\mathcal {J}(t, \cdot , z)\) and \(\mathcal {J}(t, u, \cdot )\). Accordingly, we employ the slopes
Let us consider again the alternate scheme (at time \(t^k_i\))
We remark that, given \(\tau _k\), the families \(u^{k}_{i,j}\) and \(z^k_{i,j}\) are uniquely determined (by strict separate convexity of the energy). Following [24], we interpolate and parametrize the discrete configurations \(u^k_{i,j}\) and \(z^k_{i,j}\) with respect to the energy norms \(\Vert \cdot \Vert ^2_{z}\) (for the displacement field) and \(\Vert \cdot \Vert ^2_{u}\) (for the phase field). We remark that in this case it is enough to consider piece-wise affine interpolation, which actually coincides, for both u and z, with the gradient flow in the energy norm. As a result, we get a sequence of arc-length parametrizations \( ( {\bar{t}}_k , {\bar{u}}_k, {\bar{z}}_k)\), bounded in \(W^{1,\infty } ( [ 0, R] ; [0,T] \times H^1_0(\varOmega ; \mathbb {R}^2) \times H^1 (\varOmega ))\) and of uniformly finite length, that is, with R independent of \(k \in \mathbb {N}\). Invoking [24, Lemma 4.3], there exists a subsequence (non relabelled) and a limit \(( {\bar{t}}, {\bar{u}}, {\bar{z}})\) in \(W^{1,\infty } ( [ 0, R] ; [0,T] \times H^1_0(\varOmega ; \mathbb {R}^2) \times H^1 (\varOmega ))\) such that for every sequence \(r_{k}\) converging to \(r \in [0,R]\) we have
Moreover, invoking [24, Theorem 4.2] the limit evolution satisfies the following properties:
\(({\bar{a}})\)Regularity: \(({\bar{t}},{\bar{u}},{\bar{z}})\in W^{1,\infty }([0,R];[0,T]\times H^1_0 ( \varOmega ;\mathbb {R}^{2}) \times H^{1}(\varOmega ; [0,1]))\), and for almost everywhere \(r\in [0,R]\)
$$\begin{aligned} \bar{t\,}'(r)+\Vert {\bar{u}}'(r)\Vert _{ {\bar{z}}(r)}+\Vert {\bar{z}}'(r)\Vert _{{\bar{u}} (r)}\leqq 1, \end{aligned}$$here the symbol \('\) denotes the derivative with respect to the parametrization variable r;
\(({\bar{b}})\)Time parametrization: the function \({\bar{t}}:[0,R]\rightarrow [0,T]\) is non-decreasing and surjective;
\(({\bar{c}})\)Irreversibility: the function \({\bar{z}}\) is non-increasing and \(0 \leqq {\bar{z}}(r) \leqq 1\) for every \(0\leqq r \leqq R\);
\(({\bar{d}})\)Equilibrium: for every continuity point \(r\in [0,R]\) of \(({\bar{t}},{\bar{u}},{\bar{z}})\)
$$\begin{aligned} |\partial _{u} \mathcal {J}|_{{\bar{z}}(r)} ({\bar{t}}(r),{\bar{u}}(r),{\bar{z}}(r))=0\qquad \text {and}\qquad |\partial _{z}^{-} \mathcal {J}|_{{\bar{u}}(r)} ({\bar{t}}(r),{\bar{u}}(r),{\bar{z}}(r))=0 ; \end{aligned}$$\(({\bar{e}})\)Energy-dissipation equality: for every \(r \in [0,R]\)
$$\begin{aligned} \mathcal {J}({\bar{t}}(r), \ {\bar{u}}(r), {\bar{z}}(r)) =&\mathcal {J}(0,u_0,z_0)\nonumber \\&-\int _{0}^{r}|\partial _{u} \mathcal {J}|_{{\bar{z}}(\rho )} ({\bar{t}}(\rho ),{\bar{u}}(\rho ),{\bar{z}}(\rho ))\, \Vert {\bar{u}}' (\rho ) \Vert _{{\bar{z}}(\rho )} \mathrm {d}\rho \nonumber \\&-\int _{0}^{r} \!\! |\partial _{z}^{-} \mathcal {J}|_{{\bar{u}}(\rho )} ({\bar{t}}(\rho ),{\bar{u}}(\rho ),{\bar{z}}(\rho ))\, \Vert {\bar{z}}' (\rho ) \Vert _{{\bar{u}}(\rho )} \mathrm {d}\rho \nonumber \\&+\int _{0}^{r} \!\! \mathcal {P}( {\bar{t}}(\rho ),{\bar{u}}(\rho ), {\bar{z}}(\rho )) \,\bar{t\,} '(\rho )\,\mathrm {d}\rho . \end{aligned}$$(133)
In [24] the authors showed property \(({\bar{d}})\) for every \(r\in [0,R]\) with \({\bar{t}}'(r)>0\). However, it is not difficult to see that the same equilibrium condition is verified at continuity points.
Moreover, by [24, Proposition 4.1] we have
\(({\bar{f}})\)Non-degeneracy: there exists \(C>0\) such that for almost every \(r\in [0,R]\)
$$\begin{aligned} C < \bar{t\,}'(r) + \Vert {\bar{u}}'(r) \Vert _{{\bar{z}}(r)} + \Vert {\bar{z}}'(r) \Vert _{{\bar{u}}(r)} . \end{aligned}$$(134)
Finally, note that, by the separate differentiability of the energy, the equilibrium conditions (d) can be written in an equivalent “norm-free” fashion as
\(({\bar{d}}')\)Equilibrium: for every continuity point \(r\in [0,R]\) of \(({\bar{t}},{\bar{u}},{\bar{z}})\)
$$\begin{aligned} \partial _{u} \mathcal {J} ({\bar{t}}(r), {\bar{u}}(r), {\bar{z}}(r)) [ \varphi ] = 0 , \qquad \text {and} \qquad \partial _{z} \mathcal {J} ({\bar{t}}(r),{\bar{u}}(r),{\bar{z}}(r)) [ \xi ] = 0 , \end{aligned}$$for every \(\varphi \in H^1_0 ( \varOmega ; \mathbb {R}^2)\) and every \(\xi \in H^1(\varOmega )\) with \(\xi \leqq 0\).
At this point, consider the subsequence (not relabelled) converging to (t, u, z) and let us re-interpolate the discrete configurations \(u^k_{i,j}\) and \(z^k_{i,j}\) with respect to the norms \(\Vert \cdot \Vert _{H^1}\) (for the displacement field) and \(\Vert \cdot \Vert _{L^2}\) (for the phase field) as we did in Section 5.1. In this way we get a new sequence of parametrizations \((t_k , u_k ,z_k)\) bounded in \(W^{1,\infty } ( [ 0, S] ; [0,T] \times H^1_0(\varOmega ; \mathbb {R}^2) \times L^2 (\varOmega ))\). Clearly, we can apply Proposition 6 which provides (up to a further subsequence) a limit parametrization \((t , u ,z) \in W^{1,\infty } ( [ 0, S] ; [0,T] \times H^1_0(\varOmega ; \mathbb {R}^2) \times L^2 (\varOmega ) )\) such that
for every sequence \(s_{k}\) converging to \(s\in [0,S]\). The limit (t, u, z) satisfies properties (a)–(e) of Theorem 1.
We recall that \((t_k, u_k, z_k)\) is defined in the points \(s^k_{i,j}\) and \(s^k_{i,j+\frac{1}{2}}\) (see Section 5.1). In a similar way, the interpolation \(({\bar{t}}_k, {\bar{u}}_k, {\bar{z}}_k)\) is defined in points of the form \(r^k_{i,j}\) and \(r^k_{i,j+\frac{1}{2}}\) (see Section 4.3 in [24]). Moreover, we notice that the interpolation nodes are different since the underlying parametrizations are different. However, the configurations computed by the alternate minimization scheme are the same. Therefore, we have that
The same holds for nodes of the form \(s^{k}_{i,j+\frac{1}{2}}\) and \(r^{k}_{i,j+\frac{1}{2}}\).
Since \(({\bar{t}}_k, {\bar{u}}_k , {\bar{z}}_k)\) is piecewise affine while \((t_k , u_k ,z_k)\) is not, a direct comparison of the triples \(({\bar{t}}, {\bar{u}},{\bar{z}})\) and (t, u, z) is not immediate. Nevertheless, we can show the following “equivalence” of the reparametrizations.
Lemma 15
There exist two positive constants \(C_{1}, C_{2}\) such that for every \(k\in \mathbb {N}\setminus \{0\}\) and every \(i\in \{1,\ldots , k\}\)
Proof
Using the fact that \(\Vert \cdot \Vert _z\) and \(\Vert \cdot \Vert _{H^1}\) are equivalent, by Korn’s inequality, while \(\Vert \cdot \Vert _{u}\) and \(\Vert \cdot \Vert _{H^{1}}\) are equivalent by [24, Lemma 2.3], by (90) we can write
On the other hand, by Proposition 1 and Corollary 1 we have that
Hence, again by equivalence of norms, we get
This concludes the proof of (135). \(\quad \square \)
Let us consider an interval of the form \((s^k_{i_k,j_k+\frac{1}{2}} , s^k_{i_k,j_k+1}) \subset (0,S)\) and the corresponding interval \((r^k_{i_k,j_k+\frac{1}{2}} , r^k_{i_k,j_k+1}) \subset (0,R)\). By definition, we have
for every \(r \in (r^k_{i_k,j_k+\frac{1}{2}} , r^k_{i_k,j_k+1})\) and every \(s \in (s^k_{i_k,j_k+\frac{1}{2}} , s^k_{i_k,j_k+1})\). On the contrary, the phase field interpolations coincide only in the extrema, that is,
Now, up to subsequences (non relabelled), we can assume that, as \(k \rightarrow \infty \),
Since parametrizations are different, in general we should distinguish between all the following cases: \(s^- = s^+\), \(s^- < s^+\), \(r^- = r^+\), and \(r^- < r^+\); however the situation is much simpler, thanks to the following lemma.
Lemma 16
We have \(r^- = r^+\) if and only if \(s^- = s^+\).
Proof
Assume that \(r^- = r^+\). By compactness, we know that \({\bar{z}}_k ( r^k_{i_k,j_k+\frac{1}{2}} ) = z^k_{i_k,j_k} \rightharpoonup z ( r^-)\) weakly in \(H^{1}(\varOmega )\) and that \({\bar{z}}_k ( r^k_{i_k,j_k+1} ) = z^k_{i_k,j_k+1} \rightharpoonup z ( r^+)\) weakly in \(H^{1}(\varOmega )\). Since \(r^- = r^+\), we have \({\bar{z}}(r^-) = {\bar{z}}(r^+)\) and
By (88) we know that
Thus \(s^- = s^+\).
Assume that \(s^- = s^+\). Hence, arguing as above, by (136) we have \(z (s^-) = {\bar{z}} (r^-) = z(s^+) = {\bar{z}} (r^+)\). Moreover, being \({\bar{t}}_k\), \(t_k\), \({\bar{u}}_k\) and \(u_k\) constant in the corresponding intervals, in the limit we have \(t (s^-) = {\bar{t}} (r^-) = t(s^+) = {\bar{t}} (r^+)\) and \(u (s^-) = {\bar{u}} (r^-) = u(s^+) = {\bar{u}} (r^+ )\). Then, if \(r^- < r^+\) we would contradict the non-degeneracy condition (134). \(\quad \square \)
As a consequence of Lemma 15, the solutions \(({\bar{t}}, {\bar{u}}, {\bar{z}})\) and (t, u, z) coincide in continuity points.
Proposition 9
Let r be a continuity point for \(({\bar{t}}, {\bar{u}}, {\bar{z}})\). Then, there exists a continuity point s for (t, u, z) such that \(({\bar{t}} (r), {\bar{u}} (r), {\bar{z}}(r) ) = (t (s), u (s), z (s))\).
Viceversa, if s is a continuity point for (t, u, z), then there exists a continuity point r for \(({\bar{t}},{\bar{u}},{\bar{z}})\) such that \((t (s), u (s), z (s)) = ({\bar{t}} (r), {\bar{u}} (r), {\bar{z}}(r) )\).
Proof
Fix \(\delta >0\). Since r is a continuity point, by monotonicity of \({\bar{t}}\) we have \({\bar{t}}(r + \delta ) > {\bar{t}} (r)\). Since \({\bar{t}}_k\) converges pointwise to \({\bar{t}}\), we have \( {\bar{t}}_k (r + \delta ) - {\bar{t}}_k (r) \geqq \tfrac{1}{2} ( {\bar{t}}(r + \delta ) - {\bar{t}} (r)) > 0\) for every k sufficiently large. As \({\bar{t}}_k\) changes only in parametrization intervals of the form \((r^k_{i,-1} , r^k_{i,0})\) and since \(\tau _k \rightarrow 0\), there exist two indexes \(i_k< i'_{k}\) such that,
Hence for every index \(j \in \mathbb {N}\), we have
Since \(\delta \) can be arbitrarily small, we can find a sequence \((i_{k},j_{k})\) such that
Then \({\bar{z}}_{k} ( r^k_{i_k,j_k + 1} ) \rightharpoonup {\bar{z}} (r)\) weakly in \(H^1(\varOmega )\). Since \({\bar{z}}_{k} ( r^k_{i_k,j_k + 1} ) = z_{k} ( s^k_{i_k,j_k + 1} ) \) we have \(z _{k}( s^k_{i_k,j_k + 1} ) \rightharpoonup {\bar{z}} (r)\) weakly in \(H^1(\varOmega )\). Up to a subsequence (not relabelled) \(s^k_{i_k,j_k + 1} \rightarrow s\) and thus \(z_{k} ( s^k_{i_k,j_k + 1} ) \rightharpoonup z (s)\). We conclude that \(z(s) = {\bar{z}} (r)\). In a similar way \({\bar{t}} (r) = t(s)\) and \({\bar{u}} ( r) = u (s)\).
It remains to show that s is a continuity point for (t, u, z). To this aim, let us set, up to subsequence, \(s_{\delta }:=\lim _{k} s^{k}_{i'_{k}}\), where the indexes \(i'_{k}\) have been defined in (137). Hence, applying Lemma 15 we deduce that
By definition of \(s_{\delta }\) we have that \(t(s_{\delta })= \lim _{k\rightarrow \infty } t_{k}(s^{k}_{i'_{k}}) = \lim _{k\rightarrow \infty } {\bar{t}}_{k}(r^{k}_{i'_{k}})\). By (137) we get that \(| {\bar{t}}_{k}(r^{k}_{i'_{k}}) - {\bar{t}}_{k}(r+\delta )|\leqq \tau _{k}\), from which we deduce that \(t(s_{\delta }) = {\bar{t}} ( r + \delta ) > {\bar{t}}(r) = t(s)\). This implies that s is of continuity for (t, u, z).
The viceversa can be shown in a similar way. \(\quad \square \)
On the contrary, in discontinuity points the evolution (t, u, z) and \(({\bar{t}}, {\bar{u}}, {\bar{z}})\) interpolate the same configurations but with different paths. To better understand, let us consider an interval of the form \((r^k_{i_{k},j_{k}+\frac{1}{2}}, r^k_{i_{k},j_{k}+1})\) such that \(r^k_{i_{k},j_{k}+\frac{1}{2}} \rightarrow r^-\), \(r^k_{i_{k},j_{k}+1} \rightarrow r^+\) with \(r^- < r^+\). As a consequence both \({\bar{t}}\) and \({\bar{u}}\) are constant in \((r^-, r^+)\) and thus every \(r \in (r^-, r^+)\) is not a continuity point. In this case, we have
By the non-degeneracy property of \(({\bar{t}}, {\bar{u}}, {\bar{z}})\), we deduce that \({\bar{z}} ( r^-) \ne {\bar{z}} ( r^+)\). Moreover, up to subsequence we have
Since \(z\in W^{1,\infty }([0,S]; L^{2}(\varOmega ))\), we get that \(s^- \ne s^+\). Recalling that t and u are constant in \((s^-,s^+)\), the energy balance of Theorem 1 reads as
Thus z is a (normalized) unilateral gradient flow in \(L^2\), with initial datum \(z(s^-)\). On the contrary, \({\bar{z}}\) is the affine interpolation of \({\bar{z}} ( r^-)\) and \({\bar{z}} ( r^+)\). Thus, in general, \({\bar{z}}\) and z do not coincide in the corresponding intervals \((r^-,r^+)\) and \((s^-,s^+)\) even if they coincide in the extrema.
On the Alternate Behavior in Discontinuity Points
Let us consider the set U defined in (119) and denote by \({\mathbf {1}}_{U}\) its characteristic function. From (127) we know that
Note that \(z' (\sigma ) = 0\) for every \(\sigma \in U^c\) (see Lemma 9), thus, being \(\Vert z' \Vert _{L^2} \leqq 1\) almost everywhere in [0, S], we have \(\Vert z' \Vert _{L^2} \leqq {\mathbf {1}}_{U}\) almost everywhere in [0, S]; as a consequence, the above estimate together with (22) yields
Therefore, all inequalities turn into equalitites and thus
hence \(| \partial _{z}^{-} \mathcal {F}| ( t(\sigma ), u ( \sigma ), z( \sigma )) \, ( {\mathbf {1}}_{U}(\sigma ) - \Vert z' (\sigma ) \Vert _{L^2} ) =0\) almost everywhere in [0, S].
Therefore, if \(| \partial _{z}^{-} \mathcal {F}| ( t(\sigma ), u ( \sigma ), z( \sigma )) \ne 0\) for every \(\sigma \in (\sigma _1, \sigma _2) \subset U\) then \(\Vert z' (\sigma ) \Vert _{L^2} = 1\) almost everywhere in \((\sigma _1, \sigma _2)\) and then both \(t'(\sigma )=0\) and \(u'(\sigma )=0\) almost everywhere in \((\sigma _1, \sigma _2)\), by (a) in Theorem 1. This means that in the discontinuity interval \((\sigma _1, \sigma _2)\) only z changes, following a normalized, unilateral \(L^2\) gradient flow. In view of this observation, excluding the cases in which \(| \partial _{z}^{-} \mathcal {F}| ( t(\sigma ), u ( \sigma ), z( \sigma )) = 0\), we expect that in the presence of time-discontinuties the limit evolution is still alternate, and thus it is not simultaneous in u and z. More precisely, consider a discontinuity at time t, with a transition from \((u^-,z^-)\) to \((u^+,z^+)\), parametrized in the interval \((\sigma ^-, \sigma ^+)\). With t being the limit of continuity points, the left limit \((u^-,z^-)\) is an equilibrium configuration at time t. We expect that the parametrizations of u and z, in a right neighborhood of \(\sigma ^-\), provide an alternate interpolation of sequences \(z^-_m \searrow z^-\), with \(z^-_m \ne z^-\), and \(u^-_m \rightarrow u^-\) such that
The non-degeneracy condition \(z^-_m \ne z^-\) is due to the fact that \((u^-,z^-)\) is an equilibrium configuration, and thus a separate minimizer of the energy \(\mathcal {F}( t , \cdot , \cdot )\). Indeed, if \(z^-_m = z^-\), for some index \(m \in \mathbb {N}\), then, by uniqueness of the minimizer, \(u^-_{m-1} = u^-\) and then \(z^-_{m-1} = z^-\). By induction and by monotonicity, \(z^-_m=z^-\) for every index \(m \in \mathbb {N}\) and then \(u^-_m = u^-\) for every \(m \in \mathbb {N}\); thus, there would be no transition between \((u^-,z^-)\) to \((u^+,z^+)\). In a similar way, we expect sequences \(z^+_m \nearrow z^+\) and \(u^+_m \rightarrow u^+\) in a left neighborhood \(\sigma ^+\) such that
However, in this case we cannot exclude that \(z^+_m = z^+\) for some index \(m \in \mathbb {N}\). We remark that this qualitative behavior is confirmed by numerical computations.
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Almi, S., Negri, M. Analysis of Staggered Evolutions for Nonlinear Energies in Phase Field Fracture. Arch Rational Mech Anal 236, 189–252 (2020). https://doi.org/10.1007/s00205-019-01468-4
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DOI: https://doi.org/10.1007/s00205-019-01468-4