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2015 | OriginalPaper | Buchkapitel

5. Beyond rate-independence

verfasst von : Alexander Mielke, Tomàš Roubíček

Erschienen in: Rate-Independent Systems

Verlag: Springer New York

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Abstract

Sometimes, dynamical systems host various processes, only some of which are rate-independent. These processes can be manifested depending on loading regimes, and only in some regimes (typically very slow) do the rate-independent processes dominate; cf. also Sect. 5.1.2.2 below. In other regimes, however, the modeling assumption about rate-independence is inapplicable.

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Vorheriges Kapitel Applications in continuum mechanics and physics of solids

The equidistance of the partition is now advantageously used in (5.1.30).

Similar results for the special cases can be found in [407] (for \(\mathcal{V} = 0\)) and [497]. For the estimate (5.1.76) – (5.1.77) , see also [532, Formula (11.16)].

In fact, \(\langle \frac{\mathrm{d}} {\mathrm{d}t}\partial \mathcal{R}(\dot{\boldsymbol{z}}_{\varepsilon }),\dot{\boldsymbol{z}}_{\varepsilon }\rangle \geq 0\), which allows us to ignore the \(\mathcal{R}\)-term for this estimate, is written very formally; a rigorous argument can be based on the cancellation of the \(\mathcal{R}\)-terms in (5.1.76).

For such nonlinear hyperbolic problems, see also [8, 407].

This is due to the maximum principle valid for the r-Laplacian in (5.2.14b).

Time ranges from minutes during fast rupturing and running earthquakes to thousands of years during healing phases in which the lithosphere can be considered, solid and even the small-strain concept is applicable, in contrast to hundreds of millions of years, a time scale over which the lithosphere is seen as rather fluid.

In fact, the ansatz \(\boldsymbol{u} =\zeta\), \(\boldsymbol{y} = u\), and \(\boldsymbol{z} =\pi\) would apply in (5.1.110) in this quasistatic case.

Indeed, \(\mathbb{H}\) is to be suitably qualified. For example, in addition to the invariance of \(\mathbb{C}\) where the orthogonal subspaces of deviatoric and spherical components (4.3.43) are now to be modified for \(\mathbb{C} = \mathbb{C}(\zeta )\), and we need to assume that \(\mathbb{H}\nabla e...\nabla e = \mathbb{H}_{\mbox{ D}}\nabla \mathrm{dev}\,e...\nabla \mathrm{dev}\,e + H_{\mbox{ S}}\nabla \mathrm{tr}\,e \cdot \nabla \mathrm{tr}\,e\) for some \(\mathbb{H}_{\mbox{ D}} > 0\) and \(H_{\mbox{ S}} > 0\).

For the analysis, we assume in particular that \(\epsilon \vert z\vert ^{2} \leq a(z) \leq (1+\vert z\vert ^{2})/\epsilon\) for some ε > 0 and \(\varSigma (\zeta ) = s_{_{\mathrm{Y}}}(\zeta )B_{1}\) with a yield stress \(s_{_{\mathrm{Y}}}: [0, 1] \rightarrow \mathbb{R}^{+}\) continuous and \(B_{1} \subset \mathbb{R}_{\mathrm{dev}}^{d\times d}\) convex, closed, \(0 \in \mbox{ int}\,B_{1}\). An important attribute of this model is that \(\mathbb{H}\) acts on the whole elastic strain and is not subject to damage, so that the driving force for the damage \(\frac{1} {2}\mathbb{C}(\zeta )(e(u-u_{\mathrm{D}}^{}(t))-\pi ): (e(u)-\pi )\) in the flow-rule (5.2.14b) is certainly in \(\mathrm{L}^{2}((0,T)\times \varOmega )\), and the technique from Example 5.2.3 now with the constraint \(0 \leq \zeta \leq 1\) can be used, in particular, \(\mathrm{div}((1+\vert \nabla \zeta \vert _{}^{2})_{}^{r/2-1}\nabla \zeta ) \in \mathrm{L}^{2}((0,T)\times \varOmega )\) and (5.2.14b) holds even a.e. (i.e., in the Carathéodory sense). A further important phenomenon is that assuming r > d, we have \(\zeta\) “compactly” in \(\mathrm{C}([0,T]\times \bar{\varOmega })\), and the dissipation energy \(\int _{[0,T]\times \bar{\varOmega }}[\boldsymbol{\delta }_{\varSigma (\zeta )}^{{\ast}}(\dot{\pi })](\mathrm{d}x\mathrm{d}t)\) is well defined as an integral of the continuous function \(s_{_{\mathrm{Y}}}(\zeta )\) via the measure \(\boldsymbol{\delta }_{B_{1}}(\dot{\pi })\) over the compact set \([0,T]\times \bar{\varOmega }\). Here, in limiting the time discretization, Corollary B.5.12 is employed.

More specifically, \(\mathbb{C} = \mathbb{C}(\zeta )\) was affine as a function of \(\zeta\) with \(\mathbb{C}(1)\) as in (4.1.10) with (4.1.24) with Young’s modulus \(E_{_{\mathrm{Young}}} = 27\,\) GPa and Poisson’s ratio \(\nu = 0.2\), \(\mathbb{C}(0) = \mathbb{C}(1)/10\), the elastic domain \(\varSigma (\zeta ):=\{\,\sigma \in \mathbb{R}d\times d \mathrm{dev} \;\vert \ \vert \sigma \vert \leq \zeta \sigma _{\mathrm{y}}\,\}\) with the “nondamaged” yield stress \(\sigma _{\mathrm{y}} = 2\,\) MPa, the dissipation potential a(⋅ ) in (5.2.15e) was \(a(\dot{z}):= a_{1}\dot{z}^{\,-} + a_{2}(\dot{z}^{\,-})^{2} + cb(\dot{z}^{\,+})^{2}\) with a ₁ = 10 Pa, a ₂ = 0. 1 Pa s, and c = 100 kPa s, while the damage stored energy \(b(\zeta ) = b_{0}\zeta\) in (5.2.15d) used b ₀ = 10⁻³Pa, and the damage length-scale coefficient was κ = 10⁻⁶ J/m.

For the semistability itself, it would suffice to use weak convergence of u _k and compactness of the trace operator \(v\mapsto [[v]]: \mathrm{H}^{1}(\varOmega \setminus \varGamma \mathrm{C}) \rightarrow \mathrm{L}^{2}(\varGamma \mathrm{C})\). Yet we will need strong convergence of u _k for the energy inequality.

For the quasistatic case, i.e., M = 0, a semi-implicit scheme similar to (5.2.48) combined with a spatial discretization has been used in [259], where a rate-of-convergence estimate and uniqueness of the weak solution have been proved, assuming p = 2 and at most quadratic growth of \(\gamma\).

Here the inertia plays an important role for use of the Aubin–Lions lemma (Lemma B.5.8) on strong convergence of \(\dot{u}_{\tau } -\dot{ u} \rightarrow 0\) in \(\mathrm{L}^{2}(0,T; \mathrm{H}^{1-\epsilon }(\varOmega \setminus \varGamma \mathrm{C}; \mathbb{R}^{d}))\), so that the traces on \(\varGamma \mathrm{C}\) converge to zero \(\mathrm{L}^{2}(0,T; \mathrm{L}^{p^{\sharp }-\epsilon }(\varGamma \mathrm{C}; \mathbb{R}^{d}))\) for every \(0 <\epsilon \leq p^{\sharp } - 1\), similarly as in the proof of Proposition 5.2.15.

The numerical implementation, however, used a fully implicit formula together with the AMA algorithm; cf. also Remark 3.6.16.

Cf. also [532, Example 8.104] for similar calculations.

Reversible damage or delamination (i.e., allowing for healing) has been addressed routinely in the mathematical literature; cf. the monographs [560, 567]. If not combined with an inelastic strain allowing for permanent deformation as in Figure 4.24 and Example 5.2.3, healing has a tendency to remember not only the original state of the material but also the original configuration. Such models have therefore only limited application to Robin-type but not Dirichlet loading.

A prototype is the linear heat equation \(\varDelta \theta = f\) on \(\varOmega \subset \mathbb{R}^{3}\) with zero Dirichlet boundary condition and a natural heat source \(f \in \mathrm{L}^{1}(\varOmega )\), whose nonvariational character can be seen from the fact that the formal variational problem \(\int _{\varOmega }\frac{1} {2}\vert \nabla \theta \vert ^{2} + f\theta \,\mathrm{d}x\) has the infimum \(-\infty \) if \(f\not\in \mathrm{L}^{6/5}(\varOmega )\), and the usual “variational” setting of \(\varDelta: \mathrm{H}_{0}^{1}(\varOmega ) \rightarrow \mathrm{H}^{-1}(\varOmega )\) does not work; cf. [532, Exercise 3.42].

In the C^∗-Banach-algebra theory, this ordering is defined on a complexification of \(\boldsymbol{\varTheta }\), assuming that there exists a C^∗ algebra whose real variant (i.e., a “realization” by taking self-adjoint elements as in Example A.5.4) is \(\boldsymbol{\varTheta }\). Then \(\boldsymbol{\varTheta }^{{\ast}}\) is ordered standardly by the dual ordering, and its “realization” is then the ordering considered in (5.3.1). Yet it does not seem a general property and must be assumed. Anyhow, it is easily satisfied in concrete applications.

In fact, the existence of \(\boldsymbol{\vartheta }^{+} \geq \pmb{ 0}\) and \(\boldsymbol{\vartheta }^{\,-}\geq \pmb{ 0}\) satisfying (5.3.3a) together with \(\|\boldsymbol{\vartheta }\|_{\boldsymbol{\varTheta }^{{\ast}}}^{} =\|\boldsymbol{\vartheta } ^{+}\|_{\boldsymbol{\varTheta }^{{\ast}}}^{} +\|\boldsymbol{\vartheta } ^{\,-}\|_{\boldsymbol{\varTheta }^{{\ast}}}^{}\) is a standard result from the theory of C^∗-algebras.

The first assumption is to avoid the term \(\boldsymbol{\theta }\partial _{\boldsymbol{z}\boldsymbol{\theta }}^{2}\varPsi (\boldsymbol{u},\boldsymbol{z},\boldsymbol{\theta })\dot{\boldsymbol{z}}\), which would involve a measure \(\dot{\boldsymbol{z}}\) multiplied by some functions that hardly can be continuous in time. In fact, it would suffice to assume \(\boldsymbol{\theta }\partial _{\boldsymbol{z}\boldsymbol{\theta }}^{2}\varPsi\) to be independent of \((\boldsymbol{u},\boldsymbol{z},\boldsymbol{\theta })\).

Following the scalar-valued case \(\boldsymbol{X} = \mathbb{R}\) presented in [608, Sect. XII.7], one can consider classes of equivalence of BV-functions coinciding with each other almost everywhere and then define \(\dot{\boldsymbol{z}}\) as a distributional derivative as in (B.5.13) but using the Lebesgue–Stieltjes integral, i.e., \(\langle \dot{\boldsymbol{z}},\varphi \rangle:= -\int _{0}^{T}\varphi \,\mathrm{d}\boldsymbol{z}(t)\) for all \(\varphi \in \mathcal{D}([0,T]\). Equivalently, one can consider left-continuous representatives in each such equivalence class, which would mean a possible modification of the original \(\boldsymbol{z}\) at at most a countable number of time instances.

See also [532, Rem. 12.12] for this simplified estimation technique.

For the comparison argument leading to positivity in a concrete continuous heat-transfer problem, see [184, Sect. 4.2.1] or in the time-discrete setting also [514].

Here we rely on the calculation \(\frac{1} {2}(\boldsymbol{a}^{\,-})^{2}-\frac{1} {2}(\boldsymbol{b}^{\,-})^{2}+(\boldsymbol{a}-\boldsymbol{b})\bullet \boldsymbol{a}^{\,-} = \frac{1} {2}(\boldsymbol{a}^{\,-})^{2}-\frac{1} {2}(\boldsymbol{b}^{\,-})^{2}+\boldsymbol{a}^{+}\bullet \boldsymbol{a}^{\,-}-\boldsymbol{a}^{\,-}\bullet \boldsymbol{a}^{\,-}-\boldsymbol{b}^{+}\bullet \boldsymbol{a}^{\,-}+\boldsymbol{b}^{\,-}\bullet \boldsymbol{a}^{\,-}\leq -\frac{1} {2}(\boldsymbol{a}^{\,-})^{2}-\frac{1} {2}(\boldsymbol{b}^{\,-})^{2}+\boldsymbol{b}^{\,-}\bullet \boldsymbol{a}^{\,-} = -\frac{1} {2}(\boldsymbol{a}^{\,-}-\boldsymbol{b}^{\,-})^{2} \leq 0\), where we used that \(\boldsymbol{a}^{+}\bullet \boldsymbol{a}^{\,-} =\pmb{ 0}\) and \(\boldsymbol{b}^{+}\bullet \boldsymbol{a}^{\,-}\geq \pmb{ 0}\) hold by the assumption (5.3.66).

In fact, (5.3.72) needs \((\boldsymbol{\vartheta }_{\tau }^{k})^{3} \in \boldsymbol{\varTheta }^{{\ast}}\), which in particular examples does not need to be assumed for d = 3, but in any case, the last equation in (5.3.72) holds by mollification arguments.

More specifically, heat capacity \(c_{\mathrm{v}} = 3.2\,\mathrm{M}\mathrm{J}\mathrm{m}^{-3}\mathrm{K}^{-1}\), \(\mathbb{K} =\kappa _{0}\mathbb{I}\), with heat-conduction coefficient \(\kappa _{0} = 80\,\mathrm{W}\mathrm{m}^{-1}\mathrm{K}^{-1}\), thermal-expansion coefficient \(\alpha = 2 \cdot 10^{-5}\mathrm{K}^{-1}\), Young’s modulus \(E_{_{\mathrm{Young}}} = 137\,\) GPa, Poisson ratio \(\nu = 0.3\), and plastic yield stress 450 MPa were used in the calculations.

Thus we preselect only some symmetric solutions of the original problem on \(\varOmega\). One should realize that due to lack of a rigorous uniqueness proof, only the whole set of solutions must be symmetric, and asymmetric solutions may exist. In any case, this set also contains some symmetric solutions, which can be proved just by applying the previous arguments to the reduced problem.

More specifically, for d = 3, this convergence was proved in [60], provided β > 2, \(\gamma >\max (76/17, 2\omega /(\omega -1))\), and \(\varepsilon (\tau ) = o(\tau ^{(\gamma +1)(\beta -1)/(\beta ^{2}\gamma ) })\), and the mesh parameter satisfies \(h \leq H(\tau )\) for some function H whose mere existence was shown only rather implicitly. The difficulty is that the nonnegativity of the (transformed) temperature does not seem guaranteed for spatial discretization (even on qualified triangulations), in contrast to mere time discretization based on Remark 5.3.13.

Note that for a smooth scalar-valued test function v, counting also symmetry of \(\mathbb{C}\) and \(\mathbb{E}\), we have \(\langle F(\vartheta )\bullet \dot{u},v\rangle =\langle F(\vartheta ),v\dot{u}\rangle =\int _{\varOmega }\mathcal{T} (\vartheta )\mathbb{E}: \mathbb{C}e(v\dot{u})\,\mathrm{d}x =\int _{\varOmega }\mathcal{T} (\vartheta )\mathbb{E}: \mathbb{C}(ve(\dot{u})+\frac{1} {2}\dot{u}\otimes \nabla v+\frac{1} {2}\nabla v\otimes \dot{u})\,\mathrm{d}x =\int _{\varOmega }(\mathcal{T} (\vartheta )\mathbb{C}: \mathbb{E}e(\dot{u}))v+\mathcal{T} (\vartheta )\mathbb{C}: \mathbb{E}(\dot{u}\otimes \nabla v)\,\mathrm{d}x\).

The piecewise-affine interpolant \(\lambda _{\tau }\) has classically defined piecewise-constant time derivative \(\dot{\lambda }_{\tau }\), while for its BV-limit \(\lambda\), the time derivative is meant in the sense of distributions; cf. (B.5.13). Then \(\boldsymbol{\delta }_{\varSigma _{0}}^{{\ast}}(\dot{\lambda })\) is the total variation of the measure \(\dot{\lambda }\) with respect to the (possibly asymmetric) norm \(\boldsymbol{\delta }_{\varSigma _{0}}^{{\ast}}(\dot{)}: \mathbb{R}^{N} \rightarrow \mathbb{R}\).

H.-D. Alber. Materials with Memory. Springer-Verlag, Berlin, 1998.MATH

21.

S. S. Antman. Nonlinear Problems of Elasticity. Springer-Verlag, New York, 1995.MATHCrossRef

22.

S. S. Antman. Physically unacceptable viscous stresses. Zeitschrift angew. Math. Physik, 49:980–988, 1998.MATHMathSciNetCrossRef

60.

S. Bartels and T. Roubíček. Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion. Math. Model. Numer. Anal., 45:477–504, 2011.MATHCrossRef

61.

S. Bartels and T. Roubíček. Numerical approaches to thermally coupled perfect plasticity. Numer. Meth. for Partial Diff. Equations, 29:1837–1863, 2013.MATH

63.

A. Bedford. Hamilton’s Principle in Continuum Mechanics. Pitman, Boston, 1985.MATH

73.

L. Boccardo, A. Dall’aglio, T. Gallouët, and L. Orsina. Nonlinear parabolic equations with measure data. J. Funct. Anal., 147:237–258, 1997.MATHMathSciNetCrossRef

74.

L. Boccardo and T. Gallouët. Non-linear elliptic and parabolic equations involving measure data. J. Funct. Anal., 87:149–169, 1989.MATHMathSciNetCrossRef

77.

E. Bonetti, G. Bonfanti, and R. Rossi. Global existence for a contact problem with adhesion. Math. Meth. Appl. Sci., 31:1029–1064, 2008.MATHMathSciNetCrossRef

78.

E. Bonetti, G. Bonfanti, and R. Rossi. Thermal effects in adhesive contact: modelling and analysis. Nonlinearity, 22:2697–2731, 2009.MATHMathSciNetCrossRef

79.

E. Bonetti, G. Bonfanti, and R. Rossi. Long-time behaviour of a thermomechanical model for adhesive contact. Discr. Cont. Dynam. Systems - S, 4:273–309, 2011.MATHMathSciNet

80.

E. Bonetti, G. Schimperna, and A. Segatti. On a doubly nonlinear model for the evolution of damaging in viscoelastic materials. J. Differential Equations, 218:91–116, 2005.MATHMathSciNetCrossRef

85.

E. Bouchbinder and J. S. Langer. Nonequilibrium thermodynamics of driven amorphous materials. II. Effective-temperature theory. Phys. Rev. E., 80:031132, 2009.

89.

B. Bourdin. Numerical implementation of the variational formulation for quasi-static brittle fracture. Interfaces Free Bound., 9:411–430, 2007.MATHMathSciNetCrossRef

91.

B. Bourdin, G. A. Francfort, and J.-J. Marigo. The variational approach to fracture. J. Elasticity, 91:5–148, 2008.MATHMathSciNetCrossRef

92.

B. Bourdin, C. J. Larsen, and C. L. Richardson. A time-discrete model for dynamic fracture based on crack regularization. Int. J. of Fracture, pages 133–143, 2011.

106.

M. Bulíček, E. Feireisl, and J. Málek. A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients. Nonlinear. Anal., Real World Appl., 10:992–1015, 2009.

107.

M. Bulíček, P. Kaplický, and J. Málek. An L ²-maximal regularity result for the evolutionary Stokes-Fourier system. Applic. Anal., 90:31–45, 2011.MATHCrossRef

114.

A. Chambolle, A. Giacomini, and M. Ponsiglione. Crack initiation in brittle materials. Arch. Rational Mech. Anal., 188:309–349, 2008.MATHMathSciNetCrossRef

133.

M. Comninou. Stress singularity at a sharp edge in contact problems with friction. Zeitschrift angew. Math. Physik, 27:493–499, 1976.CrossRef

149.

G. Dal Maso, G. A. Francfort, and R. Toader. Quasistatic crack growth in nonlinear elasticity. Arch. Rational Mech. Anal., 176:165–225, 2005.MATHMathSciNetCrossRef

158.

S. Demoulini. Weak solutions for a class of nonlinear systems of viscoelasticity. Arch. Rat. Mech. Anal., 155:299–334, 2000.MATHMathSciNetCrossRef

178.

L. C. Evans. Entropy and Partial Differential Equations. UC Berkeley, Lecture notes, Dept. of Math., UC Berkeley, 1996.

184.

E. Feireisl, H. Petzeltová, and E. Rocca. Existence of solutions to a phase transition model with microscopic movements. Math. Methods Appl. Sci., 32:1345–1369, 2009.MATHMathSciNetCrossRef

185.

J. R. Fernández and M. Sofonea. Variational and numerical analysis of the signorini’s conctact problem in viscoplasticity with damage. J. Appl. Math., 2:87–114, 2003.CrossRef

217.

I. M. Gelfand, D. A. Raĭkov, and G. E. Shilov. Commutative Normed Rings (In Russian, 1960). Engl. transl.: Chelsea Publ., New York, 1964.

223.

A. Giacomini and M. Ponsiglione. A Γ-convergence approach to stability of unilateral minimality properties in fracture mechanics and applications. Arch. Rational Mech. Anal., 180:399–447, 2006.MATHMathSciNetCrossRef

255.

Y. Hamiel, V. Lyakhovsky, and Y. Ben-Zion. The elastic strain energy of damaged solids with applications to non-linear deformation of crystalline rocks. Pure Appl. Geophys., 168:2199–2210, 2011.CrossRef

259.

W. Han, M. Shillor, and M. Sofonea. Variational and numerical analysis of a quasistatic viscoelastic problem with normal complience, friction and damage. J. Comput. Appl. Math., 137:377–398, 2001.MATHMathSciNetCrossRef

330.

M. Kružík, C. G. Panagiotopoulos, and T. Roubíček. Quasistatic adhesive contact delaminating in mixed mode and its numerical treatment. Math. Mech. of Solids, 2014. in print. DOI: 10.1177/1081286513507942.

340.

A. A. Kulikovski. Analytical Modelling of Fuel Cells. Elsevier, Amsterdam, 2010.

348.

C. J. Larsen. Models for dynamic fracture based on griffith’s criterion. In K. Hackl, editor, IUTAM Symp. on Variational Concepts with Appl. to the Mech. of Mater., pages 131–140. Springer, 2010. Proceedings of the IUTAM Symposium on Variational Concepts, Bochum, Germany, Sept. 22–26, 2008.

350.

C. J. Larsen, C. Ortner, and E. Suli. Existence of solution to a regularized model of dynamic fracture. Math. Models Meth. Appl. Sci., 20:1021–1048, 2010.MATHMathSciNetCrossRef

368.

V. Lyakhovsky and Y. Ben-Zion. Scaling relations of earthquakes and aseismic deformation in a damage rheology model. Geophys. J. Int., 172:651–662, 2008.CrossRef

369.

V. Lyakhovsky, Y. Ben-Zion, and A. Agnon. Distributed damage, faulting, and friction. J. Geophysical Res., 102:27,635–27,649, 1997.CrossRef

370.

V. Lyakhovsky, Y. Hamiel, and Y. Ben-Zion. A non-local visco-elastic damage model and dynamic fracturing. J. Mech. Phys. Solids, 59:1752–1776, 2011.MATHMathSciNetCrossRef

371.

V. Lyakhovsky, Z. Reches, R. Weiberger, and T. E. Scott. Nonlinear elastic behaviour of damaged rocks. Geophys. J. Int., 130:157–166, 1997.CrossRef

407.

A. Mielke, A. Petrov, and J. A. C. Martins. Convergence of solutions of kinetic variational inequalities in the rate-independent quasiastatic limit. J. Math. Anal. Appl., 348:1012–1020, 2008.MATHMathSciNetCrossRef

422.

A. Mielke, T. Roubíček, and J. Zeman. Complete damage in elastic and viscoelastic media and its energetics. Comput. Methods Appl. Mech. Engrg., 199:1242–1253, 2010.MATHMathSciNetCrossRef

466.

C. Padovani, G. Pasquinelli, and M. Šilhavý. Processes in masonry bodies and the dynamical significance of collapse. Math. Mech. Solids, 13:573–610, 2008.MATHMathSciNetCrossRef

469.

C. G. Panagiotopoulos, V. Mantič, and T. Roubíček. Two adhesive-contact models for quasistatic mixed-mode delamination problems. Math. Comput. in Simulation, 2014. Submitted.

470.

N. S. Papageorgiou. On the existence of solutions for nonlinear parabolic problems with nonmonotone discontinuities. J. Math. Anal. Appl., 205:434–453, 1997.MATHMathSciNetCrossRef

483.

P. Podio Guidugli, T. Roubíček, and G. Tomassetti. A thermodynamically-consistent theory of the ferro/paramagnetic transition. Archive Rat. Mech. Anal., 198:1057–1094, 2010.MATHCrossRef

497.

K. R. Rajagopal and T. Roubíček. On the effect of dissipation in shape-memory alloys. Nonlinear Anal., Real World Appl., 4:581–597, 2003.

508.

E. Rocca and R. Rossi. A degenerating PDE system for phase transitions and damage. Math. Models Methods Appl. Sci., 24:1265–1341, 2014.MATHMathSciNetCrossRef

514.

R. Rossi and T. Roubíček. Thermodynamics and analysis of rate-independent adhesive contact at small strains. Nonlinear Anal., 74:3159–3190, 2011.MATHMathSciNetCrossRef

515.

R. Rossi and T. Roubíček. Adhesive contact delaminating at mixed mode, its thermodynamics and analysis. Interfaces and Free Boundaries, 14:1–37, 2013.CrossRef

517.

R. Rossi and M. Thomas. From an adhesive to a brittle delamination model in thermo-visco-elasticity. ESAIM Contr. Optim. Calc. Var., 21:1–59, 2015.MATHMathSciNetCrossRef

526.

T. Roubíček. Rate independent processes in viscous solids at small strains. Math. Methods Appl. Sci., 32:825–862, 2009. Erratum Vol. 32(16) p. 2176.

528.

T. Roubíček. Thermodynamics of rate independent processes in viscous solids at small strains. SIAM J. Math. Anal., 42:256–297, 2010.MATHMathSciNetCrossRef

532.

T. Roubíček. Nonlinear Partial Differential Equations with Applications. Birkhäuser, Basel, 2nd edition, 2013.CrossRef

533.

T. Roubíček. Thermodynamics of perfect plasticity. Discr. Cont. Dynam. Systems Ser. S, 6:193–214, 2013.MATHCrossRef

546.

T. Roubíček, O. Souček, and R. Vodička. A model of rupturing lithospheric faults with re-occurringing earthquakes. SIAM J. Appl. Math., 73:1460–1488, 2013.MATHMathSciNetCrossRef

549.

T. Roubíček and G. Tomassetti. Thermomechanics of demageable materials under diffusion: modeling and analysis. In preparation, 2015.

551.

T. Roubíček and J. Valdman. Perfect plasticity with damage and healing at small strains, its modelling, analysis, and computer implementation. (Preprint arXiv 1505.01018), submitted, 2015.

560.

M. Shillor, M. Sofonea, and J. J. Telega. Models and Analysis of Quasistatic Contact. Springer, Berlin, 2004.MATHCrossRef

567.

M. Sofonea, W. Han, and M. Shillor. Analysis and approximation of contact problems with adhesion or damage. Chapman & Hall/CRC, Boca Raton, FL, 2006.MATH

608.

A. Visintin. Differential Models of Hysteresis. Springer-Verlag, Berlin, 1994.MATHCrossRef

616.

R. Vodička, V. Mantič, and T. Roubíček. Quasistatic normal-compliance contact problem of visco-elastic bodies with Coulomb friction implemented by SGBEM/QP. Comp. Meth. Appl. Mechanics Engrg., 2015, submitted.

619.

P. Wriggers. Computational Contact Mechanics. Springer, Berlin, 2 edition, 2006.MATHCrossRef

Titel: Beyond rate-independence
verfasst von: Alexander Mielke
Tomàš Roubíček
Verlag: Springer New York
Buch: Rate-Independent Systems
Print ISBN: 978-1-4939-2705-0

Electronic ISBN: 978-1-4939-2706-7

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-1-4939-2706-7_5

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