Second order approximations of quasistatic evolution problems in finite dimension

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April 2012, 32(4): 1125-1167. doi: 10.3934/dcds.2012.32.1125

Second order approximations of quasistatic evolution problems in finite dimension

Virginia Agostiniani ^1,

1.	via Bonomea 265, 34136 Trieste, Italy

Received November 2010 Revised April 2011 Published October 2011

Abstract
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In this paper, we study the limit, as $\epsilon$ goes to zero, of a particular solution of the equation $\epsilon^2A\ddot u^{\epsilon}(t)+\epsilon B\dot u^{\epsilon}(t)+\nabla_xf(t,u^{\epsilon}(t))=0$, where $f(t,x)$ is a potential satisfying suitable coerciveness conditions. The limit $u(t)$ of $u^{\epsilon}(t)$ is piece-wise continuous and verifies $\nabla_xf(t,u(t))=0$. Moreover, certain jump conditions characterize the behaviour of $u(t)$ at the discontinuity times. The same limit behaviour is obtained by considering a different approximation scheme based on time discretization and on the solutions of suitable autonomous systems.

Keywords: heteroclinic solutions., discrete approximations, saddle-node bifurcation, perturbation methods, vanishing viscosity, Singular perturbations.

Mathematics Subject Classification: Primary: 74H10, 34K18, 34C37; Secondary: 34K2.

Citation: Virginia Agostiniani. Second order approximations of quasistatic evolution problems in finite dimension. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1125-1167. doi: 10.3934/dcds.2012.32.1125

References:

[1]	F. Cagnetti, A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path, Math. Models Methods Appl. Sci., 18 (2008), 1027-1071. doi: 10.1142/S0218202508002942.
[2]	G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening, Arch. Ration. Mech. Anal., 189 (2008), 469-544. doi: 10.1007/s00205-008-0117-5.
[3]	G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling, Calc. Var. Partial Differential Equations, 40 (2011), 125-181.
[4]	G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution,, SISSA preprint 46/2010/M., ().
[5]	G. Dal Maso and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case, Netw. Heterog. Media, 5 (2010), 97-132.
[6]	M. A. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Anal., 13 (2006), 151-167.
[7]	J. Guckenheimer and P. Holme, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations on Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.
[8]	J. K. Hale, "Ordinary Differential Equations," Pure and Applied Mathematics, XX1, Krieger, Florida, 1980.
[9]	M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1976.
[10]	D. Knees, A. Mielke and C. Zanini, Crack growth in polyconvex materials, Phys. D, 239 (2010), 1470-1484. doi: 10.1016/j.physd.2009.02.008.
[11]	A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst., 25 (2009), 585-615. doi: 10.3934/dcds.2009.25.585.
[12]	A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 2011. doi: 10.1051/cocv/2010054.
[13]	F. Solombrino, Quasistatic evolution for plasticity with softening: the spatially homogeneous case, Discrete Contin. Dyn. Syst., 27 (2010), 1189-1217. doi: 10.3934/dcds.2010.27.1189.
[14]	R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth, Boll. Unione Mat. Ital. (9), 2 (2009), 1-35.
[15]	C. Zanini, Singular perturbation of finite dimensional gradient flows, Discrete Contin. Dyn. Syst., 18 (2007), 657-675. doi: 10.3934/dcds.2007.18.657.

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References:

[1]	F. Cagnetti, A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path, Math. Models Methods Appl. Sci., 18 (2008), 1027-1071. doi: 10.1142/S0218202508002942.
[2]	G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening, Arch. Ration. Mech. Anal., 189 (2008), 469-544. doi: 10.1007/s00205-008-0117-5.
[3]	G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling, Calc. Var. Partial Differential Equations, 40 (2011), 125-181.
[4]	G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution,, SISSA preprint 46/2010/M., ().
[5]	G. Dal Maso and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case, Netw. Heterog. Media, 5 (2010), 97-132.
[6]	M. A. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Anal., 13 (2006), 151-167.
[7]	J. Guckenheimer and P. Holme, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations on Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.
[8]	J. K. Hale, "Ordinary Differential Equations," Pure and Applied Mathematics, XX1, Krieger, Florida, 1980.
[9]	M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1976.
[10]	D. Knees, A. Mielke and C. Zanini, Crack growth in polyconvex materials, Phys. D, 239 (2010), 1470-1484. doi: 10.1016/j.physd.2009.02.008.
[11]	A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst., 25 (2009), 585-615. doi: 10.3934/dcds.2009.25.585.
[12]	A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 2011. doi: 10.1051/cocv/2010054.
[13]	F. Solombrino, Quasistatic evolution for plasticity with softening: the spatially homogeneous case, Discrete Contin. Dyn. Syst., 27 (2010), 1189-1217. doi: 10.3934/dcds.2010.27.1189.
[14]	R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth, Boll. Unione Mat. Ital. (9), 2 (2009), 1-35.
[15]	C. Zanini, Singular perturbation of finite dimensional gradient flows, Discrete Contin. Dyn. Syst., 18 (2007), 657-675. doi: 10.3934/dcds.2007.18.657.

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