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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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., ().   Google Scholar

[5]

G. Dal Maso and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case, Netw. Heterog. Media, 5 (2010), 97-132.  Google Scholar

[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.  Google Scholar

[7]

J. Guckenheimer and P. Holme, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations on Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.  Google Scholar

[8]

J. K. Hale, "Ordinary Differential Equations," Pure and Applied Mathematics, XX1, Krieger, Florida, 1980. Google Scholar

[9]

M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1976. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth, Boll. Unione Mat. Ital. (9), 2 (2009), 1-35.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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., ().   Google Scholar

[5]

G. Dal Maso and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case, Netw. Heterog. Media, 5 (2010), 97-132.  Google Scholar

[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.  Google Scholar

[7]

J. Guckenheimer and P. Holme, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations on Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.  Google Scholar

[8]

J. K. Hale, "Ordinary Differential Equations," Pure and Applied Mathematics, XX1, Krieger, Florida, 1980. Google Scholar

[9]

M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1976. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth, Boll. Unione Mat. Ital. (9), 2 (2009), 1-35.  Google Scholar

[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.  Google Scholar

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