Existence, uniqueness, and long-time behavior of linearized field dislocation dynamics
Authors:
Amit Acharya and Marshall Slemrod
Journal:
Quart. Appl. Math. 81 (2023), 247-258
MSC (2020):
Primary 35Q74, 37L05, 74E15; Secondary 74H20, 74H25, 74H40, 74B20
DOI:
https://doi.org/10.1090/qam/1642
Published electronically:
February 2, 2023
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Additional Information
Abstract: This paper examines a system of partial differential equations describing dislocation dynamics in a crystalline solid. In particular we consider dynamics linearized about a state of zero stress and use linear semigroup theory to establish existence, uniqueness, and time-asymptotic behavior of the linear system.
References
- R. Arora and A. Acharya, Dislocation pattern formation in finite deformation crystal plasticity, International Journal of Solids and Structures 184(2020), 2, 114–135, electronically published Feb. 26, 2019.
- Rajat Arora and Amit Acharya, A unification of finite deformation $J_2$ von-Mises plasticity and quantitative dislocation mechanics, J. Mech. Phys. Solids 143 (2020), 104050, 30. MR 4118895, DOI 10.1016/j.jmps.2020.104050
- Amit Acharya, An action for nonlinear dislocation dynamics, J. Mech. Phys. Solids 161 (2022), Paper No. 104811, 14. MR 4387914, DOI 10.1016/j.jmps.2022.104811
- A. Acharya, R. J. Knops, and J. Sivaloganathan, On the structure of linear dislocation field theory, J. Mech. Phys. Solids 130 (2019), 216–244. MR 3969201, DOI 10.1016/j.jmps.2019.06.002
- Rajat Arora, Xiaohan Zhang, and Amit Acharya, Finite element approximation of finite deformation dislocation mechanics, Comput. Methods Appl. Mech. Engrg. 367 (2020), 113076, 46. MR 4100791, DOI 10.1016/j.cma.2020.113076
- Philippe G. Ciarlet, On Korn’s inequality, Chinese Ann. Math. Ser. B 31 (2010), no. 5, 607–618. MR 2726058, DOI 10.1007/s11401-010-0606-3
- Constantine M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal. 29 (1968), 241–271. MR 233539, DOI 10.1007/BF00276727
- C. M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Functional Analysis 13 (1973), 97–106. MR 0346611, DOI 10.1016/0022-1236(73)90069-4
- S. R. Foguel, The ergodic theorem for Markov processes, Israel J. Math. 4 (1966), 11–22. MR 205323, DOI 10.1007/BF02760066
- Robert E. O’Brien Jr., Contraction semigroups, stabilization, and the mean ergodic theorem, Proc. Amer. Math. Soc. 71 (1978), no. 1, 89–94. MR 495844, DOI 10.1090/S0002-9939-1978-0495844-2
- K. Yosida, Functional analysis, 3rd ed., Springer-Verlag, New York, 1971.
- Xiaohan Zhang, Amit Acharya, Noel J. Walkington, and Jacobo Bielak, A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations, J. Mech. Phys. Solids 84 (2015), 145–195. MR 3413434, DOI 10.1016/j.jmps.2015.07.004
References
- R. Arora and A. Acharya, Dislocation pattern formation in finite deformation crystal plasticity, International Journal of Solids and Structures 184(2020), 2, 114–135, electronically published Feb. 26, 2019.
- Rajat Arora and Amit Acharya, A unification of finite deformation $J_2$ von-Mises plasticity and quantitative dislocation mechanics, J. Mech. Phys. Solids 143 (2020), 104050, 30. MR 4118895, DOI 10.1016/j.jmps.2020.104050
- Amit Acharya, An action for nonlinear dislocation dynamics, J. Mech. Phys. Solids 161 (2022), Paper No. 104811, 14. MR 4387914, DOI 10.1016/j.jmps.2022.104811
- A. Acharya, R. J. Knops, and J. Sivaloganathan, On the structure of linear dislocation field theory, J. Mech. Phys. Solids 130 (2019), 216–244. MR 3969201, DOI 10.1016/j.jmps.2019.06.002
- Rajat Arora, Xiaohan Zhang, and Amit Acharya, Finite element approximation of finite deformation dislocation mechanics, Comput. Methods Appl. Mech. Engrg. 367 (2020), 113076, 46. MR 4100791, DOI 10.1016/j.cma.2020.113076
- Philippe G. Ciarlet, On Korn’s inequality, Chinese Ann. Math. Ser. B 31 (2010), no. 5, 607–618. MR 2726058, DOI 10.1007/s11401-010-0606-3
- Constantine M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal. 29 (1968), 241–271. MR 233539, DOI 10.1007/BF00276727
- C. M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Functional Analysis 13 (1973), 97–106. MR 0346611, DOI 10.1016/0022-1236(73)90069-4
- S. R. Foguel, The ergodic theorem for Markov processes, Israel J. Math. 4 (1966), 11–22. MR 205323, DOI 10.1007/BF02760066
- Robert E. O’Brien Jr., Contraction semigroups, stabilization, and the mean ergodic theorem, Proc. Amer. Math. Soc. 71 (1978), no. 1, 89–94. MR 495844, DOI 10.2307/2042224
- K. Yosida, Functional analysis, 3rd ed., Springer-Verlag, New York, 1971.
- Xiaohan Zhang, Amit Acharya, Noel J. Walkington, and Jacobo Bielak, A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations, J. Mech. Phys. Solids 84 (2015), 145–195. MR 3413434, DOI 10.1016/j.jmps.2015.07.004
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Additional Information
Amit Acharya
Affiliation:
Department of Civil and Environmental Engineering and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, PA 15213
MR Author ID:
368246
ORCID:
0000-0002-6184-3357
Email:
acharyaamit@cmu.edu
Marshall Slemrod
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706
MR Author ID:
163635
ORCID:
0000-0002-0514-9467
Email:
slemrod@math.wisc.edu
Keywords:
Dislocations,
small deformations,
linear contraction semigroups
Received by editor(s):
August 8, 2022
Published electronically:
February 2, 2023
Additional Notes:
The work of the first author was supported by the grant NSF OIA-DMR #2021019.
Dedicated:
This paper is dedicated to our friend Costas Dafermos on the occasion of his 80th birthday
Article copyright:
© Copyright 2023
Brown University
