Numerical computations for antiplane shear in a granular flow model
Author:
Xabier Garaizar
Journal:
Quart. Appl. Math. 52 (1994), 289-309
MSC:
Primary 73G20; Secondary 35L65, 35Q72, 65C20, 73E50, 73N20, 73V20, 76A99
DOI:
https://doi.org/10.1090/qam/1276239
MathSciNet review:
MR1276239
Full-text PDF Free Access
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Abstract: We describe an algorithm for the numerical resolution of elasto-plastic deformations in the context of antiplane shear models. The algorithm is a second-order Godunov method. For these models the eigenvalues associated to the hyperbolic system are discontinuous. We test the algorithm on several examples.
S. S. Antman and W. G. Szymczak, Nonlinear elasto-plastic waves, Contemp. Math. 100, 27–54 (1989)
I.-L. Chern, J. Glimm, O. McBryan, B. Plohr, and S. Yaniv, Front tracking for gas dynamics, J. Comp. Phys. 62, 83–110 (1986)
A. Chorin, Random choice solutions of hyperbolic systems, J. Comp. Phys. 22, 517–533 (1976)
X. Garaizar and D. Schaeffer, Numerical computations for shear bands in an antiplane shear model, J. Mech. Phys. Solids, Pergamon, Oxford-Elmsford, NY, in press, 1993
J. Glimm, E. Isaacson, D. Marchesin, and O. McBryan, Front tracking for hyperbolic systems, Adv. Appl. Math. 2, 91–119 (1985)
K. Godunov, Finite-difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sbornik 47, 271–306 (1959)
E. H. Lee, A boundary value problem in the theory of plastic wave propagation, Quart. Appl. Math. 10, 335–346 (1952)
C. Moler and J. Smoller, Elementary interactions in quasilinear hyperbolic systems, Arch. Rat. Mech. Anal. 37, 309–322 (1970)
D. Schaeffer, A mathematical model for localization in granular flow: Postcritical behavior, preprint, 1991
D. G. Schaeffer and Michael Shearer, Scale-invariant initial value problems in one dimensional dynamic elasto-plasticity, with consequences for multidimensional nonassociative plasticity, European Journal of Applied Mathematics (3), 225–254 (1992)
D. G. Schaeffer and Michael Shearer, Private communications, 1992
J. Smoller, Shock-Waves and Reaction-Diffusion Equations Springer-Verlag, New York and Berlin, 1983
T. C. T. Ting, In Propagation of shock waves in solids, E. Varley, editor, pages 41–64, Amer. Soc. Mech. Eng. AMD vol. 17, 1986
John A. Trangenstein and R. B. Pember, Numerical algorithms for strong discontinuities in elastic-plastic solids, Journal of Computational Physics, 1991
B. van Leer, Towards the ultimate conservative difference scheme. V. A. second-order sequel to Godunov’s method, J. Comp. Phys. 32, 101–136 (1979)
B. van Leer, On the relation between the upwind-difference schemes of Godunov, Engquist-Osher and Roe, SIAM J. Sci. Stat. Comp. 5 (1), 1–20 (1984)
H. C. Yee, On the implementations of a class of upwind schemes for systems of hyperbolic conservation laws, Technical Report TM-86839, NASA, September 1987
H. C. Yee, A class of high-resolution explicit and implicit shock-capturing methods, Technical Report TM 101088, NASA, February 1989
S. S. Antman and W. G. Szymczak, Nonlinear elasto-plastic waves, Contemp. Math. 100, 27–54 (1989)
I.-L. Chern, J. Glimm, O. McBryan, B. Plohr, and S. Yaniv, Front tracking for gas dynamics, J. Comp. Phys. 62, 83–110 (1986)
A. Chorin, Random choice solutions of hyperbolic systems, J. Comp. Phys. 22, 517–533 (1976)
X. Garaizar and D. Schaeffer, Numerical computations for shear bands in an antiplane shear model, J. Mech. Phys. Solids, Pergamon, Oxford-Elmsford, NY, in press, 1993
J. Glimm, E. Isaacson, D. Marchesin, and O. McBryan, Front tracking for hyperbolic systems, Adv. Appl. Math. 2, 91–119 (1985)
K. Godunov, Finite-difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sbornik 47, 271–306 (1959)
E. H. Lee, A boundary value problem in the theory of plastic wave propagation, Quart. Appl. Math. 10, 335–346 (1952)
C. Moler and J. Smoller, Elementary interactions in quasilinear hyperbolic systems, Arch. Rat. Mech. Anal. 37, 309–322 (1970)
D. Schaeffer, A mathematical model for localization in granular flow: Postcritical behavior, preprint, 1991
D. G. Schaeffer and Michael Shearer, Scale-invariant initial value problems in one dimensional dynamic elasto-plasticity, with consequences for multidimensional nonassociative plasticity, European Journal of Applied Mathematics (3), 225–254 (1992)
D. G. Schaeffer and Michael Shearer, Private communications, 1992
J. Smoller, Shock-Waves and Reaction-Diffusion Equations Springer-Verlag, New York and Berlin, 1983
T. C. T. Ting, In Propagation of shock waves in solids, E. Varley, editor, pages 41–64, Amer. Soc. Mech. Eng. AMD vol. 17, 1986
John A. Trangenstein and R. B. Pember, Numerical algorithms for strong discontinuities in elastic-plastic solids, Journal of Computational Physics, 1991
B. van Leer, Towards the ultimate conservative difference scheme. V. A. second-order sequel to Godunov’s method, J. Comp. Phys. 32, 101–136 (1979)
B. van Leer, On the relation between the upwind-difference schemes of Godunov, Engquist-Osher and Roe, SIAM J. Sci. Stat. Comp. 5 (1), 1–20 (1984)
H. C. Yee, On the implementations of a class of upwind schemes for systems of hyperbolic conservation laws, Technical Report TM-86839, NASA, September 1987
H. C. Yee, A class of high-resolution explicit and implicit shock-capturing methods, Technical Report TM 101088, NASA, February 1989
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Article copyright:
© Copyright 1994
American Mathematical Society
