The singular limit of a hyperbolic system and the incompressible limit of solutions with shocks and singularities in nonlinear elasticity

Author:
Rustum Choksi

Journal:
Quart. Appl. Math. **55** (1997), 485-504

MSC:
Primary 73C50; Secondary 35L67, 73D40

DOI:
https://doi.org/10.1090/qam/1466144

MathSciNet review:
MR1466144

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Abstract: Discontinuous solutions with shocks for a family of almost incompressible hyperelastic materials are studied. An almost incompressible material is one whose deformations are not *a priori* constrained but whose stress response reacts strongly (of order ) to deformations that change volume. The material class considered is isotropic and admits motions that are self-similar, exhibit cavitation, and are energy minimizing. For the initial-value problem when considering the entire material, the solutions converge (as tends to zero) to an isochoric solution of the limit (incompressible) system with the corresponding arbitrary hydrostatic pressure being the singular limit of the pressures in the almost incompressible materials. The shocks, if they exist, disappear: their speed tends to infinity and their strength tends to zero.

**[1]**J. M. Ball,*Discontinuous equilibrium solutions and cavitation in nonlinear elasticity*, Philos. Trans. Roy. Soc. London A**306**, 557-611 (1982) MR**703623****[2]**G. Birkhoff and G. C. Rota,*Ordinary Differential Equations*, Ginn and Company, 1962 MR**0138810****[3]**P. Charrier, B. Dacorogna, B. Hanouzet, and P. Laborde,*An existence theorem for slightly compressible materials in nonlinear elasticity*, SIAM J. Math. Anal.**19**, 70-85 (1988) MR**924545****[4]**C. M. Dafermos,*Hyperbolic systems of conservation laws*, Systems of Nonlinear Partial Differential Equations (J. M. Ball, ed.), Dordrecht, Boston, 1983, pp. 25-70 MR**725517****[5]**D. G. Ebin,*Motion of slightly compressible fluids in a bounded domain*I, Comm. Pure Appl. Math.**35**, 451-485 (1982) MR**657824****[6]**M. Gurtin,*An Introduction to Continuum Mechanics*, Academic Press, 1981 MR**636255****[7]**S. Klainerman and A. Majda,*Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids*, Comm. Pure Appl. Math.**34**, 481-524 (1981) MR**615627****[8]**S. Klainerman and A. Majda,*Compressible and incompressible fluids*, Comm. Pure Appl. Math.**35**, 629-651 (1982) MR**668409****[9]**K. A. Pericak-Spector and S. J. Spector,*Nonuniqueness for a hyberbolic system: Cavitation in nonlinear elastodynamics*, Arch. Rational Mech. Anal.**101**, 293-317 (1988) MR**930330****[10]**S. Schochet,*The incompressible limit in nonlinear elasticity*, Comm. Math. Phys.**102**, 207-215 (1985) MR**820572****[11]**A. J. M. Spencer,*Finite Deformation of an Almost Incompressible Elastic Solid*, Proc. Internat. Sympos. Second-Order Effects, Hafia, 1962, pp. 200-216**[12]**C. Truesdell and W. Noll,*Nonlinear Field Theories of Mechanics*, Handbuch der Physik, III/3 (S. Flugge, ed.), Springer-Verlag, Berlin, 1965 MR**1215940**

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DOI:
https://doi.org/10.1090/qam/1466144

Article copyright:
© Copyright 1997
American Mathematical Society