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Instability in critical state theories of granular flow. (English) Zbl 0686.73038

This paper continues the study of instability in the dynamic partial differential equations in models for granular flow. Earlier [e.g.: Commun. Pure Appl. Math. 41, 879-890 (1988; Zbl 0644.73037)] we characterized the circumstances under which linear ill-posedness, an extreme form of instability, occurs in these equations. In this paper we characterize instability in the more customary sense; i.e., the circumstances under which the linear theory predicts that small deviations from a homogeneous deformation will grow exponentially in time.
Our model is of the ”Granta-gravel” type in critical state soil mechanics [e.g.: R. Jackson in: Theory of dispersed multiphase flow. R. E. Meyer (ed.) (1983; Zbl 0536.00018)], i.e., rigid- plastic, rate-independent behavior, satisfying normality, with volumetric-strain hardening. (Specific equations are given in § 1). We make a quasi-dynamic approximation that neglects all inertial terms but retains the time derivative in the continuity equation. No boundary conditions are imposed - this analysis concerns general properties of the partial differential equations analogous to the classification of equations into elliptic, hyperbolic, and parabolic. Only two-dimensional flow linearized about a homogeneous state is studied. To test for linear instability, we perform a normal mode analysis (Fourier transform) and look for an eigenvalue with a positive real part.
As discussed in § 4, our results are suggestive regarding the onset of nonuniform deformation and the formation of shear bands in various experimental tests of constitutive behavior of granular material. A mathematical definition in the paper, the distinction between instability and ill-posedness, may help clarify some of the issues regarding the formation of shear bands, even in cases not covered by the constitutive relations used in this paper.

MSC:

74H55 Stability of dynamical problems in solid mechanics
35R25 Ill-posed problems for PDEs
74B99 Elastic materials
74C99 Plastic materials, materials of stress-rate and internal-variable type
74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
35K99 Parabolic equations and parabolic systems
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