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.
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 |