An asymptotic stability condition for inhomogeneous simple shear

Chen, H. Tz.; Douglas, A. S.; Malek-Madani, R.

doi:10.1090/qam/998099

Authors: H. Tz. Chen, A. S. Douglas and R. Malek-Madani
Journal: Quart. Appl. Math. 47 (1989), 247-262
MSC: Primary 73H10; Secondary 34D10, 58E07, 73F99
DOI: https://doi.org/10.1090/qam/998099
MathSciNet review: 998099
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Abstract: Analytic steady solutions of inhomogeneous simple shear with isothermal and stress boundary conditions are found. The material is assumed to be thermoviscous and inertia and heat conduction effects are included. The basic inhomogeneous solution is spatially dependent, but time independent. Bifurcation of this solution, as the parameters vary, is analyzed. It is shown that there is a critical value of the parameter, corresponding to thermal softening, below which two steady state solutions exist for specified values of other parameters. A linear perturbation method, which gives rise to a linear set of partial differential equations (with spatially dependent coefficients), is used to distinguish the stable branch of the bifurcation diagram. After separation of variables, the existence of eigenvalues and eigenfunctions of the resulting fourth-order system is demonstrated. An asymptotic solution to the eigenvalue problem, for the case when an appropriate parameter is set equal to zero, is obtained explicitly. An integral method is then used in the general case to obtain a sufficient condition for stability.

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