On the stability of strain-rate dependent solids.: I—Structural examples

Abstract

A linear stability criterion for strain-rate sensitive solids and structures is proposed and validated with the help of two versions of Shanleys column, the first with two discrete supports and the second with a continuous distribution of supports. Linear stability transition is defined by the change in sign of the second derivative with respect to time of the columns angular position evaluated at the onset of perturbation. This criterion pertains to the initiation of instabilities but is not expected to provide information on their long term development. Two parameters influence linear stability: the dimensionless number T, defined as the ratio of the relaxation time of the viscous support to the characteristic loading time, and the perturbation size. It is found that the critical load of principal equilibria, defined for a straight column and a zero value of T, is the classical reduced modulus load, in agreement with existing stability criteria for rate-independent models based on maximum dissipation. For arbitrary values of T, two critical loads are identified at the linear stability transition. The first is named the rate-dependent tangent modulus load and is valid for perturbations sufficiently small to prevent initial unloading. That load coincides with the classical tangent modulus load for T tending to zero and is, surprisingly, a decreasing function of that dimensionless number. The second critical load is termed the rate-dependent reduced modulus load, and is applicable to columns that are partly unloaded at the onset of perturbation. This critical load approaches the classical reduced modulus load in the limit of T tending to zero, is a decreasing function of T, and depends on the imperfection size. Similar results are found for the second model, with a new insight on the role of the unloading zone extent in determining the critical load in the singular limit of T equal to zero. The proposed stability criterion is validated by comparing its predictions with the outcome of nonlinear perturbation analyses and of imperfection sensitivity studies. It is shown, in particular, that an imperfection evolves according to our stability predictions as long as the relative difference between the irreversible displacements of the supports can be disregarded. A generalization of the proposed linear stability criterion to viscoplastic continua is finally sketched.