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

A linear stability criterion for strain-rate sensitive solids and structure s is proposed and validated with the help of two versions of Shanley's colu mn, the first with two discrete supports and the second with a continuous d istribution of supports. Linear stability transition is defined by the chan ge in sign of the second derivative with respect to time of the column's an gular position evaluated at the onset of perturbation. This criterion perta ins to the initiation of instabilities but is not expected to provide infor mation on their long term development. Two parameters influence linear stab ility: the dimensionless number T, defined as the ratio of the relaxation t ime of the viscous support to the characteristic loading time, and the pert urbation 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 reduc ed modulus load, in agreement with existing stability criteria for rate-ind ependent models based on maximum dissipation. For arbitrary values of T, tw o critical loads are identified at the linear stability transition. The fir st is named the rate-dependent tangent modulus load and is valid for pertur bations sufficiently small to prevent initial unloading. That load coincide s with the classical tangent modulus load for T tending to zero and is, sur prisingly, a decreasing function of that dimensionless number. The second c ritical load is termed the rate-dependent reduced modulus load, and is appl icable to columns that are partly unloaded at the onset of perturbation. Th is 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 im perfection size. Similar results are found for the second model, with a new insight on the role of the unloading zone extent in determining the critic al load in the singular limit of T equal to zero. The proposed stability cr iterion is validated by comparing its predictions with the outcome of nonli near perturbation analyses and of imperfection sensitivity studies. It is s hown, in particular, that an imperfection evolves according to our stabilit y predictions as long as the relative difference between the irreversible d isplacements of the supports can be disregarded. A generalization of the pr oposed linear stability criterion to viscoplastic continua is finally sketc hed. (C) 1999 Elsevier Science Ltd. All rights reserved.