Keywords: deformation processes, bifurcation, instability, elastic-plastic solids, viscous solids.

In certain situations, deformation processes of solids may deviate from the expected path to a degree that endangers the purpose. The reason then is that the system (solid-constraints-loading) is unstable against disturbances, which are actually present but have been disregarded in the idealized description of the reference response. It appears useful to distinguish between consequences of initial imperfections, and the imposition of disturbances on the perfect system as pursued here. Instabilities that evolute out from imperferctions are latent in the system. They are termed static in contrast to the kinetic instability associated with the perfect system.

The issue of stability is investigated with respect to perturbations along the deformation path of the process [7]. The deformation process is classified as unstable if disturbances induce deviations that amplify. Their growth in the duration time of the deformation may be nevertheless within allowable limits. The analysis is simplified by restricting to the evolution in the neighbourhood of the reference state which defines infinitesimal or linear stability. In addition, stability is considered collectively rather than for specific disturbances. The possibility of deviations from the reference response in the absence of any disturbances marks bifurcations along the deformation path.

The stability of deformation processes of solids is examined with reference to various types of material including elastic-plastic [5,8], viscous [2,4], and quasi-brittle damaging [6]. The analytical background is exposed from the point of view of numerical computer simulation. The considerations address quasistatic deformation processes of solids represented by finite element systems. Thermal effects are assumed negligible, and thermodynamic arguments [3] are otherwise omitted.

Uniqueness of solution and global stability as discussed for the discretized representation of the deforming solid base on the momentary tangent response operator of the system. The transition to local conditions for bifurcation and instability that focuses on the constitutive level intrinsic to the material is not performed [1]. During the course of the deformation process, the incremental nonlinearity arising from the essential difference between momentary inelastic loading and elastic unloading inherent to elastic-plastic as well as brittle-damaging material behaviour, requires careful interpretation towards an assessment of states marked as critical because of a singular tangent operator associated with continuing loading. Depending on the direction in deformation space of the appertaining eigenvector the state can be unstable, or it allows for a multitude of solutions, resp. it is not critical in conjunction with the applied loading sequence. Algorithms that proceed directly along the stable deformation path are beneficial for smooth numerical simulation, but they may hide the critical states, which are of interest. In order to improve process design, the reasons for deviations from the expected path must be understood by exploring the critical states. The same applies if the evolution of a certain deformation mode is to be promoted by appropriately designing the part.

If the material is viscous, stability is studied by the amplification of deviations from the reference deformation path in the passage of time. The evolution of disturbances depends on both the momentary viscosity matrix of the system and the geometrical stiffness. The former describes the sensitivity of the stress resultants with respect to variations in velocity, the latter that with respect to variations in geometry. In addition, a dependence of the applied loads on the deforming geometry of the solid is of relevance. The quantities entering an investigation of stability are largely as for the inviscid solid, but the essential difference of the two material models implies distinct concepts. The interference between physical and numerical instability in the computer simulation of the deformation process is outlined. It is seen that definiteness of the velocity solution is prerequisite for stability against deviations from the reference path, and stability of the physics is a necessary condition for stability in numerics.

1

D. Bigoni, "Bifurcation and instability of non-associative elastoplastic solids", in H. Petryk (ed.), Material instabilities in elastic and plastic solids, CISM courses and lectures-No 414, Springer, Wien, New York, 2000.

2

M.A. Biot, "Mechanics of incremental deformations", Wiley, New York, 1965.

3

P. Glansdorff, I. Prigogine, "Thermodynamic theory of structure, stability and fluctuations", Wiley-Interscience, London, 1971, 3rd repr. 1977.

4

E.W. Hart, "Theory of the tensile test", Acta Met. 15, 351-355, 1967.

5

R. Hill, "Some basic principles in the mechanics of solids without a natural time", J. Mech. Phys. Solids, 7, 209-225, 1959. doi:10.1016/0022-5096(59)90007-9

6

D. Krajcinovic, "Damage mechanics", North-Holland, Elsevier, Amsterdam, 1996.

7

H. Petryk, "Theory of material instability in incrementally nonlinear plasticity", in H. Petryk (ed.), Material instabilities in elastic and plastic solids, CISM Courses and Lectures-No 414, Springer, Wien, New York, 2000.

8

J.R. Rice, "The localization of plastic deformation", in W.T. Koiter (ed.), Theoretical and applied mechanics, North-Holland, Amsterdam, 1976.

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