Int. 1. Engng Sci. Vol. 29, No. 2, pp. 223-235, 1991 0020-7225/91 $3.00+ 0.00 Printed in Great Britain. All rights reserved Copyright @ 1991Pergamon Press plc ASYMMETRIC STEADY-STATE SOLUTIONS FOR A CRACK IN A VISCOELASTIC FIELD OF PURE BENDING J. M. GOLDEN Environmental Research Unit, St. Martin’s House, Waterloo Road, Dublin 4, Eire G. A. C. GRAHAM? Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Eire Abstract-The problem of a crack in a viscoelastic material subjected to an alternatively positive and negative asymmetric bending moment is considered. The asymmetry refers to the fact that the bending moment spends more time and has a greater maximum in one direction, say positive, than in the other. Steady-state solutions which close at only one end of the crack are studied. This is a specificalIy viscoelastic effect, which would not occur in the elastic limit. It would seem to suggest that biased crack propagation patterns are more likely in a viscoelastic material than in an elastic material subject to the same asymmetric fluctuating bending moment. Numerical results are presented for a standard linear model. 1. INTRODUCTION The transient problem of a crack in a viscoelastic medium subject to a sinusoidal bending moment has been considered over the early history of loading [l, 21. It was found that, in contrast to the elastic case, the crack can close gradually rather than suddenly. However, a frequency-dependent class of materials was isolated for which the crack closed suddenly at one end in two consecutive cycles, without closing at the other end. It was speculated [2] that a steady-state solution might exist, consisting entirely of asymmetric closures of this kind, for certain limited types of materials. The conjecture is that such behaviour may exist under a sinusoidal loading with equal positive and negative maxima. If such were the case it would be quite surprising. One of the purposes of the present paper is to check this conjecture. However, and more importantly, it is also possible to create one-sided partial closure in a wide class of materials, presumably any material, by applying a biased loading, in other words, one that is different in the positive and negative cycles. An example would be a loading that is sinusoidal but with a constant added. The broader aim of the paper is to study behaviour under such an asymmetric loading. Steady-state solutions closing at both ends would generally be of more interest than solutions exhibiting one-sided closure. However, such problems are not of the repetitive expansion and contraction kind, dependent on only one parameter, for which a simple method of solution is applicable [3]. A more difficult, general method, based on an integral equation derived in [3], is in fact applicable to such problems but will be considered elsewhere [4]. The method we develop here for one-sided closure is in general terms the same as that applied to obtain steady-state solutions of the normal contact problem [5]. In its finer details it bears considerable resemblance to the fixed length closing crack problem under normal loading [3,6] because, in common with that problem, we shall see that only instantaneous closure is possible in the one-sided case. 2. GENERAL RELATIONS Plane strain conditions are presumed to apply and we consider the particular cross-section of the medium lying in the xy plane. In this plane, let the crack which is assumed to be of fixed t On leave from: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada VSA lS6. 223