84

inelastic and non-linear materials

The non-associative plasticity, in essence caused by frictional behaviour, may lead to non-uniqueness of solution. The equivalent viscoplastic form is, however, always unique and hence viscoplasticity is on occasion used as a regularizing procedure.

3.10 Some special problems of brittle materials 3.10.1 The no-tension material A hypothetical material capable of sustaining only compressive stresses and straining without resistance in tension is in many respects similar to an ideal plastic material. While in practice such an ideal material does not exist, it gives a reasonable approximation of the behaviour of randomly jointed rock and other granular materials. While an explicit stress-strain relation cannot be generally written, it suffices to carry out the analysis elastically and wherever tensile stresses develop to reduce these to zero. The initial stress (modified Newton-Raphson) process here is natural and indeed was developed in this context.'02 The steps of calculation are obvious but it is important to remember that the principal tensile stresses have to be eliminated. The 'constitutive' law as stated above can at best approximate to the true situation, no account being taken of the closure of fissures on reapplication of compressive stresses. However, these results certainly give a clear insight into the behaviour of real rock structures.

An underground power station Figure 3.19(a) and (b) shows an application of this model to a practical problem.'02 In Fig. 3.19(a) an elastic solution is shown for stresses in the vicinity of an underground power station with 'rock bolt' pre-stressing applied in the vicinity of the opening. The zones in which tension exists are indicated. In Fig. 3.19(b) a no-tension solution is given for the same problem, indicating the rather small general redistribution and the zones where 'cracking' has occurred.

Reinforced concrete A variant on this type of material may be one in which a finite tensile strength exists but when this is once exceeded the strength drops to zero (on fissuring). Such an analysis was used by Valliappan and Nath'03 in the study of the behaviour of reinforced concrete beams. Good correlation with experimental results for underreinforced beams (in which development of compressive yield is not important) have been obtained. The beam is one for which test results were obtained by Krahl et al.'04 Figure 3.20 shows some relevant results. Much development work on the behaviour of reinforced concrete has taken place, with various plasticity forms being introduced to allow for compressive failure and procedures that take into account the crack-closing history. References 105 and 106 list some of the basic approaches on this subject. The subject of analysis of reinforced concrete has proved to be of great importance in recent years and publications in this field are proliferating. Publications 107 to 110 guide the reader to current practice in this field.

Some special problems of brittle materials 85

Fig. 3.19 Underground power station: gravity and prestressing loads. (a) Elastic stresses; (b) 'no-tension' stresses.

86

Inelastic and non-linear materials

Fig. 3.20 Cracking of a reinforced concrete beam (maximum tensile strength 200 Ib/in2). Distribution of stresses a t various sections.'03(a) Mesh used; (b) section AA; (c) section BB; (d) section CC.

3.10.2 'Laminar' material and joint elements Another idealized material model is one that is assumed to be built up of a large number of elastic and inelastic laminae. When under compression, these can transmit shear stress parallel to their direction - providing this does not exceed the frictional resistance. No tensile stresses can, however, be transmitted in the normal direction to the laminae. This idealized material has obvious uses in the study of rock masses with parallel joints but has much wider applicability. Figure 3.21 shows a two-dimensional situation involving such a material. With a local coordinate axis x' oriented in the direction of the laminae we can write for a simple Coulomb friction joint I

~

~

j

1