Extension of limit analysis theorems to incompressible material with a non-associated flow rule

The question of extending limit analysis to materials with a non-associated flow rule is an old but important one. Many frictional soils, such as sands and drained clays, do not display the large volume change upon shearing that is predicted by an associated flow rule, thus calling into question the collapse loads that are found from the classical limit theorems. This question is important in practice, as limit analysis is often used to predict the ultimate (or limit) loads for a wide range of geotechnical structures such as slopes, tunnels, foundations and retaining walls. In 1953, Drucker proved that the limit load for any plastic material with a non-associated flow rule is less than that for the corresponding material with an associated flow rule. This result is, in fact, a minimisation property and not a bounding theorem in the limit analysis sense. For many soils, this deviation from normality is not arbitrary. Indeed, experimental evidence for some frictional soils suggests it occurs mainly in the volumetric behaviour, whereas the deviatoric behaviour is much closer to normality (Baker and Desai, 1982). This implies that if the mean stress is known, the deviatoric part of the plastic strain rate can be deduced from the normality rule. The key idea is to exploit this property. Because Hill's inequality is no longer true for a non-associated flow rule, we cannot use the limit theorems in their usual form for these materials. Furthermore, the pseudo-potential of dissipation depends on both the plastic strain rate and the stress (mean stress) and the duality property between potentials is lost. However, a weaker duality can be recovered if we consider the normality in the deviatoric plane. As a result of this approach, the static load multiplier will depend on the mean stress as well as the kinematic load multiplier, both multipliers are linked. It is well known that the collapse load for a non-associated material will be smaller than that for an associated material