With reference to practical engineering problems it is shown that considerable differences may be encountered between the results from associated and those from nonassociated plasticity theories. Next, the need for a non-associated plasticity theory is demonstrated by considering test results for sand, concrete and rock. Elementary material parameters are discussed such as Young's modulus and Poisson's ratio for the description of the elastic properties; and a cohesion and a friction angle for the determination of the strength. The salient difference from associated plasticity theory concerns the introduction of a dilatancy angle which controls the inelastic (plastic) volume changes. This dilatancy angle is not only a suitable parameter for the description of soils, but also appears to be useful for concrete and rock. Basically, the paper consists of three parts as we consider three types of models of increasing complexity. The first model is a perfectly-plastic model, which employs the five aforementioned parameters. It is based on test data rather than on Drucker's hypothesis of material stability. The consequences thereof are examined. The second model is a straightforward extension of the first model by augmenting it with friction hardening and cohesion softening. This novel idea is introduced to account for the degradation of the cohesion of cemented granular materials with increasing inelastic deformation. The model is employed in an analysis which shows that plastic deformations tend to localize in thin shear bands, which may occur even before peak strength is reached. Finally, a review is given of concepts for modelling hysteresis and strain accumulation in cyclic loading. The concept of a bounding surface in addition to a yield surface is discussed and is adapted for use in a sophisticated model for loose and cemented granular materials under cyclic loading.

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Non-Associated Plasticity for Soils, Concrete and Rock

This paper describes a new formulation, based on linear finite elements and non-linear programming, for computing rigorous lower bounds in 1, 2 and 3 dimensions. The resulting optimization problem is typically very large and highly sparse and is solved using a fast quasi-Newton method whose iteration count is largely independent of the mesh refinement. For two-dimensional applications, the new formulation is shown to be vastly superior to an equivalent formulation that is based on a linearized yield surface and linear programming. Although it has been developed primarily for geotechnical applications, the method can be used for a wide range of plasticity problems including those with inhomogeneous materials, complex loading, and complicated geometry. Copyright © 2002 John Wiley & Sons, Ltd.

Lower bound limit analysis using non-linear programming

A theoretically based design method for the thickness of the base course of unpaved roads is developed in this paper, which considers distribution of stress, strength of base course material, interlock between geosynthetic and base course material, and geosynthetic stiffness in addition to the conditions considered in earlier methods: traffic volume, wheel loads, tire pressure, subgrade strength, rut depth, and influence of the presence of a reinforcing geosynthetic (geotextile or geogrid) on the failure mode of the unpaved road or area. In this method, the required base course thickness for a reinforced unpaved road is calculated using a unique equation, whereas more than one equation was needed with earlier methods. This design method was developed for geogrid-reinforced unpaved roads. However, it can be used for geotextile-reinforced unpaved roads and for unreinforced roads with appropriate values of relevant parameters. The calibration of this design method using data from field wheel load tests and laboratory cyclic plate loading tests on unreinforced and reinforced base courses is presented in the companion paper by the authors.

Design Method for Geogrid-Reinforced Unpaved Roads. I. Development of Design Method

This paper presents the results of three-dimensional finite-element analyses of circular foundations on the surface of homogeneous, purely cohesive soil. The foundations were assumed to adhere fully to the soil, and compressive, tensile and shear stresses may develop at the interface between the footing and the soil. The predicted ultimate response of the foundations to combined vertical, moment and horizontal loading was compared with other available theoretical predictions. A three-dimensional failure locus is presented for these foundations, based on the numerical predictions. An equation that approximates the shape of the failure locus is also suggested, and this provides a convenient means of calculating the bearing capacity of circular foundations on a uniform clay and subjected to combined loading.

Numerical studies of the bearing capacity of shallow foundations on cohesive soil subjected to combined loading

A theoretical study of the dynamic plastic behavior of beams and plates with finite-deflections

A theoretical investigation is given of quasi-static axially symmetric plastic deformations in soils.The mechanical behaviour of a natural soil is approximated by that of an ideal soil which obeys Coulomb’s yield criterion and associated flow rule, with restriction to rigid, perfectly plastic deformations. There are considerable variations in the structure of the associated stress and velocity field equations for the various plastic regimes, but it is noteworthy that real families of characteristics occur in all non-trivial cases. Attention is focused on those plastic regimes agreeing with the heuristic hypothesis of Haar & von Karman as being seemingly of application to certain classes of problems, in particular to those of indentation. The stress and velocity fields are then hyperbolic with identical families of characteristics, and the stress field is statically determinate under appropriate boundary conditions. In applications of the theoretical analysis, attention is confined to situations involving only the Haar & von Karman plastic regimes. First, possible velocity fields are obtained for the incipient plastic flow of a right circular cylindrical sample of soil subjected to uni-axial compressive stress parallel to its axis. Secondly, a complete solution is obtained for the incipient plastic flow in a semi-infinite region of soil, bounded by a plane surface, due to load applied through a flat-ended,smooth, rigid, circular cylinder; numerical results obtained for this problem include the variation of yield-point load with angle of internal friction of the ideal soil. These applications relate to problems of the mechanical testing of soil samples and of load-bearing capacity in foundation engineering.

Axially Symmetric Plastic Deformations in Soils

I n this paper an analysis is given within the framework of thin plate theory of the problem of a simply-supported, square plate subjected to a uniformly-distributed rectangular pressure pulse. All effects due to elastic strain, work-hardening and strain-rate are neglected, although some approximate account of the two latter may be made. To simplify the analysis further, Johansen's yield criterion is adopted as an approximation to Tresca's. The most important results concern the maximum displacement and the total time of motion. Errors in these quantities from approximating Tresca's yield criterion are estimated to be about five per cent.

Dynamic plastic deformations of simply-supported square plates

This paper treats the propagation of stress waves through an elastic-plastic medium on the assumption of uni-axial displacement. With the further simplification to a piecewise linear stress-strain curve in terms of engineering stress and strain, wave equations are obtained for the longitudinal stress in both elastic and plastic regions, each with a distinct constant Lagrangian wave speed. The stress distribution in any region is then simply expressed in terms of two wave functions. In a general motion the medium will be divided into a sequence of alternating elastic and plastic regions separated by moving interfaces. A detailed analysis is presented for a single-interface wave interaction under general initial conditions, namely, continuous initial waves in the two directions in both elastic and plastic regions with a non-uniform yield stress in the elastic region. For different sets of initial conditions six distinct types of solution are shown to exist, and these are classified according to the direction and speed of the interface. In particular, two types involve interface speeds in excess of the elastic wave speed, not, to the authors' knowledge, demonstrated in previous plastic wave treatments, noting the absence of possible shock formation for the present linearized stress-strain laws. Further, it is shown that stress discontinuities cannot form at the interface (or elsewhere) from initially continuous stress profiles. Associated with the different types of solution are four distinct sets of interface conditions so that there is no common form for the interaction solution. Each of the six types of solution is shown to be consistent with the elastic-plastic model only under a restricted set of initial conditions, and these sets are found to be mutually exclusive for the six types, thus deciding a unique choice for the type of single-interface solution. The six sets, however, are not inclusive of all possible initial conditions, indicating a need for multi-interface solutions in the exceptional situations. Multi-interface solutions may be possible even in the non-exceptional situations, but this possibility is felt to be unlikely. Finally, it can be noted that the analysis dealing with validity of solution is, for most cases, only local in that it applies in some small neighbourhood of the current point on the interface path, being based on expansions about this point. The results of such local analysis will therefore extend to the case of non-uniform wave speeds arising from non-linear stress-strain laws, provided that no shock is formed in the neighbourhood, but a global solution can no longer be expressed simply in terms of wave functions.

A. D. Cox

Papers

Axially Symmetric Plastic Deformations in Soils

Dynamic plastic deformations of simply-supported square plates

Existence and uniqueness of solutions to uni-axial elastic-plastic wave interactions