A theoretical framework of rigid plasticity is presented that is based on optimization and includes frictional dissipation. It has been used in this paper as a foundation for existing and modified models of granular materials consisting of rigid granules. It has the major advantage that it enables yield criteria to be created numerically, which is particularly useful when analytical expressions cannot be found. This framework is constructed by first postulating: (i) a dissipation function that can depend on the current components of stress, but is always homogeneous of degree one and positive definite, (ii) a volume constraint function that is also homogeneous of degree one, and (iii) a balance of the rate of doing work and the rate at which energy needs to be dissipated. A mathematical process similar to the construction of a dual norm in convex analysis then leads to: a flow rule; a single natural representation of the yield surface; and a useful constitutive inequality involving the components of stress and strain rate. Dissipation functions appropriate for the Drucker–Prager and Matsuoka–Nakai yield surfaces are investigated when a simple dilation rule is the volume constraint. These include a case where an explicit expression for the yield function is not found and, instead, the yield function is found numerically. Such numerical yield functions have been checked graphically against carefully constructed envelopes and found to be consistent with them.

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A 2D model of a bonded granular material is presented and its properties confirmed to be that of a brittle, isotropic elastic solid. The bond stiffnesses (axial tension/compression, shear and bending) are taken from the classical solutions to the external crack problem with two half-spaces bonded by a disc of intact material. An assembly of granules is simulated using a random array of points (representing the granule locations) with a prescribed minimum separation. The bonds are then generated by a Delaunay triangulation. This produces an isotropic array of bonds giving rise to a model material with isotropic properties. Crack growth is simulated by sequentially removing the most highly stressed bond in turn. Crack paths are then produced which are shown to agree with the predictions of linear elastic fracture mechanics, in respect of both the direction of propagation and the influence of specimen size. Some well-known problems are then simulated including: the interaction of two parallel cracks; diametrical compression of a disc; the four point bending of a beam; the influence of mortar strength on the behaviour of masonry; and a flat arch.

Flash calculations are widely used and constitute an integral part of modelling vapour-liquid equilibria in compositional simulators. However, it has been discovered that during compositional simulations, flash calculations take 50-70% of the overall computational time because the procedure currently used is iterative. Hence, several methods such as the reduced variable method, compositional space adaptive tabulation (CSAT) and the tie-line table look-up (TTL) have been developed to improve on the computational speed of most flash calculations during compositional simulations. Unfortunately, most of these methods are still iterative, and pose convergence problems, even though some are developed with efficient Newton-Raphson algorithms. Non-iterative techniques may be the best option to speed up the computational time during simulations. This paper presents a non-iterative procedure for the determination of fluid phase diagrams using the convex hull method and Peng-Robinson equation of state. Convex hull is a mathematical method, and algorithmic implementations of this method are available in many software packages, of which Matlab was used in this work. Unlike the conventional flash calculation method, programs developed with convex hull does not the need an accurate start value to make fluid phase diagrams and determine phase properties for binary and ternary mixtures. The time taken to complete a simulation run using convex hull and the conventional flash calculation method were noted, and the numerical results from both methods was validated against a range of experimental data for different mixtures. The results show good agreement in all the cases investigated. From the analyses, it was shown that the convex hull method is faster than the conventional flash calculation method in achieving convergence and also gave better predictions close to the critical point. The reliability of the results and the additional time benefits are indication that the convex hull method has a promising prospect of becoming an efficient procedure for modelling vapour-liquid phase equilibria calculations for compositional simulations.