The critical state soil mechanics (CSSM) framework originally proposed by Schofield & Wroth (1968) has been shown to capture the mechanical behaviour of soils effectively. The particulate implementation of the discrete element method (DEM) can replicate many of the complex mechanical characteristics associated with sand. This research firstly shows that the CSSM framework is useful to assess whether a DEM simulation gives a response that is representative of a real soil. The research then explores the capacity of DEM to extend understanding of soil behaviour within the CSSM framework. The influence of sample size on the critical-state response observed in DEM simulations that use rigid-wall boundaries was examined. The observed sensitivity was shown to be caused by higher void ratios and lower contact densities adjacent to the boundaries. When the void ratio (e) and mean stress (p’) of the homogeneous interior regions were considered, the influence of sample size on the position of the critical state line (CSL) in e-log(p’) space diminished. A parametric study on the influence of the interparticle friction (μ) on the load-deformation response was carried out. The macro-scale stress-deformation characteristics were nonlinearly related to μ and the particle-scale measures (fabric, contact force distribution, etc.) varied systematically with μ. The limited effect of increases in μ on the overall strength at high μ values (μ>0.5) is attributable to transition from sliding-dominant to rolling-dominant contact behaviour. A μ value higher than 0.5 leads to a CSL in e-log(p’) space that does not capture real soil response. True-triaxial simulations with different intermediate stress ratios (b) were performed. The dependency of strength on b agreed with empirical failure criteria for sands and was related to a change of buckling modes of the strong force chains as b increased. DEM simulations showed that the position of the CSL in e-log(p’) space depends on the intermediate stress ratio b. This sensitivity seems to be related to the dependency of the directional fabric anisotropy on b. The link between the state parameter and both soil strength and dilatancy proposed by Jefferies & Been (2006) was reproduced in DEM simulations. A new rotational resistance model was proposed and it was shown that the new model can qualitatively capture the influence of particle shape on the mechanical behaviour of sand. However, it was shown that the effect of rotational resistance is limited and to quantitatively compare the DEM simulation results with laboratory testing data, e.g., the critical-state loci, it is necessary to use non-spherical particles. / published_or_final_version / Civil Engineering / Doctoral / Doctor of Philosophy

Over the past decades, researchers in economics, financial mathematics and actuarial science have introduced results to the concept of comonotonicity in their respective fields of interest. Comonotonicity is a very strong dependence structure and is very often mistaken as a dependence structure that is too extreme and unrealistic. However, the concept of comonotonicity is actually a useful tool for solving several research and practical problems in capital allocation, risk sharing and optimal allocation. The first topic of this thesis is focused on the application of comonotonicity in optimal capital allocation. The Enterprise Risk Management process of a financial institution usually contains a procedure to allocate the total risk capital of the company into its different business units. Dhaene et al. (2012) proposed a unifying capital allocation framework by considering some general deviation measures. This general framework is extended to a more general optimization problem of minimizing separable convex function with a linear constraint and box constraints. A new approach of solving this constrained minimization problem explicitly by the concept of comonotonicity is developed. Instead of the traditional Kuhn-Tucker theory, a method of expressing each convex function as the expected stop-loss of some suitable random variable is used to solve the optimization problem. Then, some results in convex analysis with infimum-convolution are derived using the result of this new approach. Next, Borch's theorem is revisited from the perspective of comonotonicity. The optimal solution to the Pareto optimal risk-sharing problem can be obtained by the Lagrangian method or variational arguments. Here, I propose a new method, which is based on a Breeden-Litzanbeger type integral representation formula for increasing convex functions. It enables the transform of the objective function into a sum of mixtures of stop-losses. Necessary conditions for the existence of optimal solution are then discussed. The explicit solution obtained allows us to show that the risk-sharing problem is indeed a “point-wise” problem, and hence the value function can be obtained immediately using the notion of supremum-convolution in convex analysis. In addition to the above classical risk-sharing and capital allocation problems, the problem of minimizing a separable convex objective subject to an ordering restriction is then studied. Best et al. (2000) proposed a pool adjacent violators algorithm to compute the optimal solution. Instead, we show that using the concept of comonotonicity and the technique of dynamic programming the solution can be derived in a recursive manner. By identifying the right-hand derivative of the convex functions with distribution functions of some suitable random variables, we rewrite the objective function into a sum of expected deviations. This transformation and the fact that the expected deviation is a convex function enable us to solve the minimizing problem. / published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

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published_or_final_version / abstract / toc / Business / Master / Master of Philosophy