M. Tech. (Mechanical Engineering, Faculty of Engineering and Technology): Vaal University of Technology / Presently there exist a lot of controversies about the mechanical properties of nanomaterials. Several convincing reasons and justifications have been put forward for the controversies. Some of the reasons are varying processing routes, varying ways of defining equations, varying grain sizes, varying internal constituent structures, varying techniques of imposing strain on the specimen etc. It is therefore necessary for scientists, engineers and technologists to come up with a clearer way of defining and dealing with nanomaterials’ mechanical properties. The parameters of the internal constituent structures of nanomaterials are random in nature with random spatial patterns. So they can best be studied using random processes, specifically as stochastic processes. In this dissertation the tools of stochastic processes have been used as they offer a better approach to understand and analyse random processes. This research adopts the approach of ascertaining the correct mathematical models to be used for experimentation and modelling. After a thorough literature survey it was observed that size and temperature are two important parameters that must be considered in selecting the relevant mathematical definitions for nanomaterials’ mechanical properties. Temperature has a vital role to play during grain refinement since all severe plastic deformation involves thermomechanical processes. The second task performed in this research is to develop the mathematical formulations based on the experimental observation of 2-D grains and 3-D grains deformed by Accumulative Roll-Bonding and Equal Channel Angular Pressing. The experimental observations revealed that grains deformed by Accumulative Roll-Bonding and Equal Channel Angular Pressing are elongated when observed from the rolling direction, and transverse direction, and equiaxed when observed from the normal direction. In this dissertation, the different experimental observations for the grain size variants during grain refinement were established for 2-D and 3-D grains. This led to the development of a stochastic model of grain-elongation for 2-D and 3-D grains. The third task was experimentations and validation of proposed models. Accumulative Roll-Bonding, Equal Channel Angular Pressing and mechanical testing (tensile test) experiments were performed. The effect of size on elongation and material properties were studied to validate the developed models since size has a major effect on material’s properties. The fourth task was obtaining results and discussion of theoretical developed models and experimental results. The following facts were experimentally observed and also revealed by the models. Different approaches of measuring grain size reveal different strains that cannot be directly obtained from plots of the corresponding grain sizes. Grain elongation evolved as small values for larger grains, but became larger for smaller grains. Material properties increased with elongation reaching a maximum and started decreasing as is evident in the Hall-Petch to the Reverse Hall-Petch Relationship. This was alluded to the fact that extreme plastic straining led to distorted structures where grain boundaries and curvatures were in “non-equilibrium” states. Overall, this dissertation contributed new knowledge to the body of knowledge of nanomaterials’ mechanical properties in a number of ways. The major contributions to the body of knowledge by his study can be summarized as follows: (1) The study has contributed in developing a model of elongation for 2-D grain and 3-D grains. It has been generally reported by researchers that materials deformed by Accumulative Roll-Bonding and Equal Channel Angular Pressing are generally elongated but none of these researchers have developed a model of elongation. Elongation revealed more information about “size” during grain refinement. (2) The Transmission Electron Microscopy revealed the grain shape in three directions. The rolling direction or sliding direction, the normal direction and the transverse direction. Most developed models ignored the different approaches of measuring nanomaterials’ mechanical properties. Most existing models dealt only with the equivalent radius measurement during grain refinement. In this dissertation, the different approaches of measuring nanomaterials’ mechanical properties have been considered in the developed models. From this dissertation an accurate correlation can be made from microscopy results and theoretical results. (3) This research has shown that most of the published results on nanomaterials’ mechanical properties may be correct although controversies exist when comparing the different results. This research has also shown that researchers might have considered different approaches to measure nanomaterials’ mechanical properties. The reason for different results is due to different approaches of measuring nanomaterials’ mechanical properties as revealed in this research. Since different approaches of measuring nanomaterials’ mechanical properties led to different obtained results, this justify that most published results of nanomaterials’ mechanical properties may be correct. This dissertation revealed more properties of nanomaterials that are ignored by the models that considered only the equivalent length. (4) This research has contributed to the understanding of nanomaterials controversies when comparing results from different researchers.