The application of techniques from nonlinear analysis to materials science has seen great developments in the recent years and it has really been a driving force for substantial mathematical research in the area of partial differential equations and the multi-dimensional calculus of variations. This thesis has been motivated by two recent and remarkable experimental observations of H. Seiner in shape-memory alloys which we attempt to interpret mathematically. Much of the work is original and has given rise to deep problems in the calculus of variations. Firstly, we study the formation of non-classical austenite-martensite interfaces. Ball & Carstensen (1997, 1999) theoretically investigated the possibility of the occurrence of such interfaces and studied the cubic-to-tetragonal case extensively. In this thesis, we present an analysis of non-classical austenite-martensite interfaces recently observed by Seiner et al.~in a single crystal of a CuAlNi shape-memory alloy, undergoing a cubic-to-orthorhombic transition. We show that these can be described by the general nonlinear elasticity model and we make some predictions regarding the admissible volume fractions of the martensitic variants involved, as well as the habit plane normals. Interestingly, in the above experimental observations, the interface between the austenite and the martensitic configuration is never exactly planar, but rather slightly curved, resulting from the pattern of martensite not being exactly homogeneous. However, it is not clear how one can reconstruct the inhomogeneous configuration as a stress-free microstructure and, instead, a theoretical approach is followed. In this approach, a general method is provided for the construction of a compatible curved austenite-martensite interface and, by exploiting the structure of quasiconvex hulls, the existence of curved interfaces is shown in two and three dimensions. As far as the author is aware of, this is the first construction of such a curved austenite-martensite interface. Secondly, we study the nucleation of austenite in a single crystal of a CuAlNi shape-memory alloy consisting of a single variant of stabilized 2H martensite. The nucleation process is induced by localized heating and it is observed that, regardless of where the localized heating is applied, the nucleation points are always located at one of the corners of the sample - a rectangular parallelepiped in the austenite. Using a simplified nonlinear elasticity model, we propose an explanation for the location of the nucleation points by showing that the martensite is a local minimizer of the energy with respect to localized variations in the interior, on faces and edges of the sample, but not at some corners, where a localized microstructure can lower the energy. The result for the interior, faces and edges is established by showing that the free-energy function satisfies a set of quasiconvexity conditions at the stabilized variant throughout the specimen, provided this is suitably cut. The proofs of quasiconvexity are based on a rigidity argument and are specific to the change of symmetry in the phase transformation. To the best of the author's knowledge, quasiconvexity conditions at edges and corners have not been considered before.

This master thesis aims to develop a test methodology for electromechanical linear actuators. A linear actuator acts as a linear motor, converting a power source to linear motion. The electromechanical linear actuator in this project has an electric motor as its power source and uses a rack and pinion system to transfer that power to linear motion.  The test methodology is to impose a force onto the rack of the actuator, to ensure that operation under a load scenario is within specification. To accomplish this, the design of a test rig implementation is analyzed. The test rig consists of the test unit, which is to be tested, the load unit, which is to provide the load force, and a control system for the load unit. The load unit is another linear actuator and is controlled via a load cell. The load cell gives out the load force being applied, and the controller gives out the corresponding appropriate motor command to the load unit to ensure the load force is as desired. This analysis is done through simulation of the setup. Viable options for the setup were first analyzed in order to implement the deemed promising options for a setup into a simulation environment. The simulation environment in this project was Simscape, an extension of MATLAB’s Simulink. In simulation the parameters for the test rig were rigorously analyzed, in order to determine acceptable thresholds. The primary load unit tested was another electromechanical linear actuator from Cascade Drives, the model A-100-8P. Two secondary setups, one using the same model as being tested, and another setup using two of the models being tested. Simulation found that the suggested options’ applied load force have a poor rise time, large overshoot and substantial oscillation errors. The primary source for this was determined to be the latency between load cell input, and motor command output in the controller. The poor metrics from the result could lead to problems when emergency braking, and with a long honing period, which would render most test data unusable.