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.

Recent advances in materials engineering have given rise to a new class of materials known as active materials. These materials when used appropriately can aid in development of smart structural systems. Smart structural systems are adaptive in nature and can be utilized in applications that are subject to time varying loads such as aircraft wings, structures exposed to earthquakes, electrical interconnections, biomedical applications, and many more. Materials such as piezoelectric crystals, electrorheological fluids, and shape memory alloys (SMAs) constitute some of the active materials that have the innate ability to response to a load by either changing phase (e.g., liquid to solid), and recovering deformation. Active materials when combined with conventional materials (passive materials) such as polymers, stainless steel, and aluminum, can result in the development of smart structural systems (SSS). This Dissertation focuses on characterization of SMAs and structures that incorporate SMAs. This characterization is based on a hybrid analytical, computational, and experimental solutions (ACES) methodology. SMAs have a unique ability to recover extensive amounts of deformation (up to 8% strain). NiTiNOL (NOL: Naval Ordinance Lab) is the most commonly used commercially available SMA and is used in this Dissertation. NiTiNOL undergoes a solid-solid phase transformation from a low temperature phase (Martensite) to a high temperature phase (Austenite). This phase transformation is complete at a critical temperature known as the transformation temperature (TT). The low temperature phase is softer than the high temperature phase (Martensite is four times softer than Austenite). In this Dissertation, use of NiTiNOL in representative engineering applications is investigated. Today, the NiTiNOL is either in ribbon form (rectangular in cross-section) or thin sheets. In this Dissertation, NiTiNOL is embedded in parent materials, and the effect of incorporating the SMA on the dynamic behavior of the composite are studied. In addition, dynamics of thin sheet SMA is also investigated. The characterization is conducted using state-of-the- art (SOTA) ACES methodology. The ACES methodology facilitates obtaining an optimal solution that may otherwise be difficult, or even impossible, to obtain using only either an analytical, or a computational, or an experimental solution alone. For analytical solutions energy based methods are used. For computational solutions finite element method (FEM) are used. For experimental solutions time-average optoelectronic holography (OEH) and stroboscopic interferometry (SI) are used. The major contributions of this Dissertation are: 1. Temperature dependent material properties (e.g., modulus of elasticity) of NiTiNOL based on OEH measurements. 2. Thermomechanical response of representative composite materials that incorporate NiTiNOL“fibers". The Dissertation focuses on thermomechanical characterization of NiTiNOL and representative structures based on NiTiNOL; this type of an evaluation is essential in gainfully employing these materials in engineering designs.

NiTi alloys have been the focus of Shape Memory Alloys (SMA) research and applications due their excellent ductility and shape memory properties, and these alloys have been extensively used in automotive, aerospace, and in biomedical applications. The effects of machining on the surface integrity and the corresponding material and mechanical properties of alloys can be best studied by utilizing NiTi alloys as workpiece material since their physical and mechanical properties are highly microstructure dependent. However, due to very poor machining performance of NiTi shape memory alloys, no comprehensive or systematic investigation on this topic has been conducted by researchers as yet. The current study makes a substantial and unique contribution to this area by making the first and significant contribution to studies on machining performance of NiTi shape memory alloys, and by achieving improved surface integrity and machining performance using cryogenic applications, which give significant reductions of tool-wear, cutting forces, and surface roughness. The influence of machining process conditions, including dry, MQL, preheated, cryogenic machining, and the effects of prefroze cryo machining on surface integrity characteristics such as microhardness, phase transformation, phase transformation temperature, depth of plastically deformed layer have been examined extensively, and unique findings have been obtained. The effects of machining process conditions, in particular preheated and cryogenic machining conditions, on thermo-mechanical and shape memory characteristics were identified through thermal cycling and stress-strain tests. For the first time, orthogonal cutting of NiTi shape memory alloys has been carried out in this study to investigate surface integrity comprehensively. Surface integrity and machining performance are compared for dry and prefroze cryogenic cooling conditions under a wide range of cutting speeds. Stress-induced martensitic phase transformation and deformation twinning were found in prefroze cryogenic and dry cutting conditions respectively. The existing microstructure-based constitutive models were used and modified to predict machining-induced phase transformation and resulting volume fraction. The modified model was implemented in commercial FEM software (DEFORM-2D) as a customized user subroutine. The obtained results from simulation and orthogonal cutting tests were compared considering martensitic volume fraction during cutting with various cutting speeds. The model captured the experimental trend of volume fraction induced by various cutting speeds and process variables. Overall, FEM simulation of cutting process of NiTi was successfully presented.

Ni-Co-Mn-In system is a new type of magnetic shape memory alloys (MSMAs) where the first order structural and magnetic phase transitions overlap. These materials can generate large reversible shape changes due to magnetic-field-induced martensitic transformation, and exhibit magneto-caloric effect and magnetoresistance. Ni-Co-Mn-Sn alloys are inexpensive alternatives of the Ni-Co-Mn-In alloys. In both materials, austenite has higher magnetization levels than martensite. Fe-Mn-Ga is another MSMA system, however, whose magnetization trend is opposite to those of the Ni-Co-Mn-X (In-Sn) systems upon phase transformation. The MSMAs have attracted great interest in recent years, and their magnetic and thermo-mechanical properties need to be further investigated. In the present study, the effects of indium concentration, cooling, and annealing on martensitic transformation and magnetic response of single crystalline Ni-Co-Mn-In alloys were investigated. Increasing indium content reduced the martensitic transformation start (Ms) temperature, while increasing temperature hysteresis and saturation magnetization. Increasing annealing temperature led to an increase in the Ms temperature whereas annealing at 400 degrees C and 500 degrees C led to the kinetic arrest of austenite. Cooling after solution heat treatment also notably affected the transformation temperatures and magnetization response. While the transformation temperatures increased in the oil quenched samples compared to those in the water quenched samples, these temperatures decreased in furnace cooled samples due to the kinetic arrest. The possible reasons for the kinetic arrest are: atomic order changes, or precipitate formation. Shape memory and superelastic response, and magnetic field-induced shape recovery behavior of sintered Ni43Co7Mn39Sn11 polycrystalline alloys were also examined. The microstructural analysis showed the existence of small pores, which seem to increase the damage tolerance of the sintered polycrystalline samples. The recoverable transformation strain, irrecoverable strain and transformation temperature hysteresis increased with stress upon cooling under stress. Moreover, magnetic-field-induced strain due to the field-induced phase transformation was confirmed to be 0.6% at 319K. Almost perfect superelastic response was obtained at 343K. A magnetic entropy change of 22 J kg-1 K-1 were determined at 219K from magneto-caloric effect measurements which were conducted on annealed Ni43Co7Mn39Sn11 ribbons. Magnetic characteristics and martensitic transformation behavior of polycrystalline Fe-Mn-Ga alloys were also examined. Cast alloys at various compositions were undergone homogenization heat treatments. It was verified by magnetization measurements that the alloys heat treated at 1050 degrees C shows martensitic transformation. The heat treatment time was determined to be 1 day or 1 week depending on the compositions.

Magnetic shape memory alloys (MSMAs) are a class of active materials that de- form under magnetic and mechanical loading conditions. This work is concerned with the modeling of MSMAs constitutive responses. The hysteretic magneto-mechanical responses of such materials are governed by two major mechanisms which are variant reorientation and field induced phase transformation (FIPT). The most widely used material for variant reorientation is Ni2 MnGa which can produce up to 6% magnetic field induced strain (MFIS) under 5 MPa actuation stress. The major drawback of this material is a low blocking stress, which is overcome in the NiMnCoIn material system through FIPT. This magnetic alloy can exhibit 5% MFIS under 125 MPa actuation stress. The focus of this work is to capture the key magneto-thermo-mechanical responses of such mechanisms through phenomenological modeling. In this work a detailed thermodynamic framework for the electromagnetic interaction within a continuum solid is presented. A Gibbs free energy function is postulated after identifying the external and internal state variables. Material symmetry restrictions are imposed on the Gibbs free energy and on the evolution equations of the internal state variables. Discrete symmetry is considered for single crystals whereas continuous symmetry is considered for polycrystalline materials. The constitutive equations are derived in a thermodynamically consistent way. A specific form of Gibbs free energy for FIPT is proposed and the explicit form of the constitutive equations is derived from the generalized formulation. The model is calibrated from experimental data and different predictions of magneto-thermo-mechanical loading conditions are presented. The generalized constitutive equations are then reduced to capture variant reorientation. A coupled magneto-mechanical boundary value problem (BVP) is solved that accounts for variant reorientation to investigate the influence of the demagnetization effect on the magnetic field and the effect of Maxwell stress on the Cauchy stress. The BVP, which mimics a real experiment, provides a methodology to correlate the difference between the externally measured magnetic data and internal magnetic field of the specimen due to the demagnetization effect. The numerical results show that localization zones appear inside the material between a certain ranges of applied magnetic field. Stability analysis is performed for variant reorientation to analyze these numerical observations. Detailed numerical and analytical analysis is presented to investigate these localization zones. Magnetostatic stability analysis reveals that the MSMA material system becomes unstable when localizations appear due to non-linear magnetization response. Coupled magneto-mechanical stability analysis shows that magnetically induced localization creates stress-localizations in the unstable zones. A parametric study is performed to show the constraints on material parameters for stable and unstable material responses.

No description available.

The use of high-precision automated equipment is steadily increasing due in part to the progressively smaller sizes of electronic circuits. Currently, piezoelectric transducers (piezos) dominate as the actuation device for high precision machines, but shape memory alloys (SMA) may be a viable alternative to reduce monetary costs. This work explores the implementation of a low-cost linear macro-micro positioning system. The system consists of a modified printer carriage to provide long range, macro scale linear motion (approximately 200 mm range and 200 µm precision) and a micro scale system (approximately 4 mm range and 5 µm target precision) that uses an SMA actuator. A detailed description of the design and implementation of the system is given in this research. A model of the macro-stage is then generated by first identifying and inverting a simple friction model to linearize the system, thereby allowing for modified least squares (MLS) identification of a linear model. Various controllers are attempted for the macro-stage and compared with an experimentally tuned nonlinear PD controller that is implemented in the final design. A model of the micro-stage is derived through analysis of the SMA actuator. The model for the actuator is separated into two portions, an electro-thermal model, and a hysteresis model. The hysteresis model is derived using the Preisach model, and the electro-thermal model through MLS identification. To control the micro-stage, a PI controller with antiwindup is developed experimentally. The two stages are then executed together in closed loop and the resulting coupling between the two stages is briefly examined. Experimental data used for the modelling and design is presented, along with results of the final macro-micro linear positioning system.

Shape memory alloys (SMAs) are metallic materials, which have two main stable crystalline phases: austenite, a high temperature phase and martensite, a low temperature phase. Austenite and martensite each have unique physical and mechanical properties, and transformation between these phases enables two effects known as the shape memory effect (SME) and superelasticity. When a material that displays the SME is plastically deformed at low temperature, a heat input will cause the SMA to return to its original shape before the deformation. At higher temperatures, the material displays an effect called superelasticity, where strains of up to 10% are recoverable. These characteristics of SMA allow for significant amounts of strain recovery, and enable the design of SMA actuators. The temperature in an SMA actuator is generally controlled by resistive heating, also know as joule heating, and the strain recovery capabilities are used to do work on a load, thereby creating an electro-mechanical actuator. SMA actuators have attractive properties such as high energy density, smooth and silent actuation, reduced part counts compared to traditional alternatives, and scalability down to the micromechanical level. The phase transformation in SMA actuators, however, is highly non-linear. Therefore, the use of SMA as actuators, for example in positioning systems, benefits from the development of good models to predict and control the materials. The goals of this work are to develop a model suitable for real-time implementation, and that reproduces the observed behaviour of SMA actuators. The model is then inverted and used to develop a model-based controller, used in conjunction with traditional PID control to improve the precision and robustness of SMA actuators. The modelling portion of this work consists of the development of a phenomenological SMA model. The forward model is split into three blocks: a heating block, a phase kinetics block and a mechanical block. Since joule heating is commonly used in SMA actuators to bring about an increase in temperature, the heating block presents equations to convert a current input into the temperature of the wire. The phase kinetics block equations convert the calculated temperature and applied stress to the fraction of martensite present in the SMA. Finally, the mechanical model calculates the strain in the material from the martensite fraction and the applied stress. Once the model equations are presented, experimental verification tests are shown to compare physical SMA behaviour with that predicted by the model. Each of the blocks of the forward model are then inverted in order to be used as a feedforward linearizing controller. The control section of this thesis deals with the response of two common types of SMA actuators: a constant force SMA actuator and a spring-biased SMA actuator. The response of the system to step and sinusoidal signals with period of 5 seconds is investigated using two types of controllers: a traditional PI controller and the inverse-model controller in feedforward with a PI controller in feedback. Additionally, the robustness of the system is investigated through the response of the system to transient and sinusoidal stress disturbances. The disturbance rejection is investigated on a constant force actuator both with and without the presence of a force sensor.

The use of high-precision automated equipment is steadily increasing due in part to the progressively smaller sizes of electronic circuits. Currently, piezoelectric transducers (piezos) dominate as the actuation device for high precision machines, but shape memory alloys (SMA) may be a viable alternative to reduce monetary costs. This work explores the implementation of a low-cost linear macro-micro positioning system. The system consists of a modified printer carriage to provide long range, macro scale linear motion (approximately 200 mm range and 200 µm precision) and a micro scale system (approximately 4 mm range and 5 µm target precision) that uses an SMA actuator. A detailed description of the design and implementation of the system is given in this research. A model of the macro-stage is then generated by first identifying and inverting a simple friction model to linearize the system, thereby allowing for modified least squares (MLS) identification of a linear model. Various controllers are attempted for the macro-stage and compared with an experimentally tuned nonlinear PD controller that is implemented in the final design. A model of the micro-stage is derived through analysis of the SMA actuator. The model for the actuator is separated into two portions, an electro-thermal model, and a hysteresis model. The hysteresis model is derived using the Preisach model, and the electro-thermal model through MLS identification. To control the micro-stage, a PI controller with antiwindup is developed experimentally. The two stages are then executed together in closed loop and the resulting coupling between the two stages is briefly examined. Experimental data used for the modelling and design is presented, along with results of the final macro-micro linear positioning system.

Shape memory alloys (SMAs) are metallic materials, which have two main stable crystalline phases: austenite, a high temperature phase and martensite, a low temperature phase. Austenite and martensite each have unique physical and mechanical properties, and transformation between these phases enables two effects known as the shape memory effect (SME) and superelasticity. When a material that displays the SME is plastically deformed at low temperature, a heat input will cause the SMA to return to its original shape before the deformation. At higher temperatures, the material displays an effect called superelasticity, where strains of up to 10% are recoverable. These characteristics of SMA allow for significant amounts of strain recovery, and enable the design of SMA actuators. The temperature in an SMA actuator is generally controlled by resistive heating, also know as joule heating, and the strain recovery capabilities are used to do work on a load, thereby creating an electro-mechanical actuator. SMA actuators have attractive properties such as high energy density, smooth and silent actuation, reduced part counts compared to traditional alternatives, and scalability down to the micromechanical level. The phase transformation in SMA actuators, however, is highly non-linear. Therefore, the use of SMA as actuators, for example in positioning systems, benefits from the development of good models to predict and control the materials. The goals of this work are to develop a model suitable for real-time implementation, and that reproduces the observed behaviour of SMA actuators. The model is then inverted and used to develop a model-based controller, used in conjunction with traditional PID control to improve the precision and robustness of SMA actuators. The modelling portion of this work consists of the development of a phenomenological SMA model. The forward model is split into three blocks: a heating block, a phase kinetics block and a mechanical block. Since joule heating is commonly used in SMA actuators to bring about an increase in temperature, the heating block presents equations to convert a current input into the temperature of the wire. The phase kinetics block equations convert the calculated temperature and applied stress to the fraction of martensite present in the SMA. Finally, the mechanical model calculates the strain in the material from the martensite fraction and the applied stress. Once the model equations are presented, experimental verification tests are shown to compare physical SMA behaviour with that predicted by the model. Each of the blocks of the forward model are then inverted in order to be used as a feedforward linearizing controller. The control section of this thesis deals with the response of two common types of SMA actuators: a constant force SMA actuator and a spring-biased SMA actuator. The response of the system to step and sinusoidal signals with period of 5 seconds is investigated using two types of controllers: a traditional PI controller and the inverse-model controller in feedforward with a PI controller in feedback. Additionally, the robustness of the system is investigated through the response of the system to transient and sinusoidal stress disturbances. The disturbance rejection is investigated on a constant force actuator both with and without the presence of a force sensor.