In this study, a new consistent formulation coupling kinematics, thermodynamics, and kinetics with damage using an extended multiplicative decomposition of the deformation gradient that accounts for corrosion effects is presented. The technical approach used for modeling the corrosion behavior of magnesium alloys was divided into three primary steps. First, a predictive corrosion model was developed based on experimental corrosion observations. The experimentally-observed corrosion mechanisms of pitting, intergranular, and general corrosion on the AZ31 magnesium alloy were quantified in 3.5 wt.% NaCl immersion and salt spray environments using optical microscopy and laser profilometry to document the changes in the pit characteristics. Although both environments showed similar trends, the immersion environment was more deleterious with respect to intergranular and general corrosion. On the other hand, the salt-spray environment allowed deeper pits to form throughout the entirety of the experiments, which led to a substantial thickness drop (general corrosion) compared with the immersion environment. Next, the complete corrosion model based upon the internal state variable theory was formulated to capture the effects of pit nucleation, pit growth, pit coalescence, and general corrosion. Different rate equations were given for each mechanism. Following the formulation of the model, the aforementioned experimental work and experimental work on four other magnesium alloys (AZ61, AM30, AM60, and AE44), was used to validate the model.

This dissertation proposes a multiscale Internal State Variable (ISV) inelasticity-corrosion damage model that is motivated by experimental microstructure-property relations of magnesium alloys. The corrosion damage framework was laid out based on observation of different corrosion mechanisms occurred on an extruded AM30 magnesium alloys. The extruded AM30 magnesium alloy was studied under two corrosion environments (cyclical salt spray and immersion) in order to observe the corrosion rates under different exposure environments. The coupons were examined at various times to determine the history effects of three corrosion mechanisms: (1) general corrosion; (2) pitting corrosion in terms of the nucleation rate, growth rate, and coalescence rate; and (3) intergranular corrosion. The multiscale ISV corrosion model was developed by bridging the macroscale corrosion damage to the mesoscale electrochemical kinetics, microscale material features, and nanoscale material activation energies. The corrosion testing results of Mg alloys (pure Mg, Mg-2% Al, and Mg-6% Al) were used to develop, calibrate, and validate the model, and good agreement was found between the model results and the corrosion testing data. Finally, the simultaneous effects of corrosion and cyclic loading were tested but not modelled for the extruded AM30 magnesium alloy by conducting fatigue experiments in a 3.5 wt.% NaCl solution environment. The corrosion fatigue life of the AM30 alloy was significantly reduced due to corrosion pit formation on specimen surface, hydrogen diffused into the material , and the fracture surface dissolved into the solution. The corrosion damage that arose on the fatigue specimens reduced the crack nucleation process and enhanced the crack propagation rate.

In the current study we developed an internal state variable (ISV) model based on the Bammann inelasticity internal state variable model (BIISV) to include damage, recrystallization, and texture development, which we then implemented into a mantle convection code, TERRA2D, to incorporate higher fidelity material behavior into mantle convection simulations. With experimental stress strain data found in the literature model constants for the BIISV model were determined for a number of geologic materials. The BIISV model was shown to be far superior to the steady state power law model currently used by the geologic community to capture the deformation of geologic materials. Once implemented and verified in TERRA2D the BIISV model revealed locations of hardened material that behaved like diverters in the cold thermal boundary layer that the power law model could never produce. These hardened regions could be a plausible reason for the current subduction zones present on the earth. We then altered the BIISV model equation to include the effects of damage, recrystallization, and texture development in order to model possible weakening mechanisms in the cold thermal boundary layer of the mantle. Inclusion of damage and recrystallization allowed the cold thermal boundary layer to mobilize and plunge downward into the hotter region below. Texture development increased the intensity of rotational flow within the hotter zone as cold boundary material plunged downward which aided in destabilizing the cold upper thermal boundary layer. The inclusion of an internal state variable model with damage, recrystallization, and texture development represents a significant advancement in handling deformational physics for mantle phenomena in a comprehensive, unified, and automatic manner.

A macroscale Internal State Variable (ISV) constitutive model coupling magnetism effects with thermal, elastic, and damage effects is developed. Previous models for magnetic and mechanical fields included constitutive equations describing their effects on the material system studied independently. Some models explain the mechanisms behind mechanical deformations caused by magnetization changes that are presented in the literature. They mainly focus on the nanoscale level. Other models, describe the behavior of one specific magnet that is mostly a permanent magnet. However permanent magnets are made of rare-earth elements that are subjected to a high supply risk. In attempt to find an alternative to permanent magnets, a mathematical model that captures the physical behavior of magnets is needed, to help develop a tool to create a new permanent magnet. The ISV constitutive model herein describes the macroscale mechanical deformation caused by magnetic fields on ferromagnetic materials, Iron (Fe), Cobalt (Co) and Nickel (Ni) precisely. The ISV model internally coheres the kinematic, thermodynamic, and kinetic relationships of deformation using the evolving histories of internal variables. For the kinematics, a multiplicative decomposition of deformation gradient is employed including a magnetization term, and the Jacobian that represents the conservation of mass and conservation of momentum. The First and Second Law of Thermodynamics are used to constrain the appropriate constitutive relations through the Clausius-Duhem inequality. The kinetic framework employs a stress-strain relationship with a flow rule that couples the thermal, mechanical, and damage terms. To determine the ISVs needed to mimic the behavior of magnetic materials, we conducted various magnetic experiments on three different specimens made of Iron, Nickel and Cobalt. Experiments captured the mechanical deformation of a rod sample when subjected to a magnetic field using the Michelson Interferometer. To study the magnetic hysteresis of Iron, Nickel, and Cobalt, previous literature data were used. It was shown that the magnetization equation modeled the hysteresis of Iron, Nickel, and Cobalt. The magnetostrictive strain equation shows good agreement for Nickel and Cobalt, but further investigation should be done for Iron.

This dissertation proposes a modi ed internal state variable (ISV) inelastic damage model that was motivated by experimental structure{property relations of thermoplastics. In particular, a new damage model was developed for glassy, amorphous thermoplastics. ISV evolution equations are de ned through thermodynamics, kinematics, and kinetics for isotropic damage arising from two di erent inclusion types: pores and particles. The damage arising from the particles and crazes is accounted for by three processes: damage nucleation, growth, and coalescence. Damage nucleation is de ned as the number density of voids/crazes. The associated ISV rate equation is a function of stress state, molecular weight, fracture toughness, particle size, particle volume fraction, temperature, and strain rate. The damage growth is based upon a single void growing and its growth is de ned by an ISV rate equation that is a function of stress state, strain rate sensitivity, and strain rate. The coalescence ISV equation enables interaction between voids and crazes and is a function of the nearest neighbor distance between voids/crazes, size of voids/crazes, temperature, and strain rate. The damage arising from pre-existing voids employs the Cocks{Ashby void growth rule. The total void volume fraction is a summation of the damage arising from particles, pores, and crazes. Micromechanical modeling results for a single void compare well to experimental ndings garnered from the literature. This formulation is then implemented into a nite element analysis. For damage evolution, comparisons are made between a one-dimensional material point simulator and a three-dimensional nite element (FE) simulation. Finally, good agreement is found between impact experiments and FE impact simulations using the implemented model.

The objective of this work is to develop an evolution equation for the ductile growth of a spherical void in a highly strain rate and temperature dependent material. The material considered in this work is stainless steel 304L at 982 °C. The material is characterized by a physically-based internal state variable model derived within consistent kinematics and thermodynamics — Evolving Microstructure Model for Inelasticity. Through this formulation, the degradation of the elastic moduli due to damage has been naturally acquired. An elastoviscoplasticity user material subroutine has also been developed and implemented into a commercially available finite element software ABAQUS. The subroutine utilizes a return mapping algorithm, where a purely elastic trial state (elastic predictor) is followed by a plastic corrector phase (return mapping). A conditionally stable fully-implicit scheme, derived from the backward Euler integration method, has been employed to calculate the values of the internal state variables in the elastoviscoplasticity integration routine. A repeating unit cell problem is set up by introducing a spherical void inside a matrix material that simulates a periodic array of voids in a component. Using finite element analysis, a database is generated by recording the responses of the unit cell under various combinations of loading conditions, porosity, and state variables. Functional forms of the void growth equations are constructed by utilizing normalization techniques to collapse all the data into master curves. The evolution equations are converted to a form consistent with the continuum damage variable in the complete thermal-elastic-plastic-damage version of the physically-based internal state variable model.

Mechanical structure-property relations have been quantified for AISI 4140 steel under different strain rates and temperatures. The structure-property relations were used to calibrate a microstructure-based internal state variable plasticity-damage model for monotonic tension, compression and torsion plasticity, as well as damage evolution. Strong stress state and temperature dependences were observed for the AISI 4140 steel. Tension tests on three different notched Bridgman specimens were undertaken to study the damage-triaxiality dependence for model validation purposes. Fracture surface analysis was performed using Scanning Electron Microscopy (SEM) to quantify the void nucleation and void sizes in the different specimens. The stress-strain behavior exhibited a fairly large applied stress state (tension, compression dependence, and torsion), a moderate temperature dependence, and a relatively small strain rate dependence.

In this work, the effectiveness of multi-objective design optimization using metamodeling techniques and an internal state variable (ISV) plasticity damage material model as a design tool is demonstrated. Multi-objective design optimization, metamodeling, and ISV plasticity damage material models are brought together to provide a design tool capable of meeting the stringent structural design requirements of today and of the future. The process of implementing this tool are laid out, and two case studies using multi-objective design optimization were carried out. The first was the optimization of a Chevrolet Equinox rear subframe. The optimized subframe was 12% lighter and met design requirements not achieved by the heavier initial design. The second case was the optimization of a Formula SAE front upright. The optimized upright meets all the design constraints and is 22% lighter.

This dissertation presents a numerical model constructed to investigate the dynamics and structures of the Earth’s mantle. Deformation of the Earth’s mantle, which is composed of solid silicate minerals, is strongly governed by the constitutive relation-ship among multiple length-scale structures and properties. To explain the realistic consti-tutive behavior of the silicate mantle, an Internal State Variable (ISV) theory that is an advanced and novel constitutive approach for history-dependent elastoviscoplasticity was applied. The ISV constitutive model was, in turn, implemented into a three-dimensional geodynamic code, TERRA3D, which uses the Finite Element method developed for the mantle convection problem. The sequential studies performed in this dissertation are presented in the follow-ing order: i) a comprehensive summary of the mantle material structures (compositions and microstructural features) and its mechanical properties (elasticity and rheology), ii) a development of a recrystallization and grain size dependent ISV constitutive model for the polycrystalline materials such as minerals and metals, which explains comprehensive mineral physics occurring under the conditions of pressure, temperature, and strain rate within the mantle and their history dependence, and iii) an application of the recrystalli-zation and grain size dependent ISV model to the Earth’s mantle convection problem us-ing the TERRA3D for an investigation of the grain size and dynamic recrystallization efect on the mantle dynamics. The applied ISV constitutive model within the TERRA3D Finite Element frame-work captures the subscale dynamics (dislocation density evolution, dynamic and static recrystallization, grain growth, and grain refinement) and their effect on the large-scale rheology and dynamics of the Earth’s mantle. The numerical investigations reveal that the potential for the mechanical instability and weakening within the mantle arises from the kinetics of grain size and recrystallization and their rheological effect. This mechanical instability leads to the mantle convection entering the episodic overturn regime. The TERRA3D-ISV mantle convection model herein also provides some insightful discover-ies regarding the dynamics and structures within the mantle, explaining its complex rhe-ology caused by the kinetics of recrystallization, grain size, hardening, dislocation recov-ery, and diffusion in the geological settings.

To enhance material performance at different length scales, this study strives to develop a reliable analytical and computational tool with the help of internal state variables spanning micro and macro-level behaviors. First, the practical relevance of a nonlocal damage integral added to an internal state variable (BCJ) model is studied to alleviate numerical instabilities associated within the post-bifurcation regime. The characteristic length scale in the nonlocal damage, which is mathematical in nature, can be calibrated using a series of notch tensile tests. Then the same length scale from the notch tests is used in solving the problem of a high-velocity (between 89 and 107 m/s) rigid projectile colliding against a 6061-T6 aluminum-disk. The investigation indicates that incorporating a characteristic length scale to the constitutive model eliminates the pathological mesh-dependency associated with material instabilities. In addition, the numerical calculations agree well with experimental data. Next, an effort is made rather to introduce a physically motivated length scale than to apply a mathematical-one in the deformation analysis. Along this line, a dislocation based plasticity model is developed where an intrinsic length scale is introduced in the forms of spatial gradients of mobile and immobile dislocation densities. The spatial gradients are naturally invoked from balance laws within a consistent kinematic and thermodynamic framework. An analytical solution of the model variables is derived at homogenous steady state using the linear stability and bifurcation analysis. The model qualitatively captures the formation of dislocation cell-structures through material instabilities at the microscopic level. Finally, the model satisfactorily predicts macroscopic mechanical behaviors - e.g., multi-strain rate uniaxial compression, simple shear, and stress relaxation - and validates experimental results.