In this thesis, we propose a new microstructural model to describe the rheology of entangled linear polymers. In order to reduce the number of non-linear adjustable parameters, we develop a model capable of predicting both the linear and the non-linear response, using a single set of material parameters. In a first step, a linear differential formulation of the thermal constraint release mechanism is introduced and validated against experimental results for linear polystyrene melts. In a second step, we extend the linear model to the non-linear regime by generalizing the state variables to conformation tensors and accounting for the relevant non-linear relaxation phenomena. The numerical predictions of the resulting model are then compared to experimental data for entangled polymer melts and solutions in different flow regimes. Finally, we show, on a simple reptation model, how the single generator bracket formalism of non-equilibrium thermodynamics can be used for the phenomenological improvement of microstructural constitutive models.

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No / High temperature reversion tests have revealed a state of pre-existing molecular orientation in extruded polyethylene sheet. This state is related to differences in stress-deformation behaviour when specimens of the sheet are stretched along different angles with respect to the extrusion direction. An established large deformation, rate-dependent constitutive equation has been developed to model this material, by incorporating the pre-orientation by the addition of a strained Gaussian network. The level of pre-orientation is deduced from the dimensional changes on shrinkage. The constitutive equation is incorporated into the finite element package , and the shapes and drawing forces of tensile specimens extended at various angles to the extrusion direction are modelled.

No

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Materials like asphalt, asphalt concrete and polyimides that are used in the transportation and aerospace industry show viscoelastic behavior. These materials in the working environment are subject to degradation due to temperature, diffusion of moisture and chemical reactions (for instance, oxidation) and there is need for a good understanding of the various degradation mechanisms. This work focuses on: 1) some topics related to development of viscoelastic fluid models that can be used to predict the response of materials like asphalt, asphalt concrete, and other geomaterials, and 2) developing a framework to model degradation due to the various mechanisms (such as temperature, diffusion of moisture and oxidation) on polyimides that show nonlinear viscoelastic solid-like response. Such a framework can be extended to model similar degradation phenomena in the area of asphalt mechanics and biomechanics. The thermodynamic framework that is used in this work is based on the notion that the 'natural configuration' of a body evolves as the body undergoes a process and the evolution is determined by maximizing the rate of entropy production. The Burgers' fluid model is known to predict the non-linear viscoelastic fluid-like response of asphalt, asphalt concrete and other geomaterials. We first show that different choices for the manner in which the body stores energy and dissipates energy and satisfies the requirement of maximization of the rate of entropy production that leads to many three dimensional models. All of these models, in one dimension, reduce to the model proposed by Burgers. A thermodynamic framework to develop rate-type models for viscoelastic fluids which do not possess instantaneous elasticity (certain types of asphalt show such a behavior) is developed next. To illustrate the capabilities of such models we make a specific choice for the specific Helmholtz potential and the rate of dissipation and consider the creep and stress relaxation response associated with the model. We then study the effect of degradation and healing due to the diffusion of a fluid on the response of a solid which prior to the diffusion can be described by the generalized neo-Hookean model. We show that a generalized neo-Hookean solid - which behaves like an elastic body (i.e., it does not produce entropy) within a purely mechanical context - creeps and stress relaxes when infused with a fluid and behaves like a body whose material properties are time dependent. A framework is then developed to predict the viscoelastic response of polyimide resins under different temperature conditions. The developed framework is further extended to model the phenomena of swelling due to diffusion of a fluid through a viscoelastic solid using the theory of mixtures. Finally, degradation due to oxidation is incorporated into such a framework by introducing a variable that represents the extent of oxidation. The data from the resulting models are shown to be in good agreement with the experiments for polyimide resins.

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<p dir="ltr">Magnetorheological elastomers (MREs) are a type of smart material composed of ferrous particles suspended in a solid elastic matrix [5, 6]. When an external magnetic field is applied to an MRE, the ferrous particles tend to align with the field, causing either deformation and/or a change in the mechanical properties of the system. MREs are utilized in applications such as soft robotics, actuators, sensors, vibration control systems, and mechanical metamaterials[20, 19, 27, 5, 6, 13]. Recent demand for theses technologies has motivated an increasing focus on the material properties of MRE’s over the last 20 years [6]. Multiple authors have proposed a variety of hyperelastic mechanical and magnetomechanical models to describe these materials [16, 12, 15, 25, 14, 38, 2, 6, 8, 24]. The research presented in this dissertation focuses on the modeling and characterization of MRE’s using a systematic development of the conservation and balance laws, Maxwell’s equations, and constitutive equations needed to describe the MRE as a multi-constituent system. The material parameters resulting from the derived constitutive equations are estimated using data collected from a series of compression experiments coupled with an externally applied magnetic field. The multi-constituent constitutive equations predicted the stress of the MRE in these compression experiments for a variety of ferrous particle concentrations.</p>

For years, researchers have studied the movement of cells and mathematicians have attempted to model the movement of the cell using various methods. This work is an extension of the work done by Zheltukhin and Lui (2011), Mathematical Biosciences 229:30-40, who simulated the stress and displacement of a one-dimensional cell using a model based on viscoelastic theory. The report is divided into three main parts. The first part considers viscoelastic models with a first-order constitutive equation and uses the standard linear model as an example. The second part extends the results of the first to models with second-order constitutive equations. In this part, the two examples studied are Burger model and a Kelvin-Voigt element connected with a dashpot in series. In the third part, the effects of substrate with variable stiffness are explored. Here, the effective adhesion coefficient is changed from a constant to a spatially-dependent function. Numerical results are generated using two different functions for the adhesion coefficient. Results of this thesis show that stress on the cell varies greatly across each part of the cell depending on the constitute equation we use, while the position and velocity of the cell remain essentially unchanged from a large-scale point of view.

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