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Until the twentieth century, theories of elastic rods and shells arose from collections of geometric and mechanical assumptions and approximations. These theories often lacked internal consistency and were appropriate for highly proscribed and sometimes unknown geometries and deformation sizes. The pioneering work of Truesdell, Antman, and others converted mechanical intuition into rigorous mathematical statements about the physics and mechanics of rods and shells. The result is the modern, geometrically exact theory of finite deformations of rods and shells.In the latter half of the twentieth century, biomechanics became a major focus of both experimental and theoretical mechanics. The genesis of residual stress by non-elastic growth has significant impact on the shape and mechanical properties of soft tissues. Inspired by the geometry of blood vessels and adopting a formalism found in elasto-plasticity, mechanicians have produced rigorous and applied results on the effect of growth on finite elastic deformations of columns and hollow tubes. Less attention has been paid to shells.A theory of growing elastic plates has been constructed in the context of linear elasticity. It harnessed many results in the theory of Riemann surfaces and has produced solutions that are surprisingly similar to experimental observations. Our intention is to provide a finite-deformation alternative by combining growth with the geometrically exact theory of shells. Such a theory has a clearer and more rigorous foundation, and it is applicable to thicker structures than is the case in the current theory of growing plates.This work presents the basic mathematical tools required to construct this alternative theory of finite elasticity of a shell in the presence of growth. We make clear that classical elasticity can be viewed in terms of three-dimensional Riemannian geometry, and that finite elasticity in the presence of growth must be considered in this way. We present several examples that demonstrate the viability and tractability of this approach.

This thesis presents research that was conducted to develop anatomically realistic finite element models of breast deformation under a variety of gravity loading conditions to assist clinicians in tracking suspicious tissues across multiple imaging modalities. Firstly, the accuracy of the modelling framework in predicting deformations of a homogeneous body was measured using custom designed silicon gel phantoms. The model predicted surface deformations with an average RMS error of 1.5 mm +/- 0.2 mm and tracked internal marker locations with an average RMS error of 1.4 mm +/- 0.7 mm. A novel method was then developed to determine the reference configuration of a body, when given its mechanical properties, boundary conditions and a deformed configuration. The theoretical validity of the technique was confirmed with an analytic solution. The accuracy of the method was also measured using silicon gel experiments, predicting the reference configuration surface with an average RMS error of 1.3 mm +/- 0.1 mm, and tracking internal marker locations with an average error of 1.5 mm +/- 0.8 mm. Silicon gel composites were then created to measure the accuracy of standard techniques to model heterogeneity. The models did not match the experimentally recorded deformations. This highlighted the need for further validation exercises on modelling heterogeneity before modelling them in the breast. A semi-automated algorithm was developed to fit finite element models to the skin and muscle surfaces of each individual, which were segmented from breast MR images. The code represented the skin with an average RMS error of 1.46 mm +/- 0.32 mm and the muscle with an average RMS error of 1.52 mm +/- 0.3 mm. The framework was then tested using images of the breast obtained under different gravity loading conditions and neutral buoyancy. A homogeneous model was first developed using the neutral buoyancy images as a representation of the reference configuration. The model did not accurately capture the regional deformations of the breast under gravity loading. However, the gross shape of the breast was reproduced, indicating that a biomechanical model of the breast could be useful to reliably track tissues across multiple images for cancer diagnosis. / This research was sponsored by the Top Achiever Doctoral Scholarship and the University of Auckland Doctoral Scholarship. Extra funding for travel was provided by the Graduate Research Fund and the John Logan Campbell Trust Fund.

This thesis presents research that was conducted to develop anatomically realistic finite element models of breast deformation under a variety of gravity loading conditions to assist clinicians in tracking suspicious tissues across multiple imaging modalities. Firstly, the accuracy of the modelling framework in predicting deformations of a homogeneous body was measured using custom designed silicon gel phantoms. The model predicted surface deformations with an average RMS error of 1.5 mm +/- 0.2 mm and tracked internal marker locations with an average RMS error of 1.4 mm +/- 0.7 mm. A novel method was then developed to determine the reference configuration of a body, when given its mechanical properties, boundary conditions and a deformed configuration. The theoretical validity of the technique was confirmed with an analytic solution. The accuracy of the method was also measured using silicon gel experiments, predicting the reference configuration surface with an average RMS error of 1.3 mm +/- 0.1 mm, and tracking internal marker locations with an average error of 1.5 mm +/- 0.8 mm. Silicon gel composites were then created to measure the accuracy of standard techniques to model heterogeneity. The models did not match the experimentally recorded deformations. This highlighted the need for further validation exercises on modelling heterogeneity before modelling them in the breast. A semi-automated algorithm was developed to fit finite element models to the skin and muscle surfaces of each individual, which were segmented from breast MR images. The code represented the skin with an average RMS error of 1.46 mm +/- 0.32 mm and the muscle with an average RMS error of 1.52 mm +/- 0.3 mm. The framework was then tested using images of the breast obtained under different gravity loading conditions and neutral buoyancy. A homogeneous model was first developed using the neutral buoyancy images as a representation of the reference configuration. The model did not accurately capture the regional deformations of the breast under gravity loading. However, the gross shape of the breast was reproduced, indicating that a biomechanical model of the breast could be useful to reliably track tissues across multiple images for cancer diagnosis. / This research was sponsored by the Top Achiever Doctoral Scholarship and the University of Auckland Doctoral Scholarship. Extra funding for travel was provided by the Graduate Research Fund and the John Logan Campbell Trust Fund.

This thesis presents research that was conducted to develop anatomically realistic finite element models of breast deformation under a variety of gravity loading conditions to assist clinicians in tracking suspicious tissues across multiple imaging modalities. Firstly, the accuracy of the modelling framework in predicting deformations of a homogeneous body was measured using custom designed silicon gel phantoms. The model predicted surface deformations with an average RMS error of 1.5 mm +/- 0.2 mm and tracked internal marker locations with an average RMS error of 1.4 mm +/- 0.7 mm. A novel method was then developed to determine the reference configuration of a body, when given its mechanical properties, boundary conditions and a deformed configuration. The theoretical validity of the technique was confirmed with an analytic solution. The accuracy of the method was also measured using silicon gel experiments, predicting the reference configuration surface with an average RMS error of 1.3 mm +/- 0.1 mm, and tracking internal marker locations with an average error of 1.5 mm +/- 0.8 mm. Silicon gel composites were then created to measure the accuracy of standard techniques to model heterogeneity. The models did not match the experimentally recorded deformations. This highlighted the need for further validation exercises on modelling heterogeneity before modelling them in the breast. A semi-automated algorithm was developed to fit finite element models to the skin and muscle surfaces of each individual, which were segmented from breast MR images. The code represented the skin with an average RMS error of 1.46 mm +/- 0.32 mm and the muscle with an average RMS error of 1.52 mm +/- 0.3 mm. The framework was then tested using images of the breast obtained under different gravity loading conditions and neutral buoyancy. A homogeneous model was first developed using the neutral buoyancy images as a representation of the reference configuration. The model did not accurately capture the regional deformations of the breast under gravity loading. However, the gross shape of the breast was reproduced, indicating that a biomechanical model of the breast could be useful to reliably track tissues across multiple images for cancer diagnosis. / This research was sponsored by the Top Achiever Doctoral Scholarship and the University of Auckland Doctoral Scholarship. Extra funding for travel was provided by the Graduate Research Fund and the John Logan Campbell Trust Fund.

This thesis presents research that was conducted to develop anatomically realistic finite element models of breast deformation under a variety of gravity loading conditions to assist clinicians in tracking suspicious tissues across multiple imaging modalities. Firstly, the accuracy of the modelling framework in predicting deformations of a homogeneous body was measured using custom designed silicon gel phantoms. The model predicted surface deformations with an average RMS error of 1.5 mm +/- 0.2 mm and tracked internal marker locations with an average RMS error of 1.4 mm +/- 0.7 mm. A novel method was then developed to determine the reference configuration of a body, when given its mechanical properties, boundary conditions and a deformed configuration. The theoretical validity of the technique was confirmed with an analytic solution. The accuracy of the method was also measured using silicon gel experiments, predicting the reference configuration surface with an average RMS error of 1.3 mm +/- 0.1 mm, and tracking internal marker locations with an average error of 1.5 mm +/- 0.8 mm. Silicon gel composites were then created to measure the accuracy of standard techniques to model heterogeneity. The models did not match the experimentally recorded deformations. This highlighted the need for further validation exercises on modelling heterogeneity before modelling them in the breast. A semi-automated algorithm was developed to fit finite element models to the skin and muscle surfaces of each individual, which were segmented from breast MR images. The code represented the skin with an average RMS error of 1.46 mm +/- 0.32 mm and the muscle with an average RMS error of 1.52 mm +/- 0.3 mm. The framework was then tested using images of the breast obtained under different gravity loading conditions and neutral buoyancy. A homogeneous model was first developed using the neutral buoyancy images as a representation of the reference configuration. The model did not accurately capture the regional deformations of the breast under gravity loading. However, the gross shape of the breast was reproduced, indicating that a biomechanical model of the breast could be useful to reliably track tissues across multiple images for cancer diagnosis. / This research was sponsored by the Top Achiever Doctoral Scholarship and the University of Auckland Doctoral Scholarship. Extra funding for travel was provided by the Graduate Research Fund and the John Logan Campbell Trust Fund.

Cette thèse porte sur l'étude de milieux granulaires constitués de particules molles. Elle s'appuie d'une part sur la méthode de corrélation d'images numériques (DIC) et d'autre part sur des simulations associant la méthode des éléments finis (FEM) et la méthode de Dynamique des Contacts (CD) permettant de tenir compte de la déformabilité élevée des particules et des interactions entre particules. Pour une large gamme de matériaux complexes (élastique, plastique et mousse solide), l'étude préliminaire de particules uniques comprimées radicalement valide la technique de DIC pour les mesures des champs de déformation, et pour la détermination de l'énergie de déformation pour des matériaux élastiques. Des grandeurs macroscopiques et micro-structurales tels que la compacité, la coordinence, l'énergie élastique, la densité de la probabilité de force de contact ainsi que de la densité d'énergie sont mesurées, et comparées entre les résultats numériques et expérimentaux en particulier dans le régime où les déformations des particules sont d'amplitudes finies. L'accord quantitatif entre ces approches permet de valider à la fois la méthode expérimentale pour l'étude de milieux granulaires à particules déformables, et d'autre part l'approche numérique. Forts de ces résultats, nous avons mené une étude numérique de la compression uniaxiale d'un assemblage de cylindres Néo-Hookéens. L'effet du frottement sur les paramètres macroscopiques et mico-structuraux est déterminé, y compris lorsque la compacité du système est proche de 100% / This thesis deals with the study of granular media composed of soft particles. It relies on the one hand on the method of Digital Image Correlation(DIC) and on the other hand, on simulations coupling the Finite Element Method and the Contact Dynamics method, taking into account the high deformability of particles and interactions between particles. For a wide range of complex materials (elastic, plastic and solid foam), the preliminary study of radially compressed single particles validates the DIC technique for strain field measurements, and for the determination of the strain energy for elastic materials. Macroscopic and micro-structural quantities such as packing fraction, coordination, elastic energy, probability density of the contact force as well as strain energy density, are measured and compared between the numerical and experimental results in particular in the regime where the deformations of the particles are large. The quantitative agreement between these approaches makes it possible to validate both the experimental method for the study of granular media with deformable particles, and the numerical approach. Based on these results, we conducted a numerical study of the uniaxial compression of a Neo-Hookean cylinder assembly. The effect of friction on the macroscopic and mico-structural parameters is determined, even when the packing fraction of the system is close to 100%.

Wound healing is a complex process, in which a sequence of interrelated events at both the cell and tissue levels interact and contribute to the reduction in wound size. For diabetic patients, many of these processes are compromised, so that wound healing slows down and in some cases halts. In this thesis we develop a series of increasingly detailed mathematical models to describe and investigate healing of full thickness skin wounds. We begin by developing a time-dependent ordinary differential equation model. This phenomenological model focusses on the main processes contributing to closure of a full thickness wound: proliferation in the epidermis and growth and contraction in the dermis. Model simulations suggest that the relative contributions of growth and contraction to healing of the dermis are altered in diabetic wounds. We investigate further the balance between growth and contraction by developing a more detailed, spatially-resolved model using continuum mechanics. Due to the initial large retraction of the wound edge upon injury, we adopt a non-linear elastic framework. Morphoelasticity theory is applied, with the total deformation of the material decomposed into an addition of mass and an elastic response. We use the model to investigate how interactions between growth and stress influence dermal wound healing. The model reveals that contraction alone generates unrealistically high tension in the dermal tissue and, hence, volumetric growth must contribute to healing. We show that, in the simplified case of homogeneous growth, the tissue must grow anisotropically in order to reduce the size of the wound and we postulate mechanosensitive growth laws consistent with this result. After closure the surrounding tissue remodels, returning to its residually stressed state. We identify the steady state growth profile associated with this remodelled state. The model is used to predict the outcome of rewounding experiments as a method of quantifying the amount of stress in the tissue and the application of pressure treatments to control tissue synthesis. The thesis concludes with an extension to the spatially-resolved mechanical model to account for the effects of the biochemical environment. Partial differential equations describing the dynamics of fibroblasts and a regulating growth factor are coupled to equations for the tissue mechanics, described in the morphoelastic framework. By accounting for biomechanical and biochemical stimuli the model allows us to formulate mechanistic laws for growth and contraction. We explore how disruption of mechanical and chemical feedback can lead to abnormal wound healing and use the model to identify specific treatments for normalising healing in these cases.