<p> Layers of plasticene lubricated with talcum powder were compressed in directions perpendicular to the layering. Normal kink bands oblique to the principal compression direction were produced. These structures were predicted by Cosgrove (1972) after a theory developed by Biot (1965). Other multilayers, with the layering oblique to the principal compression direction, were compressed and these developed instabilities also. </p> <p> The normal kink bands were analysed geometrically. It was found that variation in layer thickness provided the best description and also provided a statement of the state of strain of the final deformation. </p> / Thesis / Bachelor of Science (BSc)

Distributed Point Source Method (DPSM) was developed by Placko and Kundu 1, as a technique for modeling electromagnetic and elastic wave propagation problems. DPSM has been used for modeling ultrasonic, electrostatic and electromagnetic fields scattered by defects and anomalies in a structure. The modeling of such scattered field helps to extract valuable information about the location and type of defects. Therefore, DPSM can be used as an effective tool for Non-Destructive Testing (NDT). Anisotropy adds to the complexity of the problem, both mathematically and computationally. Computation of the Green's function which is used as the fundamental solution in DPSM is considerably more challenging for anisotropic media, and it cannot be reduced to a closed-form solution as is done for isotropic materials. The purpose of this study is to investigate and implement DPSM for an anisotropic medium. While the mathematical formulation and the numerical algorithm will be considered for general anisotropic media, more emphasis will be placed on transversely isotropic materials in the numerical example presented in this paper. The unidirectional fiber-reinforced composites which are widely used in today's industry are good examples of transversely isotropic materials. Development of an effective and accurate NDT method based on these modeling results can be of paramount importance for in-service monitoring of damage in composite structures.

Acoustic source localization (ASL) is pinpointing an acoustic source. ASL can reveal the point of impact of a foreign object or the point of crack initiation in a structure. ASL is necessary for continuous health monitoring of a structure. ASL in an anisotropic plate is a challenging task. This dissertation aims to investigate techniques that are currently being used to precisely determine an acoustic source location in an anisotropic plate without knowing its material properties. A new technique is developed and presented here to overcome the existing shortcomings of the acoustic source localization in anisotropic plates. It is done by changing the analysis perspective from the angular dependent group velocity of the wave and its straight line propagation to the wave front shapes and their geometric properties when a non-circular wave front is generated. Especially, 'rhombic wave front' and 'elliptical wave front' are dealt with because they are readily observed in highly anisotropic composite plates. Once each proposed technique meets the requirements of measurement, four sensor clusters in three different quadrants (recorded by 12 sensors) for the rhombus and at least three sensor clusters (recorded by 9 sensors) for the ellipse, accurate Acoustic Source Localization is obtained. It has been successfully demonstrated in the numerical simulations. In addition, a series of experimental tests demonstrate reliable and robust prediction performance of the developed new acoustic source localization technique.

Tube hydroforming is a forming method which has several advantages. By using pressure in combination with material feeding it is possible to manufacture products with high structural integration and tight dimensional tolerances. The forming method is especially suited for aluminium alloys which have a relatively low ductility. Finite Element simulations are used extensively in the sheet metal stamping industry, where the methodology has contributed to a better understanding of the process and the new prediction capability has significantly reduced costly die tryouts. Similarly, the tube hydroforming industry can benefit from Finite Element simulations, and this simulation methodology is the topic of this dissertation. Deep drawing and tube hydroforming have a basic difference, namely that the latter process essentially is a force controlled process. This fact, in combination with the anisotropic behaviour of aluminium tubes, enforces a need for accurate constitutive descriptions. Furthermore, the material testing needs to account for the specifics of tube hydroforming. The importance of proper material modelling is in this work shown for hydrobulging and hydroforming in a die with extensive feeding. The process parameters in hydroforming are the inner pressure and the material feeding, where a correct combination of these parameters is crucial for the success of the process. It is here shown, that Finite Element simulations together with an optimisation routine are powerful tools for estimating the process parameters in an automated procedure. Finally, the reliability and quality of the simulation results depend on how failure is evaluated, which in the case of hydroforming mainly oncerns wrinkling and strain localisation. Since tube hydroforming often is preceded by bending operations this fact also demands the criteria to be strain path independent. In this work, it is shown that the prediction of strain localisation depends on the ability to predict diffuse necking, which in turn is strongly related to the chosen constitutive model.

Trabecular bone has a complex and porous microstructure. This study develops approaches to determine the mechanical behaviour of this material at the macroscopic level through the use of homogenisation-based multiscale methods using micro-finite element simulations. In homogenisation-based finite element methods, a simulation involving a representative volume element of the microstructure of the considered material is performed with a specific set of boundary conditions. The macroscopic stresses and strains are retrieved as averaged quantities defined over this domain. Most of the homogenisation-based work on trabecular bone has been performed to study its macroscopic elastic regime, and therefore define its constant macroscopic stiffness tensor. The rod and plate-shaped microstructure of trabecular bone can be precisely identified with advanced scanning tools, such as micro-computed tomography devices. Taking into account the size requirements to achieve a certain independence of boundary conditions for trabecular bone in a homogenisation-based multiscale setting, the resulting stack of images can have around ten million solid voxels after binarisation. Although a completely linear finite element simulation with such a large system may be feasible with commercial packages (with the proper time and memory requirements), it is not possible to perform a nonlinear simulation for such a mesh in a reasonable time frame, and the amount of required memory may not be available. A highly scalable parallel driver program which solves finite strain elastoplastic systems was developed within the framework of the existing parallel code ParaFEM. This code was used throughout this study to evaluate the yield and post-yield properties of trabecular bone. It was run on cutting edge high performance computing platforms (BlueGene/Q at the Hartree Centre, Science and Technology Facilities Council; and ARCHER, UK National Supercomputing Service, at Edinburgh Parallel Computing Centre). Micro-finite element simulations require definition of properties at the microscopic scale and it is unclear how these properties affect the macroscopic response. This study examines the effect of compressive hydrostatic yield at the microscopic scale on the macroscopic behaviour. Two different microscopic yield criteria, one permitting yielding at compressive hydrostatic stresses and the other not, were considered. A large number of load cases were examined. It was found that these two microscopic yield criteria only influence macroscopic yield behaviour in load scenarios which are compression-dominated; for other load cases, macroscopic response is insensitive to the choice of the microscopic yield criterion, provided it has an appropriate strength asymmetry. Also, in compression-dominated load cases, high density bone is much more sensitive as it is more like a continuum, resulting in the microscopic properties being more directly upscaled. Only a few previous studies have employed homogenisation to evaluate the macroscopic yield criterion of trabecular bone. However, they either used a simplified microscopic yield surface or examined only a small number of load cases. A thorough multiaxial evaluation of the macroscopic yield surface was performed by applying a wide range of loading scenarios (160 load cases) on trabecular bone samples. Closed-form yield surfaces with different symmetries (isotropy, orthotropy and full anisotropy) were fitted to the numerically obtained macroscopic yield points in strain space, and the fitting errors were evaluated in detail for different subsets of load cases. Although orthotropy and full anisotropy showed the smallest fitting errors, they were not significantly superior to the isotropic fit. Thus, isotropy in strain space presents itself as the most suitable option due to the simplicity of its implementation. The study showed that fitting errors do depend on the chosen set of load cases and that shear load cases are extremely important as it was found that even for these highly aligned samples, trabecular bone presents some degree of shear asymmetry, i.e. different strength in clockwise and counter-clockwise shear directions. There have been no previous attempts to evaluate the post-yield behaviour of trabecular bone through homogenisation-based studies on detailed micro-finite element trabecular bone meshes. A damage and plasticity constitutive law for the microscale based on existing data in the literature was considered. A homogenisation-based multiscale approach was used to evaluate the hardening and stiffness reduction at the macroscale when uniaxial load scenarios are applied to trabecular bone samples, for a small range of plastic strain Euclidean norms. Results show that damage progression at the macroscale for trabecular bone is not isotropic, which is contrary to what has been assumed previously, and that both the evolution of the yield surface and damage are different for tension, compression and shear. Nonetheless, they can be correlated with plastic strain Euclidean norms by using linear relationships. It was also observed that macroscopic damage in a specific load case affects differently the on-axis orthotropic stiffness and the off-axis orthotropic stiffness components. The findings of this study will permit the use of a more rigorous definition of the post-elastic macroscopic behaviour of trabecular bone in finite element settings.

A popular way to reduce weight in industrial products without compromising the strength or stiffness is to replace components made of metal by plastics that have been reinforced by glass fibers. When fibers are introduced in a plastic, the resulting composite usually becomes anisotropic, which makes it much more complex to work with in simulation software. This thesis looks at modeling of such a composite using the multi-scale material modeling tool Digimat. An injection molding simulation of a brush cutter casing made of a short fiber reinforced plastic has been performed in order to obtain information about the glass fiber orientations, and thus the anisotropy, in each material point. That information has then been transferred over from the injection mesh to the structural mesh via a mapping routine. An elasto-viscoplastic material model with failure has been employed and calibrated against experimental data to find the corresponding material parameters. Lastly, a finite element analysis simulating a drop test has been performed. The results from the analysis have been compared with a physical drop test in order to evaluate the accuracy of the methodology used. The outcome has been discussed, conclusions have been drawn and suggestions for further studies have been presented.

Additive manufacturing (AM) is a particularly useful manufacturing method for components designed using topology optimization (TO) since it allows for a greater part complexity than any traditional manufacturing method. However, the AM process potentially leads to anisotropic material properties due to the layer-by-layer buildup of parts and the fast and directional cooling. For Ti6Al4V tensile specimens built using electron beam melting (EBM), it has been observed that flat built specimens show superior strength and elastic moduli compared to top built specimens. Designs with the loading direction parallel to the build layers are therefore expected to show greater reliability. In this thesis a procedure is developed to optimize the AM build orientation considering anisotropic elastic material properties. A transversely isotropic material model is used to represent the in-plane and out-of-plane characteristics of AM produced parts. Two additional design variables are added to the TO formulation in order to control the orientation of the material using a coordinate transformation. Sensitivity analysis for the material direction variables is conducted for compliance as well as maximum von-Mises stress using a -norm stress aggregation function. The procedures for the AM build orientation optimization and stress constraints are implemented in the finite element software TRINITAS and evaluated using a number of examples in 2D and 3D. It is found that the procedure works well for compliance as well as stress but that a combination of these may lead to convergence issues due to contradicting optimal material orientations. An evaluation of the -norm stress aggregation function showed that a single global stress measure in combination with a stress correction procedure works well for most problems given that the mesh is refined enough to resolve the stresses accurately.

Polycrystalline materials are widely used for industrial applications. These materials are highly anisotropic with different responses under different loading conditions. This dissertation uses a crystal plasticity scheme in the finite element framework (CPFEM) to study deformation mechanisms in alumina, aluminium and stainless steel – all polycrystalline. Four research cases in this dissertation have been presented in the form of manuscripts for publication. When possible, modelling predictions have been compared against various experimental techniques such as Diffraction Contrast Tomography (DCT), Neutron Diffraction (ND) and Electron Back Scatter Diffraction (EBSD). After an introduction (Chapter 1) and a literature review (Chapter 2) on plastic deformation and modelling techniques, the methodology and results are presented and discussed (Chapters 3 and 4). Measurements of elastic strains for individual grain families (ND) and local rotations (DCT and EBSD) are compared against corresponding predictions by the model following different loading modes. Each study reveals different degrees of agreement between predictions and measurements. The individual conclusions to each study are presented in Chapter 4. Some overall conclusions and suggestions for further work are presented in Chapter 5.

We propose an efficient wideband theory for adjoint variable sensitivity analysis of problems with general anisotropic materials. The method is formulated based on the transmission line numerical modeling technique. The anisotropic material properties of potential interest are the full tensors of permittivity, permeability, electrical conductivity, magnetic resistivity, magnetoelectric coupling, and electromagnetic coupling. The tensors may contain non-diagonal elements. Our method estimates the gradients of the desired response with respect to all designable parameters using at most one extra simulation, regardless of their number. In contrast, in the conventional sensitivity analysis method using central finite differences, the number of the required simulations scales linearly with the number of designable parameters. The theory has been implemented for sensitivity analysis of the two and three-dimensional structures. The available adjoint variable method (AVM) sensitivities enable the optimization-based design of anisotropic and dispersive anisotropic structures. We apply our AVM technique to optimization-based wideband invisibility cloak design of arbitrary-shape objects. Our method optimizes the voxel-by-voxel constitutive parameters of an anisotropic cloak. This results in a large number of optimizable parameters. The associated sensitivities of a wideband cloaking objective function are efficiently estimated using our anisotropic adjoint variable method technique. A gradient-based optimization algorithm utilizes the available sensitivity information to iteratively minimize the visibility objective function and to determine the constitutive parameters of the optimal cloak. / Thesis / Doctor of Philosophy (PhD)

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