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Throughout engineering history, metal cutting technology has been pushed to keep pace with the development of stronger and more difficult to cut materials. With the advent of robust machining technology, the finish cutting of hardened materials at high speeds and extreme conditions has become possible. As we push the thresholds of our technology, the need for a deep understanding of the processes at work in metal cutting in order to predict limit states has become clear. The prediction of machining processes began with models that described the problem qualitatively. The principals and assumptions used within these laid the foundations for further theoretically based empirical models. With the advent of computationally based numerical modeling, predictive modeling of metal cutting processes using elasto-plastic based simulation has brought a new perspective to the field. The development of finite element modeling techniques capable of modeling both continuous and discontinuous chip formation has become a subject of considerable research effort. The present work focuses on the study and development of explicit transient finite element techniques used to simulate the orthogonal machining of hardened steel under various cutting conditions producing both continuous and discontinuous chips. Metal cutting involves a diversity of complex physical phenomenon such as large strain and strain rate plasticity, high temperature contact and friction conditions, material failure, adiabatic shear localization, and thermo-mechnically based deformation. The numerical modeling of these processes in orthogonal machining operations is the primary focus of this work. The effects of cutting conditions on chip morphology and the finished workpiece are investigated and resulting phenomenon are explained in terms of numerically based stress analysis. Benchmarking comparisons with literature and commercial modeling software simulations are made and quantified in terms of strains, temperatures, residual stresses, and the various components of stress. Future directions and issues are outlined and recommendations are made. Solution stability of the finite element solver applied to machining was quite low due to the lack of an adaptive remeshing scheme and deficiencies in the contact algorithm. Thermal mechanical coupling and remeshing are currently being implemented. Future avenues include methods of surface generation without volumetric losses. the application of friction based on wear data, and the application of machining simulations to oblique three-dimensional cutting operations. / Thesis / Master of Engineering (ME)

In this thesis, we present new superconvergence properties of discontinuous Galerkin (DG) methods for two-dimensional hyperbolic problems. We investigate the superconvergence properties of the DG method applied to scalar first-order hyperbolic partial differential equations on triangular meshes. We study the effect of finite element spaces on the superconvergence properties of DG solutions on three types of triangular elements. Superconvergence is described for structured and unstructured meshes. We show that the DG solution is O(hp+1) superconvergent at Legendre points on the outflow edge on triangles having one outflow edge using three p- degree polynomial spaces. For triangles having two outflow edges the finite element error is O(hp+1) superconvergent at the end points of the inflow edge for an augmented space of degree p. Furthermore, we discovered additional mesh-orientation dependent superconvergence points in the interior of triangles. The dependence of these points on orientation is explicitly given. We also established a global superconvergence result on meshes consisting of triangles having one inflow and one outflow edges. Applying a local error analysis, we construct simple, efficient and asymptotically correct a posteriori error estimates for discontinuous finite element solutions of hyperbolic problems on triangular meshes. A posteriori error estimates are needed to guide adaptive enrichment and to provide a measure of solution accuracy for any numerical method. We develop an inexpensive superconvergence-based a posteriori error estimation technique for the DG solutions of conservation laws. We explicitly write the basis functions for the error spaces corresponding to several finite element solution spaces. The leading term of the discretization error on each triangle is estimated by solving a local problem where no boundary conditions are needed. The computed error estimates are shown to converge to the true error under mesh refinement in smooth solution regions. We further present a numerical study of superconvergence properties for the DG method applied to time-dependent convection problems. We also construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on general unstructured meshes. The global superconvergence results are numerically confirmed. Finally, the a posteriori error estimates are tested on several linear and nonlinear problems to show their efficiency and accuracy under mesh refinement. / Ph. D.

We propose a new discontinuous finite element method for higher-order initial value problems where the finite element solution exhibits an optimal convergence rate in the L2- norm. We further show that the q-degree discontinuous solution of a differential equation of order m and its first (m-1)-derivatives are strongly superconvergent at the end of each step. We also establish that the q-degree discontinuous solution is superconvergent at the roots of (q+1-m)-degree Jacobi polynomial on each step. Furthermore, we use these results to construct asymptotically correct a posteriori error estimates. Moreover, we design a new discontinuous Galerkin method to solve the wave equation by using a method of lines approach to separate the space and time where we first apply the classical finite element method using p-degree polynomials in space to obtain a system of second-order ordinary differential equations which is solved by our new discontinuous Galerkin method. We provide an error analysis for this new method to show that, on each space-time cell, the discontinuous Galerkin finite element solution is superconvergent at the tensor product of the shifted roots of the Lobatto polynomials in space and the Jacobi polynomial in time. Then, we show that the global L2 error in space and time is convergent. Furthermore, we are able to construct asymptotically correct a posteriori error estimates for both spatial and temporal components of errors. We validate our theory by presenting several computational results for one, two and three dimensions. / Ph. D.

In this dissertation we prove the superconvergence of the minimal-dissipation local discontinuous Galerkin method for elliptic problems and construct optimal immersed finite element approximations and discontinuous immersed finite element methods for the Stokes interface problem. In the first part we present an error analysis for the minimal dissipation local discontinuous Galerkin method applied to a model elliptic problem on Cartesian meshes when polynomials of degree at most <i>k</i> and an appropriate approximation of the boundary condition are used. This special approximation allows us to achieve <i>k</i> + 1 order of convergence for both the potential and its gradient in the L² norm. Here we improve on existing estimates for the solution gradient by a factor &#8730;h. In the second part we present discontinuous immersed finite element (IFE) methods for the Stokes interface problem on Cartesian meshes that does not require the mesh to be aligned with the interface. As such, we allow unfitted meshes that are cut by the interface. Thus, elements may contain more than one fluid. On these unfitted meshes we construct an immersed Q₁/Q₀ finite element approximation that depends on the location of the interface. We discuss the basic features of the proposed Q₁/Q₀ IFE basis functions such as the unisolvent property. We present several numerical examples to demonstrate that the proposed IFE approximations applied to solve interface Stokes problems maintain the optimal approximation capability of their standard counterpart applied to solve the homogeneous Stokes problem. Similarly, we also show that discontinuous Galerkin IFE solutions of the Stokes interface problem maintain the optimal convergence rates in both L² and broken H¹ norms. Furthermore, we extend our method to solve the axisymmetric Stokes interface problem with a moving interface and test the proposed method by solving several benchmark problems from the literature. / Ph. D.

The random discontinuous fiber composite has uniform properties in all directions. The wetlay process is an efficient method to manufacture random discontinuous thermoplastic preform sheets that can be molded into random composite plaques in the hot-press. Investigations were done on the molding parameters that included the set-point mold pressure, set-point mold temperature and cooling methods. The fibers used in the study included glass and carbon fiber. Polypropylene (PP) and Polyethylene Terephthalate (PET) were used as the matrix. Glass/PP and Glass/PET plaques that had fiber volume fractions ranging from 0.05 to 0.50 at an increment of 0.05 were molded. Both tensile and flexural tests were conducted. The test results showed a common pattern, i.e., the modulus and strength of the composite increased with the fiber volume fraction to a maximum and then started to descend. The test results were analyzed to find out the optimal fiber volume fraction that yielded the maximum modulus or strength. Carbon/PET composites plaques were also molded to compare their properties with Glass/PET composite at similar fiber volume fractions. Micrographs were taken of selected specimens to examine the internal structure of the material. Existing micromechanics models that predict the tensile modulus or strength of random fiber composites were examined. Predictions from some of the models were compared with test data. / Master of Science

The appeal of hybrid composites is the ability to create materials with properties which normally do not coexist such as high specific strength, stiffness, and toughness. One possible application for hybrid composites is as backplate materials in layered armor. Fiber reinforced composites have been used as backplate materials due to their potential to absorb more energy than monolithic materials at similar to lower weights through microfragmentation of the fiber, matrix, and fiber-matrix interface. Composite backplates are traditionally constructed from graphite or glass fiber reinforced epoxy composites. However, continuous alumina fiber-reinforced aluminum metal matrix composites (MMCs) have superior specific transverse and specific shear properties than epoxy composites. Unlike the epoxy composites, MMCs have the ability to absorb additional energy through plastic deformation of the metal matrix. Although, these enhanced properties may make continuous alumina reinforced MMCs advantageous for use as backplate materials, they still exhibit a low failure strain and therefore have low toughness. One possible solution to improve their energy absorption capabilities while maintaining the high specific stiffness and strength properties of continuous reinforced MMCs is through hybridization. To increase the strain to failure and energy absorption capability of a continuous alumina reinforced Nextel" MMC, it is laminated with a high failure strain Saffil® discontinuous alumina fiber layer. Uniaxial tensile testing of hybrid composites with varying Nextel" to Saffil® reinforcement ratios resulted in composites with non-catastrophic tensile failures and an increased strain to failure than the single reinforcement Nextel" MMC. The tensile behavior of six hybrid continuous and discontinuous alumina fiber reinforced MMCs are reported, as well as a description of the mechanics behind their unique behavior. Additionally, a study on the effects of fiber damage induced during processing is performed to obtain accurate as-processed fiber properties and improve single reinforced laminate strength predictions. A stochastic damage evolution model is used to predict failure of the continuous Nextel" fabric composite which is then applied to a finite element model to predict the progressive failure of two of the hybrid laminates. / Ph. D.

Discontinuous deformation analysis (DDA) is a newly developed simulation method for discontinuous systems. It was designed to simulate systems with arbitrary shaped blocks with high efficiency while providing accurate solutions for energy dissipation. But DDA usually exhibits damping effects that are inconsistent with theoretical solutions. The deep reason for these artificial damping effects has been an open question, and it is hypothesized that these damping effects could result from the time integration scheme. In this thesis two time integration methods are investigated: the forward Euler method and the Newmark method. The work begins by combining the Newmark method and the DDA. An integrated Newmark method is also developed, where velocity and acceleration do not need to be updated. In simulations, two of the most widely used models are adopted to test the forward Euler method and the Newmark method. The first one is a sliding model, in which both the forward Euler method and the Newmark method give accurate solutions compared with analytical results. The second model is an impacting model, in which the Newmark method has much better accuracy than the forward Euler method, and there are minimal damping effects. / Master of Science

Orientador: Prof. Dr. Maurício Firmino Silva Lima / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2015. / Neste trabalho estudamos generalizações do Método de Melnikov para sistemas descontínuos no plano. Neste sentido, inicialmente abordamos esse problema como uma variação do estudo [1] onde um campo Hamiltoniano que admite um ciclo heteroclínico, cujo interior é folheado de órbitas periódicas, é perturbado por um campo Hamiltoniano não autonomo. Neste trabalho estendemos esse resultado para perturbações mais gerais (não conservativas) e apresentamos funções de Melnikov nesse novo contexto. Finalmente, abordamos o problema mais geral, relativo à perturbação de campos não conservativos, onde a função de Melnikov, associada a órbita heteroclínica, é obtida. / In this work we study generalizations of Melnikov's method to planar discontinuous dynamical system. Initially we study this problem as a variation of the work [1] where a Hamiltonian vector field that admits an heteroclinic cycle with its interior foliated by a family of periodic orbits is perturbed by a Hamiltonian perturbation. In this work we extended the results to more general perturbation (non conservative) and we show the Melnikov's functions in this new context. Finally, we approach a more general problem related to a perturbation of the non-conservative vector field where we obtained the Melnikov's function that is associated with a heteroclínic orbit.