Using experimental data obtained from standard fracture test configurations, theoretical and numerical tools are developed to mathematically describe non-self-similar progression of cracks without specifying an initial crack. A cohesive-decohesive zone model, similar to the cohesive zone model known in the fracture mechanics literature as the Dugdale-Barenblatt model, is adopted to represent the degradation of the material ahead of the crack tip. This model unifies strength-based crack initiation and fracture-mechanics-based crack progression. The cohesive-decohesive zone model is implemented with an interfacial surface material that consists of an upper and a lower surface that are connected by a continuous distribution of normal and tangential nonlinear elastic springs that act to resist either Mode I opening, Mode II sliding, Mode III sliding, or a mixed mode. The initiation of fracture is determined by the interfacial strength and the progression of the crack is determined by the critical energy release rate. The adhesive is idealized with an interfacial surface material to predict interfacial fracture. The interfacial surface material is positioned within the bulk material to predict discrete cohesive cracks. The interfacial surface material is implemented through an interface element, which is incorporated in ABAQUS using the user defined element (UEL) option. A procedure is established to formulate a rate dependent model based on experiments carried out on compact tension test specimens. The rate dependent model is incorporated into the interface element approach to capture the unstable crack growth observed in experiments under quasi-static loading conditions. The compact tension test gives the variation of the fracture toughness with the rate of loading, this information is processed and a relationship between the fracture toughness and the rate of the opening displacement is established. The cohesive-decohesive zone model is implemented through a material model to be used in an explicit code (LS-DYNA). Dynamic simulations of the standard test configurations for Mode I (Double Cantilever Beam) and Mode II (End Load Split) are carried out using the explicit code. Verification of these coupon tests leads to the crash analysis of realistic structures like the square composite tube. Analyses of bonded and unbonded square tubes are presented. These tubes shows a very uncharacteristic failure mode: the composite material disintegrates on impact, and this has been captured in the analysis. Disadvantages of the interface element approach are well documented in the literature. An alternative method, known as the Extended Finite Element Method (XFEM), is implemented here through an eight-noded quadrilateral plane strain element. The method, based on the partition-of-unity, is used to study simple test configuration like the three-point bend problem and a double cantilever beam. Functionally graded materials are also simulated and the results are compared to the experimental results available in the literature. / Ph. D.

The coatings produced by an atmospheric plasma spray process (APSP) must be of uniform quality. However, the complexity of the process and the random introduction of noise variables such as fluctuations in the powder injection rate and the arc voltage make it difficult to control the coating quality that has been shown to depend upon mean values of powder particles' temperature and speed, collectively called mean particles' states (MPSs), just before they impact the substrate. Here we use a science-based methodology to develop an adaptive controller for achieving consistent MPSs. We first identify inputs into the APSP that significantly affect the MPSs, and then formulate a relationship between these two quantities. When the MPSs deviate from their desired values, the adaptive controller based on the model reference adaptive controller (MRAC) framework is shown to successfully adjust the input parameters to correct them. The performance of the controller is tested via numerical experiments using the software, LAVA-P, that has been shown to well simulate the APSP. The developed adaptive process controller is further refined by using sigma (σ) adaptive laws and including a low-pass filter that remove high-frequency oscillations in the output. The utility of the MRAC controller to achieve desired locations of NiCrAlY and zirconia powder particles for generating a 5-layered coating is demonstrated. In this case a pure NiCrAlY layer bonds to the substrate and a pure zirconia makes the coating top. The composition of the intermediate 3 layers is combination of the two powders of different mass fractions. By increasing the number of intermediate layers, one can achieve a continuous through-the-thickness variation of the coating composition and fabricate a functionally graded coating. / Doctor of Philosophy / Canned food sold in a grocery store have cans' interior surface coating with a polymer to increase the shelf life of the food. Similarly, many parts in an automobile have coatings to protect them from corrosion and possibly wear and tear. A process used to produce these coatings is rather complex and involves several variables. An undesired change these variables affects the coating quality. Automatically controlling a coating process is like a cruise control in a car. It should detect which variables have changed and either take appropriate corrective actions or shut down the process if it cannot be corrected or alert an operator to stop the process. In this work we have developed a controller to adaptively adjust the input parameters for an atmospheric plasma spray process (APSP) often used to produce thermal barrier coatings in gas turbines and blades of aircraft jet engines. These coatings hinder the flow of heat from the hot exhaust gases to the blades thereby prolonging their life span.

A key advantage of Additive Manufacturing (AM) is the opportunity to design and fabricate complex structures that cannot be made via traditional means. However, this potential is significantly constrained by the use of a facet-based geometry representation (e.g., the STL and the AMF file formats); which do not contain any volumetric information and often, designing/slicing/printing complex geometries exceeds the computational power available to the designer and the AM system itself. To enable efficient design and fabrication of complex/multi-material complex structures, several algorithms are presented that represent and process solid models as a set of voxels (three-dimensional pixels). Through this, one is able to efficiently realize parts featuring complex geometries and functionally graded materials. This thesis specifically aims to explore applications in three distinct fields namely, (i) Design for AM, (ii) Design for Manufacturing (DFM) education, and (iii) Reverse engineering from imaging data wherein voxel-based representations have proven to be superior to the traditional AM digital workflow. The advantages demonstrated in this study cannot be easily achieved using traditional AM workflows, and hence this work emphasizes the need for development of new voxel based frameworks and systems to fully utilize the capabilities of AM. / MS / Additive Manufacturing(AM) (also referred to as 3D Printing) is a process by which 3D objects are constructed by successively forming one-part cross-section at a time. Typically, the input file format for most AM systems is in the form of surface representation format (most commonly. STL file format). A STL file is a triangular representation of a 3-dimensional surface geometry where the part surface is broken down logically into a series of small triangles (facets). A key advantage of Additive Manufacturing is the opportunity to design and fabricate complex structures that cannot be made easily via traditional manufacturing techniques. However, this potential is significantly constrained by the use of a facet-based (triangular) geometry representation (e.g., the STL file format described above); which does not contain any volumetric (for e.g. material, texture, color etc.) information. Also, often, designing/slicing/printing complex geometries using these file formats can be computationally expensive. To enable more efficient design and fabrication of complex/multi-material structures, several algorithms are presented that represent and process solid models as a set of voxels (three-dimensional pixels). A voxel represents the smallest representable element of volume. For binary voxel model, a value of ‘1’ means that voxel is ‘on’ and value of 0 means voxel is ‘off’. Through this, one is able to efficiently realize parts featuring complex geometries with multiple materials. This thesis specifically aims to explore applications in three distinct fields namely, (i) Design for AM, (ii) Design for Manufacturing (DFM) education, and (iii) Fabricating models (Reverse engineering) directly from imaging data. In the first part of the thesis, a software tool is developed for automated manufacturability analysis of a part that is to be produced by AM. Through a series of simple computations, the tool provides feedback on infeasible features, amount of support material, optimum orientation and manufacturing time for fabricating the part. The results from this tool were successfully validated using a simple case study and comparison with an existing pre-processing AM software. Next, the above developed software tool is implemented for teaching instruction in a sophomore undergraduate classroom to improve students’ understanding of design constraints in Additive Manufacturing. Assessments are conducted to measure students’ understanding of a variety of topics in manufacturability both before and after the study to measure the effectiveness of this approach. The third and final part of this thesis aims to explore fabrication of models directly from medical imaging data (like CT Scan and MRI). A novel framework is proposed which is validated by fabricating three distinct medical models: a mouse skull, a partial human skull and a horse leg directly from corresponding CT Scan data. The advantages demonstrated in this thesis cannot be easily achieved using traditional AM workflows, and hence this work emphasizes the need for development of new voxel based frameworks and systems to fully utilize the capabilities of AM.

Strength and toughness are mutually exclusive mechanical properties; an increase in one result in the decline in the other. Accordingly, ceramics with superior strength have a very low toughness; likewise, metals with similar density have relatively lower strength but higher toughness. However, biological systems design lightweight materials, circumventing this limitation of conventional materials, by aggregating various multiscale toughening mechanisms. In challenging habitats, organisms evolve to produce remarkable multifunctional material systems that improve their "fit" and "survivability". Unlike traditional materials, natural materials employ special arrangements of structural elements into cellular, gradient, fibrous, layered, or overlapped "architected composites". These natural material systems are "architected" to delocalize damage and prevent defect coalescence, to avoid catastrophic failure, even though they are mainly composed of brittle building blocks (>90 vol% mineral content). Consequently, the study of natural materials has attracted the attention of scientists as the benchmark for the development of new synthetic materials. With the advent of additive manufacturing technology, the design and assessment of architected composites with bio-inspired motifs have become increasingly feasible. In this dissertation, I use multi-step fabrication methods with additive manufacturing as a key step to produce and study different biologically inspired architectures. With control over the design parameters of the architectural features, an in-depth understanding of the organization is accomplished. The case studies are primarily focused on bulk composite material systems with multiple phases and motifs inspired by various biological material systems. This dissertation aims to reveal the structure-property relationships of these structural motifs and the trade-offs to the mechanical robustness due growth-related constraints. With the help of stereolithographic additive manufacturing technique and centrifugal infiltration, we propose a bio-inspired method for preparing ceramic-metal composites. The approach allowed for flexible design, scalability, and dimensional control of individual phases. The ceramic-metal composites were fabricated with structures simplified from the mollusk shell architectures, exhibiting specific strength up to 169% higher than the base metal. The crack growth toughness of up to 12.9 MPa m1/2 was recorded, with crack deflection at ceramic-metal interfaces. Additionally, using tomographic analysis we show that the high porosities of 9% and 15% for green and sintered 3D printed parts, if improved, could further enhance the strength and fracture toughness of these composites. The outer protective layer of a bivalve mollusk exoskeleton, called the prismatic layer, is composed of normally oriented prismatic building blocks separated by soft organic matrix. The growth of the prismatic layer is regulated by the thermodynamic boundary conditions of the habitat and is directed from the exterior to the interior of the shell. A consequence of growth is a graded structure with a fine side (higher grain count with smaller grain size) and a coarse side (higher grain count with smaller grain size), however, the presence of grading results in asymmetry. Using mechanical testing we reveal that the organisms' selection of fine side as the loading face is "not the most optimized arrangement for templating". In fact, opting for the coarse side over the fine side as the loading side simultaneously enhances mutually exclusive properties such as stiffness, strength, and energy absorption. We further show that the curved prism motifs in the proximal parts of the Ostrea edulis shells result in a significant reduction in mechanical robustness due to the growth-related restrictions arising from the simultaneous normal and lateral growth of shells. Moreover, we show that although the addition of a nacre-like backing layer reduces the effects of axial directional asymmetry, the resistance of the prismatic layer to initiate damage in a coarse side-loaded hybrid composite is superior to the fine side-loaded counterpart. This part of the research highlights the need for caution when directly mimicking structural designs found in biological systems. Biological material systems are typically multifunctional, tailored to specific habitats and organism-specific needs, and often constrained by growth requirements and economic limitations. The shells of the pteropods – pelagic gastropod species, are comprised of helical or as posited by certain researchers "S-shaped" aragonite mineral motifs. These helical motifs are remarkably close packed in an organic matrix without noticeable spaces. We develop a biological process mimicking image processing technique called the "Bottom-up Sectional Morphing" to model perfectly closed packed structures with control over the radius and pitch of the helical motifs. With the developed composites we attempt to characterize the effect of the helix radius of individual motifs on the global mechanical properties. With the help of compressive tests, we characterize the delocalization of load as the radius of the helical motifs is increased. With the help of slab-shaped samples, we study the puncture resistance and interlocking behaviors due to increased helical radius. Using standardized fracture toughness tests, the toughness of the composites is determined. Additionally, the R-curve behaviors as a function of helical radiuses is characterized. On average, the fracture strength of the composite doubled as the radius of the helical motifs increased from 0 mm to 3.9 mm. Remarkably, the fracture toughness of helical composites was as high as 12-times the rule-of-mixtures estimated values. We summarize the extrinsic toughening mechanisms within the composites compared them to the mechanisms reported for helicoidal (twisted plywood) composites. Additional interlocking due to the uneven orientation of major axes in double basket weave pattern helical system are reported. Using explicit finite element simulations, we show that the curved motifs in comparison to normally oriented prisms, can help in developing localized high stress pockets, thus delocalization of damage that can help in increasing energy absorption during the progression of damage. Also, taking cues from fish scale ultrastructures, we design three-phase ceramic-epoxy-fiber composites. The fish scales feature gradient architectures with varying biomineralization extents from the distal to proximal regions (with respect to the fish body). From exterior to interior the mineralization content reduces, however, the collagen fiber count subsequently increases. To mimic the design approach, we use a 3D printed gradient ceramic lattice embedded in an epoxy matrix and backed using Kevlar fibers. With high-speed impact tests (73.5 ± 2.5 ms-1) we show that, although functionally graded composites (without Kevlar backing) show larger impact signatures compared to the similar density uniform density composites (without Kevlar backing) but absorb 35.7% higher energy during the process. High rebound velocity (22 ± 2.46 m/s) was observed for variable density composites with Kevlar backing. Additionally, using micro computed tomographic analysis of variable density composites with Kevlar backing we demonstrate that pre-stretching of fibers helps in the suppression crack. The results from this study were used in the design of polymer-elastomer composites with functionally graded material and fiber distribution. Interweaving fibers with hard solid lattices becomes challenging when one of the planar surfaces of the lattice is closed because of the functional grading. To overcome this challenge, I propose a new lattice interweaving method called "Warp-Assisted Binder-Tugging (WABT)", that can interweave the lattice using only one of the planar faces. Using WABT we refine the 3-phase composites design by incorporating strategically placed internal reinforcements. Cured photopolymer thermoset plastics are intrinsically brittle materials with mechanical properties like that of epoxy. Therefore, we choose this material along with urethane elastomer to prepare polymer-elastomer (hard-soft) composites, with and without reinforcements. We demonstrate the efficacy of strategic material distributions using dynamic puncture tests and projectile impact tests. The results show that concentrating brittle plastics towards the loading side improves energy absorption ability by 30.29% and puncture strength by 21.47%. A further 61.76% and 35.12% improvement in the energy absorption and puncture strength is recorded for slabs with backing and reinforcements. We show the response of the as-prepared composites under high speed projectile impact tests with incident projectile speeds of 151.5 ± 2.5 ms-1. The μ-CT characterization of damaged samples revealed the load delocalization and crack suppression behaviors due to the material distributions and reinforcements. / Doctor of Philosophy / It is challenging to develop materials that are strong and tough at the same time. Ceramics, for example, are very strong, but are highly sensitive to the inherent defects and subsequently, upon initiation of damage, fail catastrophically. Metals on the other hand are not as strong as ceramics but require high energy for failure. Biological materials, using ingeniously designed and organized brittle elements can combine strength and toughness into a single system. In this dissertation, I investigated various bioinspired material systems to characterize their structure-property relationships. The analysis of structures inspired by the biological materials provides valuable insights that will potentially benefit the design of new protective systems. In this dissertation, I fabricate, and study biological designs found in the bivalve mollusk shells, pteropod shells, and fish scales. Using experimental and computational methods the I studied the effects of design parameters on the mechanical robustness of the composites. Contrary to the common belief that biological systems are highly optimized, I show that the biological materials could feature "less-than-perfect" design arrangements. The case studies aim to highlight the mechanisms that help organisms to resist damage and survive in their challenging environments. These case studies allowed us to understand the design strategies as well as limitations that can help us develop mechanically robust materials based on biological materials.

The following study is part of a larger study examining techniques that might be of use in the release program of the Puerto Rican Parrot (Amazona vittata), including marking, capturing, and radio-tracking. The portion of the study reported here documents the vocal behavior of A. vittata during the reproductive season and examines the possibility of using vocalizations to identify individuals, determine the sex of individuals and determine the location of an individual's breeding territory. Objectives of this study included: 1) cataloguing and categorizing the vocal repertoire of A. vittata, 2) determining whether the vocal repertoire was sex-specific and region-specific and 3) determining if an individual's vocal repertoire could be used to identify it. The vocal repertoire was characterized using a hierarchical method and 147 calls were described. The repertoire was found contain a high percentage (76 %) of graded calls. Evolutionary strategies that may explain the complexity of such a repertoire are discussed. The vocal repertoire was found to be both sex- and region-specific. Characteristics analyzed included time and frequency parameters of sonagrams. Three methods were used to determine the feasibility of vocal recognition of individuals. These methods included: bird-call pairing, sonagraphic analysis, and linear predictive coding. Sonagraphic analyses in combination with linear predictive coding techniques show the most promise as tools in voice recognition of the parrot, however, further research will be necessary to determine how reliable voice recognition may be as a method for identifying individuals in the field. / Master of Science

Since the invention of Laser (in 1960) and low loss optical fiber (in 1966) [1], extensive research in fiber-optic sensing technology has made it a well-defined and matured field [1]. The measurement of physical parameters (such as temperature and pressure) in extremely harsh environment is one of the most intriguing challenges of this field, and is highly valued in the automobile industry, aerospace research, industrial process monitoring, etc. [2]. Although the semiconductor based sensors can operate at around 500oC, sapphire fiber sensors were demonstrated at even higher temperatures [3]. In this research, a novel sensor structure is proposed that can measure both pressure and temperature simultaneously. This work effort consists of design, fabrication, calibration, and laboratory testing of a novel structured temperature compensated pressure sensor. The aim of this research is to demonstrate an accurate temperature measurement, and pressure measurement using a composite Fabry-Perot interferometer. One interferometer measures the temperature and the other accurately measures pressure after temperature compensation using the temperature data from the first sensor. / Master of Science

Grafts currently used for the repair of anterior cruciate ligament (ACL) ruptures integrate poorly with bone due to a significant mismatch in properties between graft and bone. Specifically, conventional grafts (e.g., hamstring tendon) are unable to recapitulate intricate gradients in mechano-chemical properties and extracellular matrix (ECM) architecture found at natural bone-ligament (B-L) transitions, and thus result in stress-concentrations at the graft-bone interface leading to graft failure. In contrast, tissue-engineered scaffolds possessing gradients in properties can potentially guide the establishment of phenotypic gradients in bone marrow stromal cells (BMSCs), and thus aid the regeneration of B-L transitions in the long-term. Towards the eventual goal of regenerating complex tissue transitions, this project employs three complementary strategies to fabricate graded scaffolds. The three strategies involve the presentation of gradients in 1) mineral content, 2) scaffold architecture and 3) growth factor (GF) concentration within scaffolds to control BMSC morphology and phenotype. The first strategy involved co-electrospinning two polymers (one doped with hydroxyapatite) from offset spinnerets onto a rotating drum to produce scaffolds possessing a gradient in mineral content. Post-electrospinning, these graded scaffolds were treated with a simulated body fluid to further enhance the gradient. Analysis of mRNA expression of osteoblastic makers by BMSCs and the deposition of bone-specific ECM proteins indicated that the scaffolds could guide the formation of an osteoblastic phenotypic gradient. The second strategy involved electrospinning two polymer solutions onto a custom-designed dual-drum collector to fabricate scaffolds possessing region-wise differences in fiber alignment, diameter and chemistry. Specifically, electrospinning onto the dual-drum collector resulted in the deposition of aligned fibers from one polymer solution in the gap region between the drums, randomly oriented fibers from the other polymer solution on one of the drums and a mixture of fibers from both polymer solutions in the overlap region in between. The topographical cues within these scaffolds were shown to result in region-dependent BMSC morphology and orientation. Although the long-term goal of the third strategy was to create a co-electrospun scaffold possessing a gradient in GF concentration, a new technique to protect GF activity within electrospun scaffolds via the use of gelatin microspheres was first validated. Preliminary results from these studies indicate that microspheres can protect and deliver a model protein (lysozyme) in active conformation from electrospun scaffolds. These results further suggest that gradients of GF concentration can be achieved in the long-term by protecting GFs within microspheres and co-electrospinning as described in the first strategy. In conclusion, the results from this project suggest that graded scaffolds can help guide the formation of gradients in cell morphology, orientation and phenotype, and thus potentially promote the regeneration of B-L transitions in the long-term. The three strategies described in this project can be employed in concert to create scaffolds intended for the regeneration of complex tissue transitions. / Ph. D.

Curved beams such as arches find ubiquitous applications in civil, mechanical and aerospace engineering, e.g., stiffened floors, fuselage, railway compartments, and wind turbine blades. The analysis of free vibrations of curved structures plays a critical role in their design to avoid transient loads with dominant frequencies close to their natural frequencies. One way to increase their areas of applications and possibly make them lighter without sacrificing strength is to make them of Functionally Graded Materials (FGMs) that are composites with continuously varying material properties in one or more directions. In this thesis, we study free vibrations of FGM circular beams by using a logarithmic shear deformation theory that incorporates through-the-thickness logarithmic variation of the circumferential displacement, and does not require a shear correction factor. The radial displacement of a point is assumed to depend only upon its angular position. Thus the beam theory can be regarded as a generalization of the Timoshenko beam theory. Equations governing transient deformations of the beam are derived by using Hamilton's principle. Assuming a time harmonic variation of the displacements, and by utilizing the generalized differential quadrature method (GDQM) the free vibration problem is reduced to solving an algebraic eigenvalue problem whose solution provides frequencies and the corresponding mode shapes. Results are presented for different spatial variations of the material properties, boundary conditions, and the aspect ratio. It is found that the radial and the circumferential gradation of material properties maintains their natural frequency within that of the homogeneous beam comprised of a constituent of the FGM beam. Furthermore, keeping every other variable fixed, the change in the beam opening angle results in very close frequencies of the first two modes of vibration, a phenomenon usually called mode transition. / Master of Science / Curved and straight beams of various cross-sections are one of the simplest and most fundamental structural elements that have been extensively studied because of their ubiquitous applications in civil, mechanical, biomedical and aerospace engineering. Many attempts have been made to enhance their material properties and designs for applications in harsh environments and reduce weight. One way of accomplishing this is to combine layerwise two or more distinct materials and take advantage of their directional properties. It results in a lightweight structure having overall specific strength superior to that of its constituents. Another possibility is to have volume fractions of two or more constituents gradually vary throughout the structure for enhancing its performance under anticipated applications. Functionally graded materials (FGMs) are a class of composites whose properties gradually vary along one or more space directions. In this thesis, we have numerically studied free vibrations of FGM circular beams to enhance their application domain and possibly use them for energy harvesting.

Functionally graded structures have material properties that continuously vary in one or more directions. Examples include human teeth, seashells, bamboo stems and human organs, where the varying volume fraction of fibers and their orientations optimize functionality. Deformations of such structures typically involve bending, stretching, and shearing. An everyday example of shearing deformation is the twisting of wet fabrics to extract water. In this study, we analytically examine the large deformations of functionally graded Mooney-Rivlin circular cylinders, focusing on how radial grading of material moduli can be beneficially utilized. We investigate the finite deformations caused by pressures applied to the bounding surfaces and axial loads or twisting moments on the end faces. We also simulate residual stresses in a hollow cylinder either by inverting it inside out or by closing a longitudinal wedge opening parallel to the cylinder axis through axisymmetric deformation before other loads are applied. It is observed that the maximum shear stress in an initially stress-free Mooney-Rivlin cylinder can occur at an interior point. In the absence of axial forces on the end faces, the cylinder elongates when twisted, with the degree of elongation depending on the grading of the material moduli. These findings should aid numerical analysts in verifying their algorithms for simulating large deformations of rubber-like materials modeled by the Mooney-Rivlin relation. / Master of Science / Functionally graded materials (FGMs) are composites whose properties vary in one or more directions to exploit the functionality of the individual components. An example would be a sheet of material that is fully metallic on one side and fully ceramic on the other, with properties changing gradually through the thickness. The Mooney-Rivlin model is used to capture the stress-strain response of rubber-like materials. Therefore, functionally graded Mooney-Rivlin cylinders are rubber-like composite cylinders whose properties change throughout their thickness. Functionally graded cylinders have a wide array of applications, including in pressure vessels, vibration damping systems and tires. Therefore, having a thorough understanding of the stresses induced in these cylinders when subjected to loads is essential for safe and reliable designs. This research aims to investigate the effects of material inhomogeneity on the stresses induced in functionally graded cylinders subjected to torsion, radial expansion, eversion, and various combinations of these. Furthermore, realizing that stresses induced during the fabrication process cannot be easily quantified, we study a problem in which these induced stresses can be determined and analyze their effect on subsequent deformations of the cylinder when subjected to torsion and radial expansion. To achieve this aim, we use a member of Ericksen's third family of universal deformations, which mathematically describes torsion, inflation, and eversion, along with the Mooney-Rivlin model to determine the stress state resulting from deformation. The results show that for cylinders of the same geometry in the stress-free undeformed state subjected to identical surface tractions, material inhomogeneities greatly influence the stresses in the cylinder. It was also found that the magnitude of the normal and shear stresses, axial stretch, and the geometry of the cylinder after deformation depend on the type of deformation and functional grading. Additionally, the results indicate that the normal stresses induced in an initially stressed cylinder are much greater than those in a cylinder that is initially stress-free when subjected to the same boundary conditions.

Graded Hecke algebras are deformations of skew group algebras which arise from a group acting on a polynomial ring. Over fields of characteristic zero, these deformations have been studied in depth and include both symplectic reflection algebras and rational Cherednik algebras as examples. In Lusztig's graded affine Hecke algebras, the action of the group is deformed, but not the commutativity of the vectors. In Drinfeld's Hecke algebras, the commutativity of the vectors is deformed, but not the action of the group. Lusztig's algebras are all isomorphic to Drinfeld's algebras in the nonmodular setting. We find new deformations in the modular setting, i.e., when the characteristic of the underlying field divides the order of the group. We use Poincare-Birkhoff-Witt conditions to classify these deformations arising from the symmetric group acting on a polynomial ring in arbitrary characteristic, including the modular case.