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NASA NTRS2001

Higher-Order Theory for Functionally Graded Materials

Functionally graded materials (FGM's) are a new generation of engineered materials wherein the microstructural details are spatially varied through nonuniform distribution of the reinforcement phase(s). Engineers accomplish this by using reinforcements with different properties, sizes, and shapes, as well as by interchanging the roles of the reinforcement and matrix phases in a continuous manner (ref. 1). The result is a microstructure that produces continuously or discretely changing thermal and mechanical properties at the macroscopic or continuum scale. This new concept of engineering the material's microstructure marks the beginning of a revolution both in the materials science and mechanics of materials areas since it allows one, for the first time, to fully integrate the material and structural considerations into the final design of structural components. Functionally graded materials are ideal candidates for applications involving severe thermal gradients, ranging from thermal structures in advanced aircraft and aerospace engines to computer circuit boards. Owing to the many variables that control the design of functionally graded microstructures, full exploitation of the FGM's potential requires the development of appropriate modeling strategies for their response to combined thermomechanical loads. Previously, most computational strategies for the response of FGM's did not explicitly couple the material's heterogeneous microstructure with the structural global analysis. Rather, local effective or macroscopic properties at a given point within the FGM were first obtained through homogenization based on a chosen micromechanics scheme and then subsequently used in a global thermomechanical analysis.

Aboudi, J.↗

NASA NTRS1999

Higher-Order Theory for Functionally Graded Materials

This paper presents the full generalization of the Cartesian coordinate-based higher-order theory for functionally graded materials developed by the authors during the past several years. This theory circumvents the problematic use of the standard micromechanical approach, based on the concept of a representative volume element, commonly employed in the analysis of functionally graded composites by explicitly coupling the local (microstructural) and global (macrostructural) responses. The theoretical framework is based on volumetric averaging of the various field quantities, together with imposition of boundary and interfacial conditions in an average sense between the subvolumes used to characterize the composite's functionally graded microstructure. The generalization outlined herein involves extension of the theoretical framework to enable the analysis of materials characterized by spatially variable microstructures in three directions. Specialization of the generalized theoretical framework to previously published versions of the higher-order theory for materials functionally graded in one and two directions is demonstrated. In the applications part of the paper we summarize the major findings obtained with the one-directional and two-directional versions of the higher-order theory. The results illustrate both the fundamental issues related to the influence of microstructure on microscopic and macroscopic quantities governing the response of composites and the technologically important applications. A major issue addressed herein is the applicability of the classical homogenization schemes in the analysis of functionally graded materials. The technologically important applications illustrate the utility of functionally graded microstructures in tailoring the response of structural components in a variety of applications involving uniform and gradient thermomechanical loading.

Aboudi, Jacob↗

NASA NTRS2012

Functionally Graded Adhesives for Composite Joints

Adhesives with functionally graded material properties are being considered for use in adhesively bonded joints to reduce the peel stress concentrations located near adherend discontinuities. Several practical concerns impede the actual use of such adhesives. These include increased manufacturing complications, alterations to the grading due to adhesive flow during manufacturing, and whether changing the loading conditions significantly impact the effectiveness of the grading. An analytical study is conducted to address these three concerns. An enhanced joint finite element, which uses an analytical formulation to obtain exact shape functions, is used to model the joint. Furthermore, proof of concept testing is conducted to show the potential advantages of functionally graded adhesives. In this study, grading is achieved by strategically placing glass beads within the adhesive layer at different densities along the joint.

Stapleton, Scott E.↗

NASA NTRS2006

Low-Velocity Impact Response of Sandwich Beams with Functionally Graded Core

The problem of low-speed impact of a one-dimensional sandwich panel by a rigid cylindrical projectile is considered. The core of the sandwich panel is functionally graded such that the density, and hence its stiffness, vary through the thickness. The problem is a combination of static contact problem and dynamic response of the sandwich panel obtained via a simple nonlinear spring-mass model (quasi-static approximation). The variation of core Young's modulus is represented by a polynomial in the thickness coordinate, but the Poisson's ratio is kept constant. The two-dimensional elasticity equations for the plane sandwich structure are solved using a combination of Fourier series and Galerkin method. The contact problem is solved using the assumed contact stress distribution method. For the impact problem we used a simple dynamic model based on quasi-static behavior of the panel - the sandwich beam was modeled as a combination of two springs, a linear spring to account for the global deflection and a nonlinear spring to represent the local indentation effects. Results indicate that the contact stiffness of thc beam with graded core Increases causing the contact stresses and other stress components in the vicinity of contact to increase. However, the values of maximum strains corresponding to the maximum impact load arc reduced considerably due to grading of thc core properties. For a better comparison, the thickness of the functionally graded cores was chosen such that the flexural stiffness was equal to that of a beam with homogeneous core. The results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites.

Apetre, N. A.↗

NASA NTRS1999

Sliding Wear and Fretting Wear of DLC-Based, Functionally Graded Nanocomposite Coatings

Improving the tribological functionality of diamondlike carbon (DLC) films--developing, good wear resistance, low friction, and high load-carrying capacity-was the aim of this investigation. Nanocomposite coatings consisting of an amorphous DLC (a-DLC) top layer and a functionally graded titanium-titanium carbon-diamondlike carbon (Ti-Ti(sub x) C(sub y)-DLC) underlayer were produced on AISI 440C stainless steel substrates by the hybrid technique of magnetron sputtering and pulsed-laser deposition. The resultant DLC films were characterized by Raman spectroscopy, scanning electron microscopy, and surface profilometry. Two types of wear experiment were conducted in this investioation: sliding friction experiments and fretting wear experiments. Unidirectional ball-on-disk sliding friction experiments were conducted to examine the wear behavior of an a-DLC/Ti-Ti(sub x) C(sub y)-DLC-coated AISI 440C stainless steel disk in sliding contact with a 6-mm-diameter AISI 440C stainless steel ball in ultrahigh vacuum, dry nitrogen, and humid air. Although the wear rates for both the coating and ball were low in all three environments, the humid air and dry nitrogen caused mild wear with burnishing, in the a-DLC top layer, and the ultrahigh vacuum caused relatively severe wear with brittle fracture in both the a-DLC top layer and the Ti-Ti(sub x) C(sub y)-DLC underlayer. For reference, amorphous hydrogenated carbon (H-DLC) films produced on a-DLC/Ti-Ti(sub x) C(sub y)-DLC nanocomposite coatings by using an ion beam were also examined in the same manner. The H-DLC films markedly reduced friction even in ultrahigh vacuum without sacrificing wear resistance. The H-DLC films behaved much like the a-DLC/Ti-Ti(sub x) C(sub y)-DLC nanocomposite coating in dry nitrogen and humid air, presenting low friction and low wear. Fretting wear experiments were conducted in humid air (approximately 50% relative humidity) at a frequency of 80 Hz and an amplitude of 75 micron on an a-DLC/Ti-Ti(sub x) C(sub y)-DLC-coated AISI 440C disk and on a titanium-6 wt % aluminum-4 wt% vanadium (Ti-6Al-4V) flat, both in contact with a 9.4-mm-diameter, hemispherical Ti-6Al-4V pin. The resistance to fretting wear and damage of the a-DLC/Ti-6Al-4V materials pair was superior to that of the Ti-6Al-4V/Ti-6AI-4V materials pair.

Miyoshi, K.↗

NASA NTRS1998

HOTCFGM-1D: A Coupled Higher-Order Theory for Cylindrical Structural Components with Through-Thickness Functionally Graded Microstructures

The objective of this three-year project was to develop and deliver to NASA Lewis one-dimensional and two-dimensional higher-order theories, and related computer codes, for the analysis, optimization and design of cylindrical functionally graded materials/structural components for use in advanced aircraft engines (e.g., combustor linings, rotor disks, heat shields, blisk blades). To satisfy this objective, a quasi one-dimensional version of the higher-order theory, HOTCFGM-1D, and four computer codes based on this theory, for the analysis, design and optimization of cylindrical structural components functionally graded in the radial direction were developed. The theory is applicable to thin multi-phased composite shell/cylinders subjected to macroscopically axisymmetric thermomechanical and inertial loading applied uniformly along the axial direction such that the overall deformation is characterized by a constant average axial strain. The reinforcement phases are uniformly distributed in the axial and circumferential directions, and arbitrarily distributed in the radial direction, thereby allowing functional grading of the internal reinforcement in this direction.

Pindera, Marek-Jerzy↗

NASA NTRS2006

MODELING FUNCTIONALLY GRADED INTERPHASE REGIONS IN CARBON NANOTUBE REINFORCED COMPOSITES

A combination of micromechanics methods and molecular dynamics simulations are used to obtain the effective properties of the carbon nanotube reinforced composites with functionally graded interphase regions. The multilayer composite cylinders method accounts for the effects of non-perfect load transfer in carbon nanotube reinforced polymer matrix composites using a piecewise functionally graded interphase. The functional form of the properties in the interphase region, as well as the interphase thickness, is derived from molecular dynamics simulations of carbon nanotubes in a polymer matrix. Results indicate that the functional form of the interphase can have a significant effect on all the effective elastic constants except for the effective axial modulus for which no noticeable effects are evident.

Seidel, G. D.↗

NASA NTRS1995

Thermoelastic Theory for the Response of Materials Functionally Graded in Two Directions with Applications to the Free-Edge Problem

A recently developed micromechanical theory for the thermoelastic response of functionally graded composites with nonuniform fiber spacing in the through-thickness direction is further extended to enable analysis of material architectures characterized by arbitrarily nonuniform fiber spacing in two directions. In contrast to currently employed micromechanical approaches applied to functionally graded materials, which decouple the local and global effects by assuming the existence of a representative volume element at every point within the composite, the new theory explicitly couples the local and global effects. The analytical development is based on volumetric averaging of the various field quantities, together with imposition of boundary and interfacial conditions in an average sense. Results are presented that illustrate the capability of the derived theory to capture local stress gradients at the free edge of a laminated composite plate due to the application of a uniform temperature change. It is further shown that it is possible to reduce the magnitude of these stress concentrations by a proper management of the microstructure of the composite plies near the free edge. Thus by an appropriate tailoring of the microstructure it is possible to reduce or prevent the likelihood of delamination at free edges of standard composite laminates.

Aboudi, Jacob↗

NASA NTRS2014

Aeroelastic Tailoring of a Plate Wing with Functionally Graded Materials

This work explores the use of functionally graded materials for the aeroelastic tailoring of a metallic cantilevered plate-like wing. Pareto trade-off curves between dynamic stability (flutter) and static aeroelastic stresses are obtained for a variety of grading strategies. A key comparison is between the effectiveness of material grading, geometric grading (i.e., plate thickness variations), and using both simultaneously. The introduction of material grading does, in some cases, improve the aeroelastic performance. This improvement, and the physical mechanism upon which it is based, depends on numerous factors: the two sets of metallic material parameters used for grading, the sweep of the plate, the aspect ratio of the plate, and whether the material is graded continuously or discretely.

Dunning, Peter D.↗

NASA NTRS2003

Functionally Graded Nanophase Beryllium/Carbon Composites

Beryllium, beryllium alloys, beryllium carbide, and carbon are the ingredients of a class of nanophase Be/Be2C/C composite materials that can be formulated and functionally graded to suit a variety of applications. In a typical case, such a composite consists of a first layer of either pure beryllium or a beryllium alloy, a second layer of B2C, and a third layer of nanophase sintered carbon derived from fullerenes and nanotubes. The three layers are interconnected through interpenetrating spongelike structures. These Be/Be2C/C composite materials are similar to Co/WC/diamond functionally graded composite materials, except that (1) W and Co are replaced by Be and alloys thereof and (2) diamond is replaced by sintered carbon derived from fullerenes and nanotubes. (Optionally, one could form a Be/Be2C/diamond composite.) Because Be is lighter than W and Co, the present Be/Be2C/C composites weigh less than do the corresponding Co/WC/diamond composites. The nanophase carbon is almost as hard as diamond. WC/Co is the toughest material. It is widely used for drilling, digging, and machining. However, the fact that W is a heavy element (that is, has high atomic mass and mass density) makes W unattractive for applications in which weight is a severe disadvantage. Be is the lightest tough element, but its toughness is less than that of WC/Co alloy. Be strengthened by nanophase carbon is much tougher than pure or alloy Be. The nanophase carbon has an unsurpassed strength-to-weight ratio. The Be/Be2C/C composite materials are especially attractive for terrestrial and aerospace applications in which there are requirements for light weight along with the high strength and toughness of the denser Co/WC/diamond materials. These materials could be incorporated into diverse components, including cutting tools, bearings, rocket nozzles, and shields. Moreover, because Be and C are effective as neutron moderators, Be/Be2C/C composites could be attractive for some nuclear applications.

Voronov, Oleg A.↗

NASA NTRS2003

Functionally Graded Nanophase Beryllium/Carbon Composites

Choi, Michael K.↗

NASA NTRS1995

An evaluation of a coupled microstructural approach for the analysis of functionally graded composites via the finite-element method

A comparison is presented between the predictions of the finite-element analysis and a recently developed higher-order theory for functionally graded materials subjected to a thorough-thickness temperature gradient. In contrast to existing micromechanical theories that utilize classical (i.e., uncoupled) homogenization schemes to calculate micro-level and macro-level stress and displacement fields in materials with uniform or nonuniform fiber spacing (i.e., functionally graded materials), the new theory explicitly couples the microstructural details with the macrostructure of the composite. Previous thermo-elastic analysis has demonstrated that such coupling is necessary when: the temperature gradient is large with respect to the dimension of the reinforcement; the characteristic dimension of the reinforcement is large relative to the global dimensions of the composite and the number of reinforcing fibers or inclusions is small. In these circumstances, the standard micromechanical analyses based on the concept of the representative volume element used to determine average composite properties produce questionable results. The comparison between the predictions of the finite-element method and the higher-order theory presented herein establish the theory's accuracy in predicting thermal and stress fields within composites with a finite number of fibers in the thickness direction subjected to a thorough-thickness thermal gradient.

Pindera, Marek-Jerzy↗

NASA NTRS2001

Impact of Functionally Graded Cylinders: Theory

This final report summarizes the work funded under the Grant NAG3-2411 during the 04/05/2000-04/04/2001 period. The objective of this one-year project was to generalize the theoretical framework of the two-dimensional higher-order theory for the analysis of cylindrical functionally graded materials/structural components employed in advanced aircraft engines developed under past NASA Glenn funding. The completed generalization significantly broadens the theory's range of applicability through the incorporation of dynamic impact loading capability into its framework. Thus, it makes possible the assessment of the effect of damage due to fuel impurities, or the presence of submicron-level debris, on the life of functionally graded structural components. Applications involving advanced turbine blades and structural components for the reusable-launch vehicle (RLV) currently under development will benefit from the completed work. The theory's predictive capability is demonstrated through a numerical simulation of a one-dimensional wave propagation set up by an impulse load in a layered half-plane. Full benefit of the completed generalization of the higher-order theory described in this report will be realized upon the development of a related computer code.

Aboudi, Jacob↗

NASA NTRS1999

Analysis of a Generally Oriented Crack in a Functionally Graded Strip Sandwiched Between Two Homogeneous Half Planes

The driving forces for a generally oriented crack embedded in a Functionally Graded strip sandwiched between two half planes are analyzed using singular integral equations with Cauchy kernels, and integrated using Lobatto-Chebyshev collocation. Mixed-mode Stress Intensity Factors (SIF) and Strain Energy Release Rates (SERR) are calculated. The Stress Intensity Factors are compared for accuracy with previously published results. Parametric studies are conducted for various nonhomogeneity ratios, crack lengths. crack orientation and thickness of the strip. It is shown that the SERR is more complete and should be used for crack propagation analysis.

Shbeeb, N.↗

NASA NTRS2003

Thermal Characterization of Functionally Graded Materials: Design of Optimum Experiments

This paper is a study of optimal experiment design applied to the measure of thermal properties in functionally graded materials. As a first step, a material with linearly-varying thermal properties is analyzed, and several different tran- sient experimental designs are discussed. An optimality criterion, based on sen- sitivity coefficients, is used to identify the best experimental design. Simulated experimental results are analyzed to verify that the identified best experiment design has the smallest errors in the estimated parameters. This procedure is general and can be applied to design of experiments for a variety of materials.

Cole, Kevin D.↗

NASA NTRS1994

Thermal analysis of a functionally graded material subject to a thermal gradient using the boundary element method

The boundary element method is utilized in this study to conduct thermal analysis of functionally graded composites, materials in which the internal microstructure or properties are explicitly tailored in order to obtain an optimal response, on the micromechanical (constituent) scale. A unique feature of the boundary element formulations used here is the use of circular shape functions to convert the two-dimensional integrations of the composite fibers to one dimensional integrations. Using the computer code BEST-CMS, the through the thickness temperature profiles are computed for a representative material with varying numbers of fibers and fiber spacing in the thickness direction. The computed temperature profiles are compared to those obtained using an alternate analytical theory which explicitly couples the heterogeneous microstructure to the global analysis. The boundary element results compared favorably to the analytical calculations, with discrepancies that are explainable based on the boundary element formulation. The results serve both to demonstrate the ability of the boundary element method to analyze these types of materials, and to verify the accuracy of the analytical theory.

Goldberg, Robert K.↗

NASA NTRS2007

Freeze Tape Casting of Functionally Graded Porous Ceramics

Freeze tape casting is a means of making preforms of ceramic sheets that, upon subsequent completion of fabrication processing, can have anisotropic and/or functionally graded properties that notably include aligned and graded porosity. Freeze tape casting was developed to enable optimization of the microstructures of porous ceramic components for use as solid oxide electrodes in fuel cells: Through alignment and grading of pores, one can tailor surface areas and diffusion channels for flows of gas and liquid species involved in fuel-cell reactions. Freeze tape casting offers similar benefits for fabrication of optimally porous ceramics for use as catalysts, gas sensors, and filters.

Sofie, Stephen W.↗

NASA NTRS2003

Microstructure and Properties of Thermally Sprayed Functionally Graded Coatings for Polymeric Substrates

The use of polymer matrix composites (PMCs) in the gas flow path of advanced turbine engines offers significant benefits for aircraft engine performance but their useful lifetime is limited by their poor erosion resistance. High velocity oxy-fuel (HVOF) sprayed polymer/cermet functionally graded (FGM) coatings are being investigated as a method to address this technology gap by providing erosion and oxidation protection to polymer matrix composites. The FGM coating structures are based on a polyimide matrix filled with varying volume fractions of WC-Co. The graded coating architecture was produced using a combination of internal and external feedstock injection, via two computer-controlled powder feeders and controlled substrate preheating. Porosity, coating thickness and volume fraction of the WC-Co filler retained in the coatings were determined using standard metallographic techniques and computer image analysis. The pull-off strength (often referred to as the adhesive strength) of the coatings was evaluated according to the ASTM D 4541 standard test method, which measured the greatest normal tensile force that the coating could withstand. Adhesive/cohesive strengths were determined for three different types of coating structures and compared based on the maximum indicated load and the surface area loaded. The nature and locus of the fractures were characterized according to the percent of adhesive and/or cohesive failure, and the tested interfaces and layers involved were analyzed by Scanning Electron Microscopy.

Ivosevic, M.↗