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Discrete models for boundary value problems analysis in first strain-gradient elasticity theory, using extended finite element method
Hot section components of aircraft gas turbine engines are subjected to severe thermal-structural loading conditions, especially during the start-up and take-off portions of the engine cycle. The most severe and damaging stresses and strains are those induced by the steep thermal gradients induced during the start-up transient. These transient stresses and strains are also the most difficult to predict, in part because the temperature gradients and distributions are not well known or predictable, and also because the cyclic elasto-viscoplastic behavior of the materials at these extremes of temperature and strain are not well known or predictable. One element of the structures program will develop improved time-varying thermal-mechanical load models for the entire engine mission cycle from start-up to shutdown. The thermal model refinements will be consistent with those required by the structural code including considerations of mesh-point density, strain concentrations, and thermal gradients. Models will be developed for the burner liner, turbine vane and turbine blade.
A reference DNS database is presented, which includes third- and fourth-order moment budgets for unstrained and strained planar channel flow. Existing RANS closure models for third- and fourth-order terms are surveyed, and new model ideas are introduced. The various models are then compared with the DNS data term by term using a priori testing of the higher-order budgets of turbulence transport, velocity-pressure-gradient, and dissipation for both the unstrained and strained databases. Generally, the models for the velocity-pressure-gradient terms are most in need of improvement.
This paper describes a broad effort that is aimed at understanding the fundamental mechanisms of crack growth and using that understanding as a basis for designing materials and enabling predictions of fracture in materials and structures that have small characteristic dimensions. This area of research, herein referred to as Damage Science, emphasizes the length scale regimes of the nanoscale and the microscale for which analysis and characterization tools are being developed to predict the formation, propagation, and interaction of fundamental damage mechanisms. Examination of nanoscale processes requires atomistic and discrete dislocation plasticity simulations, while microscale processes can be examined using strain gradient plasticity, crystal plasticity and microstructure modeling methods. Concurrent and sequential multiscale modeling methods are being developed to analytically bridge between these length scales. Experimental methods for characterization and quantification of near-crack tip damage are also being developed. This paper focuses on several new methodologies in these areas and their application to understanding damage processes in polycrystalline metals. On-going and potential applications are also discussed.
The objective here is to test the Dynamic Localization (DL) model in a wall-bounded channel flow for numerical stability and accuracy of results. Algebraic stress models suggest that the model for the residual subgrid-scale (SGS) Reynolds stress and scalar flux should generally have terms comprising most of the unique products of the resolved strain (S) and rotation (R) tensors with S and the resolved scalar gradient. The standard dynamic SGS model uses a simple (Smagorinsky) base model for the residual Reynolds stress, which is made proportional to S, and down-gradient base models for residual scalar fluxes; these correspond to the lowest, 'first-order' terms in algebraic stress models. Temporal scaling terms in these base models are formed from the magnitude of the resolved strain rate. While this is appropriate for simple shear flows, it may not be appropriate for more complicated flows (relevant to geophysical and astrophysical problems) that include any combination of shear, rotation, buoyancy, etc. On the other hand, the coefficient in the dynamic SGS model readily adjusts itself to different flow conditions and may adequately take account of these effects without the need for more complicated base models. Cabot (1993) has begun to test the dynamic SGS model in buoyant flows (Rayleigh-Benard and internally heated convection) with and without buoyancy terms explicitly included in the scaling terms of the base model; no great differences were found in large eddy simulation (LES) results for the different base model scalings. The second objective in this work is to test base models with additional, 'second-order' terms (e.g., S(sup 2) and RS for the residual Reynolds stress). These terms have been found to improve large-scale flow predictions by kappa-epsilon models in the presence of rotation and shear. Second-order base models will be tested here in the LES of channel flow with and without solid-body rotation and compared with results from the standard first-order base models to determine if there are significant differences or improvements in results that would warrant the added complexity of the second-order base models.
The explicit transient dynamics technology in use today for simulating the impact and subsequent transient dynamic response of a structure has its origins in the 'hydrocodes' dating back to the late 1940's. The growth in capability in explicit transient dynamics technology parallels the growth in speed and size of digital computers. Computer software for simulating the explicit transient dynamic response of a structure is characterized by algorithms that use a large number of small steps. In explicit transient dynamics software there is a significant emphasis on speed and simplicity. The finite element technology used to generate the spatial discretization of a structure is based on a compromise between completeness of the representation for the physical processes modelled and speed in execution. That is, since it is expected in every calculation that the deformation will be finite and the material will be strained beyond the elastic range, the geometry and the associated gradient operators must be reconstructed, as well as complex stress-strain models evaluated at every time step. As a result, finite elements derived for explicit transient dynamics software use the simplest and barest constructions possible for computational efficiency while retaining an essential representation of the physical behavior. The best example of this technology is the four-node bending quadrilateral derived by Belytschko, Lin and Tsay. Today, the speed, memory capacity and availability of computer hardware allows a number of the previously used algorithms to be 'improved.' That is, it is possible with today's computing hardware to modify many of the standard algorithms to improve their representation of the physical process at the expense of added complexity and computational effort. The purpose is to review a number of these algorithms and identify the improvements possible. In many instances, both the older, faster version of the algorithm and the improved and somewhat slower version of the algorithm are found implemented together in software. Specifically, the following seven algorithmic items are examined: the invariant time derivatives of stress used in material models expressed in rate form; incremental objectivity and strain used in the numerical integration of the material models; the use of one-point element integration versus mean quadrature; shell elements used to represent the behavior of thin structural components; beam elements based on stress-resultant plasticity versus cross-section integration; the fidelity of elastic-plastic material models in their representation of ductile metals; and the use of Courant subcycling to reduce computational effort.
An X-ray technique for the measurement of internal residual strain gradients near the continuous reinforcements of metal matrix composites has been investigated. The technique utilizes high intensity white X-ray radiation from a synchrotron radiation source to obtain energy spectra from small (0.001 cu mm) volumes deep within composite samples. The viability of the technique was tested using a model system with 800 micron Al203 fibers and a commercial purity titanium matrix. Good agreement was observed between the measured residual radial and hoop strain gradients and those estimated from a simple elastic concentric cylinders model. The technique was then used to assess the strains near (SCS-6) silicon carbide fibers in a Ti-14Al-21Nb matrix after consolidation processing. Reasonable agreement between measured and calculated strains was seen provided the probe volume was located 50 microns or more from the fiber/matrix interface.
Constraints on the recent geological evolution of Venus may be provided by quantitative estimates of the rates of the principal resurfacing processes, volcanism and tectonism. This paper focuses on the latter, using impact craters as strain indicators. The total postimpact tectonic strain lies in the range 0.5-6.5%, which defines a recent mean strain rate of 10(exp -18)-10(exp -17)/s when divided by the mean surface age. Interpretation of the cratering record as one of pure production requires a decline in resurfacing rates at about 500 Ma (catastrophic resurfacing model). If distributed tectonic resurfacing contributed strongly before that time, as suggested by the widespread occurrence of tessera as inliers, the mean global strain rate must have been at least approximately 10(exp -15)/s, which is also typical of terrestrial active margins. Numerical calculations of the response of the lithosphere to inferred mantle convective forces were performed to test the hypothesis that a decrease in surface strain rate by at least two orders of magnitude could be caused by a steady decline in heat flow over the last billion years. Parameterized convection models predict that the mean global thermal gradient decreases by only about 5 K/km over this time; even with the exponential dependence of viscosity upon temperature, the surface strain rate drops by little more than one order of magnitude. Strongly unsteady cooling and very low thermal gradients today are necessary to satisfy the catastrophic model. An alternative, uniformitarian resurfacing hypothesis holds that Venus is resurfaced in quasi-random 'patches' several hundred kilometers in size that occur in response to changing mantle convection patterns.
An analytical method based on oxide/substrate intensity ratios is used to deduce the product of the atomic number density and electron mean free path as a function of depth, for the cases of device-quality gate oxides grown on Si substrates and irradiated with zero to 20 eV electrons during in situ X-ray photoelectron spectroscopy. These structures had been thinned stepwise to 25-60 A by means of a wet chemical depth-profiling procedure. Si(3+) species are formed in the Si/SiO2 interface region, and their observation during their relaxation and annihilation is correlated with strained Si-O-Si bonds. The observation of bond of cleavage and bond strain gradients in these samples is used to extend silica devitrification models for the explanation of fixed oxide charge generation and interface states.
An analytical approach for low-cycle fatigue prediction is presented. The approach combines a cyclic plasticity model with the finite element method and a damage accumulation criterion for ductile metals. The cyclic plasticity model is based on the concept of the combination of several yield surfaces. The surfaces are related to the material uniaxial stress-strain curve idealized by piecewise linear segments. The damage criterion is based on the Coffin-Manson formulae modified for the mean stress variation effect. It is extended to the multiaxial varying stress-strain field and applied for both the crack initiation and the crack growth processes. The stable slow crack growth rate is approximated by the damage accumulation gradient computed from the cracked finite element models. This procedure requires fatigue testing data of only smooth specimens under constant strain amplitudes. The present approach is illustrated by numerical examples of an aircraft wing stiffened panel subjected to compression, which causes material yielding and residual tension.
The modeling of the pressure-strain correlation of turbulence is examined from a basic theoretical standpoint with a view toward developing improved second-order closure models. Invariance considerations along with elementary dynamical systems theory are used in the analysis of the standard hierarchy of closure models. In these commonly used models, the pressure-strain correlation is assumed to be a linear function of the mean velocity gradients with coefficients that depend algebraically on the anisotropy tensor. It is proven that for plane homogeneous turbulent flows the equilibrium structure of this hierarchy of models is encapsulated by a relatively simple model which is only quadratically nonlinear in the anisotropy tensor. This new quadratic model - the SSG model - is shown to outperform the Launder, Reece, and Rodi model (as well as more recent models that have a considerably more complex nonlinear structure) in a variety of homogeneous turbulent flows. Some deficiencies still remain for the description of rotating turbulent shear flows that are intrinsic to this general hierarchy of models and, hence, cannot be overcome by the mere introduction of more complex nonlinearities. It is thus argued that the recent trend of adding substantially more complex nonlinear terms containing the anisotropy tensor may be of questionable value in the modeling of the pressure-strain correlation. Possible alternative approaches are discussed briefly.
The modeling of the pressure-strain correlation of turbulence is examined from a basic theoretical standpoint with a view toward developing improved second-order closure models. Invariance considerations along with elementary dynamical systems theory are used in the analysis of the standard hierarchy of closure models. In these commonly used models, the pressure-strain correlation is assumed to be a linear function of the mean velocity gradients with coefficients that depend algebraically on the anisotropy tensor. It is proven that for plane homogeneous turbulent flows the equilibrium structure of this hierarchy of models is encapsulated by a relatively simple model which is only quadratically nonlinear in the anisotropy tensor. This new quadratic model - the SSG model - is shown to outperform the Launder, Reece, and Rodi model (as well as more recent models that have a considerably more complex nonlinear structure) in a variety of homogeneous turbulent flows. Some deficiencies still remain for the description of rotating turbulent shear flows that are intrinsic to this general hierarchy of models and, hence, cannot be overcome by the mere introduction of more complex nonlinearities. It is thus argued that the recent trend of adding substantially more complex nonlinear terms containing the anisotropy tensor may be of questionable value in the modeling of the pressure-strain correlation. Possible alternative approaches are discussed briefly.
A review of interesting current topics in turbulence research is decorated with examples of popular fallacies about the behavior of turbulence. Topics include the status of the Law of the Wall, especially in compressible flow; analogies between the effects of Reynolds numbers, pressure gradient, unsteadiness and roughness change; the status of Kolmogorov's universal equilibrium theory and local isotropy of the small eddies; turbulence modelling, with reference to universality, pressure-strain modelling and the dissipation equation; and chaos. Fallacies include the mixing-length concept; the effect of pressure gradient on Reynolds shear stress; the separability of time and space derivatives; models of the dissipation equation; and chaos.
Experiments on graphite-epoxy laminated plates containing unloaded small holes show that these laminates are notch insensitive. That is, the uniaxial strength of these laminates with small holes exceeds the strength predicted by a point stress criterion using the stress concentration factor for the in-plane stress field. Laminates containing large holes exhibit notch sensitive behavior and consequently their strength is reasonably well predicted by the stress concentration effect. This hole size effect is manifested both in tension and in compression. Apparently, some mechanism must cause in-plane stress relief for laminates containing small holes. The purpose of this research was to study the influence of geometric nonlinearity on the micromechanical response of a filamentary composite material in the presence of a strain gradient caused by a discontinuity such as a hole. A mathematical model was developed at the micromechanical level to investigate this geometrically nonlinear effect.
The need for algebraic models and deficiencies of linear gradient models are presented. Classical algebraic Reynolds stress models are discussed with nonlinear stress-strain relations. Finally, the models are critiqued and some new developments are shown. All results are in viewgraph format.
Methodologies for understanding the plastic deformation mechanisms related to crack propagation at the nano-, meso- and micro-length scales are being developed. These efforts include the development and application of several computational methods including atomistic simulation, discrete dislocation plasticity, strain gradient plasticity and crystal plasticity; and experimental methods including electron backscattered diffraction and video image correlation. Additionally, methodologies for multi-scale modeling and characterization that can be used to bridge the relevant length scales from nanometers to millimeters are being developed. The paper focuses on the discussion of newly developed methodologies in these areas and their application to understanding damage processes in aluminum and its alloys.
Methodologies for understanding the plastic deformation mechanisms related 10 crack propagation at the nano, meso- and micro-length scales are being developed. These efforts include the development and application of several computational methods including atomistic simulation, discrete dislocation plasticity, strain gradient plasticity and crystal plasticity; and experimental methods including electron backscattered diffraction and video image correlation. Additionally, methodologies for multi-scale modeling and characterization that can be used to bridge the relevant length scales from nanometers to millimeters are being developed. The paper focuses on the discussion of newly developed methodologies in these areas and their application to understanding damage processes in aluminum and its alloys.
Structural alloys used in high temperature applications exhibit complex thermomechanical behavior that is time dependent and hereditary. Recent attention is being focused on metal matrix composite materials for high temperature applications where they exhibit all the complexities of conventional alloys and their strong anisotropy adds further complexities. Here, a proven constitutive model for isotropic materials in which the inelastic strain rate and internal state are expressible as gradients of a dissipation potential is taken to depend on invariants that reflect local transverse isotropy. Applications illustrate the capability of the theory of representing the time dependent, hereditary, anisotropic behavior typical of these materials at high temperature.