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At least 1,423 records · Page 79
All sourcesNASA NTRSDOE OSTI
NASA NTRS1936

Potential flow about elongated bodies of revolution

This report presents a method of solving the problem of axial and transverse potential flows around arbitrary elongated bodies of revolution. The solutions of Laplace's equation for the velocity potentials of the axial and transverse flows, the system of coordinates being an elliptic one in a meridian plane, are given. The theory is applied to a body of revolution obtained from a symmetrical Joukowsky profile, a shape resembling an airship hull. The pressure distribution and the transverse-force distribution are calculated and serve as examples of the procedure to be followed in the case of an actual airship. A section on the determination of inertia coefficients is also included in which the validity of some earlier work is questioned.

Kaplan, Carl↗
NASA NTRS1992

Control of low-speed turbulent separated flow over a backward-facing ramp

The relative performance and flow phenomena associated with several devices for controlling turbulent separated flow were investigated at low speeds. Relative performance of the devices was examined for flow over a curved, backward-facing ramp in a wind tunnel, and the flow phenomena were examined in a water tunnel using dye-flow visualization. Surface static pressure measurements and oil-flow visualization results from the wind tunnel tests indicated that transverse grooves, longitudinal grooves, submerged vortex generators, vortex generator jets (VGJ's), Viets' fluidic flappers, elongated arches at positive angle of attack, and large-eddy breakup devices (LEBU's) at positive angle of attack placed near the baseline separation location reduce flow separation and increase pressure recovery. Spanwise cylinders reduce flow separation but decrease pressure recovery downstream. Riblets, passive porous surfaces, swept grooves, Helmholtz resonators, and arches and LEBU's with angle of attack less than or = 0 degrees had no significant effect in reducing the extent of the separation region. Wall-cooling computations indicated that separation delay on a partially-cooled ramp is nearly the same as on a fully-cooled ramp, while minimizing the frictional drag increase associated with the wall cooling process. Dry-flow visualization tests in the water tunnel indicated that wishbone vortex generators in the forward orientation shed horseshoe vortices; wishbone vortex generators oriented in the reverse direction and doublet vortex generators shed streamwise counterrotating vortices; a spanewise cylinder located near the wall and LEBU's at angle of attack = -10 degrees produced eddies or transverse vortices which rotated with the same sign as the mean vorticity in a turbulent boundary layer; and the most effective VGJ's produced streamwise co-rotating vortices. Comparative wind-tunnel test results indicated that transferring momentum from the outer region of a turbulent boundary layer through the action of embedded streamwise vortices is more effective than by transverse vortices for the separation control application studied herein.

Lin, John C.↗
NASA NTRS1995

Analysis of the progressive failure of brittle matrix composites

This report investigates two of the most common modes of localized failures, namely, periodic fiber-bridged matrix cracks and transverse matrix cracks. A modification of Daniels' bundle theory is combined with Weibull's weakest link theory to model the statistical distribution of the periodic matrix cracking strength for an individual layer. Results of the model predictions are compared with experimental data from the open literature. Extensions to the model are made to account for possible imperfections within the layer (i.e., nonuniform fiber lengths, irregular crack spacing, and degraded in-situ fiber properties), and the results of these studies are presented. A generalized shear-lag analysis is derived which is capable of modeling the development of transverse matrix cracks in material systems having a general multilayer configuration and under states of full in-plane load. A method for computing the effective elastic properties for the damaged layer at the global level is detailed based upon the solution for the effects of the damage at the local level. This methodology is general in nature and is therefore also applicable to (0(sub m)/90(sub n))(sub s) systems. The characteristic stress-strain response for more general cases is shown to be qualitatively correct (experimental data is not available for a quantitative evaluation), and the damage evolution is recorded in terms of the matrix crack density as a function of the applied strain. Probabilistic effects are introduced to account for the statistical nature of the material strengths, thus allowing cumulative distribution curves for the probability of failure to be generated for each of the example laminates. Additionally, Oh and Finney's classic work on fracture location in brittle materials is extended and combined with the shear-lag analysis. The result is an analytical form for predicting the probability density function for the location of the next transverse crack occurrence within a crack bounded region. The results of this study verified qualitatively the validity of assuming a uniform crack spacing (as was done in the shear-lag model).

Thomas, David J.↗
NASA NTRS1995

Three-dimensional solutions for coupled thermoelectroelastic response of multilayered plates

Analytic three-dimensional solutions are presented for the coupled thermoelectroelastic response of multilayered hybrid composite plates. The plates consist of a combination of fiber-reinforced cross-ply and piezothermoelastic layers. Both the thermoelectroelastic static response and its sensitivity coefficients are computed. The sensitivity coefficients measure the sensitivity of the response to variations in different mechanical, thermal and piezoelectric material properties of the plate. A linear constitutive model is used, and the material properties are assumed to be independent of the temperature and the electric field. The plates are assumed to have rectangular geometry and special material symmetries. A mixed formulation is used with the fundamental unknowns consisting of the three transverse stress components; three displacement components; transverse component of the electric displacement field; electric potential; transverse heat flux component, and temperature change. Each of the fundamental unknowns is expressed in terms of a double Fourier series in the Cartesian surface coordinates. A state space approach is used to generate the static response and to evaluate the sensitivity coefficients. Extensive numerical results are presented showing the effects of variation in the geometric parameters of the plate on the different response quantities and their sensitivity coefficients.

Xu, Kangming↗
NASA NTRS1994

Post-impact fatigue of cross-plied, through-the-thickness reinforced carbon/epoxy composites

An experimental investigation of the post-impact fatigue response of integrally woven carbon/epoxy composites was conducted. Five different through-the-thickness (TTT) reinforcing fibers were used in an experimental textile process that produced an integrally woven (0/90/0/90/0/90/0/90/0)(sub T) ply layup with 21K AS4 carbon tow fiber. The resin was Hercules 3501-6, and the five TTT reinforcing fibers were Kevlar, Toray carbon, AS4 carbon, glass, and IM6 carbon. The purpose of this investigation was to study the post-impact fatigue response of these material systems and to identify the optimum TTT fiber. Samples were impacted with one half inch diameter aluminum balls with an average velocity of 543 ft/sec. Post-impact static compression and constant amplitude tension-compression fatigue tests were conducted. Fatigue tests were conducted with a loading ratio of R=-5, and frequency of 4 Hz. Damage growth was monitored using x-radiographic and sectioning techniques and by examining the stress-strain response (across the impact site) throughout the fatigue tests. The static compressive stress versus far-field strain response was nearly linear for all material groups. All the samples had a transverse shear failure mode. The average compressive modulus (from far-field strain) was about 10 Msi. The average post-impact static compressive strength was about 35.5 Ksi. The IM6 carbon sample had a strength of over 40 Ksi, more than 16 percent stronger than average. There was considerable scatter in the S-N data. However, the IM6 carbon samples clearly had the best fatigue response. The response of the other materials, while worse than IM6 carbon, could not be ranked definitively. The initial damage zones caused by the impact loading and damage growth from fatigue loading were similar for all five TTT reinforcing materials. The initial damage zones were circular and consisted of delaminations, matrix cracks and ply cracks. Post-impact fatigue loading caused delamination growth, ply cracking and fiber bundle failures, typically 45 deg from impact load direction. During the initial 97 percent of fatigue life, delaminations, ply cracks and fiber bundle failures primarily grew at and near the impact site. During the final 3 percent of life, damage grew rapidly transverse to the loading direction as a through-the-thickness transverse shear failure. The stress-strain response was typically linear during the initial 50 percent of life, and stiffness dropped about 20 percent during this period. During the next 47 percent of life, stiffness dropped about 34 percent, and the stress-strain response was no longer linear. The stiffness decreased about 23 percent during the final 3 percent of life. These trends were typical of all the materials tested. Therefore, by monitoring stiffness loss, fatigue failure could be accurately anticipated.

Serdinak, Thomas E.↗
NASA NTRS1994

Stars and linear dunes on Mars

A field containing 11 star and incipient star dunes occurs on Mars at 8.8 deg S, 270.9 deg W. Examples of linear dunes are found in a crater at 59.4 deg S, 343 deg W. While rare, dune varieties that form in bi- and multidirectional wind regimes are not absent from the surface of Mars. The occurence of both of these dune fields offers new insight into the nature of martian wind conditions and sand supply. The linear dunes appears to have formed through modification of a formerly transverse aeolian deposit, suggesting a relatively recent change in local wind direction. The 11 dunes in the star dune locality show a progressive change from barchan to star form as each successive dune has traveled up into a valley, into a more complex wind regime. The star dunes corroborate the model of N. Lancaster (1989), for the formation of star dunes by projection of transverse dunes into a complex, topographically influenced wind regime. The star dunes have dark streaks emanating from them, providing evidence that the dunes were active at or near the time the relevant image was obtained by the Viking 1 orbiter in 1978. The star and linear dunes described here are located in different regions on the martian surface. Unlike most star and linear dunes on Earth, both martian examples are isolated occurrences; neither is part of a major sand sea. Previously published Mars general circulation model results suggest that the region in which the linear dune field occurs should be a bimodal wind regime, while the region in which the star dunes occur should be unimodal. The star dunes are probably the result of localized complication of the wind regime owing to topographic confinement of the dunes. Local topographic influence on wind regime is also evident in the linear dune field, as there are transverse dunes in close proximity to the linear dunes, and their occurrence is best explained by funneling of wind through a topographic gap in the upwind crater wall.

Edgett, Kenneth S.↗
NASA NTRS1995

A comparison of graphite/epoxy tape laminates and 2-D braided composites mechanical properties

A comparison of the mechanical properties of unidirectional composite tape laminates and of two-dimensional triaxially braided composite was conducted. The tape laminate layups were designed to match the percentage of axial fibers and the angle of the bias tows in the braided composite. The materials system used for the laminates is AS4/3501-6 which was chosen as the closest available match to As4/1895 used for the braids. The strength and stiffness properties measured here include tension, open-hole tension, filled-hole tension, compression and open-hole compression, all of these in both the longitudinal and transverse direction. Results show that the longitudinal modulus of both material forms is quite similar, but that the transverse modulus of the braids is lower. In terms of strength, the longitudinal unnotched strength of the braids is lower than that of the laminates, while the transverse strength is significantly lower. For both strength and stiffness, the crimp in the bias tows of the braid is probably the main cause for reduced properties. On the other hand, a very significant increase in open-hole and filled-hole tension strength was observed for the braids compared to the tape laminates. However, this was not observed in compression where all the braid properties are lower than for the laminates.

Minguet, Pierre J.↗
NASA NTRS1994

Effects of Tangential Edge Constraints on the Postbuckling Behavior of Flat and Curved Panels Subjected to Thermal and Mechanical Loads

A parametric study of the effects of tangential edge constraints on the postbuckling response of flat and shallow curved panels subjected to thermal and mechanical loads is presented. The mechanical loads investigated are uniform compressive edge loads and transverse lateral pressure. The temperature fields considered are associated with spatially nonuniform heating over the panels, and a linear through-the-thickness temperature gradient. The structural model is based on a higher-order transverse-shear-deformation theory of shallow shells that incorporates the effects of geometric nonlinearities, initial geometric imperfections, and tangential edge motion constraints. Results are presented for three-layer sandwich panels made from transversely isotropic materials. Simply supported panels are considered in which the tangential motion of the unloaded edges is either unrestrained, partially restrained, or fully restrained. These results focus on the effects of the tangential edge restraint on the postbuckling response. The results of this study indicate that tangentially restraining the edges of a curved panel can make the panel insensitive to initial geometric imperfections in some cases.

Lin, W.↗
NASA NTRS1996

Probabilistic Simulation of Progressive Fracture in Bolted-Joint Composite Laminates

This report describes computational methods to probabilistically simulate fracture in bolted composite structures. An innovative approach that is independent of stress intensity factors and fracture toughness was used to simulate progressive fracture. The effect of design variable uncertainties on structural damage was also quantified. A fast probability integrator assessed the scatter in the composite structure response before and after damage. Then the sensitivity of the response to design variables was computed. General-purpose methods, which are applicable to bolted joints in all types of structures and in all fracture processes-from damage initiation to unstable propagation and global structure collapse-were used. These methods were demonstrated for a bolted joint of a polymer matrix composite panel under edge loads. The effects of the fabrication process were included in the simulation of damage in the bolted panel. Results showed that the most effective way to reduce end displacement at fracture is to control both the load and the ply thickness. The cumulative probability for longitudinal stress in all plies was most sensitive to the load; in the 0 deg. plies it was very sensitive to ply thickness. The cumulative probability for transverse stress was most sensitive to the matrix coefficient of thermal expansion. In addition, fiber volume ratio and fiber transverse modulus both contributed significantly to the cumulative probability for the transverse stresses in all the plies.

Minnetyan, L.↗
NASA NTRS1996

Plasma Heating and Flow in an Auroral Arc

We report direct observations of the three-dimensional velocity distribution of selected topside ionospheric ion species in an auroral context between 500 and 550 km altitude. We find heating transverse to the local magnetic field in the core plasma, with significant heating of 0(+), He(+), and H(+), as well as tail heating events that occur independently of the core heating. The 0(+) velocity distribution departs from bi-Maxwellian, at one point exhibiting an apparent ring-like shape. However, these observations are shown to be aliased within the auroral arc by temporal variations that arc not well-resolved by the core plasma instrument. The dc electric field measurements reveal superthermal plasma drifts that are consistent with passage of the payload through a series of vortex structures or a larger scale circularly polarized hydromagnetic wave structure within the auroral arc. The dc electric field also shows that impulsive solitary structures, with a frequency spectrum in the ion cyclotron frequency range, occur in close correlation with the tail heating events. The drift and core heating observations lend support to the idea that core ion heating is driven at low altitudes by rapid convective motions imposed by the magnetosphere. Plasma wave emissions at ion frequencies and parallel heating of the low-energy electron plasma are observed in conjunction with this auroral form; however, the conditions are much more complex than those typically invoked in previous theoretical treatments of superthermal frictional heating. The observed ion heating within the arc clearly exceeds that expected from frictional heating for the light ion species H(+) and He(+), and the core distributions also contain hot transverse tails, indicating an anomalous transverse heat source.

Moore, T. E.↗
NASA NTRS1998

Analysis of Curved Sandwich Panels Subjected to Combined Temperature Gradient and Mechanical Loads

The results of a detailed study of the nonlinear response of curved sandwich panels with composite face sheets and subjected to a temperature gradient through-the-thickness combined with mechanical loadings are presented. The analysis is based on a first-order shear-deformation Sanders-Budiansky type theory with the effects of large displacements, moderate rotations, transverse shear deformation and laminated anisotropic material behavior included. A mixed formulation is used with the fundamental unknowns consisting of the generalized displacements and the stress resultants of the panel. The nonlinear displacements, strain energy, principal strains, transverse shear stresses, transverse shear strain energy density, and their hierarchical sensitivity coefficients are evaluated. The hierarchical sensitivity coefficients measure the sensitivity of the nonlinear response to variations in the panel parameters, the effective properties of the face sheet layers and the core, and the micromechanical parameters. Numerical results are presented for cylindrical panels subjected to combined pressure loading, edge shortening or extension, edge shear and a temperature gradient through the thickness. The results show the effects of variations in the loading and the panel aspect ratio, on the nonlinear response and its sensitivity to changes in the various panel, effective layer and micromechanical parameters.

Noor, Ahmed K.↗
NASA NTRS1992

Computation of Turbulent Recirculating Flow in Channels, and for Impingement Cooling

Fully elliptic forms of the transport equations have been solved numerically for two flow configurations. The first is turbulent flow in a channel with transverse rectangular ribs, and the second is impingement cooling of a plane surface. Both flows are relevant to proposed designs for active cooling of hypersonic vehicles using supercritical hydrogen as the coolant. Flow downstream of an abrupt pipe expansion and of a backward-facing step were also solved with various near-wall turbulence models as benchmark problems. A simple form of periodicity boundary condition was used for the channel flow with transverse rectangular ribs. The effects of various parameters on heat transfer in channel flow with transverse ribs and in impingement cooling were investigated using the Yap modified Jones and Launder low Reynolds number k-epsilon turbulence model. For the channel flow, predictions were in adequate agreement with experiment for constant property flow, with the results for friction superior to those for heat transfer. For impingement cooling, the agreement with experiment was generally good, but the results suggest that improved modelling of the dissipation rate of turbulence kinetic energy is required in order to obtain improved heat transfer prediction, especially near the stagnation point. The k-epsilon turbulence model was used to predict the mean flow and heat transfer for constant and variable property flows. The effect of variable properties for channel flow was investigated using the same turbulence model, but comparison with experiment yielded no clear conclusions. Also, the wall function method was modified for use in the variable properties flow with a non-adiabatic surface, and an empirical model is suggested to correctly account for the behavior of the viscous sublayer with heating.

Chang, Byong Hoon↗
NASA NTRS2000

Asymptotic Solutions for Optical Properties of Large Particles with Strong Absorption

For scattering calculations involving nonspherical particles such as ice crystals, we show that the transverse wave condition is not applicable to the refracted electromagnetic wave in the context of geometric optics when absorption is involved. Either the TM wave condition (i.e., where the magnetic field of the refracted wave is transverse with respect to the wave direction) or the TE wave condition (i.e., where the electric field is transverse with respect to the propagating direction of the wave) may be assumed for the refracted wave in an absorbing medium to locally satisfy the electromagnetic boundary condition in the ray tracing calculation. The wave mode assumed for the refracted wave affects both the reflection and refraction coefficients. As a result, a nonunique solution for these coefficients is derived from the electromagnetic boundary condition. In this study we have identified the appropriate solution for the Fresnel reflection/refraction coefficients in light scattering calculation based on the ray tracing technique. We present the 3 x 2 refraction or transmission matrix that completely accounts for the inhomogeneity of the refracted wave in an absorbing medium. Using the Fresnel coefficients for an absorbing medium, we derive an asymptotic solution in an analytical format for the scattering properties of a general polyhedral particle. Numerical results are presented for hexagonal plates and columns with both preferred and random orientations. The asymptotic theory can produce reasonable accuracy in the phase function calculations in the infrared window region (wavelengths near 10 micron) if the particle size (in diameter) is on the order of 40 micron or larger. However, since strong absorption is assumed in the computation of the single-scattering albedo in the asymptotic theory, the single scattering albedo does not change with variation of the particle size. As a result, the asymptotic theory can lead to substantial errors in the computation of single-scattering albedo for small and moderate particle sizes. However, from comparison of the asymptotic results with the FDTD solution, it is expected that a convergence between the FDTD results and the asymptotic theory results can be reached when the particle size approaches 200 micron. We show that the phase function at side-scattering and backscattering angles is insensitive to particle shape if the random orientation condition is assumed. However, if preferred orientations are assumed for particles, the phase function has a strong dependence on scattering azimuthal angle. The single-scattering albedo also shows very strong dependence on the inclination angle of incident radiation with respect to the rotating axis for the preferred particle orientations.

Yang, Ping↗
NASA NTRS1999

Analysis of Composite Panels Subjected to Thermo-Mechanical Loads

The results of a detailed study of the effect of cutout on the nonlinear response of curved unstiffened panels are presented. The panels are subjected to combined temperature gradient through-the-thickness combined with pressure loading and edge shortening or edge shear. The analysis is based on a first-order, shear deformation, Sanders-Budiansky-type shell theory with the effects of large displacements, moderate rotations, transverse shear deformation, and laminated anisotropic material behavior included. A mixed formulation is used with the fundamental unknowns consisting of the generalized displacements and the stress resultants of the panel. The nonlinear displacements, strain energy, principal strains, transverse shear stresses, transverse shear strain energy density, and their hierarchical sensitivity coefficients are evaluated. The hierarchical sensitivity coefficients measure the sensitivity of the nonlinear response to variations in the panel parameters, as well as in the material properties of the individual layers. Numerical results are presented for cylindrical panels and show the effects of variations in the loading and the size of the cutout on the global and local response quantities as well as their sensitivity to changes in the various panel, layer, and micromechanical parameters.

Noor, Ahmed K.↗
NASA NTRS2000

Effect of Discontinuities and Uncertainties on the Response and Failure of Composite Structures

The overall goal of this research was to assess the effect of discontinuities and uncertainties on the nonlinear response and failure of composite structures subjected to combined mechanical and thermal loads. The four key elements of the study were: (1) development of simple and efficient procedures for the accurate determination of transverse shear and transverse normal stresses in structural sandwiches as well as in unstiffened and stiffened composite panels and shells; (2) study the effects of transverse stresses on the response, damage initiation and propagation in composite and sandwich structures; (3) use of hierarchical sensitivity coefficients to identify the major parameters that affect the response and damage in each of the different levels in the hierarchy (micro-mechanical, layer, panel, subcomponent and component levels); and (4) application of fuzzy set techniques to identify the range and variation of possible responses. The computational models developed were used in conjunction with experiments, to understand the physical phenomena associated with the nonlinear response and failure of composite and sandwich structures. A toolkit was developed for use in conjunction with deterministic analysis programs to help the designer in assessing the effect of uncertainties in the different computational model parameters on the variability of the response quantities.

Noor, Ahmed K.↗
NASA NTRS2001

Asymptotic Solutions for Optical Properties of Large Particles with Strong Absorption

The transverse wave condition is not applicable to the refracted electromagnetic wave within the context of geometrical optics when absorption is involved. Either the transverse magnetic (TM) or the transverse electric (TE) wave condition can be assumed for the wave to locally satisfy the electromagnetic boundary condition in a ray-tracing calculation. The assumed wave mode affects both the reflection and the refraction coefficients. As a result, nonunique solutions for these coefficients are inevitable. In this study the appropriate solutions for the Fresnel reflection-refraction coefficients are identified in light-scattering calculations based on the ray-tracing technique. In particular, a 3x2 refraction or transmission matrix is derived to account for the inhomogeneity of the refracted wave in an absorbing medium. An asymptotic solution that completely includes the effect of medium absorption on Fresnel coefficients is obtained for the scattering properties of a general polyhedral particle. Numerical results are presented for hexagonal plates and columns with both preferred and random orientations.

Yang, Ping↗
NASA NTRS2001

Optimal Design of Grid-Stiffened Panels and Shells With Variable Curvature

A design strategy for optimal design of composite grid-stiffened structures with variable curvature subjected to global and local buckling constraints is developed using a discrete optimizer. An improved smeared stiffener theory is used for the global buckling analysis. Local buckling of skin segments is assessed using a Rayleigh-Ritz method that accounts for material anisotropy and transverse shear flexibility. The local buckling of stiffener segments is also assessed. Design variables are the axial and transverse stiffener spacing, stiffener height and thickness, skin laminate, and stiffening configuration. Stiffening configuration is herein defined as a design variable that indicates the combination of axial, transverse and diagonal stiffeners in the stiffened panel. The design optimization process is adapted to identify the lightest-weight stiffening configuration and stiffener spacing for grid-stiffened composite panels given the overall panel dimensions. in-plane design loads, material properties. and boundary conditions of the grid-stiffened panel or shell.

Ambur, Damodar R.↗
NASA NTRS2001

Crack Turning Mechanics of Composite Wing Skin Panels

The safety of future composite wing skin integral stiffener panels requires a full understanding of failure mechanisms of these damage tolerance critical structures under both in-plane and bending loads. Of primary interest is to derive mathematical models using fracture mechanics in anisotropic cracked plate structures, to assess the crack turning mechanisms, and thereby to enhance the residual strength in the integral stiffener composite structures. The use of fracture mechanics to assess the failure behavior in a cracked structure requires the identification of critical fracture parameters which govern the severity of stress and deformation field ahead of the flaw, and which can be evaluated using information obtained from the flaw tip. In the three-year grant, the crack-tip fields under plane deformation, crack-tip fields for anisotropic plates and anisotropic shells have been obtained. In addition, methods for determining the stress intensity factors, energy release rate, and the T-stresses have been proposed and verified. The research accomplishments can be summarized as follows: (1) Under plane deformation in anisotropic solids, the asymptotic crack-tip fields have been obtained using Stroh formalism; (2) The T-stress and the coefficient of the second term for sigma(sub y), g(sub 32), have been obtained using path-independent integral, the J-integral and Betti's reciprocal theorem together with auxiliary fields; (3) With experimental data performed by NASA, analyses indicated that the mode-I critical stress intensity factor K(sub Q) provides a satisfactory characterization of fracture initiation for a given laminate thickness, provided the failure is fiber-dominated and crack extends in a self-similar manner; (4) The high constraint specimens, especially for CT specimens, due to large T-stress and large magnitude of negative g(sub 32) term may be expected to inhibit the crack extension in the same plane and promote crack turning; (5) Crack turning out of crack plane in generally anisotropic solids under plane deformation has been studied; (6) The role of T-stress and the higher-order term of sigma(sub y) on the crack turning and stability of the kinked crack has been quantified; (7) Asymptotic crack-tip fields including the effect of transverse shear deformation (Reissner plate theory) in an anisotropic plate under bending, twisting moments, and transverse shear loads has been presented; (8) The expression of the path-independent J-integral in terms of the generalized stress and strain has been derived; (9) Asymptotic crack-tip fields including the effect of transverse shear deformation (Reissner shallow shell theory) in a general anisotropic shell has been developed; (10) The Stroh formalism was used to characterize the crack tip fields in shells up to the second term and the energy release rate was expressed in a very compact form.

Yuan, F. G.↗