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NASA NTRS1982

Input impedance of microstrip antennas

Using Richmond's reaction integral equation, an expression is derived for the input impedance of microstrip patch antennas excited by either a microstrip line or a coaxial probe. The effects of the finite substrate thickness, a dielectric protective cover, and associated surface waves are properly included by the use of the exact dyadic Green's function. Using the present formulation the input impedance of a rectangular microstrip antenna is determined and compared with experimental and earlier calculated results.

Deshpande, M. D.↗

NASA NTRS1982

A cylindrical shell with an arbitrarily oriented crack

The general problem of a shallow shell with constant curvatures is considered. It is assumed that the shell contains an arbitrarily oriented through crack and the material is specially orthotropic. The nonsymmetric problem is solved for arbitrary self equilibrating crack surface tractions, which, added to an appropriate solution for an uncracked shell, would give the result for a cracked shell under most general loading conditions. The problem is reduced to a system of five singular integral equations in a set of unknown functions representing relative displacements and rotations on the crack surfaces. The stress state around the crack tip is asymptotically analyzed and it is shown that the results are identical to those obtained from the two dimensional in plane and antiplane elasticity solutions. The numerical results are given for a cylindrical shell containing an arbitrarily oriented through crack. Some sample results showing the effect of the Poisson's ratio and the material orthotropy are also presented.

Yahsi, O. S.↗

NASA NTRS1982

Moving cracks in layered composites

A three-layered composite with a crack spreading in the center layer has been analytically examined to evaluate the effect of material nonhomogeneity on a constant velocity crack. Two different loading characteristics are considered. In the first case, crack motion is maintained by uniform tensile stresses. In the other, crack deformation is caused by anti-plane shear stresses. Galilean transformation and Fourier sine and cosine transforms are used to determine dynamic crack tip stress fields. Standard Fredholm integral equations yield the dynamic stress intensity factors. The results show that the intensity of local dynamic stresses increases or decreases with crack length to layer thickness as a function of the relative magnitudes of the adjoining layer's material properties. Crack speed tends to increase the effect of material nonhomogeneity.

Sih, G. C.↗

NASA NTRS1983

A multi-grid method for the computation of viscid/inviscid interactions on airfoils

An improved version of the 'GRUMFOIL' code has been developed for the computation of airfoil flows. The method employs a conservative difference scheme for the potential equation, integral methods for the boundary layer, and viscous coupling conditions that fully account for the wake and strong interaction effects at trailing edges. The improved version uses Jameson's 'MAD' scheme to accelerate convergence of the inviscid solution, an improved 2nd order artificial viscosity and far field 'TARE' correction to reduce spatial truncation errors and Carter's semi-inverse method for the viscous solution. Results are presented which demonstrate a factor of ten reduction in computing cost.

Melnik, R. E.↗

NASA NTRS1982

Mechanics aspects of NDE by sound and ultrasound

Nondestructive evaluation (NDE) is considered as a means to detect the energy release mechanism of defects and the interaction of microstructures within materials with sound waves and/or ultrasonic waves. Ultrasonic inspection involves the frequency range 20 kHz-1 GHz with amplitudes depending on the sensitivity of the test instrumentation. Pulse echo systems are most frequently used in NDE. Information is extracted from the signals through measurements of the signal velocity, attenuation, the acoustic emission when stress is applied, and calculation of the acoustoelastic coefficients. Fracture properties, tensile and shear strengths, the interlaminar shear strength, the cohesive strength, yield and impact strengths, the hardness, and the residual stress can be assayed by ultrasonic methods. Finally, attention is given to analytical treatment of the derived data, with mention given to transition matrix, integral equation, and eigenstrain approaches.

Fu, L. S.↗

NASA NTRS1983

One-dimensional periodic flows with a shock transition - Application to the density wave theory of spiral structure

The problem of one-dimensional gas flow through a sinusoidal gravitational potential with a series of equally spaced shock fronts is considered. For time-independent and spatially periodic flows, an integral equation relates the flow velocity to the gravitational potential and source functions of energy and momentum densities. It is suggested that this problem simulates some of the dynamical effects of the azimuthal flow on a parcel of gas at a fixed radius in a galactic disk that supports a spiral density wave structure. It omits the radial motions. In this context, a typical azimuthal velocity across a spiral arm is specified and the resultant source functions are obtained. The implications of this calculation suggest heating of the gaseous disk by O and B stars behind the shock front, and subsequent interarm cooling is a plausible mechanism for maintaining the periodic cycling of the gaseous component of the galactic disk.

Bogdan, T. J.↗

NASA NTRS1983

Stress intensity factors in two bonded elastic layers containing cracks perpendicular to and on the interface. I Analysis. II - Solution and results

The basic crack problem which is essential for the study of subcritical crack propagation and fracture of layered structural materials is considered. Because of the apparent analytical difficulties, the problem is idealized as one of plane strain or plane stress. An additional simplifying assumption is made by restricting the formulation of the problem to crack geometries and loading conditions which have a plane of symmetry perpendicular to the interface. The general problem is formulated in terms of a coupled systems of four integral equations. For each relevant crack configuration of practical interest, the singular behavior of the solution near and at the ends and points of intersection of the cracks is investigated and the related characteristic equations are obtained. The edge crack terminating at and crossing the interface, the T-shaped crack consisting of a broken layer and a delamination crack, the cross-shaped crack which consists of a delamination crack intersecting a crack which is perpendicular to the interface, and a delamination crack initiating from a stress-free boundary of the bonded layers are some of the practical crack geometries considered. Previously announced in STAR as N80-18428 and N80-18429

Lu, M.-C.↗

NASA NTRS1983

A cylindrical shell with a stress-free end which contains an axial part-through or through crack

The interaction problem of a through or a part through crack with a stress free boundary in a semi-infinite cylindrical shell is considered. It is assumed that the crack lies in a meridional plane which is a plane of symmetry with respect to the external loads as well as the geometry. The circular boundary of the semi-infinite cylinder is assumed to be stress free. By using a transverse shear theory the problem is formulated in terms of a system of singular integral equations. The line spring model is used to treat the part through crack problem. In the case of a through crack the interaction between the perturbed stress fields due to the crack and the free boundary is quite strong and there is a considerable increase in the stress intensity factors caused by the interaction. On the other hand in the problem of a surface crack the interaction appears to be much weaker and consequently the magnification in the stress intensity factors is much less significant.

Erdogan, F.↗

NASA NTRS1983

A pressurized cylindrical shell with a fixed end which contains an axial part-through or through crack

A cylindrical shell having a very stiff and plate or a flange is considered. It is assumed that near the end the cylinder contains an axial flaw which may be modeled as a part through surface crack or a through crack. The effect of the end constraining on the stress intensity factor which is the main fracture mechanics parameter is studied. The applied loads acting on the cylinder are assumed to be axisymmetric. Thus the crack problem under consideration is symmetric with respect to the plane of the crack and consequently only the Mode 1 stress intensity factors are nonzero. With this limitation, the general perturbation problem for a cylinder with a built in end containing an axial crack is considered. Reissner's shell theory is used to formulate the problem. The part through crack problem is treated by using a line spring model. In the case of a crack tip terminating at the fixed end it is shown that the integral equations of the shell problem has the same generalized Cauchy kernel as the corresponding plane stress elasticity problem.

Yahsi, O. S.↗

NASA NTRS1984

The crack-inclusion interaction problem

The general plane elastostatic problem of interaction between a crack and an inclusion is considered. The Green's functions for a pair of dislocations and a pair of concentrated body forces are used to generate the crack and the inclusion. Integral equations are obtained for a line crack and an elastic line inclusion having an arbitrary relative orientation and size. The nature of stress singularity around the end points of rigid and elastic inclusions is described and three special cases of this intersection problem are studied. The problem is solved for an arbitrary uniform stress state away from the crack-inclusion region. The nonintersecting crack-inclusion problem is considered for various relative size, orientation, and stiffness parameters, and the stress intensity factors at the ends of the inclusion and the crack are calculated. For the crack-inclusion intersection case, special stress intensity factors are defined and are calculated for various values of the parameters defining the relative size and orientation of the crack and the inclusion and the stiffness of the inclusion.

Xue-Hui, L.↗

NASA NTRS1984

Stress intensity factors in a reinforced thick-walled cylinder

An elastic thick-walled cylinder containing a radial crack is considered. It is assumed that the cylinder is reinforced by an elastic membrane on its inner surface. The model is intended to simulate pressure vessels with cladding. The formulation of the problem is reduced to a singular integral equation. Various special cases including that of a crack terminating at the cylinder-reinforcement interface are investigated and numerical examples are given. Results indicate that in the case of the crack touching the interface the crack surface displacement derivative is finite and consequently the stress state around the corresponding crack tip is bounded; and generally, for realistic values of the stiffness parameter, the effect of the reinforcement is not very significant.

Tang, R.↗

NASA NTRS1984

Implementation of uniform perturbation method for potential flow past axisymmetric and two-dimensional bodies

The aerodynamic characteristics of potential flow past an axisymmetric slender body and a thin airfoil are calculated using a uniform perturbation analysis method. The method is based on the superposition of potentials of point singularities distributed inside the body. The strength distribution satisfies a linear integral equation by enforcing the flow tangency condition on the surface of the body. The complete uniform asymptotic expansion of its solution is obtained with respect to the slenderness ratio by modifying and adapting an existing technique. Results calculated by the perturbation analysis method are compared with the existing surface singularity panel method and some available analytical solutions for a number of cases under identical conditions. From these comparisons, it is found that the perturbation analysis method can provide quite accurate results for bodies with small slenderness ratio. The present method is much simpler and requires less memory and computation time than existing surface singularity panel methods of comparable accuracy.

Wong, T. C.↗

NASA NTRS1984

Methods for analysis of cracks in three-dimensional solids

Various analytical and numerical methods used to evaluate the stress intensity factors for cracks in three-dimensional (3-D) solids are reviewed. Classical exact solutions and many of the approximate methods used in 3-D analyses of cracks are reviewed. The exact solutions for embedded elliptic cracks in infinite solids are discussed. The approximate methods reviewed are the finite element methods, the boundary integral equation (BIE) method, the mixed methods (superposition of analytical and finite element method, stress difference method, discretization-error method, alternating method, finite element-alternating method), and the line-spring model. The finite element method with singularity elements is the most widely used method. The BIE method only needs modeling of the surfaces of the solid and so is gaining popularity. The line-spring model appears to be the quickest way to obtain good estimates of the stress intensity factors. The finite element-alternating method appears to yield the most accurate solution at the minimum cost.

Raju, I. S.↗

NASA NTRS1983

A cylindrical shell with an arbitrarily oriented crack

The general problem of a shallow shell with constant curvatures is considered. It is assumed that the shell contains an arbitrarily oriented through crack and the material is specially orthotropic. The nonsymmetric problem is solved for arbitrary self equilibrating crack surface tractions, which, added to an appropriate solution for an uncracked shell, would give the result for a cracked shell under most general loading conditions. The problem is reduced to a system to five singular integral equations in a set of unknown functions representing relative displacements and rotations on the crack surfaces. The stress state around the crack tip is asymptotically analyzed and it is shown that the results are identical to those obtained from the two dimensional in plane and antiplane elasticity solutions. The numerical results are given for a cylindrical shell containing an arbitrarily oriented through crack. Some sample results showing the effect of the Poisson's ratio and the material orthotropy are also presented. Previously annunced in STAR as N83-16783

Yahsi, O. S.↗

NASA NTRS1983

On obtaining the forward phase functions of Saturn ring features from radio occultation observations

It is noted that the near-forward scattering functions of particles in Saturn ring features are related to 3.6 cm radio occultation power spectra by a Fredholm integral equation of the first kind. The equation reduces to an algebraic system of equation whose solution by usual inversion techniques (that is, least mean squares) is ruled out by the near singularity of the forward transformation matrix. A combination of constrained linear inversion and a filtering algorithm based on eigenvector decomposition of the matrix reduces the instabilities; this yields derived phase functions valid over the range of zero to about 12 mrad. It is noted that these functions represent the collective forward diffraction lobe of particles greater than about 1 m in radius. Since multiple scattering of the signal is a significant effect, the measured phase functions must be adjusted to obtain the singly scattered component. This single-scattering correction is examined for two physical models, namely the monolayer and the classical discrete random slab; in addition, the fraction of opacity in submeter particles for each model for particular ring features is estimated.

Zebker, H. A.↗

NASA NTRS1983

The elasticity problem for a thick-walled cylinder containing a circumferential crack

The elasticity problem for a long hollow circular cylinder containing an axisymmetric circumferential crack subjected to general nonaxisymmetric external loads is considered. The problem is formulated in terms of a system of singular integral equations with the Fourier coefficients of the derivative of the crack surface displacement as density functions. The stress intensity factors and the crack opening displacement are calculated for a cylinder under uniform tension, bending by end couples, and self-equilibrating residual stresses.

Nied, H. F.↗

NASA NTRS1984

Surface roughness studies with DALLAS-detector array for laser light angular scattering

An attempt is made to develop a better mathematical description of optical scattering phenomena, in order to construct an optical scattering apparatus for reliable and routine measurements of roughness parameters without resorting to comparator standards. After a brief outline of optical scattering theory, a description is presented of an experimental instrument for measuring surface roughness which incorporates optical scattering principles. The instrument has a He-Ne laser which illuminates the test surface at a variable angle of incidence. Scattered light distribution is detected by an array of 87 fiber-optic sensors positioned in a rotating semicircular yoke. The output from the detector is digitized and analyzed in a laboratory computer. For a comparison with experimental data, theoretical distributions are calculated by substituting the roughness profiles into the operand of and integral equation for electromagnetic scattering developed by Beckmann and Spizzichino (1963). A schematic diagram of the instrument is provided and the general implications of the experimental results are discussed.

Vorburger, T. V.↗

NASA NTRS1985

Corrections for attached sidewall boundary-layer effects in 2-dimensional airfoil testing

The problems of sidewall boundary-layer effects in airfoil testing is treated by considering the changes in the flow area due to boundary-layer thinning under the influence of the airfoil flowfield. Using von Karman's momentum integral equation, it is shown that the sidewall boundary-layer thickness in the region of the airfoil can reduce to about half the undisturbed value under the conditions prevailing in testing of supercritical airfoils. A Mach number correction due to this increased width of the flow passage is proposed. Using the small disturbance approximation, the effect of the sidewall boundary-layers is shown to be equivalent to a change in the test Mach number and also in the airfoil thickness. Comparison of the results of this approach with other similarity rules and correlation of the experimental data demonstrate the applicability of the analysis presented from low speeds to transonic speeds.

Murthy, A. V.↗