Approximate Analytical Relationship Research Articles

Fatique and damage-tolerance analysis of composite laminates: Stiffness loss, damage-modelling, and life prediction

The prediction of fatigue life and evaluation of progressive damage for general composite laminates are studied analytically. From theories of damage tolerance, residual-modulus degradation, and residual-strength d degradation, a simple and approximate approach is proposed for the prediction of progressive stiffness loss, matrix-crack density, and delamination area in terms of tension-tension fatigue load and number of cycles for general laminates containing 0° plies. The proposed approach provides four choices for predicting tension-tension allowable fatigue life and for assessing fail-safety for structures made of composite laminates, namely, a residual-modulus criterion, a matrix-cracking criterion, a delamination-size criterion, and a residual-strength criterion. The approximate analytical relationship between S-N curves for general laminates containing 0° plies and for a unidirectional 0°-ply laminate is proposed. Three glass/epoxy laminates of [0/90] s, [±45/0/90] s , and [0/±45] s lay-ups, and one [0/90±45] s T300/5208 graphite/epoxy laminate under tension-tension fatigue are used to illustrate and verify the proposed approach. The analytical results are in good agreement with experimental data.

Buckling behavior of bead-stiffened composite panels

The integral formation of stiffeners in a panel is economically attractive in terms of a reduced number of parts and manufacturing operations. For thermoplastic composite panels stiffened by integral formed open-section beads, the effects of bead spacing and bead cross-section geometry on the initiation of buckling under uniaxial compression and uniform shear loading have been investigated. Approximate analytical relationships for buckling of flat, orthotropic plates have been verified by the use of classical solution methods. Finite element results for a range of stiffened panel sizes and bead geometries are presented and compared with closed-form solutions based on an effective flat plate size. Experimental verification of analytical predictions for four of the panels is described.

Estimating effective rigidity moduli of composites from electrical conductivities

An approximate analytic experimental relationship has been derived between the elasticities and electrophysical characteristics of electrically conducting composites. The theoretical-experimental relationships have been confirmed by experiment. One can determine the elasticity parameters for composites from the electrical conductivities determined by nondestructive testing.

Structure of a plane and stationary ionizing current sheath

A simple model that describes the structure of a plane and stationary current sheath moving against a neutral gas pushed by a magnetic field (like the one found in Plasma Focus experiments) is given. The numerical solutions of the equations are presented, and approximate analytical relationships between some features of the structure and the presence of the neutral gas and the magnitude of the pushing magnetic field are derived.

Mathematical model of the single-fibre action potential

A methematical model of the single-fibre extra- and intracellular action potential is proposed whereby the transition from one to the other is given by an approximate analytical relationship. The model is given in both the time and the frequency domain, and has a simple mathematical form. Based on the new model an analytical description of the transfer function of the muscle tissue is given. The proposed model compares favourably with previously published models and experimental results. The model allows a simple interpretation of the involved influence of the length and asymmetry of the transmembrane potential on the extracellular potential field. The proposed model should be helpful in the electrical modelling and simulation of neuromuscular activity.

Analytical representation of the shape of the meniscus in Czochralski growth

In order to model heat flow during Czochralski growth, a knowledge of the shape of the meniscus and its dependence on the angle of contact at the three-phase boundary and on the crystal radius is required. An approximate analytical relationship is derived for this purpose which is shown to compare well with the full iterative numerical solution of the Laplace-Young equation for the range of parameters appropriate to the Czochralski growth of germanium.

Oscillatory phenomena in a model of infectious diseases

Oscillations of the number of cases around an average endemic level are common in several infectious diseases. In this paper we study simple deterministic models, where the oscillations arise either solely from periodically varying contact rates or from the combined effect of large initial perturbation, small periodic variation of the contact rate, and the destabilizing nature of infectious and latent periods when described as time delays. The main results are: (a) For a model with a periodically varying contact rate and a recovery rate, a threshold amplitude of variation is found by numerical and analytic methods at which 2-year subharmonic resonance appears. (b) Approximate analytic relationships are derived for the amplitude and phase of the forced 1-year oscillations below this threshold and for the 2-year oscillations above it—in terms of the reproduction rate of the infection. (c) Similar calculations are performed when the recovery rate is replaced by a fixed infectious period represented by a pure time delay. The threshold amplitude of variation in the contact rate is found here to be smaller than in the recovery rate model. (d) A model with a fixed infectious period and a constant contact rate is considered. The nontrivial steady state is shown to be locally stable for the parameter range of interest. However, the ratio of the imaginary to real parts of the eigenvalues in the characteristic equation is increased as compared to the corresponding model with a recovery rate. (e) For the model with a fixed infectious period and a constant contact rate an approximation method indicates consistency in a certain range of contact rates with the existence of an unstable periodic solution about the locally stable steady state. The actual existence of such a solution is not verified. The interpretation is that the destabilizing effect of the introduction of a pure delay into the model becomes more significant as the distance in the variables space from the endemic steady state is increased. (f) For a fixed infectious period and very small subthreshold variation in the contact rate, two different types of solutions are found numerically: yearly small-amplitude oscillations about an endemic average and large-amplitude oscillations of a subharmonic period. The pattern seen depends on the initial conditions. For a sufficiently large initial deviation from the endemic level even very small seasonal variations lead to regular recurrent outbreaks of the disease. The effect of latent periods and of changing the form of the interaction are also considered.

Spherical particle populations: approximate analytic relationship between size distribution parameters and integral optical properties; erratum

Get PDF Email Share Share with Facebook Tweet This Post on reddit Share with LinkedIn Add to CiteULike Add to Mendeley Add to BibSonomy Get Citation Copy Citation Text J. Heintzenberg and M. Baker, "Spherical particle populations: approximate analytic relationship between size distribution parameters and integral optical properties; erratum," Appl. Opt. 15, 1904-1905 (1976) Export Citation BibTex Endnote (RIS) HTML Plain Text Citation alert Save article

Spherical particle populations: approximate analytic relationship between size distribution parameters and integral optical properties

The absorption and scattering efficiencies of spherical particles in a certain size range can be approximated by a three-parameter formula. With this formula, an analytic relationship is developed between the extinction, scattering, or adsorption coefficient and the parameters of a log-normal size distribution. The technique permits approximate calculation of these coefficients from the mean radius, its standard deviation, and the total number concentration. By an inversion the size distribution parameters can be estimated from measurements of any of the coefficients at three wavelengths. The calculations require only hand calculators or slide rules. It is shown that use of the technique is justified for many cases of atmospheric aerosol particles and visible radiation.

Quantitative description of spall damage

Experimental data relating to spoiling effects in organic glass (Plexiglas) and the aluminum alloy AMg-6 obtained by the shock loading of the samples are presented in the form of an approximate analytical relationship between the degree of damage, the applied stress in the rectangular pulse, and the period of action of the latter.

Approximate analytic relationship for the efficiency of a thin radiating bar

An approximate relationship between the efficiency of a thin radiating bar and the dimensionless thermal-conductivity parameter is proposed. The temperature distribution over the bar length is obtained for a wide range of values of this parameter.