Use Of Elasticity Theory Research Articles

A review of the role of underground measurements in the historical development of rock engineering in South Africa

SYNOPSIS This paper describes some important aspects associated with historical underground measurements in South African gold and coal mines. Deformation measurements were used to confirm the use of elastic theory to simulate the rock mass behaviour in the Witwatersrand gold mines in the 1960s. Although a prominent time-dependent component of stope closure was measured as early as the 1930s, it was ignored owing to the benefit of adopting elastic theory. Neglecting the time-dependent response of the rock for many decades resulted in important aspects such as the effect of mining rate, the effect of advance per blast, and the need for enhanced design criteria not being explored. Recent work is only now starting to address this gap in knowledge. In-situ measurements of large coal specimens in the 1960s and 1970s indicated that a linear formula may possibly be a better approximation of coal pillar strengths. This alternative formulation was never adopted, however, as the power law strength formula was already deeply entrenched in the industry at that stage. In spite of these apparent failures to continuously generate and adopt new knowledge, a key lesson learnt is that major advances in rock mechanics will not be possible without careful monitoring of the rock mass behaviour in experimental sites. Areas requiring further research, such as pillar strength formulae for the Bushveld Complex and enhanced design criteria for the gold mines, can only be developed using extensive underground monitoring programmes. Keywords: rock engineering, underground monitoring, elastic theory, time-dependence, pillar strength.

Assessing the composition of hetero-epitaxial islands via morphological analysis: an analytical model matching GeSi/Si(001) data

A simple, but still three-dimensional, model describing the morphological stability of realistic SiGe islands on Si(001) is presented. The experimental evolution toward steeper islands with volume can be predicted for any average composition. Despite the use of elastic theory for stress relaxation under the assumption of a uniform SiGe distribution, and of a common mean surface energy of the faceted islands, the model seems to capture the essence of the energetic balance determining the morphological evolution with volume, with no fitting parameters. This is suggested by close comparison with molecular beam epitaxy data at three different temperatures (i.e. compositions). The good agreement also allows for interpreting the minor scattering of experimental data with temperature and provides a reliable tool for extracting the average Ge content from standard atomic force microscopy analysis.

Durability of Closed-End Pressurized GRP Pipes under Hygrothermal Conditions. Part I: Monotonic Tests

Environmental factors are of extreme importance with respect to the ser-vice life of composite components based on polymeric engineering matrices. Part I deals with the role of matrix ductility and chemical environmental degradation on the composites mechanical properties. Part II is concerned with the lifetime prediction of GRP filament wound pipes under low pressure levels and hygrothermal conditions. There composites are an E glass fiber reinforced polyester resin and an ECR glass fiber reinforced vinylester resin. The mechanical behavior of the pipes before and after exposure to water for different duration is studied through internal pressurizing tests with closed end procedure. The results indicate that in the hoop direction, fibers carry the complete load while in the axial direction there is a substantial contribution from the matrix. The use of elasticity theory appears as an appropriate method to describe the mechanical behavior of the pressurized pipes under monotonic loading conditions. In the hoop direction the behavior is governed by the fibers one whereas in the axial direction damage must be accounted for in the elastic law. Chain scission effects on the mechanical properties are investigated. Experimental results show that the mechanical properties, the damage mechanisms and the failure mode of the pipes are changed by water uptake. Relations between the mechanical properties and the exposure duration based on physical considerations are assessed, allowing for the definition of a damage parameter related to hygrothermal degradation. Environmental effects (swelling and chain scission) mainly affect the yielding pressure and the pressure to failure, whereas load condition and initial defects affect the axial modulus. This leads to the definition of two independent damage parameters.

Transverse elastic moduli of unidirectional fiber composites with fiber/matrix interfacial debonding

An elastic contact model is developed to predict the transverse Young's modulus, Poisson's ratio and shear modulus of unidirectional fiber composites with interfacial debonding. The elastic deformation formulae of the fiber under contact pressure are derived by the use of elasticity theory. These results are then used in the formulation of an analytic boundaryelement method for solving the interfacial debonding problem. The two extreme cases of perfect bonding and the fiber-like void are also studied. On the basis of this theory, the upper and lower bounds of the transverse moduli for unidirectional fiber composites with imperfect fiber/matrix interfaces are provided. Numerical calculations of parametric studies are conducted for four composites, and some basic characteristics of the transverse elastic moduli of unidirectional fiber composites with interfacial debonding are presented.

Finite layer analysis of the behaviour of a raft on a consolidating soil

AbstractA numerical method is developed in this paper for the analysis of the behaviour of a raft resting on a consolidating soil. The response of the raft under an applied loading is determined using the finite layer method for the soil and the finite element method for the raft. By considering deflection compatibility on the contact surface, the distribution of contact pressure is computed at various time steps. The settlement and bending moment in the raft is then evaluated by applying the calculated contact pressure back to the raft. It is shown that, in some cases, the maximum moment in the raft occurs during consolidation and that checking the final moment in the raft by use of elastic theory may not be sufficient.

Reply to ’’Comments on ’Influence of rotary inertia and shear deformation on acoustic transmission through a plate’ ’’ [J. Acoust. Soc. Am. 59, 1500–1501 (1976)

The expression and numerical results in the original letter were based on the use of Timoshenko–Mindlin plate theory and thus have an inherent frequency–plate-thickness limitation when compared to results based on the use of elasticity theory. This limitation, which is noted by Young, was already noted, however, in the concluding section of the original letter. In addition it is noted that the second acoustic coincidence lobe was not attributed in the original letter to a dilatational mode, but was attributed to a flexural wave whose phase velocity at high frequencies approaches the velocity of a dilatational wave in an infinite elastic medium. Subject Classification: [43]30.50; [43]20.60; [43]40.24; [43]20.30.

Plane Strain Formulation for Elastic-Plastic Behavior

The finite element technique has a high level of sophistication and versatility and further progress in expanding analytical capabilities in soil mechanics depends upon a consistent mathematical formulation of general and realistic constitutive relationships. Until the present time proposed formulations used in numerical analyses have been mainly on an incremental use of elasticity theory. Experimentally it is observed that: 1. Many soils are inelastically deformed almost immediately upon application of stress; 2. there is a coupling between volume changes and changes in shear stress; 3. the deformations are stress path dependent; 4. dense soils expand in volume during pore shear; and 5. many soils, drained or undrained, experience a decrease in strength with further straining after a peak strength has been reached.

Asymptotic Interaction Energy between Pairs of Point Defects in Cubic Metals

We have calculated the asymptotic interaction energies between pairs of interstitial Cu atoms in Cu and between pairs of vacancies in Al, Na, and K. These calculations were performed using the asymptotic equations of the method of lattice statics. The exact method of lattice statics is based on the Fourier transformation of the direct-space equilibrium equations for a supercell of the lattice containing one defect surrounded by a large number of host atoms. The asymptotic interaction energies were compared with the corresponding results obtained from the exact method of lattice statics. This comparison indicates that elasticity theory cannot be used to obtain strain-field interaction energies for defects nearer than at least 26 or 27 neighbors from each other, and is probably not truly valid until much larger interdefect separations are considered. This result places a heavy restriction on the use of elasticity theory in practical defectinteraction calculations.

The Use of Elastic Theory for Settlement Prediction Under Three-Dimensional Conditions

Synopsis The results of recent research into the immediate and final settlement and rate of settlement of pad foundations on clay are given. The necessity for a three-dimensional approach is discussed and a method of predicting settlements is proposed, based the use of elastic displacement theory in combination with experimental soil parameters determined over a representative range of stress. The results computations of elastic theory suitable for this method are outlined. The limitations of an elastic approach to the calculation of immediate settlements are discussed in relation to local yield within the soil. The results of laboratory tests on model footings and predictions by the proposed method are compared. The agreement is good with respect to the total final settlement, fair with respect to the immediate settle- ment, and good with respect to the rate of settlement. Les résultats des récents travaux de recherche portant sur le tassement immédiat et final et de la vitesse de tassement des plaques de fondation sur l'argile sont donnés. La nécessité d'une méthode d'étude à trois dimensions est discutée et une méthode de prédire les tassements est proposée, basée sur l'utilisation de la théorie du déplacement élastique combinée à des paramètres relatifs au sol expérimentaux établis pour une gamme représentative des contraintes. Les résultats des calculs de théorie de l'élasticité convenant à cette méthode sont esquissés. Les restrictions d'une méthode d'étude de l'élasticité pour le calcul des tassements immédiats sont discutées par rapport à la limite élastique locale à l'intérieur du sol. Les résultats des essais de laboratoire portant sur des maquettes de semelles et les prévisions par la méthode proposée sont comparés. La concordance est bonne en ce qui concerne le tassement final total, moyenne en ce qui concerne le tassement immédiat, et excellente en ce qui concerne la vitesse de tassement.

31. The use of elastic theory for settlement prediction under three-dimensional conditions : E. H. Davis and H. G. Poulos.Géotechnique, 18, 67–91 (1968)

31. The use of elastic theory for settlement prediction under three-dimensional conditions : E. H. Davis and H. G. Poulos.Géotechnique, 18, 67–91 (1968)

The Mixed State of Thin Superconducting Films in Perpendicular Fields

The behavior of ideal thin superconducting films in perpendicular magnetic fields is studied in detail and related to that of bulk type-II superconductors. A macroscopic analysis based only on the demagnetizing factors yields the dominant effects of sample geometry on the reversible magnetization curve. The same features are also derived from Pearl's generalization of Abrikosov's microscopic model, which predicts a long-range interaction between quantized flux lines in a thin film. Comparison of the macroscopic and microscopic arguments clarifies some inaccuracies in the work of Pearl and of Maki. The dependence of critical magnetic fields on film thickness is discussed for different values of the Ginzburg---Landau parameter $\ensuremath{\kappa}$. A hydrodynamic calculation demonstrates that a triangular vortex lattice is stable against small perturbations in the long-wavelength limit ($q{n}^{\frac{\ensuremath{-}1}{2}}\ensuremath{\ll}1$); for ${n}^{\frac{1}{2}}\ensuremath{\Lambda}\ensuremath{\gg}1$, the corresponding dispersion relation is $\ensuremath{\omega}=\frac{1}{4}(\frac{\mathrm{eB}}{\mathrm{mc}}){q}^{\frac{3}{2}}{\ensuremath{\Lambda}}^{\frac{1}{2}}{(n\ensuremath{\pi})}^{\frac{\ensuremath{-}1}{2}}$, where $n$ is the vortex density, $\ensuremath{\Lambda}\ensuremath{\equiv}\frac{2{\ensuremath{\lambda}}^{2}}{d}$ is the "effective penetration depth," $\ensuremath{\lambda}$ is the actual penetration depth, and $d(\ensuremath{\ll}\ensuremath{\lambda})$ is the film thickness. This conclusion disagrees with Pearl's conjecture based on elasticity theory; the long-range interaction precludes the use of elasticity theory, as is seen from the difference between the calculated dispersion relation ($\ensuremath{\omega}\ensuremath{\propto}{q}^{\frac{3}{2}}$) and that predicted for elastic modes ($\ensuremath{\omega}\ensuremath{\propto}{q}^{2}$). The dynamics of vortex systems is contrasted with the Newtonian dynamics of point masses. In practice, thin films exhibit highly irreversible behavior, and no detailed comparison between theory and experiment is attempted.