The compression of fairly short solid cylinders under axial load is considered. A preliminary investigation examines the displacements produced by the superposition of a prescribed surface loading. This is followed by the more interesting problem in which the radial displacement is prescribed over part of the surface, the remaining part of the surface being stress free. Two types of elastic materials are considered; firstly, rubber-like materials governed by a strain energy function of the Mooney form, and secondly, metals which have a quadratic strain energy function. In the former case a finite axial compression is permitted prior to imposing any constraint on the outer curved surface of the cylinder. In all cases the irregularities introduced by the constraints are sufficiently small that they can be described by infinitesimal elasticity theory. The analysis utilizes displacement potential functions and the main problem is reduced to solving a set of dual cosine series. The particular case of the contact problem in which the cylinder height is equal to the radius is examined in detail and the contact stresses are given graphically. / Science, Faculty of / Mathematics, Department of / Graduate

Experimental results are presented on the propagation of strain waves in long rods of mild steel (type-1020) aluminum (AA2024S) and copper (soft electrolytic). Strain pulses of order 500 μ in/in amplitude were generated by mechanical impact. The average strain rate during loading was of order one per second. The length and diameter of the specimens and the dominant frequencies in the strain pulses were such that one-dimensional conditions prevailed. Budd Metalfilm resistance type strain gauges were used for recording the strain pulses. Electromagnetic disturbances were effectively eliminated by proper grounding and shielding. Elastic wave propagation in mild steel, aluminum, and copper was studied. For the steel specimen, there was no apparent attenuation or dispersion of elastic waves. However, significant attenuation and dispersion were observed in aluminum and copper specimens, a 30 percent reduction in amplitude occurring in aluminum over a distance of 8 feet. Comparison of the Fourier transforms of the strain pulses in copper and aluminum at different positions along each specimen revealed that amplitude decreased exponentially with distance and that phase angle varied linearly with distance. Furthermore, the observed attenuation and phase velocity were frequency dependent. These results conform to the behaviour of strain pulses propagating in linear visco-elastic materials. Complex compliances for aluminum AA2024S and soft electrolytic copper were derived over the frequency range 400-6000 c.p.s. from the variation of attenuation and phase angle measured in these tests. Approximate three-parameter models suitable for estimating internal damping in these two materials were also determined. Plastic wave propagation in statically prestressed rods of aluminum and copper was investigated. In copper, it was observed that strain increments propagate at constant velocity along the rod and that the velocity of propagation decreases with increasing strain. Strain-rate independent theory is thus applicable to the description of plastic wave propagation in copper, but the dynamic stress-strain curve for the material lies well above the quasi-static one. Furthermore, experimentally observed loading-unloading boundaries in copper resemble the shape predicted by Skobeev and calculations based on these boundaries are compatible with the strain-rate independent theory. It was found that annealed aluminum (AA2024S-0) does not possess a smooth quasi-static stress-strain relation and exhibits unstable behaviour under dynamic loading. / Applied Science, Faculty of / Mechanical Engineering, Department of / Graduate

An investigation of pseudoelasticity and the strain memory effect was carried out on the β’-bcc phase of alloy of composition Cu-33Zn-3.4Sn. Pseudoelasticity was found to occur by a stress-induced, martensite transformation. Maximum pseudoelasticity occurred at the Af temperature and was ~8% for single crystal specimens and ~4% for polycrystalline specimens. Calculations indicated that the large strains were due to a transformation from a bcc to a fct martensite structure. The strain memory effect was studied by deforming specimens below Af and then heating above Af. At temperatures between Ms and Af, the deformation is accommodated by the stress-induced martensite formation. At temperatures below Mf, deformation of the martensite takes place and it is suggested that there is a change in the martensite structure with an increase in the amount of orthorhombic martensite present. The pseudoelastic and strain memory effects have very similar origins and over a wide temperature range from well below Mf to well above As the combined pseudoelastic and strain memory recovery is essentially 100%. / Applied Science, Faculty of / Materials Engineering, Department of / Graduate

Elastic deformation has wide applications in medical simulations, therefore when it comes to designing physical behavior of objects for realistic training applications, determining material parameters so that objects behave in a desired way becomes a crucial. In this work we consider the problem of elasticity parameter estimation for deformable bodies, which is important for accurate medical simulations. Our work has two major steps: the first step is the data acquisition and preparation, and the second step is the parameter estimation. The experimental setup for data acquisition consists of depth and force sensors. Surface deformations are acquired as a series of images along with the corresponding applied forces. The contact point of the force sensor on the surface is found visually and the corresponding applied forces are acquired directly from the force sensor. A complete mesh deformation which is obtained from a surface tracking method is employed along with force measurements in the elasticity parameter estimation method. Our approach to estimate the physical material properties is based on an inverse linear finite element method. We have experimented with two approaches to optimize the elasticity parameters: a linear iterative method and a force-displacement error minimization method. The two methods were tested on the simulation data, and the second method was tested on three deformable objects. The results are presented for the data captured by two different depth sensors. The result is a set of two parameters, the Young's modulus and the Poisson's ratio, which represents the stiffness of the object under test.

The aim of this thesis is to extend the empirical research which has been undertaken using the GNP function approach to measuring export supply and import demand responsiveness. Exports and imports are divided into several components and detailed sets of elasticities produced. In the second part o£ the thesis imperfect adjustment is allowed for in the GNP function model. The GNP function framework treats imports as an input to the domestic technology while exports are an output. The aggregate technology can then be represented by a restricted profit function facilitating the derivation of net output supply elasticities. In this study the aggregate net outputs are exports, imports, labour and domestic sales supply. Capital is treated as a fixed input. Time-series of input-output data for Canada are used covering the period 1961 to 1980. In the first model estimated, four export and four import components are included by the use of aggregator functions and a two-stage estimation process. The recently developed Symmetric Generalised McFadden functional form which permits imposition of the correct curvature conditions while retaining flexibility is used at both the aggregator and GNP function levels. The aggregate export own-price supply elasticity was found to be 1.67 in 1970 while the aggregate import own-price demand elasticity was -1.62. Increases in the prices of both imports and labour were found to decrease the supply of exports while exports were found to be complementary to the output of domestic sales supply. The demand for labour was found to be more elastic than in earlier studies and a general trend towards increasing price responsiveness in the Canadian economy was observed. The own-price elasticities for the four export and four import components were stable and of reasonable magnitude. All the export and import components were found to be complementary. To remove the assumption of separability, modelling was extended to two larger disaggregated Generalised McFadden GNP function models containing four export (import) components, aggregate imports (exports), labour and domestic sales as net outputs. Using this procedure more substitution between the export and import components was found. A planning price model whereby the producers' notional price adjusts gradually to actual price changes indicated that imperfect adjustment is particularly important in the traded goods sector. Exports fully adjusted to price changes only over an extended period. Finally, an adjustment costs model was estimated which indicated that the main effect of allowing for imperfect adjustment was on input use. Differences between long-run and short-run export supply and import demand responsiveness were relatively small. Considerable substitutability between labour and capital in the long-run was observed and since labour was also variable in the short-run this produced overshooting of labour demand. An increase in export prices thus caused a large short-run increase in labour demand but in the long-run the capital stock was increased and substituted for much of the short-run labour increase. / Arts, Faculty of / Vancouver School of Economics / Graduate

The object of this work was to develop a computer technique for determining the characteristic parameters of spring-mass system with hysteresis effects from transient experimental data. The technique was intended to be utilized to determine the parameters of models of systems where complex geometry and rate dependent properties would make a determination of the model parameters most difficult, and where the parameters typically obtained from static test would not characterize the rate dependent properties accurately.

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