The well-known Timoshenko beam equations (which include transverse shear deformation and rotary inertia effects) are extended for a wide-flanged structure to include the additional shear lag deformation of the flanges; thus, cross-sections of the beam are allowed to distort instead of remaining plane sections. The effect of relative flange bending (bending of the flanges relative to the web) is also included and the integro-differential equations appropriate to the problem are derived. The frequency equation is given in closed form (neglecting the relative flange bending) and solutions for various values of the nondimensional parameters are given. A reduction of the elementary frequency by as much as 40 percent in the first mode is shown. / Master of Science

The steady state of deformation was analyzed for a clayey sand. The use of lubricated end platens was evaluated and proved to reduce the scatter in steady state data. The effect of different data corrections in a steady state analysis was also evaluated. For this investigation the parabolic area assumption better approximated the deformed specimen shape than the right cylinder assumption. It was concluded that the use of different area corrections greatly influences the slope and position of the steady state line. / Master of Science

The Pine Mountain window of Georgia and Alabama hosts the southernmost exposed Grenville aged basement terrane in the Appalachians. The window Is bounded on the east by the Box Ankle thrust fault which juxtaposes basement lithologies from hanging wall paragneiss, schist, and metavolcanic rocks of the Piedmont terrane. The Ocmulgee strike-slip fault separates Piedmont Terrane rocks from Avalon Terrane lithologies to the south and east of the Pine Mountain window. U-Pb ages of zircons constrain the timing of deformation along the Box Ankle and Ocmulgee faults at 304 ± 144 and 335 ± 7 Ma respectively. A contrast in zircon response to high grade deformation from both fault zones is observed. The response of zircon U-Pb systematics in these fault zones provides data on the effects of Pb loss versus U gain models, dissolution processes, and overgrowth binary mixing models from within selected mylonitized bulk rock chemistries. In the Ocmulgee fault, zircon overgrowth associated with deformation dominates U/Pb age discordancy. Isotopic re-equilibration of Sr Isotopes did not occur on a cm whole rock scale during deformation. Porphyroclasts In the Ocmulgee shear zone retained partial Sr Isotopic signatures of the shear zone protolith. In contrast, Rb-Sr Isotope systems In the Box Ankle fault were re-equilibrated during ductile deformation. Zircons from the Box Ankle fault show evidence of dissolution with no apparent overgrowth. A regional tectonic model proposed from ages obtained in this study Include transpression and doming of the basement and Piedmont cover as seen In the Box Ankle fault trace. Dextral strike-slip with right stepover displacement between the Ocmulgee-Goat Rock fault system and the Towaliga system provide a transpressional environment at the eastern end of the Pine Mountain window. / Master of Science

The approach to the problem of a variable thickness plate used in this paper is different from the usual approach in that this paper starts with general stress-strain relations and a generalized form of the position vector as used by Green and Zerna in "Theoretical Elasticity". They use R̅=L[ r̅ (θ₁,θ₂)+ λθ₃a̅₃(θ₁,θ₂)] where θ₁,θ₂, and θ₃ are curvilinear coordinates with θ₁ and θ₂ being the coordinates of the middle surface and λ=t/L being a constant for a plate of constant thickness t. This paper takes λ = λ(θ₁,θ₂) as a function of θ₁ and θ₂ so that the variable thickness may be taken into account. General tensor notation is used so as to work independent of coordinate systems. Making simplifying assumptions only when necessary, the equations of equilibrium and stress-strain relations are derived in terms of tensors connected with the middle surface as was done by Green and Zerna for a constant thickness plate. The additional terms obtained in these equations due to the variation in λ help us to evaluate the effects of the varying thickness. Expressions for stress are developed and they include the effects of transverse shear deformation and normal stress as well as the variation in thickness. These expressions are very much like those used by Essenburg and Naghdi in a paper presented at the Third U.S. National Congress of Applied Mechanics, June, 1958. However, they assumed the form for the stresses while the present paper arrived at their assumed forms with some additional terms after starting with general stress-strain relations. Using the notation of Green and Zerna, a set of nine equations involving the nine unknowns, m ^αβ, w, n^αβ, and v^α is derived and under appropriate boundary conditions, this set will yield a solution to the problem which will be better than the classical solution. Two problems are solved and numerical results are obtained and compared with the classical solutions. One of the problems involves a rectangular plate clamped on one edge with a uniform shear load on the other. The other problem involves a circular ring plate clamped on the outer edge with a uniform shear load on the inner edge. A much better correlation for the deflection of the middle surface is obtained for the rectangular than for the circular ring plate. The deflection at the inner edge of the ring plate obtained by the theory of this paper is over twice that obtained in the classical solution of the same problem. In the previously mentioned set of nine fundamental equations, we have the stress resultants n^αβ and the deflections v^α. With appropriate boundary conditions, these equations could lead to a study of in-plane forces and buckling of variable thickness plates, a field in which not much progress has been made. This paper does not include any numerical work in this direction. It is felt, however, that one of the principal contributions of this paper to the literature is that the set of nine fundamental equations includes the stress resultants in n^αβ thus enabling us to study the effect of in-plane forces as well as that of transverse shear deformation, normal stress, and surface tractions. / Ph. D.

Small-deflection plate equations are presented in terms of the midplane plate deformations and the temperature distribution within the plate, which is assumed independent of the plate deformation. The plate boundary conditions are presented in a general form and are suitable for solutions involving either fixed, free, or hinged edge conditions. The temperature distribution within the plate is assumed to be governed by Poisson's equation and a specified temperature distribution over the surfaces of the plate. Solutions for the temperature distribution are given in terms of a power series with respect to the plate thickness coordinate, the coefficients of which are dependent on the midplane temperature distribution and the midplane temperature gradient in the plate thickness direction. Out-of-plane plate deformations are discussed for plates with fixed edges. Discussions of plate deformations are also presented in which the temperature distributions result from constant heat generation within the plate and from radiation absorption. / Master of Science

In this work, a finite element formulation and associated computer program is developed for the transient large deformation analysis of laminated composite plate/shell structures. In order to satisfy the plate/shell surface traction boundary conditions and to have accurate stress description while maintaining the low cost of the analysis, a newly assumed displacement field theory is formulated by adding higher-order terms to the transverse displacement component of the first-order shear deformation theory. The laminated shell theory is formulated using the Updated Lagrangian description of a general continuum-based theory with assumptions on thickness deformation. The transverse deflection is approximated through the thickness by a quartic polynomial of the thickness coordinate. As a result both the plate/shell surface tractions (including nonzero tangential tractions and nonzero normal pressure) and the interlaminar shear stress continuity conditions at interfaces are satisfied simultaneously. Furthermore, the rotational degree of freedoms become layer dependent quantities and the laminate possesses a transverse deformation capability (i.e. the normal strain is no longer zero). Analytical integration through the thickness direction is performed for both the linear analysis and the nonlinear analysis. Resultants of the stress integrations are expressed in terms of the laminate stacking sequence. Consequently, the laminate characteristics in the normal direction can be evaluated precisely and the cost of the overall analysis is reduced. The standard Newmark method and the modified Newton Raphson method are used for the solution of the nonlinear dynamic equilibrium equations. Finally, a variety of numerical examples are presented to demonstrate the validity and efficiency of the finite element program developed herein. / Ph. D.

Thesis: Sc. D., Massachusetts Institute of Technology, Department of Civil Engineering, 1977 / Vita. / Sc. D. / Sc. D. Massachusetts Institute of Technology, Department of Civil Engineering

published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy

The purpose of the research was to develop a quantitative method which could be used to obtain a clearer understanding of the time-dependent fluid filteration and load-deformation behavior of soft, porous, fluid filled materials (e.g. biological tissues, soil). The focus of the study was on the development of a finite strain theory for multiphasic media and associated computer models capable of predicting the mechanical stresses and the fluid transport processes in porous structures (e.g. across the large blood vessels walls). The finite element (FE) formulation of the nonlinear governing equations of motion was the method of solution for a poroelastic (PE) media. This theory and the FE formulations included the anisotropic, nonlinear material; geometric nonlinearity; compressibility and incompressibility conditions; static and dynamic analysis; and the effect of chemical potential difference across the boundaries (known as swelling effect in biological tissues). The theory takes into account the presence and motion of free water within the biological tissue as the structure undergoes finite straining. Since it is well known that biological tissues are capable of undergoing large deformations, the linear theories are unsatisfactory in describing the mechanical response of these tissues. However, some linear analyses are done in this work to help understand the more involved nonlinear behavior. The PE view allows a quantitative prediction of the mechanical response and specifically the pore pressure fluid flow which may be related to the transport of the macromolecules and other solutes in the biological tissues. A special mechanical analysis was performed on a representative arterial walls in order to investigate the effects of nonlinearity on the fluid flow across the walls. Based on a finite strain poroelastic theory developed in this work; axisymmetric, plane strain FE models were developed to study the quasi-static behavior of large arteries. The accuracy of the FE models was verified by comparison with analytical solutions wherever is possible. These numerical models were used to evaluate variables and parameters, that are difficult or may be impossible to measure experimentally. For instance, pore pressure distribution within the tissue, relative fluid flow; deformation of the wall; and stress distribution across the wall were obtained using the poroelastic FE models. The effect of hypertension on the mechanical response of the arterial wall was studied using the nonlinear finite element models. This study demonstrated that the finite element models are powerful tools for the study of the mechanics of complicated structures such as biological tissue. It is also shown that the nonlinear multiphasic theory, developed in this thesis, is valid for describing the mechanical response of biological tissue structures under mechanical loadings.

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Engineering, 1988. / Mining induced fracturing and associated deformations can commonly be observed around deep gold mining excavations. As the rockmass behaviour and the stability of the excavations are directly influenced by these processes, a proper understanding of this influence would certainly improve current mining practices with respect to blasting, rock breaking, support design and mining lay-outs. The main subject of this thesis is the physics of failure and post failure behaviour of rock and similar materials. Failure is denned here as a state at which the material has been subjected to fracture and/or damage processes. The applicability of commonly used constitutive models in representing such failure and post failure processes has been investigated mainly by means of numerical simulations. Mechanisms which control fundamental fracture and damage processes have been analysed by comparing the results from relevant laboratory experiments with numerical models. Linear elastic fracture mechanics has been applied to explain and simulate the formation of large scale extension fractures which form in response to excessive tensile stresses. Using the flaw concept it is demonstrated that these fractures not only initiate and propagate from the surface of an opening in compressed rock, but that so called secondary fracturing can be initiated from within the solid rock as well. The effect of geological discontinuities such as bedding planes, faults and joints on the formation of (extension) fractures has also been investigated and it has been shown how the presence of such discontinuities can cause the formation o f additional fractures. Micro mechanical models have been, used to investigate the interaction and coalescence processes of micro fractures. It was found that the formation of large scale extension fracturing can be explained from such processes, but so called shear fractures could not directly be reproduced, although such a possibility has been claimed by previous researchers. The formation of shear fractures is of particular- interest as violent failure of rock, which is subjected to compressive stresses only, is often associated with such fractures. In an all compressive stress environment, only shear deformations would allow for the relief of excess stress and thus energy. The formation of shear fractures is associated with complex mechanisms and shear fractures can therefore not directly be represented by tingle cracks. In contrast to the propagation of tensile fractures, which can readily be explained by traditional fracture mechanics in terms of stress concentrations around the crack tip, the propagation of shear fractures requires a different explanation. In this thesis an attempt has nevertheless been made to reproduce shear fractures by direct application of fracture mechanics. This his been done by representing a shear fracture as a single crack and by assuming fracture growth criteria which are either based on critical excess shear stresses, or on a maximum energy release. Both criteria are completely empirical and require a value for the critical shear resistance in the same way as a critical tensile resistance is required to represent the formation of tensile fracture; , The determination of a critical tensile resistance ( Kk ) is relatively straight forward, as the formation of tensile fractures from a pre-existing flaw can be reproduced and observed in standard laboratory tests. The determination of a critical shear resistance is, however, not a common practice, as the formation of a shear fracture from a pre-existing flaw is very infrequently observed. The application of shear fracture growth criteria nevertheless resulted in plausible fracture patterns, which suggests that such criteria are realistic. It is argued here however that the formation of shear fractures cannot be associated with primary fracture growth, but rather with the localisation of failure and damage in an area which is subjected to plastic deformation. The application of fracture mechanics is therefore not correct from a fundamental point of view as these processes are not represented. For this reason plasticity theory has also been applied in order to simulate failure in general, and shear failure localisation in particular. It was in principle possible to reproduce the shear fractures with the use of this theory, but numerical restraints affected the results to such an extent that most of the simulations were not realistic. Plasticity theory can also be extended to include brittle behaviour by the use of so called strain softening models. The physical processes which lead to brittle failure are however not directly represented by such models and they may therefore not result in realistic failure patterns. It was in fact found that strain softening models could only produce realistic results if localisation of failure could be prevented. The effect of numerical restraints becomes even more obvious with a strain softening model in the case of failure localisation. While the plasticity models appear inappropriate in representing brittle failure, they demonstrated that plastic deformations can be associated with stress changes which may lead to subsequent brittle fracturing. Although only indirect attempts have been made to reproduce this effect, as appropriate numerical tools are not available, it is clear that many observations of extension fracturing could be explained by plastic deformations preceding the brittle fracturing processes. Many rocks are classified as brittle, but plastic deformation processes often occur during the damage processes as well. The sliding crack for instance, which is thought to represent many micro mechanical deformation processes in rock, directly induces plastic deformations when activated. A pure brittle rock, which may be defined as a rock in which absolutely no plastic deformation processes take place, may therefore only be of academic interest as it is inconceivable that such a rock materiel exists. Only in such an academic case would (linear) elastic fracture mechanics be directly applicable. As plastic deformation processes do play a role in real rock materials it is important to investigate their influence on subsequent brittle failure processes. The elastic stress distribution, which is often used to explain the onset of brittle fracturing, may be misleading as plastic deformations can substantially affect the stress distribution . -recediny fracture initiation. In an attempt to combine both plastic and brittle failure, use has been made of tessellation models, which in effect define potential fracture paths in a random mesh. The advantage of these models is that various failure criteria, with or without strain softening potential, can be used without the numerical restraints which are normally associated with the conventional continuum models. The results of these models are also not free from numerical artefacts, but they appear to be more realistic in general. One o f the m;ij, r conclusions based on these results is that shear failure does not occur in a localised fashion, but is associated with the uniform distribution and extension of damage. Shear failure, which can be related directly to plastic failure, can however induce brittle, tensile, failure due to stress redistribution. While the theories of fracture mechanics and plasticity are well established, their application to rock mechanical problems often leads to unrealistic results. Commonly observed firacture patterns in rock, loaded in compression, are most often not properly reproduced by numerical models for a combination of reasons. Either a model concentrates on the discrete fracturing processes, in which case the plastic deformation processes are ignored, or plasticity is represented, but brittle failure is pooxiy catered for. While theoretically a combination of these models might lead to better representations and simulations, numerical problems do affect all models to a certain extent and a practical solution is not immediately available. The results of numerical models can therefore only be analysed with caution and the underlying assumptions and numerical problems associated with a particular technique need to be appreciated before such results can be interpreted with any sense. Many of the problems are identified here and this may assist researchers in the interpretation of results from numerical simulations. Laboratory experiments, which have been chosen for analyses, involve specimens which have been subjected to compressive stresses and which contain openings from which failure and fracturing is initiated. Such specimens are less subjective to boundary influences and are far more representative of conditions around mining excavations than typical uni- and tri-axial tests. The uniform stress conditions in these latter tests allow boundary effects to dominate the stress concentrations, and thus failure initiation, in the specimens. The large stress gradients, which can be expected to occur around underground excavations, are not reproduced in such specimens. As a consequence failure is not u atained within a particular area, but spreads throughout the complete specimen in the uni- and tri-axial tests. Specimens containing openings are therefore far more likely to reproduce the fracture patterns which can be observed around deep level mining excavations. Numerical simulations of brittle, tensile fracturing around mining excavations resulted in consistent fracture patterns. Fracture patterns could however be strongly influenced by the presence of geological (pre-existing) discontinuities such as bedding planes. Although tensile stresses are often assumed to be absent around deej: <y vel excavations because typical hanging- and foot-walls are subjected to compressive horizontal strain and thus stress, the numerical models identified alternative locations o f Ix 'sile stress and also mechanisms which could induce secondary tensile stresses, A failure criterion has therefore been identified as the most likely cause of large scale fracturing while shear fracturing may only occur in the absence of such tensile stresses .and only as a consequence of failure localisation in damaged rock rather than fracture propagation (in solid rock). Geological discontinuities can easily induce tensile stresses vVher mobilised and may even replace the mining induced fractures by offering a more efficient meat s for energy release. The latter possibility is a true three dimensional issue which has not be en addressed any further in this study, but may be very relevant to jointed rock. Although dynamic failure has not directly been addressed, one of the micliamsms lor brittle, and thus stress relieving, failure under compressive strass conditi ons has been investigated in detail, namely shear fracturing. Shear fractures are effect vely the only discontinuities which allow for stress relief under such conditi ons', in the ibaence of preexisting, geological discontinuities, and are therefore quite rele vant to dynamic rock failure, such as rock bursts, in deep level mining conditions. Potential mechanisms for shear fracture formation and the numerical simulation of these features have been investigated and this may especially assist further research into rock bursts.