The centre of resistance and centre of rotation of a tooth are key concepts in orthodontics. Determining the coordinates of these points is essential for planning orthodontic tooth movement. The paper proposes a procedure for experimental determination of the two centres, verifies a previously developed mathematical theory through computer simulations and evaluates the effect of nonlinearity of the periodontal ligament on the parameters related to the centres. The paper relies on the mathematical theory of a rigid body embedded in an elastic medium, physical experiment, finite element method and a simple nonlinear theory of the tooth. The concept of the centre of resistance requires a revision if nonlinearity is taken into account.
The general theory of linearized elasticity with eigenstrain is considered with applications to continuous, discrete and discretized structures. It is shown that any eigenstrain can be uniquely decomposed into impotent and nilpotent constituents. The proven theorem on decomposition is based on the concepts of functional analysis, in particular, with respect to Hilbert functional spaces. This unique decomposition allows for the individual and independent control of stress, strain and displacement (e.g. shape control). The associated algorithm avoids the cumbersome solution of boundary‐value problems with eigenstrain in connection with these control problems. Decomposition of eigenstrain opens the practically important opportunity to fully separate the control of strain and stress produced by force loading.
Система уравнений сохранения количества движения и энергии, описывающая свободноконвективное течение около вертикальной поверхности в полубесконечной пористой среде при граничных условиях третьего рода, рассмотрена в приближении пограничного слоя. Построены асимптотические разложения по степеням параметра Дарси (Da) . Исследована зависимость характеристик потока от значений параметров Dа и Gr x . Построены аналитические решения, применимые во всей области течения.Ключевые слова: свободная конвекция, пористая среда, пограничный слой, асимптотические разложения, параметр Дарси
INFLUENCE OF BOUNDARY CONDITIONS ON FREE CONVECTION FLOW IN POROUS MEDIUM WITH NEWTONIAN HEATING FROM A VERTICAL SURFACE
A.A. Bocharova and I.V. Plaksina
Far East State Technical University, Vladivostok, RussiaThe system of energy and momentum conservation equations, describing a free convection flow formed near a vertical surface in a semi-infinite porous medium under third-kind boundary conditions, is considered using boundary layer approximations. Asymptotic expansions in powers of a Darcy parameter (Da) are constructed. The dependence of flow characteristics on the governing dimensionless parameters Da and Gr x is analyzed. Analytical solutions available for the entire flow domain are derived.
The method of unique decomposition of eigenstrain into two constituents, namely in impotent eigenstrain, that does not cause stress and in the complementary nilpotent eigenstrain that does not induce any deformation in the linear elastic solid is considered in detail. Decomposition is performed in the Hilbert (energy) function space. These two complete classes of eigenstrains render optimal solutions by keeping shape and stress‐control problems well separated. Such an optimization is found to be of crucial importance in the design of smart structures, which can adapt to altering conditions. Benchmark solutions of both, shape control and optimal redistribution of elastic load stresses in discretized smart structures illustrate the power of the novel method and its convenience of application. Further, the problem of residual stress control past hot rolling of I‐beams is solved: the theorem on decomposition provides the crucial tool to define the required control parameters for forced cooling‐down with the goal of producing a favourable residual stress state as close as possible to a predefined (optimal) stress state. The latter is exemplarily selected to increase the buckling strength of such columns with given slenderness ratio.
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