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Analytical solution of a linear, elliptic, inhomogeneous partial differential equation with inhomogeneous mixed Dirichlet- and Neumann-type boundary conditions for a special rotationally symmetric problem of linear elasticityEschke, Andy 30 July 2014 (has links) (PDF)
The analytical solution of a given inhomogeneous boundary value problem of a linear, elliptic, inhomogeneous partial differential equation and a set of inhomogeneous mixed Dirichlet- and Neumann-type boundary conditions is derived in the present paper. In the context of elasticity theory, the problem arises for a non-conservative symmetric ansatz and an extended constitutive law shown earlier. For convenient user application, the scalar function expressed in cylindrical coordinates is primarily obtained for the general case before being expatiated on a special case of linear boundary conditions.
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Analytical solution of a linear, elliptic, inhomogeneous partial differential equation in the context of a special rotationally symmetric problem of linear elasticityEschke, Andy 31 July 2014 (has links) (PDF)
In addition to previous publications, the paper presents the analytical solution of a special boundary value problem which arises in the context of elasticity theory for an extended constitutive law and a non-conservative symmetric ansatz. Besides deriving the general analytical solution, a specific form for linear boundary conditions is given for user convenience.
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Partial Fourier approximation of the Lamé equations in axisymmetric domainsNkemzi, Boniface, Heinrich, Bernd 14 September 2005 (has links)
In this paper, we study the partial Fourier method for
treating the Lamé equations in three-dimensional
axisymmetric domains subjected to nonaxisymmetric loads.
We consider the mixed boundary value problem of the
linear theory of elasticity with the displacement u,
the body force f \in (L_2)^3 and homogeneous Dirichlet
and Neumann boundary conditions. The partial Fourier
decomposition reduces, without any error, the
threedimensional boundary value problem to an infinite
sequence of twodimensional boundary value problems,
whose solutions u_n (n = 0,1,2,...) are the Fourier
coefficients of u. This process of dimension reduction
is described, and appropriate function spaces are given
to characterize the reduced problems in two dimensions.
The trace properties of these spaces on the rotational
axis and some properties of the Fourier coefficients u_n
are proved, which are important for further numerical
treatment, e.g. by the finite-element method.
Moreover, generalized completeness relations are described
for the variational equation, the stresses and the strains.
The properties of the resulting system of twodimensional
problems are characterized. Particularly, a priori
estimates of the Fourier coefficients u_n and of the error
of the partial Fourier approximation are given.
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Analytical solution of a linear, elliptic, inhomogeneous partial differential equation with inhomogeneous mixed Dirichlet- and Neumann-type boundary conditions for a special rotationally symmetric problem of linear elasticityEschke, Andy January 2014 (has links)
The analytical solution of a given inhomogeneous boundary value problem of a linear, elliptic, inhomogeneous partial differential equation and a set of inhomogeneous mixed Dirichlet- and Neumann-type boundary conditions is derived in the present paper. In the context of elasticity theory, the problem arises for a non-conservative symmetric ansatz and an extended constitutive law shown earlier. For convenient user application, the scalar function expressed in cylindrical coordinates is primarily obtained for the general case before being expatiated on a special case of linear boundary conditions.
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Analytical solution of a linear, elliptic, inhomogeneous partial differential equation in the context of a special rotationally symmetric problem of linear elasticityEschke, Andy January 2014 (has links)
In addition to previous publications, the paper presents the analytical solution of a special boundary value problem which arises in the context of elasticity theory for an extended constitutive law and a non-conservative symmetric ansatz. Besides deriving the general analytical solution, a specific form for linear boundary conditions is given for user convenience.
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Kugeleindruckversuch in geschichtete MaterialienSchwarzer, Norbert 08 May 1998 (has links)
Es wird das elastische Feld, das von einem Indentor mit sphärischer Oberfläche i n einem geschichtet aufgebauten Probekörper erzeugt wird, berechnet. Dabei wird neben der Normalkomponente, die bereits in der bekannten Hertzschen L ösung näherungsweise berücksichtigt wird, auch das elastische Feld der Tangentia lkomponente für kleine Winkel zwischen Indentordruckkraft und deren Normalkompon ente betrachtet. Die zwischen Indentor und Probekörper bestehende Reibung führt zu zusätzlichen Zwangskräften, deren elastische Felder ebenfalls abgeleitet werd en. Mit Hilfe eines Ansatzes für die Lösung der Laplace-Gleichung in inhomogenen Räumen, der durch Anwendung der Methode der Spiegel- oder Bildladungen der Pote ntialtheorie erhalten wird, werden unter Anwendung der Potentialmethode Lösungen für den sphärischen Indentorversuch in geschichtete Probekörper entwickelt.
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Numerical Simulation of Short Fibre Reinforced CompositesSpringer, Rolf 09 November 2023 (has links)
Lightweight structures became more and more important over the last years. One
special class of such structures are short fibre reinforced composites, produced by injection moulding. To avoid expensive experiments for testing the mechanical behaviour of these composites proper material models are needed. Thereby, the stochastic nature of the fibre orientation is the main problem.
In this thesis it is looked onto the simulation of such materials in a linear thermoelastic
setting. This means the material is described by its heat conduction tensor κ(p), its
thermal expansion tensor T(p), and its stiffness tensor C(p). Due to the production
process the internal fibre orientation p has to been understood as random variable. As
a consequence the previously mentioned material quantities also become random.
The classical approach is to average these quantities and solve the linear hermoelastic deformation problem with the averaged expressions. Within this thesis the incorpora-
tion of this approach in a time and memory efficient manner in an existing finite element software is shown. Especially for the time and memory efficient improvement several implementation aspects of the underlying software are highlighted. For both - the classical material simulation as well as the time efficient improvement of the software - numerical results are shown.
Furthermore, the aforementioned classical approach is extended within this thesis for
the simulation of the thermal stresses by using the stochastic nature of the heat conduc
tion. This is done by developing it into a series w.r.t. the underlying stochastic. For this
series known results from uncertainty quantification are applied. With the help of these
results the temperature is developed in a Taylor series. For this Taylor series a suitable
expansion point is chosen. Afterwards, this series is incorporated into the computation of the thermal stresses. The advantage of this approach is shown in numerical experiments.
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