Direct connection of rheological elements at large strains: Application to multiplicative viscoplasticity

In the field of nonlinear continuum mechanics, rheological models are often used to exemplify the structure of complex material models at large strains. For this purpose, different rheological elements are combined in series and parallel connections. Ihlemann [1] developed an innovative concept, which enables the direct connection of rheological elements within the framework of multiplicative decomposition of the deformation gradient. In the contribution at hand, this approach is applied to multiplicative viscoplasticity. Towards this end, the relations for parallel and series connections are introduced and several individual material models, i.e. the rheological elements, are defined. By analytical and numerical evaluation of the connection relations, a viscoplastic material model from the literature is reproduced. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Plane strain displacement and stress waves induced in a transversely isotropic elastic solid by a line source

The plane strain displacement and stress field induced in an unbounded, transversely isotropic, elastic solid by a uniform line load is reduced to a simple residue calculation. Explicit results on a coordinate axis are given for three representative hexagonal crystals when the line load is normal to the axis of material symmetry. The case in which the applied load is not normal to the symmetry axis is also discussed.

---

Zusammenfassung Der ebene Verformungs- und der ebene Spannungszustand, die in einem unbegrenzten transversal isotropen elastischen Körper durch eine gleichförmige Linienbelastung erzeugt werden, lassen sich auf eine einfache Residuenrechnung reduzieren. Als Anwendung werden drei repräsentative hexagonale Kristalle besprochen, wobei die Linienbelastung zunächst normal zur Symmetrieachse und dann allgemein vorausgesetzt wird.

     

The indentation of a transversely isotropic half-space by a rigid punch

Zusammenfassung Der Verfasser gibt eine allgemeine Lösung für die Verteilung des Druckes zwischen einem axialsymmetrischen Stempel und einem transversal-isotropen Halbraum. Es wird gezeigt, dass die Verteilung des Druckes für den flachen Stempel mit allgemeiner Belastung unabhängig ist von den elastischen Eigenschaften des Halbraums und auch genau dieselbe, als ob der Halbraum isotrop wäre.     

Boussinesq indentation of a transversely isotropic half-space reinforced by a buried inextensible membrane

The normal indentation of a rigid circular disk into the surface of a transversely isotropic half-space reinforced by a buried inextensible thin film is addressed. By virtue of a displacement potential function and the Hankel transform, the governing equations of this axisymmetric mixed boundary value problem are represented as a dual integral equation, which is subsequently reduced to a Fredholm integral equation of the second kind. Two important results of the contact stress distribution beneath the disk region as well as the equivalent stiffness of the system are expressed in terms of the solution of the Fredholm integral equation. When the membrane is located on the surface or at the remote boundary, exact closed-form solutions are presented. For the limiting case of an isotropic half-space the results are verified with those available in the literature. As a special case, the elastic fields of a reinforced transversely isotropic half-space under the action of surface axisymmetric patch loads are also given. The effects of anisotropy, embedment depth of the membrane, and material incompressibility on both the contact stress and the normal stiffness factor are depicted in some plots.     

Transient stresses in a transversely isotropic elastic solid caused by a moving dislocation

The plane stress field induced in an unbounded, transversely isotropic, elastic solid by a dislocation moving parallel to the material symmetry axis, is reduced to a residue calculation. The dislocation, which is suddenly applied, causes a jump in displacement across the expanding fault. The speed of the dislocation is subsonic with respect to the material propagation speeds. Explicit results are obtained for the shear stress along the axis containing the dislocation and are related to several hexogonal crystals.

---

Zusammenfassung Das ebene Verformungsfeld, das in einem unbegrenzten transversal isotropen elastischen Körper infolge einer sich parallel zur Symmetrieachse bewegten Versetzung entsteht, läßt sich mit einer einfachen Residuenrechnung ermitteln. Die Versetzung, die plötzlich entstehen soll, verursacht eine sprunghafte Veränderung der Verschiebung über sich ausdehnenden Fehler. Die Geschwindigkeit der Versetzung ist in Bezug auf die materielle Fortpflanzungsgeschwindigkeit der Unterschallgeschwindigkeit gleich. Für die Schubspannung längs der Dislokationsachse werden explizite Resultate angegeben. Diese Resultate gelten auch für einige hexagonale Kristalle.

     

Steady state stresses induced in a transversely isotropic elastic solid by a moving dislocation

An unbounded, transversely isotropic, elastic solid, is subjected to a dislocation moving at constant speed. By means of an appropriate coordinate transformation, the transient version of this problem is used to obtain the steady state solution. The solution for the plane stress field is explicit and valid for dislocation speeds which are sub-, tran-, or super-sonic with respect to the material wave speeds. The previously discovered transonic speed at which the Mach head wave was annihilated for the transient problem, is found to be present in the steady state problem also.     

Elastic buckling of isotropic triangular flat plates by finite elements

Although most of the plates in steel structures are rectangular, some triangular ones do occur. The number of available solutions for triangular plate buckling coefficients is small, and restricted to special triangle geometries and loading ratios. In this paper the range of available solutions is extended by the use of a plate bending finite element devised by Irons. A range of plate parameters is considered and results are tabulated and plotted. The program is available for the use of designers who wish to consider other values of parameters. The use of the procedures described in the Merrison report can extend these results to deal with the influence of initial imperfections and weld effects.     

Transient stresses in a transversely isotropic elastic solid caused by a dislocation moving at tran- or super-sonic speeds

A suddenly applied dislocation, moving at tran- or super-sonic speed, induces a plane stress field in an unbounded, transversely isotropic, elastic solid. Treated in detail are the-function plane waves associated with this disturbance. It is shown that at a special dislocation speed, in the tran-sonic range, these head waves disappear. The results are applied to several hexagonal crystals.

---

Zusammenfassung Eine plötzliche, sich mit transonarer- oder Überschallgeschwindigkeit ausbreitende Versetzung verursacht ein ebenes Verformungsfeld in einem unbegrenzten, transversal isotropen, elastischen Körper. Die mit dieser Störung verbundenen und mit Hilfe von Deltafunktionen beschriebenen, ebenen Wellen werden im Detail behandelt. Es wird gezeigt, dass diese Stosswellen bei einer bestimmten Geschwindigkeit im transonaren Bereich verschwinden. Die Ergebnisse werden auf einige hexagonale Kristalle angewendet.

     

Transient stresses in a transversely isotropic elastic solid caused by a moving dislocation whose path may intersect a lacuna

The-function plane waves caused by the leading edge of a dislocation moving at tran- or super-sonic speed in an unbounded, transversely isotropic, elastic solid are treated herein for situations in which a lacuna intersects the dislocation path. Generally the Mach lines emitted by the source extend backward, but if the source is located within a lacuna, forward Mach lines are also produced. Several hexagonal crystals for which this phenomenon occurs are considered.

---

Zusammenfassung Der vorliegende Artikel befasst sich mit solchen, von Delta Funktionen beschriebenen, ebenen Wellen, die sich infolge einer Versetzung in einem transversal isotropen Körper mit transsonarer, oder Überschallgeschwindigkeit ausbreiten. Besonders behandelt werden Fälle, in denen sich diese Wellen auf eine störungsfreie Zone zubewegen. Normalerweise erstrecken sich die von der Störungsquelle erzeugten Machlinien nach rückwärts. Sobald sich die Störungsquelle jedoch innerhalb der störungsfreien Zone (Stillstelle) befindet, entstehen zusätzliche, nach vorwärts gerichtete Machlinien. Bestimmte hexagonale Kristalle, auf die dieses Phänomen zutrifft, werden gesondert betrachtet.

     

Large orbits of elements centralized by a Sylow subgroup

If G has a nilpotent normal p-complement and V is a finite, faithful and completely reducible G-module of characteristic p, we prove that there exist ${v_1, v_2 \in V}If G has a nilpotent normal p-complement and V is a finite, faithful and completely reducible G-module of characteristic p, we prove that there exist v₁, v₂ ? V{v_1, v_2 \in V} such that C_G(v₁)?C_G(v₂) = P{{\bf C}_{G}{(v_1)}\cap {\bf C}_{G}{(v_2)} = P} , where P ? Syl_p(G){P \in {\rm Syl}_p(G)} . We hence deduce that, if the normal p-complement K is nontrivial, there exists v ? C_V(P){v \in {\bf C}_{V}(P)} such that |K : C _K (v)|² > |K|.     

The property of orthogonality and energy transfer by three-dimensional eigenwaves in transversely isotropic laminated plates with and without contact with a liquid

The properties of eigenwaves in laminated plates with anisotropy in a transforse directon properties are studied. The most general form of the solutions, dispersion relations, power flows and generalized orthogonality relations are analysed. The similarity and difference in the properties of the waves as compared with isotropic media and ideal fluids, as well as the extension to the case of layered spaces and half-spaces, is investigated. A method of determining the coefficients for the eigenwaves radiated in the plate is proposed in the case of a problem with dynamical sources of finite size. A method of summing the series over eigenwaves is suggested.     

A remark on the paper “elastic equilibrium of a multiply connected isotropic plate with cracks” by S. A. Kaloerov

A remark on the paper “elastic equilibrium of a multiply connected isotropic plate with cracks” by S. A. Kaloerov