Re-entrant corner flow of Phan-Thien-Tanner fluids

Results are described for the planar flow of Phan-Thien-Tanner (PTT) fluids around re-entrant corners. A stress singularity similar to the Upper Convected Maxwell (UCM) model is obtained, namely O (r^–2(1–α)) where r is the radial distance from the corner and π /α the corner angle. However, the stream function vanishes at the slower rate O (r^(1+α)α ) compared to O (r^(3–α)α ) for UCM – this feature being a consequence of the shear thinning property of the PTT model. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the constitutive relations for isotropic and transversely isotropic materials

In this work the strain and stress spaces constitutive relations for isotropic and transversely isotropic softening materials are developed. The loading surface is considered in the strain space and the normality rule; the stress relaxation is proportional to the gradient of the loading surface, is adopted. It is found that the strain space plasticity theory allows us to describe the hardening, perfectly plastic and softening materials more accurately. The validity of the strain space constitutive relation for transversely isotropic materials are confirmed by comparing with the experimental data for fiber reinforced composite materials. Some numerical examples in two and three dimensional elasto-plastic problems for various loading–unloading conditions are presented, and give a very good agreement with the existing results.     

Analytic method for solving a class of three-dimensional thermoelasticity problems for layered transversely isotropic plates with variable thicknesses

This paper examines three-dimensional boundary value problems in the theory of heat conduction and thermoelasticity for layered transversely isotropic rectangular plates with variable thicknesses acted on by a nonuniform temperature field. It is assumed that known temperature and heat flux at the surfaces of the plate or temperature of the surrounding medium allow a representation of the solution in terms of double trigonometric series. An approximate analytic method has been developed for solving this class of problems which makes it possible to reduce the initial boundary value problem for a plate of variable thickness to a recurrence sequence of the corresponding problems for plates with constant thicknesses. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 26–36, 1999.     

Equivalence of periodically layered and transversally isotropic media

The propagation of waves in isotropic media with homogeneous layers is considered. When the period decreases without bound and the number of periods grows without bound, these media go over into transversally isotropic media. The properties of the media obtained are investigated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 219–233, 1979.     

Deformation theory for gas-impregnated transversely isotropic materials

A mathematical model is constructed for a gas-impregnated transversely isotropic material. The material is modeled by a three-phase solid at each point of which there is a solid phase, and free and sorbed gas. Gas movement proceeds in two ways: by a crack and macropores system (filtration flow), and through microcracks (diffusion flow).Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 93–97, 1989.     

Stress distribution in layered composites

The mechanism governing the transmission of stresses in layer-like composite materials is analyzed on the basis of the equations of Bolotin's [2, 7] theory of layer-like media. A solution is presented for the plane problem of stress distribution in a medium under the influence of loads at the boundary of one of the reinforcing layers. Some approximate solutions based on various simplifying approximations are presented, and the limits of their applicability are discussed. Simple equations are given for the stress maxima in the binding layers. The results are used in order to discuss the mechanism underlying the transmission of stresses in layered materials.Moscow Power Institute. Translated from Mekhanika Polimerov, No. 2, pp. 319–325, March–April, 1970.     

Three-dimensional mixed problems of thermoelasticity for isotropic plates

We obtain the homogeneous thermal solutions due to a temperature field for the three-dimensional thermoelastic problem for isotropic plates on whose plane faces homogeneous thermal and mixed mechanical conditions of flat face and diaphragm type are prescribed. This makes it possible to reduce the thermoelastic boundary-value problem to the corresponding elasticity problem. Translated fromTeoreticheskaya i Prikladnaya Mekhanika. No. 25, 1995, pp. 3–8.     

Nitsche type mortaring for elliptic problems with corner singularities

The paper deals with Nitsche type mortaring as a finite element method (FEM) for treating non‐matching meshes of triangles at the interface of some domain decomposition. The approach is applied to the Poisson equation with Dirichlet conditions for the case that the interface passes re‐entrant corners of the domain and local mesh refinement is applied. Some properties of the finite element scheme and error estimates in a discrete H¹‐like and in the L₂‐norm are proved.     

The energy function for isotropic materials with couple stresses

Zusammenfassung Die Arbeit bezieht sich auf die von Toupin entwickelte Theorie der Momentenspannungen. Es werden die Invarianten der Energiefunktion eines rein elastischen isotropen Materials dieser Art hergeleitet. Die Verzerrungsenergie ihrerseits lässt sich als Polynom in diesen Invarianten ausdrücken.

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Supported by the National Science Foundation.

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Numbers in brackets refer to References, page 908.     

Optimization of properties of multicomponent isotropic composites

We suggest a method for the calculation of the extremal conductivity of composites under some natural assumptions concerning their microstructure. The method is based on the principle of consecutive assembling of binary mixtures by addition of infinitely small amounts of one of the initial compounds to the already-assembled isotropic composite. This process is assumed to produce an optimal isotropic binary mixture at each step, which is performed by the Hashin-Shtrikman procedure. We are seeking a suitable sequence of compounds to be added to the mixture in order to minimize its resultant conductivity. A solution is given to the corresponding optimization problems for both finite number and infinite number of initial compounds taken in prescribed concentrations. We also describe the microstructure of the optimal composites. The results can be used for the optimal design of elastic and heat-conducting constructions.Dedicated to G. LeitmannThe authors have accepted with pleasure the invitation to participate in the JOTA special issue dedicated to Professor G. Leitmann. The papers by Professor Leitmann are well known in the USSR. They have greatly influenced the formation of the authors' scientific viewpoints. The first author has learned much from Professor Leitmann in the course of his work in translating two books by Professor Leitmann published in the USSR; the second author has also benefited from these translations. Methods of optimization, whose development has been influenced so much by Professor Leitmann's contributions, have penetrated into a number of fields. Here, we apply these methods to a problem in the mechanics of composite materials.     

Stress channelling in transversely isotropic elastic composites

Résumé Une théorie a déjà été proposée au sujet des matériaux à fibres de renforcement où l'on supposait que ces dernières étaient inextensibles et uniformément distribuées dans un composite considéré comme incompressible. Cependant, quelques unes des prédictions de cette théorie semblent être fondamentalement en désaccord avec la théorie classique de l'élasticité. Il est démontré ici que les résultats inattendus de cette théorie correspondent en fait à des cas limites de la théorie classique de l'élasticité pour des matériaux à isotropie transversale.     

Transverse isotropic flow in biphasic materials

Biological and artificial tissues like muscles, brain material, saturated paper or other functionally graded porous materials often show a transverse isotropic permeability caused by their inner structure. The transverse isotropic permeability influences the motion of the saturating fluid and with this the overall deformation state of the mixture body. Starting with the multi-phase Theory of Porous Media, a model is presented to describe this deformation affected by the anisotropic inner permeability. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Load transmission in layered materials

Load transmission in reinforced plastic heterogeneous media is investigated on the basis of the equations of the theory proposed in [1–3]. The stress distribution problem is solved for a layered half-space with loads applied to one of the "hard" layers in the plane of that layer. Simplified stress formulas are presented. The corresponding error is estimated by working a numerical example. The results are compared with the corresponding problem of the theory of elasticity for a homogeneous orthotropic body.Mekhanika Polimerov, Vol. 4, No. 2, pp. 322–327, 1968     

Spectral parameter power series analysis of isotropic planarly layered waveguides

This work addresses the analysis of an isotropic planarly layered waveguide consisting of an inhomogeneous core that is enclosed between two homogeneous layers forming the cladding. The analysis relies on an auxiliary one-dimensional spectral problem that is intimately linked with the scalar wave equation for planarly layered media. We construct the Green function of the waveguide as an expansion involving the eigenfunctions of the continuous and the discrete spectrum of the auxiliary problem. From the eigenvalues of the discrete spectrum, we calculate the allowed propagation constants of the guided modes. The Spectral Parameter Power Series (SPPS) method [Math. Method Appl. Sci. 2010;33: 459–468] leads us to analytic expressions for the eigenfunctions of the auxiliary problem in the form of power series of the spectral parameter. In addition, we obtain an SPPS representation for the dispersion relation without making any kind of approximation or discretisation to the core of the waveguide. The SPPS analysis here presented is well suited for its numerical implementation, since all these series can be truncated due to their uniform convergence.     

Layers and corner singularities in singularly perturbed elliptic problems

Some recent work and open problems are reviewed concerning the numerical solution of singularly perturbed elliptic boundary value problems whose solutions have boundary layers and corner singularities. AMS subject classification (2000)  65N15, 65N50, 35J25     

On a least-squares formulation for hyperelastic,transversely isotropic problems

In the present work a mixed finite element based on a least-squares formulation is proposed. In detail, the provided constitutive relation is based on a hyperelastic free energy including terms describing a transversely isotropic material behavior. Basis for the element formulation is a weak form resulting from a least-squares method, see e.g. [1]. The L₂-norm minimization of the residuals of the given first-order system of differential equations leads to a functional depending on displacements and stresses. The interpolation of the unknowns is executed using different approximation spaces for the stresses (W^q (div, Ω)) and the displacements (W^1,p(Ω)), under consideration of suitable p and q. For the approximation of the stresses vector-valued shape functions of Raviart-Thomas type, related to the edges of the respective triangular element, are applied. Standard interpolation polynomials are used for the continuous approximation of the displacements. The performance of the proposed formulation will be investigated considering a numerical example. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Form of strength criterion in isotropic materials

An examination is made of the staged form of the generalized criterion of strength in isotropic media. The proposed variant for the criterion is constructed on the basis of the oriented character of the damage around the dangerous point. In this case it is proposed that the damage in some particular direction is effected with a certain probability. An example of its application is given. The process of loading the material is taken to be isothermic and short-term.