A transversely isotropic elastic model of geomaterials

Under consideration is the choice of parameters of a transversely isotropic elastic model for describing the linear deformation of geomaterials. We also discuss some analytical and numerical methods of solving the corresponding dynamic equations.     

A thermodynamic consistent model for an assumed transversely isotropic piezoceramic with nonlinear ferroelectric hysteresis

A characteristic feature of ferroelectric crystals is the appearance of a spontaneous polarisation, where its direction can be reversed by an applied electric field. This quantity, that has a maximum value at high electric‐fields, depends on the loading history of the material. In this paper we discuss a thermodynamic consistent phenomenological model for an assumed transversely isotropic ferroelectric crystal, where the history dependency is modelled by internal variables. The anisotropic behaviour is governed by isotropic tensor functions, depending on a finite set of invariants, that satisfy automatically the symmetry relationships of the considered body. The main goal of this investigation is to capture some characteristics of nonlinear ferroelectrica, such as the polarisation‐electric‐field and the strain‐electric‐field (butterfly) hysteresis loops. (© 2004 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.     

An analytical solution for bending of transversely isotropic thick rectangular plates with variable thickness

In this study, the bending solution of simply supported transversely isotropic thick rectangular plates with thickness variations is provided using displacement potential functions. To achieve this purpose, governing partial differential equations in terms of displacements are obtained as the quadratic and fourth order. Then, the governing equations are solved using the separation of variables method satisfying exact boundary conditions. The advantage of the purposed method is that there is no limitation on the thickness of the plate or the way the plate thickness is being varied. No simplifying assumption in the analysis process leads to the applicability and reliability of the present method to plates with any arbitrarily chosen thickness. In order to confirm the accuracy of the proposed solution, the obtained results are compared with existing published analytical works for thin variable thickness and thick constant thickness plate. Also, due to the lack of analytical research on thick plates with variable thickness, the obtained results are verified using the finite element method which shows excellent agreement. The results show that the maximum displacement of the plates with variable thickness is moved from the center toward the thinner plate edge. In addition, results exhibit the profound effects of both thickness and aspect ratio on stress distribution along the thickness of the plate. Results also show that varying thickness has not a profound impact on bending and twisting moments in transversely isotropic plates. Five different materials consist of four transversely isotropic and one isotropic, as a special case, are considered in this paper, which it is shown that the material properties have a more considerable impact on higher thickness plate.     

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.     

An asymmetric wave field in a transversely isotropic elastic medium

A transversely isotropic homogeneous elastic medium excited by a point force perpendicular to the anisotropic axis is considered. The wave field in this medium is constructed and investigated. The front sets of the SV and SH waves are in contact with one another at a point. The front sets in the vicinity of this point are investigated additionally. If we consider the SH wave (or the SV wave) separately, then a false plane front set arises in this region. In considering the SH and SV waves in combination, this false front set disappears. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 163–174.     

An error bound in Reissner's theory of transversely isotropic plates

Summary The report shows that Nordgren's [3] bound on the error of solutions in Reissner's theory of anisotropic plates can be improved in the case of plates with large transverse shear deformability.

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Zusammenfassung Diese Mitteilung beweist, daß Nordgren's [3] Fehlerabschätzung für Lösungen der Reissnerschen Theorie der anisotropen Platten für Platten mit schwacher Schiebungssteifigkeit der Querschnitte verbessert werden kann.

     

Thermal stresses in a plate with transversely isotropic material

In this paper the thermal stresses in a plate with transversely isotropic material have been obtained by the method of Hankel transforms. Three cases of surface temperature over a circular region of exposure with flux of heat, paraboloidal distribution and constant temperature with surface radiation have been considered. Numerical results are presented for the case of surface radiation.     

Poroelastodynamic potential method for transversely isotropic fluid-saturated poroelastic media

By introduction of two scalar potentials, an analytical method is developed for the solution of poroelastodynamic boundary value problems in transversely isotropic fluid-saturated poroelastic media. The governing equations of motion are considered in the framework of Biot's complete model without any assumption or simplification. As a case of application, solutions in three dimensions for a transversely isotropic fluid saturated porous half space loaded by an arbitrary distribution of time harmonic tractions at the free surface is derived. The free surface of the half space may be considered either permeable or impermeable. As a particular solution, Green's functions for uniform vertical and horizontal circular patch loads are presented as semi-infinite integrals which may be evaluated by means of an appropriate numerical method proposed. The accuracy of the solutions is verified both analytically and numerically against the preceding solutions. Some numerical results are also presented to clarify the influence of different degrees of anisotropy and frequency of excitation on the response of the medium.     

On anisotropic caginalp phase-field type models with singular nonlinear terms

Our aim in this paper is to study the well-posedness and the existence of the global attractor of anisotropic Caginalp phase-field type models with singular nonlinear terms. The main difficulty is to prove, in one and two space dimensions, that the order parameter remains in the physically relevant range and this is achieved by deriving proper a priori estimates.     

On convex perturbations with a bounded isotropic constant

Let be a convex body and ɛ > 0. We prove the existence of another convex body , whose Banach–Mazur distance from K is bounded by 1 + ɛ, such that the isotropic constant of K’ is smaller than , where c > 0 is a universal constant. As an application of our result, we present a slight improvement on the best general upper bound for the isotropic constant, due to Bourgain. The author is a Clay Research Fellow, and was also supported by NSF grant #DMS-0456590. Received: November 2005; Accepted: February 2006     

Stability of transversely isotropic annular plates with low shear resistance

The stability differential equations of a cylindrically orthotropic circular plate are obtained on the assumption of an axisymmetric buckling mode with allowance for transverse shears. These equations are solved for the case of a transversely isotropic material when the inner and outer edges of the plate are identically loaded by uniformly distributed radial forces. The transcendental equations for the critical load parameter are constructed for various edge conditions. The dependence of this parameter on the boundary conditions and the relative thickness of the plate, Poisson's ratio, and the ratio of the Young's and shear moduli of the material are investigated. Certain conclusions are reached concerning the design of reinforced-plastic plates.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 872–880, September–October, 1969.     

A group analysis of the equations of the dynamic transversely isotropic elastic model

Group foliation of the system of equations of motion of a transversely isotropic elastic model of geomaterials, satisfying the Gassmann conditions, is carried out. A linear system of first-order differential equations is obtained, equivalent to the equations of this model. A number of theorems describing its properties are proved. A fundamental Lie transformation and an optimum system of its subgroups are obtained, which enable all the invariant, partially invariant and differentially invariant submodels of the dynamic model of a transversely isotropic elastic medium to be obtained. Some exact solutions are derived and their physical meaning is indicated.     

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)     

Asymmetric Green's functions for a functionally graded transversely isotropic tri-material

This paper considers the elastic analysis of a functionally graded transversely isotropic tri-material solid under the arbitrary distribution of applied static loads. Using two displacement potential functions, for three-dimensional point-load and patch-load configurations, Green's functions for displacement and stress components are generated in the form of infinite line-integrals. These solutions are shown to be analytically reducible to the special cases of exponentially graded bi-material, exponentially graded half-space and homogeneous tri-material Green's functions. It also encompasses a functionally graded finite layer on a rigid base with surface loading with two cases of interfacial conditions, rigid-bonded and rigid-frictionless. Finally, for the special case of a functionally graded layer sandwiched between two homogeneous layers, using several numerical displays, the effect of material inhomogeneity on the responses is studied and the accuracy of numerical scheme is verified.