A Biphasic FEM Model for the Microperfusion in Liver Lobules

The here presented 3D-FEM model describes of microperfusion in liver lobules with the main focus on the remodeling of the microperfusion after a hepato-venous outflow obstruction caused for example by liver resection. Via a dilatation and reorientation of small vessels within the lobule, the so called sinusoids, a connection between the obstructed and normal drained areas is established. For the remodeling of the microperfusion the phenomenological approach, namely that the reorientation of the sinusoids mainly depends on the blood pressure gradient, is applied. A biphasic homogenized approach within the framework of the theory of porous media (TPM) is chosen for the modeling of the microperfusion. This work sums up the constitutive equations for the biphasic model regarding the filter velocity and transversely isotropic permeability depending on our phenomenological approach. (© 2012 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.     

Behavior of laterally and vertically loaded piles in multi-layered transversely isotropic soils

In this paper, a coupled approach of the finite element method (FEM) and the analytical layer-element method (ALEM) is proposed to conduct a research on vertically and laterally loaded piles. The FEM is used to model the pile, and the ALEM is utilized to solve the multi-layered transversely isotropic soils. Then with the assumption of force equilibrium and deformation compatibility, the interaction equation of pile and soils is obtained. Finally, the behavior of piles simultaneously subjected to lateral and vertical loads in layered transversely isotropic soils is investigated by considering the influence of lateral–vertical loads interaction and soft soil stratum.     

3D dynamic responses of a multi-layered transversely isotropic saturated half-space under concentrated forces and pore pressure

Dynamic Green's function plays an important role in the study of various wave radiation, scattering and soil-structure interaction problems. However, little research has been done on the response of transversely isotropic saturated layered media. In this paper, the 3D dynamic responses of a multi-layered transversely isotropic saturated half-space subjected to concentrated forces and pore pressure are investigated. First, utilizing Fourier expansion in circumferential direction accompanied by Hankel integral transform in radial direction, the wave equations for transversely isotropic saturated medium in cylindrical coordinate system are solved. Next, with the aid of the exact dynamic stiffness matrix for in-plane and out-of-plane motions, the solutions for multi-layered transversely isotropic saturated half-space under concentrated forces and pore pressure are obtained by direct stiffness method. A FORTRAN computer code is developed to achieve numerical evaluation of the proposed method, and its accuracy is validated through comparison with existing solutions that are special cases of the more general problems addressed. In addition, selected numerical results for a homogeneous and a layered material model are performed to illustrate the effects of material anisotropy, load frequency, drainage condition and layering on the dynamic responses. The presented solutions form a complete set of Green's functions for concentrated forces (including horizontal load in x(y)-direction, vertical load in z-direction) as well as pore pressure, which lays the foundation for further exploring wave propagation of complex local site in a layered transversely isotropic saturated half-space by using the BEMs.     

Mixed mode axisymmetric cracks in transversely isotropic infinite solid cylinders

The present study examined mixed mode cracking in a transversely isotropic infinite cylinder. The solutions to axisymmetric Volterra climb and glide dislocations in an infinite circular cylinder of the transversely isotropic material are first obtained. The solutions are represented in terms of the biharmonic stress function. Next, the problem of a transversely isotropic infinite cylinder with a set of concentric axisymmetric penny-shaped, annular, and circumferential cracks is formulated using the distributed dislocation technique. Two types of loadings are considered: (i) the lateral cylinder is loaded by two self-equilibrating distributed shear stresses; (ii) the curved surface of the cylinder is under the action of a distributed normal stress. The resulting integral equations are solved by using a numerical scheme to compute the dislocation density on the borders of the cracks. The dislocation densities are employed to determine stress intensity factors for axisymmetric interacting cracks. Finally, a good amount of examples are solved to depict the effect of crack type and location on the stress intensity factors at crack tips and interaction between cracks. Numerical solutions for practical materials are presented and the effect of transverse isotropy on stress intensity factors is discussed.     

Nonlinear transversely isotropic elastic solids: an alternative representation

A strain energy function which depends on five independent variablesthat have immediate physical interpretation is proposed forfinite strain deformations of transversely isotropic elasticsolids. Three of the five variables (invariants) are the principalstretch ratios and the other two are squares of the dot productbetween the preferred direction and two principal directionsof the right stretch tensor. The set of these five invariantsis a minimal integrity basis. A strain energy function, expressedin terms of these invariants, has a symmetry property similarto that of an isotropic elastic solid written in terms of principalstretches. Ground state and stress–strain relations aregiven. The formulation is applied to several types of deformations,and in these applications, a mathematical simplicity is highlighted.The proposed model is attractive if principal axes techniquesare used in solving boundary-value problems. Experimental advantageis demonstrated by showing that a simple triaxial test can varya single invariant while keeping the remaining invariants fixed.A specific form of strain energy function can be easily obtainedfrom the general form via a triaxial test. Using series expansionsand symmetry, the proposed general strain energy function isrefined to some particular forms. Since the principal stretchesare the invariants of the strain energy function, the Valanis–Landelform can be easily incorporated into the constitutive equation.The sensitivity of response functions to Cauchy stress datais discussed for both isotropic and transversely isotropic materials.Explicit expressions for the weighted Cauchy response functionsare easily obtained since the response function basis is almostmutually orthogonal.     

Wave propagation in an initially stressed transversely isotropic thermoelastic solid half-space

The governing equations of thermoelasticity of transversely isotropic solid with initial stresses are formulated at uniform temperature. These equations are solved analytically in two-dimensions to show the existence of three plane quasi waves, namely, Quasi-Longitudinal (QL), Thermal (T-mode) and Quasi-Transverse (QT) waves. Reflection from a thermally insulated stress free surface of an initial stressed transversely isotropic thermoelastic solid half-space is studied. A particular model is chosen for the numerical computations of the propagation speeds, attenuation coefficients and reflection coefficients. Effects of initial stress parameter and thermal disturbances are observed on speeds of propagation, attenuation coefficients and reflection coefficients.     

Non-linear radial oscillations of a transversely isotropic hyperelastic incompressible tube

The constitutive equation for a transversely isotropic incompressible hyperelastic material is written in a covariant form for arbitrary orientation of the anisotropic director. Three non-linear differential equations are derived for radial oscillations in radial, tangential and longitudinal transversely isotropic thin-walled cylindrical tubes of generalised Mooney-Rivlin material. A Lie point symmetry analysis is performed. The conditions on the strain-energy function and on the net applied surface pressure for Lie point symmetries to exist are determined. For radial and tangential transversely isotropic tubes the differential equations are reduced to Abel equations of the second kind. Radial oscillations in a longitudinal transversely isotropic tube and in an isotropic tube are described by the Ermakov-Pinney equation.     

An anisotropic phase-field model for transversely isotropic barium titanate with bounded moduli

This contribution presents a phase-field model for transversely isotropic barium titanate, which allows for the adjustment of the full set of anisotropic material parameters. It is a direct extension of the work by Schrade et al. [1] who proposed a phase-field model for ferroelectrics in the framework of invariant theory. In the present contribution, the loss of positive definiteness is avoided by formulating energetic terms that provide upper and lower bounds for all material moduli involved. We show the characteristics of the formulation by a set of numerical examples in two and three dimensions. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

A general solution for three dimensional steady-state transversely isotropic hygro-thermo-magneto-piezoelectric media

In this article, we extract the general solution of three dimensional (3D) equations using potential theory method (PTM) for steady-state, transversely isotropic, hygro-thermo-magneto-piezoelectric media (HTMPM). The governing equations are simplified by introducing the displacement functions. A general solution is completely determined by advantage of the superposition principle and operator theory, which is connected in terms of two functions, fulfilling a second-order and twelfth-order homogeneous partial differential equation (PDE), separately. With the help of Almansi’s theorem, the general solution can be further shortened, which is stated by seven harmonic functions only. The acquired general solutions are straightforward structure and helpful in boundary value problems of HTMPM. Further, we apply the 3D fundamental solutions inside an infinite and on the surface of semi-infinite of a steady point heat source united with a steady point moisture source transversely isotropic HTMPM. Comprehensive and exact solutions are given in the form of elementary functions, which appear as a standard for various types of approximate solutions and numerical codes. Some numerical simulation is conducted based on the obtained general solutions.     

Numerical simulation of transversely isotropic material using adaptive FEM

Lightweight construction plays an important role in the global task to save energy. A common approach to reduce the weight of structures is the use of composite materials like fibre reinforced polymers (FRP). FRP can be characterised by transversely isotropic material behaviour, a special case of anisotropy. In our arcticle we present the necessary efforts to include such material behaviour into an existing adaptive FEM code. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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.     

Semi-analytic modelling of transversely isotropic magneto-electro-elastic materials under frictional sliding contact

Functionally graded magneto-electro-elastic (FGMEE) materials has been increasingly used in engineering applications, particularly in smart material or intelligent structure systems. This paper proposes a semi-analytical approach for sliding frictional contact problem between a rigid insulating sphere and a transversely isotropic FGMEE film and half-space based on frequency response functions (FRFs). Multilayered approximation is used to model the functionally graded material (FGM), and the FRFs for each MEE layer are derived explicitly. The unknown coefficients in FRFs are formulated by two matrix equations, and their efficient solution process is proposed. Based on the obtained FRFs, a highly efficient semi-analytical model (SAM) is developed which is able to solve the three-dimensional frictional contact of FGMEE materials with arbitrary layer designs. The model is validated with finite element method and the literature. Furthermore, the pressure/stress distribution and electric/magnetic potential are studied in different FGM designs to investigate the influence of material layout.     

Comparison of analytical and numerical methods for homogenization of nanotube-reinforced polymers

Carbon nanotubes are increasingly getting impact as reinforcing material for polymer based nanocomposites. Hence, new modeling strategies are necessary to calculate the behavior of these materials. In the last years some attempts have been made using and developing classical micromechanical models. On the other hand numerical homogenization methods are available to tackle this problem. Examples for both types of modeling strategies are presented with focus on the nanotube geometry. The nanotubes are modeled as hollow tubes as well as as isotropic and transversely isotropic cylinders. As expected the results of numerical and analytical methods are identical for isotropic cylinder inclusions. Small deviations occur for transversely isotropic cylinders in transverse direction. In the case of hollow tube inclusions, the analytical models lead to lower stiffness values in transverse direction and for shear. The largest deviations occur for longitudinal shear with magnitudes smaller than 10%. In contrast the effort to get numerical results is enormous, so that the analytical models are still useful. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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)     

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.