Fundamental solution for transversely isotropic thermoelastic materials

We use the compact harmonic general solutions of transversely isotropic thermoelastic materials to construct the three-dimensional fundamental solutions for a steady point heat source in an infinite transversely isotropic thermoelastic material and a steady point heat source on the surface of a semi-infinite transversely isotropic thermoelastic material by three newly introduced harmonic functions, respectively. All components of coupled field are expressed in terms of elementary functions and are convenient to use. Numerical results for hexagonal zinc are given graphically by contours.     

Symmetry relations for orthotropic and transversely isotropic materials

Archive for Rational Mechanics and Analysis -     

Penny-shaped crack in transversely isotropic piezoelectric materials

Using a method of potential functions introduced successively to integrate the field equations of three-dimensional problems for transversely isotropic piezoelectric materials, we obtain the so-called general solution in which the displacement components and electric potential functions are represented by a singular function satisfying some special partial differential equations of 6th order. In order to analyse the mechanical-electric coupling behaviour of penny-shaped crack for above materials, another form of the general solution is obtained under cylindrical coordinate system by introducing three quasi-harmonic functions into the general equations obtained above. It is shown that both the two forms of the general solutions are complete. Furthermore, the mechanical-electric coupling behaviour of penny-shaped crack in transversely isotropic piezoelectric media is analysed under axisymmetric tensile loading case, and the crack-tip stress field and electric displacement field are obtained. The results show that the stress and the electric displacement components near the crack tip have (r ^−1/2) singularity. The project supported by the Natural Science Foundation of Shaanxi Province, China     

Non-linear stress-strain equations for incompressible transversely isotropic elastic materials

Non-linear stress-strain equations for incompressible, transversely isotropic elastic materials are developed. In order to obtain these equations, the expressions for a strain energy function is found. The derivation of the strain energy function follows a geometrical approach and a method suggested by Mooney. These stress-strain relations are expressed in terms of three principal stretches to the sixth order.     

Wave propagation in transversely isotropic porous piezoelectric materials

Wave propagation in porous piezoelectric material (PPM), having crystal symmetry 6 mm, is studied analytically. Christoffel equation is derived for the propagation of plane harmonic waves in such a medium. The roots of this equation give four complex wave velocities which can propagate in such materials. The phase velocities of propagation and the attenuation quality factors of all these waves are described in terms of complex wave velocities. Phase velocities and attenuation of the waves in PPM depend on the phase direction. Numerical results are computed for the PPM BaTiO₃. The variation of phase velocity and attenuation quality factor with phase direction, porosity and the wave frequency is studied. The effects of anisotropy and piezoelectric coupling are also studied. The phase velocities of two quasi dilatational waves and one quasi shear waves get affected due to piezoelectric coupling while that of type 2 quasi shear wave remain unaffected. The phase velocities of all the four waves show non-dispersive behavior after certain critical high frequency. The phase velocity of all waves decreases with porosity while attenuation of respective waves increases with porosity of the medium. The characteristic curves, including slowness curves, velocity curves, and the attenuation curves, are also studied in this paper.     

On the contact width in generalized plain strain JKR model for isotropic solids

Adhesive contact model between an elastic cylinder and an elastic half space is studied in the present paper, in which an external pulling force is acted on the above cylinder with an arbitrary direction and the contact width is assumed to be asymmetric with respect to the structure. Solutions to the asymmetric model are obtained and the effect of the asymmetric contact width on the whole pulling process is mainly discussed. It is found that the smaller the absolute value of Dundurs＇ parameter 13 or the larger the pulling angle O, the more reasonable the symmetric model would be to approximate the asymmetric one.     

The elliptic paraboloid failure surface for transversely isotropic materials off-axis loaded

The elliptic paraboloid failure surface has been well established as a potential criterion for yielding and failure of transversely isotropic materials, presenting also the strength differential effect [1]. This was done by extending well established criteria for isotropic materials presenting the strength differential effect (SDE), through an introduction process which maintained basic physical principles for the anisotropic materials. All previous literature concerned the special case where the principal axes of the external loading coincided with the principal strength axes of the material. In this paper the most general case where the two systems of frames are arbitrarily oriented relatively to each other is considered. In this situation the simplifications derived from the coincidence of the external principal stress and material principal strength axes are lost and the material should be considered as a general orthotropic one. The general properties for such types of loading of the transversely isotropic material are established by maintaining the general features of the failure locus invariant. Then, this study completes the investigation of yielding and failure mode of such materials considering the most general case of their loading.     

Representations for transversely hemitropic and transversely isotropic stress-strain relations

The sets of polynomial stress-strain relations for elastic points which are transversely hemitropic and transversely isotropic are presented as projections of free algebraic modules having 20 and 10 generators, respectively. Complete sets of relations for the projections are presented which allow the sets of interest to be identified as free submodules having 12 and 6 generators, respectively. The results are established using the Cartan decomposition of the representation of the adjoint action of the two-dimensional rotation and orthogonal groups on the space of three-by-three symmetric matrices. The results are compared to known representations for nonlinear transversely isotropic stress-strain relations and for linear, transversely hemitropic and transversely isotropic ones.     

Representations for transversely hemitropic and transversely isotropic stress-strain relations

The sets of polynomial stress-strain relations for elastic points which are transversely hemitropic and transversely isotropic are presented as projections of free algebraic modules having 20 and 10 generators, respectively. Complete sets of relations for the projections are presented which allow the sets of interest to be identified as free submodules having 12 and 6 generators, respectively. The results are established using the Cartan decomposition of the representation of the adjoint action of the two-dimensional rotation and orthogonal groups on the space of three-by-three symmetric matrices. The results are compared to known representations for nonlinear transversely isotropic stress-strain relations and for linear, transversely hemitropic and transversely isotropic ones.Work supported in part by National Science Foundation Grant INT-9106519.     

Influence of a thin isotropic coating on the edge effect zone in isotropic and transversely isotropic materials

The effect of a thin isotropic coating on the edge effect zone in a representative element of a coated material is examined. Isotropic and transversely isotropic materials are considered. The transversely isotropic material has the elastic properties of unidirectional glass-fiber-reinforced plastic. The decay of the edge effect in the directions perpendicular to the coating plane and to the plane of isotropy is studied. A boundary-value problem of elasticity for piecewise-homogeneouse orthotropic bodies and a quantitative edge effect decay criterion for normal stresses are used as a design model. The problem is solved using the finite-difference method and base schemes. The results of evaluation of the edge effect zone in homogeneous and inhomogeneous materials are presented. It is shown that the presence of a thin isotropic coating blocks the edge effect, that is, decreases the edge effect zone in both isotropic and transversely isotropic materials __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 61–67, December 2007.     

Green''s functions for transversely isotropic piezoelectric solids

Closed-form expressions are obtained for the infinite-body Green's functions for a transversely isotropic piezoelectric medium. The four Green's functions represent the coupled elastic and electric response to an applied point force or point charge. The Green's functions are obtained using a formulation where the three displacements and the electric potential are derivable from two potential functions. When piezoelectric coupling is absent, the results reduce to those for uncoupled elasticity and electrostatics.     

Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials

Summary Theoretical and numerical aspects of the formulation of electromechanically coupled, transversely isotropic solids are discussed within the framework of the invariant theory. The main goal is the representation of the governing constitutive equations for reversible material behaviour based on an anisotropic electromechanical enthalpy function, which automatically fulfills the requirements of material symmetry. The introduction of a preferred direction in the argument list of the enthalpy function allows the construction of isotropic tensor functions, which reflect the inherent geometrical and physical symmetries of the polarized medium. After presenting the general framework, we consider two important model problems within this setting: i) the linear piezoelectric solid; and ii) the nonlinear electrostriction. A parameter identification of the invariant- and the common coordinate-dependent formulation is performed for both cases. The tensor generators for the stresses, electric displacements and the moduli are derived in detail, and some representative numerical examples are presented.We thank Dipl.-Ing. H. Romanowski for his support and helpful remarks.     

Volume and surface stability of uniformly compressed transversely isotropic linearly elastic materials

The stability loss of a transversely isotropic linearly elastic medium is studied. The medium is uniformly compressed in both horizontal directions, and the initial stress in the vertical direction is equal to zero. The standard analysis based on the Hadamard condition is used. The bifurcation equation divides into two parts, and therefore, two kinds of buckling modes are possible. The critical initial compression is found, but the buckling modes remain indefinite (as the wave length so the relation between the wave numbers is arbitrary). The stability loss of a compressed half-space with a free surface is studied. Only one kind of buckling mode localized near the free surface is possible, and as for an entire space, the buckling mode and the wave length are indefinite. In these problems, linear as well as non-linear approaches are used. In the linear approach, the pre-buckling deformations are ignored. It is shown that for some values of parameters, the linear approach leads to qualitatively incorrect results. The stability loss of an uniformly compressed plate lying on a soft elastic half-space is studied. By using the non-linear post-critical analysis, it is shown that the buckling mode is a chessboard-like one.     

Study of stress concentration in short hollow cylinders of transversely isotropic materials

The axisymmetric stress state of hollow cylinders made of tranversely isotropic materials with a longitudinal axis of isotropy is examined in a three-dimesnional formulation. An analytical solution is obtained by using homogeneous solutions written in Kosmodamianskii's form. Results of calculations of stress concentration are given for different materials as a function of the relative thickness and relative external radius of the cylinder. Donetsk State University, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 7, pp. 43–48, July, 1999.     

Physically motivated invariant formulation for transversely isotropic hyperelasticity

This article discusses an invariant formulation for transversely isotropic hyperelasticity. The work is motivated by the interest of modeling materials such as tendon tissues which may exhibit drastically different characteristics in tensile, shear and volumetric responses. A multiplicative decomposition of the deformation gradient that factors out the dilation and the fiber stretch is proposed. Transversely isotropic strain invariants are constructed on the basis of the multiplicative factors. Within the framework of hyperelasticity theory, these strain invariants generate decoupled stress components in the hydrostatic pressure, the fiber tension and shear terms. An example model is suggested and is assessed against some known features of transversely isotropic solids with strong fibers.     

Rheological predictions of a transversely isotropic fluid model with extensible microstructure

A continuum extensible director theory was formulated to describe the isothermal, incompressible flow of uniaxial rodlike semiflexible liquid crystalline polymers. The model is strictly restricted to material that flow-align in shear, and that, in the absence of flow, are sufficiently far from the nematic-isotropic phase transition. The microstructure of the continuum is described by a variable length director, but the extensibility is finite. The model is an extension of the TIF (Transversely Isotropic Fluid) model of Ericksen (1960). The thermodynamic restrictions on the model parameters are found using the non-negative definiteness of the entropy production. The rheological material functions predicted by the model are calculated for steady simple shear and steady uniaxial extensional flows. In the rigid rod limit the model predictions agree with those of the TIF model, and for the finite extensibility case the model predictions are in agreement with those associated with flexible isotropic polymers: strong non-Newtonian shear viscosity, positive first normal stress differences, recoverable shear of order one, negative second normal stress differences, and a maximum in the steady uniaxial extensional viscosity.     

Allowance for complex loading in transversely isotropic solids

A model of plasticity for a transversely isotropic material with allowance for complex loading is developed, based on results of experiments with homogeneous cylindrical specimens of isotropic materials. An empirical model of plasticity for isotropic metals is constructed with allowance for vector properties of the material. The model is extended to a particular case of anisotropy. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 128–133, January–February, 2009.