An energy function for transversely-isotropic elastic material and the Ponyting Effect

On the basis of the semi-linear material of John, invoking the theory of homogenization for heterogeneous media and the theory of invariants for isotropic scalar functions, an energy function is built for a transversely-isotropic medium in finite elastic deformation. The Ponyting Effect, for material in simple shear, is reviewed for this case of transversal isotropy. It is shown that this effect is apprehended by the constructed energy function.     

A generalized polyconvex hyperelastic model for anisotropic solids

A generalized polyconvex hyperelastic model for anisotropic solids is presented. The strain energy function is formulated in terms of convex functions of generalized invariants and is given by a series with an arbitrary number of terms. The model addresses solids with orthotropic or transversely isotropic material symmetry as well as fiber-reinforced materials. Special cases of the strain energy function suitable for anisotropic elastomers and soft biological tissues are proposed. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Polyconvex energy densities with applications to anisotropic thin shells

This work shows the capability to simulate anisotropic thin shells using polyconvex energy densities by analyzing numerical examples. The variational framework is based on the enhanced assumed strain formulation. The iterative enforcement of the zero normal stress condition at the integration points allows consideration of arbitrary three–dimensional constitutive equations. We consider an additive structure of the energy decoupled in an isotropic part for a matrix material and the superposition of transversely isotropic parts for embedded fiber families and focus on polyconvex strain energy functions. In two representative examples we document anisotropy effects. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Finite homogeneous deformations of symmetrically loaded compressible membranes

The equilibrium problem of nonlinear, isotropic and hyperelastic square membranes, stretched by a double symmetric system of dead loads, is investigated. Depending on the form of the stored energy function, the problem considered may admit asymmetric solutions in addition to the expected symmetric solutions. For compressible materials, the mathematical condition allowing the computation of these asymmetric solutions is given. Moreover, explicit expressions for evaluating critical loads and bifurcation points are derived. Results and basic relations obtained for general isotropic materials are then specialized for a compressible Mooney–Rivlin material and a broad numerical analysis is performed. The qualitatively more interesting branches of asymmetric equilibrium are shown and the influence of the material parameters is discussed. Finally, using the energy criterion, some stability considerations are made.     

Finite homogeneous deformations of symmetrically loaded compressible membranes

The equilibrium problem of nonlinear, isotropic and hyperelastic square membranes, stretched by a double symmetric system of dead loads, is investigated. Depending on the form of the stored energy function, the problem considered may admit asymmetric solutions in addition to the expected symmetric solutions. For compressible materials, the mathematical condition allowing the computation of these asymmetric solutions is given. Moreover, explicit expressions for evaluating critical loads and bifurcation points are derived. Results and basic relations obtained for general isotropic materials are then specialized for a compressible Mooney–Rivlin material and a broad numerical analysis is performed. The qualitatively more interesting branches of asymmetric equilibrium are shown and the influence of the material parameters is discussed. Finally, using the energy criterion, some stability considerations are made.     

Rank-one convexity implies polyconvexity for isotropic,objective and isochoric elastic energies in the two-dimensional case

We show that in the two-dimensional case, every objective, isotropic and isochoric energy function which is rank-one convex on GL⁺(2) is already polyconvex on GL⁺(2). Thus we negatively answer Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasiconvexity. Our methods are based on different representation formulae for objective and isotropic functions in general as well as for isochoric functions in particular. We also state criteria for these convexity conditions in terms of the deviatoric part of the logarithmic strain tensor. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Implications of Shield’s inverse deformation theorem for compressible finite elasticity

Some properties of the Shield transformation on elastic strain energy functions are established. It is reflexive, it preserves objectivity and material symmetry for isotropic materials, and it also preserves infinitesimal strain response, ellipticity and Hadamard stability, and the Baker–Ericksen condition. Two new classes of strain energies for compressible isotropic materials are introduced, one of them being the image under the Shield transformation of the class of harmonic strain energies. In view of Shield’s Inverse Deformation Theorem, these new classes of strain energies will allow solution in closed form of a variety of problems in finite elastostatics.     

Implications of Shield’s inverse deformation theorem for compressible finite elasticity

Some properties of the Shield transformation on elastic strain energy functions are established. It is reflexive, it preserves objectivity and material symmetry for isotropic materials, and it also preserves infinitesimal strain response, ellipticity and Hadamard stability, and the Baker–Ericksen condition. Two new classes of strain energies for compressible isotropic materials are introduced, one of them being the image under the Shield transformation of the class of harmonic strain energies. In view of Shield’s Inverse Deformation Theorem, these new classes of strain energies will allow solution in closed form of a variety of problems in finite elastostatics. Received: January 30, 2002     

Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem

The pure azimuthal shear problem for a circular cylindrical tube of nonlinearly elastic material, both isotropic and anisotropic, is examined on the basis of a complementary energy principle. For particular choices of strain-energy function, one convex and one non-convex, closed-form solutions are obtained for this mixed boundary-value problem, for which the governing differential equation can be converted into an algebraic equation. The results for the non-convex strain energy function provide an illustration of a situation in which smooth analytic solutions of a nonlinear boundary-value problem are not global minimizers of the energy in the variational statement of the problem. Both the global minimizer and the local extrema are identified and the results are illustrated for particular values of the material parameters.      

横观各向同性多孔超弹性矩形板的单向拉伸

利用横观各向同性超弹性材料的广义neo-Hookean应变能函数研究了含有多个微孔的超弹性矩形板在单向拉伸作用下的有限变形和受力分析.给出了含有某种对称性分布的多个微孔的矩形板的变形模式,通过求解该变形模式满足的微分方程,将它用两个参数表示出来.可应用最小势能原理导出变分近似解,从而得到矩形板的变形和应力分布的解析解.分析了板中微孔的增长及微孔边缘应力的分布情况,讨论了板的各向异性程度及微孔的大小和孔间距离的影响,得到了单个、三个及五个微孔板中微孔的增长变形和孔边应力分布的一些基本规律规律,并进行了相互比较.     

A three-dimensional progressive damage model for fiber reinforced composites with an implicit-explicit integration scheme

This contribution proposes a fully three dimensional “continuum damage model” (CDM) to describe the interlaminar and intralaminar failure mechanisms of transversely isotropic elastic-brittle materials under static loading. The constitutive model is derived from an energy function with independent damage variables for each damage mode. The evolution law is based on energy dissipation within the damage process, taking into account the critical energy release rate to weaken the effect of mesh dependent outcome. The onset of damage can be predicted with Cuntze's failure mode concept [1] as well as with Hashin's failure criteria. In this model linear stress decreasing is assumed. In addition, an implicit-explicit integration scheme, first proposed by Oliver [3] for isotropic damage models, is adapted to increase the stability and robustness of numerical simulations and to decrease the computational cost of material failure analyses. By comparing the results from implicit-explicit integration schemes and standard implicit integration schemes, a high level of agreement is found. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Second-order Characteristics for Stochastic Structures Connected with Gibbs Point Processes

The points of stationary and isotropic GIBBS point process are marked by exp(corresponding local energy). For the mean mark and two mark product density functions very simple formulae are true which contain the intensity and the pair potential function of the process. Furthermore, there is a close connection between the pair correlation function of the process and the covariance function of the random field given by the conditional intensity.     

Constructing the governing equations for a layered composite of regular structure within the limits of the theory of asymmetric elasticity on the basis of energy criteria of equivalence

Within the limits of the theory of asymmetric elasticity, two (kinematic and static) approaches are offered to constructing the governing equations for a layered hybrid composite of regular structure with isotropic constituents. As a criteria of equivalence between the layered composition and a homogeneous transversely isotropic material, the equality of the specific free energy and the specific thermodynamic Gibbs energy in them is used, which allows one to determine the upper and lower bounds for the effective rigidities of the layered material. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 1, pp. 3–16, January–February, 2009.     

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.     

Radial deformation of a nonhomogeneous spherically isotropic elastic sphere with a concentric spherical inclusion

The elasticity of a spherically isotropic medium bounded by two concentric spherical surfaces subjected to normal pressures is discussed. The material of the structure is spherically isotropic and, in addition, is continuously inhomogeneous with mechanical properties varying exponentially as the square of the radius. An exact solution of the problem in terms of Whittaker functions is presented. The St. Venant’s solution in the case of homogeneous material and Lamé’s solution in the case of homogeneous isotropic material are derived from the general solution. The problem of a solid sphere of the same medium under the external pressure is also solved as a particular case of the above problem. Finally, the displacements and stresses of a composite sphere consisting of a solid spherical body made of homogeneous material and a nonhomogeneous concentric spherical shell covering the inclusion, both of them being spherically isotropic, are obtained when the sphere is under uniform compression.     

Effects of the material gradation on the SIFs of a crack in a 2-D FGM plate under thermal shock

Crack analysis of linear coupled thermoelasticity in two-dimensional, isotropic, non-homogeneous and linear elastic functionally graded materials subjected to thermal shock is performed by using a boundary-domain element method. The material parameters are assumed to be continuous functions of the spatial coordinates. Fundamental solutions of linear coupled thermoelasticity for the corresponding isotropic, homogeneous and linear elastic solids in Laplace-transformed domain are applied in the boundary-domain integral formulation. A collocation method is implemented for the numerical solution. Numerical examples for an exponential gradation of the material parameters are presented and discussed. The influences of the material gradation on the stress intensity factors are investigated. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Potential kernels,probabilities of hitting a ball,harmonic functions and the boundary Harnack inequality for unimodal Lévy processes

In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal Lévy processes. We also prove a supremum estimate and a regularity result for functions harmonic with respect to a general isotropic unimodal Lévy process.In the second part we apply the recent results on the boundary Harnack inequality and Martin representation of harmonic functions for the class of isotropic unimodal Lévy processes. As a sample application, we provide sharp two-sided estimates of the Green function of a half-space.