Some Results in Linear Theory of Thermomicrostretch Elastic Solids

The aim of this paper is to study the spatial and temporal behavior of thermoelastodynamic processes for microstretch continuum materials. The spatial behavior is described by establishing estimates of Saint–Venant type and Phragmén–Lindelöf type for bounded and unbounded bodies, respectively, with decay rate being dependent or independent of time, while the temporal behavior is studied by establishing the relations describing the asymptotic behavior of the Cesàro means of the different parts of the total energy.     

A Micropolar Theory of Porous Media: Constitutive Modelling

The extension of the classical mixture theory by the concept of volume fractions leads to the theory of porous media. In this article, the theory of porous media is generalised to micropolar constituents. The kinematic relations and the balance equations for a porous medium are developed without restricting the number of constituents. Based on the entropy inequality, the general form of the constitutive equations are derived for a binary medium consisting of a porous elastic skeleton saturated by a viscous pore-fluid. Both constituents are assumed to be compressible. Handling the saturation constraint by a Lagrangian multiplier leads to a compatibility of the proposed model to so-called hybrid and incompressible models.     

Thermal Stresses in Plane Strain of Porous Elastic Solids

This paper is concerned with the linear theory of thermoelastic materials with voids. We present a method to reduce the thermoelastic problem to an isothermal one with zero body loads and with certain known boundary data. The results are used to study the thermal stresses in a tube and the thermoelastic deformation of a cylinder subjected to a uniform temperature gradient.     

Multicomponent, Multiphase Thermodynamics of Swelling Porous Media with Electroquasistatics: II. Constitutive Theory

In Part I macroscopic field equations of mass, linear and angular momentum, energy, and the quasistatic form of Maxwell's equations for a multiphase, multicomponent medium were derived. Here we exploit the entropy inequality to obtain restrictions on constitutive relations at the macroscale for a 2-phase, multiple-constituent, polarizable mixture of fluids and solids. Specific emphasis is placed on charged porous media in the presence of electrolytes. The governing equations for the stress tensors of each phase, flow of the fluid through a deforming medium, and diffusion of constituents through such a medium are derived. The results have applications in swelling clays (smectites), biopolymers, biological membranes, pulsed electrophoresis, chromotography, drug delivery, and other swelling systems.     

Local Symmetry Group in the General Theory of Elastic Shells

We establish the local symmetry group of the dynamically and kinematically exact theory of elastic shells. The group consists of an ordered triple of tensors which make the shell strain energy density invariant under change of the reference placement. Definitions of the fluid shell, the solid shell, and the membrane shell are introduced in terms of members of the symmetry group. Within solid shells we discuss in more detail the isotropic, hemitropic, and orthotropic shells and corresponding invariant properties of the strain energy density. For the physically linear shells, when the density becomes a quadratic function of the shell strain and bending tensors, reduced representations of the density are established for orthotropic, cubic-symmetric, and isotropic shells. The reduced representations contain much less independent material constants to be found from experiments.     

Fundamental Solution of the System of Equations of Steady Oscillations in the Theory of Fluid-Saturated Porous Media

In the present paper the linear theory of the liquid-saturated porous medium consisting of a microscopically incompressible solid skeleton containing microscopically incompressible liquid is considered. The fundamental solution of the system of linear coupled partial differential equations of the steady oscillations of the porous solids is constructed in terms of elementary functions and some basic properties are established.     

Thermo-Chemo-Electro-Mechanical Formulation of Saturated Charged Porous Solids

A thermo-chemo-electro-mechanical formulation of quasi-static finite deformation of swelling incompressible porous media is derived from a mixture theory including the volume fraction concept. The model consists of an electrically charged porous solid saturated with an ionic solution. Incompressible deformation is assumed. The mixture as a whole is assumed locally electroneutral. Different constituents following different kinematic paths are defined: solid, fluid, anions, cations and neutral solutes. Balance laws are derived for each constituent and for the mixture as a whole. A Lagrangian form of the second law of thermodynamics for incompressible porous media is used to derive the constitutive restrictions of the medium. The material properties are shown to be contained in one strain energy function and a matrix of frictional tensors. A principle of reversibility results from the constitutive restrictions. Existing theories of swelling media should be evaluated with respect to this principle.     

Deformation and Filtration Near a Well in a Porous Stratum with Linear Memory

The linearized equations for saturated elastic porous media and for surrounding elastic rock are solved simultaneously; and the Volterra principle is used to derive an integro-differential filtration equation for a homogeneous weakly compressible fluid in an axisymmetric stratum with linear memory and central well. An analytical expression for porosity variation is obtained and then used to determine the permeability coefficient. The solutions are analyzed for the case where the stratum exhibits memory described by regular and singular kernels of the integral operator     

Plane Waves and Eigenfrequencies in the Linear Theory of Binary Mixtures of Thermoelastic Solids

In this paper some basic properties of wave numbers of the longitudinal and transverse plane waves are treated. The existence theorems of eigenfrequencies of the interior homogeneous boundary-value problems of steady oscillations of binary mixtures for thermoelastic solids are proved. The connection between plane waves and eigenfrequencies is established. This paper dedicated to my teacher Professor Mikheil Basheleishvili on the occasion of his 80th birthday.     

多孔硅橡胶有限变形的弹性行为

针对孔隙度较大 (孔隙度大于 5 0 % )的硅橡胶材料在压缩情况下的大变形 ,提出了可描述此类可压橡胶材料力学行为的应变能密度函数 ,推导了硅橡胶材料的本构方程。利用硅橡胶材料的单轴压缩实验进行了材料参数拟合 ,讨论了多孔硅橡胶的孔隙度和体积变形对压缩性能的影响     

Some Results in the Dynamical Theory of Porous Elastic Bodies

This paper is concerned with the linear dynamic theory of elastic materials with voids. First, a spatial decay estimate of an energetic measure associated with a dynamical process is established. Then, a domain of dependence inequality associated with a boundary-initial-value problem is derived and a domain of influence theorem is established. It is shown that, for a finite time, a solution corresponding to data of bounded support vanishes outside a bounded domain.     

有限变形下线弹性裂尖场的渐近分析

对有限变形下线弹性Ⅰ型裂纹场建立了无需分区的统一控制方程并进行了渐近分析, 利用“打靶法”得到位移场在物质描述与空间描述下的渐近阶次分别为3／4和1,Green应变、第二类P－K应力及Cauchy应力在物质描述与空间描述下的渐近阶次分数为－1／2和－2／3;对不同泊松比,裂尖有限变形线弹性场的位移均以UⅡ或u2为主导,裂纹张开角为π,现时构形中的大变形区为一垂直初始构形中裂纹表面的狭长带状区,应力则处于由σ22主导的单向拉伸状态,角分布函数U^-Ⅱ（0）及σ22^-(0)具有奇异性,但UL^-‘(Θ）／UⅡ^－‘(0)及σij^-(θ）/σ22^-(0)均趋于有限值。     

The Remarkable Nature of Radially Symmetric Deformation of Spherically Uniform Linear Anisotropic Elastic Solids

Antman and Negron-Marrero [1] have shown the remarkable nature of a sphere of nonlinear elastic material subjected to a uniform pressure at the surface of the sphere. When the applied pressure exceeds a critical value the stress at the center r=0 of the sphere is infinite. Instead of nonlinear elastic material, we consider in this paper a spherically uniform linear anisotropic elastic material. It means that the stress-strain law referred to a spherical coordinate system is the same for any material point. We show that the same remarkable nature appears here. What distinguishes the present case from that considered in [1] is that the existence of the infinite stress at r=0 is independent of the magnitude of the applied traction σ0 at the surface of the sphere. It depends only on one nondimensional material parameter κ. For a certain range of κ a cavitation (if σ0>0) or a blackhole (if σ0<0) occurs at the center of the sphere. What is more remarkable is that, even though the deformation is radially symmetric, the material at any point need not be transversely isotropic with the radial direction being the axis of symmetry as assumed in [1]. We show that the material can be triclinic, i.e., it need not possess a plane of material symmetry. Triclinic materials that have as few as two independent elastic constants are presented. Also presented are conditions for the materials that are capable of a radially symmetric deformation to possess one or more symmetry planes. This revised version was published online in August 2006 with corrections to the Cover Date.     

Linear stability analysis of supersonic axisymmetric jets

Stabilities of supersonic jets are examined with different velocities, momentum thicknesses, and core temperatures. Amplification rates of instability waves at inlet are evaluated by linear stability theory (LST). It is found that increased velocity and core temperature would increase amplification rates substantially and such influence varies for different azimuthal wavenumbers. The most unstable modes in thin momentum thickness cases usually have higher frequencies and azimuthal wavenumbers. Mode switching is observed for low azimuthal wavenumbers, but it appears merely in high velocity cases. In addition, the results provided by linear parabolized stability equations show that the mean-flow divergence affects the spatial evolution of instability waves greatly. The most amplified instability waves globally are sometimes found to be different from that given by LST.     

弹性体表面分层压缩失稳问题的数学弹性力学解

采用数学弹性力学的稳定平衡方程并结合富氏积分变换的方法研究了含表面平行裂纹的弹性体在压缩载荷下的表面分层失稳问题。导出了一级显式的精确齐次奇异积分方程组,然后.通过Gauss-Chebyshev积分公式,得到一组齐次代数方程组,从而求出临界压缩载荷。并将结果与经典的材料力学梁板稳定的研究方法所得结果进行了比较,指出经典方法误差太大而不适于求解此问题。最后,利用数学弹性力学解求出的等效弹性支承常数给出一个简单精确的临界压缩载荷计算公式。     

Topical Problems and Applications of Creep Theory

A historical review of achievements in creep theory is given. Primary attention is focused on the phenomenological approach. Different constitutive equations are discussed for primary and secondary creep as well as for creep with damage. New creep problems are examined     

A Simple Nonlinear Thermodynamic Theory of Arbitrary Elastic Beams

A ‘classical’ theory of beams (i.e., a theory in which the basic kinetic variables are a stress resultant and a stress couple) undergoing elastic, thermodynamic processes is developed by first deriving exact beamlike (one-dimensional) equations of motion and a beamlike Second Law (Clausius–Duhem inequality) by descent from three-dimensions. Then what may be considered as the three basic assumptions of a classical theory are introduced: an assumed form of the First Law (conservation of energy), a relaxed form of the Second Law, and a general form of the constitutive relations. Throughout, detailed specification of geometry, kinematics, or constitution is minimized. It is shown how the kinematic Kirchhoff hypothesis may be avoided by first introducing a mixed-energy density and then imposing a logically more satisfying constitutive Kirchhoff hypothesis. Mathematics Subject Classifications (2000) 74A15, 74B20, 74K10     

Integral Solution of a Problem with Non-Standard Boundary Conditions in an Enhanced Theory of Bending of Elastic Plates

A modified boundary integral equation method is used to solve a specific type of mixed boundary value problem in an enhanced theory of bending of elastic plates in which the effects of transverse shear deformation and transverse normal strain are taken into account. The problem considered is characterized by the fact that a combination of transverse displacement and bending and twisting moments is prescribed on the curve which bounds the middle surface of the plate. Both interior and exterior problems are formulated and the corresponding existence and uniqueness results derived.     

Thermomechanics of Porous Media - I: Thermohydraulic Model for Compacted Bentonite

A general thermomechanical model is derived for a mixture. The model describes the behavior of the mixture via proper choices of free energy and dissipation function. A model for any combination of the mixture constituents can be reduced from the general model. The theory is applied to a thermohydraulic model for a mixture of compacted bentonite, liquid water, vapor, and air with the assumption of rigid skeleton and constant uniform porosity. The free energy of the system is chosen to take into account the individual nondissipative behaviors of the constituents and their mutual interactions, namely, adsorption and mixing of the gaseous constituents. The choices for the interaction terms are based on the equilibrium conditions for the water species in different combinations of the constituents. The resulting thermodynamically consistent macroscopic model is fitted to a suction experiment and applied to a simple one-dimensional thermohydraulic simulation of the bentonite buffer of the Febex in situ test. The results calculated with finite element method are successfully compared to measurements.