Generalized variational principles of the viscoelastic body with voids and their applications

From the Boltzmann‘ s constitutive law of viscoelastic materials and the linear theory of elastic materials with voids, a constitutive model of generalized force fields for viscoelastic solids with voids was given. By using the variational integral method, the convolution-type functional was given and the corresponding generalized variational principles and potential energy principle of viscoelastic solids with voids were presented. It can be shown that the variational principles correspond to the differential equations and theinitial and boundary conditions of viscoelastic body with voids. As an application, a generalized variational principle of viscoelastic Timoshenko beams with damage was obtained which corresponds to the differential equations of generalized motion and the initial and boundary conditions of beams. The variational principles provide a way for solving problems of viscoelastic solids with voids.     

Structural function theory of generalized variational principles for linear elastic materials with voids

In this paper,we have obtained generalized variational principles for linear elasticmaterials with voids from structural function theory.Correspondent relations betweenstructural functions and generalized variational principles are given.     

Some theorems in the theory of elastic materials with voids

The linear theory of elastic materials with voids is considered. Some basic theorems concerning the existence and uniqueness of solution, the reciprocity relations and the variational characterization of the solution are presented.     

On some basic principles in dynamic theory of elastic materials with voids

According to the basic idea of dual-complementarity, in a simple and unified way proposed by the author^[1], some basic principles in dynamic theory of elastic materials with voids can be established systematically. In this paper, an important integral relation in terms of convolutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem in dynamic theory of elastic materials with voids, but also to derive systematically the complementary functionals for the eight-field, six-field, four-field and two-field simplified Gurtin-type variational principles. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly. The project supported by the Foundation of Zhongshan University Advanced Research Center     

Energy principles in theory of elastic materials with voids

According to the basic idea of dual-complementarity, in a simple and unified way proposed by the author^[1], various energy principles in theory of elastic materials with voids can be established systematically. In this paper, an important integral relation is given, which can be considered essentially as the generalized pr. inciple of virtual work. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem of work in theory of elastic materials with voids, but also to derive systematically the complementary functionals for the eight-field, six-field, four-field and two-field generalized variational principles, and the principle of minimum potential and complementary energies. Furthermore, with this appro ach, the intrinsic relationship among various principles can be explained clearly. The project supported by the National Natural Science Foundation of China     

微孔压电弹性动力学的能量原理

根据古典阴阳互补和现代对偶互补的基本思想,通过作者早已提出一条简单而统一的新途径,系统地建立了微孔压电弹性动力学的能量原理,给出一个重要的以卷积表示的积分关系式,可以认为,在力学上它是广义虚功原理的表式,从该式出发,不仅能得到微孔压电弹性动力学的虚功原理和互等定理,而且通过作者所给出的一系列广义Legendre变换,能系统地导出成互补关系的11类变量、9类变量、6类变量和3类变量简化Gurtin型变分原理的泛函,同时,通过这条新途径,还能清楚地阐明这些原理之间的内在联系。     

Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases

We review the theoretical bounds on the effective properties of linear elastic inhomogeneous solids (including composite materials) in the presence of constituents having non-positive-definite elastic moduli (so-called negative-stiffness phases). Using arguments of Hill and Koiter, we show that for statically stable bodies the classical displacement-based variational principles for Dirichlet and Neumann boundary problems hold but that the dual variational principle for traction boundary problems does not apply. We illustrate our findings by the example of a coated spherical inclusion whose stability conditions are obtained from the variational principles. We further show that the classical Voigt upper bound on the linear elastic moduli in multi-phase inhomogeneous bodies and composites applies and that it imposes a stability condition: overall stability requires that the effective moduli do not surpass the Voigt upper bound. This particularly implies that, while the geometric constraints among constituents in a composite can stabilize negative-stiffness phases, the stabilization is insufficient to allow for extreme overall static elastic moduli (exceeding those of the constituents). Stronger bounds on the effective elastic moduli of isotropic composites can be obtained from the Hashin–Shtrikman variational inequalities, which are also shown to hold in the presence of negative stiffness.     

Variational principles in elasticity with nonlinear stress-strain relation

Since 1979, a series of papers have been published concerning the variational principles and generalized variational principles in elasticity such as [1] (1979), [6] (1980), [2,3] (1983) and [4,5] (1984). All these papers deal with the elastic body with linear stress-strain relations. In 1985, a book was published on generalized variational principles dealing with some nonlinear elastic body, but never going into detailed discussion. This paper discusses particularly variational principles and generalized variational principles for elastic body with nonlinear stress-strain relations. In these discussions, we find many interesting problems worth while to pay some attention. At the same time, these discussions are also instructive for linear elastic problems. When the strain is small, the high order terms may be neglected, the results of this paper may be simplified to the well-known principles in ordinary elasticity problems.     

On the theory of elastic shells made from a material with voids

In this paper we present a theory for porous elastic shells using the model of Cosserat surfaces. We employ the Nunziato–Cowin theory of elastic materials with voids and introduce two scalar fields to describe the porosity of the shell: one field characterizes the volume fraction variations along the middle surface, while the other accounts for the changes in volume fraction along the shell thickness. Starting from the basic principles, we first deduce the equations of the nonlinear theory of Cosserat shells with voids. Then, in the context of the linear theory, we prove the uniqueness of solution for the boundary initial value problem. In the case of an isotropic and homogeneous material, we determine the constitutive coefficients for Cosserat shells, by comparison with the results derived from the three-dimensional theory of elastic media with voids. To this aim, we solve two elastostatic problems concerning rectangular plates with voids: the pure bending problem and the extensional deformation under hydrostatic pressure.     

On the Deformation of Functionally Graded Porous Elastic Cylinders

This paper is concerned with the linear theory of inhomogeneous and orthotropic elastic materials with voids. We study the problem of extension and bending of right cylinders when the constitutive coefficients are independent of the axial coordinate. First, the plane strain problem for inhomogeneous and orthotropic elastic materials with voids is investigated. Then, the solution of the problem of extension and bending is expressed in terms of solutions of three plane strain problems. The results are used to study the extension of a circular cylinder with a special kind of inhomogeneity. The influence of the material inhomogeneity on the axial strain is established.      

The reciprocal theorem and linear elastic variational principles

In this paper classical linear elastic variational principles are systematically derived from the reciprocal theorem and mixed variational principles of variations of boundary conditions are given.     

The combined energy variational principles in linear elasticity

In this paper, a new kind of mixed energy variational principles in linear elasticity—the combined energy variational principles is presented. First, we define the conjugate body of an elastic body, which is obtained by changing the boundary conditions of the elastic body. Next, we decompose the conjugate body into two component-states, construct functionals of potential energy and complementary energy, respectively, for the component-states and define the additional hybrid energy between the component-states. Thus the functionals of combined energy can be constructed. Three typical combined energy variational principles are demonstrated and the other forms of combined energy variational principles are given. The application of the proposed principles to the calculation of thin plates with complicated boundaries is shown.     

A micromechanics-based nonlocal constitutive equation incorporating three-point statistics for random linear elastic composite materials

A variational formulation employing the minimum potential and complementary energy principles is used to derive a micromechanics-based nonlocal constitutive equation for random linear elastic composite materials, relating ensemble averages of stress and strain in the most general situation when mean fields vary spatially. All information contained in the energy principles is retained; we employ stress polarization trial fields utilizing one-point statistics so that the resulting nonlocal constitutive equation incorporates up through three-point statistics. The variational structure is developed first for arbitrary heterogeneous linear elastic materials, then for randomly inhomogeneous materials, then for general n-phase composite materials, and finally for two-phase composite materials, in which case explicit variational upper and lower bounds on the nonlocal effective modulus tensor operator are derived. For statistically uniform infinite-body composites, these bounds are determined even more explicitly in Fourier transform space. We evaluate these in detail in an example case: longitudinal shear of an aligned fiber or void composite. We determine the full permissible ranges of the terms involving two- and three-point statistics in these bounds, and thereby exhibit explicit results that encompass arbitrary isotropic in-plane phase distributions; we also develop a nonlocal “Milton parameter”, the variation of whose eigenvalues throughout the interval [0, 1] describes the full permissible range of the three-point term. Example plots of the new bounds show them to provide substantial improvement over the (two-point) Hashin–Shtrikman bounds on the nonlocal operator tensor, for all permissible values of the two- and three-point parameters. We next discuss further applications of the general nonlocal operator bounds: to any three-dimensional scalar transport problem e.g. conductivity, for which explicit results are given encompassing the full permissible ranges of the two- and three-point statistics terms for arbitrary three-dimensional isotropic phase distributions; and to general three-dimensional composites, where explicit results require future research. Finally, we show how the work just summarized, treating elastostatics, can be generalized to elastodynamics, first in general, then explicitly for the longitudinal shear example.     

微孔线弹性材料的守恒律和完备性定理

本文用无穷小变换群使作用量不变的思想证明了广义Noether定理,且得到一类守恒律,对线性均匀微孔弹性材料阐明了尺度变换下守恒律的可能性,且给出了完备性定理的证明。     

带孔隙损伤的弹性固体的广义变分原理

应用弹性微结构理论,建立了具广义力场带孔隙损伤线弹性固体的基本模型．应用变积方法,同时分别建立了带孔隙损伤弹性固体四类和两类变量的广义变分原理,这些变分原理对应着带孔隙损伤弹性固体微分方程和初值边值条件．应用弹性微结构理论,建立了带孔隙损伤的弹性Timoshenko 梁的基本方程,得到带孔隙损伤的弹性Timoshenko 梁两类变量的广义变分原理．这些广义变分原理为近似求解带孔隙损伤的弹性问题提供了有效途径．     

A uniqueness theorem in the theory of elastic materials with voids

In the context of the linear theory of homogeneous and isotropic elastic materials with voids, an initial-boundary-value problem in terms of stress and volume fraction fields is formulated and the uniqueness of its solution established.     

Axially symmetric problems for a porous elastic solid

This paper is concerned with the linear theory of elastic materials with voids. The Dirichlet and Neumann problems for a half-space are studied by using the technique of integral transforms. The case of a concentrated body load is investigated in detail.     

Saint-Venant's principle for linear elastic porous materials

Toupin's version of the Saint-Venant's principle in linear elasticity is generalized to the case of linear elastic porous materials. That is, it is shown that, for a straight prismatic bar made of a linear elastic material with voids and loaded by a self-equilibrated system of forces at one end only, the internal energy stored in the portion of the bar which is beyond a distance s from the loaded end decreases exponentially with the distance s.     

Principles of minimum transformed energy and minimum principles for dynamics of plates

In the present paper, we first by Laplace transform present a derivation of principle of transformed virtual work, three principles of minimum transformed energy with influence of rotatory enertia for dynamics of anisotropic linear elastic plates with three generalized displacements. Moreover, the forms with the original in place-time domain corresponding these variational principles are presented.Then by the introduction of the set of admissible weight functions the three minimum principles for the original place-time domain are derived.In each of the preceding groups of the variational principles there are two which are the dynamic counterparts to the static principles of minimum potetial energy and minimum complementary energy; the other principles are formulated in terms of the internal force alone, but have no counterpart in elastostatics of plates.     

考虑损伤时带微孔粘弹性体的应力分布

本文根据粘弹性理论和带空沿材料的线性理论，利用Laplace变换及其逆变换，给出了粘弹性损伤材料的一种本构模型。应用这种本构关系。讨论了带损伤的圆环形板或圆筒体在内外压力作用下的平面问题，得到了圆孔边缘附近的应力场和损伤增量场的分布情况，同时根据Laplace变换的终值定理，得到圆环或圆筒的终态应力和损伤增量的分布，并分析了工程中的一个典型的例子。得到了最终损伤的一种近似分布。可为工程设计提供一定的参考。