On the generalized virial theorem for systems with variable mass

We presently extend the virial theorem for both discrete and continuous systems of material points with variable mass, relying on developments presented in Ganghoffer (Int J Solids Struct 47:1209–1220, 2010). The developed framework is applicable to describe physical systems at very different scales, from the evolution of a population of biological cells accounting for growth to mass ejection phenomena occurring within a collection of gravitating objects at the very large astrophysical scales. As a starting basis, the field equations in continuum mechanics are written to account for a mass source and a mass flux, leading to a formulation of the virial theorem accounting for non-constant mass within the considered system. The scalar and tensorial forms of the virial theorem are then written successively in both Lagrangian and Eulerian formats, incorporating the mass flux. As an illustration, the averaged stress tensor in accreting gravitating solid bodies is evaluated based on the generalized virial theorem.     

Rate form of the Eshelby and Hill tensors

Expressions are derived for the rates of change of the S and P tensors for transformed homogeneous inclusions in an anisotropic comparison medium undergoing prescribed changes of its elastic moduli. General results are obtained for ellipsoids and then reduced to yield explicit expressions in terms of the Stroh eigenvalues for cylindrical and disk-shaped inclusions in anisotropic solids and for spherical inclusions in isotropic solids. Applications are illustrated by solving the rate problem for an inhomogeneity in a large volume of a comparison medium, which is shown to be readily adaptable to standard averaging techniques for predictions of rates of change of overall moduli of composite materials experiencing evolution of phase moduli.     

Note on the virial theorem

Poincaré's formalism is used to develop a variant of the usual virial theorem in which the time average of the equation of motion of a certain function is expressed in terms of the generalized Poisson brackets. Recommended by Prof. Mei Fengxiang     

An irreducible polynomial functional basis of two-dimensional Eshelby tensors

The two-dimensional(2D) Eshelby tensors are discussed. Based upon the complex variable method, an integrity basis of ten isotropic invariants of the 2D Eshelby tensors is obtained. Since an integrity basis is always a polynomial functional basis, these ten isotropic invariants are further proven to form an irreducible polynomial functional basis of the 2D Eshelby tensors.     

Eshelby tensors for an ellipsoidal inclusion in a microstretch material

Eshelby tensors for an ellipsoidal inclusion in a microstretch material are derived in analytical form, involving only one-dimensional integral. As micropolar Eshelby tensor, the microstretch Eshelby tensors are not uniform inside of the ellipsoidal inclusion. However, different from micropolar Eshelby tensor, it is found that when the size of inclusion is large compared to the characteristic length of microstretch material, the microstretch Eshelby tensor cannot be reduced to the corresponding classical one. The reason for this is analyzed in details. It is found that under a pure hydrostatic loading, the bulk modulus of a microstretch material is not the same as the one in the corresponding classical material. A modified bulk modulus for the microstretch material is proposed, the microstretch Eshelby tensor is shown to be reduced to the modified classical Eshelby tensor at large size limit of inclusion. The fully analytical expressions of microstretch Eshelby tensors for a cylindrical inclusion are also derived.     

Eshelby tensors for a spherical inclusion in microstretch elastic fields

In the present work, microelastic and macroelastic fields are presented for the case of spherical inclusions embedded in an infinite microstretch material using the concept of Green’s functions. The Eshelby tensors are obtained for a spherical inclusion and it is shown that their forms for microelongated, micropolar and the classical cases are the proper limiting cases of the Eshelby tensors of microstretch materials.     

On the generalized Barnett–Lothe tensors for monoclinic piezoelectric materials

The three generalized Barnett–Lothe tensors L, S and H, appearing frequently in the investigations of the two-dimensional deformations of anisotropic piezoelectric materials, may be expressed in terms of the material constants. In this paper, the eigenvalues and eigenvectors for monoclinic piezoelectric materials of class m, with the symmetry plane at x₃ = 0 are constructed based on the extended Stroh formalism. Then the three generalized Barnett–Lothe tensors are calculated from these eigenvectors and are expressed explicitly in terms of the elastic stiffness instead of the reduced elastic compliance. The special case of transversely isotropic piezoelectric materials is also presented.     

On the Shape of the Eshelby Inclusions

It is shown, based on properties of analytic functions, that for inclusions of constant eigenstrain and eigenstress that the shape of the inclusion is restricted and any part of a plane (i.e. polyhedral inclusion) is prohibited. This revised version was published online in August 2006 with corrections to the Cover Date.     

The relativistic virial theorem in plasma EOS calculations

A method is presented for accurate calculation of equation of state (EOS) for warm dense matter. The method extends an approach presented recently, based on the adjustment of the correlation energy to impose consistency between two pressure representations: the volume derivative of the free energy and the relativistic virial theorem. In this work we show that the free energy of any neutral system obeys a fundamental differential equation, which bypasses the correlation specifics and serves as a basis to enhance EOS approximations. Specifically, we start with LDA calculations and improve the results significantly using this equation with a boundary condition at the zero pressure point. The method retains the emphasis on thermal excitations, but connects to the appropriate results at low temperatures. It effectively compensates for simplifications, including the use of a spherical model to account for global solid structure effects. EOS and opacities are calculated on the same footing for low to high Z elements and in large domains of density and temperature without recourse to parametric fitting procedures. Excellent agreement is obtained with experiments. Finally the method is applied successfully to calculate EOS and opacities for mixtures. Results for C–H mixture are compared with other calculations.     

On the Galerkin vector and the Eshelby solution in linear elasticity

Displacement potentials in linear static elasticity consist of three functions which can be regarded as the three components of a vector, e.g., the Galerkin vector. This research note gives an explanation as to why the biharmonic equations govern these functions in isotropic elasticity as opposed to the sixth-order partial differential equations that govern them in anisotropic elasticity. This note also shows that the Eshelby solution in two-dimensional anisotropic elasticity can be derived from the method of displacement potentials.     

Mechanical modeling of growth considering domain variation—Part II: Volumetric and surface growth involving Eshelby tensors

The growth of biological tissues is here described at the continuum scale of tissue elements. Relying on a previous work in Ganghoffer and Haussy (2005), the rephrasing of the balance laws for tissue elements under growth in terms of suitable Eshelby tensors is done in the present contribution, considering successively volumetric and surface growth. Balance laws for volumetric growth are written in both compatible and incompatible configurations, highlighting the material forces for growth associated to Eshelby tensors. Evolution laws for growth are written from the expression of the local dissipation in terms of a relation linking the growth velocity gradient to a growth-like Eshelby stress, in the spirit of configurational mechanics. Surface growth is next envisaged in terms of phenomena taking place in a varying reference configuration, relying on the setting up of a surface potential depending upon the surface transformation gradient and to the normal to the growing surface. The balance laws resulting from the stationnarity of the potential energy are expressed, involving surface Eshelby tensors associated to growth. Simulations of surface growth in both cases of fixed and moving generating surfaces evidence the interplay between diffusion of nutrients and the mechanical driving forces for growth.     

A reciprocal theorem in generalized thermoelasticity

A reciprocal theorem for initial mixed boundary value problems is obtained in the context of the linearized anisotropic thermoelasticity theory of Green and Lindsay.     

Symmetry classes of the anisotropy tensors of quasielastic materials and a generalized Kelvin approach

The anisotropy matrices (tensors) of quasielastic (Cauchy-elastic) materials were obtained for all classes of crystallographic symmetries in explicit form. The fourth-rank anisotropy tensors of such materials do not have the main symmetry, in which case the anisotropy matrix is not symmetric. As a result of introducing various bases in the space of symmetric stress and strain tensors, the linear relationship between stresses and strains is represented in invariant form similar to the form in which generalized Hooke’s law is written for the case of anisotropic hyperelastic materials and contains six positive Kelvin eigen moduli. It is shown that the introduction of modified rotation-induced deformation in the strain space can cause a transition to the symmetric anisotropy matrix observed in the case of hyperelasticity. For the case of transverse isotropy, there are examples of determination of the Kelvin eigen moduli and eigen bases and the rotation matrix in the strain space. It is shown that there is a possibility of existence of quasielastic media with a skew-symmetric anisotropy matrix with no symmetric part. Some techniques for the experimental testing of the quasielasticity model are proposed.     

On the construction of linearly independent tensors

We consider various methods for constructing linearly independent isotropic, gyrotropic, orthotropic, and transversally isotropic tensors. We state assertions and theorem that permit one to construct these tensors. We find linearly independent above-mentioned tensors up to and including rank six. The components of the tensor may have no symmetry or have symmetries of various types.     

An existence theorem of generalized Sawyer-Eliassen equation

ＡＮＥＸＩＳＴＥＮＣＥＴＨＥＯＲＥＭＯＦＧＥＮＥＲＡＬＩＺＥＤＳＡＷＹＥＲ－ＥＬＩＡＳＳＥＮＥＱＵＡＴＩＯＮＹｕＱｉｎｇ－ｙｕ（余庆余）（ＬａｎｚｈｏｕＵｎｉｖｅｒｓｉｔｙ,Ｌａｎｚｈｏｕ）ＸｕＱｉｎ（许秦）（ＣＩＭＭＳＩｎｓｔｉｔｕｔｅＵ．Ｓ．Ａ）...     

Mean Green operators and Eshelby tensors for hemispherical inclusions and hemisphere interactions in spheres. Application to bi-material spherical inclusions in isotropic spaces

This paper mainly presents an exact expression for the mean shape function of a hemispherical inclusion, from which are obtained analytical forms for the mean Green operator (GO) and Eshelby tensor of this hemi-sphere as well as for the related mean pair interaction Green operator (IGO) between the two hemi-spheres of a sphere, in media with isotropic (elastic or dielectric) properties. We secondly address the problem of bi-material inclusions, in the sense of a two-phase compact set of two or a few elementary domains, a particular inclusion pattern case for which we give an estimate of the mean stress and strain in each phase accounting for interactions. This estimate results from knowing the mean GO (or Eshelby tensor) for each pattern element plus the mean IGO between element pairs, what is rarely fulfilled analytically. The here solved case for bi-material spherical inclusions made of two different hemispherical elements adds to the recently made available solution for bi-material cylindrical inclusions made of piled coaxial finite cylinders. The obtained mean stress estimates are exemplified able to satisfactorily match with FEM calculations up to highly contrasted bi-material inclusions. Other types of bi-material spherical inclusions are mentioned for which the mean GOs for the sub-domains and their pair IGO can be obtained without calculation, owing to particular symmetries of the phase arrangement. Mean GOs and IGOs are also useful in certain homogenization frameworks yielding overall property estimates for inclusion-reinforced matrices. Further discussions and specific applications will be presented in forthcoming papers.     

On the symmetries of 2D elastic and hyperelastic tensors

A thorough investigation is made of the independent point-group symmetries and canonical matrix forms that the 2D elastic and hyperelastic tensors can have. Particular attention is paid to the concepts relevant to the proper definition of the independence of a symmetry from another one. It is shown that the numbers of all independent symmetries for the 2D elastic and hyperelastic tensors are six and four, respectively. In passing, a symmetry result useful for the homogenization theory of 2D linear elastic heterogeneous media is derived.     

On a generalized taylor theorem: A rational proof of the validity of the homotopy analysis method

IntroductionIn 1 7thcenturyIsaacNewton[1]gavesuchabinomialexpressionforfractionalandnegativeexponents(1 +t) α,i.e.,(1 +t) α =1 + +∞k=1α(α-1 ) (α-2 )… (α -k+ 1 )k !tk　　 (α≠ 0 ,1 ,2 ,… ) ,(1 )whoseconvergenceradiusisone.Furthermore ,theclassicalTaylorseries (seeRef.[2 ] )limm→+∞ mk=0f(k) (z0 )k !(z-z0 ) k (2 )ofacomplexfunctionf(z)atz=z0 isvalidmostlyinarestrictedconvergenceregion｜z-z0 ｜相似文献