Spectral decomposition of the compliance fourth-rank tensor for orthotropic materials

Summary The compliance tensor related to orthotropic media is spectrally decomposed and its characteristic values are determined. Further, its idempotent tensors are estimated, giving rise to energy orthogonal states of stress and strain, thus decomposing the elastic potential in discrete elements. It is proven that the essential parameters, required for a complete characterisation of the elastic properties of an orthotropic medium, are the six eigenvalues of the compliance tensor, together with a set of three dimensionless parameters, the eigenangles θ, ϕ and ω. In addition, the intervals of variation of these eigenangles with respect to different values of the elastic constants are presented. Furthermore, bounds on Poisson's ratios are obtained by imposing the thermodynamical constraint on the eigenvalues to be strictly positive, as specified from the positive-definite character of the elastic potential. Finally, the conditions are investigated under which a family of orthotropic media behaves like a transversely isotropic or an isotropic one. Received 5 January 1999; accepted for publication 22 June 1999     

基于磁偶极子的磁场梯度张量局部缩并及试验验证

基于磁偶极子的磁场梯度张量缩并理论虽然可较好地描述磁源,不受测量系统的朝向的影响,但在实际的磁场梯度张量测量中通常仅可测得部分磁场梯度参数,没有获得磁场梯度张量中的所有张量分量,如何有效利用这些部分磁场梯度参量,已成为目前磁场梯度张量缩并理论及在磁场定位的应用中亟需解决的问题。基于磁偶极子理论提出磁场梯度张量局部缩并的方法,给出磁场梯度张量的局部模量的概念,分析了磁场梯度张量的局部模量及相关参数的三维空间分布规律,结果表明:局部模量CXY及kXY值随着g先增大后减小,在g=39°时取最大值,在g=90°时取最小值;局部模量C_Z及k_Z值随着g先减小后增大,在g=63°时取最小值,在g=0°时取最大值。提出磁场梯度张量局部模量中重要参数的近似计算公式,并通过试验进行验证。试验结果表明,试验值与理论计算值基本一致,可为磁场梯度张量局部缩并理论的应用提供依据和参考。     

Determination of the stretch tensor for structural transformations

Structural transformations in crystalline solids are increasingly the basis of the functional behavior of materials. Recently, in diverse alloy systems, both low hysteresis and reversibility of phase transformations have been linked to the satisfaction of the nongeneric conditions of compatibility between phases. According to the Cauchy–Born rule, these conditions are expressed as properties of transformation stretch tensor. The transformation stretch tensor is difficult to measure directly due to the lack of knowledge about the exact transforming pathway during the structural change, and the complicating effects of microstructure. In this paper we give a rigorous algorithmic approach for determining the transformation stretch tensor from X-ray measurements of structure and lattice parameters. For some traditional and emerging phase transformations, the results given by the algorithm suggest unexpected transformation mechanisms.     

Effective elasticity tensor of a periodic composite

The effective elasticity tensor of a composite is defined to be the four-tensor C which relates the average stress to the average strain. We determine it for an array of rigid spheres centered on the points of a periodic lattice in a homogeneous isotropic elastic medium. We first express C in terms of the traction exerted on a single sphere by the medium, and then derive an integral equation for this traction. We solve this equation numerically for simple, body-centered and face-centered cubic lattices with inclusion concentrations up to 90% of the close-packing concentration. For lattices with cubic symmetry the effective elasticity tensor involves just three parameters, which we compute from the solution for the traction. We obtain approximate asymptotic formulas for low concentrations which agree well with the numerical results. We also derive asymptotic results for C at high inclusion concentrations for arbitrary lattice geometries. We find them to be in good agreement with the numerical results for cubic lattices. For low and moderate concentrations the approximate results of Nemat-Nasseret al., also agree well with the numerical results for cubic lattices.     

张量和张量率的投影算子

通过对张量谱分解方法的深入研究,提出了一种对于张量正、负分解在表达形式上统一的投影算子,并建议了一类投影算子的统一格式,指出Ortiz算子、Simo-Ju算子和新算子均为该统一格式的特例,澄清了已有研究在此问题上的混淆之处。对于上述三种投影算子及建议的统一格式,还给出了相应张量率惟一的投影算子,该算子同时是张量和相应张量率的公共投影算子。最后指出了投影算子的微观力学意义及各表达式可能的适用范围,并给出了投影算子在描述混凝土单边效应上的本构建模应用。     

Symmetrization of the tensor operator of the compatibility equations in stresses in the anisotropic theory of elasticity

The general form of the term whose addition to the left-hand side of the compatibility equation in stresses in anisotropic elasticity symmetrizes the rank four differential tensor operator of these equations is obtained. In the case of an arbitrary type of anisotropy, this term contains two arbitrary parameters of dimension of elastic compliances. The symmetrized compatibility equations themselves contain only one of these parameters, and the combination of the terms with this parameter can be separated from the terms containing the tensor of elastic compliances.     

Multi-dimensional self-affine fractal interpolation model in tensor form

Iterated Function System (IFS) models have been explored to represent discrete sequences where the attractor of an IFS is self-affine either in R ² or R ³ (R is the set of real numbers). In this paper, the self-affine IFS model is extended from R ³ to R ⁿ (n is an integer and greater than 3), which is called the multi-dimensional self-affine fractal interpolation model. This new model is presented by introducing the defined parameter “mapping partial derivative”. A constrained inverse algorithm is given for the identification of the model parameters. The values of this new model depend continuously on all of the variables. That is, the function is determined by the coefficients of the possibly multi-dimensional affine maps. So the new model is presented as much more general and significant. Moreover, the multi-dimensional self-affine fractal interpolation model in tensor form is more terse than in the usual matrix form.     

Algebraic PGD for tensor separation and compression: An algorithmic approach

Proper Generalized Decomposition (PGD) is devised as a computational method to solve high-dimensional boundary value problems (where many dimensions are associated with the space of parameters defining the problem). The PGD philosophy consists in providing a separated representation of the multidimensional solution using a greedy approach combined with an alternated directions scheme to obtain the successive rank-one terms. This paper presents an algorithmic approach to high-dimensional tensor separation based on solving the Least Squares approximation in a separable format of multidimensional tensor using PGD. This strategy is usually embedded in a standard PGD code in order to compress the solution (reduce the number of terms and optimize the available storage capacity), but it stands also as an alternative and highly competitive method for tensor separation.     

The elastic energy-momentum tensor

The application to continuum mechanics of the general methods of the classical theory of fields is advocated and illustrated by the example of the static elastic field. The non-linear theory of elasticity is set up in the most convenient form (lagrangian coordinates and stress tensor). The appropriate energy-momentum tensor is derived, and it is shown that the integral of its normal component over a closed surface gives the force (as the term is used in the theory of solids) on defects and inhomogeneities within the surface. Other topics discussed are Günther's and related integrals, symmetrization of the energy-momentum tensor, and the Eulerian formulation. Some further extensions, existing and potential, are indicated.     

Isotropic polynomial invariants of Hall tensor

The Hall tensor emerges from the study of the Hall effect, an important magnetic effect observed in electric conductors and semiconductors. The Hall tensor is third-order and three-dimensional, whose first two indices are skew-symmetric. This paper investigates the isotropic polynomial invariants of the Hall tensor by connecting it with a second-order tensor via the third-order Levi-Civita tensor. A minimal isotropic integrity basis with 10 invariants for the Hall tensor is proposed. Furthermore, it is proved that this minimal integrity basis is also an irreducible isotropic function basis of the Hall tensor.     

Design sensitivity analysis for the homogenized elasticity tensor of a polymer filled with rubber particles

The main purpose of this work is the computational simulation of the sensitivity coefficients of the homogenized tensor for a polymer filled with rubber particles with respect to the material parameters of the constituents. The Representative Volume Element (RVE) of this composite contains a single spherical particle, and the composite components are treated as homogeneous isotropic media, resulting in an isotropic effective homogenized material. The sensitivity analysis presented in this paper is performed via the provided semi-analytical technique using the commercial FEM code ABAQUS and the symbolic computation package MAPLE. The analytical method applied for comparison uses the additional algebraic formulas derived for the homogenized tensor for a medium filled with spherical inclusions, while the FEM-based technique employs the polynomial response functions recovered from the Weighted Least-Squares Method. The homogenization technique consists of equating the strain energies for the real composite and the artificial isotropic material characterized by the effective elasticity tensor. The homogenization problem is solved using ABAQUS by the application of uniform deformations on specific outer surfaces of the composite RVE and the use of tetrahedral finite elements C3D4. The energy approach will allow for the future application of more realistic constitutive models of rubber-filled polymers such as that of Mullins and for RVEs of larger size that contain an agglomeration of rubber particles.     

Dyadic method for tensor functions

In this paper, we discuss tensor functions by dyadic representation of tensor. Two different cases of scalar invariants and two different cases of tensor invariants are calculated. It is concluded that there are six independent scale invariants for a symmetrical tensor and an antisymmetrical tensor, and there are twelve invariants for two symmetrical tensors and an antisymmetrical tensor. And we present a new list of tensor invariants for the tensor-valued isotropic function. The project supported by the Special Funds for Major State Basic Research Project “Nonlinear Science” and the National Basic Research Project “The Several Key Problems of Fluid and Aerodynamics”     

On the derivatives of a subclass of isotropic tensor functions of a nonsymmetric tensor

This paper studies a subclass of isotropic tensor-valued functions of a nonsymmetric tensor, which satisfy the commutative condition, and their derivatives. This subclass of tensor functions includes tensor power series, exponential tensor function, etc., and is more general than those investigated before. In the case of three distinct eigenvalues, the derivatives of these tensor functions are constructed by solving a tensor equation, which is acquired by differentiating the commutative condition. By taking limits, the results are extended to the cases of repeated eigenvalues.     

Dynamical stress function tensor

This paper generalizes the concept of stress function tensor in static elasticity into the more general case of continuum dynamics, and finds out the expressions for dynamical stress function tensor.     

Random-field model for the elasticity tensor of anisotropic random media

This Note deals with the construction of a non-Gaussian positive definite matrix-valued random field whose mathematical properties allow the fourth-order elasticity tensor of random non homogeneous anisotropic three dimensional elastic media to be modelled. If the usual parametric probabilistic approach was used, then 21 mutually dependent random fields should be modelled and identified by using experimental data. Such an approach would be very difficult because the systems of the marginal probability distributions of these random fields have to be identified due to the fact that, for a boundary value problem, the displacement field of the random medium is a non-linear mapping of the random elasticity tensor. The theory presented in this paper allows such a probabilistic model of the fourth-order elasticity tensor field to be constructed and depends only of four scalar parameters: three spatial correlation lengths and one parameter allowing the level of the random fluctuations to be controlled. To cite this article: C. Soize, C. R. Mecanique 332 (2004).

Résumé

On présente la construction d'un champ aléatoire à valeurs dans les matrices définies positives dont les propriétés mathématiques permettent de modéliser le tenseur d'élasticité du quatrième ordre des mileux élastiques anisotropes tridimensionnels aléatoires. Si l'approche probabiliste paramétrique usuelle était utilisée, alors il serait nécessaire de modéliser et d'identifier à l'aide de données expérimentales 21 champs aléatoires mutuellement dépendants. Une telle approche serait très difficile de part le fait que le système de lois marginales de ces champs aléatoires doit être identifié parce que, pour un problème aux limites, le champ de déplacement est une transformation non linéaire du tenseur d'élasticité. La théorie présentée dans ce papier permet de construire une modélisation probabiliste du champ de tenseur d'élasticité qui ne dépend que de quatre paramètres scalaires : trois échelles de corrélation spatiale et un paramètre permettant de contrôler le niveau des fluctuations aléatoires. Pour citer cet article : C. Soize, C. R. Mecanique 332 (2004).     

Force flux and the peridynamic stress tensor

The peridynamic model is a framework for continuum mechanics based on the idea that pairs of particles exert forces on each other across a finite distance. The equation of motion in the peridynamic model is an integro-differential equation. In this paper, a notion of a peridynamic stress tensor derived from nonlocal interactions is defined. At any point in the body, this stress tensor is obtained from the forces within peridynamic bonds that geometrically go through the point. The peridynamic equation of motion can be expressed in terms of this stress tensor, and the result is formally identical to the Cauchy equation of motion in the classical model, even though the classical model is a local theory. We also establish that this stress tensor field is unique in a certain function space compatible with finite element approximations.     

The Eshelby stress tensor,angular momentum tensor and dilatation flux in gradient elasticity

The Eshelby (static energy momentum) stress tensor, the angular momentum tensor and the dilatation flux are derived for anisotropic linear gradient elasticity in non-homogeneous materials. The divergence of these tensors gives the configurational forces, moments and work terms in gradient elasticity. There are several types of configurational forces, acting on the dislocation density and its gradient, on the inhomogeneities, proportional to the distortion, and linear and quadratic in the distortion gradient, and on the body force.     

Fifth-order isotropic descartes tensor

Ｉｎｔｈｉｓｌｉｔｅｒａｔｕｒｅ [1 ] ,[2 ]ｏｒｄｅｒ 0～ 4ｉｓｏｔｒｏｐｉｃｄｅｓｃａｒｔｅｓｔｅｎｓｏｒ’ｓｅｘｉｓｔｅｎｃｅａｎｄｉｔｓｇｅｎｅｒａｌｒｅｐｒｅｓｅｎｔａｔｉｏｎｆｏｒｍｕｌａａｒｅｄｉｓｃｕｓｓｅｄ .Ｗｉｔｈｒｅｇａｒｄｔｏｏｒｄｅｒｋ ≥ 5(ｄｅｓｃａｒｔｅｓｔｅｎｓｏｒｏｒａｆｆｉｎｅｏｒｔｈｏｇｏｎａｌｔｅｎｓｏｒ) ,ｔｈｅｎｕｍｂｅｒ (3 ｋ)ｏｆｉｔｓｃｏｍｐｏｎｅｎｔｓｉｓｖｅｒｙｂｉｇ ,ａｎｄｉｔｓｓｔｒｕｃｔｕｒｅｉｓｖｅｒｙｃｏｍｐｌｉｃａｔｅｄ ,ｓｏｎｏｐａ…     

Analytical expressions for odd-order anisotropic tensor dimension

According to the symmetries of the matter, the number of coefficients needed to define a tensorial relation varies. It is well known that in linear elasticity the number of generic coefficients varies from 21, for a complete anisotropic material, to 2, in case of isotropy. In a previous contribution, we provided analytical expressions that give the number of generic anisotropic coefficients in any anisotropic system for an even-order tensor. In the present note, we aim at extending the previous results to the case of odd-order tensors. As an illustration, the dimension of any anisotropic system for third-order piezoelectricity tensors and of the fifth-order coupling tensors of Mindlin's strain-gradient elasticity are determined.     

Derivation of the general form of elasticity tensor of the transverse isotropic material by tensor derivate

SymbolsU--FunchonofstrainenergyQ--OrthonormaltensorE--StraintensorEar--ComponentsofthestraintensorE,i,j=l,2,3n--VectorofthesymmetricaamsofthetransverseisotropicmaterialU*,E.,n*--FormsofU,EandninanothercoordinatesystemJf--MaininvariantsofstraintensorE,i=l,2,3Jf'n--InvariantsofstraintensorEconnectingwithvectorn,i=4,5Ji--TheabbreviatedformsofJf,Jf,Jf,Jf,",Jf,",i=l,2,3,4,5fi--ConstantsindependentonE,n,i=l,2,3,4,5el,e"--Thecovariantandcontravariantofthonormalbasisoftheusedcoordinatesyste…