Basis-free expressions for derivatives of a subclass of nonsymmetric isotropic tensor functions

The present paper generalizes the method for solving the derivatives of sym- metric isotropic tensor-valued functions proposed by Dui and Chen(2004)to a subclass of nonsymmetric tensor functions satisfying the commutative condition.This subclass of tensor functions is more general than those investigated by the existing methods.In the case of three distinct eigenvalues,the commutativity makes it possible to introduce two scalar functions,which will be used to construct the general nonsymmetric tensor func- tions and their derivatives.In the cases of repeated eigenvalues,the results are acquired by taking limits.     

A tensor method for the derivation of the equations of rigid body dynamics

A tensor method for the derivation of the equations of rigid body dynamics,based onthe concepts of continuum mechanics,is presented.The formula of time derivative of theinertia tensor with zero corotational rate is used to prove the equivalences of five methods,namely,Lagrange’s equations,Nielsen’s equations,Gibbs-Appell’s equations,Kane’sequations and the generalized momentum type of Kane’s equations.Some differentialidentities on angular velocity and angular acceleration are given.     

Fifth-order isotropic descartes tensor

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

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…     

A tensor estimator for the homogenization of absolute permeability

An estimator for an effective permeability tensor based on one-phase incompressible flow is presented. Effective large-scale permeability tensors are well approximated by rough approximations to the fine-scale pressure. The estimator works for all kinds of heterogeneous reservoirs and is fairly independent of boundary conditions.     

Dyadic Green's functions for cylindrical waveguides with moving media

The dyadic Green's function for cylindrical waveguides of circular or rectangular cross section with a moving, isotropic, homogeneous medium is developed using the method of eigenfunction expansion. The orthogonality properties of the vector mode functions are discussed. In contrast to waveguides with a stationary medium, it is seen that the normalization factor in the case of the E mode introduces a pole in the integral representation for the Green's function which must be excluded from the integration contour.     

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.     

THE LINEAR BI-SPATIAL TENSOR EQUATIONφijAiXBj= C

A linear bi-spatial tensor equation which contains many often encountered equations as particular cases is thoroughly studied. Explicit solutions are obtained. No conditions on eigenvalues of coefficient tensors are imposed.     

Chemical potential tensor for a two-phase continuous medium model

Relations for jumps of thermodynamic variables with allowance for inertial terms are derived under conditions of thermal equilibrium and in the absence of dissipation on the interphase surface. The notion of the chemical potential tensor is generalized for this case within the framework of the elastic continuous medium model. A thermodynamically well-posed definition of the chemical potential tensor is proposed for a class of two-phase models of deformable solids. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 12–22, May–June, 2005.     

PRINCIPAL AXIS INTRINSIC METHOD AND THE HIGH DIMENSIONAL TENSOR EQUATION AX－XA＝C

ＰＲＩＮＣＩＰＡＬＡＸＩＳＩＮＴＲＩＮＳＩＣＭＥＴＨＯＤＡＮＤＴＨＥＨＩＧＨＤＩＭＥＮＳＩＯＮＡＬＴＥＮＳＯＲＥＱＵＡＴＩＯＮＡＸ－ＸＡ＝ＣＬｉａｎｇＨａｏｙｕｎ（梁浩云）（ＷｕｙｉＵｎｉｖｅｒｓｉｔｙ，Ｊｉａｎｇｍｅｎ５２９０２０，Ｐ．Ｒ．Ｃｈｉｎ...     

A direct proof of uniqueness of square-root of a positive semi-definite tensor

Understanding of the basic properties of the positive semi-definite tensor is a prerequisite for its extensive applications in theoretical and practical fields, especially for its square-root. Uniqueness of the square-root of a positive semi-definite tensor is proven in this paper without resorting to the notion of eigenvalues, eigenvectors and the spectral decomposition of the second-order symmetric tensor.     

A discussion about scale invariants for tensor functions

It is found that in some cases the complete and irreducible scale invariants given by Ref.[1] are not independent. There are some implicit functional relations among them. The scale invariants for two different cases are calculated. The first case is an arbitrary second order tensor. The second case includes a symmetric tensor, an antisymmetric tensor and a vector. By using the eigentensor notation it is proved that in the first case there are only six independent scale invariants rather than seven as reported in Ref.[1] and in the second case there are only nine independent scale invariants which are less than that obtained in Ref.[1].     

New method of solving Lame-Helmholtz equation and ellipsoidal wave functions

Despite the great significance of equations with doubly-periodic coefficients in the methods of mathematical physics, the problem of solving Lamé-Helmholtz equation still remains to be tackled. Arscott and Moglich method of double-series expansion as well as Malurkar nonlinear integral equation are incapable of reaching the final explicit solution.Our main result consists in obtaining analytic expressions for ellipsoidal wave functions of four species (i=1,2,3,4) including the well known Lam(α) functions E_ci(snα),E_z1(snα) as special cases. This is effected by deriving two Integra-differential equations with variable coefficients and solving them by integral transform. Generalizing Riemann’s idea of P function, we introduce D function to express their transformation properties.     

Material functions for constant-strain-rate elongational flows

The problem of determination of invariant material functions for elongational flows in which two components of the constant strain-rate tensor are equal is briefly discussed, and a method of its solution described. The method is based on simultaneous modeling of the elongational viscosity as measured in uniform uniaxial elongational flow, and the shear viscosity as measured in steady viscometric flow. A single integral model with a strain-rate dependent memory is used to correlate both viscosities over a given experimental range of strain rates. The procedure has been applied to a set of experimental data obtained for a low-density polyethylene melt by Laun and Münstedt.     

An integral representation of the surface-impedance tensor for incompressible elastic materials

A fundamental result in anisotropic elasticity and surface-wave theory is the integral representation for the surface-impedance tensor first derived by Barnett and Lothe in 1973. However, this representation is only valid for compressible materials but not valid for incompressible materials. In this paper the corresponding integral representation for the surface-impedance tensor valid for incompressible materials is derived and is used to establish the uniqueness of surface-wave speed and to obtain an expression for the tensor Green's function for the infinite space. Mathematics subject classifications (2000) 74B05, 74B15, 74B20, 74J15     

A note on spin,vorticity and the deformation-rate tensor

The validity of statements in the literature concerning the angular velocity of the principal axes of the stretching tensor is examined and an alternative expression is provided for it.     

The angular-acceleration tensor of rigid-body kinematics and its properties

Summary The angular-acceleration tensor of rigid-body kinematics is recalled, and its invariant properties are analyzed. An explicit form of its inverse is given, its eigenvalues being calculated in symbolic form. The special case in which the tensor under study becomes singular, is given due attention. Received 8 May 1998; accepted for publication 27 July 1998     

A lower bound for the invariants of the configuration tensor for some well-known differential models

A method is given to determine lower bounds for the invariants of a configuration tensor in 3D flows. For some well-known differential models these lower bounds are given. Except for the Giesekus and the FENE-P model, the lower bounds are the values in equilibrium.     

Corrigendum: incompatible deformation field and Riemann curvature tensor

The "Corollary 1" formulation in SUN, B. H. Incompatible deformation field and Riemann curvature tensor. Applied Mathematics and Mechanics (English Edition),38(3), 311-332 (2017) is corrected. It can be stated as follows:The symmetric part of the deformation gradient has no contribution to the trace of the displacement density tensor.