Non-symmetric Tensor Research Articles

The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations

In this paper we consider the equilibrium problem in the relaxed linear model of micromorphic elastic materials. The basic kinematical fields of this extended continuum model are the displacement u 2 R 3 and the non-symmetric micro-distortion density tensor P 2 R 3×3 . In this relaxed theory a symmetric forcestress tensor arises despite the presence of microstructure and the curvature contribution depends solely on the micro-dislocation tensor CurlP. However, the relaxed model is able to fully describe rotations of the microstructure and to predict non-polar size-effects. In contrast to classical linear micromorphic models, we allow the usual elasticity tensors to become positive-semidefinite. We prove that, nevertheless, the equilibrium problem has a unique weak solution in a suitable Hilbert space. The mathematical framework also settles the question of which boundary conditions to take for the micro-distortion. Similarities and differences between linear micromorphic elasticity and dislocation gauge theory are discussed and pointed out.

Massive “spin-2” theories in arbitrary [formula omitted] dimensions

Here we show that in arbitrary dimensions D≥3 there are two families of linearized second order Lagrangians describing massive “spin-2” particles via a nonsymmetric rank-2 tensor. They differ from the usual Fierz–Pauli theory in general. At zero mass one of the families is Weyl invariant. Such massless theory has no particle content in D=3 and gives rise, via master action, to a dual higher order (in derivatives) description of massive spin-2 particles in D=3 where both the second and the fourth order terms are Weyl invariant, contrary to the linearized New Massive Gravity. However, only the fourth order term is invariant under arbitrary antisymmetric shifts. Consequently, the antisymmetric part of the tensor e[μν] propagates at large momentum as 1/p2 instead of 1/p4. So, the same kind of obstacle for the renormalizability of the New Massive Gravity reappears in this nonsymmetric higher order description of massive spin-2 particles.

Geodesic mapping onto Kählerian spaces of the first kind

In the present paper a generalized K\"ahlerian space $\mathbb{G}\underset 1 {\mathbb{K}}{}_N$ of the first kind is considered, as a generalized Riemannian space $\mathbb{GR}_N$ with almost complex structure $F^h_i$, that is covariantly constant with respect to the first kind of covariant derivative. Using the non-symmetric metric tensor we find necessary and sufficient conditions for a geodesic mapping $f:\mathbb{GR}_N\to \mathbb{G}\underset 1 {\mathbb{\bar{K}}}{}_N$ with respect to the four kinds of covariant derivatives.

Differential Complexes in Continuum Mechanics

We study some differential complexes in continuum mechanics that involve both symmetric and non-symmetric second-order tensors. In particular, we show that the tensorial analogue of the standard grad-curl-div complex can simultaneously describe the kinematics and the kinetics of motions of a continuum. The relation between this complex and the de Rham complex allows one to readily derive the necessary and sufficient conditions for the compatibility of the displacement gradient and the existence of stress functions on non-contractible bodies. We also derive the local compatibility equations in terms of the Green deformation tensor for motions of $2$D and $3$D bodies, and shells in curved ambient spaces with constant curvatures.

Semidefinite Relaxations for Best Rank-1 Tensor Approximations

This paper studies the problem of finding best rank-1 approximations for both symmetric and nonsymmetric tensors. For symmetric tensors, this is equivalent to optimizing homogeneous polynomials over unit spheres; for nonsymmetric tensors, this is equivalent to optimizing multi-quadratic forms over multi-spheres. We propose semidefinite relaxations, based on sum of squares representations, to solve these polynomial optimization problems. Their properties and structures are studied. In applications, the resulting semidefinite programs are often large scale. The recent Newton-CG augmented Lagrangian method by Zhao, Sun and Toh is suitable for solving these semidefinite relaxations. Extensive numerical experiments are presented to show that this approach is practical in getting best rank-1 approximations.

New Cartan’s tensors and pseudotensors in a generalized Finsler space

In this work we defined a generalized Finsler space (GFN) as 2N-dimensional differentiable manifold with a non-symmetric basic tensor gij(x,x?), which applies that gij_?|m(x,x?)=0; ?=1,2. Based on non-symmetry of basic tensor, we obtained ten Ricci type identities, comparing to two kinds of covariant derivative of a tensor in Rund?s sense. There appear two new curvature tensors and fifteen magnitudes, we called ?curvature pseudotensors?.

SPIN FROM THE NONSYMMETRIC METRIC TENSOR

A solution to the gravitational field equations based on a nonsymmetric metric tensor is examined. Unlike Einstein's interpretation of electromagnetism, or Moffat's generalized gravity, it is shown that the nonsymmetric part of the metric tensor is the potential of the spin field, and its intimate connection to string theory is established. This formulation solves the longstanding problem of electromagnetism and torsion, naturally showing how electromagnetism, through its intrinsic spin, can create torsion.

Generalized Symmetric Formulation of Tangential Stiffness for Nonassociative Plasticity

The behavior of material nonlinearity with nonassociative plastic flow is frequently analyzed for structures made of soil and rock. It is known that for nonassociated plasticity, the elastoplastic tangential stiffness tensor (matrix) is nonsymmetric. In the FEM, this causes inconvenience for solving a system of equations using the Newton-Raphson solution procedure. In this paper, a mathematical transformation was derived for converting the nonsymmetric tangential stiffness tensor (matrix) into a symmetric tensor such that the global system of equations become unconditionally symmetric. A detailed step-by-step procedure of a stress update algorithm using the tangential stiffness method is elaborated in this paper. The paper also compares the results of the tangential stiffness method with those of the initial stiffness method using an illustrative tunnel problem for associative and nonassociative flow conditions and shows the efficacy of the proposed transformation in elastoplastic problems.

Nonuniqueness of the Fierz-Pauli mass term for a nonsymmetic tensor

Starting with a description of massive spin-2 particles in D=4 in terms of a mixed symmetry tensor $\tmnd $ without totally antisymmetric part ($T_{[[\mu\nu]\rho]}=0$) we obtain a dual model in terms of a nonsymmetric tensor $e_{\mu\nu}$. The model is of second-order in derivatives and its mass term $(e_{\mu\nu}e^{\nu\mu} + c \, e^2)$ contains an arbitrary real parameter $c$. Remarkably, it is free of ghosts for any real value of $c$ and describes a massive spin-2 particle as expected from duality. The antisymmetric part $e_{[\mu\nu]}$ plays the role of auxiliary fields, vanishing on shell. In the massless case the model describes a massless spin-2 particle without ghosts.

On spaces with non-symmetric affine connection, containing subspaces without torsion

In the works by Minčič et al. (1998, 1999, 2007) [8,9,11] we have studied the problem (started by M. Prvanović): Can a generalized Riemannian space GRN (with non-symmetric basic tensor) contain a Riemannian subspace (possessing a symmetric induced basic tensor)?In the present work the analogous problem is studied for a space LN with non-symmetric affine connection and its subspace LM. In the Section 2 it is assumed that the equations defining submanifold are given and that it is valid (2.4), i.e., induced torsion T∼=0, and the connection on the manifold MN is determined so that T≠0. At the third section we start from connection on the manifold MNT≠0. From the mentioned condition (2.4) we determine the Eq. (1.1) defining the submanifold MM⊂MN, so that induced torsion T∼=0, i.e., we get LMO⊂LN.It is proved that the answer to the question cited above is affirmative and several examples are constructed. The examinations have local character.

Dual massive gravity

The linearized massive gravity in three dimensions, over any maximally symmetric background, is known to be presented in a self-dual form as a first order equation which encodes not only the massive Klein–Gordon type field equation but also the supplementary transverse-traceless conditions. We generalize this construction to higher dimensions. The appropriate dual description in d dimensions, additionally to a (non-symmetric) tensor field hμν, involves an extra rank-(d−1) field equivalently represented by the torsion rank-3 tensor. The symmetry condition for hμν arises on-shell as a consequence of the field equations. The action principle of the dual theory is formulated. The focus has been made on four dimensions. Solving one of the fields in terms of the other and putting back in the action one obtains two other equivalent formulations of the theory in which the action is quadratic in derivatives. In one of these representations the theory is formulated entirely in terms of a rank-2 non-symmetric tensor hμν. This quadratic theory is not identical to the Fierz–Pauli theory and contains the coupling between the symmetric and antisymmetric parts of hμν. Nevertheless, the only singularity in the propagator is the same as in the Fierz–Pauli theory so that only the massive spin-2 particle is propagating. In the other representation, the theory is formulated in terms of the torsion rank-3 tensor only. We analyze the conditions which follow from the field equations and show that they restrict to 5 degrees of freedom thus producing an alternative description to the massive spin-2 particle. A generalization to higher dimensions is suggested.

Tensor Products of Type III Factor Representations of Cuntz–Krieger Algebras

We introduced a non-symmetric tensor product of any two states or any two representations of Cuntz–Krieger algebras associated with a certain non-cocommutative comultiplication in our previous work. In this paper, we show that a certain set of KMS states is closed with respect to the tensor product. From this, we obtain formulae of tensor products of type III factor representations of Cuntz–Krieger algebras which are different from results of the tensor product of factors of type III.

Geometry of integral polynomials, M-ideals and unique norm preserving extensions

We use the Aron–Berner extension to prove that the set of extreme points of the unit ball of the space of integral k-homogeneous polynomials over a real Banach space X is { ± ϕ k : ϕ ∈ X ⁎ , ‖ ϕ ‖ = 1 } . With this description we show that, for real Banach spaces X and Y, if X is a nontrivial M-ideal in Y, then ⊗ ˆ ε k , s k , s X (the k-th symmetric tensor product of X endowed with the injective symmetric tensor norm) is never an M-ideal in ⊗ ˆ ε k , s k , s Y . This result marks up a difference with the behavior of nonsymmetric tensors since, when X is an M-ideal in Y, it is known that ⊗ ˆ ε k k X (the k-th tensor product of X endowed with the injective tensor norm) is an M-ideal in ⊗ ˆ ε k k Y . Nevertheless, if X is also Asplund, we prove that every integral k-homogeneous polynomial in X has a unique extension to Y that preserves the integral norm. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given.

Dunford-Taylor Integral and the Isotropic Tensor Valued Functions Having the Commutative Property with their Tensor Argument

Isotropic tensor valued functions of tensor arguments play an important role in the formulation of the equations governing the behavior of solid materials in the field of continuum mechanics. When the tensor argument is non-symmetric, the complexity and the difficulty in dealing with the tensor functions are high. In this work, the issue of expressing an isotropic tensor valued tensor function of a non-symmetric tensor argument is attention by utilizing the Dunford-Taylor integral. An important subclass of the isotropic tensor functions is considered with the commutative property.

A Korn's inequality for incompatible tensor fields

AbstractWe prove a Korn‐type inequality for bounded Lipschitz domains in $\Omega {\rm ~in~}{\rm I\!R}^3$ and non‐symmetric square integrable tensor fields $P : \Omega \to {\rm I\!R}^{3\times 3}$ having square integrable rotation ${\rm Curl~}P : \Omega \to {\rm I\!R}^{3\times 3}$. For skew‐symmetric P or compatible $P =\nabla\;v$ our estimate reduces to non‐standard variants of Poincaré's or Korn's first inequality, respectively, for which our new estimate can be viewed as a common generalized version. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modeling Coulombic failure of sea ice with leads

[1] Sea ice failure under low-confinement compression is modeled with a linear Coulombic criterion that can describe either fractural failure or frictional granular yield along slip lines. To study the effect of anisotropy we consider a simplified anisotropic sea ice model where the sea ice thickness depends on orientation. Accommodation of arbitrary deformation requires failure along at least two intersecting slip lines, which are determined by finding two maxima of the yield criterion. Due to the anisotropy these slip lines generally differ from the standard, Coulombic slip lines that are symmetrically positioned around the compression direction, and therefore different tractions along these slip lines give rise to a nonsymmetric stress tensor. We assume that the skewsymmetric part of this tensor is counterbalanced by an additional elastic stress in the sea ice field that suppresses floe spin. We consider the case of two leads initially formed in an isotropic ice cover under compression, and address the question of whether these leads will remain active or new slip lines will form under a rotation of the principal compression direction. Decoupled and coupled models of leads are considered and it is shown that for this particular case they both predict lead reactivation in almost the same way. The coupled model must, however, be used in determining the stress as the decoupled model does not resolve the stress asymmetry properly when failure occurs in one lead and at a new slip line.

On equitorsion geodesic mappings of general affine connection spaces onto generalized Riemannian spaces

In the papers Minčić (1973) [15], Minčić (1977) [16], several Ricci type identities are obtained by using non-symmetric affine connection. Four kinds of covariant derivatives appear in these identities.In the present work, we consider equitorsion geodesic mappings f of two spaces GAN and GR¯N, where GR¯N has a non-symmetric metric tensor, i.e. we study the case when GAN and GR¯N have the same torsion tensors at corresponding points. Such a mapping is called an equitorsion mapping Minčić (1997) [12], Stanković et al. (2010) [14], Stanković (in press) [13].The existence of a mapping of such type implies the existence of a solution of the fundamental equations. We find several forms of these fundamental equations. Among these forms a particularly important form is system of partial differential equations of Cauchy type.

A posteriori error estimators for stabilized P1 nonconforming approximation of the Stokes problem

In this paper we derive and analyze some a posteriori error estimators for the stabilized P1 nonconforming approximation of the Stokes problem involving the strain tensor. This will be done by decomposing the numerical error in a proper way into conforming and nonconforming contributions. The error estimator for the nonconforming error is obtained in the standard way, and the implicit error estimator for the conforming error is derived by applying the equilibrated residual method. A crucial part of this work is construction of approximate normal stresses on interelement boundaries which will serve as equilibrated Neumann data for local Stokes problems. It turns out that such normal stresses can be simply computed by local weak residuals of the discrete system plus jumps of the velocity solution and that a stronger equilibration condition is satisfied to ensure solvability of local Stokes problems. We also derive a simple explicit error estimator based on the nonsymmetric tensor recovery of the normal stress error. Numerical results are provided to illustrate the performance of our error estimators.

A Lagrangian model for hardening behaviour of materials at finite deformation based on the right plastic stretch tensor

In this paper a modified multiplicative decomposition of the right stretch tensor is proposed and used for finite deformation elastoplastic analysis of hardening materials. The total symmetric right stretch tensor is decomposed into a symmetric elastic stretch tensor and a non-symmetric plastic deformation tensor. The plastic deformation tensor is further decomposed into an orthogonal transformation and a symmetric plastic stretch tensor. This plastic stretch tensor and its corresponding Hencky’s plastic strain measure are then used for the evolution of the plastic internal variables. Furthermore, a new evolution equation for the back stress tensor is introduced based on the Hencky plastic strain. The proposed constitutive model is integrated on the Lagrangian axis of the plastic stretch tensor and does not make reference to any objective rate of stress. The classic problem of simple shear is solved using the proposed model. Results obtained for the problem of simple shear are identical to those of the self-consistent Eulerian rate model based on the logarithmic rate of stress. Furthermore, extension of the proposed model to the mixed nonlinear isotropic/kinematic hardening behaviour is presented. The model is used to predict the nonlinear hardening behaviour of SUS 304 stainless steel under fixed end finite torsional loading. Results obtained are in good agreement with the available experimental results reported for this material under fixed end finite torsional loading.

GAUGE-EQUIVARIANT HILBERT BIMODULES AND CROSSED PRODUCTS BY ENDOMORPHISMS

C*-endomorphisms arising from superselection structures with nontrivial center define a 'rank' and a 'first Chern class'. Crossed products by such endomorphisms involve the Cuntz–Pimsner algebra of a vector bundle having the above-mentioned rank, first Chern class and can be used to construct a duality for abstract (nonsymmetric) tensor categories versus group bundles acting on (nonsymmetric) Hilbert bimodules. Existence and unicity of the dual object (i.e. the 'gauge' group bundle) are not ensured: we give a description of this phenomenon in terms of a certain moduli space associated with the given endomorphism. The above-mentioned Hilbert bimodules are noncommutative analogs of gauge-equivariant vector bundles in the sense of Nistor–Troitsky.