Symmetric Second-order Tensor Research Articles

Anisotropic Breakage Mechanics for cemented granular materials

The anisotropy of granular geomaterials is sensitive to their fabric, which exhibits anisotropic mechanical properties as a function of deposition history, microscopic fabric, and loading paths. Here, a new fabric-enriched continuum breakage model is proposed to examine the relation between elastic and inelastic anisotropy in granular materials and cemented granular materials. A microstructure model is first implemented in the framework of fabric-enriched continuum breakage mechanics (F-CBM), where the anisotropic behaviour prior to yielding is introduced through a symmetric second-order fabric tensor embedded in the expression of the elastic energy potential. The anisotropic strain energy storage prior to grain crushing leads to the rotation and distortion of the yield surface of cemented granular materials. Parametric analyses are performed to assess the overall capability of the model to characterize the anisotropic inelastic processes in cemented granular. It is shown that the proposed model can accurately predict the strong correlation between anisotropic elasticity and breakage-damage processes in cemented granular materials. When damage involving the skeleton is the dominant inelastic process, the size of the elastic domain contracts and the material exhibits augmented brittleness with the disintegration of cement. While breakage processes are predicted to dominate the response of lightly cemented granular materials resulting in hardening behaviour. This work can be further extended to dynamically capture the anisotropic response of cemented granular materials with water-sensitive mineral constituents by accounting for the evolution of microstructural anisotropy.

Building Edge Detection Technology From Remote Sensing Image Based On NSCT And Tensor Voting

An edge detection technology based on the combination of non-downsampling contour wave transform (NSCT) and tensor voting is proposed, which aims to obtain more accurate and detailed edge information of buildings in remote sensing images. Firstly, NSCT is used for image decomposition to obtain the subband frequency information of different scales and angles. Then, position encoding is performed on these subband coefficients to obtain second-order symmetric tensors at the corresponding positions. Tensors of different scales and angles at the same position are weighted and summed to complete feature fusion. Finally, the edge features of the image are obtained based on tensor voting theory. The experimental results show that compared to common edge detection technologies, such as Canny, Fast Edge (Fast edge detection using structured forests) and HED (Holistically-Nested Edge Detection), our method can more accurately and intensively reflect the boundaries of buildings and the edge information of roofs, providing better support for the analysis of building types and architectural styles. Compared to the HED method, which is based on deep learning, our method improves PSNR and SSIM metrics by 0.98 and 0.03, respectively.

A recursion formula for the integer power of a symmetric second-order tensor and its application to computational plasticity

In this paper, a recursion formula is given for the integer power of a second-order tensor in 3D Euclidean space. It can be used in constitutive modelling for approximating failure or yield surfaces with corners, and it is demonstrated for the case of Rankine failure criterion. Removing corners provides clear advantages in computational plasticity. We discuss the consequences of the approximation errors for failure analyses of brittle and quasi-brittlematerials.

A simple spectral representation of a second-order symmetric tensor and its variation

The spectral decomposition of a second-order, symmetric tensor is widely adopted in many fields of Computational Mechanics. As an example, in elasto-plasticity under large strain and rotations, given the Cauchy deformation tensor, it is a fundamental step to compute the logarithmic strain tensor.Recently, this approach has also been adopted in small-strain isotropic plasticity to reconstruct the stress tensor as a function of its eigenvalues, allowing the formulation of predictor–corrector return algorithms in the invariants space. These algorithms not only reduce the number of unknowns at the constitutive level, but also allow the correct handling of stress states in which the plastic normals are undefined, thus ensuring a better convergence with respect to the standard approach.While the eigenvalues of a symmetric, second-order tensor can be simply computed as a function of the tensor invariants, the computation of its eigenbasis can be more difficult, especially when two or more eigenvalues are coincident. Moreover, when a Newton–Raphson algorithm is adopted to solve nonlinear problems in Computational Mechanics, also the tensorial derivatives of the eigenbasis, whose computation is still more complicated, are required to assemble the tangent matrix.A simple and comprehensive method is presented, which can be adopted to compute a closed form representation of a second-order tensor, as well as their derivatives with respect to the tensor itself, allowing a simpler and numerically accurate implementation of spectral decomposition of a tensor in Computational Mechanics applications.

Construction of Invariant Relations of $n$ Symmetric Second-Order Tensors

Construction of Invariant Relations of $n$ Symmetric Second-Order Tensors

An orthotropic plasticity model at finite strains with plasticity-induced evolution of orthotropy based on a covariant formulation

We present a rate-independent model for isotropic elastic–orthotropic plastic material behaviour in a hyper-elasto-plastic setting at finite strains, which is based on a covariant formulation that includes plastic-deformation-induced evolution of orthotropy. The model relies on a treatment by Lu and Papadopoulos, who made use of the postulate of covariance for an anisotropic elasto-plastic solid and derived constitutive equations of evolving anisotropies at finite strains. The latter is tantamount to the notion of plastic spin. This treatment does not rely on a multiplicative decomposition of the deformation gradient. We test our model on in-plane sheet-metal forming processes, which are governed by the evolution of pre-existing preferred material orientations. Hence, we advocate an orthotropic yield criterion directed by evolving structural tensors to describe this material behaviour. Our formulation yields two key findings. Firstly, the covariant formulation of plasticity yields suitable evolution equations for the structural tensors characterising the symmetry group of the orthotropic yield function. Secondly, the constitutive equations for the plastic variables and the structural tensors, which are both symmetric second-order tensors, give results that are in good agreement with experimental and numerical findings from in-plane sheet forming processes.

A semidefinite programming approach for the projection onto the cone of negative semidefinite symmetric tensors with applications to solid mechanics

We propose an algorithm for computing the projection of a symmetric second-order tensor onto the cone of negative semidefinite symmetric tensors with respect to the inner product defined by an assigned positive definite symmetric fourth-order tensor {mathcal {C}}. The projection problem is written as a semidefinite programming problem and an algorithm based on a primal-dual path-following interior point method coupled with a Mehrotra’s predictor-corrector approach is proposed. Implementations based on well-known symmetrization schemes and on direct methods are theoretically and numerically investigated taking into account tensors {mathcal {C}} arising in the modelling of masonry-like materials. For these special cases, indications on the preferable symmetrization scheme that take into account the conditioning of the arising linear systems are given.

A New and Efficient Approach to Determining the Rate of the Second-Order Symmetric Tensor and Its Isotropic Function

A New and Efficient Approach to Determining the Rate of the Second-Order Symmetric Tensor and Its Isotropic Function

Fabric-enriched continuum breakage mechanics (F-CBM)

A fabric-enriched continuum breakage mechanics (F-CBM) framework is developed to examine the relation between elastic and inelastic anisotropy in granular materials subjected to high pressure. For this purpose, emphasis is given to the influence of the anisotropic fabric on the plastic cap regime – that is, a macroscopic signature of the inelasticity of granular materials widely dependent on the particle crushability. Anisotropic behaviour prior to yielding is introduced through a symmetric second-order fabric tensor embedded in the expression of the elastic energy potential. It is shown that, thanks to its energy-based formulation, the F-CBM framework provides a platform to predict the rotation and distortion of the yield cap of granular materials as an outcome of anisotropic strain energy storage prior to grain crushing. Parametric analyses indicate that both the degree of anisotropy and the orientation of the fabric have an impact on the energy release resulting from grain crushing and contribute to the resulting stress–strain behaviour. It is shown that, despite the use of very few parameters, the proposed framework can accurately predict the anisotropic yielding and inelastic stress–strain response solely on the basis of the directional properties inferred from measurements of elastic anisotropy.

On a geometry with torsion and curvature: Basic geometric structure

In this paper, we developed, in detail, the geometric structure of a certain type of geometry usually used in physical applications. This paper contains building blocks (BB), tensors of different orders, curves admitted by the geometry and a comparison between this geometry and others used in the literature. Second-order symmetric and skew-symmetric tensors, usually used for physical applications in the context of the geometrization philosophy, are given. Also, in the framework of the same philosophy, the admitted curves can be used as equations of motion of test particles for any field theory constructed in this geometry.

Damage and Fracture Analysis under Multi-axial Stress State Based on Damage Mechanics

In various press forming and deep drawing processes, it is important to anticipate the forming limit in multi-axial stress state. In the uniaxial tension test, it is necessary to take stress triaxiality into account for accurate prediction of failure strain, because after the onset of the necking, the constricted portion, which is actually subjected to loading, is in the multiaxial stress state due to the change in local cross-sectional area. Recently, several damage and fracture models have been proposed that focus on the effect of stress triaxiality on the forming limit, while most of them are phenomenological models. In this study, the constitutive equation and damage evolution equation, which are based on a second-order symmetric damage tensor within the framework of the continuum damage mechanics, are constructed as a function of the stress triaxiality to analyze damage and fracture of materials in multiaxial stress state. Then, FEM analysis of an example problem was carried out to validate this material model.

Para-Ricci-like Solitons with Vertical Potential on Para-Sasaki-like Riemannian Π-Manifolds

The objects of study are para-Ricci-like solitons on para-Sasaki-like, almost paracontact, almost paracomplex Riemannian manifolds, namely, Riemannian Π-manifolds. Different cases when the potential of the soliton is the Reeb vector field or pointwise collinear to it are considered. Some additional geometric properties of the constructed objects are proven. Results for a parallel symmetric second-order covariant tensor on the considered manifolds are obtained. An explicit example of dimension 5 in support of the given assertions is provided.

Almost Ricci-like solitons with torse-forming vertical potential of constant length on almost contact B-metric manifolds

A generalization of Ricci-like solitons with torse-forming potential, which is constant multiple of the Reeb vector field, is studied. The conditions under which these solitons are equivalent to almost Einstein-like metrics are given. Some results are obtained for a parallel symmetric second-order covariant tensor. Finally, an explicit example of an arbitrary dimension is given and some of the results are illustrated.

Visualization of 3D Stress Tensor Fields Using Superquadric Glyphs on Displacement Streamlines.

Stress tensor fields play a central role in solid mechanics studies, but their visualization in 3D space remains challenging as the information-dense multi-variate tensor needs to be sampled in 3D space while avoiding clutter. Taking cues from current tensor visualizations, we adapted glyph-based visualization for stress tensors in 3D space. We also developed a testing framework and performed user studies to evaluate the various glyph-based tensor visualizations for objective accuracy measures, and subjective user feedback for each visualization method. To represent the stress tensor, we color encoded the original superquadric glyph, and in the user study, we compared it to superquadric glyphs developed for second-order symmetric tensors. We found that color encoding improved the user accuracy measures, while the users also rated our method the highest. We compared our method of placing stress tensor glyphs on displacement streamlines to the glyph placement on a 3D grid. In the visualization, we modified the glyph to show both the stress tensor and the displacement vector at each sample point. The participants preferred our method of glyph placement on displacement streamlines as it highlighted the underlying continuous structure in the tensor field.

Effective rationality of second-order symmetric tensor spaces

Effective rationality of second-order symmetric tensor spaces

Three-dimensional trans-Sasakian manifolds and solitons

PurposeThe authors set the goal to find the solution of the Eisenhart problem within the framework of three-dimensional trans-Sasakian manifolds. Also, they prove some results of the Ricci solitons, η-Ricci solitons and three-dimensional weakly  symmetric trans-Sasakian manifolds. Finally, they give a nontrivial example of three-dimensional proper trans-Sasakian manifold.Design/methodology/approachThe authors have used the tensorial approach to achieve the goal.FindingsA second-order parallel symmetric tensor on a three-dimensional trans-Sasakian manifold is a constant multiple of the associated Riemannian metric g.Originality/valueThe authors declare that the manuscript is original and it has not been submitted to any other journal for possible publication.

Shear properties of isotropic and homogeneous beam-like solids having arbitrary cross sections

The paper addresses the theoretical framework and the numerical procedure developed to carry out the sectional analysis preliminary to the use of a recently formulated isotropic and homogeneous beam model consistently derived from the Saint Venant solid one.Specifically, in order to ensure the equivalence of the new beam model with the 3D solid one, both in terms of elastic energy and displacements of the beam axis, one requires the evaluation of a symmetric second-order tensor, accounting for the shear deformation, besides the center of twist and the torsional stiffness factor.Common property to such geometrical quantities is to be expressed as integrals of harmonic functions solution of Neumann problems that are defined over the section domain. Hence, with a view towards numerical applications, they are reformulated as integrals defined over the section boundary, assumed to be of arbitrary polygonal shape.The numerical evaluation of the resulting line integrals is obtained by exploiting a recently formulated boundary element approach in which the harmonic potential functions are expressed as polynomials defined on suitably defined subsets of each edge of the section boundary.As an outcome of the extensive numerical tests that have been carried out on compact and thin-walled sections a general criterion, established elsewhere for properly selecting the best combination of polynomial degree and edge discretization, is shown to be successful also for carrying out the shear analysis of beam sections having arbitrary shape.

Information geometry

Information geometry has emerged from the study of the invariant structure in families of probability distributions. This invariance uniquely determines a second-order symmetric tensor g and third-order symmetric tensor T in a manifold of probability distributions. A pair of these tensors (g, T) defines a Riemannian metric and a pair of affine connections which together preserve the metric. Information geometry involves studying a Riemannian manifold having a pair of dual affine connections. Such a structure also arises from an asymmetric divergence function and affine differential geometry. A dually flat Riemannian manifold is particularly useful for various applications, because a generalized Pythagorean theorem and projection theorem hold. The Wasserstein distance gives another important geometry on probability distributions, which is non-invariant but responsible for the metric properties of a sample space. I attempt to construct information geometry of the entropy-regularized Wasserstein distance.

A non-local plasticity model for porous metals with deformation-induced anisotropy: Mathematical and computational issues

A non-local (gradient) plasticity model for porous metals that accounts for deformation-induced anisotropy is presented. The model is based on the work of Ponte Castañeda and co-workers on porous materials containing randomly distributed ellipsoidal voids. It takes into account the evolution of porosity and the evolution/development of anisotropy due to changes in the shape and the orientation of the voids during plastic deformation. A “material length” ℓ is introduced and a “non-local” porosity is defined from the solution of a modified Helmholtz equation with appropriate boundary conditions, as proposed by Geers et al. (2001); Peerlings et al. (2001). At a material point located at x, the non-local porosity f(x) can be identified with the average value of the “local” porosity floc(x) over a sphere of radius R≃3ℓ centered at x. The same approach is used to formulate a non-local version of the Gurson isotropic model. The mathematical character of the resulting incremental elastoplastic partial differential equations of the non-local model is analyzed. It is shown that the hardening modulus of the non-local model is always larger than the corresponding hardening modulus of the local model; as a consequence, the non-local incremental problem retains its elliptic character and the possibility of discontinuous solutions is eliminated. A rate-dependent version of the non-local model is also developed. An algorithm for the numerical integration of the non-local constitutive equations is developed, and the numerical implementation of the boundary value problem in a finite element environment is discussed. An analytical method for the required calculation of the eigenvectors of symmetric second-order tensors is presented. The non-local model is implemented in ABAQUS via a material “user subroutine” (UMAT or VUMAT) and the coupled thermo-mechanical solution procedure, in which temperature is identified with the non-local porosity. Several example problems are solved numerically and the effects of the non-local formulation on the solution are discussed. In particular, the problems of plastic flow localization in plane strain tension, the plane strain mode-I blunt crack tip under small-scale-yielding conditions, the cup-and-cone fracture of a round bar, and the Charpy V-notch test specimen are analyzed.

Ricci-Like Solitons with Vertical Potential on Sasaki-Like Almost Contact B-Metric Manifolds

Ricci-like solitons on Sasaki-like almost contact B-metric manifolds are the object of study. Cases, where the potential of the Ricci-like soliton is the Reeb vector field or pointwise collinear to it, are considered. In the former case, the properties for a parallel or recurrent Ricci-tensor are studied. In the latter case, it is shown that the potential of the considered Ricci-like soliton has a constant length and the manifold is $\eta$-Einstein. Other curvature conditions are also found, which imply that the main metric is Einstein. After that, some results are obtained for a parallel symmetric second-order covariant tensor on the manifolds under study. Finally, an explicit example of dimension 5 is given and some of the results are illustrated.