Tensor Variables Research Articles

Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models

The quest for a consistent theory for quantum gravity is one of the most challenging problems in theoretical high-energy physics. An often-used approach is to describe the gravitational degrees of freedom by the metric tensor or related variables, and finding a way to quantise this. In the canonical tensor model, the gravitational degrees of freedom are encoded in a tensorial quantity $P_{abc}$, and this quantity is subsequently quantised. This makes the quantisation much more straightforward mathematically, but the interpretation of this tensor as a spacetime is less evident. In this work we take a first step towards fully understanding the relationship to spacetime. By considering $P_{abc}$ as the generator of an algebra of functions, we first describe how we can recover the topology and the measure of a compact Riemannian manifold. Using the tensor rank decomposition, we then generalise this principle in order to have a well-defined notion of the topology and geometry for a large class of tensors $P_{abc}$. We provide some examples of the emergence of a topology and measure of both exact and perturbed Riemannian manifolds, and of a purely algebraically-defined space called the semi-local circle.

Structured derivation of moment equations and stable boundary conditions with an introduction to symmetric, trace-free tensors

Moment equations are used to approximate kinetic equations in a physically meaningful and numerically efficient way. This paper discusses the derivation of general moment equations including various caveats that must be taken care of when moment systems get implemented in simulation tools. We suggest to use a constant Maxwellian with arbitrary constant reference values for density, velocity and temperature for the underlying expansion of the distribution. This seems restrictive in comparison to using a local Maxwellian, but full nonlinear fluid dynamics remains possible as long as the reference values do not differ too much from the local values and a sufficiently large moment vector is considered. Additionally, this approach allows to derive globally hyperbolic equations in a simple way and the transport operator becomes linear. The resulting equations are discussed in detail and compared to various existing approaches.The paper also presents boundary conditions that render the moment systems stable in an $ L^{2} $-sense. This is done both by general considerations and for the concrete case of the Maxwell accommodation model for kinetic equations, generalizing earlier stable boundary conditions for special cases of moment equations. The extensive appendix of the paper will help readers unfamiliar with tensor variables and symmetric, trace-free decompositions to understand the general concept needed for moment approximations without the need of group-theoretical expertise.

Role of energy–momentum squared gravity on the dynamics of charged dissipative plane symmetric collapse

This paper deals with the dynamics of charged plane symmetric collapse with dissipative fluid distribution in the framework of energy–momentum squared gravity. For this purpose, we consider non-static plane symmetric spacetime in the inner and static charged Vaidya spacetime in the outer regions of a star. We use Darmois junction conditions to match the interior and exterior geometries and find that masses of both spacetimes are identical if and only if their correspondence charges are same. To investigate the dynamics of the system, we apply Misner–Sharp and Müler–Israel–Stewart approaches to formulate dynamical as well as transport equations, respectively. We then couple these equations to analyze the effect of physical quantities and modified terms on the collapse rate. A relation among Weyl tensor, electromagnetic field and fluid variables, is also developed. Due to the influence of charge, anisotropic pressure and modified terms, the spacetime is not conformally flat. Further, we assume isotropic fluid and ignore the impact of electromagnetic field which yields the conformally flat spacetime and inhomogeneous energy density. We conclude that the collapse rate reduces as compared to general relativity due to the presence of a charge, effective pressure, heat flux and additional terms of this gravity.

A micromechanical approach to the mechanical characterization of 3D-printed composites

Aiming for the development of experimentally validated computational models to predict the mechanical properties of 3D-printed composites, the present study proposes a micromechanical approach by using a simplified unit cell model to characterize the material properties and behavior of 3D-printed composites manufactured through fused deposition modeling. The effective properties of the voided polymer matrix phase of the material are computed by calculating the void density as a tensorial meso-structural variable. These effective properties along with those of the fiber are input into a simplified micromechanical model to predict the material properties of the 3D-printed composite. The predictions are seen to be in very good agreement with the experimental values. The present approach is much simpler and less computationally costly compared to the finite element homogenization method. In addition, the present approach has the potential to simulate the response of the 3D-printed composite under different loading conditions.

Numerical integration in G/XFEM analysis of 2-D fracture mechanics problems for physically nonlinear material and cohesive crack propagation

PurposeThe aim of this paper is to present a novel data transfer technique to simulate, by G/XFEM, a cohesive crack propagation coupled with a smeared damage model. The efficiency of this technique is evaluated in terms of processing time, number of Newton–Raphson iterations and accuracy of structural response.Design/methodology/approachThe cohesive crack is represented by the G/XFEM enrichment strategy. The elements crossed by the crack are divided into triangular cells. The smeared crack model is used to describe the material behavior. In the nonlinear solution of the problem, state variables associated with the original numerical integration points need to be transferred to new points created with the triangular subdivision. A nonlocal strategy is tailored to transfer the scalar and tensor variables of the constitutive model. The performance of this technique is numerically evaluated.FindingsWhen compared with standard Gauss quadrature integration scheme, the proposed strategy may deliver a slightly superior computational efficiency in terms of processing time. The weighting function parameter used in the nonlocal transfer strategy plays an important role. The equilibrium state in the interactive-incremental solution process is not severely penalized and is readily recovered. The advantages of such proposed technique tend to be even more pronounced in more complex and finer meshes.Originality/valueThis work presents a novel data transfer technique based on the ideas of the nonlocal formulation of the state variables and specially tailored to the simulation of cohesive crack propagation in materials governed by the smeared crack constitutive model.

Anisotropic subloading surface Cam‐clay plasticity model with rotational hardening: Deformation gradient‐based formulation for finite strain

AbstractThis study is aimed at developing an anisotropic elastoplastic constitutive model for geomaterials at finite strain and its stress calculation algorithm based on the fully implicit return‐mapping scheme. The Cam‐clay plasticity model is adopted as a specific prototype model of geomaterials. As a pertinent representation of deformation‐induced anisotropy in geomaterials, nonlinear rotational hardening is incorporated into the model in a theoretically reasonable manner by introducing the dual multiplicative decompositions of the deformation gradient tensor. In addition to the usual decomposition into elastic and plastic parts, the plastic part is decomposed further into a part contributing to the rotational hardening and a remainder part. The former part leads to a back stress ratio tensor related to the rotational hardening via a hyperelastic‐type hardening rule. The constitutive theory is thereby formulated on proper intermediate configurations entirely in terms of deformation‐like tensorial variables possessing invariance property, without resort to any objective rates of stress or stress‐like variables. Combining the Cam‐clay plasticity with the concept of subloading surface, a class of unconventional plasticity, enables the model to be capable of reproducing complex hardening/softening accompanied by volumetric contractive/dilative responses. Basic characteristics and predictive capability of the proposed model, as well as the accuracy of the developed numerical scheme, are verified through several numerical examples including monotonic and cyclic loadings.

A Dynamic Finite-Deformation Constitutive Model for Steels Undergoing Slip, Twinning, and Phase Changes

Depending on composition, processing, microstructure, and loading mode, steels may demonstrate various inelastic deformation mechanisms, notably dislocation glide, deformation twinning, and solid-solid phase changes. Damage in the form of voids, often leading to failure by macro-cracking, is also of current interest. A finite-deformation constitutive model is constructed in order to address such mechanisms in isotropic polycrystals under static and dynamic loading, where the latter encompasses high pressures and extreme strain rates pertinent to ballistic penetration. Slip and twinning are isochoric, and their relative contributions to kinematics are not explicitly distinguished in the present application. Phase changes among, e.g., face-centered-cubic (FCC), body-centered-cubic (BCC), body-centered-tetragonal (BCT), and hexagonal (HCP) structures are admitted, with corresponding deviatoric and volumetric strains. Porosity from voids contributes to volumetric strain. A new consistent thermodynamic framework incorporating an Eulerian strain tensor and internal state variables is developed, whereby kinetic equations for slip–twinning, phase changes, and damage evolution result in contributions to dissipation. An objective rate form of the model, derived assuming small deviatoric elastic strain, is implemented numerically. The model is applied to three different primarily austenitic, medium-high Mn steels. Specifically, representations of a slip-dominated (SLIP), a TRIP (transformation-induced plasticity) and a TWIP (twinning-induced plasticity) steel are calibrated to quasi-static tension, quasi-static compression, and dynamic compression data, at both room and high temperatures. A novel functional form of material strength distinguishes hardening profiles of the different alloys. Experimental data are reasonably well represented. Model extrapolations for dynamic strength and pressure in regimes pertinent to shock compression are analyzed. Predictions for multi-axial loading of shear with simultaneous expansion or contraction demonstrate competing physical mechanisms among the alloys that could be leveraged for optimal ballistic performance.

Relaxed Euler systems and convergence to Navier-Stokes equations

We consider the approximation of Navier-Stokes equations for a Newtonian fluid by Euler type systems with relaxation both in compressible and incompressible cases. This requires to decompose the second-order derivative terms of the velocity into first-order ones. Usual decompositions lead to approximate systems with tensor variables. We construct approximate systems with vector variables by using Hurwitz-Radon matrices. These systems are written in the form of balance laws and admit strictly convex entropies, so that they are symmetrizable hyperbolic. For smooth solutions, we prove the convergence of the approximate systems to the Navier-Stokes equations in uniform time intervals. Global-in-time convergence is also shown for the initial data near constant equilibrium states of the systems. These convergence results are established not only for the approximate systems with vector variables but also for those with tensor variables.

Nonlinear fluctuations in relativistic causal fluids

In the Second Order Theories (SOT) of real relativistic fluids, the non-ideal properties of the flows are described by a new set of dynamical tensor variables. In this work we explore the non-linear dynamics of those variables in a conformal fluid. Among all possible SOTs, we choose to work with the Divergence Type Theories (DTT) formalism, which ensures that the second law of thermodynamics is fulfilled non-perturbatively. The tensor modes include two divergence-free modes which have no analog in theories based on covariant generalizations of the Navier-Stokes equation, and that are particularly relevant because they couple linearly to a gravitational field. To study the dynamics of this irreducible tensor sector, we observe that in causal theories such as DTTs, thermal fluctuations induce a stochastic stirring force, which excites the tensor modes while preserving energy momentum conservation. From fluctuation-dissipation considerations it follows that the random force is Gaussian with a white spectrum. The irreducible tensor modes in turn excite vector modes, which back-react on the tensor sector, thus producing a consistent non-linear, second order description of the divergence-free tensor dynamics. Using the Martin-Siggia-Rose (MSR) formalism plus the Two-Particle Irreducible Effective Action (2PIEA) formalism, we obtain the one-loop corrected equations for the relevant two-point correlation functions of the model: the retarded propagator and the Hadamard function. The overall result of the self-consistent dynamics of the irreducible tensor modes at this order is a depletion of the spectrum in the UV sector, which suggests that tensor modes could sustain an inverse entropy cascade.

Global weak solutions to degenerate coupled transport processes in partially saturated deformable elastic-inelastic porous media

In this work we prove the existence of global weak solutions to a degenerate and strongly coupled parabolic system arising from the transport processes through partially saturated deformable porous materials. The hygro-thermal model is coupled with quasi-static evolution equations modeling elastic and inelastic mechanical deformations. Physically relevant Newton boundary conditions are considered for water pressure and temperature of the porous system. The traction boundary condition is imposed on the deformable solid skeleton of the porous material. Degeneration occurs in both elliptic and parabolic part of the balance equation for mass of water. The coupling between water pressure, temperature, stress tensor and internal variables occurs in transport coefficients, constitutive functions and the decomposition of the total strain tensor into elastic and plastic parts due to mechanical effect and strain tensor due to thermal expansion.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/63/abstr.html

Construction of a new Lyapunov function for a dissipative gyroscopic system using the residual energy function

In a dissipative gyroscopic system with four degrees of freedom and tensorial variables in contravariant (right upper index) and covariant (right lower index) forms, a Lagrangian-dissipative model, i.e., {L, D}-model, is obtained using second-order linear differential equations. The generalized elements are determined using the {L, D}-model of the system. When the prerequisite of a Legendre transform is fulfilled, the Hamiltonian is found. The Lyapunov function is obtained as a residual energy function (REF). The REF consists of the sum of Hamiltonian and losses or dissipative energies (which are negative), and can be used for stability by Lyapunov's second method. Stability conditions are mathematically proven.

Tensor calculus in spherical coordinates using Jacobi polynomials. Part-II: Implementation and examples

We present a simulation code which can solve a broad range of partial differential equations in a full sphere. The code expands tensorial variables in a spectral series of spin-weighted spherical harmonics in the angular directions and a scaled Jacobi polynomial basis in the radial direction, as described in Vasil et al. (2018; hereafter, Part-I). Nonlinear terms are calculated by transforming from the coefficients of the spectral series to the value of each quantity on the physical grid, where it is easy to calculate products and perform other local operations. The expansion makes it straightforward to solve equations in tensor form (i.e., without decomposition into scalars). We propose and study several unit tests which demonstrate the code can accurately solve linear problems, implement boundary conditions, and transform between spectral and physical space. We then run a series of benchmark problems proposed in Marti et al. (2014), implementing the hydrodynamic and magnetohydrodynamic equations. We are able to calculate more accurate solutions than reported in Marti et al. (2014) by running at higher spatial resolution and using a higher-order timestepping scheme. We find the rotating convection and convective dynamo benchmark problems depend sensitively on details of timestepping and data analysis. We also demonstrate that in low resolution simulations of the dynamo problem, small changes in a numerical scheme can lead to large changes in the solution. To aid future comparison to these benchmarks, we include the source code used to generate the data, as well as the data and analysis scripts used to generate the figures.

Accounting for frictional contact in an anisotropic damage model based on compliance tensorial variables, illustration on ceramic matrix composites

Ceramic matrix composites have good thermomechanical properties at high or very high temperatures. The modeling of the crack networks associated to the degradation of such composites using damage mechanics is not straightforward. The main reason is the presence of a crack network mainly oriented by the loading direction, which is a priori unknown. To model this, compliance tensorial damage variables are used in a thermodynamic potential able to account for crack closure effects (unilateral contact). The damage kinematic is initially completely free and imposed by the evolution laws. The key point of the present paper is to account for friction in such cracks that can result in an apparent activation/deactivation of the shear damage. The initial model is enriched with an inelastic strain and a friction law. The plasticity criterion is expressed only using tensorial variables. The model is identified and illustrated on multiaxial data obtained at ONERA on tubes loaded in tension and torsion.

The need for a concept of shape homeostasis

A major step forward for developmental biology will be accomplished when someone figures out how to extend the concept of homeostasis to apply to shapes, in the sense of geometric properties of cells, tissues and organs. I propose that the biggest obstacle to this forward step is that biological researchers are not yet familiar with the properties of tensor variables, as compared with scalars. This key difference is that tensor properties can and usually do have different amounts in different directions, whereas scalar properties cannot vary with direction. Examples of tensor variables include stress, strain, curvature, permeability, and stiffness. Examples of scalar properties include chemical concentrations, osmotic pressure, hydrostatic pressure, adhesiveness and electrical voltage. Even D'Arcy Thompson treated mechanical tension (which is the classic example of a tensor variable) as if it were a scalar constant. This greatly reduced the number of geometric shapes that he could explain as being directly produced by forces. For example, in order to generate cylinders, surface contractions need to be twice as strong in one direction as compared with the perpendicular direction. Unless surface contractions vary with direction, only spheres can be generated.Another example of not distinguishing tensors from scalars is the use of suction pipettes to measure stresses of cell surfaces (for example, during cytokinesis). This method of measurement inescapably lumps together directional components of two different tensors (tension and stiffness) as if they were one scalar. Yet another obstacle was that certain scientists argued persuasively, but mistakenly, that attractor basins were evidence of minimization of thermodynamic free energy.Chemical concentrations have no special ability to generate gradients. Neither do any other scalar variables. As will be discussed below, repeated local equilibration of any quantitative variable will generate at least as good a gradient as diffusion can. It is misguided to think of scalars as being in any sense more quantitative than tensors. In fact, tensor variables can convey more information than chemical gradients, often faster and with less vulnerability to disturbance.

Second-invariant-preserving Remap of the 2D deviatoric stress tensor in ALE methods

For numerical simulations of impact problems or fluid–solid interactions, the ALE (Arbitrary Lagrangian–Eulerian) approach is a useful tool due to its ability to keep the computational mesh smooth and moving with the fluid. The elastic–plastic extension of the compressible fluid model requires tensor variables for the description of non-volumetric (deviatoric) mechanical stress. While Lagrangian numerical schemes based on the evolution equation of the stress tensor are well developed, tensor remap is still a relatively unexplored territory.We propose a new approach to deviatoric stress remapping, where the second invariant J2 (a conservative scalar quantity related to the strain energy) is remapped independently of the tensor components. These are re-scaled to match the remapped invariant value, effectively using only the principal directions and eigenvalue ratio from the component-wise remap. This approach is frame invariant, preserves J2 invariant bounds and conserves the total invariant. We compare our method with component-based remapping using a simple synchronized limiter or a specialized stress tensor limiter described in the literature.

Representation of Hashin–Shtrikman Bounds in Terms of Texture Coefficients for Arbitrarily Anisotropic Polycrystalline Materials

The present work generalizes the results of Bohlke and Lobos (Acta Mater. 67:324–334, 2014) by giving an explicit representation of the Hashin–Shtrikman (HS) bounds of linear elastic properties in terms of tensorial Fourier texture coefficients not only for cubic materials but for arbitrarily anisotropic linear elastic polycrystalline materials. Based on the HS bounds as given by Walpole (J. Mech. Phys. Solids 14(3):151–162, 1966) and tensor functions for the representation of the crystallite orientation distribution function, it is shown that the HS bounds are represented in terms of the exact same second- and fourth-order texture coefficients which appear in the consideration of the Voigt and Reuss bounds. The derived representations in terms of tensorial texture coefficients are valid for all symmetry classes in elasticity and present expressions highly attractive for the description of physical quantities in terms of tensorial variables. In order to make these results also accessible for the community of quantitative texture analysis, transformation relations between experimentally obtained Bunge’s or Roe’s coefficients and the tensorial texture coefficients are given. The representation of the present work offers a finite and low dimensional parametrization of the fully anisotropic Hashin–Shtrikman bounds, which can be used in inverse materials design problems in order to explore the set of possible materials properties or for the determination of optimal microstructural influence with respect to prescribed material properties. Examples for orthotropic polycrystals of cubic materials and transversely isotropic polycrystals of hexagonal materials (showing the connection and applicability of the results also to fiber orientation distributions) are discussed. Finally, an implementation in Mathematica® 11 of the HS bounds for arbitrarily anisotropic materials is offered, such that readers can reproduce all the results of this work and use them for their own purposes.

Solution of the convective heat transfer problem in a screw coordinate system

The problems of hydrodynamics and heat transfer in a screw channel flow are considered in a screw coordinate system. In particular, we consider an upward flow of liquid metal in a heated annular channel with a screw edge. The problem is related to the heat exchange in the flow around the fuel cell in the active zone of a nuclear reactor with a liquid-metal coolant. Differential equations of mass, momentum and energy conservation for a liquid metal flow are written in terms of tensor variables for a curvilinear (screw) coordinate system. The well-known equations for the cylindrical system are taken as a basis. They are supplemented with additional terms which arise due the transition to the screw coordinate system. The method of numerical calculation in the helical coordinate system greatly simplifies the description of the geometry and boundary conditions, the construction of the computational grid, the representation and subsequent processing of the results. Numerical simulations of velocity, pressure and temperature fields in a turbulent flow of liquid metal in an annular channel with a screw edge are carried out. Two problems are studied numerically and compared: channels with straight and twisted (screw) edge. Analysis of the results revealed significant influence of the edge twist on the flow and heat transfer characteristics. The presence of a gap between the edge and the outer pipe enhances the twist effects.

A velocity/stress mixed stabilized nodal finite element for elastodynamics: Analysis and computations with strongly and weakly enforced boundary conditions

A new nodal mixed finite element is proposed for the simulation of linear elastodynamics and wave propagation problems in time domain. Our method is based on equal-order interpolation discrete spaces for both the velocity (or displacement) and stress (or strain) tensor variables. The mixed form is derived using either the velocity/stress or velocity/strain pair of unknowns, the latter being instrumental in extensions of the method to nonlinear mechanics. The proposed approach works equally well on hexahedral or tetrahedral grids and, for this reason, it is suitable for time-domain engineering applications in complex geometry. The peculiarity of the proposed approach is the use of the rate form of the stress update equation, which yields a set of governing equations with the structure of a non-dissipative space/time Friedrichs’ system. We complement standard traction boundary conditions for the stress with strongly and weakly enforced boundary conditions for the velocity (or displacement). Weakly enforced boundary conditions are particularly suitable when considering complex geometrical shapes, because they do not require dedicated data structures for the imposition of the boundary degrees of freedom, but, rather, they utilize the structure of the variational formulation. We also show how the framework of weakly enforced boundary conditions can be used to develop variational forms for multi-domain simulations of heterogeneous media. A complete analysis including stability and convergence proofs is included, in the case of a space–time variational approach. A series of computational tests are used to demonstrate and verify the performance of the proposed approach.

A 3D finite-strain-based constitutive model for shape memory alloys accounting for thermomechanical coupling and martensite reorientation

The paper presents a finite-strain constitutive model for shape memory alloys (SMAs) that accounts for thermomechanical coupling and martensite reorientation. The finite-strain formulation is based on a two-tier, multiplicative decomposition of the deformation gradient into thermal, elastic, and inelastic parts, where the inelastic deformation is further split into phase transformation and martensite reorientation components. A time-discrete formulation of the constitutive equations is proposed and a numerical integration algorithm is presented featuring proper symmetrization of the tensor variables and explicit formulation of the material and spatial tangent operators involved. The algorithm is used for finite element analysis of SMA components subjected to various loading conditions, including uniaxial, non-proportional, isothermal and adiabatic loading cases. The analysis is carried out using the FEA software Abaqus by means of a user-defined material subroutine, which is then utilized to simulate a SMA archwire undergoing large strains and rotations.

Particle-based solid for nonsmooth multidomain dynamics

A method for simulation of elastoplastic solids in multibody systems with nonsmooth and multidomain dynamics is developed. The solid is discretised into pseudo-particles using the meshfree moving least squares method for computing the strain tensor. The particle’s strain and stress tensor variables are mapped to a compliant deformation constraint. The discretised solid model thus fit a unified framework for nonsmooth multidomain dynamics simulations including rigid multibodies with complex kinematic constraints such as articulation joints, unilateral contacts with dry friction, drivelines, and hydraulics. The nonsmooth formulation allows for impact impulses to propagate instantly between the rigid multibody and the solid. Plasticity is introduced through an associative perfectly plastic modified Drucker–Prager model. The elastic and plastic dynamics are verified for simple test systems, and the capability of simulating tracked terrain vehicles driving on a deformable terrain is demonstrated.