Deformation Tensor Research Articles

Symmetry invariants of higher orders in magnetoelastic energy for a trigonal easy-plane antiferromagnet

In expansion of the thermodynamic potential for a trigonal easy-plane antiferromagnet such as iron borate, FeBO3, magnetoelastic invariants of a higher order in the components of the deformation tensor and the antiferromagnetic vector with respect to those traditionally used were taken into account. Such invariants make it possible to describe new effects observed in iron borate crystals and composites based on them.

A perturbation analysis of pure torsion of incompressible hyperelastic cylinders

The stretch-based formulation of basic problems for incompressible isotropic hyperelastic materials has been the subject of renewed recent attention largely motivated by application to modelling the mechanical response of soft tissues. Here we are concerned with the classical problem of simple torsion of a circular cylinder which has in the past been primarily investigated by employing the classical approach of Rivlin in terms of invariants of the Cauchy–Green deformation tensors. Here we propose a perturbation approach to the pure torsion problem for thin (but not necessarily infinitesimally thin) cylindrical specimens motivated by potential applications to rubber-like or biological filaments and for the infinitesimal twisting of thick cylinders. The advantage of this approach is that the results are valid for all strain–energy functions whether based in terms of the principal stretches or in terms of the classical deformation invariants. The asymptotic results obtained for the resultant force and moment are applied to the special case of the one-term stretch-based Ogden model. In particular, a prediction of a transition in the Poynting effect for values of the Ogden parameter α>6 is obtained. Other illustrative examples for both invariant and stretch-based strain energies are described. In view of the complexity of a general analysis of torsion for arbitrary strain energies, the perturbation approach provides a viable alternative for assessing the resultant force and moment.

Direct numerical simulation of fluid/solid particles flow inside a channel

The modeling of moving solid particles in fluid flow has been the focus of many studies and has succeeded to attract significant attention from researchers. However, commonly used modeling approaches such as discrete element modeling (DEM) and direct numerical simulations (DNS) lack simplicity and have been computationally intensive. The aim of this paper is to develop a new approach to simulate solid transport in an incompressible Newtonian fluid flow. This method is based on the Finite element method with penalization of the deformation tensor. The fluid behavior is governed by the Navier-Stokes equations within the investigation domain. To take into account collisions, we present an algorithm which allows us to handle contacts between rigid particles. In this paper, 2D fluid/particles flow simulations are performed; the results are validated by comparison with results from other methods. We attempt to simulate the conveying of solid particles behavior of circular particles in a fluid flow inside a pipe. The numerical tests show that the present method provides a very efficient approach to directly simulate the solid transport inside the channels.

Bespoke two-dimensional elasticity and the nonlinear analogue of Cauchy’s relations

Is it possible to design an architectured material or structure whose elastic energy is arbitrarily close to a specified continuous function? This is known to be possible in one dimension, up to an additive constant (Dixon et al., Bespoke extensional elasticity through helical lattice systems, Proc. R. Soc. A. (2019)). Here, we explore the situation in two dimensions. Given (1) a continuous energy function E ( C ) , defined for two-dimensional right Cauchy–Green deformation tensors C contained in some compact set and (2) a tolerance ϵ > 0 , can we construct a spring-node unit cell (of a lattice) whose energy is approximately E , up to an additive constant, with L ∞ -error no more than ϵ ? We show that the answer is yes for affine E s (i.e., for energies E that are quadratic in the deformation gradient), but that the general situation is more subtle and is related to the generalisation of Cauchy’s relations to nonlinear elasticity.

On continuation criteria for the double-diffusive convection system in Vishik spaces

This paper proves the continuation criteria of the strong solution for the 3D double-diffusive convection system involving the deformation tensor in Vishik spaces. As a bi-product, our theorem also improve some well-known results on conditional regularity for the particular case of classical Navier–Stokes equations.

Statistics of tidal and deformation eigenvalue fields in the primordial Gaussian matter distribution: the two-dimensional case

ABSTRACT We study the statistical properties of the eigenvalues of the primordial tidal and deformation tensor for two-dimensional random Gaussian cosmic density fields. With the tidal and deformation tensors, Hessians of the gravitational and velocity potential, being Gaussian, the eigenvalue fields are distinctly non-Gaussian. We extend the Doroshkevich formula for the joined distribution of eigenvalues to two-dimensional fields and evaluate the two- and three-point correlation functions. In addition, we assess the number densities of singular points of the eigenvalue fields and find their corresponding two- and three-point correlation functions. The incentive for the present study is based on our interest in developing a fully analytical theory for the structure and dynamics of the cosmic web. The tidal forces and the resulting mass element deformation shape the prominent anisotropic wall-like and filamentary components of the cosmic web. Less well-known is that the web-like spatial pattern is already recognizable in the primordial tidal and deformation eigenvalue field. Against the full phase-space assessment of structure formation in the Universe, the caustic skeleton theory entails an analytical framework for the non-linear evolution of the cosmic web. It accomplishes this by describing the folding of the dark matter sheet and quantifying the emerging caustic singularities, which are fully specified by the spatial properties of the deformation eigenvalues and eigenvectors. Finally, the eigenvalues of the primordial tidal tensor are crucial in the generation of the angular momentum of galaxies. Understanding their spatial distribution is a critical element in predicting the resulting rotation amplitude and orientation.

On the application of generalised Newtonian fluids in the modelling of drag-reducing rigid polymers

Drag-reducing polymers with rod-like molecular structure are known to exhibit significant shear-thinning yet only weak elastic properties, making them suited to modelling by generalised Newtonian fluids (GNFs) based on the second invariant of the rate of deformation tensor, D. However only qualitative agreement with experiments is obtained when such models are employed in the direct numerical simulation of turbulent flows. In this study, we investigate GNFs to obtain guidelines for their application in predicting turbulent drag reduction, and show that the appearance of flow-type sensitivity or elastic effects, however small, places a limit on their accuracy. By incorporating a dependence on the third invariant of D, to bring in sensitivity to flow type, we demonstrate that reasonably good quantitative agreement with experimental data can be obtained.

Tensor Decompositions and Their Properties

In the present paper, we study two different approaches of tensor decomposition. The first part aims to study some properties of tensors that result from the fact that some components are vanishing in certain coordinates. It is proven that these conditions allow tensor decomposition, especially (1, σ), σ=1,2,3 tensors. We apply the results for special tensors such as the Riemann, Ricci, Einstein, and Weyl tensors and the deformation tensors of affine connections. Thereby, we find new criteria for the Einstein spaces, spaces of constant curvature, and projective and conformal flat spaces. Further, the proof of the theorem of Mikeš and Moldobayev is repaired. It has been used in many works and it is a generalization of the criteria formulated by Schouten and Struik. The second part deals with the properties of a special differential operator with respect to the general decomposition of tensor fields on manifolds with affine connection. It is shown that the properties of special differential operators are transferred to the components of a given decomposition.

Geometrically consistent nonlinear plane strain and stress constitutive models: Application to soft-material oscillations

Position-gradient constraints are used in this study to develop new geometrically consistent plane strain and stress constitutive models for the nonlinear large-displacement analysis of soft materials. The position gradients are used to define strain and stress constraints that enter into the formulation of the deformation tensor, strain-energy density function, and new constitutive models. The new approach is applied to three nonlinear hyperelastic constitutive models for incompressible materials: Neo-Hookean, Mooney Rivlin, and Yeoh Models. The underlying geometric plane-stress and plane-strain assumptions are preserved by using the finite elements (FE) of the absolute nodal coordinate formulation (ANCF). Limitations of widely used incompressibility penalty functions that produce stresses at initial stress-free configurations are discussed. Several numerical examples developed using planar ANCF shear-deformable beam and rectangular elements are used to assess the proposed formulation and verify its results using spatial solid elements implemented in commercial FE software. It is demonstrated that the ANCF plane-stress and plane-strain formulations for the incompressible materials predict accurately soft-material oscillations, demonstrating that stiffer behavior can be attributed to material-model inconsistencies and cannot always be attributed to locking.

A novel continuum mechanical framework for decoupled material behavior in thickness and in-plane directions

A novel continuum mechanical framework for decoupling the material behavior in in-plane and thickness directions is presented that is motivated by the macroscopic behavior of paper and paperboard. Due to the manufacturing process of these composite materials and the underlying microstructure resulting thereof, the material response is highly anisotropic and can be divided into in-plane and out-of-plane behavior that is independent of each other. This decoupling is mostly incorporated in existing material models by setting the corresponding material parameters to zero which limits the models to an extension to nonlinear problems and is restrictive from a continuum mechanical point of view.Therefore, the aim of this paper is to derive a continuum mechanical framework that captures the in-plane and out-of-plane decoupling in a thermodynamically consistent manner for large deformations. Thus, the right Cauchy–Green deformation tensor was split into in-plane and out-of-plane deformations by use of structural tensors that were aligned with the preferential material directions. Therefore, the Helmholtz free energy was formulated in terms of modified invariants of the deformation tensor for in-plane and out-of-plane deformations separately, which lead to a decoupled framework that could easily be modified due to the flexible formulation of the Helmholtz free energy terms. The extension to inelasticity, in particular viscoelasticity here, was straightforward due to the split of the deformation tensor. Finally, the decoupling of the in-plane and out-of-plane response of the presented framework was demonstrated by structural examples for elastic and inelastic material behavior.The presented model incorporates the split into in-plane and out-of-plane behavior in a continuum mechanical manner, and thus, is a flexible constitutive framework for decoupled material responses that can easily be adjusted to different material behavior.

Legendre–Hadamard conditions for fiber-reinforced materials with one, two or three families of fibers

Necessary conditions of the Legendre–Hadamard type for energy minimizers are derived in the setting of a special Cosserat elasticity theory for fiber-reinforced solids. These extend and generalize earlier results pertaining to fiber flexure-twist strain vectors formed from the Cosserat wryness tensors. New necessary conditions involving the associated Cosserat deformation tensors are also derived.

On the blow up criterion for the 3D nematic liquid crystal flows involving the second eigenvalue of the deformation tensor

On the blow up criterion for the 3D nematic liquid crystal flows involving the second eigenvalue of the deformation tensor

The average conformation tensor of inter-atomic bonds as an alternative state variable to the strain tensor: definition and first application — the case of nanoelasticity

Most of the mechanical models for solid state materials are in a methodological framework where a strain tensor, whatever it is, is considered as a thermodynamic state variable. As a consequence, the Cauchy stress tensor is expressed as a function of a strain tensor—and, in many cases, of one or more other state variables, such as the temperature. Such a choice for the kinematic state variable is clearly relevant in the case of infinitesimal or finite elasticity. However, one can ask whether an alternative state variable could not be considered. In the case of finite elastoplasticity, the choice of a strain tensor as the basic, kinematic state variable is not totally without issue, in particular in relation to the physical meaning of the internal state variable describing the permanent strains. In any case, this paper proposes an alternative to the strain tensor as a state variable, which is not based on the deformation (Lagrangian) gradient: the average conformation tensor of inter-atomic bonds. The purpose, however, is restricted to (1) a particular type of materials, namely the pure substances (copper or aluminum, for instance), (2) the nanoscale, and (3) the case of elasticity. The very simple case of two atoms of a pure substance in the solid state is first considered. It is shown that the kinematics of the inter-atomic bond can be characterized by a so called ``conformation'' tensor, and that the tensorial internal force acting on it can be immediately deduced from a single scalar function, depending only on the conformation tensor: the state potential of free energy (or interaction potential). Using an averaging procedure, these notions are then extended to a finite set of atoms, namely an atom and its first neighbours, which can be seen as the ``unit cell'' of a pure substance in the solid state considered as a discrete medium. They are also transposed to the Continuum case, where an expression of the Cauchy stress tensor is proposed as the first derivative of a state potential of density (per unit mass) of average free energy of inter-atomic bonds, which is an explicit function of the average conformation tensor of inter-atomic bonds. By applying a standard procedure in Continuum Thermodynamics, it is then shown that the objective part of the material derivative of this new state variable, at least in the case when the pure substance can be considered as an elastic medium, is equal to the symmetric part of the Eulerian velocity gradient, that is the rate of deformation tensor. In the case of uniaxial tension, a simple relationship is eventually set out between the average conformation tensor and a strain tensor, which is correctly approximated by the usual infinitesimal strain tensor as long as the conformation variations (from an initial state of conformation) are ``small''. From this latter result, and assuming an elastic behavior, a simple expression for the state potential of density of average free energy is inferred, showing great similarities with—but not equivalent to—the classical model of isotropic, linear elasticity (Hooke's law).

Mathematical modeling for developmental processes.

We review several mathematical models and concepts in developmental biology that have been established over the last decade. (1) Feedback vertex set: Ascidian embryos contain cells of seven types, and cell fate is controlled by ~100 interacting genes. The "feedback vertex set" of the directed graph of the gene regulatory network consists of a small number of genes. By experimentally manipulating them, we can differentiate cells into any cell type. (2) Tissue deformation: Describing morphological changes in tissues and relating them to gene expression and other cellular processes is key in understanding morphogenesis. Expansion and anisotropy of the tissue are described by a "deformation tensor" at each location. A study on chick limb bud formation revealed that both the volume growth rate and anisotropy in deformation differed significantly between locations and stages. (3) Mechanobiology: Forces operating on each cell may alter cell shape and gene expression, which may subsequently exert forces on their surroundings. Measurements of force, tissue shape, and gene expression help us understand autonomous tissue deformation. (4) Adaptive design of development: An optimal growth schedule in fluctuating environments explains the growth response to starvation in Drosophila larvae. Adaptive placement of morphogen sources makes development robust to noises.

Large deformation analysis of a hyperplastic beam using experimental / FEM/ meshless collocation method

In the present study, nonlinear numerical and experimental static bending analyses of hyperelastic beams are surveyed. Timoshenko's beam theory is used to obtain the Cauchy-Green deformation tensor. The governing equations are extracted using the neo-Hookean strain energy and Euler–Lagrange relation. Afterward, the nonlinear governing equations are solved numerically using the meshless collocation method (MCM). To validate the present investigation, a comparison study is accomplished between the acquired results from MCM and the experimental test of digital image correlation (DIC) and finite element method (FEM). The hyperelastic beam is made of natural rubber. Also, loading and boundary conditions are considered as concentrated force and clamped–clamped, respectively. In addition, the displacement contour is extracted by the DIC. The uniaxial tensile test based on the ASTM D412 standard is performed to extract the mechanical properties of natural rubber. The hyperelastic theory and neo-Hookean strain energy function are utilized to model the nonlinear behavior of rubber in the FEM. The error for the maximum displacement of system obtained from both numerical methods is less than 12% compared to the DIC. Additionally, it can be observed that in large deformations, unlike the FEM, MCM is able to predict the system's response with high accuracy.

An interface preserving and residual-based adaptivity for phase-field modeling of fully Eulerian fluid-structure interaction

In this paper, we present a novel interface preserving and residual-based adaptivity procedure for the phase-field modeling of a fully Eulerian fluid-structure interaction. The proposed adaptive fully Eulerian procedure involves a fixed background unstructured finite element mesh on which the fluid-structure boundary is treated implicitly via a diffused interface description. We employ the stabilized finite element formulation and a partitioned iterative procedure to solve the coupled system of the Allen-Cahn phase-field equation with the unified momentum equation comprising solid and fluid dynamics. To evaluate the solid stresses, the left Cauchy-Green deformation tensor is convected at each time step to trace the evolution of the solid strain in the Eulerian reference frame. We utilize a residual based error indicator and the newest vertex bisection algorithm for the adaptive refinement/coarsening of the unstructured mesh. Through the adaptive finite element method, we aim to preserve the profile of the diffused interface in the presence of convective distortion. We propose an early stopping criterion for the adaptivity of the interface dynamics by combining the effects of the phase-field interface thickness parameter and the mesh resolution. In addition, we control the size of the refined/coarsened elements by introducing a characteristic length scale governed by a weighted harmonic mean of the desired bulk and the interface mesh resolutions. The proposed stopping criterion and the characteristic length scale provide significant reductions in the computational cost while simultaneously ensuring convergence properties of the coupled fluid-structure system. We perform a detailed convergence and accuracy analysis via two benchmark problems namely, the pure solid system and a coupled fluid-solid system with an interface in a rectangular domain. We further provide a comparative study between feature- and residual based adaptive strategies. We systematically assess the performance of the adaptive procedure in terms of conservation properties and computational efficiency. Finally, we demonstrate our fully Eulerian adaptive solver to simulate the contact and bouncing phenomenon between an elastic solid and a rigid wall.

Folding Dynamics and Its Intermittency in Turbulence.

Fluid elements deform in turbulence by stretching and folding. In this Letter, by projecting the material deformation tensor onto the largest stretching direction, we depict the dynamics of folding through the evolution of the material curvature. Results from direct numerical simulation (DNS) show that the curvature growth exhibits two regimes: first, a linear stage dominated by folding fluid elements through a persistent velocity Hessian that then transition to an exponential-growth stage driven by the stretching of already strongly bent fluid elements. This transition leads to strong curvature intermittency at later stages, which can be explained by a proposed curvature-evolution model. The link between velocity Hessian to folding provides a new way to understand the crucial steps in energy cascade and mixing in turbulence beyond the classical linear description of stretching dynamics.

Universal rate-dependence in electro-magneto-active polymeric composites

Universal relations are mathematical statements for any material in a given class; the larger the class, the more powerful the universal rule. Such generic relations are to be likely explored in particle-reinforced smart polymeric composites for all possible energy functions and boundary traction forms. Such relations have mainly been developed for an isotropic material class after Ericksen’s seminal work. The present work develops an experimentally validated novel class of coupled universal relations for electro-magneto-active (EMA) polymeric composites under electromagnetic field interaction accounting for the rate-dependent inelastic effect. In such a material class, unless the preferred directions of the material are specified, there would be no general solutions. An EMA polymeric composite material class is of co-axial type if and only if the Cauchy stress (S) and deformation tensor (b) are co-axial for all types of deformation which is preserved in isotropic material class. Co-axiality in the transversely isotropic material class, on the other hand, does not follow the same rule, i.e. (Sb≠bS). Real-world materials, such as soft tissues and smart polymeric composites, do not exhibit co-axiality and instead reveal the dissipation mechanism, necessitating the current study. The proposed theory unifies the pseudo transverse isotropy and rate-dependent phenomena for EMV smart composites in the context of particle reinforcement. Additionally, the current study conducts an experimental investigation to confirm the applicability of the proposed universal relation.

An elastic–viscoplastic model with non-affine deformation and rotation of a distribution of embedded fibres

A large deformation continuum model for an elastically isotropic visco-plastic matrix reinforced by a distribution of fibres has been formulated. The fibre distribution is a symmetric positive-definite tensor with unit trace that is determined by an evolution equation. This evolution equation includes a competition between tendencies for affine evolution of the distribution and non-affine evolution causing the distribution to align with the elastic distortional deformation tensor or to become isotropic. Two elastic deformation measures are introduced associated with the fibre distribution. Specifically, non-affine measures of elastic distribution stretch and non-coaxiality of the distribution cause anisotropic stress response. The resulting model describes a number of effects relevant to growth and remodelling of biological tissues as well as to additive manufacturing of artificial materials. For example, the equations model the smooth transition from isotropic to anisotropic response seen in many biological tissues.

Spatially localized nonlinear magnetoelastic waves in an electrically conductive micropolar medium

AbstractA nonlinear model of an electrically conductive micropolar medium interacting with an external magnetic field is proposed. The deformable state of such a medium is described by two asymmetric tensors: tensor of deformations and bending‐torsion tensor. Both tensors consider both linear and nonlinear terms in rotation and displacement gradients (geometric nonlinearity). The components of the bending‐torsion tensor which have the same indices, describe torsional deformations, while the rest describe bending deformations. The stress state of the medium is described by two asymmetric tensors: stresses tensor and moment stresses tensor. It is assumed, as is customary in magnetoelasticity, that the action of an electromagnetic field on a strain field occurs through Lorentz forces. From the system of Maxwell's equations follow the equations for electric and magnetic induction, which together with electromagnetic equations of state, should be added to the group of equations which describe the dynamics of a micropolar medium. In the frames of the proposed model, a one‐dimensional nonlinear magnetoelastic shear‐rotation wave is considered. In the equations of dynamics, a nonlinear term is considered. This term makes the most significant contribution to wave processes. It is shown that two factors will influence the wave propagation: dispersion and nonlinearity. Nonlinearity leads to the emergence of new harmonics in the wave, which contributes to the appearance of sharp drops in the moving wave profile. Dispersion, on the contrary, smoothed out differences due to differences in phase velocities of harmonic component waves. The combined action of these factors can lead to the formation of stationary waves that propagate at a constant speed without changing shape. Only those cases where a constant component is absent in the deformation wave are physically realizable. Stationary waves can be either periodic or aperiodic. The latter are spatially localized waves—solitons. It is shown that the behavior of “subsonic” and “supersonic” solitons will be qualitatively different.