Terms Of Stress Tensor Research Articles

Computational aeroacoustics and phase-conjugation technique for localizing flow-induced sources

This paper presents a framework comprising computational aeroacoustics (CAA) solvers and a numerical phase-conjugation technique to localize flow-induced noise sources generated by bodies immersed in low Mach number flows. Two test-cases were considered, namely, flow over a two-dimensional (a) circular cylinder at Reynolds number Re =150 and Mach number M = 0.2 and (b) NACA-0012 airfoil at Re = 5000 and angle of attack (AoA) α = 5 deg. The CAA simulations were carried out in OpenFOAM wherein the two-dimensional unsteady, incompressible Navier–Stokes equations was solved using either the pressure-implicit splitting of operators (PISO) algorithm or large eddy simulation (LES). The near-field aerodynamic sources were extracted in terms of the unsteady Lighthill stress tensor terms, following which the far-field acoustic data were computed on boundary nodes using different methods which include the numerical solution of the Lighthill equation, Curle’s analogy, and Ffowcs Williams and Hawkings (FWH) analogy as well as linearized Euler equations (LEE). Next, the frequency-domain phase-conjugation (PC) method was implemented in the finite-element (FE) based COMSOL software to reconstruct the radiated acoustic pressure field, whereby the dipole source location was readily identified by pair of focal spots at the cylinder or at the airfoil trailing-edge (TE).

Real-space density kernel method for Kohn–Sham density functional theory calculations at high temperature

Kohn-Sham density functional theory calculations using conventional diagonalization based methods become increasingly expensive as temperature increases due to the need to compute increasing numbers of partially occupied states. We present a density matrix based method for Kohn-Sham calculations at high temperatures that eliminates the need for diagonalization entirely, thus reducing the cost of such calculations significantly. Specifically, we develop real-space expressions for the electron density, electronic free energy, Hellmann-Feynman forces, and Hellmann-Feynman stress tensor in terms of an orthonormal auxiliary orbital basis and its density kernel transform, the density kernel being the matrix representation of the density operator in the auxiliary basis. Using Chebyshev filtering to generate the auxiliary basis, we next develop an approach akin to Clenshaw-Curtis spectral quadrature to calculate the individual columns of the density kernel based on the Fermi operator expansion in Chebyshev polynomials and employ a similar approach to evaluate band structure and entropic energy components. We implement the proposed formulation in the SPARC electronic structure code, using which we show systematic convergence of the aforementioned quantities to exact diagonalization results, and obtain significant speedups relative to conventional diagonalization based methods. Finally, we employ the new method to compute the self-diffusion coefficient and viscosity of aluminum at 116 045K from Kohn-Sham quantum molecular dynamics, where we find agreement with previous more approximate orbital-free density functional methods.

Forced Axisymmetric Vibrations of an Electrically Excited Hollow Sphere Made of a Continuously Inhomogeneous Piezoceramic Material*

The forced axisymmetric vibrations of a hollow sphere made of a functionally graded radially polarized piezocelectric material are studied using the spatial theory of electroelasticity. The properties of the material vary continuously as a power function along the radial coordinate. The vibrations are excited by applying a varying electrostatic potential to the sphere surface. Solid and split electrodes are considered. After separation of variables and expression of the components of displacement and electric-flux density, electrostatic potential, and the stress tensor in terms of spherical functions, the original three-dimensional problem is reduced to a boundary-value problem for ordinary differential equations. The problem is solved by the effective method of discrete orthogonalization. The numerical solution is analyzed. Particularly, the distributions of the mechanical and electric parameters in the first modes of forced electroelastic vibrations of the sphere made of homogeneous and inhomogeneous piezoceramic materials are compared and analyzed. The effect of the loading caused by applying a variable electrostatic potential to the sphere surface on the distribution of forced vibration characteristics is studied.

Versatile mixed methods for the incompressible Navier–Stokes equations

In the spirit of the “Principle of Equipresence” introduced by Truesdell & Toupin, The Classical Field Theories (1960), we use the full version of the viscous stress tensor ν∇u+∇uT−23∇⋅uI which was originally derived for compressible flows, instead of the classical incompressible stress tensor ν∇u. (Note that, here ν is the dynamic viscosity coefficient, and u is the velocity field.) In our approach, the divergence-free constraint for the viscous stress term is not enforced ahead of discretization. Instead, our formulation allows the scheme itself to “choose” a consistent way to interpret the divergence-free constraint: i.e., the divergence-free constraint is interpreted (or enforced) in a consistent fashion in both the mass conservation equation and the stress tensor term (in the momentum equation). Furthermore, our approach preserves the original symmetrical properties of the stress tensor, e.g. its rotational invariance, and it remains physically correct in the context of compressible flows. As a result, our approach facilitates versatility and code reuse. In this paper, we introduce our approach and establish some important mathematical properties for the resulting class of finite element schemes. More precisely, for general mixed methods, which are not necessarily pointwise divergence-free, we establish the existence of a new norm induced by the full, viscous bilinear form. Thereafter, we prove the coercivity of the viscous bilinear form and the semi-coercivity of a convective trilinear form. In addition, we demonstrate L2-stability of the discrete velocity fields for the general class of methods and (by deduction) the H(div)-conforming methods. Finally, we run some numerical experiments to illustrate the behavior of the versatile mixed methods, and we make careful comparisons with a conventional H(div)-conforming scheme.

Anisotropic decay and global well-posedness of viscous surface waves without surface tension

We consider a viscous incompressible fluid below the air and above a fixed bottom. The fluid dynamics is governed by the gravity-driven incompressible Navier-Stokes equations, and the effect of surface tension is neglected on the free surface. The global well-posedness and long-time behavior of solutions near equilibrium have been intriguing questions since Beale (1981) [2]. It was proved by Guo and Tice (2013) [6] that with certain additional low horizontal frequency assumption of the initial data in 3D an integrable decay rate of the velocity is obtained so that the global unique solution can be constructed, while the global well-posedness in 2D was left open. In this paper, by exploiting the anisotropic decay rates of the velocity, which are even not integrable, we prove the global well-posedness in both 2D and 3D, without any low frequency assumption of the initial data. One of key observations here is a cancellation in nonlinear estimates of the viscous stress tensor term in the bulk by using Alinhac good unknowns, when estimating the energy evolution of the highest order horizontal spatial derivatives of the solution.

A visco-elastic-plastic constitutive model of isotropic magneto-sensitive rubber with amplitude, frequency and magnetic dependency

A three-dimensional visco-elastic-plastic constitutive model of isotropic magneto-sensitive (MS) rubber with amplitude, frequency and magnetic dependency under a continuum constitutive framework is developed. MS rubber is a kind of smart material mainly composed of a rubber matrix and magnetizable particles. Under a magnetic field, there is an increase of its dynamic modulus due to the interaction between the magnetic field and MS rubber, which is often referred to as the magnetic dependency of MS rubber. Experimental results reveal that besides the magnetic dependency, there is a frequency and amplitude dependency of its dynamic modulus. In specific, the modulus of MS rubber increases with increasing frequency and dereases with increasing strain amplitude. To depict the above properties and to consider the balance of energy in continuum mechanics framework, a new constitutive model consisting of a viscoelastic fractional derivative element, a bounding surface model in series with a neo-Hookean elastic model with magnetic sensitivity and a magnetic stress tensor term for MS rubber is proposed. The contribution of this constitutive model is that by using a free energy based method, with only eight material parameters, the amplitude, frequency and magnetic dependency of MS rubber can be reflected. After parameter identification, the simulation results show a good agreement with those of measurements. Therefore, by utilizing the model proposed, the ability of approaching the dynamic behavior of MS rubber-based vibration reduction devices in the design phase is possible which contributes to the application of MS rubber in noise and vibration reduction area.

Superluminal chaos after a quantum quench

Thermal states holographically dual to black holes in Einstein gravity display maximal Lyapunov growth as well as “butterfly effect cones”. We study these effects in highly non-equilibrium states, obtained from an initial thermal state by the sudden injection of energy. We do this by computing out-of-time-order correlators (OTOCs) in BTZ-Vaidya spacetimes, which describe transitions between black holes at different temperatures. If both pairs of boundary operators appearing in the OTOC are inserted before the energy injection, we recover standard results, with butterfly effect cones displaying a light-cone structure. But when one pair of operators is inserted before and the other pair after the energy injection, the Lyapunov growth saturates the chaos bounds set by the local temperatures and the butterfly effect cones can “open up”, becoming superluminal, albeit in a way that does not violate causality. In the limiting case, in which the initial state is the vacuum, Lyapunov growth only starts after the energy injection. Our computations of the OTOCs are phrased in terms of gravitationally interacting particles, where fields are treated in a geodesic approximation and the eikonal phase shift is expressed in terms of stress tensors and shock waves associated to geodesics.

Investigation on turbulence in the vicinity of liquid-liquid interfaces – Large eddy simulation and PIV experiment

In this work, the turbulence in the vicinity of the liquid-liquid interfaces in an incompressible interfacial flow is investigated numerically and experimentally. A new large eddy simulation formulation is employed where all the subgrid scale (SGS) terms appearing in the filtered governing equations are closed by a volume of fluid-based Approximate Deconvolution Model (ADM-VOF) (Saeedipour et al., 2018). In this structural turbulence modelling approach, the SGS terms of stress tensor and surface tension in the Navier-Stokes equations, as well as the interface dynamics in the VOF transport equation, are mathematically reconstructed from the resolved scales. An oil-water benchmark experiment was designed for validation purposes where an oil layer is poured on the top of the water into which a submerged turbulent jet is injected and interacts with the oil–water interface. The interactions of the turbulence with the liquid-liquid interfaces are then studied by shadowgraphy visualization and time-resolved PIV measurement. Two sets of LES were performed using the ADM-VOF and a conventional LES model based on eddy-viscosity approach. The statistics of interfacial turbulence including velocity fluctuations and interface dynamics are analyzed and compared with the ones obtained from the experiment. The results reveal that in the interface vicinity the ADM-VOF approach, which reconstructs more turbulent scales and includes the SGS terms of turbulence-interface interactions, is of superior performance in predicting the energy spectra and statistics of turbulence. Far from the interface no particular difference between the LES models was observed. In addition, the temporal average of the interface motion as well as the evolution of the interfacial area is studied. While both models predict the averaged temporal evolution of the interface similarly, the ADM-VOF revealed its potentials for high fluctuating regions as it accounts for the effect of SGS term interface dynamics. This study provides a perspective for connecting the interfacial turbulence to metallurgical aspects of the continuous casting process.

Bulk phase shift, CFT Regge limit and Einstein gravity

The bulk phase shift, related to a CFT four-point function, describes two-to-two scattering at fixed impact parameter in the dual AdS spacetime. We describe its properties for a generic CFT and then focus on large N CFTs with classical bulk duals. We compute the bulk phase shift for vector operators using Regge theory. We use causality and unitarity to put bounds on the bulk phase shift. The resulting constraints bound three-point functions of two vector operators and the stress tensor in terms of the gap o the theory. Similar bounds should hold for any spinning operator in a CFT. Holographically this implies that in a classical gravitational theory any non-minimal coupling to the graviton, as well as any other particle with spin greater than or equal to two, is suppressed by the mass of higher spin particles.

{\em A POSTERIORI} ERROR ESTIMATION FOR A DUAL MIXED FINITE ELEMENT METHOD FOR QUASI--NEWTONIAN FLOWS WHOSE VISCOSITY OBEYS A POWER LAW OR CARREAU LAW

A dual mixed finite element method, for quasi–Newtonian fluid flow obeying the power law or the Carreau law, is constructed and analyzed in Farhloul–Zine [13]. This mixed formulation possesses good local (i.e., at element level) conservation properties (conservation of the momentum and the mass) as in the finite volume methods. In Farhloul–Zine [12], we developed an a posteriori error analysis for a non–Newtonian fluid flow problems. The analysis is based on the fact that the equation describing the extra–stress tensor in terms of the rate of strain tensor is invertible and may give the rate of strain tensor as a function of the stress tensor. To free ourselves from this constraint of inversion of laws, and as a generalization of the obtained results in [12], we propose in this work an a posteriori error analysis to this mixed formulation.

A Framework for Characterizing Structural Uncertainty in Large-Eddy Simulation Closures

Motivated by the sizable increase of available computing resources, large-eddy simulation of complex turbulent flow is becoming increasingly popular. The underlying filtering operation of this approach enables to represent only large-scale motions. However, the small-scale fluctuations and their effects on the resolved flow field require additional modeling. As a consequence, the assumptions made in the closure formulations become potential sources of incertitude that can impact the quantities of interest. The objective of this work is to introduce a framework for the systematic estimation of structural uncertainty in large-eddy simulation closures. In particular, the methodology proposed is independent of the initial model form, computationally efficient, and suitable to general flow solvers. The approach is based on introducing controlled perturbations to the turbulent stress tensor in terms of magnitude, shape and orientation, such that propagation of their effects can be assessed. The framework is rigorously described, and physically plausible bounds for the perturbations are proposed. As a means to test its performance, a comprehensive set of numerical experiments are reported for which physical interpretation of the deviations in the quantities of interest are discussed.

Influence of Tidal Forces on the Triggering of Seismic Events

Tidal stresses are generated in any three-dimensional body influenced by an external inhomogeneous gravity field of rotating planets or moons. In this paper, as a special case, stresses caused within the solid Earth by the body tides are discussed from viewpoint of their influence on seismic activity. The earthquake triggering effects of the Moon and Sun are usually investigated by statistical comparison of tidal variations and temporal distribution of earthquake activity, or with the use of mathematical or experimental modelling of physical processes in earthquake prone structures. In this study, the magnitude of the lunisolar stress tensor in terms of its components along the latitude of the spherical surface of the Earth as well as inside the Earth (up to the core-mantle boundary) were calculated for the PREM (Dziewonski and Anderson in Phys Earth Planet Inter 25(4):297–356, 1981). Results of calculations prove that stress increases as a function of depth reaching a value around some kPa at the depth of 900–1500 km, well below the zone of deep earthquakes. At the depth of the overwhelming part of seismic energy accumulation (around 50 km) the stresses of lunisolar origin are only (0.0–1.0).103 Pa. Despite the fact that these values are much smaller than the earthquake stress drops (1–30 MPa) (Kanamori in Annu Rev Earth Planet Sci 22:207–237, 1994) this does not exclude the possibility of an impact of tidal forces on outbreak of seismic events. Since the tidal potential and its derivatives are coordinate dependent and the zonal, tesseral and sectorial tides have different distributions from the surface down to the CMB, the lunisolar stress cannot influence the break-out of every seismological event in the same degree. The influencing lunisolar effect of the solid earth tides on earthquake occurrences is connected first of all with stress components acting parallel to the surface of the Earth. The influence of load tides is limited to the loaded area and its immediate vicinity.

Motion characteristics of bilayered extensible Timoshenko microbeams

The nonlinear motion characteristics of a bilayered Timoshenko microbeam is analysed taking into account all the translational (i.e. longitudinal and transverse) and rotational motions; the effect of size is included through use of the modified couple stress theory. Considering a continuous variation through the thickness for the displacement field, the geometrically nonlinear strain terms are developed. Relating these strain terms to stress terms, via use of constitutive relations, leads to a stress tensor in terms of the displacement field; the same is applied to the symmetric curvature tensor and the deviatoric part of the symmetric couple stress tensor by means of a constitutive relation incorporating size effects. The potential energy of the bilayered microbeam is formulated via the modified couple stress theory. The kinetic energy along with the work of an external dynamic force is formulated in terms of the displacement field of the bilayered microbeam. A dynamic equilibrium state is obtained via balancing the energy and work. Three-dimensional reduced-order models for the transverse, longitudinal, and rotational motions are obtained via Galerkin's method. These coupled models are solved via a continuation method coupled with direct time-integration. The resonant responses are constructed with special consideration to the effects of the length-scale parameter and the material percentage and thickness of each layer; a comparison is also made between the motion characteristics of the bilayered and monolayered microbeam.

A bi-projection method for Bingham type flows

We propose and study a new numerical scheme to compute the isothermal and unsteady flow of an incompressible viscoplastic Bingham medium. The main difficulty, for both theoretical and numerical approaches, is due to the non-differentiability of the plastic part of the stress tensor in regions where the rate-of-strain tensor vanishes. This is handled by reformulating the definition of the plastic stress tensor in terms of a projection. A new time scheme, based on the classical incremental projection method for the Newtonian Navier–Stokes equations, is proposed. The plastic tensor is treated implicitly in the first sub-step of the projection scheme and is computed by using a fixed point procedure. A pseudo-time relaxation is added into the Bingham projection whose effect is to ensure a geometric convergence of the fixed point algorithm. This is a key feature of the bi-projection scheme which provides a fast and accurate computation of the plastic tensor. Stability and error analyses of the numerical scheme are provided. The error induced by the pseudo-time relaxation term is controlled by a prescribed numerical parameter so that a first-order estimate of the time error is derived for the velocity field. A second-order cell-centred finite volume scheme on staggered grids is applied for the spatial discretisation. The scheme is assessed against previously published benchmark results for both Newtonian and Bingham flows in a two-dimensional lid-driven cavity for Reynolds number equals 1000. Moreover, the proposed numerical scheme is able to reproduce the fundamental property of cessation in finite time of a viscoplastic medium in the absence of any energy source term in the equations. For a fixed value (100) of the Bingham number, various numerical simulations for a range of Reynolds numbers up to 200000 were performed with the bi-projection scheme on a grid with 10242 mesh points. The effect of this (physical) parameter on the flow behaviour is discussed.

Polymer Fluid Dynamics: Continuum and Molecular Approaches.

To solve problems in polymer fluid dynamics, one needs the equations of continuity, motion, and energy. The last two equations contain the stress tensor and the heat-flux vector for the material. There are two ways to formulate the stress tensor: (a) One can write a continuum expression for the stress tensor in terms of kinematic tensors, or (b) one can select a molecular model that represents the polymer molecule and then develop an expression for the stress tensor from kinetic theory. The advantage of the kinetic theory approach is that one gets information about the relation between the molecular structure of the polymers and the rheological properties. We restrict the discussion primarily to the simplest stress tensor expressions or constitutive equations containing from two to four adjustable parameters, although we do indicate how these formulations may be extended to give more complicated expressions. We also explore how these simplest expressions are recovered as special cases of a more general framework, the Oldroyd 8-constant model. Studying the simplest models allows us to discover which types of empiricisms or molecular models seem to be worth investigating further. We also explore equivalences between continuum and molecular approaches. We restrict the discussion to several types of simple flows, such as shearing flows and extensional flows, which are of greatest importance in industrial operations. Furthermore, if these simple flows cannot be well described by continuum or molecular models, then it is not necessary to lavish time and energy to apply them to more complex flow problems.

Quantum focusing conjecture

We propose a universal inequality that unifies the Bousso bound with the classical focussing theorem. Given a surface $\sigma$ that need not lie on a horizon, we define a finite generalized entropy $S_\text{gen}$ as the area of $\sigma$ in Planck units, plus the von Neumann entropy of its exterior. Given a null congruence $N$ orthogonal to $\sigma$, the rate of change of $S_\text{gen}$ per unit area defines a quantum expansion. We conjecture that the quantum expansion cannot increase along $N$. This extends the notion of universal focussing to cases where quantum matter may violate the null energy condition. Integrating the conjecture yields a precise version of the Strominger-Thompson Quantum Bousso Bound. Applied to locally parallel light-rays, the conjecture implies a Quantum Null Energy Condition: a lower bound on the stress tensor in terms of the second derivative of the von Neumann entropy. We sketch a proof of this novel relation in quantum field theory.

Proof of the quantum null energy condition

We prove the Quantum Null Energy Condition (QNEC), a lower bound on the stress tensor in terms of the second variation in a null direction of the entropy of a region. The QNEC arose previously as a consequence of the Quantum Focussing Conjecture, a proposal about quantum gravity. The QNEC itself does not involve gravity, so a proof within quantum field theory is possible. Our proof is somewhat nontrivial, suggesting that there may be alternative formulations of quantum field theory that make the QNEC more manifest. Our proof applies to free and superrenormalizable bosonic field theories, and to any points that lie on stationary null surfaces. An example is Minkowski space, where any point $p$ and null vector $k^a$ define a null plane $N$ (a Rindler horizon). Given any codimension-2 surface $\Sigma$ that contains $p$ and lies on $N$, one can consider the von Neumann entropy $S_\text{out}$ of the quantum state restricted to one side of $\Sigma$. A second variation $S_\text{out}^{\prime\prime}$ can be defined by deforming $\Sigma$ along $N$, in a small neighborhood of $p$ with area $\cal A$. The QNEC states that $\langle T_{kk}(p) \rangle \ge \frac{\hbar}{2\pi} \lim_{{\cal A}\to 0}S_\text{out}^{ \prime\prime}/{\cal A}$.

A micromechanical study on the stress rotation in granular materials due to fabric evolution

In the present study, the non-coaxiality between the axes of principal stress and strain rate tensors is investigated from micromechanical point of view. Based on the so called stress–force–fabric (SFF) relationship, which describes the macro–micro relation, an expression is derived for the principal direction of stress tensor in terms of micromechanical parameters of the fabric. In general, the rotation of principal stress axis and accordingly, the non-coaxiality angle, are influenced by both the anisotropy coefficients and directions of anisotropy of the fabric characteristics. The derived macro–micro relationship was evaluated by performing DEM simulations of 2D specimens of aggregates. It was shown that the principal directions of anisotropy parameters are almost coincident for the assemblies containing circular particles or elongated angular particles with random distribution. In such case, the principal direction of stress tensor can be regarded as the average principal direction of anisotropy. However, when the aggregate with elongated particles has inherently-anisotropic fabric, a correct estimation of the stress angle rotation requires considering all the anisotropy parameters including both anisotropy coefficients and directions.

Rate Constitutive Theories of Order Zero in Lagrangian Description for Thermoelastic Solids

This article presents constitutive theories for the stress tensor and the heat vector for homogeneous, isotropic thermoelastic solids in Lagrangian description for finite deformation. The deforming solid is assumed to be in thermodynamic equilibrium during the evolution. Since conservation of mass, balance of momenta, and balance of energy are independent of the constitution of the matter, the second law of thermodynamics must form the basis for deriving the constitutive theories. We introduce the concept of rate constitutive theory and show that for thermoelastic solids the constitutive theories are, in fact, rate theories of order zero. These theories for stress tensor consider material derivative of order zero of the conjugate strain tensor as one of the argument tensors of the stress tensor established as a dependent variable in the constitutive theory. Generalization of this concept leading to higher order rate theories in Lagrangian description are considered in followup works [1, 2]. The conditions resulting from the entropy inequality in Helmholtz free energy density permit the derivation of constitutive theory for stress tensor in terms of conjugate strain tensor or material derivative of order zero of the conjugate strain tensor. In the work presented here, it is shown that, using the conditions resulting from the entropy inequality, the constitutive theory for the stress tensor can be derived using three approaches: (i) assuming Helmholtz free energy density to be a function of the invariants of the material derivative of order zero of the conjugate strain tensor and temperature θ and then using the conditions resulting from the entropy inequality; (ii) using the theory of generators and invariants; and (iii) expanding Helmholtz free energy density in the material derivative of order zero of the conjugate strain tensor using Taylor series about a known configuration and then using the condition resulting from entropy inequality. The constitutive theories resulting from these three approaches are compared for equivalence as well as their merits and shortcomings. It is shown that the constitutive theory for heat vector can be derived: (i) using the conditions resulting from the entropy inequality; and (ii) using the theory of generators and invariants. In this approach, the argument tensors of the heat vector determine the specific form of the resulting constitutive theory. In this article, we use second Piola–Kirchhoff stress tensor and Green’s strain tensor (material derivative of the Green’s strain tensor of order zero) as a conjugate pair.

Nonaxisymmetric electroelastic vibrations of a hollow sphere made of functionally gradient piezoelectric material

The nonaxisymmetric problem of natural vibrations of a hollow sphere made of functionally gradient piezoelectric material is solved based on 3D electroelasticity. The properties of the material change continuously along a radial coordinate according to an exponential law. The external surface of the sphere is free of tractions and either insulated or short-circuited by electrodes. After separation of variables and representation of the components of the displacements and of the stress tensor in terms of spherical functions, the initially three-dimensional problem is reduced to a boundary-value problem for the eigenvalues expressed by ordinary differential equations. This problem is solved by a stable discrete-orthogonalization technique in combination with a step-by-step search method with respect to the radial coordinate. Moreover, a numerical investigation is performed based on the algorithm used for solving the problem. In particular, we investigate the influence of the geometric and electric parameters on the frequency spectrum at the nonaxisymmetry of natural vibrations of an inhomogeneous piezoceramic thick-walled sphere.