Showing papers on "Cauchy stress tensor published in 2011"

On the predictive capabilities of the shear modified Gurson and the modified Mohr–Coulomb fracture models over a wide range of stress triaxialities and Lode angles

Abstract: The predictive capabilities of the shear-modified Gurson model [Nielsen and Tvergaard, Eng. Fract. Mech. 77, 2010] and the Modified Mohr–Coulomb (MMC) fracture model [Bai and Wierzbicki, Int. J. Fract. 161, 2010] are evaluated. Both phenomenological fracture models are physics-inspired and take the effect of the first and third stress tensor invariants into account in predicting the onset of ductile fracture. The MMC model is based on the assumption that the initiation of fracture is determined by a critical stress state, while the shear-modified Gurson model assumes void growth as the governing mechanism. Fracture experiments on TRIP-assisted steel sheets covering a wide range of stress states (from shear to equibiaxial tension) are used to calibrate and validate these models. The model accuracy is quantified based on the predictions of the displacement to fracture for experiments which have not been used for calibration. It is found that the MMC model predictions agree well with all experiments (less than 4% error), while less accurate predictions are observed for the shear-modified Gurson model. A comparison of plots of the strain to fracture as a function of the stress triaxiality and the normalized third invariant reveals significant differences between the two models except within the vicinity of stress states that have been used for calibration.

Anisotropic N=4 super-Yang-Mills plasma and its instabilities.

Abstract: We present a type-IIB supergravity solution dual to a spatially anisotropic finite-temperature $\mathcal{N}=4$ super-Yang-Mills plasma. The solution is static and completely regular. The full geometry can be viewed as a renormalization group flow from an ultraviolet anti--de Sitter geometry to an infrared Lifshitz-like geometry. The anisotropy can be equivalently understood as resulting from a position-dependent $\ensuremath{\theta}$ term or from a nonzero number density of dissolved $D7$-branes. The holographic stress tensor is conserved and anisotropic. The presence of a conformal anomaly plays an important role in the thermodynamics. The phase diagram exhibits homogeneous and inhomogeneous (i.e., mixed) phases. In some regions the homogeneous phase displays instabilities reminiscent of those of weakly coupled plasmas. We comment on similarities with QCD at finite baryon density and with the phenomenon of cavitation.

Thermodynamics and instabilities of a strongly coupled anisotropic plasma

Abstract: We extend our analysis of a IIB supergravity solution dual to a spatially anisotropic finite-temperature \( \mathcal{N} = 4 \) super Yang-Mills plasma. The solution is static, possesses an anisotropic horizon, and is completely regular. The full geometry can be viewed as a renormalization group flow from an AdS geometry in the ultraviolet to a Lifshitz-like geometry in the infrared. The anisotropy can be equivalently understood as resulting from a position-dependent θ-term or from a non-zero number density of dissolved D7-branes. The holographic stress tensor is conserved and anisotropic. The presence of a conformal anomaly plays an important role in the thermodynamics. The phase diagram exhibits homogeneous and inhomogeneous (i.e. mixed) phases. In some regions the homogeneous phase displays instabilities reminiscent of those of weakly coupled plasmas. We comment on similarities with QCD at finite baryon density and with the phenomenon of cavitation.

Nearly complete stress drop in the 2011 M w 9.0 off the Pacific coast of Tohoku Earthquake

Abstract: Temporal change in the stress field after the 2011 Tohoku earthquake was observed by stress tensor inversions of focal mechanisms of earthquakes near the source region. The maximum compressive stress (σ1) axis before the earthquake has a direction toward the plate convergence, dipping oceanward at an angle of 25–30 degrees. Its dip angle significantly increased by 30–35 degrees after the earthquake, and σ1 axis came to intersect with the plate interface at a high angle of about 80 degrees. By using the observed rotation of σ1 axis, we estimated the ratio of mainshock stress drop to the background deviatoric stress Δτ/τ to be 0.9–0.95. This shows that the deviatoric stress causing the Mw 9.0 earthquake was mostly released by the earthquake, or the stress drop during the earthquake was nearly complete. Adopting the average stress drop obtained by GPS observation data, the deviatoric stress magnitude is estimated to be 21–22 MPa. This suggests the plate interface is weak. The nearly complete stress drop caused a high dip angle of σ1 axis, which is the reason why not a small number of normal fault type aftershocks have occurred.

From Navier-Stokes To Einstein

Abstract: We show by explicit construction that for every solution of the incompressible Navier-Stokes equation in $p+1$ dimensions, there is a uniquely associated "dual" solution of the vacuum Einstein equations in $p+2$ dimensions. The dual geometry has an intrinsically flat timelike boundary segment $\Sigma_c$ whose extrinsic curvature is given by the stress tensor of the Navier-Stokes fluid. We consider a "near-horizon" limit in which $\Sigma_c$ becomes highly accelerated. The near-horizon expansion in gravity is shown to be mathematically equivalent to the hydrodynamic expansion in fluid dynamics, and the Einstein equation reduces to the incompressible Navier-Stokes equation. For $p=2$, we show that the full dual geometry is algebraically special Petrov type II. The construction is a mathematically precise realization of suggestions of a holographic duality relating fluids and horizons which began with the membrane paradigm in the 70's and resurfaced recently in studies of the AdS/CFT correspondence.

Thermodynamics and Instabilities of a Strongly Coupled Anisotropic Plasma

Abstract: We extend our analysis of a IIB supergravity solution dual to a spatially anisotropic finite-temperature N=4 super Yang-Mills plasma. The solution is static, possesses an anisotropic horizon, and is completely regular. The full geometry can be viewed as a renormalization group flow from an AdS geometry in the ultraviolet to a Lifshitz-like geometry in the infrared. The anisotropy can be equivalently understood as resulting from a position-dependent theta-term or from a non-zero number density of dissolved D7-branes. The holographic stress tensor is conserved and anisotropic. The presence of a conformal anomaly plays an important role in the thermodynamics. The phase diagram exhibits homogeneous and inhomogeneous (i.e. mixed) phases. In some regions the homogeneous phase displays instabilities reminiscent of those of weakly coupled plasmas. We comment on similarities with QCD at finite baryon density and with the phenomenon of cavitation.

A reduced integration solid-shell finite element based on the EAS and the ANS concept—Large deformation problems

Abstract: In this paper we address the extension of a recently proposed reduced integration eight-node solid-shell finite element to large deformations. The element requires only one integration point within the shell plane and at least two integration points over the thickness. The possibility to choose arbitrarily many Gauss points over the shell thickness enables a realistic and efficient modeling of the non-linear material behavior. Only one enhanced degree-of-freedom is needed to avoid volumetric and Poisson thickness locking. One key point of the formulation is the Taylor expansion of the inverse Jacobian matrix with respect to the element center leading to a very accurate modeling of arbitrary element shapes. The transverse shear and curvature thickness locking are cured by means of the assumed natural strain concept. Further crucial points are the Taylor expansion of the compatible cartesian strain with respect to the center of the element as well as the Taylor expansion of the second Piola–Kirchhoff stress tensor with respect to the normal through the center of the element. Copyright © 2010 John Wiley & Sons, Ltd.

Unsteady MHD natural convection from a heated vertical porous plate in a micropolar fluid with Joule heating, chemical reaction and radiation effects

Abstract: The effects of Joule-heating, chemical reaction and thermal radiation on unsteady MHD natural convection from a heated vertical porous plate in a micropolar fluid are analyzed. The partial differential equations governing the flow and heat and mass transfer have been solved numerically using an implicit finite-difference scheme. The case corresponding to vanishing of the anti-symmetric part of the stress tensor that represents weak concentrations is considered. The numerical results are validated by favorable comparisons with previously published results. A parametric study of the governing parameters, namely the magnetic field parameter, suction/injection parameter, radiation parameter, chemical reaction parameter, vortex viscosity parameter and the Eckert number on the linear velocity, angular velocity, temperature and the concentration profiles as well as the skin friction coefficient, wall couple stress coefficient, Nusselt number and the Sherwood number is conducted. A selected set of numerical results is presented graphically and discussed.

The holographic fluid dual to vacuum Einstein gravity

Abstract: We present an algorithm for systematically reconstructing a solution of the (d + 2)-dimensional vacuum Einstein equations from a (d + 1)-dimensional fluid, extending the non-relativistic hydrodynamic expansion of Bredberg et al. in arXiv:1101.2451 to arbitrary order. The fluid satisfies equations of motion which are the incompressible Navier-Stokes equations, corrected by specific higher-derivative terms. The uniqueness and regularity of this solution is established to all orders and explicit results are given for the bulk metric and the stress tensor of the dual fluid through fifth order in the hydrodynamic expansion. We establish the validity of a relativistic hydrodynamic description for the dual fluid, which has the unusual property of having a vanishing equilibrium energy density. The gravitational results are used to identify transport coefficients of the dual fluid, which also obeys an interesting and exact constraint on its stress tensor. We propose novel Lagrangian models which realise key properties of the holographic fluid.

On the capillary stress tensor in wet granular materials

Abstract: This paper presents a micromechanical study of unsaturated granular media in the pendular regime, based upon numerical experiments using the discrete element method, compared to a microstructural elastoplastic model. Water effects are taken into account by adding capillary menisci at contacts and their consequences in terms of force and water volume are studied. Simulations of triaxial compression tests are used to investigate both macro and micro-effects of a partial saturation. The results provided by the two methods appear to be in good agreement, reproducing the major trends of a partially saturated granular assembly, such as the increase in the shear strength and the hardening with suction. Moreover, a capillary stress tensor is exhibited from capillary forces by using homogenisation techniques. Both macroscopic and microscopic considerations emphasize an induced anisotropy of the capillary stress tensor in relation with the pore fluid distribution inside the material. In so far as the tensorial nature of this fluid fabric implies shear effects on the solid phase associated with suction, a comparison has been made with the standard equivalent pore pressure assumption. It is shown that water effects induce microstrural phenomena that cannot be considered at the macro level, particularly when dealing with material history. Thus, the study points out that unsaturated soil stress definitions should include, besides the macroscopic stresses such as the total stress, the microscopic interparticle stresses such as the ones resulting from capillary forces, in order to interpret more precisely the implications of the pore fluid on the mechanical behaviour of granular materials.

Dynamics of a large scale rigid–flexible multibody system composed of composite laminated plates

Abstract: A novel computational approach for the dynamic analysis of a large scale rigid–flexible multibody system composed of composite laminated plates is proposed. The rigid parts in the system are described through the Natural Coordinate Formulation (NCF) and the flexible bodies in the system are modeled via the finite elements of Absolute Nodal Coordinate Formulation (ANCF), which can lead to a constant mass matrix for the derived system equation of motion. For modeling composite laminated plates accurately, a new composite laminated plate element of ANCF is proposed and the corresponding efficient formulations for evaluating both the elastic force and its Jacobian of the element are derived from the first Piola–Kirchhoff stress tensor. To improve computational efficiency, the sparse matrix technology and graph theory are used to solve the huge set of linear algebraic equations in the process of integrating the equations of motion by using the generalized-a method, and an OpenMP based parallel scheme is also introduced. Finally, the effectiveness of the proposed approach is validated through two numerical examples. One is the static simulation of a single composite laminated plate under gravity and the other is the dynamic simulations of unfolding process of a satellite system with a pair of complicated antennas.

From discrete particles to continuum fields near a boundary

Abstract: An expression for the stress tensor near an external boundary of a discrete mechanical system is derived explicitly in terms of the constituents’ degrees of freedom and interaction forces. Starting point is the exact and general coarse graining formulation presented by Goldhirsch in [I.Goldhirsch, Gran.Mat., 12(3):239-252, 2010], which is consistent with the continuum equations everywhere but does not account for boundaries. Our extension accounts for the boundary interaction forces in a self-consistent way and thus allows the construction of continuous stress fields that obey the macroscopic conservation laws even within one coarse-graining width of the boundary. The resolution and shape of the coarse-graining function used in the formulation can be chosen freely, such that both microscopic and macroscopic effects can be studied. The method does not require temporal averaging and thus can be used to investigate time-dependent flows as well as static and steady situations. Finally, the fore-mentioned continuous field can be used to define ‘fuzzy’ (highly rough) boundaries. Two discrete particle method (DPM) simulations are presented in which the novel boundary treatment is exemplified, including a chute flow over a base with roughness greater than a particle diameter.

Tensor Triangular Geometry

Abstract: We survey tensor triangular geometry : Its examples, early theory and first applications. We also discuss perspectives and suggest some problems. Mathematics Subject Classification (2000). Primary 18E30; Secondary 14F05, 19G12, 19K35, 20C20, 53D37, 55P42.

Kubo formulas for second-order hydrodynamic coefficients.

Abstract: At second order in gradients, conformal relativistic hydrodynamics depends on the viscosity $\ensuremath{\eta}$ and on five additional ``second-order'' hydrodynamical coefficients ${\ensuremath{\tau}}_{\ensuremath{\Pi}}$, $\ensuremath{\kappa}$, ${\ensuremath{\lambda}}_{1}$, ${\ensuremath{\lambda}}_{2}$, and ${\ensuremath{\lambda}}_{3}$. We derive Kubo relations for these coefficients, relating them to equilibrium, fully retarded three-point correlation functions of the stress tensor. We show that the coefficient ${\ensuremath{\lambda}}_{3}$ can be evaluated directly by Euclidean means and does not in general vanish.

Generalized recursion relations for correlators in the gauge-gravity correspondence.

Abstract: We show that a generalization of the Britto-Cachazo-Feng-Witten recursion relations gives a new and efficient method of computing correlation functions of the stress tensor or conserved currents in conformal field theories with an ($d+1$)-dimensional anti--de Sitter space dual, for $d\ensuremath{\ge}4$, in the limit where the bulk theory is approximated by tree-level Yang-Mills theory or gravity. In supersymmetric theories, additional correlators of operators that live in the same multiplet as a conserved current or stress tensor can be computed by these means.

Holography for asymptotically locally Lifshitz spacetimes

Abstract: We give a definition of asymptotically locally Lifshitz spacetimes, with boundary data appropriate for a non-relativistic theory on the boundary. Solutions satisfying these boundary conditions are constructed in an asymptotic expansion. We identify the boundary data with sources for dual field theory operators, and give a prescription for calculating the one-point functions of the field theory operators (including the stress tensor) in the presence of arbitrary sources. The divergences in these one-point functions can be cancelled by holographic renormalization, adding counterterms which are local functions of the boundary data.

Holographic renormalization for asymptotically Lifshitz spacetimes

Abstract: A variational formulation is given for a theory of gravity coupled to a massive vector in four dimensions, with Asymptotically Lifshitz boundary conditions on the fields. For theories with critical exponent z = 2 we obtain a well-defined variational principle by explicitly constructing two actions with local boundary counterterms. As part of our analysis we obtain solutions of these theories on a neighborhood of spatial infinity, study the asymptotic symmetries, and consider different definitions of the boundary stress tensor and associated charges. A constraint on the boundary data for the fields figures prominently in one of our formulations, and in that case the only suitable definition of the boundary stress tensor is due to Hollands, Ishibashi, and Marolf. Their definition naturally emerges from our requirement of finiteness of the action under Hamilton-Jacobi variations of the fields. A second, more general variational principle also allows the Brown-York definition of a boundary stress tensor.

Equilibrium molecular dynamics studies on nanoscale-confined fluids

Abstract: Fluid behavior within nanoscale confinements is studied for argon in dilute gas, dense gas, and liquid states. Molecular dynamics simulations are used to resolve the density and stress variations within the static fluid. Normal stress calculations are based on the Irving–Kirkwood method, which divides the stress tensor into its kinetic and virial parts. The kinetic component recovers pressure based on the ideal-gas law. The particle–particle virial increases with increased density, whereas the surface–particle virial develops because of the surface-force field effects. Normal stresses within nanoscale confinements show anisotropy primarily induced by the surface-force field and local variations in the fluid density near the surfaces. For dilute and dense gas cases, surface-force field that extends typically 1 nm from each wall induces anisotropic normal stress. For liquid case, this effect is further amplified by the density fluctuations that extend beyond the force field penetration region. Outside the wall-force field penetration and density fluctuation regions, the normal stress becomes isotropic and recovers the thermodynamic pressure, provided that sufficiently large force cut-off distances are used in the computations.

The critical-state yield stress (termination locus) of adhesive powders from a single numerical experiment

Abstract: Dry granular materials in a split-bottom ring shear cell geometry show wide shear bands under slow, quasi-static, large deformation. This system is studied in the presence of contact adhesion, using the discrete element method (DEM). Several continuum fields like the density, the deformation gradient and the stress tensor are computed locally and are analyzed with the goal to formulate objective constitutive relations for the flow behavior of cohesive powders. From a single simulation only, by applying time- and (local) space-averaging, and focusing on the regions of the system that experienced considerable deformations, the critical-state yield stress (termination locus) can be obtained. It is close to linear, for non-cohesive granular materials, and nonlinear with peculiar pressure dependence, for adhesive powders—due to the nonlinear dependence of the contact adhesion on the confining forces. The contact model is simplified and possibly will need refinements and additional effects in order to resemble realistic powders. However, the promising method of how to obtain a critical-state yield stress from a single numerical test of one material is generally applicable and waits for calibration and validation.

Nonmodal amplification of stochastic disturbances in strongly elastic channel flows

Abstract: Nonmodal amplification of stochastic disturbances in elasticity-dominated channel flows of Oldroyd-B fluids is analyzed in this work. For streamwise-constant flows with high elasticity numbers μ and finite Weissenberg numbers We, we show that the linearized dynamics can be decomposed into slow and fast subsystems, and establish analytically that the steady-state variances of velocity and polymer stress fluctuations scale as O ( We 2 ) and O ( We 4 ) , respectively. This demonstrates that large velocity variance can be sustained even in weakly inertial stochastically driven channel flows of viscoelastic fluids. We further show that the wall-normal and spanwise forces have the strongest impact on the flow fluctuations, and that the influence of these forces is largest on fluctuations in the streamwise velocity and the streamwise component of the polymer stress tensor. The underlying physical mechanism involves polymer stretching that introduces a lift-up of flow fluctuations similar to vortex tilting in inertia-dominated flows. The validity of our analytical results is confirmed in stochastic simulations. The phenomenon examined here provides a possible route for the early stages of a bypass transition to elastic turbulence and might be exploited to enhance mixing in microfluidic devices.

On a non-isothermal model for nematic liquid crystals

Abstract: A model describing the evolution of a liquid crystal substance in the nematic phase is investigated in terms of three basic state variables: the absolute temperature , the velocity field u and the director field d, representing preferred orientation of molecules in a neighbourhood of any point of a reference domain. The time evolution of the velocity field is governed by the incompressible Navier–Stokes system, with a non-isotropic stress tensor depending on the gradients of the velocity and of the director field d, where the transport (viscosity) coefficients vary with temperature. The dynamics of d is described by means of a parabolic equation of Ginzburg–Landau type, with a suitable penalization term to relax the constraint |d| = 1. The system is supplemented by a heat equation, where the heat flux is given by a variant of Fourier's law, depending also on the director field d. The proposed model is shown to be compatible with first and second laws of thermodynamics, and the existence of global-in-time weak solutions for the resulting PDE system is established, without any essential restriction on the size of the data.

Pointing the way to the products? Comparison of the stress tensor and the second-derivative tensor of the electron density

Abstract: The eigenvectors of the electronic stress tensor can be used to identify where new bond paths form in a chemical reaction. In cases where the eigenvectors of the stress tensor are not available, the gradient-expansion-approximation suggests using the eigenvalues of the second derivative tensor of the electron density instead; this approximation can be made quantitatively accurate by scaling and shifting the second-derivative tensor, but it has a weaker physical basis and less predictive power for chemical reactivity than the stress tensor. These tools provide an extension of the quantum theory of atoms and molecules from the characterization of molecular electronic structure to the prediction of chemical reactivity.

Boundary Regularity of Shear Thickening Flows

Abstract: This article is concerned with the global regularity of weak solutions to systems describing the flow of shear thickening fluids under the homogeneous Dirichlet boundary condition. The extra stress tensor is given by a power law ansatz with shear exponent p≥ 2. We show that, if the data of the problem are smooth enough, the solution u of the steady generalized Stokes problem belongs to $${W^{1,(np+2-p)/(n-2)}(\Omega)}$$ . We use the method of tangential translations and reconstruct the regularity in the normal direction from the system, together with anisotropic embedding theorem. Corresponding results for the steady and unsteady generalized Navier–Stokes problem are also formulated.