Symmetric Stress Tensor Research Articles

A thermodynamical theory of gradient elastoplasticity with dislocation density tensor. I: Fundamentals

A thermodynamical theory of gradient elastoplasticity, including kinematic hardening, is developed by introducing the concept of dislocation density tensor. The theory is self-consistent and is based on two fundamental principles, the principle of increase of entropy and the maximal entropy production rate. Thermodynamically consistent constitutive equations for plastic stretching, plastic spin and back stress are rigorously derived. Also, an expression for the plastic spin is obtained from the constitutive equation of dislocation drift rate and an expression for the back stress is given as a balance equation expressing equilibrium between internal stress and microstress conjugate to the dislocation density tensor. Moreover, it is shown that the present gradient theory yields a symmetric stress tensor. Some generalities and utility of this theory are discussed and comparisons with other gradient theories are given.

Effects of symmetric and antisymmetric stress tensors on N- and α-dielectric relaxations

We present a generalized hydrodynamic model for dielectric relaxation to describe the effect of spatial inhomogeneities in the polarization fluctuations in viscoelastic fluids. The model is a generalization of the simple Debye relaxation equation with the addition of the gradient of the total stress tensor. The two known modes of relaxation, N- and α-absorption can be incorporated in the model semiquantitatively, since we do not know the actual parameters involved. The comparison between our results and the experimental response reported for the N- and α-modes in polymers, shows that the predicted intensities of the loss peaks are well represented. However, at lower and higher frequencies the predicted curves deviate from the experimental ones.

Exact evaluation of dissipation for elastic-damage model dynamics

Some considerations about the finite strains dynamic of dissipative elasto-damage models are presented here. The dynamics of the system are treated taking care of the conservative character of the applied time-stepping algorithm in the sense of preserving the continuum balance equations. These are the linear and angular momentum laws and the expanded power of stress in respect of the Clausius-Duhem inequality. The generalised mid-point rule for time integration and the use of a particular algorithmic form of the symmetric Piola-Kirchhoff stress tensor ensure the exact evaluation of the rate of the Helmoltz free-energy. This algorithm is known as energy-momentum method. The presented constitutive model is suited to describe polymers in finite deformations and with some accomodations appears as a way to investigate strain localization problems. Some simple examples, processed using an object-oriented f.e.m. software, show the effectiveness of the algorithm in this last sense and the character of the constitutive model.

Anisotropic Damage Effect Tensors for the Symmetrization of the Effective Stress Tensor

Based on the concept of the effective stress and on the description of anisotropic damage deformation within the framework of continuum damage mechanics, a fourth order damage effective tensor is properly defined. For a general state of deformation and damage, it is seen that the effective stress tensor is usually asymmetric. Its symmetrization is necessary for a continuum theory to be valid in the classical sense. In order to transform the current stress tensor to a symmetric effective stress tensor, a fourth order damage effect tensor should be defined such that it follows the rules of tensor algebra and maintains a physical description of damage. Moreover, an explicit expression of the damage effect tensor is of particular importance in order to obtain the constitutive relation in the damaged material. The damage effect tensor in this work is explicitly characterized in terms of a kinematic measure of damage through a second-order damage tensor. In this work, tensorial forms are used for the derivation of such a linear transformation tensor which is then converted to a matrix form.

System of equations describing the flow of viscous liquid and gas in a wide range of reynolds numbers

A mathematical model is proposed for describing stationary and various self-exciting nonstationary (including turbulent) flows of viscous liquid and gas. The model does not assume that the stressed state at the points of the medium is described by a symmetric stress tensor. The model thus allows for internal moments of elementary viscous liquid and gas volumes relative to their centers of mass and for unsymmetric strains.

The existence of a symmetric stress tensor in a non-local description of continuum mechanics

Among the foundations of continuum mechanics is the description of the constitutive forces in terms of a symmetric tensor. Noll showed that this is a consequence of the axiom of material frame indifference, what in turn means a local invariance of the system under the Euclidean group. Here we will prove, in order to obtain the same result, that the assumption of locality in this axiom is redundant. We model a non-local system by means of a virtual work principle. Under the global demand that the corresponding functional does not respond on rigid infinitesimal motions, we show the existence of a symmetric stress tensor as a local result.

Spin and statistics of gravitational anyons

We determine the spin and statistics of a particle in 2 + 1 dimensions coupled to linearized gravity with a Chern-Simons term. The model and the results are compared to those of the corresponding abelian vector theory. The statistics emerge from the effective particle-particle interaction, while the spin is computed using the symmetric stress tensor of the system. We find that the spin-statistics relation is satisfied. In the gravitational case we show that while stress tensors are never gauge invariant, the Poincaré generators are; in particular, despite subtleties peculiar to two spatial dimensions, there are no “superpotential” ambiguities in these generators.

Rock: A medium with internal energy sources and sinks. Report 2

1. The stressed state of rocks is characterized by four symmetric stress tensors: σ, p, t, and τ. The tensor σ defines average stresses at the macro level; the tensor p, the stresses in the cement; and the tensors t and τ define the stresses in the grains (microstresses). All the microstresses have a special energy interpretation. 2. Microstresses are related through compatibility conditions (2.20). 3. Microstresses determine the average stresses at the macrolevel according to expressions (2.23) or (2.24).

The motion of membranes in space-time

The equation of a hypersurface interacting with external fields in space-time is established. A class of models is based on the properties of a symmetric divergence-free distribution-valued stress tensor for the interacting system. The elastodynamic properties are encoded into the induced metric and shape tensor of the immersion together with invariants constructed from the Weingarten map. Two particular examples are discussed that are relevant to recent developments in the theories of relativistic extended particles.

Variational principles for incompressible material and residual work in finite deformation.

The present paper proposes variational principles in finite deformation for a metal alloy showing predominant plastic flow or nearly incompressible material in elastic range, in order to improve the accuracy of numerical analysis. The variational theorems are presented in total Lagrangian formulation in terms of second Piola-Kirchhoff symmetric stress tensor, which can be easily reduced in an updated Lagrangian one. Attention is also focused on the unified derivation of mixed variational principles for residual work related to unbalanced forces which have not been studied previously.

Subcritical, critical, and supercritical directions of stress-rate in finite rigid plasticity

During loading of elastic-plastic materials, it has been shown previously that the direction of the Lagrangian strain-rate tensor can be either subcritical, critical, or super-critical. These three situations generalize the three types of softening behavior (subcritical, critical, and normal) observed in uniaxial compression tests on rocks. For rigid-plastic materials, the direction of the strain-rate during loading is fixed by a constitutive equation, and, as shown below, it is now the direction of the symmetric Piola-Kirchhoff stress tensor that can be either subcritical, critical, or supercritical.

Fundamental Solutions of the Linear Elastostatics Equations of the Isotropic and Homogeneous Bodies with Microstructure Having a Symmetric Stress Tensor

AbstractIn this paper the linear theory of the elastostatics of the homogeneous and isotropic bodies with microstructure having a symmetric stress tensor is considered. The matrix of fundamental solutions ℐ(x, y) of the operator A(∂/∂x) governing this theory is deduced. The stress operator ℐ(∂/∂x, n(x)) is introduced and the expression of the matrix ℐ(∂/∂x, n(x)) Γ(x, y) is given. Finally, some properties of the matrix λ(x, y) = t(ℐ(∂/∂y, n(y)) ℐ(y, x) are given.

Methods of solving spatial problems of the mechanics of a deformable solid in terms of stresses

The formulation in /1/ of a quasistatic problem of the mechanics of a deformable solid in terms of stresses is discussed, including also the variational formulation, which consists of solving six equations in six symmetric stress tensor components when six boundary conditions are satisfied. Methods of successive approximation are proposed for solving this problem and theorems on the convergence of these methods, including a “rapidly converging” method, whose rate of convergence is substantially higher than a geometric progression, are proved.

Comments on ’’Elastic wave invariants for acoustic emission’’ [J. Acoust. Soc. Am. 70, 110–115 (1981)

A symmetric radiation stress tensor is presented for elastodynamics.

Kinetic theory of reacting systems

Abstract In this paper transport processes of reacting systems are investigated, based on the Boltzmann equations. The Boltzmann equations are solved by means of Grad's moment method to thirteen moments and some formal results are obtained for transport properties. It is shown that the rate coefficients are quadratic functions of hydrodynamic fluxes and are in the form where are the scalar moments associated with the reaction and q, J, Π are heat flux, material flux and traceless symmetric stress tensor. k(0)i is the usual local equilibrium formula for reaction rate constant. Iterative solutions for the equations of change for , q, J and Π are obtained from which transport coefficients are calculated for the reacting system. It is shown that the solutions, when specialized to nonreacting mixtures, lead to results for the transport coefficients which are exactly in agreement with the Chapman-Enskog theory results. The modifications of the transport coefficients due to reactions are obtained from the iterative solutions and the bracket integrals necessary for their calculations are explicitly given in an appendix.

Stress functions in three-dimensional elastodynamics

After a historical sketch concerning the introduction of potential functions in elastodynamics, and its basic equations, a representation in terms of stresses is given with the aid of six stress functions, which are components of a symmetric stress tensor, of the second order (Schaefer's tensor) and satisfy the equation of transverse wave propagation, and with the aid of another stress function. It is also shown that this representation is equivalent to the representations given byM. Iacovache [1] andE. Sternberg andR. A. Eubanks [2] and that it is complete (each state of displacement and stress corresponding to, the problem can be expressed in this form) for a simply connected region. The case of arbitrary body forces as well as the case of conservative body forces are also considered. The corresponding particular integrals are obtained in the form of distributions with the aid of integral transforms of the Laplace and Fourier type. The results can therefore be applied to the case of concentrated loads. In particular, the case of a concentrated force as well as the case of a concentrated nucleus of space dilatation are treated. The latter case can be expressed in terms of conservative body forces.

On certain boundary-value problems with strong vanishing on the boundary

In Newtonian statics of a continuum we have a symmetric stress tensor with six components and a body-force with three components, and the divergence of the stress tensor equals the body-force vector reversed. If the body-force vector as assigned in some finite domain I I with boundary B B , we have three equations to be satisfied by six stress components. The equations of equilibrium, coupled with conditions on B B , cannot determine the stress, but they do define a class of stress distributions, provided the body-force and the conditions on B B are consistent. The purpose of this paper is to show that, if the body-force satisfies the usual conditions of equilibrium and vanishes strongly on B B in the sense that this body-force, and all its derivatives up to order N N , vanish on B B , then there exists a stress distribution which also vanishes strongly on B B , the order of vanishing being greater by one than that of the body-force.

Symmetry of the stress tensor

The stress tensor for a medium with internal angular momentum is considered, and it is shown how a symmetric stress tensor can be formed. Construction of the symmetric tensor requires that the contribution to the linear momentum of a distribution of internal angular momentum be taken into account.

NUMERICAL SIMULATION OF SEISMIC DISTURBANCES

This paper presents the results from a Lagrangian numerical scheme of calculating cylindrically symmetric, elastic‐plastic, transient disturbances. The numerical technique involves transforming the Eulerian equations of motion containing a cylindrically symmetric stress tensor into Lagrangian coordinates. These transformed equations are then differenced in the Lagrangian coordinate system. Therefore, a given instantaneous stress field determines the accelerations of the points in the Lagrangian mesh. These accelerations are allowed to act over a small time‐step so that the mesh becomes distorted. This distortion determines the change in strain at a point; the strain change is related to a stress change by Hooke’s law. This new stress is used to determine new accelerations. If the material behaves plastically, then the stresses are adjusted so that a von Mises yield condition is satisfied. Five numerical solutions of various elastic‐plastic wave propagation problems are presented; these solutions agree extremely well with their corresponding analytic and experimental solutions. A shear and tensile failure mechanism is presented that is consistent with the continuum hypothesis. This mechanism gives good results when applied to NTS alluvium. Data for hard rock are not yet available.

A theory of elasticity with spatial distribution of matter. Three-dimensional complex structure

References [1 and 2] consider a theory of elasticity with spatial distribution of matter for a medium having simple structure and for a one-dimensional medium having complex structure. In the present article the general case of a three-dimensional medium with complex structure is examined. The general scheme of the one-dimensional case [2] is retained; chief attention is directed toward the specific character of the three-dimensional problem. The original micro-model is a complex crystal lattice [3]. In Section 1 this model is generalized to the case of a continuous distribution of matter. The displacements of the mass centers of the unit cells and the micro-strains of the cells are introduced as the kinematic variables. The force variables are the micro-moments. The transition to an exact continuous representation is carried out, and the equations of an elastic medium of complex structure with spatial distribution of matter are derived. The operators corresponding to the continuous theory are expressed in terms of the original microparameters. It is shown that the well known conditions of symmetry of the tensor of elastic constants, which are usually interpreted as the condition of absence of initial stresses [3 and 4], are consequences of the invariance of the elastic energy under translation and rotation. In Section 2 some special models are examined, and the equations of a medium are obtained for the approximation of weak dispersion of matter. These equations contain as a special case the equations of linear nonsymmetric elasticity (couple-stress theory) [5 to 7]. However, in the latter it turns out that the orders of approximation are inconsistent in the various equations from the point of view of the theory of spatial distribution. In Section 3 the equations of a medium having complex structure are transformed in the acoustic range into equations, one of which contains only a single kinematic variable (the displacement of the mass centers) and the others of which are explicitly solvable for the remaining kinematic variables. The first equation of this set coincides in form with the equation for a medium with simple structure, but differs from it by the presence of a timewise dispersion which is unrelated to energy dissipation. Expressions are written for the energy density, and it is shown that it is possible to introduce a symmetric stress tensor, as in the case of a simple structure.