Showing papers on "Cauchy stress tensor published in 1990"
TL;DR: In this paper, a method for determining the reduced stress tensor with four degrees of freedom (the orientations of the three principal stress axes as well as the ratio of principal stress differences) using fault slip data (or focal mechanisms of earthquakes) is presented.
Abstract: SUMMARY
A new method for determining the reduced stress tensor with four degrees of freedom (the orientations of the three principal stress axes as well as the ratio of principal stress differences) using fault slip data (or focal mechanisms of earthquakes) is presented. From a computational point of view, the inversion of fault slip data is made in a direct way by purely analytical means; as a result, the determination process is extremely fast and adaptable on small microcomputers. From a physical point of view, the method aims at simultaneously (i) minimizing the angles between theoretical shear stress and actual slip vector and (ii) having relative magnitudes of shear stress large enough to induce slip despite rock cohesion and friction. Examples of application to actual fault slip data sets with good or poor variety of fault slip orientations are shown. The double significance of the basic criterion adopted results in a more realistic solution of the inverse problem than the single minimization of the shear-stria angle.
649 citations
TL;DR: In this paper, a 3D analysis of a semi-infinite crack embedded in a transversely isotropic piezoelectric material was performed by means of the eigenfunction expansions method.
257 citations
TL;DR: Program FMSI inverts observations of slip directions on fault planes of known orientation in order to determine the best-fitting four-parameter stress tensor, defined by three principal stress directions and R=(σ2 − σ1)/(σ3 − ρ1), and the associated uncertainty.
209 citations
TL;DR: In this article, the authors present simple analytical and graphical descriptions of the field of admissible fault geometries relative to any four-parameter stress model, which can be used to illustrate the significance of various inverse strategies.
Abstract: The shear stress direction on a fault plane depends only on four of the six components of the stress tensor. Assuming only that the slip direction marks the shear stress direction on any fault plane (and that stress is homogeneous), it is possible to estimate these four stress parameters from populations of fault planes with known slip directions, as several workers have observed. Different formulations of the problem may yield varying best-fitting stresses and estimates of uncertainty. In the simplest case, no assumptions are made regarding the orientations of fault planes relative to the stress tensor; thus the technique allows for the possibility that the fault planes may be very weak. Here we present simple analytical and graphical descriptions of the field of admissible fault geometries relative to any four-parameter stress model, which can be used to illustrate the significance of various inverse strategies. In particular, this paper explores the effects of using two alternative measures of misfit between an observed fault datum and stress model: (1) the pole rotation (the angle between the observed and predicted slip direction on the observed fault plane), and (2) the minimum rotation (the smallest angle between the observed fault geometry and any fault geometry which is consistent with the model). By allowing for variation of the fault plane as well as the slip vector, the minimum rotation procedure generally achieves a more stable and (presumably) realistic estimate of the actual discrepancy between a fault observation and stress model than the pole rotation procedure. In a test case using 17 earthquake focal mechanisms from the YuIi region of eastern Taiwan, separate inversions based on the two misfit criteria yield different optimum stress models and uncertainty estimates. Additional constraints on the stress tensor, such as the effect of friction, can be superimposed on the ones used here.
193 citations
TL;DR: In this article, an algorithm for the simultaneous estimation of the orientation and shape of the stress tensor and the individual fault plane solutions for a population of earthquakes is studied, which corresponds to a synthesis of the methods used by Brillinger et al. (1980) to obtain focal mechanisms and by Armijo and Cisternas (1978) for stress tensors analysis in microtectonics.
Abstract: An algorithm for the simultaneous estimation of the orientation and shape of the stress tensor and the individual fault plane solutions for a population of earthquakes is studied. It corresponds to a synthesis of the methods used by Brillinger et al. (1980) to obtain focal mechanisms and by Armijo and Cisternas (1978) for stress tensor analysis in microtectonics. The input data are the polarities of the P arrival and take-off angles for the set of source-station pairs. The method distinguishes, in general, which one of the nodal planes corresponds to the fault and gives the direction of the slip. The application to the aftershock sequence of the 1980 Arudy earthquake (Western Pyrenees) shows that the observations may be explained by a single stress tensor producing a N32°E extension, with a likelihood of 95 per cent.
180 citations
TL;DR: In this article, a simple polycrystalline model is constructed which precisely satisfies local kinematic constraints as well as global compatibility, and a volume-average global stress is obtained by approximating the local constraint stress as the corresponding projection of the (to be-determined) global stress.
Abstract: W e clearly elucidate the kinematic constraints, and the corresponding kinematic indeterminacy of part of the deviatoric stress tensor, in a rigid-viscoplastic single crystal lacking five independent slip systems. The indeterminate stress component is a Lagrange multiplier enforcing the kinematic constraint, and it must be determined from equilibrium considerations. A simple polycrystalline model is constructed which precisely satisfies local kinematic constraints as well as global compatibility. Volume-average global stresses are obtained by approximating the local constraint stress as the corresponding projection of the (to-be-determined) global stress. Applications of the model to hexagonal crystals without pyramidal slip, and to large deformation and texturing of orthorhombic polycrystalline materials (olivine; HDPE) are made.
162 citations
TL;DR: In this article, the authors used Boltzmann's superposition principle to express the stress as a time convolution of a fourth rank tensorial relaxation function with the strain tensor.
Abstract: SUMMARY The anisotropic linear viscoelastic rheological relation constitutes a suitable model for describing the variety of phenomena which occur in seismic wavefields. This rheology, known also as Boltzmann’s superposition principle, expresses the stress as a time convolution of a fourth rank tensorial relaxation function with the strain tensor. The first problem is to establish the time dependence of the relaxation tensor in a general and consistent way. Two kernels based on the general standard linear solid are identified with the mean stress and with the deviatoric components of the stress tensor in a. given coordinate system, respectively. Additional conditions are that in the elastic limit the relaxation matrix must give the elasticity matrix, and in the isotropic limit the relaxation matrix must approach the isotropic-viscoelastic matrix. The resulting rheological relation provides the framework for incorporating anelasticity in time-marching methods for computing synthetic seismograms. Through a plane wave analysis of the anisotropic-viscoelastic medium, the phase, group and energy velocities are calculated in function of the complex velocity, showing that those velocities are in general different from each other. For instance, the energy velocity which represents the wave surface, is different from the group velocity unlike in the anisotropic-elastic case. The group velocity loses its physical meaning at the cusps where singularities appear. Each frequency component of the wavefield has a different non-spherical wavefront. Moreover, the quality factors for the different propagating modes are not isotropic. Examples of these physical quantities are shown for transversely isotropic-viscoelastic clayshale and sandstone. As in the isotropic-viscoelastic case, Boltzmann’s superposition principle is implemented in the equation of motion by defining memory variables which circumvent the convolutional relation betweeh stress and strain. The numerical problem is solved by using a new time integration technique specially designed to deal with wave propagation in linear viscoelastic media. As a first application snapshots and synthetic seismograms are computed for 2-D transversely isotropicviscoelastic clayshale and sandstone which show substantial differences in amplitude, waveform and arrival time with the results given by the isotropic and elastic rheologies.
161 citations
TL;DR: Eshelby's elastic energy-momentum tensor is shown to satisfy a differential identity which, in the general case of a uniform elastic body with inhomogeneities, is expressible in terms of the torsion of the material connection as discussed by the authors.
Abstract: Eshelby's elastic energy-momentum tensor is shown to satisfy a differential identity which, in the general case of a uniform elastic body with inhomogeneities, is expressible in terms of the torsion of the material connection.
141 citations
TL;DR: In this paper, a numerical model is used to simulate the mechanical behavior of a small assembly of uniform, inelastic, frictional, deformable disks in a simple shear flow, where spatially and temporally averaged dimensionless stresses is the shear rate, Kn is the normal stiffness of an assumed viscoelastic contact force model, Ks/Kn is the ratio of tangential to normal stiffness, ζn is normal damping coefficient, μ is the friction coefficient, and ρs, D and m are the particle density, diameter and mass,
Abstract: Application of the kinetic theory of gases to granular flows has greatly increased our understanding of ‘rapid’ granular flows. One of the underlying assumptions is that particles interact only through binary collisions. For a given set of material and flow parameters, as the concentration increases, the transition from a binary collision mode to other modes of interaction occurs. Kinetic theory can no longer be applied. A numerical model is utilized to simulate the mechanical behaviour of a small assembly of uniform, inelastic, frictional, deformable disks in a simple shear flow. There are two objectives: to obtain the ‘empirical’ constitutive law and to gain insight into the mechanisms that operate in the transitional and quasi-static regimes. In a simple shear flow, spatially and temporally averaged dimensionless stresses is the shear rate, Kn is the normal stiffness of an assumed viscoelastic contact force model, Ks/Kn is the ratio of tangential to normal stiffness, ζn is the normal damping coefficient, μ is the friction coefficient, and ρs, D and m are the particle density, diameter and mass, respectively. The range of B from 0.001 to 0.0707 was investigated for C ranging from 0.5 to 0.9, with material constants fixed as ζn = 0.0709 (corresponding to the restitution coefficient e = 0.8 in binary impacts), Ks/Kn = 0.8 and μ = 0.5. It is found that for lower concentrations (C 0.75) τ*ij monotonically decreases as B increases. Moreover, their relationship in this regime is well approximated by power law: τ*ij ∝ B−n(C). The powers nij range from nearly zero for C = 0.775 (corresponding to the familiar square power dependency of dimensional stresses on the shear rate in the rapid flow regime), to nearly two for C = 0.9 (corresponding to shear-rate independence in quasi-static regime). The intermediate concentration range corresponds to transition. Distinct mechanisms that govern transitional and quasi-static regimes are observed and discussed.
98 citations
TL;DR: In this article, it is shown that for a rather general class of differential models, a simple condition is sufficient to prove that such tensors are positive definite (i.e., they are configuration tensors).
Abstract: Analogous to the Giesekus theory, it is possible to identify a positive definite (configuration) tensor in other theories for differential models For a rather general class of differential models, a simple condition is given which is sufficient to prove that such tensors are positive definite
93 citations
TL;DR: In this paper, a new analytic solution for the stress tensor in a homogeneous and compressible elastic sphere is presented, where the inclusion of compressibility alters stresses by several percent, and realistic failure criteria are noted to demonstrate the general failure of such ductile bodies as iron meteoroids by plastic shear, while brittle ice bodies fail by either tensile or shear fracture.
TL;DR: In this article, a compact review of the theory of flow in materials and the bulk behavior of experimentally deformed rocks is given, focusing mainly on the measured strengths of rocks and how these can be extrapolated to strengths at geological strain-rates.
TL;DR: In this paper, a nonlinear regression and robust estimation of the deviatoric stress tensor responsible for the formation of the twin lamellae has been proposed, assuming conditions of constant resolved shear stress for twinning.
Abstract: Theoretically, calcite twins analysis makes possible the full determination of the deviatoric stress tensor responsible for the formation of the twin lamellae, that is, the orientations of the principal stress axes and the three differential stress values (σi–σj) may be calculated, assuming conditions of constant resolved shear stress for twinning. Inverse method using nonlinear regression and robust estimation is proposed.
Applications to monophased synthetic polycrystals that are generated numerically by solving the direct problem and to a natural sample from the Quercy (southwest of France) have given good results. Comparison with the first method proposed by Turner (1953), modified by Nissen (1964) for principal stress orientations and by Jamison and Spang (1976) for differential stress values, clearly emphasizes the considerable interest of this new method.
01 Oct 1990
TL;DR: In this paper, a compressible generalization of the linear combination of the Smagorinsky model and scale-similarity model, in terms of Favre-filtered fields, is obtained for the subgrid-scale stress tensor.
Abstract: New subgrid-scale models for the large-eddy simulation of compressible turbulent flows are developed and tested based on the Favre-filtered equations of motion for an ideal gas. A compressible generalization of the linear combination of the Smagorinsky model and scale-similarity model, in terms of Favre-filtered fields, is obtained for the subgrid-scale stress tensor. An analogous thermal linear combination model is also developed for the subgrid-scale heat flux vector. The two dimensionless constants associated with these subgrid-scale models are obtained by correlating with the results of direct numerical simulations of compressible isotropic turbulence performed on a 96(exp 3) grid using Fourier collocation methods. Extensive comparisons between the direct and modeled subgrid-scale fields are provided in order to validate the models. A large-eddy simulation of the decay of compressible isotropic turbulence (conducted on a coarse 32(exp 3) grid) is shown to yield results that are in excellent agreement with the fine grid direct simulation. Future applications of these compressible subgrid-scale models to the large-eddy simulation of more complex supersonic flows are discussed briefly.
TL;DR: In this paper, the authors derived an expression for the stress tensor of a granular material that can exhibit normal-stress effects due to a solids fraction gradient using both the continuum and kinetic models.
TL;DR: A well-defined stress tensor is obtained that is symmetric and satisfies the force balance equation, so it provides an unambiguous route to quantities such as the surface free energy.
Abstract: Combining some results from the classical theory of elasticity with the modern functional derivative approach to nonuniform systems, we obtain a well-defined stress tensor for nonuniform equilibrium fluids and solids. This stress tensor is symmetric and satisfies the force balance equation, so it provides an unambiguous route to quantities such as the surface free energy. The ambiguities associated with earlier stress-tensor definitions are traced back to their failure to take account of Saint-Venant's strain compatibility conditions
TL;DR: In this paper, the authors present a tool for checking stability and optimal convergence of mixed finite elements (LBB-and "equilibrium" condition) under three mechanical conditions: the continuity of the normal component of the stress tensor across interelement boundaries, the second forbids spurious modes on a two element patch, and the third is to avoid zero energy-stresses on an element.
Abstract: In order to develop an efficient and manageable tool for checking stability and optimal convergence of mixed finite elements (LBB- and ‘equilibrium’ condition), three mechanical conditions are stated. The first requires the continuity of the normal component of the stress tensor across interelement boundaries, the second forbids spurious modes on a two element patch, and the third is to avoid zero-energy-stresses on an element. The mathematical proof shows that the conditions are necessary and sufficient. Finally, the hybrid implementation of two plane mixed elements is carried out, and comparisons are made with two standard displacement elements. In particular, the mixed element with constant displacement shape functions (MMC) surpasses the linear displacement element by far and also the quadratic displacement element if the computational effort is compared.
TL;DR: In this paper, a theory for the effect of stress on diffusion of a swelling penetrant into a linear viscoelastic solid is developed for the case of solute transport into a thin rectangular slab, resulting in a new expression for diffusive flux.
Abstract: A theory is developed for the effect of stress on the diffusion of a swelling penetrant into a linear viscoelastic solid A constitutive equation for the Helmholtz free energy of the mixture is used to obtain expressions for the dependence of the stress tensor on the history of the deformation and concentration fields For cases in which the solid has only a single relaxation time, the stress constitutive equation is consistent with the standard linear viscoelastic model for solid deformation The expression for the free energy is also used to obtain an equation for the difference in the chemical potential between the solute and the polymer as a function of the trace of the stress and concentration histories; as a result, the theory is thermodynamically consistent As an example, the analysis is appliedto the case of solute transport into a thin rectangular slab, resulting in a new expression for the diffusive flux
TL;DR: In this article, a macrocontinuum model for granular material is proposed to model the interparticle contact forces and contact couples. But the model does not take into account the effects of microdiscreteness and the inter particle contact properties in the system.
Abstract: Granular material perceived as a collection of particles is modeled as a macrocontinuum. The model takes into account the effects of microdiscreteness and the interparticle contact properties in the system. With consideration of particle rotation, the continuum model for granular solid is found to be of micropolar type. The derived constitutive law of the material includes variables of Cauchy stress, polar stress, deformation strain, and polar strain. Considering the effect of particle interaction, the stresses are defined in terms of interparticle contact forces and contact couples. The constitutive coefficients are derived explicitly in terms of the interparticle contact stiffness. Using the derived stress‐strain relationships, a solution procedure based on variational principle is applied to obtain solutions for boundary value problems. Examples of boundary value problems are shown for particle assemblies with elastic contact interaction. The results are compared with those obtained from a discrete ele...
TL;DR: In this paper, the disjoining pressure approach for describing a thin liquid film as a liquid layer of finite and variable thickness is applied to dynamic curved films and the results are obtained without making any assumptions about the form of the real stress tensor, are applicable to films with both uniform and uneven thickness.
TL;DR: In this paper, a three-dimensional viscoelastic-plastic model for flow of particle filled polymer melts has been formulated based upon a modification of the Leonov model by introducing a structure function describing stress generated by the presence of the dispersed phase under the assumption that the material obeys the von Mises yielding criterion before flow takes place.
Abstract: A three‐dimensional viscoelastic‐plastic model for flow of particle filled polymer melts has been formulated. The approach is based upon a modification of the Leonov model by introducing a structure function describing stress generated by the presence of the dispersed phase under the assumption that the material obeys the von Mises yielding criterion before flow takes place. The model has been applied to steady simple shear and to transient shear flows, and equations have been derived for the components of stress tensor for each flow situation. A verification of the model has been made against limited experimental data available in the literature, indicating that the model is in fair agreement with experiments.
TL;DR: In this article, a new algorithm for the computation of the coefficients of the heat kernel expansion, associated with a second-order non-negative elliptic-symmetric differential operator, defined on an N -dimensional compact riemannian manifold with boundary is presented.
TL;DR: In this article, the authors examined the self-noise produced by instability waves in a round jet, defined as the noise attributed to the nonlinear sources in Lighthill's stress tensor.
Abstract: This article is concerned with examining the self-noise produced by instability waves in a round jet. The self-noise is defined here as the noise attributed to the nonlinear sources in Lighthill’s stress tensor. The calculated self-noise is found to be proportional to the fourth order of the velocity amplitude saturation. The self-interaction of the instability waves results in a “super-directivity.” The dependency of the sound intensity on the Strauhal number and on the Mach number is in accordance with observations.
TL;DR: In this paper, the spin and statistics of a particle in 2 + 1 dimensions coupled to linearized gravity with a Chern-Simons term were determined, and the results were compared to those of the corresponding abelian vector theory.
TL;DR: In this article, Renardy et al. formulated a well-posed boundary value problem in which boundary conditions on the stresses at the inflow boundary need to be imposed in addition to the velocity boundary conditions which are familiar from the Newtonian case.
Abstract: In an earlier paper [1] M. Renardy, Rocky Mt. J. Math., 18 (1988) 445 (corrigendum 19 (1989) 561)), the author considers flow of a Maxwell fluid across a strip bounded by parallel planes. In order to formulate a well-posed boundary value problem, boundary conditions on the stresses at the inflow boundary need to be imposed in addition to the velocity boundary conditions which are familiar from the Newtonian case. However, it is not possible to prescribe all the stresses at the inflow boundary. In two dimensions, the diagonal components of the extra stress tensor can be prescribed, but in three dimensions the situation is awkward. Basically, four boundary conditions are required (this is expected from the analysis of characteristics ([2] D.D. Joseph, M. Renardy and J.C. Saut, Arch. Ration. Mech. Anal., 87 (1985) 213)), but it is not possible to identify four specific stress components. In [1], the problem is resolved by expanding the stresses at the inflow boundary in a Fourier series; certain stress components are then prescribed for each Fourier mode. It is hard to see how this procedure could be generalized to more complex geometries. In this paper, an alternative idea will be pursued. Instead of trying to prescribe components of the extra stress, we shall prescribe certain first order differential expressions acting on them. It is demonstrated how a well-posed boundary value problem can be formulated in this fashion.
TL;DR: In this paper, an incremental theory for finite plastic strain in ordered solids is described, based upon measurements to four decimal places, of deformed length, inside diameter, outside diameter, angle of twist, and helices formed by initially straight generators on the outer surface; all the measurements are made while the specimens are under load.
TL;DR: In this article, an alternative strain-temperature formulation of the evolution equations was derived, which in fact allows to describe the evolution of the plastic deformation and of the internal variables, when the motion and the temperature are given on a neighbourhood of the considered material point.
TL;DR: In this paper, a method of obtaining an analytical expression of the linear part of the pressure-strain-rate tensor in terms of the anisotropy tensor of the Reynolds stresses has been developed.
Abstract: A novel method of obtaining an analytical expression of the ‘‘linear part’’ of the pressure–strain‐rate tensor in terms of the anisotropy tensor of the Reynolds stresses has been developed, where the coefficients of the seven independent tensor terms are functions of the invariants of the Reynolds‐stress anisotropy. The coefficients are evaluated up to fourth order in the anisotropy of the Reynolds stresses to provide a guidance for development of a turbulence model.
TL;DR: In this paper, a succinct derivation of a rate equilibrium equation is presented for the direct solution of the steady state velocity field and Cauchy stress field within an elasto-plastic material.
Abstract: A succinct derivation of a rate equilibrium equation is presented. An Eulerian finite element flow formulation is then developed for the direct solution of the steady state velocity field and Cauchy stress field within an elasto-plastic material. The formulation is used for the analysis of strip drawing.
01 Jul 1990
TL;DR: In this article, a three-dimensional constitutive equation governing the flow of an isotropic rigid/perfectly plastic granular material is presented, which relates the strain-rate tensor to the Cauchy stress tensor.
Abstract: A three-dimensional constitutive equation governing the flow of an isotropic rigid/perfectly plastic granular material is presented. The equation relates the strain-rate tensor to the Cauchy stress tensor and to the co-rotational rate of the Cauchy stress. It contains scalar functions of the scalar invariants involving the stress, stress-rate and strain-rate tensors together with parameters which characterize the material. The model generalizes the double-shearing model and its relationship to existing theories is demonstrated.