Strain Rate Components Research Articles

Yield surfaces for textured polycrystals—I. Crystallographic approach

Polycrystal yield surfaces were calculated by the Bishop and Hill method for a series of idealized cube textures. The textures were specified in terms of a rotationally symmetric gaussian distribution of misorientations with scatter widths increasing from 0° (single crystal) to 45°. The corresponding polycrystal yield surfaces were determined for a sample containing up to 400 such grains; the yield surfaces are described in terms of equilibrated stress and strain rate components expressed in a five-dimensional orthonormal reference frame. The effect of increasing the scatter width on the π plane and shear stress cross-sections is illustrated. The yield stress and strain rate ratios were also calculated for these samples and the dependence on scatter width is presented.

Application of complex variables and pseudo-stress function to power-law materials and stress analysis of single rigid inclusion in power-law materials subjected to simple tension and pure shear

The two-dimensional compatibility equation for time-dependent materials described by the power law is expressed in terms of the second derivative of the stress function with respect to complex conjugate variables. The equation is solved by introducing the pseudo-stress function which satisfies the biharmonic equation resulting from the compatibility equation. The relationship between the second derivative of the stress function and the pseudo-stress function is established. The mixed derivative of the stress function associated with the dilatational stress is expressed by an integral of the complicated pseudo-stress function. The velocity and strain-rate components are expressed in terms of the pseudo-stress function. Therefore, responses of power-law creep materials subjected to a boundary condition can be obtained. Using the pseudo-stress function, the stress distribution in power-law materials containing a single rigid inclusion subjected to a uniaxial tensile and a simple shear stress is found. The results obtained by using the pseudo-stress function are compared with those obtained by another investigator. The results show that there are discrepancies between the two results in the neighborhood of the inclusion surface, and the deviation diminishes in the region away from the inclusion surface. Significantly, the discrepancies of the stress components associated with the dilatational stress, such as σ r and σ θ are predominant while σ rθ , σ r - σ θ and \\ ̄ gs agree reaso the region away from the inclusion surface, approximately 1.5 R ( R is the inclusion radius). The reasons creating the discrepancies between the two results are found, rationalized and discussed. The discrepancies are caused by the singularities involved in the integrand associated with the dilatational stress expression which is integrable deleting the singularities, and the effects of the singularities are predòminant for low strain-rate hardening exponents, m. The effects of the singularities on the stress components may not have been considered by the previous investigator and the reason is discussed. Through the analyses, it is found that: (1) the pseudo-stress function is a valid function for two-dimensional problems and it can be used for an analysis of power-law materials; and (2) the effects of the strain-rate exponent on the stress components and the stress concentration factor are found.

Creep of Anisotropic Clay: Microplane Model

Undrained constant‐volume creep of anisotropically consolidated specimens of clay is mathematically described by the microplane model, which is based on the assumption that the shear strain rates on the contact planes between mutually sliding clay platelets (the microplanes) are the resolved components of the macroscopic strain rate. Thus, the microstructure is assumed to be kinematically constrained. The rate of shear on the microplanes is assumed to be governed by activation energy (rate process theory). The matrix of the current viscosities is obtained as an integral over all spatial directions involving the shear strain rates for the microplanes. This integral, which is evaluated numerically as a summation, gives the dependence of the viscosity matrix on the applied macroscopic stress. Anisotropy of the clay is characterized by a function of the spherical angles describing the relative frequency of clay platelets of various orientations. This function can be approximately estimated from X‐ray diffract...

A flow theory for soil: Concept of multiple neutral loading surfaces

A flow theory for soils is proposed which diverts considerably from the classical theory of plasticity. The theory is based on the concept of multiple loading surfaces defining neutral states for which irreversible deformation cease to exist. The constitutive relation has an incremental nature and the ratios of the components of plastic strain rate are functions of both the current stress and the stress rate. The concept of the yield locus is no longer involved in the formulation.

Constitutive relations between stress, strain-rate and temperature.

A constitutive equation for stress, stress-rate, strain-rate and temperature effect is proposed and examined with the experimental results. The proposed equation is a combination of an Arrehnius equation and Malvern-type constitutive equation, which was proposed by the author previously, and indicated well the impact behaviors of various materials. The characteristics of the present equation are : (1) constants involved in the equation are clear and accurate physically, (2) the athermal stress component and maximum limit of strain-rate are considered, (3) at a high strain-rate, there are approximated linear relationships between stress, the reciprocal of absolute temperature and the logarithm of plastic strain-rate, (4) at a low strain-rate, there is a linear relationship between stress and the plastic strain-rate, (5) the proposed equation represents experimental results of constitutive relations of metals with a few exceptions, such as the blue brittleness range of steels.

Strain Rates On Rutford Ice Stream, Antarctica (Abstract)

In a study of the Rutford Ice Stream, strain rates were measured on a transverse section. Magnitudes ranged up to 40 × 10−3 a−1 but were typically in the order of 3 × 10−3 a−1 with an error of 0.1 χ 10−3 a−1. Variations in the strain rate between adjacent stakes of 0.2 χ 10−3 a−1 to 2 × 10−3 a−1 were matched to the thickness variations on the glacier. For each set of three adjacent stakes, the velocity gradient components of the surface strain rate tensor were calculated by assuming that the gradients were linear over the distance between adjacent stakes. When plotted against distance across the ice stream, each strain rate component revealed different aspects of the flow field. The longitudinal strain rate was compressive, with an almost constant magnitude of 10−3 a−1. The lateral strain rate is extensive, with an average value of 1.1 × 10−3 a−1 which agreed with the angle between the divergent flow lines observed on a Landsat image. Peaks in the lateral strain rate, corresponding to longitudinal bands of thicker ice, showed that these thicker bands were spreading more rapidly at the expense of thinner areas. The two velocity gradient components of the shear rate tensor also reflected differences in ice thickness.

Strain Rates On Rutford Ice Stream, Antarctica (Abstract)

In a study of the Rutford Ice Stream, strain rates were measured on a transverse section. Magnitudes ranged up to 40 × 10−3 a−1 but were typically in the order of 3 × 10−3 a−1 with an error of 0.1 χ 10−3 a−1. Variations in the strain rate between adjacent stakes of 0.2 χ 10−3 a−1 to 2 × 10−3 a−1 were matched to the thickness variations on the glacier.For each set of three adjacent stakes, the velocity gradient components of the surface strain rate tensor were calculated by assuming that the gradients were linear over the distance between adjacent stakes. When plotted against distance across the ice stream, each strain rate component revealed different aspects of the flow field. The longitudinal strain rate was compressive, with an almost constant magnitude of 10−3 a−1. The lateral strain rate is extensive, with an average value of 1.1 × 10−3 a−1 which agreed with the angle between the divergent flow lines observed on a Landsat image. Peaks in the lateral strain rate, corresponding to longitudinal bands of thicker ice, showed that these thicker bands were spreading more rapidly at the expense of thinner areas. The two velocity gradient components of the shear rate tensor also reflected differences in ice thickness.

EXPERIMENTAL ASPECTS IN POLYMER MELT ELONGATIONAL RHEOMETRY

After describing the present state-of-the-art, the open experimental problems are listed that prevent a broader application of elongational rheometry to more types of polymer melts. Small samples are preferable for the investigation of structure-properties relationships. Hints are given for the preparation of such samples and for the reduction, in simple elongation, of the error due to inhomogeneous deformations near the sample ends. Large elongations can be performed homogeneously with rotary clamps. However, it is important to characterize the quality of each test, because small errors in the strain rate lead to drastic changes in the stress-strain diagrams obtained for large strains, e.g. e > 4 (HENCKY strain) for melts of LDPE (low density polyethylene). For HDPE the main difficulty is the preparation of homogeneous samples, which requires a special cooling procedure after extrusion. The maximum elongations obtained so far are ϵmax = 5.5 for HDPE and ϵmax = 7 for LDPE. With modified rotary clamps, different modes of multiaxial elongational flows can be realized with polyisobutylene at room temperature. Eight such clamps in circular and rectangular arrangements are used for equibiaxial and planar elongations, respectively. Planar elongation gives two independent material responses. In presenting the test results, new definitions of the elongational viscosities concisely elucidate deviations from linear viscoelastic behavior. For other multiaxial flows the rotary-clamp-technique can also be applied, e.g. for flows with arbitrary ratios of the strain rate components and even with a variation of these ratios with time during an experiment. For a broader application, miniaturization of the rheomeler and operation at higher temperatures require further modifications

Application of Schwartz-Christoffel conformal mapping and complex variables to the symmetric cornered die extrusion of steady state creep materials

The two-dimensional equilibrium equation for steady creep materials described by the power law is derived in terms of the stream function, the velocity potential function and the complex potential function. The solutions of the equilibrium equation show that the potential function is independent of the rate hardening exponent ( m). The solutions are used to investigate the symmetric cornered die extrusion problem by applying the Schwartz-Christoffel conformal transformation. The horizontal and vertical velocities, the strain rate and the stress components are found for various rate hardening exponents. The analysis results illustrate that the stress components are dependent on the rate hardening exponent; however, the velocity and strain rate components are independent of the rate hardening exponent. Additionally, the constant dilatational stress (isobaric) line is found to be a constant velocity potential function.

Method of determining the rheological coefficient of the viscous component of strain rate from loading curves

The total strain of the metal is made up of the elastic strain, creep strain, and noncreep plastic strain [I, 6, 7]. The noncreep plastic strains are small at temperatures near the crystallization temperature and can be ignored [8]. Thus, the deformation of the metal during continuous teeming, i.e., at high temperatures and low strain rates, will be regarded here as the deformation of a viscoelastic body. In this case, the strain rate will be made up of the elastic component (in accordance with Hooke's law) and the viscous component [9]:

Satellite doppler determination of differential sea ice motion in the vicinity of the north pole∗

Abstract During April and May 1979 multidisciplinary observations were conducted in the central Arctic Ocean from three manned ice stations to explore the nature of the submarine Lomonosov Ridge (Weber, 1979). Positions of the LOREX (Lo‐monosov Ridge Experiment) ice camps were monitored continuously using geodetic satellite doppler receivers and the U.S. Navy Navigation Satellite System. Near real‐time doppler data reduction on the ice using broadcast satellite ephemeris provided operational horizontal positions with accuracies of about ± 250 m. A precise geodetic satellite doppler reduction program has been adapted to accommodate linear station motion in a simultaneous multistation, three‐dimensional adjustment in phases using postfitted precise satellite ephemeris, if available. Complete reprocessing of the LOREX doppler data set has produced average errors of ± 48 m for single pass solution and ± 24 m for mean three‐four horizontal station positions. The station ellipsoidal heights show strongly correlated variations in excess of 5 m, with the average error of ± 0.45 m. Station velocities are also strongly correlated, with the mean of about 225 m/h and the maximum of 1,240 m/h. Total strain and hourly strain rate components of sea ice have been evaluated for a homogeneous two‐dimensional strain model. The configuration of the ice camps provides a unique solution for mesoscale strain, indicating a major change in the pattern of pack ice deformation over the Lomonosov Ridge. The total strain reflects plastic deformation, which takes place mainly in episodic events. The strain rate also indicates nearly diurnal oscillations, with amplitudes an order of magnitude smaller than those corresponding to the major deformation events.

On a nonlinearity in flow birefringence

The nonlinearity observed in the flow-birefringent data can be explained by considering a properly formulated constitutiv relation, and is seen to be a natural consequence of the representation of a tensor function. The optical relation proposed is valid for a general threedimensional flow and, in addition, gives a meaning to the isoclinics in terms of the strain-rate components.

A finite element analysis of rigid-plastic deformation of the flange in a deep-drawing process based on a fourth-degree yield function

A general formulation for finite element analyses of very large rigid-plastic deformation is given by introducing a local system of convected co-ordinates into each element and taking the rotation of the principal axes of orthotropy of the material into consideration with respect to the two cases: (1) the case where the equivalent strain-rate ▪ is given by a function of strain-rate components ϵ ̇ ij and (2) the case where ▪ is defined by the formula ▪ = σ ij ϵ ̇ ij \\ ̄ gs , where σ ij are stress components and \\ ̄ gs is the equivalent stress. example, large deformation of the flange region in a deep-drawing process of a circular orthotropic metal sheet is analyzed on the basis of the conventional quadratic yield function and a fourth-degree one. The result shows that the effect of the axis rotation suppresses the ear-development to a certain extent, that any explicit expression of ▪ in terms of ϵ ̇ ij is not necessarily required for calculation and the definition ▪ = σ ij ϵ ̇ ij \\ ̄ gs suffices for the purpose, that the fourth-degree yield function gives us the shape of the deformed flange in good agreement with the experiment in contrast with the quadratic one which gives a too excessive ear-development, and so forth.

Slip-Line Solutions for Plain-Strain Indentation of Sandwich Metal Strips Between Flat Dies

The method of slip-lines has been used historically to solve two-dimensional problems of plane-strain plasticity in homogeneous metals. The application of the theory to obtain solutions for sandwich metals consisting of a hard core and softer clad is described here. The behavior of stress components, strain rate components, and velocity discontinuities at the interface between the two metals is examined. Some slip-line solutions for incipient plastic flow due to indentation of a sandwich strip with flat dies are presented. Three modes of deformation are considered: full penetration of the plastic zone with lateral motion of the rigid overhangs, bulging with clad flow only, and bulging with flow of clad and core. The numerical construction of the slip-line fields and hodagraphs is described and some results given. The differences between these results and the results of using homogeneous theory with an average yield strength are shown to be significant for strips of intermediate clad fraction. In addition, the results show that bulging, rather than full penetration of the plastic zone, is the expected mode of deformtion for a wide range of the strip parameters.

Inversion of flow Measurements for Stress and Rheological Parameters in a Valley Glacier

AbstractMethods are developed for determining the distributions of stress and effective viscosity in a glacier, under the assumptions: the ice is quasi-viscous, the flow is time independent, and acceleration forces are negligible. Measurements of the three-dimensional distribution of velocity are needed for their application. The differential equations of mechanical equilibrium, expressed in terms of viscosity, strain-rate components, mean stress, and their gradients, are viewed as equations to be solved for viscosity and mean stress subject to boundary conditions at the free upper surface. For certain rectilinear flow patterns, unique distributions of stress and effective viscosity can always be derived. For more complicated flow this is not necessarily so. However, it is still possible to choose the best values of rheological parameters in any trial flow law based on the requirement that the residuals to the equations of equilibrium be minimized in a mean-square sense. The techniques are applied to measurements of internal deformation made in nine bore holes on the Athabasca Glacier. At the center line the magnitude of the surface-parallel shear stress increases with depth more slowly than would be expected from a standard shape factor correction or the theoretical distribution of Nye. Correspondingly the lateral distribution of lateral shear stress shows the opposite relationships. In the lower one- to two-thirds of the depth corresponding to a range in effective stress from about 0.5 to 1.2 bars, the gross rheology of the ice is not distinguishably different from the experimentally determined flow law of Glen (n= 4.2,T= 0.02° C) as generalized by Nye. The results do not support the conclusion that the effective viscosity is higher than would be expected from Glen’s experiments as indicated by the more limited measurements of Paterson and Savage. Power-law parameters derived for the different bore holes considered separately show a spread, which suggests some rheological inhomogeneity. However, no definite conclusions can be drawn, because of direct measurement errors at the bore holes and less definable uncertainty in the interpolated distribution of velocity between the holes. The upper one- to two-thirds of the glacier constitutes an anomalous zone in which there is either a strong effect from a complex distribution of stress arising from longitudinal stress gradients or more complicated rheology than in a homogeneous power-law material.

Inversion of flow Measurements for Stress and Rheological Parameters in a Valley Glacier

AbstractMethods are developed for determining the distributions of stress and effective viscosity in a glacier, under the assumptions: the ice is quasi-viscous, the flow is time independent, and acceleration forces are negligible. Measurements of the three-dimensional distribution of velocity are needed for their application. The differential equations of mechanical equilibrium, expressed in terms of viscosity, strain-rate components, mean stress, and their gradients, are viewed as equations to be solved for viscosity and mean stress subject to boundary conditions at the free upper surface. For certain rectilinear flow patterns, unique distributions of stress and effective viscosity can always be derived. For more complicated flow this is not necessarily so. However, it is still possible to choose the best values of rheological parameters in any trial flow law based on the requirement that the residuals to the equations of equilibrium be minimized in a mean-square sense. The techniques are applied to measurements of internal deformation made in nine bore holes on the Athabasca Glacier. At the center line the magnitude of the surface-parallel shear stress increases with depth more slowly than would be expected from a standard shape factor correction or the theoretical distribution of Nye. Correspondingly the lateral distribution of lateral shear stress shows the opposite relationships. In the lower one- to two-thirds of the depth corresponding to a range in effective stress from about 0.5 to 1.2 bars, the gross rheology of the ice is not distinguishably different from the experimentally determined flow law of Glen (n = 4.2, T = 0.02° C) as generalized by Nye. The results do not support the conclusion that the effective viscosity is higher than would be expected from Glen’s experiments as indicated by the more limited measurements of Paterson and Savage. Power-law parameters derived for the different bore holes considered separately show a spread, which suggests some rheological inhomogeneity. However, no definite conclusions can be drawn, because of direct measurement errors at the bore holes and less definable uncertainty in the interpolated distribution of velocity between the holes. The upper one- to two-thirds of the glacier constitutes an anomalous zone in which there is either a strong effect from a complex distribution of stress arising from longitudinal stress gradients or more complicated rheology than in a homogeneous power-law material.

Computer-Aided Visioplasticity Solution to Axisymmetric Extrusion Through Curved Boundaries

A procedure for smoothing the experimental values of the flow function ψ in axisymmetric extrusion through curved boundaries was developed. The analysis was applied to a 45 deg conical die with a 6:1 extrusion ratio and a circular arc of 0.33-in. radius and 0.033-in. land at the exit section. An analytical expression of ψ in terms of r and z was obtained and used in the calculation of velocity and strain rate components in axisymmetric extrusion of a superplastic of the eutectic of lead–tin. The stress components were obtained from the known values of the strain rates by considering equilibrium and plasticity equations and material properties.

Fe-Si合金の下部降伏応力と流れ応力とにおよぼすひずみ速度, 温度, 結晶粒径とSi含有量の影響

The lower yield stress and flow stress of two kinds of Fe-Si alloys with various grain sizes were measured in the temperature range between 77 and 473°K with strain rates varying from 10-6 sec-1 to 33 sec-1. These data were analyzed using the Hall-Petch relation, It was found that the addition of Si to iron increased the athermal component of friction stress and decreased the thermal component. This decrease in the thermal component of friction stress caused the solid solution softening phenomena in Fe-Si alloys at low temperatures. The relations between the activation volume or activation energy of both alloys and the effective shear stress were not affected by the amount of Si added, but whole relationship was different from that of the pure iron.The differences in the low temperature deformation characteristics between iron and Fe-Si alloys were explained from the viewpoint of interaction between moving dislocation and elastic stress field in the lattice around the solute atom, Si.The strain rate dependences of the lowere yield stress of Fe-Si alloys varied continuously in the range of strain rates from 10-6 sec-1 to 33 sec-1. The relation between the effective component of lower yield stress and strain rate was well approximated by a power function at all temperatures tested.

Comments On Material Strength Model Formulation In Hydrocodes For High Loading Rate Applications

Hydrocodes are finite element/difference codes with special numerical algorithms to solve shock-wave-propagation and penetration problems. In these codes, the material description consists of (1) a strength model, and (2) an equation of state. This paper discusses only the formulation and the numerical implementation of strength models in hydrocodes. There are two types of formulations encountered. The first formulation begins with an empirical expression of the yield strength (Y) of the material as a function of strain, strain rate, and temperature. The plastic strains are calculated from classical plasticity theories. The second formulation begins with a viscoplastic description of the plastic strain rates. An expression for Y is then derived from the definition of the second invariant of the plastic strain rate components. Each type of formulation requires a particular and different method of implementation in the numerical simulation of deformation. The differences in the implementation techniques, along with examples of strength and constitutive models are, also reviewed in this paper.

Comparison of Two Complete Solutions in an Axisymmetric Extrusion With Experiment

An upper-bound and a potential solution to a forward extrusion problem were compared with experimental results obtained by the visioplasticity method. The process consisted of extruding a 2-in-dia billet of preforged lead through a conical die having a half-cone angle of 45 deg under the condition of relatively low friction. The comparison was made for steady state stream lines, velocities, strain rate components, effective strain and strain rate, grid distortion, and stress distribution. It was found that the curves were generally of similar shape and that some differences existed in magnitude only. It is suggested that the theoretical solutions can be used to advantage to a first approximation in predicting all important variables.