Total Strain Rate Research Articles

On the Generation of Discontinuous Shearing Motions of a Non-Newtonian Fluid

The system under study models unsteady, one-dimensional shear flow of a highly elastic and viscous incompressible non-Newtonian fluid with fading memory under isothermal conditions. The flow, in a channel, is driven by a constant pressure gradient, is symmetric about the center line, and satisfies a no-slip boundary condition at the wall. The non-Newtonian contribution to the stress is assumed to obey a differential constitutive law (due to Oldroyd, Johnson & Segalman), the key feature of which is a non-monotone relation between the total steady shear stress and strain rate. In a regime in which the Reynolds number is much smaller than the Deborah (or Weissenberg) number, one obtains a degenerate, singularly perturbed system of nonlinear reaction-diffusion equations. It is shown that if the driving pressure gradient exceeds a critical value (the local shear stress maximum of the steady stress vs. strain rate relation), then the solution to the governing system, starting from rest at \(\), tends as \(\) to a particular discontinuous steady state solution (the “top-jumping” steady state), except in a small neighborhood of the discontinuity. This discontinuous steady state is shown to be nonlinearly stable in a precise sense with respect to perturbations yielding smooth initial data. Such discontinuous steady states have been proposed to explain “spurting” flows, which exhibit a large increase in mean flow rate when the driving pressure is raised above a critical value.

Creep-fatigue behaviour of CoCr22Ni22W14 in the temperature range 850 °C < T < 1200 °C

The creep-fatigue behaviour of CoCr22Ni22W14 was investigated by varying the frequency, the total strain rate and the temperature. With decreasing frequency and increasing temperature, the cyclic deformation behaviour is more and more influenced by creep processes. The recorded hysteresis loops for low frequencies at 1000 and 1200 °C show that the plastic deformation takes place under nearly constant tensile and compression stresses. Compared with fatigue tests at higher frequencies, the dislocation arrangement after failure is altered, and well developed subgrain walls appear. For low frequencies the stress dependence of the plastic strain rates determined during the fatigue tests at half of the lifetime corresponded very well to the stress dependence of the stationary creep rates from creep tests. The fatigue data presented can be correlated very well with the frequency-modified Manson-Coffin relationship and with the frequency-modified Basquin relationship.

Extinction of turbulent diffusion flames by Kolmogorov microscale turbulence

The effects of turbulent straining on the structure and response of cylindrical diffusion flames were studied experimentally by using the counterflow flame configuration formed in the forward stagnation region of a porous cylinder from which propane or methane was ejected. From the shadowgraphs, it was found that the turbulence caused large-scale distortions with small amplitude on a diffusion flame which had a locally laminar structure. The time-averaged flame thickness was three times as large as the laminar flame. The turbulent flow field was measured in detail by a hot-wire anemometer. The temperature profiles were determined as functions of the applied strain rate by using a fine-wire thermocouple the thermal inertia of which was compensated electrically. The total strain rate applied to the flame was decomposed into the bulk strain rate induced by the mean flow velocity gradient and the turbulent strain rate which was modelled. In the present study, the latter was modelled by the reciprocal of the Kolmogorov time scale. The total strain rate at which the turbulent diffusion flame was extinguished coincided with the critical velocity gradient at which the extinction of a laminar flame occurred. With the increased of the total strain rate, the turbulent diffusion flame became thinner, and the maximum mean temperature gradually decreased due to the decrease of the maximum temperature within the laminar flamelet. On the other hand, the maximum rms fluctuating temperature was rather insensitive to the total strain rate over a wide range of total strain rates. However, close to extinction, nonreactive holes appeared intermittently in the flame along the stagnation line established in the forward stagnation region of the porous cylinder. These holes led to an abrupt increase in the maximum rms fluctuating temperature and, also, a decrease in the maximum mean temperature very close to the state of extinction.

Finite element analysis of viscoplastic composite materials and structures

SUMMARY A finite element solution of therm omechanical problems of composite materials and structures with viscoplastic phases is described. The solution incorporates the transformation field equations of multiphase media to relate the total and inelastic macroscopic strain rates of a composite representative volume to their local counterparts. Any micromechanical model can be used in the analysis including two-phase, and periodic models of fibrous media. Constitutive equations for the viscoplastic phases are derived from a unified formulation which combines plastic and creep strains.

Cyclic stress-strain response and dislocation substructure evolution of a ferrite-austenite stainless steel

The hardening-softening response, the cyclic stress-strain behavior and the evolution of dislocation structures of an AISI-329 ferrite-austenite stainless steel have been studied. Fatigue testing has been conducted under fully reversed total strain control and constant total strain rate. Detailed transmission electron microscopy studies have been carried out in order to determine the individual substructure evolution, as a function of increasing imposed strain amplitude, in each constitutive phase. In general, the cyclic response of the studied material may be described in terms of three different regimes within the plastic strain amplitude ( ϵ pl) range investigated, i.e. from 2 × 10 −5 to 6 × 10 −3:at ϵ pl below 10 −4 the dominant cyclic deformation mechanisms are those correlated to planar glide of dislocations within the austenite which is the phase which carries a large part of the macroscopic strain in this first regime. On the other hand, at ϵ pl higher than 6 × 10 −4 the dominant substructure evolution is observed inside the ferritic matrix. In this case, strain localization is enhanced, within the ferritic grains, through the development of veins into the wall structure. Such evolution induces a pronounced decrease of the cyclic strain hardening rate in the cyclic stress-strain curve. At ϵ pl in-between these values, the cyclic behavior is characterized by a relatively high strain hardening rate and may be classified as a mixed “ferritic/austenitic-like” behavior. In this intermediate regime substructural changes are observed in both phases and the dislocation activity in each of them seems to be strongly influenced by their particular cyclic strain hardening behaviors. Finally, the results are analyzed and compared with data from the literature in terms of volume fraction and chemical composition of the constitutive phases.

Influence of Initial Texture, Temperature and Total Strain on the Texture Development of Polycrystalline Pyrrhotite Ores in Deformation Experiments

The development of the crystallographic preferred orientation (texture) of polycrystalline pyrrhotite from Sullivan mine, Canada, and Cerro de Pasco, Peru was determined by neutron texture analyses before and after experimental deformation. The pyrrhotite ore from Sullivan has an initial texture with a distinct c‐axis maximum perpendicular to the foliation whereas the ore from Cerro de Pasco shows a weak and moreover inhomogeneous texture within the specimen.The deformation texture and deformation behaviour of samples from Sullivan were investigated as a function of temperature, total strain and the angle between the compression axis and the initial c‐axis maximum. The deformation textures show the influence of different activated glide modes as a function of the compression direction and temperature. Investigations of the deformation texture and deformation behaviour of samples from Cerro de Pasco were carried out as a function of temperature, total strain and strain rate. The inhomogeneous texture of the starting material did not allow to draw conclusions about the texture development as a function of strain rate and temperature. Nevertheless the preferred orientation of the c‐axes parallel to the compression axis becomes more evident with increasing total strain.

Deformation behavior of a rapidly solidified fine grained Al-8.5%Fe-1.2%V-1.7%Si alloy

Deformation characteristics of a rapidly solidified dispersion strengthened Al-8.5% Fe-1.2% V-1.7% Si alloy processed by planar flow casting was studied by tension testing at 25–420°C, compression testing at 25°C and hardness tests. The as-processed alloy shows non-linear elastic behavior, yield drop, low uniform and total elongation, serrated yielding, stress relaxation, flow softening and anomalous strain rate dependence of ductility. The results indicate that greater dynamic recovery due to fine grains (<0.3 μm) and essentially pure aluminum matrix in the as-processed planar flow cast alloy contribute to the observed deformation characteristics. Corroborative evidence for the recovery processes was obtained by differential scanning calorimetry and hardness measurements of cold rolled and annealed samples.

Velocity gradients in the northern Indian Ocean inferred from earthquake moment tensors and relative plate velocities

The region in the northern Indian Ocean between the Central Indian Ridge, the Sumatra Trench, and 15°S to 9°N exhibits high seismicity, along with basement and sediment deformation. This area represents a diffuse plate boundary between the distinct and separate portions of the Indian and Australian plates. We invert the spatial distribution of horizontal strain rates in this diffuse plate boundary zone to obtain the horizontal velocity field associated with the moment release from 61 earthquakes that have occurred over the past 80 years. In the inversion, the three‐dimensional rotation vector function W is determined at all corner points (i.e., knot points) of a grid such that the predicted strain rate values match the observed strain rates within the grid areas in a least squares sense. The rotation vector of Australia relative to India, defined by the earthquake data alone, is 10.1°S±9.1°, 81.5°E±5.4°, 0.1°±0.06°/m.y. (1 sigma error), which is slightly different in location from the expected vector estimated using seafloor magnetic anomalies and transform fault trends (3.2°S, 75.1°E, 0.288°/m.y.) [DeMets et al, 1994a, b]. By minimizing the rate of work within a continuous viscous medium that accommodates Australia's motion relative to India [DeMets et al, 1994a], we determine the complete horizontal velocity gradient tensor within the zone of diffuse deformation. The viscosity tensor within all of the deforming areas can be specified. When all the regions are free to strain in any direction and have the same viscosity (uniform isotropic viscosity), the predicted strain pattern is either pure extension or pure compression, which is in contrast to the style of earthquake deformation east of Chagos where both thrust and strike‐slip mechanisms are observed. However, when the isotropic viscosity of regions within the Wharton Basin is increased relative to the isotropic viscosity levels of areas along the Ninetyeast Ridge, the plate motion is accommodated by combined thrust and strike‐slip styles of strain along the Ninetyeast Ridge and within the eastern Central Indian Basin. This predicted strain that accommodates plate motion is in agreement, both in direction and style, with the earthquake‐related strains in these regions. Although most of the observed style and direction of strain can be explained by lithospheric strength differences between the Wharton Basin and Ninetyeast Ridge areas alone, the presence of strike‐slip faulting within the Wharton Basin requires a preexisting fabric that favors strike‐slip faulting there as a strain mechanism that accommodates, in part, the motion of Australia relative to India. To first order, the direction of principal strain rates, the style of strain, and the spatial distribution of strain rates from the earthquakes are in accord with the accommodation of the expected plate motion, whereas across the plate boundary zone the rates from 80 years of seismicity average out to at most 60% of the total strain rate.

An incremental elastic-viscoplastic theory indicating a reduced modulus for non-proportional buckling

The small deformation elastic-viscoplastic constitutive equations of Bodner-Partom are modified to model strong non-proportional loading paths such as experienced in corner turning tests and certain cases of inelastic buckling. An essential generalization is made to the flow rule, causing the magnitude and direction of plastic strain rate to become an explicit function of deviatoric total strain rate as well as of stress and hardening variables. With this and other modifications, the equations indicate some of the important characteristics of the response to abrupt changes in the loading direction. These are: (a) a reduced effective shear modulus; (b) a transient drop in the effective stress; and (c) a transient non-coaxiality of the plastic strain rate relative to the deviatoric stress. Predictions of the modified theory compare reasonably well with experimental values for inelastic torsional buckling of axially compressed cruciform columns and for corner turning tests.

Flow fields within the Tonga Slab determined from the moment tensors of deep earthquakes

The seismic strain rates and instantaneous velocity fields within the plane of the Tonga slab are investigated using 450 Harvard centroid moment tensor (CMT) solutions from the period 1977–1992. Down‐dip seismic strain rates at the bottom of the slab are around 1 × 10−15 s−1. Spline functions are fitted to the distribution of seismic strain rates to recover the velocity gradient tensor within the plane of the slab. Solutions show a northward motion of the 200 km depth contour in the Tonga slab at rates of 8±2 mm/yr relative to the 670 km depth contour of the slab. Relative to the deep, southern portion of the Tonga slab, down‐dip velocities increase from near zero in southern Tonga to 10±2.5 mm/yr in northern Tonga, consistent with the increased subduction rate from south to north described by the Pacific‐Australia pole of rotation. Moreover, patterns of strain and velocity obtained from all events show a remarkable similarity with the patterns of strain and velocity obtained using only events with scalar moment values less than 1 × 1025 dyn cm. Along‐strike extensional strain rates in the deep zone of northern Tonga are approximately 0.5 × 10−15 s−1. This is of the same order of magnitude as the extensional strain rates associated with changes in curvature produced by the subduction of a spherical plate, suggesting that the seismic strain rates within Tonga represent a large fraction of the total strain rate budget within the seismogenic zone. An estimated maximum of 60% of the relative motion between the surface and 670 km is absorbed as strain within the slab, indicating considerable slab thickening and penetration of the slab into the lower mantle.

Modified calculation of the dissipation term in thermoplastic finite element equations

AbstractIn the paper, a simple modification of the internal dissipation term in coupled thermoplastic finite element equations is presented. A modified form of the heat capacity and the thermomechanical coupling matrices are derived. These modifications are based on a decomposition of the internal dissipation term into two parts, which depend on the total strain rate and the rate of temperature change, respectively.

An analysis of large-strain damage elastoplasticity

The elastoplastic constitutive analysis which utilizes the model of multiplicative decomposition of the deformation gradient into its elastic and plastic parts has been mainly developed and applied to elastically isotropic materials, which remain isotropic during the process of plastic deformation. This paper extends the application of the model to materials that change their elastic properties during the deformation process as a result of the material degradation and the corresponding damage. The exact kinematic and kinetic analysis of the finite deformations leads to an additive decomposition of the total strain rate into its elastic, damage and plastic constituents. The general structure of the expression for the damage strain rate is derived, valid for utilized damage tensors of any order. The analysis of elastoplastic deformation of elastically anisotropic materials without damage is also presented, with the application to transversely isotropic materials. The relationships between the elastic and plastic strain rates and the components of the multiplicative decomposition and their rates are also given.

Modeling of hardening at very high strain rates

A modification of the Bodner–Partom elastic-viscoplastic constitutive model is proposed to account for strain rate dependence of the evolution of hardening. The suggested procedure is for the rate of hardening in the hardening evolution equation to be a direct function of total strain rate. Material constants are determined for oxygen-free-electronic copper and simulations of response behavior at very high rates of straining appear to be consistent with test observations.

Phase characteristics of pre-deformed Al–40 wt% Zn alloy

The total creep strain rate and the transient creep parameters B and n of Al–40 wt% Zn alloy are studied under constant stresses ranging from 42 to 50.3 MPa at different temperatures in the range from 503 to 623 K. Variations of creep characteristics are studied by subjecting the alloy samples to different degrees of pre-deformation in the same temperature range under a constant stress of 42.04 MPa. Pre-deformation increases the total creep strain and clears out the superplastic behaviour of the transient creep as an intensive increase for the parameters B and n and a decrease for the energies activating the creep process. The observed peaks for B and n are rendered to the softer β-phase and due to the state of the alloy being near the eutectoid temperature of the α'-phase. The integrated role of stress, pre-deformation, and temperature is cleared. The observed structural variations and the calculated activation energies suggest a dislocation mechanism involving grain boundary sliding or migration as the rate controlling mechanism for transient creep in Al-40 wt% Zn alloy.

Elastoplastic constitutive analysis with the yield surface in strain space

A common approach used in formulating elastoplastic constitutive equations is to partition the total strain rate into its elastic and plastic parts, and then develop the constitutive expression for the plastic strain rate using the concept of the yield surface in stress space. An alternative approach is to partition the stress rate into its elastic and plastic parts, and then develop the constitutive expression for the plastic stress rate using the concept of the yield surface in strain space. Both of these approaches are used in this paper to derive and compare the final structures of the corresponding constitutive equations. It is shown that the preferable choice of the yield surface may be in either stress or strain space depending on the selected strain and conjugate stress measures utilized to construct the constitutive formulation.

Strain effects on the structure of counterflowing turbulent premixed flames

Dynamic and scalar structures of turbulent premixed flames stabilized in the opposed jet configuration are investigated. The opposed jet burner facility allows control of many parameters of turbulent premixed combustion, including the bulk and turbulent strain rates. The main mixture streams are enveloped in coflowing air to reduce flame bouncing, to render the turbulence structure more isotropic and homogeneous at the stagnation plane, and, therefore, to better match the major modelling assumptions. Detailed investigation of the velocity field confirms the strong variation of the initial turbulence parameters as the flow stagnates, and the similarity between nonreacting and reacting flow fields. It is clearly demonstrated that turbulent premixed flames stabilized in the opposed jet configuration are ideal for turbulent combustion model validation, as modification of the turbulence field by the flame is minimized. Two different extinction regimes of opposed jet turbulent premixed flames are identified. For small andmoderate burner separations, the total strain rate causes extinction. For large separation distaces, mixing and dilution by the external air are the causes of flame extinction. For the first regime, it is confirmed that bulk and turbulent strain rates should be taken into account simultaneously to establish the robustness of turbulent premixed flames. For the first time, the scalar structure of opposed jet turbulent premixed flames is explored by laserinducedrayleigh light-scattering technique. All the relevant parameters of instantaneous flame fronts are determined, and estimates of the flame surface density are proposed. It is shown that, for constant mixture composition and constant initial turbulence structure, the flame surface density increases with the bulk strain rate before its extinction value is closely approached. This result is discussed by using various ingredients of flamelet models.

A unified determination of creep and strain rate sensitivity of polycrystals from the properties of constituent grains

On the basis of the observation that both creep and plasticity are fundamentally rate processes, a unified micromechanical constitutive theory is developed to predict both strain rate sensitivity and the time-dependent creep of polycrystal. The theory is established within the small-strain range for power-law creep and can provide the evolution of the microstress-strain relation and creep strain of the constituent grains. The distinctly different responses of a constituent grain and a free crystal of the same orientation are also examined under both constant-strain-rate and constant-stress creep conditions. It is also demonstrated that the higher creep rate associated with a higher stress automatically translates into the commonly observed strain rate effect on the stress-strain curves of polycrystals and that under a constant total strain rate the stress-strain curve would lead to a saturation stress. When applied to a 304 stainless steel under a thermal cycling between 600 and 650°C, the theory also yields reasonably accurate creep strains for the polycrystal. Its predictive capability is further confirmed by comparison with experiments on the development of creep strain and the strain rate effect on the stress-strain curves for this material.

Uniaxial Model for Concrete under Variable Temperature and Stress

A plasticity model using a strain-rate formulation is presented to describe the uniaxial response of concrete when subjected to combined thermal and mechanical actions. The total strain rate is resolved into three individual components: mechanical strain, thermal strain, and transient creep strain. Each component is formulated individually. The mechanical strain rate is assumed to consist of an elastic strain rate and a plastic strain rate, which itself is taken as temperature-dependent. In describing the plastic strain rate, the nonlinear part of the uniaxial stress-strain curves is assumed to be a quarter of ellipse, defined for a given temperature in the range 20°C–800°C. The model was first used to analyze uniaxial (experimental) data under variable load but isothermal conditions, and then to investigate the effect of a sustained load on the deformational response of a concrete specimen under heating. Thirdly, a set of experimental relaxation tests was studied, in which temperature and stress vary continuously under a zero total strain condition. Here, the importance of a variable temperature and stress, and the significance of the history, is demonstrated.

On the mechanics of large inelastic deformations: kinematics and constitutive modeling

In this paper we examine the complexities associated with the kinematics of finite elastoplastic deformations and other issues related to the development of constitutive equations. The decomposition of the total strain and strain rate tensors into elastic and plastic constituents is investigated by considering both a multiplicative decomposition of the deformation gradient and an additive decomposition of the deformation vector field. Physically based definitions for the elastic and plastic strain rate tensors are given and compared with other values found in the literature. Constitutive equations for the plastic flow are derived by considering both a phenomenological-energy approach and a physically motivatedmesomechanical approach based on the double-slip idealization. It is shown that by resorting to the mechanics of the double slip, specific relations for the plastic stretching and plastic spin can be rigorously derived, taking into account the effect of noncoaxiality and material rotation. Finally, the implication of such effects to large deformations is examined in connection with the localization phenomenon.

Kinematic hardening rules with critical state of dynamic recovery, part I: formulation and basic features for ratchetting behavior

Abstract Kinematic hardening rules formulated in a hardening/dynamic recovery format are examined for simulating rachetting behavior. These rules, characterized by decomposition of the kinematic hardening variable into components, are based on the assumption that each component has a critical state for its dynamic recovery to be activated fully. Discussing their basic features, the authors show that they can predict much less accumulation of uniaxial and multiaxial ratchetting strains than the Armstrong and Frederick rule. Comparisons with multilayer and multisurface models are made also, resulting in a finding that the simple one in the present rules is similar to the multilayer model with total strain rate replaced by inelastic (or plastic) strain rate. Part II of this work deals with applications to experiments.