Strain Rate Gradient Research Articles

Hexahedral mesh generation for remeshing in three-dimensional metal forming analyses

It is well known that the quality and efficiency of the design of metal forming processes can be significantly improved with the aid of effective numerical simulations. However, one of the prominent features in metal forming analysis using the finite element method is that the initial simple workpiece geometry becomes severely distorted by material flow, causing mesh degeneration due to locally concentrated deformation. Thus, an effective remeshing scheme is needed to continue the analysis and obtain more accurate analysis results. In the present study, for remeshing in three-dimensional metal forming analyses, a hexahedral mesh generator was developed. It is based on the master grid approach and octree based refinement scheme to generate uniformly sized or locally refined hexahedral mesh system. In particular, for refined hexahedral mesh generation, a modified Laplacian smoothing scheme for preserving specified mesh density conditions is proposed. Also, for the specification of mesh density conditions, distributions of effective strain rate gradient and posteriori error values based on a Z 2 error estimator were used. In order to investigate the applicability of the developed hexahedral mesh generator for remeshing, several three-dimensional metal forming analyses were carried out. From this study, it is construed that the developed hexahedral mesh generator can be effectively used for using CAMP form in three-dimensional metal forming analyses.

Transmyocardial Tissue Velocity, Strain Rate and Strain Gradients in Normal and Abnormal Myocardium with Intraoperative TEE. Nonuniform Distribution of Myocardial Layers

Transmyocardial Tissue Velocity, Strain Rate and Strain Gradients in Normal and Abnormal Myocardium with Intraoperative TEE. Nonuniform Distribution of Myocardial Layers

Remeshing for metal forming simulations—Part II: Three‐dimensional hexahedral mesh generation

AbstractIn the present study, a hexahedral mesh generator was developed for remeshing in three‐dimensional metal forming simulations. It is based on the master grid approach and octree‐based refinement scheme to generate uniformly sized or locally refined hexahedral mesh system. In particular, for refined hexahedral mesh generation, the modified Laplacian mesh smoothing scheme mentioned in the two‐dimensional study (Part I) was used to improve the mesh quality while also minimizing the loss of element size conditions. In order to investigate the applicability and effectiveness of the developed hexahedral mesh generator, several three‐dimensional metal forming simulations were carried out using uniformly sized hexahedral mesh systems. Also, a comparative study of indentation analyses was conducted to check the computational efficiency of locally refined hexahedral mesh systems. In particular, for specification of refinement conditions, distributions of effective strain‐rate gradient and posteriori error values based on a Z2 error estimator were used. From this study, it is construed that the developed hexahedral mesh generator can be effectively used for three‐dimensional metal forming simulations. Copyright © 2002 John Wiley & Sons, Ltd.

Remeshing for metal forming simulations—Part I: Two‐dimensional quadrilateral remeshing

AbstractIn this paper, a general framework of practical two‐dimensional quadrilateral remeshing, which includes the determination of remeshing time, automatic quadrilateral mesh generation, and data transfer process, will be formulated. In particular, the current work contains new algorithms of mesh density specification according to the distribution of effective strain‐rate gradients, mesh density smoothing by fast Fourier transform (FFT) and low‐pass filtering techniques, coarsening it by node placement scheme, and a modified Laplacian mesh smoothing technique. The efficiency of the developed remeshing scheme was tested through three practical two‐dimensional metal forming simulations. The results clearly indicate that the algorithms proposed in this study make it possible to simulate two‐dimensional metal forming problems efficiently and automatically. Copyright © 2002 John Wiley & Sons, Ltd.

斜め対向流場に形成されたエッジフレームの構造

Structure of premixed laminar edge-flame in counterflow field with a strain rate gradient was investigated by numerical simulation. Two-dimensional incompressible Navier-Stokes equations with full chemistry of CH4/air were solved in curvilinear coordinates. When the strain rate gradient was produced by inclining an axis of the burner, the edge-flame was observed at the side of higher strain rate. On the other hand, the structure of twin flames in the side of lower strain rate was similar to that of normal counterflow premixed flame with no strain rate gradient, and therefore temperature and mole fractions of species were almost constant along the flame surface. The local strain rates at the edge were constant for an equivalence ratio as same as the extinction strain rates for the normal counterflow flames were.

Modelling the hot plane strain compression test Part 1 – Effect of specimen geometry, strain rate, and friction on deformation

Thermomechanically coupled finite element analysis of the hot plane strain compression test has been carried out to investigate the effect of various test parameters on the measured response and deformation of specimens. The results are presented in a series of papers. In this paper (Part 1), the results of two-dimensional simulations are discussed, evaluating the effects of material type, specimen geometry, strain rate, and friction on the overall deformation behaviour. The effects of spread and friction are detailed in Part 2, and the effects of asymmetry during the test are detailed in Part 3. The present results show that the local deformation behaviour is independent of the type of material and strain rate, at least up to 50 s-1. The behaviour, however, depends strongly on friction and initial specimen geometry, with deformation becoming more uniform with decreasing initial specimen thickness, i.e. with increasing tool width w to specimen thickness h ratio. The deformation is constrained within the predicted slip line fields, and is controlled by the instantaneous geometry and not by the strain history. At low values of w/h the strain rate gradients are very high, and the raw data must be corrected for local strain rates and strains. Master curves have been produced to account for these gradients. The present results can be treated as generic, since the deformation is independent of material and the strain rate of deformation.

Influence of uniform basement extension on faulting in cover sediments

Numerical predictions are compared with the results obtained from sandbox experiments intended to reproduce the effects of uniform basement extension. For mechanically homogeneous or uniformly layered sediments that are not strain-softened, faults are not expected to develop during uniform basement extension. Even in heterogeneous or strain-softening sediments subject to uniform basement extension, the only faults to develop reveal an irregular and uncharacteristic distribution of displacements along the fault plane. It is inferred that minor and inevitable gradients in strain rate are the most important reason why faulting was observed in the sandbox models. Irregularities such as topographic variation can also lead to faulting. Numerical experiments show that with increasing thickness of a basal elastic layer (rubber membranes in sandbox models) faulting of the cover sediments can cause the strain in the underlying elastic “basement” layer to become non-uniform. Faulting during uniform basement extension is controlled by irregularities particular to each example, and thus does not give rise to a characteristic structural style in the overlying sediments. Thus, sandbox experiments of this type cannot be used as templates for structural interpretation except for the rare case when the initial distribution of sediment and basement material properties are known and their influence is explored from extensive numerical experimentation.

Experimental evidence of macroscopic plastic flow nonhomogeneity development in uniaxial tensile test samples

An experimental study of the macroscopic plastic flow nonhomogeneity in the course of a uniaxial tensile test is conducted on several aluminum alloys, nickel and 4340 steel. It was observed that the plastic flow initiates throughout the entire gage length in a nonuniform fashion, so that the growth of the deformation in the middle of the gage is faster than it is at the edges. That initial strain rate gradient almost disappeared shortly after its evolution, and the strain rate through the entire gage length became about uniform. The plastic flow nonuniformity emerged again upon further stretching, producing a gradual acceleration in the middle of the gage with corresponding slowdown toward the edges. That final development of the strain rate gradient commenced well in advance of the load maxima and was the cause of the consequent neck formation in the middle portion of the gage. It is demonstrated that the origin of plastic flow nonhomogeneity stemmed from the second elastic strain component in the transverse direction and its gradient evolution along the reduced section upon loading. It is found empirically that acceleration in the strain rate in the middle part of the reduced section was accompanied by a reduction in the apparent strainhardening exponent,n, calculated from the stress/strain chart. The maxima in the apparent strain-hardening exponent,n, obtained from the common stress/strain charts can be used to indicate the strain rate gradient onset.

Modeling Orthokinetic Coagulation in Spatially Varying Laminar Flow

An orthokinetic coagulation model including the effects of agglomeration and local stress-induced aggregate breakup was developed. This model was used to simulate coagulation in the flow between two eccentrically located and rotating cylinders. Four methods of modeling coagulation in the flow system were examined. The first technique used a volume-weighted average of the local strain rates, while a second method used an equivalent volume-weighted power (G). A third method treated each volume element as a separate batch reactor and determined a final volume-averaged floc population. The final modeling technique applied mass transfer between each of the elements. Results indicated that substantial differences in average particle diameters and populations were generated with each of the methods, especially where mass transfer between the elements was considered. It was concluded that mass transfer between regions of varying flow strain rate and/or velocity gradient should be included in accurate coagulation modeling.

Does the mechanical milieu associated with high-speed running lead to adaptive changes in diaphyseal growing bone?

Exercise during growth can be important for attaining optimal bone mass. High-intensity long-duration protocols, however, can have detrimental effects on immature bone morphology and mechanics. The underlying mechanisms are poorly understood. Here, we quantified the mechanical environment of the middiaphyseal rooster tarsometatarsus during high-speed running and examined whether short bouts of this exercise-related mechanical milieu can induce positive changes in cortical bone morphology, mechanics, and mineral ash content. At 9 weeks of age, roosters were assigned to controls (n = 9) and runners (n = 8). Treadmill running was applied in loading sessions of 5 min, three times per day (≈2600 cycles/day) for 8 weeks. Both controls and runners received double-fluorochrome labels during weeks 3 and 8 of the protocol. Middiaphyseal distributions of tarsometatarsal longitudinal normal strain, strain rate, and strain gradients engendered by walking and running were determined via in vivo strain gauges. Compared with walking, running elevated mean peak strain magnitude by 19%, peak strain rates by 136%, and peak strain gradients by approximately 18%. After 8 weeks of running, middiaphyseal areal and mechanical properties and normalized ash weight were no different between runners and controls. Transient and focal reductions in periosteal mineral apposition rates occurred during the exercise protocol. Our current data suggest that reducing the number of loading cycles can mitigate the adverse response previously observed in this model with long-duration running. This study also supports the tenet that the exercise-generated mechanical milieu must differ substantially from the habitual milieu to induce significant adaptations.

Premixed Edge-Flames in Spatially-Varying Straining Flows

Premixed-gas flames subject to steady spatially-varying straining flows were studied to examine one aspect of premixed flames in strongly turbulent flows where strain rate gradients are present and local strain rates may be high enough to cause local flame-front extinguishment. The spatially-varying straining flows were created using a counterflow slot-jet burner with slightly non-parallel jet exits. When the flow configuration was premixed combustible gas vs. cold inert gas, so that only a single flame was produced, steady flame “edges” could be created where the flame would exist in the low-strain region but would be extinguished in the high-strain region. When the flow configuration was premixed gas vs. premixed gas, twin flames would exist in the low-strain region that converged to a corner-like tip in the high-strain region. For both configurations the local strain at the location of the stationary flame edge was somewhat lower than the strain required to extinguish flames in the same mixture subject to a spatially uniform strain. The difference was greater for the twin-flame configuration, particularly at high Lewis number (Le). Due to diffusive-thermal instabilities, cellular flames were observed at low Le and travelling-wave patterns were observed at high Le. Le effects also led to the formation of isolated “flame tubes” rather than continuous fronts at sufficiently low Le and high strain rates. All of these results are consistent with recent theoretical predictions. These results indicate that “laminar flamelet” models of premixed turbulent combustion may be reasonably accurate for single flames over a wide range of Le and twin flames with Le close to unity, even at conditions approaching those where local flame quenching occurs, but may not be accurate for twin-flames except for Le near unity. This finding is somewhat different from previous experimental and theoretical results for nonpremixed edge-flames, where more substantial differences between uniform flames and edge-flames were found for all Le.

Effect of material parameters on shear band spacing in work-hardening gradient dependent thermoviscoplastic materials

We study thermomechanical deformations of a viscoplastic body deformed in simple shear. The effect of material elasticity is neglected but that of work hardening, strain-rate hardening, thermal softening, and strain-rate gradients is considered. The consideration of strain-rate gradients introduces a material characteristic length into the problem. A homogeneous solution of the governing equations is perturbed at different values t 0 of time t, and the growth rate at time t 0 of perturbations of different wavelengths is computed. Following Wright and Ockendon's postulate that the wavelength of the dominant instability mode with the maximum growth rate at time t 0 determines the minimum spacing between shear bands, the shear band spacing is computed. It is found that for the shear band spacing to be positive, either the thermal conductivity or the material characteristic length must be positive. Approximate analytical expressions for locally adiabatic deformations of dipolar (strain-rate gradient-dependent) materials indicate that the shear band spacing is proportional to the square-root of the material charateristic length, and the fourth root of the strain-rate hardening exponent. The shear band spacing increases with an increase in the strain hardening exponent and the thermal conductivity of the material.

Deformation and compatibility for elasto-plastic micropolar materials with application to geomechanical problems

The consideration of empty porous solid materials (e.g. granular skeletons) naturally falls into the macroscopic continuum mechanical concept of the Theory of Porous Media (TPM). It is well known from Various experiments that granular materials tend to localization phenomena (e. g. shear bands) as a result of local concentrations of plastic strains. The numerical simulation of these phenomena generally reveals an ill-posed problem with the consequence that the shear band width strongly depends on the chosen spatial discretization. To overcome this problem, an internal length scale must be introduced, which, in the present article, is a result of the inclusion of micropolar degrees of freedom in the sense of the Cosserat brothers. Based on this approach, the contribution of the present article is to show that the micropolar compatibility condition of the geometrically linear approach is a direct consequence of the general finite micropolar theory. Furthermore, as a result of the micropolar compatibility condition applied to elasto-plastic problems, there is no independent evolution equation for the rate of curvature tensor, since this quantity is uniquely determined by the the gradient of the plastic strain rate. Proceeding from the fact that micropolar theories are downward compatible to the usual standard approach, the numerical example is carried out on the basis of a model adaptive strategy, where an automatic switch between the standard and the micropolar approach guarantees that the additional micropolar degrees of freedom are only taken into consideration in those domains, where micropolar rotations occur, i.e. in the shear band zone.

Nonpremixed edge flames in spatially varying straining flows

Nonpremixed flames subject to steady but spatially varying straining flows were studied to examine one aspect of nonpremixed flames in strongly turbulent flows or near quenching conditions (e.g., near a burner rim), where strain-rate gradients are present and local strain rates may be high enough to cause local flame-front extinguishment. The spatially varying straining flow were created using an opposed slot-jet burner with slightly nonparallel jet exits. The most significant observation was that steady flame “edges” could be created where the flame would exist in the low-strain region but would be extinguished in the high-strain region. The local strain at the location of the stationary flame edge was almost always lower than the strain required to extinguish flames in the same mixture subject to a spatially uniform strain. The strain rate at the edge-flame location was independent of the strain-rate gradient and gradual transitions from edge-flame behavior to uniformly strained flame behavior were not observed, indicating that conventional nonpremixed flames and edge flames are quite distinct structures yet each has well-defined properties. At the flame edge, interferometer images indicate a region of locally intense burning and an abrupt transition to nonburning conditions away from the edge. These observations were found to be qualitatively consistent with recent theoretical models of flame edges. These results indicate that “laminar flamelet” models of nonpremixed turbulent combustion may require reevaluation at conditions approaching those where local flame quenching occurs. It is proposed that these models could be improved by adding edge-flame libraries to existing laminar flamelet libraries.

Adaptation of bone to physiological stimuli

The ability of bone to alter its morphology in response to local physical stimuli is predicated upon the appropriate recruitment of bone cell populations. In turn, the ability to initiate cellular recruitment is influenced by numerous locad and systemic factors. In this paper, we discuss data from three ongoing projects from our laboratory that examine how physiological processes influence adaptation and growth in the skeleton. In the first study, we recorded in vivo strains to quantify the locomotion-induced distribution of two parameters closely related to bone fluid flow—strain rate and strain gradients. We found that the magnitude of these parameters (and thus the implied fluid flow) varies substantially within a given cross-section, and that while strain rate magnitude increases uniformly with elevated speed, strain gradients increase focally as gait speed is increased. Secondly, we examined the influence of vascular alterations on bone adaptation by assessing bone blood flow and bone mechanical properties in an in vivo model of trauma-induced joint laxity. A strong negative correlation ( r 2 = 0.8) was found between increased blood flow (76%) in the primary and secondary spongiosa and decreased stiffnes (−34%) following 14 weeks of joint laxity. These data suggest that blood flow and/or vascular adaptation may interact closely with bone adaptation initiated by trauma. Thirdly, we examined the effect of a systemic influence upon skeletal health. After 4 weeks old rats were fed high fat-sucrose diets for 2 yr, their bone mechanical properties were significantly reduced. These changes were primarily due to interference with normal calcium absorption. In the aggregate, these studies emphasize the complexity of bone's normal physical environment, and also illustrate the potential interactions of local and systemic factors upon the process by which bone adapts to physical stimuli.

Control of superplastic deformation rate during uniaxial tensile tests

Precise determination of superplastic flow behavior involves imposing known and controlled strain rate during deformation of these alloys. Examination of tensile specimens after superplastic deformation has revealed variations in strain and strain rate occurring as a function of position and the difficulty of maintaining a constant strain rate during testing. To quantify these strain and strain-rate gradients within the specimens, interrupted tensile tests and tests on gridded tensile specimens were performed. It was observed that more uniform strain and strain rates could be achieved with longer gauge length specimens. While longer gauge lengths make it possible to have better control over the imposed strain rate by minimizing the effects of material flow from the specimen grip regions, it has been realized that for smaller specimen gauge lengths, typically used in most laboratories, a more complex control of crosshead speed (CHS) during a test is essential to characterize superplastic behavior. A mathematical model has been developed in order to gain better insight into this material flow and to provide an improved crosshead control schedule for constant strain-rate testing. The results of this analysis have been validated on a superplastic aluminum-magnesium alloy (5083 Al).

On the mechanisms of initiation and propagation of plastic instability in polyethylene under tensile drawing

The development of a plastic instability in a high-density ethylene-butene copolymer under tensile drawing was monitored by means of a video-controlled optical extensometer in order to study the mechanism of initiation and propagation of necking. True-stress true-strain curves measured along the neck at various times of the drawing are compared with the curve recorded as a function of time at the locus where the neck started. A strain rate gradient is shown to build up during the stage of neck initiation. Evidence is given of tensile normal stress in the concave part of the neck shoulder. X-ray diffraction reveals an oblique preferred orientation of the crystal c-axis that is governed by the lower energy-consuming pathway for the deformation. This is responsible for the plastic flow localization owing to an improved shear ability. The gradual c-axis orientation towards the draw direction as the neck grows involves a strain-hardening effect that leads to neck stabilization. The conclusion is put forward that neck propagation lies in the gradation of the preferred orientation along the neck shoulder rather than in stress triaxiality. Comparison with a parent low-density copolymer shows a better trend for oblique preferred orientation of the c-axis and a reduced propensity for localized plastic flow thanks to a more homogeneous distribution of the stress over the crystallites. © 1996 John Wiley & Sons, Inc.

On the mechanisms of initiation and propagation of plastic instability in polyethylene under tensile drawing

The development of a plastic instability in a high-density ethylene-butene copolymer under tensile drawing was monitored by means of a video-controlled optical extensometer in order to study the mechanism of initiation and propagation of necking. True-stress true-strain curves measured along the neck at various times of the drawing are compared with the curve recorded as a function of time at the locus where the neck started. A strain rate gradient is shown to build up during the stage of neck initiation. Evidence is given of tensile normal stress in the concave part of the neck shoulder. X-ray diffraction reveals an oblique preferred orientation of the crystal c-axis that is governed by the lower energy-consuming pathway for the deformation. This is responsible for the plastic flow localization owing to an improved shear ability. The gradual c-axis orientation towards the draw direction as the neck grows involves a strain-hardening effect that leads to neck stabilization. The conclusion is put forward that neck propagation lies in the gradation of the preferred orientation along the neck shoulder rather than in stress triaxiality. Comparison with a parent low-density copolymer shows a better trend for oblique preferred orientation of the c-axis and a reduced propensity for localized plastic flow thanks to a more homogeneous distribution of the stress over the crystallites. © 1996 John Wiley & Sons, Inc.

FRACTAL PLASTIC SHEAR BANDS

We present a numerical study model of shear bands in rocks with a non-associated plastic flow rule. The system drives spontaneously into a state in which the length distribution of shear bands follows a power law and where the spatial organization of the shear bands appears to be fractal. The distribution of local gradients in deviatoric strain rate has different scaling exponents for each moment which we calculate and discuss. Samples of granodiorite from the Pyrenees sheared under high confining pressure are analyzed and their properties compared with the numerical results.

Self‐organized criticality of plastic shear bands in rocks

We show that the shear bands that appear during the pure shear numerical simulations of rocks with a non‐associated plastic flow rule form fractal networks. The system drives spontaneously into a state in which the length distribution of shear bands follows a power law (self‐organized criticality) with exponent 2.07. The distribution of local gradients in deviatoric strain rate has different scaling exponents for each moment, in particular the geometrical fractal dimension is 1.7. Samples of granodiorite sheared under high confining pressure from the Pyrenees are analyzed and their properties compared with the numerical results.