Unloading Phase Research Articles

Stable critical behavior and fast field annihilation in a magnetic field reversal model

We show that the Lu (Phys. Rev. Lett. 74(13) (1995) 2511) model, which is known to exhibit some properties of a system in self-organized criticality (SOC) [Lu, 1995; Klimas et al. (J. Geophys. Res. 105 (2000) (A8), 18,765–18,780.)], can be obtained through a reduction of the resistive MHD system to an idealized one-dimensional limit. Resistivity in this model is anomalous and localized and is due to the excitation of an idealized current-driven instability at positions where large spatial gradients appear in the magnetic field distribution. We note that, by reversing the reduction to the idealized one-dimensional limit, the Lu model presents an opportunity to construct a true MHD system that incorporates kinetic phenomena when small spatial scales are generated which may evolve into SOC under some conditions. We study the evolution of this model in a driven magnetic field reversal configuration on a high-resolution spatial grid. It has been shown earlier that the behavior of several parameters that are global measures of the state of the field reversal suggests that the reversal can evolve into SOC (Klimas et al., 2000). Here, we study the internal dynamics of the field reversal during the unloading phase of a loading–unloading cycle. Unloading is due to internal, localized, dynamic field annihilation; no flux is lost by the system through its boundaries. For this continuum model, we define an “avalanche” as a group of unstable grid points that are contiguous in position and time. We demonstrate scale-free power-law size and duration distributions for these avalanches during the unloading phase of a loading–unloading cycle. We further demonstrate the stability of these distributions; they do not evolve significantly as the unloading progresses. Box counting statistics on the position–time plane show that the avalanches can be characterized as intermittent one-dimensional structures; gaps in these otherwise one-dimensional structures lower their dimension to below one. The stable scale-free avalanche size and duration distributions, plus the fractal structure of the avalanches at small scales, provide further evidence that solutions of the continuum Lu model in a field reversal configuration can evolve into SOC.

An elastic–plastic-cracking model for finite element analysis of indentation cracking in brittle materials

An ‘elastic–plastic-cracking’ (EPC) constitutive model was developed and incorporated into the commercial explicit finite element package abaqus to analyze the fracture characteristics of brittle materials subjected to indentation loads. The analysis indicated that the EPC model can capture the development of median cracks during the loading phase and the development of lateral cracks during the unloading phase of the Vickers indentation cycle. The influence of material properties on induced damage zone characteristics was analyzed by defining a non-dimensional brittleness parameter. The model predictions of hardness as well as load–depth ( P– h) relationship during an indentation cycle were found to agree well with the experimental trends presented elsewhere in the literature.

Viscoelastic inclusion I: Partial contact under Bi-directional loading

The problem of a viscoelastic material with an infinite cylindrical cavity occupied by a cylinder of another viscoelastic material and subject to varying stresses at infinity, is considered in the non-inertial approximation. The case where a constant compressive stress is applied along one axis and a varying stress along the other, is solved in detail. A condition determining the contact region is given. This region depends on material parameters in a more complex way than in the elastic case. Integral equations for the pressure and gap functions are derived and numerically solved.¶Detailed numerical results are presented for materials modelled as standard linear solids. One result of interest is that during the unloading phase, the pressure distribution is quite different from the elastic circular shape. It becomes more like a uniform distribution with a sharp fall-off near the end of the contact region. For materials with large viscoelastic losses, there is a small peak near the edge of the interval and the maximum pressure is out at the edges rather than at the centre.

Autofrettage of Open-End Tubes—Pressures, Stresses, Strains, and Code Comparisons

Autofrettage is used to introduce advantageous residual stresses into pressure vessels. The Bauschinger effect can produce less compressive residual hoop stresses near the bore than are predicted by “ideal” autofrettage solutions. A recently developed numerical analysis procedure is adopted and extended. The ratio of calculated autofrettage pressure (numerical)/ideal autofrettage pressure (Tresca criterion and plane stress) is calculated and verified against available solutions. The case of open-end conditions based upon von Mises and engineering plane strain (constant axial strain with zero net axial force) is examined in detail. The ratio in this case varies between unity and 2/3, but exhibits very significant variations from the plane stress case when the diameter ratio of the tube exceeds 1.8. Results are within 0.5 percent of available analytical, numerical, and experimental results. A simple numerical fit allows all autofrettage pressures to be replicated to within 0.5 percent. The true plane strain pressure ratio is examined and shown to be inappropriate in modeling engineering plane strain. A number of residual hoop and axial stress profiles is presented for radius ratio 2.0. Calculated pressures are used to determine residual hoop stress values for tube diameter ratios from 1.1 to 3.0 for the full range of percentage overstrain levels. These comparisons indicate that Bauschinger effect is evident when the ratio autofrettage radius/bore radius exceeds 1.2, irrespective of diameter ratio. To assist designers the important values of residual hoop stress at the bore are summarized in a composite plot and a numerical fit is provided. The accuracy of the current ASME code using pressure criteria is assessed. The code is shown to be generally and modestly conservative. A design procedure is proposed which appears capable of extending code validity beyond 40 percent overstrain (the limit of the current code) and of eliminating the small nonconservatism at very low overstrain. Hoop strain values are calculated at both the bore and outside diameter of a tube of radius ratio 2 at the peak of the autofrettage cycle using von Mises criterion with open-end, closed-end, and plane strain conditions. These are compared with available solutions; general agreement is demonstrated, with agreement within 2 percent with an accepted simple formula in the case of open ends. ASME code predictions of percentage overstrain based upon strains at the peak of the autofrettage cycle are generally within 6 percent of numerical predictions. This is in turn produces an agreement within approximately 3 percent in residual bore hoop stress calculation. This discrepancy is generally conservative, becoming non-conservative only at overstrain levels exceeding 80 percent. Strain during removal of autofrettage pressure, in the presence of Bauschinger effect, is also calculated. This shows that the difference in strain during the unloading phase is up to 8 percent (ID) and 6.3 percent (OD) compared with the predictions of elastic unloading. These latter results show similar agreement with the ASME code as in the peak-strain analysis and permit correction of estimates of percentage overstrain based upon permanent bore enlargement.

Assessment of disc injury in subjects exposed to long-term whole-body vibration.

Long-term exposure to whole-body vibration is known to increase the risk of low back problems. The chain of events leading from repeated loading of the lumbar spine to back complaints and the exact nature of the vibration-induced damage are, however, obscure. Fluid in- and outflow as well as viscoelastic deformation are important aspects of the physiological function of the lumbar disc. Precision measurement of stature, termed 'stadiometry', has previously been applied in healthy subjects to document changes in disc height in relation to the load on the lumbar spine. The purpose of this study was to explore the relation between spinal loading and stature in a cohort of 20 subjects with long-term exposure to whole-body vibration. If the change of stature (and thus the change of disc height) caused by changes in spinal loading differed between exposed and normal subjects, this would point to vibration-induced changes in structure and material properties of the discs. For this purpose, four hypotheses were tested: (1) the viscoelastic deformation and fluid exchange of intervertebral discs during phases of spinal loading and unloading differs from normal; (2) the water content of lumbar discs of subjects exposed to long-term whole-body vibration deviates from normal; (3) the mean disc height of the lumbar spine depends on the total time of vibration exposure; (4) repeated loading influences trabecular bone density of vertebrae in the lumbar spine. A cohort of 20 operators of heavy earth-moving machinery was enrolled. Back complaints suspected to be due to long-term exposure (mean 17.6 +/- 2.1 years) to whole-body vibration and application for early retirement were the selection criteria used. Change of stature during a regular 8-h shift and change of stature in standing, carrying and sitting activities were measured. The stadiometric investigations were supplemented by magnetic resonance imaging (MRI) of the lumbar spine to assess whether the water content of the discs exhibited deviations from normal. In addition, quantitative computed tomography (QCT) was performed to assess whether the trabecular bone density of the third lumbar vertebra deviated from normal. The results showed no significant difference in change of stature while standing, carrying or sitting between exposed machine operators and non-exposed operators. Likewise, MRI examinations revealed no significant differences in the water content of the discs averaged over the lumbar spine. In addition, QCT examinations revealed no significant difference in the trabecular bone density of the third lumbar vertebra. The study thus revealed no significant difference between a cohort with long-term exposure and non-exposed controls with respect to viscoelastic properties of discs as determined by stadiometry, average water content of lumbar discs and trabecular bone density of L3.

A physical model of the resistance to local failure of an isotropic elastic-fragile body under the impact of a rigid non-deforming sphere

A physical model of the resistance to local failure of an isotropic elastic-fragile body under the impact of a rigid non-deformable sphere is obtained for impact velocities that are much less than the velocities of sound in the body. The impact is considered to consist of loading and unloading phases. The derived formulae characterize stresses in the body developed during loading and unloading. The loading stresses are characterized by the duration of the loading phase only and similarly the unloading stresses are characterized by the duration of the unloading phase. It was found that, when the impact caused micro-failures of the body, the duration of the unloading phase was much larger than the duration of the loading phase, which is different from the elastic impact phenomenon. On the other hand, the unloading stresses are much smaller than loading stresses. We performed experiments to characterize the strength of concrete by measuring the durations of loading and unloading phases of the impacts on concrete specimens of aluminium and steel spheres. The calculated specimens' strengths matched the results of static loading tests well.

Resolution of the material and machine contributions to the area to height ratio obtained from force-time powder compression data II: Rotary tablet press

The slope of a plot of the area under the upper punch force-time profile versus peak upper punch force (AHratio) has been proposed as a means to characterize the compressibility of pharmaceutical tableting materials (Chilamkurti et al., 1982). The purpose of this study was to apply a blank compression technique to resolve the machine and material contributions to the AHratio in a rotary tablet press system. The rationale behind this approach is that the machine and material contributions to the AHratio must be adequately addressed in order to make comparisons between materials and across differing tableting conditions. This data analysis technique was also applied to two segments of the upper punch force-time integrals representing the loading and unloading phases of the tableting cycle. Direct tableting grades of microcrystalline cellulose, anhydrous dicalcium phosphate and lactose were studied. Polyethylene was also studied as an example of a material exhibiting negligible compactibility.

Different solutions for stress and strain fields in autofrettaged thick-walled cylinders

In the first part of the paper we will present an analytical solution to the plastic stress and strain distribution induced by a single over-pressure cycle in a cylindrical vessel. The solution is valid for materials which, during the loading phase, can be described by a linear hardening law as well as by the modified Ramberg-Osgood law. We will clarify the conditions for which the stress field expressions can be simplified, thus facilitating a parametric analysis of their increase with respect to the values one would obtain when considering an ideal elastic-perfectly plastic behaviour of the material. In the second part of the paper, residual stress fields will be numerically evaluated, taking into account, during the unloading phase, both the variability of the stress flow and the Bauschinger effect factor as a function of pre-strains.

FE simulation of the influence of die-elasticity on component dimensions in forward extrusion

The elastic behaviour of the forming die has a direct influence on the quality of the formed component. Simulation of material flow and of die deflection during forward extrusion was conducted using elastic and plastic considerations to establish the influence of die-elasticity on the form of the component under different processing conditions. The simulations show that the dimensions of the work-material are affected by die-elasticity during both, loading and unloading of the die. The change of the die-geometry does not produce obvious differences in the dimensions of the extrudate; however, it influences the geometric errors of the workpiece which is retained in the die. Larger values of friction in the injection chamber result in the corresponding increase of punch pressure which results in larger radial deflection of the die during extrusion (loading), the consequence being a greater variation in extrudate geometry. High levels of friction reduce the elastic contraction of the die during punch retraction (unloading), and hence, larger form-errors are sustained after unloading. The contraction of the die during unloading phase of the process causes plastic deformation of the work-material remaining in the injection-chamber; this plastic deformation is significant to the overall form-error of the component.

A CDM‐based computational model for the Timoshenko beam

Focuses on a finite element computational model for the Timoshenko beam which is idealized as an elasto‐plastic‐damage medium governed by Lemaitre’s continuum damage mechanics (CDM) model for ductile fracture. Response under monotonically increasing loading does not show any deviation from elasto‐plastic simulation. However, a marked difference in the residual stress field is noted by virtue of the unloading phase, in which the CDM model allows for stiffness degradation in contrast to classical elasto‐plasticity which requires unloading at the (frozen) initial stiffness of the material.

A dislocation model for the two critical stress intensities required for threshold fatigue crack propagation

We have examined the variations, with decreasing load ratio, of threshold peak and cyclic stress intensities required for fatigue crack growth in stage I (mainly mode II loading) using a simple model simulating dislocation motion near a crack tip. In this model the crack grows by dislocations running into the crack during loading and unloading phases. Initially we have studied the behaviour of a crack with a dislocation source relatively far away from the crack tip. Crack propagation rates showed a Paris regime at high ΔK, and an abrupt threshold value ΔK th below which no crack growth occurred. The variation with load ratio of the peak (K th) and cyclic (ΔK th) stress intensities at the fatigue threshold showed that two different processes controlled the behaviour. At high load ratios dislocations are generated readily during loading and the threshold is controlled by the need for sufficient unloading to allow dislocations to run back into the crack, so that the criterion ΔK ≥ ΔK∗ results. At negative load ratios it is the generation of dislocations during the loading phase that controls the threshold condition, since once generated, the large unloading and reversed loading easily forces dislocations back to the crack. Under these conditions the threshold criterion becomes K max ≥ K∗.

The indentation response of GaAs[sbnd]AlAs heterostructures

Abstract Low-load indentation has been used to investigate the deformation behaviour of a submicron layer of AlAs on a GaAs substrate. High-resolution scanning electron microscopy of cross-sections through the deformed regions under indentations into this structure reveals that the softer AlAs layer is not penetrated by the indenter and that unusual lateral cracking originates during the unloading phase. Data of intendation load against depth demonstrates the effect of the layer through the increase in composite hardness and elastic modulus with increasing load.

Modification of the sit-to-stand task with the addition of gait initiation

This work was conducted to study the common task of initiating gait from a seated position. Ten subjects performed five trials each of sit-to-stand (STS) and sit-to-walk (STW). Centre of mass displacement and momentum, centre of pressure (COP) displacement and ground reaction forces were compared between conditions. Subjects significantly increased their linear momentum in both horizontal and vertical directions during and after seat unloading in STW compared to STS. Initiation of walking started around seat-off while subjects were still rising and COM projection was close to or behind the ankles. As balance is dependent on the initial momentum gained during the unloading phase, subjects who appeared to be more concerned with such balance broke their forward momentum and emphasized initial lateral displacement of the COP. Overall, the transitional task of STW is not a serial arrangement of two separate tasks, but a merging of a discrete task with a rhythmic one around the point of seat-off and the underlying strategies are heavily dependent on balance control.

Control of grip force during restraint of an object held between finger and thumb: responses of cutaneous afferents from the digits

Unexpected pulling and pushing loads exerted by an object held with a precision grip evoke automatic and graded increases in the grip force (normal to the grip surfaces) that prevent escape of the object; unloading elicits a decrease in grip force. Anesthesia of the digital nerves has shown that these grip reactions depend on sensory signals from the digits. In the present study we assessed the capacity of tactile afferents from the digits to trigger and scale the evoked grip responses. Using tungsten microelectrodes inserted percutaneously into the median nerve of awake human subjects, unitary recordings were made from ten FA I and 13 FA II rapidly adapting afferents, and 12 SA I and 18 SA II slowly adapting afferents. While the subject held a manipulandum between a finger and the thumb, tangential load forces were applied to the receptor-bearing digit (index, middle, or ring finger or thumb) as trapezoidal load-force profiles with a plateau amplitude of 0.5-2.0 N and rates of loading and unloading at 2-8 N/s, or as "step-loads" of 0.5 N delivered at 32 N/s. Such load trials were delivered in both the distal (pulling) and proximal (pushing) direction. FA I afferents responded consistently to the load forces, being recruited during the loading and unloading phases. During the loading ramp the ensemble discharge of the FA I afferents reflected the first time-derivative of the load force (i.e., the load-force rate). These afferents were relatively insensitive to the subject's grip force responses. However, high static finger forces appeared to suppress excitation of these afferents during the unloading phase. The FA II afferents were largely insensitive to the load trials: only with the step-loads did some afferents respond. Both classes of SA afferents were sensitive to load force and grip force, and discharge rates were graded by the rate of loading. The firing of the SA I afferents appeared to be relatively more influenced by the subject's grip-force response than the discharge of the SA II afferents, which were more influenced by the load-force stimulus. The direction in which the tangential load force was applied to the skin influenced the firing of most afferents and in particular the SA II afferents. Individual afferents within each class (except for the FA IIs) responded to the loading ramp before the onset of the subject's grip response and may thus be responsible for initiating the automatic increase in grip force. However, nearly half of the FA I afferents recruited by the load trials responded to the loading phase early enough to trigger the subject's grip-force response, whereas only ca. one-fifth of the SA Is and SA IIs did so. These observations, together with the high density of FA I receptors in the digits, might place the FA I afferents in a unique position to convey the information required to initiate and scale the reactive grip-force responses to the imposed load forces.

Microstructure and deformation rate during long‐term cyclic creep of the martensitic steel X22CrMoV12‐1

The microstructure of three specimens of the martensitic steel X22CrMoV12‐1 which had been subjected to long‐term cyclic creep at 873 K with intermittent phases of unloading (stress ratio R = 0) and compression (R = −1) was quantified by electron microscopy with regard to carbides, dislocations and pores. The laws of time dependent coarsening of carbides and strain controlled growth of subgrains found for monotonic creep hold also for cyclic creep. The longer time it takes cyclic creep to reach a given strain leads to a growth advantage of carbides compared to monotonic creep. The microstructural model of plastic deformation previously developed for monotonic creep on X20(22)CrMoV12‐1 allows to calculate the cyclic creep acceleration due to this advantage in carbide growth.

Interaction of rarefaction waves with wet foam

The interaction of rarefaction waves of different shapes with wet water foams is studied experimentally. It is found that the observed values of the pressure are greater, while the surface velocity is lower than the corresponding values predicted by the pseudogas model. The foam breakdown starts as the pressure decreases by 0.3 atm relative to the initial pressure. During downstream propagation of the rarefaction-wave leading edge the propagation velocity decreases. Using of water-based foams as effective screens for damping blast waves in different technological processes has caused considerable interest in studying wave propagation in such systems. The pressure wave dynamics in a foam have been investigated in much detail, both experimentally and theoretically [1–3]. However, the interaction of rarefaction waves with foam has practically never been studied, although it was mentioned in [4] that the unloading phase following the compression wave phase is one of the factors defining the damaging action of blast waves. Besides blast-wave damping, rarefaction wave propagation takes place if such waves are used to breakup foam in oil-producing wells [5]. Below, the interaction of rarefaction waves of different shapes with wet water foams is studied. The vertical shock tube described in detail in [3] was used in these experiments.

Normalized Impact Response and Damage in a Thin Composite Laminate Supported by a Rigid Substrate

The local impact response and damage of thin composite laminates are investigated. The impact model uses a static contact law which incorporates damage effects due to local contact stresses. A major difference between this contact law and the others used previously is that the damage is accounted for during the loading rather than during the unloading phase which is expected to yield a more reasonable prediction of the maximum impact force. Normalization of the governing differential equations and initial conditions yields a single governing parameter, which is a dimensionless group of all the pertinent impact parameters (i.e., impact velocity, impactor mass, material and geometric properties). The introduction of this parameter not only simplifies the problem but also provides the necessary scaling rules for model testing.

A method of estimating linear and angular accelerations in head impacts to pedestrians

In order to investigate the relationship between impact to the head and brain injury, we have developed a method, using information obtained from reconstruction of the collisions, of estimating the peak linear and angular accelerations of the head for pedestrian impacts on a vehicle. This information includes the location of the impact on the head, the impact velocity of the head, and the stiffness of the struck surface. In developing the method we assumed that the velocity of the head on striking the vehicle was the same as the velocity of the vehicle itself, that the force vector was normal to the surface of the skull, that the force-deflection curve characterising the combined response of the impacted surfaces was linear, and that the kinetic energy of the head immediately prior to impact was converted into strain energy in deforming the head and the vehicle structure. Only the loading phase of the impact was considered, there was no assumption of an elastic unloading phase. Using cadavers, the validity of these assumptions and hence the usefulness of the method were tested by comparing the estimates of peak linear acceleration with the results of 18 pedestrian-vehicle impact reconstructions. On average, the method underestimated the experimental values by about 15%, with a range of ±35%. The results from the application of this method are currently being used to study the relationship between the magnitude and direction of the impact to the head and the distribution and severity of the brain injury resulting from actual collisions.

Cyclic stress fields ahead of tension fatigue cracks in amorphous polymers

Cyclic near-tip stress fields which evolve ahead of fatigue cracks are known to have a pronounced effect on the fracture response of materials. This region of reversed damage is responsible for the initiation of fatigue cracks from stress concentrations under fully compressive cyclic loads, crack growth retardation following tensile overloads, and transient crack growth response under spectrum fatigue loading. By recourse to photoelasticity and laser interferometry, we provide direct and in situ measurements of cyclic near-tip stresses ahead of notches and fatigue cracks in amorphous polymers subjected to cyclic tension loading. The near-tip stresses are quantified during both the loading and unloading phases of the fatigue cycle, as well as during various stages of crack advance, in an attempt to establish a link between the evolution of near-tip stress fields and the advance of the fatigue crack. In addition, transmission electron microscopy analyses of the near-tip region have been performed to characterize cyclic deformation mechanisms.

Thermodynamic Analysis of Energy Dissipation by Pharmaceutical Tablets During Stress Unloading

A thermodynamic analysis of the energy transformations occurring within pharmaceutical tablets during the unloading phase of compaction was performed on Avicel, calcium phosphate, acetaminophen, and calcium oxalate. This analysis was based on viscoelastic stress data collected using an instrumented rotary press and was conducted for the purpose of determining the extent and nature of energy release, beyond that of force displacement, during this phase of compaction. The four materials investigated were selected to reflect a range of compaction properties, and the viscous vs force‐displacement energy distributions were seen to be consistent with their brittle vs plastic character. Magnesium stearate, added in various concentrations up to 4% w/w, was found to act both as an internal lubricant for particle‐particle slipping and as a moderator of inter‐particulate bonding.