Fourth Power Law Research Articles

Modeling the Fourth Power Law Response of Viscoplastic Materials

A standard way to parametrize a steady structured shock going through a viscoplastic material, is by plotting the stress jump across the wave (Ds) as function of its maximum strain rate () plotted on a logarithmic scale. Doing this we usually get a straight line, which means that . Grady and coworkers noticed in the 1980s that for many materials b=4. Since then, several workers in the field tried to understand this unusual behavior, but they were not able to arrive at any plausible explanation. Recently, Grady revisited his fourth power law with an extensive review, including an outline of various mechanisms proposed as possible explanations of this law. But still, none of these mechanisms is able to explain the fourth power law response. In view of this background, we use here the fourth power law to calibrate the strain rate component of the constitutive response of Al 6061-T6. To this end, we integrate the steady viscoplastic equations (developed long ago) describing the propagation of a plane wave from the elastic precursor level to the stress plateau level. We find that in order to reproduce the fourth power law, the plastic deformation rate has to depend on the deviator overstress by: , with α=2.38. But for high shock levels, when the elastic precursor is overdriven, using this value of α leads to a higher value of the slope b, close to b=6. This result has yet to be verified experimentally.

Generalization of Finite Integral Transforms for Treating Nonlinear Problems in Heat Diffusion. Part II: Application to a nonlinear case.(Dept.P)

A solution methodology, based on the finite integral transform technique, appropriate for solving nonlinear problems of heat diffusion was developed by the author in previous work [1,2]. In this paper, the method is applied to solve the heat diffusion problem in a finite region subject to nonlinear boundary conditions due to radiation exchange at the interface according to the fourth power law. The results obtained form this analytic Solution are compared with those obtained from a numerical solution developed using an explicit finite difference method.

Revisiting the Power Law Characteristics of the Plastic Shock Front under Shock Loading.

Under uniaxial shock compression, the steepness of the plastic shock front usually exhibits power law characteristics with the Hugoniot pressure, also known as the "Swegle-Grady law." In this Letter, we show that the Swegle-Grady law can be described better by a third power law rather than the classical fourth power law at the strain rate between 10^{5}-10^{7} s^{-1}. A simple dislocation-based continuum model is developed, which reproduced the third power law and revealed very good agreement with recent experiments of multiple types of metals quantitatively. New insights into this unusual macroscopic phenomenon are presented through quantifying the connection between the macroscopic mechanical response and the collective dynamics of dislocation assembles. It is found that the Swegle-Grady law results from the particular stress dependence of the plasticity behaviors, and that the difference between the third power scaling and the classical fourth power scaling results from different shock dissipative actions.

Steady shock waves in porous metals: Viscosity and micro-inertia effects

The structure of steady shock waves in porous solids is a complex phenomenon involving in general the interplay of micro-inertia effects with the nonlinear elastic viscoplastic matrix response. Micro-inertia effects are due to the important acceleration of material particles in the vicinity of collapsing voids. By adopting the analytical approach recently developed for porous metals by Czarnota et al. [J. Mech. Phys. Solids 107 (2017)], we analyze the effects of matrix rate sensitivity, shock stress amplitude and micro-inertia on the structure of planar shock waves. We also analyze the relationship that links the strain rate within the shock to the jump of the stress across the shock. The fourth power law experimentally revealed for dense metals, Swegle & Grady [J. Appl. Phys. 58 (1985)] does not hold for heterogeneous materials. By considering the case of porous aluminum, we show that this relationship is characterized by two distinct regimes: (i) the first regime holds for weak shock intensities and is representative of the viscoplastic response of the dense matrix material, (ii) the second regime, that holds for shock of higher amplitude, is dominated by micro-inertia effects and is strongly influenced by the pore size. Micro-inertia effects appear to be quite beneficial since they are conducive to shock mitigation by attenuating the level of strain rate and of acceleration sustained by material particles.

Mechanisms and kinetics during reactive infiltration of molten silicon in porous graphite

Liquid silicon Infiltration (LSI) is a fast and economical process to manufacture SiC-based ceramics. For a better understanding of reactive melt infiltration of liquid silicon, the wetting and infiltration of porous graphite by molten silicon were investigated at 1450, 1500 and 1550 °C for duration comprised between 10 s and 1 h. Infiltrations tests were performed in an argon atmosphere with an inductively heated furnace operating with heating and cooling rates of 300 °C.min−1. The formation and growth of SiC grains were investigated at the outer surface and within graphitic carbon substrates with 11% porosity and a narrow pore size distribution centered at 2 μm. The length of the infiltrated zone and the SiC crystals growth were determined from scanning electron microscopy. Rapid spreading and infiltration of molten silicon are observed from the first 60 s. The growth rate of the interfacial SiC layer obeys a fourth-power law with an activation energy of 260 ± 30 kJ mol−1. Pore filling by SiC is limited by volume diffusion with an activation energy equal to 320 ± 40 kJ mol−1.

Empirical Study of Near Ground Propagation in Forest Terrain for Internet-of-Things Type Device-to-Device Communication

This paper presents continuous-wave power measurements conducted in forest terrain with a focus on path-loss in a device-to-device (D2D) communication scenario. The measurements are performed at 917.5 MHz with measurement ranges extending to more than 2.5 km, using a purpose-developed measurement system with a dynamic range of 180 dB. The impact of different antenna heights has been investigated by placing antennas at 1.5-, 2.5-, and 3.5-m elevation over the terrain. The measurements show that the antenna elevation in the given scenario has no significant impact on the received power. The measurements also show that the dominant path of propagation is through the foliage for the first approximately 1000 m, resulting in foliage excess loss being the dominant loss mechanism in this region. After approximately 1000 m, the measured received power tends to follow the fourth-power law, indicating that the dominant loss mechanism in this region is the distance-dependent path-loss. The measurement results have been compared to models proposed in the literature based on empirical data, as well as models proposed by ITU and COST. The comparison shows that only two models predicted the limit in the foliage excess loss in the measurement data. By applying these excess loss models in combination with path-loss models, it is possible to model the measured total path-loss with a root-mean-square error of less than 10 dB. This is achieved by either applying a model proposed by Tewari or a combination between the two-ray path-loss model and the ITU-R P.2108 clutter loss model.

On the homogeneous nucleation and propagation of dislocations under shock compression

The dynamic response of crystalline materials subjected to extreme shock compression is not well understood. The interaction between the propagating shock wave and the material’s defect occurs at the sub-nanosecond timescale which makes in situ experimental measurements very challenging. Therefore, computer simulation coupled with theoretical modelling and available experimental data is useful to determine the underlying physics behind shock-induced plasticity. In this work, multiscale dislocation dynamics plasticity (MDDP) calculations are carried out to simulate the mechanical response of copper reported at ultra-high strain rates shock loading. We compare the value of threshold stress for homogeneous nucleation obtained from elastodynamic solution and standard nucleation theory with MDDP predictions for copper single crystals oriented in the [0 0 1]. MDDP homogeneous nucleation simulations are then carried out to investigate several aspects of shock-induced deformation such as; stress profile characteristics, plastic relaxation, dislocation microstructure evolution and temperature rise behind the wave front. The computation results show that the stresses exhibit an elastic overshoot followed by rapid relaxation such that the 1D state of strain is transformed into a 3D state of strain due to plastic flow. We demonstrate that MDDP computations of the dislocation density, peak pressure, dynamics yielding and flow stress are in good agreement with recent experimental findings and compare well with the predictions of several dislocation-based continuum models. MDDP-based models for dislocation density evolution, saturation dislocation density, temperature rise due to plastic work and strain rate hardening are proposed. Additionally, we demonstrated using MDDP computations along with recent experimental reports the breakdown of the fourth power law of Swegle and Grady in the homogeneous nucleation regime.

Sound attenuation in amorphous silica at frequencies near the boson peak

We use molecular dynamics phonon wave packet (WP) simulations to study acoustic propagation and attenuation in amorphous silica $(a\text{\ensuremath{-}}\mathrm{Si}{\mathrm{O}}_{2})$ at frequencies near the boson peak (BP) position and compare them with the results of equilibrium molecular dynamics (EMD) simulations. The sound attenuation coefficients obtained from WP simulations are generally consistent with those from EMD predictions and have reasonable agreement with the existing experimental data. Near the BP position, we found the frequency-dependent sound attenuation coefficients for longitudinal and transverse modes both follow the Rayleigh-scattering fourth power law. Above the BP frequency, however, the propagating phonon is essentially attenuated in $a\text{\ensuremath{-}}\mathrm{Si}{\mathrm{O}}_{2}$ within a few nanometers, and the accurate determination of the sound attenuation coefficients by the WP simulation becomes challenging. The modeling results provide a reference for future experimental investigations of sound attenuation in $a\text{\ensuremath{-}}\mathrm{Si}{\mathrm{O}}_{2}$ thin film using narrow-band coherent phonons.

Feasibility of Using 4th Power Law in Design of Plastic Deformation Resistant Low Volume Roads

A low volume road (LVR) structural design has two phases: first one is selection of appropriate construction materials and second is the determination of layer thicknesses under the certain traffic and environmental conditions with considering the subgrade bearing capacity. Pavements are prompted to serve the traffic without reaching the terminal serviceability index over its design life. Rut accumulation (plastic deformation) is the most common pavement deterioration type of flexible pavements. Therefore the main goal of the design is prevent rutting. Many low volume road design manual assume that plastic deformation occurs only in subgrade. Construction of overlying layers by selective high performance materials according to the related material and construction specifications is the reason of this assumption. In fact, the assumption is not much reasonable especially for with no, or thinly overlaid low volume road pavements, where the major structural strength is comprised of unbound granular pavement materials and where the principal distress mechanism is rutting in the aggregate layers.Subgrade bearing capacity and the traffic are the main input parameters in the design stage of low volume roads. Subgrade bearing capacity is expressed with California Bearing Ratio (CBR) or Resilient Modulus (Mr). The “traffic” term is determined by Equivalent Standard Axle Load (ESAL) repetitions which is often admitted as 80kN single axle load. Although it is not too difficult to determine an axle load for an individual vehicle, it becomes quite complicated to determine the number and types of axle loads that a particular pavement will be subjected over its design life. For calculation of Load Damage Factors of different vehicle types, which have various axle load and configurations, a generalized fourth-power law has been used for more than a half century. The objective of this study is to indicate the limitations and difficulties faced on reliably applying a power law relationship in design of LVRs with no, or only thin seals. If a power law relationship to be used due to its simplicity, several parameters must be considered in selection of the power value. Such as stress dependent behavior of unbound granular materials and the selected distress type.

Highway-user fees for overweight trucks: Consequences of deviations from the fourth-power law

ABSTRACTIn updating road-user fees, highway agencies are motivated by the twin objectives of equity and efficiency. The concepts of marginal pavement consumption cost (MPCC) and the fourth-power law have been used in past research in a bid to establish efficient and equitable user fees. Over the past few decades, the use of the fourth-power law has faced some criticism that the actual power of the relationship between loading and pavement deterioration deviates significantly from 4.0. In addressing this issue, this article presents a methodology that investigates the consequences of the power variations to the estimated MPCC and the distributional (equity) impacts of road-user fees for overweight (OW) trucks. The methodology incorporates practical schedules of highway maintenance, rehabilitation, and reconstruction. The analysis suggests that there are extremely small differences between the MPCC values that result from using different values of the exponent in the power law. However, different values of the exponent in the power law result in significantly different equity impacts. This article demonstrates that the equity impacts are moderate in the case of OW trucks whose individual axles are loaded close to standard axle weights. For OW trucks that are loaded within 10% of the maximum permissible weight, this article estimates that variation of the power value from 3.0 to 5.0 results in a 1%–10% surplus or deficit of their road-user charges.

Oxidation behaviours of electron beam physical vapour deposition fabricated microcrystalline Ni–23·5Cr–2·66Co–1·44Al superalloy sheet at 1000°C

Isothermal oxidation of as-deposited microcrystalline Ni–23·5Cr–2·66Co–1·44Al (wt-%) alloy sheet fabricated by electron beam physical vapour deposition exposed at 1000°C in air up to 100 hours was investigated. Surfaces and cross-sections of the oxidised specimens were studied by scanning electron microscopy. Phase identification of the oxide scale was performed by glancing angle X-ray diffraction. The results indicated that the oxidation kinetics of the alloy sheet obeyed an approximate cubical power law at the early stage of exposure and approximate fourth power law for long oxidation terms. It was found that the oxide scales consisted mainly of an outer Cr2O3 oxide layer, and an inner NiO and NiCr2O4 mixed oxides layers. An Al2O3 layer was formed at the scale/substrate interface after a long-time oxidation. This showed that the outward diffusion rate of Cr atoms was higher than that of Ni atoms and Al atoms.

Unifying role of dissipative action in the dynamic failure of solids

A fourth-power law underlying the steady shock-wave structure and solid viscosity of condensed material has been observed for a wide range of metals and non-metals. The fourth-power law relates the steady-wave Hugoniot pressure to the fourth power of the strain rate during passage of the material through the structured shock wave. Preceding the fourth-power law was the observation in a shock transition that the product of the shock dissipation energy and the shock transition time is a constant independent of the shock pressure amplitude. Invariance of this energy-time product implies the fourth-power law. This property of the shock transition in solids was initially identified as a shock invariant. More recently, it has been referred to as the dissipative action, although no relationship to the accepted definitions of action in mechanics has been demonstrated. This same invariant property has application to a wider range of transient failure phenomena in solids. Invariance of this dissipation action has application to spall fracture, failure through adiabatic shear, shock compaction of granular media, and perhaps others. Through models of the failure processes, a clearer picture of the physics underlying the observed invariance is emerging. These insights in turn are leading to a better understanding of the shock deformation processes underlying the fourth-power law. Experimental result and material models encompassing the dynamic failure of solids are explored for the purpose of demonstrating commonalities leading to invariance of the dissipation action. Calculations are extended to aluminum and uranium metals with the intent of predicting micro-scale dynamics and spatial structure in the steady shock wave.

The Unifying Role of Dissipative Action in the Dynamic Failure of Solids

Dissipative action, the product of dissipation energy and transport time, is fundamental to the dynamic failure of solids. Invariance of the dissipative action underlies the fourth-power nature of structured shock waves observed in selected solid metals and compounds. Dynamic failure through shock compaction, tensile spall and adiabatic shear are also governed by a constancy of the dissipative action. This commonality underlying the various modes of dynamic failure is described and leads to deeper insights into failure of solids in the intense shock wave event. These insights in turn are leading to a better understanding of the shock deformation processes underlying the fourth-power law. Experimental result and material models encompassing the dynamic failure of solids are explored for the purpose of demonstrating commonalities leading to invariance of the dissipation action. Calculations are extended to aluminum and uranium metals with the intent of predicting micro-scale energetics and spatial scales in the structured shock wave.

A Continuum model for pressure-flow relationship in human pulmonary circulation

A continuum model was introduced to analyze the pressure-flow relationship for steady flow in human pulmonary circulation. The continuum approach was based on the principles of continuum mechanics in conjunction with detailed measurement of vascular geometry, vascular elasticity and blood rheology. The pulmonary arteries and veins were considered as elastic tubes and the "fifth-power law" was used to describe the pressure-flow relationship. For pulmonary capillaries, the "sheet-flow" theory was employed and the pressure-flow relationship was represented by the "fourth-power law". In this paper, the pressure-flow relationship for the whole pulmonary circulation and the longitudinal pressure distribution along the streamlines were studied. Our computed data showed general agreement with the experimental data for the normal subjects and the patients with mitral stenosis and chronic bronchitis in the literature. In conclusion, our continuum model can be used to predict the changes of steady flow in human pulmonary circulation.

Interaction forces between soft magnetic particles in uniform and non-uniform magnetic fields

The influence of the magnetization of a soft magnetic sphere on the surrounding magnetic field is measured and characterized. The interaction force between two soft magnetic particles is directly measured using an ultra precision load sensor in uniform and non-uniform magnetic fields. The interaction force largely follows an inverse fourth power law as a function of separation distance between particle centers. At small distances, the effect of magnetization of one particle on the magnetization of its adjacent particle causes the attractive (repulsive) force to be larger (smaller) than that predicted by the inverse fourth power law. The theoretical prediction based on a modified dipole model, that takes into account the coupling effect of the magnetization among soft magnetic particles, gives excellent agreement with the measured force in a uniform magnetic field. The interaction force under a non-uniform applied magnetic field can be reasonably predicted using the dipole–dipole interaction model when the local magnetic field is used to determine the magnetization.

High-frequency attenuation and backscatter measurements of rat blood between 30 and 60 MHz

There has recently been a great deal of interest in noninvasive high-frequency ultrasound imaging of small animals such as rats due to their being the preferred animal model for gene therapy and cancer research. Improving the interpretation of the obtained images and furthering the development of the imaging devices require a detailed knowledge of the ultrasound attenuation and backscattering of biological tissue (e.g. blood) at high frequencies. In the present study, the attenuation and backscattering coefficients of the rat red blood cell (RBC) suspensions and whole blood with hematocrits ranging from 6% to 40% were measured between 30 and 60 MHz using a modified substitution approach. The acoustic parameters of porcine blood under the same conditions were also measured in order to compare differences in the blood properties between these two animals. For porcine blood, both whole blood and RBC suspension were stirred at a rotation speed of 200 rpm. Three different rotation speeds of 100, 200 and 300 rpm were carried out for rat blood experiments. The attenuation coefficients of both rat and porcine blood were found to increase linearly with frequency and hematocrit (the values of coefficients of determination (r2) are around 0.82–0.97 for all cases). The average attenuation coefficient of rat whole blood with a hematocrit of 40% increased from 0.26 Nepers mm−1 at 30 MHz to 0.47 Nepers mm−1 at 60 MHz. The maximum backscattering coefficients of both rat and porcine RBC suspensions were between 10% and 15% hematocrits at all frequencies. The fourth-power dependence of backscatter on frequency was approximately valid for rat RBC suspensions with hematocrits between 6% and 40%. However, the frequency dependence of the backscatter estimate deviates from a fourth-power law for porcine RBC suspension with hematocrit higher than 20%. The backscattering coefficient plateaued for hematocrits higher than 15% in porcine blood, but for rat blood it was maximal around a hematocrit of 20% at the same rotation speed, and shifted to a hematocrit of 10% at a higher speed. The backscattering properties of rat RBCs in plasma are similar to those of RBCs in saline at a higher rotation speed. The differences in attenuation and backscattering between rat and porcine blood may be attributed to RBCs' being smaller and the RBC aggregation level being lower for rat blood than for porcine blood.

Cooling and warming laws: an exact analytical solution

This paper deals with temperature variations over time of objects placed in a constant-temperature environment in the presence of thermal radiation. After a historical introduction, the paper discusses cooling and warming laws, by taking into account first solely object–environment energy exchange by thermal radiation, and then adding object–environment heat exchange by convection. These processes are usually evaluated by approximating the law of exchange of thermal radiation by a linear relationship between power exchange and temperature difference. In contrast, in this paper an exact analytical solution considering Stefan's fourth power law is provided, under some general hypotheses, for both cases. A comparison with exponential approximations and with a historical law proposed by Dulong and Petit in 1817 is presented. Data of an experiment are used to test the analytical solution: the test has allowed evaluating the heat transfer coefficient h of the experiment and has shown that our solution provides a better fit with the measured values than any exponential function. The topic is developed in a way which can be suitable both for undergraduate students and for general physicists.

Structured shock waves and the fourth-power law

A fourth-power law relating the stress jump through a steady structured shock wave and the maximum strain rate within the shock wave has received recognition as a unifying relation over a sensibly wide range of materials and shock compression amplitudes. Less widely recognized is the complementary applicability of the invariance of the product of the energy dissipated and the time over which this energy is dissipated through a structure shock wave—a property with the dimensions of action. While the latter invariance implies the fourth-power behavior for steady shock waves, this invariance also has an apparent application to unsteady structured waves under certain conditions. A brief history of the origin of the fourth-power law is provided. Some commentary is offered on the physical principles of solid viscosity in the shock wave and the underlying invariance of the energy-time product in the shock wave event. The results for steady structured waves in porous and composite solids, which do not in general exhibit fourth-power behavior, are examined. Observations of systematic behaviors with component cell size and impedance disparities suggest acoustic scattering may contribute to wave structuring in selected materials.

Nonlinear Heat Conduction in Isotropic and Orthotropic Materials with Laser Heat Source

Although the convective cooling of simple materials undergoing laser irradiation has been well documented, the effects of radiative cooling for composite materials undergoing laser heating are not as well documented and are of increasing importance to the manufacturing and aerospace industries. The temperature distributions within one-dimensional homogeneous and two-dimensional orthotropic radiating plates subjected to a spatially decaying laser source are presented. The incident energy at the exposed external surface is partially absorbed and transferred through the plate by conduction. The temperature results are presented as a function of a wide variety of thermal and geometric dimensionless parameters. The effects of surface radiation and material conductivity are examined in detail. Results from this study are compared to data in the open literature to gain a better understanding of the effects that the fourth power law for radiation, local temperature gradient in conduction, and first power of the temperature difference in convection have on the resultant temperature profiles.

Loschmidt, Stefan, and Stigler’s Law of Eponymy

Josef Loschmidt (1821–1895) and Josef Stefan (1835–1893) were eminent scientists in the Institute of Physics at the University of Vienna during the second half of the nineteenth century but are not well known today, as their legacies have been recognized differently by the scientific community. Loschmidt first described the structure of the benzene molecule and determined the size of air molecules, from which the number of molecules per unit volume can easily be determined, yet others received the credit for these achievements. Stefan posited the fourth-power temperature radiation law, but neither he nor his student Ludwig Boltzmann (1844–1906) calculated the proportionality constant now known as the Stefan-Boltzmann constant. These are instances of Stigler’s Law of Eponymy. Besides these achievements, perhaps the greatest unheralded contribution of both Loschmidt and Stefan was the experimental evidence they provided in support of the emerging kinetic theory of gases.