Value Of Scale Parameter Research Articles

Size-dependent vibration of functionally graded curved microbeams based on the modified strain gradient elasticity theory

On the basis of the modified strain gradient elasticity theory, the free vibration characteristics of curved microbeams made of functionally graded materials (FGMs) whose material properties vary in the thickness direction are investigated. A size-dependent first-order shear deformation beam model is developed containing three internal material length scale parameters to incorporate small-scale effect. Through Hamilton’s principle, the higher-order governing equations of motion and boundary conditions are derived. Natural frequencies of FGM curved microbeams corresponding to different mode numbers are evaluated for over a wide range of material property gradient index, dimensionless length scale parameter and aspect ratio. Moreover, the results obtained via the present non-classical first-order shear deformation beam model are compared with those of degenerated beam models based on the modified couple stress and the classical theories. It is found that the difference between the natural frequencies predicted by the various beam models is more significant for lower values of dimensionless length scale parameter and higher values of mode number.

Size-dependent buckling analysis of functionally graded third-order shear deformable microbeams including thermal environment effect

Presented herein is the prediction of buckling behavior of size-dependent microbeams made of functionally graded materials (FGMs) including thermal environment effect. To this purpose, strain gradient elasticity theory is incorporated into the classical third-order shear deformation beam theory to develop a non-classical beam model which contains three additional internal material length scale parameters to consider the effects of size dependencies. The higher-order governing differential equations are derived on the basis of Hamilton’s principle. Afterward, the size-dependent differential equations and related boundary conditions are discretized along with commonly used end supports by employing generalized differential quadrature (GDQ) method. A parametric study is carried out to demonstrate the influences of the dimensionless length scale parameter, material property gradient index, temperature change, length-to-thickness aspect ratio and end supports on the buckling characteristics of FGM microbeams. It is revealed that temperature change plays more important role in the buckling behavior of FGM microbeams with higher values of dimensionless length scale parameter.

Inclusive τ lepton decay: the effects due to hadronization

Theoretical description of inclusive τ lepton hadronic decay is performed in the framework of Dispersive approach to QCD. The significance of effects due to hadronization is demonstrated. The approach on hand is capable of describing experimental data on τ lepton hadronic decay in vector and axial-vector channels. The vicinity of values of QCD scale parameter obtained in both channels testifies to the self-consistency of developed approach.

Significance of Exchanging SSURGO and STATSGO Data When Modeling Hydrology in Diverse Physiographic Terranes

The Water Availability Tool for Environmental Resources (WATER) is a TOPMODEL‐based hydrologic model that depends on spatially accurate soils data to function in diverse terranes. In Kentucky, this includes mountainous regions, karstic plateau, and alluvial plains. Soils data are critical because they quantify the space to store water, as well as how water moves through the soil to the stream during storm events. We compared how the model performs using two different sources of soils data—Soil Survey Geographic Database (SSURGO) and State Soil Geographic Database laboratory data (STATSGO)—for 21 basins ranging in size from 17 to 1564 km2. Model results were consistently better when SSURGO data were used, likely due to the higher field capacity, porosity, and available‐water holding capacity, which cause the model to store more soil‐water in the landscape and improve streamflow estimates for both low‐ and high‐flow conditions. In addition, there were significant differences in the conductivity multiplier and scaling parameter values that describe how water moves vertically and laterally, respectively, as quantified by TOPMODEL. We also evaluated whether partitioning areas that drain to streams via sinkholes in karstic basins as separate hydrologic modeling units (HMUs) improved model performance. There were significant differences between HMUs in properties that control soil‐water storage in the model, although the effect of partitioning these HMUs on streamflow simulation was inconclusive.

Application of Weibull distribution model in describing the pile salting of goat meat slices

Application of Weibull model in the pile salting of goat meat slices was studied to predict the moisture and salt contents and determine the effective water diffusion coefficient (De). The high coefficients of determination (R2 > 0.99) and low mean relative error (< 10%) indicated the acceptability of Weibull model for predicting both the moisture and salt contents and determining De. Values of scale (α) and shape (β) parameters of Weibull model for fractional amount of moisture content ranged from 0.667 to 0.873 day and from 0.527 to 0.743 day, respectively. Values of α and β parameters of Weibull model for fractional amount of salt content ranged from 0.350 to 1.192 day and from 0.578 to 0.885 day, respectively. Values of De ranged approximately from 1.66 × 10-10 to 2.44 × 10-10 m2/s. Key words: Weibull model, partial replacement, salting mixtures, effective water diffusion coefficient.

Numerical modeling of the coastal circulation off northern Baja California and southern California

A numerical study of the coastal circulation off northern Baja California and Southern California is presented. A series of numerical experiments was designed with the objective of understanding the main mechanisms that maintain and control the variability of the California Surface Current (CSC) and the California Undercurrent (CUC) off the northern Baja California coast. The coastal circulation variability in the area reflects the seasonal and mesoscale evolution of the CSC and the CUC. These currents are formed outside the model domain and are included in the open boundary conditions, which were obtained from SODA model (Simple Ocean Data Assimilation). The seasonal evolution of these currents is mainly due to wind forcing. The spring–summer intensification of the CSC is the result of density fronts caused by coastal upwelling. Upwelling produces, in spring, a strong and deep (>50m) density gradient due to an Ekman transport up to 100m3s−1×100m of coastline. The wind stress curl is strong in spring and summer, but it is more effective in strengthening the CUC during summer because during spring its effect on the CUC is inhibited due to a stronger and deeper CSC at this time. The coastal heads, especially Point Loma, enhance positive vertical relative vorticity in the near-coast part of the CSC, causing flow separation. The mesoscale variability of the currents is associated with the presence of cyclonic eddies, which interact with the seasonal currents, migrate west, and merge with each other. The high variability found in the CSC and the CUC, the low values of the baroclinic time scale parameter, and the change of sign of the potential vorticity gradient suggest that both currents are barotropically and baroclinically unstable.

Dynamic stability analysis of functionally graded higher-order shear deformable microshells based on the modified couple stress elasticity theory

Size-dependent dynamic stability response of higher-order shear deformable cylindrical microshells made of functionally graded materials (FGMs) and subjected to simply supported end supports is investigated. Material properties of the microshells vary in the thickness direction according to the Mori–Tanaka scheme. The modified couple stress elasticity theory in conjunction with the classical higher-order shear deformation shell theory is utilized to develop non-classical shell model containing additional internal length scale parameter to interpret size effect. The differential equations of motion and boundary conditions are derived by using Hamilton’s principle. The governing equations are then written in the form of Mathieu–Hill equations and then Bolotin’s method is employed to determine the instability regions. Selected numerical results are given to indicate the influences of internal length scale parameter, material property gradient index, static load factor and axial wave number on the dynamic stability behavior of FGM microshells. It is found that the width of the instability region for an FGM microshell increases with the decrease of the value of dimensionless length scale parameter. Moreover, it is shown that the classical shell model has an overestimated prediction for the width of instability region corresponding to the FGM microshells especially with lower values of material property gradient index.

Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory

In the present investigation, the bending, buckling and free vibration responses of Timoshenko microbeams made of functionally graded materials (FGMs) are studied. To take size effect into account, the most general strain gradient elasticity theory is incorporated into the classical Timoshenko beam theory to develop a size-dependent beam model containing five additional material length scale parameters. The model accommodates the beam models based on the strain gradient theory (SGT), the modified strain gradient theory (MSGT), the modified couple stress theory (MCST) and the classical theory (CT) as special cases. By using Hamilton’s principle, the governing equations and corresponding boundary conditions are derived. Afterward, the governing equations and associated boundary conditions are discretized by employing generalized differential quadrature (GDQ) method. Selected numerical results are given to demonstrate the size-dependent mechanical characteristics of FGM microbeams. Moreover, a comparison between the various beam models on the basis of MCST, MSGT and CT are presented. It is observed that the critical buckling loads and natural frequencies predicted by the beam models based on MSGT and CT are the maximum and minimum values, respectively. By increasing the value of length scale parameter, the deflection curve of FGM microbeam tends to the curve obtained by CT.

Model of geomedia containing defects: collective effects of defects evolution during formation of potential earthquake foci

This paper describes the statistical thermo-dynamical evolution of an ensemble of defects in the geomedium in the field of externally applied stresses. The authors introduce ‘tensor structural’ variables associated with two specific types of defects, fractures and localized shear faults (Fig. 1). Based on the procedure for averaging of the structural variables by statistical ensembles of defects, a self-consistency equation is developed; it determines the dependence of the macroscopic tensor of defects-induced strain on values of external stresses, the original pattern and interaction of defects. In the dimensionless case, the equation contains only the parameter of structural scaling, i.e. the ratio of specific structural scales, including the size of defects and an average distance between the defects. The self-consistency equation yields three typical responds of the geomedium containing defects to the increasing external stress (Fig. 2). The responses are determined from values of the structural scaling parameter. The concept of non-equilibrium free energy for a medium containing defects, given similar to the Ginzburg-Landau decomposition, allowed to construct evolutionary equations for the introduced parameters of order (deformation due to defects, and the structural scaling parameter) and to explore their solutions (Fig. 3). It is shown that the first response corresponds to stable quasi-plastic deformation of the geomedium, which occurs in regularly located areas characterized by the absence of collective orientation effects. Reducing the structural scaling parameter leads to the second response characterized by the occurrence of an area of meta-stability in the behavior of the medium containing defects, when, at a certain critical stress, the orientation transition takes place in the ensemble of interacting defects, which is accompanied by an abrupt increase of deformation (Fig. 2). Under the given observation/averaging scale, this transition is manifested by localized cataclastic deformation (i.e. a set of weak earthquakes), which migrates in space at a velocity several orders of magnitude lower than the speed of sound, as a ‘slow’ deformation wave (Fig. 3). Further reduction of the structural scaling parameter leads to degeneracy of the orientation meta-stability and formation of localized dissipative defect structures in the medium. Once the critical stress is reached, such structures develop in the blow-up regime, i.e. the mode of avalanche-unstable growth of defects in the localized area that is shrinking eventually. At the scale of observation, this process is manifested as brittle fracturing that causes formation of a deformation zone, which size is proportional to the scale of observation, and corresponds to occurrence of a strong earthquake. On the basis of the proposed model showing the behavior of the geomedium containing defects in the field of external stresses, it is possible to describe main ways of stress relaxation in the rock massives – brittle large-scale destruction and cataclastic deformation as consequences of the collective behavior of defects, which is determined by the structural scaling parameter. Results of this study may prove useful for estimation of critical stresses and assessment of the geomedium status in seismically active regions and be viewed as model representations of the physical hypothesis about the uniform nature of deve­lopment of discontinuities/defects in a wide range of spatial scales.

Hadronic effects in low-energy QCD: inclusive τ lepton decay

The inclusive τ lepton hadronic decay is studied within Dispersive approach to QCD. The significance of effects due to hadronization is convincingly demonstrated. The approach on hand proves to be capable of describing experimental data on τ lepton hadronic decay in vector and axial–vector channels. The vicinity of values of QCD scale parameter obtained in both channels bears witness to the self–consistency of developed approach.

Vegetated mixing layer around a finite‐size patch of submerged plants: Part 2. Turbulence statistics and structures

Dynamics of vegetated mixing layer displays analogy with canonical and shallow mixing layers. Universality of turbulence and minor influence of flexible aquatic plants on the individual turbulent eddies are anticipated as the prerequisites for the analogy. This paper explores theoretically and experimentally some aspects of analogy by examining spatial patterns of turbulence characteristics around a finite‐size patch of submerged flexible vegetation the population density of which was varied between the experimental runs [Sukhodolova and Sukhodolov, 2012]. It introduces analysis of turbulent shear stress, kinetic energy, dissipation rate, and periodicity of organized motions inside the vegetated mixing layer. The theory is compared with the field experiments. The results of the study indicate that the flow structure around the patch with dense population agrees well with the proposed theoretical scaling relations and reveals many aspects of the analogy to hydrodynamic mixing layers. Around the patch with sparse vegetation the flow structure is best represented by a perturbed boundary layer model. These conclusions on the dynamics of the flow are supported by the analysis of scaled turbulence characteristics. The patterns of turbulence profiles are approximated by scaling functions fairly well while the values of scaling parameters differ from the values reported for canonical and shallow mixing layers. This is an important result which agrees with the previous reports of laboratory research with model vegetation and is explained by a combined effect of internal shear and friction at the top of the canopy.

Analytical closed-form solutions for size-dependent static pull-in behavior in electrostatic micro-actuators via Fredholm integral equation

In this paper, a novel method is proposed for the first time to obtain static pull-in voltages with fringing field effects in electrostatically actuated cantilever and clamped-clamped micro-beams where the mid-plane stretching and the residual axial load are taken into account for clamped-clamped boundary conditions. The non-classical Euler–Bernoulli beam model containing one material length scale parameter is adopted to effectively capture the size effect. In the solution procedure, the governing fourth-order differential equation of variable coefficients is converted into a Fredholm integral equation. By adopting the first natural mode of the cantilever and clamped-clamped micro-beams as a deflection shape function, the resulting equation is solved for the static pull-in voltages. The accuracy of the present analytical closed-form solution is verified through comparing with the experimentally measured and numerical data conducted in the published works. From the experimental data available in the literature, the value of the material length scale parameter for the (poly)silicon is estimated to be in the order of magnitude of 10−1μm. Then, the effect of the material length scale parameter on the pull-in voltages of the cantilever and clamped-clamped micro-beams is investigated. The results indicate that the tensile residual stress can extend the validity range of the classical continuum mechanics to lower beam thicknesses. It is also found that microcantilever beams are more sensitive to the size effect than their corresponding clamped-clamped micro-beams.

On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory

The prime aim of the present study is to predict the free vibration behavior of microplates made of functionally graded materials (FGMs). The material properties of FGM microplates are assumed to be varied across the thickness of the microplates according to the Mori–Tanaka homogenization technique. On the basis of strain gradient elasticity theory, a non-classical higher-order shear deformable plate model containing three material length scale parameters is developed which can effectively capture the size dependencies. By using Hamilton’s principle, the size-dependent governing differential equations of motion and associated boundary conditions are derived. To evaluate the natural frequencies of FGM microplates, a Navier-type closed-form solution is carried out in which the generalized displacements are stated as multiplication of undetermined functions with known trigonometric functions so as to satisfy identically the simply-supported boundary conditions at all edges. Selected numerical results are presented to reveal the influences of dimensionless length scale parameter, material property gradient index and aspect ratio on the free vibration characteristics of FGM microplates. It is found that by approaching the thickness of microplates to the value of internal material length scale parameter, the natural frequency increases considerably.

Prediction of nonlocal scale parameter for carbon nanotubes

Based on the theory of nonlocal elasticity, a nonlocal shell model accounting for the small scale effect is developed for the bending characteristics of CNTs subjected to the concentrated load. With this nonlocal shell model, explicit expressions are derived for the bending solutions. To extract the proper values of nonlocal scale parameter, we have made molecular dynamics (MD) simulations for various radii and lengths of armchair and zigzag CNTs, the results of which are matched with those of nonlocal continuum model. It is found that the present nonlocal elastic shell model with its appropriate values of nonlocal scale parameter has the capability to predict the bending behavior of CNTs, which is comparable with the results of MD simulations. Moreover, exact closed form solutions for the nonlocal scale parameter for zigzag and armchair CNTs are obtained. The results show that nonlocal scale parameter is independent of the length of CNTs, and dependent on the radius of CNTs.

Slosh Dynamics of Liquid-Filled Rigid Containers: Two-Dimensional Meshless Local Petrov-Galerkin Approach

This paper presents some of the interesting effects arising from the nonlinear motion of the liquid-free surface, due to sloshing, in a partially filled rigid container subjected to forced excitation. A two-dimensional meshless local Petrov-Galerkin method is used for the numerical simulation of the problem. A local symmetric weak form (LSWF) for nonlinear sloshing of liquid is developed, and a truly meshless method, based on LSWF and moving least squares (MLS) approximation, is presented for the solution of the Laplace equation with the requisite time-dependent boundary conditions. The MLS approximation with linear basis as well as Gaussian type weight function is employed in the computation. At every instant of time, velocity potential is computed at each node and the nodal positions are updated. The choice of a suitable scaling parameter value in the MLS approximation is discussed in this study. The effectiveness of the developed algorithm is demonstrated through a few numerical examples. The accuracy and stability of the numerical method introduced are verified from the comparison with the existing reference solutions.

Study of Small Scale Effects on the Nonlinear Vibration Response of Functionally Graded Timoshenko Microbeams Based on the Strain Gradient Theory

In the current study, the nonlinear free vibration behavior of microbeams made of functionally graded materials (FGMs) is investigated based on the strain gradient elasticity theory and von Karman geometric nonlinearity. The nonclassical beam model is developed in the context of the Timoshenko beam theory which contains material length scale parameters to take the size effect into account. The model can reduce to the beam models based on the modified couple stress theory (MCST) and the classical beam theory (CBT) if two or all material length scale parameters are taken to be zero, respectively. The power low function is considered to describe the volume fraction of the ceramic and metal phases of the FGM microbeams. On the basis of Hamilton’s principle, the higher-order governing differential equations are obtained which are discretized along with different boundary conditions using the generalized differential quadrature method. The dimensionless linear and nonlinear frequencies of microbeams with various values of material property gradient index are calculated and compared with those obtained based on the MCST and an excellent agreement is found. Moreover, comparisons between the various beam models on the basis of linear and nonlinear types of strain gradient theory (SGT) and MCST are presented and it is observed that the difference between the frequencies obtained by the SGT and MCST is more significant for lower values of dimensionless length scale parameter.

Determination of dynamic gradient elasticity length scales

Wave dispersion is a widely recognised phenomenon that occurs when elastic waves propagate through a heterogeneous microstructured material; reflection and refraction of higher frequencies leads to an apparent reduction of the wave speed with frequency. Enhanced continua are frequently employed to capture this phenomenon efficiently. Numerical experiments are performed in this paper to establish a procedure for the determination of the length scale parameters used in dynamic gradient elasticity using spectral analysis. Suitable values of the length scale parameters are determined and verified for a one-dimensional laminated bar and for a two-dimensional chequerboard plate.

Flexural sensitivity and resonance of cantilever micro-sensors based on nonlocal elasticity theory

Sensors based on microcantilevers, especially ones with uniform structure, have ultrahigh sensitivities. The normalized natural frequencies and the sensitivity of lateral vibration of an elastic microcantilever sensor in contact with a surface are derived analytically based on the Euler–Bernoulli beam theory by taking into account the small scale effect. The interaction of the sensor with the surface is modeled by linear springs, which restricts the results to experiments involving low-amplitude excitations. The results show that the normalized natural frequencies of nonlocal microcantilever are smaller than those for its local counterpart, especially for higher values of small scale parameters. Also, each mode has a different sensitivity to variations in surface stiffness. Moreover, the most sensitivity is observed at the first mode of vibration. When the nonlocal effect is not taken into account, the natural frequencies and the sensitivity of the microcantilever in contact with the surface are compared with those obtained in previous study without considering the nonlocal effect.

Monthly and Seasonal Investigation of Wind Characteristics and Assessment of Wind Energy Potential in Al-Mokha, Yemen

The negative effects of fossil fuels have forced many countries to explore clean energy sources that are both environmenttally more suitable and renewable. In this study, wind characteristics and assessment of wind energy potential, were analyzed using the wind speed data measured during the period 1995-2001 at 10 m height of Al-Mokha, located in the west of Taiz, in the southwest of Yemen, subjected to 2-parameter Weibull analysis. Based on these data, it was found that the monthly mean wind speed ranged from 3.50 m/s in June to 7.94 m/s in November, the monthly values of Weibull shape parameter (k), ranged from 3.09 to 6.26, while the value of scale parameter (c) ranged from 3.91 to 8.63 m/s and the monthly wind power density ranged from 33.08 to 352.43 W/m2. The seasonal mean wind speed data ranged from 3.61 to 7.56 m/s, while the wind power density was ranged from 36.62 to 328.59 W/ m2 for summer and winter, respectively.

A simple damage-gradient enhanced elastoplastic formulation and its numerical implementation

This paper presents a ductile damage-gradient based nonlocal and fully coupled elastoplastic constitutive equations by adding a Helmholtz equation to regularize the initial and boundary value problem (IBVP) exhibiting some damage induced softening. First, a thermodynamically-consistent formulation of gradient-regularized plasticity fully coupled with isotropic ductile damage and accounting for mixed non linear isotropic and kinematic hardening is presented. For the sake of simplicity, only a simplified version of this model based on von Mises isotropic yield function and accounting for the single nonlinear isotropic hardening is studied and implemented numerically using an in house FE code. An additional partial differential equation governing the evolution of the nonlocal isotropic damage is added to the equilibrium equations and the associated weak forms derived to define the IBVP (initial and boundary value problem). After the time and space discretization, two algebraic equations: one highly nonlinear associated with the equilibrium equation and the second purely linear associated with the damage non locality equation are obtained. Over a typical load increment, the first equation is solved iteratively thanks to the Newton-Raphson scheme and the second equation is solved directly to compute the nonlocal damage \hbox{} D at each node. All the constitutive equations are “strongly” affected by this nonlocal damage variable transferred to each integration point. Some applications show the ability of the proposed approach to obtain a mesh independent solution for a fixed value of the length scale parameter. Comparisons between fully local and nonlocal solutions are given.