scholarly journals Scaling in Anti-Plane Elasticity on Random Shear Modulus Fields with Fractal and Hurst Effects

2021 ◽  
Vol 5 (4) ◽  
pp. 255
Author(s):  
Yaswanth Sai Jetti ◽  
Martin Ostoja-Starzewski
Keyword(s):  
Shear Modulus ◽  
Scaling Function ◽  
Plane Elasticity ◽  
Wide Sense ◽  
Plane Shear ◽  
Autocorrelation Functions ◽  
Set Up ◽  
The Hill ◽  
The Scaling Function ◽  
Stochastic Boundary

The scale dependence of the effective anti-plane shear modulus response in microstructures with statistical ergodicity and spatial wide-sense stationarity is investigated. In particular, Cauchy and Dagum autocorrelation functions which can decouple the fractal and the Hurst effects are used to describe the random shear modulus fields. The resulting stochastic boundary value problems (BVPs) are set up in line with the Hill–Mandel condition of elastostatics for different sizes of statistical volume elements (SVEs). These BVPs are solved using a physics-based cellular automaton (CA) method that is applicable for anti-plane elasticity to study the scaling of SVEs towards a representative volume element (RVE). This progression from SVE to RVE is described through a scaling function, which is best approximated by the same form as the Cauchy and Dagum autocorrelation functions. The scaling function is obtained by fitting the scaling data from simulations conducted over a large number of random field realizations. The numerical simulation results show that the scaling function is strongly dependent on the fractal dimension D, the Hurst parameter H, and the mesoscale δ, and is weakly dependent on the autocorrelation function. Specifically, it is found that a larger D and a smaller H results in a higher rate of convergence towards an RVE with respect to δ.

scholarly journals Frequency-dependent scaling from mesoscale to macroscale in viscoelastic random composites

2016 ◽  
Vol 472 (2188) ◽  
pp. 20150801 ◽  
Author(s):  
Jun Zhang ◽  
Martin Ostoja-Starzewski
Keyword(s):  
Scaling Function ◽  
Initial Boundary ◽  
Volume Element ◽  
Viscoelastic Materials ◽  
Static Properties ◽  
Frequency Dependent ◽  
Random Composites ◽  
Set Up ◽  
Initial Boundary Value ◽  
The Hill

This paper investigates the scaling from a statistical volume element (SVE; i.e. mesoscale level) to representative volume element (RVE; i.e. macroscale level) of spatially random linear viscoelastic materials, focusing on the quasi-static properties in the frequency domain. Requiring the material statistics to be spatially homogeneous and ergodic, the mesoscale bounds on the RVE response are developed from the Hill–Mandel homogenization condition adapted to viscoelastic materials. The bounds are obtained from two stochastic initial-boundary value problems set up, respectively, under uniform kinematic and traction boundary conditions. The frequency and scale dependencies of mesoscale bounds are obtained through computational mechanics for composites with planar random chessboard microstructures. In general, the frequency-dependent scaling to RVE can be described through a complex-valued scaling function, which generalizes the concept originally developed for linear elastic random composites. This scaling function is shown to apply for all different phase combinations on random chessboards and, essentially, is only a function of the microstructure and mesoscale.

Features of nonlinear in-plane shear deformation of a unidirectional and orthogonally reinforced polymer sheets of composite materials

2021 ◽  
Vol 87 (5) ◽  
pp. 47-55
Author(s):  
A. O. Polovyi ◽  
N. V. Matiushevski ◽  
N. G. Lisachenko
Keyword(s):  
Experimental Data ◽  
Composite Materials ◽  
Shear Modulus ◽  
Maximum Deviation ◽  
Linear Section ◽  
Stress Strain ◽  
Plane Shear ◽  
End Point ◽  
Reinforced Polymer ◽  
Failure Point

A comparative analysis of typical stress-strain diagrams obtained for in-plain shear of the 25 unidirectional and cross-ply reinforced polymer matrix composites under quasi-static loading was carried out. Three of them were tested in the framework of this study, and the experimental data on other materials were taken from the literature. The analysis of the generalized shear-strength curves showed that most of the tested materials exhibit the similar deformation pattern depending on their initial shear modulus: a linear section is observed at the beginning of loading, whereas further increase of the load decreases the slope of the curve reaching the minimum in the failure point. For the three parameters (end point the linear part, maximum reduced deviation of the diagram, tangent shear modulus at the failure point) characterizing the individual features of the presented stress-strain diagrams, approximating their dependences on the value of the reduced initial shear modulus are obtained. At the characteristic points of the deformation diagrams, boundary conditions are determined that can be used to find the parameters of the approximating functions. A condition is proposed for determination of the end point of the linear section on the experimental stress-strain curve, according to which the maximum deviation between the experimental and calculated (according to Hooke’s law) values of the shear stress in this section is no more than 1%, thus ensuring rather high accuracy of approximation on the linear section of the diagram. The results of this study are recommended to use when developing universal and relatively simple in structure approximating functions that take into account the characteristic properties of the experimental curves of deformation of polymer composite materials under in-plane shear of the sheet. The minimum set of experimental data is required to determine the parameters of these functions.

Assessing Model Uncertainty for the Scaling Function Inversion of Potential Fields

Geophysics ◽  
2021 ◽  
pp. 1-39
Author(s):  
Mahak Singh Chauhan ◽  
Ivano Pierri ◽  
Mrinal K. Sen ◽  
Maurizio FEDI
Keyword(s):  
Simulated Annealing Algorithm ◽  
Scaling Function ◽  
Vertical Section ◽  
Gravity Data ◽  
Geometric Model ◽  
Salt Dome ◽  
The Body ◽  
Gravity Inversion ◽  
Geometrical Parameters ◽  
The Scaling Function

We use the very fast simulated annealing algorithm to invert the scaling function along selected ridges, lying in a vertical section formed by upward continuing gravity data to a set of altitudes. The scaling function is formed by the ratio of the field derivative by the field itself and it is evaluated along the lines formed by the zeroes of the horizontal field derivative at a set of altitudes. We also use the same algorithm to invert gravity anomalies only at the measurement altitude. Our goal is analyzing the different models obtained through the two different inversions and evaluating the relative uncertainties. One main difference is that the scaling function inversion is independent on density and the unknowns are the geometrical parameters of the source. The gravity data are instead inverted for the source geometry and the density simultaneously. A priori information used for both the inversions is that the source has a known depth to the top. We examine the results over the synthetic examples of a salt dome structure generated by Talwani’s approach and real gravity datasets over the Mors salt dome and the Decorah (USA) basin. For all these cases, the scaling function inversion yielded models with a better sensitivity to specific features of the sources, such as the tilt of the body, and reduced uncertainty. We finally analyzed the density, which is one of the unknowns for the gravity inversion and it is estimated from the geometric model for the scaling function inversion. The histograms over the density estimated at many iterations show a very concentrated distribution for the scaling function, while the density contrast retrieved by the gravity inversion, according to the fundamental ambiguity density/volume, is widely dispersed, this making difficult to assess its best estimate.

THE CONSTRUCTION OF A CLASS OF TRIVARIATE NONSEPARABLE COMPACTLY SUPPORTED WAVELETS

2009 ◽  
Vol 07 (03) ◽  
pp. 255-267 ◽  
Author(s):  
YONGDONG HUANG ◽  
SHOUZHI YANG ◽  
ZHENGXING CHENG
Keyword(s):  
General Theory ◽  
Mild Condition ◽  
Scaling Function ◽  
Moment Condition ◽  
Compactly Supported ◽  
Vanishing Moment ◽  
The Scaling Function

In this paper, under a mild condition, the construction of compactly supported [Formula: see text]-wavelets is obtained. Wavelets inherit the symmetry of the corresponding scaling function and satisfy the vanishing moment condition originating in the symbols of the scaling function. An example is also given to demonstrate the general theory.

NEW AMY AND DELPHI MULTIPLICITY DATA AND THE LOGNORMAL DISTRIBUTION

1991 ◽  
Vol 06 (03) ◽  
pp. 245-257 ◽  
Author(s):  
R. SZWED ◽  
G. WROCHNA ◽  
A.K. WRÓBLEWSKI
Keyword(s):  
Energy Dependence ◽  
Lognormal Distribution ◽  
Scaling Function ◽  
Average Multiplicity ◽  
Linear Functions ◽  
The Mean ◽  
Multiplicity Distributions ◽  
The Scaling Function ◽  
Skewness And Kurtosis

Multiplicity distributions for e+e−→ hadrons recently reported by the AMY and DELPHI collaborations are compared with the data obtained at lower energies. It is proven that the new data obey the KNO-G scaling and the scaling function can be described by the lognormal distribution. The dispersions are linear functions of the mean as for the data measured at lower energies and the standardized moments (such as skewness and kurtosis) are independent of the energy. The energy dependence of the average multiplicity is described by <nch>=β sα−1.

OFF THE CRITICAL POINT BY PERTURBATION (II): THE ISING MODEL SPIN-SPIN CORRELATOR IN A MAGNETIC FIELD

1990 ◽  
Vol 04 (05) ◽  
pp. 1039-1047 ◽  
Author(s):  
Vl. S. Dotsenko
Keyword(s):  
Magnetic Field ◽  
Correlation Function ◽  
Ising Model ◽  
Scaling Function ◽  
Spin Correlation ◽  
Regularization Technique ◽  
Spin Correlation Function ◽  
Analytic Regularization ◽  
Spin Correlator ◽  
The Scaling Function

An extension of the analytic regularization technique based on the conform 1 theory is suggested for the case of the spin-spin correlation function of the Ising model in a magnetic field, <σ0σR>h=F(t)/(R)1/4, t=hR15/8. Several first terms of the expansion of the scaling function F(t) are given.

Multisingularity and scaling in partial differential approximants. I

1988 ◽  
Vol 419 (1857) ◽  
pp. 181-203 ◽  
Keyword(s):  
Scaling Function ◽  
Linear Partial Differential Equation ◽  
Linear Combinations ◽  
Partial Differential ◽  
Polynomial Coefficients ◽  
Explicit Formulae ◽  
Expansion Coefficients ◽  
Scaling Fields ◽  
The Scaling Function ◽  
Asymptotic Scaling

A partial differential approximant (or PDA), F ( x, y ) approximates a function f ( x, y ), specified by its truncated power series, in terms of a solution of a defining linear partial differential equation with polynomial coefficients. The intrinsic multisingularities of a PDA, which may approximate corresponding singularities of f ( x, y ), are analysed formally and shown to obey, in general, asymptotic scaling (as familiar in the theory of critical phenomena), i. e. F ( x, y ) ≈ C | x͠ | - γ Z ( y͠ /| x͠ | ϕ ) + B , where x͠ and y͠ are linear combinations of the deviations, ( x - x c ) and ( y - y c ), from the multisingular point ( x c , y c ). Explicit formulae, suitable for numerical computation, are derived for the characteristic exponents, γ and ϕ , for the scaling function Z ( • ), for its expansion coefficients and for the related coefficients C and B , in the case when the crossover exponent, ϕ , lies in the interval (½, 2). (Part II extends these results to general values of ϕ , which requires the introduction of the nonlinear scaling fields associated with the PDA.)

On the Analysis of Competitive Binding of Various Ligands to Cooperative and Independent Binding Sites of Macromolecules

1979 ◽  
Vol 34 (9-10) ◽  
pp. 757-769 ◽  
Author(s):  
Hans Robert Kalbitzer ◽  
Dietmar Stehlik
Keyword(s):  
Experimental Data ◽  
Binding Sites ◽  
Manganese Ions ◽  
Numerical Examples ◽  
Practical Evaluation ◽  
Multiple Binding ◽  
Calcium And Magnesium ◽  
Set Up ◽  
The Hill ◽  
Binding Structure

Abstract The paper deals with the practical evaluation of multiple binding equilibria of macromolecules and different ligands competing for the same binding sites. The necessary formalism is reviewed and set up for the equilibria involving a macromolecule with various classes of independent binding sites and/or a class of cooperative sites and up to three different ligands in competition for them. In particular, it was necessary to extend the Hill approximation to treat simultaneous competition for cooperative as well as independent binding sites, while earlier at­tempts are shown to be inadequate. Criteria are developed for a qualitative analysis of complex binding patterns using the Scatchard-plot of the experimental data in order to obtain a model of the binding structure and an adequate set of input parameters for the numerical analysis. Numerical examples refer to the binding of calcium and magnesium to the sarcoplasmic reticulum as studied by competitive re­placement of manganese ions [3].

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