Non-zero Mean Value Research Articles

A geometrical index for the group Sn and its applications in periodic solutions of differential and differential-difference equations

In this paper, we construct a new index theory, Sn index theory, which is an improvement of the S1 index theory given by Benci in [1]. Also S1 index theory is a powerful tool in the study of the multiplicity of periodic orbits of autonomous differential equations, the definition of index confined the functional space being composed by functions whose mean values must zero. Under the condition that the functional Φ is even, our new index theory can be applied to study the multiplicity of periodic orbits in Hilbert space of functions without the restriction of mean value zero. After the shift of the origin it can be applied to the multiplicity of periodic orbits with a nonzero mean value. Besides, we give in this paper a delay differential system as a complete example to show the calculation of the multiplicity of periodic orbits.

Effects of Mean Stress and Multiaxial Loading on the Fatigue Life of Springs

In this paper, the effects of mean stress and damage accumulation on the fatigue life of springs are theoretically studied. The study examines the fatigue life of homogeneously stressed material subjected to cyclic loading. The mean stress of a load cycle is non-zero. Goodman and Haigh diagrams are commonly used for estimating fatigue life in engineering applications. Alternatively, conventional hypotheses by Smith–Watson–Topper, Walker and Bergmann have been successfully used to describe uniaxial cyclic fatigue with non-zero mean value over the whole range of the fatigue life. However, the physical characteristics of the mean stress sensitivities in these hypotheses are different. The mean stress sensitivity according to Smith–Watson–Topper is identical for all materials and stress levels. This weakness reduces the applicability of the Smith–Watson–Topper parameter. At first glance, the mean stress sensitivities according to Walker and Bergmann are diverse. The mean stress sensitivities depend upon two different additional correction parameters, namely the Bergmann parameter and the Walker exponent. The possibility of fitting the mean stress sensitivity in these hypotheses overcomes the significant drawback of the Smith–Watson–Topper schema. The principal task of this actual study is to reveal the dependence between the Bergmann parameter and the Walker exponent, which leads to a certain mean stress sensitivity. The manuscript establishes the simple relationship between both fitting parameters, which causes the equivalent mean stress sensitivity for the Bergmann and Walker criteria. As known from the state of the technology, fabrication and operation yield several impacts with a significant influence on the fatigue life of springs. One effect deals with the sequence of low and high stress amplitudes and amplitude-dependent damage accumulation. Particularly, during the load cycle a certain microscopical creep occurs. This creep causes damage. The accumulation hypothesis for creep damage is introduced. The hypothesis can be verified experimentally.

Abnormal phonon angular momentum due to off-diagonal elements in the density matrix induced by a temperature gradient

Nonzero mean value of phonon angular momentum (PAM) in chiral materials can be generated when a temperature gradient is applied. We find that both diagonal and off-diagonal terms of PAM contribute to mean PAM by using the Kubo formula where both diagonal and off-diagonal elements of the heat current operator are considered. The calculation results show that the off-diagonal term is dominant when the phonon scattering is strong enough. This finding reveals that the quantum transition between different phonon modes induced by temperature gradient strongly affects the local atomic rotation. Our discovery provides an explanation of the recently observed chiral phonon activated spin Seebeck effect.

Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction.

Coupled Boussinesq equations are used to describe long weakly nonlinear longitudinal strain waves in a bi-layer with soft bonding between the layers (e.g., a soft adhesive). From a mathematical viewpoint, a particularly difficult case appears when the linear long-wave speeds in the layers are significantly different (high-contrast case). The traditional derivation of the uni-directional models leads to four uncoupled Ostrovsky equations for the right- and left-propagating waves in each layer. However, the models impose a "zero-mass constraint"; i.e., the initial conditions should necessarily have zero mean, restricting the applicability of that description. Here, we bypass the contradiction in this high-contrast case by constructing the solution for the deviation from the evolving mean value, using asymptotic multiple-scale expansions involving two pairs of fast characteristic variables and two slow time variables. By construction, the Ostrovsky equations emerging within the scope of this derivation are solved for initial conditions with zero mean, while initial conditions for the original system may have non-zero mean values. Asymptotic validity of the solution is carefully examined numerically. We apply the models to the description of counter-propagating waves generated by solitary wave initial conditions, or co-propagating waves generated by cnoidal wave initial conditions, as well as the resulting wave interactions, and contrast with the behavior of the waves in bi-layers when the linear long-wave speeds in the layers are close (low-contrast case). One local (classical) and two non-local (generalized) conservation laws of the coupled Boussinesq equations for strains are derived and used to control the accuracy of the numerical simulations.

Practical estimation method for extreme value distribution of von Mises stress in ship structure

This study presents a practical method for estimating the extreme value distribution of von Mises stress for ship structural strength evaluations. The previous methods of calculating this distribution require somewhat complicated numerical calculations, such as multi-dimensional integration. In contrast, the proposed method is based on an asymptotic approximation and can be easily calculated in a similar way to the conventional linear statistical prediction. A closed expression was derived in the case of stress components which have non-zero mean value. The formula is derived under approximations that reflect realistic stress conditions when ships are under severe sea states. Through a structural analysis of a whole ship, it was comprehensively verified that the proposed method has sufficiently high accuracy for structural strength evaluation. Furthermore, a parametric analysis was conducted to clarify its limit of applicability.

Using the Smith-Watson-Topper Parameter and Its Modifications to Calculate the Fatigue Life of Metals: The State-of-the-Art.

The Smith-Watson-Topper parameter (SWT) in its original form was designed to estimate the fatigue life of metal materials in a uniaxial load state (tension–compression) in the range up to fatigue crack initiation, with non-zero mean values. This parameter is based on the analysis of both stress and strain. Therefore, the stress–strain criterion is the focus, rather than the energy criterion. This paper presents the original SWT model and its numerous modifications. The first part presents different versions of this parameter defined by the normal parameters. Then, it presents versions defined through the tangent parameter and the most promising parameter defined through the tangent and normal parameters. It was noted that the final form of the equivalent value is defined either by stress or by an energy parameter. Therefore, the possible characteristics from which the fatigue life can be determined are also presented.

Imprints of interacting dark energy on cosmological perturbations

We investigate the characteristic modifications in the evolving cosmological perturbations when dark energy interacts with dust-like matter, causing the latter’s background energy density fall off with time faster than usual. Focusing in particular to the late-time cosmic evolution, we show that such an interaction (of a specific form, arising naturally in a scalar-tensor formulation, or a wide range of modified gravity equivalents thereof), can have a rather significant effect on the perturbative spectrum, than on the background configuration which is not expected to get distorted much from [Formula: see text]CDM. Specifically, the matter density contrast, which is by and large scale-invariant in the deep sub-horizon limit, not only gets dragged as the interaction affects the background Hubble expansion rate, but also receives a contribution from the perturbation in the (scalar field induced) dark energy, which oscillates about a nonzero mean value. As such, the standard parametrization ansatz for the matter density growth factor becomes inadequate. So we modify it suitably, and also find a numerical fit of the growth index in terms of the background parameters, in order to alleviate the problems that arise otherwise. Such a fit enables direct estimations of the background parameters, as well as the growth parameter and the reduced Hubble parameter, which we duly carry out using a redshift space distortion (RSD) subsample and its combination with the observational Hubble data. On the whole, the parametric estimates show consistency with the general observational constraints on the background level cosmology, as well as the constraints on scalar-tensor gravity from astrophysical observations, apart from having significance in the domain of cosmological perturbations.

Investigation of Changes in Fatigue Damage Caused by Mean Load under Block Loading Conditions.

The literature in the area of material fatigue indicates that the fatigue properties may change with the number of cycles. Researchers recommend taking this into account in fatigue life calculation algorithms. The results of simulation research presented in this paper relate to an algorithm for estimating the fatigue life of specimens subjected to block loading with a nonzero mean value. The problem of block loads using a novel calculation model is presented in this paper. The model takes into account the change in stress–strain curve parameters caused by mean strain. Simulation tests were performed for generated triangular waveforms of strains, where load blocks with changed mean strain values were applied. During the analysis, the degree of fatigue damage was compared. The results of calculations obtained for standard values of stress–strain parameters (for symmetric loads) and those determined, taking into account changes in the curve parameters, are compared and presented in this paper. It is shown that by neglecting the effect of the mean strain value on the K′ and n′ parameters and by considering only the parameters of the cyclic deformation curve for εm = 0 (symmetric loads), the ratio of the total degree of fatigue damage varies from 10% for εa = 0.2% to 3.5% for εa = 0.6%. The largest differences in the calculation for ratios of the partial degrees of fatigue damage were observed in relation to the reference case for the sequence of block n3, where εm = 0.4%. The simulation results show that higher mean strains change the properties of the material, and in such cases, it is necessary to take into account the influence of the mean value on the material response under block loads.

Environmental Effect on Fatigue Crack Initiation under Equi-Biaxial Loading of an Austenitic Stainless Steel

The lifetime extension of nuclear power stations is considered an energy challenge worldwide. That is why the risk analysis and the study of various effects of different factors that could potentially prevent safe long-term operation are necessary. These structures, often of great dimensions, are subjected during their life to complex loading combining varying multiaxial mechanical loads with non-zero mean values associated with temperature fluctuations under a PWR (pressure water reactor) environment. Based on more recent fatigue data (including tests at 300 °C in air and a PWR environment, etc.), some international codes (RCC-M, ASME, and others) have proposed and suggested a modification of the austenitic stainless steels fatigue curve combined with a calculation of an environmental penalty factor, namely Fen, which has to be multiplied by the usual fatigue usage factor. The determination of the field of validation of the application of this penalty factor requires obtaining experimental data. The aim of this paper is to present a new device, “FABIME2e” developed in the LISN (Laboratory of Integrity of Structures and Normalization) in collaboration with EDF (Electricity of France) and Framatome. These new tests allow the effect of a PWR environment on a disk specimen to be quantified. This new device combines structural effects such as equibiaxiality and mean strain and the environmental penalty effect with the use of a PWR environment during fatigue tests.

Stochastic analysis of asymmetric monostable harvesters driven by Gaussian white noise with moment differential equations

As an effective means to overcome the shortcomings of linear systems only performing well near the resonance frequency, monostable piezoelectric energy harvesters (MPEHs) have been investigated widely in the area of energy harvesting. However, it is difficult to achieve a perfectly symmetric potential energy function due to the asymmetries in magnets and materials. Therefore, the response characteristics of the asymmetric MPEHs with quartic potential function under Gaussian white noise excitation are explored in this paper. The method of moment differential equation is applied to approximately determine the output performance of the asymmetric potential MPEHs under the excitation of Gaussian white noise. For the symmetric MPEH, the influence of external excitation intensity and internal system parameters on the outputs is firstly analyzed theoretically and numerically. When a quadratic nonlinear coefficient is introduced and only using its variation to characterize the variation of the asymmetry, the output performance of the MPEH is enhanced and the power increases with an increase in the asymmetry. Particularly, the existence of asymmetry leads the system to have a nonzero mean value for the displacement response, which then influences the shape of the probability density function. In the condition that more coefficients are employed to represent the asymmetry, the influence of asymmetry on the output depends on the potential energy function’s shape.

Mean value theorems for a class of density-like arithmetic functions

This paper provides a mean value theorem for arithmetic functions [Formula: see text] defined by [Formula: see text] where [Formula: see text] is an arithmetic function taking values in [Formula: see text] and satisfying some generic conditions. As an application of our main result, we prove that the density [Formula: see text] (respectively, [Formula: see text]) of normal (respectively, primitive) elements in the finite field extension [Formula: see text] of [Formula: see text] are arithmetic functions of (nonzero) mean values.

Nonlinear random vibrations of functionally graded porous nanobeams using equivalent linearization method

In the present study, the nonlinear random vibration of a porous functionally graded nanobeam supported by a viscoelastic foundation has been investigated by making use of the statistical linearization method for the first time. Nonlocal Euler-Bernoulli beam theory with von Kármán type nonlinearity is utilized to derive the governing equation of the motion of the FG porous nanobeam. Two types of porosity distribution have been considered for the FG beam. To discretize the governing equation Galerkin's method is applied. The mean square value of the response of the resulted Duffing's equation in the presence of a random input with nonzero mean value has been extracted using statistical linearization method. The results of the present study are compared with the analytical results of the FPK approach and the results of the perturbation method for the classical limit case of the homogeneous beam. It is revealed through detailed numerical analysis that the statistical linearization method (in contrast to the perturbation method) gives reliable results and there is an excellent agreement between its results and the analytical results. The effect of various parameters on the mean value of the response is also investigated numerically and illustrated graphically. It is found that the mean square value of vibrations increases when porosity, power-law index, nonlocal parameter, or the mean value of the input increases.

Application of Life-Dependent Material Parameters to Fatigue Life Prediction under Multiaxial and Non-Zero Mean Loading.

This study presents the life-dependent material parameters concept as applied to several well-known fatigue models for the purpose of life prediction under multiaxial and non-zero mean loading. The necessity of replacing the fixed material parameters with life-dependent parameters is demonstrated. The aim of the research here is verification of the life-dependent material parameters concept when applied to multiaxial fatigue loading with non-zero mean stress. The verification is performed with new experimental fatigue test results on a 7075-T651 aluminium alloy and S355 steel subjected to multiaxial cyclic bending and torsion loading under stress ratios equal to R = −0.5 and 0.0, respectively. The received results exhibit the significant effect of the non-zero mean value of shear stress on the fatigue life of S355 steel. The prediction of fatigue life was improved when using the life-dependent material parameters compared to the fixed material parameters.

The Effect of Mean Load for S355J0 Steel with Increased Strength

The paper presents an algorithm for calculating the fatigue life of S355J0 steel specimens subjected to cyclic bending, cyclic torsion, and a combination of bending and torsion using mean stress values. The method of transforming cycle amplitudes with a non-zero mean value into fatigue equivalent cycles with increased amplitude and zero mean value was used. Commonly known and used transformation dependencies were used and a new model of the impact of the mean stress value on the fatigue life of the material was proposed. The life calculated based on the proposed algorithm was compared with the experimental life. It has been shown that the proposed model finds the widest application in the load cases studied, giving good compliance of the calculation results with the experimental results.

The first normal stress difference of non-Brownian hard-sphere suspensions in the oscillatory shear flow near the liquid and crystal coexistence region.

We carry out a numerical study to investigate the dynamics of non-Brownian hard-sphere suspensions near the liquid and crystal coexistence region in small to large amplitude oscillatory shear flow. The first normal stress difference (N1) and related rheological functions are carefully analyzed, focusing on the strain stiffening phenomenon, which occurs in the large strain amplitude region. Under oscillatory shear, we observe several unique behaviors of N1. A negative nonzero mean value of N1 (N1,0) is observed for the applied strain amplitudes. The change of the sign, from negative to positive, at the maximum value of N1 (N1,max) is observed at a specific point, which is not consistent with the critical strain amplitude (γ0,c) at which the modulus begins to deviate from linear viscoelasticity. The behavior of N1 in the oscillatory shear flow is different from that of N1 in steady shear flow, that is, the characteristics of N1 in strain stiffening and shear thickening are quite distinguished from each other. We also perform structural analysis to confirm the relationship between the rheological properties and microstructure of the suspension. A strong correlation is observed between the global bond order parameter (Ψ6) and the distortions in both nonlinear shear and normal stresses. The most noticeable characteristic is captured through the maximum of the global bond order parameter (Ψ6,max). The strain amplitude at the slope change of Ψ6,max corresponds to the point where a unique behavior of N1 is observed, i.e. the change of the sign in N1,max, but a strong correlation is not captured at γ0,c. This demonstrates that the normal stress responds to particle ordering more sensitively than other rheological functions based on shear stress like dynamic moduli. As far as we are concerned, the behavior of N1 has rarely been fully explored and related with the strain stiffening of non-Brownian suspensions so far. Therefore, this study has significance as the first report to strictly analyze strain stiffening along with the first normal stress difference N1.

Nonlinear vibrations of a rotating thin-walled composite piezo-beam with circumferentially uniform stiffness (CUS)

A rotating system comprising a hub and a thin-walled laminate cantilever beam with embedded nonlinear piezoelectric layers is analysed in the paper. The reinforcing fibres set-up in composite material conforms to circumferentially uniform stiffness lamination scheme. This configuration exhibits the mutual bending couplings in two orthogonal planes. Nonlinear analytical model of a piezoelectric material embedded onto the beam walls is postulated by considering the higher-order constitutive relations with respect to electric field variable. Moreover, to properly model electromechanical structural behaviour, the full two-way coupling piezoelectric effect is considered. To this aim, the assumption of a spanwise electric field variation is postulated in the mathematical model of the structure. Based on previous authors’ research, the system of mutually coupled nonlinear equations of motion is formulated. In the numerical analysis the forced response of the system under zero and nonzero mean value harmonic torque excitation is considered. In particular, the influence of hub inertia, excitation amplitude and mean rotating speed on system dynamics is investigated. The results are presented in the form of appropriate frequency response plots and bifurcation diagrams.

Mean stress effect on fatigue life estimation for Inconel 718 alloy

Uniaxial and biaxial fatigue tests related to Inconel 718 specimens are analysed using the critical plane-based multiaxial fatigue criterion by Carpinteri et al. in conjunction with a Smith-Watson-Topper (SWT) type of relationship. More precisely, specimens subjected to proportional and non-proportional loading under strain control are examined, such loadings being characterised by both zero and non-zero mean values. Fatigue life is computed through the novel formulation proposed here and compared with the experimental fatigue life in terms of number of loading cycles needed to form a surface crack whose length is equal to 1 mm.

Initial assessment of BDS-3 preliminary system signal-in-space range error

The basic constellation of the BeiDou global satellite navigation system (BDS-3) had been successfully completed by the end of 2018. It included 18 medium earth orbit satellites and 1 geostationary orbit satellite. An initial assessment of BDS-3 broadcast orbit and clock accuracy based on 55 days of broadcast message data is presented in this contribution. Satellite positions and clock offsets derived from broadcast ephemeris are compared with precise orbit determination orbits and clock offsets. Furthermore, the corresponding signal-in-space range error (SISRE), which is of most interest to navigation users, is computed. Thanks to the new inter-satellite link payloads on BDS-3 satellites, the statistics of age-of-data-ephemeris and age-of-data-clock demonstrate that more than 98% of ephemerides and 93% of clock parameters are updated within only one hour. Experimental results show that the 3D root mean square (RMS) of broadcast orbit errors is less than 0.6 m for the overall constellation. The broadcast orbit is also assessed by satellite laser ranging measurements, giving an RMS of 7.3 cm. The orbit-only SISRE is about 0.1 m. With respect to clock errors, the timescale differences between two clock products are eliminated to assess the accuracy of broadcast clock offsets. The standard deviation value of 0.25 m shows a good performance, but the RMS value is regrettably nearly 0.5 m due to a nonzero mean bias. RMS of BDS-3 SISRE amounts to approximately 0.5 m, which is largely attributed to clock errors. Finally, a positioning experiment is conducted to analyze the accuracy of single point positioning (SPP). With 95% confidence level, the horizontal accuracy is less than 5 m, and the vertical accuracy is close to 6 m. Considering that the nonzero mean bias in clock errors may affect the performance of SPP, we correct the nonzero mean value by a satellite-specific constant to analyze the influence of clock bias on the SPP performance. The results show that improvement in the 3D position accuracy can be up to 11%, especially in the up direction.

Extensions of the Novikov–Furutsu theorem, obtained by using Volterra functional calculus

The Novikov–Furutsu (NF) theorem is a well-known mathematical tool, used in stochastic dynamics for correlation splitting, that is, for evaluating the mean value of the product of a random functional with a Gaussian argument multiplied by the argument itself. In this work, the NF theorem is extended for mappings (function-functionals) of two arguments, one being a random variable and the other a random function, both of which are Gaussian, may have non-zero mean values, and may be correlated with each other. This extension allows for the study of random differential equations under coloured noise excitation, which may be correlated with the random initial value. Applications in this direction are briefly discussed. The proof of the extended NF theorem is based on a more general result, also proven herein by using Volterra functional calculus, stating that: the mean value of a general, nonlinear function-functional having random arguments, possibly non-Gaussian, can be expressed in terms of the characteristic functional of its arguments. Generalizations to the multidimensional case (multivariate random arguments) are also presented.

Mirror symmetry breaking in toy models of developed turbulence

Importance of symmetry considerations in high energy physics and statistical systems in critical region is discussed. As a concrete example, the field theoretic renormalization group technique is applied to a general A model of active vector admixture b. Parameter A represents a continuous real parameter that governs the interaction structure of the system. The model encompasses the physically most relevant magnetohydrodynamic scenario (A = 1), and the model of linearized Navier-Stokes equations (A = −1). The environment is assumed to be in a state of fully developed turbulence, i.e. Reynolds number takes an infinite value. Additionally, spatial parity breaking is incorporated via the continuous parameter ρ. Studied system exhibits instability with respect to it, and large-scale generation of non-zero mean value 〈b〉 is observed. This is then explained in terms of spontaneous symmetry breaking mechanism.