General time‐dependent C(t) and J(t) estimation equations for elastic‐plastic‐creep fracture mechanics analysis

This paper presents equations for estimating the crack tip characterizing parameters C(t) and J(t), for general elastic‐plastic‐creep conditions where the power‐law creep and plasticity stress exponents differ, by modifying the plasticity correction term in published equations. The plasticity correction term in the newly proposed equations is given in terms of the initial elastic‐plastic and steady‐state creep stress fields. The predicted C(t) and J(t) results are validated by comparison with systematic elastic‐plastic‐creep FE results. Good agreement with the FE results is found.     

Order reduction in computational inelasticity: Why it happens and how to overcome it—The ODE‐case of viscoelasticity

Time integration is the numerical kernel of inelastic finite element calculations, which largely determines their accuracy and efficiency. If higher order Runge–Kutta (RK) methods, p≥3, are used for integration in a standard manner, they do not achieve full convergence order but fall back to second‐order convergence. This deficiency called order reduction is a longstanding problem in computational inelasticity. We analyze it for viscoelasticity, where the evolution equations follow ordinary differential equations. We focus on RK methods of third order. We prove that the reason for order reduction is the (standard) linear interpolation of strain to construct data at the RK‐stages within the considered time interval. We prove that quadratic interpolation of strain based on t_n, t_{n + 1} and, additionally, t_{n ? 1} data implies consistency order three for total strain, viscoelastic strain and stress. Simulations applying the novel interpolation technique are in perfect agreement with the theoretical predictions. The present methodology is advantageous, since it preserves the common, staggered structure of finite element codes for inelastic stress calculation. Furthermore, it is easy to implement, the overhead of additional history data is small and the computation time to obtain a defined accuracy is considerably reduced compared with backward Euler. Copyright © 2011 John Wiley & Sons, Ltd.     

Internal symmetry groups for perfect elastoplasticity under axial‐torsional loadings

Abstract

In the material modeling of experimental axial‐torsional strain control tests, the hoop and radial strains are always unknown, a priori, and hence can not be viewed as inputs. This greatly complicates constitutive model analyses because the resulting differential equations become highly nonlinear. To tackle this problem, we demonstrate two new formulations. By using the two‐integrating factors idea we derive two Lie type systems in the product space M ¹⁺¹?M ¹⁺¹. The Lie algebra is the direct sum so(1, 1)?so(1, 1), and correspondingly the symmetry group is the direct product SO_o (1, 1) ?SO_o (1, 1). Then, by using the one‐integrating factor idea we convert the nonlinear constitutive equations to a Lie type system X=A(X, t)X with A?sl(2, 1, R), a Lie algebra of the special orthochronous pseudo‐linear group SL(2, 1, R). The underlying space is a cone in the pseudo‐Riemann manifold. Consistent numerical methods are also developed according to these Lie symmetries.     

On the spectral representation method in simulation

Two models X_n(t) and Y_n(t) are considered for generating samples of stationary band-limited Gaussian processes. The models are based on the spectral representation method and consist of a superposition of n harmonics. The harmonics of X_n(t) have random phase and amplitude while the harmonics of Y_n(t) only have random phase. It is shown that the two models are equal in the second-moment sense. However, Y_n(t) has stronger ergodic properties than X_n(t). On the other hand, X_n(t) is a Gaussian process for any value of n while Y_n(t) is asymptotically Gaussian as n approaches infinity. It is demonstrated that the rejection of X_n(t), because of its weak ergodic property, or of model Y_n(t), because of its nonGaussian distribution, it is not generally justified. One special case in which Y_n(t) should not be used is that of Gaussian processes with power concentrated at a few discrete frequencies.     

Dispersion law in photonic crystals in sinusoidal and quasi-relativistic approximations

ω(k) dispersion curves are calculated for electromagnetic waves in a photonic crystal (at normal incidence). The calculations rely on a model of a one-dimensional periodic layered medium with two refractive indices. The ω(k) dispersion law thus derived is used to find the frequency-dependent reflectance of the photonic crystal. Approximations are proposed that allow the dispersion law ω(k) to be obtained in explicit form.     

Linear differential-algebraic equations with properly stated leading term: B -critical points

We examine in this article so-called B-critical points of linear, time-varying differential-algebraic equations (DAEs) of the form A(t)(D(t)x(t))′ + B(t)x(t) = q(t). These critical or singular points, which cannot be handled by classical projector methods, require adapting a recently introduced framework based on Π-projectors. Via a continuation of certain invariant spaces through the singularity, we arrive at a scenario which accommodates both A- and B-critical DAEs. The working hypotheses apply in particular to standard-form analytic systems although, in contrast to other approaches to critical problems, the scope of our approach extends beyond the analytic setting. Some examples illustrate the results.     

Numerical solution of linear two-point boundary problems via the fundamental-matrix method

A numerical method is presented for the-solution of linear systems of differential equations with initial-value or two-point boundary conditions. For y ′(x) = A (x) y (x) + f (x) the domain of interest [a,b] is divided into an appropriate number L of subintervals. The coefficient matrix A (x) is replaced by its value A_k at a point x_k within the Kth subinterval, thus replacing the original system by the L discretized systems y k(x) = A _k y _k(x) + f _k(x), k = 1,2,…, L. The fundamental matrix solution Φ_k(x, x_k) over each subinterval is found by computing the eigenvalues and eigenvectors of each A _k. By matching the solutions y _k(x) at the L – 1 equispaced grid points defining the limits of the subintervals and the boundary conditions, the two-point problem is reduced to solving a system of linear algebraic equations for the matching constants characterizing the different y _k(x). The values of y ₁(a) and y _L(b) are used to calculate the missing boundary conditions. For initial-value problems this method is equivalent to a one-step method for generating approximate solutions. By means of a coordinate transformation, as in the multiple shooting method,¹ the method becomes particularly suitable for stiff systems of linear ordinary differential equations. Five examples are discussed to illustrate the viability of the method.     

The solution of two wave-diffraction problems

Summary A method for obtaining the numerical solution of first-kind integral equations with the Hankel-function kernel H ⁽¹⁾ ₀(k|x t|) is described in relation to two water-wave diffraction problems. The principal feature is the implementation of a new technique for transforming the given equations into second-kind integral equations, which have continuous kernels and from which numerical approximations can readily be determined.     

Accelerated Life Testing for a Class of Linear Hazard Rate Type Distributions

Accelerated life testing for distributions with hazard rate functions of the form r(t) = Ag(t) + Bh(t) are considered. Let V ₁, …, V _k be stress levels larger than V ₀—the stress level under normal conditions [V ₀ > 0]—and let a(v) be a nondecreasing function on (0, ∞). We discuss a generalization of the common accelerated models (the power rule model and the Arrhenius model) by assuming that the hazard rate under the stress level V, is of the form (a(V _t )) ^P (Ag(t) + Bh(t)). The maximum likelihood estimators of A, B and P for complete and censored samples are studied. The estimation procedure reduces to a solution of one equation with one unknown parameter. The estimation procedure under the assumption of aging is also described. The asymptotic variance-covariance matrix is given.     

Computation of the relaxation and creep functions of elastomers from harmonic shear modulus

The purpose of this paper is to compute the relaxation and creep functions from the data of shear complex modulus, G ^∗(iν). The experimental data are available in the frequency window ν∈[ν_min,ν_max] in terms of the storage G′(ν) and loss G″(ν) moduli. The loss factor h( n) = \fracG"( n)G￠(n)\eta( \nu) = \frac{G'( \nu )}{G'(\nu )} is asymmetrical function. Therefore, a five-parameter fractional derivative model is used to predict the complex shear modulus, G ^∗(iν). The corresponding relaxation spectrum is evaluated numerically because the analytical solution does not exist. Thereby, the fractional model is approximated by a generalized Maxwell model and its rheological parameters (G _k ,τ _k ,N) are determined leading to the discrete relaxation spectrum G(t) valid in time interval corresponding to the frequency window of the input experimental data. Based on the deterministic approach, the creep compliance J(t) is computed on inversing the relaxation function G(t).     

Fast matrix exponent for deterministic or random excitations

The solution of ż=Az is z(t)=exp(At)z₀=E_tz₀, z₀=z(0). Since z(2t)=E_2tz₀=Ez₀, z(4t)=E_4tz₀=Ez₀, etc., one function evaluation can double the time step. For an n‐degree‐of‐freedoms system, A is a 2n matrix of the nth‐order mass, damping and stiffness matrices M, C and K. If the forcing term is given as piecewise combinations of the elementary functions, the force response can be obtained analytically. The mean‐square response P to a white noise random force with intensity W(t) is governed by the Lyapunov differential equation: Ṗ=AP+PA^T+W. The solution of the homogeneous Lyapunov equation is P(t)=exp(At) P₀ exp(A^Tt), P₀=P(0). One function evaluation can also double the time step. If W(t) is given as piecewise polynomials, the mean‐square response can also be obtained analytically. In fact, exp(At) consists of the impulsive‐ and step‐response functions and requires no special treatment. The method is extended further to coloured noise. In particular, for a linear system initially at rest under white noise excitation, the classical non‐stationary response is resulted immediately without integration. The method is further extended to modulated noise excitations. The method gives analytical mean‐square response matrices for lightly damped or heavily damped systems without using modal expansion. No integration over the frequency is required for the mean‐square response. Four examples are given. The first one shows that the method include the result of Caughy and Stumpf as a particular case. The second one deals with non‐white excitation. The third finds the transient stress intensity factor of a gun barrel and the fourth finds the means‐square response matrix of a simply supported beam by finite element method. Copyright © 2001 John Wiley & Sons, Ltd.     

Weibull analysis for normal/accelerated and fatigue random vibration test

In this paper, the formula to estimate the sample size n to perform a random vibration test is derived only from the desired reliability (R(t)). Then, the addressed n value is used to design the ISO16750‐3 random vibration test IV for both normal and accelerated conditions. For the normal case, the applied random vibration stress (S) is modeled by using the Weibull stress distribution [W(s)]. Similarly, for the testing time (t), the Weibull time distribution [W(t)] is used to model its random behavior. For the accelerated case, by using the over‐stress factor fitted from the W(t) and W(s) distributions, four accelerated scenarios are formulated with their corresponding testing's profiles. Additionally, from the W(s) analysis, the stress formulation to perform the fatigue and Mohr stress analysis is given. Since the given Weibull/fatigue formulation is general, then the formulas to determine the W(s) parameters, which correspond to any principal stresses values and/or vice versa, are given. Although the application is performed to demonstrate R(t) = 0.97 by testing only n₂ = 6 parts, the guidelines to use the values given in columns n, S, and t of the Weibull analysis table to generate several accelerated testing plans are given.     

A semi-Markovian model allowing for inhomogenities with respect to process time

A non-homogeneous semi-Markov process is considered as an approach to model reliability characteristics of components or small systems with complex test resp. maintenance strategies. This approach generalizes previous results achieved for ordinary inhomogeneous Markov processes. This paper focuses on the following topics to make the application of semi-Markovian models feasible: rather than transition probabilities Q_ij(t), which are used in normal mathematical text books to define semi-Markov processes, transition rates λ_ij( ) are used, as is usual for ordinary Markov processes. These transition rates may depend on two types of time in general: on process time and on sojourn time in state i. Such transition rates can be followed from failure and repair rates of the underlying technical components, in much the same way, as this is known for ordinary Markov processes. Rather than immediately starting to solve the Kolmogorov equations, which would result in N² integral equations, a system of N integral equations for frequency densities of reaching states is considered. Once this system is solved, the initial value problem for state probabilities can be solved by straightforward integration. An example involving 14 states has been solved as an illustration using the approach.     

Three dimensional transient analysis of the installation of marine cables

Summary A numerical solution for the three dimensional transient motion of a marine cable during installation is presented for the case of a cable laying vessel arbitrarily chaning speed and direction while paying out cable with seabed slack. The cable transient behaviour is governed by the numerical solution of a set of non-linear partial differential equations with the solution methodology incorporating both spatial and temporal integration. The space integration is carried out by dividing the cable inton straight elements with equilibrium relationships and geometric compatibility equations satisfied for each element. The position of each element is described by its elevation and azimuth angles and, therefore, a system of 2n non-linear ordinary differential equations is established. The time integration of this set of equations is performed using a high order Runge-Kutta technique. Results are presented fro the cable tension and element elevation and azimuth angles as functions of time and for transient cable geometries when the cable ship executes horizontal planar manoeuvres.List of symbols A _t, A_n, A_b tangential, normal and binormal components of cable element acceleration vector - A _c (s, t) cable element acceleration vector - C _D ,C _f normal and tangential friction drag coefficients - C _m added mass coefficient - D _n n,D _b b,D _t t normal, binormal and tangential drag forces - d cable diameter - G _i (t) constant of integration - g gravitational acceleration - H hydrodynamic constant of the cable - I, J, K inertial system unit vectors - i, j, k tow vessel system unit vectors - L _i length of the cable element - N, B, T normal, binormal and tangential node forces - n number of straight elements - 2N _i /L _i , 2B _i /L _i geometric stiffness terms - OXYZ inertial system - oxyz system attached to the tow vessel - oxyz local system - R _c (s, t) vector position of a cable element - r _o(t),r _c (s, t) ship and cable element position vectors - cable element velocity vector - cable element acceleration vector - ship speed - ship acceleration - s unstretched distance along the cable - T t effective tension - T _i (s, t) axial tension in cable elementi - T _{n, 0} = –L _n H( _n ) axial tension in the beginning of elementn - T _ship(t) cable top force - 0054xx _{s, t} true axial force - t, n, b tangent, normal and binormal unit vectors - V _p ⁰ cable pay out rate - V ₀ initial ship speed - V _t ,V _n ,V _b tangential, normal and binormal components of cable element velocity vector - V _c (s, t),A _c (s, t) velocity and acceleration vectors of a cable element - w cable weight in sea water per unit length - w j self weight - X(s, t), Y(s, t) Z(s, t) functions describing cable geometric configuration - final ship azimuth direction - sea water density - _c cable physical mass per unit length - _c A _c d'Alembert force - added inertia - time taken for the ship to change heading - = (s, t), = (s, t) azimuth and elevation angles - _i , _i azimuth angle and elevation angle     

Bounds on Minimum Distance for Linear Codes over GF(5)

Let [n, k, d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). Let d ₅(n, k) be the maximum possible minimum Hamming distance of a linear [n, k, d; 5]-code for given values of n and k. In this paper, forty four new linear codes over GF(5) are constructed and a table of d ₅(n, k) k≤ 8, n≤ 100 is presented.     

Computation of the Optical Constants of a Thin Dielectric Layer from the Envelopes of the Transmission Spectrum,at Inclined Incidence of the Radiation

Abstract

A mathematical expression for the transmission of a thin dielectric layer on a transmitting substrate is found. The finite dimensions of the substrate and its absorption are taken into account. The cases of inclined incidence of non-polarised, s- and p-polarised radiation to the layer are investigated. A way of determining the optical constants n(λ) and k(λ) of the layer from the envelopes of the transmission spectrum is offered. Computation of n(λ) and k(λ) of a model optical system at different angles of radiation incidence is performed as well as analysis of the results obtained.     

On the mean residual lifetime of?consecutive k-out-of-n systems

In recent years, consecutive systems were shown to have many applications in various branches of science such as engineering. This paper is a study on the stochastic and aging properties of residual lifetime of consecutive k-out-of-n systems under the condition that n−r+1, r≤n, components of the system are working at time t. We consider the linear and circular consecutive k-out-of-n systems and propose a mean residual lifetime (MRL) for such systems. Several properties of the proposed MRL is investigated. The mixture representation of the MRL of the systems with respect to the vector of signatures of the system is also studied.     

An adaptive exponentially weighted moving average chart for the mean with variable sampling intervals

The AEWMA control chart is an adaptive EWMA (exponentially weighted moving average) type chart that combines the Shewhart and the classical EWMA schemes in a smooth way. To improve the detection performance of the FSI (fixed sampling interval) AEWMA control chart 7 in terms of the ATS(average time to signal), this paper proposes a new VSI (variable sampling interval) AEWMA control chart. A Markov chain approach is used to calculate the ATS values of the new VSI AEWMA control chart, and comparative results show that the proposed control chart performs better than the standard FSI AEWMA control chart and than other VSI control charts over a wide range of shifts.