A comparison among five equations of state in predicting the inversion curve of some fluids

Five equations of state, modified Peng-Robinson by Danesh et al. (MPR1), modified SRK equation of state by Mathias and Copeman (MSRK), Vdw11, Harmens-Knapp (HK) and modified Peng-Robinson equation of state by Ruzy (MPR2) were compared in predicting of the inversion curve of some fluids. This enable us to judge the accuracy of the results obtained from different equations of state. MSRK and HK equations of state give good prediction of the low-temperatures branch of the inversion curve and are closely matched with the experimental inversion curve. As a corollary to the present study, we have perceived that the agreement of the MPR2 and Vdw11 equations of state with the inversion curve are inadequate. We also calculated maximum inversion temperature and maximum inversion pressure for every component used in this work.     

A fuzzy approach for drawing S–N curves in the finite life region

In this work a fuzzy approach has been developed in order to estimate the probability of fatigue failure. In particular, with the proposed method the S–N curves of a material in the finite life region can be drawn. The experimental data are represented in terms of fuzzy sets and are fitted using a fuzzy linear regression. Data scattering and uncertainty in the empirical failure model are reflected in the definition of membership functions. Several examples are shown to illustrate the procedure. Failure probabilities and fatigue curves obtained by the fuzzy method are similar to those obtained by traditional statistical analysis, based on normal distributions of strength with standard deviation that remains constant with different load levels. In particular, the results obtained indicate that the possibilities offered by fuzzy systems are also applicable for estimating the Wöhler curve of a material under fatigue stresses. To evaluate its reliability, the proposed method is compared with the traditional one, with particular attention to the case in which a small amount of experimental data is available. The new fuzzy method is slightly less accurate than traditional statistical analysis to outline S–N curves in the finite life region. This is mainly due to the fact that the method is influenced by the nonuniformity of data dispersion at each level of stress.     

An equation of state for pure fluids describing the critical region

Density fluctuations of a pure nuid are treated by a cell model, in which the fluid is divided into cells containing different numbers of particles. A probability function for the particle number is derived. This function, after convolution with a classical (mean field) equation of state, leads to an improved equation of state which is valid in the critical region. The equation of state is analytical, hence not exact in the immediate vicinity of the critical point. As an example, the convolution is applied to the Carnahan-Starling/van der Waals equation of state; the resulting equation of state is used to correlate thermodynamic properties of several simple fluids.     

A corresponding-states principle for the equation of state of hard-convex-body fluids

A theoretically based corresponding-states principle is developed for the equation of state of hard-convex-body fluids. For all the fluids considered, the excess compressibility factor, reduced by means of a parameter which can be determined analytically. lies on a single curve whose analytical expression can be obtained from the equation of state of the hard-sphere fluid.     

Deriving Analytical Expressions for the Ideal Curves and Using the Curves to Obtain the Temperature Dependence of Equation-of-State Parameters

Different equations of state (EOSs) have been used to obtain analytical expressions for the ideal curves, namely, the Joule–Thomson inversion curve (JTIC), Boyle curve (BC), and Joule inversion curve (JIC). The selected EOSs are the Redlich–Kwong (RK), Soave–Redlich–Kwong (SRK), Deiters, linear isotherm regularity (LIR), modified LIR (MLIR), dense system equation of state (DSEOS), and van der Waals (vdW). Analytical expressions have been obtained for the JTIC and BC only by using the LIR, MLIR, and vdW equations of state. The expression obtained using the LIR is the simplest. The experimental data for the JTIC and the calculated points from the empirical EOSs for the BC are well fitted into the derived expression from the LIR, in such a way that the fitting on this expression is better than those on the empirical expressions given by Gunn et al. and Miller. No experimental data have been reported for the BC and JIC; therefore, the calculated curves from different EOSs have been compared with those calculated from the empirical equations. On the basis of the JTIC, an approach is given for obtaining the temperature dependence of an EOS parameter(s). Such an approach has been used to determine the temperature dependences of A ₂ of the LIR, a and b parameters of the vdW, and the cohesion function of the RK. Such temperature dependences, obtained on the basis of the JTIC, have been found to be appropriate for other ideal curves as well.     

A scaled equation of state for real fluids in the critical region

A scaled equation of state is proposed for real fluids in the critical region which incorporates asymmetry with respect to the critical isochore. In the range of reduced densities 0.65(/ _c)1.4 and for reduced temperatures (T/T _c)1.2, the equation represents P-V-T data for steam within the experimental accuracy.     

A corresponding-states principle for the equation of state of linear homonuclear fused-hard-sphere fluids

A theoretically based corresponding-states principle previously developed for the equation of state of hard-convex-body fluids is extended to rigid linear homonuclear fused-hard-sphere fluids. Effective volumes and shapes are introduced in order to account for the nonconvexity of the molecules. The excess compressibility factor, reduced by means of a parameter which can be determined analytically, is a common function of the effective packing fraction. The analytical expression for the function can be obtained from the equation of state of the hard-sphere fluid. Existing simulation data for diatomics, triatomics and tetraatomics show excellent agreement with the corresponding-states principle.     

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

The paper introduces a methodology to compute strict upper and lower bounds for linear‐functional outputs of the exact solutions of the advection–diffusion–reaction equation. The bounds are computed using implicit a posteriori error estimators from stabilized finite element approximations of the exact solution. The new methodology extends the a posteriori error estimates yielding bounds for the standard Galerkin formulation to be able to obtain bounds for stabilized formulations. This methodology is combined with both hybrid‐flux and flux‐free techniques for error assessment. The application to stabilized formulations provides sharper estimates than when applied to Galerkin methods. The best results are found in combination with the flux‐free technique. Copyright © 2012 John Wiley & Sons, Ltd.     

On the validity of Hall–Petch equation for single-phase β Ti–Al–Nb alloys undergoing stress-induced martensitic transformation

The yield strengths of Ti–Al–Nb alloys, which undergo stress-induced martensitic transformation, prior to the onset of plastic deformation during tensile testing, were found to obey the Hall–Petch relationship with grain size. The overall friction stress was observed to decrease with increase in Nb content while it remained more or less unchanged with increase in Al content in these alloys. On the other hand, the overall , the unpinning constant, which is an index of the efficiency of boundaries as obstacle to dislocation motion, was found to increase with increase in Nb and decrease with increase in Al content in these alloys.     

A modification of the Petrov–Galerkin method for the transient convection–diffusion equation

A variation of the Petrov–Galerkin method of solution of a partial differential equation is presented in which the weight function applied to the time derivative term of the transient convection–diffusion equation is different from the weight function applied to the special derivatives. This allows for the formulation of fourth-order explicit and centred difference schemes. Comparison with analytic solutions show that these methods are able to capture steep wave fronts. The ability of the explicit method to capture wave fronts increases as the amount of convective transport increases.     

An improved neural computing method for describing the scatter of S–N curves

The reliability evaluation of structural components under random loading is affected by several uncertainties. Proper statistical tools should be used to manage the large amount of causalities and the lack of knowledge on the actual reliability-affecting parameters. For fatigue reliability prediction of a structural component, the probability distribution of material fatigue resistance should be determined, given that the scatter of loading spectra is known and a suitable damage cumulating model is chosen. In the randomness of fatigue resistance of a material, constant amplitude fatigue test results show that at any stress level the fatigue life is a random variable. In this instance fatigue life is affected by a variety of influential factors, such as stress amplitude, mean stress, notch factor, temperature, etc. Therefore a hybrid neural computing method was proposed for describing the fatigue data trends and the statistical scatter of fatigue life under constant loading conditions for an arbitrary set of influential factors. To support the main idea, two examples are presented. It can be concluded that the improved neural computing method is suitable for describing the fatigue data trends and the scatter of fatigue life under constant loading conditions for an arbitrary set of influential factors, once the optimal neural network is designed and trained.     

Crack growth resistance curves of GLARE 3 5/4 0.3 fiber–metal laminates at low temperature

The objective of this work was the evaluation of crack growth resistance curves ( J–R curves) of commercial GLARE® 3 5/4 laminates at −50 °C. The experimental evaluation of these curves was performed on 50‐mm‐wide compact tension specimens through the unloading compliance technique. The tests were based on the ASTM E1820 standard with minor modifications. Additionally, tensile tests were also performed at −50 °C on dogbone specimens according to the ASTM E8M standard. Comparisons between low and room temperature properties indicate that the material preserves both its tensile strength and fracture toughness at −50 °C, although low temperature J–R curves presented smaller slopes than the room temperature ones after the onset of stable crack growth.     

An improved method for estimation of Ramberg–Osgood curves of steels from monotonic tensile properties

In the present study, first a method for estimating cyclic yield strength is improved and compared using the experimental data of 121 steels from literature. Correlations between cyclic deformation properties (K′ and n′) and monotonic tensile data are then investigated, and a simple method requiring only the monotonic tensile properties is developed for estimation of the Ramberg–Osgood curve. Prediction capability of the proposed method is not only evaluated using the aforementioned 121 steels, but also compared with several commonly used methods that are available in the literature. The proposed correlations are shown to predict the cyclic deformation properties of most of considered steels reasonable well.     

Equation of state for mixtures of nonpolar fluids: Prediction from experimental constants of the components

We present a simple procedure by which an analytical equations of state for a mixture of normal fluids can be predicted from the constantsT _c (critical temperature),p _c (critial pressure), and (Pitzer acentric factor) for each pure component. The equation covers the range from the dilute vapor or gas to the highly compressed liquid or supercritical fluid.     

A multiphase mechanical model for Ti–6Al–4V: Application to the modeling of laser assisted processing

A multiphase model for Ti–6Al–4V is proposed. This material is widely used in industrial applications and so needs accurate behaviour modeling. Tests have been performed in the temperature range from 25 °C to 1020 °C and at strain rates between 10⁻³ s⁻¹ and 1 s⁻¹. This allowed the identification of a multiphase mechanical model coupled with a metallurgical model. The behaviour of each phase is calibrated by solving an inverse problem including a phase transformation model and a mechanical model to simulate tests under thermomechanical loadings. A scale transition rule (β-rule) is proposed in order to represent the redistribution of local stresses linked to the heterogeneity of plastic strain. Finally this model is applied to two laser assisted processes: direct laser fabrication and laser welding.     

Extension of strain–life equation for low‐cycle fatigue of sheet metals using anisotropic yield criteria and distortional hardening model

The fatigue strain–life equation is in general applicable to isotropic materials. It was recently attempted to account for material anisotropy because of crystallographic texture in fatigue modelling. The proposed modification was limited to isotropic hardening. The present work is an extension of the previous work, wherein a general framework to model anisotropy using phenomenological yield criterion and anisotropic hardening is provided. Yld2004‐18p yield criterion and the so‐called homogenous anisotropic hardening model are used to demonstrate the anisotropic cyclic behaviour of low carbon steel. The proposed methodology can be utilized in applications including multiaxial fatigue modelling.     

A neural-network approach to describe the scatter of cyclic stress–strain curves

In order to predict a product’s durability in the early phases of development it is necessary to know the stress–strain behaviour of the material, its resistance to fatigue and the loading states in the material. These parameters, however, tend to exhibit a considerable degree of uncertainty. Due to a lack of knowledge of the actual circumstances in which the product is used, during the early development phase, simulations based on statistical methods are used. The results of the experiments show that the cyclic stress–strain curves demonstrate not only a large amount of scatter, but also a dependence on the temperature, the size of the cross-section, the content of alloying elements, the loading rate, etc.This article presents a method for modelling cyclic stress–strain curve scatter using a hybrid neural network for an arbitrary selection of the influencing factors. In an example of the measured data for a high pressure die-cast aluminium alloy it is clear that the suggested method is suitable for describing cyclic stress–strain curves. The main advantage of a hybrid neural network in comparison with a conventional method is the neural network’s ability to precisely describe the influence of various factors, and their combinations, based on the form and scatter of the cyclic stress–strain curve families. Defining the model parameters, i.e., training the neural network, is a procedure that does not require any additional user interventions; however, it enables us to gather knowledge that would otherwise require a lot of research. Thus, the trained neural network is a robust tool that can be used to predict cyclic stress–strain curves for random values of influencing factors. The capabilities of the presented method are only limited by the quantity of the measured data used for the neural-network training.     

Thermophysical properties from the equation of state of Mason and co-workers

The theory gives formulas for calculating the three temperature-dependent parameters of the equation of state from the intermolecular potential. But the second virial coefficient also serves to predict the entire equation of state in terms of two scaling parameters and, hence, a number of other thermodynamic properties including the Joule-Thomson inversion curve, bulk modulus. secant bulk modulus, and inverse isobaric expansivity among others. Agreement with experimental data is quite good. Paper dedicated to Professor Edward A. Mason.     

Multiplicative cases from additive cases: Extension of Kolmogorov–Feller equation to parametric Poisson white noise processes

In this paper the response of nonlinear systems driven by parametric Poissonian white noise is examined.As is well known, the response sample function or the response statistics of a system driven by external white noise processes is completely defined. Starting from the system driven by external white noise processes, when an invertible nonlinear transformation is applied, the transformed system in the new state variable is driven by a parametric type excitation. So this latter artificial system may be used as a tool to find out the proper solution to solve systems driven by parametric white noises. In fact, solving this new system, being the nonlinear transformation invertible, we must pass from the solution of the artificial system (driven by parametric noise) to that of the original one (driven by external noise, that is known). Moreover, introducing this invertible nonlinear transformation into the Itô’s rule for the original system driven by external input, one can derive the Itô’s rule for systems driven by a parametric type excitation, directly. In this latter case one can see how natural is the presence of the Wong–Zakai correction term or the presence of the hierarchy of correction terms in the case of normal and Poissonian white noise, respectively. Direct transformation on the Fokker–Planck and on the Kolmogorov–Feller equation for the case of parametric input are found.