Enrique Castillo 0000-0002-8570-0844

This paper is devoted to emphasizing the importance of human-based knowledge and to present the original property-based models. The main idea is that models are built in terms of equations that reproduce and guarantee their satisfaction, which leads to non-arbitrary parametric models. The methods based on data alone are not sufficient to be applied to fatigue S–N and GRV-N models: first, the required high number of results in machine learning is not attained in fatigue; second, a black box cannot supply a comprehension of the fatigue phenomenon; third, the absence of a supporting model impedes extrapolation beyond the scope of experimentation; and fourth, many other data are required to include the stress ratio, R, while robust models are already available. These models are artificial intelligence mixed models, where its main part is human-based whereas the parameter estimation is solved based on …

This work is a short review of the state of the art aiming to contribute to the use, disclosure, and propagation of systems of linear inequalities in real life, teaching, and research. It shows that the algebraic structure of their solutions consists of the sum of a linear subspace, an acute cone, and a polytope, and that adequate software exists to obtain, in their simplest forms, these three components. The work describes, based on orthogonality and polarity, homogeneous and complete systems of inequalities, the associated compatibility problems, and their relations with convex polyhedra and polytopes, which are the only possible solution for bounded problems, the most common in real practice. The compatibility and the observability problems, including their symbolic forms, are analyzed and solved, identifying the subsets of unknowns with unique solutions and those unbounded, important items of information with practical relevance in artificial intelligence and automatic learning. Having infinitely many solutions of a given problem allows us to find solutions when some of the assumptions fail and unexpected constraints come into play, a common situation for engineers. The linear programming problem becomes trivial when the set of all solutions is available and all solutions are obtained, contrary to the case of standard programs that provide only one solution. Several examples of applications to several areas of knowledge are presented, illustrating the advantages of solving these systems of inequalities.

The purpose of the research is to analyze the new Spanish law of Traffic, which no longer permits exceeding by up to 20 km/hour the generic speed limits when overtaking on conventional roads. In this research, deterministic and random models are developed to analyze the associated safety risks. The deterministic model highlights the importance of dimensional analysis and provides dimensionless abacuses to analyze the problem. Next, Bayesian networks and Bayesian models are used to build a random model from the previous one, providing a general method to convert deterministic into random models. In addition, the problems of ignoring the dimensions of the variables and parameters are discussed, a common mistake to be corrected. Some examples and multidimensional graphics illustrate the huge reduction in safety and the need to review the existing end of prohibition signs, most of which must be removed. Shortly, the research results show that the distance required for overtaking with safety increases drastically.

A Bayesian network approach is presented for probabilistic safety analysis (PSA) of railway lines. The idea consists of identifying and reproducing all the elements that the train encounters when circulating along a railway line, such as light and speed limit signals, tunnel or viaduct entries or exits, cuttings and embankments, acoustic sounds received in the cabin, curves, etc. In addition, since the human error is very relevant for safety evaluation, the automatic train protection (ATP) systems and the driver behaviour and its time evolution are modelled and taken into account to determine the probabilities of human errors. The nodes of the Bayesian network, their links and the associated probability tables are automatically constructed based on the line data that need to be carefully given. The conditional probability tables are reproduced by closed formulas, which facilitate the modelling and the sensitivity analysis. A sorted list of the most dangerous elements in the line is obtained, which permits making decisions about the line safety and programming maintenance operations in order to optimize them and reduce the maintenance costs substantially. The proposed methodology is illustrated by its application to several cases that include real lines such as the Palencia-Santander and the Dublin-Belfast lines.

A phenomenological approach is proposed to describe and predict the random behaviour of the total strain in plain and fiber concrete under compressive fatigue loading. Twofold viewpoints are considered in the assessment of the fatigue damage process. The first one, acknowledged as the stochastic cumulative damage evolution in terms of the number of applied cycles is represented by a sample function, physically identified with the total specimen strain caused by matrix microcracking. The second one is ascribed to the intrinsic scatter of the total lifetime referred to as the prefixed ultimate fatigue strain limit state (not necessarily failure). The sample function, once normalized to the total fatigue lifetime, is identified as a cumulative distribution function (cdf) of the generalized extreme value (GEV) family, particularly as Weibull distribution. Besides the reliable probabilistic lifetime prediction for the material under …

In this work, three classes of fatigue models are reviewed according to the fatigue regimes commonly considered in the current components design. Particular attention is devoted to the so-called Class III fatigue models, covering the three fatigue regimes, namely, LCF, HCF and VHCF. The applicability and limitations of the proposed analytical sigmoidal solutions are discussed from the viewpoint of practical design. The compatible Weibull S-N model by Castillo and Canteli is revisited and improved by considering a new reference parameter GP = E·σM ·(dε/dσ)|M as the driving force alternative to the conventional stress range. In this way, the requirement, σM ≤ σu, according to the real experimental conditions, is fulfilled and the parametric limit number of cycles, N0, recovers its meaning. The probabilistic definition of the model on the HCF and VHCF regimes is maintained and extended to the LCF regime. The strain …

This article develops a probabilistic approach to a micromechanical model to calculate the dynamic viscosity in self-compacting steel-fiber reinforced concrete (SCSFRC), which implies a paradigm shift in the approach of the deterministic models used. It builds on a previous work by the authors in which Bayesian analysis is applied to rheological micromechanical models in cement paste, self-compacting mortar, and self-compacting concrete. As a consequence of the varied characteristics of the particles in these suspensions (in terms of materials, shapes, size distributions, etc.), as well as their random nature, it seems appropriate to study these systems with probabilistic models. The Bayesian analysis, thorough Markov Chain Monte Carlo and Gibbs Sampling methods, allows the conversion of parametric-deterministic models into parametric-probabilistic models, which results in enrichment in engineering and science. The incorporation of steel fibers requires a new term in the model to account for their effect on the dynamic viscosity of SCSFRC, and this new term is also treated here with the Bayesian approach. The paper uses an extensive collection of experimental data to obtain the probability density functions of the parameters for assessing the dynamic viscosity in SCSFRC. The results obtained with these parameters’ distributions are much better than those calculated with the theoretical values of the parameters, which indicates that Bayesian methods are appropriated to respond to questions in complex systems with complex models.

Different considerations suggest that the fatigue crack growth mechanism, in particular, the threshold between propagating and non-propagating conditions is factually governed by maximum stress rather than stress range values. This propitiates the consideration of σ M, as the primary variable to which the S–N field should be referred to whereas R= σ m/σ M would act as a secondary variable. Based on physical and statistical principles, notably the compatibility condition between both distributions F (N; σ M, R) and F (σ M; N, R), a functional equation is established whose solution provided the analytical expression of the σ M–N–R field representing the more general solution of the S–N field characterized by the common fatigue endurance limit. Higher reliability in the model S–N assessment is achieved because of the pooling of all results irrespective of the R-value. The essential point is that the normalizing variable …

La influencia de la frecuencia se analiza como un efecto de endurecimiento viscoelástico del material que conduce simultáneamente a una reducción de los valores nominales de la tensión máxima, σM, y de la relación de tensiones, R, aplicados durante el ensayo para frecuencias crecientes. Esto permite la reducción del modelo σM-R-f-N, en el que la tensión máxima es la variable primaria de referencia mientras la relación de tensión, R, y la frecuencia, f, son las variables secundarias, al modelo probabilístico tridimensional σM-R-N, recientemente desarrollado. Los pares σM-R nominales aplicados en el ensayo se transforman en pares σM R efectivos que pueden evaluarse como pertenecientes a una única muestra con independencia de la frecuencia del ensayo y de la relación de tensiones. Además, el modelo define una variable normalizada V, que representa una medida del daño acumulado cuando se somete la probeta a una historia de carga variable permitiendo estimar la evolución de la probabilidad de fallo en función de la carga aplicada. El modelo se valida mediante su aplicación a un amplio programa experimental de fatiga realizado sobre alambres de aleación de aluminio de alta pureza, Al_H11, empleados en la interfaz de transistores bipolares de puerta aislada (IGBT) bajo tres frecuencias diferentes (2, 20 y 200 Hz).

We review some of the joint contributions made with Barry C. Arnold to the Probability and Statistics field during more than 30 years. First, we concentrate on those related to conditional specification, characterization of distributions and Bayesian methodologies with priors based on conditional specification. Also, we review other topics studied jointly with Barry C. Arnold such as multivariate distributions defined in terms of contours, new classes of multivariate distributions based on the Rosenblatt construction and multivariate order statistics and other applications.

The purpose of this research is to develop a model, with emphasis on compatibility conditions and model building, valid for high cycle fatigue design components such as wind turbines, automobiles, high speed railways and aeronautical material. In this work, we have added the frequency as one more variable to an existing fatigue model that already includes maximum stress, stress ratio and lifetime. As a result, a model and estimation method has been proposed and a random variable V has been identified, which, allows the accumulated damage and the probability of failure to be assessed for any load history in terms of stress levels, stress ranges and frequencies. Finally, the model is validated using a large set of real experimental data.

Sample functions, i.e., stochastic process realizations, are used to define cumulative damage phenomena which end into an observable terminal state or failure. The complexity inherent to such phenomena justifies the use of phenomenological models associated with the evolution of a physical magnitude feasible to be monitored during the test. Sample functions representing the damage evolution may be identified, once normalized to the interval [0,1], with cumulative distribution functions (cdfs), generally, of the generalized extreme value (GEV) family. Though usually only a fraction of the whole damage evolution, according to the specific problem handled, is available from the test record, the phenomenological models proposed allow the whole damage process to be recovered. In this way, down- and upwards extrapolations of the whole damage process beyond the scope of the experimental program are …

We present a new focus for the Krieger–Dougherty equation from a probabilistic point of view. This equation allows the calculation of dynamic viscosity in suspensions of various types, like cement paste and self-compacting mortar/concrete. The physical meaning of the parameters that intervene in the equation (maximum packing fraction of particles and intrinsic viscosity), together with the random nature associated with these systems, make the application of the Bayesian analysis desirable. This analysis permits the transformation of parametric-deterministic models into parametric-probabilistic models, which improves and enriches their results. The initial limitations of the Bayesian methods, due to their complexity, have been overcome by numerical methods (Markov Chain Monte Carlo and Gibbs Sampling) and the development of specific software (OpenBUGS). Here we use it to compute the probability density functions that intervene in the Krieger–Dougherty equation applied to the calculation of viscosity in several cement pastes, self-compacting mortars, and self-compacting concretes. The dynamic viscosity calculations made with the Bayesian distributions are significantly better than those made with the theoretical values.

In this work, a novel methodology for fracture characterization of metallic notched components including the effect of notch root radius and temperature is proposed based on the brittle-to-ductile transition curve. To this aim, two different regression models are derived, either by considering temperature as an influencing variable or combined with the notch radius effect. In the former case, the compatibility condition between the statistical distributions of the fracture toughness for a given temperature and of the temperature for a given fracture toughness is applied. This allows the K c-T field to be analytically defined proving that both distributions are interrelated and cannot be arbitrarily defined. The second regression model is based on the Theory of the critical distances by converting the experimental data at different notch radii to a reference value. In this way the so-called notch or apparent fracture toughness is …

This paper presents a novel methodology to characterize viscoelastic materials, allowing the limitations of current conventional models based on Time-Temperature-Superposition (TTS) principle to be avoided. It implies the definition of the temperature-time field, T-t, from short-term recorded relaxation curves at different temperatures by establishing the compatibility condition between the temperature dependent relaxation modulus at given time, E(T; t), and the time dependent relaxation modulus for a given temperature, E(t; T). The solution of the resulting functional equation allows the T-t field to be analytically defined by assuming the normalized relaxation function to be a stochastic model properly identified as a survival cumulative distribution function of certain statistical families such as normal or Gumbel ones. As a result, the corresponding master curves in the T-t field for both E-t and E-T functions are directly …

In practical design, the question of existence or non-existence of fatigue limit seems little more than an irrelevant academic discussion. Nevertheless, the answer affects many aspects of the fracture mechanics, such as, the calculation of damage accumulation; recognition of the multiplicity of fatigue mechanisms (and consequently, the multiplicity of S–N fields); interpretation of non-propagating cracks; and repercussion on the Kitagawa-Takahashi diagram. Various prestigious scientists deny the existence of a fatigue limit based on the results of failures in very high cycle fatigue (VHCF) regime, far below the high cycle fatigue (HCF) limit. However, the asymptotic extrapolation of the experimental results beyond the scope of testing with S–N models, cannot confirm or deny this hypothesis if those models do not fulfil indispensable physical and statistical requirements. Some phenomenological models based on solid …

Time and temperature, besides pressure in a lesser extent, represent the most significant variables influencing the rheological behavior of viscoelastic materials. These magnitudes are each other related through the well-known Time–Temperature Superposition (TTS) principle, which allows the master curve referred to relaxation (or creep) behavior to be derived as a material characteristic. In this work, a novel conversion law to interrelate relaxation curves at different temperatures is proposed by assuming they to be represented by statistical cumulative distribution functions of the normal or Gumbel family. The first alternative responds to physical considerations while the latter implies the fulfillment of extreme value conditions. Both distributions are used to illustrate the suitability of the model when applied to reliable derivation of the master curve of Polyvinil–Butyral (PVB) from data of experimental programs. The new approach allows not only the TTS shift factors to be estimated by a unique step, but the whole family of viscoelastic master curves to be determined for the material at any temperature. This represents a significant advance in the characterization of viscoelastic materials and, consequently, in the application of the TTS principle to practical design of viscoelastic components.

This paper provides a summary of the life and contributions of János Galambos, from Temple University. It describes the papers, conferences, books and all his contributions to Probability and Statistics, mainly to extreme value analysis. It also includes a list of his publications.