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Investigating the main determinants of the mechanical performance of metals is not a simple task. Already known physical inspired qualitative relations between 2D microstructure characteristics and 3D mechanical properties can act as the starting point of the investigation. Isotonic regression allows to take into account ordering relations and leads to more efficient and accurate results when the underlying assumptions actually hold. The main goal in this paper is to test order relations in a model inspired by a materials science application. The statistical estimation procedure is described considering three different scenarios according to the knowledge of the variances: known variance ratio, completely unknown variances, variances under order restrictions. New likelihood ratio tests are developed in the last two cases. Both parametric and non-parametric bootstrap approaches are developed for finding the distribution of the test statistics under the null hypothesis. Finally an application on the relation between Geometrically Necessary Dislocations and number of observed microstructure precipitations is shown., Comment: 18 pages, 3 figures

The martensitic transformation is one of the most important phenomena in metals science due to its essential contribution to the strength of steels and most engineering alloys. Yet the basic, atomistic mechanisms leading to martensite nucleation and twin morphology are not yet known. A detailed picture in this regard is required if the strengthening effects of martensite are to be properly understood. This work presents molecular dynamics (MD) simulations of the martensitic transformation using a model fcc/bcc semi-coherent interface with Nishiyama-Wasserman orientation relationship. Significant insight into this important phenomenon is detailed in this work which shows that the atomic displacements that cause nucleation and twin morphology formation of the martensitic phase originate at the fcc/bcc interface. The interface facilitates the initial atomic shear during the transformation which in turn causes the stress-induced homogeneous nucleation and twin morphology formation. The understanding of the atomistic processes leading to the twin morphology formation will allow the control of the twinning process for further enhancement of mechanical properties., Comment: 12 pages, 4 figures

Modeling microstructures is an interesting problem not just in Materials Science but also in Mathematics and Statistics. The most basic model for steel microstructure is the Poisson-Voronoi diagram. It has mathematically attractive properties and it has been used in the approximation of single phase steel microstructures. The aim of this paper is to develop methods that can be used to test whether a real steel microstructure can be approximated by such a model. Therefore, a general framework for testing the Poisson-Voronoi assumption based on images of 2D sections of real metals is set out. Following two different approaches, according to the use or not of periodic boundary conditions, three different model tests are proposed. The first two are based on the coefficient of variation and the cumulative distribution function of the cells area. The third exploits tools from to Topological Data Analysis, such as persistence landscapes., Comment: 29 pages, 27 figures

Although Poisson-Voronoi diagrams have interesting mathematical properties, there is still much to discover about the geometrical properties of its grains. Through simulations, many authors were able to obtain numerical approximations of the moments of the distributions of more or less all geometrical characteristics of the grain. Furthermore, many proposals on how to get close parametric approximations to the real distributions were put forward by several authors. In this paper we show that exploiting the scaling property of the underlying Poisson process, we are able to derive the distribution of the main geometrical features of the grain for every value of the intensity parameter. Moreover, we use a sophisticated simulation program to construct a close Monte Carlo based approximation for the distributions of interest. Using this, we also determine the closest approximating distributions within the mentioned frequently used parametric classes of distributions and conclude that these approximations can be quite accurate., Comment: 15 pages, 12 images (7 figures)