Theoretical-experimental method of multiscale modelling of two-component materials

Modern mechano-mathematical models of materials are multilevel ones combining macro-, mesoand micro-level models. A major problem however is how to present a clear link between the levels. Moreover, structural models should be well verified, reflecting material behavior at meso-level and its microstructure. Hence, we propose a theoretical-experimental approach to multilevel modeling, treating discs which operate under plane stress. The approach comprises the following stages: I stage: Identify a system of characteristic points on the disc surface. Employing an appropriate optical method, find point displacements using time steps in the course of deformation. II stage: Based on displacement development in time, rationally find the variation of the strain tensor components using strains directed from the main point to the closest points of its vicinity. III stage: Define a meso-representative element in the disc plane as an ellipse with diameters related to the principal strains. IV stage: Having prepared metallographic micro-sections, define systems of characteristic structural elements in the mesorepresentative ellipses. V stage: Averaging the structural systems at mesolevel, define macro-parameters involving material deformation parameters. VI stage: Find material macro-deformation parameters on the basis of an appropriate design of micro-representative elements and calculate the stressed-strained state of the latter applying representative loading. A test example is give in order to illustrate the approach.


Introduction
A modern tendency in the mechanics of materials is the design of mechano-mathematical models accounting for the material structure at three scales: micro-, meso-and macro-ones [1,2]. The arising major problems of multiscale modeling are: (1) Design of a particular model with specific parameters and development of a method of parameter identification at each scale; (2) Identification of the link between respective models and parameters, as well as verification of scale transition.
Multi-scale models with their plausibility and reliability are especially useful in modeling and calculation of safe structures and as a result-in saving energy needed for structure erection and exploitation. We focus in what follows on the analysis of materials with grain structure consisting of two components: a matrix and grain inclusions, which generally reinforce the material. Models at all the three scales are designed within the frames of continuum mechanics. Such an approach seems reasonable enough, since we do not analyse sub micro-scale of a material with a discrete atomic or molecular structure. In view of a possible application of some experimental methods we consider thin disks under plane stress undergoing static external loading. Disc thickness is much smaller than that of the grains. We treat disks as structural elements operating in exploitation conditions, i.e. they deform purely elastically without the development of damage and inelastic strains. The elastic properties are assumed to be isotropic and non-homogeneous, and they may change as a result of eventual structural changes occurring at meso-level.
The paper offers a theoretical-experimental method of multi-scale modelling of the materials concerned. It is based on generalization and further elaboration of a number of results published in refs. [3][4][5][6], adopting stages of an eventual practical realization.

First stage: Macro-experiments on disks
Consider thin elastically deformed disks undergoing static loading which generally varies in time . Select n characteristic points on the disk surface with projections on the OXYZ is introduced, plane OXY coincides with the disk middle surface and axis OZ is directed along the disk thickness, Fig. 1.   [11]) and specify vector The respective unit vector is using the selected characteristic points and the calculated distances of the least square deviation between the experimental (eq. 3) and the theoretical strains (eq. 4).
Using condition for min L we find the components of the strain tensor we should measure the change of the thickness We can calculate the increment (7). On the other hand, the increments of are found through a system of equations involving two times -1 Following a method described in [5]

Second stage: Specification of a meso-representative element and meso-structural parameters
We found the components of the macro tensor at selected characteristic points   , we grind the surface around the representative ellipse, which is a projection of the elliptical cylinder. Using an electron microscope, we identify the grain inclusions. Each grain is approximated as an elliptic cylinder, whose section cut by the middles plane is an ellipse with point

Micro-scale structural parameters
We select a micro-structural element for system (а): a prism with rectangular cross section in the disk middle plane. Its center is is loaded conditionally by a uniformly distributed load applied along the element border-see the representative loading in [5]. This loading yields uniformly distributed meso-stresses at   k a t p Q , , according to the scheme in [5].
Using nanoindentation [9] we find material micro-elasticity characteristics:   are determined via FEM [12]. Micro-strains are presented    Fig. 3, and the calculations can be performed using FEM [12].
The method presented in the study is based on available experimental data and theoretical studies and results. It provides means of the design of physically clear structural parameters at the three scales: macro-, meso-and micro-ones, specifying a link between them. This makes the mechano-mathematical model a physically well-defined tool. The employment of a multi-scale model in the mechanics of materials is an important issue enabling one to describe adequately material mechanical behavior and explicitly account for the effect of material structure. The model is exceptionally useful in structural mechanics since it allows a plausible record of the behavior of structural elements. It is also useful in modern energy saving structures compounded by surface-reinforced elements [8] providing an appropriate modeling of the reinforcement.
The study has been supported by the National Fund "Scientific Research", Project DFNI E02/10 121214.