Plastically-driven variation of elastic stiffness in green bodies during powder compaction. Part II: Micromechanical modelling

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Abstract

A micromechanical approach is set-up to analyse the increase in elastic stiffness related to development of plastic deformation (the elastoplastic coupling concept) occurring during the compaction of a ceramic powder. Numerical simulations on cubic (square for 2D) and hexagonal packings of elastoplastic cylinders and spheres validate both the variation of the elastic modulus with the forming pressure and the linear dependence of it on the relative density as experimentally found in Part I of this study, while the dependence of the Poisson's ratio on the green's density is only qualitatively explained.

Introduction

Densification of metal as well as ceramic powders is a problem connected with a strong industrial interest, so that the micromechanics of this process has been the focus of a number of investigations (almost all addressed to metal particles, while the akin problem of ceramic powders has been much less investigated). Grains have been usually assumed as spherical (or cylindrical for simplicity), so that micromechanics explains how plasticity and increase of contact areas between particles influence the overall stress/strain behaviour. The analysis of this problem sheds light on the macroscopic constitutive modelling of the powder, to be employed in the design of moulds to form green pieces with desired shape. The compaction problem is also of great academic interest in several fields, including biomechanics, where it traces back to the famous ‘Histoire Naturelle’ by the Count de Buffon, who reports on a (probably ‘thought’) experiment with peas:

Qu’on remplisse un vaisseau de pois, ou plutôt de quelqu’autre graine cylindrique, et qu’on le ferme exactement après y avoir versé autant d’eau […]; qu’on fasse boullir cette eau, tous ces cylindres deviendront des colonnes à six pans. On en voit clairement la raison, qui est purament mécanique; chaque graine, dont la figure est cylindrique, tend par son reflement à occuper le plus d’espace possible dans un espace donné, elles deviennent donc toutes nécessairement hexagones par la compression réciproque.

This is an example of compaction of a package of spheres (Fig. 1), later continued by D’Arcy Thompson in his On Growth and Form and others.

Micromechanical models of powder compaction have been developed so far for a cubic (square in 2D) geometry of spheres [1], [2], [3], [4] or cylinders [5], [6], [7], [8] in frictionless contact, and friction between grains has also been considered for the latter geometry [5]. Random packing of cylinders and spheres have been analyzed respectively in [4], [5]. All the above-mentioned investigations, in which the spheres and the cylinders are modelled within the framework of the J2-flow theory of plasticity with linear hardening or perfectly plastic behaviour (Fig. 6), are all focused on the determination of the yield surface at different stages of compaction.

The objective of the present article is to investigate how the plastic deformation of grains during compaction influences the macroscopic elastic response of the material, an aspect never considered before, but central in the development of elastoplastic coupling (see Part I of this study). To this purpose, 2D (plane strain) and 3D square/cubic and hexagonal packings of cylindrical and spherical grains are considered (Fig. 3). Although detailed information on the constitutive law valid for the grains is not available, these are modelled via von Mises perfect or linear hardening plasticity, which is typical of a basic and simple mechanical behaviour. Representative volume elements of the cylinder and sphere packings are deformed to model the state of uniaxial strain achieved in a cylindrical rigid die and the mean stress/mean strain behaviour is numerically determined using the finite element program Abaqus Unified FEA®. Once the uniaxial strain compaction has been completed, the representative element is unloaded and reloaded under uniaxial stress to evaluate the average Young modulus and Poisson's ratio of the material. In this way it is possible to determine the variation of the elastic modulus with the forming pressure and the dependence of the elastic modulus on the density. These evaluations validate the experimental results presented in the Part I of this study. The micromechanical evaluation of the Poisson's ratio is more complicated than that of the elastic modulus. In this case, the results from micromechanics correctly explain the qualitative increase of the Poisson's ratio with the forming pressure, but the values are not tight to experimental results.

The dependence of elastic stiffness on the level of plastic deformation is a crucial aspect of elastoplastic modelling of geological and granular materials, including ceramic, metal powders, and greens. Results provided in the present article explain the plastic micromechanisms inducing elastic stiffening during compaction of ceramic powders.

Section snippets

A toy mechanical model to explain elastoplastic coupling

Before to set up the micromechanical model for the qualitative and quantitative explanation of elastoplastic coupling, a simple mechanical model is presented with the aim of providing a simple explanation of the phenomenon. The model is intended only to shed light on the mechanism of increase in elastic stiffness due to plastic deformation and not to provide a quantitative evaluation.

Referring to an elastoplastic circular cylinder of initial height h0 and cross section of radius a0, this is

Micromechanical modelling

Square/cubic and hexagonal two-dimensional (grains are idealized as cylinders) and three-dimensional (grains are idealized as spheres) granule dispositions are considered as representative of ceramic powders, Fig. 3. Although at a first glance these geometries may appear far from the reality, they are usually considered to represent correctly the overall behaviour of granulates [1], [2], [3], [4], [5], [6], [7], [8]. For the considered packagings, symmetry allows the reduction into the

The evolution of elastic stiffness from micromechanics

The evolution of the elastic Young modulus E with the forming pressure is reported in Fig. 7 for the 2D simulations and in Fig. 8 for the 3D cases.

In the figures the plastically deformed shapes of the grains are reported upon unloading and before the reloading imposed to determine the elastic Young modulus.

In both cases the material parameters listed in Table 1 provide a close fit to experimental results (on alumina powder, see Part I of this study) and highlight the elastoplastic coupling

Conclusions

A micromechanical approach has been developed to explain the increase of elastic stiffness with the density of the greens observed during forming of ceramic powders (experiments have been reported in Part I of this study). Although the mechanical characteristics of the single granule are not known, a reasonable elastoplastic model for this has been employed, so that it has been possible to consider idealized 2D and 3D geometrical configurations of grains (represented as circular cylinders or

Acknowledgements

The authors gratefully acknowledge financial support from the European FP7 programs INTER-CER-2 PIAP-GA-2011-286110-INTERCER2 (L.P.A., D.C., and D.M.), ERC-2013-ADG-340561-INSTABILITIES (D.B.), and FP7-PEOPLE-2013-ITN-PITN-GA-2013-606878-CERMAT2 (A.P.).

References (8)

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