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MACROSCOPIC BEHAVIOUR OF POROUS METALS WITH INTERNAL GAS PRESSURE

By Theodore Sanders,2014-11-25 11:04
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MACROSCOPIC BEHAVIOUR OF POROUS METALS WITH INTERNAL GAS PRESSURE

MACROSCOPIC BEHAVIOUR OF POROUS METALS WITH

    INTERNAL GAS PRESSURE UNDER MULTIAXIAL LOADING

    CONDITIONS

     1,231,2Andreas Öchsner, Gennady Mishuris and José Grácio

     1Centre for Mechanical Technology and Automation 2Department of Mechanical Engineering

    University of Aveiro, Campus Universitário de Santiago,

    3810-193 Aveiro, Portugal 3Department of Mathematics

    Rzeszow University of Technology, W. Pola 2,

    35-959 Rzeszow, Poland

    ABSTRACT: The gas pressure build up within closed cells or dynamic intercellular air flow in open-cells foams influences the macroscopic behaviour during the deformation. Two dimensional numerical simulations for simple cell shapes point out that an internal pressure delays the onset of yielding in the cell walls during compressive load. Another important process affected by internal pore pressure is the roll forming of structurally porous metals. During pressing and thermo-mechanical forming the internal pressure of the pores and the temperature can raise to high values. One can find also in the literature that there is no essential influence on the forming response for gas pressure about half the yield strength of the fully dense matrix material. In this paper, the influence of internal gas pressure on the mechanical behaviour of porous metals under differing loading conditions is numerically investigated. It is found that the internal gas pressure influences rather significantly the initial yielding and the macroscopic behaviour also if the pressure is clear less than a half of the yield strength of the base material.

1. INTRODUCTION

    Porous and cellular metals, e.g. metal foams, exhibit unique properties and are currently being considered for use in lightweight structures such as cores of sandwich panels or as passive safety components of automobiles. The mechanical properties of porous and cellular metals, in particular their resistance to plastic deformation, the evolution and progress of damage and fracture within the material, are determined by the microstructure (geometry, topology, porosity/relative density) and the base/cell wall material, respectively. It is concluded from experimental investigations (1) that during deformation also the gas pressure build up within closed cells or dynamic intercellular air flow in open-cells foams influences the macroscopic behaviour. Two dimensional numerical simulations for simple cell shapes (2) point out that an internal pressure delays the onset of yielding in the cell walls during compressive load. Another important process affected by internal pore pressure is the roll forming of structurally porous metals (3). During isostatic pressing and thermo-mechanical forming the internal pressure of the gas filled pores and the temperature can raise to significant high values. However, it is stated that for gas pressures about half the yield strength of the fully dense matrix material, there is essentially no influence on the forming response. In this paper, the influence of internal gas pressure on the mechanical behaviour under uniaxial (tension and compression) and multiaxial loads is numerically investigated based on the finite element method. Three dimensional regular pore structures (cf. Fig. 1) with different relative densities and different yield

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    behaviour of the matrix material are investigated in order to clarify the influence of internal pore pressure on the macroscopic mechanical behaviour.

2. CONSTITUTIVE MODELLING OF POROUS MATERIALS

    The mechanical behaviour of a porous metal and its mathematical characterisation can be described based upon the principles of continuum mechanics, if a "representative volume element" (RVE) is considered, Fig. 1a). In the case of porous materials, this RVE needs to comprise of at least 5 to 10 unit cells (UC), in order to represent macroscopic values. The precise number depends, however, on the corresponding cell structure and should be examined separately. Since the intention of this paper is to model only isotropic solids, no further account was taken of the anisotropy caused by the particular pore distribution and only strain states having principle strain distributions parallel to the x- and y-axes were considered (cf. Fig. 1). This modelling

    technique, based on homogenous cell structures, has already been applied for the description of the plastic behaviour of porous metals (4, 5) and for the modelling of damage effects (6-8). However, the investigation of the elastic-plastic behaviour under consideration of an internal gas pressure was not a subject within those works.

    Figure 1: a) Representative volume element of a simple cubic cell structure; b) finite

    element model of a unit cell with boundary conditions and applied loads for

    uniaxial tension.

2.1 Elastic Behaviour

    Under the classical assumptions of small strains and linear relationships between the

     and the strain tensor (, the elastic stress-strain relation second order stress tensor ?ijij

    is given by the general Hooke's law

    1)?(??, (1) ?ijijijkkE1

where E is Young’s modulus and Poisson’s ratio. In a uniaxial tension or

    compression test, the only non-zero stress component ? causes axial strain ( and xxxx

    transverse strains ( = (. Thus, one can determine the elastic constants, i.e. Young’s yyzz

    modulus and Poisson’s ratio from Eq. (1) as:

    (?(yyxxzz. (2) Eand(((xxxxxx

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    We will also later make use of the ratio between the total transverse and longitudinal strains in order to characterise the plastic behaviour.

2.2 Plastic Behaviour

The three essential ingredients of plastic analysis are: the yield criterion, the flow rule,

    and the hardening rule, (9). The yield criterion relates the state of stress to the onset of yielding. The flow rule relates the state of stress ? to the corresponding increments ij

    pof plastic strain, when an increment of plastic flow occurs. The hardening rule d(ij

    describes how the yield criterion is modified by straining beyond initial yield. The yield criterion for an isotropic material can generally be expressed as

    FF(?,,)0, (3) ijij

where is the isotropic hardening parameter and are the kinematic hardening ijoparameters, respectively. The state of stress ? can be split in its spherical ? and ijij

    deviatoric part ?' = s and then be expressed in terms of the combinations of the three ijijoostress invariants J, J' and J', where J is the first invariant of the spherical stress 1231

    tensor and J' and J' are the second and third invariants of the deviatoric tensor, (10). 23

    Thus, one can replace Eq. (3) by

    o'' (4) ;;FFJ,J,J,,0.123ij

The yield condition F = 0 represents in a