MACROSCOPIC BEHAVIOUR OF POROUS METALS WITH
INTERNAL GAS PRESSURE UNDER MULTIAXIAL LOADING
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.
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
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) E，and：，？，？(((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
F，F(?,；,！)，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) ;；F，FJ,J,J,；,！，0.123ij
The yield condition F = 0 represents in a n-dimensional space a closed hypersurface
that is also called yield surface or yield loci. A direct graphical representation of the yield surface is due to its dimensionality not possible. However, a reduction of the dimensionality is possible to achieve if a principle axis transformation is applied to the argument ?. The components of the stress tensor reduce to the principle stresses ij
?, ? and ? on the principle diagonal and the non-diagonal elements are equal to IIIIII
zero. In such a principle stress space it is possible to represent the yield condition as a three-dimensional surface. This space is also called the Haigh-Westergaard stress space. A hydrostatic stress state lies in such a principle stress system on the space diagonal (hydrostatic axis). Any plane perpendicular to the hydrostatic axis is called octahedral plane. The particular octahedral plane passing through the origin is called deviatoric plane or ；-plane (10) and because of ? + ? + ? = 0 there, it follows that IIIIII
? = s, i.e. any stress state on the ；-plane is pure deviatoric. On the basis of the ijij
dependency of the yield condition on the invariants, a descriptive classification can be performed. Yield conditions independent of hydrostatic stress can be represented by the invariants J' and J'. Stress states with J' = const lie on a circle around the 232
hydrostatic axis in an octahedral plane. A dependency of the yield condition on J' 3
results in deviation from the circle shape. The yield surface forms a prismatic body owhose longitudinal axis is represented by the hydrostatic axis. A dependency on J 1
denotes a size change of the cross-section of the yield surface along the hydrostatic axis. However, the shape of the cross-section remains similar in mathematical sense. oTherefore, a dependency on J can be represented by sectional views through planes 1
along the hydrostatic axis. The geometrical interpretation of stress invariants (11) is given in Fig. 2.
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Figure 2: Geometrical interpretation of basic stress invariants: a) principal stress o = const). space; b) octahedral plane (J1
The flow rule is stated in terms of a function Q, which is described in units of stress,
and is called a plastic potential. With d， a scalar called plastic multiplier, the plastic
strain increments are given by
?Qp (5) dd.(，，ij??ij
The flow rule is said to be associated if Q = F, otherwise nonassociated. Hardening
can be modelled as isotropic (i.e. initial yield surface expands uniformly without distortion and translation as plastic flow occurs) or as kinematic (i.e. initial yield surface translates as a rigid body in stress space, maintaining its size, shape and orientation), either separately or in combination. For isotropic hardening, Eq. (3) yields F = f(?) - k(；) = 0, in which the current yield stress k represents the size of the ij
yield surface. For kinematic hardening, Eq. (3) yields F = f(? - ！) – k =0, where k ijij
and the elements of the tensor ！ are now constants. Are the elements of the tensor ！ ijij
function of the stress tensor then the shape and orientation of the yield surface can be modified which is also known as anisotropic hardening.
3. FINITE ELEMENT SIMULATION
The commercial finite element code MSC.Marc was used for the simulation of the macroscopic properties of the chosen cell structure, Fig. 1. A fine mesh with solid elements (quadratic shape functions) of such a cell structure consisting of 5 to 10 unit cells would lead, however, to a large amount of unknowns and a huge amount of computing time. Therefore, only a typical repeating portion (unit cell) was simulated. Due to the symmetry of the unit cell, only one eighth needs to be considered, Fig. 1b). It should be mentioned here that the mechanically correct constraints are of great importance in the case of uniaxial tension or compression. With the aid of so-called multiple point constraints (MPC), the used FE-system offers the possibility to realize a periodic boundary condition where all nodes on a certain surface have the same x-
displacement: u = … = u. All the following simulations were carried out with this x,ix,j
periodic boundary. Four different relative densities were realised and the influences of the different hardening behaviours of the base material on the results were investigated. Besides the realistic hardening behaviour of an aluminium alloy (AlCuMg1), the cases - ideal plasticity and linear hardening with a plastic modulus of 1/20 E - were considered. The base material (index S relates to the base material: E SS
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： = 0.34; initial yield stress k = 300 MPa) was assumed to be = 72700 MPa; SS
isotropic and to obey the von Mises yield criterion with the associated flow rule. The internal pore pressure was set to an initial value and then modified according to the ideal gas law (p?V = const.) during the deformation of the pores.
4.1 Macroscopic Uniaxial Loads
A typical stress-strain diagram for a uniaxial tension/compression test is shown in Fig. 3. The initial elastic region appears as a straight line, where Young’s modulus can be
calculated from Eq. (2). The initial (macroscopic) yield stress is recoded when the equivalent stress (von Mises) of an integration point reaches the yield stress of the base material.
Figure 3: Principle influence of internal pore pressure on uniaxial stress-strain curves.
It can be seen that the mechanical behaviour without pore pressure (p = 0 MPa) is the i
same in tension and compression. However, the presence of an internal pore pressure (p = 50 MPa) results in a different behaviour in the plastic range: the yield stress in i
tension is clearly decreased whereas yielding in compression is delayed. Also this modification is much clearer in the tensile domain. A modification of the pure elastic behaviour is not observable under the chosen problem parameters. Furthermore, vacuum in the pores (p = -0.1 MPa) has practically no influence on the macroscopic i
The behaviour of the ratio between the total transverse and longitudinal strains is shown in Fig. 4. Here, also practically no difference can be observed in the elastic region where the ratio is equal to Poisson’s ratio. However, in the plastic region the behaviour is again contrary: the ratio between the transverse and longitudinal strains increases in the compressive domain whereas this ratio decreases in the tensile domain. The absence of an internal pressure would lead to a symmetric increase of this ratio in the plastic region under compression and tension. Figure 5 illustrates the influence of different hardening behaviours of the base material on the macroscopic stress-strain diagram. Comparison with Fig. 3 also indicates that the influence of the internal pore pressure on the macroscopic stress-strain diagram is much bigger than the differing hardening behaviour. It should be emphasised here that the simulation started with an initial pressure and that the pressure is not build up from zero or
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ambient pressure through the deformation of the cells but modified according to the ideal gas law.
Figure 4: Influence of internal pore pressure on of the ratio between the total
transverse and longitudinal strains.
Figure 5: Influence of the hardening behaviour of the base material on uniaxial stress-
Figure 6: Influence of differing internal pore pressure on uniaxial stress-strain curves.
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It could be concluded form Figs. 3, 4 and 6 that there is practically no influence of the internal pressure on the elastic part of the curves for such elastic constants of the base material and chosen relative densities. Therefore, Fig. 7 summarises the variation of the normalised elastic constants, i.e. Young’ modulus and Poisson’s ratio, with the
= 0 MPa which also holds for p = 50 MPa. A clear decrease of relative density for pii
the elastic stiffness and the contraction number with decreasing relative density can be seen.
Figure 7: Relative elastic constants as a function of the relative density.
Looking at the normalised initial yield stress, one can observe that an internal pressure yields to an increase of the yield stress under compression and a decrease of the yield stress under tension. Furthermore, this effect increases slightly with decreasing relative density of the porous material. The drop between the full dense material (?/? S
= 1) and the porous material (?/? < 1) results from the fact that just the smallest pore S
acts as a stress concentrator and plastic deformation will be early initiated there.
Figure 8: Relative initial yield stress as a function of the relative density.
3.2 Macroscopic Multiaxial Loads
To investigate the shape of the yield surface it is necessary to realise multiaxial stress states and to draw the shape in the corresponding deviatoric plane according to the following transformation which projects a principle stress state first in the deviatoric
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plane (angle of transformation: 45?) and then to the Cartesian coordinate system (x,y) shown in Fig. 9.
2, (6) ;；;；y，?？0.5?？?IIIIII2
6;；. (7) x，?？?IIIII4
Here, it is now essential whether the plastic behaviour is pressure sensitive, i.e. oodepends on J, or not. If there is no dependency on J, i.e. a constant shape along the 11
hydrostatic axis, then all evaluated points can be drawn in a single octahedral plane. However, if one can expects a dependency then only stress states with the same ohydrostatic stress are allowed to be represented in the same octahedral plane where J 1oo= const holds. As a results, e.g. uniaxial tensile (J = ?) and pure shear tests (J = 0) 1I1
can not be represented in the same octahedral plane. In order to draw the shape of the yield surface for the pressure sensitive material under consideration, differing omultiaxial stress states with J = 0, e.g. ? = -? (?= 0) or ? = -2? = -2? (? > 0 1IIIIII IIIIIII
( ? < 0), were realised and the corresponding yield stresses were drawn in the I
deviatoric plane. I should be noted here that this shape changes only its size along the hydrostatic axes but remains similar in mathematical sense.
Figure 9: Graphical representation of the yield condition in the deviatoric stress plane
(p = 0 MPa). i
Figure 9 and 10 illustrate the cross section of the yield surface in the deviatoric plane. It can be seen that the shape can be bounded by the von Mises yield surface (F =
F(J’)) and the Tresca surface (lower bound) in the case of no internal pore pressure. 2
The upper and lower bounds drawn in Fig. 9 result from the convexity condition of a yield surface which is a direct result of Drucker’s stability postulate. Furthermore, it can be seen that the cross section is much better approximated by the Tresca condition (F(J’, J’)) than by the von Mises criterion. It should be mentioned here that the 23
maximum error between these both classical yield conditions is 15.47 % for ~ = 30?.
The presence of an internal pressure results in a different behaviour in the tension and compression regime and the cross section becomes now egged-shaped as shown in Fig. 10. The approximation of such a shape requires definitely the incorporation of the third invariant J’. To clarify the influence of the hydrostatic stress on the yield 3
surface, sectional views through planes along the hydrostatic axis for ~ = 0? (cf. Fig. 2)
are presented in Figs. 11 and 12.
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Figure 10: Graphical representation of the yield condition in the deviatoric stress
= 50 MPa). plane (pi
0.5oFigure 11: Graphical representation of the yield condition in the (3J')-J-21
coordinate system (p = 0 MPa; ~ = 0?). i
0.5oFigure 12: Graphical representation of the yield condition in the (3J')-J-21
coordinate system (p = 50 MPa; ~ = 0?). i
It can be seen that the yield surface is closed and symmetric to a certain octahedral plane which is in case of p = 0 identical with the deviatoric plane. Introducing an i
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internal pressure shifts significantly the yield surface along the hydrostatic axes in the compression regime.
It is found that the internal gas pressure influences significantly the initial yielding and the macroscopic stress strain behaviour also if the gas pressure is clear less than a half of the yield strength of the fully dense matrix material. Furthermore, the internal pressure results in a different influence under tension and compression. The continuum mechanical description of the yield criterion requires the incorporation of o, J’,J’), in order to accurately approximate the all three invariants, i.e. F = F(J123
shape. Further research work will concentrate on the mathematical formulation of the yield criterion and the proposition of a minimum set of experiments in order to determine the required parameters.
A. Öchsner is grateful to Portuguese Foundation of Science and Technology for financial support. G. Mishuris participated in this research during his Visiting Fellowship at LSBU granted by the Leverhulme Trust.
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