MACROSCOPIC BEHAVIOUR OF POROUS METALS WITH
INTERNAL GAS PRESSURE UNDER MULTIAXIAL LOADING
CONDITIONS
Andreas Öchsner1,2, Gennady Mishuris3 and José Grácio1,2
1
Centre for Mechanical Technology and Automation
2
Department of Mechanical Engineering
University of Aveiro, Campus Universitário de Santiago,
3810-193 Aveiro, Portugal
3
Department 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
second order stress tensor ij and the strain tensor ij, the elastic stress-strain relation
is given by the general Hooke's law
 ij 
1  


 ij kk  ,
  ij 
E 
1 

(1)
where E is Young’s modulus and  Poisson’s ratio. In a uniaxial tension or
compression test, the only non-zero stress component xx causes axial strain xx and
transverse strains yy = zz. Thus, one can determine the elastic constants, i.e. Young’s
modulus and Poisson’s ratio from Eq. (1) as:
E
 xx
 xx
and

 yy
 xx
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
 zz
.
 xx
(2)
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 ij to the corresponding increments
of plastic strain d ijp , when an increment of plastic flow occurs. The hardening rule
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 ( ij , ,  ij )  0 ,
(3)
where  is the isotropic hardening parameter and ij are the kinematic hardening
parameters, respectively. The state of stress ij can be split in its spherical ijo and
deviatoric part ij' = sij and then be expressed in terms of the combinations of the three
stress invariants J1o, J2' and J3', where J1o is the first invariant of the spherical stress
tensor and J2' and J3' are the second and third invariants of the deviatoric tensor, (10).
Thus, one can replace Eq. (3) by


F  F J1o , J 2' , J 3' , ,  ij  0.
(4)
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 ij. The components of the stress tensor reduce to the principle stresses
I, II and III on the principle diagonal and the non-diagonal elements are equal to
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 I + II + III = 0 there, it follows that
ij = sij, i.e. any stress state on the -plane is pure deviatoric. On the basis of the
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 J2' and J3'. Stress states with J2' = const lie on a circle around the
hydrostatic axis in an octahedral plane. A dependency of the yield condition on J3'
results in deviation from the circle shape. The yield surface forms a prismatic body
whose longitudinal axis is represented by the hydrostatic axis. A dependency on J1o
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.
Therefore, a dependency on J1o can be represented by sectional views through planes
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
space; b) octahedral plane (J1o = const).
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
d ipj  d
Q
.
 ij
(5)
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(ij) - k() = 0, in which the current yield stress k represents the size of the
yield surface. For kinematic hardening, Eq. (3) yields F = f(ij - ij) – k =0, where k
and the elements of the tensor ij are now constants. Are the elements of the tensor ij
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 xdisplacement: ux,i = … = ux,j. All the following simulations were carried out with this
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 ES - were considered. The base material (index S relates to the base material: ES
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= 72700 MPa; S = 0.34; initial yield stress kS = 300 MPa) was assumed to be
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. RESULTS
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 (pi = 0 MPa) is the
same in tension and compression. However, the presence of an internal pore pressure
(pi = 50 MPa) results in a different behaviour in the plastic range: the yield stress in
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 (pi = -0.1 MPa) has practically no influence on the macroscopic
curve.
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 stressstrain curves.
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
relative density for pi = 0 MPa which also holds for pi = 50 MPa. A clear decrease of
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 (/S < 1) results from the fact that just the smallest pore
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.
y  22  I  0.5 II   III  ,
(6)
x
6
4
 III   II  .
(7)
Here, it is now essential whether the plastic behaviour is pressure sensitive, i.e.
depends on J1o, or not. If there is no dependency on J1o, i.e. a constant shape along the
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
hydrostatic stress are allowed to be represented in the same octahedral plane where J1o
= const holds. As a results, e.g. uniaxial tensile (J1o = I) and pure shear tests (J1o = 0)
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
multiaxial stress states with J1o = 0, e.g. I = -II (III = 0) or I = -2II = -2III (I > 0
 I < 0), were realised and the corresponding yield stresses were drawn in the
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
(pi = 0 MPa).
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(J2’)) and the Tresca surface (lower bound) in the case of no internal pore pressure.
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(J2’, J3’)) than by the von Mises criterion. It should be mentioned here that the
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 J3’. To clarify the influence of the hydrostatic stress on the yield
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
plane (pi = 50 MPa).
Figure 11: Graphical representation of the yield condition in the (3J2')0.5-J1ocoordinate system (pi = 0 MPa;  = 0°).
Figure 12: Graphical representation of the yield condition in the (3J2')0.5-J1ocoordinate system (pi = 50 MPa;  = 0°).
It can be seen that the yield surface is closed and symmetric to a certain octahedral
plane which is in case of pi = 0 identical with the deviatoric plane. Introducing an
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internal pressure shifts significantly the yield surface along the hydrostatic axes in the
compression regime.
5. CONCLUSIONS
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
all three invariants, i.e. F = F(J1o, J2’,J3’), in order to accurately approximate the
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.
ACKNOWLEDGMENTS
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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