Acta Materialia 54 (2006) 1995–2002
www.actamat-journals.com
Yield criterion of porous materials subjected to complex stress states
D.L.S. McElwain a, A.P. Roberts b, A.H. Wilkins
a
q
b,*
School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld. 4001, Australia
b
Department of Mathematics, University of Queensland, St. Lucia, Brisbane, Qld. 4072, Australia
Received 6 October 2005; received in revised form 12 December 2005; accepted 14 December 2005
Available online 23 February 2006
Abstract
Finite-element simulations are used to obtain many thousands of yield points for porous materials with arbitrary void-volume fractions with spherical voids arranged in simple cubic, body-centred cubic and face-centred cubic three-dimensional arrays. Multi-axial
stress states are explored. We show that the data may be fitted by a yield function which is similar to the Gurson–Tvergaard–Needleman
(GTN) form, but which also depends on the determinant of the stress tensor, and all additional parameters may be expressed in terms of
standard GTN-like parameters. The dependence of these parameters on the void-volume fraction is found.
2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Porous material; Micromechanical modeling – finite-element analysis; Mechanical properties – yield phenomena
1. Introduction
The yielding and plastic flow of porous ductile materials
has received much attention over the past three decades,
mainly because the fracture of such materials is believed
to involve nucleation, growth and coalescence of voids in
the region of the crack tip [1]. Also, porous ductile materials such as foamed metals with porosities of up to 90%
have recently become popular in structural applications.
For these reasons it is necessary to construct accurate
mechanical theories of porous ductile materials.
Armed with an accurate theory of the macroscopic
mechanics, scientists and engineers may easily make predictions concerning the behaviour of materials while safely
ignoring the microscopic mechanics of individual voids
and their interactions (see, for example [2]); the latter being
computationally unfeasible when the number of voids is
large.
McClintock [3] and Rice and Tracey [4] first performed
calculations concerning the growth of voids in ductile
materials subjected to external stresses. Later, Green [5]
proposed a yield function for porous materials, and later
still Gurson [6] analytically derived an upper bound for
the yield function for a rigid, perfectly plastic, von Mises
hollow sphere subject to arbitrary external stresses. Building on early finite-element numerical work by Needleman
[7], Tvergaard [8,9] studied mesoscopic stress–strain relations, the evolution of the void volume fraction and the distribution of strain and stress throughout the macroscopic
porous material. Further numerical work by Tvergaard
and Needleman [10] led to the famous Gurson–Tvergaard–Needleman (GTN) prediction for the yield criterion
U = 0, for a ductile solid with void-volume fraction f and
matrix yield stress rM:
3
2
q Rhyd =rM 1 q3 f 2 .
U ¼ ðReqv =rM Þ þ 2fq1 cosh
2 2
ð1Þ
q
Supported by a grant from the Australian Research Council through
the Discovery Grant scheme.
*
Corresponding author. Tel.: +61 7 3365 3266; fax: +61 7 3365 1477.
E-mail addresses: [email protected] (D.L.S. McElwain), apr@
maths.uq.edu.au (A.P. Roberts), [email protected] (A.H.
Wilkins).
The notation used in this expression and below is as follows: Rij is the applied macroscopic stress tensor. Below
we will consider arrays of voids, and if rij(x) is the Cauchy
stress tensor at point x, R = Æræ, where the angled brackets
1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2005.12.028
1996
D.L.S. McElwain et al. / Acta Materialia 54 (2006) 1995–2002
indicate a volume average. The traceless part of a tensor
will be denoted by a prime: R0ij ¼ Rij 13 dij trR. The macroscopic von Mises equivalent stress and hydrostatic stresses
are, respectively,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Reqv ¼ 3R0ij R0ij =2;
1
Rhyd ¼ trR.
3
0.8
0.6
Σeqv
0.4
0.2
This paper will also use the determinant of the stress tensor, Rdet:
Rdet ¼ det R.
The parameters q1, q2 and q3 were introduced by Tvergaard
and Needleman in order that Gurson’s bound on the yield
criterion should more closely match experimental and
numerical results in the initial stages of plastic flow, and
many analyses use q3 ¼ q21 following the presentation of [8].
As discussed by Hom and McMeeking [11], Richmond
and Smelser [12] also proposed a modified yield criterion,
with q1f being replaced by f2/3, and q3f2 by f4/3.
In early studies in particular, the finite-element calculations were two-dimensional, either modelling perforated
sheet-like materials, or prismatic three-dimensional materials subject to plane stress. Some papers that fall into this
category are [7,8,10,13,14]. The three-dimensional finite-element calculations typically involve spherical or ellipsoidal
voids packed into periodic arrays (usually hexagonal or
simple cubic structures). Both axisymmetric and more general macroscopic stress configurations directed along the
principal axes of the material are used. Some papers that fall
into this category are [15,9,16,17,11,18–21,2,22–29].
The current paper builds on this work by completing a
detailed numerical study of the yield surface for almost
rigid, perfectly plastic, von Mises ductile solids containing
empty voids arranged in simple cubic (sc), body-centred
cubic (bcc) and face-centred cubic (fcc) arrays. The results
are valid for void-volume fractions ranging from f = 0% to
the percolation threshold of around f = 96%.
Various complicating factors are not included: the coalescence of voids; the associated or non-associated flow
after yielding; the effect of large strains; and, the effect of
different types of hardening. These are considered to be
beyond the scope of this paper.
It is hoped by the present authors and those cited above
that by studying materials containing regular arrangements
of voids, some insight can be gained into the mechanics of
isotropic porous materials which are costly to study
numerically. While this manuscript was in preparation, a
recent study of the isotropic case by Wicklein and Thoma
[30,31] was brought to the attention of the authors. Wicklein and Thoma simulated the behaviour of nine cubical
samples (of 39 cm3) of foamed aluminium containing
many voids of diameters between 8 and 10 mm, with void
volume fractions ranging from f = 0.5 to f = 0.65. The elastic constants, the initial yield surface, the plastic flow and
the yield-surface hardening were derived. Their initial yield
0
0
Σzz > Σxx
Σzz < Σxx
GTN (1.4, 1.1, 1.9)
GTN (1, 1, 1)
0.2
0.4
0.6
0.8
1
Σhyd
Fig. 1. Yield points for Rxx = Ryy and positive hydrostatic stress.
(Crosses) Yield points for Rzz > Rxx. (Circles) Yield points for Rzz < Rxx.
(Outer solid line) Gurson’s bound, q1 = q2 = q3 = 1. (Dotted line) Best-fit
GTN curve with q1 = 1.4, q2 = 1.1 and q3 ¼ q21 ¼ 1:9. All data collected at
f = 0.25 for voids arranged in a simple cubic pattern. These data were
obtained using the numerical procedure described below.
surface displays very similar characteristics to that
obtained here with particular emphasis being placed on
the third invariant of stress, Rdet.
Despite all the investigations already mentioned, a comprehensive study of the dependence of the yield function on
multiaxial stress states and void volume fractions has not
been performed. The purpose of this paper is to make a
comparison with the GTN proposal, quantify the effect
of Rdet, and to numerically derive the dependence of the
q-like parameters on f for the three structures, sc, bcc
and fcc.
One of the principal results of this paper is that to
describe the yield surface accurately, the yield function
must depend on Rdet. Hints of this may be seen in
[17,19,22,23,27], where the GTN q parameters appeared
to depend on the stress state used: an analogous result is
found in Ref. [32]. It may be seen most dramatically in
Richelsen and Tvergaard [21], and in the analysis of Zhang
et al. [26], who explored several arbitrary states of stress
aligned with the principal axes of a periodic voided material. It may also been seen in the data of Hom and McMeeking [11] and Goya et al. [19], where the value of stress
triaxiality did not uniquely define the yield point. Fig. 1
shows data similar to that generated by [26] – axisymmetric
with Rxx = Ryy, and with Rzz arbitrary, but the hydrostatic
stress Rhyd P 0. The data appear to describe two distinct
yield curves, meaning that yield function must depend on
more variables than simply Rhyd and Reqv, in contradistinction to Eq. (1).
2. General remarks concerning the yield function
2.1. The isotropic case
The yield function for an isotropic solid must depend
only on Rhyd, Reqv and Rdet since these are the well-known
three invariants of the rotation group.
D.L.S. McElwain et al. / Acta Materialia 54 (2006) 1995–2002
Now consider the shape of the yield curve on an arbitrary
octahedral planep
defined
ffiffiffiffiffiffiffiffi by Rhyd = h. The radial coordinate
on this plane is 2=3Reqv , while the third invariant can be
related to the angular coordinate, h, through
1
2
Rdet ¼ h3 R2eqv h þ R3eqv cos 3h.
3
27
ð2Þ
Thus the yield curve is described by 0 = Uh(Reqv, cos 3h),
which is generically a 3n-lobed shape (n is integer).
The most extreme case is the equilateral triangle, shown
in Fig. 2. It is most extreme in the sense that of all possible
non-concave shapes, the variation of Reqv is greatest. The
equilateral triangle has the property that
min
Rmax
eqv ¼ 2Reqv .
Thereby, convexity puts an upper bound on the effect of
Rdet on the yield function: the value of Reqv can vary by a
maximum of two for fixed Rhyd.
It shall be assumed that the yield function is convex
throughout this paper. While this is quite common, it is
not strictly thermodynamically necessary, and non-convex
forms have been used for porous materials by Wicklein
and Thoma [30]. Nevertheless, the numerical data generated
here suggests strongly that the yield function is convex.
2.2. Cubic symmetry
For porous materials with sc, fcc and bcc arrays of voids,
the yield function must be invariant under the cubic symme-
1997
try group – Oh in Schönflies notation – which is a subgroup
of the group of general rotations and reflections in three
dimensions [33]. Oh is finite dimensional of order 48 and
consists of rotations through 90 about the x, y and z axes,
reflections through the xy, xz and yz planes, and all combinations thereof. For the theory to be invariant under Oh, the
arguments of the yield function must be invariant combinations of components of the stress tensor.
The isotropic invariants Reqv, Rhyd and Rdet are also
invariants of Oh and so we shall use these. However, there
are further invariants, such as RxxRyyRzz (which does not
equal Rdet for general R) or R2xy þ R2xz þ R2yz which could
also be arguments of the yield function. There are six invariants in total – the yield function has six arguments – and
this makes finding an analytical expression for the yield
function that describes the finite element (FE) data
complicated.
However, here this complexity is reduced by restricting
to macroscropic stress states with
Rxy ¼ Rxz ¼ Ryz ¼ 0.
ð3Þ
Then, the independent degrees of freedom contained in the
non-zero triplet (Rxx, Ryy, Rzz) are completely captured by
the triplet (Reqv, Rhyd, Rdet), and the yield function has only
three arguments. If this paper is considered as a precursor
to a similar study of the more complicated isotropic case,
this procedure should afford some insight into the type of
results obtained there.
2.3. Final comments
Σ III
ΣI
Σ II
The yield function is also invariant under R ! R, since
the microscopic constitutive laws, in particular the von
Mises plasticity, are invariant under (R, E) ! (R, E).
When Reqv = 0 the yield hypersurface degenerates to
two points given by |Rhyd| = a(f). This follows because Reqv
is a sum of squares, so each term must be zero, and all Oh
invariant forms can be determined by Rhyd. The yield function is therefore a function of |Rhyd|. This result is evident in
Fig. 1, in which the two sets of data appear to converge
towards the hydrostatic axes as Reqv ! 0.
In the following sections, it will be useful to write the yield
function in a form which separates the contributions of Rdet
from the other invariants. This is because the size of the Rdetinduced variations around the ‘‘average’’ yielding behaviour
given by circles as in Fig. 2 are severely limited by the argument contained in Section 2.1, and because this ‘‘average’’
behaviour is described well by the GTN formula.
This is done by writing the yield function as a power-series expansion
U ¼ Uð0Þ ðReqv ; Rhyd Þ þ Uð1Þ ðReqv ; Rhyd ÞðRdet hRdet iÞ
Fig. 2. Three yield curves on an octahedral plane. The most extreme nonconcave shape is the equilateral triangle (with the maximum value of Reqv
attained at its vertices). The GTN form is necessarily a circle: the one
shown (dashed) intersects with the triangle at cos 3h = 0. The other curve
shown (in grey) is an example of a generic shape consistent with the
isotropic symmetry.
2
þ Uð2Þ ðReqv ; Rhyd ÞðRdet hRdet iÞ þ ð4Þ
with the following definitions. Introduce the function ÆReqv,
which depends on Rhyd and solves U(0)(ÆReqvæ, Rhyd) = 0.
Then define
1998
1
hRdet i R3hyd hReqv i2 Rhyd .
3
D.L.S. McElwain et al. / Acta Materialia 54 (2006) 1995–2002
ð5Þ
Thus, when cos 3h = 0 and Reqv = ÆReqvæ, we have U = 0.
Geometrically, U(0) is thus describing the circle in
Fig. 2 – it is the ‘‘average’’ behaviour. The higher-order
terms describe variations around this average behaviour,
and below it will be necessary to retain only the linear
term.
3. The numerical method
3.1. Description
The materials studied in this paper have spherical voids
arranged in sc, bcc and fcc arrays. Some examples are
depicted in Fig. 3. Various void-volume fractions are considered, from 2% up to 90% which is close to the percolation threshold of the structure where the matrix is no
longer connected.
The voids are empty and exert no pressure on the surrounding matrix material.They have zero Young’s
modulus.
Because of the periodicity of the cubic structures and the
stress ansatz of Eq. (3), only one-eighth of the unit cell need
be considered.
To stay close to the original Gurson presentation, the
matrix material is chosen to be almost rigid, perfectly plastic. The Poisson ratio of the matrix is 1/3. The initial yield
point rM = 7.5 · 105E, and the hardening is linear with
slope 7.5 · 108E, where E is the Young’s modulus.
The finite-element technique is used to generate yield
points. Each unit cell, comprising of both the matrix and
void material, is decomposed into n3 cubical elements as
illustrated in Fig. 4.
The strain rate of the one-eighth unit cell’s surfaces
(which must remain planar) is held constant (but different
for each simulation). That is,
Eij ðtÞ ¼ diagðExx ; Eyy ; Ezz Þ t;
ð6Þ
where t denotes time.
A quasi-static approximation is made: time is discretised
and the equilibrium configuration at each small time-step is
obtained iteratively by using the return-mapping algorithm
described in Chapter 12 of [34] in conjunction with solving
the infinitesimal elasticity problem with each different element having, in general, a different consistent tangent operator. A custom-built FORTRAN code is used, relying
heavily on a customised version of the 3D infinitesimal
elastic routine written and extensively bench-marked by
Garboczi and collaborators [35,36].
Fig. 3. Four different unit cells. (Top left) Simple-cubic (sc) with f = 0.11. (Top right) Body-centred cubic (bcc) with f = 0.84. (Bottom) Face-centred cubic
(fcc) with f = 0.45 and f = 0.83.
D.L.S. McElwain et al. / Acta Materialia 54 (2006) 1995–2002
1999
However, when the triaxiality (Rhyd/Reqv) is very high, it
becomes difficult to precisely ascertain when the material
has ‘‘yielded’’, because the definition of ‘‘yield’’ involves
finding a maximum in Reqv. The data becomes somewhat
suspect and sometimes may even appear to describe a
non-convex surface. This may be viewed as another sort
of error which is not properly accounted for in the above
analysis since it is not only related to the discretisation
used. Therefore, it is prudent to omit such data from the
data-fitting exercise below, although because we will
demand convexity by fitting with simple three or four
parameter models known to describe convex surfaces, their
inclusion does not affect the result substantially because the
‘‘noise’’ due to this error cancels.
4. Results
Fig. 4. A sketch of the cubical finite-element approximation to the sc unit
cell (for clarity most of the cubes are not shown, and recall that only oneeighth of the unit cell is used in the simulations).
The code was tested on non-porous materials and
bench-marked against results found in Refs. [11,21].
The strain is increased until the ‘‘yield point’’ is reached,
which is defined here following [37] as the point at which
the von Mises equivalent stress reaches a maximum.
The elements’ elastic and plastic strains are always small
(<104) because of the matrix rigidity and because the simulations are never taken past the yield point just defined.
Thus, it is safe to use the standard small-strain approximation to the mechanics.
A few hundred simulations, each with randomly chosen
(Exx, Eyy, Ezz) are performed for each void volume fraction
giving a few hundred yield points on the yield surface.
3.2. Scaling of results with number of elements and the error
4400 yield points were generated for each of the three
arrangements of voids, with f ranging from 0.02 up to 0.9
which is close to the percolation threshold of the matrix,
SC
BCC
which is approximately 0.965 ðfperc
Þ for sc, 0.994 ðfperc
Þ
FCC
for bcc, and 0.963 ðfperc
Þ for fcc. The yield points were
expressed in (Reqv, Rhyd, Rdet) space and a least-squares
fit with an extension of GTNs formula was found.
Consider the sc case with 30% void volume fraction
(f = 0.3). As is shown in Fig. 5, the average response is fitted well with GTNs formula with q1 = 1.31, q2 = 1.16 and
q3 ¼ q21 . The large amount of scatter around this curve is
due to the influence of Rdet. In Fig. 6 all FE data in the
range 0.27 6 Rhyd 6 0.33 (approximately half-way along
the hydrostatic axis in this case) is considered and a plot
of ðRdet ; R2eqv Þ is made. A linear relation is evident.
In fact, exploring other slices of hydrodynamic stress
and f results in the conclusion that R2eqv may always be
approximated as a linear function of Rdet.
Moreover, the slope of this relationship is directly proportional to Rhyd.
The yield point predicted depends on the value of n, and
it is important to choose it large enough so that an accurate
prediction is made. Numerical experiments with different n
show that the predicted value of stress, Rij, at yield scales
inversely with n:
rij
.
Rij ðnÞ ¼ R1
ij þ
n
This is quite common [38], and can be used to derive the
limiting ‘true’ value R1
ij by extrapolation, and thereby to
estimate the discretisation error at each n.
In the present case this gives an estimate of the relative
discretisation error
R ðnÞ R1 ij
ij ¼ 0:02 at n ¼ 80.
R1
ij
The dependence on the applied strain is minimal, and this
value has been obtained by averaging the results over the
void volume fraction and the types of cubic structure.
0.6
0.5
Σeqv
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Σhyd
Fig. 5. GTN yield curve (solid) and FE data points for sc voids with 30%
void volume fraction. GTN parameters are q1 = 1.31, q2 = 1.16 and
q3 ¼ q21 .
2000
D.L.S. McElwain et al. / Acta Materialia 54 (2006) 1995–2002
where
0.32
A¼1þ
0.28
2
Σeqv
0.24
0.2
-0.02
-0.01
0
0.01
0.02
Σdet
Fig. 6. R2eqv versus Rdet for those data points which lie in the band
0.27 6 Rhyd 6 0.33. Simple cubic voids with f = 0.3.
In this way, we are led to propose an extension of the
GTN yield criterion:
2
3
a2 ðf ÞRhyd =rM 1
U ¼ Reqv =rM þ 2a1 ðf Þ cosh
2
2
a1 ðf Þ sðf ÞðRhyd =rM ÞðRdet hRdet iÞ=r3M .
ð7Þ
Three new functions a1(f), a2(f) and s(f) have been introduced
2
here. Comparing
with Eq. (5), hR2eqv =r2M i ¼ 1 þ a1 ðf Þ 3
2a1 ðf Þ cosh 2 a2 Rhyd =rM . In Eq. (7), the linear relationship
between R2eqv and Rdet is evident, as is the direct proportionality between the slope of this relationship and Rhyd.
The function s(f) controls the size of the term containing
Rdet. When s(f) = 0, the yield function reduces to GTNs
formula if a1 = q1f and a2 = q2.
4.1. Monotonicity, single-valuedness and convexity
The particular forms of a1, a2 and s detailed below
ensure that U is monotonic, in that increasing f always
brings the yield function closer to the origin in stress space.
By Eq. (2), U written above may be viewed as a cubic in
Reqv, and thus it is possible that given Rhyd and h (the angle
on the octahedral plane), there are up to three possible
solutions for Reqv. This is a problem of single-valuedness.
We address this problem by simply defining yield to occur
at the smallest non-negative value of Reqv that is a root of
the equation U = 0.
A full set of conditions resulting from convexity are
algebraically extremely complicated and beyond the scope
of this paper, being largely unimportant: only the crucial
points are investigated here. Finally, for some Rhyd ¼
Rmax
hyd , the yield surface should degenerate to a point, irrespective of the value of h.
Consider the issue of convexity in further detail. Writing
Rdet in terms of h, the yield function may be expressed as
2
3
U ¼ AS þ ðReqv =rM Þ S þ ðReqv =rM Þ T ;
a21
3
a2 Rhyd =rM ;
2a1 cosh
2
1
2
ð8Þ
S ¼ 1 sðRhyd =rM Þ ;
3
2
T ¼ sðRhyd =rM Þ cos 3h.
27
It is immediately obvious that if S = 0 for some Rhyd < Rmax
hyd
then the only solution of U = 0 is Reqv = 0 which is a contradiction of the definition of Rmax
hyd . Thus, if we write
2
max
s ¼ 3x= Rhyd ;
ð9Þ
2
1 þ a21
max
Rhyd ¼
arccosh
;
3a2
2a1
the quantity x(f) 6 1 for all f. It is important that this
parameterisation of s(f) in terms of Rmax
hyd is not only convenient, but also strongly suggested by the FE results for the
sc and bcc cases: a least-squares fit with the data at fixed
volume fraction shows x(f) varies between about 0.7 for
high volume fractions and unity for low volume fractions.
(The FE data for the fcc case are not trilobal and hence
s = 0 for fcc.)
The critical point is therefore when the yield curve is an
equilateral triangle for some Rhyd, for increasing Rhyd past
this point may result in a non-convex curve.
Some straightforward algebraic manipulation shows
that triangularity is avoided if, and only if,
S2 >
1 2 2
s Rhyd A;
27
ð10Þ
for all Rhyd and f.
The cosh factor in A means a closed-form solution cannot be found. However, by writing the
pffiffiffi inequality in terms
of x and using a series expansion of A, it may be checked
that if
1
x1 ¼ 1 þ pffiffiffi max
3Rhyd
pffiffiffi
1
1 max4 pffiffiffið4Þ
2 pffiffiffi00
A
A
Að0Þ þ Rmax
ð0Þ
þ
ð0Þ
þ
.
.
.
;
R
2 hyd
4! hyd
ð11Þ
where the RHS contains
pffiffiffi a finite number of terms of the series
expansion
of
A around Rhyd = 0, (for instance
pffiffiffi00
A ð0Þ ¼ 9a1 a22 =4ð1 a1 Þr2M ), the inequality of Eq. (10) is
guaranteed to hold. Any x1 greater than this will ensure
convexity, however, this choice fits the FE data well. Note
finally that x 6 1 as required above.
4.2. Simple cubic
Applying a least-squares regression to the FE data
shows that they are well-fit by
SC 0:809
a1 ðf Þ ¼ ðf =fperc
Þ
and
SC
a2 ðf Þ ¼ 0:88 þ 1:27ðf =fperc
Þ.
ð12Þ
D.L.S. McElwain et al. / Acta Materialia 54 (2006) 1995–2002
Note that, by construction, a1(f) vanishes at f = 0 and is
SC
unity at the percolation threshold f ¼ fperc
0:965 as required. It is of a power-law form similar to that proposed
in Ref. [12], but with a different exponent. Also note that
a2(f) is close, but not equal to the values of around unity
proposed by a number of other authors such as [8,17,27].
The first five terms of the power series in Eq. (11) have been
used.
By way of example, Fig. 7 shows a slice of data from
Rhyd = 0.25rM to Rhyd = 0.5rM at f = 0.25 and the fit with
curves generated by the functions of Eq. (12) and x given
by Eq. (11). This pattern is universal: as Rhyd is increased
from zero the yield curves become more trilobal, but close
to Rmax
hyd the curves becomes circular again.
An indication of the quality of the proposed yield function is obtained by plotting the predicted value of R2eqv
(given Rhyd, Rdet and f) versus the FE result for each data
point. This is shown in Fig. 8. Included is the data from
[21] who calculated some yield points for a variety of different stress states (some with Rdet 6¼ 0) for voids at f = 0.04
and arranged in a simple cubic fashion inside a fairly stiff
matrix (rM = 0.0033E) and Poisson’s ratio of 0.3. Data
are also included from Ref. [11] in which yield points were
calculated for three different stress states for voids at
f = 0.0082 and f = 0.065 arranged in a simple cubic fashion
inside a matrix with rM = E/200, Poisson’s ratio of 0.3, and
a small power-law hardening of 0.1. In some cases data was
extracted from graphs in these two papers which may have
resulted in some error. Nevertheless, Fig. 8 shows good
agreement with an R2 of 0.998.
Σzz
Σ2eqv
2001
with Σ det
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
FE
Fig. 8. The predicted value of R2eqv versus the FE result for the 4400 yield
points generated for sc with a range of void volume fractions and stress
states. A regression with y = x has an R2 of 0.998. The FE results of [21]
are shown as triangles and the results of [11] are shown as stars.
4.3. Body-centred cubic
The FE data for the body-centred case is qualitatively
very similar to the simple cubic case. Applying a leastsquares regression to the 4400 points generated shows that
they are well-fit by
BCC
BCC
BCC 1:24
a1 ðf Þ ¼ f =fperc
þ 2:4ðf =fperc
ðf =fperc
Þ Þ;
BCC
Þ.
a2 ðf Þ ¼ 0:85 0:22ðf =fperc
ð13Þ
As for the sc case, a1(f) vanishes at f = 0 and is unity at
BCC
f ¼ fperc
0:994. This particular function fits the data
slightly better than a pure power-law. Once again, a2(f) is
close, but not equal to unity. It is sufficient to keep the first
five terms of the power series in Eq. (11). Similarly to the sc
case, this proposal has an R2 of 0.998.
4.4. Face-centred cubic
In this case, surprisingly, the 4400 data points generated
show very little dependence on the determinant. Therefore,
we set
sðf Þ ¼ 0;
Σxx
Σ yy
in contrast to the sc and bcc cases. Applying a least-squares
regression to the FE data shows that they are well-fitted by
1:43 FCC
FCC
FCC
a1 ðf Þ ¼ f =fperc þ 1:33 f =fperc f =fperc
;
ð14Þ
FCC
a2 ðf Þ ¼ 0:83 0:14 f =fperc .
Similarly to the sc case, this proposal has an R2 of 0.993.
5. Conclusions
Fig. 7. Data at f = 0.25 with Rhyd = 0.25rM (black) through to
Rhyd = 0.5rM (white) and the curves produced by Eqs. (11) and (12)
projected onto the octahedral plane. For f = 0.25, Rmax
hyd 0:6rM as can be
seen in Fig. 1. The parameters in the latter equation are produced from a
global fit to all the 4400 data points (with different f), not just those few
depicted here, and that is why the fit is not perfect. The discretisation error
is indicated by the size of the dots, and the almost-triangular nature of the
data is evident around Rhyd = 0.5rM.
Because of the role of voids in the process of ductile
fracture and the fact that modern fabrication techniques
can produce precisely engineered porous materials, it is
desirable to gain a good understanding of the yield criterion of porous materials. We have chosen to concentrate
on the simple cases where the voids are arranged in either
2002
D.L.S. McElwain et al. / Acta Materialia 54 (2006) 1995–2002
simple cubic, body-centred cubic or face-centred cubic
arrays. Previous work, in particular see Refs.
[11,19,21,26], has demonstrated that the celebrated Gurson–Tvergaard–Needleman yield criterion [6,8–10] is inaccurate for general stress states, and the calculations here
have shown that the equivalent stress at yield can be altered
by up to a factor of two by varying the determinant of the
stress tensor at fixed hydrostatic stress.
We used the finite-element technique employing the
return-mapping algorithm described in Ref. [34] to monotonically increase arbitrarily prescribed, but diagonal, macroscopic strain to find many thousand yield points for each
of the three void arrangements. A range of void volume
fractions from 2% to 90% were explored, with a linearly
hardening stiff matrix material with rM = 7.5 · 105E,
Poisson’s ratio of 1/3 and hardening slope of 7.5 · 108E,
where E is the Young’s modulus. An analysis of the relative
errors incurred by the finite-resolution of the finite-element
technique showed that they are ±2%.
The finite-element data was fitted using least-squares to
the yield function given in Eq. (7). This function’s dependence on the equivalent and hydrostatic stresses (Reqv and
Rhyd) is almost identical to GTNs formula, but there is also
an extra term which depends linearly on the determinant of
the stress tensor, Rdet. The yield function contains three
functions of void volume fraction, which are slightly different for the simple cubic, body-centred cubic and facecentred cubic cases. The first of these, a1, is well fitted by
a power-law (as initially proposed in Ref. [12]), while the
second, a2, is a slowly varying linear function of void volume fraction, and is numerically close to forms proposed
by various other authors (such as [8,17,27]). The exact
forms are given in Eqs. (12)–(14). The third, s(f), determines
the strength of the contribution of Rdet. The requirement of
yield-function convexity suggest a particular form for s(f) in
terms of a1 and a2 – given in Eqs. (8) and (9) and the first five
terms of the power series in Eq. (11) – and this compares
well with the finite-element data for the simple cubic and
body-centred cubic cases. Interestingly, for the face-centred
cubic case, s(f) = 0, and there is no dependence on Rdet.
R2 values of 0.998 (sc and bcc) and 0.993 (fcc) were
obtained, which strongly indicate that, despite a relative
error of ±2% in the data, the proposed analytic forms
describe the true shape of the yield surface.
We have thus extended GTNs formula for the yieldfunction for porous materials to a form more accurate
for general stress states without introducing any further
parameters.
The application of the techniques described herein to
isotropic random materials, where cubical discretisations
similar to Fig. 4 are particularly useful, is the goal of the
authors’ future research.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
Tvergaard V. Adv Appl Mech 1990;27:83.
Brocks W, Sun DZ, Honig A. Int J Plast 1995;11:971.
McClintock FA. J Appl Mech 1968;35:363.
Rice JR, Tracey DM. J Mech Phys Solids 1969;17:201.
Green RJ. Int J Mech Sci 1972;14:215.
Gurson AL. J Eng Mater Tech Trans ASME, Ser H 1977;99:2.
Needleman A. ASME J Appl Mech 1972;39:964.
Tvergaard V. Int J Fract 1981;17:389.
Tvergaard V. Int J Fract 1982;18:237.
Tvergaard V, Needleman A. Acta Metall 1984;32:157.
Hom CL, McMeeking RM. J Appl Mech 1989;56:309.
Richmond O, Smelser RE. ALCOA Technical Center Report; 1985.
Nagaki S, Sowerby R, Goya M. Mater Sci Eng A 1991;142:163.
Francescato P, Pastor J, Riveill-Reydet B. Eur J Mech A – Solids
2004;23:181.
Andersson H. J Mech Phys Solids 1977;25:217.
Bourcier RJ, Koss DA, Smelser RE, Richmond O. Acta Metall
1986;34:2443.
Koplik J, Needleman A. Int J Solids Struct 1988;24:835.
Worswick MJ, Pick RJ. J Mech Phys Solids 1990;38:601.
Goya M, Nagaki S, Sowerby R. JSME Int J Ser I – Solid Mech
1992;35:310.
Nagaki S, Goya M, Sowerby R. Int J Plast 1993;9:199.
Richelsen AB, Tvergaard V. Acta Metall Mater 1994;42:2561.
Kuna M, Sun DZ. Int J Fract 1996;81:235.
Faleskog J, Gao X, Shih CF. Int J Fract 1998;89:355.
Benzerga AA, Besson J. Eur J Mech A – Solids 2001;20:397.
Chien WY, Pan J, Tang SC. J Eng Mater Technol – Trans ASME
2001;123:409.
Zhang KS, Bai JB, Francois D. Int J Solids Struct 2001;38:5847.
Kim J, Gao XS, Srivatsan TS. Eng Fract Mech 2004;71:379.
Pastor J, Francescato P, Trillat M, Loute E, Rousselier G. Eur J
Mech A – Solids 2004;23:191.
Wang DA, Pan J, Liu SD. Int J Damage Mech 2004;13:7.
Wicklein M, Thoma K. Mater Sci Eng A 2005;397:291.
Wicklein M, Thoma K. In: Proceedings of MetFoam2005, Osaka,
Japan. The Japan Institute of Metals; 2005.
Leblond JB, Perrin G, Devaux J. Eur J Mech A – Solids 1995;14:499.
Liboff RL. Primer for point and space groups. New York
(NY): Springer; 2004.
Doghri I. Mechanics of deformable solids: linear nonlinear analytical
and computational aspects. New York (NY): Springer; 2000.
Garboczi EJ, Day AR. J Mech Phys Solids 1995;43:1349.
Garboczi EJ. Finite element and finite difference programs for
computing the linear electric and elastic properties of digital images of
random materials, NISTIR 6269. Available from: http://ciks.cbt.
nist.gov/garbocz/manual/man.html.
Hill R. Math Proc Cambridge Philos Soc 1979;85:179.
Roberts AJ, Garboczi EJ. J Am Ceram Soc 2000;83:3041.