A Geometric Model for the Fracture Toughness of Porous Materials

Abstract

Different models for the fracture toughness, K_IC, of porous materials have been proposed to describe K_IC as a function of the porosity P. They have in common that beside P at least one additional parameter exists that has to be adjusted to the measured data. Based on the cubic structure, we present a geometrical 3D model without any arbitrary parameter, which predicts the K_IC, the toughness, G_C, and the Young’s modulus, E, of a porous material. The model comprises three variants, depending on the material properties like open or closed porosity. It is in good agreement with a large amount of experimental data from different research groups.

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A geometric model for the fracture toughness of porous materials
H. Jelitto
*
, G.A. Schneider
Institute of Advanced Ceramics, Hamburg University of Technology, 21073 Hamburg, Germany
article info
Article history:
Received 24 November 2017
Received in revised form
27 February 2018
Accepted 12 March 2018
Available online 16 March 2018
Keywords:
Fracture toughness
Toughness
Elastic properties
Porosity
Modeling
abstract
Different models for the fracture toughness, KIC , of porous materials have been proposed to describe KIC
as a function of the porosity P. They have in common that beside Pat least one additional parameter
exists that has to be adjusted to the measured data. Based on the cubic structure, we present a
geometrical 3D model without any arbitrary parameter, which predicts the KIC , the toughness, GC, and
the Young's modulus, E, of a porous material. The model comprises three variants, depending on the
material properties like open or closed porosity. It is in good agreement with a large amount of exper-
imental data from different research groups.
©2018 Acta Materialia Inc. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-
ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
If the fracture toughness of a dense material, K
IC
, is known, it is
desirable to predict the fracture toughness of the same material,
having the porosity P. This applies for example to ceramics with low
porosity as also to polymer foams and sponges. The requirement for
our model and most other models in the literature is that the ma-
terial is isotropic and behaves linear-elastically, when it is loaded
until fracture.
One existing and widely used model was developed by Ashby,
Maiti, and Gibson [1e3]. They solved the problem on a microscopic
(cell) level. For open porosity, they got the following equation:
K
*
IC
¼c
1
s
f
ffiffiffiffiffiffiffiffi
p
w
p
r
r
s

3=2
¼c
1
s
f
ffiffiffiffiffiffiffiffi
p
w
pð1PÞ
3=2
(1)
Here, K
*
IC
is the fracture toughness of the porous material, c
1
a
normalization factor,
s
f
the fracture strength of the dense material,
wthe size of the cells,
r
the density of the porous material, and
r
s
the density of the dense material. In addition, K
*
IC
is given in an
alternative way with regard to
r
/
r
s
¼1eP. In their calculation for
open porosity, the authors used the relation
r
/
r
s
ft
2
/w
2
, in which t
is the thickness of the ligaments between two pores (cells). How-
ever, this relation is valid only for t<< w, which implies that the
porosity has to be relatively high. Consequently, Maiti et al. applied
their model to rigid polymer foams with porosities of 68%e97%,
and obtained a good agreement. In order to get such an agreement,
they had to adjust their model to the measured data with the
normalization factor c
1
¼0.65. As Maiti et al. stated, Eq. (1) shows a
slight dependence of K
*
IC
on the size wof the cells. The characteristic
of the fracture toughness according to (
r
/
r
s
)
3/2
was used, for
example, also in Refs. [4e6]. For closed porosity, Maiti et al. derived
an equation like Eq. (1) with the exponent 2 instead of 3/2 and a
normalization factor c
2
, which was not further specified [2].
Other authors worked on a more general approach, like for
example K
*
IC
f(
r
/
r
s
)
n
, where the exponent nwas adapted to the
experimental data [7e10]. Yang et al. [11], for instance, provided an
equation for the relative fracture toughness:
K
*
IC
K
IC
¼1
a
P
b
P
2
(2)
with the parameters
a
and
b
being adjusted, correspondingly. Also
finite element calculations were done to predict the dependence
between K
*
IC
and P[10,12]. It seems that all of the approaches in the
literature have more or less empirical character, since in any case at
least one parameter has to be fitted to the experimental data. Such
parameters do not have a physical meaning. (Detailed information
about cellular ceramics is given, for example, in the books of
Gibson/Ashby [3], Rice [13], Scheffler/Colombo, Eds., [14], and also
in Ashby/Br
echet [15].) In contrast, we present an analytical,
geometrical 3D model on the basis of the cubic structure which
*Corresponding author.
E-mail address: h.jelitto@tuhh.de (H. Jelitto).
Contents lists available at ScienceDirect
Acta Materialia
journal homepage: www.elsevier.com/locate/actamat
https://doi.org/10.1016/j.actamat.2018.03.018
1359-6454/©2018 Acta Materialia Inc. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-
nd/4.0/).
Acta Materialia 151 (2018) 443e453

does not include any arbitrary parameter.
2. The geometric model
The approach is based on the concept of the toughness
G
C
¼K
2
IC
E(3)
which is the energy release during crack advance per crack face
area (plane stress). Eis the elastic modulus. In the following, G
*
C
and
G
C
are the toughness of the porous and the dense material. The
main idea is that the normalized or “relative”toughness, G
*
C
=G
C
,is
given by the ratio of the substantial crack surface to the total crack
surface. The latter one includes empty spaces like pores and cracks
etc. This approach was already applied by Maiti et al. [2]. However,
they assumed high porosity, while our ansatz is valid for any
porosity. Here, “relative”or “normalized”always means “in relation
to the dense material.”The model is subdivided into three variants
AeC, described in the following. To simplify matters, they are also
named “model A”to “model C”, although they represent together
one model.
2.1. Model A eclosed porosity
For closed porosity, the pores are idealized as cubes and ar-
ranged as in Fig. 1a). The size of the unit cell (lattice constant) is w,
and the material walls between the cubic pores have the thickness
t. Now, we fracture the material in the plane of minimum fracture
surface, which is, e.g., the 100-plane. The corresponding (idealized)
crack face is shown in Fig. 1b). In a real porous material, the crack
face is not totally planar but grows along the path with the nearest
and largest pores. In our simple cubic structure, this would mean
fracture along the 100-plane with the lowest energy release.
We define the dimensionless number d¼t/wwith 0 <d1.
Thus, on the basis of Fig. 1a) we obtain for the porosity:
P¼ðwtÞ
3
w
3
¼ð1dÞ
3
(4)
and it follows
d¼1P
1=3
(5)
If we calculate the relative amount of substantial crack face area
(of the fractured cell walls) from Fig. 1b), we get
G
*
C
G
C
¼w
2
ðwtÞ
2
w
2
¼1ð1dÞ
2
(6)
The replacement of daccording to Eq. (5) yields for closed
porosity:
G
*
C
G
C
ðmod:AÞ¼1P
2=3
(7)
2.2. Model B eopen porosity
For open porosity, the material is once more idealized by a
simple cubic structure as given in Fig. 2a). The lattice constant and
the thickness of the ligaments are again wand t.
At first, we calculate the porosity as a function of d. With the
volume of the unit cell being w
3
,wefind with regard to Fig. 2a):
P¼w
3
t
3
3t
2
ðwtÞ
w
3
¼2d
3
3d
2
þ1 (8)
By considering Fig. 2b), we simply obtain the relative toughness
G
*
C
G
C
¼t
2
w
2
¼d
2
(9)
In order to get the relative toughness as a function of P,wehave
to replace din Eq. (9) by P, and hence to solve equation (8) for d. The
computing software Maple [16] provides the solution:
d¼1
2
,1þ2Pþ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
2
P
p
1=3
þ1
21þ2Pþ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
2
P
p
1=3
þ1
2(10)
Fig. 1. a) Closed porosity with the pores idealized in a cubic arrangement, b) fracture
surface with (quadratic) pores.
Fig. 2. a) Scheme of cubic structure with open porosity. The shaded part represents the
material of one unit cell. b) Cross section and crack face, respectively, along the 100-
plane.
H. Jelitto, G.A. Schneider / Acta Materialia 151 (2018) 443e453444

Because of 0 P<1, the term under the square root is negative
(except for P¼0), and therefore, both expressions in brackets are
not real but complex. Furthermore, the second and third roots
cause some ambiguities, in which most of the cases are not the
result we need. Nevertheless, it is possible to find an analytical
solution for the physically relevant part, as shown in the Appendix
A. Introducing the solution, based on Eq. (A4) in the Appendix, into
Eq. (9) gives the result for open porosity:
G
*
C
G
C
ðmod:BÞ¼"cos 2
p
cos
1
ð2P1Þ
3!þ1
2#
2
(11)
An easy way to check the validity of this equation exists inde-
pendently of Eq. (10). What we need is the toughness and the
porosity together in one diagram. With the equations (8) and (9),
we have both quantities as a function of d. So, we can use these two
equations as a parametric representation P(d) and ðG
*
C
=G
C
ÞðdÞ. The
outcome verifies the result of Eq. (11).
2.3. Model C eopen porosity with disconnections
If the porosity becomes high and approaches 100%, the liga-
ments in Fig. 2 become very thin. In many brittle materials like
ceramics, such filigree beams are unrealistic. So, with increasing
porosity it is likely that more and more filaments are disconnected,
e.g., during the sintering process (see Fig. 3).
With regard to Figs. 2 and 3, let us name the relative amount of
disconnected ligaments X. Then two extreme cases seem reason-
able. For the porosity being zero, the amount Xalso vanishes. If the
porosity becomes almost 1, also Xapproaches 1. Now, the simplest
assumption is that Xis a linear function of the porosity, P, implying
that Xand Pare not only proportional but identical. The relative
amount of existing ligaments is 1eX¼1eP. Therefore, the expres-
sion for the toughness in Eq. (11) would get an additional factor
(1eP). However, the comparison with experimental data reveals
that, for a certain porosity, the extent of disconnections is not al-
ways the same for different materials. Therefore, we add an expo-
nent n>0, which yields quantitative information about the amount
of disconnections. This creates the portion of existing ligaments
Y(P)¼(1eP)
n
, which is the simplest modification that satisfies the
boundary conditions Y(0) ¼1 and Y(1) ¼0 for positive n. Adding
this factor and replacing Pby Eq. (8) yields:
G
*
C
G
C
¼d
2
,ð1PÞ
n
¼d
2
,2d
3
þ3d
2

n
(12)
As a function of P, we get explicitly:
G
*
C
G
C
ðmod:CÞ¼"cos 2
p
cos
1
ð2P1Þ
3!þ1
2#
2
,ð1PÞ
n
(13)
Fig. 3. Schematic cross section through the ligaments in a cell structure, similar to that
one in Fig. 2, with disconnections. The relative amount of disconnections in this figure
is approximately 20%. The disconnections exist also in the third dimension, which is
not shown here.
Fig. 4. Relative toughness as a function of the porosity Pwith different parameters nof
the model C.
Fig. 5. a) SEM image of nanoporous (np) gold with ligament diameter of (43 ±5) nm
[17]. b) Schematic illustration of np structure with broken or dangling ligaments, and a
mechanically equivalent structure that is obtained by removing dangling ligaments.
Here, 4means relative density and 4
eff
is defined as the effective relative density [17].
(For the colour version of this figure, the reader is referred to the Web version of this
article.)
H. Jelitto, G.A. Schneider / Acta Materialia 151 (2018) 443e453 445

The exponent nis directly correlated with the amount of
disconnected ligaments. The higher this exponent becomes, the
larger is the latter amount, meaning that this parameter is not
arbitrary. The simplest case of n¼1 in Eq. (13) can be seen as the
basic form of model variant C. The limiting case n¼0 is identical to
variant B. As in model B, Eqs. (8) and (12) can be used for an
equivalent parametric representation. Fig. 4 represents the three
model variants AeC, where variant C is given with a few alternative
exponents n.
There might be some concern, whether the main idea of model C
is an arbitrary assumption. In their recent research, Liu et al. used
just this ansatz to explain the anomalously low strength and stiff-
ness of nanoporous gold. In Fig. 5b), taken from their publication
[17], the disconnections in the material are named “broken or
dangling ligaments.”Furthermore, this effect is also experimentally
observed (Fig. 5a). Of course, this does not mean that all materials
behave like this, but at least for some of them this seems to be a
realistic picture. So, the paper of Liu et al. confirms very well the
basic idea of variant C.
2.4. Normalized Young's modulus
The publications of experimental data concerning porous ma-
terials provide mainly the fracture toughness, K
IC
, and not the
toughness, G
C
. So, in order to compare our model with experi-
mental results, we have to convert the toughness, G
C
, accordingly.
Due to Eq. (3), we need to determine the elastic modulus, E*, of the
porous material in order to obtain K
*
IC
. This will be done on the
same geometrical basis by assuming the cubic structures like in the
Figs. 1e3and by loading the structure vertically and perpendicu-
larly to the principal horizontal plane.
Prior to the calculation, a question arises concerning the isot-
ropy of the elastic modulus within our model. Anisotropy might
exist if the loading of the ligaments changes from tension or
compression to bending, depending on the loading direction. The
following arguments suggest that, in our model, E* is almost in-
dependent from the loading direction: 1. The quadratic walls of
closed cells act as if diagonal struts exist in the walls. Therefore, the
structure behaves like a triangular framework, which is very stiff
with E* being almost isotropic. 2. For open porosity, bending is
possible mainly for thin ligaments, which means high porosity. For
medium and low porosity, bending is negligibly low. 3. If the pro-
cess zone at the crack tip is small compared to the length of the
crack front, we have plane strain condition in the middle part of the
crack front. This means that transverse strain is not possible. Even if
the square structure is loaded diagonally, the ligaments are loaded
mainly in compression or tension but not in bending. So, we as-
sume that the following estimate of the Young's modulus is more or
less valid for any loading direction.
On the basis of Figs. 1 and 2, we apply the rule of mixture two
times in each case, first for a parallel and secondly for a serial
arrangement. These basic calculations with a slight modification
are summarized in Appendix B and yield the following normalized
Young's moduli for the models AeC:
E

Eðmod:AÞ¼1d
d
2
þ2dþ1
dþ2þð1dÞ
2
d
1=2

1
(14)
E
*
Eðmod:BÞ¼ 2d
3
þ3d
2
2d
2
4dþ3(15)
E
*
Eðmod:CÞ¼ 2d
3
þ3d
2
2d
2
4dþ3
,ð1PÞ
n
¼2d
3
þ3d
2

nþ1
2d
2
4dþ3
(16)
For model A, d(P) is given by Eq. (5) and for the models B and C
by:
d¼cos 2
p
cos
1
ð2P1Þ
3!þ1
2(17)
which follows from Eq. (A4). Of course, with increasing parameter
n, in average the remaining ligaments become longer, and there-
fore, bending effects might become more important. Such bending
or buckling is not considered in the model so that the real Young's
modulus decreases even more. Nevertheless, we expect a mono-
tonic decrease of E* with respect to n, and thus, ncan be seen as a
parameter, characterizing well the microscopic structure according
to the average length of the ligaments. More information is pro-
vided in the comparison of experiments and model (chapter 4).
2.5. Normalized fracture toughness
On the basis of Eq. (3) and with Ebeing the Young's modulus of
the solid material, the normalized fracture toughness is:
K
*
IC
K
IC
¼ffiffiffiffiffiffiffiffiffiffiffiffiffi
G
*
C
E
*
G
C
E
s(18)
Note that Eq. (18) is exactly valid for the state of plane stress and
almost exactly valid for plane strain. If the Poisson ratios (
n
) for the
porous and the dense material are more or less identical, the two
factors (1e
n
2
) in the case of plain strain cancel each other. For
model variant A, we have to replace G
*
C
=G
C
and E*/Eby means of
Eqs. (7) and (14). With d¼1eP
1/3
, we obtain
K
*
IC
K
IC
ðmod:AÞ¼1P
2=3

1=2
, 1d
d
2
þ2dþ1
dþ2þð1dÞ
2
d
1=2
!
1=2
(19)
(If replacing Pby Eq. (4), the right side of Eq. (19) is dependent
only on d.) Further modification of Eq. (19) does not simplify the
result. Therefore, we leave the equation as it is. For the following
variants B and C, dis given by Eq. (17). In case of model variant B, we
replace again the right side of Eq. (18) by means of Eqs. (9) and (15),
which leads to:
K
*
IC
K
IC
ðmod:BÞ¼d
2
,2dþ3
2d
2
4dþ3
1=2
(20)
Finally, model C with Eqs. (12) and (16) and replacing Pby Eq. (8)
yields:
K
*
IC
K
IC
ðmod:CÞ¼d
2
,2dþ3
2d
2
4dþ3
1=2
,ð1PÞ
n
¼d,2d
3
þ3d
2

nþ1=2
2d
2
4dþ3
1=2
(21)
In summary, each of the three model variants yields the
normalized quantities of the toughness G
*
C
, the fracture toughness
K
*
IC
, and the Young's modulus E*. In each case and for each model A
H. Jelitto, G.A. Schneider / Acta Materialia 151 (2018) 443e453446

to C, an equation exists which is solely dependent on d. So, together
with the corresponding functions P(d) of Eqs. (4) and (8), a para-
metric representation can always be realized alternatively. In
model A, d(P) is generally given by Eq. (5) and in the models B and C
by Eq. (17), implying that all of the quantities can be expressed as a
function of P, too. The exponent nin Eq. (21) is correlated again with
the amount of disconnected ligaments. Other aspects and the limits
of this model will be examined in chapter 4.
3. Measurements of porous materials
For the comparison, the experimental data of twelve different
research groups, who tested porous materials, are considered and
briefly described. Flinn et al. [8] used data from tests of alumina,
measured by Knechtel [9], and provided directly the relative frac-
ture toughness. Deng et al. [18] measured pure alumina, too (label
A), and also two mixtures with aluminum hydroxide (labels AH60
and AH90). They presented only the absolute fracture toughness
without giving the value for the dense material. Anyway, for pure
alumina it was easy to extrapolate the K
IC
from 97% density to 100%
(see Table 1 and Fig. 6). For the mixture of Al
2
O
3
and Al(OH)
3
, the
maximum density was about 89%, and so, the extrapolation would
be less certain. However, the authors state that in this case and for
lower porosities the addition of Al(OH)
3
does not improve the K
IC
.
Furthermore, the trends of the data for pure alumina and the
mixture were almost identical so that we use the same 100%-value
for the mixed material AH60. The fracture toughness data of the
material AH90 are not included here, because a reliable extrapo-
lation to the dense material was not possible. Yang et al. [11] and
also Ohji [19] tested porous silicon nitride (
a
-Si
3
N
4
with 5 wt%
Yb
2
O
3
) and provided directly the relative densities. Hong et al. [20]
used two variants of TiB
2
, where the powder was cold isostatically
pressed at 10 and 50 MPa. The measured densities between 45%
and 95% allow for an easy extrapolation to 100%. Samborski and
Sadowski [21] worked with porous alumina and magnesia with
densities up to 96.5% and 91.5%, respectively, at static and also
dynamic, periodic loading. Only the static measurements and only
the alumina results are used, because for the magnesia tests the
extrapolation to the full density includes higher uncertainty.
Goushegir et al. [22] performed tests with alumina fibers in a RBAO
matrix (reaction bonded aluminum oxide) of different porosities.
Also here, the extrapolation of the fracture toughness for the dense
material could be done easily. For the graphical extraction of the
experimental data, the published diagrams were magnified eif
necessary eand then provided with a precise grid before deter-
mining accurately the measured quantities.
For further information and in order to get an impression about
the reliability of the results, the experimental details and the
extracted numerical data from each research group, used in this
paper, are summarized in a separate reference in “Data in Brief”
[23].
Maiti, Ashby, and Gibson [2], who presented the model of Eq.
(1), did not test ceramics but foamed polymethacrylimid (PMI-E), a
hard polymer with a very high porosity. The relative fracture
toughness was presented in a double logarithmic plot. Within their
plot, they included also some data from McIntyre/Anderton [24]as
well as from Fowlkes [25], who tested polyurethane foams (PUR).
The whole data were used here and taken from the magnified di-
agram (Fig. 3 in Ref. [2]) by creating an equidistant grid, extracting
graphically the logarithmic data, and then transforming them from
the logarithmic to the linear scale. In this case, an additional check
was performed: The numerical data were plotted again in a double
logarithmic diagram and compared with the original plot. The di-
agrams looked identical.
The extrapolation, to get the fracture toughness of the dense
material, was done by fitting a quadratic function to the measured
data and then using the function value at zero porosity. Afterwards,
the fracture toughness of the porous substances could be normal-
ized by division through the extrapolated data. Concerning K
IC
, this
was necessary for four references. The estimated values of these
references are listed in Table 1 so that they can be checked by the
reader. The error of this extrapolation should be in the range of a
few percent, like for instance 5%. An example for the data of Deng
et al. [18] is provided in Fig. 6.
Regarding the mechanical properties of porous metals, not
much literature exists. However, relating to nanoporous gold (NPG),
being a relatively new research topic, the Young's modulus was
determined by Huber et al. [26]. In this case, we got the numerical
results from the authors. Since the data were given in GPa, we
normalized them by E¼81 GPa, a value used in their reference.
From all of the references, error bars were accurately transferred if
available. If some of the used data do not have error bars, the errors
are either not provided in the reference or the error bars are smaller
than the symbol size in the diagram.
4. Comparison of experiments and model
4.1. Normalized fracture toughness
In Fig. 7, the experimental results for ceramics and polymers
[2,8,9,11,18e22,24,25] are compared with the three model variants.
It can be seen that the data for low porosities are near to model A
(closed porosity), which makes sense. With increasing porosity, P,a
transition takes place from variant A to B, and for porosities higher
than 40% and ceramic materials, the variant C (open porosity with
disconnections) yields the best agreement. This is what we would
expect. Maiti et al. normalized their experimental data by dividing
them through
s
f
√(
p
w) (Fig. 3 in Ref. [2]), which means that Eq. (1)
becomes
Fig. 6. Quadratic fit to the experimental data of the material Al
2
O
3
-A [18]. With the
nonlinear extrapolation to zero porosity, the fracture toughness of the dense material
is estimated (5.4 4 MPa√m).
Table 1
Estimated fracture toughness for the dense material, extrapolated from the exper-
imental data given in the references.
Material Reference K
IC
[MPa√m]
Al
2
O
3
eA, Al
2
O
3
eAH60 Fig. 8a) in Ref. [18] 5.44
TiB
2
(10/50 MPa) Fig. 5 in Ref. [20] 3.21/4.18
Al
2
O
3
Fig. 5 in Ref. [21] 3.63
Al
2
O
3
-fibers þRBAO-matrix Fig. 3 in Ref. [22] 4.16
H. Jelitto, G.A. Schneider / Acta Materialia 151 (2018) 443e453 447

K
*
IC
s
f
ffiffiffiffiffiffiffi
p
w
p¼0:65,
r
r
s

3=2
(22)
This dependence is included as a grey dashed line in Fig. 7. Due
to the normalization factor 0.65 [2], their model fits very well to the
polymer data. However, the theoretical relative K
IC
does not
approach 1, if the porosity becomes zero. So, for low porosities the
geometrical model, presented here, yields a better agreement with
the experiment. Of course, Maiti et al. did not intend to describe
low-porosity materials. If the experimental polymer data (open
circles) are compared with our model, it can be seen that they are
close to model B, in agreement with Maiti et al., who also used the
“open porosity”variant of their model.
With n¼1, corresponding to the “basic”model C, 30% porosity e
for example emeans 30% disconnections, 50% porosity means 50%
disconnections, etc. No disconnections means n¼0 (model B). In
Fig. 7,wefind that the low fracture toughness values from Deng
et al. for porosities between 36% and 54% can be described quite
well with exponents around n¼2 and 2.5, implying that the
amount of disconnected beams is relatively large. In contrast, the
Fig. 7. Measured relative fracture toughness for different ceramic and polymer materials in comparison with the three variants of the geometric model (black dashed lines).
Additionally, model C is provided with a set of curves for different exponents n. The materials with porosities below 60% are ceramics and above 60% polymers. The dashed grey line
illustrates the model of Maiti et al. for open porosity [2]. All of the numerical, experimental data as well as some additional figures are provided in “Data in Brief”[23].
Fig. 8. Magnification of the small frame in Fig. 7, containing only the polymer data of
Refs. [2,24,25].
Fig. 9. The amount of disconnected ligaments as a function of the exponent nfor
different porosities P, according to Eq. (23).
H. Jelitto, G.A. Schneider / Acta Materialia 151 (2018) 443e453448

high fracture toughness of silicon nitride at porosities below 30%,
measured by Ohji [19] and Yang et al. [11 ], is even above the model
predictions for closed porosity (model A). This is a hint that
toughening mechanisms exist, which are not included in the
model.
Fig. 8 represents a magnification of the small frame in the lower
right corner of Fig. 7. Here, the polymer data got different labels
according to their origins [2,24,25]. The data with porosities higher
than 80% can be best fitted with n¼0.3. The reason for this low n-
value at high porosities is that polymer materials tend to create thin
beams and thin cell walls more easily than ceramics, because of
their long molecular chains. Their amount of disconnections is
much lower than that of ceramics.
The fact that these data are below the prediction of model B is
again interpreted as missing ligaments and cell walls, respectively.
Also Gibson/Ashby/Maiti had to lower their theoretical curve for
open porosity (by the factor 0.65) to get agreement with the
measurements.
4.2. Model C and the parameter n
To get an impression about the quantitative meaning of the
exponent nin the equations of model C, some numbers are pro-
vided. The factor (1 eP)
n
, e.g. in Eq. (21), describes the relative
amount of existing ligaments in the porous material. The relative
portion of disconnections Xis given by
X¼1ð1PÞ
n
(23)
Concerning the polymer data, the highest two data points in
Fig. 8, for example, belong to ca. 70% porosity with n¼0.1. As per
Eq. (23), this gives X¼0.11, meaning 11% disconnections, which is
not much. The lowest fracture toughness of alumina (Deng et al.
[18]) around 53% porosity with n¼2.5 (Fig. 7) yields 85% discon-
nections. Of course, the numbers of these examples should not be
taken too seriously, but nevertheless, they provide an idea about
the state of the microstructure. Based on the model, an overview of
the dependence between disconnections, exponent n, and porosity
Pis provided in Fig. 9.
In ceramic materials, the transition from closed to open porosity
is often expected at around 8%. This is supported by Yang et al., who
determined separately the amounts of open and closed porosity in
silicon nitride. Below 8% porosity they obtained closed porosity and
above 15% open porosity. In between they measured a mixture of
both states [11]. (Unfortunately, the other research groups, who
tested ceramics, did not distinguish between different kinds of
porosity.) In contrast, in Fig. 7 some measured fracture toughness
values are still close to and even above the theoretical curve for
closed porosity at porosities much higher than 8%. A possible
explanation is that “isolated”pores are connected via very thin
channels, e.g., along the triple conjunctions of the grains. But also
the opposite behavior can be seen.
In Fig. 8, all of the polymer data with porosities above 65% are in
the range of the “open porosity”variant, although at least PMI-E
shows closed porosity. As already stated by Maiti et al. concern-
ing man-made foams, the reason is probably that material from the
cell walls is drawn to the cell edges by surface tension during the
manufacturing process [2]. So, not only the type of porosity is
essential but also the material distribution between the cells.
4.3. Normalized Young's modulus
4.3.1. Alumina
Comprehensive measurements of the elastic modulus with
different alumina materials were provided by Deng et al. in Fig. 4a)
of Ref. [18]. In order to be near to the original diagram, the absolute
Young's modulus is plotted as a function of the relative density
r
/
r
s
.
In this case, the theoretical curves of the model are multiplied by
the (average) elastic modulus of the dense materials of 420 GPa,
determined also by quadratic extrapolation. The comparison with
model C in Fig. 10 shows that the main trend of the experimental
data is well reproduced, in which the exponent nincreases from
approximately 1.2 at high densities to 2.5 at lower densities. This is
again in accordance with the expectation that the number of
disconnected ligaments increases with decreasing density.
It seems that for the same porosity the average lengths of the
grains between the nodes, being rigid connections between the
ligaments, can be different for different materials. This can be
denoted by an average “aspect ratio”of the ligaments, even if the
ligaments are not straight but curved. It seems that ncharacterizes
an additional material property of porous materials, being inde-
pendent of porosity and density of the parent material. So, in the
case of entangled long grains (like a haystack) and with respect to
the elastic modulus, the “ligament parameter”nyields additional
information about the microstructure.
4.3.2. Nanoporous gold
The measured Young's moduli of NPG are displayed in Fig. 11
together with the model predictions of variant C. The experi-
ments were not performed like usual, meaning that materials of
different porosities were prepared and then tested individually.
Instead, one kind of nanoporous gold was produced with a result-
ing porosity of (74 ±1)% and an average ligament diameter of
(63 ±6) nm (Huber et al. [26]). The used sample had cylindrical
shape and was successively compressed, until the sample volume
was reduced to half of the initial volume (P¼37.7%). In between,
the sample was unloaded completely, and the present Young's
modulus was determined by means of the upper half of the
unloading curves in the respective stress-strain diagram [26].
The steep initial increase of the Young's modulus, beginning at
P¼74% i n Fig. 11, was mentioned but not explained in the original
paper. According to a subsequent paper, the anomalously high
compliance can possibly be explained on an atomistic level by high
local, surface-induced prestress on the ligaments, which already
exists before loading the material [28].
At porosities between 71% and 73%, the data points reach a curve
Fig. 10. Dependence of the Young's modulus on the relative density of porous Al
2
O
3
ceramics prepared from different powders esee Fig. 4a) in the original paper of Deng
et al. [18]. Here, Deng et al. included data from Lam et al. [27], who also tested porous
alumina. Those materials were characterized by
r
0
¼0.62 and
r
0
¼0.50, representing
the initial relative densities of the two types of Al
2
O
3
compacts [18,27].
H. Jelitto, G.A. Schneider / Acta Materialia 151 (2018) 443e453 449

of model C with the minimum value n¼2.25, meaning that from
the beginning a lot of disconnected ligaments exist. This agrees
very well with [17], where the anomalously low stiffness of nano-
porous gold was explained by broken and dangling ligaments.
During further compression of the same sample, E* increases only
slightly and on a much lower rate than the model-C curves.
Consequently, the parameter nrises up to 6.5, which implies that
the number of disconnected ligaments also strongly increases. This
behavior can be understood, because when compressing a filigree
network of thin metallic ligaments down to a volume of 50%, the
structure naturally crushes successively. Additionally, bending ef-
fects reduce the Young's modulus (See also tests and simulation in
Ref. [28].). This is supported as well by stress-strain curves, given in
Ref. [26], where the overall dissipated energy due to inelastic
processes is about one to two orders of magnitude larger than the
energy, created by one cycle of elastic loading. (This follows from
considering corresponding areas under the stress-strain curves.)
4.4. Some remarks
In a paper of Ziehmer et al. [29], the microstructure of NPG was
characterized by two parameters
k
1
and
k
2
, being the two average
principal curvatures (inverse radii) on the surface of a ligament. The
first one is measured perpendicular and the second one parallel to
the alignment of the ligament. Nevertheless, they found that both
parameters are roughly correlated so that the microstructure can be
described by a single parameter. In a related paper of Hu et al. [30],
a quantity like “scaled connectivity density”within the framework
of both morphology and topology was used to determine the
microstructure. So, other approaches and parameters exist to
describe the structure in case of nanoporous gold. However, we will
not further examine whether a relation to our parameter nexists,
because with NPG we would only treat a special case, and it would
exceed the scope of this paper.
Generally, it is impossible to describe all of the experimental
fracture toughness data by a single theoretical function (see Fig. 7).
Different materials follow different paths in the diagram, which can
be best explained by different kinds of porosity and microstructure.
For one material, it seems that it is not the only way to reproduce
K
IC
(P) by a single (phenomenological) function, particularly, if the
topological properties of the microstructure change with increasing
porosity. In the given model, the experimental data are not evalu-
ated by fitting one theoretical curve to the measurement. Instead,
the measured data are plotted together with the family of curves
(reference curves) corresponding to the different model variants,
and thus, some information about the microscopic structure of the
porous material and its individual kind of porosity can be obtained.
An important question applies to the scale independence of the
given model. In principle, two effects can cause a dependence on
the pore size. A surface effect, like material segregation on the
pore's surface, can influence quantities like, e.g., the surface energy,
hardness, or strength in the surface region. So, if using the same
parent material and decreasing the pore size by leaving the porosity
constant, the total surface area increases drastically, and this can
influence the fracture toughness. The second reason is an effect,
caused by the size of the “notch root radius.”It means that a crack
would grow more easily through small pores than through large
pores. However, this effect is valid basically for 2D-structures like,
e.g., a material with cylindrical “pores,”extending through the
whole sample. Let's assume that a crack passes through such a
hollow cylinder with the crack face being parallel and the propa-
gation direction being perpendicular to the longitudinal axis of the
cylinder. Then a large radius of the cylinder acts like crack tip
blunting and impedes crack propagation more than a small radius.
However, in the three-dimensional case of spherical pores, this
effect is much less pronounced, because the amount of crack front
hitting a curved surface is considerably less. Due to a lack of cor-
responding experimental data, a potential independence (or
dependence) of the fracture toughness on the pores size, at con-
stant porosity, is a good objective for future research.
If the material characteristics show deviations regarding isot-
ropy, homogeneity of pore distribution, linearity, compactness of
pore shapes, etc., these aspects have to be considered if applying
the model. A modification of our model ansatz by a more realistic
approach, concerning shapes as well as spatial and size distribution
of the pores, would possibly cause some improvement. Neverthe-
less, we assume that this would lead only to minor changes of the
predicted results. Moreover, the model would probably become
less general and more complicated. The present model has the
advantage that it can be applied easily due to a full set of analytical
equations and that it can be used for a wide range of experiments
and materials. At least, it should be of some help for the mechanical
characterization of porous materials.
5. Conclusions
A simple 3D geometrical model is presented, which describes
the relative toughness, fracture toughness, and Young's modulus of
porous materials by analytical functions of the porosity. It is based
on the cubic structure and consists of three different variants, being
“closed porosity”(A), “open porosity”(B), and “open porosity with
disconnections”(C). The model assumes that the toughness is
proportional to the relative amount of substantial crack surface and
that fracture occurs along the path with the minimum area of
substantial crack face. It can be applied for any porosity between
0 and 1. A dependence on the cell size does not exist. The three
variants are in good agreement with a large amount of experi-
mental data. With increasing porosity in ceramic materials, the
transition from closed to open porosity is reflected well by a change
from model A to B and then to C. Instead of a variety of different
descriptions in the literature, these variants yield three funda-
mental curves, which are independent of any arbitrary parameter
and independent of material properties, except the porosity itself
and the fracture toughness of the dense material. Only model C has
a variable parameter that yields quantitative information about the
Fig. 11. Normalized Young's modulus of nanoporous gold (NPG) by analysis of the
unloading/reloading cycles [26]. These values are in the range of model variant C with
the “ligament parameter”nbetween 2 and 7.
H. Jelitto, G.A. Schneider / Acta Materialia 151 (2018) 443e453450

amount of disconnected ligaments or the aspect ratio of the
remaining ligaments.
Acknowledgement
Concerning the program Maple, we are indebted to get some
help from Henry E. Mgbemere and Diego Blaese. We thank also
Ayse Kalemtas for providing relevant literature, Jürgen Markmann
for helpful information and sending us numerical experimental
data concerning nanoporous gold, as well as Hai-Jun Jin for the
permission to use a figure (Fig. 5) from one of his publications. The
authors gratefully acknowledge financial support from the German
Research Foundation (DFG) via SFB 986 “M3”, project A6.
Appendix A. Derivation of Eq. (11)
Equation (10) presents the solution of P¼2d
3
e3d
2
þ1 (Eq. (8)).
For an easier reading, this solution ecalculated with Maple eis
given once more (So, Eqs. (A1) and (10) are identical.).
d¼1
2
,1þ2Pþ2ffiffiffiffiffiffiffiffiffiffiffiffiffi
P
2
P
p
1=3
þ1
21þ2Pþ2ffiffiffiffiffiffiffiffiffiffiffiffiffi
P
2
P
p
1=3
þ1
2
(A1)
As already said, the term under the square root is negative
(except for P¼0), and therefore, both expressions in brackets are
complex. Concerning the third root, note that any number eexcept
zero ehas three different complex cubic roots, and the square root
inside the brackets has two different solutions, namely “þ”and “e“.
This yields six different combinations for each bracket term in Eq.
(A1) and together, in principle, 36 possible combinations. Not every
combination and especially not every third root yields the solution
we need. Apart from that, a computer normally calculates in the
“real mode”and gives an error message if trying to extract the root
of a negative number. What we would like to have is the normal-
ized toughness as a simple real function of P. With the assumption
that Pis non-negative and smaller than 1, we can write Eq. (A1) in
the complex notation with √(PeP
2
) being real:
d¼1
2
,2P1þi,2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PP
2
p
1=3
þ1
2
,2P1i,2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PP
2
p
1=3
þ1
2(A2)
The bracket in the denominator of the second term in Eq. (A1)
vanishes, because the absolute value of the complex number in
each bracket is 1. (The reader might verify this easily.) This
allows for replacing the real and the imaginary part in both
brackets of the equation (A2) by trigonometric functions as
follows: 2Pe1¼cos
a
and 2√(PeP
2
)¼sin
a
. According to
cos
a
þisin
a
¼e
i
a
and
a
¼cos
1
(2Pe1), Eq. (A2) becomes
d¼1
2
,e
icos
1
ð2P1Þ

1=3
þ1
2
,e
icos
1
ð2P1Þ

1=3
þ1
2(A3)
If we now multiply the exponents in the brackets by 1/3, we get
only one solution out of three, and eapart from that ethe solution
is not that one we are looking for. We obtain the other two solutions
by replacing cos
1
(2Pe1) in both imaginary exponents by
cos
1
(2Pe1) ±2
p
. This does not change the exponential terms
themselves but their cubic roots. In principle, we can treat these
two exponents differently, like adding 2
p
in the first bracket and
leaving the second bracket unchanged. However, in most cases
dwouldn't be real anymore. If we treat both exponents in the same
way and apply cos
a
¼(e
i
a
þe
ei
a
)/2, we get
d¼cos cos
1
ð2P1Þþ2
p
k
3!þ1
2(A4)
with kbeing 1, 0, or 1. For k¼0, the ratio dwould become larger
than 1, and for k¼1, the relative toughness G
*
C
=G
C
for P¼0 would
be 0.25 with a negative d. Both cases do not describe the physical
situation. The desired solution means k¼1. So, by inserting the
corresponding dfrom Eq. (A4) into Eq. (9), we obtain directly the
target equation (11) for open porosity (variant B).
Appendix B. Calculation of normalized Young's moduli
For a parallel arrangement of two materials (constant strain),
the general relation for the combined Young's modulus is
Eðpar:Þ¼fE
1
þð1fÞE
2
(B1)
with E
1
and E
2
being the elastic moduli of the involved materials
and fbeing the normalized volume fraction of the first material (E
1
).
The Young's modulus of a serial arrangement (constant stress) is
given by
Eðser:Þ¼f
E
1
þ1f
E
2

1
(B2)
Applying these rules, we start with closed porosity. First, we
calculate the combined elasticity of the horizontal layer of thick-
ness wet(Fig. 1) in a parallel arrangement of solid material and
pores. With Ebeing the elastic modulus of the solid material and
again d¼t/w, we get for the layer, containing the pores, the elastic
modulus E
1
:
E
1
¼t
2
ðwtÞþ2tðwtÞ
2
w
2
ðwtÞ
,EþðwtÞ
3
w
2
ðwtÞ
,0¼d
2
þ2d,E
(B3)
Secondly, we determine the combined elasticity of this layer
plus the underlying solid layer of thickness tin a serial arrange-
ment. However, the latter layer cannot be treated that simply,
because the layer is not totally under homogeneous stress,
depending on the porosity. The respective volumes are illustrated
in Fig. B.1.
Fig. B.1. a) Low porosity and b) high porosity. The shaded solid areas between the
pores contribute differently to the overall elastic modulus. The arrows indicate the
mechanical loading situation. The layers (of thickness t) between the dashed lines
contain only solid material.
In case of very low porosity (Fig. B.1a), it becomes clear that the
shaded volumes between the pores act under compression as solid
material. In case, the porosity approaches 1, like in Fig. B.1b, the
corresponding shaded volumes do not contribute at all under
compression. So, in the following calculation we supply the shaded
volume with an “efficiency factor”d
m
, where mis of the order of 1.
H. Jelitto, G.A. Schneider / Acta Materialia 151 (2018) 443e453 451

For low porosity, d
m
is almost 1 and for high porosity d
m
is close to
0. Further down, it will be shown that a variation of the exponent m
does not have much influence on the final result.
With respect to Fig. 1, we get for the horizontal solid layer be-
tween the pores the following Young's modulus E
2
:
E
2
¼t
3
þ2t
2
ðwtÞ
w
2
t
,EþtðwtÞ
2
w
2
t
,d
m
,E
¼hd
2
þ2dþð1dÞ
2
d
m
i,E(B4)
The two layers of Eqs. (B3) and (B4) have relative volume frac-
tions of 1edand d, respectively. So, combining the two layers in a
serial arrangement yields the following elastic modulus E* for the
material of closed porosity (variant A) with d¼1eP
1/3
(Eq. (5)):
E
*
¼1d
E
1
þd
E
2

1
(B5)
0E
*
Eðmod:AÞ¼1d
d
2
þ2dþ1
dþ2þð1dÞ
2
d
m1

1
(B6)
For open porosity, we refer to Fig. 2 and calculate E*/E in an
analog way. Again, we start with the horizontal layer of thickness
wet. The combined Young's modulus of this layer is
E
1
¼t
2
ðwtÞ
w
2
ðwtÞ
,E¼d
2
,E(B7)
For the other layer of thickness t, we get by considering the
factor d
m
:
E
2
¼t
3
w
2
tþ2t
2
ðwtÞ
w
2
t
,d
m
,E¼hd
2
þ2ð1dÞ,d
mþ1
i,E
(B8)
The serial arrangement of both layers for open porosity (variant
B) yields:
E
*
Eðmod:BÞ¼1d
d
2
þ1
dþ2ð1dÞ,d
m

1
(B9)
Concerning the model variant C, we also have to adapt the
elastic modulus. If the ligaments vanish in proportion to (1eP)
n
,as
in Eq. (13), they cease to contribute to the elastic modulus. There-
fore, the Young's modulus, corresponding to Eq. (B7) of variant B,
would also get an additional factor (1eP)
n
. In Eq. (B8), two sum-
mands exist in the square brackets. The second summand corre-
sponds to the ligaments and clearly gets the same factor. The first
summand stands for the junction of the ligaments (nodes) with the
volume t
3
(Fig. 2). But even if these volumes do not vanish, they do
not contribute to the elastic modulus like before, if one or more
adjacent ligaments are disconnected. It follows that the final rela-
tive Young's modulus in Eq. (B9) just gets the mentioned factor, and
we obtain for model variant C:
E
*
Eðmod:CÞ¼1d
d
2
þ1
dþ2ð1dÞ,d
m

1
,ð1PÞ
n
(B10)
In order to obtain E*/E in Eqs. (B9) and (B10) as a function of P,
dhas to be replaced by Eq. (17).
Next, we have to determine the exponent mfor the models A to
C. We begin with variant B. If the wall thickness tand the “inner”
size of the pores wetare equal, which means d¼0.5, the corre-
sponding shaded volume (compare with Fig. B.1) is cubical-shaped.
Because it has contact on two side faces, which creates an inho-
mogeneous stress field in the shaded volume, we assume that the
volume participates approximately with 50% compared to the case
of homogeneous, full compression. This means that we have m¼1,
because then d
m
¼0.5. This m-value is used also for model C.
In the case of closed porosity (variant A), the shaded volume in
Fig. B.1 has contact on each of the four side faces. So, the influence
becomes stronger. Therefore, we estimate m¼0.5, which yields a
factor of 0.5
0.5
z0.71, correspondingly. These two choices, of course,
include some uncertainty, but we think that the two exponents
describe quite well the influence of the shaded volumes on the
elastic modulus.
Moreover, the influence of mon the final result K
*
IC
=K
IC
is weak,
as can be seen in Fig. B.2. For each variant AeC, the curves are given
for the exponents m¼0.5, 1, and 1.5. But although the influence is
low, the most reasonable values have been chosen. As already said,
we fix them to m¼0.5 for model A and to m¼1 for the models B
and C. By inserting these m-values into Eqs. (B6),(B9), and (B10),
we finally get the normalized Young's moduli in Eqs. (14)e(16).
Fig. B.2. Normalized fracture toughness for the models A, B, and C with three different
parameters mfor each model variant.
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