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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

pð1PÞ

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

¼1

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 <d1.

Thus, on the basis of Fig. 1a) we obtain for the porosity:

P¼ðwtÞ

3

w

3

¼ð1dÞ

3

(4)

and it follows

d¼1P

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

ðwtÞ

2

w

2

¼1ð1dÞ

2

(6)

The replacement of daccording to Eq. (5) yields for closed

porosity:

G

*

C

G

C

ðmod:AÞ¼1P

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

ðwtÞ

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

21þ2Pþ2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

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

ð2P1Þ

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

,ð1PÞ

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

ð2P1Þ

3!þ1

2#

2

,ð1PÞ

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:AÞ¼1d

d

2

þ2dþ1

dþ2þð1dÞ

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

,ð1PÞ

n

¼2d

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

ð2P1Þ

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:AÞ¼1P

2=3

1=2

, 1d

d

2

þ2dþ1

dþ2þð1dÞ

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

,ð1PÞ

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

X¼1ð1PÞ

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

21þ2Pþ2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

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

,2P1þi,2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

PP

2

p

1=3

þ1

2

,2P1i,2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

PP

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

ð2P1Þ

1=3

þ1

2

,e

icos

1

ð2P1Þ

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

ð2P1Þþ2

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 k¼1. 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

þð1fÞE

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:Þ¼f

E

1

þ1f

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

ðwtÞþ2tðwtÞ

2

w

2

ðwtÞ

,EþðwtÞ

3

w

2

ðwtÞ

,0¼d

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

ðwtÞ

w

2

t

,EþtðwtÞ

2

w

2

t

,d

m

,E

¼hd

2

þ2dþð1dÞ

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

*

¼1d

E

1

þd

E

2

1

(B5)

0E

*

Eðmod:AÞ¼1d

d

2

þ2dþ1

dþ2þð1dÞ

2

d

m1

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

ðwtÞ

w

2

ðwtÞ

,E¼d

2

,E(B7)

For the other layer of thickness t, we get by considering the

factor d

m

:

E

2

¼t

3

w

2

tþ2t

2

ðwtÞ

w

2

t

,d

m

,E¼hd

2

þ2ð1dÞ,d

mþ1

i,E

(B8)

The serial arrangement of both layers for open porosity (variant

B) yields:

E

*

Eðmod:BÞ¼1d

d

2

þ1

dþ2ð1dÞ,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:CÞ¼1d

d

2

þ1

dþ2ð1dÞ,d

m

1

,ð1PÞ

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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