International Journal of Engineering Science 130 (2018) 157–174

Contents lists available at ScienceDirect

International Journal of Engineering Science

journal homepage: www.elsevier.com/locate/ijengsci

Control of fracture at the interface of dissimilar materials

using randomly oriented inclusions and networks

Victor Birman

Global – St. Louis and Department of Mechanical and Aerospace Engineering, Missouri University of Science and Technology, 12837

Flushing Meadows Drive, St. Louis, MO 63131, USA

a r t i c l e i n f o

Article history:

Received 3 May 2018

Revised 29 May 2018

Accepted 30 May 2018

Keywords:

Fracture

Interface

Random fibers

Nanotubes

Random networks

Dundurs parameters

a b s t r a c t

Fracture at or near the interface between two isotropic materials has been a subject of

extensive research relevant to the problems encountered in aerospace, naval, electronic

packaging, biomechanical engineering and other applications. The problems of edge cracks,

semi-infinite interface cracks and substrate cracks under a thermally stressed film consid-

ered in the paper are representative of such analyses. We consider a possible improve-

ment in the fracture resistance achieved by embedding randomly distributed stiff inclu-

sions such as fibers, nanotubes, fiber or nanotube networks and ellipsoidal or spherical

particles in the more compliant material. This results in a smaller mismatch between the

stiffness of the joined materials that may prevent or alleviate fracture. Numerical examples

demonstrate both the benefits and the limitations of enhancing the stiffness of the com-

pliant material. In particular, the examples refer to the boundary between singular and

non-singular interfacial edge stresses attempting to avoid the stress singularity by embed-

ding random carbon nanotubes or networks. In the semi-infinite interfacial crack problem

it is demonstrated that a small reduction in the stiffness mismatch of two materials at the

interface causes a significant decrease in the strain energy release rate. The approach to

the analysis of substrate cracks under a thermally stressed film is outlined accounting for

the history of thermal loading and the effect of temperature on the properties of the film,

substrate and embedded inclusions. The solutions presented in the paper rely on effective

properties of randomly reinforced materials. The limitations of such approach in fracture

problems and an estimate of its validity based on a comparison of scales at the tip of

the crack and in the representative volume cell are discussed. It is suggested that frac-

ture involving nanoparticle and nanotube reinforced materials can be characterized using

effective properties.

© 2018 Elsevier Ltd. All rights reserved.

1. Introduction

Fracture along the interface between two dissimilar materials represents one of the major challenges for engineers. Pio-

neering work on the interfacial fracture between two dissimilar isotropic materials was published in England (1965) , Erdogan

(1965 ), Williams (1959) and Rice and Sih (1965) . While predicting the stress singularity at the interface crack tip in linear

isotropic materials, an additional perplexing oscillatory singularity resulting in the overlap of the sides of the interface re-

mained unexplained until later research (e.g., Comninou, 1977 ) where this phenomenon was attributed to the presence of

E-mail address: vbirman@mst.edu

https://doi.org/10.1016/j.ijengsci.2018.05.011

0020-7225/© 2018 Elsevier Ltd. All rights reserved.

158 V. Birman / International Journal of Engineering Science 130 (2018) 157–174

very short contact zones along the interface at a distance from the tip of the crack. The small length of these zones serves as

a justification for the modern interface fracture analysis employing the classical methods pioneered in the previously cited

papers. The methodology of the analyses of the interface fracture toughness of a crack between two isotropic materials and

a delamination between two composite laminae was outlined in the review ( Banks-Sills, Travitzky, & Ashkenazi, 20 0 0 ). A re-

view outlining major studies of the interface fracture mechanics in case of isotropic linear elastic materials has recently been

published ( Banks-Sills, 2018 ). A comprehensive review of tests methods evaluating a relevant problem of fracture between

adhesively bonded coatings and substrate that are also applicable to testing of the interfacial toughness was published in

Chen, Zhou, Lu, and Lam (2014 ). It is noted that a straight interface is not always a preferred solution in biology. For ex-

ample, the waviness of the tendon-to-bone interface investigated in Hu et al. (2015a; 2015b ) enhanced the toughness of

the joint, at the expense of higher stress concentrations. Such trade-off between toughness and strength observed in bi-

ology may be potentially of interest in engineering applications. Extensive studies have also been conducted investigating

toughness of anisotropic materials with interfacial cracks. Early work in this area includes the papers ( Gotoh, 1967; Qu &

Bassani, 1989 ; Wu, 1990 ). Examples of recent studies of toughness of the interface between anisotropic materials, including

numerous references to previous work, are papers ( Beom & Jang, 2012 ; Gao, Tong, & Zhang, 2003 ; Vodi cka, Kormaníková, &

Kši nan, 2018 ).

The stress singularity at the edge of the interface between two isotropic layers or solids has been investigated beginning

with the pioneering work of Williams (1952) . Dundurs demonstrated that the stresses in elastic orthogonal wedges are

dependent on two parameters that are associated with his name ( Dundurs, 1969 ). Bogy (1971 ) and Noda and Lan ( 2012 )

determined the boundaries separating the combinations of Dundurs parameters corresponding to singular and non-singular

stresses and stress intensity factors at the edge of a butt joint between two dissimilar materials in the states of plane

stress and plane strain. Most pairs of engineering materials, such as glass/ceramic, metal/metal, etc. were shown to possess

stress singularity at the edge of the interface. Furthermore, a stress singularity was found in such biological examples as the

tendon to bone insertion site ( Genin & Liu, 2013 ). Various aspects of the stress state near the edge of the interface between

two materials have been studied ( Balijepalli, Begley, Fleck, McMeeking, & Arzt, 2016; Wu & Liu, 2010 ).

Besides the edge stress singularity between isotropic materials referenced to above, the free edge singularity between two

orthotropic solids has been explored. Analytical techniques employing the anisotropic theory of elasticity and Lekhnitskii’s

stress potentials were suggested by Wang and Choi (1982a, 1982b) and Liu and Chue (2005) . The H-integral methodology

was employed to evaluate the effects of various parameters on the stress intensity factors at the edge of multi-layered

isotropic or anisotropic laminates ( Shang, Zhang, & Skallerud, 2009 ). This paper contains an extensive literature review

on the studies in the area of edge stresses and stress singularity order at both isotropic and orthotropic interface edges.

Numerical studies of the problem include papers ( Apel, Mehrmann, & Watkins, 2002 ) and ( Yosibash & Omer, 2007 ).

Besides the research on the interfacial fracture between two dissimilar materials, a somewhat relevant subject is peeling

of thin films that has been considered in such papers as Kendall (1971 , 1975 ), Wei and Hutchinson (1997 ) and Williams

and Kauzlarich(2004 ). In addition to engineering problems relevant to this subject, the application to a tendon peeled from

the bone has recently been considered Genin and Liu (2013 ) and Lipner et al. (2017 ). Particularly interesting, in light of

the present study, was the conclusion that a partially mineralized compliant tissue between tendon and bone may serve

as an energy absorbent element, toughening the joint and alleviating injury. This points to a possible engineering approach

to design, introducing a compliant layer between dissimilar materials to enhance toughness of the joint. In addition to the

peeling test, the advantage of a compliant zone enhancing toughness of the tendon-to-bone insertion side was suggested in

the previous studies ( Liu, Thomopoulos, Birman, Li, & Genin, 2012 ).

Another problem related to the interfacial fracture is the propagation of a crack parallel to the interface with a residually

stressed thin film. This problem has been extensively analyzed ( Begley, Mumm, Evans, & Hutchinson, 20 0 0; Drory, Thouless,

& Evans, 1988; Evans & Hutchinson, 1995; Yu & Hutchinson, 2003 ).

Several possible methodologies improving the interfacial fracture resistance at or near the interface between dissimilar

isotropic materials following from the previous studies include employing a functionally graded interface between dissimilar

materials (e.g., the reviews of functionally graded materials and their potential ( Birman & Byrd, 2007; Birman, Keil, & Hosder,

2012 ). Inserting a complaint layer along the interface may have a potential of improving toughness. As mentioned above,

using a wavy interface can also improve toughness, at the expense of higher stresses and reduced strength.

The methodology considered in this paper is based on embedding random reinforcements in the more compliant ma-

terial preventing or alleviating fracture. Potential reinforcements include random fibers, nanotubes, ellipsoidal or spherical

inclusions as well as fiber or nanotube networks. Three problems formulated in the paper include the edge interfacial stress

singularity, strain energy release rate in semi-infinite cracks and the approach to the analysis of substrate cracks parallel

to the interface with a residually stressed thin film. Accordingly, we outline major research conducted on these particular

problems below.

In the first problem the relationship between Dundurs’ parameters corresponding to the boundary between singular

and non-singular interface edge stresses is extrapolated to explicitly express the boundary ratio of the shear stiffness of

two materials as a function of their Poisson ratios for the plane stress and plane strain cases. As is shown in numerical

examples, adding random nanotubes or nanotube networks to eliminate the edge stress singularity may be successful if the

mismatch between the stiffness of two pristine joined at the interface materials is limited, but the desirable outcome cannot

be achieved in the joint between two significantly mismatched materials.

V. Birman / International Journal of Engineering Science 130 (2018) 157–174 159

The strain energy release rate and the interfacial fracture produced by a semi-infinite crack between two isotropic mate-

rials, such as in a four-point flexure test, is also considered in this paper. Interfacial fracture in a four-point flexure test was

first analyzed in Charalambides, Cao, Lund, and Evans, (1990 ). The comprehensive review of this problem is found in papers

( Hutchinson & Suo, 1991; Suo & Hutchinson, 1990 ). Some of the novel applications of this test for fracture investigation

along the bimaterial interface can be found in dentistry ( Chai, Lee, Mieleszko, Chu, & Zhang, 2014 ) and in testing of thermal

barrier coatings ( Chen, Wang, Yuan, & Zhu, 2008 ).

In addition to the formulation accounting for the effect of embedded random inclusions or networks on the strain energy

release rate, a simple formula is suggested demonstrating an increase in the bending moment due to embedded random

inclusions while the strain energy release rate remains unchanged from that of the original interface. Numerical results

illustrate that irrespectively of the through-the-thickness location of the interface between two isotropic materials, the strain

energy release rate can significantly be reduced as a result of a moderate increase in the stiffness of the compliant material.

The third problem discussed in the paper is a generalization of the classical solution for the strain energy release rate

of a substrate crack parallel to the interface with a thin film subject to a thermal residual stress. Previous research of this

subject ( Begley et al., 20 0 0; Drory et al., 1988; Evans & Hutchinson, 1995; Yu & Hutchinson, 2003 ) did not account for

the effect of temperature on the material properties of the film and substrate, as it is lowered from the processing to the

operational value. The modification of the structure embedding random fibers, nanotubes, ellipsoidal inclusions or random

networks in either one or both materials of the film and/or substrate potentially enhancing the toughness is presented in

this section. The history of thermal loading can be accounted for using the formulation presented in the paper. The influence

of temperature on the engineering constants of the materials and the effect of the volume fraction of inclusions are reflected

in the formulation.

All problems considered in the paper are confined to the case of isotropic or transversely isotropic materials. The method-

ologies of the evaluation of the properties of more complicated material systems, such as those discussed in Genin and

Birman (2009 ), Kanaun and Jeulin (2001 ) and Sevostianov and Kachanov (2002 ), are not included in the paper, even though

they may lead to a desirable improvement in the fracture resistance.

Besides embedding stiff random reinforcements in the compliant material, the mismatch in the properties of joined ma-

terials can be reduced embedding low-stiffness inclusions in the stiffer material. Such approach that may increase toughness

of the joint can be modelled using the methods suggested in the paper, but it is not considered in numerical examples.

The paper is organized as follows: three fracture problems are outlined in Section 2.1 , including the edge stress singu-

larity at the interface in the butt joint between two isotropic materials, the strain energy release rate in a semi-infinite

crack between two materials, and the strain energy release rate for a substrate crack parallel to the interface with a resid-

ually stressed thin film. Section 2.2 outlines the methods of the evaluation of the properties of composites with randomly

distributed fibers, nanotubes, ellipsoidal inclusions and fiber or nanotube networks. These methods can be employed to

calculate the stiffness of the compliant material with embedded random reinforcements or alternatively, the properties of

the stiffer material with compliant inclusions. This section also contains a discussion on the validity and limitations of the

solution employing effective properties of the material in fracture problems and a suggested approach to the estimate of the

appropriateness of such method. Section 3 with numerical examples concentrates on the edge interfacial stress singularity

problem. The effect of the Poisson ratios of the materials on the interfacial edge stress singularity is elucidated. Represen-

tative examples for pairs of joined materials demonstrate that random reinforcement of the more compliant material may

eliminate the stress singularity, except for the case of a large mismatch between the properties of pristine materials in the

butt joint. It is also shown that a significant reduction of the strain energy release rate in a semi-infinite crack (e.g., four-

point flexure test) can be achieved embedding a moderate volume fraction of random reinforcements in the more complaint

material.

2. Analysis

2.1. Representative fracture problems

2.1.1. Boundary between singular and non-singular stresses at the edge of the interface between two isotropic materials

Consider the interface between two different isotropic materials ( Fig. 1 ). The onset of the fracture crack can be traced to

the stress concertation at the edge of the interface. As was shown in Bogy (1971 ) and Noda and Lan (2012 ), the singularity

of stresses can be predicted considering the Dundurs parameters dependent on the engineering constants of two materials

as well as the structure being in the state of plane stress or plane strain:

α =

G 1 ( κ2 + 1 ) − G 2 ( κ1 + 1 )

G 1 ( κ2 + 1 ) + G 2 ( κ1 + 1 )

β =

G 1 ( κ2 − 1 ) − G 2 ( κ1 − 1 )

G 1 ( κ2 + 1 ) + G 2 ( κ1 + 1 ) (1)

where G i ( i = 1 , 2 ) are the shear moduli of the materials,

κi =

3 − νi

1 + ν

i

160 V. Birman / International Journal of Engineering Science 130 (2018) 157–174

Fig. 1. Problem of the stress singularity at the edge of the interface between two dissimilar materials.

and

κi = 3 − 4 νi (2)

for plane stress and plane strain, respectively, and ν i are the Poisson ratios. As noted in Noda and Lan (2012 ), it is possible

to consider only the case where β ≥ 0 since reversing materials 1 and 2 results in the reverse of the sign of both Dundurs

parameters. In the following, material 1 is assumed a stiffer material and material 2 is more complaint.

As was pointed by Hutchinson and Suo (1991 ), the first Dundurs parameter expresses a mismatch between the in-plane

moduli of elasticity across the interface. In the cases of plane stress and plane strain this parameter can be written as

α =

E 1 − E 2 E 1 + E 2

and

α =

E 1 1 −ν2

1

− E 2 1 −ν2

2

E 1 1 −ν2

1

+

E 2 1 −ν2

2

(3)

respectively. The second Dundurs parameter reflects a mismatch in the in-plane bulk moduli of two materials.

Various combinations of the Dundurs parameters correspond to the singularity of the stresses at the edge of the in-

terface, the absence of such singularity and the boundary between singular and non-singular stresses. The corresponding

combinations are

Stress singularity: α( α − 2 β) > 0 (4a)

Singularity is zero: α( α − 2 β) = 0 (4b)

Singularity vanishes: α( α − 2 β) < 0 (4c)

V. Birman / International Journal of Engineering Science 130 (2018) 157–174 161

Fig. 2. Semi-infinite crack at the interface between two materials (shown on the example of a four-point flexure test). The crack is considered semi-infinite

if its half-length several times exceeds the thickness of the cracked layer.

A singularity of stress implies that the stress intensity factors and the stress in the direction perpendicular to the inter-

face approach infinity as the distance from the edge approaches zero ( Bogy, 1971 ). In the case of non-singular stress, the

stress intensity factors and stress approach zero at the edge. At the boundary between singular and non-singular solutions

both the stress intensity factor and the interface stresses have finite values. A review of studies concerned with the order of

magnitude of stress singularity is found in Noda and Lan (2012 ).

The boundary between singular and non-singular solutions represented by Eq. (4b) is of a particular interest. It is easily

shown that this boundary corresponds to the following ratio of shear moduli of the materials:

G 1

G 2

=

3 − κ2

3 − κ1

(5)

A larger mismatch in the shear moduli than that given by Eq. (5) corresponds to singular stresses and fracture. A possible

method of avoiding fracture is to reinforce the more compliant material in the joint reducing the stiffness mismatch. In

particular, this can be achieved by embedding randomly distributed stiff fibers, nanotubes, ellipsoidal or spherical inclusions

or a random fiber or nanotube network within the compliant material. Such reinforcement can be local and limited to the

vicinity of the edge, the compliant material being transformed into a functionally graded one, although the necessary extent

of such locally stiff region is not considered in this paper. An alternative to the stiffening of the more compliant material

could be a local weakening of the stiffer material in the joint using inclusions with low stiffness, though such method may

be less practical in realistic designs.

The range of variations of the Poisson ratio in isotropic materials is between −1.0 and 0.5. While the majority of engi-

neering materials possess a positive Poisson ratio, foams with a negative Poisson ratio were developed by Lakes (1987 ) and

in subsequent studies. Such auxetic materials with a negative Poisson ratio have found application in the cores of sandwich

structures due to a potential for achieving a higher stiffness and toughness (see review Birman & Kardomateas, 2018 for

more details).

The analysis of the range of the Poisson ratios of materials 1 and 2 corresponding to physically feasible, i.e. positive,

ratios of the shear moduli in Eq. (5) in both plane stress and plane strain cases yields

ν1 ν2 > 0 (6)

implying that Poison’s ratios should have the same sign. Interestingly, this means that stress singularity can be avoided

in a butt joint of two auxetic materials. If both joined materials are incompressible, the ratio of shear moduli is equal to

1.0, i.e. for all practical purposes, we have a butt joint of either two sections of the same material or materials with the

same engineering constants. If the Poisson ratio of one of the materials is equal to −1.0, the stiffness ratio in the case of

plane stress is either infinite or equal to zero representing an unpractical situation. Likewise, in the case of plane strain, and

νi = 0 ( i = 1 or 2 ) the shear moduli ratio is either zero or infinite.

2.1.2. Strain energy release rate and load-carrying capacity in case of a semi-infinite crack between two isotropic materials (e.g.,

four-point flexure test)

The strain energy release rate in the problem depicted in Fig. 2 was obtained by Hutchinson and Suo (1991 ):

G =

M

2

E h

3

(6 η3 − 1

2 I

)(7)

2

162 V. Birman / International Journal of Engineering Science 130 (2018) 157–174

Fig. 3. Substrate crack parallel to the interface with residually stressed thin film.

where the moment per unit width of the specimen is M = P l, and E 2 =

E 2 1 −ν2

2

. The nondimensional moment of inertia is

I = �[ � − 1 /η] 2 − ( � − 1 /η) + 1 / 3

�

η( � − 1 /η) +

1

3 η3 (8)

and

� =

1 + 2�η + �η3

2 η( 1 + �η) , � =

1 + α

1 − α, η =

h

H

(9)

As is shown in numerical examples, reducing the mismatch between the stiffness of materials 1 and 2 results in a

reduction in the strain energy release rate. The benefits of embedding stiffer inclusions in the more compliant material

can also be demonstrated by evaluating the bending moment ratio while maintaining the same strain energy release rate,

reflecting on a potential for a higher load-carrying capacity of the specimen with the crack. Comparing the moment in the

specimen with the inclusions embedded in the compliant material 2 to the moment in the joint without such inclusions at

the same strain energy release rate yields

M ( V 2 )

M ( 0 ) =

√

E 2 ( V 2 ) 6 η3 − 1 / 2 I ( 0 )

E 2 ( 0 ) 6 η3 − 1 / 2 I ( V 2 ) (10)

where V 2 is the volume fraction of the inclusions (fibers, nanotubes, particles, networks). Note that adding inclusions also

increases the critical strain energy release rate ( Her & Chien, 2017 ) that is not reflected in Eq. (10) , i.e. the estimate of the

efficiency of the reinforcement presented here is conservative.

It is noteworthy that the toughness of a wavy interface may be superior to that of a flat interface, at the expense of

higher stress concentrations ( Hu et al., 2015 ). However, technological complications may be a barrier to using such approach

in industry, while embedding inclusions into the material is a well-known technology finding applications in functionally

graded materials and structures ( Birman & Byrd, 2007; Birman et al., 2012 ).

2.1.3. Minimizing strain energy release rate for substrate cracks under residually stressed thin films

The problem of cracking in a brittle substrate under thin films undergoing residual stresses has been considered in

Drory et al. (1988) and in numerous subsequent studies dealing with this type of fracture. The solution presented in

Drory et al. (1988) is concerned with the propagation of a crack parallel to the interface with the film ( Fig. 3 ). The thin

film is assumed to have a higher coefficient of thermal expansion than the substrate, so that it is subject to tensile stresses

due to a uniform decrease of temperature following the manufacturing process. Reflecting a continuous change of the en-

gineering constants of the film, substrate and random reinforcements with a decreasing temperature, the residual stress in

the film can be obtained accounting for the history of thermal loading that was disregarded in Drory et al. (1988) , i.e. the

stress in the film is

σ ( T ) =

∫ T

T 0

E f (T , V f n

)[1 − ν f

(T , V f n

)][α f

(T , V f n

)− αs ( T , V sn )

]dT (11)

V. Birman / International Journal of Engineering Science 130 (2018) 157–174 163

V

where T and T 0 are the current and initial (manufacturing) temperatures, respectively, V fn , V sn are the volume fractions of

the inclusions in the film ( f ) and substrate ( s ), αf ( T, V fn ) and αs ( T, V sn ) are the film and substrate coefficients of thermal

expansion, and E f ( T, V fn ) and ν f ( T, V fn ) are the modulus of elasticity and the Poisson ratio of the film.

If random reinforcements are embedded in the substrate only, V f n = 0 . Similarly, if only the film material is reinforced,

sn = 0 .

The nondimensional strain energy release rate is given by

G

(T , V f n , V sn

)=

G

(T , V f n , V sn

)E f

(T , V f n

)σ 2 ( T ) h

=

�(T , V f n , V sn

)2

[λ + �

(T , V f n , V sn

)][

1 +

λ2 ( λ + 1 ) 2

4

[λ + �

(T , V f n , V sn

)]I (T , V f n , V sn

)]

(12)

where h is the thickness of the film, λ is the relative crack depth,

�(T , V f n , V sn

)=

E f ( T, V f n ) E s ( T, V sn )

f or plane stress,

�(T , V f n , V sn

)=

E f ( T, V f n ) [ 1 −ν2 s ( T, V sn ) ]

E s ( T, V sn )

[ 1 −ν2

f ( T, V f n ) ] f or plane strain

The nondimensional moment of inertia of the beam per unit width is

I (T , V f n , V sn

)=

1

3

{

�(T , V f n , V sn

)[ 3

(�

(T , V f n , V sn

)− λ

)2 − 3

(�

(T , V f n , V sn

)− λ

)+ 1

] +3�

(T , V f n , V sn

)λ(�

(T , V f n , V sn

)− λ

)+ λ3

}

(14)

The position of the neutral axis of the beam relative to the crack plane normalized with respect to the thickness of the

film is

�(T , V f n , V sn

)=

λ2 + 2�(T , V f n , V sn

)λ + �

(T , V f n , V sn

)2

[λ + �

(T , V f n , V sn

)] (15)

Note that the strain energy release rate is calculated at the current temperature, i.e. while the history of thermal loading

is accounted for the stress in (11) , all other parameters in Eqs. (12 )–( 15 ) are affected only by the current temperature and

independent of the history.

The residual thermal stress given by (11) should account for the temperature dependence of the coefficients of thermal

expansion of the film, the substrate, and the material of stiff random inclusions. Accordingly, the expression for the coeffi-

cients of thermal expansion of a composite material consisting of an isotropic matrix with embedded 3-D randomly oriented

isotropic inclusions derived in Rosen and Hashin (1970 ) and Schapery (1968 ) is generalized to account for the variations of

properties with temperature:

αi ( T , V in ) = αi ( T , V in ) +

αim

( T ) − αin ( T )

1 / K im

( T ) − 1 / K in ( T )

[1 / K i ( T , V in ) − 1 /K ( T , V in )

](16)

where αi ( T , V in ) is the coefficient of thermal expansion obtained by the rule of mixtures. i = f or s , the subscript im refers to

the properties of the pristine i th “matrix” material, and the subscript in refers to the properties of the inclusions added to

the i thmaterial. The last term in Eq. (16) is

1 /K ( T , V in ) = V im

/ K im

( T ) + V in / K in ( T ) (17)

where V i j , ( j = m, n ) are the volume fractions of the matrix and inclusions and K ij ( T )are the bulk moduli of matrix and

inclusions.

The bounds for the bulk modulus of the composite with isotropic inclusions can be found following ( Hashin & Shtrik-

man, 1963 ). For example, in the case of spherical inclusions the lower bound of this modulus is the exact solution that is

extended here to account for temperature dependent properties of matrix and inclusions in the film or substrate:

K i ( T , V in ) = K im

( T ) +

V in ( K in ( T ) − K im

( T ) )

1 + V 1 m

K in ( T ) −K im ( T ) K im ( T ) +1 . 333 G in ( T )

(18)

G in being the shear modulus of the inclusions, not to be confused with the strain energy release rate.

The formulation presented here enables us to investigate the effects of temperature dependence and embedding

randomly-oriented inclusions in the substrate and/or film on the strain energy release rate. The reinforcement of the sub-

strate and/or film will also affect the fracture energy that can be determined from experiments. A detailed numerical anal-

ysis of this problem is outside the scope of the present paper.

164 V. Birman / International Journal of Engineering Science 130 (2018) 157–174

2.2. Methods of the evaluation of the properties of a composite materials with random fibers, nanotubes, inclusions or networks

embedded in an isotropic matrix

The goal of reducing the mismatch between the stiffness of the materials joined at the interface can be achieved with

stiff fibers, nanotubes or their networks, ellipsoidal or spherical inclusions incorporated into the compliant material to in-

crease its stiffness. Alternatively, it can be realized using compliant inclusions embedded within the stiffer material to reduce

its stiffness. Both the cases of stiff as well as compliant inclusions can be analyzed using the methods reviewed below. There

are several theories defining the stiffness of a composite material with two-dimensional or three-dimensional random in-

clusions or establishing the bounds for such stiffness; some of these theories are outlined below. These theories have been

validated through a comparison with experimental data as indicated in the following text.

2.2.1. Properties of a composite material with 2-D random fibers or nanotubes

A relatively simple, yet accurate, Tsai–Pagano method ( Tsai & Pagano, 1968 ) can be employed to determine the stiffness

of a 2-D random fiber-reinforced or a nanotube-reinforced material. Using this method is conditional on the knowledge

of the longitudinal and transverse moduli of the composite material with the same volume fraction of uniaxially oriented

fibers or nanotubes. These moduli are obtained by the Halpin–Tsai method that has been experimentally verified for uniaxi-

ally reinforced fiber composites in Tucker and Liang (1999 ). The analyses in Rafiee et al. (2009 ) and Erik and Tsu-Wei (2003 )

demonstrated that the Halpin–Tsai model also provides an acceptable estimate of the stiffness of uniaxial nanocompos-

ites. Notably, a close agreement between the theoretical predictions and experimental data for the longitudinal modulus of

nanocomposites ( Rafiee et al., 2009 ) demonstrated that the effect of the interphase noted in Han and Elliott (2007 ) may of-

ten be disregarded being limited to specific matrices and manufacturing processes. The accuracy of the Tsai-Pagano method

was confirmed through the comparison with experimental data for random boron/epoxy ( Halpin & Pagano, 1969 ) and for

random glass/polyester materials ( Manera, 1977 ).

According to the Tsai–Pagano method, the moduli of elasticity and shear of the material with 2-D random fibers are

determined by

E i =

3

8

E 11 i +

5

8

E 22 i G i =

1

8

E 11 i +

1

4

E 22 i (19)

where E 11 i and E 22 i are the longitudinal and transverse moduli of the uniaxial composite with the same fibers (nanotubes)

and matrix found by the semi-empirical Halpin–Tsai equations:

E 11 i =

1 +

(2 L f / d f

)ηi V f i

1 − ηi V f i

E mi E 22 i =

1 + 2 ηi V f i

1 − ηi V f i

E mi

ηi =

E f i / E mi − 1

E f i / E mi + ξ

F or E 11 i ξ = 2 L f / d f , F or E 22 i ξ = 2 (20)

In Eq. (20) , E fi and E mi are the moduli of fibers and matrix, respectively, L f , d f are the length and diameter of the fibers,

V fi is the fiber volume fraction in the matrix, and the subscript i ( = 1 , 2 ) refers to the i th material in the joint.

The stiffness of a random fiber-reinforced matrix was also determined by Christensen and Waals (1972 ) who employed

the averaging technique over the fiber orientations to derive the modulus of elasticity and the Poisson ratio of both 3-D

random fiber oriented composite as well as for the 2-D case. For the latter case, the results are:

E i =

u

2 1 i

− u

2 2 i

u 1 i

νi =

u 2 i

u 1 i

(21)

where

u 1 i =

3

8

E 11 i +

G 12 i

2

+

(3 + 2 ν12 i + 3 ν2

12 i

)G 23 i K 23 i

2 ( G 23 i + K 23 i )

u 2 i =

1

8

E 11 i −G 12 i

2

+

(1 + 6 ν12 i + ν2

12 i

)G 23 i K 23 i

2 ( G 23 i + K 23 i ) (22)

In (22) , G 12 i , G 23 i and ν12 i are the shear moduli and the Poisson ratios of the uniaxial composite with the same fibers

oriented in the 1-direction and matrix as in the randomly reinforced material, respectively, that can be determined by the

composite cylinders model ( Christensen, 2005 ) and K 23 i is the plain strain bulk modulus found following Hashin (1966 ):

K 23 i = K mi +

G mi

3

+

V f i (K f i − K mi + ( 1 / 3 )

(G f i − G mi

))−1 +

(1 − V f i

)( K mi + 4 G mi / 3 )

−1 (23)

A comparison with the experimental modulus of elasticity for glass/polyester two-dimensional random composite

demonstrated that the Christensen–Waals formula predicts the values that are in a good agreement for small fiber vol-

ume fractions. The theoretical modulus exceeded the experimental results at the fiber volume fraction larger than 0.15, but

V. Birman / International Journal of Engineering Science 130 (2018) 157–174 165

the prediction remained satisfactory even at V f i = 0 . 20 . A simple formula was suggested estimating the stiffness of a 2-D

random fiber composite in the low fiber content range 0 < V fi < 0.20:

E i =

V f i

3

E f i +

(1 + V f i

)E mi (24)

An alternative formula for 2-D random fiber composites was developed by Christensen (1976 ) employing his previous re-

sult that was simplified through the asymptotic approximation using the power series expansion of the ratio of the stiffness

of matrix to that of the fibers:

E i E mi

=

V f i E f i

3 E mi

+

1

E mi

[

˜ E 11 i

3

+

8 G 12 i

9

+

4

(3 − 2 ν f i + 3 ν2

f i

)G 23 i K 23 i

9 ( G 12 i + K 23 i )

]

+ 0

(E mi

V f i E f i

)(25)

where the last term in the right side is of a higher order of magnitude and

˜ E 11 i =

(1 − V f i

)E mi + 4 V f i

(1 − V f i

)G mi

[ (ν f i − νmi

)2

V f i G mi

K mi + G mi / 3 + 1

]

(26)

A comparison of the asymptotic formula (25) with experimental results for glass/polycarbonate yielded an excellent

agreement even for the fiber volume fraction over 40%. Furthermore, the modulus of elasticity of the material appeared

to be insensitive to the values of the Poisson ratio of the fibers and matrix. The latter observation is important since it

justifies using the rule of mixtures for the Poisson ratio.

The stiffness of random fiber reinforced material should remain within the bounds derived using variational techniques

(e.g., Hashin & Shtrikman, 1963 ) or by other methods utilizing fundamentals of mechanics. A comprehensive review of the

bounding techniques has been published by Torquato (1991 ). In the present section bounding techniques are not discussed

since it is assumed that the volume fraction of random inclusions is limited to the range where the methods presented

above have been experimentally verified.

2.2.2. Properties of a composite material with 2-D or 3-D random ellipsoidal inclusions

The Mori-Tanaka method can be employed to determine the stiffness of materials reinforced by 2-D and 3-D randomly

oriented ellipsoidal inclusions. This method is considered accurate at the volume fraction of inclusions under 30% or 35%

( Yin, Sun, & Paulino, 2004 ). Tandon and Weng (1986 a, 1986 b) suggested closed-form expressions for the engineering con-

stants of such composite materials. The results were found within the Hashin–Shtrikman bounds and a good agreement with

experimental data was observed in the case of a short-fiber glass/polystyrene composite material. In the case of randomly

distributed in the plane short fibers or ellipsoidal inclusions this method predicts the engineering constants that are within

the Willis bounds, see for details ( Pan & Weng, 1995 ).

The engineering constants of the material with an isotropic matrix and spherical inclusions were derived by Benveniste

(1987 ). In particular, the bulk and shear moduli are

K i = K 1 i +

V 2 i ( K 2 i − K 1 i )

1 + V 1 i K 2 i −K 1 i

K 1 i +1 . 333 G 1 i

G i = G 1 i +

V 2 i ( G 2 i − G 1 i )

1 + V 1 i G 2 i −G 1 i G 1 i + f 1 i

(27)

where K 1 i , G 1 i and K 2 i , G 2 i are the bulk and shear moduli of the pristine i thmaterial and inclusions, respectively. The coefi-

cient f 1 i , the modlus of elasticty of the homogeneous composite E i and its Poisson ratio ν i are evaluated from the relation-

ships

f 1 i =

G 1 i ( 9 K 1 i + 8 G 1 i )

6 ( K 1 i + 2 G 1 i )

K i =

E i 3 ( 1 − 2 νi )

=

E i G i

3 ( 3 G i − E i ) (28)

In the case of spherical inclusions, results obtained by the Mori–Tanaka method coincide with the lower Hashin–Strikman

bound.

2.2.3. Properties of the composite material with 3-D randomly distributed fibers or nanotubes

The Christensen–Waals solution developed for a 3-D random fiber distribution ( Christensen & Waals, 1972 ) provides an

estimate for the modulus of elasticity of the material that is accurate for the fiber volume fraction below 20%:

E i =

V f i

6

E f i +

[1 + ( 1 + νmi ) V f i

]E mi (29)

Note that the previously cited research of Christensen justifies using the rule of mixtures finding the Poisson ratio of the

material with random fiber reinforcements.

166 V. Birman / International Journal of Engineering Science 130 (2018) 157–174

2.2.4. Properties of a composite material with fiber or nanotube 2-D and 3-D networks distributed in the matrix

The stiffness of fiber or nanotube networks as well as references to the previous research on the subject are found in

Chen, Pan, Guo, Liu, and Zhang (2015 ), Deogekar and Picu (2017 ), Wu and Dzenis (2005 ) and Zixing, Man, and Qiang (2014 ).

In particular, explicit equations for the modulus of elasticity and the Poisson ratio of a planar (2-D) random fiber network

are presented in Wu and Dzenis (2005 ) and omitted here for brevity. The solutions for both 2-D and 3-D fiber networks

were also presented in Chen et al. (2015 ). In this paper, we do not reproduce the evaluation of the stiffness of the networks

relying in the examples on the numerical data in these references where the moduli of elasticity ( Chen et al., 2015 ; Wu &

Dzenis, 2005 ) and the Poisson ratios of the networks ( Wu & Dzenis, 2005 ) are shown as functions of the network density.

An approximate approach to the evaluation the modulus of elasticity of the material with 2-D or 3-D fiber network

embedded in an isotropic matrix is using Eqs. (24) and (31) , respectively, replacing the stiffness of the fibers in the first

term with the stiffness of the network. An alternative approach is using the Hashin–Shtrikman bounds ( Hashin & Shtrik-

man, 1963 ) that do not rely on the shape of the inclusions and should be applicable to the material consisting of a fiber

or nanotube network embedded in the matrix (these bounds that were generalized by Walpole (1966) cover the majority

of engineering materials). The Hashin-Shtrikman bounds for the bulk and shear moduli of a two-phase material of arbitrary

phase geometry are:

K im

+

V nn

1 / ( K nn − K im

) + 3 V im

/ ( 3 K im

+ 4 G im

) < K i < K nn +

V im

1 / ( K im

− K nn ) + 3 V nn / ( 3 K nn + 4 G nn )

G im

+

V nn

1 / ( G nn − G im

) + 6 V im

( K im

+ 2 G im

) / 5 G im

( 3 K im

+ 4 G im

) < G i <

G nn +

V im

1 / ( G im

− G nn ) + 6 V nn ( K nn + 2 G nn ) / 5 G nn ( 3 K nn + 4 G nn ) (30)

where i is the number of the material in the joint, nn refers to the nanotube or fiber network, and im refers to the pristine

matrix material properties. The bounds of modulus of elasticity and the Poisson ratio of the network-reinforced material are

subsequently found from the second Eq. (29) .

2.2.5. Validity of using effective material properties in fracture problems

The solutions presented above rely on the homogenization and effective properties of the material reinforced by parti-

cles or random fibers or nanotubes. These properties can successfully be employed in such macromechanical problems as

finding displacements and stresses at the macromechanical level, buckling and dynamics. However, accounting for the ef-

fect of stiff inclusions in fracture problems may become problematic if the size of the inclusions exceeds the characteristic

dimensions of the crack. For example, a related problem of the interaction between the crack surrounded with a numerous

microcrack array was considered in Montagut and Kachanov (1988 ). It was found that the effective stiffness representing the

averaged over the representative volume element properties of the material with microcracks cannot be accurately apply to

the evaluation of such quantities as the stress intensity factors. These factors appeared to be sensitive to local geometry in

the vicinity to the crack tip. The interaction between the main crack and a surrounding “cloud” of microcracks was further

rigorously investigated in Kachanov, Montagut, and Laures (1990 ). In this paper, it was demonstrated that the microcracks

close to the tip of the main crack dominate its stress intensity factor. Furthermore, the result was sensitive to the exact posi-

tion of the short-range microcracks, i.e. effective stiffness properties could not be safely applied to the analysis. The removal

of microcracks located at a distance exceeding 1.5 times the microcrack radius from the main crack tip produced negligible

effects on the stress intensity factor of the main crack. Other results demonstrating the inadequacy of the effective stiff-

ness modeling in fracture problems can be found in references cited in Kachanov et al. (1990 ), including switching between

shielding and amplification effects dependent on the use of effective stiffness versus direct integration methods. Based on

the results in Curtin and Futamura (1990 ) who employed a two-dimensional spring model, using the effective stiffness to

predict crack shielding is incorrect since the material in the immediate vicinity to the crack tip possesses the properties of

the matrix, rather than the effective properties of the material. The latter paper also stressed the importance of using the

exact distribution of microcracks and a possible statistical approach to the problem.

While the limitations of the effective properties approach in fracture problems are discussed in the previous paragraph

on the example of microcracks interacting with the main crack, it is obvious that a replacement of microcracks with stiff

inclusions should not alter the conclusions from the cited papers. Accordingly, it is advisable to assess the limitation of the

effective properties approach (e.g., such methods as those of Mori–Tanaka, self-consistent, etc.). It is hypothesized here that

the main reason for the inconsistent or inaccurate results suspected in fracture problems using the averaged heterogeneous

material properties is related to the scale mismatch. The effective averaged properties should result in accurate solutions if

the modelled phenomenon occurs at a larger scale than the scale of the representative volume elements used to derive these

properties. However, if the scale of the investigated phenomenon is smaller than the scale of the representative volume

element, the averaging procedure becomes inaccurate, e.g., the crack tip may be surrounded by the pristine unreinforced

material or in the contrary, it may encounter the reinforcing inclusion, rather than a medium with averaged properties.

In general, in the situations where the scale of the fracture problem is smaller than the scale employed in the averaging

method, the analysis should rely on statistical mechanics, as was also suggested in the above-mentioned references.

V. Birman / International Journal of Engineering Science 130 (2018) 157–174 167

The hypothesis introduced in the previous paragraph leads to the following suggested limitation of the averaging tech-

niques applicability in fracture problems: “The effective properties methods are valid in fracture problems only if the char-

acteristic dimension of the averaging model is smaller than the characteristic dimension of the crack tip.”

Let us specify the characteristic dimensions that could be considered to adopt the effective properties methods to fracture

problems. In the case of spherical particles embedded in the material, the distance between the boundaries of the adjacent

particles can be used as the characteristic dimensions. In a square particle array, this distance is the following function of

the particle radius R and its volume fraction V p :

d = R

(√

π

V sp − 2

)(31)

There is extensive data on the sizes of spherical or near-spherical sizes of nanoparticles. For example, the radius of

silicon nanoparticles was found in the range between 40 and 80 nm ( Baryshnikova, Petrov, Babicheva, & Belov, 2016 ). The

radius of carbon nanoparticles was reported in the order of 1 nm ( Rubtsov, Ten, Pruuel, & Kashkarov, 2016 ), but it can be as

large as hundreds nanometers, dependent on the manufacturing process ( Makdessi et al., 2017 ). The mean radius of silver

nanoparticles was measured at 50 nm, with the standard deviation of 4 nm ( Sokolov, Batchelor-Mcauley, Tschulik, Fletcher,

& Compton, 2015 ). Assuming that the volume fraction remains within the range from 1% to 10%, the range of possible

distance-to-radius ratio values is 3 . 605 ≤ d R ≤ 17 . 725 .

One of characteristic dimensions at the crack tip can be adopted, including the crack tip opening displacement (CTOD)

or even more conservatively, only its elastic component, the size of the plastic zone and the size of the contact zone be-

tween the faces of the crack, though using the latter factor may be somewhat superficial. For example, Zhuang, Yi, and

Xiao (2013 ) considered a substrate crack parallel to the interface with a film or coating and demonstrated that the crack tip

opening displacement varied from that calculated for the substrate material without a film effect by a factor between 0.5

and 3. Therefore, the order of magnitude of the crack tip opening displacement can be estimated using the properties of the

nanoreinforced substrate material. For mode I fracture, the elastic component of CTOD is found by the following formula

( Zhuang et al., 2013 ):

d 0 =

πa (σ 2

y

)σyield E s

(32)

where a is half-length of the crack, σ y is the applied stress acting perpendicular to the crack, and σ yield and E s are the yield

stress and the modulus of elasticity of the nanoreinforced substrate material, respectively. Denoting σy / σyield = k, k < 1 this

expression can be rewritten as

d 0 = πa k 2 (σyield

E s

)(33)

The yield stress of a polymer composite reinforced by spherical nanoparticles can be determined using one of the

strength theories cited in Fu, Feng, Lauke, and Mai (2008 ). One of the conclusion in this paper based on the study of

micro-composites Li, Helms, Pang, & Schulz, 2001 ) was that the tensile strength increases as the size of the particle de-

creases. The same conclusion was obtained for nanocomposites (e.g., Zhang & Chen, 2006 ), i.e. smaller nanoparticles result

in an enhanced strength. Furthermore, the changes in the strength were relatively small if the volume fraction of particles

remained limited to less than 10%. Accordingly, we conservatively assume that adding nanoparticles to the substrate has a

small effect on the strength that can be disregarded (if the increased strength was accounted for, the elastic CTOD calculated

by ( (33) would be larger supporting the use of the effective properties approach).

Consider for example, an epoxy substrate with embedded silica nanoparticles ( Tzetzis, Tsongas, & Mansour, 2017 ) where

the modulus of elasticity was found equal to 4.58 GPa and 5.81 GPa for the 15 wt% and 25 wt% nanoparticle silica, re-

spectively. The corresponding yield stress values were measured as 82 MPa and 98 MPa. Accordingly, the elastic CTOD are

calculated as d 0 = 56 . 2 a k 2 mm ( 15 wt% ) , d 0 = 52 . 9 a k 2 mm ( 25 wt% ) where the half-length of the crack is measured in me-

ters. If the crack is 0.1 m long and the stress ratio k = 0.1, the corresponding elastic CTOD are equal to 0.0562 mm and

0.0529 mm for the weight volume fractions of 15% and 25%, respectively. The corresponding particle volume fractions are

approximately 0.0708 and 0.118, respectively. While the size of nanoparticles was not reported in the above paper, using

the value of R = 50 nm, the distance between the particles is estimated as 233 nm and 158 nm for the weight volume frac-

tion equal to 15% and 25%, respectively. Obviously, the distance between nanoparticles is very small compared to the elastic

CTOD, justifying the use of the effective property method in this case.

The choice of a characteristic dimension for random nanotubes justifying the use of effective property approach is less

evident as compared to the case of spherical nanoparticles. It is suggested to consider a cubic cell with an embedded

nanotube and use the length of the side as the characteristic dimension. Accordingly, this dimension is found as

a =

3

√

V o l nanotube

V n (34)

where the numerator and denominator of the ratio in the right side are the volume of a single nanotube and the vol-

ume fraction of nanotubes, respectively. For example, if a nanotube has the radius of 1 nm and the length of 10 0 0 nm, the

168 V. Birman / International Journal of Engineering Science 130 (2018) 157–174

Fig. 4. Boundaries of the shear moduli ratios separating the cases of singular and non-singular stresses at the edge of the interface between two isotropic

materials. The case of plane stress is considered.

characteristic dimension is a = 31 . 6 nm , i.e. it is much smaller than the elastic CTOD justifying the use of the effective prop-

erties method. Note that as follows from Eq. (34) , increasing the nanotube volume fraction results in a smaller characteristic

dimension and a more justifiable use of the effective properties method, as could be expected.

3. Numerical examples

Example 1: Prevention of the edge stress singularity at the interface between two isotropic materials)

The boundary ratios of the shear moduli of two materials in a butt joint that separate singular and non-singular stresses

at the edge of the interface depicted in Fig. 1 are shown in Fig. 4 (plane stress) and Fig. 5 (plane strain). The range of the

Poisson ratios of the connected materials is between 0 and 0.5, excluding auxetic materials. The stress singularity is avoided

if the ratio of the shear moduli remains below the boundary curves in Figs. 4 and 5 .

As follows from Figs. 4 and 5 , in both plane stress and plane strain cases the mismatch between the stiffness of two

materials can be larger, without causing fracture, if the Poisson ratio of the compliant material is small. In the extreme, if

the Poisson ratio of the compliant material (material 2) approaches zero, the stress singularity can be avoided for practically

any stiffness ratio. On the other hand, if the compliant material is incompressible, its Poisson ratio being equal to 0.5, the

shear stiffness mismatch of two materials that prevents stress singularity is quite small. The mismatch between the stiffness

corresponding to the boundary of singular stresses increases for smaller values of the Poisson ratio of the stiffer material

(material 1). In the case of plane strain and two incompressible materials, the stiffness ratio is equal to 1.0, i.e. either the

materials in the butt joint or their mechanical properties have to be identical.

It is instructive to analyze the results in Figs. 4 and 5 in light of the plausibility of avoiding the stress singularity in

representative material joints. The following examples refer to the pairs of materials considered in literature concentrating

on the feasibility of avoiding the stress singularity by embedding nanotubes or nanotube networks in the compliant material.

Example 2: Edge stress singularity at the nickel–aluminum oxide ceramic-metal interface ( Williamson, Rabin, &

Drake, 1993 )

Using the Poisson ratios of these materials, the boundary between singular and non-singular shear stiffness should

be at G 1 / G 2 = 1 . 18 and G 1 / G 2 = 1 . 24 in the cases of plane stress and plane strain, respectively. In reality, G 1 / G 2 =4 . 22 implying the stress singularity. An attempt to stiffen the more compliant aluminum oxide with 3-D random

carbon nanotubes is unsuccessful. For example, using the properties of single wall nanotubes listed in Lu (1997 )

( E = 974 GPa , G = 465 GPa , ν = 0 . 28 ) in conjunction with the 3-D Christensen–Walls model, the shear stress ratio is

G 1 / G 2 = 1 . 61 , even if the volume fraction of nanotubes is 20%.

Consider now using networks of nanotubes, such as those analyzed in Chen et al. (2015 ), where the ratio of the modulus

of elasticity of the network to the elastic modulus of carbon nanotubes was obtained both analytically and by the finite

element method as a function of the relative network density (e.g., Fig. 17 of the referred paper). Evaluating the network

modulus of elasticity and subsequently replacing the fiber modulus in the 3-D Christensen–Walls model with the network

V. Birman / International Journal of Engineering Science 130 (2018) 157–174 169

Fig. 5. Boundaries of the shear moduli ratios separating the cases of singular from non-singular stresses at the edge of the interface between two isotropic

materials. The case of plane strain is considered.

modulus, we obtain unacceptably high shear stiffness ratios G 1 / G 2 for the entire spectrum of feasible network densities. Ac-

cordingly, it is concluded that the edge stress singularity in nickel–aluminum oxide joints cannot be avoided by embedding

stiff randomly oriented inclusions (e.g., nanotubes) in aluminum oxide due to a large mismatch in the stiffness of the joined

materials.

Example 3: Edge stress singularity at the aluminum-copper interface ( Gundrum, Cahill, & Averback, 2005 )

The mismatch between the properties of these two materials is smaller than in example 2, but it is still large, so that

G 1 / G 2 = 1 . 70 while the boundary shear stiffness ratios are G 1 / G 2 = 1 . 02 and G 1 / G 2 = 1 . 03 for plane stress and plane strain,

respectively. Adding random carbon nanotubes to the more compliant cooper is an established technology ( Daoush, Lim, Mo,

Nam, & Hong, 2009 ), but it was found that at high volume fractions of nanotubes their embedding in the matrix may result

in a lower material strength. Experimentally measured elastic modulus of the nanotube reinforced copper for the nanotube

volume fraction equal to 20% reported in Daoush et al. (2009 ) was equal to 105.9 GPa. Although it is impossible to compare

this result with the stiffness evaluated by the Christensen–Waals method since the elastic modulus of nanotubes was not

reported in Daoush et al. (2009 ), using the nanotube properties listed in Example 2, one obtains a slightly larger value

for 3-D random nanotube reinforced copper, i.e. 114.4 GPa. A discrepancy of about 7% may be explained by manufacturing

defects and a possible waviness of nanotubes. Using the stiffness evaluated by the Christensen–Waals method, the shear

moduli ratio is equal to G 1 / G 2 = 1 . 06 , i.e. it is close to the boundary between singular and non-singular stresses. Increasing

the nanotube volume fraction to 22% results in the shear moduli ratio overlapping the boundary curve in the case of plane

strain, i.e. eliminating the stress singularity.

Example 4: Edge stress singularity at the ceramic clay-ceramic clay interface ( Banks-Sills et al., 20 0 0 )

As is evident from the previous examples, the elimination of the stress singularity is challenging if the connected materi-

als have a large stiffness mismatch. An example of the interface between the materials with a very small stiffness mismatch

where the stress singularity does not possess a fracture problem even without a reinforcement of the compliant material

is found at the interface between two ceramic clays (K-142 and K-144). The difference between the stiffness of the clays

is small, G 1 / G 2 = 1 . 19 , while the boundary between singular and non-singular solution corresponds to G 1 / G 2 = 1 . 34 and

G 1 / G 2 = 1 . 45 for plane stress and plane strain, respectively. Therefore, the edge stress singularity does not occur at the

interface between these two clays.

Example 5: Edge stress singularity at the alumina-silicon carbide (SiC) interface

Layered SiC and alumina systems may find an application in coatings combining high hardness of the former with duc-

tility of the latter ( Bhushan, Gupta, & Azarian, 1995 ). The mismatch between the properties of these materials is not very

large, but the edge stress singularity is still present in the case of plane stress, the boundary shear moduli ratio being equal

to G 1 / G 2 = 1 . 41 for plane stress and G 1 / G 2 = 1 . 50 for plane strain, while the actual ratio is G 1 / G 2 = 1 . 45 . The solution to

the problem is achieved by adding a small volume fraction of random nanotubes to alumina. Even at 2% nanotube volume

fraction, the shear moduli ratio is reduced to G 1 / G 2 = 1 . 40 eliminating the singularity in both plane stress and plane strain

problems.

170 V. Birman / International Journal of Engineering Science 130 (2018) 157–174

h/H=0.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

E2/E1

Gn

h/H=0.5

Gn

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

6

E2/E1

h/H=0.7

Gn

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.902468

1012141618

E2/E1Fig. 6. Reducing a mismatch between the stiffness of two materials decreases the nondimensional strain energy release rate G n for all ratios of the interface

location h / H . The difference between the scales of G n (vertical axis) for different ratios h / H necessitates using three charts.

Example 6: Reduction of the strain energy release rate for a semi-infinite crack (four-point flexure test, Fig. 2 )

The effect of stiffening the complaint material resulting in a reduction in the strain energy release rate is demonstrated

in Fig. 6 for three different interface locations η = h/H. The nondimensional strain energy release rate is defined as

G n =

G E 1 h

3

M

2 =

(6 η3 − 1

2 I

)E 1

E (35)

2

V. Birman / International Journal of Engineering Science 130 (2018) 157–174 171

The stiffness of the stiffer material 1 was assumed constant while the change in the stiffness of the compliant material

2 can be achieved by adding random reinforcements. As follows from Fig. 6 , even a modest reinforcement of the compliant

material 2 results in a significant reduction to the strain energy release rate, particularly at small ratios E 2 / E 1 , for all values of

h / H . In particular, the feasibility of large variations in the relative stiffness embedding random nanoparticles in the compliant

material is evident from Examples 2 and 3 where adding 20% volume fraction of nanotubes to aluminum oxide and cooper,

respectively, caused a drastic change in the relative stiffness.

4. Conclusions

The paper elucidates a possible improvement of the fracture resistance at the interfaces between two dissimilar isotropic

materials by reducing the mismatch between their stiffnesses. The goal can be achieved by adding stiff random inclusions to

the more compliant material. A less feasible alternative is adding compliant inclusions to the stiffer material. The problems

considered in the paper include the elimination of the interface edge stress singularity in a butt joint and a reduction of

the strain energy release rate for a semi-infinite interfacial crack (e.g., a four-point flexure test). The approach to a reduction

of the strain energy release rate for substrate cracks parallel to the interface with a thin film undergoing residual thermal

stresses is generalized by adding random inclusions to the film or substrate and accounting for the effect of temperature

on the engineering constants of the film, substrate, and inclusions as well as the history of thermal loading. Random rein-

forcements affecting the stiffness of the complaint isotropic material include fibers or nanotubes, ellipsoidal (and spherical)

inclusions and random fiber or nanotube networks.

Reducing the mismatch between the stiffness of two materials in a butt joint by reinforcing the complaint material with

random reinforcements may eliminate the interface edge stress singularity in both plane stress and plane strain cases. The

stress singularity is also affected by the Poisson ratio of the compliant material and, to a lesser extent, by the Poisson ratio

of the stiffer material. In particular, the mismatch between the stiffness of the materials in the joint can be much higher

without triggering stress singularity if the Poisson ratio of the compliant material is small.

The applicability of the effective properties approach to the evaluation of the effect of reinforcing nanoparticles or nan-

otubes on fracture has been discussed. It is hypothesized that homogenization using the effective properties is acceptable

if the characteristic dimensions of the reinforced material are smaller than such characteristic dimensions of the crack as

the elastic crack tip opening displacement. It is demonstrated that such condition is satisfied in representative nanoparticle

and nanotube-reinforced materials, though the conclusion may be reversed in case of microscopic, rather than nanoscopic,

reinforcements.

Numerical examples presented for several material combinations were concerned with the interface edge stress singular-

ity. It is shown that if a mismatch between the properties of the connected materials is large, reinforcing the compliant ma-

terial, even using stiff carbon nanotubes or their networks, may be insufficient to eliminate the stress singularity. However,

if the difference between the properties of the connected materials is moderate, the stress singularity can be eliminated.

Remarkably, in the example where the moduli of elasticity of the materials in the joint differed by about a third, adding

just 2% volume fraction of random carbon nanotubes was sufficient to prevent stress singularity in both plane stress and

plane strain problems.

As was demonstrated in the problem of a semi-infinite crack along the interface between two isotropic materials (e.g., a

four-point flexure test) even a moderate increase in the stiffness of the compliant material results in a significant reduction

in the strain energy release rate. This conclusion remains valid irrespectively of the interface location between the stiffer

and compliant materials.

In conclusion, embedding randomly oriented fibers, nanotubes, ellipsoidal or spherical inclusions and fiber or nanotube

networks in a more compliant material results in a smaller mismatch between the stiffness of the materials in the joint. This

approach has a potential to either alleviate or eliminate fracture in such problems as the interface edge stress singularity

or a semi-infinite crack propagating along the interface. The problem of the effect of random reinforcements on a substrate

crack parallel to the interface with a thin residually stressed film-substrate interface has also been formulated accounting for

the history of thermal loading and the effect of temperature on the properties of the film, substrate and random inclusions.

Detailed numerical analysis of the latter problem will be the subject of the future investigation.

Acknowledgment

Discussions with Professor Guy M. Genin (Washington University, St. Louis, Missouri, USA) are warmly appreciated.

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijengsci.2018.05.

011 .

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