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A review of the effects of weak interfaces on crack propagation in rock: from
phenomenon to mechanism
Minghe Ju, Xiaofeng Li, Xing Li, Guanglei Zhang
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DOI:
Reference:
S0013-7944(22)00061-3
https://doi.org/10.1016/j.engfracmech.2022.108297
EFM 108297
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Engineering Fracture Mechanics
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2 September 2021
20 January 2022
31 January 2022
Please cite this article as: Ju, M., Li, X., Li, X., Zhang, G., A review of the effects of weak interfaces on crack
propagation in rock: from phenomenon to mechanism, Engineering Fracture Mechanics (2022), doi: https://
doi.org/10.1016/j.engfracmech.2022.108297
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A review of the effects of weak interfaces on crack propagation in
rock: from phenomenon to mechanism
Minghe Jua,b, Xiaofeng Lic* , Xing Lid, Guanglei Zhange
aState
Key Laboratory for Geomechanics & Deep Underground Engineering, China University of Mining
& Technology, Xuzhou 211116, China
bDepartment
of Civil Engineering, Monash University, Clayton, VIC 3800, Australia
cDepartment
of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hongkong,
China
dSchool
of Civil Engineering, Institute of Future Underground Space, Southeast University, Nanjing
211189, China
eDepartment
of Earth Science and Engineering, Imperial College London, London SW7 2BP, UK
Abstract: This article reviews the effect of weak interfaces on crack propagation in rock and
some other analogous brittle materials from phenomenon to mechanism. In this review, a
variety of fracture patterns (in terms of a propagating crack interacting with a pre-existing
interface) are classified, followed by the introduction of some critical influence factors, e.g.
strength and toughness of interface, interface angle, interlayer effect, elastic mismatch, dynamic
loading, and confining stress. Then, the effects of weak interfaces on the quasi-static and
dynamic mechanical behaviors of rock are briefly reviewed. Thereafter, the mechanism of
action of weak interfaces on crack propagation in rock is discussed in terms of crack deflection
criteria. Finally, some conclusions and outlooks on the research of the effects of weak interfaces
on crack propagation in rock are provided.
Keywords: Weak interface; crack propagation; heterogeneous rock; dynamic loading; crack
deflection criterion
*
Corresponding author: Xiaofeng Li (Email: xfli@whrsm.ac.cn; Tel: +86 19851538755)
1 / 51
1 Introduction
Weak rock structures, e.g. grain boundaries, defects, joints, and faults, are omnipresent in
rock at different scales (Fig. 1) [1-8]. As shown in Fig. 1, at the mineral scale, rock is a
microscopic heterogeneous material composed of diverse minerals and pre-existing defects (e.g.
microcracks, voids, cleavages, and grain boundaries) [7, 9-12]. The granitic rocks present a
higher index of heterogeneity than the sedimentary rocks (e.g. sandstone and limestone) at
millimeter-scale [7]. With respect to the isotropic characteristics of homogeneous rock, the
fracturing behavior of heterogeneous rock is significantly affected by the geometrical features
of mineral grains [9-12]. As shown in Fig. 1, the intrusion of dissimilar geomaterials (rock veins,
dykes, and sills) is also widely encountered in rock [13, 14]. These dissimilar inclusions could
remarkably change the failure pattern of a rock structure. Furthermore, in the exploitation and
utilization of underground resources and energy, the path of hydraulic fluid-driven cracks in
jointed rock masses may be disturbed by discontinuous geological structures [7, 15-17].
Drilling rig
Volcano
Not to scale
Rock mass
Dissimilar
inclusion
Crack deflection
Fault
Underground
cavern
Hydraulic
fractures
Dyke
Magma chamber
σv
Dynamic
load
Rock veins
Natural fractures
Granite
2 mm
σh
Natural fractures
Crack
penetration
Sill
Well
Bedding
planes
Mineral
grains
Inclusion in
granite
Rock
veins
Inclusion in
rock mass
Hydraulic fracture
propagation
Blasting in
jointed rock
Figure 1. Different types of weak rock structures at various scales [5-7, 17].
The trajectory of a propagating crack may be altered when one of the following two
scenarios is encountered: (i) a change in loading state, and (ii) the presence of a pre-existing
interface and defect [18, 19]. In this review, the term “interface” is used to include different
stress and displacement conditions of a rock fracture (a non-welded fracture: stress continuity
and displacement discontinuity, and a welded fracture: stress continuity and displacement
continuity) [20]. Loading-induced crack kinking occurs commonly in rock materials and rock
2 / 51
engineering, e.g. crack kinking from the tip of an inclined flaw towards the direction of
maximum principal stress [21], hydraulic fracture propagation in different in-situ stress states
[17, 22]. “Crack kinking/deflection” denotes crack propagation off its original path due to the
change in local fracture mode mixity (a mechanical factor) or fracture toughness (a material
factor) [19]. In a mixed loading state, cracks tend to propagate along the path of least resistance
and kink into a pure Mode-I crack in brittle rock materials [21, 23]. However, pre-existing
interfaces may change this sequence on account of their inherent weakness. The manifestation
of pre-existing interfaces on crack propagation is more intricate, not only in the elusory
fracturing network [3, 24], but the mechanical behavior of materials/structures may also be
altered [9, 10].
As with the influences of bedding planes, joints, weak zones etc. in a rock mass (Fig. 1),
the originally isotropic rock matrix exhibits significant anisotropic mechanical and physical
behaviors [25, 26]. These all are attributed to the disturbance of weak tectonic structures on the
rock stress and deformation fields [26, 27]. If subjected to external static/dynamic loads or inner
pore pressures, cracks will initiate in the rock mass and interact with such weak structures. In
that case, the trajectories of original cracks are distorted into a dispersed fracture network,
which can significantly influence the stability of underground structures and efficiency of
hydraulic performance [28]. These behaviors are related to the magnitude of the applied loads
and pre-existing crack distributions [11, 29].
In this literature review, a variety of fracture patterns (in terms of a propagating crack
interacting with a pre-existing interface) are classified, followed by the introduction of some
key influence factors, e.g. strength and toughness of interface, interface angle, interlayer effect,
elastic mismatch, dynamic loading, and confining stress in Section 2. Thereafter, in Section 3
the effects of weak interfaces on the quasi-static and dynamic mechanical behaviors of rock are
reviewed. The mechanism of action of weak interfaces on crack propagation in rock is discussed
in terms of crack deflection criteria in Section 4. Finally, some conclusions from, and outlooks
for, the research into the effects of weak interfaces on crack propagation in rock are
demonstrated in Section 5. This review work gives a detailed summary upon the effects of weak
interfaces on crack propagation in rock or rock-like materials/structures, and is of significance
for the comprehensive understanding of different fracture patterns influenced by weak
interfaces.
2 Effects of weak interface on crack propagation in rock
3 / 51
2.1 Crack propagation pattern
The problem of a crack approaching an interface between two dissimilar materials is
fundamental to the understanding of the behavior of non-homogeneous materials [30]. For two
materials in contact (Materials A and B), when a plane crack initiates under external load in
Material B, it propagates and interacts with the pre-existing jointed interface, on account of the
relative locations of the propagating crack and pre-existing interface: a series of fracture
patterns are classified as shown in Fig. 2. It should be noted that the classifications of different
fracture patterns are not limited to rock materials or structures but some other brittle and
composite materials and structures are also included due to certain similar fracturing
performances. Generally, there are four types of primary positions of the crack relative to the
interface: (i) the crack makes close perpendicular contact with the interface; (ii) the crack
contacts with the interface directly at an oblique angle; (iii) the crack normal to the interface
leaves a ligament between the crack-tip and interface; (iv) the crack is obliquely ahead of the
interface.
Perpendicular crack
(a)
B
A
Crack
Interface
(b)
B
Direct
penetration
Single
deflection
Double
deflection
Debonding
head
Shearing
head
Penetration
head
Direct
penetration
Single
deflection
Double
deflection
Curve
away
Curve
toward
Mixed
cracking
A
Crack
Interface
Inclined crack
(c)
B
A
Crack
Interface
(d)
B
Mixed
cracking
A
Crack
Interface
Figure 2. A brief summary of crack propagation patterns when approaching a bimaterial
interface. (a) and (b) are the types of a propagating crack interacting perpendicularly with
a pre-existing weak interface [31-35], while (c) and (d) are scenarios in which an inclined
crack propagates to the interface [34, 36, 37].
4 / 51
The most popular condition is shown in Fig. 2a where the crack interacts perpendicularly
with the interface. In the early stages, the T-shaped double deflection of laminated structures
was studied as it is common in composite failures [31]. Lu and Erdogan [32] summarized the
general cracking patterns of a crack normally contacting an interface, which are direct
penetration, double deflection, and mixed double deflection and penetration. These are the
widely encountered fracture patterns in brittle materials and structures [33]. Later, He and
Hutchinson [34] theoretically found that single deflection is also one of the typical fracture
patterns. From their systematic quantifications of double deflection, single deflection, and
direct penetration, single deflection was ignored on account of its low possibility of occurrence
when the mechanical properties of the two contacting materials are similar. After that, the
investigations on crack deflection and penetration at an interface were expanded across broad
areas of interest including: fiber reinforced and layered composites [38-40], laminated ceramic
composites [41-44], fluid transmission in the geological layered structures [45-49], as well as
crack propagation in layered rock structures [3, 50] (with some examples represented in Fig.
3).
(a)
(b)
Sill
Dyke
(c)
(d)
Double
deflection
Single
deflection
Direct
penetration
Double
deflection
Figure 3. Some cases of crack normal interaction with a weak interface/contact. (a) A video
record of the process of the deflection of a primary crack at a bi-layer interface in a model
system made of brittle rubber blocks [43], (b) different styles of intrusive behavior of dyke
and sill formation [45, 51], (c) T-shaped fracture between two stiff limestone layers [50],
(d) in situ photographs of direct penetration, double deflection, and single deflection in
layered sandstone [3].
5 / 51
Some researchers proposed that direct contact of the crack with an interface is a critical
condition which happens only when the interface is strong enough [35, 52-55]. Normally, if
with a weak interface, the interface should first debond when the propagating crack is about to
reach it, which is attributed to the specific stress field around the crack-tip. Cook and Gordon
[35] primarily explored different debonding patterns of the interface ahead of the crack-tip, with
the typical patterns of interface debonding in tension and shear modes shown in Fig. 2b.
Leguillon and Martin conducted a series of studies into interface debonding ahead of the crack
tip based on the findings in [35] and [34]. They systematically discussed the requirements for
each debonding state [52, 53, 56]. In addition of the patterns in [35], they found that penetration
in the target matrix ahead of the crack-tip is also possible [56-58]. Moreover, a more
complicated pattern of bifurcated penetration ahead of the crack-tip was also detected [40, 59].
Interface debonding ahead of the crack-tip was also observed in some experiments [60] and
numerical simulations [59]. These essential findings have been used in rock engineering for the
study of the interactions between hydraulic fractures and natural defects [36, 61], dyke and sill
formations [62-64], as well as layered and grain-based rock fracturing [3, 65].
The condition of a crack normal to the interface is a particular case which is somewhat
ideal in the process of crack propagation. When the propagating crack interacts closely with the
interface, at an inclined angle, as illustrated in Fig. 2c, the fracture patterns are slightly different
from those in Fig. 2a. The cracking path is no longer symmetrical and the specific profile is
related to the material properties [34, 36, 66-68]. These manifestations make the prediction of
crack trajectory more difficult but practically meaningful in the investigations of sill formation
in conditions affected by pre-existing structures [69-71], injection-induced faulting and
earthquakes [69, 72], and crack propagation in a jointed rock mass [73, 74].
The most complicated cases of cracking pattern should be those in Fig. 2d where the
inclined straight crack would curve away from, or towards, the interface depending on the
relative properties of materials A and B [34, 37]. Many studies have explored this phenomenon
as it dominates the failure of grain-based or laminated materials/structures [37, 75-80] where
the crack is attracted towards the soft materials while deviated from hard materials.
2.2 Influence factors
The cracking network in a structure with massive weak interfaces is the cumulative result
of a couple of fracture patterns shown in Fig. 2. Without the weak interfaces, crack propagation
in a quasi-homogeneous structure is predicated by the stress state. However, the defined crack
paths are disturbed causing the global failure of a structure unclear as the existence of weak
6 / 51
interfaces [81, 82]. Many factors may affect the local fracture pattern when a crack approaches
an interface, e.g. strength and toughness of each phase, geometrical properties of each
component, as well as the loading state.
(1) Strength and toughness
The strength or toughness of an interface plays a significant role on the ultimate fracture
pattern at an interface or the global failure pattern of a layered or grain-based structure. It has
been found in some experiments [83] and simulations [77, 84, 85] that with the increase in
strength or toughness of interface, the fracture pattern at an interface may shift from double or
single deflection to direct penetration. Not only given changes in the mechanical properties of
interface, changing the strength or toughness of substrate material, the fracture manifestation
will also be changed. For instance, as the matrix strength decreases, the fracture pattern is
changed from penetrating the interface instead of deflecting into the interface [86]. These local
changes of fracture pattern would affect the subsequent fracturing performance. As shown in
some laboratory tests [87, 88] and numerical simulations [89-91], by increasing the strength or
toughness ratio between the interface (or grain boundary) and matrix (or crystalline grain), the
percentage of crack deflection decreases and the fracture path becomes better defined (Fig. 4).
(a)
Crack propagation
(b)
Figure 4. (a) Intergranular and transgranular cracking performance in a grain-based brittle
material with respect to the ratio of fracture energy between the grain boundary and
crystalline grain (Rm = ГIG/ГTG). The roughness of the main fracture path is lower at a
greater Rm [90], (b) interface strength on crack propagation in laminated structure where
the interface strength decreases from left to right [87].
(2) Interface angle and grain size
Crack propagation is not only affected by the relative mechanical properties of the
interface and substrate material, but the geometrical properties of the interface also exert a
significant influence on the ultimate fracture pattern. Numerous experimental and numerical
7 / 51
studies have been conducted on this topic. With respect to the propagating crack normally
towards the interface, the inclined crack is likely to deflect into the interface [92]. Generally,
the interface debonding length decreases with the increase of interface angle [84, 93-96].
Sometimes, at a certain strength or toughness of the interface, there is a transition of the
cracking pattern from single deflection to double deflection [97] or from single deflection to
direct penetration with increasing interface angle (with some experimental observations, as
shown in Fig. 5a and b) [68, 94, 96, 98-100].
(a)
(c)
(b)
Small grain size
1 μm
1 μm
Large grain size
Figure 5. Some examples of the effect of interface angle on crack propagation. (a) An
experimental study of crack deflection at an interface [68], (b) microscopic observation of
crack deflection in a brittle material [98], (c) a phase-field modeling of grain size-effects
on crack propagation [101].
The influence of grain size on crack propagation is attributed to the intersection angle
between the grain boundary and the propagating crack. This is similar to the effects of weak
interfaces in layered structures [101, 102]. As shown in Fig. 5c, when the propagating crack
interacts with a large grain, if the direction of the propagating crack is determined, the
intersection angle between the grain boundary and the crack is greater than that with a small
grain. In addition, the interfacial crack will deflect into the target grain after a certain length of
interface debonding. Hence, transgranular cracking is more likely in the grain-based
heterogeneous structure with a larger grain size as suggested by the aforementioned two reasons.
Such a manifestation is also reflected in laminated structures (Fig. 5a and b) in which interface
debonding within a certain distance occurs followed by crack deflection into the substrate. The
influence of grain shape on crack propagation is the same (in principle) in terms of the
8 / 51
interaction angle between the crack and the tangent line at the contacting point. The material is
vulnerable to crack-penetration through the grain boundary for strip-like grains inclined at a
large interface angle [103].
(3) Interphase/interlayer
In practice, the interface between two main materials or structures sometimes cannot be
clearly identified, instead, a thin interlayer structure is sandwiched between the two main layers.
This thin interphase plays a more complicated role in the crack propagation than a single
interface [104]. Some typical profiles of the interphase in Fig. 6 indicate that the propagating
crack enters the interphase, the original direction of crack propagation is disturbed. Such local
disturbance causes the subsequent and global failure of a structure to be erratic. The interphase
can be regarded as a thin layer, dissimilar to the substrate, if it is relatively homogeneous (Fig.
6a). Thus, the crack propagation patterns relating to the geometrical and mechanical properties
of material as well as loading state can still be found from those in Fig. 2 [92, 104, 105]. On the
other hand, crack bifurcation in the interlayer may occur if the weak interlayer is thick and the
residual stress (due to the difference of material property between adjacent layers) is significant
[44, 106, 107]. Such weak interlayers cause the step-wise failure of the laminated structure and
the R-shaped apparent toughness evolution [107, 108].
(a)
(b)
(c)
(d)
Figure 6. Different types of interphases in a composite brittle structure. (a) Weak
interphase filled with pervasive hard materials, (b) interphase with a layered discontinuous
structure, (c) multilayer interphase, and (d) porous interphase [104].
If the interphase is inhomogeneous (Fig. 6b-d), e.g. in a discontinuous structure filled with
massive defects and pores analogous to the rock joints or an interphase constituted by a couple
9 / 51
of sub-interlayers, the crack trajectory in the interphase is dispersive [3, 104]. Even so, the local
competition between crack deflection and crack penetration at a weak interface is still satisfied
but the overall failure is fairly random. In addition, illustrated in Fig. 6, such weak interphase
behavior can change the crack path and often trap the propagating crack. This makes it possible
to use the interphase to protect certain vulnerable structures [83, 109, 110].
(4) Elastic mismatch
The dissimilar elastic properties between two contacting materials exert a significant
influence on crack propagation at an interface (Fig. 7). Dundurs [111] proposed two parameters
that determine the degree of elastic mismatch between two adjacent materials. Such elastic
mismatch manifestation forces the energy distribution around the crack-tip to distort when
approaching an interface [4, 112]. The possibility of deflecting into, or penetrating through, the
interface is correlated to whether the crack propagates from stiff or compliant material [4, 75,
77], where the crack is likely to deflect into the interface if propagating from a compliant
material to a stiff one (EB < EA) [45, 70, 101, 113]. In addition, the fracture patterns of curving
towards or curving away from the interface (Fig. 2d) are attributed to the elastic mismatch
between two materials if the crack is under Mode-I loading state. Some tests showed that the
propagating crack curves to the direction of interface when approaching the interface from a
compliant material to a stiff material [75, 114]. On the contrary, the crack directly penetrates
the interface if propagating from stiff to compliant materials. Such a difference is also indicated
from the stress intensity evolution of the crack-tip from stiff to compliant material as this case
is quite different from that in an inverted direction of crack propagation [115].
10 / 51
(a) 14
(b)
12
Dyke
Sill
Arrested dyke
Dyke-sill hybrid
Dyke
P0/PfU
10
Limestone
Sill
8
6
Marl
Interface
4
2
0
0.1
1
(c)
Eu/El
10
100
Limestone
Eh
Em
Em
Figure 7. Elastic mismatch of two contacted layers on crack propagation. (a) An
experiment showing different patterns of dyke and sill formation [45], (b) crack
propagation in a layered rock structure with marl sandwiched between two limestone layers
(modified from [113]), (c) a phase-field simulation of elastic mismatch of layers on crack
propagation [101].
(5) Dynamic loading
Crack propagation and interaction with a pre-existing interface under dynamic loading
manifest different phenomena from those in quasi-static conditions. If the crack has a low
propagation velocity, when it interacts with a weak interface, the propagating crack may form
T-shaped double deflection patterns and be trapped within the interface; however, with
increasing velocity of crack propagation, the interface debonding length decreases [116] and
the fracture pattern may even transform from deflection to penetration [117]. Essentially,
fracture pattern transition is the reason for fracture-surface roughness reduction when increased
crack-propagation velocity is found in heterogeneous brittle materials [65, 118]. However,
experiments by Sundaram and Tippur [119] suggest that a crack can bifurcate at the interface
even when given a fast crack-propagation velocity. This is on account of the higher magnitude
of tensile stress perpendicular to the interface when the crack is subjected to a greater
propagation velocity. Thus, the tensile stress drives the interface to debond and contributes to
crack deflection [119, 120].
Similar to those quasi-static conditions, the fracture pattern changes from crack deflection
to crack penetration with changing interface angle [18, 121, 122] but multiple cracks may
11 / 51
sometimes form under dynamic loading [123]. Furthermore, a weak interface can restrain the
dynamically propagating crack [18, 124] and the crack deflection length generally increases
with the reduction of interface strength [120, 121, 125]. The same principles as those governing
behavior in quasi-static states should make sense yet the effect of a weak interlayer on crack
propagation under dynamic loading is different from those in quasi-static conditions. When two
similar materials make close contact, but without adhesion (an open joint), the propagating
crack would be arrested within the interface. Nevertheless, the fracture pattern would be
converted from deflection to penetration if the interface is an adhesive layer [24]. Furthermore,
as the interlayer thickness increases, the penetrating mother-crack would branch into a cluster
of daughter-cracks (Fig. 8) [24, 126]. One explanation for this phenomenon is that the reduction
of the interlayer stiffness contributes to crack bifurcation at an interface [127]. These
phenomena give rise to the idea of using the weak interface or interlayer to enhance the
fragmentation intensity of rock.
(a)
5 mm
(d)
5 mm
(b)
(c)
5 mm
5 mm
(e)
5 mm
Figure 8. Crack dynamic propagation through an interface. (a) Monolithic brittle material,
(b) interface without adhesion, (c) interface having a near-zero-thick adhesive layer, (d)
interface with a 0.13 mm-thick adhesive layer, (e) interface having a 2.5 mm-thick adhesive
layer [24].
(6) Confining stress
Confining stress is also one of the significant factors affecting crack propagation at a weak
interface as both the stress fields around the crack-tip and at the interface are changed [128].
The most typical example in geological engineering work is the fluid-driven crack propagation
in jointed rocks, e.g. the interaction between a hydraulic fracture and natural fractures [22, 129,
130], and the formation of dykes and sills in layered rocks [130]. Considering the maximum
and minimum principle stresses (σ1 and σ3), the frictional properties of the interface, and the
interface angle, Blanton [129] proposed a criterion for determining whether the hydraulic
fracture will cross, or be terminated in, the interface based on the hydraulic-fracture
12 / 51
experimental results and this is verified by subsequent experiments and simulations (with some
recent typical cases in [17, 82]). In addition, another classical analysis of confining stress on
crack deflection at a frictional interface was performed by Renshaw and Pollard [131] where
they took the superimposed stress field into consideration (confining stresses and stress field at
a random point around the crack-tip). A modified version of their work was exhibited later
[132]. These analyses considering the frictional characteristics of the interface can predict the
cracking behavior of a hydraulic fracture approaching a natural interface [17, 100, 133].
Generally, in these results, for a given interface angle or coefficient of friction, the fracture
pattern may change from crack penetration to crack deflection with decreasing stress difference
(σ1-σ3).
3 Effects of crack propagation on mechanical property
3.1 Weak interfaces and their effects on rock fracture toughness
Fracture toughness is a quantitative way of expressing the resistance of a material to crack
propagation, which proves the ability of a rock structure to bear load or resist deformation in
the presence of an initiated crack [134]. Fracture toughness is remarkably related to the rock
microstructures, in both of physical and mechanical terms [9, 134-138]. In general, the
microcracking types in a grain-based heterogeneous rock are intergranular cracking along the
grain boundaries and transgranular cracking through the mineral grains in tension or shear mode
[139, 140]. Nasseri et al. undertook a series of experimental investigations of granitic
microstructures on fracture toughness. In their studies, granite is under anisotropic conditions
due to the differences in mineral composition, e.g. quartz and feldspar, in different directions.
Basically, four consequences arise: (i) fracture toughness of granite decreases inversely with
the square root of grain size (KIc~Lg-0.5) [134], which is akin to the principle of uniaxial
compression tests (σc~Lg-0.5) [135, 141, 142]; (ii) fracture toughness monotonically decreases
with pre-existing crack density Dc (KIc~Dc-0.3), crack length Lc (KIc~Lc-0.5), or crack porosity pc
(KIc~pc-0.62) [134, 143]; (iii) fracture toughness linearly increases with fracture surface
roughness θ*/C (KIc~θ*/C) [9, 144, 145], where θ* and C are the apparent dip angle on the
fracture surface and a roughness parameter characterizing the distribution of θ*, respectively
[146]; (iv) fracture toughness gradually decreases with the temperature of thermal treatment,
which may also be attributed to the difference of thermally-induced pre-existing cracks [147].
Moreover, it was found that the anisotropy index of fracture toughness of Barre granite
 (ηai~ K
 -0.06 ) [138, 148], in which ηai is defined as
monotonically decreases with loading rate K
the ratio of maximal fracture toughness to minimal fracture toughness at a determined loading
13 / 51
rate. The rate-dependent decrease of ηai is not only in the grain-based heterogeneous rocks, but
it is also apparent in bedding plane-abundant rocks [149], which is on account of the common
mechanism of pattern transition from interfacial cracking to matrix cracking with loading rate.
Although a few relationships between fracture toughness and physical properties of grainbased heterogeneous rock in laboratory tests were established, some remain speculative,
requiring further clarification. For instance, crack density/porosity and grain size should have
a positive correlation [134, 142, 150], which makes the revealed relationship to fracture
toughness suspicious. Besides, as shown in Fig. 9, the fracture surface of granite in a higher
roughness corresponds to a greater fracture toughness, but the explanation thereof is nonquantitative [9, 144, 145]. Moreover, the widely obtained experimental results of grain size to
fracture toughness of grain-based rock are in the form KIc~Lg-λ, where λ is a positive constant
dependent on rock microstructure [134, 136, 151, 152]. Nevertheless, Sabri et al. [153]
measured the fracture toughness of granite with three different grain sizes and obtained the
maximum KIc in rock of medium grain size. These debated consequences should be on account
of the differences of dissipated energy of minerals microcracking. An experiment in which
higher counts and energies of acoustic emission (AE) events were recorded in specimens of
granite with a greater KIc was demonstrated [153]. In view of these different phenomena and
findings on rock fracture toughness, the intrinsic sources of high fracture toughness in grainbased heterogeneous rock and the crucial factors on their macro fracture resistance remain to
be further investigated.
14 / 51
(b)
(a)
~1.8 mm
Crack
deflection
Deflection
Deflection
Type-1
KIc =1.89 MPa·m1/2
Penetration
~0.6 mm
Penetration
Crack
penetration
KIc (MPa·m1/2)
3.0
KIc (Tpye-1) > KIc (Tpye-2)
2.5
2.0
1.5
Deflection
dominant
1 mm
KIc =1.15 MPa·m1/2
Type-2
1.0
Type-1
Penetration
dominant
Type-2
Figure 9. (a) Experimental fracture patterns of granite observed via micro CT technique
along with the tested macro fracture toughness (modified after [9]), (b) fracture surface
profiles of African granodiorite in two microcracking types with the measured KIc arising
from use of the notched semi-circular bending method highlighted as well (reproduced
from [10]).
3.2 Loading/strain rate effect
The anisotropic behavior of transversely isotropic or mineral grain-based heterogeneous
rocks under dynamic load has been extensively studied [11, 140, 154, 155]. Generally, the
loading method can be categorized into three types. First is the notched semi-circular bending
(NSCB) testing method [140, 156-158]. In this method, the nominal Mode-I fracture toughness
of the coal and rock materials with different bedding angles was revealed [122, 156, 159]. As
summarized in Fig. 10, the difference of fracture toughness with bedding angle is found to
gradually decrease with loading rate and the anisotropy index (ηai) exponentially decreases,
tending to unity. This tendency also exists in the grain-based heterogeneous rocks under
different loading directions [160]. Second are the Brazilian disc (BD) and semi-circular bending
(SCB) tensile tests. In these two methods, similar principle was invoked [11, 154, 161-163]
wherewith the increased loading rate causes the anisotropic index of bedding-/grain-based rocks
to decrease. Another typical scenario is the uniaxial compression (UC) test. As with the
influence of weak bedding planes, the minimum uniaxial compression strength was obtained
15 / 51
when the intersection angle between the bedding and the principal loading direction was around
30° at a determined loading rate [164, 165]. These are consistent with the theoretical analysis
[25] and numerical simulation [165]. Basically, the difference in bedding angle-dependent
dynamic strength decreases at a larger loading rate (or strain rate) [164, 166].
(a) 2.0
P
1/2P
Slate
Phyllite
1/2P
SCB
(Black shale)
UC
UC
(Phyllite)
(Slate)
ηai
1/2P
ηai
(b) 2.0
P
1.5
1.5
1/2P
SCB
(Barre granite)
P
P
Bituminous
coal
1.0
10
Black
shale
Barre
granite
100
 (GPa·m1/2·s-1)
P
P
1000
1.0
10
100
BD
(Barre granite)
1000
 (GPa·s-1), ε (s-1)
Figure 10. The sensitivity of rock anisotropic index to loading/strain rate. (a) The
calculated anisotropic indexes of rocks with loading rate in the NSCB tests [149, 156, 159,
160], (b) anisotropic index with loading rate (or strain rate) of bedding-/grain-based rocks
under UC [164], BD [11], SCB [154, 161] tests.
In view of the aforementioned studies, even though using different testing methods for the
investigation of dynamic fracture of bedding-/grain-based rocks, the rate-dependent toughness
or strength is in a similar principle with loading rate (strain rate) for the NSCB [156, 160], BD
[11], SCB [154, 161], and UC [164] tests. In some papers, the anisotropic index (ηai), was used
to determine the effect of dynamic loading on the toughness/strength behavior. However, there
are two main gaps in the existing studies on the loading rate-dependent rock anisotropic
behavior: (i) the analysis of essential parameters on the transition of crack trajectory with
loading rate is sparse while most reports are limited to the phenomenological description of
experimental results; (ii) the fundamental principles of rock anisotropy and the effects of
bedding angle and loading rate on crack propagation in the layered rock materials/structures
remain to be explored.
4 Mechanism discussion from crack deflection criterion
Massive natural materials and structures are included by weak interfaces, exerting
anisotropic physical and mechanical performances. The role of crack deflection at an interface
was first realized and analyzed about half a century ago by Cook and Gordon [35]. Their
16 / 51
analysis used a strength-based fracture criterion. Later, numerous researchers used linear-elastic
fracture mechanics (LEFM) to evaluate crack deflection at an interface from the perspective of
energy evolution around the crack-tip [34, 167-171].
4.1 Strength criterion
With the loading perpendicular to the direction of crack propagation, the stress
components (σxx, σyy, and τxy) at a random point ahead of the crack-tip were derived by Inglis
[172]. If assuming the crack is elliptical where a and b are the semi-axes of the ellipse, one
example of the induced stress distributions close to the crack-tip along the major axis is shown
in Fig. 11 [35]. Along the x-axis, σyy monotonically decreases from the crack tip where initially
it has a high value [σyy ~ (1+2a/b)], while σxx increases from zero and at a distance roughly
equal to one crack-tip radius it reaches a maximum value (less than one-fifth of the maximum
value of σyy) and then gradually decreases, reaching a stable value approximately equal to the
ultimate σyy far from the crack-tip. The shear stress, τxy, around the two shoulders of the cracktip initially at a large negative value and then gradually increases to be positive along the xaxis. With the successive propagation of this tensile crack, the stress field, exhibiting a similar
pattern to that shown in Fig. 11, progressively moves forward.
σ∞
Crack
x
Crack
2b
Crack
y
2a
σ∞
σxx
σyy
τxy
Figure 11. Stress contours around the crack-tip under Mode-I tension load [35].
In case of a weak interface ahead of the propagating crack with a ligament of length la, the
defined crack trace would be disturbed under the strength difference of each phase in a specific
stress field [173, 174]. Cook and Gordon [35] distinguished the failure pattern of an interface
into three conditions (Fig. 2b) in terms of the values of σxx and τxy: (i) interface debonding on
the x-axis under direct tension and ahead of the main crack; (ii) fracture under shear of the
interface at points symmetric about the x-axis and just ahead of the main crack; (iii) penetration
of the primary crack through the interface which may or may not break afterwards. In terms of
the maximum ratio of the peak value of σxx to σyy, that was taken as 0.2, it was inferred that an
interface with a theoretical tensile strength of less than 0.2 times that of the matrix would
debond ahead of the main crack leading to the occurrence of crack deflection [174]. Later,
17 / 51
Cook and Erdogan [175] considered the stress intensity factors and the crack opening
displacement, and used the Mellin transform in conjunction with the dislocations resulting in a
fracture criterion for crack penetration or deflection given the equality of critical stress and
material strength of substrate material or interface, respectively. It is noticeable that the power
of the singularity in Irwin’s equations [176] for the crack-tip at an interface is no longer that of
the homogeneous material (λ = 1/2).
For a strong interface, debonding ahead of the growing crack will not occur, so that crack
deflection can only occur by changing its direction along the interface [174]. For an interface
with higher strength the crack grows straight through the interface into the substrate [177].
Gupta et al. [178] proposed a strength criterion whereby crack deflection is initiated if the
strength ratio of the interface ( σ i ) to matrix ( σ m ) is smaller than the maximum stress ratio in
the orthotropic directions ( σ x x
σ yy
). In this case it was predicted that, for crack deflection to
occur, the interface must have a strength of less than about 0.35 of that of the matrix when there
is no elastic mismatch [174]. The local stress criterion for crack deflection demands that, under
the probing of the crack-tip stress field, the decohesion stress for tensile or for shear separation
along the interface is reached before the cohesive strength is reached in the substrate material
at the crack tip [173]. The measurements of the stress around the crack indicated that debonding
is much more effective than slip at reducing the stress concentration ahead of the crack [179].
For a frictional interface oriented perpendicular to the propagating crack, slip along the interface
will occur when τ xy  μσ xx , where μ is the coefficient of friction of the interface. Thereafter,
a first order analysis of the stresses near a Mode-I crack upon a frictional interface results in a
simple criterion for crack deflection (
-σ xx
0.35  0.35 μ

) as shown in Fig. 12 [131]. The
σ m  σ yy
1.06
shear resistance of the interface plays the key role in development of the penetration scenario
[180].
18 / 51
Crossing stress ratio, -σxx / (σ m-σyy)
10
1
No crossing
Partial crossing
Complete crossing
Theoretical threshold
σ
T0
m shear capacity threshold
0.1
0
0.2
0.4
0.6
0.8
1
Coefficient of friction, μ
Figure 12. The crossing stress ratio criterion. Crossing occurs when stress ratios exceed
this theoretical threshold, while interface slip occurs and propagation terminates if the
stress ratio is below this threshold [131].
In consideration of the elastic mismatch of the contacted dissimilar phases, the two
Dundurs elastic constants α and β should be included [111],
α   μ A
β   μ A
1 
1 
υB

μ B 1  υ A
 
2υB

μ B 1  2 υ A
 
 μ A
1 
 μ A
υB
1 

υB
μ B 1  υ A

 
μ B 1  υ A
(1a)
 
(1b)
where μ and υ are the corresponding shear modulus and Poisson’s ratio of materials,
respectively, and indices A and B refer to the corresponding layers. Substituting the shear
modulus
μ  E  2(1+υ) into Eq. (1a), the parameter α can also be denoted as
α  ( E A*  E B* ) / ( E A*  E B* )
, where E  E (1υ ) is the elastic modulus under plane strain
*
2
condition. Stresses near the crack tip that touches the interface are found to be of the following
form: σ ij  K1 λ f ij  θ  , where λ defines the intensity of the stress singularity [181]. It was
r
shown in [175, 182] that λ can be determined as the root of the following characteristic equation:

2 λ 2  α  β  β  1   α  β  1  β
2
2
 cos  λπ   0 ， where the smallest real part is in the range
of 0  Re  λ   1 . Gupta et al. [173] proposed two dimensionless elastic constants λ and ρ for
the determination of desired interface strength required for crack deflection. The stress and
deformation fields derived for the anisotropic materials are dependent on material parameters λ
and ρ for the two media and on the two Dundurs constants α and β [178]. As shown in Fig. 13a,
the effect of β on the stress components is negligible for practical ranges of β between -1/4 and
1/4. The normal stress at the interface, σxx, is most sensitive to the variation in β for α < -0.5.
19 / 51
The transverse component of stress in the bulk material, σyy, shows a strong dependence on β
for values of α in the ranges α < -0.5 and α > 0.6. Data for the interdependence of the Dundurs
parameters were analyzed by Suga et al. [183] who discovered that β depends only weakly on
α in all cases considered, and that the associated changes in β are only approximately one tenth
of the change in α. Thus, as shown in Fig. 13a, with ρ1, ρ2, and λ1, λ2 assuming their isotropic
values, a choice of arbitrary β provides few compromises in accuracy for most of the relevant
components of stress. For a large ligament la, in consideration of the elastic mismatch between
two materials, Leguillon et al. [56] derived a strength criterion for crack deflection, respectively,
for the weak singularity (EB/EA = 0.1), no contrast (EB/EA = 1), and strong singularity (EB/EA =
10). Interface debonding has two premises: (i) a crack is nucleated at the interface ( σxx  σi ),
and (ii) substrate material does not fracture ( σ y y
 σm
) [54]. Thereafter, a consecutive strength
criterion based on Cook and Gordon mechanism [35] in consideration of the elastic mismatch
was deduced (Fig. 13b) [54]. The domain under the curve thus indicates the conditions under
which debonding occurs. Contrarily, in the domain located above the curve, debonding does
not occur.
(b) 1
(a) 1
ρ1 = ρ2 = λ1 = λ2 = 1
0.8
B
0.8
β=0
0.6
σxx / σyy
σ xx  σ i
β=-1/4
Crack
deflection
0.4
σxx / σyy = 0.35
0
-1.0
-0.5
0.6
0.4
Crack
penetration
σ i / σ mA = 0.45
σ xx / σ yy  σ i / σ m
σ yy  σ m
Crack
penetration
0.2
Crack
Interface
β=1/4
σi / σmA
σ xx / σ yy  σ i / σ m
A
0.0
0.5
α = (EA-EB)/(EA+EB)
Crack
deflection
0.2
0
-1.0
1.0
-0.5
0.0
0.5
α = (EA-EB)/(EA+EB)
1.0
Figure 13. (a) Strength criterion for crack deflection and penetration in bi-materials [173,
178]. (b) Master curve for distinguishing crack penetration and interface debonding derived
from the ratio of maximum stress components σxx and σyy (reproduced from [54]).
In terms of the strength difference between dissimilar bulk materials, crack tends to deflect
as it approaches a hard inclusion while preferentially penetrate a soft inclusion [55]. When the
crack approaches an interface from the strong material to the weak material, the crack-tip
driving force is amplified as the plastic zone spreads across the interface [184, 185]. Such
20 / 51
plasticity effect determines whether force shielding or amplification is dependent on the
distance from the crack-tip to the interface and the magnitude of remote loading. Mohamed and
Hansen [77] found that the tensile strength ratio ( σtA
σtB ) between two dissimilar materials
plays the dominant role in the fracture energy ratio (GfA/GfB) and elastic moduli ratio (EA/EB),
and proposed a strength criterion in terms of the relationship between the characteristic length
ratio (lchA/lchB) and tensile strength ratio of dissimilar materials where the characteristic length
is defined as
lch  E G f σ t 2
. A brief summary of the existing strength criteria for the
determination of crack deflection and penetration is shown in Table 1.
Table 1. Strength-based crack deflection criteria.
Calendar year
Criterion equation
σ
1964
σ
1964 & 1996
σ i σ m  0.2
max
xx
 σi ,
max
yy
References
 σm
[35]
[35, 174]
1982
 πEG 

σmax  σmin  
2
 4 1  υ  L 
1993 & 2000
σ xx
σ
 i
σ yy
σm
1995
1999
2000
-σ xx
σ t  σ yy

1
2
 2
sin 2θ 
 sin θ 

2μ 

[129]
[56, 178]
0.35  0.35 μ
1.06

l a l m  2.25 σ a σ m

[131]
 6.0
[77]
σ xx
σ
 i
σ yy σ m
( E 2 E 1  0.1 , σ i σ m  0.68 ;
E 2 E 1  1.0 , σ i σ m  0.99 ;
[56]
E 2 E 1  10 , σ i σ m  0.79 )
2004
σrr  σi
[53]
2007
σ xx  x  l , l 
σi

σ m max σ yy  x  l , l 
2010
S 0 μ -σ xx 0.35  0.35 μ

σ t  σ yy
1.06
[54]
[132]
4.2 Energy criterion
(1) Evolution of energy criterion
An alternative approach to determining the critical condition of crack deflection is to use
an energy-based analysis to consider interfacial cracking. This was first proposed by Kendall
[31] who argued that a deflection criterion based simply on tensile stresses and material
strengths is not compatible with the Griffith theory of crack growth [186]. He suggested that,
as there was no driving force for the opening of an interface ahead of the main crack, crack
21 / 51
deflection could only occur by the main crack changing its path upon reaching the interface
[174]. As shown in Fig. 11, when a pre-existing plane crack occurs under external tensile
loading, in the energy criterion, crack advance is governed by the strain energy release rate G
at the crack-tip [186, 187]. Crack nucleation occurs when the energy release rate of the cracktip is no less than the fracture energy of the matrix. As the energy release rate is correlated
directly to the stress intensity factors, KⅠ and KⅡ, of the crack-tip [176], equivalently, the crack
will grow forward if the synthetic stress intensity factor exceeds a critical value for a defined
matrix. In elastically homogeneous brittle solids, cracks are generally found to follow a
trajectory for which KⅡ = 0. This crack-path criterion is invalid when the crack advances at an
interface because, in this case, the relative magnitudes of the fracture energy between the
interface and matrix materials should also be considered [188]. If the propagating crack
approaches a weak interface, whether the crack is about to deflect into or pass through the
interface relies on the relative values of energy release rate of the crack-tip (Gd and Gp) and
fracture energy (sometimes denoted by the toughness) of interface and matrix (Гi and Гm) [34].
He and Hutchinson [34] proposed an energy criterion with the help of an asymptotic
analysis of the stress field near the tip of a crack contacting an interface in a bimaterial [182],
where the propagating crack is likely to deflect into the interface if Γi Γ m  Gd Gp . The
analysis associated with use of the HH criterion is an asymptotic one in which virtual extensions
of a crack ahead in its plane (ηp) or along the interface (ηd) are assumed [34]. The prediction of
Gd/Gp is accurate when the length ηd of the kinked crack segment is very small compared to the
parent interface itself [189]. This criterion gives a critical Gd/Gp ratio of 0.26 for a doubly
deflected crack and 0.25 for a singly deflected crack in the premise of no elastic mismatch.
Given crack deflection, the work of fracture should be less along the interface than ahead into
the substrate material [173]. Later, Gupta et al. [178] proposed two dimensionless orthotropic
parameters λ and ρ for determining the desired level of the interface strength required for crack
deflection. After that, an energy criterion for crack deflection at an interface between two
aligned, orthotropic media was established by using the method of singular integral equations,
which depends on these two parameters and the corresponding Dundurs elastic constants α and
β [168]. This criterion is similar to the HH criterion but was modified to be independent of the
virtual extensions. In consideration of the crack propagation in two opposite directions from
the crack-tips (Fig. 11), when the matrix fracture energy is small enough, H-crack formation is
again inevitable. The energy criterion for H-cracking to occur in preference to Mode-I crack
penetration was revealed by Tu et al. [190] upon the condition that Гi/Гm < 0.196, which is
22 / 51
analogous to the fundamental crack deflection criterion Гi/Гm < 0.25 in the absence of elastic
mismatch.
Those energy criteria using a local analysis based on stress intensity factors assume that
only what happens within the K-dominant field at the tip of the crack is important and therefore
geometrical effects such as crack length, specimen geometry, or loading state can be ignored
[174]. However, it was found that the finite length of branch crack and width of the target
substrate layer have profound effects on the crack deflection criterion. The deflection prediction
derived with the consideration of these two factors is in general more favorable to deflection
than the HH criterion [191]. Essentially, there are two limitations of the HH criterion: (i) its
uncertainties as to virtual extension length; (ii) it does not consider interface debonding ahead
of a primary crack. They lead to the criterion sometimes fail to predict interfacial deflection
[192, 193] or interfacial penetration [194]. If the ligament is small enough to be neglected, the
HH deflection model offers a satisfactory approximation of the interface debonding mechanism.
Nevertheless, the ligament from the crack-tip to the interface and the extension length ratio
ηd/ηp are essentially important to the crack deflection [43, 52, 56]. Leguillon et al. [56]
established an energy multi-criterion within the framework of two-dimensional linear elasticity
by taking the interface debonding ahead into account. They assumed that extension lengths of
the cracks should be related to the material and the interface microstructure, and inferred that
if assuming ηp = 2ηd can obtain the Leguillon-Sanchez (LS) criterion [195], which is slightly
different from the HH criterion where ηp = ηd is assumed. This LS criterion is based on a
numerical-analytical approach and can be carried out whatever the value of β and extends
without additional complexity to orthotropic substrates. Thereafter, an improved criterion that
does not require any assumption concerning the extension length ratio was established by using
a quasi-static approximation and assuming that the deflection mechanism occurs under constant
loading [196]. Based on a series of their outcomes, the decohesion length at an interface was
obtained using an energetic approach and thereafter an interfacial nucleation criterion coupling
strength and energetic conditions was proposed [53]. A low toughness interface is not
systematically a sufficient condition to promote the initiation of deflection. For the highest
values of the interfacial fracture length, the debonding is driven by the energy condition [57].
A summary of these quasi-static energy criteria is illustrated in Fig. 14. The discrepancy
between the theory-derived and FEM-derived criteria should be attributed to the differences in
the initial assumptions adopted for geometry and loading conditions. The assumption made in
HH criterion for computing the ratio Gd/Gp is that the two materials constitute the interface
23 / 51
form two semi-infinite planes that meet at the interface. However, the numerical result shown
in Fig. 14 accounts for the finite geometry and the β-effect [188].
Some other typical outcomes of the energy criterion derived later for the determination of
crack deflection at an interface are briefly summarized. Based on the results of numerical
simulation, a modified version of the HH energy criterion which accounts for the effects of
finite reinforcement size, phase volume fractions, phase shape, and phase distribution was
developed. The energy criterion for deflection is sensitive to material anisotropy [103, 197]. In
addition, the HH criterion was extended to give the closed-form solution for the strain energy
release rate of a hydraulic crack with arbitrary angles with respect to the in-situ stress. The
critical conditions in which the hydraulic crack deflects into weak interfaces and exhibits a
dependence on crack-surface friction and in situ stress anisotropy are given in explicit form
[198]. In their results, Mode-II fracture dominates the hydraulic fracturing process and the
intersection angle between the hydraulic crack and the weak interface is the determining factor
that accounts for crack deflection. Furthermore, the lower friction coefficient between cracked
1.5
HH criterion
Martinez and Gupta
LS criterion
1
Martin et al.
FEM assisted
Гi / Гm
planes and the greater in situ stress difference can favor hydraulic fracturing [198].
B
A
Crack
Interface
Crack penetration
0.5
Crack deflection
0
-1.0
-0.5
0.0
0.5
α = (EA-EB)/(EA+EB)
1.0
Figure 14. A brief summary of quasi-static energy criterion for the determination of crack
deflection at an interface [56, 168, 178, 188, 196].
Crack propagation behavior in two-dimensional polycrystals is a function of the fracture
toughness of the grain boundary. Whether the crack will pass through the boundary or be
deflected into it depends on the impinging angle on the grain boundary, the stress intensity
factors at crack tip and fracture toughness of both grain and grain boundary. The crack extends
only intergranularly when KICb ≤ 0.4KICg, and the percentage of transgranular fracture increases
with increasing KICb [89]. A fracture mechanics model is developed which predicts the
24 / 51
transition between the two fracture modes and provides an energy criterion suitable for the
interpretation of experimental fracture results. The proposed criterion determines intergranular
fracture occurrence when the ratio of the fracture toughness for intergranular fracture to the
fracture toughness for trangranular fracture is no larger than the critical value of 0.5 (KICb ≤
0.5KICg) [199]. A widely used alterative to simulate the transition from the intergranular to
transgranular fracture is the cohesive zone model [101]. Pro et al. [200] embedded cohesive
elements throughout the entire mesh in a finite element model (FEM) and derived a kink
nucleation criterion non-linearly related to the parameters of the fracture process zone.
Moreover, a phase-field model formulated with two-set order parameters describing the crack
field and the microstructure field, respectively, is established to study the competition between
crack penetration and deflection at an interface. The effect of the applied plastic strain is
included, resulting in the finding that the energy criterion based on LEFM is a static and critical
case of the present phase-field model [101].
(2) Elastic mismatch
A modulus mismatch changes the nature of the stress singularity at the crack tip both for
interface cracks [201, 202] and for cracks terminating at an interface [202]. These singularities
may have complex components while the resulting stress and displacement fields will have an
oscillatory nature [167].
When the crack is normal to the interface, the corresponding criterion for crack deflection
at an interface between two elastic materials with similar modulus is approximately Гi/Гm <
0.25 [34]. If the composite structure has an elastic mismatch, as shown in Fig. 14, the critical
value for the determination of crack deflection or penetration is not the defined value of 0.25
while it is significantly influenced by the modulus mismatch. For instance, in the case of a weak
singularity ( EB
EA 0.1), the LS criterion predicts deflection if Γi 0.91Γm. Without contrast
between substrates ( EB
Γi 0.27Γm.
EA 1.0),
the prediction of the LS criterion for crack deflection is
In the case of a strong singularity ( EB
deflection when
EA 10),
the LS criterion predicts
Γi 0.58Γm [56].
The near-tip driving force for fracture is influenced by whether the crack approaches the
interfaces from the compliant or stiff material. When a crack subjected to a fixed remote stress
approaches a stiffer material, the near-tip stress intensity factor KI decreases, and vice versa
[185]. The crack-tip is shielded from remote loads when it approaches the interface from the
25 / 51
compliant material. Crack tip shielding and amplification can be quantified in terms of two
contour evaluations of the J-integral where the effective J-integral at the crack tip (Jtip) is greater
than the remote J-integral (Japp) when it approaches the interface from the stiffer material [184].
Jtip/Japp < 1 implies crack tip shielding while Jtip/Japp > 1 indicates amplification [4, 184]. In
addition, the shielding and amplification effects are dependent on the thickness of the substrate
layer, the distance from the crack-tip to the interface, and the remote loading [184]. The
influence of the elastic mismatch may possibly be more pronounced for a crack with a cohesive
zone growing towards an interface [30].
In composite systems with an elastic mismatch, when EA > EB, cracks tend to deflect to
high-stress-concentration sites; conversely, where EA < EB, cracks are expected to be attracted
by the soft phase within the stiffer matrix [203]. For an inclined crack versus the existing
interface (Fig. 2), when the propagating crack approaches a more compliant material across the
interface (α < 0) KII becomes negative. If it is free to curve following a path with KII = 0, the
crack will curve toward the interface since the straight crack has KII < 0. Conversely, when the
straight crack approaches a stiffer material across the interface (α > 0), KII becomes positive
indicating that an actual crack trajectory that satisfies KII = 0 would curve away from the
interface [34].
(3) Mode mixity (phase angle)
The fracture energy (Г) for a plane crack at an interface is dependent on the relative amount
of the tensile (KI) stress field to the shear (KII) stress field at the crack tip [204]. The study in
[204] suggested that under mixed-mode loading the critical energy release rate of an interface
is a function of KII/KI ratio, depending on the roughness of the interface. It is assumed that the
source of shielding is the fact that the crack is trapped, and runs along an interface which is not
perfectly smooth. A rougher interfacial surface increases the phase angle, resulting in the
growth of interfacial fracture energy [86]. The deflected crack has a higher relative Mode-II
contribution. Interface toughness is increased with the ratio increase of shear loading relative
to that of tensile loading [188]. The mixity of crack-tip loading mode (characterized by the
phase angle, ψ, having a tangent equal to KII/KI) can vary between pure Mode-I (ψ = 0°) and
pure Mode-II (ψ = 90°), depending on the loading state as the length of the crack increases and
the ratio between the elastic properties of the two materials (Fig. 15) [174, 205]. It was found
that, in the case of wedging, the phase angle decreases as the crack grows, remains quasi1
constant in four-point bending but increases in tension [174]. When β = 0, ψ = tan  K II K I  .
26 / 51
Another finding is such that the stress intensity factors are correlated to the strength constituents
and the KI-value is related to the KII-value as follows:
K I σ 0  1  2 λβ
[181].
K II τ 0 1  2 λβ
(b)
ГIIC
(a)
45°
Ψ
0
θ
40
Interface fracture energy, Гi (J·m-2)
ГIC
Double
cantilever
beam
Ψ ~ 0.5°
90°
Four-point bend
Ψ ~ 35–60°
2a
R
Glass/Adhesive
Aluminum/Adhesive
30
Mortar/Granite
Mortar/Limestone
Steel/Epoxy
20
10
0
0
Brazilian disc
Ψ ~ 0–90°
Composite cylinder
Ψ ~ 70–90°
Tensioned push-out
Ψ ~ 0–90°
15
30
45
60
Phase angle, Ψ (°)
75
90
Figure 15. (a) Schematic illustrations of the dependence of Гi on phase angle ψ where
different testing methods for the measurement of Гi involve a wide range of ψ values [188,
205], (b) some experimental results showing variations of Гi with ψ for the adhesive
bimaterial structures [86, 188, 204, 206, 207].
The mixed-mode energy release rate at the interface could also be correlated to the phase
angle. The total energy release rate associated with an infinitesimal virtual crack extension is
written in terms of K as, G=
K
2
cosh  πε  E
2
*
, where ε 
1  1 β 
ln 
 . The energetic phase angle
2π  1  β 
ψ G is defined in a similar term to the stress intensity factors as tan 2 ψG  G II  Δa  , which
G I  Δa 
represents a Δ a -dependent measurement of fracture mode mixity. It is noticeable that when
β0, ψG  ψ K
[208].
(4) Toughening
Numerous natural materials are composites, with the utilization of crack deflection to
provide extra toughness and strength. Planes of weakness may cause toughening in composite
materials by deflecting and stopping cracks [35, 209]. One of the methods that has been used
to increase the fracture toughness of brittle ceramics is based on the addition of a dispersed
27 / 51
ductile phase [210, 211]. Other methods that have been studied by several researchers rely on
transformation toughening [212], fiber reinforcement [213], or microcrack shielding [214]. To
achieve toughness in many brittle composites, crack deflection at interfaces is essential [173].
The ability to deflect and channel a crack is an attractive means of enhancing the toughness
of brittle polycrystalline materials [97]. To maximize the degree of crack deflection and thereby
maximize the toughness of the material, large, hard inclusions should be incorporated at grain
boundaries. Toughening mechanisms at grain boundaries are associated with deflecting and
directing cracks into predetermined propagation tortuous paths [97]. The fracture toughness
loss is relatively small for large-angle intergranular cracks. However, the decrease of fracture
toughness of low-angle intergranular fracture and increasing proportion of transgranular
fracture in a large-grain-assembled system may result in a better fracture resistance [101]. A
proper balance between transgranular and intergranular failure can lead to an optimized fracture
toughness. Microstructures with refined grain sizes and balanced bonding strength in grains and
grain boundaries can enhance fracture toughness in terms of best promoting the manifestation
of more favorable failure mechanism [91]. The toughening effect of the interfaces is
significantly increased if partial debonding occurs during bridging. Thus, a limited amount of
debonding is advantageous, but too low a bond strength results in complete debonding, which
adds only a little toughness to that of the brittle matrix material [97, 209]. In addition, Hsueh et
al. [114] found that the elastic mismatch can contribute to fracture toughening in heterogeneous
materials, which is attributed to decrease of stress intensity at the crack-tip when a crack is at a
compliant-to-stiff interface. To allow crack re-nucleation, the remotely applied load should be
increased leading to a greater apparent fracture toughness. Moreover, a compliant interlayer
tends to increase the interface toughness by increasing plastic dissipation in the thin layer and
developing plastic strains in the interlayer [215].
(5) Residual stress and T-stress
The analysis under the HH criterion is asymptotic in which the prediction of Gd/Gp is
accurate when the length ηd of the kinked crack segment is very small compared to all other
lengths in the problem, including the length of the parent interface crack itself. If there is a
stress in the substrate parallel to the interface due to either residual stress caused by thermal
expansion misfit or applied loads, then an additional non-dimensional length parameter, not
considered in the HH criterion, becomes important [169, 189, 190]. The residual stress,
governed by the misfit strain between the two dissimilar bimaterials [216], affects the energy
release rates of both the deflected and penetrating cracks [169]. Relatively small values of
28 / 51
residual compression substantially enhance debonding in preference to penetration [190].
Introducing compressive residual stress in the stiffer intact layers of a composite laminate ahead
of a growing primary crack would favor crack deflection by allowing advantageous energetic
conditions [217]. These residual stresses are expected to become at least partially relaxed during
crack advance, thus contributing to the energy balance [205]; however, the residual stress effect
is negligible if the intact layer is more compliant than the cracked layer [217].
Another important feature is the T-stress, which acts parallel to the parent Mode-I crack
before deflection. This T-stress has an unusually strong influence on the energy release rate at
the tip of the deflected crack [218]. The non-singular T-stress affects the shielding and
amplification of the stress fields around the crack-tip near a bimaterial interface [184]. Simply
using the sign of the T-stress before crack kinking is not sufficient to determine crack growth
stability as observed in some experiments. A crack will propagate off its original path due to
changes in local fracture mode mixity (a mechanical factor) or fracture toughness (a material
factor). The stress intensity factor and the T-stress, have been accepted as a two-parameter
fracture criterion in predicting the crack propagation direction and the shape and size of smallscale yield zones [19].
4.3 Coupled strength-energy criterion
Historically, models of crack deflection have been developed using either a strength-based
or an energy-based fracture criterion. However, in general, crack propagation depends on both
strength and toughness [171]. Gupta et al. [173] defined two dimensionless elastic constants
determining the stress and deformation fields in each medium. In that case, the stress criterion
established allows determination of the desired interface strength required in the manufacture
of a composite to enhance the overall toughness thereof. At the fundamental level, the interface
tensile strength and intrinsic toughness are related thus:
Γ ~ σ t2
. The interface strength directly
controls the plastic dissipation and the roughness-related shielding components, while the
overall energy absorption is related to the fracturing process in terms of the energy release rate
at the crack-tip [178]. Chen [181] found that, for a mixed-mode crack, the stress intensity factors
can be correlated to the strength constituents in terms of the intensity of stress singularity (λ)
and Dundurs elastic mismatch parameter β as
K I σ t  1  2λβ
.
K II τ 1  2λβ
29 / 51
(a) 100
(b) 20
E A  EB
i Eh=1e-6
E A  EB
Crack
penetration
10
15
0.67
10
m i
Gi (J·m-2)
i
 0.33
m
Crack
deflection
10
Crack
penetration
5.0
1
1.0
2.5
1.0
5
Interface
decohesion
E i  i2h  0.01
0.1
0
0.1
1
10
Gm (J·m-2)
100
1000
0
5
10
m i
Figure 16. (a) Competition between matrix penetration and interfacial debonding with a
coupled strength-energy criterion [57], (b) determination of crack penetration or crack
deflection in toughness ratio and strength ratio space [33, 171].
For the crack with a cohesive zone, the critical stress intensity factor is related to the
cohesive zone properties as K IC  2 Eσ t ε and the length of the cohesive zone at critical load
is lcz  π 8  K IC σ t 
2
[30]. A strength criterion in terms of the relationship between the
characteristic length ratio (lchA/lchB) and tensile strength ratio of dissimilar materials was
established where the characteristic length is defined as
lch  E G f σ t 2
[77]. This characteristic
length factor was also used by Martin and Leguillon [53] to estimate the debonding length at
an interface. Low toughness and high strength interfaces are associated with low values of the
characteristic length while high toughness and low strength interfaces lead to higher values of
lch. Thereafter, the competition between the interfacial decohesion and matrix cracking was
determined in terms of a normalized energy release rate correlated to the ligament length and
interfacial debonding length. Based on the outcomes, an initiation criterion from the perspective
of applied strain was developed by Martin et al. [57] by taking both the energy and stress
conditions into account, where debonding or penetration condition can be reduced to an energy
or a stress condition depending on the relative value of some characteristic fracture lengths of
interface and matrix (Fig. 16a). Similarly, Thouless et al. [167] found that occurrence of
interfacial debonding is dependent on the stress applied to a bimaterial structure, where
σ   1.0  E Γ i a 
12
and crack penetrates to the matrix if σ   0.5  E Γ m a  . Thereafter, an
12
energy criterion Γi Γm  0.25 for crack deflection is given for two similar substrates. In
30 / 51
addition, the characteristic length is normalized with respect to the pre-existing crack length
with the criterion revealed in terms of the relationship between the toughness ratio and strength
ratio (Fig. 16b) [171].
Both energy and stress criteria are necessary conditions for the determination of interface
fracture but neither one nor the other are sufficient. Thanks to the singularity at a crack-tip, the
incremental form of the energy criterion gives a lower bound of admissible crack lengths. On
the contrary, the stress criterion leads to an upper bound [219]. Parmigiani and Thouless [171]
studied crack deflection by using a cohesive-zone model which incorporates both strength and
toughness parameters simultaneously. No matter how tough an interface is, crack deflection
can always be induced if the strength of the interface is low enough compared to the strength
of the substrate. A low toughness interface is not a sufficient condition to promote the initiation
of deflection [167]. There is a lower bound for the ratio of the substrate strength, below which
penetration is guaranteed no matter how brittle the interface is. It should be noticed that the
effect of modulus mismatch on crack deflection is very sensitive to the mixed-mode fracture
criterion for the interface [33, 171, 220]. Moreover, a fracture criterion involving both
toughness and tensile strength allows the study of the competition between, on the one hand the
crack blunting due to the pores and resulting in an apparent toughness enhancement, and on the
other hand the weakening effect caused by an increasing volume fraction of pores [221]. As a
summary of above review, a brief statistical analysis of energy criteria pertaining to crack
deflection at an interface is shown in Table 2.
Table 2. Energy-based crack deflection criteria.
Calenda
r year
1989
1989
1992
Referenc
es
Criterion equation
Γi
 0.25
Γm
[167]
Γi
 0.6
Γm
Γi
K
 d
Γm Kp
[167]
[195]
1993 &
1996
Γi  G d
[89, 205]
1996
d
Γi

4
π
d
  c
Γm
[174]
1996
2
Γ i c E 2   d  c  E 1E 2

4 π  d  c  E 12
Γm
[174]
1996
1


II
hm  
Γ m π   EΩ   h m



1




I
h B  
Γ B 16   σ A   h B
2
[190]
31 / 51
2000
2000
2000
2000
Γ i  0.43Γ m ; Γ i  0.24Γ m
Γ i  0.91 Γ m  E 1 E 2  0.1  ;
Γ i  0.27 Γ m  E 1 E 2  1.0  ;
Γ i  0 .5 8 Γ m  E 1 E 2  1 0 
Γ i K d  2ηd


Γ m K p  ηp



[52]
[56]
2 λ 1
[222]
Γ i μp K  μd   K  0

Γ m μd K  μp   K  0



2 λ 1
[56]
 K p   a1p  2 λ
 G c2   a1p 

  2λ  c  
  1  2λ  0

 G1   l 0 
  K 1   l 0 

2λ
 K d   a1d 
 G ic   a1d 
  2λ  c  
  1  2λ  0
   
 G1   l 0 
 K 1   l 0 
2001
K a
Γi
 d d
Γ m K p  a p
2004
Γ i Ai  l , d 

Γm
Am l 
2007


K Ic


acr  

2
2


β
σ
sin
α
1
σ
cos
α


 y
  x 
 
2018
 li 
ΓR  f  ψ , cz
b 
 l cz 
1992
max
Gd Gp  0.25
[203]
1992
max
Gd Gp  0.4 (Elastic mismatch α=0.57)
[203]
1996
Ki≤0.4Km
[89]
[196]

2010
RL
IC
[53]
[69]
2
i
Process zone length ratio: l czb  Γi σ b2
[200]
Γb σ i
l cz
LR
IC
K 0.5K
[199]

Gd 
 1 β 2
Gp 
1989
 K IC 
 A
 K IC  max
1998

Γi  Gd
Γm Gp
189]
[34,
1  α    d
2
 e  2 R e  de   c 2

2
[34]
B
B
B

δ
δ
K IC
 ω A  A for A  1
δ
δ
K IC


 ω  1
B
B

δ
δ
 1  A  ω  1 for A  1
δ
δ


2
1994
2

2
2
Gd d  ηn dg  dg  ηn g

Gp c2  2ηtch  η2t h2 1 α
[30]
1-2λ
 ad 
 
 ap 
[169]
 1/ 4
1994
Gd
λ1
H 22
 d 2  e 2  2 Re  de  

2
2

G p 4 cos h πε  s 11n 1 c 
2000
d
Gd

max
2
Gp
1  α  c
2007
1 2 λ1
d
d
2
2
2
2
Gd
1  c11  c 21  2  c11c12  c21c22  η   c12  c 22   η   ad 

2  p  [66]
2
2
Gp 1  α  b11
 b 221  2  b b  b b  ηp   b12
 b 222   ηp   a 
[168]
2
[191]
2
11 12
32 / 51
21 22
1  υ1 1  υ2 



μ1
μ2  2

Gd 
 K1  K 22  ,
4 cosh 2  πε 
K12  K 22  k12 a 1d 2 λ  c  h  2 Re  ch   ,


2
2013
2017

1
c 
2


1

h  4

Gα
Gθ max

2
2α
3cos  2  1  3cos α /16, when sinα>0
 


 3cos2  α  5  3cos α / 8, if sinα  0
 

2
2017
α
Gα
 cos 4  
2
G θ max
Gi θ 

[198]

2
θ 
α
sec 2  0  cos 2    6 cos α  2  s τ  2 sin α s q
2
 2 

2
2
8 4 sin θ 0 s τ s q  1  cos θ 0  s q   5  3 cos θ 0 s τ 


Gα
G θ max
Gb
[223]
3iw


1  β   iw2
2
e

e


1 α 

3iw


1  β   iw2
e  e 2 
1 α 

2017
2019
2



[198]
[198]
2
2
1 
θ
3θ   θ
3θ  
 3cos  cos    sin  sin  
16 
2
2  
2
2  
[101]
4.4 Effect of dynamic loading
A limited amount of work has been done in the analysis of crack deflection in bimaterial
structures under dynamic loading. In the framework of LEFM, Xu et al. [121] theoretically
analyzed the competition of crack deflection and penetration at an interface between two similar
isotropic elastic solids, where a crack propagates forward rapidly to reach the interface from
one side of the matrix under an impact load. This energy criterion is derived from the quasistatic LEFM crack deflection criterion in terms of the relationship between energy release rate
at the crack-tip and the fracture energy of the material. The quasi-static ratio of two energy
release rates depends on the interface angle as,
2
2
Gd  θ  1 
3θ   θ
3θ  
θ
θ 
  3cos  cos    sin  sin   =cos4   .
16 
2
2  2
2  
Gp
 2
(2)
In the counterpart of this dynamic analysis, the crack propagation velocities before (v1)
and after (v2) deflection are used to evaluate the instantaneous crack-tip energy release rate
based on the detailed results in [224]. Thereafter, the ratio of two energy release rates for the
deflected interfacial crack and the incident Mode-I crack is given as [121]:
33 / 51
2
2
θ
3θ 
3θ 
 θ

k  v 2   3cos  cos   αs2kII2  v 2   sin  sin 
2
α
Gd  θ，v 2   v 2  D1
2
2
2

 2
 
.
2
Gp  v1
16α d1kI  v1
 v1  D2
2
d2 I
(3)
The energy criterion for dynamic crack deflection is in a similar form to that in quasi-static
conditions [34]: when
Gd  θ，v 2  Γi  v 2 
, the dynamically propagating crack deflects into

Gp  v 1 
Γm  v1 
the interface. The predictions of this dynamic energy criterion are fairly consistent with their
experimental tests of brittle materials. This model was later extended to the determination of
fracture patterns in ceramic with heterogeneous microstructure [90] and crystalline rocks [65].
When the dynamically propagating Mode-I crack reaches a weak interface, it kinks out of
its original trace and shifts to propagate along the weak interface, exhibiting mixed-mode
behavior. Xu and Wang [225] performed supplementary analyses of their experiments and
theoretical model in [121], where changes in dynamic T-stress values of the incident crack over
time were determined and the effect of mode mixity of the interfacial crack was discussed. It is
found that the T-stress of the incident crack has a small positive value but exerts a significant
influence on the stress fringe pattern of crack-tip in photoelasticity. Moreover, the mode mixity
of the kinked crack depends on the kinking angle and the crack-tip velocity. A weak interface
will lead to a high Mode-II component and a fast crack-tip velocity of the kinked mixed-mode
crack.
The effect of elastic mismatch between two neighboring materials in the aforementioned
analyses on dynamic criterion of crack deflection is, however, not considered. Huang et al. [117]
evaluated the feasibility of the HH criterion in dynamic loading condition based on laboratory
tests. From their results, it is found that in dynamic loading, for a determined elastic mismatch,
the propagating Mode-I crack is likely to penetrate the interface rather than behaving in
accordance with the deflection pattern seen in quasi-static tests. They speculated that the critical
curve that determines crack deflection or penetration in quasi-static cases (Fig. 14) should be
transferred along the bottom-right direction under dynamic load. Djoković et al. [226] extended
the analysis of [121] by considering the elastic mismatch effect in terms of the two Dundurs
elastic constants (α, β) [111]. In doing so, the energy release rate at the crack-tip upon deflecting
into
the
interface
should
be
expressed
34 / 51
as
Gd 
H
2
Kd ,
2
4cosh  επ 
in
which
2
2
1  1 β 
α d1 1  α s1  α d2 1  α s2 
[227] and ε  ln 
H 

 [228]. The Dundurs elastic
2π  1  β 
μ1 D1
μ 2 D2
parameter β should have a rate-dependent change as indicated in [229]. In view of the number
of previous investigations thereof, the effect of α is much greater than that of β [34, 56, 168,
178, 196]. Therefore, the ratio of energy release rate is found to be given by [226, 230]:
Gd  θ , v 2, α
Gp  v1

1 Gd  θ , v 2 
. However, the ultimate format of this criterion looks debatable
1-α Gp  v1
as it does not give the specific deriving process and is quite similar to that of [121] with an
additional parameter α included.
The aforementioned failure/fracture mechanism of inhomogeneous rock and rock-like
brittle materials/structures in terms of crack deflection criterion has presented the theoretical
development of the effects of weak interfaces on crack propagation. With the comparison of
these criteria, it is prominent that: (i) the interaction between a propagating crack and a preexisting weak interface is associated with both the stress and energy aspects in this system; (ii)
the energy criterion is more popular in application than the strength criterion for the
determination of crack deflection or penetration at an interface as the former considers the
properties of the propagating crack. However, coupled strength-energy criteria are elusive and
difficult to use as they are sensitive to the change of each parameter therein. In addition, most
of the energy criteria apply to the quasi-static state and are mainly used in ceramic composite
materials, whereas a small number of applications of such energy criteria are found for the
interpretation of inhomogeneous rock fracturing. Moreover, few works on the energy criterion
of crack dynamic propagation upon a weak interface has been done, not to mention the
deficiencies in these energy criteria.
5 Conclusions and outlook
This literature review summarizes the effects of weak interfaces in rock-like materials and
structures on crack propagation from phenomenon description to mechanism analysis. Typical
fracture patterns in terms of the interaction between a propagating crack and a pre-existing weak
interface are elucidated firstly followed by the presentation of their effects on the mechanical
property of grain-based and bedding-abundant rocks. Based on the aforementioned review of
weak interfaces on the physico-mechanical characteristics of rock-like brittle materials, the
intrinsic mechanism is discussed in terms of crack deflection criterion. Overall, some main
conclusions are briefly presented below:
35 / 51
(1) Four general types of fracture patterns in terms of a propagating crack interaction with
a pre-existing interface are classified. They are the crack directly contacts with an interface
perpendicularly, the crack directly contacts with the interface obliquely, debonding or
fracturing ahead of the crack when approaching the interface, and the crack curves toward or
away from the interface at an inclined angle.
(2) A series of factors influence the path of a propagating crack when it goes close to a
weak interface. In this work, the key factors, e.g. strength and toughness of interface, interface
angle, interlayer effect, elastic mismatch, dynamic loading, and confining stress, are fairly
reviewed and summarized.
(3) The mechanism of action of weak interfaces on crack propagation in rock is discussed
in terms of crack deflection criteria. The evolution and progress of strength criterion, energy
criterion, and coupled strength-energy criterion are reviewed as well as the summary of limited
work on the effect of dynamic loading. Indeed, the various energy criteria are the favorable
ones in the determination of crack propagation at a weak interface.
(4) Numerous studies have been conducted upon the effects of weak interfaces on crack
propagation; to date, most of the outcomes are derived from tests on composite materials under
quasi-static loading. In rock mechanics and rock engineering, these concepts and results have
not been widely and popularly used for the investigation of heterogeneous rock fracturing
containing weak interfaces, not to mention dynamic load regimes.
This review work gives a detailed summary about the effects of weak interfaces on crack
propagation in rock or rock-like materials/structures, and is of significance for the
comprehensive understanding of different fracture patterns influenced by weak interfaces. To
rock mechanics, even though some experimental, numerical and theoretical work on the quasistatic and dynamic fracturing of brittle rocks containing weak interfaces, as reviewed above,
have been undertaken, there are some key issues remaining to be addressed: (i) most
experimental and theoretical studies are in quasi-static conditions while merely few
investigations on the strain rate-dependent rock fracturing containing weak interfaces (beddingplanes or grain/inclusion boundaries); (ii) the existing results mainly sustain discussion of the
phenomena of weak interfaces on rock fracturing under tension while the quantitative
relationship between dynamic loading and crack propagation in rocks with weak interfaces
remains insufficient; (iii) the proposed criteria for the study of quasi-static crack propagation in
ceramic and other composite materials can be used and modified in the near future for the
36 / 51
understanding and prediction of rock mass failure during underground resources/energies
exploitation; (iv) previous experimental and numerical studies on heterogeneous rock fracturing
pay more attention to mineral composition and its effects on the macroscopic failure pattern of
rock, while the essential controlling factors and the effect of mineral fracturing on macroscopic
mechanical behaviors are rarely discussed; (v) the specific processes of heterogeneous rock
dynamic fracturing have not been systematically examined and the intrinsic mechanism in
terms of the competition between propagating cracks and rock properties is scarcely
implemented for the analysis of heterogeneous rock dynamic fracturing in experiments or
simulations.
Acknowledgements
This work was supported by the China Scholarship Council-Monash University (CSCMonash) Project (No. 201606420068), the National Natural Science Foundation of China (No.
52104101), and the Monash University Postgraduate Publications Award. The authors are very
grateful to the financial contribution and convey their appreciation for supporting this basic
research.
References
[1] Hoek E, Brown ET. Practical estimates of rock mass strength. International Journal of Rock Mechanics
and Mining Sciences. 1997;34:1165-86.
[2] Zhao J, Zhou Y, Hefny A, Cai J, Chen S, Li H, Liu J, Jain M, Foo S, Seah C. Rock dynamics research
related to cavern development for ammunition storage. Tunnelling and Underground Space Technology.
1999;14:513-26.
[3] Larsen B, Gudmundsson A, Grunnaleite I, Sælen G, Talbot MR, Buckley SJ. Effects of sedimentary
interfaces on fracture pattern, linkage, and cluster formation in peritidal carbonate rocks. Marine and
Petroleum Geology. 2010;27:1531-50.
[4] Pollard DD, Aydin A. Progress in understanding jointing over the past century. Geological Society of
America Bulletin. 1988;100:1181-204.
[5] Zhang S-H, Zhao Y. Cogenetic origin of mafic microgranular enclaves in calc-alkaline granitoids: The
Permian plutons in the northern North China Block. Geosphere. 2017;13:482-517.
[6] Zhao J-J, Zhang Y, Ranjith P. Numerical simulation of blasting-induced fracture expansion in coal masses.
International Journal of Rock Mechanics and Mining Sciences. 2017;100:28-39.
[7] Ju M, Xing H. Crack propagation in jointed rock and its effect on rock macrofracture resistance: insights
from discrete element analysis. Geomechanics and Geophysics for Geo-Energy and Geo-Resources.
2022;8:1-22.
[8] Ju M, Wang D, Shi J, Li J, Yao Q, Li X. Physical and numerical investigations of bedding adhesion
strength on stratified rock roof fracture with longwall coal mining. Geomechanics and Geophysics for
Geo-Energy and Geo-Resources. 2021;7:1-31.
[9] Nasseri M, Grasselli G, Mohanty B. Fracture toughness and fracture roughness in anisotropic granitic
rocks. Rock Mechanics and Rock Engineering. 2010;43:403-15.
37 / 51
[10] Kataoka M, Obara Y, Kuruppu M. Estimation of fracture toughness of anisotropic rocks by semi-circular
bend (SCB) tests under water vapor pressure. Rock Mechanics and Rock Engineering. 2015;48:135367.
[11] Dai F, Xia K. Loading rate dependence of tensile strength anisotropy of barre granite. Pure and Applied
Geophysics. 2010;167:1419-32.
[12] Nasseri MB, Mohanty B, Robin P-Y. Characterization of microstructures and fracture toughness in five
granitic rocks. International Journal of Rock Mechanics and Mining Sciences (1997). 2005;42:450-60.
[13] Gudmundsson A. Deflection of dykes into sills at discontinuities and magma-chamber formation.
Tectonophysics. 2011;500:50-64.
[14] Gudmundsson A. Rock fractures in geological processes: Cambridge University Press; 2011.
[15] Zoback MD, Kohli AH. Unconventional reservoir geomechanics: Cambridge University Press; 2019.
[16] Scholz CH. The mechanics of earthquakes and faulting: Cambridge University Press; 2019.
[17] Llanos EM, Jeffrey RG, Hillis R, Zhang X. Hydraulic fracture propagation through an orthogonal
discontinuity: a laboratory, analytical and numerical study. Rock Mechanics and Rock Engineering.
2017;50:2101-18.
[18] Xu LR, Rosakis AJ. An experimental study of impact-induced failure events in homogeneous layered
materials using dynamic photoelasticity and high-speed photography. Optics and Lasers in Engineering.
2003;40:263-88.
[19] Li X-F, Xu LR. T-stresses across static crack kinking. Journal of Applied Mechanics. 2007;74:181-190.
[20] Wu W, Li H, Zhao J. Dynamic responses of non-welded and welded rock fractures and implications for
P-wave attenuation in a rock mass. International Journal of Rock Mechanics and Mining Sciences.
2015;100:174-81.
[21] Wong L, Einstein H. Systematic evaluation of cracking behavior in specimens containing single flaws
under uniaxial compression. International Journal of Rock Mechanics and Mining Sciences.
2009;46:239-49.
[22] Zhang X, Jeffrey RG. Reinitiation or termination of fluid ‐ driven fractures at frictional bedding
interfaces. Journal of Geophysical Research: Solid Earth. 2008;113.
[23] Horii H, Nemat‐Nasser S. Compression‐induced microcrack growth in brittle solids: Axial splitting
and shear failure. Journal of Geophysical Research: Solid Earth. 1985;90:3105-25.
[24] Park H, Chen WW. Experimental investigation on dynamic crack propagating perpendicularly through
interface in glass. Journal of Applied Mechanics. 2011;78.
[25] Tien YM, Kuo MC. A failure criterion for transversely isotropic rocks. International Journal of Rock
Mechanics and Mining Sciences. 2001;38:399-412.
[26] Li J, Rong L, Li H, Hong S. An SHPB test study on stress wave energy attenuation in jointed rock
masses. Rock Mechanics and Rock Engineering. 2019;52:403-20.
[27] Li J, Ma G, Zhao J. An equivalent viscoelastic model for rock mass with parallel joints. Journal of
Geophysical Research: Solid Earth. 2010;115.
[28] Lei Q, Latham J-P, Tsang C-F. The use of discrete fracture networks for modelling coupled
geomechanical and hydrological behaviour of fractured rocks. Computers and Geotechnics.
2017;85:151-76.
[29] Chen Z, He C, Xu G, Ma G, Wu D. A case study on the asymmetric deformation characteristics and
mechanical behavior of deep-buried tunnel in phyllite. Rock Mechanics and Rock Engineering.
2019;52:4527-45.
[30] Wäppling D, Gunnars J, Stahle P. Crack growth across a strength mismatched bimaterial interface.
International Journal of Fracture. 1998;89:223-43.
[31] Kendall K. Transition between cohesive and interfacial failure in a laminate. Proceedings of the Royal
Society of London A Mathematical and Physical Sciences. 1975;344:287-302.
38 / 51
[32] Lu M, Erdogan F. Stress intensity factors in two bonded elastic layers containing cracks perpendicular
to and on the interface. Part 2: Solution and results. 1980.
[33] Strom JL, Parmigiani JP. Transition of crack path at bi-material interfaces. Engineering Fracture
Mechanics. 2014;115:13-21.
[34] He M-Y, Hutchinson JW. Crack deflection at an interface between dissimilar elastic materials.
International Journal of Solids and Structures. 1989;25:1053-67.
[35] Cook J, Gordon J. A mechanism for the control of crack propagation in all-brittle systems. Proceedings
of the Royal Society of London Series A Mathematical and Physical Sciences. 1964;282:508-20.
[36] Gu H, Siebrits E. Effect of formation modulus contrast on hydraulic fracture height containment.
International Oil & Gas Conference and Exhibition in China: Society of Petroleum Engineers; 2006.
[37] Tambat A, Subbarayan G. Simulations of arbitrary crack path deflection at a material interface in layered
structures. Engineering Fracture Mechanics. 2015;141:124-39.
[38] Evans AG, He MY, Hutchinson JW. Interface debonding and fiber cracking in brittle matrix composites.
Journal of the American Ceramic Society. 1989;72:2300-3.
[39] Cao H, Bischoff E, Sbaizero O, Rühle M, Evans AG, Marshall DB, Brennan JJ. Effect of Interfaces on
the Properties of Fiber ‐ Reinforced Ceramics. Journal of the American Ceramic Society.
1990;73:1691-9.
[40] Kerans RJ, Hay RS, Parthasarathy TA, Cinibulk MK. Interface design for oxidation‐resistant ceramic
composites. Journal of the American Ceramic Society. 2002;85:2599-632.
[41] Morgan PE, Marshall DB. Ceramic composites of monazite and alumina. Journal of the American
Ceramic Society. 1995;78:1553-63.
[42] Sánchez‐Herencia AJ, Pascual C, He J, Lange FF. ZrO2/ZrO2 layered composites for crack bifurcation.
Journal of the American Ceramic Society. 1999;82:1512-8.
[43] Lee W, Yoo Y-H, Shin H. Reconsideration of crack deflection at planar interfaces in layered systems.
Composites Science and Technology. 2004;64:2415-23.
[44] Bermejo R, Baudín C, Moreno R, Llanes L, Sánchez-Herencia A. Processing optimisation and fracture
behaviour of layered ceramic composites with highly compressive layers. Composites Science and
Technology. 2007;67:1930-8.
[45] Kavanagh JL, Menand T, Sparks RSJ. An experimental investigation of sill formation and propagation
in layered elastic media. Earth and Planetary Science Letters. 2006;245:799-813.
[46] Zhang X, Jeffrey RG. The role of friction and secondary flaws on deflection and re-initiation of hydraulic
fractures at orthogonal pre-existing fractures. Geophysical Journal International. 2006;166:1454-65.
[47] Zhang X, Jeffrey RG, Thiercelin M. Deflection and propagation of fluid-driven fractures at frictional
bedding interfaces: a numerical investigation. Journal of Structural Geology. 2007;29:396-410.
[48] Gudmundsson A. Toughness and failure of volcanic edifices. Tectonophysics. 2009;471:27-35.
[49] Gudmundsson A, Løtveit IF. Sills as fractured hydrocarbon reservoirs: examples and models. Geological
Society, London, Special Publications. 2014;374:251-71.
[50] Gudmundsson A, Simmenes TH, Larsen B, Philipp SL. Effects of internal structure and local stresses
on fracture propagation, deflection, and arrest in fault zones. Journal of Structural Geology.
2010;32:1643-55.
[51] Gudmundsson A, Loetveit IF. Dyke emplacement in a layered and faulted rift zone. Journal of
Volcanology and Geothermal Research. 2005;144:311-27.
[52] Carrère N, Martin E, Lamon J. The influence of the interphase and associated interfaces on the deflection
of matrix cracks in ceramic matrix composites. Composites Part A: Applied Science and Manufacturing.
2000;31:1179-90.
[53] Martin E, Leguillon D. Energetic conditions for interfacial failure in the vicinity of a matrix crack in
brittle matrix composites. International Journal of Solids and Structures. 2004;41:6937-48.
39 / 51
[54] Pompidou S, Lamon J. Analysis of crack deviation in ceramic matrix composites and multilayers based
on the Cook and Gordon mechanism. Composites Science and Technology. 2007;67:2052-60.
[55] Zhou R, Li Z, Sun J. Crack deflection and interface debonding in composite materials elucidated by the
configuration force theory. Composites Part B: Engineering. 2011;42:1999-2003.
[56] Leguillon D, Lacroix C, Martin E. Interface debonding ahead of a primary crack. Journal of the
Mechanics and Physics of Solids. 2000;48:2137-61.
[57] Martin E, Poitou B, Leguillon D, Gatt J. Competition between deflection and penetration at an interface
in the vicinity of a main crack. International Journal of Fracture. 2008;151:247-68.
[58] Leguillon D, Martin E. The strengthening effect caused by an elastic contrast—part I: the bimaterial
case. International Journal of Fracture. 2013;179:157-67.
[59] Mitts C, Naboulsi S, Przybyla C, Madenci E. Axisymmetric peridynamic analysis of crack deflection in
a single strand ceramic matrix composite. Engineering Fracture Mechanics. 2020:107074.
[60] Osman TM, Hassan HA, Lewandowski JJ. Interface effects on the quasi-static and impact toughness of
discontinuously reinforced aluminum laminates. Metallurgical and Materials Transactions A.
2008;39:1993-2006.
[61] Dahi-Taleghani A, Olson JE. Numerical modeling of multistranded-hydraulic-fracture propagation:
accounting for the interaction between induced and natural fractures. SPE journal. 2011;16:575-81.
[62] Gudmundsson A. Strengths and strain energies of volcanic edifices: implications for eruptions, collapse
calderas, and landslides. Natural Hazards and Earth System Sciences. 2012;12:2241.
[63] Duraiswami RA, Shaikh TN. Geology of the saucer-shaped sill near Mahad, western Deccan Traps,
India, and its significance to the flood basalt model. Bulletin of volcanology. 2013;75:731.
[64] Barnett ZA, Gudmundsson A. Numerical modelling of dykes deflected into sills to form a magma
chamber. Journal of Volcanology and Geothermal Research. 2014;281:1-11.
[65] Zhang QB, Zhao J. Effect of loading rate on fracture toughness and failure micromechanisms in marble.
Engineering Fracture Mechanics. 2013;102:288-309.
[66] Zhang Z, Suo Z. Split singularities and the competition between crack penetration and debond at a
bimaterial interface. International Journal of Solids and Structures. 2007;44:4559-73.
[67] Gu H, Siebrits E, Sabourov A. Hydraulic fracture modeling with bedding plane interfacial slip. SPE
Eastern Regional/AAPG Eastern Section Joint Meeting: Society of Petroleum Engineers; 2008.
[68] Alam M, Parmigiani JP, Kruzic JJ. An experimental assessment of methods to predict crack deflection
at an interface. Engineering Fracture Mechanics. 2017;181:116-29.
[69] Gaffney ES, Damjanac B, Valentine GA. Localization of volcanic activity: 2. Effects of pre-existing
structure. Earth and Planetary Science Letters. 2007;263:323-38.
[70] Menand T. The mechanics and dynamics of sills in layered elastic rocks and their implications for the
growth of laccoliths and other igneous complexes. Earth and Planetary Science Letters. 2008;267:93-9.
[71] Menand T. Physical controls and depth of emplacement of igneous bodies: A review. Tectonophysics.
2011;500:11-9.
[72] Browning J, Gudmundsson A. Caldera faults capture and deflect inclined sheets: an alternative
mechanism of ring dike formation. Bulletin of Volcanology. 2015;77:4.
[73] Taleghani AD. Fracture re-initiation as a possible branching mechanism during hydraulic fracturing.
44th US Rock Mechanics Symposium and 5th US-Canada Rock Mechanics Symposium: American
Rock Mechanics Association; 2010.
[74] Dahi Taleghani A, Olson JE. How natural fractures could affect hydraulic-fracture geometry. SPE
Journal. 2013;19:161-71.
[75] Suresh S, Sugimura Y, Tschegg E. The growth of a fatigue crack approaching a perpendicularly-oriented,
bimaterial interface. Scripta Metallurgica et Materialia. 1992;27:1189-94.
[76] Sugimura Y, Grondin L, Suresh S. Fatigue crack growth at arbitrary angles to bimaterial interfaces.
Scripta Metallurgica et Materialia. 1995;33.
40 / 51
[77] Mohamed AR, Hansen W. Micromechanical modeling of crack-aggregate interaction in concrete
materials. Cement and Concrete Composites. 1999;21:349-59.
[78] Kolednik O. The yield stress gradient effect in inhomogeneous materials. International Journal of Solids
and Structures. 2000;37:781-808.
[79] Kitey R, Phan A-V, Tippur H, Kaplan T. Modeling of crack growth through particulate clusters in brittle
matrix by symmetric-Galerkin boundary element method. International Journal of Fracture.
2006;141:11-25.
[80] Nguyen T-T, Yvonnet J, Waldmann D, He Q-C. Phase field modeling of interfacial damage in
heterogeneous media with stiff and soft interphases. Engineering Fracture Mechanics. 2019;218:106574.
[81] Abedi R, Clarke PL. A computational approach to model dynamic contact and fracture mode transitions
in rock. Computers and Geotechnics. 2019;109:248-71.
[82] Wang H. Hydraulic fracture propagation in naturally fractured reservoirs: Complex fracture or fracture
networks. Journal of Natural Gas Science and Engineering. 2019;68:102911.
[83] Blaese D, Garcia DE, Guglielmi P, Hotza D, Fredel MC, Janssen R. ZrO2 fiber-matrix interfaces in
alumina fiber-reinforced model composites. Journal of the European Ceramic Society. 2015;35:1593-8.
[84] Lei J, Wang Y-S, Gross D. Numerical simulation of crack deflection and penetration at an interface in
a bi-material under dynamic loading by time-domain boundary element method. International Journal
of Fracture. 2008;149:11.
[85] Carolan D, Tuković Ž, Murphy N, Ivanković A. Arbitrary crack propagation in multi-phase materials
using the finite volume method. Computational Materials Science. 2013;69:153-9.
[86] Buyukozturk O, Hearing B. Crack propagation in concrete composites influenced by interface fracture
parameters. International Journal of Solids and Structures. 1998;35:4055-66.
[87] Ma J, Wang H, Weng L, Tan G. Effect of porous interlayers on crack deflection in ceramic laminates.
Journal of the European Ceramic Society. 2004;24:825-31.
[88] Kovar D, Thouless M, Halloran JW. Crack deflection and propagation in layered silicon nitride/boron
nitride ceramics. Journal of the American Ceramic Society. 1998;81:1004-112.
[89] Kim B-N, Wakayama S, Kawahara M. Characterization of 2-dimensional crack propagation behavior
by simulation and analysis. International journal of fracture. 1996;75:247-59.
[90] Mousavi ST, Richart N, Wolff C, Molinari J-F. Dynamic crack propagation in a heterogeneous ceramic
microstructure, insights from a cohesive model. Acta Materialia. 2015;88:136-46.
[91] Li Y, Zhou M. Effect of competing mechanisms on fracture toughness of metals with ductile grain
structures. Engineering Fracture Mechanics. 2019;205:14-27.
[92] Virgo S, Abe S, Urai JL. Extension fracture propagation in rocks with veins: Insight into the crack‐
seal process using Discrete Element Method modeling. Journal of Geophysical Research: Solid Earth.
2013;118:5236-51.
[93] Becher PF, Hwang S-L, Hsueh C-H. Using microstructure to attack the brittle nature of silicon nitride
ceramics. MRS Bulletin. 1995;20:23-7.
[94] Becher aP, Sun E, Hsueh C-H, Alexander K, Hwang S-L, Waters S, Westmoreland C-G. Debonding of
interfaces between beta-silicon nitride whiskers and Si Al Y oxynitride glasses. Acta Materialia.
1996;44:3881-93.
[95] Neuendorf R, Saiz E, Tomsia A, Ritchie R. Adhesion between biodegradable polymers and
hydroxyapatite: Relevance to synthetic bone-like materials and tissue engineering scaffolds. Acta
Biomaterialia. 2008;4:1288-96.
[96] Kruzic J, Satet R, Hoffmann M, Cannon R, Ritchie R. The utility of R ‐ curves for understanding
fracture toughness‐strength relations in bridging ceramics. Journal of the American Ceramic Society.
2008;91:1986-94.
41 / 51
[97] Ebrahimi M, Balint D, Sutton A, Dini D. A discrete crack dynamics model of toughening in brittle
polycrystalline material by crack deflection. Engineering Fracture Mechanics. 2019;214:95-111.
[98] Sun EY, Becher PF, Plucknett KP, Hsueh CH, Alexander KB, Waters SB, Hirao K, Brito ME.
Microstructural design of silicon nitride with improved fracture toughness: II, effects of yttria and
alumina additives. Journal of the American Ceramic Society. 1998;81:2831-40.
[99] Tavallali A, Vervoort A. Failure of layered sandstone under Brazilian test conditions: effect of microscale parameters on macro-scale behaviour. Rock mechanics and rock engineering. 2010;43:641-53.
[100] Zhang F, Dontsov E, Mack M. Fully coupled simulation of a hydraulic fracture interacting with natural
fractures with a hybrid discrete‐continuum method. International Journal for Numerical and Analytical
Methods in Geomechanics. 2017;41:1430-52.
[101] Chen H, Zhang C, Lu Q, Chen H, Yang Z, Wen Y, Hu S, Chen L. A two-set order parameters phasefield modeling of crack deflection/penetration in a heterogeneous microstructure. Computer Methods in
Applied Mechanics and Engineering. 2019;347:1085-104.
[102] Roy U, Zhou M. A computational framework for predicting the fracture toughness of metals as function
of microstructure. Journal of the Mechanics and Physics of Solids. 2020:103955.
[103] Li Y, Zhou M. Prediction of fracturess toughness of ceramic composites as function of microstructure:
II. analytical model. Journal of the Mechanics and Physics of Solids. 2013;61:489-503.
[104] Naslain RR. The design of the fibre-matrix interfacial zone in ceramic matrix composites. Composites
Part A: Applied Science and Manufacturing. 1998;29:1145-55.
[105] Marshall DB, Ratto JJ, Lange FF. Enhanced fracture toughness in layered microcomposites of Ce‐
ZrO2 and Al2O3. Journal of the American Ceramic Society. 1991;74:2979-87.
[106] Pavlacka R, Bermejo R, Chang Y, Green DJ, Messing GL. Fracture behavior of layered alumina
microstructural composites with highly textured layers. Journal of the American Ceramic Society.
2013;96:1577-85.
[107] Chang Y, Bermejo R, Messing GL, Marshall D. Improved fracture behavior of alumina microstructural
composites with highly textured compressive layers. Journal of the American Ceramic Society.
2014;97:3643-51.
[108] Brach S, Hossain M, Bourdin B, Bhattacharya K. Anisotropy of the effective toughness of layered
media. Journal of the Mechanics and Physics of Solids. 2019;131:96-111.
[109] Blanks K, Kristoffersson A, Carlström E, Clegg W. Crack deflection in ceramic laminates using porous
interlayers. Journal of the European Ceramic Society. 1998;18:1945-51.
[110] Nam KH, Park IH, Ko SH. Patterning by controlled cracking. Nature. 2012;485:221-4.
[111] Dundurs J. Discussion:“Edge-bonded dissimilar orthogonal elastic wedges under normal and shear
loading”(Bogy, DB, 1968, ASME J. Appl. Mech., 35, pp. 460–466). 1969.
[112] Saied M, Lloyd I, Haller W, Lawn B. Joining dental ceramic layers with glass. Dental Materials.
2011;27:1011-6.
[113] Guo J, Luo B, Lu C, Lai J, Ren J. Numerical investigation of hydraulic fracture propagation in a layered
reservoir using the cohesive zone method. Engineering Fracture Mechanics. 2017;186:195-207.
[114] Hsueh C, Avellar L, Bourdin B, Ravichandran G, Bhattacharya K. Stress fluctuation, crack renucleation
and toughening in layered materials. Journal of the Mechanics and Physics of Solids. 2018;120:68-78.
[115] Kim JW, Bhowmick S, Hermann I, Lawn BR. Transverse fracture of brittle bilayers: Relevance to
failure of all ‐ ceramic dental crowns. Journal of Biomedical Materials Research Part B: Applied
Biomaterials: An Official Journal of The Society for Biomaterials, The Japanese Society for
Biomaterials, and The Australian Society for Biomaterials and the Korean Society for Biomaterials.
2006;79:58-65.
[116] Singh RP, Parameswaran V. An experimental investigation of dynamic crack propagation in a brittle
material reinforced with a ductile layer. Optics and lasers in engineering. 2003;40:289-306.
42 / 51
[117] Huang Z, Pan Z, Li H, Wei Q, Li X. Hidden energy dissipation mechanism in nacre. Journal of
Materials Research. 2014;29:1573.
[118] Zimmermann EA, Gludovatz B, Schaible E, Busse B, Ritchie RO. Fracture resistance of human cortical
bone across multiple length-scales at physiological strain rates. Biomaterials. 2014;35:5472-81.
[119] Sundaram BM, Tippur HV. Dynamic Penetration and Bifurcation of a Crack at an Interface in a
Transparent Bi-Layer: Effect of Impact Velocity. Dynamic Behavior of Materials, Volume 1: Springer;
2017. p. 69-74.
[120] Sundaram BM, Tippur HV. Dynamics of crack penetration vs. branching at a weak interface: an
experimental study. Journal of the Mechanics and Physics of Solids. 2016;96:312-32.
[121] Xu LR, Huang YY, Rosakis AJ. Dynamic crack deflection and penetration at interfaces in
homogeneous materials: experimental studies and model predictions. Journal of the Mechanics and
Physics of Solids. 2003;51:461-86.
[122] Wang Y, Yang R. Study of the dynamic fracture characteristics of coal with a bedding structure based
on the NSCB impact test. Engineering Fracture Mechanics. 2017;184:319-38.
[123] Chalivendra V, Rosakis A. Interaction of dynamic mode-I cracks with inclined interfaces. Engineering
fracture mechanics. 2008;75:2385-97.
[124] Xu LR, Rosakis AJ. Impact failure characteristics in sandwich structures. Part II: Effects of impact
speed and interfacial strength. International Journal of Solids and Structures. 2002;39:4237-48.
[125] Sundaram B, Tippur H. Dynamic crack growth normal to an interface in bi-layered materials: an
experimental study using digital gradient sensing technique. Experimental Mechanics. 2016;56:37-57.
[126] Parab ND, Chen WW. Crack propagation through interfaces in a borosilicate glass and a glass ceramic.
International Journal of Applied Glass Science. 2014;5:353-62.
[127] Paggi M, Reinoso J. Revisiting the problem of a crack impinging on an interface: a modeling
framework for the interaction between the phase field approach for brittle fracture and the interface
cohesive zone model. Computer Methods in Applied Mechanics and Engineering. 2017;321:145-72.
[128] Elser M, Tschegg E, Stanzl-Tschegg S. Fracture behaviour of polypropylene-fibre-reinforced concrete
under biaxial loading: An experimental investigation. Composites science and technology.
1996;56:933-45.
[129] Blanton TL. An experimental study of interaction between hydraulically induced and pre-existing
fractures. SPE unconventional gas recovery symposium: Society of Petroleum Engineers; 1982.
[130] Philipp SL, Afşar F, Gudmundsson A. Effects of mechanical layering on hydrofracture emplacement
and fluid transport in reservoirs. Frontiers in Earth Science. 2013;1:4.
[131] Renshaw C, Pollard D. An experimentally verified criterion for propagation across unbounded
frictional interfaces in brittle, linear elastic materials. International Journal of Rock Mechanics and
Mining Sciences & Geomechanics Abstracts: Elsevier; 1995. p. 237-49.
[132] Gu H, Weng X. Criterion for fractures crossing frictional interfaces at non-orthogonal angles. 44th US
rock mechanics symposium and 5th US-Canada rock mechanics symposium: American Rock
Mechanics Association; 2010.
[133] Xu W, Zhao J, Rahman SS, Li Y, Yuan Y. A comprehensive model of a hydraulic fracture interacting
with a natural fracture: analytical and numerical solution. Rock Mechanics and Rock Engineering.
2019;52:1095-113.
[134] Nasseri M, Mohanty B, Robin P-Y. Characterization of microstructures and fracture toughness in five
granitic rocks. International Journal of Rock Mechanics and Mining Sciences. 2005;42:450-60.
[135] Fredrich JT, Evans B, Wong TF. Effect of grain size on brittle and semibrittle strength: Implications
for micromechanical modelling of failure in compression. Journal of Geophysical Research: Solid Earth.
1990;95:10907-20.
43 / 51
[136] Huang J, Wang S. An experimental investigation concerning the comprehensive fracture toughness of
some brittle rocks. International Journal of Rock Mechanics and Mining Sciences & Geomechanics
Abstracts: Elsevier; 1985. p. 99-104.
[137] Hatzor YH, Zur A, Mimran Y. Microstructure effects on microcracking and brittle failure of dolomites.
Tectonophysics. 1997;281:141-61.
[138] Dai F, Xia K. Laboratory measurements of the rate dependence of the fracture toughness anisotropy of
Barre granite. International Journal of Rock Mechanics and Mining Sciences. 2013;60:57-65.
[139] Li X, Li H, Zhao J. The role of transgranular capability in grain-based modelling of crystalline rocks.
Computers and Geotechnics. 2019;110:161-83.
[140] Ju M, Li J, Li X, Zhao J. Fracture surface morphology of brittle geomaterials influenced by loading
rate and grain size. International Journal of Impact Engineering. 2019;133:103363.
[141] Olsson WA. Grain size dependence of yield stress in marble. Journal of Geophysical Research.
1974;79:4859-62.
[142] Wong RH, Chau K, Wang P. Microcracking and grain size effect in Yuen Long marbles. International
Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts: Elsevier; 1996. p. 47985.
[143] Nasseri M, Schubnel A, Young R. Coupled evolutions of fracture toughness and elastic wave velocities
at high crack density in thermally treated Westerly granite. International Journal of Rock Mechanics
and Mining Sciences. 2007;44:601-16.
[144] Nasseri M, Grasselli G, Mohanty B, Wirth J, Braun M. Experimental relationship between fracture
toughness and fracture roughness in anisotropic granitic rocks. 1st Canada-US Rock Mechanics
Symposium: American Rock Mechanics Association; 2007.
[145] Nasseri M, Rezanezhad F, Young R. Analysis of fracture damage zone in anisotropic granitic rock
using 3D X-ray CT scanning techniques. International Journal of Fracture. 2011;168:1-13.
[146] Grasselli G, Wirth J, Egger P. Quantitative three-dimensional description of a rough surface and
parameter evolution with shearing. International Journal of Rock Mechanics and Mining Sciences.
2002;39:789-800.
[147] Nasseri M, Tatone B, Grasselli G, Young R. Fracture toughness and fracture roughness
interrelationship in thermally treated Westerly granite. Pure and Applied Geophysics. 2009;166:801-22.
[148] Dai F, Xia K, Nasseri M. Micromechanical model for the rate dependence of the fracture toughness
anisotropy of Barre granite. International Journal of Rock Mechanics and Mining Sciences.
2013;63:113-21.
[149] Ju M, Li J, Li J, Zhao J. Loading rate effects on anisotropy and crack propagation of weak bedding
plane-rich rocks. Engineering Fracture Mechanics. 2020:106983.
[150] Přikryl R. Some microstructural aspects of strength variation in rocks. International Journal of Rock
Mechanics and Mining Sciences. 2001;38:671-82.
[151] Iqbal M, Mohanty B. Experimental calibration of ISRM suggested fracture toughness measurement
techniques in selected brittle rocks. Rock Mechanics and Rock Engineering. 2007;40:453.
[152] Yu M, Wei C, Niu L, Li S, Yu Y. Calculation for tensile strength and fracture toughness of granite
with three kinds of grain sizes using three-point-bending test. PloS One. 2018;13.
[153] Sabri M, Ghazvinian A, Nejati HR. Effect of particle size heterogeneity on fracture toughness and
failure mechanism of rocks. International Journal of Rock Mechanics and Mining Sciences. 2016;81:7985.
[154] Dai F, Xia K, Zuo JP, Zhang R, Xu NW. Static and dynamic flexural strength anisotropy of barre
granite. Rock Mechanics and Rock Engineering. 2013;46:1589-602.
[155] Li X, Zhang Q, Li H, Zhao J. Grain-based discrete element method (GB-DEM) modelling of multiscale fracturing in rocks under dynamic loading. Rock Mechanics and Rock Engineering. 2018;51:3785817.
44 / 51
[156] Shi X, Yao W, Liu Da, Xia K, Tang T, Shi Y. Experimental study of the dynamic fracture toughness
of anisotropic black shale using notched semi-circular bend specimens. Engineering Fracture Mechanics.
2019;205:136-51.
[157] Xu Y, Dai F, Xu N, Zhao T. Numerical investigation of dynamic rock fracture toughness determination
using a semi-circular bend specimen in split Hopkinson pressure bar testing. Rock Mechanics and Rock
Engineering. 2016;49:731-45.
[158] Oh S-W, Min G-J, Park S-W, Kim M-S, Obara Y, Cho S-H. Anisotropic influence of fracture toughness
on loading rate dependency for granitic rocks. Engineering Fracture Mechanics. 2019;221:106677.
[159] Zhao Y, Gong S, Hao X, Peng Y, Jiang Y. Effects of loading rate and bedding on the dynamic fracture
toughness of coal: Laboratory experiments. Engineering Fracture Mechanics. 2017;178:375-91.
[160] Dai F, Xia KW. Laboratory measurements of the rate dependence of the fracture toughness anisotropy
of Barre granite. International Journal of Rock Mechanics and Mining Sciences. 2013;60:57-65.
[161] Shi X, Liu Da, Yao W, Shi Y, Tang T, Wang B, Han W. Investigation of the anisotropy of black shale
in dynamic tensile strength. Arabian Journal of Geosciences. 2018;11.
[162] Qiu J, Li D, Li X. Dynamic failure of a phyllite with a low degree of metamorphism under impact
Brazilian test. International Journal of Rock Mechanics and Mining Sciences. 2017;94:10-7.
[163] Zhao Y, Zhao G-F, Jiang Y, Elsworth D, Huang Y. Effects of bedding on the dynamic indirect tensile
strength of coal: Laboratory experiments and numerical simulation. International Journal of Coal
Geology. 2014;132:81-93.
[164] Zhang X, Ou X, Gong F, Yang J. Effects of Bedding on The Dynamic Compressive Properties of Low
Anisotropy Slate. Rock Mechanics and Rock Engineering. 2018;52:981-90.
[165] Zhou C, Xu C, Karakus M, Shen J. A particle mechanics approach for the dynamic strength model of
the jointed rock mass considering the joint orientation. International Journal for Numerical and
Analytical Methods in Geomechanics. 2019.
[166] Liu X, Dai F, Zhang R, Liu J. Static and dynamic uniaxial compression tests on coal rock considering
the bedding directivity. Environmental Earth Sciences. 2015;73:5933-49.
[167] Thouless M, Cao H, Mataga P. Delamination from surface cracks in composite materials. Journal of
Materials Science. 1989;24:1406-12.
[168] Martinez D, Gupta V. Energy criterion for crack deflection at an interface between two orthotropic
media. Journal of the Mechanics and Physics of Solids. 1994;42:1247-71.
[169] He MY, Evans AG, Hutchinson JW. Crack deflection at an interface between dissimilar elastic
materials: role of residual stresses. International Journal of Solids and Structures. 1994;31:3443-55.
[170] Lu M-C, Erdogan F. Stress intensity factors in two bonded elastic layers containing cracks
perpendicular to and on the interface. Part 1: Analysis. 1980.
[171] Parmigiani J, Thouless M. The roles of toughness and cohesive strength on crack deflection at
interfaces. Journal of the Mechanics and Physics of Solids. 2006;54:266-87.
[172] Inglis CE. Stresses in a plate due to the presence of cracks and sharp corners. Trans Inst Naval Archit.
1913;55:219-41.
[173] Gupta V, Argon A, Suo Z. Crack deflection at an interface between two orthotopic media. Journal of
Applied Mechanics. 1992;59:S79-S87.
[174] Lee W, Clegg W. The deflection of cracks at interfaces. Key Engineering Materials: Trans Tech Publ;
1996. p. 193-208.
[175] Cook T, Erdogan F. Stresses in bonded materials with a crack perpendicular to the interface.
International Journal of Engineering Science. 1972;10:677-97.
[176] Irwin GR. Analysis of stresses and strains near the end of a crack transversing a plate. Trans ASME,
Ser E, J Appl Mech. 1957;24:361-4.
[177] Siegmund T, Fleck N, Needleman A. Dynamic crack growth across an interface. International Journal
of Fracture. 1997;85:381-402.
45 / 51
[178] Gupta V, Yuan J, Martinez D. Calculation, measurement, and control of interface strength in
composites. Journal of the American Ceramic Society. 1993;76:305-15.
[179] Shaw M, Marshall D, Dalgleish B, Dadkhah M, He M, Evans A. Fatigue crack growth and stress
redistribution at interfaces. International Journal of Fatigue. 1996;3:213.
[180] Vahab M, Khoei A, Khalili N. An X-FEM technique in modeling hydro-fracture interaction with
naturally-cemented faults. Engineering Fracture Mechanics. 2019;212:269-90.
[181] Chen D-H. A crack normal to and terminating at a bimaterial interface. Engineering Fracture Mechanics.
1994;49:517-32.
[182] Zak A, Williams ML. Crack point stress singularities at a bi-material interface. California Institute of
Technology, Pasadena, CA. 1962.
[183] Suga T, Elssner G, Schmauder S. Composite parameters and mechanical compatibility of material
joints. Journal of Composite Materials. 1988;22:917-34.
[184] Sugimura Y, Lim P, Shih C, Suresh S. Fracture normal to a bimaterial interface: effects of plasticity
on crack-tip shielding and amplification. Acta Metallurgica et Materialia. 1995;43:1157-69.
[185] Kim A, Suresh S, Shih C. Plasticity effects on fracture normal to interfaces with homogeneous and
graded compositions. International Journal of Solids and Structures. 1997;34:3415-32.
[186] Griffith AA. VI. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal
Society of London Series A, Containing Papers of a Mathematical or Physical Character. 1921;221:16398.
[187] Faber KT, Evans AG. Crack deflection processes—I. Theory. Acta Metallurgica. 1983;31:565-76.
[188] Kwang ML, Oral B, Ayad O. Fracture analysis of mortar-aggregate interfaces in concrete. Journal of
Engineering Mechanics. 1992;118:2031-46.
[189] He MY, Bartlett A, Evans AG, Hutchinson JW. Kinking of a crack out of an interface: role of in‐
plane stress. Journal of the American Ceramic Society. 1991;74:767-71.
[190] Tu WC, Lange FF, Evans AG. Concept for a damage ‐ tolerant ceramic composite with “strong”
interfaces. Journal of the American Ceramic Society. 1996;79:417-24.
[191] He M, Hsueh C, Becher P. Deflection versus penetration of a wedge-loaded crack: effects of branchcrack length and penetrated-layer width. Composites Part B: Engineering. 2000;31:299-308.
[192] Warrier S, Majumdar B, Miracle D. Interface effects on crack deflection and bridging during fatigue
crack growth of titanium matrix composites. Acta Materialia. 1997;45:4969-80.
[193] Kovar D, King BH, Trice RW, Halloran JW. Fibrous monolithic ceramics. Journal of the American
Ceramic Society. 1997;80:2471-87.
[194] Ahn B, Curtin W, Parthasarathy T, Dutton R. Criteria for crack deflection/penetration criteria for fiberreinforced ceramic matrix composites. Composites Science and Technology. 1998;58:1775-84.
[195] Leguillon D, Sanchez-Palencia E. Fracture in heterogeneous materials, weak and strong singularities.
New Advances in Computational Structural Mechanics: Elsevier Amsterdam; 1992. p. 423-34.
[196] Martin E, Leguillon D, Lacroix C. A revisited criterion for crack deflection at an interface in a brittle
bimaterial. Composites Science and Technology. 2001;61:1671-9.
[197] Li Y, Zhou M. Prediction of fracture toughness scatter of composite materials. Computational Materials
Science. 2016;116:44-51.
[198] Zeng X, Wei Y. Crack deflection in brittle media with heterogeneous interfaces and its application in
shale fracking. Journal of the Mechanics and Physics of Solids. 2017;101:235-49.
[199] Rupp D, Weygand S. Anisotropic fracture behaviour and brittle-to-ductile transition of polycrystalline
tungsten. Philosophical Magazine. 2010;90:4055-69.
[200] Pro JW, Sehr S, Lim RK, Petzold LR, Begley MR. Conditions controlling kink crack nucleation out
of, and delamination along, a mixed-mode interface crack. Journal of the Mechanics and Physics of
Solids. 2018;121:480-95.
[201] Sih G, Rice J. Plane problems of cracks in dissimilar materials. J Appl Mech. 1965;32:418-23.
46 / 51
[202] Bogy DB. Two edge-bonded elastic wedges of different materials and wedge angles under surface
tractions. Journal of Applied Mechanics. 1971;38:377-386.
[203] Rao KV, Soboyejo W, Ritchie R. Ductile-phase toughening and fatigue-crack growth in Nb-reinforced
molybdenum disilicide intermetallic composites. Metallurgical Transactions A. 1992;23:2249-57.
[204] Cao H, Evans A. An experimental study of the fracture resistance of bimaterial interfaces. Mechanics
of Materials. 1989;7:295-304.
[205] Howard S, Phillipps A, Clyn T. The interpretation of data from the four-point bend delamination test
to measure interfacial fracture toughness. Composites. 1993;24:103-12.
[206] Jensen HM. Mixed mode interface fracture criteria. Acta metallurgica et materialia. 1990;38:2637-44.
[207] Hutchinson JW, Suo Z. Mixed mode cracking in layered materials. Advances in Applied Mechanics:
Elsevier; 1991. p. 63-191.
[208] París F, Correa E, Mantič V. Kinking of transversal interface cracks between fiber and matrix. 2007.
[209] Tvergaard V. Effect of ductile particle debonding during crack bridging in ceramics. International
Journal of Mechanical Sciences. 1992;34:635-49.
[210] Krstic VV, Nicholson PS, Hoagland RG. Toughening of glasses by metallic particles. Journal of the
American Ceramic Society. 1981;64:499-504.
[211] Sigl LS, Mataga P, Dalgleish B, McMeeking R, Evans A. On the toughness of brittle materials
reinforced with a ductile phase. Acta Metallurgica. 1988;36:945-53.
[212] McMeeking R, Evans A. Mechanics of transformation‐toughening in brittle materials. Journal of the
American Ceramic Society. 1982;65:242-6.
[213] Budiansky B, Hutchinson JW, Evans AG. Matrix fracture in fiber-reinforced ceramics. Journal of the
Mechanics and Physics of Solids. 1986;34:167-89.
[214] Hutchinson JW. Crack tip shielding by microcracking in brittle solids. Acta Metall. 1987;35:1605-19.
[215] Depinoy S, Strepenne F, Massart T, Godet S, Pardoen T. Interface toughening in multilayered systems
through compliant dissipative interlayers. Journal of the Mechanics and Physics of Solids. 2019;130:120.
[216] Beyerle DS, Spearing SM, Zok FW, Evans AG. Damage and failure in unidirectional ceramic–matrix
composites. Journal of the American Ceramic Society. 1992;75:2719-25.
[217] Lee W, Myoung J-M, Yoo Y-H, Shin H. Effect of thermal misfit stress on crack deflection at planar
interfaces in layered systems. Composites Science and Technology. 2006;66:435-43.
[218] Lu TJ. Crack branching in all‐oxide ceramic composites. Journal of the American Ceramic Society.
1996;79:266-74.
[219] Leguillon D. Strength or toughness? A criterion for crack onset at a notch. European Journal of
Mechanics-A/Solids. 2002;21:61-72.
[220] Alam M, Grimm B, Parmigiani JP. Effect of incident angle on crack propagation at interfaces.
Engineering Fracture Mechanics. 2016;162:155-63.
[221] Leguillon D, Piat R. Fracture of porous materials–Influence of the pore size. Engineering Fracture
Mechanics. 2008;75:1840-53.
[222] Leguillon D, Lacroix C, Martin É. Matrix crack deflection at an interface between a stiff matrix and a
soft inclusion. Comptes Rendus de l'Académie des Sciences-Series IIB-Mechanics-Physics-Astronomy.
2000;328:19-24.
[223] Chamat A, Aden-Ali S, Gilgert J, Petit E, Nasri K, Abbadi M, Azari Z. Crack behaviour in zinc coating
and at the interface zinc-hot galvanised TRIP steel 800. Engineering Fracture Mechanics. 2013;114:1225.
[224] Freund LB. Dynamic fracture mechanics: Cambridge University Press; 1990.
[225] Xu LR, Wang P. Dynamic fracture mechanics analysis of failure mode transitions along weakened
interfaces in elastic solids. Engineering Fracture Mechanics. 2006;73:1597-614.
47 / 51
[226] Djoković JM, Nikolić RR, Šumarac DM, Bujnak J. Analysis based on the energy release rate criterion
of a dynamically growing crack approaching an interface. International Journal of Damage Mechanics.
2016;25:1170-83.
[227] Deng X. Complete complex series expansions of near-tip fields for steadily growing interface cracks
in dissimilar isotropic materials. Engineering Fracture Mechanics. 1992;42:237-42.
[228] Rice J. Elastic fracture mechanics concepts for interfacial cracks. Journal of Applied Mechanics.
1988;55:98-103.
[229] Yang W, Suo Z, Shih C. Mechanics of dynamic debonding. Proceedings of the Royal Society of
London Series A: Mathematical and Physical Sciences. 1991;433:679-97.
[230] Djoković JM, Nikolić RR, Bujňák J, Hadzima B. Criteria for prediction the interfacial crack growth in
concrete. Solid State Phenomena: Trans Tech Publ; 2017. p. 514-7.
48 / 51
Highlights
1. The effects of weak interfaces in rock-like materials and structures on crack propagation
are systematically reviewed.
2. The intrinsic mechanism of interface effects on crack propagation is discussed in terms
of crack deflection criterion.
3. Weak interface effects on rock dynamic failure are reviewed and the remaining issues
are extracted.
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Phenomenon: Weak interface effects on crack propagation
Single
deflection
Double
deflection
Direct
penetration
Mechanical feedback: Crack propagation on mechanical property
3.0
Deflection
Penetration
Deflection Penetration
2.5
2.0
Deflection
dominant
1.5
1.0
Type-2
Type-1
Mechanism: Crack deflection criterion
1.5
HH criterion
Martinez and Gupta
LS criterion
1
Martin et al.
FEM assisted
Гi / Гm
Type-1
KIc (MPa·m1/2)
KIc (Tpye-1) > KIc (Tpye-2)
B
A
Crack
Interface
Crack penetration
0.5
Crack deflection
0
-1.0
-0.5
0.0
0.5
α = (EA-EB)/(EA+EB)
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1.0
Penetration
dominant
Type-2
Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal
relationships that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be
considered as potential competing interests:
None.
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