A Microstructure-Based Model to Characterize ... - Springer Link

Rock Mech Rock Eng (2014) 47:435–452 DOI 10.1007/s00603-013-0402-y

ORIGINAL PAPER

A Microstructure-Based Model to Characterize Micromechanical Parameters Controlling Compressive and Tensile Failure in Crystallized Rock T. Kazerani • J. Zhao

Received: 7 December 2012 / Accepted: 18 March 2013 / Published online: 10 April 2013 Ó Springer-Verlag Wien 2013

Abstract A discrete element model is proposed to examine rock strength and failure. The model is implemented by UDEC which is developed for this purpose. The material is represented as a collection of irregular-sized deformable particles interacting at their cohesive boundaries. The interface between two adjacent particles is viewed as a flexible contact whose stress–displacement law is assumed to control the material fracture and fragmentation process. To reproduce rock anisotropy, an innovative orthotropic cohesive law is developed for contact which allows the interfacial shear and tensile behaviours to be different from each other. The model is applied to a crystallized igneous rock and the individual and interactional effects of the microstructural parameters on the material compressive and tensile failure response are examined. A new methodical calibration process is also established. It is shown that the model successfully reproduces the rock mechanical behaviour quantitatively and qualitatively. Ultimately, the model is used to understand how and under what circumstances micro-tensile and micro-shear cracking mechanisms control the material failure at different loading paths. Keywords Rock microstructure Microparameter Micro-fracture Orthotropic cohesive interface Discrete element method T. Kazerani (&) Civil Engineering Department, Faculty of Engineering, University of Nottingham, Nottingham, UK e-mail: [email protected] J. Zhao Laboratoire de Mećanique des Roches (LMR), Ecole Polytechnique Fe´de´rale de Lausanne (EPFL), Lausanne, Switzerland

List of Symbols a Crack half-width ac Contact surface area b Multiplier parameterizing fracture process zone thickness BTS Tensile strength of rock C Internal cohesion of model cc Contact cohesion D Contact damage variable dp Particle edge size dcs Contact critical displacement in shear dct Contact critical displacement in tension deff Contact effective displacement dn Normal separation over the contact surface ds Shear sliding over the contact surface dut Contact ultimate displacement tension e Base of the natural logarithm E Young’s modulus of intact rock Fc Concentrated force acting over the first contact next to the crack tip Fn Normal component of contact force Fs Shear component of contact force U Internal friction angle of model /c Contact friction angle G Shear modulus of intact rock Gf Fracture energy KI Mode-I stress intensity factor KIC Mode-I fracture toughness KII Mode-II stress intensity factor KIIC Mode-II fracture toughness ks Contact initial stiffness coefficients in shear kt Contact initial stiffness coefficients in tension l Length of fracture process zone m Integer parameterizing Lennard-Jones’ potential n Integer parameterizing Lennard-Jones’ potential

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436

m P Pc P r r r3 rc rf rn rres rt tc UCS w X x1 x2 z z0 zm f

T. Kazerani, J. Zhao

Poisson’s ratio of intact rock Inter-molecular force Cohesive force Lennard-Jones’ potential Distance from crack tip Stress applied on contact surface Lateral (confining) stress in triaxial testing Uniaxial compressive strength of model Remote tensile stress acting normal to the crack Induced stress acting on crack plane near the crack tip Post-failure residual strength of contact Tensile strength of model Contact tensile strength Uniaxial compressive strength of rock Thickness of fracture process zone Molecular bond energy Coded factors representing contact cohesion Coded factors representing contact friction angle Separation distance between two adjacent molecules Equilibrium spacing between two adjacent molecules z at maximum intermolecular force Parameter of Lennard-Jones’ potential

1 Introduction Igneous rock is in nature constituted by a microstructure composed of crystalized minerals where their individual mechanical properties and interaction to each other characterize failure behaviour of the material. Macroscopic strength, brittleness and failure pattern of a rock have been usually explained through developing constitutive laws. However, micromechanical studies suggest that these properties are in fact the extrinsic reflections of the responses that the material microstructure gives to the applied loading. In this context, material failure can be studied as the consequence of tensile and/or shear microfailure events progressing within the microstructure. The role of micro-tensile cracks in global compressive failure of brittle rock has been vastly explored. Investigations have indicated that the initiation and progression of fracture in a compressive sample are very much affected by the presence of grain-scale stress heterogeneity that introduces local tensile stress concentrations within the microstructure. These stress concentrations create micro-scale tensile fractures that propagate and coalesce ultimately into macroscopic failure surfaces (e.g. Van de Steen et al. 2003; Gallagher et al. 1974; Blair and Cook 1998; Tapponier and

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Brace 1976; Wong 1982; Fredrich et al. 1995; Wong et al. 2006). Therefore, tensile mechanisms are dominantly accepted as the primary process of damage in rocks, and it is believed that shear mechanisms become dominant merely after tensile damage sufficiently proceeds (Fairhurst and Cook 1966; Brace et al. 1966; Hallbauer et al. 1973; Fonseka et al. 1985; Martin and Chandler 1994; Diederichs 2003). This research aims to create a numerical model to study the role of the micro-tensile and micro-shear cracking in global failure of igneous rock. The model involves a 2D microstructural simulation based on the discrete element method (DEM) coupled with the fracture process zone (FPZ) theory and establishing a straightforward calibration process. The universal distinct element code (UDEC) has been employed to impellent the model. The software is developed to incorporate a new constitutive law for particle boundary behaviour. This is achieved by establishing a user-defined model through creating a dynamic link library (DLL) and attaching it to the code. In addition, a preprocessor program based on the Delaunay triangulation algorithm (Du 1996) is created to generate arbitrarily sized triangular particles. This development was needed to enhance the model efficiency as the UDEC imbedded Voronoi algorithm is too slow that makes it very hard to generate small-sized particle assemblages. The model assumes rock microstructure as an assemblage of deformable particles whose boundary behaviour is dominated by the so-called cohesive contact law. In this simulation, the particles are assumed to be of the same elastic properties with the rock, and the interfaces between the particles account for potential fracture. Particles are individually discretized by CST elements, i.e., constantstrain triangles as presented in Fig. 1. The particle assemblage is generated randomly to capture rock heterogeneity and diverse fracture patterns. The novelty of this research is to develop a DEM model which can predict the compressive, tensile, and shear behaviours of rock concurrently. Since this study assumes deformable particles, a calculation for the inter-particle contact stiffness has been developed. The study also involves establishing a definite framework for model

Fig. 1 Assemblage of particles reproducing rock microstructure and particle composition

A Microstructure-Based Model

calibration. Extendable to a variety of similar simulations, the calibration process is carried out through implementing a disciplined set of laboratory test simulations and the comparison of the numerical outputs (i.e., strength, elastic constants, and fracture pattern) with the laboratory measurements. However, laboratory data fit is not the goal of this research. It actually aims at the exploration of physical interpretations for the model microparameter in terms of the material macroscopic properties. This objective is achieved through adopting statistical methods and developing closed-form equations. Using the model, the relation between macroscopic failure behaviour of rock and its microstructure properties, i.e., grain-scale tensile and shear strength and grain boundary friction is investigated. At last, the role of tensile and shear failure micro-mechanisms in terms of precedence, contribution and dominance is compared and discussed.

2 Discontinuum-Based Modelling of Failure Representation of fracturing process by separation through the inter-element boundaries began with developing the elements boundary breaking approaches in finite element method (FEM), which were based on either the fracture mechanics theory or failure criteria governing the detachment of the inter-element boundaries. The first approach is used in several FEM codes like ABAQUS (ABAQUS 2005), FRANC (Agrawal and Sun 2004) and MARC (MSC Software 2007). The second method inserts interface elements along the inter-element boundaries. That has been popularly applied in concrete and rock modelling (Alfaiate et al. 1997; Cho et al. 2003; Cho and Kaneko 2004). The most successful development of the element boundary breaking approach is the cohesive zone modelling (CZM) which dates back to the work of Hillerborg et al. (1976) and Belytschko et al. (1976). The CZM has been successfully used in the simulation of fracture and fragmentation in brittle materials, multiple discrete crack propagation, and dynamic crack growth in ceramic materials (Block et al. 2007; Camacho and Ortiz 1996; Elmarakbi et al. 2009; Karedla and Reddy 2007; Li et al. 2004; Molinari et al. 2007; Murphy and Ivankovic 2005; Pinho et al. 2006; Remmers et al. 2008; Tomar et al. 2004; Yang and Chen 2005; Zhai et al. 2004; Zhou and Molinari 2004; Zhou et al. 2005). 2.1 DEM as Numerical Approach The discrete element method has been vastly used to capture the sequences of separation and reattachment observed in the fragmentation process of brittle materials.

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Formulation and development of the DEM have progressed over years since the pioneering study of Cundall (1971). Jing and Stephansson (2007) have comprehensively provided the fundamentals of the DEM and its application in rock mechanics. According to the solution algorithm used, the DEM implementations can be divided into two groups of explicit and implicit formulations. The most popular representations of the explicit DEM are the computer codes of PFC and UDEC (ITASCA Consulting Group and 2008). Cundall and Strack (1979) showed how the DEM could be employed to simulate behaviour of granular media and Potyondy and Cundall (2004) showed how a similar approach could be used to model rock material as a dense packing of particles interacting at their contact points. The significant advantage of this approach is to model crack as a real discontinuity. In addition, complicated empirical constitutive behaviour can be replaced with simple particle/ contact logic. In this context, stress–displacement relation of contact, i.e. micromechanical constitutive law characterizes macroscopic behaviour of the model. Generally speaking, two types of particle geometry have been adopted to reproduce rock texture, i.e. rounded grains, vastly examined by particle flow code (PFC) (e.g. Potyondy and Cundall 2004; Yoon 2007; Wang and Tonon 2009; Scho¨pfer et al. 2009; Lobo-Guerrero et al. 2006; Lobo-Guerrero and Vallejo 2010), and polygonal particles usually produced through the Voronoi diagram generator or the Delaunay triangulation algorithm (e.g. Lorig and Cundall 1987; Kazerani and Zhao 2010; Kazerani et al. 2012; Lan et al. 2010; Mahabadi et al. 2010). Geometrically, it may sound that the polygonal configuration may be the most representative of the mineral structure observed in crystalline rock. However, the vast majority of micromechanical models have been carried out by rounded (disc-shaped) particles. As discussed by Potyondy and Cundall (2004), modelling using rounded particles fails in accurate reproduction of rock mechanical behaviour. They showed that calibrating PFC to the uniaxial strength gave a very low triaxial strength. In addition, predicted Brazilian tensile strength of rock was approximately 0.25 of uniaxial compressive strength. Comparing various types of rocks, this value is unacceptably high as the ratio of tensile to compressive strength is typically reported around 0.05–0.1 (Hoek and Brown 1998). It is generally argued that these shortcomings are because the rounded particles cannot represent the irregular-shaped and interlocked grains of rock appropriately. To resolve this problem, different solutions have been proposed, e.g. flat joint formulation (two contact point per bond) in PFC provided by ITASCA (ITASCA Consulting Group and 2008), controlled bond density configuration in YADE (Kozicki and Donze´ 2008), and in earlier time the so-called

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cluster (Potyondy and Cundall 2004) and clump logics (Cho et al. 2007). The clumped and cluster models are more convincing in reproducing the random geometry of igneous rock grains; however, inter-grain contact is still puncticular (no physical contact area). In addition, some PFC microparameter, e.g. coefficient of friction, contact modulus and parallel bond modulus show no effect on the model response, and thus are deprived of physical sense. As Kazerani and Zhao (2010), Mahabadi et al. (2010) and Lan et al. (2010) mentioned, using polygonal particles enhances the simulation as they can represent mineral interaction efficiently.

3 Numerical Model The solution scheme in UDEC is identical to that used by the explicit finite difference method for continuum analysis. The details can be found in the software documentation (ITASCA Consulting Group 2008). Briefly speaking, solving procedure alternates between the application of a stress–displacement law at all the contacts and Newton’s second law for all the particles. The contact stress–displacement law is used to find the contact stresses from the known and fixed displacements. Newton’s second law gives the particles’ motion resulting from the known and fixed forces acting on them. The motion is calculated at the grid points of the triangular constant-strain elements within the elastic particle. Then, application of the material constitutive relations gives new stresses within the elements. 3.1 Micromechanical Constitutive Law In this study, an orthotropic cohesive law has been developed for contact to capture strength, brittleness and anisotropy of rock. Orthotropy has been provided by assuming contact to have different tensile and shear behaviours in terms of strength, stiffness and ultimate displacement. The fracture process zone theory has been also introduced into modelling by assuming contacts to follow a decaying stiffness at pre-failure to represent the damage behaviour of the FPZ. At post-failure, contact endures different stress softening depending on whether it undergoes tension or shear. The stress r applied on the contact surface is defined as r ¼ rðdeff ; kt ; ks ; tc ; cc ; /c Þ

ð1Þ

where deff is the contact effective displacement; kt and ks denote the contact initial stiffness coefficients in tension and shear, respectively, and the parameters tc, cc and /c characterize the strength of contact which represent strength parameters of rock microstructure. They

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respectively referred to as contact tensile strength, contact cohesion, and contact friction angle. deff is defined as ( qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2n þ d2s dn 0 deff ¼ ð2Þ dn \0 ds where dn and ds are the normal separation and shear sliding over the contact surface. dn is assumed positive where contact undergoes opening (tension). As Eq. 2 implies, by assumption when two bonded grains are being detached the total elongation that the bond endures is taken into account to calculate the total stress acting upon the grains boundary. This stress then is decomposed to produce normal and shear components. On the other hand, if the grains slide past each other under compression, normal and shear components of the boundary stress are calculated separately. Each of the stress components is controlled only by the corresponding contact displacement component, i.e., shear stress by ds and normal compressive stress by dn. dn is then the amount of numerical overlapping at the grains touch point (see Eq. 13). 3.1.1 Definition of Contact Stiffness Coefficient The majority of the discontinuum-based models for microstructure assume rigid particle and a decoupled massstiffness system, i.e. the mass of the model is distributed discontinuously lumped at the rigid particles while its global stiffness is provided by the springs assumed at the zero-mass contact points. Therefore, these models offer mathematical relations to consider the contact (or spring) stiffness in terms of the material elastic properties (e.g. Zhao et al. 2011). Alternatively, material can be modelled as a structure involving deformable particles in contact with zero-thickness interfaces to let the model incorporate the structural mass with the stiffness. As the particles must have the same density and elastic properties with the physical material, the contact stiffness must be set infinite to avoid reduction in the global stiffness of the system. However, this ideal assumption is practically impossible to carry out as it makes the numerical solver algorithms fail due to inconsistencies called the effect of ill-conditioning (e.g. Babuska and Suri 1992; Chilton and Suri 1997). In practice, contact (or interface) stiffness is arbitrarily reduced but not so much as the system global stiffness is altered noticeably (e.g. Zhai et al. 2004; Pinho et al. 2006; Elmarakbi et al. 2009; Mahabadi et al. 2010). These simulations actually assume the contact stiffness as an arbitrary parameter which is only needed to be large enough. In order to provide an interpretation for the contact stiffness, Tomar et al. (2004) developed a mathematical


439

model which was implemented in FEM. Our study also provides an explanation for the contact stiffness but more physical and based on the FPZ theory. The FPZ theory suggests that fracture should not be regarded only as a material detachment, but the role of complex damage mechanisms at the crack-tip area must be also taken into account. Since the model does not assume damage in particle, contact must represent material deterioration within the PFZ. Thus, prior fracture initiation, the contact (initial) stiffness coefficient is considered as kt ¼

E w

and

ks ¼

G w

ð3Þ

where E and G are the Young’s and shear modulus of the intact material and w is the thickness of the process zone ahead the fracture. Contact is assumed to lose stiffness gradually upon opening to represent local material deterioration as fracture initiates. Hence, a non-linear relation is adopted to describe the contact pre-peak stress–displacement behaviour. The slope of the relation should gradually decay from the initial value at the origin as suggested by Eq. 3 to zero at the peak strength (see Fig. 2). Two closed-form expressions are provided in Appendix 1 for the FPZ thickness. Using them, the contact initial stiffness coefficients in plane-stress are expressed as follows. kt ¼

E2 rt 2 4KIC

and

ks ¼

GErt 2 4KIIC

ð4Þ

E2 rt 2 4ð1 m2 ÞKIC

and

ks ¼

GErt 2 4ð1 m2 ÞKIIC

ð5Þ

The ratio of the initial stiffness coefficients is ks 1 KIC 2 ¼ kt 2ð1 þ mÞ KIIC

The stress–displacement relation for a contact undergoing separation is expressed through 8 kt deff expðdeff ðÞdct Þ deff dct > > < tc ð1 DÞ dct \deff dut K deff ¼ dMax r¼ k d dct \deff dut K deff \dMax > red eff > : 0 deff [ dut ð7Þ where dct is the critical tensile displacement of contact beyond which cohesive softening happens, and dut is the ultimate tensile displacement of contact at which contact loses its entire strength. As illustrated in Fig. 2a, at the peak point r = tc and deff = dct. Substituting these values in Eq. 7 and solving it for dct yields dct ¼ e

ð6Þ

tc kt

ð8Þ

where e is the base of the natural logarithm. When deff B dct, stress–displacement behaviour is assumed non-linear but elastic, i.e. the unloading and reloading paths are the same and no energy dissipates through contact opening; the governing non-linear equation is the exponential traction–separation law described by Xu and Needleman (1996). As deff exceeds dct, contact failure happens and it is permitted to release energy in unloading– reloading cycles. The damage variable is then defined by D¼

And in plane-strain condition, they are kt ¼

3.1.2 Stress–Displacement Relation of Contact

dMax dct dut dct

ð9Þ

where dmax is the maximum effective displacement that contact has undergone (see Fig. 2a). Fracture healing is therefore avoided as D either increases or remains constant. When deff \ dmax (unloading–reloading cycles), contact follows a linear stress–displacement path where kred is the ratio of stress to effective displacement at dmax (see Fig. 2a).

Fig. 2 Stress–displacement behaviour of a cohesive contact (arrows denote loading, unloading and reloading paths)

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When contact is sheared under compression, the stress– displacement law is described as ks deff expðdeff =dcs Þ deff dcs r¼ ð10Þ rres ¼ kt dn tanð/c Þ deff [ dcs

When the FPZ is described by a linear-decaying cohesive law, the length of the process zone ahead of a stationary mode-I crack under plane-stress loading condition can be estimated by Rice’s formula (Rice 1980).

where rres is the post-failure residual strength of contact which is supplied by the particle boundary friction. In shearing, no post-failure softening is considered as it is assumed that the frictional effects appear instantly after contact failure. Like tension, the critical displacement of contact in shear is calculated by cc dcs ¼ e ð11Þ ks

l¼

In the post-peak region, contact follows a linear unloading–reloading behaviour (Fig. 2b) where the stress increment at each deformation step is calculated as ks Ddeff r\rres Dr ¼ ð12Þ 0 r ¼ rres Ultimately, the normal and shear components of contact force are obtained through ds r ddeffn ac dn 0 Fn ¼ ac ð13Þ and Fs ¼ r kt dn ac dn \0 deff where ac is the contact surface area. 3.1.3 Contact Fracture Energy The area under the curve in Fig. 2a represents the energy needed to fully open the unit area of contact surface. Since contact is the numerical representation of fracture, this area should be equal to the fracture energy, Gf. Thus, Gf ¼

Zdut

rddeff ¼ tc

dut þ dct ð2e 5Þ 2

ð14Þ

0

3.2 Model Validity with Regard to Rock Grain Size To avoid reduction in the model global stiffness, contacts must be at least one order (i.e. 10 times) stiffer than particle. As a measure for particle stiffness, E/dp (or G/dp) can be compared with the contact initial stiffness coefficient kt (or ks) to check this condition, where dp is the largest dimension of particle. This defines a lower bound for dp as 10 w. On the other hand, particle size should be sufficiently (say 3 times) smaller than the FPZ length, l, to allow it to be resolved at least by three particles and to minimize the influences of particle arrangement (mesh topology) on crack propagation pattern. Given Eq. 2, particle size should thus meet the following condition 10w\dp \

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l 3

ð15Þ

2 9p KIC 32 r2t

ð16Þ

Using this and Eq. 34 for w, the valid range for the particle edge size is obtained as 40

2 2 KIC 3p KIC \dp \ 32 r2t Ert

ð17Þ

This means that in modelling an igneous rock the presented model and the suggestions for kt and ks are valid only if the rock grains’ size falls within the above range.

4 Model Calibration Table 1 lists modelling parameters which are referred to as microparameter beside the analogous material properties. A micromechanical investigation by the model requires proper selection of the microparameter by means of a calibration process in which responses of the model are compared directly to the observed responses of the physical material. These comparisons are made at the laboratory scale and include tensile and compressive test results. For this study, a group of simulations are generated and calibrated to Augig Granite (AG). They comprise modelling samples of the Brazilian tension, uniaxial compression and triaxial compression tests. AG is the main rock type near the Grimsel Pass in the Canton of Berne, Switzerland which has been vastly tested in the rock mechanics laboratory of the Swiss federal institute of technology (EPFL). As direct tensile tests are usually engaged with pre-mature failure due to gripping, the Brazilian tensile test has been adopted which resorts to compression-induced tension to evaluate material tensile strength. The average standard mechanical properties and mean grain size of AG are listed in Table 2. According to the specimens’ geometry and loading condition, a plane-strain and a plane-stress analysis are adopted for the compressive and Brazilian models, respectively. Samples are placed between two steel platens whose interfacial friction angle is assumed 5°. The contact initial stiffness coefficients obtained through Eqs. 4 and 5 are listed in Table 3. The model validity condition described in Eq. 17 is met as the AG grains size range (i.e. 2–6 mm) falls within the range suggested by the equation (i.e. 0.4–8.6 mm). The samples are built by particles with a mean edge size equal to the AG grains average size, i.e. 4.0 mm. The coefficient


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Table 1 Material properties and model microparameter Material property

Model microparameter

Tensile strength (rt)

Tensile strength of contact (tc)

Internal cohesion (C)

Cohesion of contact (cc)

Internal friction angle (/)

Friction angle of contact (/c)

In the following, the tensile and the uniaxial compressive strength obtained from the modelling are denoted by rt and rc, respectively, versus the laboratory measurements for the Brazilian tensile strength (BTS) and uniaxial compressive strength (UCS) of AG. 4.1 Parametric Study

Table 2 Mechanical properties AG Property

AG

Young’s modulus, E (MPa)

25,800

Poisson’s ratio, m

0.23

Mode-I fracture toughness, KIC (MPam1/2)

1.5

Mode-II fracture toughness, KIIC (MPam1/2)

3.0

Internal cohesion, C (MPa)

21.0

Internal friction angle, / (°)

53.0

Brazilian tensile strength, BTS (MPa)

8.8

Uniaxial compressive strength, UCS (MPa)

122.0

Grain size range (mean) (mm)

2–6 (4)

of variation in particle generation process is taken 0.1 mm which ends up assemblages containing particles whose dimension is between 3.4 and 4.6 mm. This implies that the model FPZ is resolved at least by six particles. The generated compressive and tensile samples are 80 9 160 mm and 80 9 80 mm, respectively, and include 1,122 and 452 particles, respectively. Figure 3 presents two representative compressive and tensile samples. An axial-strain increment is applied to the upper and lower planets of the systems by setting a very low axial displacement rate, i.e. 10-4 m/s for a number of steps, e.g. 5,000 steps. The compression is then stopped by setting the displacement rate to zero, and the systems are cycled until quasi-static equilibrium is reached. During this process, the reaction force at both the upper and lower supports is continuously recorded to generate stress–strain curves. The model tensile strength is measured through the following equation rt ¼

2FMax ptD

ð18Þ

where Fmax is the maximum axial force recorded. D and t denote the sample diameter and thickness where t = 1 for 2D simulation. Table 3 Contact initial stiffness coefficients for simulation of AG samples

Sample Compressive Brazilian

A parametric study is carried out to determine which microparameters have the largest impacts on which model macroscopic response. cc and /c values are initially given the uniaxial compressive strength and the internal frictional angle of AG, respectively, while tc is changing. Considering Eq. 32, the mixed-mode fracture energy, Gf, of AG is calculated as 413.0 and 436.0 J/m2 for plane-strain and plane-stress, respectively. dut is then calculated by Eq. 14 for each assumed value of tc as listed in Table 4. As expected the model global strength is dependent of tc. Figure 4a indicates a linear relation between tc and rt. This result is in agreement with the prediction by Eq. 44 in Appendix 2. The relationship for the uniaxial compressive strength, rc, is non-linear. Note that the axes are normalized to BTS = 8.8 MPa, and UCS = 122 MPa. By establishing a linear regression fit to the data, tc is estimated 24.15 MPa to have rt = BTS. Repetition of the simulation using this value results in rt = 8.73 MPa which is very near the AG BTS. Given obtained tc, the sensitivity of the model to cc and /c are examined as presented in Fig. 4b and c. As seen rc is highly influenced by cc and /c, but it does not change with /c when /c \ 0.75/. On the other hand, rt is independent of cc and /c. This implies that the model global tensile strength is controlled only by the contact tensile strength and the obtained tc (=24.15 MPa) is thus the target value. 4.2 Design of Experiment The calibration process is carried out using a group of statistical techniques known as the design of experiment (DOE). The DOE is an efficient, structured and organized discipline to quantitatively evaluate the relations between the measured responses of an experiment and the given input variables called factors (NIST/SEMATECH 2003). The objective of the DOE is to observe how and to what

Property

Value (MPa/mm)

Contact normal stiffness coefficient in plane-stress, kt

6.51 9 105

Contact shear stiffness coefficient in plane-stress, ks

6.61 9 104

Contact normal stiffness coefficient in plane-strain, kt

6.87 9 105

Contact shear stiffness coefficient in plane-strain, ks

6.98 9 104

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Fig. 3 Compressive and tensile samples generated by the model

Table 4 Values of contact tensile strength and corresponding ultimate displacements for AG simulation tc (MPa) (tc/BTS)

8.8 (1)

17.6 (2)

26.4 (3)

35.2 (4)

dut for compressive test (mm)

0.094

0.047

0.031

0.023

dut for Brazilian test (mm)

0.099

0.050

0.033

0.025

extent changes in the factors influence the response variables. There are many different DOE methods. The best choice depends on the number of factors involved and the accuracy level required. Kennedy and Krouse (1999) presented the details for different DOE methods and categorized them based on the experimental objectives they meet. The DOE begins with the definition of the experiment objectives and the selection of the input/output variables. In our purpose, the unknown microparameter (i.e. cc and /c) are chosen as the factors; and the assemblage macroscopic responses in terms of internal cohesion (C) and internal friction angle (/) are considered as the responses. 4.2.1 Estimation of Factors Range To carry out the DOE we need to estimate the range of the factors. As Fig. 4b suggests should the contact cohesion be about 0.5 9 UCS, the model demonstrates a uniaxial compressive strength close to that of AG. Hence, target cc is guessed to be between 0.25 9 UCS and 0.75 9 UCS, i.e. 30.50–91.50 MPa. On the other hand, as the model response does not vary for /c \ 0.75/, 0.85/ and 1.15/ (45.05° and 60.95°) are taken the lower and upper bounds for /c. 4.2.2 Central Composite Design Depending on the level of accuracy required, a complete description of the response behaviour might need a linear, a

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Fig. 4 Normalized tensile and compressive strength of AG models versus normalized contact microparameters

quadratic or even a higher-order DOE. Under some circumstances, a design involving only main effects and interactions may be appropriate to describe a response surface when analysis of the results reveals no evidence of pure quadratic curvatures in the response of interest. As Fig. 4 implies, there is, however, a probability of existing interaction between the factors. Hence, a quadratic model is necessary to satisfy our objective. As an efficient quadratic model, the central composite design (CCD) is applied for the estimation of non-linear relations between the microparameter and the model macroscopic responses. CCD provides high quality prediction of a response surface over the entire design space, including linear, quadratic, and interaction effects. It contains an imbedded factorial or fractional factorial design with centre points that are augmented with a group of star points that allows estimation of curvature (see Table 6). If the distance from the design space centre to a factorial point is assumed ±1,


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Table 5 Definition of factors and numerical value of microparameter at each coded level Coded factor

Corresponding microparameter

Value of microparameter at coded levels

Transformation formula

-a

-1

0

?1

?a

x1

Contact cohesion (cc)

17.87

30.50

61.00

91.50

104.13

x2

Contact friction angle (/c)

41.76

45.05

53.00

60.95

64.24

cc = 30.50 9 x1 ? 61.00 /c = 7.95 9 x2? 53.00

Table 6 Complete design matrix for central composite design

CCD offers a limited number of combinations for the factors. These combinations are collected in a matrix called design matrix as listed in Table 6. This matrix is converted to the matrix of the real factors, i.e., microparameter by the transformation formula expressed at the last column in Table 5. The uniaxial and triaxial compressive tests are then simulated using each set of the CCD-suggested microparameter, and the predictions for confining pressures of 0, 4 and 8 MPa are recorded in Table 7. Using the outputs of each run, internal cohesion and internal friction angle of the model are calculated as the DOE responses.

the distance from the design space centre to a star point will be ±a. The precise value of a depends on the number of factors involved. Since there are two factors in the model (i.e. cc and /c), a = 21/2 & 1.414, and the number of factorial runs will be four (NIST/SEMATECH 2003). The levels ±1 represent the upper and lower bounds of the factors. The value of each factor at the centre point is defined as the arithmetic mean of the upper and lower bound values. Given the lower and upper bounds, the centre, factorial, and star points are calculated as listed in Table 5. Table 7 CCD-suggested design matrix and obtained results Run

Microparameter matrix cc

/c

DOE responses C

Numerical results for compressive models /

r3 = 0 MPa

r3 = 4 MPa

r3 = 8 MPa

1

30.53

45.05

7.2

57.6

55.98

84.10

151.04

2

91.58

45.05

35.7

38.9

147.32

171.56

182.38

3

30.53

60.95

15.1

73.7

181.74

460.30

570.00

4

91.58

60.95

13.7

74.1

180.57

433.50

590.34

5

17.88

53.00

3.9

69.8

35.87

183.62

287.03

6

104.22

53.00

26.4

56.4

181.09

205.94

268.65

7

61.05

41.76

35.3

21.9

104.08

114.09

121.63

8

61.05

64.24

17.3

74.8

245.97

510.31

694.55

9

61.05

53.00

11.4

67.0

115.40

200.89

308.71

10 11

61.05 61.05

53.00 53.00

12.1 10.1

64.6 69.5

107.89 114.47

183.33 227.16

264.81 358.74

12

61.05

53.00

10.2

70.2

106.14

269.88

369.24

13

61.05

53.00

11.8

63.6

103.95

165.91

248.86

123

444


For each run, the particle assemblage is separately created. In addition, the simulations are repeated for five times with the same microparameter at the centre points (see runs 9–13 in Table 7). It is because the particle assemblage is generated arbitrarily and two numerical runs might produce slightly different results. Hence, the CCD predictively carries out this repetition to minimize the variability in modelling. The targeted response parameters are statistically analyzed by applying the above data in the statistical software of JMP (Sall et al. 2007). The data are evaluated using the Fischer test, and quadratic models are generated for each response parameter using multiple linear regression analysis, analysis of variance and a backward elimination procedure. A numerical optimization procedure using desirability approach is ultimately used to locate the optimal settings of the formulation variables in view to obtain the desired response (Park and Park 2010). Using the data presented in Table 7, the following equations between the model macroscopic response and the coded factors are constructed eventually. C ¼ 11:12 þ 7:36x1 4:94x2 7:48x1 x2 þ 1:32x21 þ 6:89x22 / ¼ 66:98 4:66x1 þ 15:76x2 þ 4:78x1 x2 0:60x21 7:98x22 ð19Þ Comparison of the multipliers in the above equations justifies the necessity of CCD as a quadratic model. Solving the equations for C = 21 MPa and / = 53° of AG gives x1 = 0.443 and x2 = -0.533. These are coded factors that should be transformed to uncoded values using the transformation equations. Eventually, cc = 74.57 MPa and /c = 48.76° are obtained as the target microparameter.

5 Solution Verification 5.1 Quantitative Comparison The obtained microparameter are used as input for new simulations which are expected to give the closest match with the laboratory test in terms of the Young’s modulus, Poisson’s ratio, Brazilian tensile strength, uniaxial compressive strength, internal cohesion, and internal frictions angle. Table 8 Experimental properties of AG versus model predictions

123

Since different assemblage arrangements result in different model strength, five models with different particle arrangement are created. Table 8 lists the results of mean, standard deviation and relative error percentage that show fair agreement with the experimental measurements where the relative error is always less than 5 %. Though the target microparameters are expected to provide a close match, little variations in the numerical results are unavoidable because of the inherent randomness of particle placement in the model generation. Note that this is not regarded as a disadvantage as two separate experimental tests on rock material necessarily do not lead to identical results due to rock heterogeneity. In fact, the randomness of the particle arrangement comparably represents rock heterogeneity when material anisotropy is introduced into calculation by the adopted orthotropic contact law. Comparisons between the curves of axial stress versus axial and lateral strain for the laboratory test and a representative AG simulation are presented in Fig. 5a. Note that some special aspects for rock behaviour such as closure of initial flaws and pores are not captured in the modelling. This causes the stress–strain curves obtained in the simulation to be slightly different from those of the laboratory tests particularly where the initial nonlinearity is not reflected in the modelling. The elastic constants for the laboratory testing were obtained from the middle portions of the curves where relatively linear relation between stress and strain is maintained. For numerical simulation, those are computed using stress and strain increments occurring between the start of the test and the point at which one-half of the peak stress is obtained (tangent method). Figure 5b also plots the model predictions for the compressive and tensile strengths versus the laboratory measurements. As seen, the numerical results follow nearly the same pattern with the laboratory data. The Hoek– Brown failure envelops are also drawn for both the numerical and experimental data where fair agreement is observed. 5.2 Qualitative Comparison Some specific features of the simulation, e.g. failure mode and fracturing pattern cannot be quantified efficiently. A

Property

E (GPa)

m

rt (MPa)

rc (MPa)

C (MPa)

/ (°)

Experimental value

25.8

0.23

8.8

122.0

21.0

53.0

Numerical mean

25.2

0.24

8.7

125.4

20.9

53.5

Standard deviation

0.21

0.01

0.16

2.35

0.85

1.12

Relative error (%)

2.33

4.35

1.14

2.79

0.48

0.94


445

Fig. 5 Comparison of laboratory data to model predictions

Fig. 6 Experimental stress– strain response of AG compared to model predictions in triaxial compression

qualitative comparison is therefore required to complete the modeming validation where the solution correctness is examined through comparing those features from laboratory results to simulation outputs. As shown by Paterson (1987), rocks exhibit higher ductility under triaxial circumstances than uniaxial. Figure 6 offers that this phenomenon is fairly captured by the model. The simulation gives further yielding and plastic deformation with confinement, whilst an abrupt softening at post-peak region is observed in the laboratory results. As Wawersik and Fairhurst (1970) described, rock failure in unconfined circumstances occurs in two distinct modes of axial splitting (cleavage failure) and shear rupture (faulting). Axial cleavage generally precedes faulting for hard crystalized rocks and characterizes failure initiation. Figure 7a shows that the modelling is able to capture these phenomena where the predicted failure mode shows the typical cleavage observed in the AG laboratory tests. The post-failure picture for the AG Brazilian sample is plotted in Fig. 7b. In the modelling, tensile failure starts at about the centre of the sample, then the induced fracture propagates toward loading points rapidly, and the sample is

split into half ultimately. Further loading causes contacts located beneath the loading platens break. As seen, the failure features in terms of the major fault induced into the sample and the wedge-shaped zone created beneath the platen are fairly captured.

6 Micromechanics of Failure 6.1 Tensile and Shear Micro-Cracking Figure 8 presents the modelling predictions for initiation and progression of failure where the contacts failed under shear and tension are plotted in black and red, respectively. The general tendency is that tensile micro-fractures occur in parallel with the loading, whilst the shear ones are oblique. The triaxial samples endure more micro-failures at peak strength than the uniaxial ones. In addition, confinement gives rise to the contribution of micro-shear fractures in failure because lateral compression makes the microtensile failures accruing in parallel with loading and due to rock geometrical inhomogeneities harder.

123

446

Fig. 7 Comparison of laboratory failure modes with model predictions


The predictions show that with or without confinement failure (peak strength) is associated with shear microfractures except for the uniaxial circumstances (see Fig. 8a, d, g) where the tensile micro-cracks are dominant and their coalescence create the vertical major macro-cracks and causes the sample cleavage as presented in Fig. 7a. This is in agreement with the existing understating about rock failure mechanism (Fairhurst and Cook 1966; Brace et al. 1966; Hallbauer et al. 1973; Tapponier and Brace 1976; Fonseka et al. 1985; Martin and Chandler 1994; Diederichs 2003). At low triaxiality, i.e. r3 = 4 MPa, the shear microcracks join together and at the peak stress they form an oblique failure surface (Fig. 8d). However, as the axial deformation progresses due to the loading the tensile micro-cracks develop and link the shear ones together at post-failure stage (Fig. 8f). At high triaxiality, i.e., r3 = 8 MPa, the confinement makes the creation of the tensile micro-failures very hard. Failure behaviour is thus dominated by the shear micromechanisms (Fig. 8g) where the shear micro-cracks continue to disseminate throughout the sample and cause it to smash ultimately (Fig. 8i). These results indicate the dominant role of the shear micro-fractures in controlling rock failure mechanism. The

Fig. 8 Progression of micro-tensile (red) and micro-shear (black) fractures in AG samples under different levels of triaxiality (color figure online)

123


447

Fig. 9 Some of particle assemblages created to explore particle size effects

Fig. 10 Variation of normalized model compressive and Brazilian strength with particle size

dominancy of the tensile micro-fractures is limited to the uniaxial compression circumstances. 6.2 Effect of Rock Grain Size As particle is the numerical representation of rock grain, the model can be used to investigate the effect of grain size on strength in igneous rock by changing the model particles size and comparing the results. This will show how much material macro-scale strength is a matter of microstructure geometry than microstructural strength. Equation 44 (Appendix 2) suggests that whilst rt predicted by the model changes with tc linearly (as discussed in Sect. 4.1), its variation with particle size follows a square root relationship. A series of simulations are carried out to examine this numerically, which consist of particles with an average edge size between 2 and 7.5 mm. This range is within the valid limit for grain size (Eq. 17). The details for some of the simulations are presented in Fig. 9.

The results presented in Fig. 10 indicate that although the compressive strength has nearly no change, the Brazilian strength decreases with the particle size decrease, and follows a trend similar to that predicted by Eq. 44. The numerical results are compared to Eq. 44 by fitting a power-law curve with an exponent of 0.39 to them. However, the equation suggests a power of onehalf. This discrepancy may be explained through the argument that the power will be one-half provided the linear elastic fracture mechanics (LEFM) conditions apply. However, such conditions are not valid here because model’s contacts follow a cohesive behaviour in the post-peak region. Note that this result is based on the assumption that micro-failure only takes place at the particles boundary not inside them, i.e. no particle crushing is assumed. 7 Conclusion A numerical model was created to represent the microstructure of igneous crystalized rock by considering grainscale geometric heterogeneity, contact orthotropy and micro-failure mechanisms. The grain-scale heterogeneity was captured by developing a Delaunay triangle generator. The discrete element program was then used to calibrate the model such that it reproduces a variety of standard laboratory testing data. Meanwhile statistical disciplines and mathematical developments have been employed to provide a physical understanding for the model microparameters.

123

448

The grain-scale microparameters were shown how to control the rock macroscopic failure response. Microtensile strength and grain size were shown to have impact on peak tensile strength for the crystalized rock investigated where micro-scale cohesion and grain boundary friction were seen to have no significant effect. Uniaxial and triaxial compressive strengths were observed to be controlled by micro-tensile strength, micro-cohesion and grain boundary friction angle but independent of grain size. The presented model is a research tool to aid in understanding brittle failure processes. It was shown to provide a proper reproduction of the rocks’ macroscopic behaviour quantitatively and qualitatively. The model suggests that micro-shear failure mechanisms control both failure and post-failure behaviours in igneous rock under a variety of loading conditions except for failure at peak uniaxial compressive strength where micro-tensile failures dominate the mechanism. It also shows that in triaxial loading micro-shear fractures progress oblique to the direction of loading and a micro-tensile fracture joins two micro-shear fractures when they reach the vicinity of each other. This process eventually leads to an inclined macroscopic failure surface as observed in laboratory. Beside, this study offers a straightforward and disciplined calibration procedure which is easily extendable to use in other discontinuum-based microstructural simulations. It also provides close-formed expressions for the stiffness coefficient, tensile strength and cohesion of the contact showing how the material macroscopic properties are related to the microstructural parameters. Acknowledgments The laboratory data used have been supplied by the LMR test room. The authors would like to thank Mr. JeanFrançois Mathier, the Head of the Laboratory, for providing the test data. They also would like to thank the anonymous reviewer for his/ her valuable suggestions to improve the paper.


Appendix 1 An estimation for thickness of fracture process zone A material cracks when sufficient stress and energy are applied to break the inter-molecular bonds. These bonds hold the molecules together and their strength is supplied by the attractive forces between the molecules. Many equations have been proposed to formulate this force and its potential energy. The Lennard-Jones’ potential (Griebel et al. 2007) is a simple and extensively used function: n m f f PðzÞ ¼ aX ð20Þ z z where z denotes the separation distance between two adjacent molecules, and 1 n nm 1 n a¼ ð21Þ n m mm The depth of the potential, X, describes the energy needed to break the bond and thereby the strength of the molecular force (Fig. 11). It is called bond energy. The value f parameterizes the zero crossing of the potential; the integers m and n depend on the material molecular nature and are more commonly among 6–16. On close inspection, all real materials show a multitude of heterogeneities even if they macroscopically appear to be homogeneous. These deviations from homogeneity may exist in the form of cracks, voids, particles or regions of a foreign material, layers or fibres in a laminate, grain boundaries or irregularities in a crystal lattice. Heterogeneities of any kind can locally act as stress concentrators and thereby lead to the formation and coalescence of micro-cracks or voids as a source of progressive material damage. To take these microstructural defects into account, a homogenization approach is adopted by assuming the

Fig. 11 Lennard-Jones’ potential for real and homogenized material (left and middle), and homogenized inter-molecular force (right)

123


449

process zone as the representative volume element across which the fine-scale heterogeneous microstructure is ‘‘smeared out’’ and the material is described as homogeneous with spatially constant effective properties. The latter then accounts for the microstructure in an averaged sense. They, for our purpose, include the bound energy and zero crossing of the potential. As illustrated in Fig. 11, the effective potential of the homogenized process zone is thus formulized by n m f f P ðzÞ ¼ aX ð22Þ z z where the parameters superscripted by * denote the effective ones. As the potential derivative with respect to z, the homogenized inter-molecular force P ðzÞ is written as n m oP aX f f ¼ n þm P ðzÞ ¼ ð23Þ oz z z z The peak value of the homogenized inter-molecular force, which is called effective cohesive force, Pc , takes place at zm as shown in Fig. 11. Note that Pc is significantly smaller than the actual peak molecular force in the physical material as it includes an average effect of the entire material micro-defects. Solving the derivative of P ðzÞ for z, 1 nðn þ 1Þ nm zm ¼ f ð24Þ mðm þ 1Þ Substituting zm into Eq. 23 leads to " # n1 m1 nm nm X n ð n þ 1 Þ n ð n þ 1 Þ þm Pc ¼ a n mðm þ 1Þ m ð m þ 1Þ f

Table 9 Values of b for common values of m and n m

n = 13

n = 14

n = 15

n = 16

n = 17

n = 18

8

0.30

0.29

0.28

0.27

0.26

0.26

9

0.28

0.27

0.26

0.25

0.25

0.24

10

0.26

0.25

0.24

0.24

0.23

0.22

11

0.25

0.24

0.23

0.22

0.22

0.21

12

0.24

0.23

0.22

0.21

0.21

0.20

energy, Gf, defined as the rate of energy release per unit cracked area, is expressed as Gf ¼

The homogenized equilibrium spacing between two molecules (z0 ) occurs when the potential energy is at a minimum or the force is zero. Solving Eq. 23 for z provides 1 h n inm ð26Þ z0 ¼ f m A tensile force is required to increase the separation distance from the homogenized equilibrium value. If this force exceeds the effective cohesive force, the bond is completely severed. The homogenized material then cracks and stress in a width equal to z0 is released. This means that z0 (which is significantly larger than real molecular equilibrium spacing) represents the homogenized process zone thickness (w), i.e. w ¼ z0 . When a bond breaks, a quantity of energy equal to X* is dissipated. The accumulation of these energies over the process zone surface supplies the energy dissipation through fracturing. Therefore, the Griffith’s fracture

ð27Þ

Substituting X* obtained from Eq. 25 into the above relation yields " n1 m1 #1 f Pc nðn þ 1Þ nm nðn þ 1Þ nm Gf ¼ 2 n þm mðm þ 1Þ mðm þ 1Þ az0 ð28Þ On the other hand, the effective tensile strength of the homogenized material which represents the actual tensile strength of the material is estimated by Pc z2 0

rt ¼

ð29Þ

Substituting Pc from Eq. 28 and z0 from Eq. 26 into Eq. 29 and solving it for z0 , which actually represents the homogenized fracture process zone thickness, w, yields w¼

ð25Þ

X z2 0

1 Gf b rt

ð30Þ

where 1n nþ1 mþ1 nm mþ1 nm mþ1 nþ1 nþ1

b¼

1 m

ð31Þ

depends on the integers m and n. Table 9 shows that b is relatively constant at 0.25 for common values of m [ [8,12] and n [ [13, 18]. In mixed-mode fracturing, Gf ¼

2 KIC K2 þ IIC ~ ~ E E

ð32Þ

~ ¼ E for plane-stress, and E ~ ¼ E ð1 m2 Þ for where E plane-strain. Given b = 0.25, for fracturing under pure tension, w¼

2 4KIC ~ t Er

ð33Þ

and under pure sliding

123

450


For a regularly packed assemblage loaded along packing direction, at the incipient failure or crack extension, rc = tc and KI = KIC. Substituting them into Eq. 39 yields qffiffiffiffiffiffiffiffiffi 2KIC d

p ð40Þ tc ¼ p dp

Fig. 12 Crack representation within a particle assemblage

w¼

2 4KIIC ~ t Er

ð34Þ

Appendix 2 Relation between micro- and macro-tensile strength Figure 12 presents a cutout of a representative collection of isotropic linear elastic triangles where a finite number of contacts are already broken to form a cracked surface. Assuming that the boundary is sufficiently far from the crack, LEFM suggests that for a through-thickness crack of half-width a the induced stress, rn, acting on the crack plane near the crack tip, i.e. r \\ a, is (Anderson 1995) qffiffiffiffiffiffiffiffi rn ¼ rf a=2r ð35Þ where rf denotes the remote tensile stress acting normal to the crack and r is the distance from the crack tip. The concentrated force Fc acting over the first contact right adjacent the crack tip is given by Z dp 2 pffiffiffiffiffiffiffi Fc ¼ ð36Þ rn dr ¼ rf adp 0

As soon as Fc exceeds the assumed strength of the contact, it breaks and the crack expands. Since the mode-I stress intensity factor for the system is defined as pffiffiffiffiffiffi KI ¼ rf p a ð37Þ Equation 36 can be re-written as qffiffiffiffiffiffiffiffi

ffi Fc ¼ KI dp p

ð38Þ

Therefore, the tensile stress created at the first contact point is evaluated by qffiffiffiffiffiffiffiffiffi Fc 2KI d

p rc ¼ ¼ ð39Þ p dp dp 2

123

Equation 40 indicates that material fracture toughness is in fact the macroscopic representation of the contact tensile strength and the adopted particle size. This result is anticipated as the concept of fracture toughness implies an internal length scale whereby the ratio of fracture toughness to material strength has the dimension of square root of length. The particle size supplies this internal length scale in the modelling. The same approach can be followed to express KIIC in terms of contact cohesion, cc. Ultimately, pffiffiffiffiffi KIC / tc dp ð41Þ pffiffiffiffiffi ð42Þ KIIC / cc dp It is generally accepted that KIC and rt, measured by the Brazilian testing, are related together in a vast range igneous, metamorphic and sedimentary rocks (e.g. Zhang 2002; Gunsallus and Kulhawy 1984; Harison 1994). Zhang (2002) suggested the following empirical relation which provides good estimations with a coefficient of determination r2 = 0.94. rt ¼ 6:88KIC

ð43Þ

where the parameters involved are in the SI units. Substituting Eq. 43 in 41 leads to pffiffiffiffiffi ð44Þ rt / tc dp :

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