Research Article On the Tensile Strength of Granite at ...downloads.hindawi.com/journals/amse/2016/6279571.pdf · Research Article On the Tensile Strength of Granite at High Strain

Research ArticleOn the Tensile Strength of Granite at High Strain Ratesconsidering the Influence from Preexisting Cracks

Mahdi Saadati,1,2 Pascal Forquin,3 Kenneth Weddfelt,2 and Per-Lennart Larsson1

1Department of Solid Mechanics, KTH Royal Institute of Technology, 10044 Stockholm, Sweden2Atlas Copco, 70225 Orebro, Sweden33SR Laboratory, Grenoble-Alpes University, 38041 Grenoble, France

Correspondence should be addressed to Per-Lennart Larsson; [email protected]

Received 4 February 2016; Accepted 14 June 2016

Academic Editor: Konstantinos I. Tserpes

Copyright © 2016 Mahdi Saadati et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The dynamic tensile behavior of granite samples, when some preexisting cracks are introduced artificially, is investigated. Spallingtests using a Hopkinson bar are performed and strain rates of ∼102 1/s are obtained in both specimen types (with and without initialcracks).This experimental technique is employed being of the same order as strain rates in rockmaterials during percussive drilling,the application of interest here.The dynamic tensile responses of both sample-sets are compared using the velocity profilemeasuredon the free-end of the sample. Furthermore, an anisotropic damagemodel based on the concept of obscuration probability describesthe response without preexisting cracks. Here, a term of cohesive strength in obscuration zones is added to accurately handle thesoftening behavior of thematerial in tension. Results from the spalling tests are used to validate themodel prediction of the dynamictensile strength and also to calibrate the cohesive model parameters. Damaged elements are numerically introduced in the finiteelement calculations simulating the spalling experiments performed on predamaged samples. The results are compared with theexperimental ones. Good agreement is obtained showing that a two-scale approach may constitute a suitable method to simulatenumerically the tensile response of predamaged granite.

1. Introduction

The strain rate dependency of the mechanical response inbrittle materials has been widely investigated in the literature.Considerable rate dependency is reported especially in thecase of tensile strength [1–4]. There exists a threshold levelof strain rate beyond which the tensile strength increases byincreasing the strain rate. This threshold level is explained tobe connected to the size of heterogeneity in the material [1, 5]and material defects size, population, and distribution [6, 7].Direct tension test with split-Hopkinson bar (SHB) has beenused to study the tensile strength for strain rates between 10−1and 101 1/s [3, 8]. In some cases, indirect tensile strength ismeasured by means of a Brazilian disc together with split-Hopkinson pressure bar (SHPB) apparatus [9, 10]. Inspiredfrom the classical SHPB method, spalling test with Hopkin-son bar is a suitable technique to measure the tensile strengthof brittlematerials at strain rates between 101 and 102 1/s [2, 4].The main idea in this test is that the impact of the projectile

induces a compressive wave that propagates through the barand is mainly transferred to the specimen. This wave is ref-lected as a tensile wave from the free surface of the specimenthat leads to damaging of the material.

There is a wide range of applications pertinent to thedynamic tensile behavior of brittle materials from blastingin open quarries and concrete structures exposed to impactloading to screen rupture of cell phones due to free fall. Per-cussive drilling, which is the application of interest in thisinvestigation, is just one of them. The main goal in this workis to develop a reliable numerical tool for simulating therock fragmentation mechanism during percussive drilling.In modeling such problems, a constitutive model is neededto cover both the tensile behavior of the brittle materials athigh strain rate, because of the rapid indentation, and alsoconfined compression behavior that occurs underneath theindenter. The Krieg, Swenson, and Taylor- [11, 12] (KST-)Denoual, Forquin, andHild [6, 7] (DFH)model is adopted toperform such analysis. This material model is composed of a

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2016, Article ID 6279571, 9 pageshttp://dx.doi.org/10.1155/2016/6279571

2 Advances in Materials Science and Engineering

plasticity model (KST) to simulate the compressive behaviorof geomaterials accounting for the effect of hydrostatic anddeviatoric parts of the stress tensor. Also the fragmentationprocess in tensile loading, due to the opening of cracks,is defined by using an anisotropic damage model (DFH).This model is based on a probabilistic approach describingthe dynamic fragmentation of the brittle materials. It shouldbe mentioned that the original DFH model (with no cohe-sion strength in obscuration zones) enables predicting thedynamic tensile strength of the brittlematerials, and themainmodel parameters are obtained from a set of quasi-staticexperiments [6, 7]. Saadati et al. [13, 14] applied the KST-DFHmodel on granite and investigated the rock fragmentationprocess and the force-penetration response at percussivedrilling. The KST-DFH model parameters for Bohus granitehave been determined based on previously reported exper-imental results [15]. It was shown that preexisting cracks ingranite have a significant effect on its mechanical responseand the fracture pattern at impact loading.

Spalling tests are performed to investigate the dynamictensile behavior of Bohus granite. As the DFHmodel predictsthe dynamic tensile strength of the brittle materials at highstrain rates, this work can be seen as a validation step for themodel prediction of the granite tensile strength. The strainrate in the rock during the percussive drilling process is inorder of 102 1/s based on previous numerical simulations [13].This is the same order as the strain rates in the rock materialduring the spalling tests with Hopkinson bar performed inthis investigation and therefore makes the spalling experi-ment an appropriate tool to verify the rate dependency of thematerial response at dynamic tensile loading. Furthermore, acohesive strength is added to the original DFHmodel [16, 17]to more accurately deal with the postpeak behavior of thematerial at tensile loading.Hence the results from the spallingtests are used to calibrate the cohesive model parameters in adynamic situation.

The experimental results from the spalling tests on Bohusgranite are presented. The dynamic tensile strength of thematerial (without preexisting cracks) is measured and com-pared with the quasi-static results. When there are cracks inthe specimens that are introduced in addition to the materialdefault cracks and defects, called structural cracks in thiswork, the material response changes considerably. The effectof the structural cracks on the mechanical response andthe fracture pattern in Edge-On Impact (EOI) tests, that is,impact of an aluminum projectile onto a rock slab, was previ-ously studied [13, 14]. When it comes to drilling application,these cracks can be introduced in the rock during drillingdue to either former impact of the drill bit or use of methodssuch asmicrowave and laser. It was shown that the preexistingstructural cracks facilitate the drilling process regardless oftheir orientation [14]. In order to investigate the effect of thesecracks on the dynamic tensile response of the material, someof the spalling specimenswere exposed to coarsermechanicalloading during the cutting process in order to introduce newcracks to the material. In the present work spalling tests havebeen performed also on such specimens with preexistingstructural cracks and it is shown that the dynamic tensilebehavior changes considerably. Numerical modeling of the

Projectile Input bar Rock specimen

G1 G2G3

Laserinterferometer

Rear face of the specimen

Figure 1: Spalling experiment setup with Hopkinson bar.

spalling tests is performed and the results are comparedwith the experimental ones. The DFH model together with acohesive strength is used for this purpose.The cohesivemodelparameters are calibrated based on the experimental resultsperformed on granite without preexisting cracks. In the caseof specimens with preexisting structural cracks, these cracksare introduced to the numerical samples by means of setsof predamaged finite elements randomly distributed in thesample. It is shown that the numerical results are in goodagreement with the experimental ones.

2. Experimental Investigation and ResultsPertinent to the Tensile Strength of Graniteat High Strain Rates

2.1. Experimental Setup with Hopkinson Bar. Spalling testwith Hopkinson bar is suggested as a suitable technique tomeasure the tensile strength of brittle materials at strainrates between 101 and 102 1/s [2, 4]. As is stated before, thenumericalmodeling of percussive drilling that was previouslyperformed in [13] gives the tensile strain rates in the rockin order of 102 1/s. It should be mentioned that, in order toobtain this tensile strain rate from the numerical simulationof percussive drilling, the maximum positive principal stresswas used from the simulation with only the KST plasticitymodel (in order to evaluate the level of tensile strain rate in afinite element without influence from damage).

Accordingly, the spalling test with Hopkinson bar isan appropriate method to investigate the dynamic tensilebehavior of the material in the interesting range of strainrates pertinent to percussive drilling.The experimental setupused in this work is shown in Figure 1. It consists of a Hop-kinson bar made of high-strength aluminum, an aluminumprojectile with a spherical cap-end, which is optimized toinduce more homogenous tensile stress [4], and the rocksample that is well attached to the bar to increase the wavetransmission. The rock specimen is instrumented with straingauges to record the strain data and also a laser interferometeris directed toward the free surface of the specimen tomeasurethe rear face velocity. The main idea is that the impact of theprojectile induces a compressive pulse that travels along thebar and is mainly transferred to the rock specimen.When thecompressive wave reaches the free surface of the sample, itis reflected as a tensile wave that propagates in the oppositedirection and leads to failure if the tensile strength of thematerial is passed. Using the laser velocity profile, one canobtain the dynamic tensile strength of the material from the

Advances in Materials Science and Engineering 3

40 60 80 100 120 140 1600

1

2

3

4

5

6

7

8Re

ar fa

ce v

eloci

ty (m

/s)

Time (𝜇s)

(a)

100 120 140 160 180 200 220 240 260 280 300

0

0.05

0.1

0.15

0.2

Axi

al st

rain

(%)

Gauge G1Gauge G2

Gauge G3

Time (𝜇s)

−0.05

−0.1

(b)

Figure 2: Spalling test results for a specimen without preexisting cracks. Rear face velocity profile from the laser (a) and strain data from thethree strain gauges (b).

Novikov formula [18] assuming a linear-elastic behavior ofthe tested material prior to the peak strength.

2.2. Spalling Test on Granite. The results from the spallingtests are presented in this section. A typical result for both therear face velocity (obtained from the laser) and the strain data(obtained from the strain gauges) is shown in Figure 2.Threestrain gauges (G1, G2, and G3) of length 20mm are placed ata distance of 116mm, 45mm, and 35mm from the rear faceof the specimen, respectively. The projectile impact velocityin this case was 7.3m/s. Based on the Novikov et al. [18]approach, one can relate the dynamic tensile strength of thematerial to the pullback velocity (the difference between themaximum speed and the speed corresponding to rebound;see Figure 2) assuming a linear-elastic behavior of thematerialbefore reaching the tensile ultimate strength:

𝜎dyn =1

2

𝜌𝐶

0Δ𝑉pb, (1)

where 𝜌 = 2660 kg/m3 is the density, 𝐶0= 4050m/s is the

one-dimensional wave velocity in the material, and Δ𝑉pb =

3.5m/s is the pullback velocity. A dynamic tensile strengthof 18.9MPa is obtained (the strain rate is about 70 1/s) in thistest using the Novikov et al. [18] approach. The quasi-statictensile strength of the specimenwith the same size is reportedas about 8MPa [15].

Furthermore, the nominal stress level (𝐸𝜀) obtained fromthe incident wave data at strain gauge G1 is shifted in thetime direction and compared to the one from the rear facevelocity data using (1); see Figure 3. The results are very closein the prepeak and early postpeak regions which indicatethat the assumption of linear-elastic behavior before damageinitiation is valid. This assumption can be further validated

0 10 20 30 40

0

Nom

inal

stre

ss (M

Pa)

−10

−20

−30

−40

−50−10−20−30−40

Time (𝜇s)

Gauge G1Laser

Figure 3: Nominal stress obtained from gauge G1 (𝐸𝜀) and basedon rear face velocity ((1/2)𝜌𝐶

0𝑉) in a specimen without preexisting

cracks.

by looking at the stress-strain data obtained earlier from theflexural test on the material [15].

2.3. Effect of Preexisting Structural Cracks on the Results. Theeffect of the preexisting cracks on the mechanical responseand the fracture pattern in Edge-On Impact (EOI) tests, thatis, impact of an aluminum projectile onto a rock slab, waspreviously studied [13, 14]. In order to investigate the effect ofthese cracks on the dynamic tensile response of the material,some of the spalling specimens were subjected to coarser


40 60 80 100 120 140 1600

1

2

3

4

5

6

7

8

9

10Re

ar fa

ce v

eloci

ty (m

/s)

Time (𝜇s)

(a)

100 120 140 160 180 200 220 240 260 280 300

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Axi

al st

rain

(%)

−0.1

−0.2

Gauge G1Gauge G2

Gauge G3

Time (𝜇s)

(b)

Figure 4: Spalling test results for a specimen with preexisting structural cracks. Rear face velocity from the laser (a) and strain data from thethree strain gauges (b).

mechanical loading during the cutting process. This intro-duced new cracks in the specimens in addition to the cracksand defects that are present in the material by default. Thesenewly introduced cracks in the specimen are hereafter calledstructural cracks as they are not part of an intact material.To quantify the amount of these cracks, one can perform,for example, quasi-static tensile test andmeasure the effectivestiffness reduction of the specimen. Quasi-static tensile testswere performed on some of the samples and average effectivestiffness of about 30GPa was obtained instead of 52GPa thatis the average effective stiffness of the intact specimens.

When preexisting structural cracks are present in thespecimen, the dynamic response of the material during aspalling test changes considerably. A typical result of thespalling test with preexisting structural cracks is shown inFigures 4 and 5. Strain gauges G1 (𝐿 = 20mm), G2 (𝐿 =

30mm), and G3 (𝐿 = 20mm) are placed in a distanceof 120mm, 61mm, and 41mm from the rear face of thespecimen, respectively. The projectile impact velocity in thiscase was 8.0m/s and the specimen density and 1Dwave speedwere 𝜌 = 2671 kg/m3 and 𝐶

0= 4400m/s, respectively. There

are three main differences in these results compared to theintact specimen presented in the previous section. First, inthe prepeak region (see Figure 5, region I), the structuralcracks in the material are partly closed when the compressivestress travels through the specimen leading to slightly largerdifference between gaugeG1 and rear face velocity (convertedinto stress level in Figure 5) contrary to the case of an intactspecimen. This is mainly due to nonlinear material behav-ior in compression that is related to presence of the struc-tural cracks. Secondly, in the early postpeak region (seeFigure 5, region II), the nominal stress level obtained fromthe incident wave at gauge G1 (calculated as 𝐸𝜀) is not closeto the one obtained from the rear face velocity (calculatedas (1/2)𝜌𝐶

0𝑉). It is most probably due to the fact that

0 10 20 30 40

0

Nom

inal

stre

ss (M

Pa)

Region I

Region II

Region III

−10

−20

−30

−40

−50

−60−10−20−30−40

Time (𝜇s)

Gauge G1Laser

Figure 5: Nominal stress obtained from gauge G1 (𝐸𝜀) and basedon rear face velocity ((1/2)𝜌𝐶

0𝑉) in a specimen with preexisting

structural cracks.

the reflected incident wave from the free surface of thespecimen experiences some partial reflection when it reachesthe surfaces of the structural cracks. It also indicates thatthe Novikov linear-elastic assumption is not valid in thiscase. Thirdly in the postpeak region of rear face velocity(Figures 4(a) and 5, region III), the rebound phenomenonis not seen or is negligible in these specimens and the rearface velocity curve is more plateau-like in this section.This ismost probably due to the fact that when rebounding shouldoccur due to generated waves from initiated final fracture


planes, the waves are blocked between the structural cracksand cannot reach the free surface. Accordingly, the structuralcracks generate a damping effect in this case that prevents therebounding phenomenon.

It should bementioned that themeasurement from gaugeG2 seems to be extremely high in the tensile part. The reasonfor this could be that this gauge was glued near one of thestructural cracks in which the tensile axial strain gets locallylarge. The pullback velocity in this test is Δ𝑉pb = 3.0m/s anda dynamic tensile strength of 17.4MPa is obtained (the strainrate is about 100 1/s) if one uses the Novikov et al. approach[18]. However, it should be mentioned that the Novikov et al.approach [18] is invalid in this case as the tensile behaviorprior to the ultimate stress is not linear-elastic.

3. Numerical Modeling

3.1. Constitutive Model and Results. The fragmentation pro-cess in brittle materials exposed to dynamic loading, withparticular application to percussive drilling, is of mostinterest in this investigation. The stress state in the materialbeneath the drilling tool consists of both compressive andtensile stresses. It is well known that brittle materials such asrocks behave differently in compression and tension. There-fore the constitutive model for these materials, to account forsuch types of phenomena, should include this difference andshould be able to distinguish between the two different stresssign dependent responses. For this reason and based on aprevious investigation [13], the KST-DFH constitutive modelis selected to deal with the fragmentation modeling in brittlematerials due to dynamic loading.

The KST-DFH material model is composed of two sep-arate parts in order to deal with both compressive andtensile responses of the material. A plasticity model (KST) isemployed to simulate the compressive behavior of geomate-rials accounting for the effect of hydrostatic and deviatoricparts of the stress tensor. The fragmentation process, due tothe opening of cracks, is defined by using a damage model(DFH), which is explained in detail in [6, 7]. In the spallingtest, however, the level of compressive stresses is not highenough to induce any plastic deformation.Therefore themainemphasis in this investigation with spalling tests is to validatethe DFH part of the constitutive model that deals with thedynamic tensile behavior of the material.

In the DFH model, defects with different sizes andorientations are assumed to be randomly distributed withinthe brittle material. Under static loading the weakest defectis triggered leading to a rapid failure of the sample. Conse-quently the failure stress is a random variable. Accordingly, aprobabilistic approach may be employed to explain the mate-rial response to tensile loading at high strain rates.The weak-est link theory andWeibullmodel are adopted as a frameworkfor the damagemodel [7, 19, 20].Themodel covers the tensilebehavior of brittlematerials from low to high loading rates. Atlow loading rate conditions, the fracture process is generallythe consequence of the initiation and growth of a single crack.This single crack is created from the activation and propaga-tion of the weakest defect in thematerial.Therefore aWeibullmodel is adopted to explain this probabilistic response at

low loading rates. Using a Poisson point-process framework,the weakest link assumption, and aWeibull model, the failureprobability 𝑃

𝐹is given by

𝑃

𝐹= 1 − exp [−𝑍eff𝜆𝑡 (𝜎𝐹)] , (2)

where 𝑍eff is the effective volume [21] and 𝜆𝑡is the initiation

density defined by

𝜆

𝑡(𝜎

𝐹) = 𝜆

0(

𝜎

𝐹

𝑆

0

)

𝑚

, (3)

where 𝑚 is the Weibull modulus, 𝑆𝑚0/𝜆

0is the Weibull scale

parameter, and 𝜎

𝐹is the maximum principal stress in the

whole domain. The effective volume, 𝑍eff , is expressed as

𝑍eff = 𝑍𝐻

𝑚, (4)

where 𝑍 is the size of the whole volume and 𝐻

𝑚the stress

heterogeneity factor [22] written as

𝐻

𝑚=

1

𝑍

∫

Ω

(

⟨𝜎

1⟩

𝜎

𝐹

)

𝑚

𝑑𝑍, when 𝜎

𝐹> 0. (5)

In (5), 𝜎1is the local maximum principal stress and ⟨∙⟩

Macaulay’s brackets. The stress heterogeneity factor char-acterizes the effect of the load pattern on the cumulativefailure probability. Last, the average failure stress 𝜎

𝑤and the

corresponding standard deviation 𝜎sd are written as

𝜎

𝑤= 𝑆

0(𝜆

0𝑍𝐻

𝑚)

−1/𝑚

Γ (1 +

1

𝑚

) , (6)

𝜎sd = 𝑆

0(𝜆

0𝑍𝐻

𝑚)

−1/𝑚√Γ(1 +

2

𝑚

) − Γ

2(1 +

1

𝑚

),(7)

where D is the Euler function of the second kind

Γ (1 + 𝑥) = ∫

∞

0

exp (−𝑢) 𝑢𝑥𝑑𝑢. (8)

Under high strain rate conditions, such as the situation dur-ing a spalling test, several cracks are initiated and propagatefrom the initial defects leading to multiple fragmentation.In that case, while the weakest defect is activated and theresulting crack propagates, several other cracks are initiatedduring this time. In contrast to the low loading rate conditionsthat characterize the fracture process in general (with theconsequence of the initiation and growth of a single crackleading to fracture of the whole structure), when the loadingrate is high there is enough time for the stress to reach highlevels and activate smaller (or stronger) defects. When acrack is propagating at a very high velocity (a portion of thestress wave velocity), it relaxes the stresses in its vicinity. Themultiple fragmentation process with multiple cracks growingat the same time stops when the whole structure is covered bythese relaxed stress regions (see Figure 6).

The interaction law between cracks already initiated andthe critical defects of the material is given by the concept ofprobability of nonobscuration 𝑃no [6, 7]. In the case of multi-ple fragmentations, the interaction between the horizon (the


The stress level is increasing by time

Figure 6: Obscuration phenomenon and multiple fragmentationprocess.

region around a crack where stresses are relaxed due to crackopening) and the boundary of the domain Ω is small and ifa uniform stress field is assumed, the obscuration probability𝑃

𝑜is written as [6]

𝑃

𝑜(𝑇) = 1 − 𝑃no (𝑇)

= 1 − exp(−∫𝑇

0

𝑑𝜆

𝑡

𝑑𝑡

[𝜎 (𝑡)] 𝑍

𝑜(𝑇 − 𝑡) 𝑑𝑡) .

(9)

In (9),𝑍𝑜is the obscured zone, 𝜎 the local eigenstress compo-

nent, 𝑇 the current time, and 𝑡 the crack initiation time. Theprobability of obscuration is defined for each eigendirection𝑖 and the change of 𝑃

𝑜𝑖is expressed in differential form, in

order to be employed in an FE code using (9), as

𝑑

2

𝑑𝑡

2(

1

1 − 𝑃

𝑜𝑖

𝑑𝑃

𝑜𝑖

𝑑𝑡

) = 3!𝑆 (𝑘𝐶

0)

3

𝜆

𝑡[𝜎

𝑖(𝑡)] ,

when𝑑𝜎

𝑖

𝑑𝑡

> 0, 𝜎

𝑖> 0,

(10)

where 𝜎

𝑖is the local eigenstress component, 𝑆 is a shape

parameter (equal to 4𝜋/3 when the obscuration volume issimilar to a sphere in 3D), 𝑘 is a constant parameter (𝑘 = 0.38

when the crack length becomes significantly larger than theinitial size), and 𝐶

0is the 1D wave speed.

More recently, Erzar and Forquin [17] have investigatedthe postpeak tensile behavior of concrete by means of MonteCarlo calculations and tensile experiments performed ondamaged but unbroken spalled samples. An improvement ofthemodelingwas proposed based on the following statement:despite the propagation of the unstable cracks in the spec-imen, there is still a cohesive stress in the vicinity of thesetriggered cracks that controls the whole softening behaviorof the material. Therefore a cohesive model is combined withtheDFHmodel to describe the cohesive stress in the obscuredzone and the softening behavior of geomaterials in dynamictension [19, 20]. In the cohesivemodel, an extra term is addedto the macroscopic stress Σ

𝑖as

Σ

𝑖= (1 − 𝑃

𝑜𝑖) 𝜎

𝑖+ (𝑃

𝑜𝑖)

𝛼𝐷𝜎coh (𝜀) = (1 − 𝐷

𝑖) 𝜎

𝑖, (11)

where 𝜎coh is the residual strength in the obscuration zoneyielding

𝜎coh = 𝜎

𝑑

𝑜exp(−( 𝜀

𝜀

𝑑

0

)

𝑛𝑑

) . (12)

Table 1: Material parameters used in the DFH material model.

Mechanical parameters] 0.15DFH model parametersWeibull parameters𝑚 23𝜎

𝑤(MPa) 18.7

𝜎sd (MPa) 1.0𝑍eff (mm3) 195

Obscuration volume parameters𝑆 3.74𝑘 0.38

In (12) 𝛼𝐷, 𝜎𝑑𝑜, 𝜀𝑑0, and 𝑛

𝑑are material-dependent parameters

and𝐷𝑖is the damage variable defined for each principal direc-

tion. The cohesive term can be seen as an extra contributionrelated to the fracture energy of the material. It enforces thefinal failure of an element to occurwhen the dissipated energydue to the damage process reaches the fracture energy of thematerial.

In the eigenstress frame, the compliance tensor is definedby

[

[

[

𝜀

1

𝜀

2

𝜀

3

]

]

]

=

1

𝐸

[

[

[

[

[

[

[

1

1 − 𝐷

1

−] −]

−]1

1 − 𝐷

2

−]

−] −]1

1 − 𝐷

3

]

]

]

]

]

]

]

[

[

[

Σ

1

Σ

2

Σ

3

]

]

]

, (13)

where 𝜀

1, 𝜀2, and 𝜀

3are the principal strains and 𝐸 and ]

are Young’s modulus and Poisson’s ratio of the undamagedmaterial, respectively.

In the numerical analysis discussed below, the Bohusgranite rock characterized in [15] is considered. Explicitvalues of thematerial parameters are presented in Table 1.Thecohesionmodel parameters are calibrated based on the resultsof the spalling tests and are reported later in the next section.

At high strain rates, the ultimate strength is deterministicand is obtained from the DFH model as a function of theWeibull parameters as

Σ

𝑢= 𝜎

𝑐(

1

𝑒

(𝑚 + 𝑛 − 1)!

𝑚!𝑛!

)

1/(𝑚+𝑛)

,(14)

where 𝑛 is the medium dimension (𝑛 = 3 in 3D) and 𝜎

𝑐a

characteristic stress as

𝜎

𝑐= (𝑆

0𝜆

0

−1/𝑚)

𝑚/(𝑚+𝑛)

(

∙

𝜎)

𝑚/(𝑚+𝑛)

(𝑆

1/𝑛𝑘𝐶)

−𝑛/(𝑚+𝑛)

.(15)

It should be mentioned that the DFH model predicts thegranite tensile strength at the strain rate of 70 1/s (the samestrain rate as in the spalling test discussed earlier) as 19.5MPawhich is fairly close to the experimental result, 18.9MPa.Thematerial parameters used for this calculation are taken fromTable 1. It should also be mentioned that the effective volumein this calculation is the volume of the cylindrical specimen


that is used in the spalling tests and therefore the averagefailure stress and the standard deviation should be scaledbased on the Weibull size effect.

The material strain rate sensitivity can be described bythe DFH model using a multiscale description that is prob-abilistic at low strain rates and deterministic at high strainrates [7]. Equation (10) is valid for a multiple fragmentationphenomenon at a high stress rate. In order to cover bothsingle and multiple fragmentation processes using the samefinite element (FE) code, the defect density function ismodified as [6, 7]

∧

𝜆𝑡[𝜎

𝑖(𝑡)] =

{

{

{

{

{

0, if 𝜎𝐹(𝑡) ≤ 𝜎

𝑘,

𝜆

0(

𝜎

𝑖(𝑡)

𝑆

0

)

𝑚

, otherwise,(16)

where 𝜎𝑘is the random stress generated for each finite ele-

ment according to Weibull law. A more detailed descriptionof the multiscale model can be found in [7]. To show theprediction of the model in the whole range of strain rates,three values of 𝜎

𝑘are considered and the model prediction

is plotted in Figure 7.

3.2.Modeling of the Spalling Test andDetermining the CohesiveModel Parameters. The equation of motion is discretizedusing the FEmethod and the explicit time integration schemeis employed. The numerical simulation of the spalling tests iscarried through with the DFH material model implementedas a VUMAT subroutine in the ABAQUS/Explicit software[23]. The finite element mesh used in the simulation of spall-ing tests is shown in Figure 8 using 8-node linear elementswith reduced integration.

First the original DFH model with no cohesion isemployed and the results are comparedwith the experiments.Later on, a cohesive strength is added to the original modelto more realistically deal with the softening behavior of thematerial at dynamic loading. A parameter study is performedto obtain the cohesive model parameters that forms a bestfit to the experimental results. It can be seen that adding thecohesive model makes the results more realistic and closer tothe experimental results (see Figure 9). The cohesive modelparameters are summarized inTable 2. Furthermore, the axialstrain from the numerical model is compared with the gaugesmeasurement and good agreement is obtained (see Figure 10).

3.3. Modeling of the Spalling Test with Preexisting StructuralCracks. The numerical modeling of the spalling tests withpreexisting structural cracks is performed. As the state of theinitial damage in each specimen is not completely clear, aset of numerical analyses is needed to define this state foreach test. This calibration stage mainly includes changing theamount of the preexisting cracks to obtain the similar stiffnessreduction as the specimen that reflects itself mainly in thepostpeak part in the rear face velocity profile.

Figure 11 shows the initial damage state used in thenumerical simulation of the specimenwith preexisting cracks(the corresponding experimental results for this specimen arediscussed in Section 2.3). The initial cracks in the numerical

12

14

16

18

20

22

24

Strain rate (1/s)

Ulti

mat

e ten

sile s

treng

th (M

Pa)

Spalling test

10−2 10−1 100 101 102 103

Σu(𝜎w − 𝜎sd) Σu(𝜎w + 𝜎sd)Σu(𝜎w)

Figure 7: Ultimate strength in granite as a function of the strain ratein logarithmic scale based on the DFH model using the multiscaledescription and three values of random stress 𝜎

𝑘.

Figure 8: FE mesh used in the simulations of spalling tests with38,000 8-noded elements.

40 60 80 100 120 140 1600

1

2

3

4

5

6

7

8

Rear

face

velo

city

(m/s

)

Time (𝜇s)

ExperimentDFH model

DFH model with cohesion

Figure 9: Finite element and experimental results from the spallingtest for rear face velocity.


Table 2: Material parameters used in the cohesion model.

Cohesion model parameters𝛼

𝐷1.5

𝜎

𝑑

𝑜(MPa) 12

𝜀

𝑑

00.01

𝑛

𝑑1

150 170 190 210 230 250 270 290

0

0.05

0.1

0.15

0.2

0.25

Axi

al st

rain

(%)

−0.05

−0.1

Time (𝜇s)

G1-exp.G1-modelG2-exp.

G2-modelG3-exp.G3-model

Figure 10: Finite element and experimental results from the spallingtest for axial strain.

model consist of sets of elements with negligible tensilestrength that leads to their immediate failure (the damagevalue becomes unity) when loaded in tension. They can stillcarry compressive loads when crack closure occurs due tocompressive stresses. The numerical approach is describedin detail in [14]. It should be mentioned that the quantityand length of these structural cracks do not necessarilycorrespond to the real situation in the specimen. Thereforeonly the total initial damage is calibrated based on thestiffness reduction occurring in the experimental results.Thecomparison between the experimental and numerical resultsis shown in Figure 12. The results with the preexisting cracksare in a better agreement with the experimental ones.

Furthermore, a numerical (FEM) quasi-static tensile testis performed on the specimen with initial damage stateaccording to Figure 11 and an effective stiffness of about35GPa is obtained. This is fairly close to the average stiffnessfrom the quasi-static tensile tests that is performed on someof the specimens with preexisting cracks as discussed earlier.This further indicates that the state of initial damage in thenumerical simulation is of the correct order.

4. Conclusions

The rate dependency of tensile strength in granite with andwithout preexisting cracks is investigated bymeans of spalling

Figure 11: Preexisting structural cracks in the analyzed specimen.The finite element mesh is also shown.

40 60 80 100 120 140 1600

1

2

3

4

5

6

7

8

9

10

Rear

face

velo

city

(m/s

)

Time (𝜇s)

ExperimentDFH model with cohesionDFH model with cohesion and st. cracks

Figure 12: Finite element and experimental spalling test results.Preexisting structural cracks are present in the simulations.

experiments. Considerable strain rate dependency of thetensile strength is obtained at strain rates of about 102 1/s, thisloading rate being pertinent to the situation of rock materialsat percussive drilling, which is the application of interest inthis investigation. For instance, a dynamic tensile strengthof 18.9MPa is obtained at a strain rate of 70 1/s in a samplewithout preexisting cracks.This is more than twice the tensilestrength of the specimen (with the same size) at quasi-staticconditions, which is 8MPa.

TheDFHanisotropic damagemodel is used to explain thematerial response at dynamic loading.TheDFHmodel allowspredicting the dynamic tensile strength at strain rate of 70 1/sof 19.5MPa which is fairly close to the experimental results.

Some specimens are exposed to coarser mechanical load-ing during the cutting process and new cracks, called struc-tural cracks in this work, are introduced in addition to thedefault material cracks and defects. It is shown that themechanical response of the material changes dramaticallyduring spalling test due to such preexisting cracks. The lowereffective stiffness of these specimens, in tension, reflects itselfin the asymmetric postpeak part in the rear face velocityprofile. Also the rebounding phenomenon is not seen or is


negligible in these specimens and the rear face velocity curveis more plateau-like in the postpeak section.

Numerical modeling (finite element modeling) of thespalling tests is performed. First the original DFHmodel withno cohesion is employed and the results are compared withthe experiments. Later on, a cohesive strength is added to theoriginal model to more realistically deal with the softeningbehavior of the material at dynamic loading. It is shown thatadding the cohesive strength makes the results more realisticand closer to the experimental results.

Furthermore, two-scale numerical modeling (FEM) ofthe spalling tests accounting for preexisting structural cracksis performed. As the state of the initial damage in each speci-men is not completely clear, a first set of numerical analyses isconducted to define this state for each test. The initial cracksare introduced in the numerical model by selecting sets ofelements and allocating them negligible tensile strength thatleads to their immediate failure when loaded in tension.Theycan still carry compressive loads when crack closure occursdue to compressive stresses. It is shown that adding suchcracks leads to results more similar to the experimental ones.

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper.

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Research Article On the Tensile Strength of Granite at ...downloads.hindawi.com/journals/amse/2016/6279571.pdf · Research Article On the Tensile Strength of Granite at High Strain