Fracture of PBX notched specimens: Experimental research andnumerical prediction

Y.L. Liu a, D.A. Cendón b,⇑, P.W. Chen a,⇑, K.D. Dai aa State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 10081, ChinabUniversidad Politécnica de Madrid, Departamento de Ciencia de Materiales, E.T.S de Ingenieros de Caminos, Canales y Puertos, calle Profesor Aranguren 3, 28040 Madrid, Spain

a r t i c l e i n f o

Article history:Received 14 May 2017Revised 6 June 2017Accepted 6 June 2017Available online 11 June 2017

Keywords:PBXCohesive crack modelTheory of Critical Distances

a b s t r a c t

Polymer-bonded explosives (PBXs) are being increasingly applied for both military and civil applications,especially when high performances are required. From a material structure point of view, they can beconsidered as a kind of composite material since they are made of a polymer matrix filled with a highcontent of explosive granules. They are intended for applications in which common explosives are noteasily melted into the required final shape or are difficult to machine. For this reason, their use is nor-mally associated to complex shapes with presence of corners and many different stress concentrators.Since unexpected failure and crack propagation in these materials may lead to malfunctioning and evento safety issues, the study of their fracture behavior is of paramount importance.In this paper the results of an experimental campaign on PBX semi-circular notched specimens sub-

jected to mode-I bending are presented. In the tests different notch lengths were considered while thenotch tip geometry was preserved. Besides this, the critical loads experimentally obtained were also eval-uated through different criteria, such as the Embedded Cohesive Crack Model or the Theory of CriticalDistances, in order to study the best approach to the fracture assessment of notched components madePBX.

! 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Polymer-bonded explosives (PBXs) are highly particle-filledcomposites, which contain a high percentage of explosive granules(usually more than 85% by weight) distributed in soft polymericcomposites (typically 2–10% by weight). PBXs offer a wide varietyof applications, ranging from rocket propellants to explosivecharges in conventional munitions. As PBXs usually containnumerous cracks, flaws and natural fractures inside, mechanicaldamage and fracture may appear during the process of manufac-ture, transport and storage. Like other quasi-brittle materials, PBXswith low strength are likely to fracture under tensile loading.Therefore research [1–6] on the mode-I (tension/opening mode)fracture damage characteristics is one of the main endeavors insecurity and reliability issues of PBXs.

However, due to brittle behavior, the difficulty in preparationand the risk of handling of explosives of PBX samples, the studyof notched PBX components poses challenges. From a few publica-tions [6–8] on measurement of fracture toughness, three point

bending (TPB) test with a single notch was conducted to measureMode-Ⅰ fracture toughness of PBXs. However, for brittle or quasi-brittle materials such as explosives, it is difficult to prepare thespecimens for TPB test due to its low strength, brittle nature andsafety concern. Therefore, some new specimen geometries and testmethods have been developed for measuring the fracture tough-ness. Chong and Kuruppu [9] firstly used the semi-circular bending(SCB) test to determine the fracture toughness of rocks and devel-oped a formula for KIC through both the strain energy release rateand the elliptical displacement methods. Zhou [6] applied SCB testto PBX mock materials to measure fracture toughness which has agood consistency with the ones measured by TPB tests and flat-tened Brazilian disc test. Since the geometry of SCB samples isadvantageous for its convenient specimen preparation and for itsrelative ease of experimental operation, the better choice to studythe fracture behavior and failure assessment of PBX is SCB test.

In the case of brittle materials under mode I, the assumption oflinear fracture behavior is satisfactorily accurate for materials thathave low strength and fail in a brittle manner. However, when thestress concentration is caused by notches, the analysis becomesmore complex and a different framework must be used, even ifthe material behaves the linear elastic type. Thus differentapproaches have been proposed during the last decades for the

http://dx.doi.org/10.1016/j.tafmec.2017.06.0040167-8442/! 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding authors.E-mail addresses: dcendon@mater.upm.es (D.A. Cendón), pwchen@bit.edu.cn

(P.W. Chen).

Theoretical and Applied Fracture Mechanics 90 (2017) 268–275

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analysis of the critical load for notched brittle materials, such asthose based on critical virtual cracks [10], the strain energy density[11], the Theory of Critical Distances (TCD) [12] or the cohesivezone model (CZM) [13]. Beside these methods, the Theory of Crit-ical Distances is particularly useful for predicting fracture staticfailures in notched brittle components even if the applied systemof forces is complexed. The advantage of cohesive zone model isthat crack may form anywhere at the continuum and not onlyahead of a pre-existing crack or stress concentration. Therefore,both TCD and CZM could be applied for the critical load assessmentof brittle materials with different geometries. However, the appli-cations to study on notched PBX components are still rare.

The motivation of this work is to apply experimental andnumerical approaches to study the tensile deformation and frac-ture behavior of PBX mock material. The paper is organized as fol-lows. First, the experimental results of SCB tests are summarizedand fracture mechanics parameters such as tensile strength andfracture energy were measured. The second part provides theimplementation and numerical prediction of embedded cohesivecrack model (ECCM) based on CZM theory. The third is devotedto the application achieved with the Theory of Critical Distances.Finally, experimental results and numerical predictions are com-pared and discussed.

2. Material and specimen

The PBX mock material used in this paper is comprised of Ba(NO3)2 particles (90–95% by weight) held together by fluoro-rubber (5–10% by weight). The samples with a diameter of20 mm were made via hot pressing with 200 MPa at 100 "C for1 h. The measured specimen density was 1.829 g/cm3. The SCBsamples were notched with a 0.2 mm thickness steel blade andthe initial notch was cut along the line of symmetry at the sampleedge and oriented along the specimen thickness. The specimenswere prepared with length of initial crack: 0.80 mm, 0.96 mm,1.52 mm, 2.30 mm, 3.14 mm, 3.66 mm, 4.30 mm, and 5.00 mm.The radius of the initial crack tip was 0.15 mm.

2.1. Fracture mechanics characterization

Considering the geometry of SCB specimen, as shown in Fig. 1, aMode-Ⅰ crack is propagated, so it is suited to study the fracturebehavior of materials. Fracture toughness is an intrinsic propertyof brittle materials to resist crack initiation and propagation. Andthe nominal critical stress intensity factor KIC measured by SCB testis given [9]

KIC ¼ PQffiffiffiffiffiffipa

p

DBYK ð1Þ

YK ¼ 4:47þ 7:4aD% 106

aD

" #2þ 433:3

aD

" #3ð2Þ

where a is notch length; D and B are diameter and the thickness ofthe specimen respectively; YK is a function related to crack lengthratio a/D, non-dimensional parameter; PQ is the critical failure load,which is determined by Pmax/PQ = 1.05, Pmax is external load. When0.25 & a/D & 0.35 and 2S/D = 0.8, the equation to calculate stressintensity factor is valid [14]. KIC value could be obtained from sam-ples with notch length 5 mm if the diameter of samples is 20 mm.Consequently, a KIC magnitude of 0.50 ± 0.02 MPa m1/2 wasobtained [6]. It must be pointed out that this is a nominal valueof plane strain fracture toughness, since it has been obtained fromspecimens notched instead of being cracked.

According to Irwin’s theory, it is possible to obtain the specificfracture energy from the fracture toughness. For two-dimensional problems (Using the classical linear elastic fracturemechanics (LEFMs) for an edge-cracked beam under involvingcracks that move in a straight path in the mode I, the fractureenergy GF is related to fracture toughness KIC,

GF ¼K2

IC

E0 ð3Þ

E0 = E for plane stress and E0 ¼ E1%t2 for plane strain, where E is

the Young’s modulus. However such expression is valid only underelastic conditions and therefore it leads to reasonable results forbrittle materials. A Young’s modulus of 10.2 GPa was estimatedfor this material from the load-displacement results and Poisson’sratio of 0.36 was obtained from Zhou’s previous work [15]. There-fore, by applying Eq. (3), it results in GF = 0.0224 N/mm.

2.2. Tensile strength

For the determination of the tensile strength in brittle materi-als, the splitting (or Brazilian) test is one of the preferred by manyauthors. However, there are other indirect methods that can pro-vide a reliable estimation of the tensile strength. For example, Hof-mand [16], Molenaar [17] and Baymoy [18] proposed formulas to

Nomenclature

a notch lengthD diameterB thicknessPQ critical failure loadPmax maximum loadrmax maximum stressGF fracture energyE young’s modulusKIC nominal critical stress intensity factort traction forces vector between the crack lips

w crack opening vector.r Cauchy’s stress tensor~w equivalent crack openingn unitary vector normal to the crack orientatione strain tensorD the fourth order elastic moduli tensorn parameter defining the shape of the softening curver0 failure stressL characteristic material lengthry yield stress

(a) SCB test of notched specimen

P

Notch

P/2 P/2D2S

P

(b) SCB test of unnotched specimen

Fig. 1. Sketch of SCB test.

Y.L. Liu et al. / Theoretical and Applied Fracture Mechanics 90 (2017) 268–275 269

calculate the critical stress respectively by SCB test while the dis-tance between supporting points is 0.8 of the diameter of thespecimen.

According to these authors, the tensile strength can be obtainedthrough the following expressions respectively:

rmax ¼4:263Pmax

DBð4Þ

rmax ¼4:8Pmax

BDð5Þ

rmax ¼4:888Pmax

DBð6Þ

where rmax is the maximum stress of the specimen, Pmax is the max-imum load, D is diameter and B is the thickness.

In this work, the SCB tests of unnotched specimens were con-ducted and maximum stress could be calculated, as list in Table 1,resulting in an average value of 13.86 ± 1.36 MPa.

3. Experimental campaign on notched specimens

All SCB specimens were gradually loaded on a MTS testingmachine and controlled by displacement mode at a speed of0.05 mm/min at room temperature. The experiment loadingarrangement was as shown in Fig. 2. The distance of the two bot-tom support rollers, 2S in Fig. 1 is 16 mm. The dimensions of thedifferent specimens are listed in Table 2. As the thickness of each

specimen is not identical, the results have been presented in termsof unitary critical load, which is defined as the failure load dividedby the thickness as shown in Table 2. The unitary critical loaddecreases with the increase of notch length.

All samples even the unnotched were fractured under mode Ⅰcrack propagation. As shown in Fig. 3, the sample (notchlength = 0) failed by central through crack and for the notchedsamples as shown in Fig. 3(b), the crack extends along the orienta-tion of notch, and the samples fractured into two parts finally.

4. Numerical predictions

Modelling damage around notches is very difficult and stronglydependent on the microstructural aspects of PBX. In this part, twoapproaches for the numerical simulation of the experimental cam-paign presented in the previous section are discussed. The SCBsamples were modelled as macroscopic homogenous and isotropicmaterial. The critical failure loads under various notch lengthshave been evaluated through: the Embedded Cohesive CrackModel (ECCM) and the Theory of Critical Distances (TCD), assumingplane strain conditions in both cases.

4.1. Embedded cohesive crack model (ECCM)

All cohesive crack approaches rely on the theory of CZM whichwas firstly proposed by Barenblatt [19] and Dugdale [20]. The

Table 1Tensile strength calculated by three methods from unnotched specimens.

DiameterD/mm

ThicknessB/mm

Failure loadPmax/N

Maximum stress rmax/MPa(Hofmand [15])

Tensile strength rmax/MPa(Molenaar [16])

Tensile strength rmax/MPa(Bayomy [17])

20 8.50 499.8 12.5 14.1 14.420 8.36 504.9 12.9 14.5 14.820 8.57 509.5 12.7 14.3 14.5

Fig. 2. SCB test.

Table 2Results of notched SCB specimens.

Notch length a/mm Diameter D/mm Thickness B/mm Failure load Pmax/N Unitary failure load/N

0 20.0 8.47 504.73 59.550.80 20.0 8.06 263.96 32.750.96 20.0 10.00 334.30 33.431.52 20.0 6.76 258.64 37.662.30 20.0 7.48 145.11 19.403.14 20.0 7.08 152.74 21.823.66 20.0 9.48 142.81 15.064.30 20.0 7.52 120.50 16.055.00 20.0 9.00 116.40 12.93

270 Y.L. Liu et al. / Theoretical and Applied Fracture Mechanics 90 (2017) 268–275

assumption of CZM is that a fracture process zone develops the tipof a preexisting crack in the material. Among all the possibleapproaches to cohesive cracks, we have opted for the discrete crackapproach, developed from the view of Hillerborg [21]. Thisapproach has been implemented in the framework of the finite ele-ment method, by means of embedded cohesive crack model [22].This method has been successfully applied to analyze the proper-ties of concrete, ceramics, steel, PMMA and other brittle andquasi-brittle materials [22–25]. The model can be briefly describedas a linear elastic model with no failure under loading condition,while failure in tension is modeled once a threshold value for themaximum principal stress is exceeded. The cohesive crack isinserted in the finite elements, in other words, the displacementjumps are embedded in the corresponding finite element displace-ment field.

4.1.1. Material modelThe generalization of the cohesive crack to mixed mode pre-

sented in [25] is used here, which assumes that the traction vectort transmitted across the crack surfaces is parallel to the crack dis-placement vector w in a central forces model.

t ¼ f ð ~wÞ~w

'w ð7Þ

where f( ) is the softening function, which relates the stress actingacross the crack faces to the corresponding crack opening; ~w is anequivalent crack opening displacement defined as the historicalmaximum of the magnitude of ~w ¼ maxðjwjÞ. According the equa-tion, the softening curve unloads to the origin.

For the continuum, linear elastic behavior of the material out-side the crack has been used. At initial stages of the loading pro-cess, before the maximum principal stress reaches the value ofthe tensile strength, the material behaves as a linear elastic one.Therefore, the stresses are evaluated through

r ¼ De ð8Þ

where r is the Cauchy’s stress tensor, e is the strain tensor and D isthe fourth order elastic moduli tensor.

Once the maximum principal stress exceeds the tensilestrength, a crack is inserted inside the element in the direction nor-mal to the maximum principal stress. Since the crack traction vec-tor must be in equilibrium with the continuum stress tensor, thefollowing equation must be satisfied:

rn ¼ f ð ~wÞ~w

'w ð9Þ

where n is the unitary vector normal to the crack orientation. This isthe fundamental equation for ECCM, since it cannot be explicitlysolved; theNewton-Raphson iterative schemehas been programmed.More details about the implementation can be found in [25].

In particular, the softening curve is a property that must beinput to any of the continuum-based models. A family of softeningcurves is used [26] in ECCM model and defined by followingexpression:

f (ðnÞ ¼ 1% enn1þ ðen % 1Þn

ð10Þ

where f ( is the dimensionless cohesive strength, n is the dimension-less crack opening and n is a real number that modifies the shape ofthe softening curve. By changing the parameter n, it is possible tochange the shape of the softening curve: negative values of n leadto a quasi-rectangular curve; while positive values lead to aquasi-exponential softening curve. This family of curves makes itpossible to modify the softening curve of the material smoothly justby changing the parameter n.

The embedded cohesive crack model has been implemented asa user defined material in FORTRAN code with explicit time inte-gration LS-DYNA 971.

4.1.2. Numerical simulations of ECCMThrough a lot of trial and previous research on quasi-brittle

materials, the exponential-like curves are preferred in simulationsfor a better description of the real softening behavior in most brit-tle and quasi-brittle materials. For this reason, the value of param-eter n in formula (10) has been set to 1.0 in this research.

For a complete definition of the softening curve, the tensilestrength, ft and the specific fracture energy, GF, which is the exter-nal energy, required to create and fully break a unit surface area ofa cohesive crack. The tensile strength and the specific fractureenergy are material properties and can be measured by standardtests. According to Section 3, the specific fracture energy was setto 0.0224 N/mm. Based on the experimental results, the tensilestrength was adjusted slightly for a better agreement, but alwaysinside the experimental scaler of 13.86 ± 1.36 MPa. The materialproperty was based on linear elastic material model and all param-eters in the present model for SCB test are presented in Table 3.

Fig. 4 shows the experimental results when notch length is0.96 mm (Experimental result from [4]) and the shows a character-istic brittle response of PBX mock material. With the increase ofexternal force, the specimen deforms almost lineally. After thepeak load, the crack will initiate. The external load reaches its peakload. After this, the load rapidly decreases and behaves brittle.

In the case of notch length 0.96 mm, Fig. 4 compares the exper-imental records load vs. displacement with the numerical predic-tion. The tail of the numerical curve slightly differs from theexperimental one, probably because the softening curve is too sim-ple. However, the loading curve has a good agreement with theresult from the test. Fig. 5 presents the crack path of the numericalprediction, i.e., a single dominant crack is generated centrally oralong the orientation of preset notch tip, which is a sufficientlyaccurate approximation of the crack path as compared with the

Fig. 3. Fracture morphology of SCB sample (a) unnotched; (b) notch length is 0.96 mm.

Table 3Model parameters for SCB test.

E/GPa t ft/MPa GF/N'mm%1 n

10.2 0.36 14 0.0224 1.0

Y.L. Liu et al. / Theoretical and Applied Fracture Mechanics 90 (2017) 268–275 271

failure morphology as shown in Fig. 3. So the parameters aboveand ECCM model turns out to be effective and valid for PBX mockmaterials.

4.2. The Theory of Critical Distances (TCD)

The Theory of Critical Distances (TCD) [12], proposed by Taylor,is considered to be an effective technique in fracture and fatiguelife prediction of the materials and structures [27,28] as well asstress concentration features created by notches and cracks. Thetheory of TCD assumes that failure of the body containing a crackor notch can be predicted using elastic stress information in a crit-ical region close to the notch tip. This approach now could beapplied successfully to predict static failures in brittle (or quasi-brittle) materials weakened by any kind of geometrical feature[29,30]. However, the relevant research on PBXs is rare.

4.2.1. Basic theory of TCDThe Theory of Critical Distances generally contains the line

method (LM), the point method (PM), the area method (AM) andthe volume method (VM). In all those methods, the critical regionis defined in terms of a characteristic material length L, which isassumed to be a material property and could be calculated by fol-lowing formula.

L ¼ 1p

Kc

r0

$ %2

ð11Þ

where Kc is fracture toughness, r0 is failure stress. To be precise, ifthe TCD is applied to perform the static assessment of brittlenotched materials, failure stress r0 can be taken equal to the mate-

rial ultimate tensile strength and the fracture toughness Kc equalsto the plane strain fracture toughness KIc [12].

The point method and line method were first proposed andused in fatigue by Peterson [31] and Neuber [32] respectively. Asshown in Fig. 6, the point method (PM) is the simplest approach.The criterion for fatigue failure is that if the stress at a distanceL/2 from the notch tip equals the plain-specimen endurance limit,failure will occur. The line method (LM) states that failure willoccur once the averaged stress ahead of the notch reaches a criticalvalue around the maximum stress point. It is also possible to aver-age the stress over an area (AM) and a volume (VM) around thenotch-tip. In this paper, critical point method (PM) is chosen topredict the critical load of PBX with different SCB samples.

4.2.2. Application of TCD to SCB testsFor the sake of simplicity and given that the information about

the actual geometry of the notch tip was limited, the way to deter-mine the inherent material strength [33] is briefly depicted inFig. 6, the critical distance L/2 and r0 result from the intersectionof the linear elastic stress vs. distance curves determined in incip-ient failure condition and generated by testing notched samples.The linear-elastic stress fields in the vicinity of the notch tips weredetermined numerically by using commercial Finite Element (FE)software LS-DYNA. The SCB models were simulated with fully inte-grated solid element with the mesh density around the notch tipsas shown in Fig. 7.

To apply the reasoning explained in the previous paragraph, thefirst step of the prediction is to make the finite element analysis ofSCB samples. The linear-elastic material behavior is used andYoung’s modulus and Poisson’s ratio is the same with the one usedin ECCM method. For the tensile failure model, the crack propaga-tion path is perpendicular to the direction of the maximum tensile

Fig. 4. External load and displacement curves when notch length is 0.96 mm.

Fig. 5. Numerical prediction of the crack path of the specimen (a) unnotched; (b) notch length is 0.96 mm.

Linear-Elastic StressDistribution

L/2

2L

σ0σnom

σnoma

Stress

Distance

Fig. 6. TCD methods: the PM uses the stress at a point r = L/2 from the notch root;the LM uses the average stress over a distance r = 0 to 2 L [29]

272 Y.L. Liu et al. / Theoretical and Applied Fracture Mechanics 90 (2017) 268–275

stress. The process to predict the critical load with TCD method isshown below:

Step 1: Make the finite element analysis of SCB model with dif-ferent notch lengths respectively.Step 2: Make linear-elastic stress vs. distance curve at theendurance limit which is the limit load value obtained byexperimental test and list in Table2.Step 3: Collect the intersections and calculate the average val-ues of stress and distance which is the critical stress r0 and crit-ical distance L/2.Step 4: Calculate critical load. In this case, linear-elastic materialbehavior is applied, whichmeans that the stress and the load arelinearly related for a given point. There’s a scale factor g betweenthe critical stress and critical load in following formula.

g ¼ r0

Pcritical¼ rx

Pxð12Þ

where rx is the extension stress at the point of average critical dis-tance from the notch tip; Px is the external load which can beobtained from simulation results. Thus, the critical load accordingto TCD theory is predicted.

According to Step 2 above, Fig. 8 shows that linear elastic stressvs. distance curves in which X-stress is the stress of X direction atthe time when the load reaches to the failure load from experi-ments; the distance is from the notch root to the edge. The tensilestress declines along the distance which agrees with the predictionin TCD theory [29]. From all intersection, the average critical dis-tance and corresponding stress could be obtained. The values ofthese two parameters are 0.55 mm and 7.82 MPa. By applyingthe scale factor to the load at the same time with rx, the criticalload according to TCD theory could be predicted.

From the finite simulations, r0 has been obtained to be7.82 MPa and the characteristic material length L could be givenby Eq. (11). Therefore, it results in L = 1.3 mm. That is, the theoret-ical value of critical distance is 0.65 mm, which is not far from theFE result 0.55 mm. The difference of critical load between the the-oretical value and FE result will be discussed in the followingsection.

5. Results

5.1. Effect of the thickness

From the point of view of fracture mechanics, it is possible todetermine the minimum thickness required to ensure plane strainconditions:

B > 2:5K IC

ry

$ %2

ð13Þ

where B is the thickness of specimen, KIC is the fracture toughness,ry is the yield stress. According to experimental data, KIC is0.50 MPa m1/2 and the yield stress ry could be approached by themaximum stress, 13.86 MPa. With these parameters, a minimumthickness of 3.25 mm to satisfy plane strain conditions is calculated.As from Tables 1and 2, the thickness in SCB specimens is higherthan the minimum thickness for plane strain conditions, it seemsreasonable to assume plane strain conditions for the analysis ofresults. However, for a deeper research on the effect of thicknesson the prediction results, a set of numerical simulations have beendone for the 3.14 mm notch length specimens in order to comparethe differences among plane strain, plane stress and the full thick-

Fig. 7. Simulation model of SCB test (notch length = 0.96 mm).

Fig. 8. Stress of X direction vs. distance along notch tip and enlarged plot about theinteraction zoom (frame area).

Y.L. Liu et al. / Theoretical and Applied Fracture Mechanics 90 (2017) 268–275 273

ness of the specimen. The stress of X direction vs. distance alongnotch tip under these three conditions is shown in Fig. 9. The differ-ences between the three models are scarce, so it’s possible to vali-date the hypothesis of plane strain conditions made in thesimulations. For this reason, the experimental results and thenumerical predictions will be analyzed in terms of unitary criticalload.

5.2. Comparison between the ECCM and the TCD predictions

The unitary failure loads vs. pre-crack lengths obtained fromexperiments, ECCM and TCD method are compared in Fig. 10. BothECCM method and TCD method show good agreement with theactual results. The unitary failure load of ECCMmethod sufficientlydecreases with the increase of notch length approximately in a pat-tern close to exponential one. TCDmethod seems have a closer pre-diction to each point of experimental result and follows a lineardecreasing with increasing notch. The unitary failure loadsobtained from FE result 0.55 mm and theoretical value 0.65 mmrespectively are shown in Fig. 10 with blue and red dots. The uni-tary failure loads obtained by the theoretical value is a bit higherthan the FE results but very close, which implies PBX material isquite brittle and the FE results are acceptable. Another conclusionderived from this result is that the TCD can be used in this materialwith predictive purposes, since the material parameters requiredto feed the model can be obtained directly from characterizationtests, without the need of conducting tests on notched specimensto obtain the TCD parameters, as done in this research.

The deviation could be calculated according to the followingtrivial relationship:

Deviation ¼ Psimulation % Pexperiment

Pexperiment½%* ð14Þ

The use of both the TCD and ECCM resulted in predictions fall-ing within an error interval of ±20%. The level of accuracy is cer-tainly satisfactory, since in the presence of stress concentrators,it is not possible to distinguish between an error of ±20% and anerror of 0% as a consequence of those problems that are usuallyencountered when performing the experimental tests as well asthe numerical analyses [12].

However, LEFM is not able to predict the behavior of shortcracks or of notches [34,35], that is, if the cracks less than somecritical length or the opening angles of notches are smaller thansome critical value, in a manner which LEFM is unable to predict.From the results of Fig. 10, the prediction failure load of TCDmethod is beyond the experimental data when the notch length

is smaller than or equal to 0.4 mm. Meanwhile, ECCM methodgives a good prediction of failure load of unnotched specimens.From the view of practical applications in PBXs, TCD method hasaccurate predictions when the notch length is larger than 0.4 mm(or the critical distance 0.55 mm) while the ECCM method is notlimited to the notches or cracks features.

From an operational perspective, TCD method is easy and sim-ple to implement in simulations with a few parameters (E, t), whilethe ECCM is fed solely by material characterization tests (E, ft andGF).

6. Conclusions

Based on semi-circular bending test of various notch lengths,the mechanical parameters relevant to notch-fracture mechanics,such as tensile strength, rupture load, fracture toughness as wellas specific fracture energy were measured. The load displacementplots showed a characteristic brittle response and the fractureresponse was mostly elastic with only minimal non-linearbehavior.

Both Embedded Cohesive Crack Model and the Theory of CriticalDistances are capable of predicting tensile stress concentrationphenomena, showing good agreements with experimental results.It is possible to conclude that reliable static assessment could beperformed without the need to invoke complex non-linear consti-tutive laws of PBX. It is also possible to transfer the techniques andmethods to other explosive formulations and propellants.

Future work will extend the application of these approachesbased on the TCD and ECCM to dynamic loading conditions andmixed failure mode. Additional experiments and analyses will beperformed to examine the influence of the notch tip geometry.

Acknowledgement

This work was supported by Santander Scholarship, the NSFC ofChina (Grant No. 11521062) and the NSAF of China (Grant No.U1330202). The authors appreciate the Institute of Chemical Mate-rials of CAEP for providing the powder of PBX mock material.

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