Geometry Effects in Four-Point Bending Test for Thin SheetStudied by Finite Element Simulation

Xiaolong Dong1, Hongwei Zhao1,+, Lin Zhang1,2, Hongbing Cheng1 and Jing Gao1

1School of Mechanical Science and Engineering, Jilin University, Changchun 130025, China2Department of Integrated System Engineering, The Ohio State University, Columbus, OH 43210, U.S.A

Some experimental geometry parameters of four-point bending tests, such as span ratio and support radius, obtained from experience,likely lead systematic errors into the final results. In this paper, some typical geometry effects of four-point bending tests for thin sheet areinvestigated by employing finite element analysis (FEA). In order to assure the reasonability and accuracy of the FEA results, the standard tensiletests and four-point bending tests are carried out by commercial instruments. Based on the simulation results, it is noted that the stressdistribution between the inner supports and outer supports is different with the variation of span ratio. The maximum wedging stress is locatednear the inner supports. According to the study, the optimum value of the span ratio is suggested to be around 1/3 at plastic stage. In addition, theeffects of support radius and contact roughness are also discussed to make a beneficial reference for designing four-point bending experimentsand devices. [doi:10.2320/matertrans.M2015178]

(Received May 7, 2015; Accepted November 4, 2015; Published February 25, 2016)

Keywords: four-point bending, geometry effect, finite element analysis, span ratios, stress distribution

1. Introduction

Four-point bending test is a standard widely used methodfor determining materials mechanical properties, such asflexural strength and flexural modulus.1) The standardspecimens and process employed in four-point bending testsare simple to operate, leading it to a convenient way for bothplastic and brittle materials. It is well known that the four-point bending fixture consists of two inner supports and twoouter supports, symmetrically arranged relative to specimen.The four-point bending most remarkable difference fromthree-point bend is that the load is applied by two innersupports. Compared with three-point bending tests, themoment along the specimen between the inner supports isuniformly distributed. Although four-point bending testsare widely used and some relevant standards have beenestablished, our understanding on the deformation mechan-ism of materials in four-point bending test is still far fromcomplete.2,3) For example, the span ratio between innersupports and outer supports is often treated as an empiricalparameter. Up to now, it is hardly to accurately decide thespan ratio when designing four-point test experiments orinstruments. As a result, the span of inner supports is usuallyset to be 1/2 or 1/3 of the outer supports span just because itis easy to operate, but different span ratios between the innersupports and outer supports can easily lead to somesystematic errors in determining flexural strength.4,5)

For this reason, a lot of studies based on experimental andvarious theoretical models have been carried out to makedeep understanding of the geometry effects in four-pointbending tests. All those previous studies have provided muchhelp in understanding such material performance character-ization method.6­8) Theobald et al. conducted an experimen-tal research to analyze the influence of geometric config-uration on four-point bending tests about flexural strengthand modulus. They found that flexural strength is dependenton span ratios and decreases as the inner span increases.6)

Hiroshi Yoshihara and Hiroki Kondo suggested that the shearmodulus may change according to different depth/span ratioin asymmetric four-point bending tests.7)

As the four-point bending test is a continuous and simplemethod to measure the physical performance of materials,it is often carried out as a kind of fatigue test methodfor measuring fatigue properties of metallic and ceramicmaterials.8­13) Grabowski and Yates conducted four-pointbending fatigue tests on waspaloy (a Ni-based superalloy) toinvestigate the geometry effect on the specimen fatiguebehavior.9) The span ratio between inner supports and outersupports is discovered to have a significant impact onmaterial’s fatigue life. It is necessary to make sense of thegeometry effect on four-point bending test and to optimizethe test method.

As the mentioned above, four-point bending test gives auniform maximum tensile stress between two inner supportsbeneath the specimen surface, which is also known as thepure-bending section length. The nominal maximum stress ·on the surface of specimen can be obtained by the beammechanical theory.14)

· ¼ 3Fðl� tÞ=ðbh2Þ ð1Þwhere F is the resultant bending force of inner supports; t isdistance between inner supports; l is the distance betweenouter supports; h and b is the thickness and width of thespecimen, respectively.

Based on some previous studies, the tensile stress betweentwo inner supports is not completely consistent.4,15,16)

Discovering the rules of stress distribution will be helpfulto account for the materials mechanical behaviors under thebending moment. T. Zhai et al. conducted a systematic studyof stress distributions for the specimen with different spanratios of inner supports and outer supports and different ratiosof support span and specimen thickness in four-point bendthrough finite element simulation. They found that the valueof t/h between 1.2³1.5 and l/t between 4³5 leads torelatively uniform stress distribution.4) However, comparedwith the outer span, the specimen thickness selected in their+Corresponding author, E-mail: hwzhao@jlu.edu.cn

Materials Transactions, Vol. 57, No. 3 (2016) pp. 335 to 343©2016 The Japan Institute of Metals and Materials

study is too large. The results are not practical in determinemost of the mechanical properties of thin sheets. The four-point bending test standards (ASTM D790-791 and ASTMD6272-02) recommend the outer span/specimen thickness²16.17) For most thin sheet four-point bending tests, the outerspan is always above 50 times of specimen thickness. Thus,there is an actual demand to make clear the geometry effecton thin sheet four-point bending tests.

Finite element analysis (FEA) is a powerful tool inmechanical researches and could monitor many phenomenawhich cannot be easily obtained by experiments.18) In thispaper, the research of different span ratios of inner supportsand outer supports for 6061 aluminium alloy thin sheet infour-point bending test is carried out via FEA. In addition,the effects of support radius and contact roughness on four-point bending test are also studied. This investigation isaimed to find out the optimum geometry parameter ofdifferent situations for thin sheet four-point bending test andmake a beneficial reference for the future experimentalresearches and instrument designs.

2. Experiments

2.1 Material mechanical propertiesThe mechanical properties of 6061 aluminium alloy were

obtained by standard tensile tests (DNS type universal testingmachine produced from CRIMS). The specimen with thick-ness of 2.0mm is shown in Fig. 1. The specimens were madeby wire electrical discharge machining from a flat cold-rolledsheet. In order to avoid random error in one singleexperiment, five uniaxial tensile experiments were conductedwith the same experimental conditions. The mechanicalproperties are shown in Table 1.

In the uniaxial tensile tests, the deformation of materialwas measured by high precision extensometer to make surethat the strain was measured as accurate as possible, shown asFig. 2(a). What calls for attention is that the tested stress andstrain should be converted to true stress and strain. Therelationship between true stress-strain and nominal stress-strain is as follows,

· true ¼ ·nomð1þ ¾nomÞ ð2Þ¾true ¼ lnð1þ ¾nomÞ ð3Þ

where ·true and ¾true represents the true stress and true strainrespectively; ·nom and ¾nom represents the nominal stress andnominal strain respectively. The curve of true stress againsttrue strain is shown in Fig. 2(b).

2.2 Four-point bending experimentsIn general, the commercial four-point bending instruments

are either too large or easily bring about errors due toinsufficient stiffness. A small sized scientific device wasdeveloped to conduct relative four-point bending experi-ments. The self-developed device is integrated with highprecision sensors, flexible structures, feedback control andcompensation algorithm, which enable to avoid the dis-advantages of current commercial instruments. The sche-matic diagram of the experiment setup is shown as Fig. 3(a).One of the main concerns was the stiffness of the four-pointtesting device. To minimize the error due to the stiffness of

the supports during four-point bending tests, the supportswere made of tool steel, which was much higher stiffnessthan the specimen. The inner supports and outer supportswere fastened to the fixture and device base. Compared withstandard measurement results, the stiffness of the device wasmeasured to be 3,540N/mm and this parameter was takeninto consideration in the tests to modify the experimentalresults.

In order to make sure the finite element analysis accurateand reasonable, a serious of four-point bending experiments

Fig. 1 Specimen for standard tensile test.

Table 1 Properties of 6061 aluminium alloy.

Young’s modulus E (GPa) 72.0

Poisson’s ratio v 0.33

Yield stress ·s (MPa) 158

Elongation A (%) 17

(a)

(b)

Fig. 2 Tensile tests of 6061 aluminium alloy: (a) tensile test machine;(b) stress-strain curve.

X. Dong, H. Zhao, L. Zhang, H. Cheng and J. Gao336

of 6061 aluminium alloy were conducted, shown asFig. 3(b). The loading speed was set to be 0.24mm/min asa quasi-static loading process. The dimensions of two kindsthin sheet specimen were 84.0mm © 5.0mm © 1.0mm and84.0mm © 5.0mm © 2.0mm. According to the experimentalresults, the results consisted well with that from the standardtensile tests.

3. FE Simulation

3.1 FE models and validationThe finite element analysis was adopted using the software

ABAQUS to investigate geometry effects in four-pointbending test.19) Because of the symmetry of specimens andsupports, only half part of the whole model was established.The symmetric boundary conditions were applied on the mid-plane of specimen. The specimen was modeled as a three-dimension deformable body and the material parameters usedare obtained from the standard tensile tests.19) The size ofspecimen was as same as the specimen size in the experi-ments. The supports were idealized as rigid bodies becausethe hardness and strength of supports were much larger thanthe bending materials. The radius of supports was 2.5mmjust as the situation of experiments. Static analysis wasimplemented in the FE model and C3D8R elements wereused for the specimen. The region between inner supportswas of interest. Different mesh sizes were used in two regionsto simplify the analysis, shown as Fig. 4. Along the specimenlength direction, the size of each element was set to be0.15mm in the pure-bending region and 0.40mm in the otherregion. The size was 0.10mm and 0.25mm along the

specimen thickness direction and width direction, respec-tively. Coulomb friction was defined between the contactsurfaces of specimen and supports, and the friction coefficientwas set to be 0.02 in view of good surface treatments. Theboundary conditions of inner support are allowed with singledirection along y-axial while the outer support is fully fixed.The freedom of specimen is not limited as the actualexperimental conditions.

In the first case, the inner span was selected to be 20.0mmand outer span was set to be 75.0mm in the FE model.Figure 5 illustrates the stress distribution along specimenlength direction. It can be seen that the material maximumstress have exceeded the yield stress and both specimenshave experienced plastic strain. In addition, the region aboveneutral layer is subjected to pressure stress and the regionbelow neutral layer is subjected to tensile stress. Themaximum stresses are mainly concentrated on the top andbottom surface of the specimen near the inner supports. Thelocal stress is concentrated in the region between the innersupports when material is squeezed by support. The stressconcentration in pressure is higher than the maximum stressin tension. Local failures beneath the inner supports arealways observed in bending experiments.20) However,experimental results show that the deficiency of tensilestrength is the main reason for fatigue and failure for

(a)

(b)

Fig. 3 Four-point bending experiments of 6061 aluminium alloy:(a) schematic diagram of the experiment setup; (b) four-point bendingtest apparatus.

Fig. 4 Schematic diagram of four-point bend FE model.

(a)

(b)

Fig. 5 Stress distribution of the specimen in four-point bending tests withspan ratio of 20/75: (a) specimen thickness of 1.0mm; (b) specimenthickness of 2.0mm.

Geometry Effects in Four-Point Bending Test for Thin Sheet Studied by Finite Element Simulation 337

aluminum alloy in four-point bending tests.4) Furtherexplanation of stress on the tension surface was describedin the following statement.

The load-displacement curves from simulation results werecompared with those obtained in the four-point bendingexperiments, shown as Fig. 6. Since the specimen stiffness inthe thickness was small, vibrations could be seen in the load-displacement curve during loading process for the case of1.0mm specimen thickness. The comparison of maximumloads from simulation and experiments is listed in Table 2.The error is 6.0% and 1.7% for specimen thickness of 1.0mmand 2.0mm, respectively. The simulation load-displacementcurves basically consisted with experimental results, whichverified the parameters setting in the FE model was feasibleand reasonable.

3.2 Simulation resultsThe finite element models with different span ratios

between inner supports and outer supports were calculated.The outer span l was fixed on 75.0mm, while the inner spant varied from 10.0mm to 40.0mm. For each specimen withthe same thickness, six inner spans were set as 10.0mm,16.0mm, 20.0mm, 26.0mm, 30.0mm, and 40.0mm. Theload-displacement curves of different span ratios are plottedin Fig. 7. With the inner span longer, the moment arm offour-point bend is larger. The load of long inner spanincreases faster than those with short inner spans.

The flexure modulus E could be calculated from linearfitting of load-displacement curve when the material was inelastic stage. Flexure modulus E with respect to inner span tand outer span l is,

E ¼ ðlþ 2tÞðl� tÞ2=ðbh3Þ � ð�F=�dÞ ð4Þwhere d is the displacement of inner support.

Experiment results of Theobald et al. showed that theflexure modulus was independent of span ratios.6) The flexuremodulus listed in Table 3 from simulation results also showthat no obvious difference appears in flexure modulus withdifferent span ratios and the maximum error with standardvalue is below 2.5%. It means that the flexure modulusobtained from four-point bending test with different spanratios is credible and accurate when the bending deflection ofspecimen is not large enough to change contact position inthe supports.1)

(a)

(b)

Fig. 6 Comparison of load-displacement curves of the simulations andexperiments: (a) specimen thickness of 1.0mm; (b) specimen thickness of2.0mm.

Table 2 Comparison of maximum loads from simulation and experiments.

Specimen thickness,t/mm

Experiment Simulation Error, ¤/%

1 15.0 14.1 6.0

2 47.5 48.3 1.7

(a)

(b)

Fig. 7 Load-displacement curves of different span ratios: (a) specimenthickness of 1.0mm; (b) specimen thickness of 2.0mm.

X. Dong, H. Zhao, L. Zhang, H. Cheng and J. Gao338

3.3 Analysis of different span ratiosWhen materials are in the elastic stage, the stress in tension

tends to increase towards the ends of pure-bending region forthin sheet. Figure 8 shows the stress distribution of thetension surface corresponding to different span ratios t/lwhen middle point stress is around 60.0MPa. The simulationresults are similar with previous studies. Zhai et al. foundthat there is a wedging stress near inner supports for four-point bending test with large ratio of t/h. Thus, it isunderstandable that cracks were generally formed at the endof inner span when applied with high-circle fatigue tests.10,11)

With the same outer span, the wedging stress is larger whenthe inner span of four-point bend gets longer.

Interestingly, the tress distribution on tension surfacechanges when the load increases. Figure 9 shows the stressdistribution corresponding to various span ratios t/l whenmiddle point stress is around 165.0MPa. Plastic deformationof the material has occurred. For the case of 1.0mm specimenthickness, the maximum stress is still at the end of pure-

bending region for short inner spans, such as the situation oft/l = 10/75, 16/75. The maximum stress occurs at middle ofpure-bending region when the inner span is long enough,such as the situation of t/l = 30/75, 40/75, although there isa minor stress lift in the region near inner support. The viewof stress distribution with different values of t/l is shown inFig. 10.

Figure 11 illustrates the maximum deviation of stressfrom middle point along pure-bending region with differentspan ratios between inner support and outer support from10/75 to 40/75. The minimum stress deviation appears atthe span ratio of 26/75, while the maximum stress deviationappears at the span ratio of 40/75. It can be seen that thenon-uniformity in pure-bending region remains the highestfor t/l value of 40/75, no matter at elastic stage or plasticstage. It makes sense that the inner span is always less than1/2 of outer span recommended by four-point bendingstandards.

However, the rule of stress distribution of the 2.0mmspecimen thickness is different from that of the 1.0mmspecimen thickness when the stress reaches yield stage.Figure 12 shows the stress distribution corresponding todifferent span ratios t/l when middle point stress is around164.0MPa. The wedging stress remains a dominant factor indetermining how stresses are distributed and the maximumstress occurs at the end sides of pure-bending region, just asthe situation at elastic stage. The difference is that non-uniformity is getting obvious and the point at which stressstarts to rise towards inner supports gets closer to the middlepoint of the specimen. For a shorter distance between theinner supports, the stress rise-point becomes closer to middleof specimen. Although the wedging stress with span ratio of10/75 is smaller than other wedging stresses, it is still notconvinced that the uniformity with span ratio of 10/75 is thebest considering the position of stress rise-point. Consideringall the situations and explanations above, the best uniformityof stress distribution belongs to the span ratio of 26/75. Thus,it suggests that the fatigue life may be longer when the outerspan is around three times of inner span while the material isapplied with low-circle fatigue tests in which material oftenexperienced plastic deformation.

Table 3 Flexure modulus with different span ratios.

Specimen thickness,t/mm

Flexure modulus, E/GPa

10/75 16/75 20/75 26/75 30/75 40/75

1 70.9 70.8 70.2 70.8 71.8 71.3

2 71.0 70.7 72.3 71.8 72.0 72.0

(a)

(b)

Fig. 8 Stress distribution corresponding to different span ratios t/l whenmiddle point stress is around 60.0MPa: (a) specimen thickness of 1.0mm;(b) specimen thickness of 2.0mm.

Fig. 9 Stress distribution corresponding to different span ratios t/l whenmiddle point stress is around 165.0MPa for specimen thickness of1.0mm.

Geometry Effects in Four-Point Bending Test for Thin Sheet Studied by Finite Element Simulation 339

It is noted that the stress distribution changes as the loadincreases. The view of stress distribution on the tensionsurface with different maximum strains for specimen thick-ness of 1.0mm and span ratio of 40/75 is shown in Fig. 13. Itis suggested that the strain is one of the main factors thatinfluence stress distribution. The experimental research fromHassan and Liu also demonstrated that difference of strain

range would influence material’s bending fatigue life.21) Thestress in middle region tends to become larger compared withother regions between two inner supports when the strain ofspecimen is large enough. The strain effect becomes moresignificant for thinner specimen and longer inner span withthe same outer span in 6061 aluminum alloy.

(a) (b)

(c) (d)

(e) (f)

Fig. 10 The view of stress distribution of the tension surface for specimen thickness of 1.0mm: (a) t/l = 10/75; (b) t/l = 16/75;(c) t/l = 20/75; (d) t/l = 26/75; (e) t/l = 30/75; (f ) t/l = 40/75.

Fig. 11 Maximum deviation of stress from middle point with different spanratios for specimen thickness of 1.0mm.

Fig. 12 Stress distribution corresponding to different span ratios t/l whenmiddle point stress is around 164.0MPa for specimen thickness of2.0mm.

X. Dong, H. Zhao, L. Zhang, H. Cheng and J. Gao340

3.4 Analysis of different support radiusesThe effect of support radius on four-point bending tests is

analyzed by changing the radius of inner support from1.0mm to 5.0mm. The inner span and outer span were fixedon 20.0mm and 75.0mm. The load-displacement curves ofdifferent support radiuses are shown in Fig. 14(a). There isno difference at initial phase. However, the load increasespeed becomes faster with larger support radius when thedisplacement exceeds 1.5mm. Figure 14(b) shows the stressdistribution of different support radiuses on the tensionsurface. The wedging stress gets closer to inner support withthe larger support radius and the maximum tension stress mayget over the initial pure-bending region.

The support contact position varies along with the bendingrotation of specimen. The true inner span increases as thespecimen rolls over the support. On the pressure surface, astress concentration region is located on the contact position,as shown in Fig. 15. It should be noted that the mesh sizes ofspecimens are same in Fig. 15, and the only difference is theradius of support. If the contact radius does not changesignificantly, the stress concentrations have only minorvariations from photo-elastic studies.22) Simulation resultswere consistent with the literature. But the stress concen-tration region moves to end side of the specimen which canrepresent the change trend of contact position. When thesupport radius gets larger, move speed of stress concentrationregion gets faster. Thus, large support radius should beavoided to select for thin sheet four-point bending tests tominimize the move trend of contact position.

(a) (b)

Fig. 13 The view of stress distribution of the tension surface with different maximum strains for specimen thickness of 2.0mm and spanratio of 40/75: (a) ¾max = 0.003865; (b) ¾max = 0.006884.

(a)

(b)

Fig. 14 Four-point bending test results with different support radiuses:(a) load-displacement curves; (b) stress distribution.

(a) (b)

(c)

(d)

Fig. 15 Stress concentration on pressure surface with different supportradius: (a) r = 1.0mm; (b) r = 2.5mm; (c) r = 5.0mm; (d) stressdistribution.

Geometry Effects in Four-Point Bending Test for Thin Sheet Studied by Finite Element Simulation 341

3.5 Analysis of different contact roughnessThe specimen is loaded directly by the supports during

four-point bending tests. The contact roughness betweenspecimen and supports must have some influence on the four-point bending results. The surface qualities between thesupports and specimens largely effect the interaction relation-ship.23) Different contact roughness is modeled by settingdifferent friction coefficients in FE simulations. The supportsin the experiments are made of tool steel and the test materialis 6061 aluminum alloy. The friction coefficients on thecontact surfaces are set at 0.02, 0.10 and 0.20 in threedifferent situations.

The lateral force in four-point bending tests increasesalong with the deformation of specimen. Different contactroughness directly influences the level of lateral force, shownas Fig. 16(a). The lateral force with rough face could be timesof which with smooth surface. Although the lateral force offour-point bending test is not included in the resultant forceof two inner supports, the stress distributions with differentlateral forces are remarkably different. The stress distribu-tions on tension surface with different friction coefficients areshown in Fig. 16(b). As mentioned above, the stress tends todecrease from middle to end sides of specimen. This stressattenuation trend slows down for large friction coefficientwhich means that the whole stress level in pure-bendingregion is higher for poor contact roughness.

4. Conclusions

In this paper, a series of FE simulations were conducted to

investigate the influences of span ratio, support radius andcontact roughness on four-point bending tests. The geometryeffects in thin sheet four-point bending test were discussedthrough the view of stress distribution. The main results areconcluded as follows:(1) The stresses were not completely uniform in the region

of pure-bending in four-point bending tests. Thewedging stress existed in thin sheet four-point bendingtests and remained the maximum stress during elasticstage. The wedging stress got larger when the innerspan got longer, while the stress distribution wouldchange when the load increased. The best span ratio foruniformity of stress distribution between inner supportsand outer supports belonged to 26/75 at plastic stage. Itsuggested that the 1/3 may be the optimum value ofspan ratio for thin sheet applied with low-circle four-point bending fatigue tests.

(2) The strain may be one of the main factors that influencethe stress distribution in four-point bending tests. Themiddle stress tended to become the maximum stressalong pure-bending region while large deformation hadoccurred. For thinner specimen and for longer innerspan, the strain effect was more significant.

(3) Different support radiuses mainly effected the contactpositions between support and specimen and changedthe magnitudes of contact position movement. Thus,large support radius should be avoided to minimize themove trend of contact position for thin sheet four-pointbending tests.

(4) The lateral force in four-point bending test was muchlarger for rough contact surface compared with smoothcontact surface. For poor contact roughness, the stressattenuation trend slowed down and the whole stresslevel in pure-bending region was higher.

Acknowledgments

This research is funded by the National Hi-tech Researchand Development Program of China (863 Program) (No.2012AA041206), National Natural Science Foundation ofChina (NSFC) (No. 51275198), and Program for NewCentury Excellent Talents in University of Ministry ofEducation of China (No. NCET-12-0238).

REFERENCES

1) F. Mujika: Polym. Test. 25 (2006) 214­220.2) T. Lube, M. Manner and R. Danzer: Fatigue Fract. Eng. Mater. Struct.

20 (1997) 1605­1616.3) G. D. Quinn and R. Morrell: J. Am. Ceram. Soc. 74 (1991) 2037­2066.4) T. Zhai, Y. G. Xu, J. W. Martin, A. J. Wilkinson and G. A. D. Briggs:

Int. J. Fatigue 21 (1999) 889­894.5) L. Wang, W. Liu, H. Fang and L. Wan: J. Compos. Mater. (2014) DOI:

10.1177/0021998314554124.6) D. Theobald, J. McClurg and J. G. Vaughan: Int. Compos. Expo.

(Washington, 1997) pp. 1­9.7) H. Yoshihara and H. Kondo: Bio Resour. 8 (2013) 3858­3868.8) L. K. Zhang, Z. H. Chen, D. Chen, X. Y. Zhao and Q. Zheng: J. Non-

Cryst. Solid. 370 (2013) 31­36.9) L. Grabowski and J. R. Yates: Int. J. Fatigue 14 (1992) 227­232.10) J. R. Yates, W. Zhang and K. J. Miller: Fatigue Fract. Eng. Mater.

Struct. 16 (1993) 351­362.11) S. Gungor and L. Edwards: Mater. Sci. Eng. A 160 (1993) 17­24.

(a)

(b)

Fig. 16 Four-point bending test results with different contact roughness:(a) lateral force-displacement curves; (b) stress distribution.

X. Dong, H. Zhao, L. Zhang, H. Cheng and J. Gao342

12) J. Qian and A. Fatemi: Fatigue Fract. Eng. Mater. Struct. 19 (1996)1277­1284.

13) G.-D. Zhan, M. J. Reece, M. Li and J. M. Calderon-Moreno: J. Mater.Sci. 33 (1998) 3867­3874.

14) T.-K. Kang: Measurement 44 (2011) 871­874.15) K. H. Vepakomma, S. H. Carley, J. T. Westbrook and G. V. Morgan:

J. Am. Ceram. Soc. (2015) DOI: 10.1111/jace.13473.16) K. H. Vepakomma, J. Westbrook, S. Carley and J. Kim: J. Disp.

Technol. 9 (2013) 82­86.17) G. Williams, R. Trask and I. Bond: Composites A 38 (2007) 1525­

1532.

18) K.-Y. Chen, C.-W. Huang, M. Wu, W.-C. J. Wei and C.-H. Hsueh:J. Am. Ceram. Soc. 97 (2014) 1170­1176.

19) Hibbitt: ABAQUS. version 6.9 (Karlsson and Sorensen, Inc.Pawtucket, RI 2011).

20) G. Caprino, M. Durante, C. Leone and V. Lopresto: Composites B 71(2015) 45­51.

21) T. Hassan and Z. Liu: Int. J. Press. Vessels Piping 78 (2001) 19­30.22) E. R. C. Draper and A. E. Goodship: J. Biomech. 36 (2003) 1497­

1502.23) C.-C. Yu, J. P. Chu, C.-M. Lee, W. Diyatmika, M. H. Chang, J.-Y. Jeng

and Y. Yokoyama: Mater. Sci. Eng. A 633 (2015) 69­75.

Geometry Effects in Four-Point Bending Test for Thin Sheet Studied by Finite Element Simulation 343