# NUMERICAL VERIFICATION OF THE RELATIONSHIP BETWEEN THE IN-PLANE GEOMETRIC CONSTRAINTS USED IN FRACTURE MECHANICS PROBLEMS.

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5 Marci Graba Numerical Verificatio of the Relatioship Betwee he I-Plae Geometric Co-Straits used i Fracture Mechaics Problems a) a) b) b) c) Fig. 6. he MBLA model which was used i the umerical program a); Crack tip regio usig i the MBLA model b) he J-itegral were calculated usig two methods. he first method, called the virtual shift method, uses cocept of the virtual crack growth to compute the virtual eergy chage. he secod method is based o the J itegral defiitio: J = [wdx 2 t ( u x1 )ds ], (17) C where w is the strai eergy desity, t is the stress vector actig o the cotour C draw aroud the crack tip, u deotes displacemet vector ad ds is the ifiitesimal segmet of cotour C. ab. 2. he mechaical properties of the materials used i umerical aalysis ad the HRR parameters for plae strai [MPa] E [MPa] 26 ν ε=/e α ~θθ (θ = ) I RESULS OF HE NUMERICAL ANALYSIS Fig. 7 presets the ifluece of the stress parameter value o the shape ad size (deoted i Fig. as rp) of the plastic zoe ear crack tip. For smaller value of the stress parameter, the greater plastic zoe is observed, if the same level of the J-itegral was used to calculate the boudary coditios ad the same material characteristic was established. he bigger plastic zoe is observed for MBLA model characterized by smaller yield stress (see Fig. 8). 42 Fig. 7. Ifluece of the stress parameter o shape ad size of the plastic zoe for MBLA model characterized by =315MPa, =1, J=1kN/m: a) =.5, rp=.29m; b) =, rp=.33m; c) =-.5, rp=.9m (brighter regio is the plastic zoe) For smaller values of the stress parameter the, smaller value of the stress are observed. he ifluece of the work hardeig expoet o =() trajectories should be cosidered for a umber of cofiguratios. For the cases characterized by yield stress 5MPa ad for J-itegral values betwee 1 ad 25kN/m (which are used to determie the boudary coditios), the lower values of the stress are observed for weakly stregthe materials (see Fig. 9). For theses cases almost parallel arragemet of the =() trajectories is observed. he highest o the chart are situated curves for strogly stregthe material (=3). I other cases, whe the J-itegral value is equal to or greater tha 5kN/m, it ca be see the itersectig of the =() curves for differet values of the work hardeig expoet (see Fig. 1). For small values of the stress parameter (which meas that we are dealig with a case of high-level of the flat geometric costraits), the lower values of parameter are observed for weakly hardeig materials. Icrease value of the stress parameter makes cuttig curves ad the reversal of the tred o the chart - the a smaller values of the parameter are observed for the case of the strogly hardeig materials. Numerical aalysis show, that the ifluece of the yield stress o =() trajectories is quite complex. For small values of the J-itegral (J=1kN/m or J=25kN/m), it ca be cocluded, that =() curves characterized by regularity of arragemet, especially for strog hardeig materials (=3 ad =5). he lowest o the charts are arraged the =() curves, described by small value of the yield stress. his meas that for the materials characterized by higher yield stress, the higher values of the parameter are obtaied (see Fig. 11).

6 acta mechaica et automatica, vol.6 o.2 (212) a) -.4 b) c) -.8 =315MPa J=5kN/m E=26GPa ν=.3 =3 =5 =1 = Fig. 1. he ifluece of the work hardeig expoet o the =() trajectories for MBLA characterized by =315MPa, J=5kN/m.5 Fig. 8. Ifluece of the yield stress o shape ad size of the plastic zoe for MBLA model characterized by =, =1, J=1kN/m: a) =5MPa, rp=.13m; b) =1MPa, rp=.3m; c) =15MPa, rp=.1m (brighter regio is the plastic zoe) =5MPa J=1kN/m E=26GPa ν=.3 =3 =5 =1 = Fig. 9. he ifluece of the work hardeig expoet o the =() trajectories for MBLA characterized by =5MPa, J=1kN/m he ifluece of the yield stress o =() trajectories is sigificat for the low level of the J-itegral which was used to determiatio of the boudary coditios. Sometimes the ifluece of the yield stress o stress value is egligible, whe material characterized by big work hardeig expoet ad the level of the J-itegral is quite large (J 5kN/m). For weakly hardeig materials, =() trajectories are parallel, whe J- itegral characterized by very high level (for example J=25kN/m or J=5kN/m) =3 J=25kN/m E=26GPa ν=.3 =315MPa =5MPa =1MPa =15MPa Fig. 11. he ifluece of the yield stress o the =() trajectories for MBLA characterized by =3, J=25kN/m he most importat coclusio cocers the ifluece of the J-itegral (which is used to determiatio of the boudary coditios for MBLA model) o stress value. As show by umerical calculatios, the value of the parameter as a fuctio of the stress parameter i a very small extet depeds o the J-itegral value adopted to determie the boudary coditios at MBLA issue. A very little impact (hardly isigificat), or the lack of impact is characterized for MBLA models, for which the J-itegral level used to determiatio of the boudary coditios was equal to or greater tha 5kN/m (see Fig. 12). Fig. 13 presets the ifluece of the work hardeig expoet o stress value for differet level of the stress parameter, which may be cosidered as a measure of the i-plae costrait parameter. For the case of low costraits (low value of the stress, equal to -.5 or -1.), stress value decreases whe the value of the work hardeig expoet icreases. Whe the value of the stress parameter is greater tha -.25, it ca be see that stress value is costat or slightly icreases if value of the work hardeig expoet icreases. 43

7 Marci Graba Numerical Verificatio of the Relatioship Betwee he I-Plae Geometric Co-Straits used i Fracture Mechaics Problems a) a).4 J=5kN/m =315MPa E=26GPa ν= b) J [kn/m] =3 =315MPa E=26GPa ν=.3 =.5 =.25 = =-.25 =-.5 =-1. b) J=1kN/m =5MPa E=26GPa ν=.3 =.5 =.25 = =-.25 =-.5 =-1. c) J [kn/m] =5 =315MPa E=26GPa ν=.3 =.5 =.25 = =-.25 =-.5 =-1. c) J=25kN/m =1MPa E=26GPa ν=.3 =.5 =.25 = =-.25 =-.5 = Fig. 12. he ifluece of the J-itegral value which was used to determiatio of the boudary coditios, o stress value for differet stress parameter: a) =3, =315MPa; b) =5, =315MPa; b) =1, =5MPa 5. CONCLUSIONS J [kn/m] =1 =5MPa E=26GPa ν=.3 =.5 =.25 = =-.25 =-.5 =-1. I the paper, catalogue of the umerical solutios based o Modify Boudary Layer Approach to determie the relatioship betwee -stress ad -stress were preseted. Based o method proposed by Larsso ad Carlsso, the -stress value were calculated for sixtee elastic-plastic materials for differet value of -stress ad exteral load expressed by J-itegral both parameter were used to determie the boudary coditios, which are ecessary to carry out the MBLA aalysis. he ifluece of the -stress parameter ad material properties o -stress value were tested =.5 =.25 = =-.25 =-.5 =-1. Fig. 13. he ifluece of the work hardeig expoet o stress value for differet level of the stress parameter: a) J=5kN/m, =315MPa; b) J=1kN/m, =5MPa; c) J=25kN/m, =1MPa Obtaied umerical results lead to followig coclusios: for smaller value of the stress parameter, the smaller value of the stress are observed; the ifluece of yield stress o the stress value is sigificat for the case characterized by low level of the J-itegral, adopted to determie the boudary coditios at MBLA issue; i case of a higher level of the J-itegral, it ca be observed a little impact of the yield stress o the stress value, sometimes this effect is egligible; for differet stress value, stress value depeds o work hardeig expoet ; this ifluece should be discussed for each MBLA models separately; the stress parameter very weak depeds (or i geeral does ot deped) o the level of the J-itegral adopted 44

9 Marci Graba Numerical Verificatio of the Relatioship Betwee he I-Plae Geometric Co-Straits used i Fracture Mechaics Problems J=1kN/m =1MPa J=1kN/m =15MPa ab. A.2. Results for MBLA models, characterized by J=25kN/m J=25kN/m =315MPa J=25kN/m =5MPa J=25kN/m =1MPa J=25kN/m =15MPa ab. A.3. Results for MBLA models, characterized by J=5kN/m J=5kN/m =315MPa J=5kN/m =5MPa J=5kN/m =1MPa J=5kN/m =15MPa ab. A.4. Results for MBLA models, characterized by J=1kN/m J=1kN/m =315MPa

10 acta mechaica et automatica, vol.6 o.2 (212) J=1kN/m =5MPa J=1kN/m =1MPa J=1kN/m =15MPa ab. A.5. Results for MBLA models, characterized by J=25kN/m J=25kN/m =315MPa J=25kN/m =5MPa J=25kN/m =1MPa J=25kN/m =15MPa ab. A.6. Results for MBLA models, characterized by J=5kN/m J=5kN/m =315MPa J=5kN/m =5MPa J=5kN/m =1MPa J=5kN/m =15MPa

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