Stress triaxiality to evaluate the effective distance in the volumetric approach in fracture mechanics

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1 IOSR Journal of ehanial and Civil Engineering (IOSR-JCE) e-issn: ,p-ISSN: X, Volume 11, Issue 6 Ver. IV (Nov- De. 014), PP 1-6 Stress triaxiality to evaluate the effetive distane in the volumetri approah in frature mehanis B. El Hadim 1, H. El inor 1 (Department of ehanis / CREF, Settat, oroo) (Researh Team S / ENSA, Agadir, oroo) Abstrat: The distribution of stress triaxiality is determined in the viinity of noth defet performed in the speimen submitted to ompression load. The alulations are arried out by using 3D finite element analysis. The evaluation of noth stress intensity fator is made by using the volumetri approah. A new method to evaluate the effetive distane in this approah is proposed. Keywords: Stress triaxiality, noth defet, stress intensity fator, volumetri approah and effetive distane. I. Introdution In frature mehanis, the ritial stress intensity fator or so alled frature toughness and other mehanial proprieties are measured in some onditions of geometry, loading mode and onstraint. So, they have to be applied in the same onditions. The transferability of these proprieties is imperative in strutural design beause hange of geometry or onstraint onditions may promote brittle frature. This transferability is based on the stress triaxiality whih takes into aount the tridimensional stress field. In this paper, β is used as a measure of stress triaxiality. This parameter is defined as the ratio of the hydrostati stress over the equivalent Von ises stress: h (1) Where: And: 1 xx h eq, V 3 eq, V ( 1 ) ( 1 ) ( 1 ) It is well known that dutile frature is sensitive to the stress triaxiality. There are several indiators to quantify the state of onstraints at defet tip in the literature. Over the list of these indiators, one an quote the T-Stress [1], the Q parameter [], the multiaxiality parameter [3] and the p parameter [4]. The evaluation of noth stress intensity fator is made by using the volumetri approah whih supposes that the frature proess requires a ertain frature volume. This volume is assumed as a ylinder with effetive distane at its diameter. In this paper, we propose a new method based on stress triaxiality to evaluate the effetive distane. II. aterial The material studied is high-strength steel 45CroSi6 aording to the Frenh standard. Its mehanial and hemial proprieties are listed in Table I and Table II. Table I Chemial proprieties of 45CroSi6 C n Si Cr o Table II ehanial proprieties of 45CroSi6 E Y U Density A% (GPa) (Pa) (Pa) (kg/m 3 ) III. Geometrial Charateristis Tests were performed using U-nothed irular speimens (Fig.1) with the external radius Re=0 mm, internal radius Ri=10 mm, thikness B=7 mm, and the noth length a=4 mm. Different noth radii were obtained using a wire-utting eletrial disharge mahine (ED) [5]. Four noth radii are onsidered: ρ = 0.3, ρ = 0.5, zz 1 Page

2 Stress triaxiality to evaluate the effetive distane in the volumetri approah in frature mehanis ρ = 1.0 and ρ =.0 mm. The speimens are submitted to ompression load in order to determine the ritial loads when the frature ours. These loads are introduing to the simulation omputation to finally evaluate the ritial noth stress intensity fator and the stress triaxiality evolution. Compressed speimen Fig.1. Speimen geometry and ompression load. IV. Noth Stress Intensity Fator And Volumetri ethod Stress distributions around the noth defet have been onverted into so alled noth stress intensity fator using the noth frature mehanis and partiularly the volumetri method. The volumetri method [6] is a loal frature riterion, whih supposes that the frature proess requires a ertain frature volume. This volume is assumed as a ylinder with effetive distane at its diameter. The elasti-plasti stress distribution along the ligament is plotted in the bi-logarithmi diagram as an be seen in Fig.: Three distint zones in the diagram an be distinguished: - Zone I: the elasti-plasti stress opening stress inreases and attains a peak value. - Zone II: the elasti-plasti stress drops gradually in the elasti regime. - Zone III: starts at a ertain distane whih is named the effetive distane. It represents linear behaviour in the bi-logarithmi diagram. Fig.. Shemati elasti-plasti stress distribution along noth ligament and stress intensity onept. The noth stress intensity fator [N.S.I.F] is defined as funtion of effetive distane and effetive stress: K eff X eff () The effetive distane orresponds to the minimum of the relative stress gradient whih given as: 1 ( r) ( r) (3) ( r) r The effetive stress is onsidered as the Page

3 Stress triaxiality to evaluate the effetive distane in the volumetri approah in frature mehanis average volume of the stress distribution over the effetive distane. However stresses are multiplied by a weight funtion in order to take into aount the influene of stress gradient due to geometry and loading mode. The effetive stress is defined as: eff 1 X 0 eff X eff ( r) (1 r ( r)) dr (4) V. Finite Element Analysis The part modelled in 3D analysis is meshed by quadrangular elements with eight nodes. Computing was arried out on Castem software 014 [7]. By using the maximum hoop stress as failure riterion [8], the stress distribution was simulated front of the noth (Fig.3) in the opening frature mode. Fig.3. esh density and stress onentration in speimen. The results of numerial simulations exhibit that for all noth radii, the hoop stress whih is maximal at the middle of the noth, is predominant (Fig.5). Then, we will be interested in the radial diretion in the middle of speimen where the rak is expeted to our first [9] and grow up radially (Fig.4). Fig.4. Studied ligament in the noth tip of speimen. The stress evolutions are plotted versus the noth tip distane for eah radius (Fig.5). The maximum stress values derease when the noth radius inreases. 3 Page

4 Stress triaxiality to evaluate the effetive distane in the volumetri approah in frature mehanis Fig.5. Stress distributions in the noth tip ligament. VI. Results The ritial noth stress intensity fator (C.N.S.I.F.) noted K ρ is omputed by using the ritial loads leading to the frature speimens [5]. The ritial effetive distane X eff and the assoiated ritial effetive stress σ eff are given by numerial simulations. These two harateristis are taken into aount to determine the C.N.S.I.F. K ρ by using the relation (). The results are summarized in table III. Noth radius ρ (mm) Table III Critial Noth Stress Intensity Fator for various noth radii Critial Critial effetive stress effetive distane σ eff (Pa) X eff (mm) Critial noth stress intensity fator K ρ (Pa.m 0.5 ) The evolution of the relative stress gradient along the noth tip ligament is given in Fig.6 for eah onsidered radius. The ritial effetive distane X eff orresponds to the minimum of relative stress gradient, as mentioned in setion IV. All effetive distanes and assoiated effetive stresses are reported above, in table III. 4 Page

5 Stress triaxiality to evaluate the effetive distane in the volumetri approah in frature mehanis Fig.6. Relative stress gradient distributions in the noth tip ligament. The stress triaxiality evolutions along the ligament are shown in Fig.7. These urves present two onvex forms with an intersetion in a point (). It is proven that this intersetion point is at a distane on the ligament whih orresponds to the effetive distane X eff (Table IV). One an onlude that the stress triaxiality an be used to determine the effetive distane as that an be made with the relative stress gradient. The effetive distane is then: X eff = (5) Where the point is loated at a distane and defined as: lim dβ r + dβ lim r dr > 0 dr < 0 (6) Table IV Comparison of distane with effetive distane X eff. Angle ρ (mm) effetive distane X eff (mm) distane (mm) Aording to the figure below, we an also note that the amplitude of the ritial maximum of triaxiality dereases when the noth radius inreases. 5 Page

6 Stress triaxiality to evaluate the effetive distane in the volumetri approah in frature mehanis Fig.7. Critial stress triaxiality evolutions in the noth tip ligament. VII. Conlusion The volumetri approah requires the determination of the diameter of frature elaboration zone and the assoiated stress in the bilogarithmi diagram to evaluate the noth stress intensity fator. That is made by numerial omputation of the effetive distane X eff and effetive stress σ eff. The maximum tangential stress is used as frature riterion. In order the take into aount the tridimensional stresses; the evolution of stress trixiality β is studied along the noth tip ligament. The stress triaxiality urves reveal two onvex forms whose point of intersetion is loated at a distane noted whih proved to be equal to the effetive distane X eff. One an onlude that stress triaxiality an be used to determine the diameter of frature elaboration zone in volumetri approah. In other words, the value of the effetive distane an be obtained from the stress triaxiality is equal to. Referenes [1] C. Betegon, J. W. Hanok, Two parameter harateristis of elasti-plasti rak tip fields, J. Appl. eh. 58, pp , (1991). [] N.P. O Dowd, C.F. Shih, Family of rak tip fields haraterized by a triaxiality parameter I: Struture of fields, J. eh. Phys. Solids; 39, pp , (1991). [3] H. Clausmeyer, K. Kussmaul, E. Roos, Influene of stress state on the failure behaviour of raked omponents made of steel, ASE, Appl. eh. Rev. 44 (), pp. 77-9, (1991). [4] O. Akourri, I. Elayahi, Stress Triaxiality as Frature Toughness Transferability Parameter for Nothed Speimens, International Review of ehanial Engineering, (007). [5] H. El inor, Rupture fragile en mode mixte amorée à partir d entaille, Thèse de Dotorat, Université de Rabat, oroo, (00). [6] G. Pluvinage, Frature and Fatigue emanating from stress onentrators (Dordreht: Kluwer Aademi Publishers; 003).). [7] Castem 014, Logiiel d éléments finis développé par le département des études méaniques et thermiques du ommissariat français de l énergie atomique C.E.A, Frane. [8] B. El Hadim and ol., Evaluation of Noth Stress Intensity Fator in pipe with external oriented defet, International Review of ehanial Engineering, Vol.5 N1, pp. 7-11, January (011). [9] B. El Hadim and ol., Frature toughness transferability in pipe with external oriented defet, International Review of ehanial Engineering, Vol.5 N5, pp , July (011). 6 Page