Fracture of underwater notched structures

Transcription

1 Egieerig Solid Mechaics 4 (2016) Cotets lists available at GrowigSciece Egieerig Solid Mechaics homepage: Fracture of uderwater otched structures M.M. Mirsayar a* ad Behrouz Takabi b a Zachry Departmet of Civil Egieerig, Texas A&M Uiversity, College Statio, TX , USA b Departmet of Mechaical Egieerig, Texas A&M Uiversity, College Statio, TX 77843, USA A R T I C L E I N F O A B S T R A C T Article history: Received 6 September, 2015 Accepted 20 December 2015 Available olie 21 December 2015 Keywords: Uderwater structures V-otch Fracture Hydrostatic pressure Uderwater structures are subjected to hydrostatic pressure durig their service life. Sharp V- otched compoets ca be see as a part of may uderwater structures. For example, welded compoets, machied parts, gears, screws ad bolts are amog the well-kow elemets that cotai sharp V-otches. The otch tip is a likely zoe for iitiatio of cracks due to high stress/ strai cocetratio. The reliability aalysis of the V-otched compoets requires a good uderstadig of stress/ strai distributio ear the otch tip. The fracture iitiatio of the V- otched compoets ca be cotrolled by the tagetial strai field ear the otch tip. The tagetial strai distributio ad fracture iitiatio coditios are studied i this paper for otched compoets subjected to hydrostatic pressure. The effect of each tagetial strai term o fracture iitiatio agle as well as tagetial strai distributio aroud the otch tip is ivestigated usig fiite elemet simulatio of a V-otched semi-circular specime. It is show that ot oly the sigular terms, but also the costats trai field sigificatly ifluece o tagetial strai distributio ad fracture iitiatio agle aroud the otch tip. The results of this paper ca be used for stadardizatio of the fracture i uderwater structures cotaiig V-otched compoets Growig Sciece Ltd. All rights reserved. 1. Itroductio Oe of the most crucial difficulties o desigig uderwater structures is limitig their stresses uder allowable stress criteria; because they are uder hydrodyamic loads i their etire life. For this purpose, some researchers (Nayebi & Asari, Sadrabadi, 2013; Takabi, 2016) have utilized moder ad advaced materials such as FG materials to decrease stresses of structures. However, whe it comes to otches, especially i uderwater structures, material ehacemet could ot work out per se, ad a meticulous study should also be performed o otched compoets. Notched compoets are see i may egieerig structures ragig from micro scale to macro scale. Whe the radius of the otch tip is very small compared to the legth of its sides, the otch is cosidered as sharp. Cracks are itroduced as a special case of otches mathematically, i which the otch agle is zero. I a V-otched compoet * Correspodig author. Tel.: +1 (979) addresses: mirmilad@tamu.edu (M.M. Mirsayar) 2016 Growig Sciece Ltd. All rights reserved. doi: /j.esm

2 44 subjected to exteral mechaical loadig, the state of the stress/ strai becomes sigular at the otch tip which meas the stress/ strai field approaches ifiity adjacet the otch tip, mathematically. However, it is impossible to reach ifiite stress/strai i a real material ad hece, micro-cracks develop i a area, called fracture process zoe, aroud the otch tip. The stress/ strai field evaluatio at the otch tip is very importat because fracture may iitiate from the weakeed area at the otch tip by icreasig exteral mechaical loadig which may fially result i catastrophic failure of the whole structures. The elastic stress ad strai fields aroud V-otch tip ad crack tip have bee extesively bee ivestigated by may researchers i the past (Sewery, 1994; Qia, 2001; Sewery & Lukaszewicz, 2002; Susmel & Taylor, 2008; Ayatollahi et al., 2006, 2010a,b, 2011a, 2013; Ayatollahi & Mirsayar, 2011; Abd-Elhady, 2013; Ayatollahi & Aliha, 2007, 2008 a, b, 2009, 2011; Ameri et al., 2012, Arabi et al., 2013; Mirsayar & Samaei, 2013, 2014, 2015; Lazzari et al., 2010; Gomez et al., 2008; Mirsayar, 2013, 2014a,b,c, 2015a; Aliha et al., 2015a, b; Torabi, 2013; Mirsayar et al., 2014). Amog them, Mirsayar et al. (2014) ad Ayatollahi et al. (2011a) have studied the effect of first o-sigular stress term o the stress distributio as well as the fracture iitiatio agle aroud the tip of a otch existig at the iterface of two dissimilar materials. However, these studies are maily focused o V-otches subjected to remote mechaical loadig ad the V-otches subjected to surface tractios have rarely bee studied. Qia (2001) obtaied the stress field equatios for a iterface otch with surface shear ad ormal tractios. He poited out that surface tractios could be preseted as a costat stress field aroud the otch tip ad may ifluece o the stress distributio aroud the iterface otch tip. However, the effect of surface tractios o the strai field ad fracture iitiatio coditio aroud the otch tip has ot bee ivestigated yet. Several fracture criteria have bee proposed so far to ivestigate fracture iitiatio coditios at the otch tip. These criteria ca basically be classified ito three differet groups: eergy based, stress based ad strai based approaches (Erdoga & Sih, 1963; Sih, 1974; Chag, 1981; Selvarathiam & Goree, 1998; Mirsayar, 2015b). Amog the eergy based facture criteria, the maximum strai eergy desity criterio (Sih, 1974) is widely used by the researches ad claims that fracture iitiates at the otch tip i directio where the strai eergy reaches its maximum value. The maximum tagetial stress criterio (Erdoga & Sih, 1963) is a well-established stress based criterio which says that fracture propagates at the critical distace (rc) from otch tip i directio where tagetial stress () reaches the ultimate stregth ( c) of the material. Similarly, the maximum tagetial strai (MTSN) criterio claims that fracture occurs at the critical distace from the otch tip i directio where tagetial strai () attais its ultimate tesile strai ( c)(chag, 1981). However, all of these traditioal fracture criteria oly cosider the sigular stress/ strai field ad the effects of o-sigular terms are igored. I this paper, the role of costat strai term, iduced from ormal surface tractios (hydrostatic pressure), o the tagetial strai field as well as fracture iitiatio agle aroud the otch tip is studied. It is show that the costat strai field sigificatly iflueces o the tagetial strai field aroud otch tip. Also, the sigificat effect of costat strai field o the fracture iitiatio directio is highlighted. 2. Elastic field equatios aroud V-otch tip Fig. 1 shows the schematic of a sharp V-otch subjected to both remote mechaical loadig ad ormal surface tractios (hydrostatic pressure) alo\g its edges. Williams (1952) studied the stress field equatio aroud the V-otch tip usig liear elastic fracture mechaics (LEFM) assumptios. Accordig to this work, the stress field aroud the otch tip subjected to remote exteral loadig ca be expressed i terms of the so-called Williams series expasios(williams, 1952; Ju & Chug, 2007) as;

3 M.M. Mirsayar ad B. Takabi / Egieerig Solid Mechaics 4 (2016) 45 I I II II rr frr (,, ) frr (,, ) I II A I I B II II I f (,, ) II (,, ) 1 f 1 1 r I I 1 r II II r fr (,, ) fr (,, ) (1) where r ad are the polar coordiates with the origi at the otch tip(see Fig. 1), is the order of term i the ifiite series ad is the wedge agle. Fig. 1. Schematic of a sharp V-otch subjected to remote mechaical loadig together with a surface hydrostatic pressure. The parameters I ad II are mode I ad mode II eigevalues, obtaied from positive roots of the followig characteristic equatio, which respectively represet opeig ad slidig mode of deformatio at the otch tip. I I si( ) si( ) 0 II II si( ) si( ) 0 The Eq. (1) is oly valid for real values of the eigevalues ad as a result, the correspodig stress k k itesity factors, A ad B, are real umbers. The fuctios fij (,, ) are kwo fuctios which represet variatio of the stress field i tagetial directio ad are give i Ayatollahi ad Nejati (2011) (( i, j) (r, ) ad k (, I II) ). For = 1, the coefficiets A1 ad B1 are related to the sigular terms ad are called mode I ad mode II stress itesity factors respectively. Qia (2001) idicated that for a iterface otch problem subjected to combied shear ad ormal surface tractios, a particular solutio should be added to the covetioal solutio. Accordig to this (2)

4 46 work, the particular solutio is correspodig to = 1 which leads a stress term idepedet from radial distace (-1=0), called costat stress term. For otch problem i a homogeeous media subjected to oly hydrostatic pressure (p) boudary coditios, the radial, tagetial ad shear compoets of costat stress term ca be give as; 0 rr 1 cos[ 2 ] cos[ 2 ] 0 1cos[ 2 ] A01 cos[ 2 ] p 0 r si[ 2 ] si[ 2 ] where, the coefficiet A01 is preseted as; A01 4p 1 si(2 ) cos(2 ). (4) Therefore, the stress field aroud the otch tip i presece of the hydrostatic pressure o the edges of the otch ca be preseted i terms of summatio of the geeral solutio, Eq.(1), ad the particular solutio, Eq. (3). I Eq. (1), the terms correspodig to eigevalues III, 1 are called sigular terms because these terms approach ifiity whe r 0. That meas the sigular terms play more importat role tha higher order terms i stress ad strai distributio aroud the otch tip. 3. Maximum tagetial strai fracture criterio I this study, the maximum tagetial strai fracture criterio (MTSN), proposed by Chag (1981), is employed to ivestigate fracture iitiatio agle aroud the otch tip. Accordig to MTSN criterio, fracture occurs i the directio, 0, where the tagetial strai,, reaches its maximum value, at a critical distace, rc, from the otch tip. The criterio could simply be represeted as: 2 2 rrc ; 0 rrc ; (3) (5) (6) ( r, ) / E, c 0 T T where, T is the tesile fracture stregth ad E is the Youg s modulus of material. The critical distace rc could be cosidered as damaged zoe aroud the otch tip which is a material property (Taylor, 2008; Ayatollahi & Aliha 2008). Because we are usig LEFM assumptios (which is a valid for brittle materials), the strai field aroud otch tip ca be obtaied usig Hook s law as follows; 1 ( rr ), E (7) where is the Poisso s ratio of the material. Usig Eq. (1), Eq. (3) ad Eq. (7), it ca be cocluded that costat stress term leads to a costat strai term. 4. Numerical example ad discussio I order to study the effect of costat strai term, iduced from surface hydrostatic pressure o the tagetial strai distributio as well as fracture propagatio agle, a fiite elemet (FE) simulatio is performed o a V-otched semi-circular specime. The specime, show i Fig. 2(a), is a cetrally otched semicircular disc of radius R = 40mm with otch depth of h = 20mmad otch agle of ( = 360 o - ). The specime is subjected to hydrostatic pressure, p = 1 MPa, ormal to the otch edges as well as oe of its sides. The detailed loadig ad boudary coditios is show i Fig. 2 (b).

5 M.M. Mirsayar ad B. Takabi / Egieerig Solid Mechaics 4 (2016) 47 Fig. 2. Semicircular otched specime a) geeral cofiguratio, b) loadig ad boudary coditios Fig. 3 shows the typical FE mesh used for modelig the specime ad the closer view of the elemets adjacet the otch tip. A very fie mesh was utilized i the regio ear the tip due to high stress/ strai gradiet i that area. I the FE aalyses, 6-ode quadratic triagular elemets were used for the first row of elemets surroudig the otch tip ad 8-ode quadratic quadrilateral elemets were utilized for the rest of the model. The plae strai coditios ad the thickess of uity were assumed. The stress itesity factors correspodig to the sigular term associated with ( = 0.51) was calculated usig fiite elemet over-determiistic (FEOD) approach. This method has successfully bee used for calculatio of the stress itesity factors of cracks ad otches i both homogeeous ad bi-material media (Ayatollahi et al., 2010; Ayatollahi & Nejati, 2011). Fig. 3. A typical FE mesh used for modelig of V-otched semicircular specime with = 30 o

6 48 Fig. 4 shows the distributio of the tagetial strai field throughout deformed shape of the specime. The closer view of this cotour of plot reveals that a high cocetratio i strai field exists at the otch tip. Fig. 4. Cotour of plots for tagetial strai distributio throughout the deformed shape of semicircular otched disc specime together with a closer view aroud otch tip The asymmetry i deformatio of odes at the otch tip is i agreemet with asymmetry i loadig ad boudary coditio. It ca be see from Fig. 4 that maximum tagetial strai occurs aroud otch tip i directio 0<0 o. However, accordig to Eq. (5), the directio of the fracture iitiatio depeds o the critical distace, rc, where the tagetial strai compoet is evaluated. Accordig to the literature (Ayatollahi & Aliha, 2008, 2011; Aliha et al., 2010, 2012; Aliha & Ayatollahi, 2012, 2013, 2014, Akbardoost et al., 2014; Ayatollahi et al., 2006, 2011b), the critical distace parameter may have differet values for differet brittle ad quasi-brittle materials like, ceramics (withi the rage of mm ad 0.5 mm), brittle polymers (e.g. PMMA with r c i the rage of 0.05 ad 0.2 mm), graphite (rc = 0.35 mm ad rc = 0.47 mm for graphite 7477ad graphite SA, respectively) or rocks (rc = 2 to 6 mm). I the preset work, a typical critical distace of 0.18 mm is selected. I other words, it is assumed that crack propagates at a critical distace of 0.18 mm from otch tip. Fig. 5 illustrates the variatios of tagetial strai versus tagetial directio of at a circular path of radius r = 0.18 mm from the otch tip. It ca be see that cosiderable errors exist betwee the strai distributios calculated by cosiderig oly sigular solutio ad those of directly obtaied from FE aalysis (which cosiders all terms of the Williams series expasio give i Eq. (1)). The predicted values for agle of the fracture iitiatio, 0, at the critical distace of r = 0.18 mm are also show i Fig. 5. A substatial differece

7 M.M. Mirsayar ad B. Takabi / Egieerig Solid Mechaics 4 (2016) 49 ca be see betwee fracture iitiatio agles estimated by oly cosiderig sigular strai field (-1.7 o ) ad FE results (-19.2 o ). However, by addig the costat strai field to the sigular solutio, a very good cosistecy ca be observed betwee the theoretical predictios ad FE aalysis for distributio of the tagetial strai field as well as the estimated fracture iitiatio agle. The distributio of the tagetial strai field alog radial directio of 0 = o is also show i Fig. 6. It is see that costat strai term sigificatly iflueces o the strai field eve i the very ear the otch tip. It ca also be realized that the higher osigular strai terms (correspodig to >1) have o effect o the strai distributio for radial distaces less tha 1 mm from otch tip. Fially, it ca be cocluded that the costat strai term may play a importat role i predictio of the fracture iitiatio directio as well as the strai field distributio aroud otch tip Sigular strai field Sigular + first osigular strai field Fiite elemet solutio r c = 0.18 mm , degrees Fig. 5. Variatio of tagetial strai fields versus at a circular path of radius r = rc = 0.18 mm from the otch tip Sigular strai field Sigular + first osigular strai field Fiite elemet solutio = o r (mm) Fig. 6. Variatio of tagetial strai fields versus radial distace alog = o 5. Coclusios The fracture iitiatio i V-otched compoets subjected to hydrostatic pressure was studied i this paper. It was show that the costat strai term, which causes as a result of hydrostatic pressure alog the otch edge, sigificatly iflueces o the tagetial strai distributio aroud the otch tip. The remarkable effect of costat strai term o the evaluatio of the fracture iitiatio agle usig the

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