Crack onset assessment near the sharp material inclusion tip by means of modified maximum tangential stress criterion

Transcription

1 Focuse on Mechanical Fatigue of Metals Crack onset assessment near the sharp material inclusion tip by means of moifie maximum tangential stress criterion Onřej Krepl, Jan Klusák CEITEC IPM, Institute of Physics of Materials AS CR, Žižkova 22, Brno , Czech Republic ABSTRACT. In the case of particle reinforce composites, where the particles are in a form of sharp material inclusions, singular stress concentration exists on each tip of each inclusion. This is ue to the geometric an material iscontinuities between matrix an particle. These points of stress concentration are susceptible of crack initiation an thus often responsible for failure of the whole structure. The moifie maximum tangential stress criterion is employe in orer to preict crack onset conitions. KEYWORDS. Sharp Material Inclusion; Singular an Non-singular Stress Terms; Generalize Fracture Mechanics. Citation: Krepl, O., Klusák, J., Crack onset assessment near the sharp material inclusion tip by means of moifie maximum tangential stress criterion, Frattura e Integrità Strutturale, 42 (217) Receive: Accepte: Publishe: Copyright: 217 This is an open access article uner the terms of the CC-BY 4., which permits unrestricte use, istribution, an reprouction in any meium, provie the original author an source are creite. INTRODUCTION C omposites fin application in a variety of engineering structures. Overall excellent properties, which can only be achieve by combination of two or more homogenous materials, bring solution for high emans given by contemporary avance technologies. owever, the very nature of composites also causes a ifficulty in their assessment in terms of fracture mechanics. Common composite types, i.e. fibre reinforce composites or particle reinforce composites give rise to points of singular stress concentration. These points are foun in loci where fibre ens are embee in matrix or at each tip of each sharp particle an are referre as General Singular Stress Concentrators (GSSCs). This paper eals with the case of particle embee in matrix which is in a form of Sharp Material Inclusion (SMI). The strength of singularity in this case is ifferent an lower than in the case of a crack. owever, the GSSCs are points often susceptible for crack initiation an thus they can be responsible for failure of the whole structure. A thorough unerstaning of crack initiation conitions in GSSC leas to improve estimation of the critical failure loa. Stress, energy or couple criteria are commonly use in fracture mechanics analyses of many GSSC types such as sharp notches. These criteria can be employe in orer to preict crack onset conitions to the case of SMI. The GSSC characterize by weak singularity often require consieration of higher orer non-singular terms of the asymptotic stress series to escribe the stress state an thus to receive consistent critical loa preictions. Suggestion of moifie maximum tangential stress stability criterion for the SMI with consieration of non-singular terms is the main objective of this paper. 66

2 FRACTURE MECANICS OF SARP MATERIAL INCLUSION T he SMI is moelle in 2D as a special case of multi-material junction, the bi-material junction [1, 2] as shown in Fig. 1. Both material regions are consiere to consist of linear elastic material an fully escribe by Young s mouli E 1 an E 2 an Poisson s ratios ν 1 an ν 2. The geometry is characterize by the opening angle α. Perfect boning (traction an isplacement continuity) is assume at both interfaces Γ an Γ 1. The problem is consiere either in a state of plane stress or plane strain. The stress fiel in the vicinity of bi-material junction tip, i.e. when r, is escribe by following asymptotic series: (1) ij r fij 1m r fij 2m r fij 3m where k is the kth Generalize Stress Intensity Factor (GSIF) which correspons to the kth eigenvalue k, that forms the stress singularity exponent (1 k ). The terms of the series can be either singular, when < ( k) < 1 or non-singular when 1 < ( k ). As r the singular terms become unboune, while the non-singular terms vanish. f ijkm (θ) is the imensionless angular eigenfunction constructe for ijth component of the stress tensor, kth eigenvalue an mth material as in [2, 1]. Note that the series above is in sake of simplicity written for real eigenvalues an real GSIFs. Such form of the series provies satisfactory escription for most of the cases. Figure 1: Sharp material inclusion moel. In majority of the fracture mechanics analyses of GSSCs only the singular terms are use for escription of stress fiel [3, 4] an following etermination of crack onset conitions. In [5, 6] the effect of first non-singular stress term on stress istribution in case of sharp V-notch is stuie. The effect of the first non-singular term in case of bi-material notches is stuie in [7, 8]. In [9] Klusák et al. have shown the significance of consieration of the first non-singular term in the case of sharp bi-material orthotropic plate. A stuy which has shown the effect of non-singular terms in the cases of SMI has been conucte in [1]. In orer to ientify singular an non-singular terms for given configuration the epenence of the eigenvalues 1, 2 an 3 of chosen geometric SMI cases on Young s mouli ratio E 1/E 2 is presente in Fig. 2. ere, material region 1 an Young s moulus E 1 belongs to inclusion an the region 2 with moulus E 2 correspons to the matrix. In the case of rectangular SMI characterize by = 9 two singular an one non-singular terms for all E 1 /E 2 ratios are foun. In the case of sharper SMI with = 6 there are two singular terms together with one non-singular terms for cases where inclusion is more compliant than matrix, i.e. E 1 /E 2 < 1 an one singular term together with two non-singular terms for cases of inclusion stiffer than matrix, i.e. E 1 /E 2 >1. The SMI with a blunt opening angle of = 12 shows reverse tren, as the cases of more compliant inclusion have one singular term together with two non-singular terms an cases with stiffer inclusion have two singular together with one non-singular term. 67

3 Figure 2: Depenence of 1, 2 an 3 on E 1 /E 2 shown for 3 SMI geometries = 6, 9, 12. To preict crack initiation conitions, a moifie maximum tangential stress criterion is use. The tangential stress epens on raial istance from the singular concentrator tip. To mitigate the raial istance epenence the mean value can be calculate as: 1 m r, r (2) The parameter is relate to the fracture mechanism, e.g. in case of cleavage fracture it can be set as = 2-5 grain size of the material [13]. The criterion states that the crack will initiate in the irection θ,m of maximum value of mean tangential stress. The extreme value is foun as: 2 ; 2, m, m (3) Let s consier the first 3 terms (singular an non-singular) of the stress series (1). By averaging it over specific istance as in (2) an substituting it into Eq. (3) we obtain: f f f m 2m 3m (4) The first GSIF 1 is factore out an the consequent ratios are enote as k1 = k/ 1. Equation has now the only unknown, the crack initiation irection θ,m. f f f m 2m 3m (5) 68

4 In general, global an local extremes can be foun in bi-material problem. The crack will be initiate in the irection of maximum of an corresponing minimum value of generalize fracture toughness [14]. Note that the crack initiation irection oes not epen on the absolute values of GSIFs but rather on their ratios k1. The general form of the equation for n terms of the stress series is: n k1 k k1 k f km (6) The crack initiation conitions of the SMI are escribe by means of critical stress quantity. In this way, Knésl in [13] assesses a stability of V-notch. e proposes comparison of such critical quantity of the crack an the V-notch. Since in both cases (crack tip an the V-notch tip) the stress is singular, he supposes that crack initiation mechanism in V-notch will be the same as the crack propagation mechanism. Then crack initiation conitions in case of GSSCs can be assesse by means of critical average stress ascertaine for a crack. For a crack in homogeneous material an loae in normal moe I, the critical mean tangential stress value is: IC C 2K (7) 2 Whereas for the GSIF with consieration of first three singular an non-singular terms of the series (1), it is: C, m C, m 11 f 1m 21 f 2m 31 f 3m (8) By comparison of the two relations above the generalize fracture toughness 1C,m is: 2 IC, m 1C, m K f ( ) f ( ) f ( ) 11 1m, m 21 2m, m 31 3m, m (9) which in general form for n terms of the stress series is written: 2K IC, m 1C, m 1 k 2 n k1k1 fkm, m k ( ) The generalize fracture toughness 1C,m epens on fracture toughness K IC,m of the material m. It is obvious that in case of SMI two materials m = 1 an 2 shoul be consiere an possibility of crack initiation into the matrix an into the inclusion shoul be evaluate. If the value 1C,1 is lower than 1C,2 crack initiation is expecte into the inclusion, otherwise it occurs into matrix. In sake of completeness, the case of crack initiation into the interface shoul be also evaluate. In that case, the value of 1C,interface has to be calculate base on fracture toughness of the interface K IC,interface. Note that for all the critical values 1C,1, 1C,2, an 1C,interface the shape functions f,m shoul contain corresponing angle of potential crack initiation,m (m = 1, 2, interface). The crack initiation occurs if the following stability criterion is violate (11): min,, 1 1C,1 1C,2 1C,interface Crack is not initiate in SMI tip if the value of GSIF 1 is lower than its critical value. Finally, the critical applie loa σ crit can be calculate as: (1) (11) 69

5 crit appl 1 appl min 1C,1,1, 1C,2,2, 1C,interface 1,2 ) (12) NUMERICAL EXAMPLE T he SMI is moelle by FEM in 2D. The moel geometry an bounary conitions are shown in Fig. 3. The rectangular inclusion with α = 9 in a plane strain conitions is stuie. The moelle specimen size is mm. The moel is loae with unit tension of σ appl = 1 MPa. In this numerical example the inclusion more compliant than matrix is consiere, which represents sanstone inclusion in cement paste of various stiffness. The moelle bimaterial configurations are liste in Tab. 1. The Poisson s ratios are ientical for both materials, ν 1 = ν 2 =.2. The eigenvalues λ k are etermine as a solution of an eigenvalue problem [2, 1]. The GSIFs k are calculate semi-analytically by means of overeterministic metho [2, 1], with resulting values as shown in Tab. 1. The iameter which provies inputs (noal isplacements) for GSIFs calculation is chosen as r 1 = 1 mm. The characteristic length which correspons to the fracture mechanism is for the case of cement paste composites chosen as = 1 mm. Fig. 4 shows the tangential stress istribution for the case E 1 /E 2 =.5, nevertheless istribution of this kin is characteristic to the cases of inclusion more compliant than matrix in general. The istribution isposes of two extremes, global which is foun in the matrix θ,2 = 18 an local foun in the inclusion θ,1 =. The values of crack initiation angles θ,m are given by the symmetry of the problem. For the same reason the o term, the skew relate GSIF 2 = MPa m 1-λ2. It is necessary to asses potential crack initiation in all possible ways, i.e. in the irection of the global maximum, local maximum an the interface. The crack initiation angle in the latter case is θ,interface = 45. In Tab. 1 the generalize fracture toughness is calculate for global function extremes, the local one an the interface. The fracture toughness value of matrix, inclusion an the interface is chosen equally, K IC,m = 1 MPa m 1/2 for m = {1, 2, interface}. Generalize fracture toughness for unit K IC,m can be unerstoo as normalize value an in Tab. 1 is enote as. * 1C,m E 1 [GPa] E 2 [GPa] E 1 /E 2 1 [MPa m 1-λ1 ] 3 [MPa m 1-λ3 ] K IC [MPa m 1/2 ] * 1C,2 [MPa m 1-λ1 ] * 1C,1 [MPa m 1-λ1 ] * 1C,interface [MPa m 1-λ1 ] Table 1: Bi-material configurations with resulting GSIFs an values of generalize fracture toughness. Figure 3: The numerical moel. 7

6 1, 3 It is evient that the stress escription (Fig. 4) by means of the first an the thir stress terms ( ) is more precise than 1 the escription by the first stress term only ( ). It can be seen especially in irections θ = 9 an 27 corresponing to regions of lower tangential stresses. The stability criterion employing critical value 1C following from all three stress terms (9), (1) is use. Fig. 5 shows results of the values 1 an the critical values * 1C corresponing to crack initiation into matrix, into the inclusion an into the interface. The values 1 follow from FEM numerical solution of moel with unit loaing 1 MPa. The critical values * 1C,m are ascertaine for unit fracture toughness K IC,m = 1 MPa m 1/2 for m = {1, 2, interface}. Particular values 1C,m for given fracture toughness of the matrix, the inclusion an the interface are: K (13) * 1C, m 1C,m IC, m Figure 4: Tangential stress istribution for the case E 1 /E 2 =.5. Yellow an magenta curve stay for the analytical tangential stress solution on r = 1 mm obtaine by one term 1 an by two terms 1 an 3 respectively. The cyan curve is the mean tangential stress obtaine by two terms 1 an 3. Black ots represent the FEM solution on r = 1 mm. The yellow vertical lines enote the interfaces an black vertical lines the function local an global extreme. The results show that crack initiation conitions epen on the ratio of materials Young's mouli, an on the fracture toughness of material components (matrix, inclusion) an the interface. In the numerical example E 1 correspons to Young's moulus of the inclusion. The inclusion here is consiere to be a sanstone aggregate with E 1 = 2 GPa. Corresponing fracture toughness of sanstone is between.28 an.52 MPa m 1/2. On the other han, E 2 correspons to Young's moulus of matrix. In the numerical example it varies from 26.6 to 8 GPa. The fracture toughness of common harene cement paste is between.1 an.8 MPa m 1/2, where higher values of E 2 usually match to higher values of K IC,2. The fracture toughness of interface can wiely vary. In case of silicate base composites it is epenent on the manufacturing process an evelopment of interfacial transition zone. The values 1C,interface are higher than 1C,1 an 1C,2 thus it seems that the crack kink to the interface is not probable, but min that usually K IC,interface is lower than K IC,matrix an K IC,inclusion. When consiering particular values of K IC,m for particular ratios E 1/E 2, the curves of the critical values 1C,m for matrix, the inclusion an the interface will change by multiplying as inicate in (13). Thus every particular case will show if crack initiation occurs to matrix, to the inclusion or the interface, i.e. min{ 1C,1, 1C,2, 1C,interface}. Let us note that stuie material moel is universal. Although it supposes sharp concave inclusion embee in matrix, at the same time it can escribe convex corner of stiffer inclusion fille with matrix. In this case E 1 woul match to matrix an E 2 to the inclusion. Principally, the approaches will be the same an the results very similar. 71

7 Figure 5: GSIF 1 for unit applie loa an the critical values 1C neee for crack initiation to matrix an to the inclusion. CONCLUSIONS T he article eals with generalize fracture mechanics of a crack initiation in a tip of polygonal material inclusion embee in matrix. Stability criterion for etermination of crack initiation conitions is propose by means of average value of tangential stress. The stability criterion is written for stress escribe by singular an non-singular (higher orer) terms. The criterion employing higher orer terms is necessary for the assessment of sharp material inclusion, because singular stress terms only o not escribe stress state satisfactorily in some material an geometrical configurations. The propose criterion allows inicating whether crack initiation occurs into matrix, into the inclusion or into the interface. Its general form allows such assessment for particular known fracture toughness of the matrix, inclusion or the interface. Description of stability of stress concentrators of this kin contribute to better unerstaning of toughening mechanisms in particle composites. This kin of unerstaning leas to composite esign optimisation or composite structure failure prevention. ACKNOWLEDGEMENTS T his research has been financially supporte by the Ministry of Eucation, Youth an Sports of the Czech Republic uner the project CEITEC 22 (LQ161). The authors woul like to thank the Czech Science Founation for financial support through the Grant 16/1872S. REFERENCES [1] Paggi, M., Carpinteri, A., On the Stress Singularities at Multimaterial Interfaces an Relate Analogies With Flui Dynamics an Diffusion, Applie Mechanics Reviews, 61 (28) [2] Krepl, O., Klusák, J., Reconstruction of a 2D stress fiel aroun the tip of a sharp material inclusion, Proceia Structural Integrity, 2 (216)

8 [3] Williams, M.L., Stress singularities resulting from various bounary conitions in angular corners of plates in extension. J Appl Mech,19 (1952) [4] Seweryn, A., Molski, K., Elastic stress singularities an corresponing generalize stress intensity factors for angular corners uner various bounary conitions, Engineering Fracture Mechanics, 55(4) (1996) [5] Kim, J.K., Cho, S.B., Effect of secon non-singular term of moe I near the tip of a V-notche crack, Fatigue an Fracture of Engineering Materials an Structures, 32 (29) [6] Ayatollahi, M.R., Dehghany, M., Nejati, M., Fracture analysis of V-notche components Effects of first non-singular stress term, International Journal of Solis an Structures, 48 (211) [7] Ayatollahi, M.R., Mirsayar, M.M., Nejati, M., Evaluation of first non-singular stress term in bi-material notches, Computational Materials Science, 5 (21) [8] Mirsayar, M.M., Aliha, M.R.M., Samaei, A.T., On fracture initiation angle near bi-material notches Effects of first non-singular stress term, Engineering Fracture Mechanics, 119 (214) [9] Klusák, J., rstka, M., Profant, T., Krepl, O., Ševeček, O., Kotoul M., The influence of the first non-singular stress terms on crack initiation irection in an orthotropic bi-material plate, Theoretical an Applie Fracture Mechanics, 71 (214) [1] Krepl, O., Klusák, J., The influence of non-singular terms on the precision of stress escription near a sharp material inclusion tip, Theoretical an Applie Fracture Mechanics, in press. DOI: 1.116/j.tafmec [11] Sih, G.C., A special theory of crack propagation, in: G.C. Sih (E.), Mechanics of Fracture Methos of analysis an solutions of crack problems, Noorhoff International Publishing, Leyen, The Netherlans, (1973) XXI-XLV. [12] Sih, G.C., Mechanics of Fracture Initiation an Propagation, Kluwer Acaemic Publishing, Boston, (1991). [13] Knésl, Z., A criterion of V-notch stability, Int J of Fracture, 48 (1991) [14] Klusák, J. Profant, T., Knésl, Z, Kotoul M., The influence of iscontinuity an orthotropy of fracture toughness on conitions of fracture initiation in singular stress concentrators, Engineering Fracture Mechanics, 11 (213)