A. Campagnolo et alii, Frattura ed Integrità Strutturale, 4 (017) 181-188; DOI: 10.31/IGF-ESIS.4.19 Mode II brittle frature: reent developments A. Campagnolo Department of Industrial Engineering, University of Padova, Via Venezia 1, 35131, Padova, Italy S.M.J. Razavi, M. Peron, J. Torgersen, F. Berto Department of Mehanial and Industrial Engineering, Norwegian University of Siene and Tehnology (NTNU), Norway javad.razavi@ntnu.no, miro.peron@ntnu.no, jan.torgersen@ntnu.no, filippo.berto@ntnu.no ABSTRACT. Frature behaviour of V-nothed speimens is assessed using two energy based riteria namely the averaged strain energy density (SED) and Finite Frature Mehanis (FFM). Two different formulations of FFM riterion are onsidered for frature analysis. A new formulation for alulation of the ontrol radius R under pure Mode II loading is presented and used for predition of frature behaviour. The ritial Noth Stress Intensity Fator (NSIF) at failure under Mode II loading ondition an be expressed as a funtion of noth opening angle. Different formulations of NSIFs are derived using the three riteria and the results are ompared in the ase of sharp V-nothed brittle omponents under in-plane shear loading, in order to investigate the ability of eah method for the frature assessment. For this purpose, a bulk of experimental data taken from the literature is employed for the omparison among the mentioned riteria. Citation: Campagnolo, A., Razavi, S.M.J., Peron, M., Torgesen, J., Berto, F., Mode II brittle frature: reent developments, Frattura ed Integrità Strutturale, 4 (017) 181-188. Reeived: 8.06.017 Aepted: 5.07.017 Published: 01.10.017 Copyright: 017 This is an open aess artile under the terms of the CC-BY 4.0, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original author and soure are redited. EYWORDS. Frature strength; Sharp V-noth; mode II frature; Noth stress intensity fator; Strain energy density. INTRODUCTION N umerous failure riteria have been proposed by sholars dealing with the brittle frature of V-nothes [1]. Aording to the intensified linear elasti stress at the noth tip, introduing a stress field parameter is useful. Considering the Linear Elasti Noth Mehanis (LEFM) onept, NSIFs an be suessfully employed for the frature assessment of brittle materials when weakened by sharp V-nothes [,3]. Lazzarin et al. [4] ompared two widely employed riteria based on energy onsidering V-nothed omponents under pure Mode I loading; the first riterion is based on the loal SED [5] while the seond riterion is the so-alled FFM approah. Two different formulations of the FFM riterion is available in the literature: the first formulation was proposed by Leguillon [6,7] and the seond one was proposed by Carpinteri et al. [8]. The main aim of the urrent paper is to extend the previous omparison to the ase of Mode II loading ondition. 181
A. Campagnolo et alii, Frattura ed Integrità Strutturale, 4 (017) 181-188; DOI: 10.31/IGF-ESIS.4.19 The FFM riteria have been reently extended to the mixed Mode I/II and Mode II loading onditions [9-11]. Instead, onsidering the SED riterion, a new formulation for estimating the ontrol radius under in-plane shear loading is proposed here and is ompared with results obtained from the expression whih is valid for Mode I. Lazzarin and Zambardi [5] proposed the loal SED riterion whih is based on the strain energy density averaged over a ontrol volume surrounding the noth tip. The size of ontrol volume is dependent on the material property and the loading onditions [1,13]. The average SED value is independent of the mesh size [14]. Additionally, a new method for average SED alulations for raks under mixed mode I+II loading by adopting very oarse meshes has been reently proposed [15,16] whih is based on the maximum stresses obtained from FE analyses. In this paper the analytial formulation of the three ompared riteria [5,9-11] is introdued to obtain the ritial Mode II Noth Stress Intensity Fator using different approahes. After all, the mentioned riteria are applied to sharp V-nothed plates subjeted to in-plane shear loading, in order to investigate the ability of eah method to analysis the frature behaviour of brittle materials. The experimental data available in the literature were used for the analyses [17]. (a) R W R (b) l r noth bisetor () r noth bisetor Figure 1: Referene system for: (a) averaged SED riterion; (b) Leguillon et al. riterion and () Carpinteri et al. riterion. FAILURE CRITERIA FOR SHARP V-NOTCHES UNDER PURE MODE II LOADING Averaged strain energy density (SED) riterion A ording to Lazzarin and Zambardi [5], the frature of a brittle material takes plae when the strain energy density averaged over a ontrol volume haraterized by a radius R (Fig. 1a), beomes equal to the ritial value W (Eq. 1). In the ase of a smooth omponent under nominal shear loading ondition, employing Beltrami s hypothesis, the following expression an be derived: τ 1 ντ W G E (1) where τ is the ultimate shear strength, G the shear modulus and E the Young's modulus, while ν represents the Poisson s ratio. Considering a V-nothed plate subjeted to nominal pure Mode II loading, the relationship W W is verified under ritial onditions. Aordingly, one an obtain the expression for, whih is the ritial NSIF at failure: 1 1 1 1 e E R E e R () The ontrol radius R an be evaluated by onsidering a set of experimental data that provides the ritial value of the Noth Stress Intensity Fator for a given noth opening angle. If the V-noth angle is equal to zero (α = 0, λ = 0.5), the ase of a raked speimen under nominal pure Mode II loading is onsidered, so that under ritial onditions oinides with the Mode II frature toughness II. Then, taking advantage of Eq. (), with II, and following the 18
A. Campagnolo et alii, Frattura ed Integrità Strutturale, 4 (017) 181-188; DOI: 10.31/IGF-ESIS.4.19 same proedure proposed by Yosibash et al. [18] for obtaining the ontrol radius under Mode I loading ondition, the expression of R turns out to be: R II, 198 9 8 II II II e ( 0) (1 ) 8 (1 ) 8 (3) Moreover, it is useful to express the NSIF at failure as a funtion of the Mode I material properties ( I and σ ), whih are simpler to determine or to find in the literature than Mode II material properties. For this purpose, it is possible to approximately estimate the Mode II frature toughness ( II ) as a funtion of I, aording for example to Rihard et al. [19]. In the same manner, it is possible to approximately estimate the ultimate shear strength (τ ) as a funtion of the tensile one (σ ). With referene to brittle materials with linear elasti behaviour (as for example polymethylmetharylate, graphite, ), it has been observed experimentally [0] that the most appropriate riterion is that of Galileo-Rankine. Aordingly the following expressions are valid: II 3 I (4) 0.80 1.00 on an experimental basis (5) Finally, substitution of Eqs. (3)-(5) into Eq. () gives the NSIF at failure in a more useful form: 1 1 3 98 1 1 1 I e 4 8 (6) It should be noted that the ontrol radius R ould be in priniple different under Mode I and Mode II loading ondition, this means that it depends on the material properties but also on the loading onditions. Finite Frature Mehanis: Leguillon et al. formulation By using Leguillon et al. riterion, it is thought that at failure an inremental rak of length l initiates at the tip of the noth. Aording to Leguillon et al. [7,11], two onditions an be imposed on stress omponents and on strain energy and both are neessary for frature. They have to be simultaneously satisfied to reah a suffiient ondition for frature. On the basis of the stress ondition, the failure of the nothed element happens when the singular stress omponent normal to the frature diretion is higher than the material tensile stress σ all along the rak of length l just prior to frature. On the basis of the ondition imposed on strain energy, the failure ours when the SERR reahes a value higher than, whih is the ritial value for the material. is the ratio between the potential energy variation at rak initiation (δw p ) and the new rak surfae reated (δs). These two onditions an be formalised as follows providing a general riterion for the frature of omponents in presene of pointed V-nothes. (7a) 1 () Stress riterion: ( l, ) kl * W, p kh l d (7b) S l d Energy riterion: In Eqs. (7a,b) the length of the inremental rak is l (see Fig. 1b). λ is the Mode II Williams eigenvalues quantities [1], () is a funtion of the angular oordinate, while d is the that is a funtion of the V-noth opening angle 183
A. Campagnolo et alii, Frattura ed Integrità Strutturale, 4 (017) 181-188; DOI: 10.31/IGF-ESIS.4.19 * thikness of the nothed element. Finally, H(, ) is a geometrial fator funtion of the loal geometry (α) and of the frature diretion ( ). Leguillon et al. riterion requires that onditions (7a) and (7b) must be simultaneously satisfied. The length of the inremental rak an be determined by solving the system of two equations (Eqs. (7a,b)), then by substituting it into Eq. (7a) or (7b), the frature riterion an be expressed in the lassial Irwin form ( I I). In this ase the ritial value of the NSIF k an be provided as a funtion of the material properties (σ and ), the V-noth angle α and the ritial rak propagation angle. 1 1 k H, k * () (8) Yosibash et al. [11] have omputed the funtion H for a range of values of the noth opening angle α and of the frature diretion, taking into aount a material haraterized by a Young s modulus E = 1 MPa and a Poisson s ratio * ν = 0.36. The funtion. H.for any other Young s modulus E and Poisson s ratio ν an be easily obtained aording to the following expression: H * 1 1, H, E 1 0.36 (9) A more useful expression for k, as a funtion of the Mode I frature toughness I and of the ultimate tensile stress σ, an be derived by substituting Eq. (9) and the link between and I into Eq. (8). Then, by employing Gross and Mendelson s [] definition for the ritial NSIF, the following expression an be obtained: 1 1 1 0.36 1 1 1 () I H, (10) Finite Frature Mehanis: Carpinteri et al. formulation In a similar manner to Leguillon, a frature riterion for brittle V-nothed elements based on FFM onept has been proposed by Carpinteri et al. in [8-10]. Under ritial onditions, a rak of length Δ is thought to initiate from the noth tip. Again, a suffiient ondition for frature an be ahieved from the satisfation of both a stress riterion and an energy-based one. On the basis of the averaged stress riterion, the failure of the omponent at the V-noth tip happens when the singular stress omponent normal to the rak faes, averaged on the rak length Δ, beomes higher than the tensile stress σ of the material under investigation. The energy-based ondition, instead, requires for the failure to happen that the strain energy released at the initiation of a rak of length Δ is higher than the material ritial value, whih depends on. By onsidering the relationship between the SERR and the SIFs I and II of a rak under loal mixed mode I+II loading, it is possible to derive a more useful formulation. This is valid under plane strain hypotheses and onsidering that the rak propagates in a straight diretion. The ontemporary verifiation of the onditions given by Eq. (11a) and (11b) allows to formalize a riterion for the brittle frature of sharply V-nothed elements: * (11a) II Average stress riterion: r, dr ( ) 1 dr r 0 0 dw da a da p Energy riterion:, da (11b) 0 0 184
A. Campagnolo et alii, Frattura ed Integrità Strutturale, 4 (017) 181-188; DOI: 10.31/IGF-ESIS.4.19 * 1 I a, II a, da II a 1,, da I 0 0 In Eqs. (11a) and (11b), Δ represents the length of the rak initiated at the V-noth tip (see Fig. 1), while λ is the Mode II Williams eigenvalue [1]. (r, θ) are the polar oordinate system entred at the noth tip and a represents a generi rak length. With the aim to employ the energy-based approah, the knowledge of the SIFs I and II of the tilted rak nuleated at V-noth tip, as a funtion of the rak length a is stritly required. In order to this, the expressions of the SIFs I and II derived by Beghini et al. [3], on the basis of approximate analytial weight funtions, an be used. They are funtions of the rak length a, the V-noth angle α, the frature diretion θ and the NSIF II *. It is useful to introdue a simplified notation aording to Eq. (1), in whih the relationship between the parameter aording to Sapora et al. [10] and the parameter H aording to Yosibash et al. [11] is shown, as highlighted also in [9]., 1,, H, 1 0.36 (1) On the basis of Carpinteri et al. approah, the frature of the omponent happens when both onditions provided by Eqs. (11a) and (11b) are simultaneously satisfied. By solving the system of two equations, the length of the initiated rak Δ and the ritial NSIF II * an be expliitly found. Moreover, by adopting the lassial definition of due to Gross and Mendelson [], the following expression results to be valid: 1 1 1 1 1 1 1 I, ( ) (13) ANALYTICAL COMPARISON T he riteria taken into onsideration in the present ontribution, an be ompared on the basis of the final relationships of the Mode II ritial Noth Stress Intensity Fator aording to Gross and Mendelson s definition [], see Eqs. (6), (10) and (13). The same proportionality relation is ommon to all riteria: 1 1 I (14) Proportionality fators 3.00.50.00 1.50 1.00 SED Leguillon et al. Carpinteri et al. 0.50 0.00 0 0 40 60 80 100 Noth opening angle α, [ ] Figure : Comparison among proportionality fators given by Eqs. (6), (10) and (13). 185
A. Campagnolo et alii, Frattura ed Integrità Strutturale, 4 (017) 181-188; DOI: 10.31/IGF-ESIS.4.19 The only differene in the final expressions is represented by the proportionality fator. Indeed, the latter is a funtion of the V-noth angle (α) in Leguillon et al. and Carpinteri et al. formulations. On the other hand, the proportionality fator of the SED approah is a funtion also of the Poisson's ratio ν. It should be noted that the V-noth tip singularity (1-λ ), tied to Mode II loading, disappears for opening angles higher than 10 degrees, aording to Williams [1]. Therefore the analytial omparison of Eqs. (6), (10) and (13) has been arried out for noth opening angles α between 0 and about 100 degrees. The proportionality fators are very similar in their trends (Fig. ). In partiular, the fator of Leguillon et al. approah is always a bit greater than those obtained by using the other two riteria. The fators aording to Leguillon et al. and SED riteria assume the same values for α= 0 and 100 degrees: 0.87 and.7, respetively. The fator based on Carpinteri et al. approah, instead, does not math that of the other riteria for any value of α. COMPARISON BASED ON EXPERIMENTAL DATA T he assessment apability of the onsidered riteria is ompared here by employing experimental data reported in the literature. The data are all relevant to omponents weakened by sharp V-nothes and onstituted by polymethylmetharylate (PMMA). The material properties and the experimental details are summarised in the following. A more extended omparison an be found in [15]. The experimental results and the theoretial preditions obtained from Eqs. (6), (10) and (13) have been ompared on the basis of the ritial value of the Mode II NSIF,. The ritial NSIFs are plotted as a funtion of the noth angle α. If not expliitly given in the original ontributions, the NSIFs to failure were alulated from D FE analyses, by adopting FE meshes onsisting of eight node solid elements (PLANE 183) and by applying to the FE models the ritial loads experimentally obtained. All FE analyses have been performed by means of ANSYS, version 14.5. It should be noted that is oinident with the NSIF evaluated on the noth bisetor line aording to Gross and Mendelson 1 definition [] ( r r with r 0 ), provided that the FE model is loaded with the experimental ritial load. 1000.0 NSIF to failure, [MPa m (1-λ ) ] 100.0 10.0 1.0 0.1 Experimental data SED Leguillon et al. Carpinteri et al. 0 0 40 60 80 100 Noth opening angle α, [ ] Figure 3: Plot of the NSIF to failure (Log-sale) and omparison with the experimental data from PMMA double V-nothed speimens. The onsidered set of experimental results has been derived from Aran tests arried out by Seweryn et al. [3] on PMMA double V-nothed speimens. The tested V-nothed omponents were haraterized by a length l = 00 mm, a width w = 100 mm, a noth depth a = 5 mm and a thikness t = 5 mm. Seweryn et al. [3] took into aount speimens with four different V-noth angles, α = 0, 40, 60 and 80 degrees, being the loading angle to obtain pure Mode II loading equal to ψ = 90 degrees [4]. Three PMMA samples have been tested for eah geometry. The relevant properties of the onsidered material have been reported in the original ontribution [4] as follows: the Young s modulus E was equal to 3000 MPa, the Poisson s ratio equal to 0.30, while the Mode I frature toughness and the ritial tensile stress were I = 1.37 MPa.m 0.5 and σ = 115 MPa, respetively. 186
A. Campagnolo et alii, Frattura ed Integrità Strutturale, 4 (017) 181-188; DOI: 10.31/IGF-ESIS.4.19 The experimental results relevant to the PMMA V-nothed omponents are shown in Fig. 3 in terms of the ritial NSIF, along with the theoretial preditions based on the frature approahes under onsideration. It an be observed from Fig. 3 that the agreement between theoretial preditions and experimental results, in terms of ritial Mode II NSIF, is very good for all onsidered riteria. Some reent developments dealing with reep stresses are reported in [5]. CONCLUSIONS T hree different frature riteria for brittle and quasi-brittle materials weakened by sharp V-nothes have been onsidered in the present ontribution. The attention has been foused on in-plane shear loading onditions (Mode II). The averaged strain energy density (SED) riterion and two different formulations of the Finite Frature Mehanis (FFM) theory, aording to Leguillon et al. and to Carpinteri et al. respetively, have been aurately ompared. With referene to the riterion based on the averaged SED, a new expression for estimating the ontrol radius R under pure Mode II loading has been proposed. First, the riteria have been ompared analytially by providing the expressions of the ritial value of the Noth Stress Intensity Fator. The same proportionality relation has been found to exist between and two key material properties: the Mode I frature toughness I and the ultimate tensile stress σ. The only differene between the analysed riteria is represented by the proportionality fator. Finally, the approahes taken into onsideration in the present ontribution have been adopted for the frature assessment of brittle V-nothed omponents subjeted to pure Mode II loading. This has allowed to investigate the assessment apability of eah approah under in-plane shear loading. A set of experimental data reported in the literature has been employed for the omparison. The agreement between experimental data and theoretial preditions has been found very good for all riteria onsidered in the present investigation. REFERENCES [1] Berto, F., Loal approahes for the frature assessment of nothed omponents: The researh work developed by Professor Paolo Lazzarin, Frattura ed Integrita Strutturale, 9(34) (015) 11-6. [] nesl, Z., A riterion of V-noth stability, Int. J. Frat., 48 (1991) R79 R83. [3] Seweryn, A., Brittle frature riterion for strutures with sharp nothes, Eng. Frat. Meh., 47 (1994) 673 681. [4] Lazzarin, P., Campagnolo, A., Berto, F., A omparison among some reent energy- and stress-based riteria for the frature assessment of sharp V-nothed omponents under Mode I loading, Theor. Appl. Frat. Meh., 71 (014) 1 30. [5] Lazzarin, P., Zambardi, R., A finite-volume-energy based approah to predit the stati and fatigue behavior of omponents with sharp V-shaped nothes, Int. J. Frat., 11 (001) 75 98. [6] Leguillon, D., A riterion for rak nuleation at a noth in homogeneous materials, C. R. Aad. Si. II B, 39 (001) 97-10. [7] Leguillon, D., Strength or toughness? A riterion for rak onset at a noth, Eur. J. Meh. A Solids, 1 (00) 61 7. [8] Carpinteri, A., Cornetti, P., Pugno, N., Sapora, A., Taylor, D., A finite frature mehanis approah to strutures with sharp V-nothes, Eng. Frat. Meh., 75 (008) 1736 175. [9] Sapora, A., Cornetti, P., Carpinteri, A., A Finite Frature Mehanis approah to V-nothed elements subjeted to mixed-mode loading, Eng. Frat. Meh., 97 (013) 16 6. [10] Sapora, A., Cornetti, P., Carpinteri, A., V-nothed elements under mode II loading onditions, Strut. Eng. Meh., 49 (014) 499 508. [11] Yosibash, Z., Priel, E., Leguillon, D., A failure riterion for brittle elasti materials under mixed-mode loading, Int. J. Frat., 141 (006) 91 31. [1] Berto, F., Ayatollahi, M.R., Frature assessment of Brazilian dis speimens weakened by blunt V-nothes under mixed mode loading by means of loal energy, Mater. Des., 3 (5) (011) 858-869. [13] Radaj, D., Berto, F., Lazzarin, P., Loal fatigue strength parameters for welded joints based on strain energy density with inlusion of small-size nothes, Eng. Frat. Meh., 76 (8) (009) 1109-1130. [14] Lazzarin, P., Berto, F., Zappalorto, M., Rapid alulations of noth stress intensity fators based on averaged strain energy density from oarse meshes: Theoretial bases and appliations, Int. J. Fatigue, 3 (010) 1559 1567. 187
A. Campagnolo et alii, Frattura ed Integrità Strutturale, 4 (017) 181-188; DOI: 10.31/IGF-ESIS.4.19 [15] Campagnolo, A., Meneghetti, G., Berto, F., Rapid finite element evaluation of the averaged strain energy density of mixed-mode (I+II) rak tip fields inluding the T-stress ontribution, Fatigue Frat. Eng. Mater. Strut., 39 (016) 98 998. [16] Meneghetti, G., Campagnolo, A., Berto, F., Atzori, B., Averaged strain energy density evaluated rapidly from the singular peak stresses by FEM: raked omponents under mixed-mode (I+II) loading, Theor. Appl. Frat. Meh., 79 (015) 113 14. [17] Campagnolo, A., Berto, F., Leguillon, D., Frature assessment of sharp V-nothed omponents under Mode II loading: a omparison among some reent riteria, Theor. Appl. Frat. Meh., 85 (016) 17 6. [18] Yosibash, Z., Bussiba, A., Gilad, I., Failure riteria for brittle elasti materials, Int. J. Frat., 15 (004) 307 333. [19] Rihard, H.A., Fulland, M., Sander, M., Theoretial rak path predition, Fatigue Frat. Eng. Mater. Strut., 8 (005) 3 1. [0] Berto, F., Lazzarin, P., Reent developments in brittle and quasi-brittle failure assessment of engineering materials by means of loal approahes, Mater. Si. Eng. R, 75 (014) 1 48. [1] Williams, M.L., Stress singularities resulting from various boundary onditions in angular orners of plates in tension, J. Appl. Meh., 19 (195) 56 58. [] Gross, B., Mendelson, A., Plane elastostati analysis of V-nothed plates, Int. J. Frat., 8 (197) 67 76. [3] Beghini, M., Bertini, L., Di Lello, R., Fontanari, V., A general weight funtion for inlined raks at sharp V-nothes, Eng. Frat. Meh., 74 (007) 60 611. [4] Seweryn, A., Poskrobko, Sł., Mróz, Z., Brittle Frature in Plane Elements with Sharp Nothes under Mixed-Mode Loading, J. Eng. Meh., 13 (1997) 535 543. [5] Gallo, P., Berto, F., Glinka, G Generalized approah to estimation of strains and stresses at blunt V-nothes under non-loalized reep, Fatigue Frat. Eng. Mater. Strut, 39 (016) 9-306. 188