DESIGN OF FRACTURE RESISTANT COMPOSITES BY UTILIZING SPATIAL MATERIAL PROPERTY VARIATIONS

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1 0 th Internatonal Conference on Composte aterals Copenhagen, 19-4 th ul 015 DESIGN OF FRACTURE RESISTANT COPOSITES BY UTIZING SPATIAL ATERIAL PROPERTY VARIATIONS. Sstanna 1,, O. Kolednk 1 1 Erch Schmd Insttute of aterals Scence, Austran Academ of Scences, ahnstrasse 1, A-8700 Leoben, Austra Emal: masoud.sstanna@mcl.at, web page: aterals Center Leoben Forschung GmbH, Roseggerstrasse 1, A-8700 Leoben, Austra Emal: masoud.sstanna@mcl.at, web page: Kewords: ultlaered compostes, Crack drvng force, Confguratonal forces, Fracture toughness ABSTRACT Recent theoretcal, numercal and expermental nvestgatons have shown that the fracture resstance of materals can be strongl mproved b utlzng spatal varatons of the mechancal propertes. The reason for ths effect s that the crack drvng force s reduced, f the crack enters a regon where the Young s modulus strongl ncreases; the reducton of the crack drvng force can lead to crack arrest. Ths mechansm s responsble for the hgh fracture resstance of certan bologcal materals, such as deep-sea glass sponges, although the consst to 95% of brttle bo-glass. It has been demonstrated that the crack drvng force s also reduced, f the crack enters a regon where the eld stress strongl ncreases. Therefore, the dea s to enhance the fracture resstance of brttle hgh-strength materals b ntroducng thn, soft nterlaers. From the numercal results, mum nterlaer confguratons are derved,.e. for a gven matrx materal and load, the magntudes of thckness and eld stress of the soft nterlaer are determned so that the crack drvng force exhbts a mnmum. Ths knowledge s then used to derve crtera for the mal desgn of fracture resstant compostes based on the eld stress nhomogenet effect. 1 INTRODUCTION ultlaer structures wth soft nterlaers can reach ver hgh fracture resstance compared to the homogenous materal, due to spatal varatons of the mechancal propertes [1-7]. The reason for the effect s that n materals wth spatal varatons of the mechancal propertes, the crack drvng force s locall reduced, compared to a homogenous materal, whch leads to a strong ncrease of the fracture resstance. It s worth to menton that ths effect occurs ndependentl of delamnatng nterfaces. an fracture resstant bologcal materals, such as the skeleton of deep-sea glass sponges, also have a multlaer mcrostructure consstng of hgh stffness and strength laers connected b thn, soft nterlaers [4, 8]. Numercal modelng and applcaton of the concept of confguratonal forces has recentl revealed that the strong spatal varaton of the Young s modulus between the hard and brttle bo-glass and the thn, soft proten laers s the domnant mechansm for the hgh fracture resstance of these structures [4]. The reason s that the crack drvng force strongl decreases when the crack enters the soft laer. It has been shown that the fracture resstance of a composte becomes much hgher than that of the homogenous materal, f the composte archtecture fulflls certan desgn rules, whch were derved n [4]. The dea of enhancng the fracture resstance b ntroducng soft nterlaers shall be transferred to engneerng materals where both the hard and the soft materal behave elastc plastc. A problem s, however, that the crteron proposed n [4] for the desgn of fracture resstant laered compostes can be onl used for elastc materals. In order to work out desgn rules of fracture resstant elastc plastc multlaered materals, t s necessar frst to quantf the nfluence of a sngle soft nterlaer on the crack drvng force.

2 . Sstanna, O. Kolednk The present paper frst quantfes the nfluence of a sngle soft nterlaer wth eld stress nhomogenet on the crack drvng force usng numercal modelng and applcaton of the concept of the confguratonal forces. From the numercal results, mum nterlaer confguratons are derved,.e. for a gven matrx materal and load, the magntudes of thckness and eld stress of the soft nterlaer are determned so that the crack drvng force exhbts a mnmum. Fnall, the knowledge of the mum nterlaer confguratons wll be used to derve crtera for the mal desgn of fracture resstant compostes wth the varatons of the eld stress. CONCEPT OF CONFIGURATIONAL FORCES FOR THE EVALUATION OF CRACK DRIVING FORCE IN INHOOGENOUS ATERIALS The concept of confguratonal forces s approprate for analzng the behavor of cracks n nhomogenous materals. Based on ths concept, the crack sheldng or ant-sheldng effect due to materal nhomogenetes s quantfed b the materal nhomogenet term, C nh [9-11]. For a bod contanng a crack and a sharp nterface Σ where the materal propertes exhbt a jump, the materal nhomogenet term can be calculated for small stran theor b Cnh e [[ ]] [[ u]] ndl, (1) where ϕ s the stran energ denst, I the dentt tensor, σ the Cauch stress, the Lagrangan gradent operator, u the dsplacement, e the unt vector n the drecton of crack extenson and n s the unt vector normal to the nterface Σ. In Eq. (1), the jump of a quantt a at the nterface s denoted as [[ a]] a a and the average of a quantt a across the nterface s denoted as / a a a, where a and a denote the lmtng values of the quantt on ether sde of the nterface. The crack drvng force, expressed n terms of the near- -ntegral, s the sum of the -feld -ntegral and the materal nhomogenet term, C. () nh The -feld -ntegral, whch s a loadng parameter, measures the drvng force nduced b the appled loads nto the specmen. In a hard/soft or stff/complant transton, C nh becomes postve and leads to a crack ant-sheldng effect. In the opposte case, n a soft/hard or complant/stff transton, C nh becomes negatve and leads to a crack sheldng effect. 3 INFLUENCE OF A SINGLE, SOFT INTERLAYER ON THE CRACK DRIVING FORCE In ths secton, the nfluence of the eld stress varaton due to a sngle soft nterlaer on the crack drvng force s quantfed usng numercal modelng and applcaton of the concept of the confguratonal forces. 3.1 Numercal modelng The FE analses are mplemented for a standard Compact Tenson specmen (Fg. 1) that has an nterlaer wth two sharp nterfaces, nterface 1 (IF1) and nterface (IF), where the materal propertes exhbt a jump. Both the base materal and the nterlaer are assumed to be sotropc elastc plastc materals wth deall plastc behavor (wthout hardenng), and the bondng at the nterfaces s assumed to be perfect. A straght crack s assumed lng perpendcular to the nterfaces, where L 1 and L are the dstances between the crack and IF1 and IF, respectvel. When the crack s stuated left of an nterface, L (the dstance between the crack and the nterface) s negatve. A postve L denotes that the crack s stuated rght of an nterface. L 1 and L are related b where t s the nterlaer thckness. L L1 t, (3)

3 0 th Internatonal Conference on Composte aterals Copenhagen, 19-4 th ul 015 The numercal analses are performed wth a commercal mplementaton of the fnte element method (ABAQUS, The analses are performed for two-dmensonal FE models under the assumpton of plane stran condton. 4 node elements are used for dscretzng the FE models and the loadng s controlled b prescrbng the load-lne dsplacement. After the FE analss, the materal nhomogenet term, C nh, b Eq. (1) and the -feld -ntegral,, are evaluated for each ncrement of dsplacement b a post-processng process, t s recommended to see [9-1] for more nformaton regardng the numercal evaluaton of and C nh. Once the -feld -ntegral and the materal nhomogenet term of IF1, C nh1, and the materal nhomogenet term of IF, C nh, are calculated for each crack poston, the crack drvng force can be determned from the relaton C C. (4) nh1 nh 3. The materal nhomogenet effect due to a sngle soft nterlaer Snce the elastc propertes of nterlaer and matrx are equal, a materal nhomogenet effect onl appears f plastc deformaton occurs,.e. as long as the crack plastc zone touches the nterface. The radus of the crack plastc zone, r, can be determned b Irwn s model [13] as r E, (1 ) where E s the Young s modulus, σ the eld stress, ν Posson s rato and β = 1/6π for plane stran condtons. When the nterlaer has a lower eld stress than the matrx, there s a hard/soft transton at IF1 and therefore, C nh1 s postve as long as the crack plastc zone touches IF1. Due to the soft/hard transton at IF, C nh s negatve as long as the crack plastc zone touches IF. The materal nhomogenet effect of the nterlaer on the crack drvng force can be quantfed b the summaton of the materal nhomogenet terms of the nterfaces IF1 and IF, C C C. (6) nh1 nh The term C s called the nterlaer nhomogenet term. B nsertng Eq. (6) nto Eq. (4), the crack drvng force can be determned from C. (7) Dependng on the crack poston, the nterlaer can exert a sheldng or ant-sheldng effect on the crack. If C s negatve, the nterlaer exerts a sheldng effect on the crack and f C s postve, an ant-sheldng effect occurs. Fg. a shows, as an example, the magntude of the nterlaer nhomogenet term C for four dfferent values of, where Young s modulus E = 70 GPa, t = 0.3 mm, Posson s rato ν = 0.3, the eld stresses of the matrx ( ) and nterlaer ( ) are 500 Pa and 00 Pa, respectvel. The absolute sze of the nterlaer nhomogenet term, C, ncreases wth because, accordng to the HRR-feld soluton, the stran energ denst s proportonal to. Fg. b shows the varaton of the crack drvng force,, for four dfferent values of between 30 and 10 k/m. As t can be seen, when the crack approaches the soft nterlaer, the crack drvng force frst ncreases at IF1 and then decreases at IF. The mnmum value of s reached mmedatel when the crack has crossed IF. Ths poston s the crtcal poston for possble crack arrest. In the followng, ths poston s referred to as the poston. The crack drvng force for a crack located at poston s referred to (5). Consequentl, a soft nterlaer works as a ver effcent crack arrester, f becomes ver low. In the next subsecton, we explore the mum thckness and eld stress of the soft nterlaer so that the maxmum reducton of the crack drvng force can be acheved.

4 . Sstanna, O. Kolednk Fgure 1: CT-specmen wth an nterlaer (two sharp nterfaces) perpendcular to the crack. Fgure : C and plotted aganst L 1 for an elastc plastc sngle nterlaer specmen wth four dfferent values of. 3.3 Optmum effectveness of a sngle, soft nterlaer A soft nterlaer exerts the maxmum sheldng effect on a crack when the crack s located at the poston. The rato of the crack drvng force vs. the -feld drvng force, /, ndcates the reducton of the crack drvng force due to a soft nterlaer, compared to a homogeneous materal whch s loaded to the same. Clearl, the mum effectveness of a soft nterlaer can be acheved f / becomes a mnmum. Therefore, n order to fnd the mum thckness and the

5 0 th Internatonal Conference on Composte aterals Copenhagen, 19-4 th ul 015 eld stress of the soft nterlaer, the nfluence of the thckness and the eld stress of the soft nterlaer on / should be quantfed. For smplct, the rato / s referred to, therefore, (8) s a dmensonless parameter and called the -reducton factor. In Fg. 3, / s plotted aganst the loadng parameter for a sngle nterlaer specmen wth t = 0.3 mm, E = 70 GPa, 00 Pa, and seven dfferent eld stress ratos / between 0.7 and 0. The later value s related to the case of an elastc plastc nterlaer n an elastc matrx. For all cases, the plastc zone frst touches IF and hence decreases wth ncreasng. The reducton of wth ncreasng contnues untl the plastc zone reaches IF1. After that, wth further ncreasng, ncreases due to the ant-sheldng effect of IF1. Therefore, reaches a mnmum value when the plastc zone begns to touch IF1. It can be concluded that the mn maxmum effectveness of a soft nterlaer can be acheved, f the plastct begns to spread over the whole nterlaer. Therefore, the mum nterlaer thckness can be estmated as where 1/ 6 for plane stran condtons and where The magntude of r E t r, (1 ) mn for dfferent eld stress ratos t. These values from Fg. 3 are plotted aganst r s the sze of the plastc zone n the nterlaer. (9) /, can be extracted from Fg. 3, / n Fg. 4. As can be seen, mn decreases wth decreasng /, untl / 0.. After that, mn remans constant. As Fg. 4 shows, the mum eld stress of a soft nterlaer becomes 0.. (10) Fgure 3: plotted aganst for a sngle nterlaer specmen wth seven dfferent eld stress ratos / between 0.7 and 0.

6 . Sstanna, O. Kolednk Fgure 4: Curve mn vs. /. mn reaches ts mnmum value when / 0.. oreover Fg. 4 demonstrates that a soft nterlaer wth mum thckness and mum eld stress can reduce the crack drvng force of a homogenous bulk materal b a factor of 10, mn 0.1, whch s the maxmum possble reducton of the crack drvng force. 4 DESIGN CRITERIA FOR FRACTURE RESISTANT ULTAYER COPOSITES WITH VARIATIONS OF THE YIELD STRESS The prevous secton focused on the nfluence of a soft nterlaer on the crack drvng force and the mum nterlaer confguratons were derved so that the maxmum reducton of the crack drvng force can be reached. In ths secton, the knowledge of the mum nterlaer confguratons s used to derve crtera for the mum desgn of fracture resstant multlaers. Fracture resstant multlaer compostes can be constructed b nsertng soft nterlaers at regular dstances nto the matrx materal, see Fg. 5a. The wavelength of the eld stress varaton λ s the decsve desgn parameter of the multlaer. The am of ths secton s to derve a crteron for fndng the mum wavelength λ. Fgure 5: (a) A multlaer composte wth soft nterlaers and a crack n the matrx materal. The crack n arrested at poston. (b) Comparable homogenous specmen made of matrx materal.

7 0 th Internatonal Conference on Composte aterals Copenhagen, 19-4 th ul An archtectural crteron for preventng fracture We assume a small crack located somewhere n the mddle of a laer of matrx materal, Fg. 5a. atrx- and nterlaer materals have equal elastc propertes, Young s modulus E and Posson s rato ν, but dfferent eld stresses, and, respectvel. The soft nterlaers are thn, t. The composte s loaded b the appled stress σ appl. The condton for the ntaton of crack growth s where, (11) denotes the fracture ntaton toughness of the matrx materal determned b a fracture mechancs experment [14, 15]. If the crack drvng force of the crack located at poston, fracture ntaton toughness, becomes smaller than the, the crack s arrested at poston. It can be assumed that crack growth of the arrested crack s re-ntated, f, due to the ncrease of the appled stress σ appl, the crack drvng force s ncreased so that. Consder a homogeneous specmen made of matrx materal wth an nteror crack of length a and subjected to an appled stress σ appl (Fg. 5b). The -feld -ntegral can be wrtten as [16, 17] hom appl (1 ) a. (1) E Eq. (1) s vald for lnear elastc and small-scale eldng condtons. Snce the materal s hom hom homogeneous, the -ntegral s path ndependent,. Note that, f Eq. (11) s fulflled, the crack can grow wthout hndrance n the homogeneous materal untl the whole specmen s broken. The wavelength of the multlaer composte λ determnes the length of the arrested crack, a t /, (13) (Fg. 5a). The -feld -ntegral of the arrested crack n the multlaer,, can be evaluated from Eq. (1), wth the crack length a replaced b the length of the arrested crack a /. The crack drvng force of the arrested crack, (1), eldng, can be estmated b nsertng Eq. (13) and Eq. (8) nto Eq. appl (1 ). (14) E In order to prevent the re-ntaton of growth of the arrested crack, the crack drvng force must reman smaller than the fracture ntaton toughness of the matrx materal,. (15) Thus, b nsertng Eq. (14) nto Eq. (15), we get a crteron for preventng growth of the arrested crack n the form, E appl (1 ). (16) Eq. (16) shows that an arrested crack n a composte wth soft nterlaers s unable to grow, f the wavelength of the composte s small enough. The necessar wavelength s nversel proportonal to the square of the appled stress,.e. a hgh appled stress requres a small wavelength λ, and vce versa.

8 . Sstanna, O. Kolednk Eq. (16) can be used as an archtectural crteron for the desgn of fracture resstant multlaered compostes, utlzng the eld stress nhomogenet effect. 4. Fndng the mum confguraton of a multlaer composte We assume that the matrx materal s known wth ts propertes, E, ν, select the approprate nterlaer materal, whch has the mum eld stress, and. Then we 0. (10). If the appled stress σ appl s known, Eq. (16) gves the requred wavelength λ of the composte. In order to get the mum wavelength, one should consder a stress safet factor, e.g. s 1.3, (1 ) mn s appl E, Eq.. (17) Note that the mum -reducton factor Ψ mn, mn 0.1, has been nserted nto Eq. (17). Ths s correct, f the nterlaer also has the mum nterlaer thckness t from Eq. (9). In order to do ths, we frst nsert the length of the arrested crack, a, Eq. (13), nto Eq. (1), and then nsert Eq. (1) nto Eq. (9). The transformaton results t s appl. (18) The pre-factor n Eq. (18) reduces to 1 1 for plane stran condtons. If the appled stress σ appl changes n tme, the queston arses, whch stress should be nserted nto Eq. (17) and Eq. (18)? Ths queston s not trval, snce the -reducton factor Ψ ncreases wth decreasng load, see Fg. 3. The answer to ths queston has been gven n our forthcomng paper [18]. The maxmum appled stress appl,max should be nserted. If the crack cannot grow at appl,max, t s also not able to grow at a lower stress, see [18]. 5 CONCLUSIONS In the current paper, the nfluence of the soft nterlaer on the crack drvng force s examned and the man fndngs are: Soft nterlaers can enhance greatl the fracture resstance of materals. The reason s that, when the crack has crossed a soft nterlaer, the crack drvng force strongl decreases and the crack s arrested b the nterlaer. The parameters nfluencng the mechansm are worked out. From these the condtons for the maxmum effectveness of a soft nterlaer as crack arrester are derved (mum nterlaer thckness, mum nterlaer strength). B nsertng mum soft nterlaers, the crack drvng force s reduced b a factor of 10, whch s the maxmum possble reducton of the crack drvng force. A smple desgn crteron s derved for fracture resstant multlaered structures Load-dependent mum multlaer archtecture s derved. ACKNOWLEDGEENTS Fnancal support b the Austran Federal Government and the Stran provncal Government wthn the research actvtes of the K Competence Center Integrated Research n aterals, Processng and Product Engneerng, under the frame of the Austran COET Competence Center Program, s gratefull acknowledged (Project A4.0-WP1).

9 0 th Internatonal Conference on Composte aterals Copenhagen, 19-4 th ul 015 REFERENCES [1] F.D. Fscher,. Predan, P. Fratzl and O. Kolednk, Sem-analtcal approaches to assess the crack drvng force n perodcall heterogeneous elastc materals, Internatonal ournal of Fracture, 173, 01, pp [] P. Fratzl, H.S. Gupta, F.D. Fscher and O. Kolednk, Hndered crack propagaton n materals wth perodcall varng oung's modulus - Lessons from bologcal materals, Advanced aterals, 19, 007, pp [3] O. Kolednk, The eld stress gradent effect n nhomogeneous materals, Internatonal ournal of Solds and Structures, 37, 000, pp [4] O. Kolednk,. Predan, F.D. Fscher and P. Fratzl, Bonspred desgn crtera for damageresstant materals wth perodcall varng mcrostructure, Advanced Functonal aterals, 1, 011, pp [5] S. Suresh, Y. Sugmura and T. Ogawa, Fatgue crackng n materals wth brttle surface coatngs, Scrpta etallurgca et aterala, 9, 1993, pp [6]. Zechner and O. Kolednk, Fracture resstance of alumnum multlaer compostes, Engneerng Fracture echancs, 110, 013, pp [7] O. Kolednk,. Predan, F.D. Fscher and P. Fratzl, Improvements of strength and fracture resstance b spatal materal propert varatons, Acta aterala, 68, 014, pp [8].W.C. Dunlop and P. Fratzl, Bologcal compostes, Annual Revew of aterals Research, 40, 010, pp [9] O. Kolednk,. Predan and F.D. Fscher, Reprnt of "Cracks n nhomogeneous materals: Comprehensve assessment usng the confguratonal forces concept", Engneerng Fracture echancs, 77, 010, pp [10] N.K. Smha, F.D. Fscher, O. Kolednk and C.R. Chen, Inhomogenet effects on the crack drvng force n elastc and elastc-plastc materals, ournal of the echancs and Phscs of Solds, 51, 003, pp [11] N.K. Smha, F.D. Fscher, O. Kolednk,. Predan and G.X. Shan, Crack sheldng or antsheldng due to smooth and dscontnuous materal nhomogenetes, Internatonal ournal of Fracture, 135, 005, pp [1]. Sstanna and O. Kolednk, Effect of a sngle soft nterlaer on the crack drvng force, Engneerng Fracture echancs, 130, 014, pp [13] G.R. Irwn, Plastc zone near a crack and fracture toughness, In Sagamore Research Conference Proceedngs, 4, [14] AST E , Standard Test ethod for easurement of Fracture Toughness, In: Annual Book of AST Standards, Vol. 3, AST Internatonal, West Conshohocken, 005. [15] ESIS P-9, ESIS Procedure for Determnng the Fracture Behavour of aterals, European Structural Integrt Socet, Delft, 199. [16] T.L. Anderson, Fracture mechancs: fundamentals and applcatons, 3rd Edton, Talor & Francs, Boca Raton, 005. [17] D. Gross and T. Seelg, Fracture echancs: Wth an Introducton to cromechancs, Sprnger Scence & Busness eda, Darmstadt, 007. [18]. Sstanna and O. Kolednk, Desgn of fracture resstant multlaer compostes wth spatal varatons of the eld stress, Acta aterala, to be publshed.