Effect of fiber volume fraction on fracture mechanics in continuously reinforced fiber composite materials

Transcription

1 University o South Florida Scholar Coons Graduate Theses and Dissertations Graduate School 2005 ect o iber volue raction on racture echanics in continuously reinorced iber coposite aterials Thoas Wasik University o South Florida Follow this and additional works at: Part o the Aerican Studies Coons Scholar Coons Citation Wasik, Thoas, "ect o iber volue raction on racture echanics in continuously reinorced iber coposite aterials" (2005). Graduate Theses and Dissertations. This Thesis is brought to you or ree and open access by the Graduate School at Scholar Coons. It has been accepted or inclusion in Graduate Theses and Dissertations by an authorized adinistrator o Scholar Coons. For ore inoration, please contact scholarcoons@us.edu.

2 ect o Fiber Volue Fraction on Fracture Mechanics in Continuously Reinorced Fiber Coposite Materials by Thoas Wasik A thesis subitted in partial ulillent o the requireents or the degree o Master o Science in Mechanical ngineering Departent o Mechanical ngineering College o ngineering University o South Florida Major Proessor: Autar K. Kaw, Ph.D. Glen H. Besterield, Ph.D. Thoas ason, Ph.D. Date o Approval: March 25, 2005 Keywords: Finite leent, Crack Propagation, Coposite Interace, Ansys, Interace Failure Modes Copyright 2005, Thoas Wasik

3 TABL OF CONTNTS LIST OF TABLS...iii LIST OF FIGURS... iv LIST OF SYMBOLS... viii ABSTRACT... x CHAPTR INTRODUCTION.... Overview....2 Literature Review... 3 CHAPTR 2 FINIT LMNT MODL DSIGN Geoetry and Boundary Conditions Fundaental quations Isotropic-Fiber, Isotropic-Matrix Transversely Isotropic Fiber, Isotropic Matrix... 6 CHAPTR 3 FINIT LMNT MODL VALIDATION... 8 CHAPTR 4 FINIT LMNT MODL ANALYSIS Fiber-Volue Fraction Criterion Orthotropic Fiber Criterion Theral Stress Criterion CHAPTR 5 RSULTS AND DISCUSSION Fiber-Volue Fraction lastic Moduli Ratio = lastic Moduli Ratio = i

4 5..3 lastic Moduli Ratio = lastic Moduli Ratio = Silicon Carbide/poxy Coposite Fiber Orthotropy Theral Stress CHAPTR 6 CONCLUSIONS RFRNCS APPNDICS Appendix : Ansys Input File Appendix 2: Maple Instructions Appendix 3: Mathcad File ii

5 LIST OF TABLS Table : Percentage rror Values o Five Nodal Locations Table 2: Table 3: Table 4: Table 5: Table 6: Table 7: Material Properties o Constituents in the Fiber-to-Matrix Moduli Ratio Analysis Material Properties o Fiber and Matrix in Silicon Carbide/poxy Coposite Displaceents and Interace Strengths Used in Silicone Carbide/poxy Analysis. 25 Material Properties o Orthotropic Fiber and Isotropic Matrix in Graphite/poxy Material Properties o Fiber and Matrix in Graphite/poxy Coposite Material Properties o Graphite/poxy Used or Theral Stress Analysis iii

6 LIST OF FIGURS Figure : Modes o Failure o Unidirectional Laina Under a Longitudinal Tensile Load..2 Figure 2: The Representative Volue leent... 6 Figure 3: Scheatic Representation o Finite leent Model... 8 Figure 4: Plane 2 leent... 9 Figure 5: Deored and Undeored Shapes o Finite leent Model... Figure 6: The Crack Tip in the Finite leent Model... 2 Figure 7: Cross-Sections o Coposites with Hexagonal and Rando Fiber Arrangeent... 3 Figure 8: Principle o Superposition... 9 Figure 9: Stress Ratios o ( zz ) ax in the Matrix as Function o Noralized Crack Length, a r Figure 0: Figure : The Stress Ratio, Length, r Stress Ratio, r ( rr ) ax as a Function o Noralized Crack a or lastic Moduli Ratio o =.. 32 ( rz ) ax as a Function o Noralized Crack Length, a or lastic Moduli Ratio o = iv

7 Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: The Stress Ratio, Length, r Stress Ratio, r ( rr ) ax as a Function o Noralized Crack a or lastic Moduli Ratio o = ( rz ) ax as a Function o Noralized Crack Length, a or lastic Moduli Ratio o = The Stress Ratio, Length, r Stress Ratio, r ( rr ) ax as a Function o Noralized Crack a or lastic Moduli Ratio o = ( rz ) ax as a Function o Noralized Crack Length, a or lastic Moduli Ratio o = The Stress Ratio, Length, r Stress Ratio, r ( rr ) ax as a Function o Noralized Crack a or lastic Moduli Ratio o = ( rz ) ax as a Function o Noralized Crack Length, a or lastic Moduli Ratio o = Figure 8: The Inluence o Fiber-Matrix-Moduli Ratio and Fiber-Volue ( ) Fraction on Tensile Stress Ratio rr ax v

8 Figure 9: The Inluence o Fiber-Matrix Modulus Ratio and Fiber-Volue ( ) Fraction on the Shear Stress Ratio rz ax... 4 Figure 20: Stress Ratio, ( rr ) ax as a Function o Noralized Crack Length, a or Silicon Carbide/poxy at 00% Displaceent r Figure 2: Stress Ratio, ( rz ) ax as a Function o Noralized Crack Length, a or Silicon Carbide/poxy at 00% Displaceent r Figure 22: Stress Ratio, ( rr ) ax as a Function o Noralized Crack Length, a or Silicon Carbide/poxy at 00% and 50% Displaceents r Figure 23: Stress Ratio, ( rz ) ax as a Function o Noralized Crack Length, a or Silicon Carbide/poxy at 00% and 50% Displaceents.. 45 r Figure 24: Stress Ratio, a r ( rr ) ax as a Function o Noralized Crack Length, or Graphite/poxy Coposite with Orthotropic and Isotropic Fibers vi

9 Figure 25: Stress Ratio, a r ( rz ) ax as a Function o Noralized Crack Length, or Graphite/poxy Coposite with Orthotropic and Isotropic Fibers Figure 26: Stress Ratio, a r ( rr ) ax as a Function o Noralized Crack Length, or Graphite/poxy Coposite With and Without Theral Load Present Figure 27: Stress Ratio, a r ( rz ) ax as a Function o Noralized Crack Length, or Graphite/poxy Coposite With and Without Theral Load Present... 5 vii

10 LIST OF SYMBOLS Modulus o lasticity G, µ Modulus o Rigidity α V ν K I u ε Coeicient o Theral xpansion Fiber Volue Fraction Poisson s ratio Stress Intensity Factor Displaceent Strain, τ Stress a r k Γ p J 0 b Crack Length Radius Plane-Strain Bulk Modulus Maxiu Principle Stress nergy Release Rate Pressure Bessel Function o zero order Width o Fiber and Matrix Subscripts Matrix Fiber r, z, θ Cylindrical Coordinates d Delection p Penetration viii

11 c ic in t ib at Mode I Toughness Toughness o Interace Interace Fiber Matrix ix

12 FFCT OF FIBR VOLUM FRACTION ON FRACTUR MCHANICS IN CONTINUOUSLY RINFORCD FIBR COMPOSIT MATRIALS Thoas Wasik ABSTRACT The application o advanced coposite aterials, such as graphite/epoxy, has been on the rise or the last our decades. The echanical advantages, such as their higher speciic stiness and strength as copared to onolithic aterials, ake the attractive or aerospace and autootive applications. Despite these advantages, coposites with brittle ibers have lower ductility and racture toughness than onolithic aterials. One way to increase the racture toughness o coposites is to have a weak iber-atrix interace that would blunt crack tips by crack delection into the interace and hence enhance racture toughness. However, this also reduces the transverse properties o the coposite. Thereore, an optiu iber-atrix interace would be the one that is just weak enough to cause crack delection into interace. This study investigates the eect o iber-to-atrix oduli ratio, ibervolue raction, iber orthotropy, and theral stresses on the possibility o crack x

13 delection. A inite eleent odel is used to analyze a 2-D axisyetric representative volue eleent- a three-phase coposite cylinder ade o iber, atrix, and coposite. A penny shaped crack is assued in the iber. To deterine whether the crack would delect into the interace or propagate into the atrix, axiu stresses at the iber-atrix interace and in the atrix are copared to the interace and atrix strengths. As opposed to ost studies in the literature, this study ound that ibervolue ractions do have an ipact on crack delection and this ipact increases with large iber-to-atrix oduli ratios. The presence o orthotropic iber in the coposite increases the possibility o crack delection with increasing ibervolue raction in the early and iddle stages o the iber crack growth. The theral stresses decrease the likelihood o crack delection when the theral expansion coeicient o the atrix is larger than that o the iber. xi

14 CHAPTR INTRODUCTION. Overview The use o coposite aterials has been steadily increasing in the past several decades. The high strength, high stiness and lightweight ake the particularly attractive to designers in a variety o industries. The coposite aterial consists o a atrix and one o the reinorcing phases such as particulate, lake and iber. In the continuous iber coposites, due to its large surace area, the iber-atrix interace inluences the behavior o a coposite. In addition to providing a echanis to transer loads ro atrix to ibers, the interace also plays an iportant role in deterining the coposite toughness. In spite o any advantages, the coposite aterials suer ro lower ductility and toughness when copared to coonly used etals. A unidirectional coposite with brittle ibers and a crack propagating perpendicular to the ibers can ail in at least three odes under longitudinal tensile load. These odes are:(a) brittle ailure, (b) brittle ailure with iber pullout, (c) brittle ailure with iber pullout and interace shear ailure or interace tensile ailure []. This is illustrated in Figure. The tensile or shear interace ailure is a prerequisite or phenoena such as crack delection into the interace, crack bridging by ibers, and iber pullout [2].

15 All o these are energy-dissipating phenoena during crack propagation process and help enhance toughness o the iber-reinorced coposites. Figure : Modes o Failure o Unidirectional Laina Under a Longitudinal Tensile Load (a) (b) (c) By controlling the strength o the interace bond between atrix and the iber, the designer is able to inluence the echanical properties o the coposite. To take a ull advantage o the iber properties and to obtain high strength and high stiness coposite, a strong iber-atrix bond is very desirable. Moreover, a strong interace bond results in high shear strength o the coposite and an eective load transer to the ibers under longitudinal tensile load. However, a strong interace bond will signiicantly decrease the ability o the iber to debond ro atrix during racture process and lowering the coposite toughness. This ability is very beneicial especially in brittle iber 2

16 coposites because the debonding process can act as crack arrestor and prevent urther propagation o the crack. This study presents an axisyetric inite eleent analysis o a pennyshaped crack in a brittle iber approaching a iber-atrix interace. The ain goal o this study was to deterine the inluence o iber-volue raction or various iber-to-atrix elastic oduli ratios on possibility o the interace ailure either in shear or in tension. Furtherore, the inluence o residual stresses and the iber orthotropy were also exained. The residual stresses arise ro the theral expansion isatch between iber and atrix as the coposite is cooled down ater processing..2 Literature Review The iber-volue raction is one o the paraeters eployed in analyzing coposites. There have been several odels developed to address the ailure o the coposites as unction o this paraeter. These odels are: iber cracks in dilute iber-volue raction coposites by Gupta [3], periodic cracks in higher iber volue raction coposites by rdogan and Bakioglu [4], and nonhoogenous interaces and nondilute iber-volue ractions by Bechel and Kaw [5]. In addition, a nuber o criteria have been presented in the past by various authors in order to explore the phenoenon o crack delection at the iber-atrix interace. He and Hutchinson [6] exained the tendency o the transverse crack ipinging on the interace joining two dissiilar aterials to penetrate the interace or to delect into the interace. The aterials on either 3

17 side o the interace are elastic and isotropic. They presented criteria that copared the energy release rate or the delected crack to the axiu energy release rate or a penetrating crack Γ d. This result can be copared to ratio o Γ p the toughness o the interace to the ode I toughness o uncracked aterial Γ ic. The ipinging crack is ost likely to be delected into the interace i Γ c Γ Γ ic c Γ < Γ d p () because the condition or propagation into the interace will be et at a lower load than that or penetration across the interace. The crack will tend to penetrate the interace when the inequality is reversed. Swenson and Rau [7] studied the plain strain proble o a crack terinating perpendicular to the interace between two isotropic hal spaces with dierent elastic constants. They concluded that the probability o an interace ailure in shear or in tension is very highly inluenced by odulus ratio o the two isotropic hal spaces. A crack in the stier aterial will likely cause the interace to ail in shear, whereas the crack in soter aterial will lead to tensile splitting o the interace. Cornie et al. [8] cae up with the criteria that addressed the iber-atrix debonding. The debonding can be expressed in ters o cohesive strength o the interace, shear strength o the interace, and iber racture stress. They ound that i the ratio o the interace cohesive strength (noral or shear) to the 4

18 iber strength is less than the ratio o the noral (or shear) stress at the interace to axial stress at the crack tip the tendency o the crack to delect along the interace is higher. Pagano [9] investigated the transverse atrix crack ipinging on the iberatrix interace in a brittle atrix coposite. In this study, he constructed general aterial design curves or iber penetration and interace debonding or ultiple iber-to-atrix ratios. These curves allow a coparison between potential energy release rate and a aterial toughness value to ake initial assessent o the success o ailure o a coposite ade ro a particular cobination o aterials. 5

19 CHAPTR 2 FINIT LMNT MODL DSIGN 2. Geoetry and Boundary Conditions The analysis o a penny-shape crack located in a brittle iber was perored using the inite eleent sotware package ANSYS 8.0. To siulate a racture behavior o the cracked iber and the resulting stresses, a representative volue eleent (RV) consisting o a single iber surrounded by cylindrical tubes o atrix and coposite, respectively, was used as illustrated in Figure 2. The RV is considered to represent the coposite and to respond in the sae way as the whole coposite [0]. z r MATRIX FIBR INTRFAC COMPOSIT CRACK Figure 2: The Representative Volue leent 6

20 The inite eleent odel was designed as 2-D axisyetric structure in the r-z plane. The use o an axisyetric odel greatly reduced the odeling and analysis tie copared to that o an equivalent 3-D odel. The geoetry and boundary conditions o the inite eleent odel are scheatically represented in Figure 3. Due to syetry in the geoetry and the boundary conditions the inite eleent calculations were perored on the right upper quadrant o the representative volue unit shown in Figure 2. The boundary conditions or the inite eleent odel were taken as:. at z = 0 a) u z =0 or a r W 2. at z = L b) rz =0, zz =0 or 0 < r < a a) u z=prescribed unior displaceent, 0 r W b) rz =0, 0 r W 3. at r = 0 a) u r =0 or 0 z L b) rz =0, 0 z L 4. at r = W a) rz =0, 0 z L b) rr =0, 0 z L 7

21 r Figure 3: Scheatic Representation o Finite leent Model 8

22 Also, the ode o deoration is axisyetric so the non-zero stress and displaceent coponents depend only on r and z and are independent o θ. The 6-node triangular eleent (Plane 2) with a quadratic displaceent behavior that was used or all the analyses perored in this study is shown in Figure 4. The diensions o the inite eleent odel were 0 units wide and 30 units high and were kept constant throughout the entire study. The inite eleent sotware used to carry out the inite eleent coputations in this study supported only a liited nuber o nodes (28,000). Figure 4: Plane2 leent Consequently, the odel was subdivided into ive separate areas to allow greater concentration o eleents in the regions in which the stress gradient was expected to be high, such as the crack tip and iber-atrix interace (Area I and Area II). The reaining areas had signiicantly lower concentrations o eleents. On average, there were 20,000 nodes and 60,000 eleents in each odel. 9

23 The iber o unit radius ( r ) is coprised o Area I and Area III. Also, the Area I contains crack o radius a. The radius b o the two concentric cylinders representing iber and atrix was calculated based on the iber-volue raction given by 2 2 r V = (2) b The three iber-volue ractions used in the analysis were: 0.25, 0.50, 0.75 and the corresponding b values were: 2,.44 and.55, respectively. The iber containing the penny-shaped crack is parallel to the longitudinal axis (z axis) and the crack plane z=0 is oriented perpendicular to that axis. The iber-atrix interace was odeled as perectly bonded. Furtherore, the coposite was subjected to unior and constant longitudinal tensile strain in the positive z direction and thereore was displaceent controlled. As a consequence, the crack experiences Mode I loading. Figure 5 shows the shapes o deored and undeored inite eleent odel. The Linear lastic Fracture Mechanics (LFM) approach was used as a eans to obtain stress ield caused by the presence o the crack. This approach was justiied due to the brittle nature o the iber. Because the stresses are singular in the region iediately surrounding the crack tip and vary as, d where d is the distance ro the crack tip, the triangular quadratic eleents were eployed with their idside nodes shited by a quarter toward the crack tip. 0

24 The eleents were arranged in seicircle around the crack tip, one eleent every 30 degrees. Figure 6 illustrates the eleent arrangeent in the crack tip vicinity. Figure 5: Deored and Undeored Shapes o Finite leent Model

25 Second Row o leents Crack Tip First Row o leents Figure 6: The Crack Tip in the Finite leent Model 2.2 Fundaental quations The ajority o a unidirectional iber-reinorced coposites are classiied either as an orthotropic or transversely isotropic aterials. This classiication is based on the geoetric iber arrangeent in the atrix. A unidirectional iberreinorced coposite with ibers arranged in hexagonal or rando anner in the plane perpendicular to the ibers axes, as shown in Figure 7, is considered to be transversely isotropic. 2

26 HXAGONAL RANDOM Figure 7: Cross-Sections o Coposites with Hexagonal and Rando Fiber Arrangeent The transversely isotropic aterial requires only ive engineering constants to ully describe its elastic behavior. The engineering constants are: zz, rr, ν zr, G zr, G rθ. By considering ibers to be along z-axis in the cylindrical coordinate syste, then the r-θ plane becoes isotropic and there is no preerred direction in that plane. The ollowing subsections list equations [] that were used to calculate engineering constants needed to describe coposite aterial. The equations are part o Input Files written or inite eleent sotware. The saple o an Input File is located in Appendix. 3

27 Isotropic Fiber, Isotropic Matrix The ollowing are the equations used or calculating aterial properties o the coposite consisting o isotropic iber and isotropic atrix..lastic Moduli a. Longitudinal ( ) = zz G k V k V V V V V 4 2 ν ν (3) where, V is the iber volue raction V is the atrix volue raction is the elastic odulus o iber is the elastic odulus o atrix G is the shear odulus o atrix ν is the Poisson s ratio o iber ν is the Poisson s ratio o atrix k is the plane-strain bulk odulus o iber k is the plane-strain bulk odulus o atrix b. Transverse = = rr V θθ (4)

28 5 2. Poisson s Ratios a. ( ) = = z zr G k V k V k k V V V V ν ν ν ν ν ν θ (5) b. + = k zz zr rr r ν ν θ (6) where, k the is plane-strain bulk odulus 3. Shear Moduli a. ( ) ( ) = = z zr V G V G V G V G G G G θ (7) where, G is the shear odulus o iber b. ( ) θ θ ν r rr r G = + 2 (8) where, ( ) G = +ν 2 (9) ( ) G = +ν 2 (0)

29 4. Bulk Modulus where, ( k + G ) + V G ( k k ) k + G V ( k k ) k2 = () k = ( 2ν )( +ν ) 2 (2) k = ( 2ν )( +ν ) 2 (3) Transversely Isotropic Fiber, Isotropic Matrix The ollowing are the equations used or calculating aterial properties o the coposite consisting o transversely isotropic iber and isotropic atrix..lastic Moduli a. Longitudinal zz = V + V (4) b. Transverse rr = θθ = (5) V Shear Moduli a. G zr G = Gz θ = (6) G V G zr 6

30 b. G rθ = V G G G r θ (7) 3. Poisson s Ratios a. vzr = vz θ = v V + vv zr (8) rr b. v rθ = (9) 2G rθ 4. Coeicients o Theral xpansion a. Longitudinal α V + α V zz zz α zz = (20) V + V zz α = V α + ν + V α + ν V ν + V ν α (2) b. Transverse rr ( ) ( ) ( ) zz rr zr zr where, α is the coeicient o theral expansion o iber α is the coeicient o theral expansion o atrix 7

31 CHAPTR 3 FINIT LMNT MODL VALIDATION Solving the sae test proble with analytical and inite eleent ethod assessed the accuracy o the inite eleent odel. The values o u z, u r,, zz rz, and θθ obtained ro the analysis o the inite eleent odel at chosen locations (r, z) were copared to the values obtained at the sae locations by using analytical analysis o the sae odel. Linear elastic and isotropic aterial behavior was assued or the inite eleent and analytical rr, 6 odel (= 30 0 and ν =0.3). Moreover, the assuption o perect iber-atrix interace was ade. The stress ield in the analytical ethod was deterined by superposition o two boundary value probles, one hosting a crack, the other being uncracked as illustrated in Figure 8. In Figure 8a, an uncracked cylinder is subjected to a unior boundary traction p in the z direction. This created the ollowing stresses: =p, zz rr =0, θθ =0, and rz =0. Because o the isotropic aterial assuption, the Hooke s law in cylindrical coordinates was used as a basis or displaceents derivation in r and z directions, respectively. 8

32 Figure 8: Principle o Superposition ε rr = rr zz + [ v( )] θθ (22) and because p, = 0, = 0 zz = ε rr sipliies to Thereore, rr θθ vp ε rr = (23) u r = ε r (24) rr Siilarly, u = ε z zz (25) z 9

33 or u z, u z, Figure 8b shows traction p applied on the crack ace. To obtain the values, rr, zz rz, and θθ, a syste o dierential equations [2] was solved using a code written in Maple 9.0 (see Appendix 2). Due to the syetry, the proble can be reduced to the hal space ( z >, 0 r < ) conditions on the z = 0 plane: rz ( r, 0,0) = 0 r 0 (r=cylinder radius) zz ( r,0,0 ) = p 0 r < a (a=hal crack length) ( r, 0,0) = 0 u z r > a 0 with the ollowing A single potential unction ( r, z) was eployed which autoatically rees plane z = 0 ro shear stress rz. The displaceent and stress coponents are then written in ters o that unction: u r u z 2 = ( 2ν ) + z r r z (26) 2 = 2( ν ) + z 2 z z (27) ( 2ν ) 2 + z 2 2 rr = 2µ ν (28) 2 r z r z where, µ is the shear odulus o elasticity ν is the Poisson s ratio 20

34 2 2 z θθ = 2µ + 2ν + (29) 2 r r r r r z 2 3 = zz 2µ + z 2 (30) z z 3 3 rz = 2µ z (3) 2 r z By using Fourier-Hankel transor, the unction ( r, z) o two variables is expressed in ters o the unction A (s), which depends only on the variable s. The unction A (s) is ound by solving the ollowing dual integral: A πµ () s = sin( st) a dt t rp ( r) dr 2 2 ( t r ) (32) where, p is the noral traction The solution o the above equation is inserted into the equation or ( r, z). A( s) ( sz) ds ( r, z) = J0 ( rs) e (33) 0 s where, ( r, z) is the potential unction o two variables J 0 is the Bessel unction o order zero This, in turn, enables us to ind the two displaceents and our stress coponents. The inal values o the analytical analysis are obtained by adding results ro part a and b as shown in Figure 8c. This was done using Mathcad 8.0 (Appendix 3). 2

35 The inite eleent odel was constructed as described in chapter 2. In order to achieve desired accuracy o results, the convergence o a inite eleent solution was conducted. The purpose o the convergence study was to reine the esh size so that the relative error between analytical and inite eleent solutions was less than one percent. Table lists the percentages o relative errors at dierent locations along the interace and in the atrix o the coposite. The highest error was 0. percent. Table : Percentage rror Values o Five Nodal Locations Nodal Coordinates u r u z rr θθ zz rz r= z= r= z= r=.0329 z=0.038 r= z= r=.0307 z=

36 CHAPTR 4 FINIT LMNT MODL ANALYSIS The inite eleent odel analysis was divided into several separate parts. ach part o the analysis investigated the inluence o a single criterion on the possibility o the interace ailure. The ollowing are the criteria used in the analysis:. Fiber-Volue Fraction (FVF) 2. Fiber Orthotropy 3. Theral Stress For each criterion, the noralized length o the iber crack, a, was progressively r increased ro 0.6 to Furtherore, each criterion was analyzed at 0.25, 0.50, and 0.75 iber-volue ractions. 4. Fiber-Volue Fraction Criterion The irst part o the FVF analysis ocused on how the iber-volue raction aects the interace tensile and shear ailure or dierent iber-to-atrix elastic oduli ratios. There were our oduli ratios used in the analysis as ollows: 23

37 . = represents coposite with iber and atrix ade o the sae aterial =6 represents typical ceraic atrix coposite =20 represents typical polyer atrix coposite such as glass/epoxy =80 represents typical polyer atrix coposite such as graphite/epoxy ach oduli ratio was analyzed at 0.25, 0.50 and 0.75 iber-volue ractions. The iber and atrix were assued to be linear elastic and isotropic with the sae Poisson s ratios. Table 2 lists iber and atrix properties used in the analysis. Table 2: Material Properties o Constituents in the Fiber-to-Matrix Moduli Ratio Analysis PROPRTY SYMBOL FIBR MATRIX Modulus o lasticity, 6, 20, 80 Poisson s Ratio ν The displaceent u z was taken as 0., which constitutes 0% o the iber radius. The second part o the FVF analysis involved exaining the inluence o iber-volue raction and two dierent longitudinal displaceents on interace ailure in silicon carbide/epoxy coposite. The two displaceents used in the 24

38 above analysis were calculated based on the iber ultiate tensile strength. The irst displaceent was obtained by calculating displaceent value needed to create tensile stress in the iber equal to the ultiate tensile strength o that iber, henceorth called 00% displaceent. The second displaceent was taken as a hal o the irst one, henceorth called 50% displaceent. Table 3 lists the properties o iber and atrix or silicon carbide/epoxy coposite [3,4]. Table 4 contains displaceents and interace strengths used in peroring the analysis o silicon carbide/epoxy coposite [5]. Table 3: Material Properties o Fiber and Matrix in Silicon Carbide/poxy Coposite PROPRTY SYMBOL FIBR MATRIX lastic Modulus 400 [GPa] 3.44 [GPa] Poisson s Ratio ν Fiber Ultiate Tensile Strength Matrix Ultiate Tensile Strength Coeicient o Theral xpansion ib 3450 [MPa] at [MPa] o α 0 60 [ µ C ] Table 4: Displaceents and Interace Strengths Used in Silicone Carbide/poxy Analysis 00 % Displaceent applied u z 50 % Displaceent applied u z Interace Noral Strength Interace Shear Strength int 35 [MPa] τ int 32.5 [MPa] 25

39 4.2 Orthotropic Fiber Criterion To deterine the inluence o the orthotropic iber on the interace ailure, two graphite/epoxy coposites were used or analysis with the iber and atrix properties listed in Tables 5 and 6 [6]. For coparison purposes, isotropic and orthotropic ibers were used. The analysis was perored or 0.25, 0.50, and 0.75 iber-volue ractions. The applied displaceent u z was calculated based on ultiate tensile strength o the graphite iber. The applied displaceent, u z, in both cases was Table 5: Material Properties o Orthotropic Fiber and Isotropic Matrix in Graphite/poxy PROPRTY SYMBOL FIBR MATRIX Longitudinal lastic Modulus Transverse lastic Modulus Shear Modulus zz 260 [GPa] 3.5 [GPa] rr, θθ 4 [GPa] 3.5 [GPa] G zr, G z θ [GPa] Shear Modulus G rθ 8.27 [GPa] Poisson s Ratio Poisson s Ratio Ultiate Tensile Strength ν zr, ν zθ ν r θ ib 4038 [MPa] Table 6: Material Properties o Isotropic Fiber and Matrix in Graphite/poxy Coposite PROPRTY SYMBOL FIBR MATRIX lastic Modulus 260 [GPa] 3.5 [GPa] Poisson s Ratio ν Ultiate Tensile Strength ib 4038 [MPa] 26

40 4.3 Theral Stress Criterion The theral stresses are created as a result o a isatch in theral expansion coeicients o iber and atrix. The graphite/epoxy coposite having dierent iber and atrix theral expansion coeicients was analyzed. The obtained results were then copared to the results or the sae coposite analyzed without theral expansion coeicients. ach coposite was analyzed at 0.25, 0.50, and 0.75 iber-volue ractions. The displaceent uz was calculated based on ultiate tensile strength o a graphite iber. Table 7 lists aterial properties or iber and atrix in the theral stress analysis o graphite/epoxy coposite. The applied displaceent, u z, was Table 7: Material Properties o Graphite/poxy Used or Theral Stress Analysis PROPRTY SYMBOL FIBR MATRIX Longitudinal lastic Modulus Transverse lastic Modulus zz 260 [GPa] 3.5 [GPa] rr, θθ 4 [GPa] 3.5 [GPa] Shear Modulus G zr, G z θ [GPa] Shear Modulus G rθ 8.27 [GPa] Poisson s Ratio Poisson s Ratio Coeicient o Theral xpansion Coeicient o Theral xpansion Ultiate Tensile Strength ν zr, ν zθ ν r θ o o α [ µ C ] 90[ µ C ] zz o o α rr,αθθ 3.24 [ µ C ] 90[ µ C ] ib 4038 [MPa] 27

41 In order to deterine how each criterion plays a role in inluencing the possibility o shear and tensile interace ailure, the ollowing stress ratios were calculated: rr ( ax) ( ax) and rz or each crack length. Those ratios were then plotted as a unction o noralized crack length, a, or the three iber-volue r ractions. The preceding ratios provide us with the qualitative eans to deterine and to copare the inluence o dierent paraeters on the two types o the interace ailures. This coparison is not only possible between dierent iber-volue ractions o the sae coposite, but also between coposites with various elastic oduli ratios. The rr( ax), and rz( ax) stresses represent the axiu tensile and shear stresses along iber-atrix interace. In turn, the represents the largest principal stress present in the atrix. The choice o using principal stress instead o zz( ax) in the above ratios was ade based on the act that principal stress was increasing at a higher rate than zz( ax) as the crack was approaching iber-atrix interace. To illustrate the dierence in values between and ( zz ) ax with increasing crack length, the stress ratios, ( zz ) ax were calculated or 0.50 and 0.75 iber-volue ractions and plotted as a unction o noralized crack length. Figure 9 clearly shows that the stress ratios, ( zz ) ax, or both iber-volue ractions are higher than one when the crack is close to the iber-atrix interace. 28

42 /( zz ) ax 80: (0.75 FVF) /( zz ) ax 80: (0.50 FVF).60 Stress Ratio, /( zz ) ax, ( zz ) ax / [non-di] Noralized Crack Length, a/r [non-di] Figure 9: Stress Ratios o Length, a r ( zz ) ax in the Matrix as Function o Noralized Crack 29

43 CHAPTR 5 RSULTS AND DISCUSSION As entioned in the previous chapters, in this study, we want to assess how the iber-volue raction, iber orthotropy, and theral stresses inluence the crack propagation path o a cracked iber does the crack propagate across the interace to the atrix, or does the crack propagate along the interace. We understand that debonding o iber-atrix interace causes the blunting o the crack tip, acts as a crack arrestor, and hence contributes to the overall increase in coposite toughness. This can be easily accoplished by aking a weak iber-interace, but such weak interaces decrease transverse copressive and shear strength. Hence, to be able to quantiy to build a iber-atrix interace that is just weak enough to allow interace debonding requires us to ully understand the echaniss o crack propagation. The stress ratios used to understand propagation paths o a iber crack were described in chapter 4 and are used to deine the conditions necessary or debonding o iber-atrix interace in the iber reinorced coposite subjected to longitudinal tensile strain. The debonding at the interace will occur i:. The ratio o the axiu tensile stress at the interace, ) ax, to the largest principal stress in the atrix, is greater than the ratio o the interace noral strength, int 30 ( rr, to the ultiate atrix strength, at, that is,

44 ( rr ) ax int > (34) at 2. The ratio o the axiu shear stress at the interace, ( rz ) ax, to the largest principal stress [8],, is greater than the ratio o the interace shear strength, τ int, to the ultiate atrix strength, at, that is, ( rz ) ax τ int > (35) at For a speciic coposite, the two strength ratios int τ, and int at at are aterial properties o a particular iber and atrix cobination. These strength ratios are not dependent on iber-volue raction. In contrast, the two stress ratios ( rr ) ax ( ) and rz ax on the let side o the inequalities (quations 34 and 35) are inluenced by several variables such as: crack length, iber-volue raction, iber-to-atrix elastic oduli ratio, iber orthotropy, and theral stresses. The presentation and discussion o the results is divided into three separate parts to study the inluence o iber-volue raction, iber orthotropy and theral stress. 3

45 5. Fiber-Volue Fraction 5.. lastic Moduli Ratio = When a coposite is ade o iber and atrix that have identical elastic oduli, all iber-volue ractions represent the sae geoetry o a iber surrounded by a atrix o ininite radius. So the noralized stress ratios, ( rr ) ax ( ) and rz ax a as a unction o noralized crack length, are the sae r or all iber-volue ractions as given in Figures 0 and. Note the single nuber given or the noralized crack length o unity. Stress Ratio, ( rr ) ax / [non-di] FVF 0.50 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interace or all FVFs Noralized Crack Length, a/r [non-di] Figure 0: The Stress Ratio, a or lastic Moduli Ratio o = r rr ) ax as a Function o Noralized Crack Length, ( 32

46 0.40 Stress Ratio, ( rz ) ax / [non-di] FVF 0.25 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interace or all FVFs Noralized Crack Length, a/r [non-di] Figure : Stress Ratio, or lastic Moduli Ratio o = rz ) ax a as a Function o Noralized Crack Length, r ( 5..2 lastic Moduli Ratio = 6 Now let us exaine how the iber-volue raction aects the crack propagation path or coposites where the iber and atrix elastic oduli are not the sae. Figures 2 shows the noralized stress ratio, ( rr ) ax as a unction o noralized crack length. The trends or iber-volue ractions o up to 0.5 are 33

47 the sae, and only or large iber-volue ractions, the noralized stress, ( rr ) ax shows arkedly higher values. Figure 3 show the noralized stress ratios, ( rz ) ax, as a unction o noralized crack length. The trends or all iber-volue ractions look the sae Stress Ratio, ( rr ) ax / [non-di] FVF 0.50 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interace or all FVFs Noralized Crack Length, a/r [non-di] Figure 2: The Stress Ratio, a or lastic Moduli Ratio o = 6 r rr ) ax as a Function o Noralized Crack Length, ( 34

48 0.70 Stress Ratio, ( rz ) ax / [non-di] FVF 0.50 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interace or all FVFs Noralized Crack Length, a/r [non-di] Figure 3: Stress Ratio, or lastic Moduli Ratio o = 6 rz ) ax a as a Function o Noralized Crack Length, r ( 5..3 lastic Moduli Ratio = 20 Higher iber-to-atrix oduli ratios, like = 20 representing a typical glass/epoxy give results in a siilar behavior as the case o = 6 except the dierences between stress ratio values are ore pronounced. This is illustrated in Figures 4 and 5. 35

49 Stress Ratio, ( rr ) ax / [non-di] FVF 0.50 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interace or all FVFs Noralized Crack Length, a/r [non-di] Figure 4: The Stress Ratio, a or lastic Moduli Ratio o = 20 r rr ) ax as a Function o Noralized Crack Length, ( 36

50 0.90 Stress Ratio, ( rz ) ax / [non-di] FVF 0.50 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interace or all FVFs Noralized Crack Length, a/r [non-di] Figure 5: Stress Ratio, or lastic Moduli Ratio o = 20 rz ) ax a as a Function o Noralized Crack Length, r ( 5..4 lastic Moduli Ratio = 80 Higher iber-to-atrix oduli ratios, like = 80 representing a typical graphite/epoxy coposite, give results in a siilar behavior as the cases o 37

51 = 6 and = 20 except the dierences between stress ratio values are ore pronounced. This is illustrated in Figures 6 and Stress Ratio, ( rr ) ax / [non-di] FVF 0.50 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interace or all FVFs Noralized Crack Length, a/r [non-di] Figure 6: The Stress Ratio, a or lastic Moduli Ratio o = 80 r rr ) ax as a Function o Noralized Crack Length, ( 38

52 .00 Stress Ratio, ( rz ) ax / [non-di] FVF 0.50 FVF 0.75 FVF Stress Ratio at Fiber-Matrix Interace or all FVFs Noralized Crack Length, a/r [non-di] Figure 7: Stress Ratio, or lastic Moduli Ratio o = 80 rz ) ax a as a Function o Noralized Crack Length, r ( The preceding results clearly indicate a substantial inluence o ibervolue raction on the crack propagation path. The eect is zero or iber-to- atrix oduli ratio, = and becoes ore pronounced as the iber-to-atrix oduli ratio increases. For large iber-volue ractions, we see that the possibility o crack propagating along the interace increases, as was observed in experiental studies []. This is contrary to recent studies [6,7,8] where crack propagation paths are considered to be independent o iber-volue ractions. 39

53 Stress Ratio, ( rr ) ax / [non-di] FVF (6: Fiber-Matrix Ratio) 0.50 FVF (20: Fiber-Matrix Ratio) 0.50 FVF (80: Fiber-Matrix Ratio) 0.75 FVF (6: Fiber-Matrix Ratio) 0.75 FVF (20: Fiber-Matrix Ratio) 0.75 FVF (80: Fiber-Matrix Ratio) Noralized Crack Length, a/r [non-di] Figure 8: The Inluence o Fiber-Matrix-Moduli Ratio and Fiber-Volue Fraction on Tensile Stress Ratio rr ) ax ( While the eect o iber-volue raction on tensile interace ailure was rather straightorward, the sae cannot be said about the eect o iber-volue raction on interacial shear ailure. 40

54 Stress Ratio, ( rz ) ax / [non-di] FVF (6: Fiber-Matrix Ratio) 0.50 FVF (20: Fiber-Matrix Ratio) FVF (80: Fiber-Matrix Ratio) 0.75 FVF (6: Fiber-Matrix Ratio) 0.75 FVF (20: Fiber-Matrix Ratio) 0.75 FVF (80: Fiber-Matrix Ratio) Noralized Crack Length, a/r [non-di] Figure 9: The Inluence o Fiber-Matrix Modulus Ratio and Fiber-Volue Fraction on the Shear Stress Ratio rz ) ax ( 5..5 Silicon Carbide/poxy Coposite The purpose o analyzing a particular coposite syste is that we wanted to deterine crack propagation path under dierent reote loading values. We apply strain equal to and then hal o the ultiate longitudinal strain o the iber. The corresponding longitudinal displaceents were derived in chapter 4, and are called 00% and 50% displaceents, respectively. 4

55 Also, we know the ultiate shear and noral strength o the interace or this particular coposite syste (Table 4). Hence we cannot only ind whether the interace ails but also whether it ails due to shear or noral stress in the interace Stress Ratio, ( rr ) ax / [non-di] FVF 0.50 FVF 0.75 FVF 0.0 Interace Strength Ratio Stress Ratio at Fiber-Matrix Interace or all FVFs Noralized Crack Length, a/r [non-di] rr ) ax ( Figure 20: Stress Ratio, or Silicon Carbide/poxy at 00% Displaceent a as a Function o Noralized Crack Length, r The Figures 20 and 2 illustrate the behavior o a coposite that was subjected to a displaceent that created stress in the iber equivalent to ultiate strength o that iber. As it can be seen, the interace tensile ailure would take place at noralized crack lengths o 0.7 or 0.75 iber-volue raction and o 0.8 or 0.25 and 0.50 iber-volue ractions. Because the 42 ( rz ) ax stress ratio values

56 or all three iber-volue ractions are below the interace strength ratio value throughout the entire crack propagation process, the interace shear ailure would not take place..00 Stress Ratio, ( rz ) ax / [non-di] FVF 0.5 FVF 0.75 FVF Interace Strength Ratio Stress Ratio at Fiber-Matrix Interace or all FVFs Noralized Crack Length, a/r [non-di] rz ) ax ( Figure 2: Stress Ratio, or Silicon Carbide/poxy at 00% Displaceent a as a Function o Noralized Crack Length, r To exaine the ipact o dierent longitudinal displaceents on stress ratios and interace ailure ode, the results o two displaceents (00% and 50%) were plotted on the sae graph, Figures 22 and 23. The graphs clearly show that the reduction in longitudinal displaceent by 50% did not aect the stress ratio distribution and interace ailure in a signiicant way. 43

57 Stress Ratio, ( rr ) ax / [non-di] FVF (50% DISPL) 0.50 FVF (50% DISPL) 0.75 FVF (50% DISPL) 0.25 FVF (00% DISPL) 0.50 FVF (00% DISPL) 0.75 FVF (00% DISPL) Interace Strength Ratio rr ) ax Noralized Crack Length, a/r [non-di] a as a Function o Noralized Crack Length, r ( Figure 22: Stress Ratio, or Silicon Carbide/poxy at 00% and 50% Displaceents The interace tensile ailure or 0.75 iber-volue raction would initiate at 0.75 crack length and 0.25 and 0.50 iber-volue ractions at 0.85 crack length. The interace shear ailure as beore would not take place because the shear strength ratio is signiicantly larger than stress ratios present at the iber-atrix interace. 44

58 Ater running several additional analyses with other saller displaceents, it was ound that the shear strength ratio was always higher than the corresponding shear stress ratios. Moreover, the displaceents in a 4% to 00% range show that the crack propagation would be along the interace and would be caused by interace tensile ailure Stress Ratio, ( rz ) ax / [non-di] VFR (50% DISPL) 0.50 VFR (50% DISPL) 0.75 VFR (50% DISPL) VFR (00% DISPL) 0.50 VFR (00% DISPL) 0.75 VFR (00% DISPL) Interace Strength Ratio Noralized Crack Length, a/r [non-di] rz ) ax a as a Function o Noralized Crack Length, r ( Figure 23: Stress Ratio, or Silicon Carbide/poxy at 00% and 50% Displaceents 5.2 Fiber Orthotropy Up to this point, the coposites used in this study were assued to be ibers with isotropic aterial properties. This section exaines what ipact iber 45

59 orthotropy have on interace ailure. This is accoplished by coparing analysis results o the coposite with the orthotropic iber (transversely isotropic) to the analysis results o the coposite with isotropic iber. In both cases, graphite/epoxy coposite is used. As can be seen in Figure 24, the iber orthotropic aterial properties have a unique ipact on tensile interace ailure. The possibility o interace ailure in tension increases with increasing iber-volue raction between noralized crack lengths. During this crack growth, the coposite with the highest (0.75) iber-volue raction is ost likely to experience tensile interace ailure. However, when the noralized crack reaches 0.9, the possibility o interace ailure in tension or a coposite with orthotropic iber becoes copletely independent o iber-volue raction. That is, all three iber-volue ractions generate the sae tensile stress ratios. Also, at that point in crack growth, the coposite with isotropic iber and 0.75 iber-volue raction has the sae chance o experiencing interace tensile ailure as the coposite with orthotropic iber. 46

60 .0.00 Stress Ratio, ( rr ) ax / [non-di] FVF (ORTHOTROPIC) 0.50 FVF (ORTHOTROPIC) 0.75 FVF (ORTHOTROPIC) 0.25 FVF (ISOTROPIC) 0.50 FVF (ISOTROPIC) 0.75 FVF (ISOTROPIC) Noralized Crack Length, a/r [non-di] Figure 24: Stress Ratio, ( rr as a Function o Noralized Crack Length, ) ax or Graphite/poxy Coposite with Orthotropic and Isotropic Fibers a r Figure 25 illustrates the eect o orthotropic iber on interace ailure in shear. It can be clearly seen that the presence o orthotropic iber diinishes the possibility o interace shear ailure. As the crack propagates and approaches iber-atrix interace, the shear stress ratio get progressively saller. In act, when the noralized crack length reaches 0.97, the coposite with 0.25 ibervolue raction has the largest possibility to experience an interace ailure in shear. 47

61 Also, during the inal stage o crack growth, the coposites with isotropic ibers are ore prone to undergo an interace ailure in shear than those with orthotropic ibers Stress Ratio, ( rz ) ax / [non-di] FVF (ORTHOTROPIC) 0.50 FVF (ORTHOTROPIC) FVF (ORTHOTROPIC) 0.25 FVF (ISOTROPIC) 0.50 FVF (ISOTROPIC) 0.75 FVF (ISOTROPIC) Noralized Crack Length, a/r [non-di] ( ) Figure 25: Stress Ratio, rz ax a as a Function o Noralized Crack Length, r or Graphite/poxy Coposite with Orthotropic and Isotropic Fibers 5.3 Theral Stress The theral stresses in the coposite arise due to a isatch between theral expansion coeicients o a iber and a atrix. This isatch puts the 48

62 iber-atrix interace either in tension or in copression depending which constituent has larger coeicient o theral expansion. The theral strain and the corresponding theral stress were calculated based on the ollowing equation: where, ε = α( T ) (36) T ε T is the theral strain α is the coeicient o theral expansion, and T is the dierence between the abient and processing teperatures, T = ( T TRF ) The processing was taken to be T RF =70º C and the inal teperature was assued to be a roo teperature at T = 20º C. The resulting negative T indicates shrinkage o both coponents during the cooling process. To exaine the inluence o theral stress on interace ailure, two identical coposites, one in presence and other in absence o theral stresses, were analyzed and the results were copared. The coposite used in the analysis was graphite/epoxy with aterial properties listed in section 4.3. By looking at Figure 26, it can be concluded that theral stresses reduce the possibility o interace ailure in tension. Because the analyzed coposite had α > α, the resulting copressive stress noral to the interace akes debonding ro crack in iber less likely. 49

63 Stress Ratio, ( rr ) ax / [non-di] FVF (Theral Stress) FVF (Theral Stress) FVF (Theral Stress) FVF (w/o Theral Stress) 0.50 FVF (w/o Theral Stress) FVF (w/o Theral Stress) Noralized Crack Length, a/r [non-di] ( ) Figure 26: Stress Ratio, rr ax a as a Function o Noralized Crack Length, r or Graphite/poxy Coposite With and Without Theral Load Present The Figure 27 illustrates the ipact o theral stress on interace ailure in shear. For the coposite with theral stress present, the shear stress ratios are signiicantly lower during crack propagation between noralized crack lengths o 0.6 to 0.8 but or larger cracks, the dierences between the shear stress ratios aong the sae iber-volue ractions are alost negligible. 50

64 Stress Ratio, ( rz ) ax / [non-di] FVF (Theral Stress) FVF (Theral Stress) 0.75 FVF (Theral Stress) 0.25 FVF (w/o Theral Stress) FVF (w/o Theral Stress) 0.75 FVF (w/o Theral Stress) Noralized Crack Length, a/r [non-di] ( ) Figure 27: Stress Ratio, rz ax a as a Function o Noralized Crack Length, r or Graphite/poxy Coposite With and Without Theral Load Present 5

65 CHAPTR 6 CONCLUSIONS The conclusions gathered ro the results o this study can be suarized as ollows:. The iber-volue raction has a proound inluence on interace ailure a) The possibility o interace tensile ailure increases with higher iber-volue raction b) The interace tensile ailure is ore likely to occur or coposites with high iber-volue raction and high iber-to-atrix oduli ratio c) The increase o iber-volue raction ro ediu to high akes the interace shear ailure ore likely during initial crack growth but the possibility diinishes as the crack approaches the iber-to-atrix interace. 2. The interace in the silicon carbide/epoxy will never ail in shear regardless o the iber-volue raction and displaceent applied. 3. The tensile interace ailure in the silicon carbide/epoxy will take place between 4% and 00% o ultiate longitudinal strain. 4. In the early and iddle stages o crack growth, the presence o orthotropic iber in the graphite/epoxy coposite increases the likelihood o tensile interace ailure with increasing iber-volue raction. During the inal stage 52