Availableonlineatwww.sciencedirect.com e Science Direct Acta mATERIaLiA ELSEVIER Acta Materialia 56(2008)5783-5795 Fibrous composite with threshold strengths in multiple directions Design and fabrication X.H. Jin. L. gao. Sun. L.H. gui State Key Laboratory on High Performance Ceramics and Superfine Microstructures, of ceramics, Received 17 July 2008: accepted 3 August 2008 Available online 2 September 2008 Abstract This work reports the design of a composite consisting of square fibers separated by thin compressive layers. Fracture mechanics anal- sis indicated that this composite showed radial and axial threshold strengths corresponding to applied stresses in the direction perpen dicular to the fiber side face and parallel to the fiber central axis, respectively. In accordance with the above designing concept, Si3N fibrous composites with distinctive threshold strengths were prepared through a simple double -laminating procedure. The measured radial threshold strength of the Si3 N/TiN composite agreed quite well with the theoretical prediction, while a substantial discrepancy existed between the measured and predicted axial threshold strength due to the difference in the configuration of the cracks used in testing and theoretical modeling of the axial threshold strength o 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved Keywords: Residual stresses; Ceramic matrix composites; Mechanical properties: Nitride; Threshold strength 1. Introduction Recently, Lange and colleagues [9-12]found that a lam inar composite could be designed to possess threshold The strength of brittle material such as ceramics and strength by introducing a thin compressive layer between glass is highly sensitive to the flaw size and shows a statis- the adjacent thick layers, that is, a strength below which ical distribution. For this reason, a brittle material exhibits failure will not occur despite the presence of very large a high probability of catastrophic failure when it is under cracks. This discovery casts a new light on solving the reli- externally applied stress. Traditional ways of solving the ability problem of brittle material. It allows the engineer to problem are mostly to reduce the flaw sensitivity of design structural components with the knowledge that they strength by increasing the toughness and fracture work of will not fail below the threshold strength, as claimed by the material. And in accordance with the above thinking, Rao [9]. However, restricted by its layered structure, such various methods have been developed, such as the addition a laminar composite exhibits threshold strength only under of toughening agents [1, 2], self-toughening through micro- a load parallel to the layers. The strong dependence of structure tailoring [3, 4], crack arresting by surface com- threshold strength on load direction is undesirable if the pressive stress [5, 6]and laminar or fibrous architecture material is to be used under complex loading conditions design with weak interface [7, 8] etc. But low reliability Fair [13] attempted to reduce the sensitivity of threshold associated with poor toughness still poses an obstacle for strength to the load direction by three-dimensional architec the wide application of ceramics in structural fields, despite ture design of a polyhedral composite, but failed because of great improvement having been achieved in some cases the great difficulty in controlling the microstructure unifor- mity and periodicity. In the current work, a composite with 1359-6454$34.00@ 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:l0.1016 actant200808.010
Fibrous composite with threshold strengths in multiple directions: Design and fabrication X.H. Jin, L. Gao *, J. Sun, L.H. Gui State Key Laboratory on High Performance Ceramics and Superfine Microstructures, Shanghai Institute of Ceramics, Chinese Academy of Sciences, 1295 Ding Xi Road, Shanghai 200050, China Received 17 July 2008; accepted 3 August 2008 Available online 2 September 2008 Abstract This work reports the design of a composite consisting of square fibers separated by thin compressive layers. Fracture mechanics analysis indicated that this composite showed radial and axial threshold strengths corresponding to applied stresses in the direction perpendicular to the fiber side face and parallel to the fiber central axis, respectively. In accordance with the above designing concept, Si3N4/TiN fibrous composites with distinctive threshold strengths were prepared through a simple double-laminating procedure. The measured radial threshold strength of the Si3N4/TiN composite agreed quite well with the theoretical prediction, while a substantial discrepancy existed between the measured and predicted axial threshold strength due to the difference in the configuration of the cracks used in testing and theoretical modeling of the axial threshold strength. 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Residual stresses; Ceramic matrix composites; Mechanical properties; Nitride; Threshold strength 1. Introduction The strength of brittle material such as ceramics and glass is highly sensitive to the flaw size and shows a statistical distribution. For this reason, a brittle material exhibits a high probability of catastrophic failure when it is under externally applied stress. Traditional ways of solving the problem are mostly to reduce the flaw sensitivity of strength by increasing the toughness and fracture work of the material. And in accordance with the above thinking, various methods have been developed, such as the addition of toughening agents [1,2], self-toughening through microstructure tailoring [3,4], crack arresting by surface compressive stress [5,6] and laminar or fibrous architecture design with weak interface [7,8], etc. But low reliability associated with poor toughness still poses an obstacle for the wide application of ceramics in structural fields, despite great improvement having been achieved in some cases. Recently, Lange and colleagues [9–12] found that a laminar composite could be designed to possess threshold strength by introducing a thin compressive layer between the adjacent thick layers, that is, a strength below which failure will not occur despite the presence of very large cracks. This discovery casts a new light on solving the reliability problem of brittle material. It allows the engineer to design structural components with the knowledge that they will not fail below the threshold strength, as claimed by Rao [9]. However, restricted by its layered structure, such a laminar composite exhibits threshold strength only under a load parallel to the layers. The strong dependence of threshold strength on load direction is undesirable if the material is to be used under complex loading conditions. Fair [13] attempted to reduce the sensitivity of threshold strength to the load direction by three-dimensional architecture design of a polyhedral composite, but failed because of the great difficulty in controlling the microstructure uniformity and periodicity. In the current work, a composite with a fibrous architecture is proposed. This composite shows threshold strengths in multiple directions, and the load 1359-6454/$34.00 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2008.08.010 * Corresponding author. Tel.: +86 21 5241 2718; fax: +86 21 5241 3122. E-mail address: liangaoc@online.sh.cn (L. Gao). www.elsevier.com/locate/actamat Available online at www.sciencedirect.com Acta Materialia 56 (2008) 5783–5795
5784 X.H. Jin et al / Acta Materialia 56(2008)5783-5795 m+32+2 A△TE1Ed 2. Modeling of residual stresses in fibrous composite E2d(1+v1)+E1(1+v2) (E2d+2E1)[E2d+(1-v2)E1-(E2 Fig. I shows the structure model of the fibrous compos A△TE1Ed layer with a lower thermal expansion coefficient (CTE). Suppose that failure is controlled by the flaws within the fi- Owing to thermal mismatch, the fiber is under tensile stres- bers; then two threshold strengths can be expected in this ses of GzH in the direction perpendicular to the fiber side material due to the crack-arresting effect of the compressive face and azL in the direction parallel to the fiber central stresses in the thin layer. The first one corresponds to ap- axis, while the thermal stresses in the thin layer are a little plied stress perpendicular to the fiber side face, the second complicated. Apart from the compressive stresses of gh one corresponds to applied stress parallel to the fiber cen- and oIL, which are in the opposite direction to azH and tral axis, and they are denoted as radial and axial threshold G2L, it is under a tensile stress of gir in its thickness direc tion. When the thin layer thickness t is sufficiently small in comparison with the fiber diameter d or t/d-+0, OIR can 3. Fracture mechanics modeling of the composi be neglected, as found in a thin-walled pressure cylinder and testified by Fair in a fibrous Al, 0,/mullite composite 3.1. Radial threshold strength analysis [13, 14]. At this time, the following relations exist according to the linear elastic mechanics theory Fig. 2 shows the fracture mechanics models used to derive the radial threshold strength (i. e, Ihr). These models 0=vl⊥(1-v2)oH-v0=△△T E (la) are developed using the superimposition of stress intensity factors in a similar manner to that used to derive the threshold strength for a laminar composite [9]. In the G1L-v1G1.G2-2v22 △x△T E (1b) model shown in Fig 2a, a composite containing a slit crack 2a in length (d< 2a s d+ 2n) is considered. This crack gIh d (lc) spans the fiber in the direction perpendicular to the com- pressive thin layer with the crack plane aligning in the direction parallel to the fiber central axis. When a tensile (1d) stress o perpendicular to the fiber side face is applied on the composite, the crack will be under a stress state shown on the left-hand side of Fig. 2a. There, an and o2 represent where vi and v2 represent Poisson's ratio of the thin layer the actual stresses applied on the thin layer and fiber as a and the fiber, E1 and E2 are their Young's moduli, Az is result of the applied stress redistribution, which is caused which thermal stress builds up in the composite In accor. by the elastic modulus ditterence between the two structure dance with the above equations, the compressive stresses within the thin layer can be derived El(t +d) 01=F,t+E2d (3a) E This stress state can be well produced by the superimpose- tion of the two stress states shown on the right-hand side In the first stress state, the crack is subjected to a tensile stress of o1-G1H that is imposed on the whole materia in the second state, the same crack is subjected to a tensile stress of aIH O2H o2-oI that acts only over the pro- portion of the crack that spans the fiber core. Adding up the stress intensity factors for the two stress states, the total stress intensity factor Krd is given [15] IR Krd=(01-0IHVIa+(oIH+O2H +02-a1) Fig. 1. Schematic illustration showing the architecture and residual thermal stresses within the fibrous composite
direction sensitivity of threshold strength is greatly reduced in comparison with a laminar composite, which gives greater flexibility for reliable structural component design. 2. Modeling of residual stresses in fibrous composite Fig. 1 shows the structure model of the fibrous composite. It consists of square fibers separated by a thin uniform layer with a lower thermal expansion coefficient (CTE). Owing to thermal mismatch, the fiber is under tensile stresses of r2H in the direction perpendicular to the fiber side face and r2L in the direction parallel to the fiber central axis, while the thermal stresses in the thin layer are a little complicated. Apart from the compressive stresses of r1H and r1L, which are in the opposite direction to r2H and r2L, it is under a tensile stress of r1R in its thickness direction. When the thin layer thickness t is sufficiently small in comparison with the fiber diameter d or t/d ? 0, r1R can be neglected, as found in a thin-walled pressure cylinder and testified by Fair in a fibrous Al2O3/mullite composite [13,14]. At this time, the following relations exist according to the linear elastic mechanics theory: r1H m1r1L E1 þ ð1 m2Þr2H m2r2L E2 ¼ DaDT ð1aÞ r1L m1r1H E1 þ r2L 2m2r2H E2 ¼ DaDT ð1bÞ r1H r2H ¼ d t ð1cÞ r1L r2L ¼ d 2t ð1dÞ where m1 and m2 represent Poisson’s ratio of the thin layer and the fiber, E1 and E2 are their Young’s moduli, Da is the CTE difference, and DT is the temperature range at which thermal stress builds up in the composite. In accordance with the above equations, the compressive stresses within the thin layer can be derived: r1H ¼ E2dð1 þ m1Þ þ 2E1tð1 þ m2Þ ðE2d þ 2E1tÞ½E2d þ ð1 m2ÞE1tðE2dm1 þ 2E1tm2Þ 2 DaDTE1E2d ð2aÞ r1L ¼ E2dð1 þ m1Þ þ E1tð1 þ m2Þ ðE2d þ 2E1tÞ½E2d þ ð1 m2ÞE1tðE2dm1 þ 2E1tm2Þ 2 DaDTE1E2d ð2bÞ Suppose that failure is controlled by the flaws within the fi- bers; then two threshold strengths can be expected in this material due to the crack-arresting effect of the compressive stresses in the thin layer. The first one corresponds to applied stress perpendicular to the fiber side face, the second one corresponds to applied stress parallel to the fiber central axis, and they are denoted as radial and axial threshold strength, respectively. 3. Fracture mechanics modeling of the composite 3.1. Radial threshold strength analysis Fig. 2 shows the fracture mechanics models used to derive the radial threshold strength (i.e., rI thr). These models are developed using the superimposition of stress intensity factors in a similar manner to that used to derive the threshold strength for a laminar composite [9]. In the model shown in Fig. 2a, a composite containing a slit crack 2a in length (d 6 2a 6 d + 2t) is considered. This crack spans the fiber in the direction perpendicular to the compressive thin layer with the crack plane aligning in the direction parallel to the fiber central axis. When a tensile stress r perpendicular to the fiber side face is applied on the composite, the crack will be under a stress state shown on the left-hand side of Fig. 2a. There, r1 and r2 represent the actual stresses applied on the thin layer and fiber as a result of the applied stress redistribution, which is caused by the elastic modulus difference between the two structure elements. It is easy to prove that r1 ¼ E1ðt þ dÞ E1t þ E2d r ð3aÞ r2 ¼ E2 E1 r1 ð3bÞ This stress state can be well produced by the superimposition of the two stress states shown on the right-hand side. In the first stress state, the crack is subjected to a tensile stress of r1 r1H that is imposed on the whole material; in the second state, the same crack is subjected to a tensile stress of r1H + r2H + r2 – r1 that acts only over the proportion of the crack that spans the fiber core. Adding up the stress intensity factors for the two stress states, the total stress intensity factor Krd is given [15]: Krd ¼ ðr1 r1HÞ ffiffiffiffiffi pa p þ ðr1H þ r2H þ r2 r1Þ ffiffiffiffiffi pa p 2 p sin1 d 2a ð4aÞ σ1R σ1R σ1H σ1H σ1L σ1L σ2H σ2H σ2H σ2H σ2L σ2L d t Fig. 1. Schematic illustration showing the architecture and residual thermal stresses within the fibrous composite. 5784 X.H. Jin et al. / Acta Materialia 56 (2008) 5783–5795
Y.H. Jin et al /Acta Materialia 56(2008 )5783-5795 OIH+O2H+O2-01 f fftf11tt 02H Fiber cross section O1-O1H O1H+O2H+o2-01 b 2+62H ttttttttttttttttttftt 52H 2+02H O1-O2-O1H-O GH+O2H+02-0 ↑ttt ↑ fett tttt 01o2-1H-02H G1H+02H+02-O1 2. Fracture mechanics models used to derive the radial threshold strength function, with the red arrows representing the stresses that act only over the portion of the crack that spans the fiber. For interpretation of the references to color in this figure legend, the reader is referred to the web version of Substituting Eqs. (Ic),(3a), and (3b) into Eq. (4a), then layer. In this case, 2t+d<2a 3d+ 2t and the total stress rearranging, Krd can be expressed as intensity factor Krd can be expressed as E2-E12 d Ei(t +d) kd=(02+o2)√ra-(2-01+o+a) 2a」Et+E2 2t+d +(02-01+O1H+2H) + OHV (+2m( The second term is always negative. When GIH is suffi ciently large, the stress intensity factor becomes gradually obtained ng the above equation, the followin g equation is smaller(i.e, dKnd/da <0)as the crack extends deeper into the compressive layers. This causes a stable crack growth, Krd=(o2+02H)/Ia-2(a2-a1+IH+ 2H) and the applied stress increases with the extension of the crack. However, when the crack penetrates through the (4d) compressive layer, something different occurs t Fig. 2b shows the fracture mechanics model for the When dIH is sufficiently large, dkr /da is always positive it penetrates through the thin compressive Supposing that the toughness of the compressive layer
Substituting Eqs. (1c), (3a), and (3b) into Eq. (4a), then rearranging, Krd can be expressed as Krd ¼ r ffiffiffiffiffi pa p 1 þ E2 E1 E1 2 p sin1 d 2a E1ðt þ dÞ E1t þ E2d þ r1H ffiffiffiffiffi pa p 1 þ t d 2 p sin1 d 2a 1 ð4bÞ The second term is always negative. When r1H is suffi- ciently large, the stress intensity factor becomes gradually smaller (i.e., dKrd/da < 0) as the crack extends deeper into the compressive layers. This causes a stable crack growth, and the applied stress increases with the extension of the crack. However, when the crack penetrates through the compressive layer, something different occurs. Fig. 2b shows the fracture mechanics model for the crack when it penetrates through the thin compressive layer. In this case, 2t + d < 2a 6 3d + 2t and the total stress intensity factor K0 rd can be expressed as K0 rd ¼ ðr2 þ r2HÞ ffiffiffiffiffi pa p ðr2 r1 þ r1H þ r2HÞ ffiffiffiffiffi pa p 2 p sin1 2t þ d 2a þ ðr2 r1 þ r1H þ r2HÞ ffiffiffiffiffi pa p 2 p sin1 d 2a ð4cÞ Rearranging the above equation, the following equation is obtained: K0 rd ¼ ðr2 þ r2HÞ ffiffiffiffiffi pa p 2ðr2 r1 þ r1H þ r2HÞ ffiffiffi a p r sin1 2t þ d 2a sin1 d 2a ð4dÞ When r1H is sufficiently large, dK0 rd=da is always positive. Supposing that the toughness of the compressive layer σ1-σ1H σ1 σ2 σ1 σ1 σ2 σ1 2a σ1H σ2H σ1H σ1-σ1H = + σ1H+σ2H+σ2-σ1 σ1H+σ2H+σ2-σ1 Fiber cross section = σ1 σ2 σ1 σ1 σ2 σ1 2a σ2H σ1H σ2H σ1H σ2H σ2 σ2 σ2 σ2 σ2+σ2H σ2+σ2H + σ1-σ2-σ1H -σ2H σ1-σ2-σ1H -σ2H σ1H +σ2H +σ2 -σ1 σ1H +σ2H +σ2 -σ1 + Fig. 2. Fracture mechanics models used to derive the radial threshold strength function, with the red arrows representing the stresses that act only over the proportion of the crack that spans the fiber. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) X.H. Jin et al. / Acta Materialia 56 (2008) 5783–5795 5785
5786 X.H. Jin et al / Acta Materialia 56(2008)5783-5795 (Kid) and the fiber(kid) are constants (i.e, dIc/da= extend at an externally applied stress of e, which is gov- dic/da=0), the crack in Fig. 2a will become unstable erned by Eq (7a), as can be deduced according to the frac when it breaks through the compressive layer. And the ap- ture mechanics theory plied stress should show a maximum(omax) at the critical E1t exd point when the crack penetrates through the compressive ac layer. At this time, 2a=d+ 2t and Krd= Knd=kc, the E(+0k1(ma)-1202n oughness of the thin layer or fiber, while omax=h, which Substituting thr for ae then rearranging, a critical crack can be given by size aHc is established E2+E1 E,-E;2 E(+分12+E1m r Elstad o hr +, When a< aHc, the stress needed for crack extension(o) V(+1-(1+2sm/1 will be higher than ah. At this time, the crack extend through the whole material without being arrested by the thin compressive layer, causing catastrophic failure. The (5) residual strength of the material (oles)is higher than oihrt When Kic is larger than Kfc, Kc=kIc; otherwis and decreases with an increase in the crack length following Eq.(7a,i.e,dls=e. However, when aHo≤a<d∥2,a Kc=kic. One can see that keeping the t/d ratio constant, three-step crack propagation occurs. First, the crack ex- the radial threshold strength increases linearly with the tends across the fiber at a stress o smaller than dl. and compressive stress oIH and the toughness of the thin layer then is arrested in the thin compressive layer.Afterwards material or the fiber, while it decreases with the fiber diam- it grows in a stable er with an increase in the applied eter. It should be noted here that Eq (5) is only valid when stress. When the applied stress increases to oh, the crack penetrates through the compressive layer and catastrophic ensure that the crack extends through the whole thickness failure occurs. In this case, the residual strength should of the compressive layer in a stable manner(dkrd/da o show a constant value of oh, independent of the crack at d< 2a 2t t d) but becomes unstable after it penetrates length d<2a< 2t+ 3d). Substituting o thr for o in Eq(4b), then 3.2. Axial threshold strength analysis making Kc=Kic and dKrd/da=0 at 2a= 2t+d, oHc can be derived. Fig. 3a shows one of the fracture mechanics models used to derive the axial threshold strength (i.e, olh). In this model, a square plane crack rather than a slit crack is con- OHC= sidered. This square crack is 2a in length(d< 2a< d+ 2n) 3+d)√2+2」 and cuts through a fiber in the direction perpendicular to 2(E1-E2) sin" E its central axis. with the crack front embedded in the com- pressive layer surrounding the fiber. Mechanics analysis 2(E2-Ei)sin"(z+)+rE proves that (6)a1=2+aE Similarly, substituting ohr for a in Eq.(4d), then making 2t +d, dHc can be where a1, 02 and o hold the same meaning as in Fig 2a, but their directions are parallel to the fiber central axis 四2+可(件m()-1) In theory, the stress intensity factor for the stress state shown on the left-hand side of Fig 3a can be produced E2-E1 y superimposing the intensity factors for the two stress E1+(E2-E1)sin-( states shown on the right-hand side. Yet, the stress inten- sity factor for either of the latter two str (6b) complicated; it varies from nt on the crack front, and no specific stress intensity factor function can Until now, only a crack with a length of d 2a< 2t+d be given for either of them. This makes it difficult to derive is considered. However, when the crack is totally embedded the axial threshold strength function in the fiber, i.e., 2a <d, the following phenomenon will In order to overcome the difficulty, the mechanics model occur,depending on the crack size. The crack begins to shown in Fig. 3a is transformed to the model shown in
ðKl ICÞ and the fiber ðKf ICÞ are constants (i.e., dKl IC=da ¼ dKf IC=da ¼ 0), the crack in Fig. 2a will become unstable when it breaks through the compressive layer. And the applied stress should show a maximum (rmax) at the critical point when the crack penetrates through the compressive layer. At this time, 2a = d + 2t and Krd ¼ K0 rd ¼ KC, the toughness of the thin layer or fiber, while rmax ¼ rI thr, which can be given by rI thr ¼ E2 þE1 t d E1 1þ t d 1þ E2 E1 E1 2 p sin1 1 1þ2t d " # ! 1 Kc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pd 2 1þ2t d q þr1H 1 1þ t d 2 p sin1 1 1þ2t d " # ! 8 >< >: 9 >= >; ð5Þ When Kl IC is larger than Kf IC; KC ¼ Kl IC; otherwise, KC = Kf IC. One can see that keeping the t/d ratio constant, the radial threshold strength increases linearly with the compressive stress r1H and the toughness of the thin layer material or the fiber, while it decreases with the fiber diameter.It should be noted here that Eq. (5) is only valid when r1H is above two critical values, namely rHC and r0 HC, to ensure that the crack extends through the whole thickness of the compressive layer in a stable manner (dKrd/da 6 0 at d 6 2a 6 2t + d) but becomes unstable after it penetrates through the compressive layer (dK0 rd=da 0 at 2t + d < 2a 6 2t + 3d). Substituting rI thr for r in Eq. (4b), then making KC ¼ Kl IC and dKrd/da = 0 at 2a = 2t + d, rHC can be derived: rHC ¼ A p 2 ð2t þdÞ 1=2 Kl IC dþt d 2 p sin1 d 2tþd 2d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2tþdÞ 2d2 p A 1dþt d 2 p sin1 d 2tþd h i 1 A ¼ 2ðE1 E2Þ sin1 d 2tþd 2d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2tþdÞ 2d2 p pE1 2ðE2 E1Þsin1 d 2tþd þpE1 ð6aÞ Similarly, substituting rI thr for r in Eq. (4d), then making KC ¼ Kf IC and dKrd=da ¼ 0 at 2a ¼ 2t þ d; r0 HC can be derived: r0 HC ¼ A0 Kf IC ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 ð2t þ dÞ p A0 tþd d 2 p sin1 d 2tþd 1 dþt d h i A0 ¼ E2 E1 E1 þ 2 p ðE2 E1Þsin1 d 2tþd ð6bÞ Until now, only a crack with a length of d 6 2a 6 2t + d is considered. However, when the crack is totally embedded in the fiber, i.e., 2a < d, the following phenomenon will occur, depending on the crack size. The crack begins to extend at an externally applied stress of re, which is governed by Eq. (7a), as can be deduced according to the fracture mechanics theory: re ¼ E1t þ E2d E2ðt þ dÞ Kf ICðpaÞ 1=2 r2H h i ð7aÞ Substituting rI thr for re then rearranging, a critical crack size aHC is established: aHC ¼ ðKf ICÞ 2 p E2ðtþdÞ E1tþE2d rI thr þ r2H 2 ð7bÞ When a < aHC, the stress needed for crack extension (re) will be higher than rI thr. At this time, the crack extends through the whole material without being arrested by the thin compressive layer, causing catastrophic failure. The residual strength of the material (rI res) is higher than rI thr, and decreases with an increase in the crack length following Eq. (7a), i.e., rI res ¼ re. However, when aHC 6 a < d/2, a three-step crack propagation occurs. First, the crack extends across the fiber at a stress re smaller than rI thr, and then is arrested in the thin compressive layer. Afterwards, it grows in a stable manner with an increase in the applied stress. When the applied stress increases to rI thr, the crack penetrates through the compressive layer and catastrophic failure occurs. In this case, the residual strength should show a constant value of rI thr, independent of the crack length. 3.2. Axial threshold strength analysis Fig. 3a shows one of the fracture mechanics models used to derive the axial threshold strength (i.e., rII thr). In this model, a square plane crack rather than a slit crack is considered. This square crack is 2a in length (d 6 2a 6 d + 2t) and cuts through a fiber in the direction perpendicular to its central axis, with the crack front embedded in the compressive layer surrounding the fiber. Mechanics analysis proves that r1 ¼ ð2t þ dÞE1 dE2 þ 2tE1 r ð8aÞ r2 ¼ E2 E1 r1 ð8bÞ where r1, r2 and r hold the same meaning as in Fig. 2a, but their directions are parallel to the fiber central axis. In theory, the stress intensity factor for the stress state shown on the left-hand side of Fig. 3a can be produced by superimposing the intensity factors for the two stress states shown on the right-hand side. Yet, the stress intensity factor for either of the latter two stress states is very complicated; it varies from point to point on the crack front, and no specific stress intensity factor function can be given for either of them. This makes it difficult to derive the axial threshold strength function. In order to overcome the difficulty, the mechanics model shown in Fig. 3a is transformed to the model shown in 5786 X.H. Jin et al. / Acta Materialia 56 (2008) 5783–5795
Y.H. Jin et al /Acta Materialia 56(2008 )5783-5795 +o2L+o21 ↓↓↓↓I 11L +o2+o2-0 +o2+o201 +o2+201 2+o2H O1-O2-O1H-O2H G1H+o2H+0201 O1-O2-01H-O2H d1H+o2H+021 Fig 3. Fracture mechanics models used to derive the axial threshold strength function. The red arrows represent the that act only over the proportion of the crack that spans the fiber, and models (b)and(c) are converted from model (a) through a structure trar sfo ation. In model (c), the compressive layer is treated approximately as a thin-walled tube embedded in a homogeneous matrix of the fiber material. For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article. Fig. 3b, based on the following fact: the stress intensity fac- stresses in both the fiber and compressive thin layer are tor of a circular penny crack is nearly equal to the stress kept unchanged as in Fig. 3a. At this time, the stress inten intensity factor's maximum of a square plane crack when sity factor for the circular penny crack can be given as [16] the diameter of the former crack is equal to the diagonal length of the latter[16]. This allows the square crack prob- Kax=2(G1-GuLVF+2(01L+02+02-G1 lem in Fig. 3a to be converted into a circular crack problem in Fig. 3b by substituting the square crack with a circular penny crack and the square fiber with a round fiber. The radius r of the circular crack and the diameter d of the round fiber are 2a and v2d, respectively, while the com pressive layer thickness T between the round fibers is v2t The same model transformation is used for a square This transformation leads to a slight overestimation of the plane crack with a length of d+ 2t< 2a 3d+ 2t, as stress intensity factor for the circular crack in comparison shown in Fig. 3c. The stress intensity factor in this case with the square plane crack when the magnitudes of the can be given as
Fig. 3b, based on the following fact: the stress intensity factor of a circular penny crack is nearly equal to the stress intensity factor’s maximum of a square plane crack when the diameter of the former crack is equal to the diagonal length of the latter [16]. This allows the square crack problem in Fig. 3a to be converted into a circular crack problem in Fig. 3b by substituting the square crack with a circular penny crack and the square fiber with a round fiber. The radius r of the circular crack and the diameter D of the round fiber are ffiffiffi 2 p a and ffiffiffi 2 p d, respectively, while the compressive layer thickness T between the round fibers is ffiffiffi 2 p t. This transformation leads to a slight overestimation of the stress intensity factor for the circular crack in comparison with the square plane crack when the magnitudes of the stresses in both the fiber and compressive thin layer are kept unchanged as in Fig. 3a. At this time, the stress intensity factor for the circular penny crack can be given as [16] Kax ¼ 2ðr1 r1LÞ ffiffiffi r p r þ 2ðr1L þ r2L þ r2 r1Þ ffiffiffi r p r 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 D2 4r2 s2 4 3 5 ð9aÞ The same model transformation is used for a square plane crack with a length of d þ 2t < 2a 3d þ 2t, as shown in Fig. 3c. The stress intensity factor in this case can be given as σ1 σ2 σ1 σ1 σ2L σ1L σ1-σ1L σ1-σ1L σ1L+σ2L+σ2-σ1 σ1L+σ2L+σ2-σ1 = + Square plane crack σ1L+σ2L+σ2-σ1 σ1L+σ2L+σ2-σ1 σ2 σ1 σ2L σ1L σ1 σ1-σ1L σ1-σ1L = + Circular penny crack σ1 σ2 σ1 σ2L σ1L Circular penny crack σ2L σ2 σ2 σ2+σ2H σ2+σ2H = σ1-σ2-σ1H -σ2H σ1-σ2-σ1H -σ2H σ1H +σ2H +σ2 -σ1 σ1H +σ2H +σ2 -σ1 + + c b a Fig. 3. Fracture mechanics models used to derive the axial threshold strength function. The red arrows represent the stresses that act only over the proportion of the crack that spans the fiber, and models (b) and (c) are converted from model (a) through a structure transformation. In model (c), the compressive layer is treated approximately as a thin-walled tube embedded in a homogeneous matrix of the fiber material. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) X.H. Jin et al. / Acta Materialia 56 (2008) 5783–5795 5787