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
X.H. Jin et al / Acta Materialia 56(2008)5783-5795 Kax=2(02+021)1/=-2(a2-01+ou+a) Similarly, substituting Eqs.( 8a)and(8b)into Eq(9b),and making o=othr, Kc=KIc, dKa / da=0 at 2a=2t+d, dc can be derived (2T+D)2 When oIL is sufficiently large, dKax /dr0atd≤2a≤d+2 t and d+2t<2a≤3d+21 respectively. As discussed above, this means that the applied stress will reach its maximum(omax)at the critical E2+(E1-E2)1/1-(x4 point when the crack penetrates through the compressive layer. At this 2r=2T+D(or 2a =d+21) (11b) Kax =kc, while omax=athr, which can be expressed In addition, as in the case of radial threshold strength. there also exists a critical crack size alc for the crack com- 2tE +dE, pletely embedded in the fiber (2+)+(-Ey1-() x(k1)2 (12a) 4√2(+2) √2x 2t+d When a<alc, the residual strength of the material(dlls)is 2(2+d)2 larger than dthr and decreases with an increase in the crack size following Eq(12b): (10) 2E1+E2d「KC/m) When Kl is larger than Ki E2(2+d)2 Kc=Kic. Just like the radial threshold strength, the axial threshold strength also increases with the compressive In contrast, when 2aLc 2a< d, the residual strength stress GIL and the toughness of the thin layer or fiber, while shows a constant value of olh it decreases with the fiber diameter. Furthermore similar to the case for radial threshold strength, Eq (10)is only valid 4. Experimental procedure when oIL is larger than two critical values, oLc and dLc ubstituting Eqs ( 8a) and( 8b)into Eq (9a), and making To test the above design concept, fibrous composite a=ohr, Kc =(Kic), dAx/da=0 at 2a=2t+ d, TLc composed of Si3N4/TiN square fibers and Si3N4 thin layer can be derived were fabricated, in the hope that the thermal mismatch 2π(2+d between the fiber and the thin layer would induce a high compressive stress in the latter, leading to the appearance ±_2(21+4)2+d of threshold strength phenomenon (E2-E1)2+d2+1-(+d)V(2+d32-dE2 4. 1. Material preparation (E1-E2(2+d)2-1+(2+d)V(21+d2-d2E2 Three Si3 N4/TiN fibrous composites with a tiN content of 30 vol. %0, 35 vol %and 40 vol %in the fiber were prepared, (lla) and they were denoted by F30, F35 and F40, respectively utting plane SiNTiN Cutting Densified fibrous composite Laminar plate Fibrous green body Laminar green compact Fig 4. Flow chart showing the fabrication procedure of Si3 N4/TiN fibrous composite
K0 ax ¼ 2ðr2 þ r2LÞ ffiffiffi r p r 2ðr2 r1 þ r1L þ r2LÞ ffiffiffi r p r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 D2 4r2 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2T þ DÞ 2 4r2 2 s 4 3 5 ð9bÞ When r1L is sufficiently large, dKax=dr 0 at d 6 2a 6 d + 2t and d + 2t < 2a 6 3d + 2t, respectively. As discussed above, this means that the applied stress will reach its maximum (rmax) at the critical point when the crack penetrates through the compressive layer. At this time, 2r ¼ 2T þ D (or 2a ¼ d þ 2t) and Kax ¼ K0 ax ¼ KC, while rmax = rII thr, which can be expressed as rII thr ¼ 2tE1 þ dE2 ð2t þ dÞ E2 þ ðE1 E2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 d 2tþd 2 " # r ffiffiffiffiffiffiffiffiffi ffiffiffi 2 p p p 2ð2t þ dÞ 1=2 KC þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2t þ d d 2 1 s 2t d 2 4 3 5 r1L 8 < : 9 = ; ð10Þ When Kl IC is larger than Kf IC, KC = Kl IC; otherwise, KC = Kf IC. Just like the radial threshold strength, the axial threshold strength also increases with the compressive stress r1L and the toughness of the thin layer or fiber, while it decreases with the fiber diameter. Furthermore, similar to the case for radial threshold strength, Eq. (10) is only valid when r1L is larger than two critical values, rLC and r0 LC. Substituting Eqs. (8a) and (8b) into Eq. (9a), and making r = rII thr, KC = ðKl ICÞ, dKax/da = 0 at 2a = 2t + d, rLC can be derived: rLC ¼ ffiffiffiffiffiffiffiffiffi ffiffiffi 2 p p p ð2t þ dÞ 1=2 AKl IC 4t d 2½ð2tþdÞ 2þd2 d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2tþdÞ 2d2 p 2A ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2tþdÞ 2d2 p 2t d A ¼ ðE2 E1Þ½ð2t þ dÞ 2 þ d2 ð2t þ dÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2t þ dÞ 2 d2 q E2 ðE1 E2Þ½ð2t þ dÞ 2 d2 þð2t þ dÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2t þ dÞ 2 d2 q E2 ð11aÞ Similarly, substituting Eqs. (8a) and (8b) into Eq. (9b), and making r = rII thr, KC = Kf IC, dK0 ax=da ¼ 0 at 2a ¼ 2t þ d, r0 LC can be derived: r0 LC ¼ ffiffiffiffiffiffiffiffiffi ffiffiffi 2 p p p 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 2t þ d p dA0 Kf IC A0 2t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2t þ dÞ 2 d2 q 2t d A0 ¼ E2 E1 E2 þ ðE1 E2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 d 2tþd 2 r ð11bÞ In addition, as in the case of radial threshold strength, there also exists a critical crack size aLC for the crack completely embedded in the fiber: aLC ¼ pðKf ICÞ 2 4 ffiffiffi 2 p E2ð2tþdÞ 2E1tþE2d rII thr þ r2L 2 ð12aÞ When a < aLC, the residual strength of the material (rII res) is larger than rII thr and decreases with an increase in the crack size following Eq. (12b): rII res ¼ 2E1t þ E2d E2ð2t þ dÞ Kf IC 25=4 p a 1=2 r2L ð12bÞ In contrast, when 2aLC 6 2a < d, the residual strength shows a constant value of rII thr. 4. Experimental procedure To test the above design concept, fibrous composites composed of Si3N4/TiN square fibers and Si3N4 thin layer were fabricated, in the hope that the thermal mismatch between the fiber and the thin layer would induce a high compressive stress in the latter, leading to the appearance of threshold strength phenomenon. 4.1. Material preparation Three Si3N4/TiN fibrous composites with a TiN content of 30 vol.%, 35 vol.% and 40 vol.% in the fiber were prepared, and they were denoted by F30, F35 and F40, respectively. Laminating Si3N4/TiN Si3N4 d t Cutting plane Cutting Laminar green compact Laminar plate HPing Fibrous green body Densified fibrous composite t2 d2 t d Fig. 4. Flow chart showing the fabrication procedure of Si3N4/TiN fibrous composite. 5788 X.H. Jin et al. / Acta Materialia 56 (2008) 5783–5795
Y.H. Jin et al /Acta Materialia 56(2008 )5783-5795 5789 Fig 4 shows the flow chart of the fabrication procedure. direction perpendicular to the Si3N4 compressive layer First, Si, N,TiN and Si3 N4 green sheets both containing 6 was produced on the sample surface by the vickers inden wt%Y2O3 and 4 wt % Al2O3 as sintering additives were pre- tation method at an indenter load of 49, 98 or 196N. This pared by the tape casting method, using butanone as the sol- artificial crack was located in either the center of the fiber's vent, glycerol trioleate as the dispersant and PvB as the cross-section region or the center of the fiber's side face, binder. Then, a laminar green compact composed of alter- depending on either the radial or the axial threshold nating Si N4/TiN thick layer and Si3N4 thin layer was fabri- strength was tested. The residual strength of the pre- cated by laminating the Si3 N4/TiN and Si3 N4 sheets at cracked sample was measured in four-point bending at a 100 MPa in a steel die, followed by cold isostatic pressing crosshead speed of 0.05 mm min. The microstructure at 300 MPa. Afterwards, the laminar compact was cut into was characterized using optical and scanning electron plates of designated thickness in the direction perpendicular microscopy to the layers, softened in ethanol vapor at 50C for 2 h and laminated again with a second set of Si3 N4 sheet at 200 MPa. 5. Results and discussion The thickness of the plate(d2)and the second set of Si3N4 sheet(t2)were controlled following the equations d2= d/p 5.1. Fabrication of Si, N, /TiN fibrous composites and t2=t/P, where d and t are the thicknesses of the p and p are the relative densities of the laminar green com- d Owing to the existence of organic binder in the Si3 N4/ Si3 N/TiN and Si3 N4 layers in the laminar green compact, pact and the second set of Si3 N4 sheet, respectively. After flexibility and could be readily cut into Si N,/TiN plates debindering in vacuum at 850C, the green compact formed 1. 5 mm thick. Fig. 5a and b demonstrate the optical was hot pressed in a size- fitted graphite mold at 1800C for micrographs of one of the Si, N4/TiN plates. The micro- I h under 30 MPa During hot pressing, the restriction from structure is reasonably uniform with a thin Si3 N4 layer the mold wall allowed the sample only to shrink uniaxially sandwiched between thick Si3 N4/TiN layers. The thickness along the direction of the applied load. And this unidirec- of the Si,N4 and Si3 N4/TiN layer in the plate are 50 and tional shrinking of the sample was exactly compensated by 450 um, respectively; and the relative density of the plate is the thickness difference between d2 t f2 and d+ t. As a 50%. These plates were carefully thinned to a thickness of result, a fibrous composite composed of square fibers sepa- 900-910 um, and laminated with Si3 N4 green sheets of den rated with thin compressive layer was obtained. The diame- sity m35% and thickness 160 um. After debindering and ter of the fiber and the thin layer thickness were d and t, hot pressing, Si, N/TiN fibrous composites with an archi respectively tecture shown in Fig. 5c and d were obtained. The majority of Si3 N4/TiN fibers in the material show a nearly perfect 4.2. Mechanical testing and microstructure characterization square cross-section and are separated by a Si3 N4 layer of uniform thickness. No significant deviation from the The sintered samples were cut into bars 3 x 4 x 35 mm, designed architecture is observed except for some mismatch mirror polished, then an artificial crack aligning in in the position of the Si3 N4 thin layer. The diameters of the (b) 1 0mm Fig. 5. Optical micrographs of (a and b) Si N/TiN plates cut from a laminar green compact, and the cross-sections of (c)F30 and(d)F40 fibrous composites densified through hot pressing.( For interpretation of the references to color in this figure legend, the reader is referred to the web version this article
Fig. 4 shows the flow chart of the fabrication procedure. First, Si3N4/TiN and Si3N4 green sheets both containing 6 wt.% Y2O3 and 4 wt.% Al2O3 as sintering additives were prepared by the tape casting method, using butanone as the solvent, glycerol trioleate as the dispersant and PVB as the binder. Then, a laminar green compact composed of alternating Si3N4/TiN thick layer and Si3N4 thin layer was fabricated by laminating the Si3N4/TiN and Si3N4 sheets at 100 MPa in a steel die, followed by cold isostatic pressing at 300 MPa. Afterwards, the laminar compact was cut into plates of designated thickness in the direction perpendicular to the layers, softened in ethanol vapor at 50 C for 2 h and laminated again with a second set of Si3N4 sheet at 200 MPa. The thickness of the plate (d2) and the second set of Si3N4 sheet (t2) were controlled following the equations d2 = d/q and t2 = t/q0 , where d and t are the thicknesses of the Si3N4/TiN and Si3N4 layers in the laminar green compact, q and q0 are the relative densities of the laminar green compact and the second set of Si3N4 sheet, respectively. After debindering in vacuum at 850 C, the green compact formed was hot pressed in a size-fitted graphite mold at 1800 C for 1 h under 30 MPa. During hot pressing, the restriction from the mold wall allowed the sample only to shrink uniaxially along the direction of the applied load. And this unidirectional shrinking of the sample was exactly compensated by the thickness difference between d2 + t2 and d + t. As a result, a fibrous composite composed of square fibers separated with thin compressive layer was obtained. The diameter of the fiber and the thin layer thickness were d and t, respectively. 4.2. Mechanical testing and microstructure characterization The sintered samples were cut into bars 3 4 35 mm, mirror polished, then an artificial crack aligning in direction perpendicular to the Si3N4 compressive layer was produced on the sample surface by the Vickers indentation method at an indenter load of 49, 98 or 196 N. This artificial crack was located in either the center of the fiber’s cross-section region or the center of the fiber’s side face, depending on either the radial or the axial threshold strength was tested. The residual strength of the precracked sample was measured in four-point bending at a crosshead speed of 0.05 mm min1 . The microstructure was characterized using optical and scanning electron microscopy. 5. Results and discussion 5.1. Fabrication of Si3N4/TiN fibrous composites Owing to the existence of organic binder in the Si3N4/ TiN laminar green compact, it showed good strength and flexibility and could be readily cut into Si3N4/TiN plates 1.5 mm thick. Fig. 5a and b demonstrate the optical micrographs of one of the Si3N4/TiN plates. The microstructure is reasonably uniform with a thin Si3N4 layer sandwiched between thick Si3N4/TiN layers. The thickness of the Si3N4 and Si3N4/TiN layer in the plate are 50 and 450 lm, respectively; and the relative density of the plate is 50%. These plates were carefully thinned to a thickness of 900–910 lm, and laminated with Si3N4 green sheets of density 35% and thickness 160 lm. After debindering and hot pressing, Si3N4/TiN fibrous composites with an architecture shown in Fig. 5c and d were obtained. The majority of Si3N4/TiN fibers in the material show a nearly perfect square cross-section and are separated by a Si3N4 layer of uniform thickness. No significant deviation from the designed architecture is observed except for some mismatch in the position of the Si3N4 thin layer. The diameters of the (b) 2mm (a) (d) 1.0mm (c) Fig. 5. Optical micrographs of (a and b) Si3N4/TiN plates cut from a laminar green compact, and the cross-sections of (c) F30 and (d) F40 fibrous composites densified through hot pressing. (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 5789
5790 X.H. Jin et al / Acta Materialia 56(2008)5783-5795 Table 1 and, in comparison with the axial threshold strength, the d for threshold calculation [7-21] radial threshold strength is significantly lower, despite ne bulk modulu ted according to the hile the elastic me he average of the lues in literature aIH being higher than oIL. The authors believe that the di for bulk materials ference in crack configuration assumed for the derivation CTE Elastic modulus Poisson,s Bulk modulus of the threshold strength functions is the fundamental rea (x/×10-6cC-)(E/GPal son for such a great difference between the two threshold Si3N43.25 strengths. Under the same applied stress, the stress inten- sity factor of a square plane crack is much smaller than that of a slit crack [16]. In consequence, the axial threshold strength becomes higher than the radial one. The validity of the above predictions is evaluated by measuring the Si3 N4/TiN fibers in F30, F35 and F40 are 455+ 25, threshold strengths of the materials 1+20 and 447+ 23 um, and the thicknesses of the Fig 6a-d shows the variation in residual strength with Si3N4 layers in them are 54+3.5, 55+4.1 and pre-crack length during the radial and axial threshold 52+3.2 um, respectively. The architectures of these com- strength testing of F30 and F40 composites. For compari- posites are almost identical in terms of fiber diameter and son, the residual strength of a pre-cracked Si3N4/TiN thin layer thickness, indicating good reproducibility of monolithic composite containing 30 vol. TiN but without the fabrication technique the SigNa compressive layers is also reported in Fig. 5a. It is found that the residual strength of the Si3 N/TiN mono- 5.2. Threshold strength calculation and measurement lithic composite is strongly sensitive to the pre-crack size and declines steadily as the pre-crack length increases, Using the physical properties listed in Table 1 [17-21], which is typical for a brittle material. Conversely, the resid- the elastic modulus, CTE and Poissons ratio of the ual strengths of the F30 and F40 composites show little Si,N4/TiN fiber within the composites were estimated variation with the pre-crack length, and distinctive thresh according to the calculation methods given by Fair [13]. old strength phenomena are observed during both the On this basis, the residual compressive stresses within the radial and the axial residual strength testing. The same Si3N4 thin layer were calculated using Eqs.(2a) and(2b) threshold strength phenomena are also observed for F35 During the residual stress calculations, a AT value of composite, although they are not given here. The occur- 1200C was assumed, according to the high temperature rence of threshold strength phenomenon in these fibrous strength and creep behavior of Si3 N4/TiN composites composites is in agreement with the theoretical prediction [22]. Furthermore, the critical compressive stresses, includ- that the residual strength of the material will remain con- ing HC, HC, OLc and dic were also estimated using Si3N4 stant as long as the crack size falls into a certain range, toughness of 5.5 MPa m"and Si3 N4/TiN toughness of namely 2aHc 2a 2t+d for radial threshold strength 6.0, 6.3 and 6.5 MPa m/2 for the F30, F35 and F40 com- testing or 2aLc 2a 21+d for axial threshold strength posite, respectively. All the toughnesses were measured testing(Table 2) through the vickers indentation method on monolithic materials containing no compressive layers 5.3. Theoretical analysis of the influence of indentation field Table 2 shows the results of the calculations. It can be on the residual strength seen that the residual compressive stresses within the Si3N4 thin layers are all much higher than the critical ones. It must be pointed out that an indented crack rather Therefore, the threshold strengths of the composites can be than a natural crack was used to measure the residual estimated using Eqs. (5)and (10). The calculated threshold strength in the present work, and this is different from trengths are listed in Table 2, together with the calculated the fracture mechanics modeling in Section 3, where a nat residual compressive stresses within the Si3n4 layer and the ural crack is used for the analysis of stress intensity factor. critical compressive stresses. It is found that the threshold The difference between a natural crack and an indented strengths increase with the residual compressive stresses crack in the stress state may lead to a difference in the resid- within the Si3N4 layer and the tin content of the fiber ual strength as discussed below Residual compressie.an stresses within the SigN. compressive layer and the predicted threshold strengths of the SiN/TiN fibrous composites Calculated residual compress Threshold strength Critical crack size(um) OIH, Eq gHC, Eq d1L,上q OLC, Eq alc, Eq. 25.1 F401079 22.0
Si3N4/TiN fibers in F30, F35 and F40 are 455 ± 25, 451 ± 20 and 447 ± 23 lm, and the thicknesses of the Si3N4 layers in them are 54 ± 3.5, 55 ± 4.1 and 52 ± 3.2 lm, respectively. The architectures of these composites are almost identical in terms of fiber diameter and thin layer thickness, indicating good reproducibility of the fabrication technique. 5.2. Threshold strength calculation and measurement Using the physical properties listed in Table 1 [17–21], the elastic modulus, CTE and Poisson’s ratio of the Si3N4/TiN fiber within the composites were estimated according to the calculation methods given by Fair [13]. On this basis, the residual compressive stresses within the Si3N4 thin layer were calculated using Eqs. (2a) and (2b) During the residual stress calculations, a DT value of 1200 C was assumed, according to the high temperature strength and creep behavior of Si3N4/TiN composites [22]. Furthermore, the critical compressive stresses, including rHC, r0 HC, rLC and r0 LC, were also estimated using Si3N4 toughness of 5.5 MPa m1/2 and Si3N4/TiN toughness of 6.0, 6.3 and 6.5 MPa m1/2 for the F30, F35 and F40 composite, respectively. All the toughnesses were measured through the Vickers indentation method on monolithic materials containing no compressive layers. Table 2 shows the results of the calculations. It can be seen that the residual compressive stresses within the Si3N4 thin layers are all much higher than the critical ones. Therefore, the threshold strengths of the composites can be estimated using Eqs. (5) and (10). The calculated threshold strengths are listed in Table 2, together with the calculated residual compressive stresses within the Si3N4 layer and the critical compressive stresses. It is found that the threshold strengths increase with the residual compressive stresses within the Si3N4 layer and the TiN content of the fiber and, in comparison with the axial threshold strength, the radial threshold strength is significantly lower, despite r1H being higher than r1L. The authors believe that the difference in crack configuration assumed for the derivation of the threshold strength functions is the fundamental reason for such a great difference between the two threshold strengths. Under the same applied stress, the stress intensity factor of a square plane crack is much smaller than that of a slit crack [16]. In consequence, the axial threshold strength becomes higher than the radial one. The validity of the above predictions is evaluated by measuring the threshold strengths of the materials. Fig. 6a–d shows the variation in residual strength with pre-crack length during the radial and axial threshold strength testing of F30 and F40 composites. For comparison, the residual strength of a pre-cracked Si3N4/TiN monolithic composite containing 30 vol.% TiN but without the Si3N4 compressive layers is also reported in Fig. 5a. It is found that the residual strength of the Si3N4/TiN monolithic composite is strongly sensitive to the pre-crack size and declines steadily as the pre-crack length increases, which is typical for a brittle material. Conversely, the residual strengths of the F30 and F40 composites show little variation with the pre-crack length, and distinctive threshold strength phenomena are observed during both the radial and the axial residual strength testing. The same threshold strength phenomena are also observed for F35 composite, although they are not given here. The occurrence of threshold strength phenomenon in these fibrous composites is in agreement with the theoretical prediction that the residual strength of the material will remain constant as long as the crack size falls into a certain range, namely 2aHC 6 2a 6 2t + d for radial threshold strength testing or 2aLC 6 2a 6 2t + d for axial threshold strength testing (Table 2). 5.3. Theoretical analysis of the influence of indentation field on the residual strength It must be pointed out that an indented crack rather than a natural crack was used to measure the residual strength in the present work, and this is different from the fracture mechanics modeling in Section 3, where a natural crack is used for the analysis of stress intensity factor. The difference between a natural crack and an indented crack in the stress state may lead to a difference in the residual strength as discussed below. Table 1 Physical property data used for threshold strengths calculation [17–21]; the bulk modulus is calculated according to the equation K = E/3(1 2m), while the elastic modulus is the average of the reported values in literature for bulk materials CTE (a/ l06 C1 ) Elastic modulus (E/GPa) Poisson’s ratio (m) Bulk modulus (K/GPa) Si3N4 3.25 310 0.27 225 TiN 8.0 460 0.25 307 Table 2 Calculated residual compressive stresses within the Si3N4 compressive layer and the predicted threshold strengths of the Si3N4/TiN fibrous composites Residual compressive stress (MPa) Threshold strength (MPa) Critical crack size (lm) r1H, Eq. (2a) rHC, Eq. (6a) r0 HC, Eq. (6b) r1L, Eq. (2b) rLC, Eq. (11a) r0 LC, Eq. (11b) rI thr, Eq. (5) rII thr, Eq. (10) aHC, Eq. (7b) aLC, Eq. (12a) F30 835 58 20 762 63 25 478 641 33.7 28.7 F35 960 56 25 878 59 30 530 711 29.8 25.1 F40 1079 54 29 989 55 35 576 775 26.4 22.0 5790 X.H. Jin et al. / Acta Materialia 56 (2008) 5783–5795
Y.H. Jin et al /Acta Materialia 56(2008 )5783-5795 000 a°鄙爵圉 d300F 200 100F Si N/TIN composite 100 100150200250300350400450 Prerack length, 2a/um precrack length, 2a /um 700c 700 F40 500F 品苏 400 100 100150200250300350400 100150200250300350 Precrack length, 2a/ um Precrack length, 2a/ um Fig. 6. Plot residual strength vs pre-crack length for F30 and F40 fibrous composite during(a and c)radial and (b and d) axial threshold strength testing. usin acks made at indenter loads of 49N(D), 98N()and 196N(). For comparison, the residual strength of a conventional SiN/TiN ing 30 vol. TiN is also reported in(a). n comparison with a natural crack that is residual stress old strength testing, K=Kax, K=Kar, or=G2L and free in itself, the residual stress associated with an indented K,=2(02+a2La/)/2 crack will cause an additional increase in the stress inten- When aIH and oIL are suficiently large dK/ da is nega sity factor at the crack tip [23], which can be approximately tive at d< 2a 2t+d and dk, /da is positive at expressed as Eq. (13a)after the crack expands to size a 2t+d< 2a 2t+3d for both radial and axial residual under the externally applied stress strength testing independent of the indenter load. Under E2\2P such a circumstance, a critical initial crack size appears, △K=Kmd (13a) similar to the case where a natural crack is used for residual strength testing. This critical initial crack size is denoted by Kfc-ar(ao/ E (13b) ual strength testing. When the initial size of the indented crack is smaller than the critical value, the compressiv where ao is the initial size of the indented crack(ao d/2) stress within the thin layer fails to stabilize the crack during P is the indenter load, H2 is the hardness of the fiber, s is a residual strength testing, and catastrophic failure occurs dimensionless constant with a value of 0.016, and or is the before the crack extends into the compressive layer. Mak tensile thermal stress in the fiber Thus. the total stress ing dK1/da=0 and KI=Kic, the radial(ores )and axial intensity factor at the crack tip can be expressed by eqs. (oIl)residual strengths at this time can be derived (14a-14c) depending on the size of the extending crack E1t+ ed K1=K1+m=+s/E2)2P E1(t+d) 2H7 ≤ k2=K+km=K+:(2)Pn,d≤20≤2+4(4 (15a) 2E1t+E2d 3 n(kic E2(2t+d) 0≤a (14c) (15b) For radial threshold strength testing, K K=K In contrast. when the initial size of the indented crack is Or=O2H and K,=(02+O2H)(a) while for axial thresh- larger than the critical value, the crack can be effectively
In comparison with a natural crack that is residual stress free in itself, the residual stress associated with an indented crack will cause an additional increase in the stress intensity factor at the crack tip [23], which can be approximately expressed as Eq. (13a) after the crack expands to size a under the externally applied stress: DK ¼ Kind ¼ n E2 H2 1=2 P a3=2 ð13aÞ P ¼ Kf IC rrðpa0Þ 1=2 n H2 E2 1=2 a 3=2 0 ð13bÞ where a0 is the initial size of the indented crack (a0 6 d/2), P is the indenter load, H2 is the hardness of the fiber, n is a dimensionless constant with a value of 0.016, and rr is the tensile thermal stress in the fiber. Thus, the total stress intensity factor at the crack tip can be expressed by Eqs. (14a–14c) depending on the size of the extending crack: K1 ¼ Kr þKind ¼ Kr þn E2 H2 1=2 P a3=2 ; 2a d ð14aÞ K2 ¼ K þKind ¼ K þn E2 H2 1=2 P a3=2 ; d 2a 2t þd ð14bÞ K0 2 ¼ K0 þKind ¼ K0 þn E2 H2 1=2 P a3=2 ; 2t þd 2a 2tþ3d ð14cÞ For radial threshold strength testing, K = Krd, K0 = K0 rd, rr = r2H and Kr = (r2 + r2H)(pa)1/2; while for axial threshold strength testing, K = Kax, K0 = K0 ax, rr = r2L and Kr = 2(r2 + r2L)(a/p) 1/2. When r1H and r1L are sufficiently large, dK2/da is negative at d 6 2a 6 2t + d and dK0 2/da is positive at 2t + d : 9 >= >; ; 0 : 9 >= >; ; 0 < a0 aII 0 ð15bÞ In contrast, when the initial size of the indented crack is larger than the critical value, the crack can be effectively 150 200 250 300 350 400 0 100 200 300 400 500 600 700 800 Si3 N4 /TiN composite F30 Residual Strength/ MPa Prerack length, 2a/ μm 100 150 200 250 300 350 400 450 0 100 200 300 400 500 600 700 800 F30 Residual strength / MPa precrack length, 2a /μm 100 150 200 250 300 350 400 0 100 200 300 400 500 600 700 800 F40 Residual Strength/ MPa Precrack length, 2a/ μm 100 150 200 250 300 350 0 100 200 300 400 500 600 700 800 F40 Residual Strength/ MPa Precrack length, 2a/ μm a b c d Fig. 6. Plots of the residual strength vs. pre-crack length for F30 and F40 fibrous composite during (a and c) radial and (b and d) axial threshold strength testing, using pre-cracks made at indenter loads of 49N (h), 98N (}) and 196 N ( ). For comparison, the residual strength of a conventional Si3N4/TiN composite containing 30 vol.% TiN is also reported in (a). X.H. Jin et al. / Acta Materialia 56 (2008) 5783–5795 5791
X.H. Jin et al / Acta Materialia 56(2008)5783-5795 stabilized by the compressive stress within the thin layer, strength becomes indented-flaw controlled, and decreases and catastrophic failure occurs at 2a= 2t +d In this case, with an increase in the indenter load. In the present work, the radial(on )and axial (olld)residual strengths are given the indenter load is far above 10 N, and the indented crack size is large. Therefore, the invariance of residual strength ad in -2什,[+2m(4)mP四mmm 4P4 In fact. the invariance of residual strength with the (16a) indenter load is also frequently observed in many laminar r/2(2r+d a≤ao≤d/2 composites that contain thin layers with a high compressive teL+ de residual stress [9, 29]. The fundamental reason for that is still unclear, but can probably be explained in terms of E2(21+d)+(E1-E2 the relaxation of the indentation field under externally ()“ applied stress [30]. It is well known that the restricted vol- a0≤ao≤d/2 (16b) ume expansion of the plastic deformation zone is the origin of the residual stresses associated with an indented crack Eqs.(15a)and(16a)show the same value at ao=ao, and this [23]. Owing to the strong crack-arresting effect of the thin also true for Eqs. (15b)and(16b)at ao=ao compressive layer, a high tensile stress is needed for the It can be seen from the residual strength functions above indented crack to extend through it. This causes a sig that, unlike the residual strengths tested using a natural cant increase in the crack opening displacement, which allows the plastic deformation zone of the indented crack decrease monotonically with an increase in the indenter load to expand more freely and leads to a nearly complete relax and, correspondingly, the initial size of the indented crack, the thin layer is sufficiently high. As a result, the influence because of the increase in the stress intensity factor at the crack tip caused by the indentation field. And the residual of the indentation field on the residual strength is mini- strengths tested using an indented crack should be lower mized, and the residual strength becomes almost the same than their counterparts tested using a natural crack as that tested using a natural crack, showing a threshold strength phenomenon. In any case, further investigation The residual strengths of F30 and F40 composites were should be carried out to prove the above hypothesis calculated using Eqs.(16a)and(16b), and are listed in Table 3, together with the critical initial crack sizes that 5.4. Comparison between measured and predicted threshold were estimated according to Eqs.(15a),(15b),(16a) (16b)and(13b). Both materials show a significant degrada streng tion in the radial and axial residual strengths as the inden ter load increases from 49 to 196N. This is in contradiction The measured threshold strengths are reported in Fig. 7 to the experimental result shown in Fig. 6, where little var- along with the predicted ones using Eqs. (5) and (10). The iation in residual strength with indenter load is observed measured threshold strengths varies with the tin content of the fiber in the same trend as the predicted ones, because Although the invariance of residual strength with indenter of the increase in the compressive stresses within the Si3N4 load is sometimes observed in coarse-grained ceramics. it appears only when the indenter load is sufficiently small layer, which leads to an enhancement in the crack-arresting [24-28] For Si3 N4 and Si3 N4/TiN ceramics, the effect. And for the same material. the measured axial load is typically 10N, the residual threshold strength agrees generally quite well with the the- oretical prediction, the measured axial threshold strength much lower than the predicted strength The authors believe that the large discrepancy between Table 3 the measured and predicted axial threshold strengths has Calculated residual strengths and the critical initial crack sizes for F30 and its origin primarily in the testing method used. As testified F40 composite when an indented crack is used for residual strength testing by the threshold strength measurement for a laminar com- ab(um) a0(um) P(N) es (MPa) ane(MPa) posite [ll] it is suitable to use a surface indentatio 10.2 for the radial threshold strength testing owing to the fol- 98422 lowing fact: under the restriction of the compressive layers, material without significant transverse crack growth during 98518 the radial threshold strength testing. which makes it show 737 close similarity to a slit crack in the crack configuration
stabilized by the compressive stress within the thin layer, and catastrophic failure occurs at 2a ¼ 2t þ d. In this case, the radial (rI res) and axial (rII res) residual strengths are given by rI res ¼ rI thr E1t þ E2d E1ðt þ dÞ 1 þ E2 E1 E1 2 p sin1 d 2t þ d 1 E2 H2 1=2 4Pn p1=2ð2t þ dÞ 2 ; aI 0 a0 d=2 ð16aÞ rII res ¼ rII thr 2tE1 þ dE2 E2ð2t þ dÞþðE1 E2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2t þ dÞ 2 d2 q E2 H2 1=2 23=4 p1=2Pn ð2t þ dÞ 2 ; aII 0 a0 d=2 ð16bÞ Eqs. (15a) and (16a) show the same value at a0=aI 0, and this is also true for Eqs. (15b) and (16b) at a0 = aII 0 . It can be seen from the residual strength functions above that, unlike the residual strengths tested using a natural crack, the residual strengths tested using an indented crack decrease monotonically with an increase in the indenter load and, correspondingly, the initial size of the indented crack, because of the increase in the stress intensity factor at the crack tip caused by the indentation field. And the residual strengths tested using an indented crack should be lower than their counterparts tested using a natural crack. The residual strengths of F30 and F40 composites were calculated using Eqs. (16a) and (16b), and are listed in Table 3, together with the critical initial crack sizes that were estimated according to Eqs. (15a), (15b), (16a). (16b) and (13b). Both materials show a significant degradation in the radial and axial residual strengths as the indenter load increases from 49 to 196 N. This is in contradiction to the experimental result shown in Fig. 6, where little variation in residual strength with indenter load is observed. Although the invariance of residual strength with indenter load is sometimes observed in coarse-grained ceramics, it appears only when the indenter load is sufficiently small [24–28]. For Si3N4 and Si3N4/TiN ceramics, the indenter load is typically 10 N, the residual strength becomes indented-flaw controlled, and decreases with an increase in the indenter load. In the present work, the indenter load is far above 10 N, and the indented crack size is large. Therefore, the invariance of residual strength with indented crack size (and indenter load) in Fig. 6 cannot be interpreted as a microstructural-flaw-controlled phenomenon. In fact, the invariance of residual strength with the indenter load is also frequently observed in many laminar composites that contain thin layers with a high compressive residual stress [9,29]. The fundamental reason for that is still unclear, but can probably be explained in terms of the relaxation of the indentation field under externally applied stress [30]. It is well known that the restricted volume expansion of the plastic deformation zone is the origin of the residual stresses associated with an indented crack [23]. Owing to the strong crack-arresting effect of the thin compressive layer, a high tensile stress is needed for the indented crack to extend through it. This causes a signifi- cant increase in the crack opening displacement, which allows the plastic deformation zone of the indented crack to expand more freely and leads to a nearly complete relaxation of the indentation field when the compressive stress in the thin layer is sufficiently high. As a result, the influence of the indentation field on the residual strength is minimized, and the residual strength becomes almost the same as that tested using a natural crack, showing a threshold strength phenomenon. In any case, further investigation should be carried out to prove the above hypothesis. 5.4. Comparison between measured and predicted threshold strengths The measured threshold strengths are reported in Fig. 7 along with the predicted ones using Eqs. (5) and (10). The measured threshold strengths varies with the TiN content of the fiber in the same trend as the predicted ones, because of the increase in the compressive stresses within the Si3N4 layer, which leads to an enhancement in the crack-arresting effect. And for the same material, the measured axial threshold strength is also higher than the radial strength. However, a comparison between the measured and the predicted results finds that, although the measured radial threshold strength agrees generally quite well with the theoretical prediction, the measured axial threshold strength is much lower than the predicted strength. The authors believe that the large discrepancy between the measured and predicted axial threshold strengths has its origin primarily in the testing method used. As testified by the threshold strength measurement for a laminar composite [11], it is suitable to use a surface indentation crack for the radial threshold strength testing owing to the following fact: under the restriction of the compressive layers, the surface indentation crack will extend deep into the material without significant transverse crack growth during the radial threshold strength testing, which makes it show close similarity to a slit crack in the crack configuration. Table 3 Calculated residual strengths and the critical initial crack sizes for F30 and F40 composite when an indented crack is used for residual strength testing aI 0 (lm) aII 0 (lm) P (N) rI res (MPa) an res (MPa) F30 10.2 9.8 196 367 492 98 422 566 49 450 603 F40 8.2 8.0 196 460 622 98 518 698 49 547 737 5792 X.H. Jin et al. / Acta Materialia 56 (2008) 5783–5795