Seramik kompozitlerde yük aktarımı ve köprülenme

Abstract

Bu çalışmanın, birinci bölümünde, eksenel yüklü liflerin matrise elastik yük transferi analizi yapılmış ve lifteki gerilme formülü çıkarılmıştır. İkinci bölümde, ortotropik kompozit levhanın kısmi çatlaması halinde, çatlak yüzeyi yerdeğiştirmesinin fonksiyonu olan sırasıyla elastik, sürtünmeli ve sıyrılma köprülenmeleri elde edilmiştir. Üçüncü bölümde, kısmi çatlakh seramik kompozit levhanın Airy diferensiyel denklemi çıkarılmıştır. Bu diferensiyel denklem, Fourier dönüşümleriyle çözülerek, çatlak yüzeyi eğimi, f(t), bilinmeyen fonksiyonu cinsinden, normal ve kayma gerilmeleri ifadeleri çıkarılmıştır. Kısmi çatlakh bir kompozit levha için problemin formülasyonu oluşturulmuştur. İlk olarak çatlağın tamamında elastik köprülerime olduğu düşünülerek, örnek bir malzeme için çatlak uçlarındaki gerilme şiddeti çarpanı bulunmuş ve çatlak ekseni boyunca çatlak açıklığı değerleri grafik olarak gösterilmiştir, integral denklemin çözümü için Gauss-Chebyshev sayısal integrasyon yöntemi kullanılmıştır. İkinci ve üçüncü olarak sürtünmeli köprülerime ve sıyrılma köprülenmesi için aynı işlemler yapılmıştır. Sürtünmen köprüleme çözümüde elde edilen lineer olmayan deklem sistemi, Newton-Raphson iterasyon yöntemi ile çözülmüştür.

During fracture of fiber-matrix composites, at least three fiber bridging processes can be operating. In this work, first, the elastic load transfer from partially embedded axialry loaded fibre to matrix is studied. It is assumed that the fiber and the matrix both behave elastic and are izotropic, the composite is orthotropic, there is no sliding between, the interfâcial stress is transfered by constant t" and the crack surface is perpendicular to the fiber axis. To simplify the problem, the model in Figure-1 is usedfl]. matrix Figure-1 A shematic drawing showing load transfer from the partially embedded, axialry loaded fibre in the matrix[15]. vu For elastic bridging/when the partially embedded fiber is subjected to an axial stress, a", the axial stress distribution in the fiber, Cf(y), and the interfacial shear stress, Zi(y), are; E"h crf(y) = Ef a (-T-1) sinh(wy) sinh(m/) + 1 i + ^(4-D Ef V (D r.Cy) = -f(y-D^JO+v.)i t2 ~t\( rr 1.2 "N 2 A -£/ " y cosh(my) ( sinh(w/) (2) where; m2 = 1 + ^4-1) (l + vm)[bzln(b/a)-^-^- (3) a is the radius of the fiber, 1 is the embedded length of the fiber( 1> 10a), [5] E and v are Young's modulus and Poisson's ratio, respectively. The subscripts f, m and c denote the matrix, the fiber and the composite, respectively. The displacement in the fiber can be obtained from integration of the strain in the fiber. With the condition that eml))\))e~ml for /»10a, [5] the half opening displacement, u, is; o=av (4) where; a = 2EfEc aVaEa\{\+vmX-\nVf-Vm) (5) vin Second,the frictional bridging is studied. For partially unbonded fiber- matrix interfaces, frictional stretching-sliding occurs at the interface along the unbonded length of the fiber when the fiber is subjected to tension and remains intact during fracture of the composite. Along the sliding length of the fiber, the interfacial stress is not constant, but, to simplify the analysis, a constant interfacial frictional stress is assumed. The stress versus half opening displacement relationship is; °0=PJ^ (6) where; 0=2\HHr (7) Third, the pull-out bridging is studied. For unbonded interfaces, fibers are pulled out when they are fractured within the matrix during cracking of the composite. With the assumption of a constant frictional stress, x, within the pull-out zone, the stress in the fiber at the crack surface is (for two sided pull-out); a=mzA (8) p is the pull-out length of the fiber. If pull-out occurs at one side of the crack surface, u should be replaced by 2u in Eq. (8). The bridging stress along the crack surface P(x), can be determined by multiplying the stress in the fiber,Oo, with the volume fraction, W [5]. P(x) = V,a0 = Kv(x) (9) Where, K is called the spring coefficient and u(x) is the opening of the crack. Thus, the bridging stress in elastic and pull-out bridging is linear but the bridging stress in frictional bridging is in a nonlinear relation with the crack opening. The governing equation of an infinite orthotropic plate with a central crack extending from -c to c, in terms of the stress function F(x,y) is as follow, IX 1 â4F 1 â4F ?+ - + Ex ây* Ey âx4 1 2v *y *y â4F Ex J âx2dy2 = O (10) where Gm and Gf are the shear modulus of the matrix and fiber, respectively. To formulate the crack problem, the crack surface derivative is taken as the unknown function, jy">0 ax (11) After solving the diferencial equation in Eq.(9) using Fourier transforms, the stresses are written in terms of the unknown function f(t) as follows; 2,-2 c wx(t-x) w2(t-x) a =± w^2 f W-*> ^~x> \mdt ' 7ca22(w\-w22)> Wy+it-xf w22y2+(t-x)2)JKJ (12) " * «22 (W - w2) J {w2y2 +(t- x)2 w,V+(r-*)a/w (13) 2...2 c. ^ = i ^1^2 r ^y * an (W - w22 ) J I w2/ + (* - x) "^ 2...2..2 * (14) Finally, the unknown function, f(t), is solved for elastic bridging. Consider a finite plate under a uniform tension, o, having a central crack. Applying the bridging stresses, which are the functions of the half opening displacement, the integral equation can be written as; 1 E.wtw% \ fit) 1 £.,W,W2 e Til) j, ",. x(wl+w2)S(t-x) (15) To complete the formulation of the problem, single- valuedness condition for the displacement should be written as follow; f/<0* = 0 (16) The crack opening displacement can be determined by integrating Eq.(l 1) from x to c. i*x) = -]f{t)dt (17) To solve single integral equation and the single-valuedness condition togather in (15) and (16), first they must be normalized by the following transforms: x-cx -c(x,t(c t = ct - \{x,t{\ /(>) = /*(') The unknown normalized function has singularity at both ends, thus following can be written as: f\t) = J^= (18) VI-*2 Eqn.(17) is substituted in Eq.(15),then the transforms in Eq.(18) are substituted in Eq.(15). And Eq.(18) is substituted in Eq.(15),then two sides of the normalized integral equations are divided by o0 and c to make them out of units. To solve the singular integral equation and the single valuedness condition Gauss-Chebyshev numerical integration method is used. After these operations, obtained system of algebraic equations is as follows: S JİT? - =f +~ ZF('/K = -1 i=l,2,...,(n-l);j=l,2,...,n n a (w,+w2)^(0-xf) af } J XI J^Fitj^j = O i=n ; j=l,2,...,n (19) where; y = 2,...,»-l *'"{*l£% (20) J = !,...,« i = !,...,»-! Here, ©j is the weight function of Chebyshev Polynominals of the first kind; Xj is the zeros of Chebyshev Polynominals of the second kind; t,, is the zeros of Chebyshev Polynominals of the first kind. But, the zeros used in this study are the ones that F. Erdoğan has developed[12]. First, an elastic bridging part-through crack problem is considered and Ku(x) in Eq.(16) is replaced by the expression in Eq.(4), yields; \EC wtw2 fFjtj)^ iCV,Ee 2EfEe * i=l,2,...,(n-l);j=l,2 n JtF{tj)a>j = 0 i=n ;rU,.,n (21) Second, a frictional bridging part-through crack problem is considered, Ku(x) in Eq.(16) is replaced by the expression in Eq.(6), yields; i=l,2,...,(n-l)j=l,2,...,n xn £F6)û>,.=0 i=n;j=lA-,n (22) Third, a pull-out bridging part-through crack problem is considered, Ku(x) in Eq.(16) is replaced by the expression in Eq.(8), yields; i=l,2,...,(n-l);H,2,...,n ^fOj^uj =0 i=n ; j=l,2,...,n (23) These systems of algebraic equations in (21), (22) and (23) are solved by the Fortran programs written in EkA, EkB and EkC for the normalized stress intesity factors at crack tips and the crack opening displacements. For frictional bridging, in Eq.(22) the system of algebraic equation is non-linear and is solved by applying Newton-Raphson iteration method[13]. For each bridging, normalized half crack opening due to normalized crack length graphics are shown in Figure- 1, Figure-2 and Figure-3. And the normalized stress intesity factors at crack tips are shown in Table-1. In this study, SiC-fiber reinforced AI2O3 composite is used[S]. The properties of materials used are; E* « 400 GPa, Ef = 500 GPa, vf = 0.3 vm = 0.25; the volume fraction of fiber, Vf = 0.6; total bridging zone length is lmm, (i.e. c=0.5mm); constant shear stress, x = 50MPa; radius of fibers, a = 0.3 um; pull-out length, p - 1.2um; and the stress applied to the composite, a - 4GPa[5]. The stress intensity factor at crack tips is defiend as follow[10]; k(c) = Km, _, J2(x-c)<7(x,0) (24) Substituting Eq.(13) for y = 0 and Eq.(18) in Eq.(24), the stress intensity factor at crack tip; k(c) = -^ T*w* F(l) (25) an{wl +w2) xin 0,000007t u(x)'c 0,000008 0,000005 0,000004 0,000003 0,000002 0,000001 Elastic bridging -? x/c -1 -0,5 0.5 Figure-2 Normalized half crack opening along normalized crack length for elastic bridging. 0iooi i »Wc 0,0012- 0,0010 0,0008 0,0006 0,0004 0.0002- Frictional bridging _? x/c -0,5 0,5 Figure-3 Normalized half crack opening along normalized crack length for frictional bridging.