COMPUTATIONAl ATERIALS SCIENCE ELSEVIER Computational Materials Science 4(1995)249-262 FE-modelling of the toughening mechanism in whisker reinforced ceramIc-matrIx-composites A Mukherjee"and H s Rac Received 14 August 1995; accepted 31 August 1995 Ceramic-matrix-composites(CMCs)are fast replacing other materials in many applications where the higher production costs can be offset by significant improvement in performance. However due to their lack of toughness they are prone to catastrophic failures. Hence, the important consideration in the design of CMcs is to achieve satisfactory toughening. Recent experimental studies of whisker-reinforced ceramics have shown that substantial improvements in fracture toughness can be achicved via the mechanisms of whisker bridging, whisker pull-out and crack deflection. This paper demonstrates the use of finite element method to model the whisker/ matrix interface through isoparametric interface elements. A micro mechanical finite element model which uses isoparametricformu non-linear force-displacement response of a-SiC ceramic whisker embedded in Al 2 O, ceramic matrix and coin s the ation has been presented. The micro mechanical finite clement model is then validated by characterisin ing it with the simplified analytical solutions, The FE model is then used to demonstrate the effect of whisker/ma trix interface shear strength on the toughening behaviour, and failure modes of SiC whisker reinforced Al, O, (m trix)/SiC (whisker) ceramic composite 1. Introduction hiker/ matrix interface has an important bear The ceramic-matrix-composites (CMCs)are ing on the mechanical properties of the CMcs ideal as structural material in many respects. Due Recent experimental studies of whisker-rein to their lack of toughness, however, they are forced ceramics have shown that substantial im prone to catastrophic failures. Therefore, the rovements in fracture toughness can be achieved main consideration in development and design of he mechanisms of whisker bridging, whisker ceramic-matrix-composites is to toughen them, so pull-out and crack deflection [1-5]. These mecha that, the attractive high temperature strength and nisms and hence the mechanical properties of environmental resistance offered by these materi- ceramic-matrix-composites are greatly influenced by the whisker/matrix interface strength whisker/ matrix interface should be weak and Analytical studies of toughening mechanisms should allow debonding at the interface Thus the cated due to the involvement of numerous pa rameters. Such analytical studies [6, 7] often rely (e.g. steady 0927-0256/95/$0950 9 1995 Elsevier Science B V. All rights Icseived SSD0927-0256(95)00049.6
ELSEVIER Computational Materials Science 4 (1995) 249-262 COMPUTATIONAL MATERIALS SCIENCE FE-modelling of the toughening mechanism in whisker reinforced ceramic-matrix-composites A. Mukherjee * and H.S. Rao Department of Civil Engineering, Indian Institute of Technology, Powai, Bombay 400 076, India Received 14 August 1995; accepted 31 August 1995 Abstract Ceramic-matrix-composites (CMCs) are fast replacing other materials in many applications where the higher production costs can be offset by significant improvement in performance. However, due to their lack of toughness they are prone to catastrophic failures. Hence, the important consideration in the design of CMCs is to achieve satisfactory toughening. Recent experimental studies of whisker-reinforced ceramics have shown that substantial improvements in fracture toughness can be achieved via the mechanisms of whisker bridging, whisker pull-out and crack deflection. This paper demonstrates the use of finite element method to model the whisker/matrix interface through isoparametric interface elements. A micro mechanical finite element model which uses isoparametricformulation has been presented. The micro mechanical finite element model is then validated by characterising the non-linear force-displacement response of a-Sic ceramic whisker embedded in Al,O, ceramic matrix and comparing it with the simplified analytical solutions. The FE model is then used to demonstrate the effect of whisker/matrix interface shear strength on the toughening behaviour, and failure modes of SIC whisker reinforced Al,O,(matrix)/SiC (whisker) ceramic composite. 1. Introduction The ceramic-matrix-composites (CMCs) are ideal as structural material in many respects. Due to their lack of toughness, however, they are prone to catastrophic failures. Therefore, the main consideration in development and design of ceramic-matrix-composites is to toughen them, so that, the attractive high temperature strength and environmental resistance offered by these materials can be exploited. To achieve this the whisker/matrix interface should be weak and should allow debonding at the interface. Thus the * Corresponding author. whisker/matrix interface has an important bearing on the mechanical properties of the CMCs. Recent experimental studies of whisker-reinforced ceramics have shown that substantial improvements in fracture toughness can be achieved via the mechanisms of whisker bridging, whisker pull-out and crack deflection [l-5]. These mechanisms and hence the mechanical properties of ceramic-matrix-composites are greatly influenced by the whisker/matrix interface strength. Analytical studies of toughening mechanisms in ceramic-matrix-composites are rather complicated due to the involvement of numerous parameters. Such analytical studies [6,7] often rely on many simplifying assumptions (e.g. steady state 0927-0256/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0927-0256(95)00049-6
AMukherjee, H.S. Rao/ Computational Materials Science 4(1995)249-262 cracking, one dimensional strain analysis etc. )Lo 2. The finite element model ake the problem amenable to solution. Finite element studies of wake toughening mechanisms In this paper, a two dimensional plane are scant [8] and they address only the behavi stress/strain finite element model is developed of a single fibre near the crack tip in a centrally for the analysis of CMCs. Though, these 2D cracked plate under mode I loading at loads less models are approximations of the real 3D com- than the failure load of the fibre. Laird Il and posites, they can still provide a good insight into Kennedy [9] used discrete spring slider elements the toughening mechanisms and their depen to model the interfacial connectivity between the dence on the whisker/matrix interface strength matrix and whisker in the finite element analysis The salient features of the model have been of CMCs. This approach models the connectivity between the whisker and matrix at discrete nodal Developing an adequate Fe model for the points only. As a result, the chances of overlap- analysis of ceramic-matrix-composites, involves ping of the nodes along the interface is not elimi- modelling of: (a) the matrix (b) fibre or whisker lated. Further, this type of idealisation of inter- and (c) the interface between the two materials face considers only the shear failure of the inter- (Fig. elling the mechanics of whisker/matrix face and cannot consider the tensile (normal) failure of the interface which is also possible interface is rather complex. The complexity is due CMCS. Therefore, this approach cannot model to the possible relative displacements at the bi the interface taking into account the interfacial material interface further, if the interface is in mechanics realistically compression, a frictional force between the two In the present work a llicIOnechanical finite Inaterial exists. This frictional grip betweell the element model is developed to analyse the Cmos, matrix and the whisker resists the sliding of the taking the whisker/ matrix interface into effect. whisker against a cracked matrix. If this frictional The matrix, and fibre or whisker are modelled by force is directly considered in the analysis, the eight node isoparametric quadrilateral elements. system becomes non-conservative and the global The whisker/matrix interface is modelled by six stiffness matrix in finite element analysis becomes noded isoparametricinterface elements. Eigh un-symmetric. Such un-symmetric matrices pose noded isoparametric quadrilateral elements are difficulties in the solution routines, resulting ill used for modelling the matrix as well as the numerical instability. On the other hand, if the hiker. As elements of compatible shape func- interface is in tension, the contact between the tions are used to model all the components, i.e., matrix and the whisker ceases to exist and hence whisker, matrix and the interface the interaction there will not be any frictional force. Hence it has between them is modelled realistically. moreover, this model the whisker/matrix interface is modelled all along the interface instead of dis hiker Whisker crete nodal points. The model is capable of con sidering both shear and normal failures of the interface. The possibility of overlapping is also MatrixMatr effectively eliminated. The model is then vali- dated by characterising the non-linear force-dis placement response of a-SiC ceramic whisker cmbcddcd in Al,O3 ceramic matrix and compar- ing it with the simplified analytical solutions. The FE model is then used to demonstrate the effect of whisker/ matrix interface strength on the toughening behaviour of the Al2O,(matrix)/Sic Fig. 1. Longitudinal section of a typical CMC showing the hiker )ceramic-matrix-composite whisker, matrix and interface
250 A. Mukherjee, H.S. Rae/Computational Materials Science 4 (1995) 249-262 cracking, one dimensional strain analysis etc.) to make the problem amenable to solution. Finite element studies of wake toughening mechanisms are scant 181 and they address only the behaviour of a single fibre near the crack tip in a centrally cracked plate under mode I loading at loads less than the failure load of the fibre. Laird II and Kennedy [9] used discrete spring slider elements to model the interfacial connectivity between the matrix and whisker in the finite element analysis of CMCs. This approach models the connectivity between the whisker and matrix at discrete nodal points only. As a result, the chances of overlapping of the nodes along the interface is not eliminated. Further, this type of idealisation of interface considers only the shear failure of the interface and cannot consider the tensile (normal) failure of the interface which is also possible in CMCs. Therefore, this approach cannot model the interface taking into account the interfacial mechanics realistically. In the present work a micromechanical finite element model is developed to analyse the CMCs, taking the whisker/matrix interface into effect. The matrix, and fibre or whisker are modelled by eight node isoparametric quadrilateral elements. The whisker/matrix interface is modelled by six noded isoparametricinterface elements. Eight noded isoparameteric quadrilateral elements are used for modelling the matrix as well as the whisker. As elements of compatible shape functions are used to model all the components, i.e., whisker, matrix and the interface, the interaction between them is modelled realistically. Moreover, in this model the whisker/matrix interface is modelled all along the interface instead of discrete nodal points. The model is capable of considering both shear and normal failures of the interface. The possibility of overlapping is also effectively eliminated. The model is then validated by characterising the non-linear force-displacement response of a-Sic ceramic whisker embedded in Al,O, ceramic matrix and comparing it with the simplified analytical solutions. The FE model is then used to demonstrate the effect of whisker/matrix interface strength on the toughening behaviour of the Al,O, (matrix)/SiC (whiskerjceramic-matrix-composite. 2. The finite element model In this paper, a two dimensional plane stress/strain finite element model is developed for the analysis of CMCs. Though, these 2D models are approximations of the real 3D composites, they can still provide a good insight into the toughening mechanisms and their dependence on the whisker/matrix interface strength. The salient features of the model have been described below. Developing an adequate FE model for the analysis of ceramic-matrix-composites, involves modelling of: (a) the matrix, (b) fibre or whisker, and (c) the interface between the two materials (Fig. 1). Modelling the mechanics of whisker/matrix interface is rather complex. The complexity is due to the possible relative displacements at the bimaterial interface. Further, if the interface is in compression, a frictional force between the two material exists. This frictional grip between the matrix and the whisker resists the sliding of the whisker against a cracked matrix. If this frictional force is directly considered in the analysis, the system becomes non-conservative and the global stiffness matrix in finite element analysis becomes un-symmetric. Such un-symmetric matrices pose difficulties in the solution routines, resulting in numerical instability. On the other hand, if the interface is in tension, the contact between the matrix and the whisker ceases to exist and hence there will not be any frictional force. Hence it has Whisker Fig. 1. Longitudinal section of a typical CMC whisker, matrix and interface. showing the
A. Mukherjee, H.S. Rao/Computational Materials Science 4(1995) 249-262 been rather difficult to model the whisker/matrix displacement based FE formulation uf isopard interface in the finite element analysis of ce- metric interface element, the compatibility condi mic-matrix-composites. In the present work the tions are satisfied along the entire length of the ceramic matrix and the ceramic whisker are mod- interface. moreover, the formulation of isopara elled by standard eight node isoparametric metricinterface element is an integral part of the quadrilateral elements (see Fig. 2(a). However, FE model. to model the whisker/ matrix interface, the me Thus, in the present finite element model of chanics of behaviour at the interface are treated the CMCs, the matrix and whisker are Modelled to be similar to that of rock joints, where relative by eight node isoparametric quadrilateral ele displacements occur across a thin discontinuity. ments and the whisker/matrix interface is mod Interface elements were introduced in the area of elled by the six noded isoparametric interface ock mechanics by Goodman et al. [10] and they element. These isoparametric interface elements have been used in a variety of problems ever model the connectivity between the ceramic ma since. The present work uses the isoparametric- trix and ceramic fibre or whisker allowing slip formulation presented by Buragohain and Shah when the inter face shear stress exceeds the inter [11] for interface elements(see Appendix). In this face shear strength(s)of the ceramic composite work, the idea of using isoparametricinterface At each load increment iterative FE analysis has elements for FE modelling of the whisker /matrix to be carried out to account for shear failure or interface in ceramic-matrix-composites has been separation at the interface. Initially, the value of demonstrated. The whisker/matrix interface has shear stiffness(Ks) is set to a very large value to been modelled by six node isoparametricinterface avoid any non-slip displacement. When the inter- elements(see Fig. 2(b)). This isopararmetricinter- face shear stress (os)at a point exceed face line element with quadratic variation of both interface strength, the shear stiffness(K )of that geometry and slip uses the relative displacements point is set to zero, thus, allowing slip. If the between the adjacent matrix and whisker ele- normal stress(on) at the interface is compressive ments as its degrees of freedom. therefore, the there is a possibility of overlapping of the nodes use of interface element for modelling the of the adjacent elements. To avoid this, the value whisker/matrix interface does not introduce any of normal stiffness(K, )of the interface elements editional degrees of freedom in the probleM. is set to be about thousand Lines the ImlaximuIml Further, the present model of CMCs is superior value of Youngs modulus of the bordering ele- to other discrete spring models [9] because in the ments, in compression. When the normal stress Element B 2.(a)Eight node iso-p tric quadrilateral element used for modelling the whisker and matrix.(b) Six noded isoparametric ce element used for modelling the bi-material i
A. Mukherjee, H.S. Rao / Computational Materials Science 4 (1995) 249-262 251 been rather difficult to model the whisker/matrix interface in the finite element analysis of ceramic-matrix-composites. In the present work the ceramic matrix and the ceramic whisker are modelled by standard eight node isoparametric quadrilateral elements (see Fig. 2(a)). However, to model the whisker/matrix interface, the mechanics of behaviour at the interface are treated to be similar to that of rock joints, where relative displacements occur across a thin discontinuity. Interface elements were introduced in the area of rock mechanics by Goodman et al. [lo] and they have been used in a variety of problems ever since. The present work uses the isoparametricformulation presented by Buragohain and Shah [ll] for interface elements (see Appendix). In this work, the idea of using isoparametricinterface elements for FE modelling of the whisker/matrix interface in ceramic-matrix-composites has been demonstrated. The whisker/matrix interface has been modelled by six node isoparametricinterface elements (see Fig. 2(b)). This isoparametricinterface line element with quadratic variation of both geometry and slip uses the relative displacements between the adjacent matrix and whisker elements as its degrees of freedom. Therefore, the use of interface element for modelling the whisker/matrix interface does not introduce any additional degrees of freedom in the problem. Further, the present model of CMCs is superior to other discrete spring models [9] because in the displacement based FE formulation of isoparametric interface element, the compatibility conditions are satisfied along the entire length of the interface. Moreover, the formulation of isoparametricinterface element is an integral part of the FE model. Thus, in the present finite element model of the CMCs, the matrix and whisker are modelled by eight node isoparametric quadrilateral elements and the whisker/matrix interface is modelled by the six noded isoparametric interface element. These isoparametric interface elements model the connectivity between the ceramic matrix and ceramic fibre or whisker allowing slip when the interface shear stress exceeds the interface shear strength (7,) of the ceramic composite. At each load increment iterative FE analysis has to be carried out to account for shear failure or separation at the interface. Initially, the value of shear stiffness (K,) is set to a very large value to avoid any non-slip displacement. When the interface shear stress (a,) at a point exceeds the interface strength, the shear stiffness (KS1 of that point is set to zero, thus, allowing slip. If the normal stress (a,) at the interface is compressive there is a possibility of overlapping of the nodes of the adjacent elements. To avoid this, the value of normal stiffness (K,) of the interface elements is set to be about thousand times the maximum value of Young’s modulus of the bordering elements, in compression. When the normal stress lb) Fig. 2. (a) Eight node iso-parametric quadrilateral element used for modelling the whisker and matrix. (b) Six noded isoparametric interface element used for modelling the bi-material interface
252 A Mukherjee, H.S. Rao /Computational Materials Science 4(1995)249-262 (o)of a point exceeds the interface normal strength, the normal stiffness(K)and the shear stiffness(K, of that point is set to zero to allow debonding at the interface. Thus the algorithm of the interface element takes the value of interface stiffness as NO SLIP 1)K,=0, if o> interface shear strength(slip condition): )Kn=10 Em, if o <0(overlapping con dition) 3)K=0, and K.=0 if o> interface normal strength(debonding condition) The iterative finite element analysis has to be carried out to account for shear failure or separa tion at the interface. Though this FE solution is computationally expensive it is still a powerful ool for performing numerical experiments and Fig. 3. Model of whiskers embedded in a matrix with an sensitivity studies on CMCs. However, the pre sent author gaged in the devclopment or where of the whisker, T is the interface shear an artificial neural network approach which gen cralises the limited responses obtained from the strength finite element analysis, by adaptively learning the Fig. 3 shows such a case with slip occurring constitutive relation. This will provide a fast and over a distance L along the matrix/whisker inte computationally economical tool for conducting face causing a displacement u of the atrix due sensitivity studies on CMCs. to some applied stress. Marshall and Cox [6] developed a simple analytical model to calculate this displacement, u, for infinitely long fibres 3. Validation of the Fe model based on the premise that a region of strain continuity or no-slip exists above the slipped in The FE model has been validated by charac terface. Their expression is given below sponse of the a-siC ceramic long and short p=2LuTVTEr(1+n)/R whiskers embedded in Al,o, ceramic matrix with wher EVI/Emm, p is the force per unit the present finite element model. These force- length, T is the interface shear stress, Er is displacement responses have been compared with Young s modulus of the fibre, em is the Youngs a simplified analytical solution, and also with the solution presented by Laird II and Kennedy [9]. whiskers, Vm is the volume fraction of the matrix, R is the radius of the fibre and v is the poisson's 3. Analytical model Based on the above expression Laird Il and For the case of whiskers bridging a crack, slip Kennedy [9] derived the following relation be ill occur for some distance along the tween the force on the whisker (F)and displace whisker/matrix interface, provided the whisker ment u for plane strain has minimum length required for load transfer to W EH(1-)2+1, the whisker by interfacial shear. The critical F=2Erm-v2)IEm m(I-m) length required for such load transfer can be calculated from the following equation. L=01r/7, (1) where
252 A. Mukherjee, H.S. Rao /Computational Materials Science 4 (1995) 249-262 (a,) of a point exceeds the interface normal strength, the normal stiffness (K,) and the shear stiffness (K,) of that point is set to zero to allow debonding at the interface. Thus the algorithm of the interface element takes the value of interface stiffness as: 1) K, = 0, if a, > interface shear strength (slip condition); 2) K, = 10’ X E,, if a, < 0 (overlapping condition); 3) K, = 0, and K, = 0 if a,, > interface normal strength (debonding condition). The iterative finite element analysis has to be carried out to account for shear failure or separation at the interface. Though this FE solution is computationally expensive, it is still a powerful tool for performing numerical experiments and sensitivity studies on CMCs. However, the present authors are engaged in the development of an artificial neural network approach which generalises the limited responses obtained from the finite element analysis, by adaptively learning the constitutive relation. This will provide a fast and computationally economical tool for conducting sensitivity studies on CMCs. 3. Validation of the FE model The FE model has been validated by characterising the non linear force-displacement response of the a-Sic ceramic long and short whiskers embedded in Al,O, ceramic matrix with the present finite element model. These forcedisplacement responses have been compared with a simplified analytical solution, and also with the solution presented by Laird II and Kennedy [9]. 3.1. Analytical model For the case of whiskers bridging a crack, slip will occur for some distance along the whisker/matrix interface, provided the whisker has minimum length required for load transfer to the whisker by interfacial shear. The critical length required for such load transfer can be calculated from the following equation. L, = u,r/r, (1) f \ F F Fig. 3. Model of whiskers embedded in a matrix with an applied stress at a distance far from the crack plane. where a, is the whisker fracture stress, r is the radius of the whisker, r is the interface shear strength. Fig. 3 shows such a case with slip occurring over a distance L along the matrix/whisker interface causing a displacement u of the matrix due to some applied stress. Marshall and Cox [6] developed a simple analytical model to calculate this displacement, U, for infinitely long fibres based on the premise that a region of strain continuity or no-slip exists above the slipped interface. Their expression is given below. p = 2[ U7V;2Ef(1 + 7))/R]“*, (2) where 77 = E,V,/E,V,, p is the force per unit length, r is the interface shear stress, E, is Young’s modulus of the fibre, E, is the Young’s modulus of matrix, V, is the volume fraction of whiskers, V, is the volume fraction of the matrix, R is the radius of the fibre, and v is the Poisson’s ratio. Based on the above expression Laird II and Kennedy [9] derived the following relation between the force on the whisker (F) and displacement cf for plane strain. where
A. Mukherjee, H.S. Rao/ Computational Materials Science 4(1995)249-26 w is the width of the whisker; and Wm is the width of the matrix between whiskers is believed to be cable only for long whiskers having length greater than Le, since a no slipped interface occurs above he slipped interface before the whisker fractures However, this requirement of no-slip does not nterface model the actual behaviour, in cases where slip can occur along the entire length of the whisker/matrix interface, before the whisker fiske pull-out occurs. Further this analytical model is developed for a purely frictional bond between the matrix and the whisker. as a result, the model is incapable of modelling a whisker debonding effect. This is because whiell a whisker lebonds, the contact between the whisker and matrix ceases to exist, resulting into a zero fric tional force. Thus, there is a need to develop a model which alleviates and capable of analysing CMCs more realistically 3.2. Finire element models Yy 09 Hm 0.3 Hm To validate the developed FE model, the non linear force-displacement response of the long Fig 4. Quarter-symmetry finite element model of long a-SiC and short a-SiC ceramic whiskers embedded in whisker embedded Al2O, ceramic matrix have been obtained from the finite element analysis of the Al,O3(matrix)/ a-SiC (whisker) ceramic composite. Finite Ele- where, vr, Vm are the Poisson's ratios of the ment Models(FEM) of long and short whiskers whisker and matrix, respectively have been constructed using (2-D)plane strain With these values the critical length of the analysis. Though this 2-D analysis is an approxi a-SiC whiskers is calculated(Eq (1)as 2.58 ul mation of the real three-dimensional situation If the length of the whisker is greater than 2. 58 where the whiskers may be randomly oriented, um, (long whisker)it will be able to bridge the nevertheless this analysis should still provide a matrix crack till the stress in whisker reaches o meaningful insight into the upper limits of frac For whiskers that are shorter than 2.58 ull(short ture toughness achievable in this class of compos- whisker), the stress in the whisker will not reach ites with whiskers aligned in one principle direc- r, hut they will fail by pulling out of the matrix Material properties [9] used in this study are 3.2.1. Long whisker mode given belov Fig. 4 shows the quarter symmetry finite ele- Em=380 GPa, vm=0.22 for the Al2O3 matrix ment model used to develop the non- linear force/displacement responses of long c-SiC Er-=689 GPa, v=0. 22 for the a-SiC whisker whisker embedded in Al,0, matrix. The use of E:= 380 GPa, v=0.22 for the Al, O,/a-Sic quarter symmetry required that the nodes along of symmetry (x=1.2 um) be fixed in x-direction. The whisker is fixed at the bottom Ts -Interfacc shear strength=800.0 MPa (y=0). As this study is aimed to obtain the
A. Mukherjee, H.S. Rao / Computational Materials Science 4 (1995) 249-262 253 W, is the width of the whisker; and W, is the width of the matrix between whiskers. However, this expression is believed to be applicable only for long whiskers having length greater than L,, since a no slipped interface occurs above the slipped interface before the whisker fractures. However, this requirement of no-slip does not model the actual behaviour, in cases where slip can occur along the entire length of the whisker/matrix interface, before the whisker pull-out occurs. Further this analytical model is developed for a purely frictional bond between the matrix and the whisker. As a result, the model is incapable of modelling a whisker debonding effect. This is because when a whisker debonds, the contact between the whisker and matrix ceases to exist, resulting into a zero frictional force. Thus, there is a need to develop a model which alleviates the above shortcomings and capable of analysing CMCs more realistically. 3.2. Finite element models To validate the developed FE model, the nonlinear force-displacement response of the long and short a-Sic ceramic whiskers embedded in Al,O, ceramic matrix have been obtained from the finite element analysis of the Al,O, (matrix)/ a-Sic (whisker) ceramic composite. Finite Element Models (FEM) of long and short whiskers have been constructed using (2-D) plane strain analysis. Though this 2-D analysis is an approximation of the real three-dimensional situation where the whiskers may be randomly oriented, nevertheless this analysis should still provide a meaningful insight into the upper limits of fracture toughness achievable in this class of composites with whiskers aligned in one principle direction 191 Material properties [91 used in this study are given below: E, = 380 GPa, v, = 0.22 for the Al,O, matrix E, = 689 GPa, vf = 0.22 for the a-Sic whisker E, = 380 GPa, V, = 0.22 for the AIZO,/cr-Sic composite rs = Interface shear strength = 800.0 MPa Fig. 4. Quarter-symmetry finite element model of long cu-Sic whisker embedded in Al,O, matrix. where, vr, v, are the Poisson’s whisker and matrix, respectively. With these values the critical ratios of the length of the (Y-Sic whiskers is calculated (Eq. (1)) as 2.58 n,rn. If the length of the whisker is greater than 2.58 Frn, (long whisker) it will be able to bridge the matrix crack till the stress in whisker reaches or. For whiskers that are shorter than 2.58 urn (short whisker), the stress in the whisker will not reach af, but they will fail by pulling out of the matrix. 3.2.1. Long whisker model Fig. 4 shows the quarter symmetry finite element model used to develop the non-linear force/displacement responses of long a-Sic whisker embedded in Al,O, matrix. The use of quarter symmetry required that the nodes along the axis of symmetry (x = 1.2 r*.rn) be fixed in x-direction. The whisker is fixed at the bottom (y = 0). As this study is aimed to obtain the