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 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, 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
254 A. Mukherjee, H.S. Rao/ Computational Materials Science 4(1995)249-262 I-linear force-displacement Response of whiskers as they pull-out from the cracked ma trix, the matrix portion at y=0 has been kept free The long whisker has been modelled as havin length of 30 um and half width of 0.3 um and is connected to a matrix that is 0.9 um wide by o-node isoparametric interface elements. These 148μm dimensions give a 25 percent whisker reinforce ment. The 150 matrix and 50 whisker elements are 8-node isoparametric quadrilateral elements and have side dimensions of0.6×0.3μm.The50 Whisker interface elements model the connectivity be- tween the matrix and the whisker allowing slip 2.59μm when the interface shear stress exceeds the inter face shear strength(800 MPa)of the composite Initially the values of shear stiffness(, of these interface elements is set Em x10 to limit any non-slip displacement. The 03m force has been applied on the composite in steps and the stresses at the interface have been moy tored. when the interface shear stress exceeds he interface strength(800 MPa), Ks is reduced to zero to allow slip. to avoid overlapping of the adjacent nodes in compression, the value of not Fig. 5. Quarter-symmetry finite element model of short a-SiC mal stiffness(K) of these interface elements whisker embedded in Al2O3 matrix. aken to be about 1000 times the maximum value of Youngs modulus of the bordering element. explained for long whisker model Mesh conver Loads were applied incrementally and the re- gence was checked in a similar manner as in the sponse has been obtained Mesh convergence for long whisker which indicated the present mesh is his long whisker model has been investigated by a stable configuration. refining the grid shown in Fig 4 from the present 50 to 100 elements along the whisker/matrix 3. 3. Comparison with the existing models interface. The displacements of selected nodes were found to change less than 3 as the mesh Fig 6 shows the comparison of the computed was refined, indicating that the coarse mesh rep- force/displacement curve for long whisker with resented a stable configuration the analytical solution(Eq. (3)) and with that obtained by Laird II and Kennedy [9]. The dis 3. 2. Short whisker model placement values along the y-axis were calculated as the difference of displacements of nodes at the The short whisker shown in Fig. 5 has a whisker whisker/matrix interface(x=0.9 um) and at the length of 2.58 um. This model uses 210 matrix matrix axis of symmetry (x=0.0 um). The diver and 70 whisker eight node quadrilateral clemen gence of the force/displacement curve obtained with element dimensions of 0.074 x 0.15 um along from the present study from the analytical solu he whisker/matrix interface. Thirty-five 6-node tion(Eq. (3)) is due to simplifying assumptions isoparametric interface elements have been used made in the analytical solution i.e. strains/dis- to model the whisker/matrix interface Iterations placements are 1-D, with the matrix having a are carried out for convergen ce similar to that uniform displacement along its bottom edge
254 A. Mukherjee, H.S. Rao / Computational Materials Science 4 (1995) 249-262 non-linear force-displacement response of whiskers as they pull-out from the cracked matrix, the matrix portion at y = 0 has been kept free. The long whisker has been modelled as having a length of 30 urn and half width of 0.3 pm and is connected to a matrix that is 0.9 p,rn wide by 6-node isoparametric interface elements. These dimensions give a 25 percent whisker reinforcement. The 150 matrix and 50 whisker elements are &node isoparametric quadrilateral elements and have side dimensions of 0.6 x 0.3 urn. The 50 interface elements model the connectivity between the matrix and the whisker, allowing slip when the interface shear stress exceeds the interface shear strength (800 MPa) of the composite. Initially the values of shear stiffness (KS) of these interface elements is set equal to a value of E, x lo7 to limit any non-slip displacement. The force has been applied on the composite in steps and the stresses at the interface have been monitored. When the interface shear stress exceeds the interface strength (800 MPa), K, is reduced to zero to allow slip. To avoid overlapping of the adjacent nodes in compression, the value of normal stiffness (K,) of these interface elements is taken to be about 1000 times the maximum value of Young’s modulus of the bordering element. Loads were applied incrementally and the response has been obtained. Mesh convergence for this long whisker model has been investigated by refining the grid shown in Fig. 4 from the present 50 to 100 elements along the whisker/matrix interface. The displacements of selected nodes were found to change less than 3% as the mesh was refined, indicating that the coarse mesh represented a stable configuration. 3.2. Short whisker model The short whisker shown in Fig. 5 has a whisker length of 2.58 km. This model uses 210 matrix and 70 whisker eight node quadrilateral elements with element dimensions of 0.074 X 0.15 km along the whisker/matrix interface. Thirty-five 6-node isoparametric interface elements have been used to model the whisker/matrix interface. Iterations are carried out for convergence similar to that > Matrix >Interface L x . LI Fig. 5. Quarter-symmetry finite element model of short a-Sic whisker embedded in Al,O, matrix. explained for long whisker model. Mesh convergence was checked in a similar manner as in the long whisker which indicated the present mesh is a stable configuration. 3.3. Comparison with the existing models Fig. 6 shows the comparison of the computed force/displacement curve for long whisker with the analytical solution (Eq. (3)) and with that obtained by Laird II and Kennedy 191. The displacement values along the y-axis were calculated as the difference of displacements of nodes at the whisker/matrix interface (X = 0.9 km) and at the matrix axis of symmetry (X = 0.0 km). The divergence of the force/displacement curve obtained from the present study from the analytical solution (Eq. (3)) is due to simplifying assumptions made in the analytical solution i.e. strains/displacements are l-D, with the matrix having a uniform displacement along its bottom edge
A Mukherjee, H.S. Rao/Computational Materials Science 4(1995)249-4 short whisker long whisker Ref 0.0030 00035 0.0020 0008 0.0020 Displacement, um 6. fea derive of long SiC whisker embedded in Al2O3 matrix 00015 0.0000.0020.0040.0060.0080010 which is not true. The present FE model predicts Disp LaiceaenL. HIn Fig 8. FEA derived force-displacement curves for long and stiffer behaviour than the analytical solution short a-sic whisker embedded in al o. mat which is expected as the analytical solution is developed for 1-d plane strain conditions. The marginal difference between the force/displace ment curve obtained in the present study and that obtained from reference [9] may be due to discrete spring model employed in reference [9] to model the whisker/matrix interface Fig. 7 shows the comparison of the computed force/displacement curve for short whisker with that obtained by Laird and Kennedy [9 There is no analytical solution available for short whisker 0.0040 Again the difference between the two curves may be attributed to the discrete spring connection between the matrix and whisker employed in their [9] study. From Fig. 6 and Fig. 7 it can be observed that the force-displacement response obtained from this study is softer than that de rived in reference [9] for both long and short 0.0025 Fig. 8 shows the comparison of force/displace 0.0020 ment curves for long and short whiskers with analytical solution for a long whisker(Eg. (3).It 0.0015 is apparent froilI Fig. 8 that the long and short whiskers have almost identic: force/displacement responses, except at values very near the force that would initiate fracture of Fig. 7. FEA derived non-linear force-displacement curve of the whisker(of). The same observation was re short a-SiC whisker embedded in Al2O3 matrix
A. Mukherjee, H.S. Rao /Computational Materials Science 4 (1995) 249-262 255 t 00020 t 0 000 0002 0 004 0006 0008 0010 Displacement, pm Fig. 6. FEA derived non-linear force-displacement curves of long SIC whisker embedded in Al,O, matrix. which is not true. The present FE model predicts stiffer behaviour than the analytical solution which is expected as the analytical solution is developed for 1-D plane strain conditions. The marginal difference between the force/displacement curve obtained in the present study and that obtained from reference [91 may be due to 0.0045 0.0040 z 0.0035 & :! 2 0.0030 0.0025 0.0020 0.00 I5 0 00 I I , I 0 0.002 0.004 0.006 0.008 Displncement, pm Fig. 7. FEA derived non-linear force-displacement curve of short U-SIC whisker embedded in Al,O, matrix. z 0.0030 6 2 9 0.0025 0.0020 0.0015 L I * 1 , I , I , I n.uuu U.UU? 0.004 0.006 0.008 0.010 Displacement. pm Fig. 8. FEA derived force-displacement curves for long and short a-Sic whisker embedded in Al,O, matrix. discrete spring model employed in reference [9] to model the whisker/matrix interface. Fig. 7 shows the comparison of the computed force/displacement curve for short whisker with that obtained by Laird and Kennedy [9]. There is no analytical solution available for short whisker. Again the difference between the two curves may be attributed to the discrete spring connection between the matrix and whisker employed in their [91 study. From Fig. 6 and Fig. 7 it can be observed that the force-displacement response obtained from this study is softer than that derived in reference [9] for both long and short whiskers. Fig. 8 shows the comparison of force/displacement curves for long and short whiskers with analytical solution for a long whisker (Eq. (3)). It is apparent from Fig. 8 that the long and short whiskers have almost identical non-linear force/displacement responses, except at values very near the force that would initiate fracture of the whisker (af). The same observation was reported by Laird and Kennedy 191
A. Mukherjee, H.S. Rao/Computational Materials Science 4(1995)249-262 4. ETect uf interface strength on toughening of CMC Material properties used for the present Al( iC CMC Property Matrix As discussed earlier, the whisker/matrix inter- face strength has an important bearing on the Youngs modulus )0.0 Gpa 600.0GPa oughening mechanics of the ceramic composites. 0.22 700 MPa The interface should not be too weak to effect a Tensile strength Radius 03μm tal whisker pull out at a very low stress. It should not be too strong to cause a brittle frac- ture. Therefore, the interface shear strength is to be engineered to an optimum level to ensure a crack deflection mechanism and a good work of matrix of 0.6 um width. These dimensions give a fracture. To demonstrate the effect of interface 33.3 percent of whisker reinforcement. The top shear strength on the toughening of CMCs, a matrix layer of 15 um length is provided to min single whisker quarter symmetry finite element imisc the local disturbances arising in the vicinity model of Al,O,(matrix)/SiC(whisker) ceramic of the load. on the mechanics of the interface. matrix-composite has been constructed(see Fig. The use of quarter symmetry required that the 9). This model has been analysed in(2-D) plane nodes along the axis of symmetry be fixed in x strain. Though the 2D analysis is an approxima- direction. This FE mesh consists of 150 matrix tion to the real 3D situation, but it still provides a and 75 whisker eight node quadrilateral ele- good insight. Material properties used in this ments. The whisker/matrix interface has been are modelled by 50 six-node isoparamctricintcrfacc The whisker has been modelled in quarter elements. Three interface elements are used at symmetry as having a half length of 30 um, and a the bottom axis of symmetry (y=0)as control half width of 0.3 um, and is connected to a elements. When the normal stress in these con- 个个个↑个 30 H m Matris whisker 06日l 0.3 Fig9(a)Quarter symmetry FE model of the Al,,/SiC ceramic composite (b)Finite element discretization of the CMc model
256 A. Mukherjee, H.S. Rao /Computational Materials Science 4 (1995) 249-262 4. Effect of interface strength on toughening of Table 1 CMC Material properties used for the present Al,O? /Sic CMC Property Matrix Whisker As discussed earlier, the whisker/matrix inter- (Al,O,) (SIC) face strength has an important bearing on the Young’s modulus 400.0 Gpa 600.0 GPa toughening mechanics of the ceramic composites. Poission’s ratio 0.22 0.22 The interface should not be too weak to effect a Tensile strength 700 MPa 6.89 GPa Radius total whisker pull out at a very low stress. It should not be too strong to cause a brittle fracture. Therefore, the interface shear strength is to be engineered to an optimum level to ensure a crack deflection mechanism and a good work of fracture. To demonstrate the effect of interface shear strength on the toughening of CMCs, a single whisker quarter symmetry finite element model of Al,O, (matrix)/SiC (whisker) ceramicmatrix-composite has been constructed (see Fig. 9). This model has been analysed in (2-D) plane strain. Though the 2D analysis is an approximation to the real 3D situation, but it still provides a good insight. Material properties used in this study are given in Table 1. The whisker has been modelled in quarter symmetry as having a half length of 30 km, and a half width of 0.3 pm, and is connected to a 0.3 pm Length 60 b matrix of 0.6 pm width. These dimensions give a 33.3 percent of whisker reinforcement. The top matrix layer of 15 pm length is provided to minimise the local disturbances arising in the vicinity of the load, on the mechanics of the interface. The use of quarter symmetry required that the nodes along the axis of symmetry be fixed in x direction. This FE mesh consists of 150 matrix and 75 whisker eight node quadrilateral elements. The whisker/matrix interface has been modelled by 50 six-node isoparametricinterface elements. Three interface elements are used at the bottom axis of symmetry (y = 0) as control elements. When the normal stress in these conFig. 9. (a) Quarter symmetry FE model of the Al,O, /Sic ceramic composite. (b) Finite element discretization of the CMC model
A. Mukherjee, H.S. Rao/ Computational Materials Science 4(1995)249-262 257 3000 IFS 3800 MPa FS 3000 MPa IFS 2600 MP 2500 IFS 2200 MPa 2000 0 IFS 800 MPa IFS= interface shear strength 0.0000.00200040.0060.0080.0100.012 STRAIN, E Fig. 10. FEA derive stress-strain curves of Al2 O,/SiC CMC at different inter face shear strengths. acked) IMrad Fig. 11.(a)Magnified deflected shape of the CMC with an IFS=800 MPa at an applied stress=1333.3 MPa depicting crack deflection mechanism. (b)Magnified deflected shape of the CMC with IFS=2200 MPa, at an applied stress=1556 MPa, depicting e whisker bridging the matrix crack. Observe the intac rface.(c)Magnified deflected shape of the CMC with IFS=3800 MPa at an applied stress= 2777. 8 MPa, depicting fibre fracture with very little debonding at the interface
A. Mukherjee, H.S. Rao / Computational Materials Science 4 (1995) 249-262 257 (a) 3000 2500 IFS 3800 MPa 1FS3000MPaIFS2600MP~ IFS 2200 MPa “z = 2000 ,. 2 IS00 E IO00 IFS 800 MPa 5 00 .-i IFS = Interface shear strenglh / I I I I I I 0 f 0.000 0.002 0.004 0.006 0.008 0.010 0.012 STRAIN, E Fig. 10. FEA derived stress-strain curves of Al,O, /Sic CMC at different interface shear strengths Fig. 11. (a) Magnified deflected shape of the CMC with an IFS = 800 MPa at an applied stress = 1333.3 MPa depicting crack deflection mechanism. (b) Magnified deflected shape of the CMC with IFS = 2200 MPa, at an applied stress = 1556 MPa, depicting the whisker bridging the matrix crack. Observe the intact interface. (c) Magnified deflected shape of the CMC with IFS = 3800 MPa at an applied stress = 2777.8 MPa, depicting fibre fracture with very little debonding at the interface
A. Mukherjee, H.S. Rao/ Computational Materials Science 4(1995)249-262 trol elements exceeds the matrix cracking stress 1333. 3 MPa. In order to observe the crack deflec (700 MPa)this elements stiffness is reduced to tion and the slip the displacements are magnified zero thus simulating the occurrence of matrix by a factor of 25. This magnified deflected shape cracking. In the other 50 interface elements used of the CMC (see Fig. 11 (a)), clearly shows the to model the whisker /matrix interface the shear crack deflection mechanism. For the CMcs with stiffness(K is set to a high value initially to IFS ranging from 2200 to 3800 MPa, the debond limit any non-slip displacement. Once the inter- ing of the interface has not occurred even when face shear stress exceeds the interface shear he Matrix crack reaches the whisker. The whisker strength, the shear stiffness Ks is set to zero momentarily halts the further propagation of the allowing slip. Loads were applied incrementally crack, thus, bridging the matrix crack (i.e. the and stress-strain relationships have been ob- whisker bridging mechanism). The whisker bridg- tained. Stress-strain curves for five different in- ing continues till the interface starts debonding terface shear strengths of 800 MPa, 2200 MP or the whisker fractures, depending on the rela 2600 MPa, 3000 MPa and 3800 MPa have been tive strengths of the interface and the whisker. blaine fruil the finite elenent allalysis. In real cpicted in magnified dclc ity increase of interface strength increases the the CMcs in Figs. 11(b)and(c). Fig. 11(b)shows matrix cracking stress. In the present work this the deflected shape of the CMc with an IFS of effect has not been considered, simply to avoid 2200 MPa, at an applied stress of 1556 MPa. It is e problem. The em- seen from the figure that, the debonding of the phasis of this work is to numerically demonstrate nterface does not initiate at this stress, though, he effect of interface strength on the toughening the matrix cracks at 922 MPa. It can be further of the CMCs observed from Fig. 10 that at this stress level, the stress-strain relation is linear. The matrix crack 41. Results and discussion has been bridged by the strong whisker till the interface debonding initiated at a stress of 1611 Fig 10 shows the stress-strain curves obtained MPa, thus deflecting the crack along the inter from the finite element analysis of the chosen face. On the other hand in the case of Cmc with ceramic-matrix-composite, for different interface an IFS of 3800 MPa, whisker bridging continued strength parameters. The matrix crack originated till the whisker stress bccame ncarly cqual to its from the free end A(see Fig 9)at an applied fracture value, and finally the CMc failed with stress of 922.2 MPa, and the entire matrix portion whisker fracture, with very little debonding of the AB cracked at this stress level, thus transferring interface(see Fig. 11(d)). This figure depicts, the the load on to the whisker. when the interface magnified deflected shape of the composite with shear strength(IFS )is 800.0 MPa, the interface IFS= 3800 MPa at the instant of failure. It can started debonding readily thus deflecting the be seen from the figure, that very little debonding crack parallel to the whisker (crack deflection has taken place at the interface, bcforc the hanism). At this point, the stress-strain curve whisker fractured. This is due to the relatively leviates from the linear path with further in- trong whisker /matrix interface. crease in load, the crack continues to propagate With increasing interface strength, it can be along the whisker/matrix interface. This is indi- observed that the stress-strain curves(see Fig 10) cated in the present model by the failure of the deviated from the linear path at higher stress interface clements. The deviation of the stress- levels. This is due to the relatively stronger strain curve from the lincar path indicates the whisker/matrix interface not debonding readily, nitiation of the toughening exhibited by the Cmc. thus contributing to lesser toughening. when the This crack deflection mechanism is depicted in interface shear strength was increased to 3800.0 Fig. 11a. The figure shows the magnified de- MPa, very little debonding has occurred at the flected shape of the CMC with an interface shear whisker/matrix interface, resulting in a brittle strength(IFS)of &00 MPa, at an applied stress of failure, characterised by the near linear stress-
258 A. Mukherjee, H.S. Rao /Computational Materials Science 4 (1995) 249-262 trol elements exceeds the matrix cracking stress (700 MPa) this elements stiffness is reduced to zero thus simulating the occurrence of matrix cracking. In the other 50 interface elements used to model the whisker/matrix interface the shear stiffness (K,) is set to a high value initially to limit any non-slip displacement. Once the interface shear stress exceeds the interface shear strength, the shear stiffness K, is set to zero allowing slip. Loads were applied incrementally and stress-strain relationships have been obtained. Stress-strain curves for five different interface shear strengths of 800 MPa, 2200 MPa, 2600 MPa, 3000 MPa and 3800 MPa have been obtained from the finite element analysis. In reality increase of interface strength increases the matrix cracking stress. In the present work this effect has not been considered, simply to avoid too many parameters in the problem. The emphasis of this work is to numerically demonstrate the effect of interface strength on the toughening of the CMCs. 4.1. Results and discussion Fig. 10 shows the stress-strain curves obtained from the finite element analysis of the chosen ceramic-matrix-composite, for different interface strength parameters. The matrix crack originated from the free end A (see Fig. 9) at an applied stress of 922.2 MPa, and the entire matrix portion AB cracked at this stress level, thus transferring the load on to the whisker. When the interface shear strength (IFS) is 800.0 MPa, the interface started debonding readily thus deflecting the crack parallel to the whisker (crack deflection mechanism). At this point, the stress-strain curve deviates from the linear path. With further increase in load, the crack continues to propagate along the whisker/matrix interface. This is indicated in the present model by the failure of the interface elements. The deviation of the stressstrain curve from the linear path indicates the initiation of the toughening exhibited by the CMC. This crack deflection mechanism is depicted in Fig. lla. The figure shows the magnified deflected shape of the CMC with an interface shear strength (IFS) of 800 MPa, at an applied stress of 1333.3 MPa. In order to observe the crack deflection and the slip, the displacements are magnified by a factor of 25. This magnified deflected shape of the CMC (see Fig. 11(a)), clearly shows the crack deflection mechanism. For the CMCs with IFS ranging from 2200 to 3800 MPa, the debonding of the interface has not occurred even when the matrix crack reaches the whisker. The whisker momentarily halts the further propagation of the crack, thus, bridging the matrix crack (i.e. the whisker bridging mechanism). The whisker bridging continues till the interface starts debonding or the whisker fractures, depending on the relative strengths of the interface and the whisker. This is depicted in magnified deflected shapes of the CMCs in Figs. 11(b) and cc>. Fig. 11(b) shows the deflected shape of the CMC with an IFS of 2200 MPa, at an applied stress of 1556 MPa. It is seen from the figure that, the debonding of the interface does not initiate at this stress, though, the matrix cracks at 922 MPa. It can be further observed from Fig. 10 that at this stress level, the stress-strain relation is linear. The matrix crack has been bridged by the strong whisker, till the interface debonding initiated at a stress of 1611 MPa, thus deflecting the crack along the interface. On the other hand, in the case of CMC with an IFS of 3800 MPa, whisker bridging continued till the whisker stress became nearly equal to its fracture value, and finally the CMC failed with whisker fracture, with very little debonding of the interface (see Fig. 11(d)). This figure depicts, the magnified deflected shape of the composite with IFS = 3800 MPa at the instant of failure. It can be seen from the figure, that very little debonding has taken place at the interface, before the whisker fractured. This is due to the relatively strong whisker/matrix interface. With increasing interface strength, it can be observed that the stress-strain curves (see Fig. 10) deviated from the linear path at higher stress levels. This is due to the relatively stronger whisker/matrix interface not debonding readily, thus contributing to lesser toughening. When the interface shear strength was increased to 3800.0 MPa, very little debonding has occurred at the whisker/matrix interface, resulting in a brittle failure, characterised by the near linear stress-
