MCI—r! mu W“ F 0. .- . . L-xh # ‘ IL. 39“.?” ‘75»: ~C (V'I‘F' I.“ - N. M. ‘ M. 919. - ‘4 ‘ A..n 1.7 ‘ . u r ‘ ‘ J! ’ “‘01:” 1‘ 'Iué‘i . a. 3 . g .. x , .1. .A “A. n 7 ' 1:52: /; . 1..., . J. ' l ‘ ‘ , . .i . ‘ 3 LA“. '5 _a l‘ i. ‘1 .‘t .-'.‘ 'fi : r...‘ I A 1 1:33: rise . .. ,7. ”.7, _ . '"'»--... , ., v -'$«»... . -.7_-_ .., . ‘ , .. 5'!“- 4:9; aw- - é {3' HI» 1' . in motile ill llllll T"\ \i 1\\i93 00900 97 ‘7 '1 ill This is to certify that the thesis entitled Finite Element Method Applied to Deformation of Metallic Glass Ribbon Reinforced Glass - Ceramic Matrix Composite Material presented by John Dung Ninh has been accepted towards fulfillment of the requirements for Master's 'degree in Metallurgy 1’) ‘ ‘k - [\f- C'LWLMMOIM f , 7 Major professor Date4/167 /q ‘- o-7639 MS U is an Affirmative Action/Equal Opportunity Institution LKBMRY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before dde due. DATE DUE DATE DUE DATE DUE l li—TL I! MSU Is An Affirmetlve ActlorVEquel Opportunliy Institution emma-m .—————g i” 7 l FINITE ELEMENT METHOD APPLIED TO DEFORMATION OF METALLIC - GLASS RIBBON REINFORCED GLASS - CERAMIC MATRIX COMPOSITE MATERIAL BY John Dung Ninh A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for degree of MASTER OF SCIENCE Department of Metallurgy Mechanics and Material Science 1991 557—— 35;»; ABSTRACT FINITE ELEMENT METHOD APPLIED TO DEFORMATION OF METALLIC GLASS RIBBON REINFORCED GLASS - CERAMIC MATRIX COMPOSITE MATERIAL BY John Dung Ninh The deformation characteristics of ceramic composite material was studied numerically with the objective of investigating the dependence of bending properties on size, shape and distribution of the ribbon reinforcing phase. The model chosen for comparision was metallic glass ribbon reinforced glass ceramic matrix composite. The overall constitutive response of the composite and the evolution of stress and strain field quantities in the matrix of composite was computed using Finite Element Method within the context of axisymetric and plane strain unit cell formulations. The results indicate that the development of significant stresses within a composite matrix provides an important contribution to strengthening. The numerical results provide a mechanistic rationale for the experimentally observed trend for the effect of ribbon on stress distribution within the composite. AKNOWLEDGEMENTS I would like to take this opportunity to sincerely thank my advisor, Dr. K. N. Subramanian, but for whose guidance and encouragement this endeavor would not have ended successfully. A special thanks to Dr. Rajendra U. Vaidya and my colleague Gregory Fell, for having helped me during this work. This work could not have been completed without the help of all the aforementioned. iii TABLE OF CONTENTS List of figures .............. ....... ....... List of tables .. ........ ...... ....... ..... I. Introduction 1. Finite Element Method ( FEM ) ..... 2. Historical background ............. 3. ANSYS program ..................... a. Preprocessing phase ......... b. Solution phase .............. c. Postprocessing phase .. ...... 4. Description of analysis .. ......... II. Procedure of Finite Element Analysis 1. Composite system ................... 2. Finite Element Method ( FEM ) ..... 3. Finite element models ............. III. Results and discussion 1. The analysis of stress distribution .. .................... a. Specimen reinforced with one ribbon ......... ......... b. Specimen reinforced with two ribbons ................ iv Page 11 12 18 32 32 32 IV. V. c. Specimen reinforced with four ribbons .......... d. Specimen reinforced with eight ribbons ............... e. Specimen reinforced with sixteen ribbons ....... 2. Analysis of the plots ....... Summary References 33 34 36 47 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 1. 2. 10. 11. 12. LIST OF FIGURES Specimen model ....... . ......... . ....... Finite element model of specimen reinforced with one ribbon .... ......... Finite element model of specimen reinforced with two ribbons ............ Finite element model of specimen reinforced with four ribbons ........... Finite element model of specimen reinforced with eight ribbons .......... Finite element model of specimen reinforced with sixteen ribbons ........ Stress distribution for specimens with one ribbon .. .................. .... Stress distribution for specimens with two ribbons ....................... Stress distribution for specimens with four ribbons ...................... Stress distribution for specimens with eight ribbons ............. ...... . Stress distribution for specimens with sixteen ribbons .................. The plot of fraction of cracked matrix vi Page 17 22 24 26 28 3O 38 4O 42 44 Figure Figure Figure Figure Figure Figure Figure Figure 13. 14. 15. 16. 17. 18. 19. 20. segments versus number of ribbons ..... The plot of number of cracked matrix segments versus number of ribbons ..... Schematic illustrating location of middle nodes relative to the origin The plot of stress versus distance of nodes to the origin for specimen with two ribbons ................ ...... The plot of stress versus distance of nodes to the origin for specimen with four ribbons .............. ....... The plot of stress versus distance of nodes to the origin for specimen with eight ribbons ........... . ........ The plot of stress versus distance of nodes to the origin for specimen with sixteen ribbons ...... ...... ...... The plot of stress versus distance of nodes to the origin for specimens with two, four, eight, and sixteen ribbons . .......... . ...... ...... ....... The plot of load versus displacement of composite specimen ................. vii 49 51 53 59 68 Table 1. Table 2. LIST OF TABLES page The properties of metallic-glass ribbon ............................. ..... 14 The properties of glass-ceramic matrix ......................... . ........ 15 Viii INTRODUCTION The effect of the ribbons on the crack growth in a brittle glass matrix has been investigated [ 1 ]. These composites, have been shown to have high longitudinal strength coupled with good off-axis properties [ 1 ]. Ribbon reinforcements unlike fiber reinforcements, possess the ribbon width as an additional geometrical parameter. Hence, the load transfer characteristics in the ribbon reinforced composites are expected to be influenced by ribbon's width. The ribbons can also be oriented differently with respect to their long and short transverse faces normal to the opening crack front. Metallic materials, when reinforced with brittle fibers offer the potential for significant improvements in strength and toughness, particularly in mechanical performance over monolithic alloys. While metal matrix composites exhibit higher stiffness and strength than matrix alloys, they often suffer from lower ductility and inferior fracture toughness. Arguments for the strength and failure in the metal matrix composite are difficult, due to the lack of complete information on the processing, characterization and properties of the 2 materials. In a significant number of prior investigations, with which the composite properties are compared, the unreinforced matrix alloys, are obtained from different routes. In this work, a chosen thickness of the metallic glass reinforcement in a glass ceramic matrix composite was studied by numerical analysis to gain a perspective on mechanics of strengthening. An understanding of this effect will be useful in optimizing the properties of such brittle matrix composites, in the attempt to make them suitable for structural applications. The purpose of the finite element calculations is to measure changes in geometrical variables associated with the shape and distribution of the reinforcement. 1. Finite Element Method FEM. FEM analysis was carried out using a Prime 750 or Unix computer system, and the program ANSYS, which is used for the solutions of several classes of engineering problems. The ANSYS element library offers 95 different element types to carry out a wide range of engineering analyses. Element type STIFF 42 was used for analysis of metallic glass ribbon reinforced glass ceramic matrix composites. A mesh of elements is presented for ANSYS and FEM analysis, which is carried out with various numbers of ribbons. Dividing the area into a greater 3 number of elements increases the degree of accuracy. 2. Historical background. Fischmeister and Sundstrom [ 2 ] used the FEM technique to calculate stress-strain curves for different hardness ratios between different phases present in plain carbon steels ( 0.11 and 0.21 weight percent carbon ) and aluminum bronzes (8 to 12 percent aluminum). They also studied the deformation of individual phase regions in the case of plain carbon steel by using a two dimensional model of the ferrite-martensite microstructure. Karlsson and Sundstrom [ 3 ] have also studied the plastic deformation in ferritic-martensitic steels. They used FEM to study inhomogeneity in the corresponding two phase model. Sundstrom [ 4 ] has used FEM to study the elastic plastic behavior of WC-Co ( tungsten carbide-cobalt ) alloys. He showed that the continuum mechanics can be applied to a two dimensional model of real microstructure. His results indicated that the plastic strain distribution calculated by FEM is very inhomogeneous on a microscale and FEM can not give the high resolution of stress and strain fields in different phases. Jinoch et. a1. [ 5 ] used FEM to study the stress and strain of alpha beta Ti-8Mn alloy. They used a uniform mesh of 392 triangular two dimensional ( plane stress ) plate elements. The volume fraction, particle size and shape could be varied to designate each triangle as either alpha or beta. The shapes of particles used in meshes were not the same as those in actual specimens but were idealized to make calculations simpler while maintaining the volume fraction constant. Jinoch et. al. showed that an FEM calculated stress-strain curve attained lower stress levels for similar strain levels as compared to an experimentally calculated stress- strain curve . This difference may be due to the fine grain sizes in Ti-8Mn alloy and the contributions of the interface phase which were not considered in the FEM calculations. Margolin et. al. [ 6 ] studied the influence of particle size, matrix and volume fraction of phases on the stress and strain relationship of alpha beta Ti alloys. They found that for a given volume fraction of the second phase, the calculation of stress and strain were higher for specimens with a finer particle size as compared to those of specimens with coarser particle size. 3. ANSYS program. ANSYS is a specific computer program, developed and maintained by Swanson Analysis System Inc. It has the ability to analyse static, dynamic, elastic, creep, swelling, buckling, heat transfer, fluid, and current 5 flow problems . The program is based on the Finite Element Method. ANSYS program gives results with high degree of accuracy in a short period of computer time. The program is very flexible and can be applied to many engineering problems. ANSYS program includes three phases: the preprocessing phase, the solution phase, and the postprocessing phase. a. Preprocessing phase. The preprocessing phase is carried out interactively; the input is given directly and it may immediately provide plots. The preprocessing phase begins with PREPn where n is the number which dependes on the kinds of problems; PREP7 for general purposes, PREP4 is for piping problem etc, and the preprocessing phase may include many PREPn as the subroutines. b. Solution phase. The solution phase checks the data before program progressed to the actual execution. The solution phase does not provide the solutions: it interprets the data input and, checks elements, and various levels of program. If there were errors, it would produce the errors in this phase. c. postprocessing phase. The postprocessing phase consists of several modules. Each module performs a different operation and contains a unique set of commands. The postprocessing phase will give the solutions, the plots and it stores data to use for next time. 4. Description of analysis. The overall equilibrium equations for static analysis are [ K ] { u } II A F } or i K ] { u } { F1 } + { F2 } where [ K ] is total stiffness matrix = total element stiffness matrix. { u } is nodal displacement vector { F1 } is reaction load vector, and { F2 } is total applied load vector. The stress sigma is related to the strain epsilon by { sigma } = [ D ]({ epsilon } - { epsilonl }) where [ D ] is elasticity matrix. { epsilonl } is thermal strain vector. The element stresses are computed by { sigma } = [ D ]({ B ]{ u } - { epsilonl }) because { epsilon } = [ B ]{ u } where 7 [ B ] is strain displacement matrix which must be specified for each stress calculation point, and { u } is nodal displacement. 5. The aim of this project. The aim of this project is to study the effect of different ribbon configurations on the stress distribution in ribbon reinforced brittle ceramic matrix composite by using two dimensional Finite Element Method. PROCEDURE OF FINITE ELEMENT ANALYSIS 1. Composite system. The specimens considered in this study were fabricated using Corning glass code 7572 ( code numbers of products of Corning Glass Company ) as the matrix material. An iron based metallic glass METGLAS 2605 S - 2 ( registered trademark of Allied Signal Inc. for amorphous metallic alloys and brazing alloys ) alloy was used as the reinforcement [ 1 ]. The physical properties and chemical compositions of metallic glass and ceramic glass matrices are presented in tables I and II. The volume fraction of the ribbons ( 0.8 ) was kept constant in all of the samples studied by FEM. This was achieved by changing the thickness ( subdividing ) but keeping the width and length the same. The residual stress induced by the thermal expansion coefficient between the matrix and reinforcement was small and could be neglected since coefficient of matrix 95 x 10E-7 per degree coefficient of ribbon 76 x 10E-7 per degree. Strong bonding was observed between ribbon and matrix. The ribbon geometries used in the modelling varied as 40 mm 40mm 40 mm 40 mm 40 mm X X X X X 50 50 50 50 50 mm mm X X 8 mm for one ribbon. 4 mm for two ribbons. 2 mm for four ribbons. 1 mm for eight ribbons. 0.5 mm for sixteen ribbons. The dimension of the composite specimen was 40 mm X 50mm X 10mm. Experimental verification of the predictions based on FEM analysis were carried out with specimens prepared by using the following steps [ 7 ]: a. Specimen was wet pressed in a steel die, And then compacted at 3000 Psi ( 21 MPa ) The specimen was next preheated to 250 degree C for 15 minutes to drive off the organic binder, Then sintered at 400 degree C for 90 minutes. Then it was treated at 450 degree C for 20 minutes for matrix crystallization as recommended by manufacturer, And cooled to room temperature within the furnace in order to minimize thermal shock. The main crystalline phase was 2Pb0.Zn0, and the specimens were tested in three point bending using an Instron machine with a crosshead speed of 0.05 cm / min. 10 2. Finite Element Method ( FEM ). FEM was used to calculate the stress distributions in the specimens. The meshes were provided to the ANSYS program. In the finite element model, the nodes A and B can move only in the X-direction. Node C can move in the Y- direction. The other nodes can move in both X and Y— directions. A force of 125 N ( load corresponding to fracture strength of the matrix of the selected size ) was applied to node C, normal to the ribbons, as shown in Figure 1. 11 TABLE I Properties of metallic glass ribbon [ 7 ]. Property : Metglas 2605 5-2 Chemical : Fe 78 composition w/o : B 13 ° Si 9 Crystallization : : 550 temperature : Elastic modulus : 85 ( GPa ) ° Yield strength > 700 ( MPa ) : -7 Coefficient of thermal : 76 X 10 -1 : expansion (degree C ) : 12 TABLE II Properties of ceramic glass matrix [ 7 ]. Property : Corning glass 7572 Softening point (degree C) : 375 : -7 Coefficient of thermal : 95 X 10 -1 expansion ( degree C ) -3 Density ( g cm ) : 6 Elastic modulus ( GPa ) 33 4 Continuous service : 450 temperature ( degree C ) Chemical composition : PbO 70 ( w/o ) : B 0 5 - 10 2 3 SiO 2 - 5 2 13 FIGURE 1. Specimen model. - Darker segments are ribbons. — Lighter segments are matrix. - Nodes A and B move in X direction only. - Node C moves in Y direction only. 14 CA‘25" V ', J” 11.1.. «Vinny! w.” .‘: . -3 e .f 'f‘ \' .- . I,’ "‘2‘." 3"“ . N’S‘AUN'B‘fl' ’. A. I; v.» :fliv f; . . . . ~ _ - ~ 2 . '- ‘ .- rum-,1; -' "'t ’- ‘~-_...,.‘vu‘~l I n‘» J» \ . . . ~ ‘ ‘. .‘ > . '. . ‘ , .7 a -" ‘ ' ' ‘ «7.4,; ‘ t 'c . ‘_, . ~-. _-I —'—-. "-V‘L.’ _ “Kl...” [(4 '. yr ‘ ' l. _ I. -5‘x,» .- .o ”wk. -.~‘ is. L: y - u. - .. '_(-.~ - .~- .l‘ »"~“ ‘ , n . . > . .‘ Jr _. . ’7‘. ,3 Q 1 '“rbbon ~~ < i ‘ i ‘ ll. '- .._ - , . .- 1 . ' _ . 3"} . . —‘ .-. v . ' ' ’ ' - 3’31: I . ‘ r," ' «‘ 'J“ . f“. n ‘ . ‘ '-' . 3.3-... - — "z . L; 40 mm 15 3. Finite elements models. In the first case, the specimen was reinforced with one ribbon ( 8 mm thick ), as shown in Figure 2. There were two symmetric parts, the dark and light parts. Each part included one matrix segment that was divided into smaller triangular elements, and another half of thickness of the ribbon which was divided into larger triangular elements. The composite specimen was analysed by using 256 triangular elements and 177 nodes. Both the matrix segments, and the ribbon segments consisted of 128 elements. The thickness of the matrix segment was divided into two parts located on either side of ribbon, each part having a thickness of 1 mm. The ribbon segment was 8 mm thick . The length of the specimen was 40 mm, and was divided into 17 equi-distant nodes. The nodes were numbered in an increasing order in the horizontal direction. The number of the node increased by 20 for each step in the vertical direction. For the second case, the specimen was reinforced with two ribbons as shown in Figure 3. There were three matrix segments and two ribbon segments. As in the first case, the matrix segments were divided into the smaller triangular elements and the ribbon segments were divided into larger triangular elements. The composite specimen 16 was analysed by using 320 triangular elements and 217 nodes. In this case, each matrix segment was 0.66 mm thick, while each ribbon segment had a thickness 4 mm. The length of the specimen was 40 mm, and was divided into 17 equi-distant nodes. Node numbering was carried out in a fashion similar to the one described in the previous case. In the third case, the specimen was reinforced with four ribbons. There were five matrix segments and four ribbon segments, as shown in Figure 4. The composite specimen was analysed using 576 elements and 377 nodes. In this case, each matrix segment was 0.4 mm thick, while each ribbon segment was 2 mm thick. The length of the specimen was 40 mm, and was divided into 17 equi-distant nodes. For the fourth case, the specimen was reinforced with eight ribbons as shown in Figure 5. There were nine matrix segments and eight ribbon segments. In this case, the composite specimen was analysed by using 1088 elements and 697 nodes. Each matrix segment at either surface of the specimen was 0.3 mm thick, while the other matrix segments had a thickness of 0.2 mm. Each ribbon segment was 1 mm thick. The length of the specimen was 40 mm and was divided into 17 equi-distant nodes. In the last case, the specimen was reinforced with 17 sixteen ribbons; there were 17 matrix segments and 16 ribbon segments, as shown in Figure 6. The specimen was analysed by using 528 elements and 677 nodes. Each matrix segment at either surface of the specimen was 0.175 mm thick. The other matrix segments had a thickness of 0.11 mm. Each ribbon segment was 0.5 mm thick. The length of the specimen was 40 mm and was divided into 17 equi-distant nodes. 18 FIGURE 2 . Finite element model of specimen reinforced with one ribbon. - Small triangles are in matrix element. - Large triangles are in ribbon element. NUM thh baa><> rim j‘lEfi E RIBBON 20 FIGURE 3 . Finite element model of specimen reinforced with two ribbons. - Small triangles are in matrix element. - Large triangles are in ribbon element. 21 252 mm>B mhzmzmam hammm H .02 Foam ‘HO‘ r~ U) int—mm zm--omo. atgmqm» Hand-r DIST=15 . 557 =20 =5 XF YF 25 FOUR R I BBONS 24 FIGURE 5. Finite element model of specimen reinforced with eight ribbons. - Small triangles are in matrix element. - Large triangles are in ribbon element. 25 u h» on" LX www.mduewmo an >N Hu >x 292 mm»& mbzmzmam hmmmm H .02 Foam mvnmmuma Hmma Va m<2 <¢.v mxmz< mzommHm HIOHM. 26 FIGURE 6. Finite element model of specimen reinforced with sixteen ribbons. - Small rectangles are in matrix element. - Large rectangles are in ribbon element. 27 mu my own bx hmm.msuBmHo an >N H“ >x 2:2 mm>s mbzmzmqm hmmmm — .02 Foam vauofludm amma cu m4: m2< szmmH¢ zmweme RESULTS AND DISCUSSIONS The maximum tensile stress in the outer fiber ( before failure of the matrix ) of the sample ( in three point bending ) is given by 2 sigma = 3PL / 2bd where sigma is fracture strength of the matrix P is required force L is length of composite 40 mm b is width of composite 50 mm, and d is thickness of composite 10 mm. The first crack in the matrix will form when the outer fiber stress reaches the matrix fracture stress. The load for failure of the specimen considered was calculated as 2 P = 2bd sigma / 3L = 125 N . The stress in the matrix segments was determined by the FEM analysis. The stress in each matrix segment was determined at the load where the first matrix crack would be initiated on the tensile surface corresponding to the fracture stress of the matrix ( = 15 MPa ). 28 29 1. The analysis of the stress distribution. a. Specimen reinforced with one ribbon. The stress distribution in the specimen reinforced with one ribbon is shown in Figure 7. The stress was compressive in the upper half, while it was tensile in the lower half. The stress was close to zero ( 2.014 MPa ) at the neutral surface. The maximum tensile stress is in the darkest region which has the value of 59.846 MPa. The maximum compressive stress is in the lightest region, and had a value -70.276 MPa. The stress slowly increased from the maximum tension to maximum compression from the lower to the upper part of specimen. The stresses are also reduced at the nodes laterally far—distanced from the middle nodes. with the values of stress distribution given by ANSYS, and the fracture strength of the matrix ( 15 MPa ) one can predict the number of cracked matrix segments. The matrix segments would be cracked when the value of stress distribution on it is greater than 15 MPa. The matrix segment will be uncracked when the value of stress distribution is lower than 15 MPa. In Figure 7, the crack might occur in the lower matrix segment, and the shortest length of cracked line is 1 mm. 30 b. Specimen reinforced with two ribbons. The stress distribution in the specimen reinforced ‘with two ribbons is shown in Figure 8. The stress is compressive in the upper half, while it is tensile in the lower half. The maximum tensile stress is 60.892 MPa in the darkest region, and the maximum compressive stress is -73.68 MPa in the lightest region. The stress was close to zero ( 1.082 MPa ) at the neutral surface. The values of stress slowly decreased from the darkest region to lightest region; stress also decreased at the nodes laterally far-distanced from the middle nodes. With the values of stress distribution given by ANSYS, and the fracture strength of the matrix ( 15 MPa ), one can predict the number of cracked matrix segments. The matrix segments would be cracked when the value of stress on it is greater than 15 MPa. They will be uncracked when the value of stress is lower than 15 MPa. In Figure 8, The crack might occur in the lowest matrix segments and the shortest length of cracked line is 0.66 mm. c. Specimen reinforced with four ribbons. The stress distribution in the specimen reinforced with four ribbons is shown in Figure 9. The stress is compressive in the upper half, while it is tensile in the lower half. The maximum tensile stress is 63.644 MPa in the darkest region, and the maximum compressive stress 31 is -78.962 MPa in the lightest region. The values of tensile and compressive stresses reduced from the darkest region to the lightest region, and at the nodes laterally far-distanced from the middle nodes. The stress was close to zero ( 0.253617 MPa ) at neutral surface. With the value of stress distribution given by the ANSYS, and the fracture strength of the matrix ( 15 MPa ), one can predict the number of cracked matrix segments. The matrix segments would be cracked when the value of stress distribution on it is greater than 15 MPa. They will be uncracked when the value of stress distribution is lower than 15 MPa. In Figure 9, the crack might occur in the two lowest matrix segments, and the total shortest length of the cracked line will be 0.8 mm. d. Specimen reinforced with eight ribbons. The stress distribution in the specimen reinforced with eight ribbons is shown in Figure 10. The stress is compressive in the upper half, while it is tensile in the lower half. The maximum tensile stress is 64.474 MPa in the darkest region. The maximum compression stress region is - 80.555 MPa in the lightest region. The values of stress were reduced from the darkest region to the lightest region, and at the nodes laterally far-distanced from the middle nodes. The stress was close to zero ( 0.0166 MPa ) at the neutral surface. With the value of stress distribution given by 32 ANSYS, and the fracture strength of the matrix ( 15 MPa ), one can predict the number of cracked matrix segments. The matrix segements would be cracked when the value of stress distribution on it is greater than 15 MPa. They will be uncracked when the value of stress distribution is lower than 15 MPa. In Figure 10, the crack might occur in the four lowest matrix segments, and the total shortest length of cracked line will be 0.9 mm. e. Specimen reinforced with sixteen ribbons. The stress distribution in the specimen reinforced with sixteen ribbons is shown in Figure 11. The stress is compressive in the upper half, while it is tensile in the lower half. The maximum tensile stress is 70.299 MPa in the darkest region, and the maximum compressive stress is - 95.524 MPa in the lightest region. The value of stress reduced from the darkest region to the lightest region, and at the nodes laterally far-distanced from the middle nodes. The stress is close to zero at the neutral surface. With the values of stress distribution given by ANSYS, and the fracture strength of the matrix ( 15 MPa ), one can predict the number of cracked matrix segments. The matrix segment would be cracked when the value of stress distribution on it is greater than 15 MPa. They will be uncracked when the value of stress is lower than 15 MPa. In Figure 11, the crack might occur in the lowest 33 seven matrix segments, and the total shortest length of the cracked line will be 0.835 mm. A comparison of specimens with one, two, four, eight, and sixteen ribbons, provides the following results: - The magnitude of stress increases at the regions far- distanced from the neutral surface of the specimen, and decreases at the nodes laterally far-distanced from the middle nodes. - The maximum value of outer fiber tensile stress was lowest in the specimen with one ribbon 59.846 MPa; this value increased in the specimens having more ribbons, as in the specimen with sixteen ribbons the maximum tensile stress was 70.299 MPa. - The maximum value of compressive stress region was lowest at the specimen with one ribbon - 70.276 MPa; this value increased in the specimens having more ribbons as in the specimen with sixteen ribbons, the maximum compressive stress was - 95.524 MPa. So the maximum values of tensile and compressive stresses increase with increasing number of ribbons. 34 FIGURE 7 . Stress distribution for specimen reinforced with one ribbon. - Positive number is the value of tensile stress. - Negative number is the value of compressive stress. - Small triangles are in matrix element. - Large triangles are in ribbon element. - MN is middle nodes. 35 www.mm www.mauhmuo an >N an >x mmN.mwnmx2m www.mmn XZm mnm.~oalum22m whN.chIn 22m Hummem mmmmam games 3 .oz yoga tv.v m*mz< zommnm mzo 36 FIGURE 8 . Stress distribution for specimen reinforced with two ribbons. - Positive number is the value of tensile stress. - Negative number is the value of compressive stress. - Small triangles are in matrix element. - Large triangles are in ribbon element. - MN is middle nodes. 37 www.mwumxzm mmmmfim HsmOm m .02 Hoqm HmanHNH Mama m m4: <¢.v m>m2< mzommHm 039 38 FIGURE 9 . Stress distribution for specimen reinforced with four ribbons. Positive number is the value of tensile stress. — Negative number is the value of compressive stress. — Small triangles are in matrix element. - Large triangles are in ribbon element. - MN is the middle nodes. 39 www.mw mmh.nv vmm.Hm N h N F v I >1 >N HI >x avo.Hhumx:m www.mwn xxm mNmNho.on x20 Atmoqu m AU>w2< mzomnnm finch 40 FIGURE 10 . Stress distribution for specimen reinforced with eight ribbons. - Positive number is the value of tensile stress. - Negative number is the value of compressive stress. - Small triangles are in matrix element. - Large triangles are in ribbon element. - MN is the middle nodes. 41 3.4.3 I en 3 . I new «A I ann.os 838.0 I . - I 26 3 I ~H~.~n- e I an 3 II 3.3- I mmm.os- u E, ONu bx www.msuemHo T >N T >x vmv.~pumxzm .2.in xzm wra.waalnm22m mmm.omnn 22m Hmhmho.on x20 wmmmhm «Ewen H .02 804m mmnvouha Hmma va m<2 m2< mzommwm FIUHN FIGURE 11 . Stress distribution for specimen reinforced with sixteen ribbons. - Positive number is the value of tensile stress. — Negative number is the value of compressive stress. - Small rectangles are in matrix element. - Large rectangles are in ribbon element. - MN is the middle nodes. 43 «8.3 I . .ll mv mm 393 I e.n- n.- mwméw- II mm.ov- . - Ill «#6 mm . - nll mac as 43.3- I n hr ON“ bx hmm.msnemno H >N nu >x Ham.msumxzm mmm.osn xzw mHh.¢leumzxw www.mmiu 22m mamomo.ou x2e wwMMBm Hamom H .02 Foam ovuwouom Hmma v” m4: > sigma3 A typical load displacement curve for such a composite is shown in the Figure 20. Modelling studies can be carried out incorporating practical conditions. The results based on such studies can affect the decision to use or discard any given component once a certain fraction of the matrix has cracked. Another important point which has to be accounted for is the constantly changing position of the neutral axis in a dynamic test. The results obtained in the present study are only useful under static conditions ( constant load ). Any dynamic loading can alter the conditions dramatically and change the end results. 64 FIGURE 20. The plot of load versus displacement of composite specimen. - Number 1 indicates the position where the first matrix segment crack occurs. - Number 2 indicates the position where the load is carried by ribbons and uncracked matrix segments. - Number 3 indicates the position where the composite failure occurs. 6'5 LOA D DISPLACEMENT SUMMARY For a given volume fraction of ribbon reinforcement, the ribbon distribution ( number of ribbons ) affects the stress distribution; however, it does not significantly change the fraction of cracked matrix segments. This behavior indicates that the number of the ribbons would not alter the end results, provided the volume fraction is kept constant. Hence, based on practical constraints, it would be beneficial to incorporate fewer ribbons rather than more narrower ones. 66 IV . REFERENCES Rajendra U. Vaidya and K. N. Subramanian, " Effect of Ribbon Orientation on Fracture Toughness of Metallic-Glass Ribbon Reinforced Glass- Ceramic Matrix Composite, " J. Am. Ceram. Soc., 73, 2952 ( 1990 ). H. Fischmeister, J. Hjalmered, B. Karlsson, G. Lindon and B. Sundstrom, Third international conference on strength of materials and alloys, Institute Metals, and Iron and Steel Institute, Cambridge and London, 1, 621 ( 1973 ). B. Karlsson and B. Sundstrom, " Inhomogeneity in Plastic Deformation of Two Phase Steel, " Mat. Sci. Eng., 17, 161 ( 1974 ) B. Sundstrom, " Elastic Plastic Behavior of WC-Co Analyzed by Continuum Mechanics, " Mat. Sci. Eng., 12, 265 ( 1973 ). J. Jinoch, S. Ankem and H. Margolin, " Calculation of Stress-Strain Curve and Stress- Strain Distribution for Alpha-Beta Ti-8Mn Alloy, " Mat. Sci. 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