Part B: engineering ELSEVIER Composites: Part B 37(2006)530-541 Structure, nonlinear stress-strain state and strength of ceramic lltilayered O.N. Grigoriev a, *, A V. Karoteev ,E.N. Maiboroda I.L. Berezhinsky ,B K. Serdega DYu. ostrovo.vg. piskunoy Institute for Problems of Materials Science, NAS of Ukraine, 3 Krghizhanovskii Str, 03142 Kiev, Ukraine Institute of Physics of Semiconductors, NAS of Ukraine, Kiev, Ukraine Institute for Problems of Strength, NAS of Ukraine, Kiev, Ukraine d National University of Transport, Kiev, Ukraine Received 4 April 2005: received in revised form 7 September 2005: accepted 15 September 2005 Available online 27 April 2006 Abstract The effect of structure and residual stresses on the mechanical behavior of the SiC/TiB, multilayer composite ceramic materials was studied. The multilayered ceramics were obtained using the following scheme: a slip casting of thin films followed by a packet rolling and hot pressing. The use of B-SiC powders allowed to obtain Sic layers with porous structure reinforced by crystals. Such structures possess the relaxation ability of thermal strains that excludes formation of cracks during material production and provides enhanced strength of the SiC/TiB2 composite. Mechanical response of the laminated ceramic composites to static bending was studied experimentally. a procedure for solving the inverse problem using experimental data on the deformation of a laminated ceramic composite specimen in the form of a beam was developed. This approach allows the mechanical characteristics of the laminates to be predicted. The nonlinear stress-strain dependencies for the laminate as a quasi-homogeneous structure and for each of the two separate materials of the layers were obtained. The modeling of the stress-strain state of t aminate was performed. c 2006 Elsevier ltd. all rights reserved Keywords: A. Layered structures; A Laminates; B Residual/intemal stress; B Strength 1. Introduction zirconium borides to silicon carbide allows increasing strength for and fracture toughness by 50-100%.However, corrosion Laminated ceramic composites offer the best prospect for resistance of these ceramics decreases significantly which is rational use of the unique physical-mechanical properties of undesirable for the majority of high-temperature applications monolithic ceramics and ceramic-matrix composites providing Therefore, a design of the multilayer composites with a way to improve their durability, fracture toughness, corrosion external layers of corrosion resistive SiC and internal layers of and thermal resistance, wear, etc. [1-7 improved mechanical performance such as SiC/MeB2 may b Silicon carbide is one of the most promising ceramic promising. Moreover, in this type of material the external Sic materials for structural applications because of its unique layers are under thermal compression stresses due to their thermomechanical properties and high corrosion resistance. lower coefficient of thermal expansion compared with internal However, low fracture toughness and reliability of silicon SiC/MeB, layers. This will also increase apparent fracture carbide significantly limit its potential applications. The toughness as well as strength and reliability of laminates. The improvement of mechanical properties is possible under the studies over the last few years have shown that the increase in careful control of structure and is due to transition from strength and/or fracture toughness of multilayer ceramic monolithic ceramics to composites. In particular, it is well composites may provide increased tolerance against damages known [8-10) that the additions of 15-30% titanium and However, the production of such composites requires a solution to layer bonding problems. Also there is the possibility of generating new defects in the thermal stresses fields [11] E-mail address: oleggrig@ipms keiv. ua(O N. Grigoriev ) Therefore, optimization of both composite manufacturing 1359-8368/- see front matter o 2006 Elsevier Ltd. All rights reserved. conditions and its structure are very important to ensure elastic doi: 10.1016/j- composites. 2006.02.009 strain relaxation. The development and design of these
Structure, nonlinear stress–strain state and strength of ceramic multilayered composites O.N. Grigoriev a,*, A.V. Karoteev a , E.N. Maiboroda a , I.L. Berezhinsky a , B.K. Serdega b , D.Yu. Ostrovoi c , V.G. Piskunov d a Institute for Problems of Materials Science, NAS of Ukraine, 3 Krzhizhanovskii Str., 03142 Kiev, Ukraine b Institute of Physics of Semiconductors, NAS of Ukraine, Kiev, Ukraine c Institute for Problems of Strength, NAS of Ukraine, Kiev, Ukraine d National University of Transport, Kiev, Ukraine Received 4 April 2005; received in revised form 7 September 2005; accepted 15 September 2005 Available online 27 April 2006 Abstract The effect of structure and residual stresses on the mechanical behavior of the SiC/TiB2 multilayer composite ceramic materials was studied. The multilayered ceramics were obtained using the following scheme: a slip casting of thin films followed by a packet rolling and hot pressing. The use of b-SiC powders allowed to obtain SiC layers with porous structure reinforced by prismatic crystals. Such structures possess the relaxation ability of thermal strains that excludes formation of cracks during material production and provides enhanced strength of the SiC/TiB2 composite. Mechanical response of the laminated ceramic composites to static bending was studied experimentally. A procedure for solving the inverse problem using experimental data on the deformation of a laminated ceramic composite specimen in the form of a beam was developed. This approach allows the mechanical characteristics of the laminates to be predicted. The nonlinear stress–strain dependencies for the laminate as a quasi-homogeneous structure and for each of the two separate materials of the layers were obtained. The modeling of the stress–strain state of the laminate was performed. q 2006 Elsevier Ltd. All rights reserved. Keywords: A. Layered structures; A. Laminates; B. Residual/internal stress; B. Strength 1. Introduction Laminated ceramic composites offer the best prospect for rational use of the unique physical–mechanical properties of monolithic ceramics and ceramic-matrix composites providing a way to improve their durability, fracture toughness, corrosion and thermal resistance, wear, etc. [1–7]. Silicon carbide is one of the most promising ceramic materials for structural applications because of its unique thermomechanical properties and high corrosion resistance. However, low fracture toughness and reliability of silicon carbide significantly limit its potential applications. The improvement of mechanical properties is possible under the careful control of structure and is due to transition from monolithic ceramics to composites. In particular, it is well known [8–10] that the additions of 15–30% titanium and zirconium borides to silicon carbide allows increasing strength and fracture toughness by 50–100%. However, corrosion resistance of these ceramics decreases significantly which is undesirable for the majority of high-temperature applications. Therefore, a design of the multilayer composites with external layers of corrosion resistive SiC and internal layers of improved mechanical performance such as SiC/MeB2 may be promising. Moreover, in this type of material the external SiC layers are under thermal compression stresses due to their lower coefficient of thermal expansion compared with internal SiC/MeB2 layers. This will also increase apparent fracture toughness as well as strength and reliability of laminates. The studies over the last few years have shown that the increase in strength and/or fracture toughness of multilayer ceramic composites may provide increased tolerance against damages. However, the production of such composites requires a solution to layer bonding problems. Also there is the possibility of generating new defects in the thermal stresses fields [11]. Therefore, optimization of both composite manufacturing conditions and its structure are very important to ensure elastic strain relaxation. The development and design of these Composites: Part B 37 (2006) 530–541 www.elsevier.com/locate/compositesb 1359-8368/$ - see front matter q 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2006.02.009 * Corresponding author. E-mail address: oleggrig@ipms.keiv.ua (O.N. Grigoriev)
O.N. Grigoriev et aL./Composites: Part B 37(2006)530-541 531 composites requires the determination of their mechanical Grinding and mixing of the batch components were carried characteristics such as of elastic moduli, strength, stiffness, out in a planetary ball mill. The powder particles after mixing stress-strain diagrams, distribution of phases of different have a sufficiently small grain size for hot pressing(2.5 and materials in the structure of a composite, and the magnitude 1. 1 m, respectively), ensuring optimum dispersion. The B4C and distribution of residual stresses in their volume. There are additives and in some cases. TiO, additives were introduced several analytical and experimental methods to solve these into SiC-and TiB2-based mixtures In the presence of TiO2, the problems [12-21] etc. The work presented here is based on the reactionary hot pressing with formation of secondary TiB analysis of experimentally obtained physical-mechanical during reaction T1O2+B. 2+Co took place. The characteristics of a composite [14-22]. This approach creates additives were introduced for reducing and matching of hot possibilities for theoretical investigation of a deformation pressing temperatures of various composition layers behavior of the developed composites We used a slip casting method for manufacturing lamina The effect of structure and residual stresses on strength of tapes. The ceramic tapes with a thickness 50 um were multilayer SiC(B. C)/MeB, composites was explored. A layer prepared from powders of various compositions. To remove structure (grain size, porosity etc. )was altered by using casting defects the tapes were folded in rolls and then rolled up different sintering additives, various raw materials (a- and to the thickness of 400 um. Fro B-SiC powders) as well as by changing the manufacturing required size were cut out and the packages containing 11-13 conditions. Residual thermal stresses are controlled by the pairs of alternating layers were obtained for the chosen composition of layers. Experimental studies on the mechanical compositions strength of the ceramic composite specimen were performed Hot pressing was carried out using a pilot induction hot using procedures and equipment described in [23-24] ress in graphite dies without chamber. The 2. Materials and procedures 1600-2150C, pressure 26-30 MP of isothermal densification 7-20 min, and heating rates were up to Two kinds of a-Sic powders were used: (1) technical 100%/min. Samples for testing of the mechanical properties abrasive powders, M5 grade, produced by the Zaporo abrasive plant, Ukraine, and(2)powders of UFO5 and UF10 sectioned dies grades from the Starck company, Germany. Both powders were ne specimens for mechanical tests in the form of a mixtures of polytypes: mainly 6H, 15R and 3C. TiB2 powders rectangular beam(Fig. 1a) were produced by sawing the (TC 6-09-03-7-75)from the Donetsk factory of chemical package into billets and polishing them with a diamond tool in reagents (Ukraine), and abrasive B4C powders from such a way that the layers throughout the thickness of the beam ere located as symmetrically as possible relative to the Zaporozhye abrasive plant(GOST 5744-74), were used as middle plane(Fig. 1b). Specimens of preset sizes were tested in sintering additives. Some properties of as-received powders are given in four-point bending(pure flexure, Fig. la)at room temperature Table 1 with a deformation rate(displacement of the cross-head of a The B-SiC powder, produced by Institute for Problems of testing machine)of 0.005 mm/min Vickers hardness was determined under the load of 5n. the Materials Science, had the content of 3C-polytype up to 100%o Sic powders were very different in their defectiveness and microstructure of composites was investigated by optical and sinterability. The powders UFO5 and M5 were characterized scanning electron microscopy(SEM), and the phase compe with low width of X-ray diffraction peaks and good resolution sIton--by X-ray diffraction(XRD) of Ka doublets and, therefore, had a high degree of structural perfection. XRD peaks of UF10, as well as of B-Sic powder 3. Results and discussion were very broad due to high density of defects(stacking faults, 3. 1. Characteristics of monolithic ceramics polytypes interlayer, and nonhomogeneous microstrains accordance with TEM data) which, apparently, facilitated an increase in their activity during sintering The bending strength of single-phase and heterogeneous ramics with composition similar to the ones in the layered composites is shown in Table 2. Single-phase silicon carbide Table 1 ceramics had high porosity (5-10%), average grain size 5-10 The characteristic of powder and up to 100 um for raw powders of a-SiC and B-SiC, Powder Size of particles, Content of oxy- Free carbon respectively. In the latter case the high grain size is due to grain dso(um) gen(wt%) (wt%) growth during B-a transformation of silicon carbide at hot pressing Hot pressing of pure silicon carbide without sintering 0.17 additives leads to the formation of porous coarse-grained B-Sic 0.1-0.2 ≤0.5×10 materials with the low strength(110-190 MPa). Introduction <0.1 of boron carbide allows to reduce porosity of ceramics to 1-3% with the relevant increasing of strength up to 300-370 MPa
composites requires the determination of their mechanical characteristics such as of elastic moduli, strength, stiffness, stress–strain diagrams, distribution of phases of different materials in the structure of a composite, and the magnitude and distribution of residual stresses in their volume. There are several analytical and experimental methods to solve these problems [12–21] etc. The work presented here is based on the analysis of experimentally obtained physical–mechanical characteristics of a composite [14–22]. This approach creates possibilities for theoretical investigation of a deformation behavior of the developed composites. The effect of structure and residual stresses on strength of multilayer SiC(B4C)/MeB2 composites was explored. A layer structure (grain size, porosity etc.) was altered by using different sintering additives, various raw materials (a- and b-SiC powders) as well as by changing the manufacturing conditions. Residual thermal stresses are controlled by the composition of layers. Experimental studies on the mechanical strength of the ceramic composite specimen were performed using procedures and equipment described in [23–24]. 2. Materials and procedures Two kinds of a-SiC powders were used: (1) technical abrasive powders, M5 grade, produced by the Zaporozhye abrasive plant, Ukraine, and (2) powders of UF05 and UF10 grades from the Starck company, Germany. Both powders were mixtures of polytypes: mainly 6H, 15R and 3C. TiB2 powders (TC 6-09-03-7-75) from the Donetsk factory of chemical reagents (Ukraine), and abrasive B4C powders from the Zaporozhye abrasive plant (GOST 5744-74), were used as sintering additives. Some properties of as-received powders are given in Table 1. The b-SiC powder, produced by Institute for Problems of Materials Science, had the content of 3C-polytype up to 100%. SiC powders were very different in their defectiveness and sinterability. The powders UF05 and M5 were characterized with low width of X-ray diffraction peaks and good resolution of Ka-doublets and, therefore, had a high degree of structural perfection. XRD peaks of UF10, as well as of b-SiC powder, were very broad due to high density of defects (stacking faults, polytypes interlayer, and nonhomogeneous microstrains, in accordance with TEM data) which, apparently, facilitated an increase in their activity during sintering. Grinding and mixing of the batch components were carried out in a planetary ball mill. The powder particles after mixing have a sufficiently small grain size for hot pressing (2.5 and 1.1 mm, respectively), ensuring optimum dispersion. The B4C additives and, in some cases, TiO2 additives were introduced into SiC- and TiB2-based mixtures. In the presence of TiO2, the reactionary hot pressing with formation of secondary TiB2 during reaction TiO2CB4C/TiB2CCO took place. The additives were introduced for reducing and matching of hot pressing temperatures of various composition layers. We used a slip casting method for manufacturing lamina tapes. The ceramic tapes with a thickness w50 mm were prepared from powders of various compositions. To remove casting defects the tapes were folded in rolls and then rolled up to the thickness of 400 mm. From the sheets, plates of the required size were cut out and the packages containing 11–13 pairs of alternating layers were obtained for the chosen compositions. Hot pressing was carried out using a pilot induction hot press in graphite dies without a vacuum chamber. The temperature of isothermal sintering was in the range of 1600–2150 8C, pressure 26–30 MPa, time of isothermal densification 7–20 min, and heating rates were up to 1008/min. Samples for testing of the mechanical properties having sizes 45!45!5 mm were produced in the multisectioned dies. The specimens for mechanical tests in the form of a rectangular beam (Fig. 1a) were produced by sawing the package into billets and polishing them with a diamond tool in such a way that the layers throughout the thickness of the beam were located as symmetrically as possible relative to the middle plane (Fig. 1b). Specimens of preset sizes were tested in four-point bending (pure flexure, Fig. 1a) at room temperature with a deformation rate (displacement of the cross-head of a testing machine) of 0.005 mm/min. Vickers hardness was determined under the load of 5 N. The microstructure of composites was investigated by optical and scanning electron microscopy (SEM), and the phase composition—by X-ray diffraction (XRD). 3. Results and discussion 3.1. Characteristics of monolithic ceramics The bending strength of single-phase and heterogeneous ceramics with composition similar to the ones in the layered composites is shown in Table 2. Single-phase silicon carbide ceramics had high porosity (5–10%), average grain size 5–10 and up to 100 mm for raw powders of a-SiC and b-SiC, respectively. In the latter case the high grain size is due to grain growth during b/a transformation of silicon carbide at hot pressing. Hot pressing of pure silicon carbide without sintering additives leads to the formation of porous coarse-grained materials with the low strength (110–190 MPa). Introduction of boron carbide allows to reduce porosity of ceramics to 1–3% with the relevant increasing of strength up to 300–370 MPa. Table 1 The characteristic of powders Powder Size of particles, d50, (mm) Content of oxygen (wt%) Free carbon (wt%) SiCM5 5 1.5 1–2 SiCUF05 1.47 0.55 – SiCUF10 0.7 1.2 0.17 b-SiC 0.1–0.2 %0.5!10K2 – TiB2 30 0.3 !0.1 B4C 20 1.5 2 O.N. Grigoriev et al. / Composites: Part B 37 (2006) 530–541 531
O.N. Grigoriev et al./Composites: Part B 37(2006)530-541 删斗 多缓汤0 多多 b多多 Fig. 1. Bending configuration(a)and structure of the cross-section of the specimen(b)(sizes in mm) Simultaneous introduction of 5% BC and 12% TiB, results in should follow from the Eshelby model. There are the edge an increase of strength up to 408 MPa(Table 2)and up to effects of redistribution of stresses, inhomogeneity of a stress 650 MPa at TiB 2 content of 20-25%0 The ceramics TiB2-B4C distribution across the thickness of layers. In particular, the has the strength of 415 MPa with a grain size of 5-10 um and extensive zone with tension stresses directed perpendicular to practically no porosity. The results of a detailed study of the the layers plane(ou>0) which are located near the edges. structure and mechanical behavior of Sic-TiB2-B4C ceramic Within this zone the delamination may occur. Moreover, system are presented in [9-10 various types of fracture and microcracking in composites may take place under the joint effect of both thermal and applied 3. 2. Laminated composites stresses Usually we know the parameters of materials for stress 3.2.1. Distribution of the internal stresses calculation with insufficient accuracy and calculated results At the first stage of work we studied the layered composites have only a qualitative nature. There is the need for with the maximum thermal expansion misfit between layers. experimental measurement of the stress fields in layered These systems have the highest probability of uncontrollable composites. Traditionally the tasks of the stress-strain fracture under the influence of thermal stresses measurements are performed by the diffraction methods, and In the temperature range 20-1500C the effective coeffi- first by XRD. These methods are well developed and universal cients of thermal expansion are 5.8 and 8.9/C for Sic in many cases. However, they usually require long measure- and TiB2, respectively, [25]. Within the framework of Eshelby ments and have other problems caused by the low intensity of model, there are the average stresses in a plane of layers [ 22I= X-ray peak lo331=1.4 GPa, tension of TiB2 and compression of Sic Also there is a problem of internal stress determination in According to accepted orientation of axes, the components microcrystals or between layers in layered composites, when 022 and 033 of principal stresses are in a plane of layers, the thickness of alternating layers is made from several microns whereas the au- component is perpendicular to the plane of up to 100s of microns. Therefore search and development of layers. As one can see, for TiB2-SiC laminates calculated alternative methods for determination of internal stresses is residual thermal stresses exceed a possible level of strength and should result in fracture. In practice, the level of thermal Table 2 stresses will be lower as a result of the viscoelastic relaxation Compositions and mechan due to the segregation of impurities on layer's boundaries, reinforced composites cal properties of monolithic ceramics and particles especially in composites based on the a-SiC powders, and also due to presence of the phase(B4C) with intermediate No. Composition of coefficient of thermal expansion (a=6.05X10/C in the ceramics(vol o) (SD, MPa) factor(%) temperature range of 20-1000C [251). l10(5 Finite element methods give more accurate estimation of 3 thermal stresses and the character of their distribution 26. The 4 M5+10%B4C calculations of the residual stresses in this work were done for 5 a- Siculo5+10%BC306(56 five-layer ABCBA symmetric configuration, where A-C are 6 a-SICUFO5+12% layers of ceramics: SiC, SiC +20% TiB2 and TiB2, respect- TiB, +5% B C TiB2+42% BC 415(53) ively(Fig. 2). The stress distribution is more complex, so it
Simultaneous introduction of 5% B4C and 12% TiB2 results in an increase of strength up to 408 MPa (Table 2) and up to 650 MPa at TiB2 content of 20–25%. The ceramics TiB2–B4C has the strength of 415 MPa with a grain size of 5–10 mm and practically no porosity. The results of a detailed study of the structure and mechanical behavior of SiC–TiB2–B4C ceramic system are presented in [9–10]. 3.2. Laminated composites 3.2.1. Distribution of the internal stresses At the first stage of work we studied the layered composites with the maximum thermal expansion misfit between layers. These systems have the highest probability of uncontrollable fracture under the influence of thermal stresses. In the temperature range 20–1500 8C the effective coeffi- cients of thermal expansion are 5.8 and 8.9!10K6 /8C for SiC and TiB2, respectively, [25]. Within the framework of Eshelby model, there are the average stresses in a plane of layers js22jZ js33jy1.4 GPa, tension of TiB2 and compression of SiC. According to accepted orientation of axes, the components s22 and s33 of principal stresses are in a plane of layers, whereas the s11-component is perpendicular to the plane of layers. As one can see, for TiB2–SiC laminates calculated residual thermal stresses exceed a possible level of strength and should result in fracture. In practice, the level of thermal stresses will be lower as a result of the viscoelastic relaxation due to the segregation of impurities on layer’s boundaries, especially in composites based on the a-SiC powders, and also due to presence of the phase (B4C) with intermediate coefficient of thermal expansion (aZ6.05!10K6 /8C in the temperature range of 20–1000 8C [25]). Finite element methods give more accurate estimation of thermal stresses and the character of their distribution [26]. The calculations of the residual stresses in this work were done for five-layer ABCBA symmetric configuration, where A–C are layers of ceramics: SiC, SiCC20% TiB2 and TiB2, respectively (Fig. 2). The stress distribution is more complex, so it should follow from the Eshelby model. There are the edge effects of redistribution of stresses, inhomogeneity of a stress distribution across the thickness of layers. In particular, the extensive zone with tension stresses directed perpendicular to the layers plane (s11O0) which are located near the edges. Within this zone the delamination may occur. Moreover, various types of fracture and microcracking in composites may take place under the joint effect of both thermal and applied stresses. Usually we know the parameters of materials for stress calculation with insufficient accuracy and calculated results have only a qualitative nature. There is the need for experimental measurement of the stress fields in layered composites. Traditionally the tasks of the stress–strain measurements are performed by the diffraction methods, and first by XRD. These methods are well developed and universal in many cases. However, they usually require long measurements and have other problems caused by the low intensity of X-ray peaks. Also there is a problem of internal stress determination in microcrystals or between layers in layered composites, when the thickness of alternating layers is made from several microns up to 100s of microns. Therefore search and development of alternative methods for determination of internal stresses is Fig. 1. Bending configuration (a) and structure of the cross-section of the specimen (b) (sizes in mm). Table 2 Compositions and mechanical properties of monolithic ceramics and particles reinforced composites No. Composition of ceramics (vol%) Bending strength (SD, MPa) Fluctuation factor (%) 1 b-SiC 190(40) 21 2 a-SiCM5 110(57) 52 3 a-SiCUF05 170(48) 28 4 a-SiCM5C10% B4C 372(71) 19 5 a-SiCUF05C10% B4C 306(56) 18 6 a-SiCUF05C12% TiB2C5% B4C 408(64) 17 7 TiB2C42% B4C 415(53) 13 532 O.N. Grigoriev et al. / Composites: Part B 37 (2006) 530–541
O.N. Grigoriev et aL./Composites: Part B 37(2006)530-541 604B91自 01p304560;:9:0z2 33 01203045060708090100Z2 aaea3n405060790地2 Fig. 2. Distribution of principal stresses o11, 022. 33(aHc)as well as o1(d)in laminated composites SiC/SiC-TiB2/TiB2 rgent. One of such alternative methods is optic-polarization Since, the magnitude of the signal is defined by phase traditionally used in crystallography to study transparent changes of a light wave reflected from a sample it provides materials and for simulation of the mechanic behavior a very high sensitivity for stress measurements. It was (polymers, glasses)[27] shown [30], that changes of anisotropic dielectric properties In the present work the polarization-modulation method is caused by stresses are comparable in magnitude with considered for study of internal stresses in composite materials. anisotropy of properties changes caused by the deformation It is suitable for research of transparent materials in of sample by its own weight due to gravity forces. The distribution of tensile residual stresses with a spatial resolution 10% atati, E transmittance mode and for opaque materials in reflection relative anisotropy of refractive index, arising from such mode. With this method it is possible to obtain local gravitational deformation in silicon crystal, does not exceed In this case the phase difference between two orthogonal polarized components of light with wavelength The optical scheme of experimental setup is shown in a of I cm is about 2X10 which corresponds approxi Fig. 3. The scheme performed as a Michelson interferometer mately one angular second however its peculiarity is the measurement of the phase In the case of polycrystalline ceramics the reflected beam polarized light wave cause by An=nx-y and arising at its result of interaction of reflections from 1 to 100 crystalgrana changes between two orthogonal components of linearly from accident beam of a diameter about 10 um is formed as a transmission (or reflection) through the sample. For this The state of polarization of the reflected light wave will be purpose the photoelastic modulator(PM)[28] is added into characterized by some average effective parameters of the scheme. This element represents a dynamic phase plate The alternating mechanical load of a suitable frequency of wo was applied to plate. During the period of one vibration the plate becomes the quarter wave or half wave plate depending on the magnitude of load. In the first case the linearly polarized light wave after passing through the plate is transformed in LG.126 circular polarized light, and in the second case it is transformed in linear orthogonally polarized light The principle of setup operation consists in the followin [29]. Radiation of the laser LG-126 (rad 0.63 or 1.15 um), polarized at the angle of 45 to the axis Y in the plane Xor equally divided on two light beams by the splitter. One of them is directed to a anisotropic reflector(), and another one is focused on a sample(S)by the lens(O1). The beams reflected from the reflector(R)and the sample are combined together and directed to the photodetector(PD)through the photoelastic Fig. 3. Optical scheme of polarization-modulated setup: LG-126-He-Ne modulator(PM) and polarizer(P). In the transmittance mode laser, S, sample: R, anisotropic reflector; PM, photoelastic modulator: the mirror is places behind the sample darizer pd
urgent. One of such alternative methods is optic-polarization traditionally used in crystallography to study transparent materials and for simulation of the mechanic behavior (polymers, glasses) [27]. In the present work the polarization-modulation method is considered for study of internal stresses in composite materials. It is suitable for research of transparent materials in transmittance mode and for opaque materials in reflection mode. With this method it is possible to obtain local distribution of tensile residual stresses with a spatial resolution of about 3 mm. The optical scheme of experimental setup is shown in a Fig. 3. The scheme performed as a Michelson interferometer however its peculiarity is the measurement of the phase changes between two orthogonal components of linearly polarized light wave cause by DnZnxKny and arising at its transmission (or reflection) through the sample. For this purpose the photoelastic modulator (PM) [28] is added into the scheme. This element represents a dynamic phase plate. The alternating mechanical load of a suitable frequency of u0 was applied to plate. During the period of one vibration the plate becomes the quarter wave or half wave plate depending on the magnitude of load. In the first case the linearly polarized light wave after passing through the plate is transformed in circular polarized light, and in the second case it is transformed in linear orthogonally polarized light. The principle of setup operation consists in the following [29]. Radiation of the laser LG-126 (lrad 0.63 or 1.15 mm), polarized at the angle of 458 to the axis Y in the plane XOY equally divided on two light beams by the splitter. One of them is directed to a anisotropic reflector (R), and another one is focused on a sample (S) by the lens (O1). The beams reflected from the reflector (R) and the sample are combined together and directed to the photodetector (PD) through the photoelastic modulator (PM) and polarizer (P). In the transmittance mode the mirror is places behind the sample. Since, the magnitude of the signal is defined by phase changes of a light wave reflected from a sample it provides a very high sensitivity for stress measurements. It was shown [30], that changes of anisotropic dielectric properties caused by stresses are comparable in magnitude with anisotropy of properties changes caused by the deformation of sample by its own weight due to gravity forces. The relative anisotropy of refractive index, arising from such gravitational deformation in silicon crystal, does not exceed 10K10. In this case the phase difference between two orthogonal polarized components of light with wavelength of 1 cm is about 2p!10K6 which corresponds approximately one angular second. In the case of polycrystalline ceramics the reflected beam from accident beam of a diameter about 10 mm is formed as a result of interaction of reflections from 1 to 100 crystal grains. The state of polarization of the reflected light wave will be characterized by some average effective parameters of Fig. 3. Optical scheme of polarization-modulated setup: LG-126—He–Ne laser; S, sample; R, anisotropic reflector; PM, photoelastic modulator; P, polarizer; PD, photodiode. Fig. 2. Distribution of principal stresses s11, s22, s33 (a)–(c) as well as s12 (d) in laminated composites SiC/SiC–TiB2/TiB2. O.N. Grigoriev et al. / Composites: Part B 37 (2006) 530–541 533
O.N. Grigoriev et al./Composites: Part B 37(2006)530-54 a) 1234 5 0.5mm (e)p 6 吕2 Fig. 4. Morphology of surface of double layer ceramic (a) and the curve of its scan by laser beam(b), the signal of photodiode versus the press applied to the sample double layer ceramic (c). ceramics, which depends on the structure and its physical state, zero represents the conditionally chosen origin of stresses including stresses or strains counting off. The vertical lines represent the conditionally Reflection of linearly polarized radiation from the sample drawn borders between the structurally different zones results the elliptically polarized light. The parameters of this including a porous one, near layers boundary, that was created ellipticity are defined by internal stresses or caused by external due to chemical interaction of layer components during loading. Both PM and P carry out dynamic analysis of sintering. The obtained curve of the stress distribution near polarization state of the light. The intensity of a circular boundary qualitatively fits more complex distribution than we component of elliptically polarized light is proportional to expected for simple case of two-layer configuration value of elastic stresses or deformations. Anisotropy of An Signal of the photodiode was calibrated against the pressure n-y is connected with stresses or deformations by ratio: applied to sample at the same conditions, at which the stresses An=T(ox-Oy) and An=p(Er-Ey), where T and p are- were registered. Using such calibration curve(Fig. 4c)the piezooptic and elastooptic coe fficients accordingly for each numerical magnitude of stresses was obtained for entire scan structural component of a material Results presented in Fig 4 show that the maximum stress drop For example, Fig. 4 shows the results of quantitati near boundary of layers is about 200 MP measurements carried out on a sample of Sic and Sic+ 20%TiB2 two-layered ceramics. The powders Sic and TiB2 3.2.2. Strength of laminates with the diameter of 5-10 um were taken to fabricate a sample Our studies have shown that in the composites containing The rectangular bar with the sizes 5x5X8 mm was cut out and the a-SiCMs Powders(the composites 1 and 2, Table 3)both polished for measurements types of layers(SiC and TiB 2) have a low porosity, which are Fig. 4b shows stresses registered at the scanning of a sample formed during sintering(Fig. 5a). Low relaxation ability of by the laser beam(=0.63 um and a diameter 50 um) along an such structures results in microcracking of composites with axis x according to Fig. 3. The horizontal line passing through types of fracture described in [11]
ceramics, which depends on the structure and its physical state, including stresses or strains. Reflection of linearly polarized radiation from the sample results the elliptically polarized light. The parameters of this ellipticity are defined by internal stresses or caused by external loading. Both PM and P carry out dynamic analysis of polarization state of the light. The intensity of a circular component of elliptically polarized light is proportional to value of elastic stresses or deformations. Anisotropy of DnZ nxKny is connected with stresses or deformations by ratio: DnZp(sxKsy) and DnZp(3xK3y), where p and p are— piezooptic and elastooptic coefficients accordingly for each structural component of a material. For example, Fig. 4 shows the results of quantitative measurements carried out on a sample of SiC and SiCC 20%TiB2 two-layered ceramics. The powders SiC and TiB2 with the diameter of 5–10 mm were taken to fabricate a sample. The rectangular bar with the sizes 5!5!8 mm was cut out and polished for measurements. Fig. 4b shows stresses registered at the scanning of a sample by the laser beam (lZ0.63 mm and a diameter 50 mm) along an axis x according to Fig. 3. The horizontal line passing through zero represents the conditionally chosen origin of stresses counting off. The vertical lines represent the conditionally drawn borders between the structurally different zones, including a porous one, near layer’s boundary, that was created due to chemical interaction of layer components during sintering. The obtained curve of the stress distribution near boundary qualitatively fits more complex distribution than we expected for simple case of two-layer configuration. Signal of the photodiode was calibrated against the pressure applied to sample at the same conditions, at which the stresses were registered. Using such calibration curve (Fig. 4c) the numerical magnitude of stresses was obtained for entire scan. Results presented in Fig. 4 show that the maximum stress drop near boundary of layers is about 200 MPa. 3.2.2. Strength of laminates Our studies have shown that in the composites containing the a-SiCM5 powders (the composites 1 and 2, Table 3) both types of layers (SiC and TiB2) have a low porosity, which are formed during sintering (Fig. 5a). Low relaxation ability of such structures results in microcracking of composites with types of fracture described in [11]. Fig. 4. Morphology of surface of double layer ceramic (a) and the curve of its scan by laser beam (b), the signal of photodiode versus the press applied to the sample of double layer ceramic (c). 534 O.N. Grigoriev et al. / Composites: Part B 37 (2006) 530–541
O.N. Grigoriev et aL./Composites: Part B 37(2006)530-541 Table 3 Compositions and mechanical properties of laminated composites Compositions of layers in composite(vol%) hess HV of composite Bending strength Fluctuation s(MPa) (SD, MPa) factor(%) 2)TiB2+42%B4C 24(7) 2)TB2+42%BC 1)阝-SiC 2)TiB2+42%B4C 324 1)B-SiC+10%B4C+4%TO2 41034) ②2)TiB2+42%B4C 324 Layers TiB2 are of double thickness, interlayer TiO2 on phase boundaries. Three samples are tested, in brackets is shown minim The tensile stresses in TiB, layers form cellular structur The structure and properties of composites obtained using with microcracks(Fig. 5b). In such layers the transverse cracks B-Sic powders(composites 3 and 4, Table 3)are essentially approaching a boundary propagate either into the near- different from others. SiC layers are coarse grained due to boundary volumes in the case of strong inter-layer boundaries the high recrystallization activity of these powders during or into boundaries in case of weak inter-layer bonds. It was hot pressing. The porosity of these layers is exceptionally found, that the boundary strength is reduced with a formation high-up to 40%. The prismatic columnar SiC crystals with of TiO, inter-layers. The delamination cracks were observed at a size about thickness of a layer (100 um)form composite butt-ends because of the redistribution of thermal 'engineered arch structure binding together the dense stresses at the edges and in the volume of the material. strong TiB 2 layers(Fig. 6) Considerable damage of structure results in low bending The microscopic studies of composites have not revealed strength of composites(20-100 MPa). any microcracking. The measured bending strength is high TiB2 TiB2 过 50μm TiB2 I TiB2 Fig. 5. Structure and crack formation in composites with low strength (<100 MPa)(a) microphoto of composite in the field of phase boundaries of layers a SiC/TiB2; (b)cracks in(TiB2) layer with tension stresses; (c)cracks on the weak phase boundaries: (d) longitudinal cracks along the pivotal zone of compressed Sic
The tensile stresses in TiB2 layers form cellular structure with microcracks (Fig. 5b). In such layers the transverse cracks approaching a boundary propagate either into the nearboundary volumes in the case of strong inter-layer boundaries or into boundaries in case of weak inter-layer bonds. It was found, that the boundary strength is reduced with a formation of TiO2 inter-layers. The delamination cracks were observed at composite butt-ends because of the redistribution of thermal stresses at the edges and in the volume of the material. Considerable damage of structure results in low bending strength of composites (20–100 MPa). The structure and properties of composites obtained using b-SiC powders (composites 3 and 4, Table 3) are essentially different from others. SiC layers are coarse grained due to the high recrystallization activity of these powders during hot pressing. The porosity of these layers is exceptionally high—up to 40%. The prismatic columnar SiC crystals with a size about thickness of a layer (w100 mm) form ‘engineered’ arch structure binding together the dense strong TiB2 layers (Fig. 6). The microscopic studies of composites have not revealed any microcracking. The measured bending strength is high Table 3 Compositions and mechanical properties of laminated composites No. Compositions of layers in composite (vol%) Hardness HV of composite layers (MPa) Bending strength (SD, MPa) Fluctuation factor (%) 1 (1) a-SiCM5C10% B4C 28 100(14) 14 (2) TiB2C42% B4C 33 2a (1) a-SiCM5C10% B4C 32 24(7) 29 (2) TiB2C42% B4C 33.5 3 (1) b-SiC 4 584(560–601)b 3 (2) TiB2C42% B4C 32.4 4 (1) b-SiCC10% B4CC4% TiO2 11 410(34) 8 (2) TiB2C42% B4C 32.4 a Layers TiB2 are of double thickness, interlayer TiO2 on phase boundaries. b Three samples are tested, in brackets is shown minimum maximum significances of strength. Fig. 5. Structure and crack formation in composites with low strength (%100 MPa) (a) microphoto of composite in the field of phase boundaries of layers aSiC/TiB2; (b) cracks in (TiB2) layer with tension stresses; (c) cracks on the weak phase boundaries; (d) longitudinal cracks along the pivotal zone of compressed SiC layers. O.N. Grigoriev et al. / Composites: Part B 37 (2006) 530–541 535
O.N. Grigoriev et al. / Composites: Part B 37(2006)530-54 解新A Fig. 6. Structure of strong(400-600 MPa)composites with porous SiC layers. 400-600 MPa), and the composites strength exceeds the further t that u in response to mechanical Taking into trength of monolithic ceramics. There is an unusual inverse account the numerical values of al-mechanical relationship between strength of a composite and hardness of characteristics for each layer of the Ite were not iC layers(Fig. 7). The decrease in hardness is determined by determined experimentally, the following estimates of the the relevant growth of porosity in silicon carbide. Hence, the elastic moduli (E)of the layers were assumed as a first increase of composite strength with the growth of porosity of approximation for further calculations: EA=50 GPa(material one of its component is stipulated by a high relaxation ability A) and EB=50 GPa(material B), which is justified by the and, apparently, high magnitude of critical fracture strains. object of this study The composition and strength of composites prepared using SiCUFos powders with small amount of additives and perspective for the high-temperature applications are given 3.3.1. Linear approach Table 4 Test results for the laminated specimen were used to obtain The same as in case of SiCMs powders, materials with large the experimental load F--deflection w curve for its middle mismatch of thermal expansion between layers(the composites cross-section (Fig. 8). This nonlinear curve exhibits the and 2)have low strength--60-65 MPa Decrease in thermal portions of drastic relaxation of applied loads xpansion mismatch between layers(composites 3 and 4) The analysis of the diagram demonstrated that elastic eliminates microcracking and allows for increasing strength of moduli EA and Eb of the layers were initial ones and they composites up to 300-400 MPa. It should be noted that for all cannot be used for getting analytical results fitted to layered composites, a standard deviation and fluctuation factor experimental data. At the same time, the comprehensive 2-4). It clearly demonstrates that the laminated structure makes based on g-e diagrams for each of those materials. This would geneities)which should result in the increase of Weibull procedure for calculating composite beams [21, 221 parameters and reliability Let us first consider the calculations for the composite specimen based on initially preset elastic moduli, i.e. let us 3.3. Experimental and theoretical studies on the nonlinear make calculations by linear approximation. Their results are stress-strain state of a laminated ceramic composite Composites designed as a package of twelve alternating layers of more rigid(material A) and less rigid (material B) HV SiC layers materials were used in investigations. Material A is a dense TiB2-B4C layer, and material B is a porous SiC layer(relative 5 HHV TiB porosity up to 40%o) Previous experimental investigations of the above laminated composite demonstrated the inelastic pattern of its defor- mation, which is probably caused by microcracking in the structure as a result of internal stresses in composite layers [6] At the present stage of investigation, the mechanical behavior of the composite was analyzed regardless of the initial stress- strain state determined by the above residual stresses and its Fig. 7. Relations between strength of composites and hardness of their layers
(400–600 MPa), and the composites strength exceeds the strength of monolithic ceramics. There is an unusual inverse relationship between strength of a composite and hardness of SiC layers (Fig. 7). The decrease in hardness is determined by the relevant growth of porosity in silicon carbide. Hence, the increase of composite strength with the growth of porosity of one of its component is stipulated by a high relaxation ability and, apparently, high magnitude of critical fracture strains. The composition and strength of composites prepared using SiCUF05 powders with small amount of additives and perspective for the high-temperature applications are given in Table 4. The same as in case of SiCM5 powders, materials with large mismatch of thermal expansion between layers (the composites 1 and 2) have low strength—60–65 MPa. Decrease in thermal expansion mismatch between layers (composites 3 and 4) eliminates microcracking and allows for increasing strength of composites up to 300–400 MPa. It should be noted that for all layered composites, a standard deviation and fluctuation factor of strength is much lower, than for monolithic ceramics (Tables 2–4). It clearly demonstrates that the laminated structure makes it possible to control effectively the size of defects (inhomogeneities) which should result in the increase of Weibull parameters and reliability. 3.3. Experimental and theoretical studies on the nonlinear stress–strain state of a laminated ceramic composite Composites designed as a package of twelve alternating layers of more rigid (material A) and less rigid (material B) materials were used in investigations. Material A is a dense TiB2–B4C layer, and material B is a porous SiC layer (relative porosity up to 40%). Previous experimental investigations of the above laminated composite demonstrated the inelastic pattern of its deformation, which is probably caused by microcracking in the structure as a result of internal stresses in composite layers [6]. At the present stage of investigation, the mechanical behavior of the composite was analyzed regardless of the initial stress– strain state determined by the above residual stresses and its further change in response to mechanical loading. Taking into account that the numerical values of physical–mechanical characteristics for each layer of the composite were not determined experimentally, the following estimates of the elastic moduli (E) of the layers were assumed as a first approximation for further calculations: EAZ50 GPa (material A) and EBZ50 GPa (material B), which is justified by the object of this study. 3.3.1. Linear approach Test results for the laminated specimen were used to obtain the experimental load F—deflection w curve for its middle cross-section (Fig. 8). This nonlinear curve exhibits the portions of drastic relaxation of applied loads. The analysis of the diagram demonstrated that elastic moduli EA and EB of the layers were initial ones and they cannot be used for getting analytical results fitted to experimental data. At the same time, the comprehensive theoretical investigation of composite deformation should be based on s–3 diagrams for each of those materials. This would allow for the evaluation of the stress–strain state by a known procedure for calculating composite beams [21,22]. Let us first consider the calculations for the composite specimen based on initially preset elastic moduli, i.e. let us make calculations by linear approximation. Their results are Fig. 7. Relations between strength of composites and hardness of their layers. Fig. 6. Structure of strong (400–600 MPa) composites with porous SiC layers. 536 O.N. Grigoriev et al. / Composites: Part B 37 (2006) 530–541
O.N. Grigoriev et aL./Composites: Part B 37(2006)530-541 Table 4 Compositions and strength of laminated composites based on SiCUPos powders Bending strength(SD MPa) Fluctuation factor (%) = +42%B4C UFo5+12% TiB2+5% BC a-SiCUFO5+12% TiB 2+5% BC (1)a-SiCu05+10%B4C 397(19) (2)a-SiCUFO5+12% TiB2+5% BC 5 shown in Fig. 8, where the curve obtained by calculation of a corresponding to the portion of the F-w curve up to a loading multilayer beam with the initial moduli is superimposed on the level of F=50 N is presented in Fig. 9, by approximating the experimental F-w diagram. At a loading level of F=50N, equation of the diagram which corresponds to the first significant step on the diagram The obtained Eeff values give no way of determining real (this step is assumed to characterize the initial structural stress distributions across the thickness of the specimen changes of the composite), the deflection w by linear approach accounting for the elastic moduli of the layers would be equal to wc=4.941 um at an experimental value We=7.850 um, i.e. we/w=1.588. The calculated stress in the upper layer (Fig. 1b)is 0.=85.3 MPa. Since experimental 3.3.3. Stress-strain curves for the materials of the layers strains and corresponding stresses were not determined In connection with the above, the object of the second stage because of technical reasons, they should be estimated is to plot a-E curves for each layer of the material. This process theoretically from nonlinear specimen deformation patterns can be divided into two sub stages. First, several assumptions for the behavior of the specimen upon loading(change of physicomechanical and geometrical characteristics, longitudi 3.3.2. Plotting of the nonlinear a-E curve for a quasi nal strain distribution patterns across the thickness of the cross- homogeneous specimen section, etc )are introduced; thus the first approximation a-E At the first stage of theoretical investigations, the effective diagrams for each layer may be plotted. Second, the stress- a-e curve for the composite specimen as some quasi- strain state of th e laminated spec is determined from the homogeneous material should be plotted. From this curve, obtained a-E diagrams for successive loading points.Then the known relations of applied mechanics, for determining the accordance with experimental data, and the stress-strain state is determined points of the beam [22], the effective elastic modulus is found To plot a-E curves for each layer, let us following E (1) 1. The ratio EB/EA=m is maintained in each loading point up where l is the length of the specimen, 1=0.04 m, and In2 is the to the largest experimental value of F=50N distance between the support points, 12=0.02 m If experimental deflections we that correspond to certain loading levels are taken as w and the cross-section is considered to be uniform. with the moment of inertia I= bh/12, we obtain the final equation for the effective elastic modulus (secant modulus) (2) where b is the width of the specimen, b=6.21 m, and h is the thickness of the specimen, h=3. 1X10m From this equation, moduli values summarized in Table 5 are calculated. Let us determine the bending moment and maximum stresses M/W=3F/2bh in each loading point with account of the corresponding effective elastic Deflection w,um noduli. As a result, strains Emax=0max/Ey are obtained. This Fig 8 Experimental load F--deflection w curve()and curve derived from data is also summarized in Table 5, and the a-e diagram initial elastic moduli (2)
shown in Fig. 8, where the curve obtained by calculation of a multilayer beam with the initial moduli is superimposed on the experimental F–w diagram. At a loading level of FZ50 N, which corresponds to the first significant step on the diagram (this step is assumed to characterize the initial structural changes of the composite), the deflection w by linear approach would be equal to wcZ4.941 mm at an experimental value weZ7.850 mm, i.e. we/wcZ1.588. The calculated stress in the upper layer (Fig. 1b) is scZ85.3 MPa. Since experimental strains and corresponding stresses were not determined because of technical reasons, they should be estimated theoretically from nonlinear specimen deformation patterns. 3.3.2. Plotting of the nonlinear s–3 curve for a quasihomogeneous specimen At the first stage of theoretical investigations, the effective s–3 curve for the composite specimen as some quasihomogeneous material should be plotted. From this curve, effective elastic moduli can be obtained. In accordance with known relations of applied mechanics, for determining the deflection in the middle of the span relative to the support points of the beam [22], the effective elastic modulus is found wmax Z Fl3 128Ey3 I 0Ey3 Z Fl3 128wmaxI ; (1) where l is the length of the specimen, lZ0.04 m, and l/2 is the distance between the support points, l/2Z0.02 m. If experimental deflections we that correspond to certain loading levels are taken as wmax and the cross-section is considered to be uniform, with the moment of inertia IZ bh3 /12, we obtain the final equation for the effective elastic modulus (secant modulus) Ey3 Z 3Fl3 32webh3 (2) where b is the width of the specimen, bZ6.21!10K3 m, and h is the thickness of the specimen, hZ3.1!10K3 m. From this equation, moduli values summarized in Table 5 are calculated. Let us determine the bending moment and maximum stresses smaxZM=WZ3Fl=2bh2 in each loading point with account of the corresponding effective elastic moduli. As a result, strains 3maxZsmax=Ey3 are obtained. This data is also summarized in Table 5, and the s–3 diagram corresponding to the portion of the F–w curve up to a loading level of FZ50 N is presented in Fig. 9, by approximating the equation of the diagram. The obtained Eeff values give no way of determining real stress distributions across the thickness of the specimen accounting for the elastic moduli of the layers. 3.3.3. Stress–strain curves for the materials of the layers In connection with the above, the object of the second stage is to plot s–3 curves for each layer of the material. This process can be divided into two sub stages. First, several assumptions for the behavior of the specimen upon loading (change of physicomechanical and geometrical characteristics, longitudinal strain distribution patterns across the thickness of the crosssection, etc.) are introduced; thus the first approximation s–3 diagrams for each layer may be plotted. Second, the stress– strain state of the laminated specimen is determined from the obtained s–3 diagrams for successive loading points. Then the initial diagrams are corrected by several iterations in accordance with experimental data, and the stress–strain state is determined. To plot s–3 curves for each layer, let us assume the following: 1. The ratio EB/EAZm is maintained in each loading point up to the largest experimental value of FZ50 N. Table 4 Compositions and strength of laminated composites based on SiCUF05 powders No. Compositions of layers (vol%) Bending strength (SD MPa) Fluctuation factor (%) 1 (1) a-SiCUF05 65(13) 20 (2) TiB2C42% B4C 2 (1) a-SiCUF05C12% TiB2C5% B4C 61(6) 10 (2) TiB2C42% B4C 3 b-SiC 314(20) 6 a-SiCUF05C12% TiB2C5% B4C 4 (1) a-SiCUF05C10% B4C 397(19) (2) a-SiCUF05C12% TiB2C5% B4C 5 Fig. 8. Experimental load F—deflection w curve (1) and curve derived from initial elastic moduli (2). O.N. Grigoriev et al. / Composites: Part B 37 (2006) 530–541 537
O.N. Grigoriev et al./Composites: Part B 37(2006)530-541 Preliminary results of calculation on the base the experimental dependence F- F(w) We(MKM (MIla) MIla) Enax(M/M) D(HM) EA (MIla) EB(MIla) EA(MM) EB(M/M) GA(MIla) B(MIla 3.55X1 5.50×105500×1040 0.5 3.12×105503 1.61×10-54808 523×105475×104163×10-51.59×10-58.53×10°0.76×10 1.10 295×105 527×10-54546 3.41×10 494×105449×104345×10-53.37×10-517.1×10°151×10 4.80X10 436×104533×10-55.20×10-5256×10°227×10 50505050 m 744×10-54.117 453×105412×104753×10-57.35×10-5341×10°3.03×10 977 431×105392×1049.88×10-5965×10-5426×10° 78×10 243×105 124×1 408×1053.71×10412.1×10-5123×10-5512×10° .54X10 232×1053519 3.88×105353×104154×10-515.0×10-5597×10°5.30×10° 3.78×105344×104180×10-5176×10-5682×10°605×10° 5×105452411×10-533093.60×103327×10521.3×10-520.8×10-5767×1°6.81×10° 7.8 207×10550.27 243×10-53185346×1053.5×10246×10-5240×10-5853×10°1756×10° 2. Longitudinal strains E of the specimen are linearly Let us rewrite the second formula in Eq. ( 3)using the elastic distributed across its thickness, i.e. the hypothesis of moduli Ea and EB and allowing for the third assumption: plane cross-sections is fulfilled(this is determined by the ortion between support points ) and the total thickness of D=E∑3(-a)+E∑3(a-动) the specimen and the layer thicknesses do not change 3. We assume (not finally) that in each loading point, the The terms under the summation signs are the inertia elastic moduli of all layers do not depend on longitudinal moments for the A and B layers relative to the center of strains E, i.e. the elastic moduli of materials A and b do not stiffness C of the specimen cross-section [ 19] change across the thickness of the cross-section 4. The elastic moduli of the layers in compression and tension b b IB <. Let us come to the real heterogeneous laminated system; for With account of the first assumption, let us write down the his system, the following equations for the coordinate of the final equation for stiffness in any loading point as: center of stiffness C and bending stiffness [11](Fig. 10)are D=EACA mlB) The largest deflections of the specimen can be calculated by ∑E(=2-dm) D=3∑E(-)(3 2∑Ek(a06-xom,) Wmax128D where k is the number of a layer calculated from the bottom If experimental we values are taken as the largest deflections, stiffnesses for each loading point can be determined from Eq (7): D= o=1.14*+102*e3-7,34*108*g2+3.18°10°e 20n Strain s,。10m/m Fig.9. Effectiveσ re for the case of a quasi-homogeneous bear Fig. 10. Calculation scheme for the specimen cross-section
2. Longitudinal strains 3 of the specimen are linearly distributed across its thickness, i.e. the hypothesis of plane cross-sections is fulfilled (this is determined by the fact that the beam is in the state of pure flexure over the portion between support points), and the total thickness of the specimen and the layer thicknesses do not change. 3. We assume (not finally) that in each loading point, the elastic moduli of all layers do not depend on longitudinal strains 3, i.e. the elastic moduli of materials A and B do not change across the thickness of the cross-section. 4. The elastic moduli of the layers in compression and tension are equal. Let us come to the real heterogeneous laminated system; for this system, the following equations for the coordinate of the center of stiffness C and bending stiffness [11] (Fig. 10) are used zC Z P 12 kZ1 Ek z 2 06kKz 2 0Hk 2 P 12 kZ1 Ekðz06kKz0Hk Þ ; D Z b 3 X 12 kZ1 Ek z 3 6k Kz 3 Hk (3) where k is the number of a layer calculated from the bottom upwards. Let us rewrite the second formula in Eq. (3) using the elastic moduli EA and EB and allowing for the third assumption: D Z EA X 6 kZ1 b 3 z 3 62kKz 3 H2k CEB X 6 kZ1 b 3 z 3 62kK1Kz 3 H2kK1 (4) The terms under the summation signs are the inertia moments for the A and B layers relative to the center of stiffness C of the specimen cross-section [19]: IA Z b 3 X 6 kZ1 z 3 62k Kz 3 H2k ; IB Z b 3 X 6 kZ1 z 3 62kK1Kz 3 H2kK1 : (5) With account of the first assumption, let us write down the final equation for stiffness in any loading point as: D Z EAðIA CmIBÞ: (6) The largest deflections of the specimen can be calculated by the equation: wmax Z Fl3 128D (7) If experimental we values are taken as the largest deflections, stiffnesses for each loading point can be determined from Eq. (7): D Z Fl3 128we (8) Fig. 10. Calculation scheme for the specimen cross-section. Table 5 Preliminary results of calculation on the base the experimental dependence F–w F (w) We (MKM) Ey3 (MPa) smax (MPa) 3max (M/M) D (H M2 ) EA (MPa) EB (MPa) 3A (M/M) 3B (M/M) sA (MPa) sB (MPa) 0 0 3.55!105 0 0 5.060 5.50!105 5.00!104 0000 5 0.52 3.12!105 5.03 1.61!10K5 4.808 5.23!105 4.75!104 1.63!10K5 1.59!10K5 8.53!106 0.76!106 10 1.10 2.95!105 10.05 3.41!10K5 4.546 4.94!105 4.49!104 3.45!10K5 3.37!10K5 17.1!106 1.51!106 15 1.70 2.86!105 15.08 5.27!10K5 4.412 4.80!105 4.36!104 5.33!10K5 5.20!10K5 25.6!106 2.27!106 20 2.40 2.70!105 20.11 7.44!10K5 4.117 4.53!105 4.12!104 7.53!10K5 7.35!10K5 34.1!106 3.03!106 25 3.15 2.57!105 25.13 9.77!10K5 3.968 4.31!105 3.92!104 9.88!10K5 9.65!10K5 42.6!106 3.78!106 30 4.00 2.43!105 30.16 12.4!10K5 3.750 4.08!105 3.71!104 12.1!10K5 12.3!10K5 51.2!106 4.54!106 35 4.90 2.32!105 35.19 15.2!10K5 3.571 3.88!105 3.53!104 15.4!10K5 15.0!10K5 59.7!106 5.30!106 40 5.75 2.26!105 40.22 17.8!10K5 3.478 3.78!105 3.44!104 18.0!10K5 17.6!10K5 68.2!106 6.05!106 45 6.80 2.15!105 45.24 21.1!10K5 3.309 3.60!105 3.27!104 21.3!10K5 20.8!10K5 76.7!106 6.81!106 50 7.85 2.07!105 50.27 24.3!10K5 3.185 3.46!105 3.15!104 24.6!10K5 24.0!10K5 85.3!106 7.56!106 Fig. 9. Effective s–3 curve for the case of a quasi-homogeneous beam. 538 O.N. Grigoriev et al. / Composites: Part B 37 (2006) 530–541
O.N. Grigoriev et aL./Composites: Part B 37(2006)530-541 539 Table 7 The calculation results of first stage on the base a-e dependence Calculation results on the base of corrected a-e dependence F(H) w(MKM) GA(MIla) oB (MIla) D(HM) F(H) w(MKM) A (MIla) oB(MIla) D(HM) 844×100.75×10°4.79 842×100.75×10° 4.579 1.48×10° 166×10147×10° 248×10220×10p 175 46×10°2.16×10° 4.287 32.6×10°290×10° 2.82×1 40.3×10°3.58×10°4277 98×10° 45×1 3.982 47.8×10° 0°4147 47.1×10°405×1 3.825 52×10° 10°4017 542×10°462×1 2.5×10°5.54×10° 61.1×105.16×10°3.509 Stiffnesses calculated by this equation are summarized in Table 5. Maximum normal stresses oa and oB arising in the upper and lower layers, respectively, are found by Hooke's law 4. Let us equate the stiffnesses by Eqs.(6)and( 8); as a result o=Ee with account of calculated elastic moduli and strains get the equation for the elastic modulus of material A Stress values are also cited in Table 5 To plot a-E diagrams for the materials under study, EA Material A: 0A The elastic modulus of material B is determined by the equation EB=mEA Calculated elastic moduli(secant moduli) =1.83×1012g3-1.21×10°c2+5.35 are summarized in Table 5 To obtain longitudinal strains in the middle cross-section of X 10E(MPa) the beam, let us use the equation of applied mechanics 174×10g3-1.13×10c2+487 where z is the coordinate of the which is examined relative to the cross-sectional The ab X10*E(MPa) assumptions allow one to take the arbitrary z coordinate of the cross-section for the plotting of a-E diagrams. Let us take the These relations became the basis for determining the stress- coordinate za=1.569x10m(upper point of the cross- strain state section)for A layers and the coordinate zB=1.531X10 m (lower point of the cross-section) for B layers relative to the 3. 3. 4. Stress-strain state of the composite specimen center of stiffness. Strains EA and EB in those points are given in The procedure for determining the stress-strain state (second substage)taking into account the nonlinear properties of the layers(Eqs.(11)and(12)is as follows: Fig. 11. Experimental (points), nonlinear(solid line), and linear(dash line)F-w Fig. 12. Stress-strain curves for A and B layers and effective deformation diagram for the beam as a quasi-homogeneous material
Stiffnesses calculated by this equation are summarized in Table 5. Let us equate the stiffnesses by Eqs. (6) and (8); as a result we get the equation for the elastic modulus of material A: EA Z Fl3 128ðIA CmIBÞwe (9) The elastic modulus of material B is determined by the equation EBZmEA. Calculated elastic moduli (secant moduli) are summarized in Table 5. To obtain longitudinal strains in the middle cross-section of the beam, let us use the equation of applied mechanics 3 1 2 ;z Z Fl 4D z; (10) where z is the coordinate of the point which is examined relative to the cross-sectional stiffness center. The above assumptions allow one to take the arbitrary z coordinate of the cross-section for the plotting of s–3 diagrams. Let us take the coordinate zAZ1.569!10K3 m (upper point of the crosssection) for A layers and the coordinate zBZ1.531!10K3 m (lower point of the cross-section) for B layers relative to the center of stiffness. Strains 3A and 3B in those points are given in Table 5. Maximum normal stresses sA and sB arising in the upper and lower layers, respectively, are found by Hooke’s law sZE3 with account of calculated elastic moduli and strains. Stress values are also cited in Table 5. To plot s–3 diagrams for the materials under study, calculated data was fitted by cubic polynomials: Material A :sA Z1:83!1012 3 3K1:21!109 3 2 C5:35 !105 3 ðMPaÞ; (11) Material B :sB Z1:74!1011 3 3K1:13!108 3 2 C4:87 !104 3 ðMPaÞ: (12) These relations became the basis for determining the stress– strain state. 3.3.4. Stress–strain state of the composite specimen The procedure for determining the stress–strain state (second substage) taking into account the nonlinear properties of the layers (Eqs. (11) and (12)) is as follows: Table 6 The calculation results of first stage on the base s–3 dependence F (H) w (MKM) sA (MPa) sB (MPa) D (H M2 ) 0 0 0 0 5.060 5 0.52 8.44!106 0.75!106 4.796 10 1.07 16.7!106 1.48!106 4.668 15 1.65 24.8!106 2.20!106 4.538 20 2.27 32.6!106 2.90!106 4.408 25 2.92 40.3!106 3.58!106 4.277 30 3.62 47.8!106 4.24!106 4.147 35 4.36 55.2!106 4.90!106 4.017 40 5.14 62.5!106 5.54!106 3.891 45 5.97 69.8!106 6.19!106 3.770 50 6.84 77.2!106 6.84!106 3.656 Fig. 11. Experimental (points), nonlinear (solid line), and linear (dash line) F–w curves. Table 7 Calculation results on the base of corrected s–3 dependence F (H) w (MKM) sA (MPa) sB (MPa) D (H M2 ) 0 0 0 0 5.060 5 0.55 8.42!106 0.75!106 4.579 10 1.13 16.6!106 1.47!106 4.435 15 1.75 24.6!106 2.16!106 4.287 20 2.42 32.3!106 2.82!106 4.136 25 3.14 39.8!106 3.45!106 3.982 30 3.92 47.1!106 4.05!106 3.825 35 4.77 54.2!106 4.62!106 3.667 40 5.70 61.1!106 5.16!106 3.509 45 6.70 68.0!106 5.68!106 3.356 50 7.79 75.1!106 6.21!106 3.211 Fig. 12. Stress–strain curves for A and B layers and effective deformation diagram for the beam as a quasi-homogeneous material. O.N. Grigoriev et al. / Composites: Part B 37 (2006) 530–541 539