DETERMINATION OF MATERIAL PARAMETERS OF A TEXTILE REINFORCED COMPOSITE USING AN INVERSE METHOD

Transcription

1 DETERMINATION OF MATERIAL PARAMETERS OF A TEXTILE REINFORCED COMPOSITE USING AN INVERSE METHOD J. Blom, H. Cuypers, P. Van Itterbeek and J. Wastiels VUB in Brussels -Faulty of Engineering, Department of Mehanis of Materials and Construtions, Pleinlaan 2, 1050 Brussels, Belgium, johan.blom@vub.a.be ABSTRACT A great diversity of different ement based fibre reinfored onrete (FRC) materials an be found today either in pratial use or under development in researh laboratories. The main goal of using fibres in a ement based matrix is to improve the dutility. A FRC is alled a high performane material when the use of fibres leads to tensile strain hardening. This an only be obtained by using a fibre volume fration whih is exeeding the ritial volume fration. If so the fibres an ensure strength and stiffness at applied loads far exeeding the range in whih matrix multiple raking ours. Textile reinfored ementitious omposites with glass fibres as reinforement exhibit relatively high strength and dutility and thus provide and interesting new material for onstrution purposes. The textile reinfored onrete (TRC) under study in this paper is essentially omposed of two brittle materials. In this ase a random distributed glass fibre textile is used in ombination with an inorgani phosphate ement matrix. The material itself is haraterised by a linear behaviour in ompression and a non linear tensile behaviour. However, in order to design and optimize onstrutions it is not only neessary to develop effetive analysis and design guidelines but also a method to obtain the neessary model parameters. In this paper an inverse method is presented to derive the model parameters whih are normally obtained from a tensile test. The model parameters an be defined as the ultimate matrix stress and the effiieny of the fibres and matrix. The proposed inversed method uses the experimental fore- defletion urves obtained from a four point bending test to derive the model parameters. 1. INTRODUCTION Textile reinfored ementitious omposites with glass fibre reinforement are new omposite materials whih an be tailored to the needs and speifiations of the onstrutive elements. By using a fibre volume fration whih is exeeding the ritial fibre volume fration, the fibres an ensure strength and stiffness at applied loads far exeeding the range in whih matrix multiple raking ours. The TRC omposites are interesting new high performane materials. Freeform arhiteture and lightweight onstrution elements (e.g. sandwih panels with TRC faes [1] and hypar shells [2]) only represent a small part of the onstrution types where the advantages of this material ould be exploited. If however we want to design and optimize a onstrution with a new material the development of a material model is neessary. Over the years muh effort has been direted into the modelling of the non linear tensile behaviour. The behaviour in tension and ompression of the above mentioned material is well doumented in literature. The Aveston Cooper and Kelly (ACK) theory is one of the earliest developed models [3, 4]. In ompression, this material exhibits a linear elasti behaviour up to failure. Sine the use of fibre reinforement in the form of textiles 1

2 allows the introdution of a high fibre volume fration, a distint strain hardening behaviour an be obtained in tension [5]. In an initial step based on the ACK theory, the bending behaviour of slender TRC beams an be analytially desribed [6, 7]. For eah ross setion the equilibrium of fores and moments an be alulated. Subsequent integration of the bending stiffness for eah differential beam element with a ertain height and a unit width, over the total length, will result in a omputed fore displaement diagram. The main goal of this paper is to propose a model whih enables the user to derive the model parameters whih enable the user to derive the parameters whih are normally obtained from a tensile test, from data obtained from a bending test. The model parameters are the effiieny of the matrix ( matrix ), the effiieny of the fibres ( ) and the ultimate matrix stress ( fibre mu ). The material parameters an be defined as the stiffness in the unraked stage, the omposite multiple raking stress and the stiffness of the omposite in raked stage. The proposed inverse method uses the experimental fore defletion urves obtained from a four point bending test. To determine the influene of model parameters on the omputed bending behaviour a parametri study was preformed. 2. MODEL 2.1 TRC in tension and ompression In this paper the stress strain behaviour in tension will be modelled using the ACK theory. In this model, the mehanial behaviour of unidiretional (UD) ontinuous fibre reinfored brittle matrix omposites is desribed. Basi assumptions of the ACK theory are: - The fibres an only ope with loads along their longitudinal axis, they provide no bending stiffness. - There is a weak bond between fibre and matrix - One the matrix and fibres are debonding a pure fritional shear replaes the existing adhesion shear bond. - The fritional shear is assumed to be onstant - Poisson effets of the fibre and matrix are negleted. - The load distribution in the ross setion perpendiular to longitudinal axis is assumed to be uniform ACK theory. Aording to the ACK theory, three distint stages an be deteted in the stress-strain urve of a unidiretional reinfored brittle matrix omposite. In the first stage the material behaves linear elasti. The omposite stiffness in stage I (E1) an be derived from the law of mixtures see equation (1). In this first stage a prefet elasti bond between matrix and fibres is assumed. The failure strain of the matrix is lower than the failures strain of the fibres. At the ultimate matrix strain the omposite will rak. If the fibre volume fration is higher than the ritial fibre volume fration, the fibres will be able to sustain the additional loading. Aording to the ACK theory this stage is alled the multiple raking stage. In the third stage post raking the matrix is ompletely rak. The fibre will arry the load in this final stage, until failure. Figure 1 illustrates a theoretial stress-strain urve with these three distint stages (used in the ACK theory): linear elasti stage (I), multiple raking stage (II) and post raking stage (III). 2

3 E 3 E 1 Fig.1: experimental and theoretial stress-strain urve, aording to the ACK theory [3] In the linear elasti stage (stage I), aording to the ACK theory, the stiffness of the omposite E 1 is funtion of the fibre volume fration V f, the volume fration of the matrix V m, the stiffness of the fibres E f and the stiffness of the matrix E m. The matrixfibre interfae bond is assumed to be elasti and the omposite stiffness E 1 an be determined by the law of mixtures: E E V E V (1) 1 f f m m The ACK theory is modified, due to imperfet matrix-fibre adhesion, warping or misalignment of the unidiretional fibres, inlusion of air voids, et. the fibre volume fration has to be lowered with a fibre effiieny fator f. Also the stiffness of the matrix has to be redued with a matrix effiieny fator m. The modified law of mixtures an be written as followed. E E V E V (2) 1 f f m m The effetive fibre volume fration an be alulated by using the following equation : V f V f f (3) Taking the effiieny of the matrix in aount results in the following equation: V V (4) f m m When the first rak is introdued the ultimate strain of matrix ( mu ) and omposite multiple raking strain ( m ) are equal. The omposite multiple raking stress ( m ) an be determined by the following equation (5) with ( mu ) defined as the ultimate matrix stress. mue1 m (5) Em When the first rak appears and reahes a fibre, debonding of the matrix-fibre interfae ours and further matrix-fibre interation ours through frition. The fritional interfae stress ( o ) is assumed to be onstant along the debonding interfae. The debonding length 0 an be alulated from the equilibrium of the fores along the rak. In ase of irular fibres with a radius (r) the following expression (6) an be used to derive the debonding length. mu. r. Vm 0 (6) 2.. V 0 f Inreasing the load will lead to multi raking, if the fibre volume fration is inreasing the ritial volume fration. Aording to the ACK theory at a ertain unique omposite stress m, multiple parallel raks are introdued in the matrix. Craks are introdued 3

4 until saturation is reahed. The distane between neighbouring raks are situated between o and 2 o, with an average of 1,337 o. After multiple raking, the strain of stageii the omposite an be alulated as follows(7).(from integration of the strain field between two raks with distane 1,337) stageii ( ) mu (7) with EmVm EV f f (8) One full multiple raking has ourred, only the fibres further dediate to the stiffness in stage III (post-raking stage). The stiffness of the omposite E 3 in this stage is thus: E E V (9) 3 f f 2.3 TRC beam element in bending Sine the behaviour of TRC in tension shows three stages, three expressions are needed to define the equilibrium of fores and moments if a speimen is loaded in bending. By expressing the equilibrium of fores and moments, the position of the neutral axis (whih will be further written as a ) and the maximum ourring tensile strain ( t ) in the setion an be alulated for eah ross setion along the beam. Subsequent integration of the bending stiffness for eah differential beam element with height h and a unit width, over the total length, will lead to a fore displaement diagram. For eah ross setion one of the following sets of equilibrium equations an be used depending on the value of the maximum tensile stress. As long as the omposite behaves linear elasti along the whole beam setion the following expressions an be used, with M e defined as external moment: a h / 2 (10) 2 Me a ² E1 t (11) 3 One the omposite maximum tensile strain is situated in the multiple raking stage the equilibrium equations an be based on the internal strains and stresses in figure 2 a. a Neutral axis h Mi dx t t t (2a) (2b) Fig. ( 2 ) distribution of stress and strain over ross setion (2a) distribution of stress strain over ross setion in the multiple raking stage (2b) distribution of stress strain over ross setion in the post raking stage By expressing the equilibrium of fores (12) and moments (13), the position of the neutral axis and the maximum ourring tensile strain ( t ) in the setion an be alulated for eah ross setion along the beam. 4

5 E a² = + 2( h a) 2 m 1 t m mu with ( ) and t h a h a a³ m E1 t ² m ( ) Me (13) 3( h a) 3 2 Finally the omposite maximum tensile strain will be situated in the post-raking stage. By expressing the equilibrium of fores (14) and moments (15), the position of the neutral axis and the maximum ourring tensile strain ( t ) in the setion an be alulated for eah ross setion along the beam. (12) E a² = + + ( ) E V 2( h a) 2 2 m stageii 1 t m t f f with E h a stageii m 1 t ² m ( ) mu a³ 3( h a) 3 2 stageii ( h a ) ( t ) E fv f Me 3 2 One the equilibriums of all setions are established, the defletion in any setion an be determined by double integration of M e /EI setion along the length of the beam. EI setion is the bending stiffness as alulated for eah setion of the beam. (14) (15) 2.4 Sensitivity analysis: The sensitivity analysis showed that ertain model parameters (e.g. the ultimate matrix stress, fibre effiieny and matrix stiffness) have a onsiderable influene on the modelled bending behaviour. Figure (3 a) is representing the tensile stress strain urve obtained from the analytial solution based on the ACK theory. The parameters used in this alulation are listed in the table below. Em 18 GPa matrix stiffness Ef 72 GPa fibre stiffness max 1.5 % omposite ultimate strain 0.27 Effiieny of the fibres fibre matrix 1 Effiieny of the matrix l 200 mm Length B 50 mm Width H 7.56 mm Thikness nfl 10 Number of layers mu 2 up to 12 (step 2) MPa matrix ultimate stress Table1: input parameters used in the parametri study The bending model simulated a 4 point bending test with a span of 200 mm between the supports and a distane of 100 mm between the applied fores. For eah tensile stress strain urve the resulting fore defletion urve is plotted (3 b). Inreasing the ultimate matrix stress ( mu ) will not only enlarge the linear elasti stage in bending stage, but will also aentuate the non linear behaviour. 5

6 inreasing mu inreasing mu Figure 3 a: omputed tensile stress the influene of mu strain urve, showing Figure 3 b: omputed fore displaement urves in 4 point bending The stiffness in the third stage is inreasing by the influene of the effiieny of the fibre reinforement ( f ). This is learly shown in figure (4 a) whih is representing the tensile stress strain urve obtained from the analytial solution based on the ACK theory. The fore displaement urve shows the same trend as the tensile stress strain urve. Changing the fibre effiieny from 0.05 up to 0.3 with a step of 0.05 mainly affet the last part of the fore displaement, this is shown in figure (4 b). The fibre effiieny was limited to 0.3 aording value obtained for 2 D random fibres (6). inreasing fibre inreasing fibre Fig:ure 4 a: omputed tensile stress the influene of fibre effiieny strain urve, showing Figure 4 b: omputed fore displaement urves in 4 point bending The effet of hanging the matrix stiffness between 5 up to 25 GPa with a step of 5 GPa is plotted in the stress strain urve figure (5 a). When observing the urves it is lear that the effet of hanging the matrix stiffness is small. When observing the tensile stress urves it is lear that the effet of hanging the matrix stiffness only affets the first zone. inreasing E m inreasing E m Figure 5 a: omputed tensile stress influene of matrix stiffness strain urve, showing the Figure 5 b: omputed fore displaement urves in 4 point bending 6

7 Figure 5 b shows the influene of the matrix stiffness on the defletion in 4 point bending when the maximum tensile strain is kept onstant (1.5%). It s lear that the matrix stiffness influenes the omplete behaviour of the omposite. When inreasing the matrix stiffness, the effet on the nonlinear behaviour on the fore defletion urves is inreasing. 3. INVERSE METHOD The inverse method developed within the sope of this paper allows the user to determine a set of model parameters ( matrix,, fibre mu ) with a ertain degree of auray. The set of model parameters ontains the effiieny of the matrix ( matrix ), the effiieny of the fibres ( ) and the ultimate matrix stress ( fibre mu ). At first the experimental fore defletion urve obtained from a four point bending test will be loaded in the program. The inverse method will use the bending model based on the ACK theory. The program will generate for eah set of three parameters ( matrix,, fibre mu ) an analytial fore displaement urve. The proess is repeated until the differene between the experimental and analytial fore displaement urve is minimized, using the oeffiient of determination (R²) [10]. 4. EXPERIMENTS The experimental part was preformed to obtain fore defletion urves from 4 point bending test whih were used in the inverse method as input data. To verify the proposed inverse method, tensile tests were also preformed. 4.1 Speimen preparation The speimens used in this paper were glass fibre- reinfored ementitious laminates. The ementitious matrix used for all the speimens is an Inorgani Phosphate Cement (IPC)[11]. IPC has been developed at the Vrije Universiteit Brussel and shows a neutral ph after hardening. Therefore, the fibres are hardly attaked by the onrete matrix with time. The material properties of hardened IPC are similar to those of other ement-based materials. Although the strength in ompression is rather high, the tensile strength is low. The tensile strength of an IPC speimen is about one tenth of the ompressive strength. One standard IPC mixture is hosen in this work, without use of any fillers or retards or aelerating omponents. All IPC laminates are made by hand lay-up and ured in ambient onditions for 24 hours. Post-uring is performed at 60 C for 24 hours. Like most ementitious mixtures, the strength of IPC inreases with time. This effet an however be aelerated by the post-uring at 60 C. During the uring and post-uring proess, both sides of the laminate are overed with plasti to prevent early evaporation of water. The E-glass fibre reinforements used in the speimens was hopped glass fibre mats ( 2D-random ) with a fibre density of 300g/m² (Owens Corning M ). The dimension of the plate is 50mx50m. The plates were produed with an average matrix onsumption of 1200 g/m² for eah layer. The plates were ut with a water ooled diamond saw. About four ut strip speimens had a width of 25mm and six others had a width of 50mm. The speimens with a width of 25mm are used for tensile testing and the speimens with a width of 50mm were tested in bending. 7

8 4.2 Bending test For eah laminate, several speimens were loaded in a 4 point bending test. Two supports were plaed with a span of 200 mm, the rosshead was equipped with horizontal bars were two ylinder were mounted with a distane of 100 mm. By moving the ross head at a onstant speed the ylinders will transfer a fore to the speimen. The fore on the ross head and the displaement in the entre of the laminate were measured. The testing mahine was displaement ontrolled; the loading rate was set to 1 mm/min. 4.3 Tensile testing The tensile test was arried out to obtain the ACK-model parameters for all tested laminates. The stress strain urve data was generated using a tensile testing mahine with a apaity of 100 kn. The rate of ross head displaement was set at 1 mm/min. The strain was measured with an extensometer. The resulting stress-strain urves for a 6 layers random reinfored TRC are plotted and disussed here. Figure 8: shows the stress-strain urves obtained on the speimens from plate 2 (with 6 layers of fibres) The stiffness in the first (E1) and the third stage (E3) an be obtained from the experimental data by determination of the slope of the urves. The values of E1 and E3 are listed below in table 2. The model values obtained from the experimental data are printed ursive. Plate number Number of layers Thikness (mm) V f (%) E 1 (GPa) experimental E 3 (GPa) experimental matrix fibre avg stdev Table 2: overview of the experimental data mu(gpa) 5. COMPARISON BETWEEN MODEL PREDICTIONS AND EXPERIMENTS To illustrate the inverse method the tensile stress urves are generated with the material parameters ( matrix, fibre, mu ) obtained from the inverse method whih uses the experimental fore defletion urves. In this ase the fore defletion urves (fig. 9) of a six layer random speimen are used in the inverse method.. 8

9 6 layers Random 2 D fibre mats 300 g/m² Stress MPa Figure 9: experimental obtained fore displaement urves of a 6 layers random speimen and the stress strain urves omputed with the inverse method ,5 1 1,5 2 strain % For eah fore displaement urve as set of parameters ( matrix, shown in table below. fibre, mu ) is derived as F- urve s mu (MPa) matrix fibre E1 (GPa) E3 (GPa) R² avg stdev Table 3: values obtained from the inverse method (6 layers random) The inverse method was used to obtain the parameter of the other speimens. For eah set of speimens with the same amount of layers the average of parameters were plotted in the table below (4). The values in the thiker retangle are derived by the inverse method. n fl matrix fibre mu matrix m fibre f mu mu MP a MP a MP a , a vg stdev Table 4: average values obtained from the inverse method When omparing the experimental model parameters and the values obtained from the inverse method it is lear that the influene of the thikness an be negleted. The values of the model parameters ( matrix,, fibre mu ) derived from the inverse method are lose to the experimental. In table 4 the differene between experimental and analytial model parameters is listed (,, matrix fibre mu ). 9

10 6. CONCLUSIONS The inverse method developed within the sope of this paper allows the user to determine a set of model parameters ( matrix,, fibre mu ). Comparing the experimental and analytial values of the model parameters showed that the differene is smaller than the standard deviation whih means that the proposed inverse model is apable to ompute the parameters with auray. When omparing the experimental model parameters and the values obtained from the inverse method it is lear that the influene of the thikness on the model parameters an be negleted. REFERENCES (1) Cuypers H., Analysis and Design of Sandwih panels with Brittle matrix omposite faes for building appliations, Phd. Thesis, VUB 2002 (2) E. De Bolster, H. Cuypers, K. Watzeels, G. Mosselmans, M. Alshaaer,Numerial analysis of a modular system onsisting of hyperboli paraboloids, made of a ementitious omposite, Department of mehanis and onstrutions, Vrije Universiteit Brussel, Belgium (3) Aveston, J.; Cooper, G.A.; Kelly, A.: Single and multiple frature. The properties of fibre omposites Proeedings Conf. National Physial Laboratories, IPC Siene & Tehnology Press Ltd., London, UK, Nov. 1971, p15. (4) Aveston J., Merer R.A. and Sillwood J.M., The mehanism of fibre reinforement of ement and onrete,si No 90/11/98, January 1975, DMA 228, February 1976 (5) Naaman A. E., Toughness, dutility, surfae energy and defletion-hardening FRC omposites, Proeedings of the JCI international Workshop on Dutile Fiber Reinfored Cementitious Composites (DFRCC) - Appliation and Evaluation (DFRCC-02), Takayama, Japan, Otober, 2002, pp (6) Cox H;L;, The elastiity and strength of paper and other fibrous material, British Journal of Applied Physis, Vol.3,1952,p (7) J. Blom, H. Cuypers P,. Van Itterbeek, J. Wastiels, Modelling the behaviour of Textile Reinfored Cementitious omposites under bending, Prague, Fibre Conrete 4 th International Conferene, p , 2007 (8) Maalej M., Li V.C., Flexural/Tensile-Strength Ratio in Engineered Cementitious Composites, Journal of Materials in Civil Engineering Vol 6.no4, November 1994 (9) G.V. Reklaitis, A. Ravindran, K.M. Ragsdell, Engeneering Optimalization Methods and Appliations, A Wiley Intersiene Publiation, hap 2, hap 3. (10) Everitt, B.S, Cambridge Ditionary of Statstis, Cup. ISBN x, 2002 (11) Vubonite, tehnial dat sheet, 10

11 This doument was reated with Win2PDF available at The unregistered version of Win2PDF is for evaluation or non-ommerial use only. This page will not be added after purhasing Win2PDF.