Finite Element Modeling of Residual Thermal Stresses in Fiber-Reinforced Composites Using Different Representative Volume Elements

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1 Proceedings o the World Congress on Engineering 21 Vol II WCE 21, June 3 - July 2, 21, London, U.K. Finite Element Modeling o Residual Thermal Stresses in Fiber-Reinorced Composites Using Dierent Representative Volume Elements M. M. Shokrieh and A. R. Ghanei Mohammadi Abstract In this paper, inite element analysis o residual thermal stresses in iber-reinorced composites has been carried out. For a more realistic simulation o the microstructure o these materials subjected to dierent loadings, a representative volume element (RVE) may be used. In this paper, three dierent types o RVE conigurations, circular, square and hexagonal are modeled and the eects o each type o iber packing are studied. A mono iber circular unit cell is considered using Finite Element (FE) method. Extending the mono iber model, FE models with dierent arrays o ibers have been created to investigate the eects o neighboring ibers on the results. The results obtained are used to introduce new boundary conditions or the mono iber model to make it able to predict the macro behavior in an eicient way. In all steps, the results are also compared with theoretical results presented in the literature. The boundary conditions presented in this research are proved to model the overall behavior eiciently. Index Terms ibrous composite material, inite element, representative volume element, inhomogeneous interphase, boundary conditions. I. INTRODUCTION Composite materials are becoming an essential part o present engineered materials because they oer advantages such as higher speciic stiness and strength, better atigue strength and improved corrosion resistance compared to conventional materials. These high perormance composites consist o dierent constituents. Subjected to thermal or thermo-mechanical loads, dierent deormations occur in dierent constituents leading to large dierences o deormations and stresses between these constituents, which are known as residual deormations and stresses. However, the use o composite materials is limited by the lack o eicient tools to predict their degradation and lietime under service loads, environment and the process induced residual stresses. For a more realistic simulation o the microstructure o these materials subjected to mechanical, thermal or thermo-mechanical loadings, a representative volume element (RVE) may be used. In the investigations applying FE method, one common procedure is the numerical generation o a RVE or a unit cell (UC) o the material being studied. A RVE or a UC is a statistical representation o the material [1]. Two major groups o these models are the composite cylinder models (CCM) [2] in which the RVE consist o two or more concentric cylinders and unit cell models (UCM) [3]-[5]. Because o its ability in reproducing the real stress and strain evolution, the simulation through a RVE may provide the understanding o the composite thermal behavior. This understanding is a need or the proposal o macroscopical results. In the analysis o the microstructure, the periodicity hypothesis o the iber within the composite has been traditionally employed. This hypothesis reduces the analysis o the microstructure to the analysis o a single unit cell (the simpler RVE) and may lead to analytical solutions [6]. Three representative volume elements (Fig. 1) based on the 3-D elasticity theory have been proposed in (Liu and Chen, 22) or the study o iber reinorced composites. They are the cylindrical RVE (Fig. 1), square RVE (Fig. 1) and hexagonal RVE (Fig. 1(c)). The cylindrical RVE can be applied to model the models including dierent diameters (Hyer, 1998). Under axisymmetric as well as antisymmetric loading, a 2-D axisymmetric model can be applied or the cylindrical RVE, which can signiicantly reduce the computational work (Liu and Chen, 22). The square RVE models can be applied when the conventional iber-reinorced composites are arranged evenly in a square array, while the hexagonal RVE models can be applied when they are in a hexagonal array, in the transverse direction. These RVEs can be used to study the interactions with the matrix, such as the load transer mechanism and stress distributions along the interaces (Liu and Chen, 22) or to evaluate the eective material properties o the composites [7]. Manuscript received March 23, 21. M. M. Shokrieh is with Iran University o Science and Technology, Composite Research Laboratory, Center o Excellence in Solid Mechanics and Dynamics, Mechanical Engineering Department, Tehran, , Iran (corresponding author to provide phone: ; ax: ; shokrieh@ iust.ac.ir). A. R. Ghanei Mohammadi is a graduate student in Iran University o Science and Technology Composite Research Laboratory, Center o Excellence in Solid Mechanics and Dynamics, Mechanical Engineering Department, Tehran, , Iran ( amirreza.ghanei@gmail.com). ISBN: ISSN: (Print); ISSN: (Online) Fig. 1 three representative volume elements [7] WCE 21

2 Proceedings o the World Congress on Engineering 21 Vol II WCE 21, June 3 - July 2, 21, London, U.K. Although these unit cells can be useul or some purposes and can be employed successully in two-scale methods to reproduce macroscopical behavior [8, 9], they do not relect the reality o composite materials, in which the iber is randomly distributed or it is placed among many other neighboring ibers, and consequently, they are not usable to simulate some o the complex mechanisms which take place in long iber reinorced polymers and which may cause microscopic ailure [1]. The selection o appropriate boundary conditions has been a challenge or many researchers. Since the RVEs must be modeled in a way that results obtained can eiciently describe the overall the composite material behavior, these boundary conditions have an important role in modeling the eect o the neighboring ibers, which are not considered in a mono iber RVE cell. Many researchers have used micromechanical method to provide overall behavior o the composites rom known properties o their constituents (iber and matrix) through an analysis o a periodic representative volume element (RVE) or a unit-cell model [11,12]. In the macro-mechanical approach, on the other hand, the heterogeneous structure o the composite is replaced by a homogeneous medium with anisotropic properties. The advantage o the micromechanical approach is not only the global properties o the composites but also various mechanisms such as damage initiation and propagation, can be studied through the analysis [13, 14]. In the previous works, square and hexagonal arrays or RVEs have been studied extensively, but RVEs with curved boundaries, such as circular, have not been studied very much. In this paper, a new unit cell inite element model has been presented to investigate iber-reinorced composites subjected to thermal loading. An inhomogeneous interphase region has been assumed in the model. A mono iber circular unit cell model is presented and then the model is generalized to include dierent geometrical conigurations, such as a square unit cell, square cell arrays and hexagonal arrays. In order to present a mono iber model which is eiciently able to model the macro thermal behavior o a ibrous composite material, the mono iber model is extended and FE models with dierent arrays o ibers are created to investigate the eects o neighboring ibers on the results. The results obtained in these models are used to introduce new boundary conditions in the mono iber model to make it able to predict the macro behavior in an eicient way. In all steps, the results are also compared with theoretical results available in the literature. The boundary conditions selected in the present work are proved to model the overall behavior o the mono iber model eiciently. II. FINITE ELEMENT MODELING In this paper, a circular RVE, containing a iber, interphase region and surrounding matrix, is presented and considered in the FE model. In order to reduce the FE problem size and the runtime, one ourth o the circular model is considered here. The interphase region is taken to be inhomogeneous. I the material properties o the interphase keep unchanged, the interphase is called homogeneous. Otherwise, it is called ISBN: ISSN: (Print); ISSN: (Online) inhomogeneous. In the inhomogeneous interphase considered in this research, the mechanical properties o this region undergo an exponential variation with the radial coordinate. The mathematical representation o this property variation can be written as [15]: 1 r 1 re r r t = tm + ( t tm ) where r =, r r = 1 1 r r e i ri (1) where t = ( E L, E T, ν 12, ν 13, α L, α T ) which it represents the transverse and longitudinal Young s moduli E T and E L, Poisson s ratios ν 12 and ν 13, and thermal expansion coeicients in the transverse and longitudinal directions α T and α L, respectively. Also, r is the radial distance rom the iber center and r and r i are the iber and interphase outer radii. To apply the property variation above to the FE model, the interphase region is assumed to consist o some layers and the material properties are kept constant in each layer. To calculate the properties o each layer, a simple mathematical code is used to generate the properties as a unction o the radial coordinate. The average value o two neighboring radial distances is calculated and taken to be the property value or the layer. The Young s moduli, Poisson s ratios and thermal expansion coeicients o the iber are taken to be E T = GPa, E L = GPa, ν 12 =.258, ν 13 =.2, α T =6.25e-6/ C and α L =5.9e-6/ C. And those o the matrix are set to E m =2 GPa, ν m =.3, and α m =12.5e-6/ C. The outer radii o the iber, interphase and matrix are taken as 5, 6 and 1 μm, respectively. The externally applied thermal load is a uniorm temperature drop o -5 C. ABAQUS FE package has been used or modeling. Fig. 2 shows the circular mono iber model used in this research. Fig. 2 a) One ourth mono iber circular RVE model. b) inite In order to investigate the neighboring eects on stress and displacement results around the iber and interphase, extended RVEs are modeled. These models are the mono iber square RVE (Fig. 3), multi iber 2x2 square array RVE (Fig. 4), multi iber 3x3 square array RVE (Fig. 5), multi iber 4x4 square array RVE (Fig. 6) and multi iber 5x5 square array RVE (Fig. 7). WCE 21

3 Proceedings o the World Congress on Engineering 21 Vol II WCE 21, June 3 - July 2, 21, London, U.K. Fig. 3 a) One ourth mono iber square RVE model. b) inite Fig. 8 a) hexagonal unit cell. b) inite Fig. 4 a) multi iber 2x2 square array RVE model. b) inite Fig. 9 a) extended hexagonal unit cell. b) inite Fig. 5 a) multi iber 3x3 square array RVE model. b) inite Fig. 6 a) multi iber 4x4 square array RVE model. b) inite Fig. 7 a) multi iber 5x5 square array RVE model. b) inite As the third coniguration possible, hexagonal packing o ibers in the matrix is modeled, too. Unlike the circular and square coniguration, the simplest hexagonal RVE contains 2 ibers (see Fig.8). To investigate the eect o neighboring ibers and the convergence study o the results, an extended hexagonal unit cell (Fig.9) and a hexagonal array (Fig.1) are modeled. Fig. 1 a) hexagonal array. b) inite These extended RVEs can help us have a better and more realistic understanding o overall macroscopic behavior o composites subjected to thermal loadings. The results o the extended models are compared with the results obtained rom the mono iber circular RVE to investigate the eect o neighboring ibers existence on its stress and displacement results and these results are used to select appropriate boundary conditions or the mono iber model to make it able to predict the macro behavior o the ibrous composite material eiciently. Such a model can be used to represent all the ibers in the composite material. Considering dierent theoretical models available [14,15] and based on a repetitive procedure, a new boundary condition is obtained and introduced in which generalized plane strain assumption (with a strain equal to e-3) is made and also mechanical symmetry conditions are applied to the one ourth model o the circular RVE. III. RESULTS AND DISCUSSION The results o residual radial and circumerential and axial stresses and radial displacement around the interphase region are presented in this section. COMPARISON OF RESULTS OF MONO FIBER CIRCULAR AND SQUARE AND HEXAGONAL MODELS ISBN: ISSN: (Print); ISSN: (Online) WCE 21

4 Proceedings o the World Congress on Engineering 21 Vol II WCE 21, June 3 - July 2, 21, London, U.K. In Figs 11, 12, 13 and 14, the results o residual radial and circumerential and axial stresses and radial displacement around the interphase region are compared or the mono iber square and circular and hexagonal models. (See Fig. 2,3 and 8) Fig. 11 Comparison o residual radial stresses or mono iber square and circular and hexagonal models residual circumerential stress (MPa) e-5-3.e-5 Fig. 14 Comparison o residual radial displacements or mono iber square and circular and hexagonal models As is shown, the results o residual axial stresses and radial displacements in all mono iber models are quite similar. In the case o residual radial stresses, hexagonal unit cell shows less magnitude o these stresses and the residual radial stresses in square model are more than the other two models. In the interphase region, the results o hexagonal unit cell and the circular model are very close to each other. In the case o residual circumerential stresses in the iber, the results o the square and hexagonal unit cell and in the matrix, the results o the circular and square models are very close. In all the regions, the circular model shows higher level o these stresses and in the interphase the results o all models are similar. Fig. 12 Comparison o residual circumerential stresses or mono iber square and circular and hexagonal models CONVERGENCE OF RESULTS FOR DIFFERENT FE ARRAYS In Figs 15 to 22, the results o residual radial and circumerential and axial stresses and radial displacement around the interphase region are compared or the dierent square and hexagonal array FE models. (See Figs 4, 5, 6, 7, 9 and 1). As is shown, increasing the number o ibers in the model and creating a larger RVE, leads to more accurate results. Since the inite is chosen to be ine enough, the results converge very ast and increasing the number o ibers rom 9 to 16 and 25 does not aect the results signiicantly Fig. 13 Comparison o residual axial stresses or mono iber square and circular and hexagonal models Fig. 15 Convergence o residual radial stresses or dierent square arrays ISBN: ISSN: (Print); ISSN: (Online) WCE 21

5 Proceedings o the World Congress on Engineering 21 Vol II WCE 21, June 3 - July 2, 21, London, U.K. 6 4 residual circumerential stress (MPa) 4 2 residual circumerential stress (MPa) Fig. 16 Convergence o residual circumerential stresses or dierent square arrays Fig. 2 Convergence o residual circumerential stresses or dierent hexagonal models Fig. 17 Convergence o residual axial stresses or dierent square arrays -2.e-5-3.e-5 Fig. 18 Convergence o residual radial displacements or dierent square arrays Fig. 19 Convergence o residual radial stresses or dierent hexagonal models Fig. 21 Convergence o residual axial stresses or dierent hexagonal models -2.e-5-3.e-5 Fig. 22 Convergence o residual radial displacements or dierent hexagonal models COMPARISON OF RESULTS OF MONO AND MULTI FIBER FE MODELS AND A THEORETICAL MODEL In Figs 23, 24, 25 and 26, the residual radial and circumerential and axial stresses and radial displacement results obtained rom the mono iber circular RVE model and the multi iber 5x5 array model and the hexagonal array are compared with You s [15] theoretical results. A good agreement is observed between the results, which proves that the presented mono iber circular RVE model with the applied boundary conditions is capable o modeling the macro stress and displacement behavior o the composite material with an acceptable accuracy. ISBN: ISSN: (Print); ISSN: (Online) WCE 21

6 Proceedings o the World Congress on Engineering 21 Vol II WCE 21, June 3 - July 2, 21, London, U.K You [15] FE (5x5 multi iber model) Fig. 23 Comparison o residual radial stresses or dierent models residual circumerential stress (MPa) You[15] FE (5x5 multi iber model) Fig. 24 Comparison o residual circumerential stresses or dierent models You[15] FE (5x5 multi iber model) Fig. 25 Comparison o residual axial stresses or dierent models -2.e-5-3.e-5 You [15] FE (5x5 multi iber model) Fig. 26 Comparison o residual radial displacements or dierent models IV. CONCLUSIONS Following an extensive study, appropriate boundary conditions are selected or the mono iber unit cell model and the results o the unit cell model show good agreement with You s theoretical model [15]. As the next step, multi-iber square arrays o ibers, such as 2x2, 3x3, 4x4 and 5x5 arrays and also extended hexagonal arrays are modeled using FEM. Comparing the results o the mentioned arrays, a convergence is observed in the results. Also, the results o array modeling are compared with the theoretical results and a good agreement is observed between the results in this part. These comparisons oered the chance to investigate the behavior o each possible coniguration o RVEs and iber packing and its eect on the residual stress and displacement distribution around the interace region o the iber and matrix, incorporating an inhomogeneous interphase. The results show that each coniguration gives slightly dierent results in dierent analyses and regions and one single model cannot be chosen as the most appropriate and it was shown that all these models do not have very much dierent behavior. REFERENCES [1] R. Hill, J. Mech. Phys. Solids 11 (1963) [2] Mikata Y, Taya M. 1985, Stress ield in a coated continuous iber composite subjected to thermo-mechanical loadings. J Comp Mater; 19: [3] J. Aboudi, Micromechanical analysis o composites by the method o cells. Appl Mech Rev; 42: 1989, pp [4] RP. Nimmer Fiber matrix interace eects in the presence o thermally induced residual stress. J Comp Tech Res; 12: 199, pp [5] D.D. Robertson, S. Mall Micromechanical relations or iber-reinorced composites using the ree transverse shear approach. J Comp Tech Res; 15: 1993, pp [6] D. Trias, J. Costa, A. Turon and J. E. Hurtado, Determination o the critical size o a statistical representative volume element (SRVE) or carbon reinorced polymers, Acta Materialia 54, 26, pp [7] Y.J. Liu, X.L. Chen, Evaluations o the eective material properties o carbon nanotube-based composites using a nanoscale representative volume element, Mechanics o Materials 35, 23, pp [8] E. Car, F. Zalamea, S. Oller, J. Miquel and E. On ate, Numerical simulation o composite materials: two procedures. Inter J Solid Struct.; 39 (7), 22, pp [9] F. Feyel, A multilevel inite element method (FE2) to describe the response o highly non-linear structures using generalized continua. Comput. Meth. Appl. Mech. Eng, 192, (28 3): 23, pp [1] T. Matsuda, N. Ohno, H. Tanaka andt. Shimizu, Eects o ibre distribution on elastic viscoplastic behavior o long ibre-reinorced laminates. Int. J. Mech. Sci. 45, 23, pp [11] J. Aboudi, Mechanics o Composite Materials, A Uniied Micromechanical Approach. Elsevier Science Publishers, Amsterdam, [12] S. Nemat-Nasser, M. Hori,. Micromechanics: Overall Properties o Heterogeneous Materials. Elsevier Science Publishers, Amsterdam, [13] Z. Xia, Y. Chen and F. Ellyin, A meso/micro-mechanical model or damage progression in glass iber/epoxy cross-ply laminates by inite-element analysis. Composite Science and Technology, 6, 2, pp [14] F. Ellyin, Z. Xia and Y. Chen,. Viscoelastic micromechanical modeling o ree edge and time eects in glass iber/epoxy cross-ply laminates. Composites, Part A, 33, 22, pp [15] L.H. You and X.Y. You, A uniied numerical approach or thermal analysis o transversely isotropic iber-reinorced composites containing inhomogeneous interphase, Journal o composites, Part A: applied science and manuacturing, 36, 25, pp ISBN: ISSN: (Print); ISSN: (Online) WCE 21