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Comparative study on the effect of representative volume element (RVC) boundary conditions on the elastic properties of unidirectional composite model based on Micromechanics

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https://www.eduzhai.net International Journal of Composite Materials 2017, 7(2): 51-71 DOI: 10.5923/j.cmaterials.20170702.03 A Comparative study on the Effect of Representative Volume Cell (RVC) Boundary Conditions on the Elastic Properties of a Micromechanics Based Unidirectional Composite Material Model Sandeep Medikonda1,*, Ala Tabiei1, Rich Hamm2 1Department of Mechanical and Materials Engineering, University of Cincinnati, Cincinnati, OH, USA 2P&G, Cincinnati, OH, USA Abstract A micromechanical model based on the physically viable sub-cell boundary conditions is developed and implemented for use with uni-directional composite laminates in the explicit finite element method. Stress-strain relations have been presented in a three-dimensional context and hence can be used with solid elements. The objective of this work is to study the effect of boundary conditions in accurately estimating the elastic properties of a uni-directional composite lamina. In order to achieve this, the developed micro-model has been studied alongside 2 other models with different boundary conditions specified in the literature. Numerical results are generated for engineering constants by the considered models and compared against each other for different laminas. In particular, transverse and shear modulus have been analyzed in detail alongside popular analytical methods and verified against available experimental results for various volume fractions. Good agreements have been observed for the presented model in comparison with the experimental results. Keywords Unidirectional Composites, Micro-mechanical model, Representative Volume Cell (RVC), Iso-stress boundary conditions, Iso-strain boundary conditions, Elastic constants 1. Introduction Composite materials have a wide range of applications, their use as structural components in automobiles, air and space vehicles is an attractive option because of their high impact tolerance, high stiffness-to-weight and strength-to-weight ratios. As a consequence, over the last decade, an increased usage of composite materials has been observed. The constitutive relations used to numerically model composite materials can be characterized into two groups, macro-mechanical models and micro-mechanical models. Micro-mechanical models provide the overall behavior of the composite from the properties of the individual constituents (e.g. fibers and matrix) and their interactions. Macro-mechanical models, on the contrary, replace the heterogeneous structure of the composite with a homogeneous medium consisting of anisotropic properties (which need to be determined). While, macro-mechanical material models require less computational work, they fall * Corresponding author: medikosp@mail.uc.edu (Sandeep Medikonda) Published online at https://www.eduzhai.net Copyright © 2017 Scientific & Academic Publishing. All Rights Reserved short in accurately capturing composite response especially for complex load histories. On the other hand, micro-mechanical material models despite requiring more computational work can overcome these short comings. Sophistications in computing technology have made it feasible to implement the micro-mechanical models alongside non-linear finite element (FE) codes. The popular transient non-linear FE code LS-DYNA [1] has been chosen in the current work as it offers a very simple interface to implement user material models. Micromechanical analyses can be further divided into mechanics of materials (MoM) and theory of elasticity (ToE) approaches. While the MoM approach is based on simplifying assumptions regarding the hypothesized behavior of the composite, the ToE apporach is characterized by getting rid of some of these assumptions and more rigorously satisfying the equilibrium, continuity and compatibility conditions. Because the focus of this paper is on the implementation of the model in a finite element framework, it needs to be a reasonable mix of simplicity and accuracy; hence the mechanics of materials approach was used. In the explicit solution scheme of the finite element analysis, the accelerations are solved for as the inverse of the diagonal mass matrix times the force vector. Once 52 Sandeep Medikonda et al.: A Comparative study on the Effect of Representative Volume Cell (RVC) Boundary Conditions on the Elastic Properties of a Micromechanics Based Unidirectional Composite Material Model accelerations are known at time ‘n’, velocities are calculated at time ‘n+1/2’, and displacements at time ‘n+1’. This displacement increment causes an increment of strain at a material point. The strain is passed into the material model to calculate the incremental stress. Stiffness, strains, and stresses are tracked at the material points within each element. This information is provided by the micromechanical composite material model, which interfaces with the nonlinear explicit finite element code. The heterogeneous nature of the composite material is hidden from the main analysis code. Micromechanical theory discussed in this work predicts the average behavior of the lamina as a function of the constituent properties and the local boundary conditions; this is an important advantage as no prior knowledge of the lamina response is required. In one of the earliest attempts at including micro-mechanics into the FE analysis of composite structures, Adams and Crane [2] developed a micro-mechanical model using the representative volume element. However, this model has been observed to computationally expensive. Pecknold and Rahman [3] proposed a simpler micro-model which has been a basis for the current study. A significant advantage of the Pecknold and Rahman model is its capability to analyze elastic as well as in-elastic constituents (e.g. visco-elastic and visco-plastic), thus forming a unified approach in predicting the overall behavior of the composite material. Figure 1. Modeling strategy and representative volume cell of the unidirectional composite International Journal of Composite Materials 2017, 7(2): 51-71 53 The objectives of the current work are: 1. Use physically viable sub-cell boundary conditions and modify the Pecknold and Rahman micro-model. Subsequently, derive constitutive relations for the RVC based on these modified boundary conditions. 2. Implement this material model as a user subroutine in LS-DYNA and compare the elastic properties predicted against similar numerical models (same RVC but different boundary conditions), analytical models and experimental results presented in the literature for various laminae. 2. Geometric Description of the Micro-mechanical Model A schematic of the representative volume cell (RVC) used to develop the micro-mechanical relations is shown in Figure. 1. The basic structure of the RVC which was originally proposed by Pecknold and Rahman [3] has been used by Tabiei et al. [4-6] in several studies to accurately capture different behaviors of uni-directional composites. The underlining premise of the micro-model is based on the assumption that the internal micro-structure of the lamina consists of periodic square fibers. In addition, the following assumptions are made regarding behavior of the constituents and the composite as a whole: 1. The fiber material is homogeneous and linearly elastic. 2. The matrix material is homogeneous and linearly elastic. 3. The fibers are positioned in the matrix such that the composite lamina is a homogenous material with linearly elastic behavior. 4. There is a complete and strong bond at the interface of the constituent materials. The unit cell is divided into three sub-cells: one fiber sub-cell, denoted as f, and two matrix sub-cells, denoted as MA and B respectively. The three sub-cells are grouped into two parts: material part A consists of the fiber sub-cell f and the matrix sub-cell MA, and material part B consists of the remaining matrix B. The dimensions of the unit cell are 1 ×1 unit square. The dimensions of the fiber and matrix sub-cells are denoted by Wf and Wm. Wf  Vf ; Wm  1Wf (1) where Vf is the fiber volume fraction. As explained in a later section, effective stresses in the RVC are determined from the sub-cell values in two phases: first, stresses in fiber f and matrix MA are combined to obtain effective stresses in part A which are then combined with stresses in matrix B to obtain the effective RVC stresses. 3. Boundary Conditions of the Micro-mechanical Model Once the finite element solver passes the average strains of the RVC into the user material model sub-routine, micromechanical relations are used to obtain the average stresses using Parallel (Iso-strain) – Series (Iso-stress) assumptions. The RVC is divided into three sub-cells and sub-cells are either in parallel or in series for the various components of strains and stresses. If two sub-cells are in parallel for a particular directional component, then the corresponding strains are equal and the average RVC stress in that direction is given by the rule of mixture applied on the average stresses in those sub-cells. On the other hand, if two sub-cells are in series for a particular component, then the corresponding stresses are equal and the average RVC strain in that direction is given by the rule of mixture applied on the strains in those sub-cells. One of the distinguishing aspects of the current work as compared to the micro-model presented by Pecknold and Rahman [3] and subsequently used by Tabiei et al. [4-6] is the manner in which iso-stress and iso-strain boundary conditions have been applied on individual components. In order to simplify the complexity of the micro-mechanical relations, Tabiei and Babu [6] have used the iso-strain boundary conditions in both phases of homogenization. Whereas, Pecknold and Rahman [3] and Tabiei et al. [4, 5] have used iso-strain boundary condition in the 11-direction and iso-stress boundary conditions for the rest of the directions in phase-1 homogenization i.e., part A and iso-strain boundary conditions for all the components in the phase-2 homogenization i.e., part A and matrix B. Both of these assumptions are physically in-accurate. Hence, in the following work, an effort has been made to derive the stress-strain relations based on the corrected Parallel-Series boundary conditions. In the equations shown below, the symbols ‘  * ij ’ and ‘ Ci*j ’ represent the stress tensor components and the elastic stiffness constants, whereas the symbols ‘  * ii ’ an ‘  * ij ’ represent the axial strain tensor and engineering shear strain components respectively. Note that in all of these symbols the superscript ‘ *’ corresponds to one of the following sub cells: fiber f, matrix MA, matrix B or part A and for the occasions in which this superscript is missing, the whole RVC values are being represented. The sub-cells f and MA in part A, are subjected to iso-strain boundary conditions in directions 11, 22, and 12 and iso-stress boundary conditions in directions 23, 33, 31 (as shown in figure 2). Hence the homogenized stresses and strains in part A are represented by the following relations: 54 Sandeep Medikonda et al.: A Comparative study on the Effect of Representative Volume Cell (RVC) Boundary Conditions on the Elastic Properties of a Micromechanics Based Unidirectional Composite Material Model Figure 2. A schematic illustration of the loading and boundary conditions on Part A International Journal of Composite Materials 2017, 7(2): 51-71 55 Figure 3. A schematic illustration of the loading and boundary conditions on the RVC 56 Sandeep Medikonda et al.: A Comparative study on the Effect of Representative Volume Cell (RVC) Boundary Conditions on the Elastic Properties of a Micromechanics Based Unidirectional Composite Material Model     A   f   M A   11   11   11       A 22    f 22    MA 22        A   f   MA           12A23      12f23      12 MA 23         A   f   M A  33  33  33       A 31   f 31   M A 31   (2a – 2f)  A   11  f   11  M A   11   A   22  f   22   MA 22        A  f  MA       12 A     W f    12 f     W m    12  M A       23   23   23   A   33   f   33   M A   33   A   31   f   31    MA 31   (3a – 3f) As shown in figure 3, the sub-cell B and part A, are subjected to iso-strain boundary conditions in the 11, 33, 23 and 31 directions and iso-stress boundary conditions in the rest of the directions. The homogenized stresses and strain in the unit cell are hence given by the following equations:  11     33      A 11 A 33       B 11 B 33      23 31       A 23 A 31       B 23 B 31      22 12       A 22 A 12       B 22 B 12    (4a – 4f) 11   33    A 11   A 33    B 11   B 33     23 31     W f   A 23 A 31     Wm   B 23 B 31        22 12       A 22 A 12       B 22 B 12    (5a – 5f) Based on the parallel-series assumptions presented in equations (2) – (5), it is possible to eliminate the intermediate sub-cell A and express the stresses and strains of the RVC in terms of the corresponding constituent sub-cell’s (f, MA and B): For Strains: 11  f 11  M 11 A  B 11  22  W f  f 22  Wm B 22  W f  M 22 A  Wm B 22  33  W f  f 33  Wm M 33 A   B 33  12  W f  f 12  Wm B 12  W f  M 12 A  Wm B 12  23  W f  f 23  Wm M 23 A   B 23  31  W f  f 31  Wm M 31 A   B 31 (6a – 6f) For Stresses:  11  W f2 f 11  W f Wm M 11 A  Wm B 11  22  W f  f 22  Wm M 22 A   B 22  33  W f  f 33  Wm B 33  W f  M 33 A  Wm B 33  12  W f  f 12  Wm M 12 A   B 12  23  W f  f 23  Wm B 23  W f  M 23 A  Wm B 23  31  W f  f 31  Wm B 31  W f  M 31 A  Wm B 31 (7a – 7f) We have 18 unknowns of strains which correspond to components in each sub-cell (f, MA and B). However, since the average strains in the RVC are known (provided by main FE program), the number of unknowns reduce to 10 with the help of equations provided in (6). Solving for these unknowns becomes necessary in order to calculate the stresses in the RVC. The next section focuses on doing just this. 4. Constitutive Relations of the Micro-mechanical Model The RVC and the intermediate sub-cell ‘A’ are heterogeneous in nature. However, the base sub-cells (f, MA and B) are considered to be homogenous. The fiber is assumed to be transversely isotropic with the axis of transverse isotropy along the corresponding fiber axis (which is 1-direction for this case). Constitutive relations of the fiber after applying the known values of strain given in equations (6) are: Fiber Sub-Cell (f):   f 11 f 22    CC11ff21 C1f2 C2f2 C1f2 C2f3 0 0 0 0 0  11  0    f 22     f 33 f 12     C1f2 0 C2f3 0 C2f2 0 0 C4f4 0 0 0 0    13ff23       f 23 f 31    0   0 0 0 0 0 0 0 C5f5 0 0 C4f4      f 23 f 31    (8a – 8f) International Journal of Composite Materials 2017, 7(2): 51-71 57 Where, C1f1  E1f (1  f 23 ) (1  21f2 f 21  f 23 ) ;  C1f2  E1f E2f 1f2 f 21 (1  21f2 f 21  f 23 ) ; C2f2  E2f (11f2 2f1) (1  f 23 )(1  21f2 f 21  f 23 ) ; C2f3  E2f ( f 23 1f2 2f1) (1  f 23 )(1  21f2 f 21  f 23 ) ; C4f4  G1f2 & C5f5  C2f2  C2f3 2  G2f3  E2f 2(1  f 23 ) Ei f is the stiffness in the axial directions (i.e., subscript i 1, 2 or 3) whereas f ij , Gijf are the poisson’s ratio and the stiffness in the shear directions (i.e., subscript ij  12, 23 or 31). The matrix material is assumed to be isotropic and hence the constitutive relations of the sub-cells (MA and B) after applying the known strains from equation (6) are: Matrix Sub-Cell (MA): 1M1 132MMM232 A A A A         C1R1 C1R2 C1R2  0 C1R2 C1R1 C1R2 0 C1R2 C1R2 C1R1 0 0 0 0 C4R4 0 0 0 0 0   11  0      f 22   0    M 33 A   0     1f2     M 23 A     0 0 0 0 C4R4 0    M 23 A    M 31 A    0 0 0 0 0 C 4R4   M 31 A   (9a – 9f) Matrix Sub-Cell (B): Where, 1B1   C1R1 C1R2 C1R2 0 0 123BBB232       C1R2 C1R2  0 C1R1 C1R2 0 C1R2 C1R1 0 0 0 C4R4 0 0 0 0 0 0 0        11   B 22 13B23        B 23     0 0 0 0 C4R4 0    23    B 31    0 0 0 0 0 C4R4   31  C1R1  (1 E R (1  R )(1 R) 2 R ) ; C1R2  (1  E R R R )(1 2 R) ; C4R4  G R  ER 2(1  R) (10a – 10f) The elastic stiffness constants for sub-cells MA and B are the same since they are made out of the same resin material; hence these constants are equivalently represented by the super-script R in the equations (9) and (10). The unknown strains in equation (8), (9) and (10) are denoted by the super-scripts ‘f’ ‘MA’ ‘B’. Hence, in order to calculate the effective RVC stresses it is essential to solve for the following sub-cell strains (10 in total):  f 22 ,  f 33 ,  B 22 ,  M 33 A , f 12 , f 23 , 3f1, M 23 A ,  B 12 and  MA 31 . From equation (3c), we have the following relation, Wf  f 12  Wm M 12 A  B 12 . Substituting the relations from (8d) and (9d) into this equation, the shear strain  B 12 can be expressed in terms of  f 12 (as shown in (11a)), which can then be used in equation (6d) to solve for  f 12 . 58 Sandeep Medikonda et al.: A Comparative study on the Effect of Representative Volume Cell (RVC) Boundary Conditions on the Elastic Properties of a Micromechanics Based Unidirectional Composite Material Model 1B2  (W f C4f4 WmC4R4 )1f2 C4R4 ; 1f2  Wf C4R4 C4R412 Wf WmC4f4  Wm2C4R4 (11a – 11b) Next, using the relations given in (2d), (8d) and (9d), the shear strain  MA 23 can be expressed in terms of  f 23 as shown in equation (12a). This can then be used to solve for  f 23 by substituting (12a) in equation (6d).  M 23 A  C5f5 f 23 C4R4 ;  f 23  Wf C4R4 23 C4R4  WmC5f5 & hence  M 23 A  W f C5f5 23 C4R4  WmC5f5 (12a – 12c) Since the boundary conditions between the sub-cells for the components in directions 23 and 31 are the same, the procedure for calculating the unknown strains  f 31 and  MA 31 are similar. Utilizing equations (8f), (9f) in (2f), the shear strain  f 31 can be expressed in terms of  MA 31 (as shown in (13a)), which can then be used in equation (6f) to solve for  f 31 .  M 31 A  C4f4 f 31 C4R4 ;  f 31  Wf C4R4 31 C4R4  WmC4f4 & hence  M 31 A  W f C4f4 31 C4R4  WmC4f4 (13a – 13c) Equations (11) - (13) help reduce the number of unknowns to 4, i.e.,  f 22 ,  f 33 ,  B 22 and  MA 33 . Furthermore,  MA 33 can be expressed in-terms of f 22 and  f 33 by using (8c) and (9c) in (2e).  MA 33  (C1f2  C1R2 )11  (C2f3  C1R2 ) f 22 C1R1  C2R2 f 33 (14) Similarly, using equations (8b), (9b), (10b) and (14) in (7b),  B 22 can be expressed in-terms of f 22 and  f 33 as shown below:  B 22    W f  (C1f2  C1R2 ) C1R1  WmC1R2 (C1f2  C1R2 ) (C1R1)2    11    C1R2 C1R1    33     (W f C2f2 WmC1R1) C1R1  WmC1R2 (C2f3  C1R2 ) (C1R1)2     f 22    W f C2f3  C1R1  WmC1R2C2f2 (C1R1)2    3f3 (15) Equation (14) can be further used in (6c) to generate an equation in f 22 and  f 33 as shown below: a f 22  b3f3  c where'a','b' and 'c' are given as : a  Wm (C2f3  C1R2 ) b  W f C1R1 WmC2f2 c  C1R133  Wm (C1R2  C1f2 )11 Similarly, using equation 15 in (6b) helps generate the following equation: ….. (16) International Journal of Composite Materials 2017, 7(2): 51-71 59 d f 22  e3f3  f where'd','e' and 'f' are given as :       Wf C1R1 2  W f WmC2f2C1R1  WmC1R1 2  Wm 2 C1R2 (C2f3  C1R2 ) d  C1R1 2    e  W f WmC2f3C1R1  C1R1 Wm 2 2 C1R2C2f2 (17)     f  22  C1R1 2 Wm (W f C1R1  WmC1R2 )(C1f2  C1R2 )11  WmC1R2C1R133 C1R1 2 Values 'a','b','d' and'e' are constants whereas 'c' and 'f' can be easily calculated since the average RVC strains are provided by the Finite Element solver at each time step. Lastly, the linear equations (15) and (16) can hence be solved to calculate f 22 and f 33 : 2f2  ce ae  fb  bd 3f3  af ae  cd  bd (18) Thus, the strains in all 3 constituent sub-cells are available at this point, which can then be used in equations (8), (9) and (10) and subsequently in equations provided in (7) to calculate the average stresses in the RVC. 5. Results and Discussion The micro-mechanical equations discussed in the previous section (Current Model), along with the numerical models discussed in the [6] and [4], [5] (Numerical Models 1 and 2 respectively) have been implemented as user material subroutines (UMAT) in LS-DYNA. All three micro-models use the same homogenization scheme; however, employ different sub-cell boundary conditions.  Current (Curr.) Model: Numerical Model based on physically viable sub-cell boundary conditions (Section 4):  Sub-cell A: Iso-strain b.c between fiber (‘f’) and Matrix (‘MA’) sub-cells in 11, 22, and 12 directions and Iso-stress b.c in 33, 23 and 31 directions.  Total RVC: Iso-strain b.c between sub-cell (‘A’) and sub-cell (‘B’) in 11, 33, 23 and 31 directions and Iso-stress b.c in 22, 12 directions.  Numerical (Num.) Model 1: Numerical Model based on the sub-cell boundary conditions used by Tabiei and Babu [6]:  Sub-cell A: Iso-strain b.c between fiber (‘f’) and Matrix (‘MA’) sub-cells in all directions.  Total RVC: Iso-strain b.c between sub-cell (‘A’) and sub-cell (‘B’) in all the directions.  Numerical (Num.) Model 2: Numerical Model based on the sub-cell boundary conditions used by Tabiei and Chen, Tabiei and Yi [4], [5]:  Sub-cell A: Iso-strain b.c between fiber (‘f’) and Matrix (‘MA’) sub-cells in 11 and Iso-stress b.c in 22, 33, 12, 23 and 31 directions.  Total RVC: Iso-strain b.c between sub-cell (‘A’) and sub-cell (‘B’) in all the directions. For the micro-mechanical relations provided in [5] and [6], the in-elastic behavior of the resin has been taken into account. However, since the current work focuses on studying only the elastic constants, it would be apt to derive these equations just for an elastic resin material. Appendix ‘A’ provides the elastic strains of the sub-cells and the constitutive response of the RVC based on the boundary conditions considered in [4] or [5] and [6]. These relations have been derived based on the procedure highlighted in the previous section. 5.1. Analytical Models In an effort to accurately predict the elastic properties of composite materials based on the constituent properties, a wide range of analytical micro-mechanical models have been discussed in literature over the last century. Younes et al. [7] reviewed some of the most known, readily available analytical models and discussed their performance. The investigated models belonged to the following categories: 1. Phenomenological Models: Rule of Mixture (ROM) / Inverse Rule of Mixture (IROM). 2. Semi-empirical Models: Modified Rule of Mixtures (MROM), Halpin-Tsai model and Chamis model [8]. 3. Elasticity approach Models (EAM): Hashin-Rosen [9] composite cylinder assemblage model (CCA) coupled with Christensen model [10]. 4. Homogenization Models: Mori-Tanaka Model, Self-Consistent model (S-C) and Bridging model [11], [12]. The Chamis model has been observed to yield good results for all the cases studied. Whereas, the rest of the models have been observed to predict elastic properties well for only 60 Sandeep Medikonda et al.: A Comparative study on the Effect of Representative Volume Cell (RVC) Boundary Conditions on the Elastic Properties of a Micromechanics Based Unidirectional Composite Material Model certain cases. In this study, the best working model in each of these categories (ROM / IROM, Chamis, EAM and Bridging) has been compared against the numerical models. The motive for using some of the widely popular analytical methods is that in the absence of experimental data, they help decide how good the numerical prediction really is. In addition, since the constituent properties and some of the available experimental data have been collected from various sources, there is a good chance that that the results might not completely match. Therefore, the analytical results help generate confidence in the data generated by the numerical models as there may be abnormal data points in the experimental values. 5.3. Comparative Study, Analysis and Discussion In this section a comparison of the analytical models and numerical results with available experimental data is presented. A total of 6 composite laminas have been studied and the constituent properties of these laminas have been provided in Table’s (2) – (7): 5.2. Numerical FE Model Numerical FE models have been widely used in predicting the mechanical properties of composites. For the current case, a single element model (Figure 4) has been subjected to a displacement constant ‘c’ on various faces, with different boundary conditions (as shown in Table 1). A total of six simulations have been conducted for each lamina considered with the value of the displacements (u, v and w) varying depending on the property that is being calculated. The stresses and strains used in these property calculations are obtained from the integration point at the center of the element. Figure 4. An un-deformed single solid element FE model Table 1. Deformed state’s, Boundary Conditions on the X, Y and Z faces of the Solid Element Boundary Conditions Properties calculated X- Face: u=0 X+ Face: u=c E11   11 11 ;  12   22 11 ;  13   33 11 Y- Face: v=0 Y+ Face: v=c E22   22  22 ;  23    33  22 Z- Face: w=0 Z+ Face: w=c X- Face: u, v, w=0; Y- Face: u, v, w=0; X+ Face: u=0, v=c Y+ Face: u=0, v=0 & w=0 & w=c Z- Face: u, v, w=0; Z+ Face: u=c, v=0 & w=0 E33   33  33 G12   12  12 G23   23  23 G31   31  31 Graphite Epoxy Table 2. Properties of Graphite/Epoxy (AS/3501) [4] E1 GPa E2 GPa v12 v23 213.7 13.8 0.2 0.25 3.45 3.45 0.35 0.35 G12 GPa 13.8 1.3 Boron Epoxy Table 3. Properties of Boron/Epoxy (B4/5505) [4] E1 GPa 400 3.45 E2 GPa v12 v23 400 0.2 0.2 3.45 0.35 0.35 G12 GPa 166.7 1.3

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