Computational Materials Science 43(2008)1193-1206 Contents lists available at ScienceDirect Computational Materials Science ELSEVIER journalhomepagewww.elsevier.com/locate/commatsci A micromechanical characterization of angular bidirectional fibrous composites Nabi abolfathi, Abhay Naik, Ghodrat Karami Chad Ulven Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Fargo, ND 58105-5285, USA ARTICLE INFO A BSTRACT Article history Received 24 December 2007 A micromechanical numerical algorithm to efficiently determine the homogenized elastic properties of bidirectional fibrous composites is presented. A repeating unit cell (RUC) based on a pre-determined bidi- Received in revised form 7 March 2008 rectional fiber packing is assumed to represent the microstructure of the composite For angular bidirec- Available online 24 April 2008 tional fiber distribution, the symmetry lines define a parallelepiped unit cell, representing the periodic crostructure of an angular bidirectional fiber composite. The lines of symmetry extrude a volume to capture a three dimensional unit cell. Finite element analysis of this unit cell under six possible indepen 6143.Bn dent loading conditions is carried out to study and quantify the homogenized mechanical property of the cell. A volume averaging scheme is implemented to determine the average response as a function of load- of stresses and strains. The individual elastic properties of the constituents'materials, as well as, the composite can be assumed to be completely isotropic to completely anisotropic. The output of the analysis can determine this degree. The logic behind the selection of the unit cell and the implementation of the periodic boundary conditions as well as the constraints are presented. To verify this micromechan- ics algorithm, the results for four composites are presented. The results in this paper are mainly focused n the impact of the fiber cross angles on the stiffness properties of the ites chosen. The accuracy of the results from this micromechanics modeling procedure has been compared with the stiffness/ com pliance solutions from lamination theory. The methodology is to be accurate and efficient to the extent that periodicity of the composite material is maintained. In addition, the results will show the impact of fiber volume fraction on the material properties of the composite. This micromechanics tool could make a powerful viable algorithm for determination of many linear as well as nonlinear properties in continuum mechanics material characterization and analysis e 2008 Elsevier B V. All rights reserved 1. Introduction entation In another study, the change of angle in bidirectional composites was studied for variation in permeability through composites against complicated loadings are determined by insert- ferent temperature for bidirectional ceramic composite has been ing fibers inside the matrix in different directions Composites with studied for fracture behavior [ 4. Domnanovich et al. [10] studied bidirectional and multidirectional fibers produce stiffness against the elastic module of bidirectional carbon/carbon composite under complicated structural and thermal loading scenarios. Mechanical heat treatment process. They also examined the shear strength as properties of mono-directional fiber reinforced composite have well as elastic modulus using resonant beam method. een extensively studied [1-7]. however, a detailed modeling ef- n application, a composite composed of a matrix with rein- fort investigation of the mechanical properties of bidirectional forced multidirectional fibers is a basic structural material in most composites as a function of the reinforcing fiber cross angles and aircraft constructions. glass fabric made with multi bidirectional entation has not been fully undertaken Studies related to bidi- fibers are used as stiffening materials at many applications. the rectional composites are usually focused on [ 0/90 or [0/45 ply use of glass in aerostructures, particularly, sandwich composite orientations. Among the many experimental efforts and proce- structures is a recent development. Glass fabric as a new infra- dures conducted for characterization of multidirectional fibrous structure composite material is now available commercially in composites, multi layer bidirectional composites made by vetrotex hundreds of different weights, weaves, strengths and working has been used as a lap joint by Ferreira et al. 8. This lap joint com- properties. Multiple layers of glass fabric oriented in different posite was studied for fatigue loading for [0/45] and [o/90 ply ori- directions are laminated together to form the panels for various applications In biomechanics applications and biological systems 4 Corres g author.Tel:+17012315859ax:+17012318913 nd organs, different loading directions and scenarios need to pre mail address: G Karami@ndsu. edu (G Karami) vide a material with proper strength in multiple directions. a bidi- s- see front matter o 2008 Elsevier B v. All rights reserved. doi: 10.1016/j-commatsci200803.017
A micromechanical characterization of angular bidirectional fibrous composites Nabi Abolfathi, Abhay Naik, Ghodrat Karami *, Chad Ulven Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Fargo, ND 58105-5285, USA article info Article history: Received 24 December 2007 Received in revised form 7 March 2008 Accepted 11 March 2008 Available online 24 April 2008 PACS: 61.43.Bn 62.20.D 62.20.-x 81.05.Ni 02.70.Dh Keywords: Bidirectional fibrous composites Micromechanics Finite element method Repeating unit cell Periodic boundary conditions abstract A micromechanical numerical algorithm to efficiently determine the homogenized elastic properties of bidirectional fibrous composites is presented. A repeating unit cell (RUC) based on a pre-determined bidirectional fiber packing is assumed to represent the microstructure of the composite. For angular bidirectional fiber distribution, the symmetry lines define a parallelepiped unit cell, representing the periodic microstructure of an angular bidirectional fiber composite. The lines of symmetry extrude a volume to capture a three dimensional unit cell. Finite element analysis of this unit cell under six possible independent loading conditions is carried out to study and quantify the homogenized mechanical property of the cell. A volume averaging scheme is implemented to determine the average response as a function of loading in terms of stresses and strains. The individual elastic properties of the constituents’ materials, as well as, the composite can be assumed to be completely isotropic to completely anisotropic. The output of the analysis can determine this degree. The logic behind the selection of the unit cell and the implementation of the periodic boundary conditions as well as the constraints are presented. To verify this micromechanics algorithm, the results for four composites are presented. The results in this paper are mainly focused on the impact of the fiber cross angles on the stiffness properties of the composites chosen. The accuracy of the results from this micromechanics modeling procedure has been compared with the stiffness/compliance solutions from lamination theory. The methodology is to be accurate and efficient to the extent that periodicity of the composite material is maintained. In addition, the results will show the impact of fiber volume fraction on the material properties of the composite. This micromechanics tool could make a powerful viable algorithm for determination of many linear as well as nonlinear properties in continuum mechanics material characterization and analysis. 2008 Elsevier B.V. All rights reserved. 1. Introduction Improvements in mechanical properties of fiber reinforced composites against complicated loadings are determined by inserting fibers inside the matrix in different directions. Composites with bidirectional and multidirectional fibers produce stiffness against complicated structural and thermal loading scenarios. Mechanical properties of mono-directional fiber reinforced composite have been extensively studied [1–7], however, a detailed modeling effort investigation of the mechanical properties of bidirectional composites as a function of the reinforcing fiber cross angles and orientation has not been fully undertaken. Studies related to bidirectional composites are usually focused on [0/90] or [0/45] ply orientations. Among the many experimental efforts and procedures conducted for characterization of multidirectional fibrous composites, multi layer bidirectional composites made by Vetrotex has been used as a lap joint by Ferreira et al. [8]. This lap joint composite was studied for fatigue loading for [0/45] and [0/90] ply orientation. In another study, the change of angle in bidirectional composites was studied for variation in permeability through change of fiber angles [9]. Creep loading of composites under different temperature for bidirectional ceramic composite has been studied for fracture behavior [4]. Domnanovich et al. [10] studied the elastic module of bidirectional carbon/carbon composite under heat treatment process. They also examined the shear strength as well as elastic modulus using resonant beam method. In application, a composite composed of a matrix with reinforced multidirectional fibers is a basic structural material in most aircraft constructions. Glass fabric made with multi/bidirectional fibers are used as stiffening materials at many applications. The use of glass in aerostructures, particularly, sandwich composite structures is a recent development. Glass fabric as a new infrastructure composite material is now available commercially in hundreds of different weights, weaves, strengths and working properties. Multiple layers of glass fabric oriented in different directions are laminated together to form the panels for various applications. In biomechanics applications and biological systems and organs, different loading directions and scenarios need to provide a material with proper strength in multiple directions. A bidi- 0927-0256/$ - see front matter 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.03.017 * Corresponding author. Tel.: +1 701 231 5859; fax: +1 701 231 8913. E-mail address: G.Karami@ndsu.edu (G. Karami). Computational Materials Science 43 (2008) 1193–1206 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
N. Abolfathi et aL/ Computational Material Science 43(2008)1193-1206 rectional fiber composite as a medical texture is a promising appli- boundary conditions where a repeating cell represents a homoge- cation for such applications. In this respect, biocompatible textures nized continuum point at the macroscale. Other enforced types of can be applied in a wide range of applications from polymeric boundary conditions(symmetry, homogenous boundary condi- valves through woven or knitted artificial ligaments, to polymeric tions )can introduce some form of local impact and therefore can- wound closure devices. Implantable cardiac support devices, vas-not nogenized property. The analysis cular prosthesis and heart valves are other examples [11]. procedure is based on FEM solution with a mechanized processing During the last decade, fiber reinforced composite was intro- algorithm which be extended to nonlinear and time-dependen duced as a new material for dentistry and orthodontic application. material behavior. There has been growing interest in utilizing fiber reinforced com- Micromechanics approach replaces the heterogeneous structure posites for load-bearing applications such as dental crowns, fixed of the composite by a homogeneous medium with anisotropic partial dentures, and implant-supported prostheses. Metal free properties [17, 18 In fact, the rve or the RUC should simulate a composite materials can be used for the fabrication of single crown continuum point behavior of the domain. The advantage of the and coverage of the fixed dentures as well as adhesive fixed partial micromechanical approach is not only the evaluation of the overall dentures and post core systems [12-14]. Fibrous composites have global properties of the composites but also determining the values replaced the traditional metal reinforced bridges for better bond- for various mechanisms such as damage initiation, propaga ng. In fact, the bond strength between the prostheses and the and failure can be studied through the algorithm [19-21. Many butment teeth obtained when using fiber reinforced composite micromechanical methods have been brought forward for analysis naterials is 50-100% higher than the bond strength achieved when and prediction of the overall behavior of composite materials [22- using metal framework [15]. Tezvergil et al. [16] studied to evalu- 32]. In particular, methods for upper and lower bounds of elastic ate the bond strength and fracture pattern of fiber-reinforced com- moduli have been derived using energy variational principles by posite with two different fiber orientations and matrix closed-form analytical expressions [1]. Based on an energy balance ompositions to dentine and enamel. They used two bidirectional approach with the aid of elasticity theory, whitney and Riley [3 and random distribution in their studies obtained closed-form analytical expressions for composite 's elastic Fibrous composites are often composed of a matrix reinforced moduli. The generalization of these methods to viscoelastic, elasto- with multidirectional fibers. The mechanical properties of compos- plastic and nonlinear behavior are very difficult. Aboudi 17] devel ites are functions of the individual properties of the constitutive oped a unified micromechanical theory based on the study of materials, their volume ratios, and the microstructural arrange- interacting repeating cells which was implemented to predict the ment. To obtain desired properties for composites, a microstruc- overall behavior of composite materials both for elastic and inelas tural analysis is required to determine the influence of tic constituents. A micromechanics model called the finite-volume parameters such as the arrangements of the fibers within the ma- direct averaging micromechanics (FVDAM)theory with inelastic trix and their angular orientations, along with other geometrical response capability for the individual phases has been provided and material parameters. An efficient characterization algorithm by Bansal and Pindera [33]. following the re-construction of the is the micromechanical approach, in which the response of a repre- elastic version of the"high-fidelity"generalized method of cells sentative volume element(RVE)most often in the form of repeat- [34. As discussed in Ref [33 the original method of cells [17] is ng unit cells(RUCs)of the composite should be studied and a spring-like model based on periodicity concepts applied in a sur- examined under various loading conditions to conclude and deter- face-averaged sense. In micromechanics period mine the overall or homogenized property of the composite. In this should be utilized to replicate the material response of the unit cell paper, a micromechanical modeling approach is introduced and throughout the continuum domain. a simple explanation for the employed to study bidirectional fiber composites. This character- outcome of these conditions will be that the adjacent unit cells also ization tool can be employed to determine microstructural effects deform in the same manner as the analyzed ruc does. There motivation behind the desired thermo-mechanical property. The many published data in which physical boundary conditions of composites for rk presented is as follows simulated as periodic boundary conditions which are incorrect regardless of whether the results are close for the special simple Proper micromechanical characterization of angle-ply and bidi- cases under consideration. rectional fibrous composites is essential in accurate character Through periodicity assumptions, many investigators have used tion, design and selection of composite materials for finite element analysis in elastic and thermoelastic analyses of the applications in industry so-called RUCs [25 to determine the mechanical properties and Theoretical characterization and most computational schemes damage mechanisms of composites [6, 19, 26-28, 35). In most of are based on simulating composites as unidirectional fibrous these cases, the applications are limited to the unidirectional lam- composites and therefore, their extensions to angle-ply and inates. Micromechanical analysis has been extended to thermal bidirectional fibrous composites introduce rough estimates in residual stresses [33, crack initiation and propagation [22] and many situations. Approximate theories such as lamination the- viscoplastic or viscoelastic behaviors [26, 27, 29, 30, 36]. In particu lar, Brinson and Lin [29 and Fisher and Brinson 30 used microm- Stiffness/compliance transformation rules and lamination the- echanics for periodic structures but under physical boundary ory are limited to the situations when laminas homogenized conditions. Their results have been compared to Mori-Tanaka properties are known at least along the principal material direc- method with a fair degree of success. ns Micromechanics characterization is needed to develop the In the present study the fem micromechanical analysis method stiffness/compliance of the lamina from the lamina's micro- is applied to bidirectional fibers at different cross angles to deter- structure and constituents'materials in any direction. mine the homogenized elastic properties of a composite. The RUC is subjected to six load scenarios, under which the stresses The micromechanics model presented in this paper is thus and strains will be recorded. The six load cases are categorized to established based on the microstructure and properties of constit- three axial loadings in three directions and two longitudinal shears nts,with no introduction of approximation in geometry. The and one transverse shear for a complete set of independent load- are analyzed under six load types to determine the general ings. Proper periodic boundary conditions are implemented any angle fro, properties. The cross angles of fibers can take with the necessary physical constraints to stop rigid body mo 0 to 90 :(e)The RUCs are exposed to periodic of the RUC. The volume averaged responses under the spe
rectional fiber composite as a medical texture is a promising application for such applications. In this respect, biocompatible textures can be applied in a wide range of applications from polymeric valves through woven or knitted artificial ligaments, to polymeric wound closure devices. Implantable cardiac support devices, vascular prosthesis and heart valves are other examples [11]. During the last decade, fiber reinforced composite was introduced as a new material for dentistry and orthodontic application. There has been growing interest in utilizing fiber reinforced composites for load-bearing applications such as dental crowns, fixed partial dentures, and implant-supported prostheses. Metal free composite materials can be used for the fabrication of single crown and coverage of the fixed dentures as well as adhesive fixed partial dentures and post core systems [12–14]. Fibrous composites have replaced the traditional metal reinforced bridges for better bonding. In fact, the bond strength between the prostheses and the abutment teeth obtained when using fiber reinforced composite materials is 50–100% higher than the bond strength achieved when using metal framework [15]. Tezvergil et al. [16] studied to evaluate the bond strength and fracture pattern of fiber-reinforced composite with two different fiber orientations and matrix compositions to dentine and enamel. They used two bidirectional and random distribution in their studies. Fibrous composites are often composed of a matrix reinforced with multidirectional fibers. The mechanical properties of composites are functions of the individual properties of the constitutive materials, their volume ratios, and the microstructural arrangement. To obtain desired properties for composites, a microstructural analysis is required to determine the influence of parameters such as the arrangements of the fibers within the matrix and their angular orientations, along with other geometrical and material parameters. An efficient characterization algorithm is the micromechanical approach, in which the response of a representative volume element (RVE) most often in the form of repeating unit cells (RUCs) of the composite should be studied and examined under various loading conditions to conclude and determine the overall or homogenized property of the composite. In this paper, a micromechanical modeling approach is introduced and employed to study bidirectional fiber composites. This characterization tool can be employed to determine microstructural effects of composites for any desired thermo-mechanical property. The motivation behind the work presented is as follows: Proper micromechanical characterization of angle-ply and bidirectional fibrous composites is essential in accurate characterization, design and selection of composite materials for applications in industry. Theoretical characterization and most computational schemes are based on simulating composites as unidirectional fibrous composites and therefore, their extensions to angle-ply and bidirectional fibrous composites introduce rough estimates in many situations. Approximate theories such as lamination theory are also too approximate at many situations. Stiffness/compliance transformation rules and lamination theory are limited to the situations when laminas’ homogenized properties are known at least along the principal material directions. Micromechanics characterization is needed to develop the stiffness/compliance of the lamina from the lamina’s microstructure and constituents’ materials in any direction. The micromechanics model presented in this paper is thus established based on the microstructure and properties of constituents, with no introduction of approximation in geometry. The RUCs are analyzed under six load types to determine the general material elastic properties. The cross angles of fibers can take any angle from 0 to 90; (e) The RUCs are exposed to periodic boundary conditions where a repeating cell represents a homogenized continuum point at the macroscale. Other enforced types of boundary conditions (symmetry, homogenous boundary conditions) can introduce some form of local impact and therefore cannot be regarded as a homogenized property. The analysis procedure is based on FEM solution with a mechanized processing algorithm which be extended to nonlinear and time-dependent material behavior. Micromechanics approach replaces the heterogeneous structure of the composite by a homogeneous medium with anisotropic properties [17,18]. In fact, the RVE or the RUC should simulate a continuum point behavior of the domain. The advantage of the micromechanical approach is not only the evaluation of the overall global properties of the composites but also determining the values for various mechanisms such as damage initiation, propagation, and failure can be studied through the algorithm [19–21]. Many micromechanical methods have been brought forward for analysis and prediction of the overall behavior of composite materials [22– 32]. In particular, methods for upper and lower bounds of elastic moduli have been derived using energy variational principles by closed-form analytical expressions [1]. Based on an energy balance approach with the aid of elasticity theory, Whitney and Riley [3] obtained closed-form analytical expressions for composite’s elastic moduli. The generalization of these methods to viscoelastic, elastoplastic and nonlinear behavior are very difficult. Aboudi [17] developed a unified micromechanical theory based on the study of interacting repeating cells which was implemented to predict the overall behavior of composite materials both for elastic and inelastic constituents. A micromechanics model called the finite-volume direct averaging micromechanics (FVDAM) theory with inelastic response capability for the individual phases has been provided by Bansal and Pindera [33], following the re-construction of the elastic version of the ‘‘high-fidelity” generalized method of cells [34]. As discussed in Ref. [33], the original method of cells [17] is a spring-like model based on periodicity concepts applied in a surface-averaged sense. In micromechanics periodicity, constrains should be utilized to replicate the material response of the unit cell throughout the continuum domain. A simple explanation for the outcome of these conditions will be that the adjacent unit cells also deform in the same manner as the analyzed RUC does. There are many published data in which physical boundary conditions are simulated as periodic boundary conditions which are incorrect regardless of whether the results are close for the special simple cases under consideration. Through periodicity assumptions, many investigators have used finite element analysis in elastic and thermoelastic analyses of the so-called RUCs [25] to determine the mechanical properties and damage mechanisms of composites [6,19,26–28,35]. In most of these cases, the applications are limited to the unidirectional laminates. Micromechanical analysis has been extended to thermal residual stresses [33], crack initiation and propagation [22] and viscoplastic or viscoelastic behaviors [26,27,29,30,36]. In particular, Brinson and Lin [29] and Fisher and Brinson [30] used micromechanics for periodic structures but under physical boundary conditions. Their results have been compared to Mori–Tanaka method with a fair degree of success. In the present study the FEM micromechanical analysis method is applied to bidirectional fibers at different cross angles to determine the homogenized elastic properties of a composite. The RUC is subjected to six load scenarios, under which the stresses and strains will be recorded. The six load cases are categorized to three axial loadings in three directions and two longitudinal shears and one transverse shear for a complete set of independent loadings. Proper periodic boundary conditions are implemented along with the necessary physical constraints to stop rigid body motions of the RUC. The volume averaged responses under the specified 1194 N. Abolfathi et al. / Computational Materials Science 43 (2008) 1193–1206
N. Abolfathi et aL/ Computational Materials Science 43(2008)1193-1206 load cases are analyzed simultaneously (inverse analysis)to pre- with cross angles 0. 45 and 90. For the parallel distributed fibers dict the material characteristics of the unit cell. The methodology the symmetry lines define the periodic microstructure of a straight has verified itself for cases of known solutions, i.e. when the unit fiber composite(Fig. 1). The geometrical parameters of the unit cell is assumed of the same homogenized pure elastic and thus cells are the cross-sectional width w, height h and length L corre- the input data should be expected from the reverse analysis. The lated to the diameters of fibers to maintain fiber/matrix volume illustrative analysis presented in the current study is limited to fractions. elastic materials; however, the methodology can be used for vari- Different unit cell models were developed to study the effect of ous circumstances of composite characterization procedures, such changing cross angles of the fibers, as well as, different fiber vol- as viscoelastic materials [36] with straight or wavy fibers [191. ume fractions of the bidirectional fibers on the overall material property of the composite. In order to verify the accuracy of the 2. Repeating unit cell(RUC)of the bidirectional fibrous modeling procedure, pure unit cells made of the same material type for both fiber and matrix at different cross angles were exam- ined under the six loading conditions to yield the input character- As shown in Fig. 1, bidirectional fibers with a crossing angle of o istics of the materials bers is assumed to remain constant so that a periodic unit cell 2.1. Loading and periodicity constraints can be defined. The periodicity of microstructure determines the geometry of the unit cell. As shown, a parallelplied geometry made Load cases: To determine the compliance and stiffness coeffi- of the matrix and fibers creates the cross-sectional view of the f- cients of the composite, each individual model was analyzed under brous composite As a bidirectional fibrous composite it is assumed six load scenarios. The six types of loadings include three axial and that the microstructure of the composite along the third direction three shear forces(two direct shears, and one shear due to torsion) (perpendicular to plane of cross-section) remains constant. The Referring to Fig. 2, the directions 1, 2, and 3 correspond to the netric shape of the RUC is shown in Fig. 1c. The fibers are all longitudinal, transverse in the plane of fiber, and transverse normal for such a RUC is shown in Fig. 1d. Fig. 2 shows three different RUC Fig 3)were defined as the following. ctively. Six load cases(see ssumed straight and of circular cross sections. a typical FEM mesh to the plane of fiber directions, res b d Periodic Unit Cod Fig. 1.(a)The bidirectional fibers at cross angles of o embedded in matrix, (b)the cross-sectional view of a unit cell. (c)the periodic 3-D unit cell volume, and(d)the FEm discretized of the ruc
load cases are analyzed simultaneously (inverse analysis) to predict the material characteristics of the unit cell. The methodology has verified itself for cases of known solutions, i.e., when the unit cell is assumed of the same homogenized pure elastic and thus the input data should be expected from the reverse analysis. The illustrative analysis presented in the current study is limited to elastic materials; however, the methodology can be used for various circumstances of composite characterization procedures, such as viscoelastic materials [36] with straight or wavy fibers [19]. 2. Repeating unit cell (RUC) of the bidirectional fibrous composite As shown in Fig. 1, bidirectional fibers with a crossing angle of u as embedded in a matrix are shown. The crossing angle of the fi- bers is assumed to remain constant so that a periodic unit cell can be defined. The periodicity of microstructure determines the geometry of the unit cell. As shown, a parallelplied geometry made of the matrix and fibers creates the cross-sectional view of the fi- brous composite. As a bidirectional fibrous composite it is assumed that the microstructure of the composite along the third direction (perpendicular to plane of cross-section) remains constant. The volumetric shape of the RUC is shown in Fig. 1c. The fibers are all assumed straight and of circular cross sections. A typical FEM mesh for such a RUC is shown in Fig. 1d. Fig. 2 shows three different RUC with cross angles 0, 45 and 90. For the parallel distributed fibers, the symmetry lines define the periodic microstructure of a straight fiber composite (Fig. 1). The geometrical parameters of the unit cells are the cross-sectional width w, height h and length L correlated to the diameters of fibers to maintain fiber/matrix volume fractions. Different unit cell models were developed to study the effect of changing cross angles of the fibers, as well as, different fiber volume fractions of the bidirectional fibers on the overall material property of the composite. In order to verify the accuracy of the modeling procedure, pure unit cells made of the same material type for both fiber and matrix at different cross angles were examined under the six loading conditions to yield the input characteristics of the materials. 2.1. Loading and periodicity constraints Load cases: To determine the compliance and stiffness coeffi- cients of the composite, each individual model was analyzed under six load scenarios. The six types of loadings include three axial and three shear forces (two direct shears, and one shear due to torsion). Referring to Fig. 2, the directions 1, 2, and 3 correspond to the longitudinal, transverse in the plane of fiber, and transverse normal to the plane of fiber directions, respectively. Six load cases (see Fig. 3) were defined as the following: Fig. 1. (a) The bidirectional fibers at cross angles of u embedded in matrix, (b) the cross-sectional view of a unit cell, (c) the periodic 3-D unit cell volume, and (d) the FEM discretized of the RUC. N. Abolfathi et al. / Computational Materials Science 43 (2008) 1193–1206 1195
N. Abolfathi et aL/ Computational Material Science 43(2008)1193-1206 Fig. 2. Three RUCs at fiber cross angles of 0, 45 and 90- 2 Load Case I Load Case 2 Load Case 3 oad Case 4 Load Case 5 Load case 6 Fig 3. The load cases under which the response analyses are carried Load case 1, 2 and 3: are direct concentrated forces in direction 1 Load case 6: is a twist load produced by two concentrated shear 2 and 3 each being applied at the center-node of faces 1, 3 and 5. forces tangential to the faces in 1-and 3-direction being applied simulate microstresses and at the center-nodes of faces 3, and 5. The magnitudes of the pairs strains associated with a condition of uniform uniaxial normal of loads produce equal and opposite torques around the unit stress 011, 022, and 33 as in a tensile coupon. cell. This load case simulates the microstresses for the condition Load case 4 and 5: are concentrated shear forces in direction 1 on of pure transverse shear, 23, in a lamina the faces 3 and 5 each being applied at one of the center-nodes of faces 3 and 5. These load cases simulates microstresses asso- For load cases 1-5 the application of a single load at the center iated with a condition of uniform(pure)longitudinal shear of the face is required as the periodic boundary conditions enforce stress 12 and t13 in a lamina. the three face pairs to deform and to act as they are under a uni-
Load case 1, 2 and 3: are direct concentrated forces in direction 1, 2 and 3 each being applied at the center-node of faces 1, 3 and 5, respectively. These load cases simulate microstresses and strains associated with a condition of uniform uniaxial normal stress r11, r22, and r33 as in a tensile coupon. Load case 4 and 5: are concentrated shear forces in direction 1 on the faces 3 and 5 each being applied at one of the center-nodes of faces 3 and 5. These load cases simulates microstresses associated with a condition of uniform (pure) longitudinal shear stress s12 and s13 in a lamina. Load case 6: is a twist load produced by two concentrated shear forces tangential to the faces in 1- and 3-direction being applied at the center-nodes of faces 3, and 5. The magnitudes of the pairs of loads produce equal and opposite torques around the unit cell. This load case simulates the microstresses for the condition of pure transverse shear, s23, in a lamina. For load cases 1–5 the application of a single load at the center of the face is required as the periodic boundary conditions enforce the three face pairs to deform and to act as they are under a uniFig. 2. Three RUCs at fiber cross angles of 0, 45 and 90. 1 3 2 Load Case 1 Load Case 2 Load Case 3 Load Case 4 Load Case 5 Load Case 6 Fig. 3. The load cases under which the response analyses are carried. 1196 N. Abolfathi et al. / Computational Materials Science 43 (2008) 1193–1206
N. Abolfathi et al/ Computational Materials Science 43(2008)1193-1206 formly distributed load at infinity For load case 6 two forces at the u=-uf, uf=-u1, 5=-u's(i=1, 2, 3) (1 center of faces 3 and 5 are necessary to produce a required tor- sional load Periodicity constraints: Periodicity requires that opposite faces 2. 2. Node pairs on faces of the unit cell deform identically. This requires certain constraint For all nodes on face 2 except the center node and aints requires the number and distribution of nodes on oppo- edges e24 and ez6 and the corner node n246. One ha site faces to be identical. it is also convenient to have a node located at the geometric center of each face. To show how these achieved, consider again the solid model shown in Applying Eq (8)one gets Fig 4. On this geometry therefore six faces, 12 edges, six center- uf2-uf1-2uf1=0 face nodes and eight corner nodes. The displacement degrees of freedom for the nodes on half of the faces(2, 4 and 6) edges Ing constraInt r 4, 26, 46,.)and corners(246,. . must be written in terms of the degrees of freedom of the nodes on the other half. These u 24=u m-2(uf1+uf) algebraic relations are such that they force opposite faces to de- u 26=u ts-2(u 1+ form to the same shape though they may have a rigid body trans- lation between them At corner node n246, one has To enforce the repeating behavior, the following constraint u 2=u 35-2(u1+u2+u5) (5 relations are enforced. In the following, u(i=1, 2, 3)represents the displacement in the ith-direction, c(i=1, 2,..., 6)represents he center face nodes, ny and n represent the node pair on oppo 2.3. Node pairs ing faces i and j, ey stands for the edge i, sharing the faces i and j. sents the corner node sharing faces i,j and k. The constraint equa- 246 and the nodes on edges ez4 and e46, one has e ny stands for the nodes located on edge e, and finally nik repre- For all nodes on face 4 except the center node, the corner node ions are defined such that displacement components of ea node on faces 2, 4 and 6 are removed in terms of the respective components for the pair node on faces 1, 3 and 5. To enforce For the nodes on edge eas the following relation n is enforced: deformation compatibility between opposite faces yet still allow rigid body motion between the two faces, the displacements for the nodes on each face are expressed relative to the center node on that face. Because edge and corner nodes are shared be- 2. 4. Node pairs on faces 5 and tween multiple faces, care must be taken to avoid redundant (over)constraints For all nodes on face 6 except the center node, the corner node Additional constraints on the center nodes of the opposite faces n246 and the nodes on edges e26 and eas one has are applied. The slave nodes on the center of faces 2, 4 and 6 are related to the active nodes on the faces 1.3 and 5 as shown below: u=ul5-2ur5 Face 2: Opposite to Face 1 Face 4: Opposite to Face 3 Face 6: Opposite to Face 5 Face 3 Edge 23 Edge 36 Comer 136 er235 Face 6 Comer 14 Edge 14 Face 5 Face 4 Comer 245 Edge 15 Fig. 4. The unit cell faces, edges, and comers designations related to periodic constraint descriptions
formly distributed load at infinity. For load case 6 two forces at the center of faces 3 and 5 are necessary to produce a required torsional load. Periodicity constraints: Periodicity requires that opposite faces of the unit cell deform identically. This requires certain constraint relations between the nodes on the faces. Invoking these constraints requires the number and distribution of nodes on opposite faces to be identical. It is also convenient to have a node located at the geometric center of each face. To show how these constraints are achieved, consider again the solid model shown in Fig. 4. On this geometry therefore six faces, 12 edges, six centerface nodes and eight corner nodes. The displacement degrees of freedom for the nodes on half of the faces (2, 4 and 6), edges (24, 26, 46, ...) and corners (246, ...) must be written in terms of the degrees of freedom of the nodes on the other half. These algebraic relations are such that they force opposite faces to deform to the same shape though they may have a rigid body translation between them. To enforce the repeating behavior, the following constraint relations are enforced. In the following, ui (i = 1, 2, 3) represents the displacement in the ith-direction, ci (i = 1, 2, ... , 6) represents the center face nodes, ni and nj represent the node pair on opposing faces i and j, eij stands for the edgeij, sharing the faces i and j, nij stands for the nodes located on edge eij, and finally nijk represents the corner node sharing faces i, j and k. The constraint equations are defined such that displacement components of each node on faces 2, 4 and 6 are removed in terms of the respective components for the pair node on faces 1, 3 and 5. To enforce deformation compatibility between opposite faces yet still allow a rigid body motion between the two faces, the displacements for the nodes on each face are expressed relative to the center node on that face. Because edge and corner nodes are shared between multiple faces, care must be taken to avoid redundant (over) constraints. Additional constraints on the center nodes of the opposite faces are applied. The slave nodes on the center of faces 2, 4 and 6 are related to the active nodes on the faces 1, 3 and 5 as shown below: uc2 i ¼ uc1 i ; uc4 i ¼ uc3 i ; uc6 i ¼ uc5 i ði ¼ 1; 2; 3Þ ð1Þ 2.2. Node pairs on faces 1 and 2 For all nodes on face 2 except the center node and the nodes on edges e24 and e26 and the corner node n246, one has, un2 i ¼ un1 i uc1 i þ uc2 i ð2Þ Applying Eq. (8) one gets, un2 i un1 i 2uc1 i ¼ 0 ð3Þ On the edges e24 and e26, the following constraint relations apply, respectively, un24 i ¼ un13 i 2ðuc1 i þ uc3 i Þ un26 i ¼ un15 i 2ðuc1 i þ uc5 i Þ ð4Þ At corner node n246, one has un246 i ¼ un135 i 2ðuc1 i þ uc3 i þ uc5 i Þ ð5Þ 2.3. Node pairs on faces 3 and 4 For all nodes on face 4 except the center node, the corner node n246 and the nodes on edges e24 and e46, one has un4 i ¼ un3 i 2uc3 i ð6Þ For the nodes on edge e46 the following relation is enforced: un46 i ¼ un35 i 2ðuc3 i þ uc5 i Þ ð7Þ 2.4. Node pairs on faces 5 and 6 For all nodes on face 6 except the center node, the corner node n246, and the nodes on edges e26 and e46 one has un6 i ¼ un5 i 2uc5 i ð8Þ Fig. 4. The unit cell faces, edges, and corners designations related to periodic constraint descriptions. N. Abolfathi et al. / Computational Materials Science 43 (2008) 1193–1206 1197
N Bolat Other than the periodic constraints enforced on the node-pairs on composites as a means of comparison with the results from the pposite surfaces of the unit cell, translational and rotational con- developed algorithm Assuming 123 to be the principal(orthotro- straints to prevent the singularity of rigid body modes are also pic) directions and xyz be general anisotropic directions in a lami- enforced nate. The stress and strains components in these two coordinate Rigid body constraints: The central node at the center of body is systems are correlated by a geometrical transformation matrix fixed in all directions. To prevent translations and rotations the [T] as center-nodes on faces 1 and 2 are fixed in directions 2 and 3 and finally a node on one of the edges of face 1 is fixed in direction 2 Exx Eyy Ezz xy 7x y) or 3 depending on the edge to prevent rigid rotation about the =[T1E11 222 233 712 713 723] length of the model 3. The constitutive micromechanical material approach 833712713723 (11) where [T], the transformation matrix is written as nFig2, three rucs at fiber cross angles of0°,45°and90°are shown. If the fibers are arranged at zero crossing angles, i.e. parallel mi ni Pi 2m,n 2p, m, 2n,P, fibers, the homogenized properties of the composite should follow m3 n2 p2 2m2n2 2p, m2 2n2p2 those of unidirectional fibrous composites. If the fibers are ar- m=I m? n, p2 2m3, ranged at 90 cross angles, i.e., perpendicular to each others, due mim2 nn2 p1P2 1P2+n2P, p, m2+p2m1 min2+m2n1 creation of three perpendicular planes of material symmetry. m,m, nan, P3P, naP,+, P,m,+p, m, m,n+m, n,I the composite material is orthotropic. At any other cross angle m2m3 nzn3 pp3 nip,+n2p3 p3m2 +p2m m2n3+m3n2 the composite has only one plane of symmetry and therefore the (12) clinic isotropic with the material chosen. If the material is aniso- m ni pi(i=1, 3)are the cosine directions of the coordinate system tropic, then the compliance matrix is fully populated. Thus the xyz with respect to 123. The elastic material constitutive relations constitutive relations in an anisotropic coordinate system 1-2-3 cal an be utilized to correlate the elastic constants in different direc- can be written in the following form: tions or coordinate systems. Therefore, the following relations be tween the material compliance or stiffness in principal directions S111 S1122 S1133 S1112 S1113 S11231(a1 and in generalized anisotropic directions can be concluded S3311 S3322 S3333 S3312 S3313 S3323 S1211S1 (9)Syox Syyy Sya Syyay Sym Syyyz S2311S232S23S2312S233S2323 Sxyax Sxyyy Sxyzz Sxyay Sxya sxyz Sxxx Sxy Sxzn Sxzxy Sxzxz Sxz where Sijk represents are elements of the 6 x 6 matrix of compli- Syax Syzyy Syzz Syzxy Syzxz Sy ance coefficients for a completely anisotropic material. For an S1l Fuzz S anisotropic material one has 21 independent coefficients. In the haracterization procedure, the rUC of the material will S221S222S223000 ected to six independent loading scenarios. The outcome six scenarios of loading yield six sets of stresses and strains, and therefore, thirty six equations needed to find the 36 0000S130 mponents of the compliance or stiffness matrix in general. The 00000S2323 mpliance and stiffness components and their relations in terms of engineering constants can be found in most composite text- For the bidirectional angle-ply laminate under consideration in this books 38]. paper, the equivalent compliance/stiffness of the laminate can be For each load scenario, the stress-strain information obtained considered a combination of the stiffness/compliance of two lami- from the analysis are volume averaged over the volume of the nas(shown in Fig. 1), that is Cleg=vi[Ch+ vlc (10) where [c [S] and V(i= 1, 2)are the stiffness, compliance matri- where 'V is the volume of the ruc these correlated ces and the volume fractions of the laminas 1 and 2, respectively provide the base for determination of t I indepen- The equivalent compliance/stiffness can then be measured elements of compliance coefficients ing the as- employing Eqs.(12)-(14)to be compared with the solutions of constitutive relations response analysis under different loading scenarios will be carried principal directions, and are orthotropic in their respective prind out by finite elements. The volume average data and inverse pal directions. One lamina remains unidirectional in the global characterization analysis will be interfaced with the finite ele- 123 directions(shown in Fig. 1). and the second one is only ment package orthotropic along the principal directions oriented at a cross an gle of o with respect to direction 1. The cosine directions become 3.1. Determination of angle-ply elastic constants from lamination mpler for the transformation, ie, m,=COSp, m 2=sine, m3=0 1=-sin, n2=COS, n3=0: p1=0. P2=0, P3=1. In the modeling procedure one laminate is unidirectional and therefore the co As a means of characterization of fibrous composites, lamina- pliance and stiffness remain constant in 123 directions. The sec- tion theory can be used with a degree of mation. Here this ond laminate has a principal directions oriented along the fiber approximated theory is tailored for the solution of the angle-ply directions
Other than the periodic constraints enforced on the node-pairs on opposite surfaces of the unit cell, translational and rotational constraints to prevent the singularity of rigid body modes are also enforced. Rigid body constraints: The central node at the center of body is fixed in all directions. To prevent translations and rotations the center-nodes on faces 1 and 2 are fixed in directions 2 and 3 and finally a node on one of the edges of face 1 is fixed in direction 2 or 3 depending on the edge to prevent rigid rotation about the length of the model. 3. The constitutive micromechanical material approach In Fig. 2, three RUCs at fiber cross angles of 0, 45 and 90 are shown. If the fibers are arranged at zero crossing angles, i.e. parallel fibers, the homogenized properties of the composite should follow those of unidirectional fibrous composites. If the fibers are arranged at 90 cross angles, i.e., perpendicular to each others, due to creation of three perpendicular planes of material symmetry, the composite material is orthotropic. At any other cross angle, the composite has only one plane of symmetry and therefore, the homogenized properties of the composite are considered monoclinic isotropic with the material chosen. If the material is anisotropic, then the compliance matrix is fully populated. Thus the constitutive relations in an anisotropic coordinate system 1–2–3 can be written in the following form: e11 e22 e33 e12 e13 e23 8 >>>>>>>>>>>>>>>: 9 >>>>>>>>= >>>>>>>>; ¼ S1111 S1122 S1133 S1112 S1113 S1123 S2211 S2222 S2233 S2212 S2213 S2223 S3311 S3322 S3333 S3312 S3313 S3323 S1211 S1222 S1233 S1212 S1213 S1223 S1311 S1322 S1333 S1312 S1313 S1323 S2311 S2322 S2333 S2312 S2313 S2323 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 r11 r22 r33 s12 s13 s23 8 >>>>>>>>>>>>>>>: 9 >>>>>>>>= >>>>>>>>; ð9Þ where Sijkl represents are elements of the 6 6 matrix of compliance coefficients for a completely anisotropic material. For an anisotropic material one has 21 independent coefficients. In the material characterization procedure, the RUC of the material will be subjected to six independent loading scenarios. The outcome of these six scenarios of loading yield six sets of stresses and strains, and therefore, thirty six equations needed to find the 36 components of the compliance or stiffness matrix in general. The compliance and stiffness components and their relations in terms of engineering constants can be found in most composite textbooks [38]. For each load scenario, the stress–strain information obtained from the analysis are volume averaged over the volume of the RUC, i.e., rij ¼ 1 V Z v rij dv; eij ¼ 1 V Z v eij dv ð10Þ where ‘V’ is the volume of the RUC. These averaged correlated data provide the base for determination of the 36 (21 independent) elements of compliance coefficients Sijkl or Cijkl using the assumed constitutive relations eij ¼ Sijklrkl or rkl ¼ Cijkleij. The response analysis under different loading scenarios will be carried out by finite elements. The volume average data and inverse characterization analysis will be interfaced with the finite element package. 3.1. Determination of angle-ply elastic constants from lamination theory As a means of characterization of fibrous composites, lamination theory can be used with a degree of approximation. Here this approximated theory is tailored for the solution of the angle-ply composites as a means of comparison with the results from the developed algorithm. Assuming 123 to be the principal (orthotropic) directions and xyz be general anisotropic directions in a laminate. The stress and strains components in these two coordinate systems are correlated by a geometrical transformation matrix [T] as, f exx eyy ezz cxy cxz cyz gT ¼ ½T 1 f e11 e22 e33 c12 c13 c23 gT f exx eyy ezz cxy cxz cyz gT ¼ ½T 1 f e11 e22 e33 c12 c13 c23 gT ð11Þ where [T], the transformation matrix is written as ½T ¼ m2 1 n2 1 p2 1 2m1n1 2p1m1 2n1p1 m2 2 n2 2 p2 2 2m2n2 2p2m2 2n2p2 m2 1 n2 1 p2 1 2m3n3 2p3m3 2n3p3 m1m2 n1n2 p1p2 n1p2 þn2p1 p1m2 þp2m1 m1n2 þm2n1 m3m1 n3n1 p3p1 n3p1 þn1p3 p3m1 þp1m3 m3n1 þm1n3 m2m3 n2n3 p2p3 n3p2 þn2p3 p3m2 þp2m3 m2n3 þm3n2 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ð12Þ mi, ni, pi (i = 1, 3) are the cosine directions of the coordinate system xyz with respect to 123. The elastic material constitutive relations can be utilized to correlate the elastic constants in different directions or coordinate systems. Therefore, the following relations between the material compliance or stiffness in principal directions and in generalized anisotropic directions can be concluded. Sxxxx Sxxyy Sxxzz Sxxxy Sxxxz Sxxyz Syyxx Syyyy Syyzz Syyxy Syyxz Syyyz Szzxx Szzyy Szzzz Szzxy Szzxz Szzyz Sxyxx Sxyyy Sxyzz Sxyxy Sxyxz Sxyyz Sxzxx Sxzyy Sxzzz Sxzxy Sxzxz Sxzyz Syzx Syzyy Syzzz Syzxy Syzxz Syzyz 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ¼ ½T 1 S1111 S1122 S1133 000 S2211 S2222 S2233 000 S3311 S3322 S3333 000 000 S1212 0 0 0000 S1313 0 00000 S2323 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ½T ð13Þ For the bidirectional angle-ply laminate under consideration in this paper, the equivalent compliance/stiffness of the laminate can be considered a combination of the stiffness/compliance of two laminas (shown in Fig. 1), that is, ½C eq ¼ V1½C 1 þ V2½C 2 ½S eq ¼ ½C 1 eq ; ½S 1 ¼ ½C 1 1 ; ½S 2 ¼ ½C 1 2 ð14Þ where [C]i, [S]i and Vi (i = 1,2) are the stiffness, compliance matrices and the volume fractions of the laminas 1 and 2, respectively. The equivalent compliance/stiffness can then be measured employing Eqs. (12)–(14) to be compared with the solutions of the micromechanics formulations. Note that both laminas have principal directions, and are orthotropic in their respective principal directions. One lamina remains unidirectional in the global 123 directions (shown in Fig. 1), and the second one is only orthotropic along the principal directions oriented at a cross angle of u with respect to direction 1. The cosine directions become simpler for the transformation, i.e., m1 = cosu, m2 = sinu, m3 = 0; n1 = sinu, n2 = cosu, n3 = 0; p1 = 0, p2 = 0, p3 = 1. In the modeling procedure one laminate is unidirectional and therefore the compliance and stiffness remain constant in 123 directions. The second laminate has a principal directions oriented along the fiber directions. 1198 N. Abolfathi et al. / Computational Materials Science 43 (2008) 1193–1206
4. Numerical results transversely isotropic. The elastic properties of these materials are presented in Table 1. Depending on the constitutive materials 4. 1. The material input nd directions, the composite can be transversely isotropic to com pletely anisotropic. The algorithm will determine the degree of our composite materials chosen consist of four sets of constitu- isotropy by counting the independent number of constants result- tive materials. The composite #1 consists of an epoxy matrix rein- ing from the analysis. It should be noted that the accuracy of the forced with glass fibers, composite #2 is made of ceramic fibers method is dependent on the accuracy of the FEM procedure and embedded in glass matrix, composite #3 contains a ceramic matrix therefore, the solutions are mesh-dependent. However, in the re- reinforced by another ic fiber, and finally composite #4 is sults that are presented here, the solution convergence has been made of an epoxy matrix reinforced with carbon fibers. All the reached with the mesh employed as shown in Fig. 1d materials chosen are isotropic except the carbon fibers which are 4.2 Stress distributions in ruts For each of the composites and at fiber cross angles between 0 Elastic properties of the constituents of the matrix and 90, a periodic unit cell of the assumed fiber packing(shown in Constituent Figs. 1 and 2) was analyzed under six loading conditions using ABAQUS 37. Results are stored for all six load cases at each cross Glass fiber [39 E=72.9 v=022 angle Using the volume averaging utine program interfaced Epoxy matrix [35 E=4.5 v=0.45 with ABAQUS, the volume-averaged responses of the RUCs are stored and used to obtain the resultant composite compliance Textron SCS-6 SiC E=423 and stiffness properties. In Figs. 5 and 6. the cross-sectional view fiber [40] v=0.2 stress distribution(over half of the cell) in the off-axis fibers due to load cases1.and2 for angles of 0°.45°and90° are plotted. iC matrix [411 E=251 In see in Fig. 5 that for load case 1, stresses Sn1 induced in iC fiber [411 he off-axis fibers decreases with increase in fiber angles wherea or axial fibers stresses Snl increase with increase in fiber angle. AS4 Carbon fiber E=2010, E2=E3=13.5. v12=v13=0.22and Opposite behavior is seen in Fig. 6 for load case 2, where S22 in- G12=G13=95,G23=49 v23=025 creases for off-axis fiber with increase in fiber angle and the same Epoxy matrix [35] E=4.5 v=45 decreases for axial fibers with increase in fiber angle Also looking at Fig. 7, it can be observed that for shear cases, the composite Fig. 5. Stress distribution contours for the cross-section of the off-axis fiber inside the ruc under load case 1, for fiber cross angles of o=0. 45. 90 and with the constitutive aterials of the composite#1.(a)φ=0°,(b)=45°(c)φ=90°
4. Numerical results 4.1. The material input Four composite materials chosen consist of four sets of constitutive materials. The composite #1 consists of an epoxy matrix reinforced with glass fibers, composite #2 is made of ceramic fibers embedded in glass matrix, composite #3 contains a ceramic matrix reinforced by another ceramic fiber, and finally composite #4 is made of an epoxy matrix reinforced with carbon fibers. All the materials chosen are isotropic except the carbon fibers which are transversely isotropic. The elastic properties of these materials are presented in Table 1. Depending on the constitutive materials and directions, the composite can be transversely isotropic to completely anisotropic. The algorithm will determine the degree of isotropy by counting the independent number of constants resulting from the analysis. It should be noted that the accuracy of the method is dependent on the accuracy of the FEM procedure and therefore, the solutions are mesh-dependent. However, in the results that are presented here, the solution convergence has been reached with the mesh employed as shown in Fig. 1d. 4.2. Stress distributions in RUCs For each of the composites and at fiber cross angles between 0 and 90, a periodic unit cell of the assumed fiber packing (shown in Figs. 1 and 2) was analyzed under six loading conditions using ABAQUS [37]. Results are stored for all six load cases at each cross angle. Using the volume averaging subroutine program interfaced with ABAQUS, the volume-averaged responses of the RUCs are stored and used to obtain the resultant composite compliance and stiffness properties. In Figs. 5 and 6, the cross-sectional view stress distribution (over half of the cell) in the off-axis fibers due to load cases 1, and 2 for angles of 0, 45 and 90 are plotted. One can see in Fig. 5 that for load case 1, stresses S11 induced in the off-axis fibers decreases with increase in fiber angles whereas, for axial fibers stresses S11 increase with increase in fiber angle. Opposite behavior is seen in Fig. 6 for load case 2, where S22 increases for off-axis fiber with increase in fiber angle and the same decreases for axial fibers with increase in fiber angle. Also looking at Fig. 7, it can be observed that for shear cases, the composite Fig. 5. Stress distribution contours for the cross-section of the off-axis fiber inside the RUC under load case 1, for fiber cross angles of u = 0, 45, 90 and with the constitutive materials of the composite #1. (a) u = 0, (b) u = 45, (c) u = 90. Table 1 Elastic properties of the constituents of the matrix Constituent E or G (GPa) v Composite # 1 E-Glass fiber [39] E = 72.9 v = 0.22 Epoxy matrix [35] E = 4.5 v = 0.45 Composite # 2 Textron SCS-6 SiC fiber [40] E = 423 v = 0.15 F-glass matrix [40] E = 59 v = 0.2 Composite # 3 SiC matrix [41] E = 251 v = 0.16 SiC fiber [41] E = 200 v = 0.25 Composite # 4 AS4 Carbon fiber [35] E1 = 201.0, E2 = E3 = 13.5, G12 = G13 = 95, G23 = 4.9 v12 = v13 = 0.22 and v23 = 0.25 Epoxy matrix [35] E = 4.5 v = .45 N. Abolfathi et al. / Computational Materials Science 43 (2008) 1193–1206 1199
N. Abo b ig. rials es the composite ono ur) for te oss-s4stioc the or- xis nibe inside the RUc under load case 2. for nber cross angles of p-0. 45, 90 and with the constitutive attends highest stress values at an angle of 45 with minimum at 0 sumed to be 40%. In addition, to study the impact due to fiber vol- and 90% ume fractions, the analysis are extended to another two values Although the magnitude of the stress is a function of the load- 20%, and 60% The results at a 45 fiber angle for the different fiber ing, but the constitutive parameters are evaluated based on the lin- volume fractions are plotted in Figs. 12a and 12b for composite# ear stress-strain relations, therefore the deformation is linearly dependent on the load. For convenience, the amount of applied 43. Material property change with fiber cross angle loads will be calculated based on an assumption that the ruc will be under a uniform stress distribution of magnitude 1(S11= 1 for In the following, unless otherwise specified, the fiber volume load case 1, S22=1 for load case 2, S33=1 for load case 3, S12=1 fraction(Vf/) is 40% Tables 2 and 3 show the compliance coeffi- for load case 4, S13=1 for load case 4 and S23=1 for load case 6) cients for the composite #1. Although the constituent materials if the ruc is made of a purely uniform elastic material. Therefore, for this case are isotropic, the composite becomes transversely iso- isotropic at all other angles. This conclusion is made foo monoclinic the average stress in each of the contour plots in Figs 5-7 should tropic for fiber cross angle of o orthotropic for 90 an We have calculated the compliance and stiffness coefficients for ber of symmetric planes observed and from the nonzero indepen each case which are tabulated in Tables 2, 4, 6 and 8. In these ta- dent coefficient values evaluated from the methodology as bles, the numerical solutions are compared with the solutions from presented in Table 2. The material properties of this composite in the lamination theory solution procedure developed in previous terms of Youngs modulus and Poisson s ratio are also presented section. In general the agreement is good especially for the cases in Table 3. As expected, the Youngs modulus in direction 1 at 0 when the material approaches orthotropic material symmetry(o is higher than that of 90 fiber cross angle. For the direction 2 and 90). This is expected as the lamination theory will yield more opposite behavior to direction 1 is observed. Similar data and con- accurate solutions as the material symmetry is observed. Also, clusions were made for the composite #2 as only the numerical when the material properties of the two constituents become clo- values for the constitutive material properties have been changed. ser and closer the agreement between the two solution procedures The resulted data for composite #2 are shown in Tables 4 and 5. mproves. Again this is expected from the lamination theory to The stiffness coefficients for the composite #3 are tabulated ld more accurate results for closer constituents'materials In for the three cross angles of 0, 45 and 90 in Table 6. In addition these tables the solutions for the lamination theory at 0 cross an the material properties in terms of Young s modulus and Poissons gle are taken from the micromechanics solutions. This is because ratios are presented in Table 7. Due to close material properties of he needs the homogenized material properties of the layers at a the constituent matrix and fiber and the 40% fiber volume fraction, cross angle (say 0 here)to go ahead with the lamination theory a small change in material properties of the composite due to and the transformation materials rules involved. The microme- change in cross angle is resulted. The material is orthotropic at chanics solutions at 0 cross angle agree with the solutions of the 0 and 90, and remains anisotropic at other angles. For composite #4, the stiffness properties are shown in tables 8 and The compliance and stiffness parameters are also plotted in the carbon fibers are transversely isotropic and the epe Figs. 8-11. The fiber volume fractions for all these figure are as- isotropic, the composite is monolithic isotropic at 0o
attends highest stress values at an angle of 45 with minimum at 0 and 90. Although the magnitude of the stress is a function of the loading, but the constitutive parameters are evaluated based on the linear stress–strain relations, therefore the deformation is linearly dependent on the load. For convenience, the amount of applied loads will be calculated based on an assumption that the RUC will be under a uniform stress distribution of magnitude 1 (S11 = 1 for load case 1, S22 = 1 for load case 2, S33 = 1 for load case 3, S12 = 1 for load case 4, S13 = 1 for load case 4 and S23 = 1 for load case 6) if the RUC is made of a purely uniform elastic material. Therefore, the average stress in each of the contour plots in Figs. 5–7 should be 1. We have calculated the compliance and stiffness coefficients for each case which are tabulated in Tables 2, 4, 6 and 8. In these tables, the numerical solutions are compared with the solutions from the lamination theory solution procedure developed in previous section. In general the agreement is good especially for the cases when the material approaches orthotropic material symmetry (0 and 90). This is expected as the lamination theory will yield more accurate solutions as the material symmetry is observed. Also, when the material properties of the two constituents become closer and closer the agreement between the two solution procedures improves. Again this is expected from the lamination theory to yield more accurate results for closer constituents’ materials. In these tables the solutions for the lamination theory at 0 cross angle are taken from the micromechanics solutions. This is because one needs the homogenized material properties of the layers at a cross angle (say 0 here) to go ahead with the lamination theory and the transformation materials rules involved. The micromechanics solutions at 0 cross angle agree with the solutions of the mixture rules very well. The compliance and stiffness parameters are also plotted in Figs. 8–11. The fiber volume fractions for all these figure are assumed to be 40%. In addition, to study the impact due to fiber volume fractions, the analysis are extended to another two values of 20%, and 60%. The results at a 45 fiber angle for the different fiber volume fractions are plotted in Figs. 12a and 12b for composite #1. 4.3. Material property change with fiber cross angle In the following, unless otherwise specified, the fiber volume fraction (Vf/V) is 40%. Tables 2 and 3 show the compliance coeffi- cients for the composite #1. Although the constituent materials for this case are isotropic, the composite becomes transversely isotropic for fiber cross angle of 0, orthotropic for 90 and monoclinic isotropic at all other angles. This conclusion is made from the number of symmetric planes observed and from the nonzero independent coefficient values evaluated from the methodology as presented in Table 2. The material properties of this composite in terms of Young’s modulus and Poisson’s ratio are also presented in Table 3. As expected, the Young’s modulus in direction 1 at 0 is higher than that of 90 fiber cross angle. For the direction 2 opposite behavior to direction 1 is observed. Similar data and conclusions were made for the composite #2 as only the numerical values for the constitutive material properties have been changed. The resulted data for composite #2 are shown in Tables 4 and 5. The stiffness coefficients for the composite #3 are tabulated for the three cross angles of 0, 45 and 90 in Table 6. In addition, the material properties in terms of Young’s modulus and Poisson’s ratios are presented in Table 7. Due to close material properties of the constituent matrix and fiber and the 40% fiber volume fraction, a small change in material properties of the composite due to change in cross angle is resulted. The material is orthotropic at 0 and 90, and remains anisotropic at other angles. For composite #4, the stiffness properties are shown in Tables 8 and 9. Although the carbon fibers are transversely isotropic and the epoxy matrix is isotropic, the composite is monolithic isotropic at 0 and 90. The Fig. 6. Stress distribution contours for the cross-section of the off-axis fiber inside the RUC under load case 2, for fiber cross angles of u = 0, 45, 90 and with the constitutive materials of the composite #1. (a) u = 0, (b) u = 45, (c) u = 90. 1200 N. Abolfathi et al. / Computational Materials Science 43 (2008) 1193–1206
N. Abolfathi et al Computational Materials Science 43(2008)1193-1206 儈 b 8,12 Fig. 7. Stress distribution contours of the ruC under load case 4 and 6 for fiber cross angles of o=0. 45 and 90 with the constitutive materials of the composite #1(a Table 2 The 13 independent compliance coefficients(x10-2, 1/GPA) for the composite #1 at various fibers cros at 40% fiber volume fraction Fibers cross Method S111 S1122 and S133 and S112 and Sxz S233 and sz1 3.160-09200920 0000 0000 15 MM3.580-1.2500.95822501020-391 10.100.793 256230.20-06363500 4.360-2.090-0930 391010813.57 0875 5.270-2940 0950 2550 11.5 -3.51 0.75 660.740 7.5730.39 1.51 9.30-323 17.8731.30-1.270 5.540-3.090-10100.660 1440 0.004 7.612-2.64 210032.00-1.10033.20 5.510-2.760 10.24307 9.551.681 8912319551.50533.69 MM 4860-0963 6.3732.40-0634 5.150-1.750 754 37 8.9532 52 0865 MM4790-0669 4.79-1.76 9.540.000 29.203260 32.60 4.750-0.680 0000 0000 MM. current micromechanics method: Lt
Fig. 7. Stress distribution contours of the RUC under load case 4 and 6 for fiber cross angles of u = 0, 45 and 90 with the constitutive materials of the composite #1. (a) u = 0, (b) u = 45, (c) u = 90. Table 2 The 13 independent compliance coefficients (102 , 1/GPA) for the composite #1 at various fibers cross angles at 40% fiber volume fraction Fibers cross angle Method S1111 S1122 and S2211 S1133 and S3311 S1112 and S1211 S2222 S2233 and S3322 S2212 and S1222 S3333 S3312 and S1233 S1212 S1313 S1323 and S2313 S2323 0 MM 3.160 0.920 0.920 0.000 9.68 3.60 0.000 9.66 0.000 29.40 29.40 0.000 36.40 15 MM 3.580 1.250 0.958 2.250 10.20 3.91 0.44 10.10 0.793 25.62 30.20 0.636 35.00 LT 4.360 2.090 0.930 3.910 10.81 3.57 3.04 9.68 0.350 16.87 29.81 0.875 36.34 30 MM 4.300 1.770 1.050 2.429 10.10 3.66 0.938 10.10 1.505 19.72 30.70 1.100 34.50 LT 5.270 2.940 0.950 2.550 11.51 3.51 0.75 9.66 0.740 7.57 30.39 1.51 35.62 45 MM 4.790 1.900 1.210 1.270 9.30 3.23 3.24 9.96 2.020 17.87 31.30 1.270 33.90 LT 5.540 3.090 1.010 0.660 11.32 3.38 4.45 9.63 1.193 5.76 31.175 1.74 34.65 60 MM 4.930 1.540 1.440 0.004 7.612 2.64 4.64 9.79 2.090 21.00 32.00 1.100 33.20 LT 5.510 2.760 1.130 1.07 10.24 3.07 5.17 9.55 1.681 8.912 31.955 1.505 33.69 75 MM 4.860 0.963 1.660 0.447 5.69 2.04 3.57 9.62 1.400 26.37 32.40 0.634 32.80 LT 5.150 1.750 1.400 2.080 7.54 2.37 6.52 9.37 1.820 18.95 32.52 0.865 32.98 90 MM 4.790 0.669 1.760 0.000 4.79 1.76 0.000 9.54 0.000 29.20 32.60 0.000 32.60 LT 4.750 0.680 1.670 0.000 4.75 1.67 0.000 9.20 0.000 29.60 32.73 0.000 32.73 MM, current micromechanics method; LT, lamination theory. N. Abolfathi et al. / Computational Materials Science 43 (2008) 1193–1206 1201
N. Abolfathi et aL/ Computational Material Science 43(2008)1193-1206 C1111一C2222 0.12 s2222 S3333 E8n9 006 8 10 Fiber Cross Angle(deg) Fig 8. The variation of the compliance and the stiffness components of the composite #l in longitudinal directions as a function of the fiber cross angle (ol C2323 0.20 8生苏 Fiber Cross Angle(deg) Fig 9. The variation of the compliance and the stiffness components of the composite #1 in shear directions as a function of the fber cross angle (ol composite then becomes completely anisotropic at all other angles. in direction 2(S2222)decreases with an increase in angle, while the The compliance coefficients presented in Table 8 for a cross angle stiffness(C2222)increases. Such changes show the natural behavior of 45 show this. In the Tables 3, 5 and 7 where the data for Youngs of a composite material when it transforms from a unidirectional moduli and Poissons ratios were presented where E11, E22, E33, V12,(0)to a completely bidirectional(90)composite For composite V21, V13 are the elastic constants of the composite, and Em, vm, Ef, Vr #1 the fiber is much stiffer than the matrix and therefore, the stiff are the elastic constants of the matrix and fiber Vf and Vm are the ness contribution in direction 2 should increase with an increase in volume fractions As shown in Figs. 8-11, the variations in the stiff- cross fiber angle. In addition, as the results show the contribution ness and compliance coefficients of composites considered with of stiffness in direction 1(Cull) for unidirectional (0 angle)com cross fiber angles are obviously functions of the constituent mate- posite is higher than in 2-direction(C2222)of 90 bidirectional. This rial properties of the composite as one set of fibers change their is obvious as the percentage of fiber in direction 1, in case of uni- orientations within the nd therefore contribute to the directional composite is 100%, whereas in case of 90 bidirectional l directions. For example, in directo rameters in different composite it is less than 100% Since the fiber will change direction 1(Cu111)for the com- only in the plane of directions 1 and 2, the coefficients in direction posite #1 as plotted in Fig 8 decreases with an increase in fiber 3 should not be affected much by the fiber angles. This has been cross angle, while the compliance(S1111)increases The compliance observed from the presented data in the tables 2, 4 and 6
composite then becomes completely anisotropic at all other angles. The compliance coefficients presented in Table 8 for a cross angle of 45 show this. In the Tables 3, 5 and 7 where the data for Young’s moduli and Poisson’s ratios were presented where E11, E22, E33, v12, v21, v13 are the elastic constants of the composite, and Em, vm, Ef, vf are the elastic constants of the matrix and fiber. Vf and Vm are the volume fractions. As shown in Figs. 8–11, the variations in the stiffness and compliance coefficients of composites considered with cross fiber angles are obviously functions of the constituent material properties of the composite as one set of fibers change their orientations within the matrix, and therefore contribute to the changes in the stiffness and compliance parameters in different directions. For example, stiffness in direction 1 (C1111) for the composite #1 as plotted in Fig. 8 decreases with an increase in fiber cross angle, while the compliance (S1111) increases. The compliance in direction 2 (S2222) decreases with an increase in angle, while the stiffness (C2222) increases. Such changes show the natural behavior of a composite material when it transforms from a unidirectional (0) to a completely bidirectional (90) composite. For composite #1 the fiber is much stiffer than the matrix and therefore, the stiffness contribution in direction 2 should increase with an increase in cross fiber angle. In addition, as the results show the contribution of stiffness in direction 1 (C1111) for unidirectional (0 angle) composite is higher than in 2-direction (C2222) of 90 bidirectional. This is obvious as the percentage of fiber in direction 1, in case of unidirectional composite is 100%, whereas in case of 90 bidirectional composite it is less than 100%. Since the fiber will change direction only in the plane of directions 1 and 2, the coefficients in direction 3 should not be affected much by the fiber angles. This has been observed from the presented data in the Tables 2, 4 and 6. 0.00 0.03 0.06 0.09 0.12 0 10 20 30 40 0 15 30 45 60 75 90 Compliance Coefficients (1/GPa) Stiffness Coefficients (GPa) Fiber Cross Angle (deg) C1 111 C2222 C3 333 S1111 S2222 S3333 Fig. 8. The variation of the compliance and the stiffness components of the composite #1 in longitudinal directions as a function of the fiber cross angle (u). 0.00 0.10 0.20 0.30 0.40 0 2 4 6 8 0 15 30 45 60 75 90 Compliance Coefficients (1/GPa) Stiffness Coefficients (GPa) Fiber Cross Angle (deg) C1212 C1313 C2323 S1212 S1313 S2323 Fig. 9. The variation of the compliance and the stiffness components of the composite #1 in shear directions as a function of the fiber cross angle (u). 1202 N. Abolfathi et al. / Computational Materials Science 43 (2008) 1193–1206