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