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 sperectional fiber composite as a medical texture is a promising appli￾cation 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, vas￾cular prosthesis and heart valves are other examples [11]. During the last decade, fiber reinforced composite was intro￾duced as a new material for dentistry and orthodontic application. There has been growing interest in utilizing fiber reinforced com￾posites 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 bond￾ing. 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 evalu￾ate the bond strength and fracture pattern of fiber-reinforced com￾posite 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 compos￾ites are functions of the individual properties of the constitutive materials, their volume ratios, and the microstructural arrange￾ment. To obtain desired properties for composites, a microstruc￾tural analysis is required to determine the influence of parameters such as the arrangements of the fibers within the ma￾trix 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 repre￾sentative volume element (RVE) most often in the form of repeat￾ing unit cells (RUCs) of the composite should be studied and examined under various loading conditions to conclude and deter￾mine 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 character￾ization 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 bidi￾rectional fibrous composites is essential in accurate character￾ization, 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 the￾ory are also too approximate at many situations. Stiffness/compliance transformation rules and lamination the￾ory are limited to the situations when laminas’ homogenized properties are known at least along the principal material direc￾tions. Micromechanics characterization is needed to develop the stiffness/compliance of the lamina from the lamina’s micro￾structure and constituents’ materials in any direction. The micromechanics model presented in this paper is thus established based on the microstructure and properties of constit￾uents, 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 homoge￾nized continuum point at the macroscale. Other enforced types of boundary conditions (symmetry, homogenous boundary condi￾tions) can introduce some form of local impact and therefore can￾not 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, elasto￾plastic and nonlinear behavior are very difficult. Aboudi [17] devel￾oped 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 inelas￾tic 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 sur￾face-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 lam￾inates. 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 particu￾lar, Brinson and Lin [29] and Fisher and Brinson [30] used microm￾echanics 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 deter￾mine 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 load￾ings. 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