CENTRAL REFLECTION AND ITS USE IN FORMULATION OF UNIT CELLS FOR - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS CENTRAL REFLECTION AND ITS USE IN FORMULATION OF UNIT CELLS FOR MICROMECHANICAL FEA S. Li 1 *, Z. Zou 2 1 Dept of M3, Faculty of Engineering, University of Nottingham, Nottingham NG7 2RD, UK 2 School of MACE, University of Manchester, Manchester M60 1QD, UK * Corresponding author (shuguang.li@nottingham.ac.uk) Keywords : Unit cells ( UC ) ; Central reflection ( CR ) ; Symmetry; Micromechanics; Periodic boundary conditions 1 Introduction Mathematically, these three types of symmetries Symmetry as a geometric property is a well are mutually independent and collectively compre- understood topic. There are three generic types of hensive so that any other type of symmetry, such symmetries: translations, reflections and rotations. reflection about the central point, etc., can be All of them have been used to different extents in obtained as a simple combination of some of these structural analyses, with reflections being most generic symmetries and is hence not independent. attended to such an extent that they are sometimes The lack of independence of such derived perceived as the symmetry, as transpired through the symmetries has probably deterred users from fact that it is the only type of symmetries that has exploiting them. As a result, they have never been been incorporated in most commercial FE codes. To employed in structural analyses. implement other types, such as translations and A central reflection is a combination of a plane reflection and a 180 ° rotation. Taking advantage of rotations, the user will have to improvise using the facility of equation boundary conditions or multiple this when it is available in a unit cell, it will produce point constraints. a set of boundary conditions which is applicable to The three generic types of symmetries have also all loading cases as well any combination of been utilised extensively in formulating unit cells for macroscopic stresses, while halving the size of the micromechanical analyses of materials of regular unit cell. Neither a plane reflection nor a rotation microstructures [1-7]. Of them, translations result in alone delivers such a property. the commonly called periodic conditions. The benefits of using translational symmetries alone 2 Central reflection have been elaborated in reasonable length in Mechanical applications of geometric symmetries [2,3,4,6] that a single set of boundary conditions are complicated by the existence of two different natures, viz . symmetry and anti-symmetry, which are applies for any loading condition in terms of a single macroscopic stress component or a combination of related to applied loads, as well as the internal several of them. The existence of further sym- stresses and strains, in presence of geometric metries, such as reflections and rotations, can be symmetry. Loading conditions for micromechanical used to reduce the size of the unit cell, as has been analyses of unit cells are typically expressed in illustrated in analyses of UD composites [1,4] and terms of macroscopic stresses or strains. They more recently plain weave textile composites [5]. preserve symmetric nature under translational However, the use of reflectional and rotational symmetry transformations. This is responsible for symmetries tends to restrict the applicability in terms the fact that a single set of boundary conditions are of combined loading conditions. Also, different sets applicable for all loading conditions if only of boundary conditions have to be imposed when the translational symmetries are used. unit cell is under different loading conditions. While Reflectional and rotational symmetries only a single set of boundary conditions offers conve- preserve the senses of some stress and strain niences, reduced unit cell size is computationally components, typically those direct stresses, but not attractive, especially when dealing with complicated others, typically some shear stresses and strains, for unit cells, e.g. for textile composites where high which the concept of antisymmetry has to be demand on computing power often arises [6]. A resorted to. Involvement of any of these symmetries compromise has not been available. splits applied loads, as well as stresses and strains,

into two mutually exclusive categories, symmetric of boundary conditions which are applicable to all and antisymmetric. As a result, different boundary loading conditions and any of their combinations. σ z conditions will have to be prescribed under different n z loading conditions. This prohibits any combination τ yz τ xz of loads of different natures, undermining the use of z n σ y reflectional and rotational symmetries. Unit cell y τ xy x users have been left in a dilemma in which one can n σ x have either a full sized unit cell under a single set of Fig. 2 Stresses showing perfect symmetry under CR boundary conditions for all loading conditions or a unit cell with reduced size, which has to be analysed The mapping for displacements is using separate sets of boundary conditions under ( ) ( ) − − − → − ′ − ′ − ′ − (3) CR u u v v w w u u v v w w : , , , , 0 0 0 0 0 0 different loading conditions, stripping its capability ′ ′ ′ are the displacements at where ( , u v w and ( u v w , ) , , ) of dealing with combined loading conditions. P and P’ , respectively, and those at O . u v w A centrally reflectional symmetry (CR) is often ( , , ) 0 0 0 observed in popular objects. However, patterns and This should be dealt with properly and made use of shapes possessing this symmetry often show other correctly in order to obtain boundary conditions symmetries, such as reflections and rotations, rationally since boundary conditions for unit cells making the recognition of CR either redundant or have to be described in terms of displacements. unobvious. A simple but distinctive shape of CR is Consider the application of CR in structural a triclinic crystal, Fig. 1, which is a hexahedron of analysis. The triclinic crystal as shown in Fig. 1 has three parallel, unnecessarily orthogonal, pairs of been taken as a symbolic representation of CR faces and three independent side lengths. In this structure loaded in a symmetric manner under the geometry, CR is the only symmetry available. same symmetry, Fig. 3(a). Applying the symmetry, the structure can be analyses with only half of it, Fig. 3(b), if appropriate boundary conditions are c β P’ imposed on the shaded section plane. O γ a ≠ b ≠ c O b P α α ≠ β ≠ γ ≠ 90 ° c a F P’ Fig. 1 A triclinic crystal b /2 O F F P b The analytical description of CR can be given as a a (a) (b) mapping Fig. 3 Application of reflectional symmetry P ′ → . (1) CR P (a) A centrally symmetric structure (b) The symmetric : where P ( , , ) half of the structure x y z is an arbitrary point in the triclinic body as origin and P’ ( x’,y’,z’ ) as the image of P , or Since relative displacements (to point O ) are all ′ ′ ′ − − − = − − − − x x y y z z x x y y z z (2) ( , , ) ( , , 0 ) 0 0 0 0 0 centrally reflected to opposite directions according being the coordinates of the centre O . x y z ( , , ) to the symmetry transformation (5), one has 0 0 0 ′ + = ( ) u u u ′ All stresses and strains preserve their senses under − = − − 2 u u u u 0 0 0 or (4) ′ + = a CR, see Fig. 2, where CR would not cause any ( ) v v v ′ − = − − 2 v v v v 0 0 0 ′ change and the same applies to the outward normals + = ( ) w w w − = − ′ − 2 w w w w 0 0 0 to the cube, also shown in Fig. 2. With the outward ′ ′ ′ are displacements at P where u v w and u v w ( , , ) ( , , ) normals, surface traction can be obtained as a part of and P’ on the section surface as origin and image the periodic boundary conditions, although they are under the symmetry transformation. (4) above not required in conventional FE analyses [7]. CR deliver the desirable boundary conditions for the breaks the aforementioned dilemma straightaway, section surface, when P takes positions of all points i.e. the size of the unit cell having such symmetry on any half of the section plane, e.g. the half on the can now be halved without any penalty and the new left hand side of the dash-dot chain in Fig. 3(b). (4) unit cell of reduced size will come with a single set are based on the relative displacements to point O .