� 18 TH I�TER�ATIO�AL CO�FERE�CE O� COMPOSITE MATERIALS A MULTI"SCALE MODEL FOR THE COMPUTER"AIDED DESIG� OF POLYMER COMPOSITES Namin Jeong*, David W. Rosen 1 School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA * Corresponding author ( specialnamin@gatech.edu ) Keywords �������������������������������������������������������������������������������� ������ where � is a scaling (dilation) factor and � is a 1. � Introduction translation factor. In the geometric modeling domain, the wavelet transforms were used to In engineering design, geometry and material can be describe planar curves with multiple resolutions. separately specified at the traditional macroscale. Part and material microstructure boundaries can be However, at the micro� and meso�scales, material viewed as surface singularities that are discon� compositions become important in functional tinuous in one direction while continuous in the realization, such as in composites and functionally other two directions in 3D space. Therefore, we graded materials. A novel CAD system is under propose new surfacelet basis functions for multiscale development that supports multiscale geometry and modeling [3]. Particularly, a 3D ridgelet (type of materials modeling which enables concurrent surfacelet) that represents plane singularities is product�material design. We proposed a new defined as [4] multiscale geometric and materials modeling method β α + � cos cos that uses an implicit representation based on = − 1/2 − 1 � ( ) � � � � wavelets and their extension to efficiently capture � � , , , α β β α + β − � cos sin � sin � internal and boundary information. This new (2) approach enables integration of structure�property where is the location in the domain relationships for materials design. We call our in the Euclidean space, is a wavelet modeling approach dual representation or dual�Rep [1]. In this paper, the surfacelet transform is defined, function, is a surface function so that which consists of the Radon and wavelet transforms, implicitly defines a surface, with in order to develop structure�property relationships. factor � and the shape parameter vector � ∈ R m We demonstrate the methods with an example determining the location and shape of surface polymer nanocomposite material and illustrate singularities, respectively, and and structure�property model integration. � ∈ [� � /2, � /2] are angular parameters corresponding to rotations. 2. � Geometric Modeling We propose the dual�Rep model that uses both Our objective is to develop a geometric model that wavelet and surfacelet basis functions in order to can represent both part macroscale geometry and model external part shapes as well as internal material microstructure; i.e., a multiscale geometry microstructural geometry boundaries. The approach for computer�aided design of composite materials. to generating dual�Rep models of microstructure is Wavelets are the most common representation for to recognize microstructure features from stacks of multi�resolution modeling in the domain of 2D 2D micro�scale images. The Radon transform is an shape representation. Similar to Fourier analysis, effective method for representing line singularities wavelet analysis represents and approximates signals in images [3]. The Radon transform was developed (or functions). The functional space for wavelet to reconstruct images from CT scans [5], which analysis is decomposed based on a scaling function consist of sets of parallel scans where the source and � ( � ) and a wavelet function � ( � ) with one�dimensional sensor rotate around the target. We use this variable � for multi�resolution analysis [2]: transform to fit surfacelet models to microstructures. ( ) ( ) ( ) (1) = − 1/2 − 1 − � � � � � � � � � ,
Then, by applying a wavelet transform to the results of the Radon transform, an image representation is produced that is potentially sparse and enables many image processing techniques to be applied. Mathematically, the Radon transform in a domain > is the integral along the plane (represented as the dash line in 2D), which is perpendicular to a line at angle α , as illustrated in Fig. 5. The plane and the line intersect at a point which has the radial distance � from the origin. Varying � results in a vector of integral values, � α ( � ) in 2D and � α,β ( � ) in 3D: ( ) ( ) ( ) ∫∫ α � = δ α + α − � � � � � � � ��� � ��� ���� (3) Fig 2. SEM images of nanofiber micro�structure (nano� where δ is the Dirac delta function. The simplest fibers shown at tips of arrows). surfacelet is the ridgelet transform The strain�stress relationship for a fiber in a polymer ( ) ( ) Ψ = � α β � � ψ � (4) matrix microstructure can be derived readily from � � � � � α β � � � � basic composite materials models for a single fiber In general, our generic surfacelet transform is the 1D in a polymer matrix [7]: wavelet transform of the surface integrals. σ � � � 0 11 12 1 (5) ε = σ = σ S � � 0 12 22 2 γ 0 0 2 � � 66 12 � where the � �� are four independent constants in the α elastic compliance tensor of a unidirectional laminate in its local frame ( � � = elastic modulus of matrix, � �� = elastic modulus of fiber in direction � , � > = shear moduli, ν = Poisson’s ratio, � = volume fraction of fiber): Fig. 1. Geometric interpretations of parameters in ( ) surfacelet transform. − − 1 1 � � ( ) ( ) − 1 = − + � � � 2 � � 1 � = � 11 � � � 1 � 22 � 3. � Polymer �anocomposite � ( ) ( ) − − ν − + ν 1 1 � � 1 (6) We are interested in a nanocomposite system � � � 12 = � � � 2 � = � � ( ) 12 − + 66 � 1 � � consisting of calcium�phosphorus (CaP) nanofibers � � � 2 � � The local compliance tensors can be transformed to in a polyhydroxybutyrate (PHB) matrix, (CaP/PHB). global coordinates using the fourth�rank coordinate Figure 2 is the scanning electron microscopy image of CaP/PHB which was characterized for dispersion transformation law � ���� � = ! �� ! �" ! �� ! �� � � and distribution, thermal properties, and �"�� (7) thermomechanical properties [6]. We use two where each ! �� is a standard rotation matrix. methods to develop structure�property relationships. 4. � Simple Fiber"Reinforced Composite Example The multiscale microstructure model of CaP/PHB was represented using the surfacelet transform. We The surfacelet representation and its hierarchical use the surfacelet model to recognize microstructural modeling capabilities are illustrated with a simple features, such as fibers, so that effective mechanical example of a fiber�reinforced composite material. properties can be computed. By using the surfacelet Fig. 3 shows the sample microstructure, with vertical transform, we can develop structure�property� and horizontal fibers spaced 100 � m apart. We geometry relationships. assume a typical carbon�epoxy composite material with property values of � � = 2.94 GPa, � � 1 = 234.6 GPa, � � 2 = 13.8 GPa. The sample’s elastic modulus
Recommend
More recommend
