- Email: [email protected]
Computer-aided design and modeling of composite unit cells
Composites Science and Technology 61 (2001) 289±299
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Computer-aided design and modeling of composite unit cells W. Sun *, F. Lin, X. Hu Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104, USA Received 12 June 2000; received in revised form 29 August 2000; accepted 10 October 2000
Abstract A novel modeling approach for designing and constructing CAD-based composite unit cells is presented. The modeling technique is developed based on a reasoning Boolean operation algorithm which consists of merging and extracting operation to construct composite unit cells at the heterogeneous material level. Owing to its CAD-based nature, the unit cell models can be readily implemented with advanced CAD/CAE/CAM systems for integrated design, simulation, and manufacturing of advanced composites. A seamless integration between the CAD model and ®nite element analysis, and results of stress and deformation in composite unit cells and in the ®ber and matrix is demonstrated. The proposed modeling technique is able to capture the designed composite geometry, reinforced ®ber architecture, and constituent material heterogeneity. Sample unit cells representing 2D woven fabric, 2D basket weave, 3D tri-axial braided, and 3D uni-directional composite are also presented. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: C. Finite element analysis (FEA); Composites; Unit Cell; CAD; Heterogeneous modeling
1. Introduction Advancements in design and manufacturing using integrated computer-aided design/computer-aided engineering/computer-aided manufacturing (CAD/CAE/CAM) have brought dramatic changes in today's industry [1±3]. A major factor in these changes is the utilization of modeling-based design, particularly the use of solid modeling in computer-aided mechanical design systems. Solid modeling has been used extensively in design representation, analysis, and manufacturing. However, these powerful modeling techniques and advanced CAD/CAE/ CAM systems are rarely used to design and manufacture advanced composites. Factors contributing to this situation include the inherent complexities of geometrical con®guration and diculties in the fabrication of composite structures, and more important, the lack of a capable CAD modeling tool that can be utilized to model and to characterize composite structural complexity, material heterogeneity, and manufacturing diculties. For example, designing textile composite structures requires knowledge of ®ber and matrix material properties, representation of internal ®ber architecture, and an eective model that encompasses all above for composite structural representation, manipulation, and * Corresponding author. Fax: +1-215-895-1478. E-mail address: [email protected] (W. Sun).
optimization. Commonly available solid modelers, such as B-rep, CSG, or hybrid B-rep/CSG based modeling architectures, contain only geometrical and topological information without material attribute [4], and therefore are not suitable to model heterogeneous composites. Research on modeling heterogeneous objects has been reported only recently. Cavalcanti, Carvalho and Martha [5] used intersecting multi-loop simple patches and geometric complex completeness for spatial decomposition of heterogeneous objects. Sun and Lau [6,7] proposed a framework for developing a knowledge-enriched CAD model for solid freeform realization of heterogeneous material structures. Kumar and Dutta [8,9] presented a rmset modeling approach to construct heterogeneous objects and functional degraded materials, and de®ned layered manufacturing processing algorithm. However, techniques for CAD-based design and modeling for composite materials and structures have not yet been seen. This paper presents a reasoning Boolean operationbased CAD modeling technique for designing and constructing heterogeneous composite unit cell structures. The unit cells form part of a multi-volume modeling database containing the designed composite geometry, reinforced ®ber architecture, and constituent material heterogeneity. In addition, the CAD-based models can retain distinctive constituent phase properties. Because of this, the unit cell model can be implemented directly with advanced CAD/CAE/CAM systems for integrated
0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(00)00218-9
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design, simulation, and manufacturing of advanced composites. The presentation of this paper is organized as follows: . Section 2 presents a description of the principle of reasoning Boolean operation based CAD modeling technique and introduces a reasoning Boolean operation algorithm containing reasoning merging and reasoning extracting operations. . Section 3 describes an approach to modeling heterogeneous composite unit cells. Examples of 2D woven fabric, 2D basket weave, 3D tri-axial braided, and 3D uni-directional composite unit cells are given in this section. . Section 4 demonstrates a seamless integration between a CAD-based unit cell model and ®nite element analysis. This process allows prediction of
unit cell properties, constituent mechanical properties, stress, and deformation. . Section 5 includes a conclusion and discussion. 2. Principle of reasoning Boolean operation based heterogeneous CAD modeling Our modeling database consists of multi-volume topological elements (solids) and reasoning Boolean operations. To account for material heterogeneity, material identi®cation is associated within each topological element (volume) as an attribute in the design database. Reasoning Boolean operation algorithm is developed to construct multi-volume composite unit cell models. This algorithm contains the reasoning merging operation and the reasoning extracting operation. As shown in Fig. 1a, conventional Boolean operation only
Fig. 1. Comparison of conventional and reasoning Boolean operations.
Fig. 2. Reasoning extracting operation for subtract and union.
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Fig. 3. Reasoning extracting operation for intersect and complex_union.
Fig. 4. Procedure of constructing heterogeneous composite unit cell model.
Fig. 5. Heterogeneous composite unit cells.
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manipulates the geometry of topological primitives. As the result of the operation, a single-volume object is constructed by the operation of union, subtraction, or intersection. While in the proposed reasoning Boolean
operation, the algorithm manipulates multi-attributed topological elements through two steps: (1) the merging operation; and (2) the extracting operation. The reasoning merging operation identi®es the material attributes assigned to the topological elements and compares them to decide whether they are identical and need to be merged. The reasoning extracting operation follows the Boolean merging operation to generate the needed intersecting surfaces, edges, and/or the splitting of volumes for merged elements. Material identi®cation is consolidated for topological elements with the same material attribute and is retained within the new extracting volumes if they are dierent. For a heterogeneous composite object C consisting of two-volume topological elements A and B, with material identi®cation MA and MB, as shown in Fig. 1b, the reasoning Boolean operation algorithm can be described as follows: If MA=MB, Then (single volume object with identical material identi®cation) C=A(MA)+B(MA) OR C=A(MA) ÿB(MA) OR C=A(MA)\B(MA)
(conventional union) (conventional subtract) (conventional intersect)
Otherwise (multi-volume object with dierent material identi®cation)
Fig. 6. Processing ¯owchart for modeling and analysis of composite unit cell.
C=A(MA)ÿB(MB) (MA dominant subtract) or C=B(MB)ÿA(MA) (MB dominant subtract) OR C=A(MA)+{B(MB)ÿA(MA)} (MA dominant union) or C=B(MB)+{A(MA)ÿB(MB)} (MB dominant union)
Fig. 7. Example of unit cell CAD model and FE meshing of 2D woven composite.
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OR C=A(MA) \ B(MB) (MA dominant intersect) or C=B(MB) \ A(MA) (MB dominant intersect) OR C={B(MB)ÿA(MA)}+{A(MA) \ B(MB)} (MA dominant complex_union) or C={A(MA)ÿB(MB)}+{B(MB) \ A(MA)} (MB dominant complex_union) end if
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Unlike conventional Boolean operations, the reasoning Boolean operation needs to be executed according to the dominant material information. In the example shown in Fig. 1b, the dominant material information is de®ned either MA dominant or MB dominant union, subtract, and intersect according to the design intent. It needs to be pointed out that two new set operations are introduced in the reasoning Boolean operation algorithm: (1) MA dominant complex_union; and (2) MB
Fig. 8. (a) Boundary condition and loading in 2D woven composite unit cell; (b): Maximum principal stress in unit cell under in-plane tension (unit: MPa); (c): Maximum principal stress in unit cell ®bers and matrix under in-plane tension (unit: MPa) (continued on next page).
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Fig. 8. (continued).
dominant complex_union. The operation of complex union ``assembles'' the results of the intersect operation and the subtract operation to form a heterogeneous assembly. In the database of the assembled heterogeneous model, each of the solid elements retains its original material identi®cation. Results of the reasoning extracting operation for manipulating two-volume objects with dierent material identi®cations are presented in Figs. 2 (for subtract and union) and 3 (for intersection and complex_union). 3. Heterogeneous modeling of composite unit cells The reasoning Boolean operation algorithm is applied to construct CAD-based heterogeneous composite unit
cell model. Advanced CAD software, Pro/Engineer 2000 (Pro/E) by Parametric Technology Corporation [9], was used in the model realization. The procedure of constructing the ``assembled'' heterogeneous composite unit cell model from the constituent ®ber and matrix topological elements is described in Fig. 4. As shown in the ®gure, the matrix (MA)-dominant subtract operation cuts the ®ber element (B) from the matrix (A) to form a matrix-dominant subtraction (matrix with cavity), while the ®ber (MB)-dominant intersect operation produces a geometrically ®tted ®ber elements (®tted reinforcement). The ®ber-dominant Boolean complex_union operation assembles the results of the matrix-dominant subtraction and the ®ber-dominant intersection to form a heterogeneous composite unit cell model.
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One of the advantages in the proposed modeling technique is that the geometry and orientation of ®ber architecture can be designed and built in the CAD model so that the constructed model characterizes both material and geometry properties of the unit cell constituents. In addition, through computer rendering techniques, the constructed unit cell model can be used as a powerful tool to allow design engineer to exam the composite internal structure, pattern of the reinforcement, designed architecture and the ®ber orientation in
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a CAD-based virtual unit cell model. Such a tool can also greatly facilitate the communication between the design and fabrication of composite structures. The proposed modeling technique and its ability of visualization is extremely useful in the design of complicated composite structures, for example, three-dimensional textile composites with complex ®ber architectures. Examples of using the reasoning Boolean operation algorithm to construct composite unit cells representing 2D woven fabric, 2D basket weave, 3D tri-axial braided,
Fig. 9. (a). Unit cell under in-plane shear; (b) unit cell deformation (top-view); (c) maximum principal stress in unit cell ®bers and matrix under inplane shear (unit: MPa) (continued on next page).
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Fig. 9. (continued).
and 3D uni-directional ®ber-reinforced composite structures are presented in Fig. 5. 4. Finite-element analysis for stress and deformation of composite unit cells As indicated in Section 3, the heterogeneous composite unit cell model constructed through the reasoning Boolean operation algorithm retains distinctive constituent geometrical and material information in the unit cell modeling database. This means that the unit cell database can directly provide distinctive geometric
information and material information for each constituent in the composite unit cell and generate the ®nite-element model at the constituent level. So the integration of the unit cell CAD design with ®nite-element analysis and prediction of the unit cell properties at both unit cell structural level and at the constituent level becomes possible. This is a unique feature of the proposed CAD-based heterogeneous composite unit cell model in comparing with conventional methods which are limited to calculate the composite eective properties from the smeared reinforced constitute and matrix material. This section introduces a seamless integration between the CAD model and ®nite element analysis
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Fig. 10. Maximum principal stress in unit cell ®bers and matrix (3D tri-axial composite unit cell under uni-axial tension, unit: MPa).
through using advanced Pro/Engineer and Pro/ MECHANICA (Pro/M) software. In the examples shown below, the material heterogeneity and 3D ®ber architecture were de®ned using the heterogeneous modeling technique described in the previous section. The ®nite-element analysis model inherits the geometric and material information de®ned in the unit cell CAD model. The processing ¯owchart for modeling and analysis of heterogeneous composite unit cell structure is brie¯y described in Fig. 6.
After generating the CAD ``assembly'' of the heterogeneous composite unit cell by Pro/E assembly module, The Pro/E CAD model was transferred seamlessly into a Pro/M model for ®nite-element analysis. Finite-element meshing was separately generated through Pro/ MECHANICA's AutoGEM function for ®ber and matrix and the generated meshes automatically match at the ®ber and matrix interfaces. To match meshes at the bi-material interface is a basic requirement in order to successfully conduct ®nite-element analysis for heterogeneous material
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Fig. 11. Unit cell-based integrated design and analysis for composite structures.
structure. Often times, conventional ®nite element approach fails to generate ®nite-element meshes to match at the material interface, particularly in the analysis of composites with complex 3D ®ber architectures. Therefore, the proposed modeling technique will provide a powerful tool in the numerical analysis of composite structures. In the ®nite-element analyses presented in this paper, the mechanical properties of the ®ber and the matrix are assumed to be linear elastic with either isotropic or orthotropic constitutive properties. The interface between the ®ber and the matrix is assumed to be ``perfectly'' bonded throughout the model, which means that the traction and displacements are assumed to be continuous across the interface. The P-version high order interpolating polynomials was used in the Pro/M for solution convergence in the ®nite element analysis. In our case studies, six ploop passes were performed before all elements satis®ed the convergence condition. Fig. 7 presents a rendered image of heterogeneous ®ber and matrix assembled 2D woven fabric composite unit cell solid model and a display of the ®nite-element meshes obtained from AutoGEM. AutoGEM in Pro/M produced 5896 tetrahedral elements for ®bers and matrix in the ®nite element model. Results of ®nite-element analyses for the unit cell under in-plane tension loading are presented in a series of ®gures. In these, Fig. 8a presents unit cell in-plane loading and constraint condition. Fig. 8b displays a contour of the maximum
principal stress distributed in the 2D woven fabric composite unit cell structure. Fig. 8c presents the maximum principal stress distribution in the unit cell ®bers and in the matrix, respectively. The results obtained from the ®nite-element analyses can be used to predict unit cell mechanical properties and 3D eective material constants. Also, the results of stress and deformation predicted at the composite ®ber and matrix level, as shown in Fig. 8c, can be used for detail micromechanics analyses of the strength and failure of ®bers, matrix, unit cell structure, and interfacial mechanism of heterogeneous composites. In a similar approach, we used ®nite-element analysis to predict stress and deformation of a 2D woven-fabric composite unit cell under in-plane shear loading (Fig. 9a). Due to a limited space, only the result of the unit cell deformation is presented in Fig. 9b. In this ®gure, the deformed and undeformed unit cell structures were compared. Examples of using ®nite-element analysis to predict the stress and deformation for a 3D braided composite unit cell are shown in Fig. 10. Again, due to a limited space, only results of the maximum principal stress in unit cell ®bers and in the matrix are presented. A diagonal section was cut for unit cell matrix and the stress distribution contour is displayed in order to exam the stress distribution and stress concentration inside of the matrix.
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5. Conclusion and discussions This paper presents a new approach for constructing CAD-based heterogeneous composite unit cell model and to demonstrate the applicability of computer-aided technologies in designing and manufacturing of composite materials. In the modeling construction, a reasoning Boolean operation algorithm is developed to manipulate ®ber and matrix topological elements and to construct heterogeneous composite unit cells. Examples of using the developed technique to construct unit cell models representing 2D woven fabric, 2D basket weave, 3D triaxial braided, and 3D uni-directional ®ber-reinforced composite structures are given. The implementation of the developed unit cell model with available CAD/CAE/ CAM software for ®nite-element analysis are demonstrated and the numerical calculation for stress and deformation in unit cells and in the constituent ®bers and matrix are also presented. We would like to point out that the purpose of the ®nite-element analysis presented in current paper is to demonstrate the ability to calculate stress and deformation in unit cell structure and in the individual ®bers or matrix through the seamless integration of the CADbased composite unit cell model (Pro/E) and ®nite element analysis (Pro/M), rather than to predict the accurate composite unit cell properties. Therefore, we did not compare our numerical results with any real composite unit cell property data. However, we believe that the proposed unit cell model is capable of performing such predictions and will be particularly useful in design and simulation of composites with complex ®ber architecture, such as 3D textile composite structures [10]. Currently, no eective tool for design and analysis of such composites has been seen. The future work includes developing a unit cell library which contains various designed ®ber geometric, threedimensional ®ber architectures, and/or processing con®gurations of composite structures, such as dierent textile ®ber geometric pattern shown in Fig. 11. In the database of the unit cell library, the composite design intent and performance requirement can be built into the unit cell CAD model through the section of materials and designed reinforcement pattern. Those information
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can be further extracted and used to compare with the results of ®nite element analysis and for the fabrication of composite structures. For example, the automatic generation of ®nite element meshing in a heterogeneous structure is becoming possible from the developed unit cell CAD models, and the results of ®nite element analyses can be further used to guide the selection of unit cells from the unit cell library or to alter ®ber orientations in order to achieve an optimal composite design con®guration or to meet the performance criteria. Such a concept of the unit cell model-based composite design is shown in Fig. 11.
References [1] Sun W, Lau A. Design for manufacturability and concurrent engineering in integrated product development process. In: Billatos SB, Zhang HC, editors. Proceeding of Concurrent Product Design and Environmentally Conscious Manufacturing, the ASME Winter Annual Meeting, 16±21 November, 1997, Dallas, TX. p. 113±24. [2] Jo HH, Parsaei HR, Sullivan WG. Principal of concurrent engineering. In: Parsaei HR, Sullivan, WG. editors. Concurrent engineeringÐcontemporary issues and modern design tools. Chapman and Hall, 1993. p. 3±23. [3] Bedworth, DD, Henderson, MR, and Wolfe, PM. Computerintegrated design and manufacturing McGraw-Hill, Inc., 1991. [4] Cavalcanti PR, Carvalho PCP, Martha LF. Non-manifold modeling: an approach based on spatial subdivision. Computer-Aided Design 1997;29(3):209±20. [5] Sun W, Lau A. A knowledge-enriched CAD modeling and solid free-form realization for heterogeneous material structures. In: Proceeding of The Seventh International Conference on Rapid Prototyping, 31 March ± 3 April, 1997, San Francisco, CA. p. 79±87. [6] Sun W. Multi-volume CAD modeling for heterogeneous object design and fabrication. Journal of Computer Science and Technology, 15, (1), 27-:36 [7] Kumar V, Dutta D. An approach to modeling and representation of hetrogeneous objects. J Mechanical Design 1998;120:659- 67. [8] Kumar V. Solid modeling and algorithm for heterogeneous objects. Ph.D. dissertation, Department of Mechanical Engineering, The University of Michigan, 1999. [9] Pro/Engineer 2000. Waltham, (MA): Parametric Technology Corporation 2000. [10] Sun W, Dwivedi A. Heterogeneous CAD modeling based ®nite element analysis for eective properties of 3D textile composites in preparation.
www.elsevier.com/locate/compscitech
Computer-aided design and modeling of composite unit cells W. Sun *, F. Lin, X. Hu Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104, USA Received 12 June 2000; received in revised form 29 August 2000; accepted 10 October 2000
Abstract A novel modeling approach for designing and constructing CAD-based composite unit cells is presented. The modeling technique is developed based on a reasoning Boolean operation algorithm which consists of merging and extracting operation to construct composite unit cells at the heterogeneous material level. Owing to its CAD-based nature, the unit cell models can be readily implemented with advanced CAD/CAE/CAM systems for integrated design, simulation, and manufacturing of advanced composites. A seamless integration between the CAD model and ®nite element analysis, and results of stress and deformation in composite unit cells and in the ®ber and matrix is demonstrated. The proposed modeling technique is able to capture the designed composite geometry, reinforced ®ber architecture, and constituent material heterogeneity. Sample unit cells representing 2D woven fabric, 2D basket weave, 3D tri-axial braided, and 3D uni-directional composite are also presented. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: C. Finite element analysis (FEA); Composites; Unit Cell; CAD; Heterogeneous modeling
1. Introduction Advancements in design and manufacturing using integrated computer-aided design/computer-aided engineering/computer-aided manufacturing (CAD/CAE/CAM) have brought dramatic changes in today's industry [1±3]. A major factor in these changes is the utilization of modeling-based design, particularly the use of solid modeling in computer-aided mechanical design systems. Solid modeling has been used extensively in design representation, analysis, and manufacturing. However, these powerful modeling techniques and advanced CAD/CAE/ CAM systems are rarely used to design and manufacture advanced composites. Factors contributing to this situation include the inherent complexities of geometrical con®guration and diculties in the fabrication of composite structures, and more important, the lack of a capable CAD modeling tool that can be utilized to model and to characterize composite structural complexity, material heterogeneity, and manufacturing diculties. For example, designing textile composite structures requires knowledge of ®ber and matrix material properties, representation of internal ®ber architecture, and an eective model that encompasses all above for composite structural representation, manipulation, and * Corresponding author. Fax: +1-215-895-1478. E-mail address: [email protected] (W. Sun).
optimization. Commonly available solid modelers, such as B-rep, CSG, or hybrid B-rep/CSG based modeling architectures, contain only geometrical and topological information without material attribute [4], and therefore are not suitable to model heterogeneous composites. Research on modeling heterogeneous objects has been reported only recently. Cavalcanti, Carvalho and Martha [5] used intersecting multi-loop simple patches and geometric complex completeness for spatial decomposition of heterogeneous objects. Sun and Lau [6,7] proposed a framework for developing a knowledge-enriched CAD model for solid freeform realization of heterogeneous material structures. Kumar and Dutta [8,9] presented a rmset modeling approach to construct heterogeneous objects and functional degraded materials, and de®ned layered manufacturing processing algorithm. However, techniques for CAD-based design and modeling for composite materials and structures have not yet been seen. This paper presents a reasoning Boolean operationbased CAD modeling technique for designing and constructing heterogeneous composite unit cell structures. The unit cells form part of a multi-volume modeling database containing the designed composite geometry, reinforced ®ber architecture, and constituent material heterogeneity. In addition, the CAD-based models can retain distinctive constituent phase properties. Because of this, the unit cell model can be implemented directly with advanced CAD/CAE/CAM systems for integrated
0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(00)00218-9
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design, simulation, and manufacturing of advanced composites. The presentation of this paper is organized as follows: . Section 2 presents a description of the principle of reasoning Boolean operation based CAD modeling technique and introduces a reasoning Boolean operation algorithm containing reasoning merging and reasoning extracting operations. . Section 3 describes an approach to modeling heterogeneous composite unit cells. Examples of 2D woven fabric, 2D basket weave, 3D tri-axial braided, and 3D uni-directional composite unit cells are given in this section. . Section 4 demonstrates a seamless integration between a CAD-based unit cell model and ®nite element analysis. This process allows prediction of
unit cell properties, constituent mechanical properties, stress, and deformation. . Section 5 includes a conclusion and discussion. 2. Principle of reasoning Boolean operation based heterogeneous CAD modeling Our modeling database consists of multi-volume topological elements (solids) and reasoning Boolean operations. To account for material heterogeneity, material identi®cation is associated within each topological element (volume) as an attribute in the design database. Reasoning Boolean operation algorithm is developed to construct multi-volume composite unit cell models. This algorithm contains the reasoning merging operation and the reasoning extracting operation. As shown in Fig. 1a, conventional Boolean operation only
Fig. 1. Comparison of conventional and reasoning Boolean operations.
Fig. 2. Reasoning extracting operation for subtract and union.
W. Sun et al. / Composites Science and Technology 61 (2001) 289±299
Fig. 3. Reasoning extracting operation for intersect and complex_union.
Fig. 4. Procedure of constructing heterogeneous composite unit cell model.
Fig. 5. Heterogeneous composite unit cells.
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manipulates the geometry of topological primitives. As the result of the operation, a single-volume object is constructed by the operation of union, subtraction, or intersection. While in the proposed reasoning Boolean
operation, the algorithm manipulates multi-attributed topological elements through two steps: (1) the merging operation; and (2) the extracting operation. The reasoning merging operation identi®es the material attributes assigned to the topological elements and compares them to decide whether they are identical and need to be merged. The reasoning extracting operation follows the Boolean merging operation to generate the needed intersecting surfaces, edges, and/or the splitting of volumes for merged elements. Material identi®cation is consolidated for topological elements with the same material attribute and is retained within the new extracting volumes if they are dierent. For a heterogeneous composite object C consisting of two-volume topological elements A and B, with material identi®cation MA and MB, as shown in Fig. 1b, the reasoning Boolean operation algorithm can be described as follows: If MA=MB, Then (single volume object with identical material identi®cation) C=A(MA)+B(MA) OR C=A(MA) ÿB(MA) OR C=A(MA)\B(MA)
(conventional union) (conventional subtract) (conventional intersect)
Otherwise (multi-volume object with dierent material identi®cation)
Fig. 6. Processing ¯owchart for modeling and analysis of composite unit cell.
C=A(MA)ÿB(MB) (MA dominant subtract) or C=B(MB)ÿA(MA) (MB dominant subtract) OR C=A(MA)+{B(MB)ÿA(MA)} (MA dominant union) or C=B(MB)+{A(MA)ÿB(MB)} (MB dominant union)
Fig. 7. Example of unit cell CAD model and FE meshing of 2D woven composite.
W. Sun et al. / Composites Science and Technology 61 (2001) 289±299
OR C=A(MA) \ B(MB) (MA dominant intersect) or C=B(MB) \ A(MA) (MB dominant intersect) OR C={B(MB)ÿA(MA)}+{A(MA) \ B(MB)} (MA dominant complex_union) or C={A(MA)ÿB(MB)}+{B(MB) \ A(MA)} (MB dominant complex_union) end if
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Unlike conventional Boolean operations, the reasoning Boolean operation needs to be executed according to the dominant material information. In the example shown in Fig. 1b, the dominant material information is de®ned either MA dominant or MB dominant union, subtract, and intersect according to the design intent. It needs to be pointed out that two new set operations are introduced in the reasoning Boolean operation algorithm: (1) MA dominant complex_union; and (2) MB
Fig. 8. (a) Boundary condition and loading in 2D woven composite unit cell; (b): Maximum principal stress in unit cell under in-plane tension (unit: MPa); (c): Maximum principal stress in unit cell ®bers and matrix under in-plane tension (unit: MPa) (continued on next page).
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Fig. 8. (continued).
dominant complex_union. The operation of complex union ``assembles'' the results of the intersect operation and the subtract operation to form a heterogeneous assembly. In the database of the assembled heterogeneous model, each of the solid elements retains its original material identi®cation. Results of the reasoning extracting operation for manipulating two-volume objects with dierent material identi®cations are presented in Figs. 2 (for subtract and union) and 3 (for intersection and complex_union). 3. Heterogeneous modeling of composite unit cells The reasoning Boolean operation algorithm is applied to construct CAD-based heterogeneous composite unit
cell model. Advanced CAD software, Pro/Engineer 2000 (Pro/E) by Parametric Technology Corporation [9], was used in the model realization. The procedure of constructing the ``assembled'' heterogeneous composite unit cell model from the constituent ®ber and matrix topological elements is described in Fig. 4. As shown in the ®gure, the matrix (MA)-dominant subtract operation cuts the ®ber element (B) from the matrix (A) to form a matrix-dominant subtraction (matrix with cavity), while the ®ber (MB)-dominant intersect operation produces a geometrically ®tted ®ber elements (®tted reinforcement). The ®ber-dominant Boolean complex_union operation assembles the results of the matrix-dominant subtraction and the ®ber-dominant intersection to form a heterogeneous composite unit cell model.
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One of the advantages in the proposed modeling technique is that the geometry and orientation of ®ber architecture can be designed and built in the CAD model so that the constructed model characterizes both material and geometry properties of the unit cell constituents. In addition, through computer rendering techniques, the constructed unit cell model can be used as a powerful tool to allow design engineer to exam the composite internal structure, pattern of the reinforcement, designed architecture and the ®ber orientation in
295
a CAD-based virtual unit cell model. Such a tool can also greatly facilitate the communication between the design and fabrication of composite structures. The proposed modeling technique and its ability of visualization is extremely useful in the design of complicated composite structures, for example, three-dimensional textile composites with complex ®ber architectures. Examples of using the reasoning Boolean operation algorithm to construct composite unit cells representing 2D woven fabric, 2D basket weave, 3D tri-axial braided,
Fig. 9. (a). Unit cell under in-plane shear; (b) unit cell deformation (top-view); (c) maximum principal stress in unit cell ®bers and matrix under inplane shear (unit: MPa) (continued on next page).
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Fig. 9. (continued).
and 3D uni-directional ®ber-reinforced composite structures are presented in Fig. 5. 4. Finite-element analysis for stress and deformation of composite unit cells As indicated in Section 3, the heterogeneous composite unit cell model constructed through the reasoning Boolean operation algorithm retains distinctive constituent geometrical and material information in the unit cell modeling database. This means that the unit cell database can directly provide distinctive geometric
information and material information for each constituent in the composite unit cell and generate the ®nite-element model at the constituent level. So the integration of the unit cell CAD design with ®nite-element analysis and prediction of the unit cell properties at both unit cell structural level and at the constituent level becomes possible. This is a unique feature of the proposed CAD-based heterogeneous composite unit cell model in comparing with conventional methods which are limited to calculate the composite eective properties from the smeared reinforced constitute and matrix material. This section introduces a seamless integration between the CAD model and ®nite element analysis
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Fig. 10. Maximum principal stress in unit cell ®bers and matrix (3D tri-axial composite unit cell under uni-axial tension, unit: MPa).
through using advanced Pro/Engineer and Pro/ MECHANICA (Pro/M) software. In the examples shown below, the material heterogeneity and 3D ®ber architecture were de®ned using the heterogeneous modeling technique described in the previous section. The ®nite-element analysis model inherits the geometric and material information de®ned in the unit cell CAD model. The processing ¯owchart for modeling and analysis of heterogeneous composite unit cell structure is brie¯y described in Fig. 6.
After generating the CAD ``assembly'' of the heterogeneous composite unit cell by Pro/E assembly module, The Pro/E CAD model was transferred seamlessly into a Pro/M model for ®nite-element analysis. Finite-element meshing was separately generated through Pro/ MECHANICA's AutoGEM function for ®ber and matrix and the generated meshes automatically match at the ®ber and matrix interfaces. To match meshes at the bi-material interface is a basic requirement in order to successfully conduct ®nite-element analysis for heterogeneous material
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Fig. 11. Unit cell-based integrated design and analysis for composite structures.
structure. Often times, conventional ®nite element approach fails to generate ®nite-element meshes to match at the material interface, particularly in the analysis of composites with complex 3D ®ber architectures. Therefore, the proposed modeling technique will provide a powerful tool in the numerical analysis of composite structures. In the ®nite-element analyses presented in this paper, the mechanical properties of the ®ber and the matrix are assumed to be linear elastic with either isotropic or orthotropic constitutive properties. The interface between the ®ber and the matrix is assumed to be ``perfectly'' bonded throughout the model, which means that the traction and displacements are assumed to be continuous across the interface. The P-version high order interpolating polynomials was used in the Pro/M for solution convergence in the ®nite element analysis. In our case studies, six ploop passes were performed before all elements satis®ed the convergence condition. Fig. 7 presents a rendered image of heterogeneous ®ber and matrix assembled 2D woven fabric composite unit cell solid model and a display of the ®nite-element meshes obtained from AutoGEM. AutoGEM in Pro/M produced 5896 tetrahedral elements for ®bers and matrix in the ®nite element model. Results of ®nite-element analyses for the unit cell under in-plane tension loading are presented in a series of ®gures. In these, Fig. 8a presents unit cell in-plane loading and constraint condition. Fig. 8b displays a contour of the maximum
principal stress distributed in the 2D woven fabric composite unit cell structure. Fig. 8c presents the maximum principal stress distribution in the unit cell ®bers and in the matrix, respectively. The results obtained from the ®nite-element analyses can be used to predict unit cell mechanical properties and 3D eective material constants. Also, the results of stress and deformation predicted at the composite ®ber and matrix level, as shown in Fig. 8c, can be used for detail micromechanics analyses of the strength and failure of ®bers, matrix, unit cell structure, and interfacial mechanism of heterogeneous composites. In a similar approach, we used ®nite-element analysis to predict stress and deformation of a 2D woven-fabric composite unit cell under in-plane shear loading (Fig. 9a). Due to a limited space, only the result of the unit cell deformation is presented in Fig. 9b. In this ®gure, the deformed and undeformed unit cell structures were compared. Examples of using ®nite-element analysis to predict the stress and deformation for a 3D braided composite unit cell are shown in Fig. 10. Again, due to a limited space, only results of the maximum principal stress in unit cell ®bers and in the matrix are presented. A diagonal section was cut for unit cell matrix and the stress distribution contour is displayed in order to exam the stress distribution and stress concentration inside of the matrix.
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5. Conclusion and discussions This paper presents a new approach for constructing CAD-based heterogeneous composite unit cell model and to demonstrate the applicability of computer-aided technologies in designing and manufacturing of composite materials. In the modeling construction, a reasoning Boolean operation algorithm is developed to manipulate ®ber and matrix topological elements and to construct heterogeneous composite unit cells. Examples of using the developed technique to construct unit cell models representing 2D woven fabric, 2D basket weave, 3D triaxial braided, and 3D uni-directional ®ber-reinforced composite structures are given. The implementation of the developed unit cell model with available CAD/CAE/ CAM software for ®nite-element analysis are demonstrated and the numerical calculation for stress and deformation in unit cells and in the constituent ®bers and matrix are also presented. We would like to point out that the purpose of the ®nite-element analysis presented in current paper is to demonstrate the ability to calculate stress and deformation in unit cell structure and in the individual ®bers or matrix through the seamless integration of the CADbased composite unit cell model (Pro/E) and ®nite element analysis (Pro/M), rather than to predict the accurate composite unit cell properties. Therefore, we did not compare our numerical results with any real composite unit cell property data. However, we believe that the proposed unit cell model is capable of performing such predictions and will be particularly useful in design and simulation of composites with complex ®ber architecture, such as 3D textile composite structures [10]. Currently, no eective tool for design and analysis of such composites has been seen. The future work includes developing a unit cell library which contains various designed ®ber geometric, threedimensional ®ber architectures, and/or processing con®gurations of composite structures, such as dierent textile ®ber geometric pattern shown in Fig. 11. In the database of the unit cell library, the composite design intent and performance requirement can be built into the unit cell CAD model through the section of materials and designed reinforcement pattern. Those information
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can be further extracted and used to compare with the results of ®nite element analysis and for the fabrication of composite structures. For example, the automatic generation of ®nite element meshing in a heterogeneous structure is becoming possible from the developed unit cell CAD models, and the results of ®nite element analyses can be further used to guide the selection of unit cells from the unit cell library or to alter ®ber orientations in order to achieve an optimal composite design con®guration or to meet the performance criteria. Such a concept of the unit cell model-based composite design is shown in Fig. 11.
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