### Mechanics of Curved Composites (Solid Mechanics and Its Applications)

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Ciaran K. Earl H. Hornsen HS Tzou. Peter W. Esben Byskov. Grigore Gogu. Zengtao Chen. Natalia I. Reza Eslami. Sri Namachchivaya. Hornsen Tzou. Home Contact us Help Free delivery worldwide. Free delivery worldwide. Bestselling Series. Harry Potter. Popular Features. New Releases. Notify me. Description This book is the frrst to focus on mechanical aspects of fibrous and layered composite material with curved structure. By mechanical aspects we mean statics, vibration, stability loss, elastic and fracture problems.

By curved structures we mean that the reinforcing layers or fibres are not straight: they have some initial curvature, bending or distortion. This curvature may occur as a result of design, or as a consequence of some technological process.

## Design of Reinforcement in Nano- and Microcomposites

During the last two decades, we and our students have investigated problems relating to curved composites intensively. These investigations have allowed us to study stresses and strains in regions of a composite which are small compared to the curvature wavelength.

These new, accurate, techniques were developed in the framework of continuum theories for piecewise homogeneous bodies. We use the exact equations of elasticity or viscoelasticity for anisotropic bodies, and consider linear and non-linear problems in the framework of this continuum theory as well as in the framework of the piecewise homogeneous model.

For the latter the method of solution of related problems is proposed. We have focussed our attention on self-balanced stresses which arise from the curvature, but have provided sufficient information for the study of other effects. We assume that the reader is familiar with the theory of elasticity for anisotropic bodies, with partial differential equations and integral transformations, and also with the Finite Element Method. Other books in this series.

The results of calculations for a population of 50 individuals are shown in Figure In the calculations, it has been assumed that functions must always pass through a point with coordinates 4. Furthermore, the possible coordinate variations in the vertical direction were limited, i. As can be seen, in Figure 11 and Table 7 , there is a good agreement of theoretical and numerical solutions, both in the case of using genetic algorithms GA as well as a modified evolutionary strategy MEA.

The number of generations necessary to obtain optimal solutions is small—see Figure In this way, generated curves that do not meet this condition are not taken into account in the results presented in Figure Unidirectional composites are not reinforced with a single fiber but a bundle of fibers embedded in the matrix. Using the numerical homogenization see Section 2 and [ 31 ] , we can determine the mechanical properties of an elementary cell made of fiber and matrix.

Thus, it also affects the stiffness of the laminate. Equation 15 can be written in the following explicit form:. The condition in the form of Equation 16 will be written in the dimensionless form during the generation of key points. Therefore, additional restrictions on the derivative value at the ends of the curve were imposed. On the edge of the elementary cell, external exertion was applied in the form of a unit displacement acting along the x -axis. A circle was assumed as the original shape of the fiber bundle. The results of solving Equations 15 and 16 are shown in Figure In the numerical computations, the algorithm of the modified evolutionary strategy MEA was used.

Planar representative volume element—initial circular and optimal shapes of fibre bundles. The presented numerical solution draws new possibilities to optimize the mechanical properties of the material by forming a bundle of reinforcement fibers. It is possible from a technological point of view. However, attention should be paid to the issue of the interfacial surface, which was not included in the considerations. The functionally gradient materials FGM distribute the material functions throughout the material body to achieve the maximum mechanical properties.

FGMs are characterized as structures having different properties in the thickness z -direction in the form proposed by Qian et al. For FGM, the function V z is defined as:. Other definitions of the V z can be found, e. Other propositions consider the use of Equations 18 and 19 for degradation in a different direction than thickness.

First density grading in the width direction is as follows:. Second density grading in the length direction is as follows:. Next definition of density diagonal grading across the xy plane in two directions is as follows:. A model for characterizing the mechanical properties of functionally graded materials FGMs with the regular polygonal cross-section is also developed by introducing the power-law rule—see Ref. However, modelling of their mechanical behaviour still leads to many problems, particularly in the description of 1D or 2D structures, such as beams, plates or shells.

The broader discussion of those problems and solutions can be found in Muc et al. The present methodology is similar to the FGM description. In the current problem, the density grading in the length direction is presented. Bending of a simply supported beam having a variable fibre volume fraction distribution along the length. The objective of the optimisation is as follows: to find the minimal mass of the beam satisfying the equality constraint imposed on the displacement parameter and inequality upper and lower bounds constraint—on the fibre volume density fraction V f —the strict formulation is given by Banichuk et al.

The optimisation problem deals with the topology optimisation since we are looking for the optimal material distribution. The results are presented in Figure 15 and show a good correlation between numerical and analytical studies. Upper values red lines in Figure 15 were calculated for the lower value of the assumed beam deflection.

Comparison of fibre volume fraction distribution along the beam length: analytical a dotted line Banichuk et al. The present paper discussed the application of numerical homogenization and optimization in the studies of micro- and nanocomposites. The issue of boundary conditions, reinforcement shape and distribution, and form of representative volume element is presented taking into account also the topic of 2D curves optimization. The method of description representation of the 2D curve related to the number of design variables included in the optimization process was presented.

There was a good agreement of theoretical and numerical solutions, both in the case of using genetic algorithms GA as well as a modified evolutionary strategy MEA for the shape optimization of the single fibre. In shape optimization of fibre bundles, a circle was assumed as the original shape. In the numerical computations, the algorithm of modified evolutionary strategy MEA was used.

The composite beam reinforced by short fibres, loaded at the centre and having a variable density of fibres along the length, was considered, being an example of FGM. The optimization problem deals with topology optimization since we were looking for the optimal material distribution. The results showed a good correlation between numerical and analytical studies.

Some examples of numerical homogenization of various 2D and 3D RVE have been studied to present the influence of boundary conditions, form of RVE, shape and distribution of the reinforcement on the effective material properties. The comparison of numerical homogenization and micromechanical models showed that micromechanical models are appropriate in the case of the simple shapes of reinforcement.

Conceptualization, formal analysis, methodology, visualization, writing—original draft, M. National Center for Biotechnology Information , U. Journal List Materials Basel v. Materials Basel. Published online May 7. Author information Article notes Copyright and License information Disclaimer. Received Apr 18; Accepted May 2. This article has been cited by other articles in PMC. Abstract The application of numerical homogenization and optimization in the design of micro- and nanocomposite reinforcement is presented. Keywords: optimization, homogenization, microcomposites, nanocomposites, FEA, evolutionary algorithms, the shape of the reinforcement.

Introduction The group of non-homogeneous materials includes, e. In this paper, the following issues influencing the reinforcement design in homogenization problem are discussed: Boundary conditions of the representative volume element, Form of the representative volume element, Shape of the reinforcement, Distribution of the reinforcement.

Preliminary Remarks In homogenization, a heterogeneous body is replaced with a homogeneous one based on physical relationships between stresses and strains— Figure 1. Open in a separate window. Figure 1. Boundary Conditions The three classical boundary conditions applied in the analysis of RVE include: Linear displacement boundary condition Dirichlet condition , Constant traction boundary condition Neumann condition , Periodic boundary condition.

Figure 2. Table 1 Boundary displacements of the quarter of representative volume element. Form of the Representative Volume Element The distribution of reinforcement in micro- and nanocomposites is of high attention and has a significant influence on their mechanical behavior. Figure 3. Table 2 Material properties of carbon nanotube and epoxy resin used in finite element analysis.

### Bibliographic Information

Figure 4. Optimization Problems with the Use of 2D Curves In optimization problems, distributions of arbitrary design variables can be expressed through n-dimensional curves. Figure 5. Figure 6. Generation of Key Point Population The number of key points and the way they are arranged in a two-dimensional coordinate system equidistant or not are directly dependent on the type of optimization problem.

Figure 7. Depending on the values of prescribed angles, we distinguish two cases: a. Figure 8. The area of acceptable positions of base vectors inside a quadrangle. Shape of the Reinforcement 6. Figure 9. Table 5 Elastic properties of microcomposite constituents. One of the most well-known issues in this field is the following problem: Find a closed plane curve of a given perimeter L refers to the length of the boundary connecting two points A and B which encloses the maximal Area.

Figure Table 7 Comparison of the numerical results. Shape of Fibre Bundles Unidirectional composites are not reinforced with a single fiber but a bundle of fibers embedded in the matrix.

Distribution of the Reinforcement The functionally gradient materials FGM distribute the material functions throughout the material body to achieve the maximum mechanical properties. Conclusions The present paper discussed the application of numerical homogenization and optimization in the studies of micro- and nanocomposites. The conducted analyses and comparisons are summarized as follows: The method of description representation of the 2D curve related to the number of design variables included in the optimization process was presented. Author Contributions Conceptualization, formal analysis, methodology, visualization, writing—original draft, M.

Funding This research received no external funding. Conflicts of Interest The author declares no conflict of interest. References 1. Tsai S. Theory of Composites Design. Aboudi J. Micromechanics of Composite Materials. A Generalized Multiscale Analysis Approach. Elsevier; Amsterdam, The Netherlands: Nemat-Nasser S.

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Micromechanics: Overall Properties of Heterogeneous Materials. Voigt W. On the relation between the elasticity constants of isotropic bodies. Reuss A. Determination of the yield point of the polycrystals based on the yield conditions of single crystals. Qin Q. Springer; Berlin, Germany: Eshelby J. The determination of the elastic field of an ellipsoidal inclusion, and related problems.

Hill R. A self-consistent mechanics of composite materials. Christensen R. Mechanics of Composite Materials.

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Hashin Z. A variational approach to the theory of the elastic behaviour of multiphase materials.

Halpin J. Stiffness and expansion estimates for oriented short fiber composites. The Halpin-Tsai equations: A review. Mori T. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metal. Benveniste Y. Mura T. Micromechanics of Defects in Solids. Bensoussan A. Asymptotic Analysis for Periodic Structure. Suquet P. Elements of homogenization for inelastic solid mechanics. In: Sanchez-Palencia E. Homogenization Techniques for Composite Media.

A Unified Micromechanical Approach. Chen X. Square representative volume elements for evaluating the effective material properties of carbon nanotube-based composites. Muc A. Homogenization models for carbon nanotubes. Georgantzinos S. Shokrieh M. A review of the mechanical properties of isolated carbon nanotubes and carbon nanotube composites.

Design and identification methods of effective mechanical properties for carbon nanotubes. Key Eng. Numerical evaluation of effective material constants for CNT-based polymeric nanocomposites. Arash B. Malagu M. Diameter-dependent elastic properties of carbon nanotube-polymer composites. B Eng. Tserpes K. Alian A. Multiscale modeling of carbon nanotube epoxy composites. Gupta A. Analysis of mechanical properties of carbon nanotube reinforced polymer composites using multi-scale finite element modeling approach.

Deformations and tensile fracture of carbon nanotubes based on the numerical homogenization. Acta Phys. Radue M. Savvas D. Effect of waviness and orientation of carbon nanotubes on random apparent material properties and RVE size of CNT reinforced composites.

Choice of design variables in the stacking sequence optimization for laminated structures. Analytical discrete stacking sequence optimization of rectangular composite plates subjected to buckling and FPF constraints. Description of the resin curing process—formulation and optimization. Simplified optimal modeling of resin injection molding process. Qian L. Static and dynamic deformations of thick functionally graded elastic plate by using higher-ordershear and normal deformable plate theory and meshless local Petrov—Galerkin method.

Jin P.

### New & Forthcoming Titles

Flutter characteristic study of composite sandwich panel with functionally graded foam core. A model for functionally graded materials. Bendsoe M. Generating optimal topologies in structural design using a homogenization method. Methods Appl. Hassani B. A review of homogenization and topology optimization: I—Homogenization theory for media with periodic structure.

A review of homogenization and topology optimization: II—Analytical and numerical solution of homogenization equations. Chen Y. Multiobjective topology optimization for finite periodic structures. Groen J. Homogenization-based topology optimization for high-resolution manufacturable microstructures. Gao J. Topology optimization for multiscale design of porous composites with multi-domain microstructures. Sanchez-Palencia E. Non-Homogenous Media and Vibration Theory. Lecture Notes in Physics. Xia Z. On selection of repeated unit cell model and application of unified periodic boundary conditions in micro-mechanical analysis of composites.

Nguyen V. Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation. Sun C. Prediction of composite properties from a representative volume element.

## PUBLICATIONS - walterlacarbonara_eng

The elastic moduli of fiber-reinforced materials. Elastic properties of reinforced solids: Some theoretical principles. Jones R. Vasiliev V. Mechanics and Analysis of Composite Materials. Free vibrations of carbon nanotubes with defects. Nonlocal analysis of natural vibrations of carbon nanotubes. Buckling and free vibrations of nanoplates—Comparison of nonlocal strain and stress approaches. Vanin G. Micro-Mechanics of Composite Materials. Nauka Dumka; Kiev, Ukraine: Ramakrishnam C. Structural shape optimization using penalty functions.

Barthelemy B. Accuracy analysis of the semi-analytical method for shape sensitivity calculation. Lee M. Shape optimization in laminated composite plates. Pedersen P. Design for minimum stress concentration—Some practical aspect. In: Rozvany G. Structural Optimization. Kluwer; Dordrecht, The Netherlands: Shapes of orthotropic plates for minimum energy concentration. Genetic algorithms and finite element analysis in optimization of composite structures.

An evolution strategy in structural optimization problems for plates and shells. Evolutionary design of engineering constructions. Solids Struct. Kristiansen E. On the optimum shape of fillets in plates subjected to multiple in-plane loading cases. Methods Eng. Homogenization of random heterogeneous media with inclusions of arbitrary shape modeled by XFEM.

Fujii T. Damage Mechanics of Composite Materials. Mir; Moscow, Russia: In Russian [ Google Scholar ]. Cox H. The elasticity and strength of paper and other fibrous materials. Strzelecki T. Mechanics of Nonhomogeneous Media. Theory of Homogenization. DWE; Wroclaw, Poland: