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Composite Structures Design, Mechanics, Analysis, Manufacturing, and Testing Manoj Kumar Buragohain This article was downloaded by: 10.3.98.104 On: 26 Sep 2021 Access details: subscription number Publisher: CRC Press Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: 5 Howick Place, London SW1P 1WG, UK Composite Structures Design, Mechanics, Analysis, Manufacturing, and Testing Manoj Kumar Buragohain Micromechanics of a Lamina Publication details https://www.routledgehandbooks.com/doi/10.1201/9781315268057-3 Manoj Kumar Buragohain Published online on: 20 Sep 2017 How to cite :- Manoj Kumar Buragohain. 20 Sep 2017, Micromechanics of a Lamina from: Composite Structures, Design, Mechanics, Analysis, Manufacturing, and Testing CRC Press Accessed on: 26 Sep 2021 https://www.routledgehandbooks.com/doi/10.1201/9781315268057-3 PLEASE SCROLL DOWN FOR DOCUMENT Full terms and conditions of use: https://www.routledgehandbooks.com/legal-notices/terms This Document PDF may be used for research, teaching and private study purposes. Any substantial or systematic reproductions, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The publisher shall not be liable for an loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. 3 Micromechanics of a Lamina 3.1 CHAPTER ROAD MAP A laminate is a laminated composite structural element, and laminate design is a cru- cial aspect in the overall design of a composite structure. As mentioned in Chapter 1, laminae are the building blocks in a composite structure; knowledge of lamina behav- ior is essential for the design of a composite structure and analysis of a lamina is the starting point. Figure 3.1 presents a schematic representation of the process of compos- ite laminate analysis (and design) at different levels. A lamina is a multiphase element and its behavior can be studied at two levels—micro level and macro level. For micro- mechanical analysis of a lamina, the necessary input data are obtained from the experi- mental study of its constituents, viz. reinforcements and matrix, and lamina behavior is estimated as functions of the constituent properties. The lamina characteristics are then used in the analysis of the lamina at the macro level and subsequent laminate design and analysis. Alternatively, the input data for the macro-level analysis of a lamina and subsequent laminate design and analysis can be directly obtained from an experimental study of the lamina. Thus, in the context of product design, the micromechanics of a lamina can be considered as an alternative to the experimental study of the lamina. In this chapter, we provide an introductory remark followed by a brief review of the basic micromechanics concepts. There are many micromechanics models in the litera- ture. Our focus is not a review of these models; instead, we dwell on the formulations of some mechanics of materials-based models for the evaluation of lamina thermoelastic parameters and briefly touch upon the elasticity-based models and semiempirical models. 3.2 PRINCIPAL NOMENCLATURE A Area of cross section of a representative volume element Ac, Af, Am Areas of cross section of composite, fibers, and matrix, respectively, in a representative volume element bc, bf, bm Widths of composite, fibers, and matrix, respectively, in a representa- tive volume element d Fiber diameter Ec Young’s modulus of isotropic composite E1c, E2c Young’s moduli in the longitudinal and transverse directions, respec- tively, of transversely isotropic composite Ef Young’s modulus of isotropic fibers E1f, E2f Young’s moduli in the longitudinal and transverse directions, respec- tively, of transversely isotropic fibers Em Young’s modulus of matrix Fc Total force on composite (representative volume element) Ff, Fm Forces shared by the fibers and matrix, respectively Gf Shear modulus of isotropic fibers Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 79 80 Composite Structures Analysis of composite structure Analysis of composite laminate Macromechanical analysis of lamina Micromechanical Experimental Or analysis of lamina study of lamina Characterization of constituents FIGURE 3.1 Schematic representation of composite laminate analysis process. G12f, G23f Shear moduli in the longitudinal and transverse planes, respectively, of transversely isotropic fibers Gm Shear modulus of matrix l, b, t Length, width, and thickness, respectively, of a representative volume element lc, lf, lm Lengths of composite, fibers, and matrix, respectively, in a represen- tative volume element s Fiber spacing tc, tf, tm Thicknesses of composite, fibers, and matrix, respectively, in a repre- sentative volume element Vf, Vm, Vv Fiber volume fraction, matrix volume fraction, and voids volume fraction, respectively (Vf )cri, (Vf )min Critical fiber volume fraction and minimum fiber volume fraction, respectively vc Total volume of composite vf, vm, vv Volumes of fibers, matrix, and voids, respectively Wf, Wm Mass fraction of fibers and mass fraction of matrix, respectively wc Total weight of composite wf, wm Mass of fibers and mass of matrix, respectively αc Coefficient of thermal expansion of isotropic composite α1c, α2c Longitudinal and transverse coefficients of thermal expansion, respectively, of transversely isotropic composite α1f, α2f Longitudinal and transverse coefficients of thermal expansion, respectively, of transversely isotropic fibers αm Coefficient of thermal expansion of matrix βc Coefficient of moisture expansion of isotropic composite β1c, β2c Longitudinal and transverse coefficients of moisture expansion, respectively, of transversely isotropic composite β1f, β2f Longitudinal and transverse coefficients of moisture expansion, respectively, of transversely isotropic fibers βm Coefficient of moisture expansion of matrix γ12c, γ23c Longitudinal (in a longitudinal plane) and transverse (in a transverse plane) shear strains, respectively, in composite Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 Micromechanics of a Lamina 81 (γc)ult Ultimate shear strain in isotropic composite (γ12c)ult, (γ23c)ult Ultimate longitudinal (in a longitudinal plane) and transverse (in a transverse plane) shear strains, respectively, in transversely isotropic composite γ12f, γ23f Longitudinal (in a longitudinal plane) and transverse (in a transverse plane) shear strains, respectively, in fibers (γf )ult Ultimate shear strain in isotropic fibers (γ12f)ult, (γ23f)ult Ultimate longitudinal (in a longitudinal plane) and transverse (in a transverse plane) shear strains, respectively, in transversely isotropic fibers γ12m, γ23m Longitudinal (in a longitudinal plane) and transverse (in a transverse plane) shear strain, respectively, in matrix (γm)ult Ultimate shear strain in matrix Δc, Δf, Δm Deformations in composite, fibers, and matrix, respectively ΔCc, ΔCf, ΔCm Changes in moisture content in composite, fibers, and matrix, respectively Δl Change in length of a representative volume element Δlc, Δlf, Δlm Changes in length of composite, fibers, and matrix, respectively, in a representative volume element ΔT Change in temperature T T εε12c , c Longitudinal and transverse tensile strains, respectively, in composite C C εε12c , c Longitudinal and transverse compressive strains, respectively, in composite T ()εc ult Ultimate tensile strain in isotropic composite T T ()εε12c ultc,( )ult Ultimate longitudinal and transverse tensile strains, respectively, in transversely isotropic composite C C ()εε12c ultc,( )ult Ultimate longitudinal and transverse compressive strains, respec- tively, in transversely isotropic composite T T εε12f , f Longitudinal and transverse tensile strains, respectively, in fibers C C εε12f , f Longitudinal and transverse compressive strains, respectively, in fibers T ()ε f ult Ultimate tensile strain in isotropic fibers T T ()εε12f ultf,( )ult Ultimate longitudinal and transverse tensile strains, respectively, in transversely isotropic fibers C C ()εε12f ultf,( )ult Ultimate longitudinal and transverse compressive strains, respec- tively, in transversely isotropic fibers T T εε12m , m Longitudinal and transverse tensile strains, respectively, in matrix C C εε12m , m Longitudinal and transverse compressive strains, respectively, in matrix T ()εm ult Ultimate tensile strain in matrix η Fiber packing factor (in Halpin–Tsai equations) νf Poisson’s ratio of isotropic fibers ν12f, ν23f Major Poisson’s ratios (in the longitudinal plane and transverse plane, respectively) of transversely isotropic fibers νm Poisson’s ratio of matrix ξ Reinforcing factor (in Halpin–Tsai equations) ρc, ρf, ρm Density of composite, fibers, and matrix, respectively T T σσ12c , c Longitudinal and transverse tensile stresses, respectively, in composite C C σσ12c , c Longitudinal and transverse compressive stresses, respectively, in composite T ()σc ult Ultimate tensile stress in isotropic composite (i.e., tensile strength of isotropic composite) Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 82 Composite Structures T T ()σσ12c ultc,( )ult Ultimate longitudinal and transverse tensile stresses, respectively, in transversely isotropic composite
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