Click here to load reader

View

0Download

0

Embed Size (px)

UntitledThis article was downloaded by: 10.3.98.104 On: 10 Jan 2022 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

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: 10 Jan 2022 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.

D ow

nl oa

de d

B y:

1 0.

3. 98

.1 04

A t:

05 :5

4 10

J an

2 02

2; F

or : 9

78 13

15 26

80 57

, c ha

pt er

3, 1

0. 12

01 /9

78 13

15 26

80 57

-3

79

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

3 Micromechanics of a Lamina

D ow

nl oa

de d

B y:

1 0.

3. 98

.1 04

A t:

05 :5

4 10

J an

2 02

2; F

or : 9

78 13

15 26

80 57

, c ha

pt er

3, 1

0. 12

01 /9

78 13

15 26

80 57

-3 80 Composite Structures

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

Analysis of composite structure

Analysis of composite laminate

Macromechanical analysis of lamina

Micromechanical analysis of lamina

Experimental study of laminaOr

D ow

nl oa

de d

B y:

1 0.

3. 98

.1 04

A t:

05 :5

4 10

J an

2 02

2; F

or : 9

78 13

15 26

80 57

, c ha

pt er

3, 1

0. 12

01 /9

78 13

15 26

80 57

(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 ε ε1 2c

T c

T, Longitudinal and transverse tensile strains, respectively, in composite ε ε1 2c

C c

( )εc T

ult Ultimate tensile strain in isotropic composite ( ) , ( )ε ε1 2c

T ult c

T ult Ultimate longitudinal and transverse tensile strains, respectively, in

transversely isotropic composite ( ) , ( )ε ε1 2c

C ult c

tively, in transversely isotropic composite ε ε1 2f

T f

T, Longitudinal and transverse tensile strains, respectively, in fibers ε ε1 2f

C f

( )ε f T

ult Ultimate tensile strain in isotropic fibers ( ) , ( )ε ε1 2f

T ult f

T ult Ultimate longitudinal and transverse tensile strains, respectively, in

transversely isotropic fibers ( ) , ( )ε ε1 2f

C ult f

tively, in transversely isotropic fibers ε ε1 2m

T m

T, Longitudinal and transverse tensile strains, respectively, in matrix ε ε1 2m

C m

( )εm T

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 σ σ1 2c

T c

T, Longitudinal and transverse tensile stresses, respectively, in composite σ σ1 2c

C c

( )σc T

ult Ultimate tensile stress in isotropic composite (i.e., tensile strength of isotropic composite)

D ow

nl oa

de d

B y:

1 0.

3. 98

.1 04

A t:

05 :5

4 10

J an

2 02

2; F

or : 9

78 13

15 26

80 57

, c ha

pt er

3, 1

0. 12

01 /9

78 13

15 26

80 57

( ) , ( )σ σ1 2c C

σ σ1 2f T

σ σ1 2f C

( )σ f T

ult Ultimate tensile stress in isotropic fibers (i.e., tensile strength of iso- tropic fibers)

( ) , ( )σ σ1 2f T

( ) , ( )σ σ1 2f C

σ σ1 2m T

m T, Longitudinal and transverse tensile stresses, respectively, in matrix

σ σ1 2m C

matrix ( ) , ( )σ σm

T ult m

C ult Ultimate tensile and compressive stresses, respectively, in matrix (i.e.,

tensile and compressive strengths of matrix) τ12c, τ23c Longitudinal (in a longitudinal plane) and transverse (in a transverse

plane) shear stresses, respectively, in composite (τc)ult Ultimate shear stress (i.e., shear strength) of isotropic composite (τ12c)ult, (τ23c)ult Ultimate longitudinal and transverse shear stresses, respectively, in

transversely isotropic composite (i.e., longitudinal and transverse shear strengths)

τ12f, τ23f Longitudinal (in a longitudinal plane) and transverse (in a transverse plane) shear stresses, respectively, in fibers

(τf)ult Ultimate shear stress (i.e., shear strength) of isotropic fibers (τ12f)ult, (τ23f)ult Ultimate longitudinal and transverse shear stresses, respectively, in trans-

versely isotropic fibers (i.e., longitudinal and transverse shear strength) τ12m, τ23m Longitudinal (in a longitudinal plane) and transverse (in a transverse

plane) shear stresses, respectively, in matrix (τm)ult Ultimate shear stress in matrix (i.e., shear strength of matrix)

3.3 INTRODUCTION A composite lamina is made up of two constituents—reinforcements and matrix. As we know, these constituents combine together and act in unison as a single entity. Micromechanics is the study in which the interaction of the reinforcements and the matrix is considered and their effect on the gross behavior of the lamina is determined. Toward this, we need to determine several thermoelastic parameters of the lamina in terms of constituent properties. These parameters include

Elastic moduli Strength parameters Coefficients of thermal expansion (CTEs) Coefficients of moisture expansion (CMEs)

Extensive work, as reflected by numerous research papers available in the literature, has been done in the field of micromechanics. The subject is also discussed at different

D ow

nl oa

de d

B y:

1 0.

3. 98

.1 04

A t:

05 :5

4 10

J an

2 02

2; F

or : 9

78 13

15 26

80 57

, c ha

pt er

3, 1

0. 12

01 /9

78 13

15 26

80 57

-3 83Micromechanics of a Lamina

levels of treatment in many texts on the mechanics of composites [1–5]. Micromechanics models have been of keen research interest and several approaches have been adopted to develop models for the prediction of various parameters, especially elastic moduli, of a unidirectional lamina. A detailed survey of various approaches is provided by Chamis and Sendeckyj [6]; these approaches are netting analysis, mechanics of materi- als, self-consistent models, bounding techniques based on variational principles, exact solutions, statistical methods, finite element methods, microstructure theories, and semiempirical models. The netting models and mechanics of materials-based models involve grossly simplifying assumptions. The rest of the approaches are based on the principles of elasticity and they, barring the semiempirical models, are typically asso- ciated with rigorous treatment and complex mathematical and graphical expressions. Thus, for the sake of convenience of discussion, the micromechanics models can be put into a simple classification as follows:

Netting models Mechanics of materials-based models Elasticity-based models Semiempirical models

Netting models are highly simplified models in which the bond between the fibers and the matrix is ignored for estimating the longitudinal stiffness and strength of a unidirectional lamina; it is assumed that longitudinal stiffness and strength are pro- vided completely by the fibers. On the other hand, transverse and shear stiffness and Poisson’s effect are assumed to be provided by the matrix. These models typically underestimate the properties of a lamina but due to their simplicity they are still used in the preliminary ply design of pressure vessels [7].

The mechanics of materials-based models too involve grossly simplifying assump- tions (see, for instance, References 8–10). Averaged stresses and strains are used in force and energy balance in a representative volume element (RVE) to derive the desired expressions for elastic parameters. Typically, the continuity of displacement across the interface between the constituents is maintained. Some of the common assumptions in micromechanics (see Section 3.4.1) are relaxed/modified suitably and a number of mechanics of materials-based models have been proposed in the past. Several of these models relate to different assumed geometrical array of fibers (square, rectangular, hexagonal, etc.), fiber alignment, inclusion of voids, etc.

Elasticity-based models involve more rigorous treatment of the lamina behavior (see, for instance, References 11–20). In an exact method, an elasticity problem within the general frame of assumptions (see Section 3.4.1) is formulated and solved by various techniques, including numerical methods such as the finite element method. A variation of the exact method is the self-consistent model. Variational principles are employed to obtain bounds on the elastic parameters. In the statistical methods, the restrictions of aligned fibers in regular array are relaxed and the elastic parameters are allowed to vary randomly with position. All these models, however, are somewhat complex and they have limited utility in the design of a product. Also, many variables that actu- ally influence the lamina elastic behavior are ignored, leading to unreliable estimates. In semiempirical models, the mathematical complexity is reduced and the effects of process-related variables are taken into account by incorporating empirical factors [21].

An exhaustive discussion of the models available in the literature is beyond the scope of this book; for in-depth reviews, interested readers can refer to References 6, 22, and 23 and the bibliographies provided therein. In this chapter, we shall attempt to provide an overall idea required in a product design environment. With this in mind, we shall discuss the mechanics of materials models in detail for all the parameters listed above.

D ow

nl oa

de d

B y:

1 0.

3. 98

.1 04

A t:

05 :5

4 10

J an

2 02

2; F

or : 9

78 13

15 26

80 57

, c ha

pt er

3, 1

0. 12

01 /9

78 13

15 26

80 57

-3 84 Composite Structures

A brief discussion is also provided on the elasticity approach and the semiempirical approach for the elastic moduli.

3.4 BASIC MICROMECHANICS

3.4.1 Assumptions and Restrictions

Micromechanics models are based on a number of simplifying assumptions and restric- tions in respect of lamina, its constituents, that is, fibers and matrix, and the interface. These assumptions and restrictions are as follows:

The lamina is (i) macroscopically homogeneous, (ii) macroscopically ortho- tropic, (iii) linearly elastic, and (iv) initially stress-free.

The fibers are (i) homogeneous, (ii) linearly elastic, (iii) isotropic, (iv) regularly spaced, (v) perfectly aligned, and (vi) void-free.

The matrix is (i) homogeneous, (ii) isotropic, (iii) linearly elastic, and (iv) void-free. The interface between fibers and matrix has (i) perfect bond, (ii) no voids, and

(iii) no interphase, that is, fiber–matrix interaction zone.

Some of the restrictions are not realistic and some of them are relaxed in the deriva- tions of various models. For example, glass fibers are isotropic, but carbon and aramid fibers are highly anisotropic. They can be considered as transversely isotropic and their elastic moduli and strengths are direction-dependent. As we shall see in the next section, the mechanics of materials-based models discussed here can accommodate anisotropic (transversely isotropic) fibers. Fibers are generally randomly spaced and their align- ment is not perfect. Similarly, the matrix can have voids and the lamina can have initial stresses. Also, an interphase is present at the interface between the fibers and the matrix.

3.4.2 Micromechanics Variables

The general procedure, irrespective of the micromechanics model used, is to express the desired parameter in terms of a number of basic micromechanics variables. These variables are as follows:

Elastic moduli of fibers and matrix Strengths of fibers and matrix Densities of fibers and matrix Volume fractions of fibers, matrix, and voids Mass fractions of fibers and matrix

3.4.2.1 Elastic Moduli and Strengths of Fibers and Matrix

The elastic moduli and strengths of fibers and matrix are determined experimentally. The number of these parameters to be determined experimentally for use in micromechan- ics would depend on the restriction in respect of behaviors of fibers and matrix. Certain fibers such as carbon are highly anisotropic and they can be considered as transversely isotropic. For these fibers, we need five stiffness parameters: E1f , E2f , G12f , ν12f , and ν23f . For isotropic fibers such as glass, the number of stiffness parameters reduces to three— Ef , Gf , and νf . On the other hand, all common matrix materials are isotropic for which we need the three stiffness parameters—Em , Gm , and νm . Further, under the restriction of homogeneousness, all of these parameters are uniform across the fibers or matrix.

3.4.2.2 Volume Fractions

As we know, a composite material is made up of primarily two constituents—fibers and matrix. However, during the manufacture of a composite laminate, deviations do

D ow

nl oa

de d

B y:

1 0.

3. 98

.1 04

A t:

05 :5

4 10

J an

2 02

2; F

or : 9

78 13

15 26

80 57

, c ha

pt er

3, 1

0. 12

01 /9

78 13

15 26

80 57

-3 85Micromechanics of a Lamina

occur and voids are introduced. Thus, the total volume of a composite material consists of three parts—fibers, matrix, and voids. Fiber volume fraction is defined as the ratio of the volume of fibers in the composite material to the total volume of composite. Similarly, matrix volume fraction is defined as the ratio of the volume of matrix to the total volume of composite, and voids volume fraction is defined as the ratio of the vol- ume of voids to the total volume of composite. Thus,

V

v

v

c

= = =, , and

(3.1)

where Vf fiber volume fraction Vm matrix volume fraction Vv voids volume fraction vf volume of fibers vm volume of matrix vv volume of voids vc total volume of composite material

It is clear that

V V Vf m v+ + =1 (3.3)

For an ideal composite material, vv = Vv = 0 and we get

V Vf m+ =1 (3.4)

We shall see in the subsequent sections that fiber volume fraction is a key param- eter that greatly influences lamina properties such as longitudinal modulus and major Poisson’s ratio. It is useful to know the theoretical maximum fiber volume fraction of a lamina. In a composite material, fibers are packed in a random fashion. However, with a view to determining the maximum theoretical fiber volume fraction, as shown in Figure 3.2, let us consider two regular arrays of fibers—square array and triangular array. Fiber volume fractions can be expressed as

For square array, V

d(b)…

Composite Structures Design, Mechanics, Analysis, Manufacturing, and Testing Manoj Kumar Buragohain

Micromechanics of a Lamina

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: 10 Jan 2022 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.

D ow

nl oa

de d

B y:

1 0.

3. 98

.1 04

A t:

05 :5

4 10

J an

2 02

2; F

or : 9

78 13

15 26

80 57

, c ha

pt er

3, 1

0. 12

01 /9

78 13

15 26

80 57

-3

79

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

3 Micromechanics of a Lamina

D ow

nl oa

de d

B y:

1 0.

3. 98

.1 04

A t:

05 :5

4 10

J an

2 02

2; F

or : 9

78 13

15 26

80 57

, c ha

pt er

3, 1

0. 12

01 /9

78 13

15 26

80 57

-3 80 Composite Structures

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

Analysis of composite structure

Analysis of composite laminate

Macromechanical analysis of lamina

Micromechanical analysis of lamina

Experimental study of laminaOr

D ow

nl oa

de d

B y:

1 0.

3. 98

.1 04

A t:

05 :5

4 10

J an

2 02

2; F

or : 9

78 13

15 26

80 57

, c ha

pt er

3, 1

0. 12

01 /9

78 13

15 26

80 57

(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 ε ε1 2c

T c

T, Longitudinal and transverse tensile strains, respectively, in composite ε ε1 2c

C c

( )εc T

ult Ultimate tensile strain in isotropic composite ( ) , ( )ε ε1 2c

T ult c

T ult Ultimate longitudinal and transverse tensile strains, respectively, in

transversely isotropic composite ( ) , ( )ε ε1 2c

C ult c

tively, in transversely isotropic composite ε ε1 2f

T f

T, Longitudinal and transverse tensile strains, respectively, in fibers ε ε1 2f

C f

( )ε f T

ult Ultimate tensile strain in isotropic fibers ( ) , ( )ε ε1 2f

T ult f

T ult Ultimate longitudinal and transverse tensile strains, respectively, in

transversely isotropic fibers ( ) , ( )ε ε1 2f

C ult f

tively, in transversely isotropic fibers ε ε1 2m

T m

T, Longitudinal and transverse tensile strains, respectively, in matrix ε ε1 2m

C m

( )εm T

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 σ σ1 2c

T c

T, Longitudinal and transverse tensile stresses, respectively, in composite σ σ1 2c

C c

( )σc T

ult Ultimate tensile stress in isotropic composite (i.e., tensile strength of isotropic composite)

D ow

nl oa

de d

B y:

1 0.

3. 98

.1 04

A t:

05 :5

4 10

J an

2 02

2; F

or : 9

78 13

15 26

80 57

, c ha

pt er

3, 1

0. 12

01 /9

78 13

15 26

80 57

( ) , ( )σ σ1 2c C

σ σ1 2f T

σ σ1 2f C

( )σ f T

ult Ultimate tensile stress in isotropic fibers (i.e., tensile strength of iso- tropic fibers)

( ) , ( )σ σ1 2f T

( ) , ( )σ σ1 2f C

σ σ1 2m T

m T, Longitudinal and transverse tensile stresses, respectively, in matrix

σ σ1 2m C

matrix ( ) , ( )σ σm

T ult m

C ult Ultimate tensile and compressive stresses, respectively, in matrix (i.e.,

tensile and compressive strengths of matrix) τ12c, τ23c Longitudinal (in a longitudinal plane) and transverse (in a transverse

plane) shear stresses, respectively, in composite (τc)ult Ultimate shear stress (i.e., shear strength) of isotropic composite (τ12c)ult, (τ23c)ult Ultimate longitudinal and transverse shear stresses, respectively, in

transversely isotropic composite (i.e., longitudinal and transverse shear strengths)

τ12f, τ23f Longitudinal (in a longitudinal plane) and transverse (in a transverse plane) shear stresses, respectively, in fibers

(τf)ult Ultimate shear stress (i.e., shear strength) of isotropic fibers (τ12f)ult, (τ23f)ult Ultimate longitudinal and transverse shear stresses, respectively, in trans-

versely isotropic fibers (i.e., longitudinal and transverse shear strength) τ12m, τ23m Longitudinal (in a longitudinal plane) and transverse (in a transverse

plane) shear stresses, respectively, in matrix (τm)ult Ultimate shear stress in matrix (i.e., shear strength of matrix)

3.3 INTRODUCTION A composite lamina is made up of two constituents—reinforcements and matrix. As we know, these constituents combine together and act in unison as a single entity. Micromechanics is the study in which the interaction of the reinforcements and the matrix is considered and their effect on the gross behavior of the lamina is determined. Toward this, we need to determine several thermoelastic parameters of the lamina in terms of constituent properties. These parameters include

Elastic moduli Strength parameters Coefficients of thermal expansion (CTEs) Coefficients of moisture expansion (CMEs)

Extensive work, as reflected by numerous research papers available in the literature, has been done in the field of micromechanics. The subject is also discussed at different

D ow

nl oa

de d

B y:

1 0.

3. 98

.1 04

A t:

05 :5

4 10

J an

2 02

2; F

or : 9

78 13

15 26

80 57

, c ha

pt er

3, 1

0. 12

01 /9

78 13

15 26

80 57

-3 83Micromechanics of a Lamina

levels of treatment in many texts on the mechanics of composites [1–5]. Micromechanics models have been of keen research interest and several approaches have been adopted to develop models for the prediction of various parameters, especially elastic moduli, of a unidirectional lamina. A detailed survey of various approaches is provided by Chamis and Sendeckyj [6]; these approaches are netting analysis, mechanics of materi- als, self-consistent models, bounding techniques based on variational principles, exact solutions, statistical methods, finite element methods, microstructure theories, and semiempirical models. The netting models and mechanics of materials-based models involve grossly simplifying assumptions. The rest of the approaches are based on the principles of elasticity and they, barring the semiempirical models, are typically asso- ciated with rigorous treatment and complex mathematical and graphical expressions. Thus, for the sake of convenience of discussion, the micromechanics models can be put into a simple classification as follows:

Netting models Mechanics of materials-based models Elasticity-based models Semiempirical models

Netting models are highly simplified models in which the bond between the fibers and the matrix is ignored for estimating the longitudinal stiffness and strength of a unidirectional lamina; it is assumed that longitudinal stiffness and strength are pro- vided completely by the fibers. On the other hand, transverse and shear stiffness and Poisson’s effect are assumed to be provided by the matrix. These models typically underestimate the properties of a lamina but due to their simplicity they are still used in the preliminary ply design of pressure vessels [7].

The mechanics of materials-based models too involve grossly simplifying assump- tions (see, for instance, References 8–10). Averaged stresses and strains are used in force and energy balance in a representative volume element (RVE) to derive the desired expressions for elastic parameters. Typically, the continuity of displacement across the interface between the constituents is maintained. Some of the common assumptions in micromechanics (see Section 3.4.1) are relaxed/modified suitably and a number of mechanics of materials-based models have been proposed in the past. Several of these models relate to different assumed geometrical array of fibers (square, rectangular, hexagonal, etc.), fiber alignment, inclusion of voids, etc.

Elasticity-based models involve more rigorous treatment of the lamina behavior (see, for instance, References 11–20). In an exact method, an elasticity problem within the general frame of assumptions (see Section 3.4.1) is formulated and solved by various techniques, including numerical methods such as the finite element method. A variation of the exact method is the self-consistent model. Variational principles are employed to obtain bounds on the elastic parameters. In the statistical methods, the restrictions of aligned fibers in regular array are relaxed and the elastic parameters are allowed to vary randomly with position. All these models, however, are somewhat complex and they have limited utility in the design of a product. Also, many variables that actu- ally influence the lamina elastic behavior are ignored, leading to unreliable estimates. In semiempirical models, the mathematical complexity is reduced and the effects of process-related variables are taken into account by incorporating empirical factors [21].

An exhaustive discussion of the models available in the literature is beyond the scope of this book; for in-depth reviews, interested readers can refer to References 6, 22, and 23 and the bibliographies provided therein. In this chapter, we shall attempt to provide an overall idea required in a product design environment. With this in mind, we shall discuss the mechanics of materials models in detail for all the parameters listed above.

D ow

nl oa

de d

B y:

1 0.

3. 98

.1 04

A t:

05 :5

4 10

J an

2 02

2; F

or : 9

78 13

15 26

80 57

, c ha

pt er

3, 1

0. 12

01 /9

78 13

15 26

80 57

-3 84 Composite Structures

A brief discussion is also provided on the elasticity approach and the semiempirical approach for the elastic moduli.

3.4 BASIC MICROMECHANICS

3.4.1 Assumptions and Restrictions

Micromechanics models are based on a number of simplifying assumptions and restric- tions in respect of lamina, its constituents, that is, fibers and matrix, and the interface. These assumptions and restrictions are as follows:

The lamina is (i) macroscopically homogeneous, (ii) macroscopically ortho- tropic, (iii) linearly elastic, and (iv) initially stress-free.

The fibers are (i) homogeneous, (ii) linearly elastic, (iii) isotropic, (iv) regularly spaced, (v) perfectly aligned, and (vi) void-free.

The matrix is (i) homogeneous, (ii) isotropic, (iii) linearly elastic, and (iv) void-free. The interface between fibers and matrix has (i) perfect bond, (ii) no voids, and

(iii) no interphase, that is, fiber–matrix interaction zone.

Some of the restrictions are not realistic and some of them are relaxed in the deriva- tions of various models. For example, glass fibers are isotropic, but carbon and aramid fibers are highly anisotropic. They can be considered as transversely isotropic and their elastic moduli and strengths are direction-dependent. As we shall see in the next section, the mechanics of materials-based models discussed here can accommodate anisotropic (transversely isotropic) fibers. Fibers are generally randomly spaced and their align- ment is not perfect. Similarly, the matrix can have voids and the lamina can have initial stresses. Also, an interphase is present at the interface between the fibers and the matrix.

3.4.2 Micromechanics Variables

The general procedure, irrespective of the micromechanics model used, is to express the desired parameter in terms of a number of basic micromechanics variables. These variables are as follows:

Elastic moduli of fibers and matrix Strengths of fibers and matrix Densities of fibers and matrix Volume fractions of fibers, matrix, and voids Mass fractions of fibers and matrix

3.4.2.1 Elastic Moduli and Strengths of Fibers and Matrix

The elastic moduli and strengths of fibers and matrix are determined experimentally. The number of these parameters to be determined experimentally for use in micromechan- ics would depend on the restriction in respect of behaviors of fibers and matrix. Certain fibers such as carbon are highly anisotropic and they can be considered as transversely isotropic. For these fibers, we need five stiffness parameters: E1f , E2f , G12f , ν12f , and ν23f . For isotropic fibers such as glass, the number of stiffness parameters reduces to three— Ef , Gf , and νf . On the other hand, all common matrix materials are isotropic for which we need the three stiffness parameters—Em , Gm , and νm . Further, under the restriction of homogeneousness, all of these parameters are uniform across the fibers or matrix.

3.4.2.2 Volume Fractions

As we know, a composite material is made up of primarily two constituents—fibers and matrix. However, during the manufacture of a composite laminate, deviations do

D ow

nl oa

de d

B y:

1 0.

3. 98

.1 04

A t:

05 :5

4 10

J an

2 02

2; F

or : 9

78 13

15 26

80 57

, c ha

pt er

3, 1

0. 12

01 /9

78 13

15 26

80 57

-3 85Micromechanics of a Lamina

occur and voids are introduced. Thus, the total volume of a composite material consists of three parts—fibers, matrix, and voids. Fiber volume fraction is defined as the ratio of the volume of fibers in the composite material to the total volume of composite. Similarly, matrix volume fraction is defined as the ratio of the volume of matrix to the total volume of composite, and voids volume fraction is defined as the ratio of the vol- ume of voids to the total volume of composite. Thus,

V

v

v

c

= = =, , and

(3.1)

where Vf fiber volume fraction Vm matrix volume fraction Vv voids volume fraction vf volume of fibers vm volume of matrix vv volume of voids vc total volume of composite material

It is clear that

V V Vf m v+ + =1 (3.3)

For an ideal composite material, vv = Vv = 0 and we get

V Vf m+ =1 (3.4)

We shall see in the subsequent sections that fiber volume fraction is a key param- eter that greatly influences lamina properties such as longitudinal modulus and major Poisson’s ratio. It is useful to know the theoretical maximum fiber volume fraction of a lamina. In a composite material, fibers are packed in a random fashion. However, with a view to determining the maximum theoretical fiber volume fraction, as shown in Figure 3.2, let us consider two regular arrays of fibers—square array and triangular array. Fiber volume fractions can be expressed as

For square array, V

d(b)…