Volume Weight Fractions Density Void Fraction - USF

Chapter 3 Micromechanical Analysis of a LaminaVolume Fractions, Weight Fractions,Density, and Void ContentDr. Autar KawDepartment of Mechanical EngineeringUniversity of South Florida, Tampa, FL 33620Courtesy of the TextbookMechanics of Composite Materials by Kaw

vfV f , andvcV f V m 1vf v m vcvmV m vc

Wf wf, andwcW f W m 1w f wm wcwmW m wc

wc ρ c vc ,w f ρ f vf ,wm ρm vm , andwc w f wm

ρfV f , andWf ρcρmW m Vmρc

Wf W m ρfρmρfV f V mρmVf1ρf(1 - V m) V mρmVm

ρ c vc ρ f vf ρ m v mρc ρ fvfv ρm mvcvcρ c ρ f V f ρ mV m1ρc Wfρf Wmρm

Example 3.1A Glass/Epoxy lamina consists of a 70% fiber volume fraction. Useproperties of glass and epoxy from Tables 3.1 and 3.2, respectively todetermine thea) density of laminab) mass fractions of the glass and epoxyc) volume of composite lamina if the mass of the lamina is 4 kg.d) volume and mass of glass and epoxy in part (c).

Table 3.1 Typical Properties of Fibers (SI system of units)PropertyUnitsGraphiteGlassAramidAxial modulusGPa23085124Transverse modulusGPa22858Axial Poisson's ratio---0.300.200.36Transverse Poisson's ratio---0.350.200.37Axial shear modulusGPa2235.423Axial coefficient of thermal expansionμm/m/0C-1.35-5.0Transverse coefficient of thermal expansionμm/m/0C7.054.1Axial tensile strengthMPa206715501379Axial compressive strengthMPa19991550276Transverse tensile strengthMPa7715507Transverse compressive strengthMPa4215507Shear strengthMPa363521Specific gravity---1.82.51.4

Table 3.2 Typical Properties of Matrices (SI system of units)PropertyUnitsEpoxyAluminumPolyamideAxial modulusGPa3.4713.5Transverse modulusGPa3.4713.5Axial Poisson's ratio---0.30.300.35Transverse Poisson's ratio---0.30.300.35Axial shear modulusGPa1.308271.3Coefficient of thermal expansionμm/m/0C632390Coefficient of moisture expansionm/m/kg/kg0.330.000.33Axial tensile strengthMPa7227654Axial compressive strengthMPa102276108Transverse tensile strengthMPa7227654Transverse compressive strengthMPa102276108Shear strengthMPa3413854Specific gravity---1.22.71.2

Table 3.3 Typical Properties of Fibers (USCS system of units)PropertyUnitsGraphiteGlassAramidAxial ModulusMsi33.3512.3317.98Transverse modulusMsi3.1912.331.16Axial Poisson's ratio---0.300.200.36Transverse Poisson's ratio---0.350.200.37Axial shear modulusMsi3.195.1360.435Axial coefficient of thermal expansionμin/in/0F-0.72222.778-2.778Transverse coefficient of thermal expansionμin/in/0F3.8892.7782.278Axial tensile strengthksi299.7224.8200.0Axial compressive strengthksi289.8224.840.02Transverse tensile strengthksi11.16224.81.015Transverse compressive strengthksi6.09224.81.015Shear strengthksi5.225.083.045Specific gravity---1.82.51.4

Table 3.4 Typical Properties of Matrices (USCS system of units)PropertyUnitsEpoxyAluminumPolyamideAxial modulusMsi0.49310.300.5075Transverse modulusMsi0.49310.300.5075Axial Poisson's ratio---0.30.300.35Transverse Poisson's ratio---0.30.300.35Axial shear modulusMsi0.18973.9150.1885Coefficient of thermal expansionμin/in/0F3512.7850Coefficient of moisture expansionin/in/lb/lb0.330.000.33Axial tensile strengthksi10.4440.027.83Axial compressive strengthksi14.7940.0215.66Transverse tensile strengthksi10.4440.027.83Transverse compressive strengthksi14.7940.0215.66Shear strengthksi4.9320.017.83Specific gravity---1.22.71.2

Example 3.1a)ρ f 2500 kg/ m3 , andρ m 1200 kg/ m3ρ c (2500)(0.7 ) (1200)(0.3) 2110 kg/ m3

ρ f 2500 kg/ m3 .Example 3.1b)2500Wf 0.3, and2110 0.82941200 0.32110 0.1706Wm W f W m 0.8294 0.1706 1.000

Example 3.1c)νc wcρc42110 1.896 10 3 m 3

Example 3.1d)ν f V fν c (0.7 )(1.896 10 3 ) 1.327 10 3 m 3

Example 3.1d)ν m Vmν c (0.3)(0.1896 10 -3 ) 0.5688 10 -3 m 3

Example 3.1d)wf ρ f ν f (2500)(1.327 10 3 ) 3.318 kg

Example 3.1d)wm ρ m ν m 3 (1200)(0.5688 10 ) 0.6826 kg

FIGURE 3.2 Photomicrographs of cross-section of a lamina with voids.

νvV v νcνc νf ν m ν vνc wcρ ce, andνf ν m wcρ ct

wcρ ce wcρ ct νν , thenwc ρ ct - ρ ce νν ρ ce ρ ct

ννVν νcρ ct - ρ ce ρ ct

Example 3.2A Graphite/Epoxy cuboid specimen with voids hasdimensions of and its mass is Mc. After putting it in amixture of sulphuric acid and hydrogen peroxide, theremaining graphite fibers have a mass Mf. Fromindependent tests, the densities of graphite and epoxy areρf and ρm, respectively. Find the volume fraction of thevoids in terms of a, b, c, Mf, Mc, ρf, and ρm.

νc νf ν m ν v

νf Mfρf, and ν m Mc- M fρm

νc abc

abc Mfρf Mc- M fρm νν1 M f M c - M f νν 1 Vν abc ρ fabcρ m

Alternative Solutionρ ct ρ f V 'f ρ m (1 - V f′ )volume of fibersV f ′ volume of fibers volume of matrixMf′fV ρfMfρf Mc- M fρm

Alternative Solutionvolume of fibersV f′ volume of fibers volume of matrixMf′fV ρfMfρf Mρ ce cabcMc- M fρm

Alternative Solution1Vv 1abc M f M c - M f ρ m ρ f

Alternative Solutionwcρc ρwwc - wi

Alternative Solutionwcρc ρwwc ws - ww

A Glass/Epoxy lamina consists of a 70% fiber volume fraction. Use properties of glass and epoxy from Tables3.1 and 3.2, respectively to determine the. a) density of lamina b) mass fractions of the glass and epoxy. c) volume of composite lamina if the mass of the lamina is 4 kg . d) volume and mass of glass and epoxy in part (c).