Chapter 3 Micromechanical Analysis of a Lamina - USF

1 Chapter 3 Micromechanical Analysis of a Lamina Elastic Moduli Dr. Autar Kaw Department of Mechanical Engineering University of South Florida, Tampa, FL 33620. Courtesy of the Textbook Mechanics of Composite Materials by Kaw A f = t f h, Am = t m h, and Ac = t c h A m Af V m=. Vf= Ac Ac tm tf =. = tc tc = 1-V f 3 2. 1. h tc Lc h tm/2. tm/2. tf FIGURE tc Representative volume element of a unidirectional Lamina . h c c tm/2. tf tm/2. tc FIGURE A longitudinal stress applied to the representative volume element to calculate the longitudinal Young's modulus for a unidirectional Lamina .

2 Fc= F f + Fm F c = c Ac , c = E1 c , F f = f A f , and f = E f f , and F m = m Am m = Em m E 1 c Ac = E f f A f + E m m Am If ( c = f = m), then : Af Am E1 = E f + Em E1 = E f V f + E m V m Ac Ac E1 = E f V f + E m V m F c = c Ac , c = E1 c , F f = f A f , and f = E f f , and F m = m Am m = Em m Ff Ef = Vf Fc E 1. FIGURE Fraction of load of composite carried by fibers as a function of fiber volume fraction for constant fiber to matrix moduli ratio. Example Find the longitudinal elastic modulus of a unidirectional Glass/Epoxy Lamina with a 70% fiber volume fraction.

3 Use the properties of glass and epoxy from Tables and , respectively. Also, find the ratio of the load taken by the fibers to that of the composite. Example Ef = 85 Gpa Em = GPa E 1 = (85) ( ) + ( ) ( ). = GPa Example FIGURE Longitudinal Young's modulus as function of fiber volume fraction and comparison with experimental data points for a typical glass/polyester Lamina . Example Ff 85. = ( ) = F c c h c tm/2. tf tm/2. tc FIGURE A transverse stress applied to a representative volume element used to calculate transverse Young's modulus of a unidirectional Lamina .

4 C= f = m c = f + m . c = c, c = t c c , E2. f f = t f f , and f= , and Ef m = t m m m . = m Em 1 1 tf 1 tm = + , and E2 E f tc Em tc 1 V f Vm = +. E2 E f Em Example Find the transverse Young's modulus of a Glass/Epoxy Lamina with a fiber volume fraction of 70%. Use the properties of glass and epoxy from Tables and , respectively. Example E f = 85 GPa Em = GPa 1 = +. E 2 85 E 2 = GPa FIGURE Transverse Young's modulus as a function of fiber volume fraction for constant fiber to matrix moduli ratio. 1. d 4V f 2. = . s . 1.

5 D 2 3 V f 2. =. s . s d (a). s FIGURE d Fiber to fiber spacing in (a) square packing (b). geometry and (b) hexagonal packing geometry. FIGURE Theoretical values of transverse Young's modulus as a function of fiber volume fraction for a boron/epoxy unidirectional Lamina (Ef = 414 GPa, vf =. , Em = GPa, vm = ) and comparison with experimental values. Figure (b) zooms figure (a) for fiber volume fraction between and ( experimental data from Hashin, Z., NASA tech. rep. contract no. NAS1-8818, November 1970.). h 1. 1. tm/2.

6 Tf tm/2. (a) tc tm/2. tc + cT tf + fT tf tc tm/2. Lc (b). FIGURE A longitudinal stress applied to a representative volume element to calculate Poisson's ratio of unidirectional Lamina . f m T. =. c T. + T. Tf =. T. f , tf T. = , and tc = t f + tm . T m T T T. m c f m tm T. =. T c c tc T. f =- L , f f T. - t c 12 cL = - t f f Lf - t m m mL. m = - , and m L. m T. 12 = - c L. c - t c 12 = - t f f - t m m . L. c L. f L. m If c = f = m , then : L L L. t c 12 = t f f + t m m tf tm 12 f = + m 12 = f V f + m V m tc tc Example Find the Major and Minor Poisson's ratio of a Glass/Epoxy Lamina with a 70% fiber volume fraction.

7 Use the properties of glass and epoxy from Tables and , respectively. Example f = m = 12 = ( ) ( ) + ( ) ( ). = Example E1 = Gpa E2 = GPa Example E2. 21 = 12. E1. = = h c c tm/2. tf tm/2. tc FIGURE An in-plane shear stress applied to a representative volume element for finding in-plane shear modulus of a unidirectional Lamina . c = f + m c = c tc , f = f t f , and m = m tm c = f + m . c= c , G12 c = c tc , f f= , and f = f t f , and Gf . m = m m = m tm Gm c f m tc = tf + tm G12 Gf Gm c f m tc = tf + tm G12 Gf Gm If c = f = m , then : 1 1 tf 1 tm = +.

8 G12 G f t c G m t c 1 V f Vm = +. G12 G f G m Example Find the in-plane shear modulus of a Glass/Epoxy Lamina with a 70% fiber volume fraction. Use properties of glass and epoxy from Tables and , respectively. Example E f = 85 GPa f = Ef Gf =. 2 (1 + f ). 85. =. 2 (1 + ). = GPa Example E m = GPa m = Em Gm =. 2 (1 + m). =. 2 (1 + ). = GPa Example 1 = +. G12 G12 = GPa FIGURE Theoretical values of in-plane shear modulus as a function of fiber volume fraction and comparison with experimental values for a unidirectional glass/epoxy Lamina (Gf = GPa, Gm = GPa).

9 Figure (b) zooms figure (a) for fiber volume fraction between and ( experimental data from Hashin, Z., NASA tech. rep. contract no. NAS1-8818, November 1970.). E1 = E f V f + E m V m E2 = 1+ V f Em 1 - V f ( E f / E m) - 1. =. ( E f / E m) + . Example Find the transverse Young's modulus for a Glass/Epoxy Lamina with a 70%. fiber volume fraction. Use the properties for glass and epoxy from Tables and , respectively. Use Halphin-Tsai equations for a circular fiber in a square array packing geometry. 2. Example 2. a b FIGURE Concept of direction of loading for calculation of transverse Young's modulus by Halphin Tsai equations.

10 Example =2 Ef = 85 GPa Em = GPa (85 ) - 1. =. (85 ) + 2. = Example E 2 = 1 + 2( )( ). 1 ( )( ). E 2 = GPa FIGURE Theoretical values of transverse Young's modulus as a function of fiber volume fraction and comparison with experimental values for boron/epoxy unidirectional Lamina (Ef = 414 GPa, f = , Em = GPa, m = ). Figure (b). zooms figure (a) for fiber volume fraction between and ( experimental data from Hashin, Z., NASA tech. rep. contract no. NAS1-8818, November 1970.). Ef/Em = 1 implies = 0, (homogeneous medium). Ef/Em implies = 1, (rigid inclusions).