ANALYSIS_OF_COMPOSITE_MATERIALS-1.pptx
- 1. ANALYSIS OF COMPOSITE MATERIALS (UNIT- 2) Introduction and Mechanics and Material Approach (Strength of Material Approach)
- 2. Basic Definitions Micromechanics and Macro-mechanics Micromechanics is the study of the behavior of fibers and filaments, matrices, interfaces, and interphases in a composite upon the application of stress and strain. Micromechanics (or, more precisely, micromechanics of materials) is the analysis of composite or heterogeneous materials on the level of the individual constituents that constitute these materials. Macro-mechanics is the study of the behavior of the lamina or laminate when stresses or strains are applied. Macro-mechanics is the study of composite material behavior wherein the material is presumed to be homogeneous, and the effects of the constituent materials are detected only as averaged apparent macroscopic properties of the composite materials.
- 3. Lamina A lamina (also called a ply or layer) is a single flat layer of unidirectional fibers or woven fibers arranged in a matrix. Laminate A laminate is a stack of plies of composites. Each layer can be laid at various orientations and can be made up of different material systems.
- 4. Micromechanical Analysis 1. Mechanics of materials (simplifying assumptions unnecessary to specify the details of the stress and strain distribution at the micromechanical level, geometry of the microstructure is arbitrary) 2. Elasticity theory (elasticity theory models involves the solution of actual local stress and strain fields and actual geometry of the microstructure is taken into account) 3. Empirical solutions (curve fitting to elasticity solutions or experiments)
- 5. Analysis Assumptions 1. The fibers are: (i) continuous, (ii) straight, (iii) infinitely long in the X1 direction, (iv) perfectly aligned with X1 axis, (v) circular in cross section, and (vi) arranged in a periodic square array. 2. The fiber and matrix materials are (i) homogeneous, (ii) isotropic, and (iii) linearly elastic. 3. The fiber and matrix are perfectly bonded at their interface. 4. Mechanical loads are applied at infinity. 5. Loads and material properties do not vary along the X1 direction.
- 6. Void Content During the manufacture of a composite, voids are introduced in the composite This causes the theoretical density of the composite to be higher than the actual density. Also, the void content of a composite is detrimental to its mechanical properties. These detriments include lower • Shear stiffness and strength • Compressive strengths • Transverse tensile strengths • Fatigue resistance • Moisture resistance The volume of void is given by
- 7. The volume fraction of the voids is Where, Vv = volume of voids, v_c = Total volume for composite, ρ_ce= experimental density, ρ_ct=theoretical density, wc = weight of composite
- 8. Elastic Moduli The Elastic Modulus is the measure of the stiffness of a material. In other words, it is a measure of how easily any material can be bend or stretch. It is the slope of stress and strain diagram up to the limit of proportionality. There are four elastic moduli of a unidirectional lamina: • Longitudinal Young’s modulus, E1 • Transverse Young’s modulus, E2 (Young's modulus is a measure of the ability of a material to withstand changes in length when under lengthwise tension or compression) • Major Poisson’s ratio, ν12 • In-plane shear modulus, G12 Shear modulus, also known as Modulus of rigidity, is the measure of the rigidity of the body, given by the ratio of shear stress to shear strain.
- 10. Mechanics of Material Approach Strength of Materials Approach Elasticity Approach Semi-Empirical Models
- 11. STRENGTH OF MATERIALS APPROACH The following assumptions are made in the strength of materials approach model: • The bond between fibers and matrix is perfect. • The elastic moduli, diameters, and space between fibers are uniform. • The fibers are continuous and parallel. • The fibers and matrix follow Hooke’s law (linearly elastic). • The fibers possess uniform strength. • The composite is free of voids.
- 13. From a unidirectional lamina, take a representative volume element that consists of the fiber surrounded by the matrix. The fiber, matrix, and the composite are assumed to be of the same width, h, but of thicknesses t_f , t_m, and t_c, respectively. The area of the fiber is given by
- 14. The two areas are chosen in the proportion of their volume fractions so that the fiber volume fraction is defined as And the matrix fiber volume fraction V_m is
- 15. Longitudinal Young’s Modulus From the given figure under a uniaxial load Fc on the composite VE, the load is shared by the fiber F_f and the matrix F_m so that
- 16. The loads taken by the fiber, the matrix, and the composite can be written in terms of the stresses in these components and cross-sectional areas of these components as ( σ = F/A )
- 17. Assuming that the fibers, matrix, and composite follow Hooke’s law and that the fibers and the matrix are isotropic, the stress–strain relationship for each component and the composite is
- 18. The strains in the composite, fiber, and matrix are equal (εc = εf = εm); then, from Equation Substitute above equation into volume fraction -------A
- 19. Equation A gives the longitudinal Young’s modulus as a weighted mean of the fiber and matrix modulus. It is also called the rule of mixtures. The ratio of the load taken by the fibers F_f to the load taken by the composite F_c is a measure of the load shared by the fibers.
- 20. Problem: Determine the longitudinal elastic modulus of a unidirectional glass/epoxy lamina with a 70% fiber volume fraction. Use the properties of glass and epoxy from E_f =85 GPA and E_m = 3.4 GPA, V_f=0.7 and V_m =0.3 respectively. Also, find the ratio of the load taken by the fibers to that of the composite.
- 21. Solution: Given Longitudinal elastic modulus of the unidirectional lamina is
- 22. The ratio of the load taken by the fibers to that of the composite is
- 23. Transverse Young’s Modulus The composite is stressed in the transverse direction. The fibers and matrix are again represented by rectangular blocks as shown.
- 24. The fiber, the matrix, and composite stresses are equal. Thus, -----A
- 26. By using Hooke’s law for the fiber, matrix, and composite, the normal strains in the composite, fiber, and matrix are --------C
- 27. Substituting Equation (B) and Equation (C) in Equation (A) and using Equation (A) gives -----(D)
- 28. Major Poisson’s Ratio The major Poisson’s ratio is defined as the negative of the ratio of the normal strain in the transverse direction to the normal strain in the longitudinal direction, when a normal load is applied in the longitudinal direction. The deformations in the transverse direction of the composite is the sum of the transverse deformations of the fiber and the matrix
- 29. Because the thickness fractions are the same as the volume fractions, per Equation
- 30. Problem: Determine the major and minor Poisson’s ratio of a glass/epoxy lamina with a 70% fiber volume fraction. Use the properties of glass and epoxy from V_f =0.7, V_m = 0.3, v_f =0.2 v_m=0.3, E_1 =60.52 and E_2 = 10.37, respectively. Solution: Given
- 31. The major Poisson’s ratio is: The longitudinal Young’s modulus is The transverse Young’s modulus is The minor Poisson’s ratio