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Introduction to Composite Materials Due Feb 15 11:59 pm Homework 2 A composite material is made with an epoxy matrix ( E ! = 3.5 GPa , ν ! = 0.33 ) and 40 vol% of unidirectional continuous glass fibers ( E " = 72 GPa , ν " = 0.22 ). Assume that the fiber and matrix constituents are isotropic and homogeneous. In this homework, f and m denote properties associated with the fiber and the matrix, respectively. 1. Estimate the elastic constants of the composite E # , E $ , ν #$ , and G #$ . From the results, for each property, mention which phase is dominant. Use rule of mixtures and Reuss models for this problem. 2. State the assumptions (in terms of stress/strain) adopted to derive the expressions of E # , E $ and G #$ . 3. A rod is made of the composite material above with fibers aligned in the axial direction. The cross-sectional area of the rod is A = 100 mm $ . Determine the loads carried by the fibers and the matrix upon a tensile longitudinal strain of ε = 0.01 (assuming strains in the matrix and the fibers are the same). Which phase is taking most of the load and what is the percentage of the load it is taking? 4. Consider the same rod with A = 100 mm $ . The tensile strength of matrix and fiber are σ !% = 60 MPa and σ "% = 1 GPa . Compute the critical fiber volume fraction for composite strengthening. Also, calculate the tensile load such that the composite will reach its tensile strength. 5. Suppose the volume fractions of fiber V " is a variable ranging from zero to 100%. Plot curves (using plot tools of your choice such as Matlab, Excel, Python) of E # versus V " , and E $ versus V " , on the same graph, and comment on the findings (for the epoxy-glass composite). Do the same for a composite with a copper matrix ( E ! = 130 GPa , ν ! = 0.34 ) and SiC fibers ( E " = 380 GPa , ν ! = 0.22 ). Compare the shapes of curves of epoxy-glass and copper- SiC composites, what conclusions can you draw regarding the contribution of V " to E $ as a function of the relative stiffness between fibers and matrix? Use rule of mixtures and Reuss models for this problem. 6. The above epoxy-glass composite is now made of discontinuous fibers with a fiber aspect ratio & ! ’ ! = 10 . Calculate E # , E $ for a continuous, a unidirectional and a randomly oriented lamina. What insights can you provide after comparing E # , E $ of composites with continuous fibers (take ζ = 100 for E # and ζ = 2 for E $ ), unidirectional discontinuous fibers, and randomly oriented discontinuous fibers? Use Halpin-Tsai equations here for all the comparisons. 7. The fiber aspect ratio, & ! ’ ! , can be viewed as a variable characterizing the length of discontinuous fibers, plot E # versus & ! ’ ! for epoxy-glass composites ( E ! = 3.5 GPa , ν ! = 0.33 , E " = 72 GPa , ν " = 0.22 ) with unidirectional discontinuous fibers (the volume fraction of fibers is still 40%). Based on the results in Problem 1, how does E # converge as & ! ’ ! increases? 8. (ME227 only) Plot lower and upper Hashin-Shtrikman (H-S) bounds, as well as Voigt and Reuss models for Young’s modulus (plot relationship E vs. V " ). Based on your plot, describe the relationship between H-S bounds and Voigt and Reuss models. Use E ! = 3.5 GPa , ν ! = 0.33 , E " = 72 GPa , ν " = 0.22 . Hint: H-S bounds are given by K &()*+ = K ! + V " 1 K " − K ! + 3V ! 3K ! + 4G !

Introduction to Composite Materials Due Feb 15 11:59 pm K %,,*+ = K " + V ! 1 K ! − K " + 3V " 3K " + 4G " G &()*+ = G ! + V " 1 G " − G ! + 6(K ! + 2G ! )V ! 5G ! (3K ! + 4G ! ) G %,,*+ = G " + V ! 1 G ! − G " + 6(K " + 2G " )V " 5G " (3K " + 4G " ) We can use E = 9KG/(3K + G) to obtain lower and upper bounds for Young's modulus. Check [ Hashin, Zvi, and Shmuel Shtrikman. Journal of the Mechanics and Physics of Solids 11.2 (1963): 127-140] for more details.

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