Micromechanics of Composites

Abstract

In this chapter we consider the results of incorporating a reinforcement (fibers, whiskers, particles, etc.) in a matrix to make a composite. It is of great importance to be able to predict the properties of a composite, given the properties of the components and the geometric arrangement of the components in the composite. We examine various micromechanical aspects of composites. A particularly simple case is the rule-of-mixtures, a rough tool that considers the composite properties as volume-weighted averages of the component properties. It is important to realize that the rule-of-mixtures works in only certain simple situations. Composite density is an example where the rule-of-mixtures is applied readily. In the case of mechanical properties, there are certain restrictions to its applicability. When more precise information is desired, it is better to use more sophisticated approaches based on the theory of elasticity.

Problems

• 10.1. Describe some experimental methods of measuring void content in composites. Give the limitations of each method.

• 10.2. Consider a 40 % V _f SiC whisker-reinforced aluminum composite. E _f = 400 GPa, E _m = 70 GPa, and (l/d) = 20. Compute the longitudinal elastic modulus of this composite if all the whiskers are aligned in the longitudinal direction. Use Halpin-Tsai-Kardos equations. Take ξ = 2(l/d).

• 10.3. A composite has 40 % V _f of a 150 μm diameter fiber. The fiber strength is 2 GPa, the matrix strength is 75 MPa, while the fiber/matrix interfacial strength is 50 MPa. Assuming a linear build up of stress from the two ends of a fiber, estimate the composite strength for (a) 200 mm long fibers and (b) 3 mm long fibers.

• 10.4. Derive the load transfer expression Eq. (10.62) using the boundary conditions. Show that average tensile stress in the fiber is given by Eq. (10.63).

• 10.5. Consider a fiber reinforced composite system in which the fiber has an aspect ratio of 1,000. Estimate the minimum interfacial shear strength τ _i, as a percentage of the tensile stress in fiber, σ _f, which is necessary to avoid interface failure in the composite.

• 10.6. Show that as ξ → 0, the Halpin-Tsai equations reduce to

  $$\begin{array}{ccccccc}l/p = {V_{{m}}}/{p_{{m}}} + {V_{{f}}}/{p_{{f}}}\\ {\hbox{while}}\;\,{\hbox{as}}\,\;\xi \to \infty, \;\,{\hbox{they}}\;\,{\hbox{reduce}}\;\,{\hbox{to}}\end{array} $$
  $$p = V_{{m}}{p_{{m}}} + {V_{{f}}}{p_{{f}}}. $$

• 10.7. Consider an alumina fiber reinforced magnesium composite. Calculate the composite stress at the matrix yield strain. The matrix yield stress 180 MPa, E _m = 70 GPa, and ν = 0.3. Take V _f = 50 %.

• 10.8. Estimate the aspect ratio and the critical aspect ratio for aligned SiC whiskers (5 μm diameter and 2 mm long) in an aluminum alloy matrix. Assume that the matrix alloy does not show much work hardening.

• 10.9. Alumina whiskers (density = 3.8 g/cm³) are incorporated in a resin matrix (density = 1.3 g/cm³). What is the density of the composite? Take V _f = 0.35. What is the relative mass of the whiskers?

• 10.10. Consider a composite made of aligned, continuous boron fibers in an aluminum matrix. Compute the elastic moduli, parallel, and transverse to the fibers. Take V _f = 0.50.

• 10.11. Fractographic observations on a fiber composite showed that the average fiber pullout length was 0.5 mm. If V _fu = 1 GPa and the fiber diameter is 100 μm, calculate the strength of the interface in shear.

• 10.12. Consider a tungsten/copper composite with following characteristics: fiber fracture strength = 3 GPa, fiber diameter = 200 μm, and the matrix shear yield strength = 80 MPa. Estimate the critical fiber length which will make it possible that the maximum load bearing capacity of the fiber is utilized.

• 10.13. Carbon fibers (V _f = 50 %) and polyimide matrix have the following parameters:

  $$ \begin{array}{ccccccc} {{E_{{f}}} = 280\,{\hbox{GPa}}} \hfill & {{E_{{m}}} = 276\,{\hbox{MPa}}} \hfill \\{{v_{{f}}} = 0.2} \hfill & {{v_{{m}}} = 0.3} \hfill \\\end{array} $$

  • (a) Compute the elastic modulus in the fiber direction, E ₁₁, and transverse to the fiber direction, E ₂₂.

  • (b) Compute the Poisson ratios, ν ₁₂ and ν ₂₁.

• 10.14. Copper or aluminum wires with steel cores are used for electrical power transmission. Consider a Cu/steel composite wire having the following data:

  inner diameter = 1 mm

  outer diameter = 2 mm

  $$ \begin{array}{ccccccc} {{E_{\rm{Cu}}} = 150\,\,{\hbox{GPa}}} \hfill & {{\alpha_{\rm{Cu}}} = 16 \times {{10}^{{ - 6}}}\,{{\hbox{K}}^{{ - 1}}}} \hfill \\{{E_{\rm{steel}}} = 210\,\,{\hbox{GPa}}} \hfill & {{\alpha_{\rm{steel}}} = 11 \times {{10}^{{ - 6}}}\,{{\hbox{K}}^{{ - 1}}}} \hfill \\{{\sigma_{{y{\rm{Cu}}}}} = 100\,\,{\hbox{MPa}}} \hfill & {{v_{\rm{Cu}}} = {v_{\rm{steel}}} = 0.3} \hfill \\{{\sigma_{\rm{ysteel}}} = 200\,\,{\hbox{MPa}}} \hfill & {} \hfill \\\end{array} $$

  • (a) The composite wire is loaded in tension. Which of the two components will yield plastically first? Why?

  • (b) Compute the tensile load that the wire will support before any plastic strain occurs.

  • (c) Compute the Young’s modulus and CTE of the composite wire.

• 10.15. A composite is made of unidirectionally aligned carbon fibers in a glass-ceramic matrix. The following data are available:

  $$ \begin{array}{ccccccc} {E_{{{\rm{f}}1}}} = 280\,\,{\hbox{GPa,}} & \,\,{E_{\rm{f2}}} = 40\,\,{\hbox{GPa,}}\,\,\,{E_{{m}}} = 70\,\,{\hbox{GPa}} \hfill \\{\nu_{{{\rm{f}}1}}} = 0.2\,\,\,\,\,\,{\nu_{{m}}} = 0.3 \hskip -70pt\hfill \\{G_{{{\rm{f}}12}}} = 18\,\,{\hbox{GPa}}\hskip -57pt \hfill \end{array}$$

  • (a) Compute the elastic modulus in the longitudinal and transverse directions.

  • (b) Compute the two Poisson’s ratios.

  • (c) Compute the principal shear modulus, G ₁₂.