Commercial-purity titanium reinforced with 50 % uniaxial continuous silicon carbide fibers has been proposed for use as a fan-blade material in gas turbines, because the high elastic modulus will reduce blade deflection. The yield strength of the commercial-purity titanium is 414 MPa, and the elastic modulus is 170 GPa. Assume that the titanium is perfectly plastic after yield, thus the yield and tensile strength are equal. The strain at fracture of the titanium is 25 %. The tensile strength of the silicon carbide fibers is 3.9 GPa, and the elastic modulus is 400 GPa.
(a) Calculate the strain where the first change in elastic modulus occurs in this composite material for a uniaxial tensile stress parallel to the fibers.
(b) Calculate the composite stress where the change in elastic modulus occurs.
(c) Calculate the elastic modulus at low strain, and if the titanium yields before the fibers fail, calculate the elastic modulus after the titanium yields.
(d) Calculate the fracture strength of the composite.
(a) First it is necessary to determine the strain at fracture for both the fibers and for the yield of the matrix: The strain in the fibers at failure is
(b) The stress in the fibers at the yield of the matrix can be calculated from the elastic modulus of the fibers of 400 GPa and the yield strain of the matrix of 0.0024
(c) The elastic modulus before yield of the matrix can be calculated from the rule of mixtures:
After yield of the matrix the elastic modulus is due only to the fibers, because the matrix contributes no increase in stress with strain. Thus the composite elastic modulus is equal to 200 GPa.
(d) Since the strain at fracture of the matrix is 0.25 and the strain at fracture of the fibers is 0.01, the fibers fracture before fracture of the matrix. The composite fracture stress at fracture of the fibers is then:
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