A composite material is made from a 5052 aluminum matrix with 60 % uniaxial continuous boron fibers. The yield strength of the aluminum is 200 MPa, and the elastic modulus is 70 GPa. Assume that the matrix material is perfectly plastic after yield (elastic-perfectly plastic); therefore the ultimate tensile strength for the aluminum is also 200 MPa, and the strain at fracture of the matrix material is 15 percent. The tensile strength of the boron fibers is 2.8 GPa, and the elastic modulus is 400 GPa. Assume that the boron fibers are brittle.

Question

(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 initial change in elastic modulus occurs.
(c) If the matrix yields before the fibers fracture, calculate the elastic modulus before and after yield of the matrix.
(d) Calculate the fracture strength of the composite.

Answer 1

(a) First it is necessary to determine the strain at fracture for 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 is calculated from the elastic modulus of the fibers of 400 GPa and the yield strain of the matrix of 0.003







(c) The elastic modulus before yield of the matrix is calculated with the rule of mixtures:









After yield of the matrix the effective modulus is due only to the fibers, because the matrix does not increase in stress with increased strain in a perfectly plastic material. The composite effective modulus is equal to 240 GPa.



(d) Since the strain at fracture of the matrix is 0.15, and the strain at fracture of the fibers is 0.007, the fibers fracture before fracture of the matrix. The composite fracture stress at fracture of the fibers is then:







The matrix contribution to the stress when the fibers fracture is 200 MPa, because the matrix is perfectly plastic after yield.