Provide the 2D rule-of-mixtures expression for the in-plane shear modulus Gbar12 using iso-strain.

• A. Gbar12 ≈ (Vf/Gf12) + (Vm/Gm12)
• B. Gbar12 ≈ Gf12^Vf + Gm12^Vm
• C. Gbar12 = Vf Gf12 - Vm Gm12
• D. Gbar12 ≈ Vf Gf12 + Vm Gm12

Answer: (D)

Under iso-strain, both phases experience the same in-plane shear strain ε. Each phase carries a stress proportional to its modulus: σ_f = Gf12 ε and σ_m = Gm12 ε. The composite stress is the weighted sum of these stresses: σ_bar = Vf σ_f + Vm σ_m = (Vf Gf12 + Vm Gm12) ε. The overall shear modulus is σ_bar/ε, so Gbar12 = Vf Gf12 + Vm Gm12. This linear combination by volume fractions is the correct iso-strain rule of mixtures for in-plane shear.