Abstract: Composite materials with doubly periodic rigid line inclusions under antiplane shear in an infinite medium are discussed.Its representative cell contains four rigid lines of unequal size.By using the elliptic function and conformal transformation theory,a close form solution to this problem is obtained.The effective antiplane shear modulus of the composite material is derived using the periodicity of the microstructure and the average stress/strain theorem.A series of meaningful solutions for various arrays of periodic rigid lines can be obtained as special cases.Numerical results demonstrate the variations of the effective antiplane shear modulus of such heterogeneous materials with microstructure parameters.The present close form solution can also provide benchmark solution for other numerical or approximate methods.