The numerical manifold method (NMM) is a more generalnumerical method than finite element method with mathematical and physicalmeshes. The mathematical mesh provides the nodes to form a finite coveringof the solution domain and the partition of unity functions, while thephysical mesh provides the domain of integration. The numerical manifoldmethod has some advantages, for example, the solution domain is discretizedas an arbitrary mesh which is independent on the complex geometry of theboundary of the solution domain or the interface of bi-materials. However,the generalized degrees of freedom in the NMM will be enhanced to inducemore computing cost when the higher polynomial interpolating function isused as the displacement function. In this paper, based on the complexvariables theory, the approximation function of a two-dimensional problem isdeveloped with one-dimensional basis function, and the approximationfunction is applied to the NMM for two-dimensional elasticity. Then thecomplex variable numerical manifold method (CVNMM) for 2D elasticity ispresented, and the formulae of the CVNMM are obtained. The influences ofboundary conditions and initial stress on the final linear algebra equationssystem are discussed. In addition, two numerical cases were carried out.When the composite material structure is simulated using the CVNMM, thecomputing meshes can be easily generated along the interfaces of materials,then the CVNMM is more flexible than the finite element method. Furthermore,bi-material interface crack problem is analyzed using the CVNMM, and thestress intensity factor of the interface crack is obtained with thenumerical extrapolation method. The CVNMM has greater computationalefficiency and precision validated with the numerical cases.