Summary: Due to the differences in Young’s moduli under the tensile and compressive loadings for a rock-like and asphalt-mixture materials, the analysis becomes difficult as all coefficients in Hook’s law are dependent on the stress state in the field. In this paper, the meshless method is formulated for the first time to evaluate a cracked structure made of bimodular materials. In general, there are four stress zones in the field determined by the principal stresses. It is known that for classical elasticity (single modulus material), the stresses are singular at the crack tip described by William’s series. For the pure mode I fracture, it is proved that the solutions for the stress and displacement by William’s series are still valid for bimodular materials as there is only first kind zone in the vicinity of the crack tip. With the utilization of the Lagrange series, the meshless Finite Block Method (FBM) is demonstrated to deal with cracked structures with bimodular materials. The numerical solutions of the partial differential equations with variable coefficients are solved in a strong form by using the iterative technique. Numerical examples are demonstrated to verify the degree of accuracy and convergence with Finite Element Method (FEM) with very fine mesh.