The constitutive relation is the weakest link in the mathematical model for any body. Traditionally, explicit constitutive relations defining stress in terms of strain have been used. Recently, a new class of elastic bodies have been developed, where the stress and linearized strain are related through an implicit function satisfying the second law of thermodynamics, material symmetry and material frame indifference. Such implicit constitutive relations perform better for concrete, rubber, and gum metal materials. In 2016, Shankar et al. attempted to address issues in solving plane boundary value problems involving implicit constitutive relations. The method proposed by Shankar et al. required continuous basis functions. It is impossible to generate conforming continuous basis functions for arbitrarily shaped quadrilaterals. Moreover, when materials crack under the load, discontinuities appear in the body's geometry, which distorts the topological mapping and requires re-meshing. Therefore, this study is focused on developing a scheme involving appropriate meshless basis functions to solve plane boundary value problems in Mode-I fracture loading and utilizing the framework proposed by Shankar et al. for implicit constitutive theories. In this work, both the stress and displacement fields are treated as unknown variables. The stress is estimated using Airy's potential function, which satisfies the equilibrium equation a priori. The linearized strain is obtained from the assumed displacement field using the strain-displacement relationship. The Airy's potential function is written as the linear combination of Hermite-type Moving Least Squares (MLS) basis functions and the displacement field using Lagrangian MLS basis functions. Of all the divergence-free stress fields satisfying traction boundary conditions and admissible displacement fields satisfying displacement boundary conditions, the stress and displacement field satisfying the constitutive relation is sought. Due to a finite-dimensional approximation of the basis functions, the implicit constitutive relation between the stress and strain is satisfied in a weak integral sense, and the error in satisfying the constitutive relation is estimated using the Frobenius norm of the constitutive relation. The viability of the numerical scheme is established by solving simple plane boundary value problems with different boundary conditions and thin plate problems with square hole subjected to in-plane extension for a non-linear non-dissipative concrete model developed by Gokulnath et al. In a subsequent study, an analytical solution will be derived for a thin plate problem with a square hole using complex variable representation for an implicit constitutive model, and the accuracy of the numerical scheme will be ascertained. Finally, the dependence of stress and displacement field in Mode-I fracture on constitutive relation is quantified.