One of the major challenges of multiscale investigation of granular material is to establish the continuum-scale stress-strain constitutive model directly based on the particle scale material constants and material state parameters[1]. The elastic modulus of granular materials have been derived by taking the uniform strain assumption [2,3,4]. However, its accuracy is under question [5,6,7]. Here we show the reason is the ignorance of structural heterogeneity. To solve this challenge, the micro-structural definition of the stress tensor is revisited and generalized to account for the contribution of external forces imposed on granular particles. The newly developed formula expresses the macro stress tensor as the summation of two components: one is contributed by the inter-particle contact forces associated with contact vectors, and the other by the imposed particle forces associated with particle locations. Based on the newly developed stress expression, the elastic response of granular assembly has been investigated, and the stiffness matrix is derived. Its accuracy has been demonstrated by the numerical experimental data from Discrete Element simulations.
The statistical characteristics of geometrical variables associated with two stress components are studied. Applying the directional statistical theory [9], the influence of material anisotropy on the contact-force related stress component is quantified and expressed in terms of tensor product of contact stiffness matrix and anisotropic matrix parameters. The spatial correlation of particle unbalanced forces is explored, demonstrating that the heterogeneity of the internal structure is the cause of non-affine particle displacement, leading to the expression of elastic stiffness matrix with considering internal structural heterogeneity. Discrete element simulation of material elastic behaviour with different friction coefficients and loading directions are carried out. Results are used to demonstrate the accuracy of the proposed theory and provide an effective method for quantitatively developed material constitutive modeling with consideration of structure heterogeneity.

Homogenisation theory, Stiffness matrix, Structural heterogeneity, Material anisotropy, Non-affine displacement