Initially stressed strain gradient elasticity: A constitutive model incorporates size effects and initial stresses

Unlike ordinary solid materials, underground nano-materials such as kerogen, have relatively small dimensions and suffer from unavoidable in-situ stresses. The coexistence of size effects and initial stresses poses a great challenge to the constitutive modeling of deeply buried nano-inclusions. Despite the theories of strain gradient elasticity (SGE) and initially stressed elasticity (ISE) have been separately developed, the phenomenological model that fully considers the impact of the two ingredients remains unexplored. This paper proposes a strain gradient elasticity constitutive model for kerogen with size effects and in-situ stresses. Based on the decomposition of strains and strain gradients, the initially stressed strain gradient elasticity (ISSGE) framework is established. Then, a new form of the volumetric response function for kerogen is derived utilizing the density and porosity independence of the Poisson ratio. On this basis, we construct the corresponding hyperelastic and higher-order strain energy densities embedded with the given initial stress. The new constitutive model is applied to investigate the spherical pore contraction problem. Theoretical analysis and experimental results indicate that combining the in-situ stress and the size effect strengthens the elastic stiffness. Such enhancement cannot be comprehensively described by the existing theories. The model presented here provides the first constitutive relation of initially stressed strain gradient elasticity and lays the foundation for further incorporating more mechanical behaviors of underground nano-materials.