Abstract:
Governing equations in classical continuum mechanics utilise derivative operators applied to functions that have spatial derivatives at every point in the domain. Derivative operators, therefore, cannot be directly used in governing equations to model spatial discontinuous such as those found in solid fracture and crack propagation. Non-ordinary state-based peridynamics (NOSB-PD) replaces partial derivative operators with their integro-differential counterparts which are appropriate for material failure simulations. Nevertheless, NOSB-PD suffers from the presence of spurious oscillations on the boundary of solid as well as at the crack tip. Inadequate approximation of the deformation gradient tensor and internal forces are the main reasons for these fictitious oscillations. Moreover, use of NOSB-PD with non-uniform discretisation results in unreliable solutions. To improve the accuracy of the deformation gradient tensor, generalised derivative operators (GDO) are defined for any square integrable L 2 function. The generalised derivative operators are determined using a bi-orthogonal basis corresponding to the monomial basis which is utilised in the finite truncation of a Taylor series expansion. These new operators are verified and/or validated by comparing the total derivative of a vector valued function and the divergence of a tensor of rank two with their corresponding approximations obtained from the GDO. Furthermore, a second order linear differential equation and also Laplace’s equation are solved in order to verify GDO’s application in solving ordinary differential equations and partial differential equations. The classical form of the governing equation in reference configuration is reformulated using the derivative operators (GDO based governing equation). Moreover, the GDO based governing equation is numerically solved to obtain the displacement and damage ratio index of each point of the material. In addition, the proposed method can be applied to a wide range of damage modelling scenarios to capture various aspects of solid fracture. The capabilities of the GDO based governing equation are assessed through considering an isotropic linear elastic plate with/without an initial crack which is subjected to velocity boundary conditions. Firstly, the formulation is validated comparing the GDO results with a FEM solutions. Secondly, the stress concentration at the crack tip is successfully simulated because the equivalent von Mises stress at the crack tip is significantly more than the other points. Lastly, the proposed formulation is capable of capturing damage evolution and mixed-mode crack growth.