The objective of this paper is threefold: (a) to determine the validity of applying the continuum-based linear elastic fracture mechanics (LEFM) to investigate fracture processes at the discrete atomic level in amorphous materials, such as polymers, (b) to quantify the contribution from entropic effects to the J-integral at the nanoscale due to the thermal motion of atoms, and (c) to demonstrate that the atomistic J-integral can be applied to extract the cohesive traction-separation law at a nanoscale notch-tip as a material property in a polymer when specific modeling criteria are obeyed. Verification of the atomistic J-integral algorithm was performed using toughness data for a monocrystalline graphene sheet, before application to a polymer. A generic epoxy polymer (EPON 862-DETDA) was selected for this study, and the simulations were carried out using molecular dynamics (MD). Furthermore, atomistic J-integral was employed as a nanoscale fracture metric to explore flaw tolerance at the nanoscale and to build a methodology to evaluate and predict the material’s initiation fracture toughness. For this purpose, three MD models of edge-notched specimens that are geometrically similar to the ASTM standard compact tension (CT) specimens were developed, each with a different nano-size notch length to study the effect of notch size on toughness and strength. Significant deviation of peak atomistic J-integral values from LEFM-based macro-scale fracture toughness values for epoxy was observed at notch lengths below a critical threshold crack length, indicating that LEFM is not applicable below this length scale. Entropic effects arise in materials at the micro/nano scale due to the random thermal motion of atoms and molecules, especially near crack faces. Results from our MD simulations demonstrated a significant entropic contribution to the atomistic J-integral, especially at elevated temperatures. Notch-size independence of the cohesive traction-separation law near a nanoscale subcritical notch and the effect of temperature were also demonstrated using the atomistic J-integral.