Conventional Continuum Damage Mechanics (CCDM) typically functions by smearing the damage evolution over the element volume resulting in a large amount of energy release when the mesh resolution is coarse and may lead to numerical instabilities. This is because, physically, energy release due to crack propagation in cases such as fibre failure typically occur over a small volume and thus to obtain accurate results with CCDM, mesh resolution needs to be closer to the size of the cracks. This limiting mesh dependency is a bottleneck to model large structures that typically requires coarse meshes. Furthermore, linear elements fail to provide accurate damage initiation stress when the mesh resolution is coarser. To overcome the inherent mesh dependency in damage propagation, a Localised Continuum Damage Mechanics (LCDM) is developed using higher-order elements in explicit integration. Here, the localisation band is introduced within the continuum elements such that elastic and damage regions can be separated using kinematic enhancements without using geometric discontinuities. This separation enables cracks to be modelled as localised features and thus the stress concentrations and damage propagation are modelled more accurately than CCDM. The higher-order continuum element adopted in this framework provides improved predictions for the damage initiation stress compared to linear elements. This combined method is demonstrated using Over-height Compact Tension (OCT) and Open-Hole Tensile (OHT) tests where the dominant mechanism is fibre failure. The results are compared against experiments as well as ply-level discrete crack models.