Thermodynamic irreversibility can be imposed on empirical material behaviour by using an appropriate algorithm which takes the path-dependence of the degradation process into account. This new algorithm, Algorithmically Imposed Mechanics (AIM), for algorithmically irreversible mechanics, is described, and the convergence and unicity of the solutions obtained are proven. AIM is applicable to a range of mechanical behaviour and is demonstrated to work in conjunction with non-local damage with rotating cracks as well as a mixed plastic and damage behaviour. Copyright (c) 2015 John Wiley & Sons, Ltd.