A new meshfree approach to continuum mechanics is proposed. The approach uses a discretisation based on particles similar to well established smooth particle hydrodynamics methods. However, constructing corresponding particle shape functions with local maximum entropy interpolation schemes a consistent first order approximation is obtained, which is also amendable to rigorous analysis. The advantage of local maximum entropy interpolation schemes over moving least square interpolation schemes - often used in particle hydrodynamics frameworks - is the straight forward realization of essential boundary conditions. Thus the discretisation is well suited for flows with a variety of boundary conditions as well as fluid-structure interaction problems. Although from a computational point of view local maximum entropy schemes are slightly more complex than classical smooth particle hydrodynamics schemes, they can be efficiently realised on GP-GPU architectures.