We propose a new phase-field damage formulation which takes into account anisotropic damage evolution in solids. Such anisotropy projects itself in fracture energy values which depend on the direction of the crack surface. Therefore, instead of one constant scalar parameter for the fracture energy value, we use a direction-dependent fracture energy function. By incorporating a direction-dependent fracture energy function, only a single damage variable as well as a first order damage gradient need to be used within the standard phase-field damage model. (nhnhanh)