The formulation of a phase-field continuum theory for brittle fracture in elastic-plastic solids and its computational implementation are presented in this contribution. The theory is based on a virtual-power formulation in which two additional and independent kinematical descriptors are introduced, namely the phase-field and the accumulated plastic strain. Further, it incorporates irreversibility of both phase-field and plastic strain evolutions by introducing suitable constraints and by carefully heeding the influence of those constraints on the kinetics underlying microstructural changes associated with plasticity and fracture.The numerical implementation employs the finite-element method for spatial discretization and a splitting scheme with sub-stepping for the time integration. To illustrate its potential utility, we apply the model to a number of well known linear, as well as non-linear, fracture mechanics problems.The described phase-field model, coupled with plasticity, provides a feasible technique to analyzing crack initiation and the subsequent crack growth resistance only if the length scale parameter included in the phase-field model is finite and treated as a material parameter which should be properly characterized.

To gain control over the diffusive heat flux in a given domain, one needs to engineer a thermal metamaterial with a specific distribution of the generally anisotropic thermal conductivity throughout the domain. Until now, the appropriate conductivity distribution was usually determined using transformation thermodynamics. By this way, only a few particular cases of heat flux control in simple domains having simple boundary conditions were studied. Thermal metamaterials based on optimization algorithm provides superior properties compared to those using the previous methods. As a more general approach, we propose to define the heat control problem as an optimization problem where we minimize the error in guiding the heat flux in a given way, taking as design variables the parameters that define the variable microstructure of the metamaterial. In the present study we numerically demonstrate the ability to manipulate heat flux by designing a device to concentrate the thermal energy to its center without disturbing the temperature profile outside it.