Field solution and uniformity condition in heterogeneous materials for linear multi-physical problems

For linear physical problems of equilibrium or steady-state phenomena in uncoupled and coupled cases, their fundamental equations are essentially similar so they can be treated on an equal footing. This work provides a unified formulation of the field solution in heterogeneous materials for linear multi-physical problems. Based on a modified Eshelby's equivalent inclusion model, the fields in the whole domain of the two-phase heterostructure are expressed by a non-uniform multi-physical Eshelby tensor for the inhomogeneity of a general shape. It is found that uniform fields could be created by applying eigenfields and boundary loads under a uniformity condition, which is also derived by an inverse approach. Furthermore, the uniformity condition for multi-phase heterogeneous materials is also found without restrictions on the constituent geometry and statistical homogeneity. This might be helpful for the design and fabrication of heterogeneous materials that could lead to some novel applications in certain scenarios.