Si is a promising anode material for Li-ion batteries, since it absorbs large amounts of Li. However, insertion of Li leads to 334% of volumetric expansion, huge stresses, and fracture; it can be suppressed by utilizing nanoscale anode structures. Continuum approaches to stress relaxation in LixSi, based on plasticity theory, are unrealistic, because the yield strength of LixSi is much higher than the generated stresses. Here, we suggest that stress relaxation is due to anisotropic (tensorial) compositional straining that occurs during insertion-extraction at any deviatoric stresses. Developed theory describes known experimental and atomistic simulation data. A method to reduce stresses is predicted and confirmed by known experiments. Chemical potential has an additional contribution due to deviatoric stresses, which leads to increases in the driving force both for insertion and extraction. The results have conceptual and general character and are applicable to any material systems.