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SUMMARY The mechanical heterogeneity of Earth's lithosphere leads to significant amplification of stresses across spatial scales ranging from mineral grains to tectonic plates. These stress amplifications play a key role in mechanical and chemical processes within the rock that affect bulk rock strength. Identifying the most effective causes of stress amplification is critical for understanding processes such as strain localization and fluid transport at scales ranging from microshear zones to tectonic plate boundaries. However, studies quantifying and predicting stress heterogeneities and amplifications are limited. We used numerical modelling of two-phase isotropic viscous systems to explore the factors influencing and controlling stress amplification and the potential magnitude of stress amplification in viscous regimes. We found the most geologically relevant amplification factors to be weak-phase spacing, rheological contrast and loading type. Our results indicate that stress amplification can reach a factor of ∼9 under specific conditions, but most of our experiments suggest amplifications at or below a factor of 2. Pressure differences across the model domains generally do not exceed ∼55 MPa, but some are as high as ∼110 MPa. The stress and pressure amplifications resulting from our analyses are large enough to drive a variety of geologically important processes such as failure and strain localization, as well as transient permeability and fluid migration.