Recent studies have suggested that injection of solid particles such as alumina and calcite particles for stratospheric aerosol injection (SAI) instead of sulfur-based injections could reduce some of the adverse side effects of SAI such as ozone depletion and stratospheric heating. Here, we present a version of the global aerosol-chemistry-climate model SOCOL-AERv2 and the Earth system model (ESM) SOCOLv4 which incorporate a solid-particle microphysics scheme for assessment of SAI of solid particles. Microphysical interactions of the solid particle with the stratospheric sulfur cycle were interactively coupled to the heterogeneous chemistry scheme and the radiative transfer code (RTC) for the first time within an ESM. Therefore, the model allows simulation of heterogeneous chemistry at the particle surface as well as feedbacks between microphysics, chemistry, radiation and climate. We show that sulfur-based SAI results in a doubling of the stratospheric aerosol burden compared to the same mass injection rate of calcite and alumina particles with a radius of 240 nm. Most of the sulfuric acid aerosol mass resulting from SO₂ injection does not need to be lifted to the stratosphere but is formed after in situ oxidation and subsequent water uptake in the stratosphere. Therefore, to achieve the same radiative forcing, larger injection rates are needed for calcite and alumina particle injection than for sulfur-based SAI. The stratospheric sulfur cycle would be significantly perturbed, with a reduction in stratospheric sulfuric acid burden by 53 %, when injecting 5 Mtyr^-1 (megatons per year) of alumina or calcite particles of 240 nm radius. We show that alumina particles will acquire a sulfuric acid coating equivalent to about 10 nm thickness if the sulfuric acid is equally distributed over the whole available particle surface area in the lower stratosphere. However, due to the steep contact angle of sulfuric acid on alumina particles, the sulfuric acid coating would likely not cover the entire alumina surface, which would result in available surface for heterogeneous reactions other than the ones on sulfuric acid. When applying realistic uptake coefficients of 1.0, 10^-5 and 10^-4 for H₂SO₄, HCl and HNO₃, respectively, the same scenario with injections of calcite particles results in 94 % of the particle mass remaining in the form of CaCO₃. This likely keeps the optical properties of the calcite particles intact but could significantly alter the heterogeneous reactions occurring on the particle surfaces. The major process uncertainties of solid-particle SAI are (1) the solid-particle microphysics in the injection plume and degree of agglomeration of solid particles on the sub-ESM grid scale, (2) the scattering properties of the resulting agglomerates, (3) heterogeneous chemistry on the particle surface, and (4) aerosol-cloud interactions. These uncertainties can only be addressed with extensive, coordinated experimental and modelling research efforts. The model presented in this work offers a useful tool for sensitivity studies and incorporating new experimental results on SAI of solid particles.