Physical simulation relies on spatially-varying mechanical properties, typically laboriously hand-crafted. We present the first feed-forward model to predict fine-grained mechanical properties, Young’s modulus ($E$), Poisson’s ratio ($\nu$), and density ($\rho$), throughout *the volume* of 3D objects. Our model supports any 3D representation that can be rendered and voxelized, including Signed Distance Fields (SDFs), Gaussian Splats and Neural Radiance Fields (NeRFs). To achieve this, we aggregate per-voxel multi-view features for any input, which are passed to our trained Geometry Transformer to predict per-voxel material latent codes. These latents reside on the trained manifold of physically plausible materials, which we train on a real-world dataset, guaranteeing the validity of decoded per-voxel materials. To obtain object-level training data, we propose an annotation pipeline combining knowledge from segmented 3D datasets, material databases, and a vision-language model. Experiments show that VoMP estimates accurate volumetric properties and can convert 3D objects into simulation-ready assets, resulting in realistic deformable simulations and far outperforming prior art.