Graph neural networks (GNNs) have been shown to be a successful surrogate model for finite element method (FEM) simulations. Such neural networks make predictions on unstructured graphs and can therefore be trained on mesh structures with arbitrary numbers of nodes and elements. However, applications have focused on simple material models such as linear elasticity and, more recently, plasticity. We extend such work to consider viscoplastic materials undergoing cyclic loading with damage. First, we use GNNs to predict the entire plastic strain field in representative volume element viscoplastic simulations of porous microstructures. We study the performance of the network, how performance scales with data, and whether network architecture affects results. Next, we investigate single-variable prediction using graph convolutional networks by mapping mesh geometry to cycles to failure. Here we determine whether significantly reducing the dimension of the output space to a single variable of interest improves model accuracy. Finally, we focus on graph super-resolution by utilizing GNNs to map from low spatial resolution simulations of bolts to high fidelity results with the aim of achieving considerable speedup compared to conventional simulation. SNL is managed and operated by NTESS under DOE NNSA contract DE-NA0003525