The identification of heterogeneous shear modulus (SM) distributions in solids using displacement or strain fields is of significant importance for applications such as nondestructive medical diagnosis and structural damage detection. In recent years, deep learning methods have been widely used to achieve elastography by utilizing large amounts of training data based on supervised learning techniques. However, the acquisition of sufficient and high-quality full-field training data remains a significant challenge. To address this issue, we propose an unsupervised learning method based on deep image priors (DIP) that directly solves the elastic inverse problem without relying on training data. We use a backpropagation-based DIP network to reconstruct the shear modulus distribution of the domain of interest and integrate it with a finite element solver to optimize the reconstruction results. This method does not require pretraining of the network model and does not depend on any training datasets, enabling image reconstruction of arbitrary shear modulus distributions. From the simulated and experimental data, we observe that the proposed method is capable of achieving excellent reconstruction results, demonstrating the effectiveness of this inverse method. Overall, the unsupervised learning method we propose addresses the limitation of deep learning in elastography regarding the demand for a large number of training datasets and shows great potential on the improvement of the accuracy of shear modulus reconstruction.