  Accurately modeling the mechanical behavior of materials is crucial for
numerous engineering applications. The quality of these models depends directly
on the accuracy of the constitutive law that defines the stress-strain
relation. Discovering these constitutive material laws remains a significant
challenge, in particular when only material deformation data is available. To
address this challenge, unsupervised machine learning methods have been
proposed. However, existing approaches have several limitations: they either
fail to ensure that the learned constitutive relations are consistent with
physical principles, or they rely on a predefined library of constitutive
relations or manually crafted input features. These dependencies require
significant expertise and specialized domain knowledge. Here, we introduce a
machine learning approach called uLED, which overcomes the limitations by using
the input convex neural network (ICNN) as the surrogate constitutive model. We
improve the optimization strategy for training ICNN, allowing it to be trained
end-to-end using direct strain invariants as input across various materials.
Furthermore, we utilize the nodal force equilibrium at the internal domain as
the training objective, which enables us to learn the constitutive relation
solely from temporal displacement recordings. We validate the effectiveness of
the proposed method on a diverse range of material laws. We demonstrate that it
is robust to a significant level of noise and that it converges to the ground
truth with increasing data resolution. We also show that the model can be
effectively trained using a displacement field from a subdomain of the test
specimen and that the learned constitutive relation from one material sample is
transferable to other samples with different geometries. The developed
methodology provides an effective tool for discovering constitutive relations.
