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Nonlinear materials are often difficult to model with classical state modeltheory because they have a complex and sometimes inaccurate physical andmathematical description or we simply do not know how to describe suchmaterials in terms of relations between external and internal variables. Inmany disciplines, Neural Network methods have arisen as powerful tools toidentify very complex and non-linear correlations. In this work, we use thevery recently developed concept of Physically Guided Neural Networks withInternal Variables (PGNNIV) to discover constitutive laws using a model-freeapproach and training solely with measured force-displacement data. PGNNIVsmake a particular use of the physics of the problem to enforce constraints onspecific hidden layers and are able to make predictions without internalvariable data. We demonstrate that PGNNIVs are capable of predicting bothinternal and external variables under unseen load scenarios, regardless of thenature of the material considered (linear, with hardening or softening behaviorand hyperelastic), unravelling the constitutive law of the material henceexplaining its nature altogether, placing the method in what is known aseXplainable Artificial Intelligence (XAI).

Predicting and explaining nonlinear material response using deep Physically Guided Neural Networks with Internal Variables