Rediscovering the Mullins Effect With Deep Symbolic Regression

Abstract

The Mullins effect represents a softening phenomenon observed in rubber-like materials and soft biological tissues. It is usually accompanied by many other inelastic effects like for example residual strain and induced anisotropy. In spite of the long term research and many material models proposed in literature, accurate modeling and prediction of this complex phenomenon still remain a challenging task. In this work, we present a novel approach using deep symbolic regression (DSR) to generate material models describing the Mullins effect in the context of nearly incompressible hyperelastic materials. The two step framework first identifies a strain energy function describing the primary loading. Subsequently, a damage function characterizing the softening behavior under cyclic loading is identified. The efficiency of the proposed approach is demonstrated through benchmark tests using the generalized the Mooney-Rivlin and the Ogden-Roxburgh model. The generalizability and robustness of the presented framework are thoroughly studied. In addition, the proposed methodology is extensively validated on a temperature-dependent data set, which demonstrates its versatile and reliable performance.

Figures (6)

• Figure 1: Visualization of the deep symbolic regression process. The environment consists of a traversal of tokens where the last entry is sampled through the RNN environment. The neural network receives as input observations of the sibling and parent of the current token. The output of the agent is a probability distribution function that is used to sample the next action.
• Figure 2: Visualization of the implemented two step procedure. In the first step, the strain energy function $\Psi$ is expressed in terms of the invariants and other parameters using the data of the primary loading curve. In the second step, the damage function $\eta$ is determined using all inputs from the first step as well as the values of $\Psi$ and $\Psi_m$. The fit is performed on the cyclic loading data.
• Figure 3: Comparison of the mean stress-strain response of five models (\ref{['tab:gMRPredictions']}) obtained by DSR with the corresponding training and test data for UT, PS and EBT based on the generalized Mooney-Rivlin model for all three sets of material constants. Color surroundings of the curves reflect $6\sigma$ confidence intervals of the predictions.
• Figure 4: Comparison of the mean stress-strain response of five models obtained by DSR with the corresponding training and test data for UT, PS and EBT for the two different volumetric strain energy functions \ref{['vol-energies']}. Color surroundings of the curves reflect $6\sigma$ confidence intervals of the predictions.
• Figure 5: Comparison of the mean stress-strain response of five models (\ref{['tab:OR']}) obtained by DSR with the corresponding training and test data for UT, PS and EBT based on the generalized Mooney-Rivlin of case 1 for the primary loading and the Ogden-Roxburgh model for the softening (unloading and further reloading.) Color surroundings of the curves reflect $6\sigma$ confidence intervals of the predictions.