Building an accurate constitutive model for soft materials is essential for better understanding its rate-dependent deformation characteristics and improving the design of soft material devices. To establish a concise constitutive model with few parameters and clear physical meaning, a variable-order fractional model is proposed to accurately describe and predict the rate-dependent mechanical behavior of soft materials. In this work, the discrete variable-order fractional operator enables the predicted stress response to be entirely consistent with the whole stress history and the fractional order’s path-dependent values. The proposed model is further implemented in a numerical form and applied to predict several typical soft materials’ tensile and compressive deformation behavior. Our research indicates that the proposed variable-order fractional constitutive model is capable of predicting the nonlinear rate-dependent mechanical behavior of soft materials with high accuracy and more convinced reliability in comparison with the existing fractional models, where the fractional order contains a constant initial order to depict the initial elastic response and a linear variable-order function to account for the strain hardening behavior after acrossing the yield point.

This study proposes an uncertainty quantification method based on deep neural networks and Catmull–Clark subdivision surfaces for vibroacoustic problems. The deep neural networks are utilized as a surrogate model to efficiently generate samples for stochastic analysis. The training data are obtained from numerical simulation by coupling the isogeometric finite element method and the isogeometric boundary element method. In the simulation, the geometric models are constructed with Catmull–Clark subdivision surfaces, and meantime, the physical fields are discretized with the same spline functions as used in geometric modelling. Multiple deep neural networks are trained to predict the sound pressure response for various parameters with different numbers and dimensions in vibroacoustic problems. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

The temperature–sensitive mechanical responses of thermoplastics, including nonlinear elasticity, yielding, strain softening, and hardening, present significant challenges for constitutive modeling. Existing phenomenological models often suffer from ambiguous physical interpretations and exhibit discontinuous transitions between deformation regions. To address these limitations, this study proposes a fractional-order modeling framework based on continuous transition functions. Innovatively, a modified dual-Sigmoid function is employed to achieve seamless coupling between small-strain viscoelastic and large-strain viscoplastic behaviors. The model is validated through uniaxial tensile and compressive experiments on representative thermoplastics. The results demonstrate that the developed fractional model exhibits superior prediction accuracy for the mechanical responses of PC under static or high-strain-rate loading conditions over a broad temperature range. The proposed fractional-order modeling framework provides an effective approach for simulating the thermo-mechanical behavior of thermoplastics across wide temperature ranges.