PINN-Based Solution of the Fourth-Order Warping Equation for Size-Dependent Torsion of Rectangular Micro-Bars

Size-dependent torsional behavior in micro-scale structures cannot be captured by classical elasticity, motivating the use of strain gradient elasticity with intrinsic material length-scale parameters. However, the resulting higher-order governing equations and boundary conditions make conventional analytical and numerical methods inefficient, particularly for complex cross-sections. In this study, the Saint-Venant torsion of micro-bars is formulated within the strain gradient framework following Tong et al., adopting the equal length-scale assumption. A Physics-Informed Neural Network (PINN) is developed as a mesh-free solver for the resulting fourth-order boundary value problem by embedding the governing PDE and boundary conditions directly into the loss function. The proposed PINN accurately predicts the warping field and torsional constant while capturing the nonlinear size-dependent stiffening induced by couple-stress effects. Validation against Tong’s analytical solution shows excellent agreement, with maximum discrepancies below ۱%. Parametric investigations further demonstrate how the material length scale and cross-sectional aspect ratio influence warping suppression and effective torsional rigidity. The results confirm that PINNs provide a robust and efficient alternative for solving strain-gradient torsion problems in micro- and nano-scale structural analysis.

کلیدواژه ها:

Physics-Informed Neural Networks (PINNs) ، Torsion ، Couple stress theory ، Strain gradient elasticity ، Microstructure