Local Stiffness Modeling Method for Thin-Walled Blades Based on Abaqus Secondary Development

Flexural–Torsional Buckling Behavior of FGP Curved Beams Based on a Modified Variational Method

This study presents a modified variational modeling method for the flexural–torsional buckling of functionally graded porous (FGP) curved beams. Unlike conventional methods that rely on strictly admissible functions, the proposed framework accommodates arbitrary orthogonal polynomial basis functions and segment-wise discretization. Lagrange multipliers and least-squares residuals are introduced to rigorously enforce interfacial and boundary constraints, which enables the proposed method to adapt to arbitrary boundary conditions. The energy functional accounts for shear deformation and material gradient effects, with the critical buckling loads determined numerically through a discretized matrix formulation. Due to the high orders of admissible functions employed, the accuracy of the responses in stress fields is enhanced and, in turn, the solution accuracy of the buckling properties is also increased. The accuracy and computational efficiency of the method are validated against existing results in the literature. A comprehensive parametric investigation is conducted to examine the influences of the power-law exponent, porosity distribution, slenderness ratio, boundary conditions, and loading configurations on buckling behaviors of the curved beams. This research provides a unified and high-fidelity framework for the stability design of advanced FGP structures in civil, mechanical, and aerospace engineering applications.

Position-Dependent Milling Stability Modeling and Experimental Validation for Thin-Walled Components

Thin-walled structures with weak stiffness are widely applied in aerospace, precision machinery, and mold manufacturing; however, their machining processes are commonly challenged by insufficient rigidity, complex dynamic characteristics, and a high susceptibility to chatter. Due to the differentiated dynamic parameters of these structures at various spatial positions, the compliant interaction between the tool and workpiece during milling significantly increases the risk of chatter, thereby degrading machining precision, surface integrity and productivity. To this end, this research establishes a three degree of freedom (3-DOF) milling process dynamics model based on the full discretization method (FDM), which systematically obtains the modal parameters of the thin-walled component at different locations. Building upon this, position-dependent stability lobe diagrams for milling prediction are constructed to theoretically reveal the influence of local structural regions on milling stability. Furthermore, this paper proposes a multivariate nonlinear regression method to establish a nonlinear identification model for milling force coefficients. Key parameters were effectively identified through experiments, predicting the variation trends of force coefficients under different cutting conditions. Subsequently, cutting experiments were conducted across different spatial regions of the thin-walled structure to comparatively analyze stability performance under various combinations of cutting parameters. The results demonstrate that the established dynamic model and force coefficient identification method can effectively predict the milling stability of weak-stiffness structures at different physical locations and can well explain the spatial distribution characteristics of cutting chatter. This research proposes a novel method for position-dependent milling stability prediction, providing a theoretical foundation and experimental data support for resolving the issue of frequent chatter at different locations on weak-stiffness structures in practical machining, which holds significant engineering value for the efficient and stable processing of complex thin-walled components.

Research on Machining Deformation Simulation and Prediction of Low-Stiffness Structural Components under Cutting Force

During the cutting process of low-stiffness structural components, the coupling effect between dynamic deformation and cutting forces presents a significant challenge in accurately predicting machining-induced deformations, thereby complicating quality control in the manufacturing of such parts. To address this issue, a cutting force-structural coupling simulation method that combines experiment and finite element is proposed, which takes into account the low-stiffness characteristics of structural components. Focusing on thin-plate parts as the research object, an orthogonal experimental scheme is designed considering workpiece thickness that serves as an indicator of rigidity. A milling force prediction model correlated with workpiece thickness is established. Based on the predicted cutting forces, a multi-analysis-step simulation method is introduced to analyze the machining deformation of structural parts. Additionally, a theoretical analytical model for the machining deformation of thin-plate workpieces is developed. A comparison between the theoretical and simulation results shows a relative error of less than 1.03%, validating the accuracy of the proposed simulation method. Finally, the exponential regression model for the machining deformation is constructed using training data obtained from the simulations. The prediction error of the regression model is less than 15%. The findings of this study are also applicable to predicting machining deformations in other large and low-stiffness structural components.

Forced vibration of functionally graded sandwich plates with surface irregularities under moving vehicle loads on elastic foundations

Static analysis of functionally graded porous thin-walled I-beams based on higher order shear deformation beam theory

This paper presents a static analysis of functionally graded porous (FGP) thin-walled beams based on higher-order shear deformation beam theory. The material properties are assumed to vary through the thickness of the thin-wall. Three porosity distribution models, namely uniform porosity distribution (UPD) and symmetric porosity distribution with stiffer (SPD-1) and softer at the surface (SPD-2), are considered. A mono-symmetrical thin-walled I-beam profile is analyzed under static loading conditions. The beam kinematics are formulated within the framework of higher-order thin-walled beam theory. The governing equations for the static problem are derived using Lagrange’s equations. Based on the Bézier approximation function, the Ritz method is employed to solve these equations. Additionally, the numerical results examine the effects of higher-order shear deformation, porosity coefficient, porosity distribution, and length-to-height ratio on the vertical displacement and twist angle of the beams. These outcomes aim to evaluate the model’s accuracy and establish benchmarks for future investigations.

Deep Energy Method for Large Deformation Analysis of Isotropic and Inhomogeneous Hyperelastic Ellipsoidal Pressurized Structures

The accurate prediction of displacement and stress fields in pressure vessels is essential for the safe and reliable design of these structures, particularly when dealing with nonlinear behavior such as that of hyperelastic functionally graded materials (FGMs). Conventional finite element method (FEM) approaches, while robust, often require extensive meshing and may encounter challenges in graded or strongly nonlinear domains. In this work, we present a Deep Energy Method (DEM) framework, based on physics‐informed neural networks, to analyze the axisymmetric deformation of ellipsoidal vessels subjected to internal and external pressure loading. The formulation employs automatic differentiation and a hyperelastic constitutive model to evaluate the strain‐energy functional, with pressure boundary conditions incorporated through natural work terms. Both homogeneous and exponentially graded materials are studied, enabling direct assessment of how material gradation influences the structural response. Three neural network architectures with varying depth and width are examined to evaluate the effect of model complexity on accuracy and convergence. The DEM solution is obtained by minimizing the total potential energy, comprising the internal strain energy and pressure work, to train a neural network representation of the displacement field. Convergence analyses confirm stable optimization for all cases, with graded vessels showing fast and smooth loss reduction due to the stabilizing effect of material gradation. DEM predictions of displacement agree closely with FEM benchmarks in both expansion‐type (internal pressure) and contraction‐type (external pressure) scenarios, accurately capturing global deformation modes as well as local responses near the inner and outer boundaries. To further ensure numerical robustness, a Monte Carlo sampling strategy is also employed for a selected test case, yielding displacement and von Mises stress fields fully consistent with structured‐grid DEM and FEM solutions. von Mises stress fields derived from DEM match FEM references for both homogeneous and graded vessels. While stress concentrations appear near loaded boundaries in homogeneous materials, gradation leads to smoother stress transitions, and DEM successfully reproduces these patterns. Qualitative comparisons and magnitude‐wise agreement across all network architectures demonstrate the method's capability to reflect nonlinear constitutive behavior and spatial material variation. The findings highlight several advantages of DEM: it provides smooth mesh‐free solutions, naturally accommodates material gradation, and embeds the variational energy principle directly within the learning process. The method shows strong potential for extension to more complex geometries and nonlinear multiphysics problems, although sensitivity to optimization parameters and training cost remain important considerations. It also offers a powerful and reliable alternative to FEM for the nonlinear analysis of homogeneous and functionally graded pressure vessels. The method not only reproduces FEM accuracy but also provides additional flexibility and robustness across different network architectures and sampling schemes, positioning DEM as a promising tool for next‐generation computational mechanics.