Analytical Study of Stress and Wall Thickness During Stable Internal Inversion Forming of Pipe Fittings with Fillet Dies

Literature Overview

This paper by Niu Weizhong from Lanzhou Jiaotong University, published in the Journal of Shanghai Jiaotong University in 2008 (Volume 42, Issue 5, pages 752-756), presents an analytical study of the stress state and wall thickness distribution during the stable internal inversion forming of metal thin-walled cylindrical tubes using a right-angle fillet die. The research combines theoretical analysis based on axisymmetric plastic thin-shell theory with experimental validation, providing analytical formulas for predicting forming parameters that are essential for industrial application.

Process Description and Forming Mechanism

Internal inversion forming, also known as inside-inversion or internal turning, is a metal forming process in which a cylindrical tube is axially compressed by a punch through a die with a fillet radius, causing the tube to invert into itself. The process is analogous to external inversion but occurs on the inside of the tube, producing a double-wall section that can be used for manufacturing various pipe fittings and components.

The stable forming regime is characterized by a uniform inversion length along the tube, with the inversion front advancing at a constant velocity without wrinkling or buckling. Achieving stable forming requires careful control of the punch force, die geometry, and friction conditions. The authors focus on the right-angle fillet die configuration, which is the most common industrial configuration for this process.

Theoretical Analysis Framework

The analysis is based on the thickness-change axisymmetric plastic thin-shell theory, which accounts for the through-thickness stress variation and the change in wall thickness during forming. This is a more advanced approach than the classical membrane theory, which assumes constant thickness and through-thickness stress uniformity.

The key variables in the analysis include the punch force, the stress state at various locations in the tube, the final wall thickness distribution, and the required forming parameters. The authors establish relationships between these variables and the geometric parameters of the tube and die, as well as the friction conditions.

Stress State Analysis

The authors identify several critical stress regions in the forming process. The inversion zone, where the tube wall bends over the die fillet, experiences the highest stress concentration and is the critical region for failure. The compressed zone, where the tube is being pushed inward by the punch, experiences compressive axial stress and tensile circumferential stress. The free zone, the uninverted portion of the tube, experiences a complex stress state influenced by the boundary conditions at the inversion front.

The analytical formulas derived by the authors provide expressions for the radial stress, circumferential stress, and axial stress at any point in the forming zone as functions of the local geometry, material properties, and friction conditions. These expressions account for the thickness change during forming, which is a significant improvement over simplified analyses that assume constant thickness.

Wall Thickness Distribution

One of the most important practical aspects of internal inversion forming is the prediction of the final wall thickness distribution. The inversion process causes significant thinning in the bend region due to the combined effects of bending and stretching. The authors derive analytical expressions for the thickness at the bend apex, the bend surface, and the flat sections, as functions of the initial wall thickness, die fillet radius, and material properties.

The thickness reduction is most severe at the outer surface of the bend, where the material undergoes the greatest strain. The analytical model predicts that the thickness reduction increases with decreasing die fillet radius and increasing friction, which is consistent with experimental observations. The model also captures the effect of material strain hardening, which influences the thickness distribution through the stress state.

Experimental Validation

The authors conducted experiments on steel tubes with various diameters and wall thicknesses to validate the analytical predictions. The experimental setup included a hydraulic press with load measurement, a die with a right-angle fillet, and instruments for measuring the punch force and the final wall thickness. The experimental results showed good agreement with the analytical predictions, with deviations typically within 10-15%.

The experiments also revealed the influence of process parameters on the forming quality. The optimal die fillet radius was found to be in the range of 2 to 5 times the initial wall thickness, which provides a good balance between forming force and thickness reduction. Friction reduction through lubrication was shown to significantly reduce the required punch force and improve the thickness uniformity.

Engineering Practice Implications

Internal inversion forming is used in the production of various pipe components including double-wall tubes, heat exchanger tubes, and certain types of pipe fittings. The analytical formulas provided by the authors enable engineers to predict the required punch force, estimate the final wall thickness, and optimize the process parameters before conducting expensive trials. This is particularly valuable for new product development and process qualification.

The analysis also provides insight into the failure modes of the process. Excessive punch force can cause tensile fracture at the bend apex, while insufficient force can lead to wrinkling or unstable inversion. The analytical model helps identify the safe operating window and provides guidelines for avoiding these failure modes.

Key Insights and Reflections

The analytical approach taken by the authors is well-suited to the problem and provides valuable physical insight into the forming mechanics. The use of thickness-change theory is a significant improvement over simplified models and captures the essential physics of the thickness reduction during forming. The analytical formulas are practical and can be used for quick engineering estimates without requiring complex numerical simulation.

However, the analysis assumes axisymmetric deformation and uniform material properties, which may not hold in all practical situations. The presence of weld seams, material anisotropy, and thickness variations can affect the forming behavior and should be considered in detailed process design. The analytical model serves as a first-order approximation that captures the dominant effects, but numerical simulation may be required for detailed analysis of specific applications.

The work also highlights the importance of understanding the fundamental mechanics of metal forming processes. Despite the availability of advanced finite element analysis tools, analytical models remain valuable for providing physical insight, enabling rapid parameter estimation, and serving as a basis for process design and optimization. The combination of analytical theory with experimental validation provides a robust understanding that is essential for reliable industrial application.

In summary, this paper presents a rigorous analytical framework for the stable internal inversion forming of pipe fittings with fillet dies. The theoretical analysis based on thickness-change plastic thin-shell theory provides practical formulas for predicting forming parameters, and the experimental validation confirms the accuracy of the predictions. The analytical approach offers valuable physical insight and practical engineering value, making it a useful tool for process design and optimization in pipe fitting manufacturing.