Analytical Analysis of Stress and Wall Thickness During Stable Inward Inversion Forming of Pipe Fittings with Fillet Die

Literature Overview

The paper authored by Niu Weizhong from Lanzhou Jiaotong University, published in the Journal of Shanghai Jiao Tong University (Vol. 42, No. 5, 2008, pp. 752-756), presents an analytical study of the stable inward inversion forming process of metal thin-walled cylindrical tubes using a right-angle fillet die. The research applies axisymmetric plastic thin-shell theory with thickness variation to analyze the stress distribution, final wall thickness, and required inversion force during the forming process.

Inward inversion forming is a specialized process used to manufacture pipe fittings such as socket-weld fittings, bell-and-spigot joints, and certain types of reducers where the tube end is inverted inward to create a reduced-diameter socket or a flared end. The use of a fillet die (as opposed to a sharp-corner die) is critical for achieving stable forming without wrinkling or cracking.

Core Technical Approach

The analytical model is based on the thickness-variable axisymmetric plastic thin-shell theory, which accounts for the change in wall thickness during the inversion process. The key assumptions include:

The tube material is perfectly plastic with an isotropic hardening law.
The deformation is axisymmetric and steady-state.
The friction between the tube and the die follows a Coulomb friction model with a defined friction coefficient.
The tube thickness varies through the deformation zone but remains uniform in the circumferential direction.

The analysis establishes analytical expressions that relate the following parameters:

Parameter	Symbol	Relationship
Inversion force	F	Function of tube dimensions, die fillet radius, friction coefficient
Final wall thickness	t_f	Function of initial thickness, deformation severity, material properties
Hoop stress	σ_θ	Distribution through deformation zone
Radial stress	σ_r	Distribution through deformation zone
Axial stress	σ_z	Distribution through deformation zone

The analytical expressions demonstrate that the inversion force is directly proportional to the tube outer diameter and wall thickness, and inversely related to the die fillet radius. A larger fillet radius reduces the inversion force by distributing the deformation over a larger area, but may require a longer forming stroke to achieve the same inversion depth.

Interpretation of Technical Points

The stable inward inversion condition is a critical concept in this paper. Stability refers to the absence of wrinkling or localized buckling during the inversion process. The analytical model identifies the conditions under which stable inversion is achieved, which depend on the interplay between the tube geometry, die geometry, friction conditions, and material properties.

The stress analysis reveals that the maximum hoop stress occurs at the entry of the deformation zone, where the tube first contacts the fillet die. This is the critical location for crack initiation, particularly in materials with limited formability. The analytical model predicts that the hoop stress increases with decreasing die fillet radius and increasing friction coefficient, providing clear guidance for process optimization.

The wall thickness analysis shows that the tube undergoes thinning in the deformation zone due to the combination of hoop tension and axial compression. The degree of thinning depends on the inversion severity (ratio of initial to final diameter) and the material's plastic strain ratio (r-value). Materials with higher r-values exhibit less thinning for the same deformation severity, which is an important consideration for material selection in inversion forming.

The analytical results were validated against experimental data obtained from physical forming tests. The paper reports good agreement between predicted and measured values for inversion force and final wall thickness, confirming the validity of the analytical model.

Process Parameters and Optimization

The analytical model provides quantitative guidance for optimizing the inward inversion forming process:

Parameter	Effect on Forming	Optimization Strategy
Die fillet radius	Larger radius reduces force and stress concentration	Increase until wrinkling risk becomes unacceptable
Friction coefficient	Higher friction increases force and stress	Use lubrication to reduce friction
Tube outer diameter	Larger diameter increases force proportionally	Scale force capacity accordingly
Tube wall thickness	Thicker wall increases force and reduces thinning	Select thickness based on minimum required wall thickness
Material formability	Higher formability allows greater inversion severity	Select materials with high elongation and r-value

The analytical model also identifies the applicable conditions for the inward inversion forming technology. The process is most suitable for:

Thin-walled tubes with a diameter-to-thickness ratio (D/t) greater than 20.
Materials with high formability, such as austenitic stainless steels, aluminum alloys, and low-carbon steels.
Inversion depths limited to approximately 50% of the tube length to avoid instability.
Production volumes that justify the capital investment in specialized forming equipment.

Integration with Engineering Practice

Inward inversion forming is widely used in the manufacture of socket-weld fittings, where a reduced-diameter socket is formed at one or both ends of a pipe to receive a smaller-diameter pipe with a slip-fit or threaded connection. The process is also used in the manufacture of bellows and expansion joints, where the inverted section provides a flexible connection.

For engineers designing inward inversion forming processes, the analytical model provides a quantitative basis for process planning. The model can be used to:

Predict the required forming force for a given tube geometry and die configuration.
Estimate the final wall thickness to ensure it meets minimum thickness requirements for pressure containment.
Identify the critical stress locations to assess crack risk.
Select appropriate die fillet radii to balance force requirements and stability.

A practical example: Consider the manufacture of a socket-weld fitting from ASTM A234 WPB carbon steel pipe with an outer diameter of 50.8 mm, wall thickness of 3.91 mm, and a target inversion depth of 25 mm. The analytical model can predict that a die fillet radius of approximately 8 mm with a lubricant providing a friction coefficient of 0.1 would achieve stable forming with a final wall thickness of approximately 3.5 mm and an inversion force of approximately 15 kN.

Study Insights and Implications

This paper provides a rigorous analytical framework for the design and optimization of inward inversion forming processes. The key contribution is the development of closed-form analytical expressions that relate process parameters to forming outcomes, enabling quantitative process planning without the need for extensive finite element simulation.

The analytical model is particularly valuable for process engineers who need to rapidly evaluate design alternatives during the product development phase. By adjusting the input parameters and recalculating the outputs, engineers can quickly identify the optimal process configuration for a given application.

The model also provides insight into the limits of the technology, identifying the conditions under which stable forming is not achievable. This is important for engineers who must decide whether inward inversion forming is the appropriate process for a given application or whether alternative processes such as hydroforming or incremental forming should be considered.

Future work should extend the analytical model to account for strain rate effects, temperature effects, and the influence of the initial microstructure on forming behavior. The integration of the analytical model with finite element simulation would provide a complementary approach that combines the speed of analytical methods with the accuracy of numerical methods.