# Dealing with internal pressure in free hydraulic bulging

## Predicting results with FEA

*For hydraulic tube bulging, direct pressure control is the most commonly used process. Pressure control allows engineers to determine the correct capacity hydraulic system and, more importantly, prevent tube rupture. However, inflow control, or control of the volume of fluid inside the tube, theoretically could be another viable hydroforming process. Finite element analysis has shown that inflow control could allow engineers to more accurately predict deformation behavior and therefore enhance the hydroforming process.*

Finite element analysis (FEA) is a numerical process that treats a single item, such as a metal tube, as a large number of small, individual components, or finite elements. It performs calculations on each of the elements and uses the result to predict the deformation behavior of the entire item. FEA is used to create a visual representation of the item under analysis. This process is the finite element method, or FEM.

FEA is a convenient method of analyzing the tube hydroforming process.

However, achieving an accurate analysis of tube hydroforming is a complicated task because hydroforming isn't a single, simple process. It comprises several other metalworking processes, such as bending, forming, and expanding. Each of these processes has unique challenges and poses its own difficulties in getting reasonable results from FEM simulation. Improvements in tube hydroforming, therefore, can be achieved after making improvements in its element forming methods.The conventional method of controlling the tube hydroforming process is direct pressure control. Direct pressure control focuses on manipulating the pressure of the forming medium. However, it doesn't always provide accurate FEA results. Another method, inflow control, focuses on the volume of forming medium.

In experiments with free hydraulic bulging (FHB), which expands a tube without dies, inflow control has been shown to have several advantages over direct pressure control. Specifically, inflow control can be advantageous for evaluating the effects of material properties on tube deformation; is more suitable for an FEM simulator based on the dynamic explicit method; and can lead to new ways to design the loading paths in tube hydroforming.

It is usually assumed that manufacturing speed has no effect on the material's deformation behaviors in forming metal sheet and tube. However, solutions provided by FEM simulators based on the dynamic explicit method innately include the effect of manufacturing speed. This is because dynamic simulators originally were intended for analyzing the effect of impact loading on a material's deformation. Therefore, researchers who know the principles of FEM simulators are cautious about using dynamic simulators to analyze metal forming processes.

On the other hand, the robust calculation procedures of dynamic simulators are attractive in many practical metal forming applications. In addition, it is possible with dynamic simulators to get the calculated results in much less calculation time than conventional FEM simulators based on the static implicit method.

In any case, researchers need to consider the effect of manufacturing speed, especially when using dynamic FEM simulators.

In the research that led to this article, the effect of speed with the dynamic simulator appears mainly in the manufacturing force or, more specifically, internal pressure. The higher the speed is set, the higher the pressure is estimated in the calculated results. Material properties, k value (plastic coefficient in the nth power hardening law), and r value (coefficient of anisotropy) also have an effect on the amount of pressure under a constant n value. Hence, the effect of the manufacturing speed interferes in the effort to get accurate solutions for the material's properties.

When conventional static simulators are used to analyze FHB, the problem caused by manufacturing speed disappears. However, another problem remains—determining the maximum pressure that can load the tube.

Using inflow control of the forming medium can solve the problems associated with both dynamic and static simulators.

## Inflow Control Versus Direct Pressure Control

The pressure medium pulls double duty inside the tube during hydraulic bulging. One job is to pressurize the tube, and the other is to fill it up. It is impossible to control both pressure and volume simultaneously, because their relationship is like two sides of the same coin (see **Figure 1**).

Engineers tend to focus on the amount of pressure, because controlling the pressure prevents tube bursting or leakage. Engineers pay attention to the pressure for other reasons, such as determining the hydraulic pump size needed to run the hydroforming press and preventing accidents caused by overpressurizing the tube.

Inflow control is another option. It focuses on the amount of forming medium inside the tube. Inflow control's advantage is that it can be easier to estimate the total volume of hydroformed product than optimum forming pressure.

When direct pressure control is used, the time-pressure relationship is used as the input data for the FEM simulator. For the inflow control process, the time-volume relationship is used.

What Is Free Hydraulic Bulging? FHB is a simple forming method in which tube is pressurized without forming dies. Internal pressure attains the maximum value during FHB. The maximum value of internal pressure and other values, such as radial expansion and change in wall thickness, are important because they reflect both strain-hardening properties and the size of the bulged tube.

Experimental Conditions. An FEM simulator that is based on the dynamic explicit method was used in this study. Tube materials are virtual ones that follow the von Mises yield criterion and nth power hardening law. The tubes have an OD of 40 millimeters and a wall thickness of 2 mm. Their properties are:

- n = 0.2, 0.4, and 0.6
- k = 200 newtons per mm
^{2}, 400 N/mm^{2}, and 600 N/mm^{2}

Using the formula s = k en results in nine sets of properties.

The tubes are expanded freely by internal pressure only. Axial displacement and rotational movement are not allowed at either end of the tube during the forming process. The modulus of volume elasticity for modeling the pressure medium is set the same as for the tube material.

For inflow control, inflow speed is generalized using the initial tube volume (see **Figure 2**).

## Evaluating the Effects of Material Properties on Tube Deformation

The relation between internal pressure and radial expansion is determined, in part, by the effects of the material's properties on tube deformation. Theoretically, this relationship can be obtained without computer calculations, using an assumption—that the tube has infinite length—and several concepts, such as the membrane theory, total strain theory, the von Mises yield criterion, and nth power hardening law.

It would be expected that an FEM simulator would produce the same results as the simple theoretical analysis. However, the calculated relationships depend on the loading speed of the internal pressure in direct pressure control, especially when FEA is based on the dynamic explicit method. Because the internal pressure has a maximum value, it is difficult to determine the relationship between time and pressure in advance of the calculation. Because of this difficulty, inflow control is more useful than conventional direct pressure control.

An FEM model for a tube with infinite length shows that all the elements are stretched in their circumferential direction without longitudinal (X direction) strain during tube deformation (see **Figure 3**).

When using the direct pressure control method, the FEM simulator obtains results that are similar to the theoretical results (see **Figure 4**). On the other hand, when using inflow control, the simulator replicates theoretical results accurately at any of the given speeds.

The values of the maximum pressure depend on the bulged length. It is difficult to obtain theoretical results when conceptualizing tubes with finite length. However, the necessary equilibrium equations were solved in 1982.^{1}

Maximum values of internal pressure and radial expansions at the maximum pressure are determined by theoretical results in the figures for tube with n = 0.4 and n = 0.6 (see **Figure 5**).^{2}These results were generated by using an FEM model for a tube with finite length (see **Figure 6**). Tube deformation and thickness distribution at the maximum pressure
depend on the ratio of length to initial OD (see introductory illustration).

## FEM Simulator Based on the Dynamic Explicit Method

An FEM simulator based on the dynamic explicit method has become a commonly used tool for analyzing metal forming processes. The simulator solves dynamic equations. However, metal forming processes should be modeled as a static process.

The ratio between internal energy and kinetic energy determines the static degree of the calculated results. A higher ratio indicates that internal energy is much larger than kinetic energy, which, in turn, ensures the calculated results are obtained under nearly static deformation conditions (see **Figure 7**).

For a tube with infinite length, the relationship between energy ratio and radial expansion for direct pressure control is significantly different than that for inflow control. In the case of direct pressure control, the energy ratio steadily decreases during tube deformation. This tendency indicates that the deformation behavior under the conventional pressure control gradually becomes dynamic and unstable. Although the same relationship for inflow control varies, it is generally more stable than the conventional pressure control.

A theoretical large-scale expansion using direct pressure control with the dynamic explicit method is a result of suddenly inflating a tube (see **Figure 8**). Soon after this deformation, the calculation reaches compulsory termination (the program crashes).

## New Methods of Designing Loading Paths

The loading path, which consists of internal pressure and axial feeding of the tube, is the decisive factor that determines the success or failure when hydroforming tube. Axial feeding is useful for applications that have a large degree of tube expansion. Finding the optimum balance between internal pressure and axial feeding is necessary for successful hydroforming. Excessive axial feeding causes buckling and folding of tube wall. On the other hand, excessive internal pressure causes tube bursting.

Regardless of the application, the FEM simulator must have enough functions to follow deformation through the forming process. All the tube deformation, including failures, is the result of tube deformation history. Tracing tube deformation with direct pressure control, especially using FEM simulation based on the dynamic explicit method, doesn't predict future results accurately.

However, direct pressure control still is the only practical control method. Although inflow control shows potential for improving hydroforming, it is still a theoretical idea and will not be implemented until a valid, practical loading path that corresponds to inflow control is established.

*Atsushi Shirayori is a research associate of the Mechanical Systems Engineering Department and a member of the engineering faculty of Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi 321-8585, Japan, phone and fax 81-28-689-6059, shira@cc.utsunomiya-u.ac.jp,
www.mech.utsunomiya-u.ac.jp.*

*The author acknowledges the Grant-in-Aid for Encouragement of Young Scientists #13750665 (by Japan Society for the Promotion of Science) for funding a part of this research.*

*Notes*

*1. Sadakatsu Fuchizawa, Kyoji Kamei, and Michiharu Narazaki, "A Numerical Analysis on Bulge Forming of Thin Tube of Fixed Total Length," (in Japanese), Journal of the Japan Society for Technology of Plasticity, Vol. 23, No. 255 (1982), pp. 351-356.*

*2. Ibid.*

*This article is adapted from a conference paper, "Free Hydraulic Bulging of Tube by Inflow Control of Pressure Medium," written by Atsushi Shirayori, Sadakatsu Fuchizawa, Michiharu Narazaki, and Haruhisa Ishigure, in conference proceedings from the 5th International Conference and Workshop on Numerical Simulation of 3D Sheet Forming Processes, Vol. 1 (2002), pp. 441-446.*

### Atsushi Shirayori

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