Description of the Expansion of a Two-Layer Tube: An Analytic Plane-Strain Solution for Arbitrary Pressure-Independent Yield Criterion and Hardening Law

Abstract

The main objective of the present paper is to provide a simple analytical solution for describing the expansion of a two-layer tube under plane-strain conditions for its subsequent use in the preliminary design of hydroforming processes. Each layer’s constitutive equations are an arbitrary pressure-independent yield criterion, its associated plastic flow rule, and an arbitrary hardening law. The elastic portion of strain is neglected. The method of solution is based on two transformations of space variables. Firstly, a Lagrangian coordinate is introduced instead of the Eulerian radial coordinate. Then, the Lagrangian coordinate is replaced with the equivalent strain. The solution reduces to ordinary integrals that, in general, should be evaluated numerically. However, for two hardening laws of practical importance, these integrals are expressed in terms of special functions. Three geometric parameters for the initial configuration, a constitutive parameter, and two arbitrary functions classify the boundary value problem. Therefore, a detailed parametric analysis of the solution is not feasible. The illustrative example demonstrates the effect of the outer layer’s thickness on the pressure applied to the inner radius of the tube.

Latest Hydroforming Technology of Metallic Tubes and Sheets

“…The contributions are generally divided into three basic groups according to the workpiece geometry and forming methods. The first group is tube hydroforming (THF) [1][2][3][4][5][6][7][8][9], in which a hydraulic media and dies are used to deform a tube workpiece. The second group is tube forming [10][11][12][13][14], in which only dies are used to deform a tube workpiece.…”

“…Han and Feng [8] investigated the circumferential material flow using overlapping blanks with axial constraints in tube hydroforming of a variable-diameter part. Alexandrov et al [9] proposed a simple analytical solution for describing the expansion of a two-layer tube under plane-strain conditions with an arbitrary pressure-independent yield criterion and a hardening law. The results can be applied to the preliminary design of hydro-expansion processes.…”

“…Alexandrov et al [9] proposed a simple analytical solution for describing the expansion of a two-layer tube under plane-strain conditions. Each layer's constitutive equations consist of an arbitrary pressure-independent yield criterion, its associated plastic flow rule, and an arbitrary hardening law.…”

A New Semi-Analytical Solution for an Arbitrary Hardening Law and Its Application to Tube Hydroforming

“…It has been demonstrated in [ 9 ] that a plane strain solution for an expanding two-layer hollow cylinder can be useful for analyzing tube hydroforming processes. However, the loading path is very important for designing such processes (for example, [ 10 ]).…”

An Analytic Solution for an Expanding/Contracting Strain-Hardening Viscoplastic Hollow Cylinder at Large Strains and Its Application to Tube Hydroforming Design

“…Despite the wide use of numerical methods for hydroforming process designs [20,21], simple semi-analytic solutions are essential for estimating the optimal values of the process parameters [22,23]. In particular, it has been shown in [24] that a simple plane-strain solution for the expansion of a two-layer tube supplies an accurate prediction for the maximum internal pressure value. Apart from the intrinsic utility above, the present solution is also useful for testing the accuracy of numerical solutions.…”