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# Evolution of internal variables in an expanding hollow cylinder at large plastic strains

- Sergei Alexandrov
^{1}Email author, - Nguyen Dinh Kien
^{2}, - Yaroslav Erisov
^{3}and - Fedor Grechnikov
^{3}

**Received:**30 August 2015**Accepted:**17 March 2016**Published:**29 March 2016

## Abstract

An efficient method for calculating the evolution of internal variables in an expanding hollow cylinder of rigid/plastic material is proposed. The conventional constitutive equations for rigid plastic, hardening material are supplemented with quite an arbitrary set of evolution laws for internal variables assuming that the material is incompressible. No restriction is imposed on the hardening law. The problem is solved in Lagrangian coordinates. This significantly facilitates a numerical treatment of the problem. In particular, the initial/boundary value problem is reduced to a system of equations in characteristic coordinates. A finite difference scheme is used for solving these equations. An illustrative example is presented assuming that the internal variables are the equivalent plastic strain and a damage parameter.

## Keywords

- Internal variables
- Hollow cylinder
- Expansion
- Large strains
- Lagrangian coordinates
- Rigid plastic material
- Finite difference method

## Background

Expansion of a hollow cylinder is one of the classical problems of elasticity and plasticity. The first strict elastic/plastic solution for the expansion of a hollow cylinder of elastic/plastic material at large strains has been provided by Hill et al. (1947). Subsequent limit analysis has been used by Leu (2007, 2009), Leu and Li (2012) to find solutions for several rigid plastic models. A constitutive equation for nonlinear viscoelasticity has been adopted by Wineman and Min (1996). The location of yield in rotating thick-walled cylindrical shells made of functionally graded materials has been determined by Fatehi and Nejad (2014). A great number of solutions at small elastic/plastic strains have been proposed for various material models (see, for example, Bland 1956; Chen 1986; Megahed and Abbas 1991; Rees 1987, 1990; Stacey and Webster 1988; Lazzarin and Livieri 1997; Livieri and Lazzarin 2002; Loghman and Wahab 1994; Farrahi et al. 2013). However, in many cases elasticity is not so important at large strains, i.e. the elastic portion of the strain rate tensor may be neglected. In particular, rigid plastic models are used even in conjunction with finite element methods (see, for example, Guo and Kamitani 2010; Cheong et al. 2014; Eom et al. 2014). On the other hand, it is important to predict the evolution of internal variables since these variables control material properties. A typical model capable of such predictions is based on the conventional system of equations of plasticity theory (i.e. a yield criterion and its associated flow rule) and evolution equations for internal variables. These equations should be solved simultaneously. Commonly used internal variables are the equivalent strain, various damage parameters (Lemaitre 1985; Chandrakanth and Pandey 1993; Besson 2010 among many others), dislocation density (Ganapathysubramanian and Zabaras 2004; Lin and Dean 2005; He et al. 2012; Hore et al. 2015) and many other parameters that characterize the microstructure of material.

It is shown in the present paper that for a generic set of evolution equations for internal variables the initial/boundary value problem for an expanding hollow cylinder is reduced to a hyperbolic system of equations. The characteristic curves of this system are coordinate lines of a Lagrangian coordinate system. Therefore, a very high accuracy of numerical solutions can be easily achieved using a finite difference method and these solutions can serve as simple benchmark tests for numerical packages that deal with the prediction of the evolution of internal variables in metal forming processes. A necessity of such tests has been pointed out by Roberts et al. (1992). Note that even in the case of linear elasticity numerical results often depend on a particular method adopted to solve the problem (Helsing and Jonsson 2002). Therefore, reliable benchmark tests are important for large strain plasticity models that include internal variables.

The method proposed is an extension of the method developed in Alexandrov and Jeng (2011). In this paper, the evolution of damage in an expanding/contracting hollow sphere has been investigated.

## Statement of the problem

*u*is the radial velocity and \(u_{0}\) is constant.

*t*is the time, \({d \mathord{\left/ {\vphantom {d {dt}}} \right. \kern-0pt} {dt}}\) denotes the convected derivative and \(\xi_{eq}\) is the equivalent strain rate. In the case under consideration, the latter is expressed in terms of the physical components of the strain rate tensor in the cylindrical coordinate system as

*F*is quite an arbitrary function of its arguments.

## Analytic treatment

*R*is a Lagrangian coordinate and \(\rho = R\) at \(b = 1\) (or \(b_{c} = b_{0}\)). It follows from Eq. (18) that

*b*,

*R*and \(\alpha_{p}\) \((1 \le p \le k).\) In the Lagrangian coordinates Eq. (10) is

*F*is determined from Eqs. (5), (6) and (11) as

*b*,

*R*, \(\sigma_{r}\) and \(\alpha_{p}\) \((1 \le p \le k)\). Thus Eqs. (23) and (24) constitute the system of equations for \(\sigma_{r}\) and \(\alpha_{p}\) in the Lagrangian coordinates. It is evident that the characteristic curves of this system are \(R = {\text{constant}}\) and \(b = {\text{constant}}\). The boundary condition (1) becomes

*F*. Analogously, Eq. (23) can be integrated along the characteristic curve \(b = 1\). In particular, using Eqs. (12) and (13)

The solution of Eqs. (27) and (29) facilitate a numerical treatment of Eqs. (23) and (24).

## Numerical treatment

*N*+ 1 nodes are chosen on the spatial coordinate and

*T*+ 1 nodes on the time-like coordinate. Therefore, the mesh used is

The value of \(\beta\) has been introduced in Eq. (15) and the value of \(b_{m}\) should be prescribed.

Replacing the derivatives in Eq. (31) by the values given in Eq. (32) a more accurate approximation of \(\sigma_{r}\) and \(\alpha_{p}\) at the point \(j = N\) and \(i = 2\) is determined. It is obvious that this procedure can be extended to \(N - 1 \ge j \ge 1.\) As a result, the distribution of \(\sigma_{r}\) and \(\alpha_{p}\) along the line \(i = 2\) is obtained. The line \(i = 3\) can be treated in a similar manner since the solution of Eq. (27) is available for any *b*. This procedure can be extended to any number *i*.

## Illustrative example

*D*is a damage parameter and \(D_{0}\) is its initial value. In this case typical functions

*F*and \({\Phi}\) are given by Hartley et al. (1997)

*M*is the Ramberg–Osgood hardening exponent, and \(q\) and

*A*are material constants. Then, Eq. (27) becomes

The values \(M = 2.4\), \(A = 1.52\) and \(q = 1\) were used in all calculations. These values are representative for 2024-T351 aluminum alloy (Hartley et al. 1997). It was initially assumed that \(N = 100\) and \(T = 200\). The accuracy of calculations was controlled by solving the boundary value problem at \(N = 200\) and \(T = 400\). The maximum difference was less than 1 %.

*D*is depicted for several values of

*b*. The maximum value of

*b*corresponds to the initiation of ductile fracture at the inner radius of the tube.

## Conclusions

The initial/boundary value problem for calculating the evolution of internal variables in an expanding hollow tube at large strains has been reduced to simple equations in characteristic coordinates. An advantage of the approach proposed is that an arbitrary hardening law and an arbitrary set of internal variable evolution equations are included in the formulation. The equilibrium equation has been integrated analytically at \(b = 1\) [see Eq. (29)]. The internal variable evolution equations can be integrated analytically or, in the most general case, reduced to numerical integration at \(R = 1\) independently of the general solution of the initial/boundary value problem [see Eq. (27)]. These solutions have been used in the numerical scheme proposed. They can also be used in conjunction with any other numerical scheme to increase the accuracy of calculation and to reveal possible errors in numerical solutions.

The illustrative example deals with the evolution of damage. It is seen from Figs. 1, 2 and 3 that the initiation of ductile fracture occurs at the inner radius of the tube in all the cases considered. It is worth noting here that the fracture initiation in an expanding thick ring may occur at mid-annulus (Tomkins and Atkins 1981). It is evident that the damaged model adopted in the present paper is not appropriate in such cases. However, the general solution found and the general numerical scheme developed are independent of the specific form of the functions \({\Phi}\) and *F*. The present general solution can be used to test various function involved in Eqs. (5) and (10) to find functions \({\Phi}\) and *F* that are in agreement with the experimental result presented by Tomkins and Atkins (1981).

The accuracy of the numerical solution is rather high. Therefore, the final result is useful for verifying more general numerical codes, which can be used to solve arbitrary initial/boundary value problems.

## Declarations

### Authors’ contributions

All the authors have participated and prepared the manuscript. All authors read and approved the final manuscript.

### Acknowledgements

The research described in this article has been supported by the Grants RFBR-14-01-93000 (Russia) and VAST.HTQT.NGA.07/14-15 (Vietnam).

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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