Plane strain bending under tension of a functionally graded sheet at large strains as an ideal flow process

Ideal plastic deformations (ideal flows) have been defined as solenoidal smooth deformations in which an eigenvector field associated everywhere with the greatest (major) principal rate of deformation is fixed in the material. In the case of plane strain deformation of rigid perfectly plastic material obeying an arbitrary isotropic yield criterion and its associated flow rule, it is always possible to find an equilibrium stress field which is compatible with an ideal deformation. It is shown in the present paper that an ideal deformation is possible for functionally graded sheets in the process of plane strain bending under tension. In contrast to the general process, the tensile force and bending moment cannot be prescribed arbitrary but should be found from the solution.


Introduction
Ideal plastic flows are those for which all material elements follow minimum work paths. The ideal flow theory has long been associated with the Tresca yield criterion [1,2]. Recently, it has been demonstrated that ideal flow solutions exist in anisotropic plasticity [3]. In particular, the theory of anisotropic plasticity proposed in [4] has been adopted. A comprehensive overview of the ideal flow theory and corresponding solutions has been provided in [5]. This theory has been widely used as the basis for inverse methods for the preliminary design of bulk metal forming processes driven by minimum plastic work (see, for example, [6]). The process of bending under tension is widely used in metal forming technologies. Using a number of assumptions concerning the through-thickness distribution of strains analytic plane-strain solutions at large strains have been proposed in [7,8]. Ignoring the transverse stress a solution has been obtained in [9]. An efficient semi-analytic approach to analysis of plane strain bending under tension has been proposed in [10]. This approach is an extension of the corresponding approach for the process of pure bending [11]. In the present paper, the approach developed in [10] is used in conjunction with the ideal flow theory to design the process of plane strain bending under tension of functionally graded sheets. Other ideal flow solutions for this process have been proposed in [12,13]. For incompressible materials an approach to analysis of plane strain pure bending at large strains has been proposed in [11]. This approach has been extended to bending under tension in [10]. For isotropic material models kinematics of the process of bending under tension is independent of constitutive equations others than the equation of incompressibility. For completeness, the basic equations derived in [10] are presented in this section. The process of bending under tension is described by the following mapping between Eulerian Cartesian coordinates  

Kinematics
where H is the initial thickness of the sheet, s is an arbitrary function of a, a is a function of the time, t, and 0 a  at for 0 a  . Using (1) and (2) it is possible to verify by inspection that xH   and yH   as 0 a  . The initial shape of the sheet is a rectangular. With no loss of generality it is possible to assume that the sides of this rectangular are given by the equations xH  (or 1 ). It is convenient to introduce a moving plane polar coordinate and 0 y  . Then, it is possible to find from (1) Here h is the current thickness of the sheet. The straight boundaries of this sector are given by

.
aL H   The coordinate curves of the Lagrangian coordinate system are principal strain rate trajectories. The principal strain rates can be found from (1)  It will be shown in subsequent sections that this description of kinematics is compatible with stress equations for the model chosen.

Material model
The material is supposed to be rigid plastic. In particular, the elastic portion of the strain tensor is neglected. It is assumed that the principal axes of the strain rate and stress tensors coincide. Then, the coordinate curves of the Lagrangian coordinate system are principal stress trajectories. The plane strain yield criterion at the initial instant is   where   and   are the deviatoric stresses in the Lagrangian coordinate system. Moreover, it is seen where r  and   are the deviatoric stresses in the polar coordinate system. The associated flow rule is automatically satisfied because the mapping (1) satisfies the equation of incompressibility and, by assumption, the principal axes of the strain rate and stress tensors coincide. Thus the only constitutive equation to be satisfied is the yield criterion.

General stress solution
The sheet is loaded by bending moment M and force F per unit length. A consequence of the equilibrium equations is that some pressure P should be applied to the concave surface of the sheet in the process of deformation. Assuming that the pressure is uniformly distributed over the surface it is possible to find that 1 F r P  . Using equation (4)  Note that 1 m  in the process of pure bending of a homogeneous rigid perfectly plastic sheet if 0  is the yield stress in uniaxial tension [14]. Replacing in (13) integration with respect to r with integration with respect to  by means of (3) results in 0 0 1 3 .
Taking into account (10) The boundary condition to this equation is P    for 1   where P is given in (11). The solution of equation (16)  by means of (7). If f is prescribed then (22) is an ordinary differential equation for s as a function of a. This equation should satisfy the boundary condition (2). However, in the case of ideal deformation f should be found from the solution.

Ideal deformation
It follows from the conditions imposed on ideal deformation (see, for example, [5]) that The initial value of F should be prescribed,

Conclusions
A new ideal flow solution for the process of bending under tension of functionally graded sheets has been presented. The through -thickness distribution of the yield stress is characterized by the function   gx involved in (8) and is quite arbitrary. A review of material property distributions in functionally graded sheets used in the mechanics of functionally graded material structures has been provided in [15]. Any of these distributions can be combined with the new solution. The solution is practically analytic. A numerical technique is only necessary to evaluate ordinary integrals in (14), (17), (20) and (24).