May 2021 Volume 3

FORGING RESEARCH

The approximations and limitations in the modeling software and applications may create inaccuracy or undesirable results that contradict what was sought from the simulation, and consequently may cause a dilemma for the engineers in interpreting the simulation results. These limitations could also make it sometimes difficult or uncertain to provide a sound, reliable reference for decision making in a process development. Hence, the modeling can only be used as an assistive tool. Other analytical or experimental tools must be adopted as well to calibrate and validate the simulation results so that a trustworthy conclusion can be reached. 5.2 Brief introduction to finite element formulations The finite element method is a numerical procedure for stress, thermal and many other types of analyses. It is applied to problems that are too complex to be solved by classical or closed-form analytical methods. The finite element concept came about because the object to be studied needed to be discretized into a group of small parts or finite elements on which the calculations were performed. Understandably, when an object with complex geometry is divided into small elements, the analysis becomes feasible, versatile and easy. It is also understood that because of this discretization, the solution provided by the analysis is approximate – although the approximation can approach the true solution very closely if the right element type is selected, a large enough number of elements is assigned and all modeling parameters are representative of the actual conditions. The centerpiece of the finite element method is the finite element formulation, from which the relationship of the loads and displacements (deformations) is established and the stresses, strains and other state variables can be calculated. The finite element formulation can have different forms, but only two basic and commonly used forms, namely the implicit formulation and An implicit (or “standard,” as it is usually called) finite element algorithm, commonly used in quasi-static cases, is formulated from the virtual power principle on the pseudo-time increment basis. [8] (Be aware that many formulations are from the virtual work principle on the displacement basis. The only difference between the two is that the former is virtual velocity-based and the latter is virtual displacement-based.) It is constructed from the law of energy conservation, which states that the work rate causing plastic deformation must equal the work rate imposed by the body force (i.e., gravity force) and the external forces acting on the body’s surfaces (“work rate” is equivalent to “work” in the virtual work formulations). The algorithm can be applied in the Lagrangian space, meaning that the mesh points are always coincident with the material particles or in the Eulerian space, meaning that the mesh is fixed while the material particles flow through it. The general rate form of the standard finite element algorithm is: explicit formulation, are discussed in this article. 5.2.1 Implicit finite element formulation

where and are the body forces and surface tractions, respectively, and v i are nodal velocities. If no body force is considered, Eq. (15) becomes:

(16)

If the deformable body is made of an almost incompressiblematerial, the volumetric strain rate will be very small in comparison to the deviatoric strain rate. Substituting into Eq. (16) yields:

(17)

where is the Kronecker delta, are the deviatoric stresses, are the hydrostatic stresses, is the bulk modulus, and are the volumetric strain rates. Because , Eq. (17 becomes:

(18)

In this equation, the admissible velocities satisfy the conditions of compatibility and incompressibility, as well as the velocity boundary conditions. The bulk modulus k will be infinity if the material is completely incompressible (Poisson’s ratio v = 0.5). Thus, the equation has the same format as the penalty method. After substituting the constitutive relationship (Levy-Mises Eqs. (8) and (9)), Eq. (18) becomes:

(19)

The surface traction boundary condition in the metal forming problem is difficult to handle because of the unknown magnitudes and directions of the force and velocity acting on the interface between the workpiece and the tool. Even worse, there is always a velocity discontinuity along the tool-workpiece interface, which results in numerical difficulties in solving nonlinear equations. To deal with this situation, one relationship was proposed by Kobayashi ([8] and [9]) where a velocity dependent frictional stress is used as an approximation of the condition of constant frictional stress. Eq. (19) applies to a domain in which the entire body deforms plastically. In metal forming, however, situations do arise in which rigid zones exist and unloading occurs. The rigid zones are characterized by a very small value of effective strain rate in comparison with that of the deforming body. If these portions are included within the control volume V, the value of the first term of

(15)

FIA MAGAZINE | MAY 2021 75