Finite Element Analysis

Process modeling for the determination of deformation mechanics has been a major concern in modern metalworking technology. Proper design and control of metal forming processes requires global as well as local knowledge of the mechanics during deformation. In this regard, the finite-element method has a central role in metalforming process modeling.

To simulate metal flow during deformation processes, the most promising technique is the finite-element method. The concept of the finite-element method is one of discretization. The finite-element model is constructed in the following manner. A number of finite points are identified in the domain of the function, and the values of the function and its derivatives, when appropriate, at these points are specified. These points are called nodal points. The domain of the function is represented approximately by a finite collection of subdomains called finite elements. The domain then is an assemblage of elements connected together appropriately on their boundaries. The function is approximated locally within each element by continuous functions which are uniquely described in terms of the nodal point values associated with the particular element [132].

The path to the solution of a finite-element problem consists of five specific steps: (a) the problem, (b) the element, (c) the element equation, (d) the assemblage of element equations, and (e) the numerical solution of the global equations [132]. The formation of element equations is accomplished from one of four directions: (1) direct approach; (2) variational method; (3) method of weighted residuals; and (4) energy balance approach [132]. The basis of finite-element metal-flow modeling, for example, using the variational approach is to formulate proper functionals, depending upon specific constitutive relations. The solution of the original boundary value problem is obtained by the solution of the dual variational problem in which the first-order variation of the functional vanishes. Choosing an approximate interpolation function (or shape function) for the field variable in the elements, the functional is expressed locally within each element in terms of the nodal-point values. The local equations are then assembled into the overall problem. Thus, the functional is approximated by a function of global nodal-point values. The condition for this function to be stationary results in the stiffness equations. These stiffness equations are then solved under appropriate boundary conditions, with mathematical solution techniques [52].

In most hot forming processes, plastic strains usually outweigh elastic strains and the idealization of rigid-plastic or rigid-viscoplastic material behavior is acceptable. Thermo-viscoplastic analysis is preferred for the hot forming. During cold forming, on the other hand, elastic deformation generally needs to be considered. Typical examples are the prediction of spring-back for bending process, and an analysis of elastic deformation of rolls in order to determine the minimal thickness of the sheet that can be rolled. The drawback for the elastic-plastic formulation is a much higher computational cost. The nature of elastic-plastic constitutive equations require short time steps in nonsteady-state analysis; this requirement is severe when the workpiece goes from elastic to plastic deformations.

If an existing general-purpose FEM program (e.g., MARC, ABAQUS) is used, selection of material law is an important issue. With MARC, for example, an algorithm based on multiplicative decomposition of deformation gradient captures the correct physical behavior under large elastic deformation and yields numerically more accurate solutions for elastic-plastic deformations [134].

The formulation based on rigid-plastic or rigid-viscoplastic material behavior is often regarded as flow formulation, while the modeling based on elastic-plastic or elastic-viscoplastic behavior is called solid formulation. The original problem associated with the deformation process of materials is a boundary-value problem. For flow formulation, at a certain stage in the process of quasistatic distortion, the shape of the body, the internal distribution of temperature, the state of inhomogeneity, and the current values of material parameters are supposed to be given or to have been determined already. The velocity vector **u** is prescribed on a part of surface S_{u} together with traction **F** on the remainder of the surface, S_{F}. Solutions to this problem are the stress and velocity distributions that satisfy the governing equations and the boundary conditions. In the solid approach, the boundary value problem is stated such that, in addition to the current states of the body, the internal distribution of the stress also is supposed to be known and the boundary conditions are prescribed in terms of velocity and traction-rate. Distributions of velocity and stress-rate (or displacement and stress-increment) are the solutions to the problem [132]. For hot forming processes, particular attention should be paid to coupling of deformation analysis and temperature calculation. In order to solve coupled thermal plastic deformation problem, it is necessary to solve simultaneously the flow problem for a given temperature distribution and to solve the thermal equations.

To be noted is that for the metal forming, particular attention should be paid not only to the process analysis and constitutive equations, but also to the boundary conditions (friction/lubrication, heat transfer) as well as ductile fracture and plastic instability. Further, the residual stresses, and the relationship between microstructure and deformation mechanics, are often important issue for analysis.

Interactions with Metallurgical Processes

Classic thermomechanical analysis only handles thermal and mechanical parameters: strains, stresses, temperature, friction, heat transfer, etc. This may also include the residual stresses. Currently, the knowledge on microstructure and mechanical properties is often needed, too. It is of great interest to use FEM to analyze various metallurgical phenomena besides mechanical and thermal ones. **Fig. 1** presents all the interactions between phenomena.

Two types of interactions occur when phase transformation takes place under an applied stress. The first type is a kinetic modification and sometimes leads to a different morphology in the phase produced. The second one (commonly called "transformation plasticity") is a mechanical modification related to the progress of transformation: a plastic deformation occurs even under stresses lower than the yield stress of the material [265].

Two practices can be introduced to address the metallurgical phenomena during finite element analysis: direct solution and indirect solution. Indirect solution is the most popular one. Parameters related to the microstructure transformation, such as strain, strain rate, temperature, etc., are determined with general FEM simulation. Microstructure dependence on those parameters is then considered in a material model and entered into the finite element program as, say, subroutine. Many general-purpose FEM program, such as MARC, allow user to display customized parameter, such as grain size, or any user-defined parameters.

Example of direct solution is the work by A.M. Habrakem [265]. In the constitutive laws of finite element method, additional terms arise from the thermal expansion and from temperature dependence of material parameters. The latent heat due to phase transformation was included in an enthalpic formulation. A similar method was used for the volumetric dilatation during transformation. In addition, transformation plasticity, mechanical effects on transformation kinetics and microstructural analysis, and so on, are considered.

Fig. 1: Couplings between phenomena [265]

References

[52] Shiro Kobayashi, Soo-Ik Oh, Taylan Altan: Metal forming and the finite-element method. 1989. ISBN 0-19-504402-9.

[132] Shiro Kobayashi: A Review on the Finite-Element Method and Metal Forming Process Modeling. Applied Metalworking, Jul. 1982.

[134] A. Choudhry and T.B. Wertheimer: Comparison of Finite Strain Plasticity Algorithms in MARC. Marc Analysis Research Corp.

[265] A. M. Habraken: Coupled thermo-mechanical analysis with microstructural computation of steel pieces. Numiform 89, Thompson et al. (eds.), 1989, ISBN 9061918

[267] P. Funke, Y. Cai, G. Heinemann: Über die Genauigkeit von Finite-Elemente-Berechnungen in der Umformtechnik, dargestellt am Beispiel des Stangenzuges. BLECH ROHRE PROFILE 37 (1990) 10. S.681-683