Finite Element Method in Metal Forming - Methods of Analysis for Forging Operations

Chapter 9 Methods of Analysis for Forging Operations

9.4 Finite Element Method in Metal Forming

Method in Metal Forming

The basic approach of the ﬁnite element (FE) method is one of discretization. The FE model is constructed in the following manner [Kobay- ashi et al., 1989]:

● A number of ﬁnite points are identiﬁed in the domain of the function and the values of

the function and its derivatives, when appropriate, are speciﬁed at these points.

● The domain of the function is represented approximately by a ﬁnite collection of sub- domains called ﬁnite elements.

● The domain is then an assemblage of elements connected together appropriately on their boundaries.

● The function is approximated locally within each element by continuous functions that are uniquely described in terms of the nodal point values associated with the particular element.

The path to the solution of a finite element problem consists of five specific steps: (1) iden- tification of the problem, (2) definition of the element, (3) establishment of the element equation, (4) the assemblage of element equations, and (5) the numerical solution of the global equations. The formation of element equations is accomplished from one of four directions: (a) direct approach, (b) variational method, (c) method of weighted residuals, and (d) energy balance approach.

The main advantages of the FE method are:

● The capability of obtaining detailed solutions of the mechanics in a deforming body, namely, velocities, shapes, strains, stresses, temperatures, or contact pressure distribu- tions

● The fact that a computer code, once written, can be used for a large variety of problems by simply changing the input data

9.4.1 Basis for the

Finite Element Formulation

The basis for the ﬁnite element metal forming formulation, using the “variational approach,” is to formulate the proper functional (function of functions) depending on speciﬁc constitutive relations. The variational approach is based on one of two variational principles. It requires that among admissible velocities uithat satisfy the

conditions of compatibility and incompressibility, as well as the velocity boundary conditions, the actual solution gives the following functional a stationary value [Kobayashi et al., 1989]:

p ⳱

冮

r ¯edV ⳮ¯

冮

F u dSi i (Eq 9.35a)

V SF

(for rigid-plastic materials) and:

p ⳱

冮

E(˙e )dV ⳮi j

冮

F u dSi i (Eq 9.35b)

V SF

(for rigid-viscoplastic materials) wherer¯ is the effective stress, ˙¯e is the effective strain rate, Fi

represents surface tractions, E(˙e )ij is the work function, and V and S the volume and surface of the deforming workpiece, respectively. The solution of the original boundary-value problem is then obtained from the solution of the dual- variational problem, where the ﬁrst-order vari- ation of the functional vanishes, namely:

dp ⳱

冮

rd ¯edV ⳮ¯

冮

Fidu dS ⳱ 0i (Eq 9.36)

V SF

wherer ⳱ ¯r(¯e) and ¯r ⳱ ¯r(¯e,¯e)¯ ˙ for the rigid- plastic and rigid-viscoplastic materials, respectively.

For an accurate ﬁnite element prediction of material ﬂow in a forging process, the formulation must take into account the large plastic deformation, incompressibility, workpiece-tool contact, and, when necessary, temperature cou- pling.

The basic equations to be satisﬁed are the equilibrium equation, the incompressibility condition, and the constitutive relationship. When applying the penalty method, the velocity is the primary solution variable. The variational equation is in the form [Li et al., 2001]:

dp(v) ⳱

冮

rd¯edV Ⳮ K¯

冮

e d˙e dV˙v v

V V

ⳮ

冮

Fidu dS ⳱ 0i (Eq 9.37)

where K, a penalty constant, is a very large posi- tive constant.

An alternative method of removing the incompressibility constraint is to use a Lagrange multiplier [Washizu, 1968] and [Lee et al., 1973] and modifying the functional by adding the term is the volumetric 兰k˙e dV, where ˙e ⳱ ˙e ,v v ii

strain rate. Then: ˙

dp ⳱

冮

rd ¯edV Ⳮ¯

冮

kd˙e dVv

V V

Ⳮ

冮

edkdV ⳮ˙

冮

Fidu dS ⳱ 0i (Eq 9.38)

V SF

In the mixed formulation, both the velocity and pressure are solution variables. They are

solved by the following variational equation [Li et al., 2001], ˙ dp(v,p) ⳱

冮

rd ¯edV Ⳮ¯

冮

pd ˙e dV Ⳮv

冮

e dp˙v V V V ⳮ

冮

Fidu dS ⳱ 0i (Eq 9.39) SF

where p is the pressure.

Equations 9.37 to 9.39 can be converted into a set of algebraic equations by utilizing the standard FEM discretization procedures. Due to the nonlinearity involved in the material properties and frictional contact conditions, this solution is obtained iteratively.

The temperature distribution of the workpiece and/or dies can be obtained readily by solving the energy balance equation rewritten by using the weighted residual method as [Li et al., 2001]:

˙ kTijdT dV Ⳮij qcTdTdV

冮

V ˙ ⳮ

冮

␣¯r ¯edTdV ⳱

冮

qndTdS (Eq 9.40) V S

where k is the thermal conductivity, T the temperature,q the density, c the speciﬁc heat, ␣ a fraction of deformation energy that converts into heat, and qnthe heat ﬂux normal to the boundary,

including heat loss to the environment and friction heat between two contacting objects. Using FEM discretization, Eq 9.40 can also be converted to a system of algebraic equations and solved by a standard method. In practice, the solutions of mechanical and thermal problems are coupled in a staggered manner. After the nodal velocities are solved at a given step, the deformed conﬁguration can be obtained by up- dating the nodal coordinates [Li et al., 2001].

9.4.2 Computer Implementation of the Finite Element Method

Computer implementation of the basic steps in a standard ﬁnite element analysis consists of three distinct units, viz. the preprocessor, processor, and the postprocessor.

Preprocessor. This operation precedes the analysis operation. It takes in minimal information from the user (input) to generate all necessary problem parameters (output) required for a ﬁnite element analysis.

The input to the preprocessor includes information on the solid model, discretization re-

quirements, material identiﬁcation and parameters, and boundary conditions; whereas the output includes coordinates for nodes, element connectivity and element information, values of material parameters for each element, and boundary loading conditions on each node.

In the preprocessing stage, a continuum is di- vided into a ﬁnite number of subregions (or elements) of simple geometry (triangles, rectan- gles, tetrahedral, etc.). Key points are then selected on elements to serve as nodes where problem equations such as equilibrium and compatibility are satisﬁed.

In general, preprocessing involves the following steps to run a successful simulation using a commercial FE package:

1. Select the appropriate geometry/section for simulation based on the symmetry of the component to be analyzed.

2. Assign workpiece material properties in the form of the ﬂow stress curves. These may be user deﬁned or selected from the database provided in the FE package.

3. Mesh the workpiece using mesh density windows, if necessary, to reﬁne mesh in critical areas.

4. Define the boundary conditions (velocity, pressure, force, etc.). Specify the movement and direction for the dies. Specify the interface boundary and friction conditions. This prevents penetration of the dies into the workpiece and also affects the metal flow depending on the friction specified. In massive forming processes such as forging, extrusion, etc., the shear friction factor “m” is specified. Note: If die stress analysis or thermal analysis is required, one would need to mesh the dies and specify the material properties for them. Also die and workpiece temperatures and interface heat transfer coefficients would need to be specified. Material data at elevated temperatures would be required for the tools and the workpiece. The die stress analysis procedure is described in Chapter 16 on process modeling in impression die forging.

Processor. This is the main operation in the analysis. The output from the preprocessor (including nodal coordinates, element connectivity, material parameters, and boundary/loading conditions) serves as the input to this module. It establishes a set of algebraic equations that are to be solved, from the governing equations of the boundary value problem. The governing

Fig. 9.4 Flowchart of ﬁnite element implementation

equations include conservation principles, kinematic relations, and constitutive relations. The FE engine/solver then solves the set of linear or nonlinear algebraic equations to obtain the state variables at the nodes. It also evaluates the ﬂux quantities inside each element.

The following steps are pursued in this operation:

1. A suitable interpolation function is assumed for each of the dependent variables in terms of the nodal values.

2. Kinematic and constitutive relations are sat- isﬁed within each element.

3. Using work or energy principles, stiffness matrices and equivalent nodal loads are es- tablished.

4. Equations are solved for nodal values of the dependent variables.

Postprocessor. This operation prints and plots the values of state variables and ﬂuxes in the meshed domain. Reactions may be evalu- ated. Output may be in the form of data tables or as contour plots.

Figure 9.4 shows a ﬂowchart, which shows the sequence of ﬁnite element implementation in analysis of metal forming processes.

In interpreting FE results, careful attention is required since the idealization of the physical problem to a mathematical one involves certain assumptions that lead to the differential equation governing the mathematical model. As illus- trated in Fig. 9.5, reﬁnement of the mesh size, alteration of the boundary conditions, etc., may be needed to increase the accuracy of the solution.

9.4.3 Analysis of Axisymmetric Upsetting by the FE Method

The use of the FE method in analysis of forging processes is discussed with the help of a simple cylindrical compression simulation. Figure 9.6(a) shows the cutaway of a cylinder, which is to be upset. Since the cylinder is symmetric about the central axis, i.e., axisymmetric, a 2-D simulation is adequate to analyze the metal ﬂow during deformation. Figure 9.6(b) shows the half model selected for simulation as a result of ro- tational symmetry, with the nodes along the axis of symmetry (centerline) restricted along the R- direction. This model can be further simpliﬁed to yield a quarter model (Fig. 9.6c) with velocity/displacement boundary conditions restricting the movement of the nodes along the Z-direction

as shown. The boundary conditions necessary to be prescribed for this problem under isothermal conditions can be summarized as:

● Velocity/displacement boundary conditions depending on the symmetry of the workpiece, to restrict to movement of nodes along the planes of symmetry as required

● Die movement and direction according to the process being analyzed

● Interface friction conditions at the surfaces of contact between the tools and the workpiece

9.4.4 Analysis of

Plane Strain Deformation

Figure 9.7 shows the setup of an FE model for a plane strain condition. In this case the part has a uniform cross section along the length and

Fig. 9.5 The process of ﬁnite element analysis. [Bathe, 1996]

the strain along the Z or length direction is negligible; i.e.,ezis negligible. Hence, only the two-

dimensional cross section shown in Fig. 9.7(b) is selected for process simulation.

The procedure for simulation is the same as mentioned for the axisymmetric case. However, the velocity boundary conditions are different since the metal is free to ﬂow in the X direction (Fig. 9.7b). If, however, the cross section is uniform, a quarter model can be used for simulation as shown in Fig. 9.7(c) with velocity boundary conditions similar to those used for the axisymmetric case discussed in the previous section. It should be noted, however, that the results obtained from the plane strain simulation have to be multiplied by the length of the billet used for the forging process to get the ﬁnal results.

9.4.5 Three-Dimensional Nonisothermal FE Analysis

Figure 9.8 shows the FE model of the hot forging process of an automotive component. The starting preform geometry is not axisymmetric; hence a 2-D simulation could not be used to analyze the metal ﬂow. The preform is, however, symmetric, and hence a quarter model could be used for modeling the process. Since the forging process is performed at elevated temperatures, a nonisothermal simulation is con- ducted. The simulation input differs from those of an isothermal simulation in the following ways:

● Tool and workpiece temperatures have to be considered, and thermal properties such as

Fig. 9.7 Analysis of plane strain upsetting by the FE method. (a) Part for plane strain upsetting. (b) Full model. (c) Quarter model

Fig. 9.6 Analysis of axisymmetric cylinder upsetting using the FE method. (a) Cutaway of the cylinder. (b) Half model. (c) Quarter model

thermal conductivity, heat capacity, interface heat transfer coefﬁcients, etc. have to be speciﬁed.

● Die and workpiece material properties are speciﬁed as a function of temperature and strain rate. Dies (still considered as rigid) have to be meshed since heat transfer between workpiece and tools is simulated. The boundary conditions in a 3-D simulation are similar to those in a 2-D process:

● Velocity/displacement boundary conditions need to be speciﬁed depending on the work-

piece geometry. In Fig. 9.8(b), movement of the boundary nodes is restricted along the planes of symmetry.

● Interface friction is speciﬁed depending on the type of lubrication at the tool/workpiece interface. In addition to the friction, the interface heat transfer coefﬁcient is also a critical input to the simulation.

9.4.6 Mesh Generation in FE

Simulation of Forging Processes

Bulk forming processes are generally char- acterized by large plastic deformation of the

Fig. 9.8 Three-dimensional non-isothermal FE simulation of a forging process. (a) FE model. (b) Final part with temperature contour.

workpiece accompanied by significant relative motion between the deforming material and the tool surfaces. The starting mesh in the FE simulation is generally well defined with the help of mesh density windows to refine the mesh in critical areas. However, as the simulation pro- gresses the mesh tends to get significantly dis- torted. This calls for (a) the generation of a new mesh taking into consideration the updated geometry of the workpiece and (b) interpolation of the deformation history from the old mesh to the new mesh. Current commercial FE codes, such as DEFORM娂 accomplish this using automatic mesh generation (AMG) subroutines. AMG ba- sically determines the optimal mesh density distribution and generates the mesh based on the given density. Once the mesh density was defined, the mesh generation procedure considers the following factors [Altan et al., 1979]:

● Geometry representation.

● Density representation. The density speciﬁ- cation is accommodated in the geometry.

● Identiﬁcation of critical points (like sharp corner points) for accurate representation of the geometry.

● Node generation. The number of nodes between critical points are generated and re- positioned based on the density distribution.

● Shape improvement and bandwidth mini- mization. The shape of the elements after the mesh generation procedure has to be smoothed to yield a usable mesh. Also the bandwidth of the stiffness matrix has to be improved to obtain a computationally efﬁ- cient mesh.

The FE modeling inputs and outputs are discussed in detail in Chapter 16 on process modeling.

FE simulation is widely used in the industry for the design of forging sequences, prediction of defects, optimization of ﬂash dimensions, die stress analysis, etc. A number of commercial FE codes are available in the market for simulation of bulk forming processes, viz. DEFORM娂, FORGE娂, QFORM娂, etc. Some practical applications of FE simulation in hot and cold forging processes are presented in Chapters 16 and 18.

REFERENCES

[Altan et al., 1979]: Altan, T., Lahoti, G.D., “Limitations, Applicability and Usefulness of Different Methods in Analyzing Forming Problems,” Ann. CIRP, Vol 28 (No. 2), 1979, p 473.

[Avitzur, 1968]: Avitzur, B., Metal Forming:

Processes and Analysis, McGraw-Hill, 1968.

[Bathe, 1996]: Bathe, K.J., Finite Element Pro-

cedures, Prentice Hall, 1996.

[Becker, 1992]: Becker, A.A., The Boundary

Element Method in Engineering, McGraw- Hill International Editions, 1992.

[Hoffman et al., 1953]: Hoffman, O., Sachs, G.,

Introduction to the Theory of Plasticity for Engineers, McGraw-Hill, 1953.

[Kobayashi et al., 1989]: Kobayashi, S., Oh, S.I., Altan, T., Metal Forming and the Finite Element Method, Oxford University Press, 1989.

[Lee et al., 1972]: Lee, C.H., Altan, T., “Inﬂu- ence of Flow Stress and Friction upon Metal Flow in Upset Forging of Rings and Cylin- ders,” ASME Trans., J. Eng. Ind., Aug 1972, p 775.

[Lee et al., 1973]: Lee, C.H., Kobayashi, S., “New Solutions to Rigid-Plastic Deformation Problems using a Matrix Method,” Trans. ASME, J. Eng. Ind., Vol 95, p 865.

[Li et al., 2001]: Li, G., Jinn, J.T., Wu, W.T., Oh, S.I., “Recent Development and Applica- tions of Three-Dimensional Finite Element Modeling in Bulk Forming Processes,” J. Ma- ter. Process. Technol., Vol 113, 2001, p 40– 45.

[Thomsen et al., 1954]: Thomsen, E.G., Yang, C.T., Bierbower, J.B., “An Experimental In-

vestigation of the Mechanics of Plastic De- formation of Metals,” Univ. of California Pub. Eng., Vol 5, 1954.

[Thomsen et al., 1965]: Thomsen, E.G., Yang, C.T., Kobayashi, S., Mechanics of Plastic De- formation in Metal Processing, Macmillan, 1965.

[Washizu, 1968]: Washizu, K., Variational

Methods in Elasticity and Plasticity, Perga- mon Press, Oxford, 1968.

SELECTED REFERENCE

[Altan et al., 1983]: Altan, T., Oh, S.I., Gegel, H., Metal Forming: Fundamentals and Appli- cations, ASM International, 1983.

In document Cold and Hot Forging Fundamentals and Applications (Page 101-109)