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CHAPTER 5

FINITE ELEMENT ANALYSIS AND AN ANALYTICAL

APPROACH OF WARM DEEP DRAWING OF AISI 304

STAINLESS STEEL SHEET

5.1 INTRODUCTION

Nowadays, the finite element based simulation is very widely used

by many researchers to analyze the sheet metal processes successfully.

Accurate prediction of the effects of various process parameters on the

detailed metal flow became possible only recently, when the finite element

method was developed for the analyses (Shiro Kobayashi et al 1989).The

finite element based simulations are carried out in order to investigate the

maximum drawing loads, the thickness, radial and hoop strains all expressed

in percentages, in warm deep drawing of circular cups from AISI 304

stainless steel sheets. The finite element results at room temperature and at

different experimented warm temperatures are compared with those of the

experimental values for validation.

Many researchers tried the analytical method to find out the

thickness distribution in the deep drawn cup, LDR value, height of the drawn

cup and the drawing force in the conventional deep drawing process. Very

few researchers used the analytical methods in warm deep drawing process

and still much to be developed in this regard. To name the few researchers

used analytical methods are: Korhonen (1982), Ramaekers et al (1994), Cho

et al (2002), Cwiekala et al (2011), Bai et al (2011) in conventional deep

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drawing ; Swadesh Kumar Singh and Ravi Kumar (2005), Azodi et al (2008)

in hydro-mechanical deep drawing; Chandra and Kannan (1992) (i & ii),

Jung-Ho Cheng (1996), Hwang et al (1997) in super plastic forming and

Hong Seok Kim et al (2008) in warm deep drawing.

From the literature survey, it is observed that most of the current

research work in warm deep drawing are concentrated on the experimental

and FE simulations alone and very little focus has been made on analytical

methods. In this research work, an attempt is made to analytically find out

some of the parameters like thickness distribution in the deep drawn cup,

LDR value, drawing force in warm deep drawing process of AISI stainless

steel sheet at different temperatures ranging from room temperature to 300oC.

The calculated values by analytical methods are compared with those of the

experimental and FE simulations results for its accuracy of prediction.

5.2 FINITE ELEMENT ANALYSIS ON WARM DEEP DRAWING

In metal forming technology, proper design and control requires,

among other things, the determination of deformation mechanics involved in

the processes. Without the knowledge of the influences of variables such as

material properties, workpiece geometry, friction and contact conditions on

the process mechanics, it would not be possible to design the dies and

equipments adequately, or to predict or prevent the occurrence of defects in

the components produced. Thus, process modeling for computer simulation

has been a major concern in modern metal forming technology.

5.2.1 Commercially Available Software Packages for Finite Element

Analysis on Warm Deep Drawing

There are many commercially available software packages for

finite element method such as NUMISHEET’93, DEFORM, AUTOFORM,

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DD3IMP, LS-DYNA, MSC.MARC, PAM-STAMP, ANSYS and ABAQUS

etc. 2-Dimensional (2D) and 3-dimensional (3D) simulation for finite element

analysis is possible in these software packages. Recently, Mark Colgan and

John Monaghan (2003) have combined experimental and finite element

analysis using the program AUTOFORM and Padmanabhan et al (2007) have

performed the finite element method combined with Taguchi technique using

deep drawing 3 dimensional implicit codes (DD3IMP) to analyze the deep

drawing operation.

5.2.2 ABAQUS Software Package

The ABAQUS software is a product of Dassault systèmes simulia

corporation, USA. ABAQUS/CAE is a complete ABAQUS environment that

provides a simple, consistent interface for creating, submitting, monitoring

and evaluating results from ABAQUS/Standard or ABAQUS/Explicit

simulations. Different modules are available in ABAQUS/CAE such as

defining the geometry, defining the material and generating the mesh etc., and

each module defines a logical aspect of modeling process. Once all or

required modules are defined, a model is built from which the ABAQUS/CAE

generates an input file which is submitted to ABAQUS/Standard or

ABAQUS/ Explicit for analysis. Analysis is performed and the information is

sent to the ABAQUS/CAE so that the user can know the progress of the job

and any error indicated can be rectified. Once the input is accepted

successfully, the job is analyzed and the result database is generated. Finally

the visualization module helps to read the output database and to view the

results of the analysis.

The meshing of parts of the model is very important in any analysis

by finite element method. In ABAQUS, there are many types of elements that

are available for meshing. To name a few, the widely used softwares are:

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CAX4R- 4 node, reduced integration, axisymmetric quadrilateral element,

SAX1- first order, axisymmetric shell element, S4R- first order, finite strain

quadrilateral shell element, MAX1 first order, axisymmetric membrane

element etc (ABAQUS user’s manual 2009).

5.2.3 Finite Element Analysis of Warm Deep Drawing of AISI 304

Stainless Steel Using ABAQUS Software Package

The finite element method (FEM) based simulations of deep

drawing using ABAQUS/CAE at room temperature, 100oC, 200oC, and 300oC

are carried out for a circular shaped cups which are drawn from AISI 304

stainless steel sheet. The results from the simulations are compared with the

experimental values with respect to the maximum drawing load, strains like

thickness strains, radial strains and hoop strains.

In deep drawing, the metal is held between the die and the blank

holder and the punch forces the material into the die to form a component

with the desired size and shape. The ratio of drawing against stretching is

controlled by the force on the blank holder and the friction conditions at the

interfaces between blank-die and blank holder- blank. Higher blank holder

force and friction at these interfaces limit the slip at the interface and

increases the radial stretching of the blank. So, it is essential to control the slip

at these interfaces in order to deep draw successfully. Rupture or necking

occurs, if the slip is restrained too much, due to the severe stretching of the

material, whereas, wrinkles will form, if the material flows very easily into

the die and so proper interface conditions are very much important for the

satisfactory results during deep drawing process simulation (ABAQUS user’s

manual 2009).

The flow chart of methodology used for the FEM based simulation

of deep drawing of circular cups is shown in Figure 5.1.

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Figure 5.1 Flowchart for FEM simulation methodology

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5.2.3.1 Finite element model and geometry

All finite element models are created using ABAQUS/CAE preprocessor which are analyzed in this study and investigations. Theaxisymmetric FEM model created for analysis is shown in the Figure 5.2 andthe 3D model is shown in the Figure 5.3.

Figure 5.2 Finite element model of circular cup deep drawing

Figure 5.3 3D Model for FEM simulation of deep drawing process

Ø 21.85

Ø 20

R 51.0

R 6

Ø 42

Ø 21.75

PUNCH

DIE

BLANK HOLDER

BLANK

All dimensions in mm

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For the analysis in ABAQUS/CAE, the punch, die, and the blank

holder are modeled as analytically rigid surfaces whereas only the blank is

defined as deformable body. The blank is meshed by the element CAX4R, a

four node bilinear axisymmetric quadrilateral elements with reduced

integration. These elements belong to the family of solid elements and are of

the first order, which means that the strain is computed as an average over the

element volume instead of the first order gauss point (Magnus Söderberg

2006). The feature of reduced integration used in the CAX4R element causes

the integration order to be lower than full integration; in this case only one

integration point in the centre of the element is used. With the use of reduced

integration, the number of constraints which are introduced by the elements is

reduced, and this prevents locking in the elements causing a stiff response.

The drawback of this technique is that no energy is registered in the

element integration point for certain modes of deformation and these modes

are usually referred to as hourglass mode which is addressed in ABAQUS

using hourglass control algorithm (Magnus Söderberg 2006). The blank is

modeled using 20 elements of type CAX4R in order to match with the grid

pattern used in the experimental analysis. These meshes are coarser for this

analysis. However, since the primary interest in this problem is to study the

membrane effects, the analysis will still provide a fair indication of stresses

and strains occurring in the process. Thickness changes and membrane effects

are modeled properly with CAX4R element however, the bending stiffness of

the element is very low. The element does not exhibit locking due to

incompressibility and the element is very cost- effective due to lesser

computational time when compared to other elements (ABAQUS user’s

manual 2009).

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5.2.3.2 Material properties

The material used in the simulation of deep drawing process and

the important properties of the material are shown in the Table 5.1.

Table 5.1 Important material properties of AISI304 austenitic stainlesssteel used in FEM simulations

S.No. Property Value

1 Density 7.8 g/cc

2 Young’s modulus 210 GPa

3 Poisson ratio 0.3

The plastic stress-strain values used in this analysis are from the

flow curves of stainless steel 304 obtained experimentally up to the

temperatures of 200oC by Eren Billur et al (2009). The stress values for the

corresponding strain values for 300oC are extrapolated by numerical method.

The material model used in these analyses is isotropic Von Mises hardening

model.

5.2.3.3 Contact and boundary conditions

The contact between the blank and the tools is enforced by a

kinematic contact condition, using pure master-slave surface pairs established

in the first step of the solution. The surfaces of the analytically rigid bodies

are defined as the master surfaces and the surfaces defined on the blank form

the slave surfaces. The friction between the contact surfaces is implemented

with a coulomb model. The boundary conditions are defined for each step of

the simulation which defines the displacement of the blank, punch, die, and

blank holder and the type of loading.

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5.2.3.4 Loading conditions

The entire finite element analysis is carried out in five steps. In the

first step, the blank holder is moved onto the blank with the prescribed

displacement to establish the contact. The second step involves the removal of

the boundary condition and application of the blank holder force of 100 KN

and this force is kept constant for step 2 and 3. The third step is the actual

deep drawing process in which the punch pushes the blank with the defined

punch force of 300 KN into the die through a total distance of 32 mm, that is,

the height of the cup (30 mm) plus the initial clearance (2 mm) between the

punch and the top surface of the blank. The important process parameters

used during the deep drawing step is shown in the Table 5.2. In the fourth

step, all the nodes of the model are fixed in their current position and the

contact pairs are removed from the model and the last step is to withdraw the

punch back to its original position.

Table 5.2 Important process parameters used in FEM simulations

S.No. Process parameter Value

1 Punch speed 60 m/min

2 Friction coefficient (ABAQUSuser’s manual 2009)

(a) Blank-punch 0.25

(b) Blank-die 0.10

(c) Blank-blank holder 0.10

5.2.3.5 Assumptions Made in the Simulations

(i) The material is assumed to be isotropic which means that it

has similar properties in all directions.

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(ii) The material is assumed to satisfy the relationship between the

true stress and true strain given by Hollomon (1945) which is

mathematically expressed by the equation (5.1).

= K n (5.1)

(iii) The mechanical interaction between the contact surfaces is

assumed to be the frictional contact.

(iv) For shells and membranes, the thickness change is calculated

from the assumption of incompressible deformation of the

material.

(v) It is assumed that no reverse loading occurs during simulation

and so the Bauchinger effect is not modeled.

5.3 APPLICATION OF ANALYTICAL METHOD IN WARM

DEEP DRAWING

5.3.1 Flow Stresses and Strains in Warm Deep Drawing of StainlessSteel Sheet

The flow stress and strain of the material is very important

parameter in deciding the forming characteristics of the material especially in

deep drawing operation. There are many constitutive material equations are

available to relate the flow stress and the flow strain which is known as the

flow curve equation and depending on the situation, the appropriate equation

may be used for accurate results.

The flow stress and strain values of AISI 304 stainless steel sheet

material with 1.0 mm thickness for the analytical and FEM simulations in this

research work are used from the experimental values obtained by hydraulic

bulge test at various temperatures and strain rates by Eren Billur et al (2009).

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In this work, it is assumed that the material obeys the Hollomon

strain hardening equation (5.1)

= K n

The parameters K and n are determined by fitting the equation (5.1)

using least square method and the flow stresses are calculated for the different

strains and also for different temperatures up to 200oC.

5.3.2 Thickness Distribution in the Warm Deep Drawn Cup

The change in the thickness of the material, when it is deep drawn

from the blank into a desired shape and dimensions, occurs due to plastic

deformation and also due to the influence of temperature in warm deep

drawing. The prediction of the amount and region of maximum reduction of

thickness is the primary concern of the designer in order to design a part

without the occurrence of fracture either during manufacturing or while in use

in future.

A new methodology is developed to calculate the thickness

distribution in the warm deep drawn cup of AISI 304 stainless steel material

and the steps involved are as follows:

(i) For the elements/nodes on the blank which moves on the top

surface of the die before reaching the die corner radius while

deep drawing, the thickness at the die entry (te) is calculated

by the equation (5.2) which is derived by Ramaekers et al

(1994).

= (5.2)

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(ii) When the element bends over the die radius, the change in

thickness is calculated using the equation (5.3) given by

Marciniak et al (2002).

(5.3)

where,

T0 0 t0 (5.4)

Ty y t0 (5.5)

(iii) When the element leaves the die radius, it unbends and gets

straighten and the change in thickness is again calculated

using the equation (5.3)

(iv) When the element wrap around the punch corner radius also

the equation (5.3) is used for calculating the thickness value.

(v) Identify the elements which undergo the types of deformation

as mentioned above and apply the appropriate equations to

determine the final thickness of the element of the deep drawn

cup.

(vi) The same procedure is adopted for warm deep drawing also by

using the corresponding material constants at that temperature.

In the experimental study of the present work, the measurements

are made at the positions of 0, 6, 12, 18, 24, 30, 36 and 42 mm from the center

of the blank. For the analytical prediction also, the same nodes/ elements are

considered in order to compare the calculated values with those of

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experimental and FEM simulation results. The types of deformation that the

nodes/elements undergo are stated below:

(i) The node/element at 42 mm and 36 mm move along the top

surface of the die and bend at the die corner radius.

(ii) The node/ element at 30 mm and 24 mm move along the top

surface of the die and bend as well as unbend to straighten at

the die radius.

(iii) The node/ element at 18 bend and unbend at the die radius and

also bend at the punch corner radius.

(iv) The node/ element at 12 mm, 6mm and center of the blank

theoretically do not undergo any deformation and the

thickness remains unchanged.

It is assumed that the value of 0 = 0.01, since the pre strain, in most

of the cases, is less than 0.01 (Ramaekers et al 1994).

For AISI 304 stainless steel, R = 1 ; y = 262 MPa

Initial thickness of the blank (t0) = 1.0 mm

The value of 0 is calculated using the values of 0, appropriate K

and n from the equation (5.1).

5.3.3 Analytical Method of Determination of LDR Values and Height

of the Deep Drawn Circular Cup

The LDR values at different temperatures are calculated using the

equation (5.6) from the literature of Swadesh Kumar Singh and Ravikumar

(2005).

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= + 1

(5.6)

The drawing efficiency ( ) for different temperatures are initially

assumed and finally checked with the experimental drawing efficiency values

by using the equation 5.7 from George E. Dieter (1987). Since the flow stress

values are decreased when the temperature is increased, the assumed drawing

efficiencies are 70% at room temperature (Kurt Lange 1985), 80% at 100oC,

90% at 200oC, and 95% at 300oC.

LDR e (5.7)

The deep drawn height of the cup is determined by the

equation (5.8) (Marciniak et al 2002).

1 (5.8)

5.3.4 Analytical Method of Determining the Punch Force

The punch force excluding the blank holding force, force required

to overcome the friction, die cushion force and consideration of the factor of

safety is calculated from the equation (5.9) given by Korhonen (1982).

Fp = (5.9)

Since the flow stress, yield stress and ultimate tensile strength are

decreased, when the temperature is increased, it is assumed that the ultimate

tensile strength decreased by 15% when the temperature is increased from

room temperature to 100oC; further decreased by 10% of the stress value at

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100oC, when the temperature is increased from 100oC to 200oC; and finally,

decreased by 10% of the stress value at 200oC, when the temperature is

increased from 200oC to 300oC. The punch force is calculated at different

temperatures and compared with the punch force obtained in the experiments.

5.4 SUMMARY

Finite element based simulations of deep drawing of stainless steel

AISI304 circular cups are carried out using ABAQUS/CAE software at

different temperatures from room temperature (30oC) to 300oC at an

increment of 100oC. The results of FEM simulations on drawing loads, the

maximum thinning region location and thickness, radial and hoop strain

measurements are compared with those of experimental results for validation.

A new methodology for the determination of thickness distribution

using analytical method in the warm deep drawn cup is proposed and the

LDR values, height of the deep drawn cups and the punch force at different

temperatures are calculated using the analytical methods which are used for

conventional deep drawing process by determining the materials constants of

the strain hardening equation at each temperature. The results of analytical

methods are compared with those of experimental results for its accuracy of

predictions.