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ISSN 2167-1273 Volume 3, Issue 4, April 2014

FEA Information Engineering Journal

Hot Forming - Hot Stamping Volume I

6th – 8th Conference Publications Publications by Arthur. B. Shapiro, LSTC


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Cover: 6th European LS-DYNA Users’ Conference LS-DYNA Features for Hot Forming 3. Positioning - The blank is placed on the lower die and begins to cool due to contact with the colder die.

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TABLE OF CONTENTS

Publications are © to LS-DYNA 2013 9th European Users‘ Conference Author: Arthur B. Shapiro, LSTC [email protected] 6th European LS-DYNA Users’ Conference LS-DYNA Features for Hot Forming 7th European LS-DYNA Conference Using LS-Dyna for Hot Stamping 8TH 7th European LS-DYNA Conference How to Use LS-OPT for Parameter Estimation – hot stamping and quenching applications Available Course - Modeling Warm Forming & Hot Stamping

All contents are copyright © to the publishing company, author or respective company. All rights reserved.

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6th European LS-DYNA Users’ Conference

2.5.1 2.127

LS-DYNA Features for Hot Forming

Authors: Arthur Shapiro

LSTC

Correspondence: Arthur Shapiro

Livermore Software Technology Corp. 7374 Las Positas Rd.

Livermore, CA 94550, USA Phone +1 925 449 2500 Email [email protected]

ABSTRACT: LS-971 has several features to model the hot forming process. A thick thermal shell formulation for the blank allows modeling a temperature gradient through the thickness. The keyword, *MAT_ADD_THERMAL_EXPANSION, allows calculating thermal strains for all the mechanical material models. A user-defined flag is available to turn off thermal boundary conditions when part surfaces come in contact. A thermal one-way contact algorithm is available to more accurately calculate contact between a die zoned with a CAD type surface mesh when in contact with a uniform meshed blank. Thermal-mechanical contact user defined parameters allow modeling the coefficients of friction as a function of temperature and thermal contact resistance as a function of interface pressure. A new feature models bulk fluid flow through the die cooling passages.

Keywords: Thermal analysis, hot forming, thermal-stress modelling


2.128 2.5

INTRODUCTION This paper presents features in LS-DYNA useful in modelling hot forming. The hot forming process has 5 steps [1]:

1. Heating 2. Transfer 3. Positioning 4. Forming 5. Quenching

1. Heating The blank is heated and held at the austenization temperature of

950C.

2. Transfer The blank cools by convection and radiation during transfer from the oven to the forming press.

3. Positioning The blank is placed on the lower die and begins to cool due to contact with the colder die.

4. Forming The blank is formed.

5. Quenching The blank is held in the press and cooled to induce a solid-solid phase transition from austenite to martensite.

The finite element model is shown below.

Heater

Bottom die with cooling passage

Blank Top die


2.5.1 2.129

Step 1 – Heating to Austenization Temperature The first step is to heat the blank from room temperature (~25C) to the austenization temperature (~950C) as shown in the computational sequence below.

This figure shows the blank increasing in temperature from 25C to 450C to 950C. The red trapezoidal shaped part represents the furnace at 1100C. The easiest modelling technique is to define the initial temperature condition of the blank to be 950C. However, if the time required to heat the blank is desired, then a boundary condition must be defined. LS-DYNA allows flux, convection, and radiation boundary conditions. The blank in this model is heated by convection. The convection heat transfer coefficient is enhanced to include radiation heat transfer affects. This is a technique that will linearize the problem and decrease CPU time.

Step 2 - Transfer The second step is to transfer the hot blank from the oven or heating zone to the dies as shown in the computational sequence below. The blank loses heat by convection heat transfer to the environment at 25C. In this model, the blank cools by 50C during the transfer operation.


2.130 2.5

3. Positioning The blank is placed on the lower die and begins to cool due to contact with the colder die as shown in the following figure. The attribute flag “thickness on” is enabled in LS-PrePost so that blank thickness can be displayed. An important modelling technique is to turn any thermal boundary condition off when the surface of the blank comes in contact with the die. This is accomplished by setting a flag on the *CONTACT_THERMAL keyword.


2.5.1 2.131

Note the temperature gradient through the thickness of the blank. The blank is modelled by a thin mechanical shell and flagged to be treated as a thick thermal shell by the parameter TSHELL on the *CONTROL_SHELL keyword. The thick thermal shell is a 12 node element – 4 nodes in the plane of the shell and 3 nodes through the thickness. The 3 nodes through the thickness allow the use of quadratic shape functions to more accurately calculate the through thickness temperature gradient.

4. Forming The forming operation is depicted in the following figure. In the plant, the forming operation completes in a fraction of a second, but requires several hours of CPU time on

MPP computers to simulate. Nothing of interest happens in this hypothetical simulation since the die surfaces are flat. However, in a real simulation, the answers drive the design of the dies and the manufacturing process parameters. LS-DYNA has several

features useful in modelling the forming process.

*MAT_ADD_THERMAL_EXPANSION - The coefficient of thermal expansion as a function of temperature can be defined using this keyword. This allows the calculation of thermal strains for any of the mechanical material models in LS-DYNA. LS-DYNA.

*CONTACT_(option)_THERMAL_FRICTION - The suffix FRICTION enables the capability to model the mechanical coefficients of friction as a function of temperature and the thermal contact resistance as a function of interface pressure. There are currently 4 formulas to calculate the contact resistance, h, as a function of interface pressure, P:


2.132 2.5

load curve h(P) is defined by a load curve

polynomial h(p) = a + bP + cP2 + dP3

Shvets formula [2] ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+=

8.0

85.14

)(σλ

π PkPh gas

Sellers formula [3] d

cPbaPh ⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−−= exp.1)(

Additionally, a user subroutine allows the creation of other models.

5. Quenching The quenching process, which lasts for 10 to 30 seconds, is shown in the computational sequence below.

In the above figure, notice that the blank cools down from the left figure to the right figure and the dies increase in temperature. The cooling rate of the blank affects the hardening properties of the material. Material type 113 features a special hardening law [4] aimed at modelling the temperature dependent hardening behaviour of TRIP-steels. TRIP stands for Transformation Induced Plasticity. The material gains ultra high strength through the hot forming process. In this material, a phase transformation from austenite to martensite occurs during forming, an effect which is sensitive not only to the stain level, but also to strain rate and temperature. The material model is composed of 2 basic equations:

1. TRIP kinetics rate equation

( )

( )( )DTCVV

VTQ

ABV P

m

BB

m

mm +−⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎠⎞

⎜⎝⎛=

∂∂

+

tanh1211

exp1

ε


2.5.1 2.133

2. yield stress equation

( ) ( ){ }( ) mn

HSHSHSy VHTKKmABB αγεσ →Δ++−−−= 21exp

Additionally, a user subroutine is available that allows the creation of more sophisticated material models with the interchange of material history variables between the mechanical material user subroutine and the thermal material user subroutine.

6. Die Cooling Although not previously mentioned as a process step, the dies must be cooled during the quenching process. Also, the dies must be cooled to some operating temperature before the next hot blank can be formed. The computational sequence below shows fluid entering the lower die cooling passage at 25C. As the fluid passes through the die it increases in temperature as it absorbs heat from the hotter die. The die in turn decreases in temperature as shown in the following figure.

Cooling passage fluid flow is defined by beam elements that trace the centreline of the passage through the die. The keyword *BOUNDARY_THERMAL_BULKFLOW assigns a mass flow rate to the beam. The beam formulation is unique in that it includes the advection terms to model fluid flow. The assumption is that the flow is turbulent and therefore well mixed. Additionally, the flow is inviscid. These are typical assumptions made in designing heat exchangers.

Each beam is wrapped with a thin walled cylinder made of shell elements. The radius of the cylinder is the radius of the pipe. This cylinder can be thought of as the outer surface of the fluid. The keyword *BOUNDARY_THERMAL_BUKNODE associates nodes on the beam with the surrounding shells. This keyword also includes the heat transfer parameters to model the fluid-structure interaction.


2.134 2.5

References 1. Figure courtesy of David Lorenz, DaimlerChryler, Mercedes Car Group,

Germany. 2. I.T. Shvets, “Contact Heat Transfer Between Plane Metal Surfaces”, Int. Chem.

Eng., Vol. 4, No. 4, p621, 1964. 3. Li & Sellers, Proc. Of 2nd Int. Conf. Modeling of Metals Rolling Processes,

The Institute of Materials, London, 1996. 4. D. Hilding & E. Schedin, “Experience from Using a New Material Model for

Stainless Steels with TRIP-effect”, 5th European LS_DYNA Users Conference, Birmingham, UK, 2005.

7th European LS-DYNA Conference

© 2009 Copyright by DYNAmore GmbH

Using LS-Dyna for Hot Stamping Arthur B. Shapiro

LSTC

Livermore, CA, USA

Summary: Presented is a methodology for finite element modeling of the continuous press hardening of car components using ultra high strength steel. The Numisheet 2008 benchmark problem BM03 [1] is selected as the model problem to be solved. LS-DYNA [2] has several features that are useful to numerically model hot sheet metal stamping, such as: (1) modeling high rate dynamics for press forming; (2) conduction, convection and radiation heat transfer; (3) tool-to-part contact conductance as a function of interface pressure; (4) material models that account for temperature dependent properties, phase change, phase fractions, and Vickers hardness prediction; and (5) a CFD solver for tool cooling. Keywords: Hot Stamping



1 Model Problem The continuous press hardening of a car B-pillar shown in Figure 1 was proposed by Audi as Numisheet 2008 benchmark problem BM03[1]. The process steps are:

1. Heating of the blank to 940 C. 2. Transport from the oven into the tool (6.5 s). Temperature of the blank at the beginning of the

die movement, 810 C. 3. Temperature of the tools, 75 C. 4. Forming time 1.6 s 5. Quench time 20 s.

Figure 1. The problem to be solved was proposed by Audi as Numisheet 2008 benchmark problem BM03. Shown are the actual tools and the FE model.

2 Material Data and Constitutive Model There are 2 material models in LS-DYNA that are relevant to hot stamping.

1. Material model 106 (MAT-106) which is an elastic visco-plastic material model with thermal effects.

2. Material model 244 (MAT-244) which is specific to ultra high strength steels and can model the phase transformation kinetics [3,4].

Material properties used for these models are presented in the following figures and tables. Figure 2 shows stress versus strain data as a function of temperature for 22MnB5 steel at a strain rate of 0.1s-1. The Numisheet 2008 BM03 should be consulted for material property data at 2 additional strain rates. Viscous effects can be accounted for using the Cowper-Symonds [5] coefficients c and p by which the

yield stress is scaled by ( ) pp c 11 ε&+ . C and p have strong temperature dependence (see Table 2) but are weak functions of strain rate. MAT-244 requires values for the latent heat of transformation of austenite into ferrite, pearlite, and bainite (590 MJ/m3), and the latent heat for the transformation of austenite into martensite (640 MJ/m3). Table 1. Nomenclature and parameter values used in this paper.

Blank material dimensions

l, thickness [m] length [m] width [m]

properties ρ, density [kg/m3] Cp, heat capacity [J/kgK] k, thermal conductivity [W/mK] α, linear expansion, [1/C] E, Young’s modulus, [GPa] ν, Poisson’s ratio

22MnB5 0.00195 1 0.25 7830. 650. 32. 1.3e-05 100. 0.30



Air properties at 483 C ρ, density, [kg/m3] Cp, heat capacity, [J/kg C] k, thermal conductivity, [W/m C] μ, viscosity, [kg/m s] β, volumetric expansion, [1/C]

0.471 1087. 0.055 3.48e-05 1.32e-03

Figure 2. Stress versus strain data [1] as a function of temperature for 22MnB5 steel at a strain rate of 0.1s-1.

Table 2. Thermal-mechanical material properties for 22MnB5 steel [6].

Temp [C] 20 100 200 300 400 500 600 700 800 900 1000E [MPa] 212 207 199 193 166 158 150 142 134 126 118 ν 0.284 0.286 0.289 0.293 0.298 0.303 0.310 0.317 0.325 0.334 0.343p 4.28 4.21 4.10 3.97 3.83 3.69 3.53 3.37 3.21 3.04 2.87c 6.2e9 8.4e5 1.5e4 1.4e3 258. 78.4 35.4 23.3 22.2 30.3 55.2k [W/mC] 30.7 31.1 30.0 27.5 21.7 23.6 25.6 27.6Cp [J/kg] 444. 487. 520. 544. 561. 573. 581. 586. 590. 596. 603.

3 Heating the Blank The first step is to heat the blank from room temperature (25C) to the austenization temperature (940C). The easiest modeling technique is to define the initial temperature condition of the blank to be 940C. However, doing this will not calculate the thermal expansion of the blank between 25 C and 940 C. Therefore, the blank is heated in the FE model resulting in a thickness change from 1.95 mm to 1.97 mm.

4 Transport from the Oven to the Tool The next step is to transfer the hot blank from the heating oven to the tools. The blank loses heat by convection and radiation heat transfer to the environment at 25C. The convection coefficient can be entered into LS-DYNA as a function of temperature defined by a data curve, or as an equation. Entering an equation allows the calculation of convection coefficients using standard empirical equations from the literature such as:



Blank top surface ( )[ ]LkGrhtop

33.0Pr14.0 ∗= (eq. 1)

Blank bottom surface ( )[ ]LkGrhbot

25.0Pr27.0 ∗= (eq. 2)

These equations are for turbulent free convection from a hot horizontal plate. The convection from the top surface is greater because the buoyancy driven flow is free to rise from the surface, where as it stagnates on the bottom surface. The properties for air used in calculating the Grashof number, Gr, and the Prandtl number, Pr, are evaluated at the film temperature, Tfilm. L is a length scale.

( ) ( ) CTTT sfilm 483

225940

2=

+=

+= ∞ (eq. 3)

( ) m

widthlengthwidthlengthL 4.02

=+∗

= (eq. 4)

( ) 8

2

32

1039.1 ∗=−

= ∞

μβρ LTTg

Gr s (eq. 5)

687.0Pr ==k

C pμ (eq. 6)

The convection heat transfer coefficient is htop = 8.3 W/m2 C using equation 1. A significant concern is that these empirical formulas were developed for heat transfer at temperatures below 400C. However, a quick hand calculation reveals that radiation transport dominates at 810C to 940C and any inaccuracies in the convection coefficient will not significantly alter the results. A radiation conductance can be calculated using

( )( )

( )( )( )( ) 107

298121329812138.00867.5 44

21

42

41 =

−−−

=−

−=

eTT

TThradσε

W/m2 C (eq. 7)

This shows that radiation heat loss is more than 10 times greater than the convection loss. We can perform an energy balance on the blank by equating its change in internal energy to the heat loss by radiation from both sides.

( )442 surfp TTAdtdTVC −= ∞σερ (eq. 8)

This ordinary differential equation can be solved by integration between the (time, temperature) limits of (0, Ti) and (t,Tf) resulting in

( ) ( )( ) ( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+

−+

−+=

∞

−

∞

−

∞∞∞

∞∞

∞ TT

TT

TTTTTTTTT

TAVC

t if

ii

ffp 1133 tantan

21ln

41

2 σερ

(eq. 9)

Using equation 9 and noting that V/A is the blank thickness, it takes 6.6 s for the blank to drop in temperature from Ti=940C to Tf=810C. This is in agreement with the benchmark specification that the transfer time is 6.5 s.



A useful modeling technique is to define an “effective” heat transfer coefficient that combines both convection and radiation effects.

radconveff hhh += (eq. 10) This is a linearization technique that will decrease computer computation time by reducing the number of nonlinear iterations that are required to achieve a converged solution. Solving the radiation transport equation is highly nonlinear due to the T4 terms. However, making use of equation 10 we solve

( )∞−= TTAhq eff (Eq. 11) which is linear in T and is only nonlinear in h. This is easier to solve and can be modeled as a convection boundary condition with the convection heat transfer coefficient defined by (heff, T) data pairs as shown in Table 3. Table 3: Convection, radiation and effective heat transfer coefficients

T [C] hconv [W/m2C] by eq. 1

hrad [W/m2C] by eq. 7

heff [W/m2C] by eq 11

50 5.68 5.31 11.0 100 6.80 6.8 13.6 200 7.80 10.8 18.6 300 8.23 16.3 24.5 400 8.43 23.6 32.0 500 8.51 33.0 41.5 600 8.52 44.8 53.3 700 8.50 59.3 67.8 800 8.46 76.6 85.1 900 8.39 97.2 106. 1000 8.32 121. 129.

5 Positioning and Forming The hot blank loses heat to the environment by convection and radiation until it contacts the tools. When the hot blank (~810C) contacts the lower tool (~75C), its lower surface begins to cool due to contact as shown in Figure 3. The metal-to-metal thermal contact conductance (~2000 W/m2C) is much greater than the convection (8.3 W/m2C) and radiation (107 W/m2C) coefficients and these modes of heat transfer become negligible. However for increased accuracy, the analysis code should have the feature to turn off thermal boundary conditions for regions in contact. There will be a through thickness temperature gradient in the blank due to the large difference in heat loss rates from the top and bottom surfaces. This is calculated in LS-DYNA using the 12 node thick thermal steel formulation developed at Lulea University [8]. This shell has 4 nodes in the plane and 3 nodes through the thickness. The 3 nodes through the thickness allows the use of quadratic shape functions to accurately calculate the through thickness temperature gradient.

Figure 3. Contact heat transfer from the blank to the tool is the dominant mode of heat transfer.

The top surface loses heat to the environment by convection and radiation

The bottom surface loses heat to the tool. The contact heat transfer to the tool is at least 10x greater than heff.



The contact conductance is the most critical parameter controlling cooling of the blank during forming and quenching, and has the greatest uncertainty. Values are presented in [9] for several metals with various surfaces roughness as a function of interface pressure. Also, included are values when lubricants are used. Merklein [10] presents data for 22MnB5 steel at various temperatures and pressures. This data for T=550C is shown in Table 4. Shvets [11] presents the following correlations

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

8.0

8514 r

air PkPhσλ

π (eq. 12)

Shvets’ formula contains a roughness parameter, λ, and a rupture stress, σr. The two end points of Merklein’s data can be substituted into equation 15 to calculate λ and σr. Then, this equation can be used to calculate h at other pressures as shown in Table 4. Table 4. Values for contact heat transfer conductance as a function of interface pressure

P [MPa] h [W/m2C] at 550C Merklein data

h [W/m2C] Shvets formula

h [W/m2C] Numisheet BM03

0 750 750 1300 5 1330 1330 10 1750 1770 20 2500 2520 4000 35 4500 40 3830 3830 LS-DYNA has the capability to model the mechanical coefficient of friction as a function of interface temperature and the thermal contact conductance as a function of interface pressure. Data pairs of (μ,T) and (h,T) can be entered in a table, defined by an inline function in the input file (e.g., Shvets’ formula), or by a user friction subroutine for more complex models. A data table was used in this analysis with h versus pressure data pairs defined in the Numisheet BM03 benchmark specification as shown in Table 4. There are 2 analysis techniques with increasing complexity and computer run time for the forming and quenching analyses:

1. All parts are modeled using shells. The tools are held at 75 C. The blank is modeled with a 4 node thin shell formulation that does not allow the calculation of a thru-thickness temperature gradient. Blank thickness changes are calculated. Contact conductance is a function of interface pressure.

2. The tools are modeled with solids, figure 4, allowing calculation of tool temperature changes. The blank is modeled with a 12 node thick shell formulation [8] allowing the calculation of a thru-thickness temperature gradient and thickness changes. Contact conductance is a function of interface pressure.

Table 5 contains the element count for each model and typical run times. The run times are for a double precision run on a single 2.40GHz Intel CPU DELL workstation. The number of elements used for the blank increases during the run because of mesh adaptivity to accurately calculate sharp angle changes of the mesh during deformation. The tools in both analyses use a rigid material constitutive model. Approximating a deformable body as rigid is a preferred modeling technique in many real world applications. For example, in sheet metal forming problems the tooling can properly and accurately be treated as rigid. Elements which are rigid are bypassed in the LS-DYNA element processing algorithm and no storage is allocated for storing history variables. Consequently, the rigid material type is very CPU cost efficient. As shown in Table 5, although the tool element count increases by a factor of 7.8 between the two analyses, the CPU time only increases by a factor of 1.2.



Figure 4. FE model using solid elements for the tools

Table 5. The number of elements and computer run times.

Analysis 1 Analysis 2 Elements used: Tool mesh Initial blank mesh Final blank mesh

68268 shells 3096 shells 11739 shells

532,927 solids 2096 shells 11739 shells

CPU time: Forming Quenching

5.1 h 20 min

5.9 h 25 min

The blank was modeled using LS-DYNA mechanical material MAT-106 which is an elastic visco-plastic thermal model. The primary reason for using this model is the ability to enter data tables of stress versus strain as a function of temperature as shown in Figure 2. These data tables were provided in the benchmark specification. Also, all other material properties (e.g., Young’s modulus, coefficient of thermal expansion) can be entered as a function of temperature. Forming results for the 2 analysis variants are presented in Table 6. There is about a 5% difference in thickness and a 2.5% difference in temperature between the 2 analyses. Figure 5 shows the thickness distribution in the blank with a range from 1.43 mm (light color) to 2.19 mm (dark color). The lightest colored regions are susceptible to tearing. Table 6. Blank thickness and temperature results after forming

Analysis 1 Analysis 2 Thickness minimum maximum

1.36 mm 2.19 mm

1.43 mm 2.19 mm

Temperature minimum maximum

633 C 828 C

650C 808 C



Figure 5. Shown is the thickness distribution in the blank after forming. The range is from 1.43 mm (blue) to 2.19 mm (red).

6 Quenching The blank is held in the tools for 20 s for the quenching process. The cooling rate of the blank affects the microstructure and hardness properties of the material. Figure 6 overlays the two analysis results on a CCT diagram for a single location on the blank. Curve 1 is for the case where the tools are modeled with shells and held at 75C per the benchmark specification. Curve 2 is for the case where the tools are modeled with solids and allowed to change temperature. The results are significantly different between the two analysis variants. The reason is that in analysis 1, the tools do not change temperature and are held at 75C. This results in a faster cooling rate. The LS-DYNA thermal material model allows specification of the latent heat, 640 MJ/m3, and the phase change temperature interval, 230-410 C for the austenite to martensite transition. Including the latent heat further slows the cooling rate. This is shown by curve 3 in Figure 6. Figure 6. Shown is the quenching temperature history for 3 modeling scenarios overlaid on a CCT diagram for 22MnB5 steel. (1) tools modeled with shells, (2) tools modeled with solids, (3) including latent heat.

During quenching, a higher cooling rate increases the amount of martensite whereas a slower rate gives a higher content of ferrite and pearlite in the blank. LS-DYNA material mode MAT-244 [3,4] is specific to ultra high strength steels such as 22MnB5. This material model calculates the material phase fractions (Figure 7) and Vickers hardness (Figure 8) throughout the part.



Figure 7. Martensite fraction after quenching (light grey scale=100%, dark grey scale=80%)

Figure 8. Vickers hardness after quenching (light grey scale=497, dark grey scale=422)

7 Literature

1. Nunmisheet 2008, The Numisheet Benchmark Study, Benchmark Problem BM03, Interlaken, Switzerland, Sept. 2008.

2. LS-DYNA Keyword User’s Manual, Version 971, Livermore Software Technology Corp., Livermore, CA, USA, May 2007.

3. P. Akerstrom & M. Oldenburg, “Austenite Decomposition During press hardening of a Boron Steel – Computer Simulation and test”, Journal of Material Processing Technology, 174(2006) 399-406.

4. T. Olsson, A LS-DYNA Material Model for Simulations of Hot Stamping Processes of Ultra High Strength Steels, Engineering Research Nordic AB, Sweden.

5. R.E. Cowper and P.S. Symonds, “Strain Hardening and Strain Rate Effects in the Impact Loading of Cantilever Beams”, Brown University, Applied Mathematics Report, 1958.

6. D. Lorenz, private communication, DYNAmore GmbH, Stuttgart, Germany. 7. Shapiro, “Mysteries behind the Coefficient of Thermal Expansion (CTE) Revealed”, FEA

Information News, www.feainformation.com, May 2008. 8. G. Bergman and M. Oldenburg, “A Finite Element Model for Thermo-mechanical Analysis of

Sheet-metal Forming”, Int. J. Numer. Meth. Eng., 59, 1167-1186, 2004. 9. N. Fitzroy, Ed., Heat Transfer Data Book, General Electric Corp., Schenectady, NY, USA,

1970. 10. M. Merklein and J. Lechler, “Determination of Material and Process Characteristics for Hot

Stamping Processes of Quenchenable Ultra High Strength Steels with Respect to a FE-based process Design”, SAE International, SAE Technical Paper Series, 2008-01-0853, 2008.

11. I.T. Shvets, “Contact Heat Transfer Between Plane Metal Surfaces”, Int. Chem. Eng., Vol. 4, No. 4, p621, 1964.

How to Use LS-OPT for Parameter Estimation – hot stamping and quenching applications

Arthur Shapiro, LSTC, Livermore, CA, USA. [email protected]

The “direct” heat transfer problem is one in which material properties and boundary conditions are specified, and LS-DYNA [1] is used to calculate the temperature response of the nodes in the mesh. The “inverse” heat transfer problem is one in which the temperature response of a node point in the mesh (e.g., a surface node) is specified from experimental measurements, and the objective is to calculate material properties and boundary conditions that cause this temperature response. This paper describes how to use LS-OPT [2] to solve the “inverse” heat transfer problem. Applications include:

• calculating material parameters for austenite-to-martensite phase change kinetics – fitting material properties to experimental data

• calculating contact heat transfer coefficients as a function of temperature and pressure during hot stamping – fitting a function to experimental data

• calculating boiling heat transfer coefficients for quenching in liquids – fitting a load curve to experimental data

MAT_UHS_STEEL Phase Change Kinetics

This section describes how to use LS-OPT to calculate material properties for MAT_UHS_STEEL (MAT_244). The methodology shows how to calculate the phase transformation activation energies by matching numerical results with experimental measurements of Vickers hardness. The ultra high strength steel material model requires specification of the ferrite (Qf), pearlite (Qp) and bainite (Qb) activation energies used in the phase change kinetic equations:

Where the phase hardness values, Hi , are a function of the cooling rate at 700C.

The functions f(xf), f(xp), and f(xb) account for the effect of the current fraction formed on the reaction rate. The function f(G) accounts for the grain size and f(T) accounts for undercooling. The material hardness is a function of the individual phase fractions (xi) and hardness (Hi)

8th European LS‐DYNA Users Conference, Strasbourg ‐ May 2011

Table 1 presents experimental data of Vickers hardness versus cooling rate [3] for USIBOR 1500P with an ASTM grain size of 6.8. The third column in Table 1 is calculated results after the

the experimental and calculated hardness values versus cooling rate. Values in the shaded boxes were used in the optim

optimization. Each experimental hardness value in Table 1 is for a different cooling rate. TheLS-DYNA model consists of 10 separate parts, one for each cooling rate. Each part is a single shell element. A minimum of 3 parts are required to obtain a global minimum because there are 3 parameters to be optimized, Qf, Qp, and Qb. Since the data was available, 2 extra parts were added to span the experimental data set and obtain a better answer. The 5 points selected for the parameter optimization are shaded in Table 1.

Table 1. Shown in the table are

ization response functions

Cooling rate [C/sec]

Experimental Vickers hardness

Calculated Vic

[ref. 3] kers Hardness

1 181 181 5 244 244 10 331 331

12.5 375 372 15.0 407 406 17.5 435 429 20 451 443 25 458 457 30 464 465 40 469 472

The optimization problem is addressed by mini ng the relative err Vickers hardness. LS-YNA calculates Vickers hardness and prints it out as element history variable #6 in the d3plot

hv61(t) = d3plot elem ., history variable 6)

The rem -OPT input parameters are shown in Table 2. The res 035. An alternate approach is to define a

mizi or in Dfile. For the first data point in Table 1, the response function is

ResExp1=abs(Final(“hv61(t)-181)/181.

Where,

ent time history of Vickers hardness (i.e

Final = lsopt script to use the final value

abs = lsopt script to take the absolute value

aining 4 response functions and other LSults are Qf = 11335, Qp = 16431, and Qb = 15


composite mean squared error response function as demonstrated by the “System Parameter Identification” problem in the LS-OPT training manual [4].

Table 2. LSOPT input parameters Strategy SRSM – sequential with domain reduction Variables QR2(start, min, max) = (13022, 10000, 15000)

00) QR3(start, min, max) = (15570, 14000, 180QR4(start, min, max) = (15287, 14000, 18000)

Sampling Polynomial, linear, D-Optimal Histories ing rate 1

part 2 with cooling rate 5

hv61(t) = History variable 6 for part 1 with coolhv62(t) = History variable 6 forhv63(t) = History variable 6 for part 3 with cooling rate 10hv64(t) = History variable 6 for part 4 with cooling rate 15hv65(t) = History variable 6 for part 5 with cooling rate 30

Response ResExp1 = abs(Final(“hv61(t)-181)/181 ResExp2 = abs(Final(“hv62(t)-244)/244 ResExp3 = abs(Final(“hv63(t)-331)/331 ResExp4 = abs(Final(“hv64(t)-407)/407 ResExp5 = abs(Final(“hv65(t)-465)/465

Objective ResExp1 = 1 ResExp2 = 1 ResExp3 = 1 ResExp4 = 1 ResExp5 = 1

Constraints None Algorithm LFOP Run 20 iterations

culate hot stamping tool-to-blank contact heat terface pressure. The described methodology shows how

, nd

Contact Heat Transfer Coefficients

This section describes how to use LS-OPT to caltransfer coefficients as a function of into fit a function (e.g., *DEFINE_FUNCTION) to experimental data. Experimental data [5] consists of temperature-time histories for thermocouples mounted below the surface of the blanktop tool, and bottom tool at several applied pressures. Data was recorded for P=0, 20MPa, a30MPa. Figure 1 shows experimental data for P=0.


Figure 1. Experimental temperature versus time for P=0.

The contact heat flux from the hot blank to the cooler tools is calculated by ( )toolblank TThAq −=&onditions,

as shown in figure 2. This is similar to the equation used for convection boundary c

(= ThAq blan& , instead of defining the model as a coupled thermal-stress problem, it can be defined as a thermal only problem using a convection boundary conditiis specified for the top and bottom blank surfaces using the experimental tool temperature datBy this methodology, we avoid the numerical application of a pressure boundary conditiothe tool surfaces and any calculated numerical noise in the pressure and temperature at the top and bottom contact surfaces. This makes the problem well behaved for the LS-OPT optimization.

) Thereforeon.

a. n on

Figure 2.

∞−Tk .

hvets [6] provides the following function to calculate the heat transfer coefficient. S

⎥⎥⎦

⎤⎡ 8.0

⎢⎢⎣

⎟⎠⎞

⎜⎝⎛+= 0 851σPhh

where P is the applied pressure, h0 is the heat transfer coefficient at P=0, and σ is a material hardness metric. h0 and σ are parameters to be determined using LS-OPT. The LS-DYNA keywords used to define the pressure dependence of the convection heat transfer coefficient are


*PARAMETER rho 1000. rsigma 1000. $====== CONVECTION BOUNDARY CONDITIONS ===== $ *BOUNDARY_CONVECTION_SET $# sid 1 $# fid lcidh lcidt mult 1 0. 11 1. $ $================ FUNCTIONS ================ $ *DEFINE_FUNCTION $# fid definition 1 top surface coefficient h1=h0*(1.+85.*(20./sigma)**0.8)

The LS-DYNA input defines 2 parts (see Fig. 3) because there are 2 parameters (h0 and σ) to be etermined. One part is for P=0 and the other part is for P=20. Table 3 presents the LS-OPT arameters used for the optimization. The results are h0=634.5 and σ=1193. Figure 4 shows a

dpcomparison between the numerical answers and the experimental data for the two cases of P=0 and P=20.

Table 3. LS-OPT input parameters

Strategy SRSM – sequential with domain reduction Variables h0(start, min, max) = 1000, 500, 1500)

= (1000, 500, 4000) hard(start, min, max) Sampling Polynomial, linear, D-Optimal Histories T_numerical, T_experimental for P=0 @ node point 1

@ node point 5 T_numerical, T_experimental for P=20 Response

xp for P=20 MeanSqrErr for T_num vs T_exp for P=0 MeanSqrErr for T_num vs T_e

Objective MeanSqrErr for P=0 MeanSqrErr for P=20

Constraints None Algorithm LFOP Run 10 iterations


igure 3. Th DYNA s of parts, one for P=0 and the other for =20.

F2

e LS- model consist

P

where .

Figure 4. Comparison of numerical vs. experimental temperatures using the optimization results h0=634.5, σ=1193.

P=0 P=20

uenching Heat Transfer Coefficients

late boiling heat transfer coefficients as a nction of temperature for quenching in liquids as shown in Figure 5a. Instead of installing

t plate is used with thermocouples mounted on the top 2

Q

This section describes how to use LS-OPT to calcufuthermocouples on the real part, a small flaand bottom surfaces. The experimental data [7] consists of temperature-time histories for the thermocouples, Figure 5b. The objective is to use LSOPT to determine the temperature dependent heat transfer coefficients such that the LS-DYNA calculated temperatures match the measured temperatures.

Fig 5a. Water quench Fig 5b. Measured surface temperatures


Fifteen heat transfer coefficients to be determined where specified as shown in table 5 spanning the temperature range of 75C to 1125C. The LSOPT starting value, lower, and upper bound are also shown in Table 5. The LFOP optimization algorithm was used with a mean squared error objective function between the measured and calculated temperatures on the bottom and top surfaces. Figure 6a shows the calculated temperature history using the optimized heat transfer coefficients in Figure 7a. Note the kink in curve 6a at time=175sec and the noise in the h values in curve 7a at temperature=1100C. Figures 6b and 7b show the results using the GA algorithm. Note that the noise is attenuated. Next, the GA optimized h values were used as the starting point for a 2nd optimization using the LFOP algorithm. The lower and upper bounds where set to ±25% of the starting point values. The results are shown in Figures 6c and 7c. Figure 8 shows a comparison between the measured temperatures and those calculated using the heat transfer coefficients from Figure 7c. The agreement is very good.

Table 5. LSOPT parameters to be determined with their starting point values, lower

bound and upper bound. The last column is the optimized answer. *DEFINE_CURVE temperature, parameter

LSOPT parameter value start lower upper

Optimized answer (see fig. 7c) h [W/m2C]

75. h1 10 10 500 14 175. h2 100 100 500 236 275. h3 100 100 500 248 375. h4 100 100 500 289 475. h5 100 100 500 319 575. h6 100 100 500 310 675. h7 100 100 500 258 775. h8 100 100 500 242 825. h9 100 100 500 225 875. h10 100 100 500 209 925. h11 100 100 500 193 975. h12 100 100 500 159 1025. h13 100 100 500 132 1075. h14 50 50 200 78 1125. h15 10 10 100 49

Fig usin 2nd optimiza

6a. Calculated results g the LFOP algorithm

Fig 6b. Calculated results Fig 6c. calculated results after using GA algorithm

tion



Fig7usin

Fig 7c. Calculated h values after the 2nd optimization

a. Calculated h values g the LFOP algorithm

Fig 7b. Calculated h values using the GA algorithm

Fig 8. A comparison between the measured temperatures and those calculated using the heat transfer coefficients from Figure 7c.

eferences

1. LS 10

2. LS

3. A. Bardelcik, S. Winkler, M.J. Worswick, M.A. Wells, "Investigation of Experimental and Numerical Predictions of Phase Transformations in Hot Stamping at Sub-Critical Cooling Rates", (in Preparation), University of Waterloo, 2011.

4. LS-OPT Training Class manual, LSTC, December 2010.

5. Experimental data courtesy of David Lorenz, DYNAmore GmbH, Stuttgart-Vaihingen, Germany.

6. I.T. Shvets, Contact Heat Transfer Between Plane Metal Surfaces, Int. Chem. Eng., Vol. 4, No. 4, p621, 1964.

7. R.A. Wallis, "Application of process Modelling to Heat treatme Superalloys", Cameron Forge Co., Houston, TX, Industrial heating, January 1988.

R

-DYNA Keyword User’s Manual, LSTC, Version 971/Rev5, May 20

-OPT User’s Manual, LSTC, Version 4.1, August 2010.

nt of

Hot

Stamping Process

fringes of

temperature

Sections covered during the course

Getting Started – Learn to create a KEYWORD input file to solve for the thermal expansion of an aluminum block. Lean how to interpret LS-PrePost temperature fringe plots to gain knowledge of the physical process.

Equation Solvers & Nonlinear Solution Method - Learn the advantages and disadvantages between the Gauss direct solvers & conjugate gradient iterative solvers in LS-DYNA. Learn the nonlinear heat transfer keyword parameters and how Newton’s nonlinear method works.

Time Step Control – Learn how to select a thermal and mechanical time step size, and understand the difference between explicit and implicit solution methods.

Initial and Boundary Conditions – Learn how to define temperature, flux, convection, and radiation boundary conditions. Learn how to hand calculate a convection heat transfer coefficient, which is the parameter with the greatest uncertainty in your model.

Thermal-Mechanical Contact – Learn thermal-mechanical contact modeling issues with sheet metal forming applications.

Thermal-stress coupling – An introduction to coupled thermal stress modeling. Topics include conversion of plastic work to heat, conversion of sliding friction to heat, and calculation of thermal expansion. Thermal-mechanical material constitutive models are also presented.

Modeling Hot Stamping - The Numisheet 2008 B-pillar hot stamping benchmark problem BM03 is presented and solved.

Modeling Warm Forming - The Numisheet 2011 magnesium warm forming benchmark problem BM02 is presented and solved.

Class Information: Class Starts at 9AM. Lunch will be provided.

Getting Started with

April 12th, 2013 Using the heat transfer capabilities in LS-DYNA. LSTC Michigan Office Instructor: Dr. Arthur Shapiro LSTC 1 Day Series Registration: $100.00 Students $ 50.00 Contact: [email protected]

Description: This class provides guidelines in using the heat transfer capabilities in LS-DYNA to model coupled thermal-stress problems with a focus on warm forming and hot stamping manufacturing operations. It is intended for people with a background in using LS-DYNA for computational mechanics, but who are not familiar with modeling heat transfer or coupled thermal-stress. Class Material: A 30-day demo LS-DYNA license will be authorized after the class to continue your learning experience. Course Notes will be distributed the morning of the class.

LSTC One Day Class Series Modeling Warm Forming & Hot Stamping

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