MODELLING OF STRAIN HARDENING AND STRAIN RATE HARDENING OF DUAL PHASE STEELS IN FINITE ELEMENT...

MODELLING OF STRAIN HARDENING AND STRAIN RATE HARDENING OF DUAL PHASE STEELS IN FINITE ELEMENT

ANALYSIS OF ENERGY ABSORBING COMPONENTS

MODELLING OF STRAIN HARDENING AND STRAIN RATE HARDENING OF DUAL PHASE STEELS IN FINITE ELEMENT ANALYSIS OF ENERGY-ABSORBING COMPONENTS

Anders Söderberg and Ulf Sellgren

KTH - Machine Design, Royal Institute of Technology (KTH), Sweden

SUMMARY

The choice of constitutive equations affects the ability to accurately predict the behaviour of dual phase steel components during a crash event, when strain hardening and strain rate hardening effects have a significant influence on the behaviour of a steel member. Consequently, both strain hardening and strain rate hardening must be thoroughly modelled in order to accurately predict overall crash and component behaviour from finite element analysis (FEA).

Four different models for strain hardening and four models that include strain rate hardening are presented in this paper and the relations between the continuum level model differences and the results from a crash simulation are analysed. Finally, a model that includes thermal softening effects as well as strain and strain rate hardening effects is analysed.

The constitutive equations were fitted to experimental data for two cold rolled dual phase steels, Docol 600 DP and Docol 800 DP. To establish the differences at the continuum level and to study how post-necking behaviour is treated, the strain hardening equations were implemented in a commercial FE code and a FEA of a quasi-static tensile test was performed. Finally, the constitutive equations were used in a FEA of axial crushing of square beams in order to study their influence on the outcome of a crash simulation. The results obtained from the simulations are compared to experimental results.

Despite significant differences at the continuum level, the choice of strain hardening equation does not have any notable effect on the outcome of the crash simulations. The constitutive models’ ability to capture strain rate hardening has a more significant influence on the result.



1: Introduction

Due to their exceptional properties as regards strength, fatigue and formability, dual phase steels are quickly replacing more conventional steels in a wide range of automotive applications, including safety critical components such as front and rear rails, crush cans, cross members, door intrusion beams and wheel rims. High-strength dual phase steel that is a mixture of a ferrite matrix and martensite islands is commonly used in crash energy absorbing members.

During a crash event such components have to absorb the kinetic energy in less then one second. Under such conditions, strain hardening and strain rate hardening effects have a significant influence on the crash behaviour of a steel member. The collapse mode of a single component, as well as the overall crash behaviour, may change if the strain rate effect is not incorporated, (e.g. [1]). Furthermore, an FEA that neglects strain rate effects is non-conservative. Consequently, both strain hardening and strain rate hardening must be thoroughly modelled in order to accurately predict overall crash and component behaviour from an FEA.

To get reliable and accurate results from crash simulations, the model should include forming effects and thermal softening caused by the extensive heating of the material due to extensive plastic deformation,( see [2],[3]). Forming effects are incorporated by a pre-simulation of the forming process. The forming analysis is often performed using the same FE code as that used for the crash simulation. Although strain rate effects may influence the forming process, the strain rates are not of the same magnitude as in a crash event, which places different demands on the constitutive equations. The work reported in this paper focused on developing a material model of strain hardening and strain rate effects for FE-based crash simulations of vehicles.

Numerous material models that incorporate strain hardening and strain rate effects have been presented in the literature. Some of the constitutive relations are based on empirical observations on a macro level while others are motivated by micromechanical considerations. Many of these models are implemented in commercial and proprietary FE codes

This paper examines the constitutive equations from an engineering design point of view and focuses on how the ability to accurately predict the behaviour of a dual phase steel component during a crash event is affected by the choice of constitutive equations. It presents four different models for strain hardening and four models that include strain rate hardening, and analyses the relations between the continuum level model differences and the results from a crash simulation. Finally, a model that includes thermal softening effects as well as strain and strain rate hardening effects is analysed.



The paper is structured as follows: in the first sections the studied constitutive equation are described and fitted to experimental data for two cold rolled dual phase steels, Docol 600 DP and Docol 800 DP. To establish the differences at the continuum level and to study how post-necking behaviour is treated, the strain hardening equations are implemented in a commercial FE code and a FEA of a quasi-static tensile test is performed. Finally, the implemented constitutive equations are used in FEA of axial crushing of square beams in order to study their influence on the outcome of a crash simulation. In the last section of the paper, the results are discussed and conclusions drawn.

2: Strain hardening equations

One of the most commonly used strain hardening equations is the power law named after Hollomon:

nBεσ = (1)

This equation models the relation between yield stress, σ, and total strain, ε. The parameters B and n are usually determined from experimental data.

It is often more convenient to use the relation between yield stress and plastic strain, εp. This can be modelled by another power law, known as the Ludwik equation

npBA εσ += (2)

Here A is the initial yield stress of the material and is experimentally determined along with B and n. For ferritic steels, Bergström proposes that the strain hardening should be modelled by

( )pkA εσ 5.0exp1 −−+= (3)

The initial yield stress A and the parameter k are determined from experiments.

None of the above equations are capable of describing the strain hardening of steels at low and high strain levels. Consequently, one has to choose what strain range to ‘tune’ the equation for. With this in mind Engberg, (e.g. [4]) suggests that the strain hardening of ferritic steels can be modelled by a Ludwik relation at low strains and a Bergström relation at high strains. This method has proved to give good correlation with experimental data for a number of steels, among them dual phase steels. The method has been included in this work as a complement to the other strain hardening models.



3: The studied strain rate hardening equations

One of the more frequently used constitutive equations that include strain rate hardening is the empirically based Johnson-Cook equation, (e.g. [5]):

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛++=

0

ln1εε

εσ&

& pnp CBA (4)

The strain hardening at a chosen reference strain rate is described with a Ludwik equation. The strain rate term includes the parameter C that is determined from experiments. The Johnson-Cook model is only valid at high strain rates. Kang et al. [5] address this limitation by introducing a quadratic strain rate relation:

[ ]⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛++=

2

02

01 lnln1

εε

εε

εσ&

&

&

& ppnp CCBA (5)

Another disadvantage of the Johnson-Cook equation is that it is only capable of describing material with diverging flow curves (i.e., those where the strain rate hardening increases with the strain level). Xu et al. [6] have therefore suggested the following modified version:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

0

' ln1εε

εεσ&

&nn CB (6)

Here the strain hardening term is the Hollomon equation and the second term includes both strain and strain rate.

Another often used relation is the Cowper-Symond equation, (e.g. [7]):

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛++=

ppn

p DBA

/1

1ε

εσ&

(7)

In this model, strain hardening is modelled with the Ludwik equation and the strain rate term includes two parameters, D and p, which are determined from experiments.

All the equations presented so far have been based on empirical observations and include parameters that have to be determined from experimental tensile tests. The equations are therefore never more correct then the experimental data to which they have been fitted. As high speed tensile testing is complicated, a



constitutive model that minimises the need for high speed measurements is highly desirable.

Engberg has developed a theoretical model of the strain rate hardening of ferritic steels, (e.g. [4]). The model is micromechanically based and assumes that flow stress is the sum of an athermal and a thermal contribution. The athermal contributions cause strain hardening, which is modelled by a Ludwik equation at low strains and a Bergström equation at high strains. The thermal contribution links strain rate hardening and thermal softening. The model has nine material parameters. Engberg has found that many ferritic steels can be modelled using the same set of parameters.

The Engberg model is implemented in the SSAB proprietary software DoCrash. The software fits the strain hardening relation to quasi-static experimental data and uses the strain rate relation to calculate the strain hardening at other strain rates. This approach makes it possible to obtain high strain rate data for a material even if experimental data is missing. DoCrash can predict the material’s behaviour in the cases of an isothermal or an adiabatic deformation process. In the former case, it is assumed that all heat arising from plastic deformation is conducted away from the deformation zone, while in the latter case there is no heat conduction. The isothermal case is directly comparable to the other constitutive equations in this work in that the thermal softening due to heat generation is neglected. The adiabatic model is included in this work to indicate how heat generation can influence the result of a simulated crash with large plastic deformations.

4: Fitting the equations to experimental data

All the investigated relations, except for the Engberg strain rate relation, include parameters that are determined by fitting the equations to experimental data. In this work the data from tensile tests conducted by Peixihno et al. [7] was used. The tensile tests were performed on sheet specimens of the two dual phase steels at four different stain rates in the range 0.0001 s–1 to 200 s–1.

The strain hardening parameters were determined by least square fitting of the equations to the measured flow curves including data until onset of necking. The flow curves were assumed to be isothermal. This assumption is commonly made and is valid at low strain rates, but can be questioned at high strain rates and strain levels, see [3]. Quasi-static flow curves generated with the fitted strain hardening equations are shown in figure 1. The strain hardening terms in the three variations of the Johnson-Cook equation were fitted to measured data at the rate 1 s–1.



Figure 1: Comparison of quasi-static strain hardening extrapolated to 30 % plastic

strain. Right: Docol 600 DP; Left: Docol 800 DP (H-Hollomon, L-Ludwik E-Engberg, B-Bergström)

Some differences between the equations can be seen at low strains, but the major differences are found when extrapolating the equations beyond the onset of necking. Since the tensile tests do not give reliable information about post-necking behaviour, no conclusions can be drawn about which of the equations best describes the real situation.

The strain rate parameters were determined from the measured relation between yield stress and strain rate at fixed strain. In order to minimise the influence of load fluctuations at low strains in the measured data and thermal softening at high strains at an intermediate strain. The least square fitted strain hardening relations are shown in figure 2. The isothermal and adiabatic Engberg relations are also included in the figure.

Figure 2: Comparison of fitted strain rate hardening equations. Right:

Docol 600 DP; Left: Docol 800 DP (JC-Johnson-Cook, K-Kang et al, X-Xu et al, CS-Cowper-Symonds, E-Engberg, Eadi-Adiabatic Engberg)

0

200

400

600

800

1 000

0.10 0.20 0.30 0.40 True plastic strain [ - ]

1 400

1 200

True

Stre

ss [M

Pa]

E

B L

H

0

100

200 300 400 500 600

700

800

0.00 0.10 0.20 0.30 0.40

900 L

E B H

True

stre

ss [M

Pa]

True plastic strain [ - ]

400

500

600

700

800

900

0.00001 0.001 0.1 10 1000

True

stre

ss [M

Pa]

Strain rate [s-1]

JC, X

Eadi

E

CS

K

1000

400

800

1000

1200

1400

0.00001 0.001 0.1 10 1000

True

stre

ss [M

Pa]

Strain rate [s-1]

JCX

Eadi

E

CS

K

600

0.00



Neither the original Johnson-Cook model nor the modified version by Xu et al. is capable of reproducing the material behaviour at strain rates below 1 s–1, but the modification by Kang et al. gives an acceptable approximation at low strain rates. The Cowper-Symonds equation seems to capture the strain rate dependence at both low and high strain rates, whereas the isothermal Engberg relation underestimates the strain rate hardening at high strain rates. The deviation is even more noticeable if the adiabatic relation is used. This is an indication that dual phase steels may show higher strain rate hardening then other ferritic steels.

5: Complete material models of the dual phase steels for FEA

Two different types of material models were set up: quasi-static models in which only strain hardening was included, and dynamic models in which the strain rate hardening was also integrated. Quasi-static models for each of the four investigated strain hardening relations and dynamic models for each of the four combined strain and strain rate hardening relations were constructed for both steels. In all models the steels were assumed to be homogenous and isotropic with a Young’s modulus of 210 GPa and a Poisson’s ratio of 0.3, and were described with von Mises yield criteria.

In commercial FE codes, the strain and strain rate hardening can often be included in the material model by using an implemented material model or by defining the flow curve in tabular form. If a tabular model is used, the data can be taken either directly from measurements or calculated from a theoretical model. The latter option was used in this work. The strain hardening was described up to 70 % plastic strain, and in the dynamic model the strain hardening was described at five different strain rates: 0.0001 s–1, 1 s–1, 10 s–1, 100 s–1 and 1000 s–1.

6: FEA of tensile tests

To investigate how well the equations model post-necking strain hardening, the quasi-static material models were implemented and the quasi-static tensile tests described above were simulated.

The FEA tensile tests were performed as non-linear quasi-static analyses with the commercial implicit solver Abaqus/Standard [8]. Because of symmetry, only one quarter of the specimen was modelled, using the four-node shell element S4. To be consistent with the results from the experimental tensile tests, engineering stress-strain curves were calculated from the reaction forces at the displaced boundary of the FE model and from the elongation of the uniform section of the specimen.



The mesh was refined near the centre of the specimen where the strain gradient was large. Models with different mesh density were analysed by comparing the obtained stress-strain curves to determine numerical convergence. A mesh with an element side length of 0.157 mm at the centre and 2.5 mm far from the centre was considered adequate for this study. The sequence in figure 3 shows the strain levels in the specimen during the simulations.

Figure 3: Contour of equivalent plastic strain in the specimen at different stages of

the tensile test: (a) Unloaded; (b) Elastic elongation; (c) Uniform plastic elongation; (d) Diffuse necking; (e) Localised necking

The deformation process agrees well with experimental observations of ductile sheet metals. The initial phase is characterised by elastic elongation in the whole specimen, until the elastic limit is reached. It is followed by a phase with uniform plastic elongation of the specimen. When the tensile load reaches its maximum, diffuse necking occurs. Before fracture, a second instability process, localised necking, is initiated. In this model the neck is a narrow band inclined at an angle to the specimen’s axis. Once localized necking has started, there is little change in the width of the specimen, but the thickness of the necking band shrinks rapidly, leading to fracture.

Examples of engineering stress-strain curves obtained from simulations with the different strain hardening equations are shown in figure 4.

Figure 4: Comparison of stress-strain curves for Docol 600 DP from experiments

and simulations. (E-Engberg, H-Hollomon, B-Bergström, L-Ludwik)

The choice of equation has a large influence on the prediction of the onset of necking. The best agreement between simulated and measured results was achieved with the Engberg relation. The other constitutive equations seemed to

0

200

400

600

0,00 0,05 0,10 0,15 0,20 Engineering Strain [ - ]

E

Experiment H B

L

Engi

neer

ing

Stre

ss [

MPa

]

(a) (b) (c) (d) (e)



overestimate the strain hardening at high strains, shifting the onset of necking towards higher strain levels.

Other simulations in which the measured flow curve was integrated directly in the material model were also performed. The strain hardening until onset of necking was included. At higher strains, ideal plastic behaviour was assumed. In these simulations, diffused and localised necking occurred simultaneously at lower strains then in the experiments, indicating that the strain hardening at higher strains should not be neglected in the material model.

7: FEA of crushing of square beams

In order to study how the choice of constitutive equations influences predictions of the behaviour of an energy-absorbing component, FEA of axial square beam crushing was performed.

The FE models were set up to simulate experiments performed in a previous study by Sjöström and Gunnarsson [9]. In the experiments square beams of Docol 800 DP were axially crushed under both quasi-static and dynamic conditions. The length of a beam was 300 mm with a cross-section of 60 x 60 mm and a thickness of 1.2 mm. To ensure that the beams collapsed by progressive buckling, triggers were stamped on two opposite sides on their upper part. The quasi-static experiments were performed in a hydraulic press at a constant piston speed of 2 mm/s. In the dynamic experiments, a piston with a mass of 51.2 kg hit the top of the beam with an impact velocity of 18 m/s, causing it to collapse. The piston was then decelerated by the plastic deformation of the beam. In both types of experiments the reaction force of the beam on the piston and the tangential displacement were continuously measured. The energy absorbed by the beam could then be computed as a function of the axial deformation.

The simulations were performed with the explicit solver Abaqus/explicit [10]. Due to symmetry one quarter of the beam was modelled. The material domain was meshed with the four-node shell element S4R. Based on an analysis of results from models with different mesh densities, a uniform mesh with an element length of 2 mm was considered adequate for the simulations. Smaller elements yielded only negligible changes in the results. Self-contact of the beam surfaces was modelled by a penalty formulation. The piston hitting the beam was modelled as a rigid surface and the contact between the beam and the surface was modelled by a penalty formulation. The coefficient of friction was 0.1 for all contacts. The dynamic crushing was simulated by hitting the beam with a rigid surface with a point mass of 12.8 kg and an impact velocity of 18 m/s. The simulations were evaluated in the same way as the experiments (i.e., by studying the reaction force, the axial deformation and the absorbed energy).



0

50

100

150

200

250

0 50 100 150

Deformation [mm]

Rea

ctio

n fo

rce

[kN

]

ExperimentSimulation

Figure 5: (a) Comparison of force-deformation curve from dynamic experiment

and simulation with Johnson-Cook equation. (b) The deformation of the beam by progressive buckling (simulation)

The simulations showed good agreement with both the quasi-static and the dynamic experiments regardless of the choice of constitutive equation in the material model. Some dissimilarity from the experiments was observed in the initial phase of the quasi-static simulations. These deviations were believed to derive from differences in the form of the triggers between the models and the real beams. In order to minimise the influence of these variations, the absorbed energy was only evaluated for deformations occurring after the initial deformation phase (i.e., for deformations between 40 and 120 mm).

The quasi-static models were implemented in quasi-static simulations, and the dynamic material models in dynamic simulations. Furthermore, quasi-static simulations were performed where the measured flow curve was integrated directly in the material model, ignoring the post-necking strain hardening. The energy absorbed in the quasi-static simulations is shown in figure 6 and the corresponding result from the dynamic simulations in figure 7.

2,22 2,24 2,23 2,21 2,162,93 2,88 2,92 2,91 2,85 2,76

0,00

1,00

2,00

3,00

4,00

Experiment H L B E Real

Abs

orbe

d en

ergy

[kJ]

Docol 600 DPDocol 800 DP

Figure 6: Comparison of absorbed energy from quasi-static experiments and

simulations (H-Hollomon, L-Ludwik E-Engberg, B-Bergström, Real-Simulation with measured flow data)

2,59 2,58 2,63 2,77 2,49 2,40

3,73 3,60 3,59 3,59 3,893,29 3,21

0,00

1,00

2,00

3,00

4,00

5,00

Experiment JC K X CS E Eadi

Abso

rbed

ene

rgy

[kJ]

Docol 600 DP

Docol 800 DP

Figure 7: Comparison of absorbed energy from dynamic experiments and

simulations (JC-Johnson-Cook, K-Kang et al, X-Xu et al, CS-Cowper-Symonds, E-Engberg, Eadi-Adiabatic Engberg)

(a) (b)



The choice of strain hardening relation had a very small influence on both the force-deformation curve and the absorbed energy level. The simulation with measured flow data, neglecting the strain hardening at high strains, resulted in a 6 % lower energy prediction for a beam of Docol 600 DP and a 12 % lower prediction for a beam of Docol 800 DP.

The choice of strain rate hardening relation had a somewhat greater influence on the crash behaviour. The three relations based on the Johnson-Cook equation gave equivalent energy predictions. In the simulations with the Cowper-Symonds equation, the absorbed energy was 7 % higher than with the Johnson-Cook equation, and with the Engberg relation it was 4 % lower for Docol 600 DP and 8 % lower for Docol 800 DP. For both steels, the absorbed energy was 4% lower with the adiabatic Engberg relation then with the isothermal relation.

8: Conclusions and discussion

In this paper different constitutive equations have been integrated in FE simulations of a crash event in order to evaluate their influence on the result. The commercial FE code ABAQUS has been used and simulations have been compared with experiments. Since constitutive equations are never more correct then the experimental data to which they are fitted, a constitutive model that minimises the need for high-speed measurements, like the Engberg model studied in this paper, is highly desirable.

High strain levels are present in deformed crash components. Ignoring the post-necking strain hardening in simulations may lead to an underestimation of the absorbed energy. Although the four studied strain hardening equations demonstrated significant differences for post-necking strains at a continuum level, the choice of equation did not have any noteworthy influence on a global level in crash simulations.

A constitutive equation that includes strain rate effects is necessary to get accurate results from simulations of a high-speed crash event. The choice of strain rate equation proved to have a greater influence than that of strain hardening equation. The differences between the four equations at a continuum level correspond well to the results observed in the crash simulations.

Finally it should be pointed out that the importance of modelling strain rate effects should be set in relation to other phenomena, such as forming effects and thermal softening due to heat development during high-rate deformations. Including thermal softening in the simulations led to a notable decrease in the absorbed energy. Forming effects were not included in the models, which may explain the dissimilarities observed between the simulations and the experiments.



ACKNOWLEDGMENT

This work was financed by SSAB Tunnplåt AB. The supervision of Joachim Larsson at SSAB Tunnplåt AB is gratefully acknowledged. Professor Göran Engberg at the same company is acknowledged for permission to use his material model in this work. Professor A.C. Marques Pinho at the University of Miho, Portugal, provided the data from the tensile tests.

REFERENCES

1. SIMUNOVIC, S, SHAW, J, and ARAMAYO, G A, –Material modelling effects on impact deformation of ultralight steel auto body, SAE paper no 2000-01-2715, 2000

2. ZENG, D, LIU, S D, MAKAM, V, SHETTY, S, ZHANG, L and ZWENG, F –Specifying steel properties and incorporating forming effects in full vehicle impact solutions, SAE paper no 2002-01-0639, 2002

3. DUCROCQ, P, MARKIEWICZ, E, HARMAD, S, DE LUCA, P and DRAZETIC, P – Thermal influences on mild steel during crash event, Int. J. Crashworthiness, Vol. 3, No 2, pp.163–190, 1998

4. CARLSSON, B –Material considerations in the design of high strength steel crash members, Proceeding of ESDA: 6th Biennial Conference on Engineering System Design and Analysis, Istanbul, Turkey, July 8-11, 2002

5. KANG, W J, CHO, S S, HUH, H and CHUNG, D T– Modified Johnson-Cook model for vehicle body crashworthiness simulations, Int. J. Vehicle Design, Vol. 21 Nos 4/5 pp.424–435, 1999

6. XU, K, WONG, C, YAN, B and ZHU, H – A high strain rate constitutive model for high strength steels, SEA paper no 2003-01-0260, 2003

7. PEIXINHO, N, PIHNO, A and JONES, N –Determination of crash relevant material properties for high strength steels and constitutive equations, SAE paper no 2002-01-2332, 2002

8. Abaqus/Standard user’s manual v 6.3, Hibbit, Karlsson & Sorenson, Inc, Rhode Island, USA, 2002

9. SJÖSTRÖM, E and GUNNARSSON, L – Crash testing of welded square beams made of carbon and stainless steel sheets, IM-2001-534, Swedish Institute of Metals Research, Stockholm, Sweden, 2001

10. Abaqus/Explicit user’s manual v 6.3, Hibbit, Karlsson & Sorenson, Inc., Rhode Island, USA, 2002

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MODELLING OF STRAIN HARDENING AND STRAIN RATE HARDENING OF DUAL PHASE STEELS IN FINITE ELEMENT...

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