johnson-cook empirical models

7/23/2019 Johnson-Cook Empirical Models

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A brief introduction to

Johnson-Cook Empirical Mode


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Introduction

Structural impact involves events such as plastic flow at high strain rates, possible local increase of tempfracture.

The usual approach in numerical simulations is to operate with two different models, one representingrepresenting fracture. These two models can be coupled or uncoupled.

To describe the various phenomena taking place during ballistic penetration, it is necessary to charact

materials under impact-generated high strain rate loading conditions.

The characterization involves not only the stress-strain response at large strains, different strain rates and t

the accumulation of damage and the mode of failure. Such complex material behavior involving fracture is

analytical models.

Another important feature is the physical difference between plastic flow and fracture. In ductile metal

viewed macroscopically as a visible shape change, microscopically as the appearance of slip lines, and at t

movement of dislocations.

It is also known that plastic flow is driven by the deviatoric stress state in the material. Initiation of damag

arrests of dislocations by micro defects or micro stress concentrations giving de-cohesion and subsequent

and coalescence of micro cracks and micro voids. The damage evolution is strongly influenced by the hydr

the material. Accordingly, different mathematical models are needed to describe plastic flow and fracture, J

them.

J-C model is a coupled material model capturing viscoplasticity(rate-dependent plasticity) and ductile d

impact and penetration problems. This model is analytically verified by Hopperstad et al. [1] with

viscoplasticity and continuum damage mechanics [2], allowing for large plastic strains, high strain rates and a


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Gordon R Johnson works for US defense systems Division in Edina while William H Cook was engaged

Laboratory.

Together they have created a plasticity model in 1983 [3] which is a particular type of Mises plasticity m

forms of the hardening law and rate dependence. It is suitable for high-strain-rate deformation of many ma

metals. Moreover, It is typically used in adiabatic transient dynamic simulations.

In Johnson-Cook plasticity model equivalent stress () is assumed to be of the form

= + + + (1)

Where A, B, C, n and m are constants, is accumulated plastic strain and = / is dimensionless stra

plastic stain rate and user defined reference strain rate.

The non-dimensional temperature

=

Where is the current temperature, is the ambient temperature, and is the melting temperature.

Clearly, the von Mises equivalent flow stress is the product of three factors representing strain harde

temperature. This facilitates the calibration of the model because each of the parentheses in Eq. (1) can b

in three series with uniaxial tensile tests.

Johnson-Cook Plasticity model


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To describe ductile fracture, two years after plasticity model, in 1985, Johnson and Cook also proposed a moof stress triaxiality, temperature, strain rate and strain path on the failure strain [4] which is as follows :

= + +

+ (2)

Where

-

are material constants, =

/

is the stress triaxiality ratio and

is the mean stress.

The first set of brackets in the Johnson-Cook fracture model is intended to represent the observation thadecreases as the hydrostatic tension increases. The second set of brackets in the strain to failure expressionan increased strain rate on the material ductility while the third set of brackets represent the effect of thmaterial ductility.

The model assumes that damage accumulates in the material element during plastic straining, and thimmediately when the damage reaches a critical value (). In other words, the damage has no effect on thfracture has not taken place. But in light of continuum damage mechanics [2] , stress field depends on damprinciple) and hence damage degrades material strength during deformation. So, equation (1) needs to be cou

= + + + (3)

Where D is damage variable, r is damage accumulated plastic strain =(1-D) , D = 0 virgin material and D =and hence D=Dc

Stress state is an important factor in determining when fracture occurs. In particular stress triaxiality plays governing the tendency for ductile fracture. Stress triaxiality is used to describe the portion of stress tensordefined as the ratio of hydrostatic stress to equivalent Von-Mises stress. In other words the stress staapproaches the completely hydrostatic stress while with lower triaxiality stress state deviatoric strinvestigations have revealed that increased triaxiality reduces ductility and thus failure strain

The model describes linear elasticity, initial yielding, strain hardening, strain-rate hardening, damage evmaterial constants can be identified from uniaxial tensile tests.

Johnson-Cook damage model


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Borvik et. al.[5] has calibrated the Johnson cook model experimentally for Weldox 460 E and fo

parameters by doing some simple experiments as taught in [3,4]. This as a case study which is presented

Step 1. (, ) Quasi-static tensile tests were performed at a strain rate of5 1 04

to determine the elatests in three different directions (0 deg., 45 deg. and 90 deg.) were done to check for the anisotropy in th

the results the material were kept isotropic keeping in mind that the damage after necking is direct

anisotropy at large plastic strains.

Step 2.(A) Again the Quasi-static tensile tests were performed at large strains. Diameter reduction was ob

steps by vernier caliper and plastic strain in necked region is obtained by

= 2 ln (

)

where is initial and is current diameter respectively. True stress is obtained as

=F

A

where F is the applied force and A is the cross-sectional area instantaneous steps

Before necking,

= f (F)

Determination of J-C Parameters(acasestudy)


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After necking, becomes three dimensional because components of hydrostatic

tension tends to make net tensile stress greater than . In view of this Bridgman [6]

correction was applied to true stress and the curve was redrawn.

= (1 +

2

) ln 1 +

2

where R is the curvature radius of the neck, and a is the radius of the specimen in thenecked zone.

The true stress strain curve becomes more steeper after applied Bridgman correction.

As at large strains correction is considerable hence only yield stress constant (A) is

determined from shown figure.

Step 3. (B, n, Dc) In a similar way as for the smooth specimen, the applied load and cross-sectional area in the notch specimens are measured during testing.

It is assumed that the stress triaxiality ratio is approximately constant during plasticstraining in each notched specimen while it varies in smooth specimen tensile test.

The different notches used gives a concentration of hydrostatic tension in the testspecimen.



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For a specified true strain, the stress is seen to increase when the notched radius R isreduced. It is also seen that the presence of hydrostatic tension significantly decreases thestrain at which the material fractures.

The model constants B, n and Dc are determined from eq. 3 for =1,T=To and =

=

+

(4)

Where , and are discrete values of the variables

, and, and is the measuredfracture strain for the different notched specimens and is the measured fracture strain fordifferent notched specimen.

An artificial notch produces an initial triaxiality different from that in the case of a smoothspecimen where triaxiality is initiated only after the commencement of necking.Bridgmans relationship was used to correlate initial notch radius R and maximum stress

triaxiality ratio

=1

3+ ln 1 +

2

The damaged accumulated plastic stain is calculated as :

=

(5)

The method of least squares is then used to minimize the difference between theexperimental determined curves and the model by varying B, n and Dc simultaneously

As seen, good correlation is obtained for the notched specimens. However, the fit is

somewhat poorer for the smooth specimen. This is probably caused by the variation instress triaxiality during testing.



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Step 4. (C) The viscous effect is obtained by means of uniaxial tensile tests at a range ofdifferent strain rates from 104 to 10

From these tests an average effect of strain rate can be obtained. The figure showsthat the strain rate sensitivity of the material is moderate and almost unaffected bythe level of plastic strain.

To have an average value of the strain rate sensitivity C, the data was fitted to theproposed model using the method of least squares for a plastic strain of =0 = 0.1 = . The mean value of C is then used to describe theviscous effect.

= +

Step 5. (m) Effect of thermal softening on true stress is shown in figure. yield stressshows a linier decrease with increasing temperature while ultimate stress shows a localminima at about 300 deg. C

By assuming adiabatic condition at high strain rates, the effect of thermal softening isincluded in the model by fitting the material constant m to the decreasing yield stress for

( = 0,

= / = 1 ) = +



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Step 6. (, , ) The strain to fracture used in the damage evolution rule is given by :

= +

The expression in the first set of brackets gives the effect of hydrostatic stress on thestrain to fracture for quasi-static conditions.

Figure shows the dimensionless triaxiality ratio in the center of the specimen iscalculated based on Bridgman's analysis

=

=

1

3+ ln 1 +

2

where a and R are the initial radius of the specimen in the neck and the initial neckradius, respectively, since the triaxiality ratio is assumed constant during plasticstraining.

The three material constants, , , in the model have been fitted to thegiven data, and the curve has been extrapolated into the hydrostatic compressionregion as seen in Figure.

cannot assume negative values because it corresponds to the purely hydrostaticalstress state.



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Step 7. () The effect of strain rate on the strain to failure is given in the second set ofbrackets as :

= +

The effect of strain rate is isolated from the temperature effect by considering themeasured fracture strains for smooth specimens at =1/3 and = 0

This is shown in Figure, where the strain rate constant is fitted to the measuredfracture strains indicating that the fracture strain decreases slightly with increasingstrain rate.

Step 8. () In a similar way, the effect of temperature on the fracture strain is isolatedfrom the strain rate as :

= +

Keeping = / = 1 and the temperature constant is obtained from figureshown.



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Some Physical Aspects of J-C Paramete

Positive aspects:

Few material constants (5+5) needs to be evaluated.

Since the strain rate and temperature effects on the flow stress are uncoupled ( eq. 1) , the Johnsomodel is easy to calibrate with a minimum of experimental data.

Damage evolution can be coupled with the model via strain equivalence principle of damage mechanic

Negative aspects:

Strain rate sensitivity (step 4) is found experimentally to increase with increasing temperature, while thstress is decreasing as observed by Harding [7] which is in contrast to the model.

According to Harding [7] this uncoupling between thermal and viscous effects (step 7 and 8) used in model may not capture the correct physics observed in experiments.


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One of the important parameter to accurately calibrate J-C model is stress-triaxiality. As seen, to obtain stress triaxiality for different notched specimen isa tedious and costly affair. Alternatively, it can be evaluated in Abaqus-CAE byrequesting the TRIAX command in field output for different notchedspecimen.

J-C elasticity model is : = +

+ +

and all the parameters A, B, etc. can be effectively evaluated in a single shot justby having a engineering stress strain data (may be or may not be time dependent)for the material in use. How ? By using Mcalibration software developed byJorgen Bergstrom [8]. It uses the regression analysis and fits the full equation in asingle go to the given material data.

Borvik [5] suggested some improvements in the plasticity model like

approaches minus infinity for very small strain rates. So he modified thecorresponding bracket by +

Value of quantifies . As is related to deviatoric state of stress during

the failure it governs the deformed shape of the material as shown in figure inwhich creation of petals during an impact by conical projectile is governed bydeviatoric state of the Cauchys stress tensor [9]. Changing the value of by

keeping all others constants same will change the way material deforms.

Closing Remarks


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1. Hopperstad OS, Berstad T, Borvik T, Langseth M. Computational model for viscoplasticity and ductile dama

"fifth International LS-DYNA User Conference, Michigan, USA, 21-22 September 1998.

2. Lemaitre J. A Course on damage mechanics. Berlin: Springer, 1992.

3. Johnson GR, Cook WH. A constitutive model and data for metals subjected to large strains, high stra

temperatures. Proceedings of Seventh International Symposium on Ballistics, The Hague, The Netherlands,

4. Johnson GR, Cook WH. Fracture characteristics of three metals subjected to various strains, strain rates,

pressures. Eng. Fracture Mech. 1985; 21: 31-48.

5. Borvik T, Langseth M, Hopperstad OS, Malo KA. Ballistic penetration of steel plates. Int J Impact Eng 1999;2

6. Bridgman PW. Studies in large plastic fow fracture. Cambridge, MA: Harvard Univ. Press, 1964.

7. Harding J. The development of constitutive relationships for material behaviour at high rates of strain, Inst

102: Session 5, Oxford, UK, 1989.

8. http://polymerfem.com/content.php?9-MCalibration

9. M.A. Iqbal, S.H. Khan, R. Ansari, N.K. Gupta, Experimental and numerical studies of double-nosed pro

aluminum plates, International Journal of Impact Engineering, Volume 54, April 2013, Pages 232-245,

10.1016/j.ijimpeng.2012.11.007.

References
http://polymerfem.com/content.php?9-MCalibrationhttp://polymerfem.com/content.php?9-MCalibrationhttp://polymerfem.com/content.php?9-MCalibrationhttp://polymerfem.com/content.php?9-MCalibration


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johnson-cook empirical models

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