Johnson-Cook Empirical Models

A brief introduction to

Johnson-Cook Empirical Models Presented by : Sanaan H Khan IIT Kanpur

Introduction • Structural impact involves events such as plastic flow at high strain rates, possible local increase of temperature, and material fracture. • The usual approach in numerical simulations is to operate with two different models, one representing plastic flow and one representing fracture. These two models can be coupled or uncoupled. • To describe the various phenomena taking place during ballistic penetration, it is necessary to characterize the behavior of 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 temperatures, but also the accumulation of damage and the mode of failure. Such complex material behavior involving fracture is difficult to describe in analytical models. • Another important feature is the physical difference between plastic flow and fracture. In ductile metals, plastic flow may be viewed macroscopically as a visible shape change, microscopically as the appearance of slip lines, and at the atomic level as the movement of dislocations. • It is also known that plastic flow is driven by the deviatoric stress state in the material. Initiation of damage or fracture is due to arrests of dislocations by micro defects or micro stress concentrations giving de-cohesion and subsequently nucleation, growth and coalescence of micro cracks and micro voids. The damage evolution is strongly influenced by the hydrostatic stress state in the material. Accordingly, different mathematical models are needed to describe plastic flow and fracture, Johnson-cook is one of them.

• J-C model is a coupled material model capturing viscoplasticity (rate-dependent plasticity) and ductile damage, developed for impact and penetration problems. This model is analytically verified by Hopperstad et al. [1] within the framework of viscoplasticity and continuum damage mechanics [2], allowing for large plastic strains, high strain rates and adiabatic heating.

Johnson-Cook Plasticity model •

Gordon R Johnson works for US defense systems Division in Edina while William H Cook was engaged in the US Air Force Laboratory.

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Together they have created a plasticity model in 1983 [3] which is a particular type of Mises plasticity model with analytical forms of the hardening law and rate dependence. It is suitable for high-strain-rate deformation of many materials including most metals. Moreover, It is typically used in adiabatic transient dynamic simulations.

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In Johnson-Cook plasticity model equivalent stress (𝜎𝑒𝑞 ) is assumed to be of the form 𝝈𝒆𝒒 = 𝑨 + 𝑩 𝜺𝒑

𝒏

𝟏 + 𝑪 𝒍𝒏 𝜺∗

𝟏 + 𝑻∗

𝒎

…

(1)

Where A, B, C, n and m are constants, 𝜀 𝑝 is accumulated plastic strain and 𝜀 ∗ = 𝜀 /𝜀𝑜 is dimensionless strain rate. 𝜀 and 𝜀𝑜 are plastic stain rate and user defined reference strain rate. •

The non-dimensional temperature 𝑇∗ =

𝑇 − 𝑇𝑟𝑜𝑜𝑚 𝑇𝑚𝑒𝑙𝑡 − 𝑇𝑟𝑜𝑜𝑚

Where 𝑇 is the current temperature, 𝑇𝑟𝑜𝑜𝑚 is the ambient temperature, and 𝑇𝑚𝑒𝑙𝑡 is the melting temperature. •

Clearly, the von Mises equivalent flow stress 𝜎𝑒𝑞 is the product of three factors representing strain hardening, strain rate and temperature. This facilitates the calibration of the model because each of the parentheses in Eq. (1) can be handled separately in three series with uniaxial tensile tests.

Johnson-Cook damage model • To describe ductile fracture, two years after plasticity model, in 1985, Johnson and Cook also proposed a model including the effects of stress triaxiality, temperature, strain rate and strain path on the failure strain [4] which is as follows : 𝜺𝒇𝒂𝒊𝒍𝒖𝒓𝒆 = 𝑫𝟏 + 𝑫𝟐 𝒆𝒙𝒑 𝑫𝟑 𝝈∗

𝟏 + 𝑫𝟒 𝒍𝒏 𝜺∗

𝟏 + 𝑫𝟓 𝑻∗ … (2)

Where 𝐷1 -𝐷5 are material constants, 𝜎 ∗ = 𝜎𝑚 /𝜎𝑒𝑞 is the stress triaxiality ratio and 𝜎𝑚 is the mean stress. • The first set of brackets in the Johnson-Cook fracture model is intended to represent the observation that the strain to fracture decreases as the hydrostatic tension increases. The second set of brackets in the strain to failure expression represent the effect of an increased strain rate on the material ductility while the third set of brackets represent the effect of thermal softening on the material ductility. • The model assumes that damage accumulates in the material element during plastic straining, and that the material breaks immediately when the damage reaches a critical value (𝐷𝑐 ). In other words, the damage has no effect on the stress field as long as fracture has not taken place. But in light of continuum damage mechanics [2] , stress field depends on damage (strain equivalence principle) and hence damage degrades material strength during deformation. So, equation (1) needs to be coupled with damage i.e.

𝝈𝒆𝒒 = 𝟏 − 𝑫 𝑨 + 𝑩 𝒓

𝒏

𝟏 + 𝑪 𝒍𝒏 𝒓∗

𝟏 + 𝑻∗

𝒎

… (3)

Where D is damage variable, r is damage accumulated plastic strain 𝑟=(1-D) 𝜀 , D = 0 virgin material and D = 1 fully broken material and hence D=Dc • Stress state is an important factor in determining when fracture occurs. In particular stress triaxiality plays an important role [4] in governing the tendency for ductile fracture. Stress triaxiality is used to describe the portion of stress tensor that is hydrostatic. It is defined as the ratio of hydrostatic stress to equivalent Von-Mises stress. In other words the stress state with high triaxiality approaches the completely hydrostatic stress while with lower triaxiality stress state deviatoric stress dominates. Several investigations have revealed that increased triaxiality reduces ductility and thus failure strain • The model describes linear elasticity, initial yielding, strain hardening, strain-rate hardening, damage evolution and fracture all material constants can be identified from uniaxial tensile tests.

Determination of J-C Parameters (a case study) •

Borvik et. al.[5] has calibrated the Johnson cook model experimentally for Weldox 460 E and found different model parameters by doing some simple experiments as taught in [3,4]. This as a case study which is presented as follows :

Step 1. (𝑬, 𝝂) Quasi-static tensile tests were performed at a strain rate of 5 × 10−4 𝑠 −1 to determine the elastic Constants. Tensile tests in three different directions (0 deg., 45 deg. and 90 deg.) were done to check for the anisotropy in the material. Based on the results the material were kept isotropic keeping in mind that the damage after necking is directional and introduces anisotropy at large plastic strains.

Step 2.(A) Again the Quasi-static tensile tests were performed at large strains. Diameter reduction was observed till fracture in steps by vernier caliper and plastic strain in necked region is obtained by

𝜀 𝑝𝑙

𝑑𝑜 = 2 ln ( ) 𝑑

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

𝜎𝑒𝑞

F = A

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

𝜎𝑒𝑞 = f (F)

Determination of J-C Parameters (a case study) • After necking, 𝜎𝑒𝑞 becomes three dimensional because components of hydrostatic tension tends to make net tensile stress greater than 𝜎𝑒𝑞 . In view of this Bridgman [6] correction was applied to true stress and the curve was redrawn. 𝜎𝑥 2𝑅 𝑎 = (1 + ) ln 1 + 𝜎𝑒𝑞 𝑎 2𝑅 where R is the curvature radius of the neck, and a is the radius of the specimen in the necked 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 crosssectional area in the notch specimens are measured during testing. • It is assumed that the stress triaxiality ratio 𝜎 ∗ is approximately constant during plastic straining in each notched specimen while it varies in smooth specimen tensile test.

• The different notches used gives a concentration of hydrostatic tension in the test specimen.

Determination of J-C Parameters (a case study) • For a specified true strain, the stress is seen to increase when the notched radius R is reduced. It is also seen that the presence of hydrostatic tension significantly decreases the strain at which the material fractures. • The model constants B, n and Dc are determined from eq. 3 for 𝜀 ∗ =1,T=To and 𝜎 ∗ = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝝈𝒆𝒒 = 𝟏 −

𝑫𝒄 𝜺 𝜺𝒇 𝝈∗𝒊 𝒊

𝐀 + 𝐁𝒓𝒏𝒊 … (4)

Where 𝜎𝑖∗ , 𝜀𝑖 and 𝑟𝑖 are discrete values of the variables 𝜎 ∗ , 𝜀 and, and 𝑟 is the measured fracture strain for the different notched specimens and 𝜀𝑓 is the measured fracture strain for different notched specimen.

• An artificial notch produces an initial triaxiality different from that in the case of a smooth specimen where triaxiality is initiated only after the commencement of necking. Bridgman’s relationship was used to correlate initial notch radius R and maximum stress triaxiality ratio 1 𝑎 ∗ 𝜎𝑚𝑎𝑥 = + ln 1 + 3 2𝑅 • The damaged accumulated plastic stain is calculated as : 1 𝐷𝑐 2 𝜀 ∗ 𝑓𝜎

𝑟 = 𝜀 − 2𝜀

…(5)

• The method of least squares is then used to minimize the difference between the experimental 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 in stress triaxiality during testing.

Determination of J-C Parameters (a case study) Step 4. (C) The viscous effect is obtained by means of uniaxial tensile tests at a range of different strain rates from 10−4 to 103 𝑠 −1 •

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

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To have an average value of the strain rate sensitivity C, the data was fitted to the proposed model using the method of least squares for a plastic strain of 𝜀 = 0 𝑎𝑛𝑑 𝜖 = 0.1 𝑤ℎ𝑒𝑛 𝑇 = 𝑇𝑜. The mean value of C is then used to describe the viscous effect. 𝝈𝒆𝒒 = 𝟏 + 𝑪 𝒍𝒏 𝜺∗

Step 5. (m) Effect of thermal softening on true stress is shown in figure. yield stress shows a linier decrease with increasing temperature while ultimate stress shows a local minima at about 300 deg. C By assuming adiabatic condition at high strain rates, the effect of thermal softening is included in the model by fitting the material constant m to the decreasing yield stress for ( 𝜀 = 0, 𝜀 ∗ = 𝜀/𝜀0 = 1 )

𝝈𝒆𝒒 = 𝟏 + 𝑻∗

𝒎

Determination of J-C Parameters (a case study)

Step 6. (𝑫𝟏 , 𝑫𝟐 , 𝑫𝟑 ) The strain to fracture used in the damage evolution rule is given by : 𝜺𝒉𝒚𝒅𝒓𝒐 = 𝑫𝟏 + 𝑫𝟐 𝒆𝒙𝒑 𝑫𝟑 𝝈∗ The expression in the first set of brackets gives the effect of hydrostatic stress on the strain to fracture for quasi-static conditions. •

Figure shows the dimensionless triaxiality ratio in the center of the specimen is calculated based on Bridgman's analysis 𝜎𝑚 1 𝑎 𝜎∗ = = + ln 1 + 𝜎𝑒𝑞 3 2𝑅

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

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The three material constants, 𝑫𝟏 , 𝑫𝟐 𝒂𝒏𝒅 𝑫𝟑 , in the model have been fitted to the given data, and the curve has been extrapolated into the hydrostatic compression region as seen in Figure.

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𝑫𝟑 cannot assume negative values because it corresponds to the purely hydrostatical stress state.

Determination of J-C Parameters (a case study)

Step 7. ( 𝑫𝟒 ) The effect of strain rate on the strain to failure is given in the second set of brackets as : 𝜺𝒔𝒕𝒓𝒂𝒊𝒏 𝑹 = 𝟏 + 𝑫𝟒 𝒍𝒏 𝜺∗ The effect of strain rate is isolated from the temperature effect by considering the measured fracture strains for smooth specimens at 𝜎 ∗ =1/3 and 𝑇 ∗ = 0 This is shown in Figure, where the strain rate constant 𝑫𝟒 is fitted to the measured fracture strains indicating that the fracture strain decreases slightly with increasing strain rate. Step 8. (𝑫𝟓 ) In a similar way, the effect of temperature on the fracture strain is isolated from the strain rate as : 𝜺𝒕𝒆𝒎𝒑 = 𝟏 + 𝑫𝟓 𝑻∗ Keeping 𝜀 ∗ = 𝜀/𝜀0 = 1 and the temperature constant 𝑫𝟓 is obtained from figure shown.

Some Physical Aspects of J-C Parameters Positive aspects: •

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

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

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Damage evolution can be coupled with the model via strain equivalence principle of damage mechanics [2].

Negative aspects: •

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

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

Closing Remarks • One of the important parameter to accurately calibrate J-C model is stresstriaxiality. As seen, to obtain stress triaxiality for different notched specimen is a tedious and costly affair. Alternatively, it can be evaluated in Abaqus-CAE by requesting the “TRIAX” command in field output for different notched specimen. • J-C elasticity model is : 𝝈𝒆𝒒 = 𝑨 + 𝑩 𝜺𝒑

𝒏

𝟏 + 𝑪 𝒍𝒏 𝜺∗

𝟏 + 𝑻∗

𝒎

and all the parameters A, B, … etc. can be effectively evaluated in a single shot just by 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 by Jorgen Bergstrom [8]. It uses the regression analysis and fits the full equation in a single go to the given material data.

• Borvik [5] suggested some improvements in the plasticity model like 𝒍𝒏 𝜺∗ approaches minus infinity for very small strain rates. So he modified the corresponding bracket by 𝟏 + 𝜺∗ 𝑪 • Value of 𝑫𝟑 quantifies 𝝈∗ . As 𝝈∗ is related to deviatoric state of stress during the failure it governs the deformed shape of the material as shown in figure in which creation of petals during an impact by conical projectile is governed by deviatoric state of the Cauchy’s stress tensor [9]. Changing the value of 𝑫𝟑 by keeping all others constants same will change the way material deforms.

References

1.

Hopperstad OS, Berstad T, Borvik T, Langseth M. Computational model for viscoplasticity and ductile damage. Proceedings of "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 strain rates and high temperatures. Proceedings of Seventh International Symposium on Ballistics, The Hague, The Netherlands, April 1983

4.

Johnson GR, Cook WH. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures, and 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;22:855–86.

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 Phys Conf Ser No. 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 projectile impact on aluminum plates, International Journal of Impact Engineering, Volume 54, April 2013, Pages 232-245, ISSN 0734-743X, 10.1016/j.ijimpeng.2012.11.007.

Thank you