JOHNSON-COOK material model
Abstract
Object Obtain Johnson-Cook material model parameter through a serial of experiments 。 Explain the yield rule 、 hardening rule 、 strain rating effect 、 temperature effect and failure model 。 Constitutive relation under simple stress state is obtained through experiment ， and then it is extended to multi-axial stress state. Main content Theory part Experiment part
Theory part Main content ： Stress tensor and its invariants Decomposition of stress tensor Several common state equation Typical yield criterion of isotropic material Hill yield criterion of orthotropic material Key point Physical meaning of plane How to establish the connection between uniaxial stress state and complex stress state.
Stress tensor Stress tensor ： stress state of any point within the object can be expressed by symmetric tensors ：
σ x τ xy τ xz σ y τ yz σ ij = τ yx τ zx τ zy σ z
Principal plane and principal stress For the stress state of one point, the section which only has normal stress(no shearing stress )is called principal plane. The normal stress in the principal plane is called principal stress. Eigenvalue Equation of stress tensor ： σ 3 − I 1σ 2 − I 2σ − I 3 = 0
This equation has three real roots, which represents the three principal stress ， respectively.
I1 ， I2 ， I3 is called the first 、 second and third invariant respectively. They are independent from the selection of coordinate axis. I1 = σ x + σ y + σ z = σ 1 + σ 2 + σ 3 I 2 = −(σ xσ y + σ y σ z + σ zσ x ) + τ + τ + τ = −(σ 1σ 2 + σ 2σ 3 + σ 3σ 1 ) 2 xy
σx
τ xy
τ xz
I 3 = τ yx
σy
τ yz
τ zx
τ zy
σz
2 yz
2 zx
= σ x σ y σ z + 2τ xyτ yzτ zx − σ τ − σ τ − σ τ = σ 1σ 2σ 3 2 z yz
2 y zx
2 z xy
Decomposition of stress tensor average stress ： 1 1 1 σ m = I1 = ( σ 1 + σ 2 + σ 3 ) = (σ x + σ y + σ z ) 3 3 3 stress tensor is decomposed into spherical tensor and deviatoric stress tensor ：
σ ij = σ mδ ij + sij
σmδij is spherical tensor of stress ， indicates the principal stress which is equal in three direction ； sij is deviatoric stress tensor
δij is Kronecker symbol written in matrix form σ x τ xy τ xz σ m 0 0 σ x − σ m τ xy τ xz + τ τ σ τ = 0 σ 0 σ − σ τ m y m yz yx y yz yx τ τ σ 0 0 σ τ τ zy σ z − σ m m zx zy z zx
s x s xy s xz σ x − σ m τ xy τ xz sij = s yx s y s yz = τ yx σ y − σ m τ yz s s s τ τ σ − σ zy z m zx zy z zx
deviatoric stress tensor has its own invariants. J 1 = s x + s y + s z = σ x + σ y + σz − 3σ m = 0 2 2 2 J 2 = −(s x s y + s y s z + s zs x ) + τ xy + τ yz + τ zx
1 2 2 2 [(σ x − σ y )2 + (σ y − σz )2 + (σz − σ x )2 + 6(τ xy + τ yz + τ zx )] 6 1 1 = [(σ1 − σ 2 )2 + (σ 2 − σ 3 )2 + (σ 3 − σ1 )2 ] = sij sij 6 2 2 2 2 J 3 = s x s y s z + 2τ xy τ yzτ zx − s zτ yz − s y τ zx − s zτ xy = s1s 2s 3 =
Decomposition meaning ： Since only the deviatoric stress tensor affects plastic deformation, the spherical tensor can be neglected during plastic deformation.
Constitutive relation
State of equation of solid under high pressure
Several detail expression of p
• Murnagham equation a state equation under isentropy condition .
• Where A ， n is material constant ， can be difined by wave spreading experiment under isentropy condition ， the value of n for most metal material is around 4.
constitutive model of Hydro-elasto-plastic body
Yield rule
stress and principal stress space • Stress space ： The six components of stress tensor can be treated as the coordinate of six dimensional space. So each stress state corresponds to one point in the six dimensional space. The yield condition means a hyper-surface in the six dimensional space. • Principal stress space ： stress tensor can be expressed by three principal stresses. Take the three principal stressσ1 、 σ2 、 σ3 as the coordinate of three dimensional space ， which is called principal stress space. Yield condition is expressed as one hook face in principal stress space, which is called yield surface. The yield surface is the interface between elastic zone and plastic zone.
hydrostatic pressure axis
Î plane
Tresca yield criterion
Figure 2 Tresca yield locus in π plane
Figure 3 Tresca yield surface in principal stress space
determined by uniaxial tensile experiment ďźš
determined by pure shear experiment ďźš
Von Mises yield criterion
Figure 2 Von Mises yield locus in π plane
Figure 3 Von Mises yield surface in principal stress space
determined by uniaxial tensile experiment ďźš
determined by pure shear experiment ďźš
experiment designing uniaxial tensile experiment Executive Standard ： GB228-Metallic materials-Tensile testing at ambient temperature Strip specimen ， material is Q235 ， thickness is 3mm ， detailed dimension is shown below.
figure ： Strip specimen of tensile testing
Mises
Tresca uniaxi al Pure shear Figure 2 Tresca yield locus and Von Mises yield locus in π plane
ďƒ˜ pure shear experiment Thin-walled cylinder sample of torsion test is shown below, and GB10128 Metallic materials-Torsion test at room temperature is excuted.
Tresca
Tresca
uniaxial
Mises
Pure shear
Mises Figure 2 Tresca yield locus and Von Mises yield locus in π plane
• generally speaking • Using uniaxial tensile testing to determine yield condition, Mises yield condition and Tresca yield condition are identical under uniaxial tensile and compressive stress condition. For pure shear testing ， there is a big deviation ， about 15%. • Using pure shear testing to determine yield condition, Mises yield condition and Tresca yield condition are identical under pure shear stress condition. Under uniaxial tensile stress condition ， there is a big deviation ， about 13%. • The effect of yield strength caused by middle principal stress is considered by Mises yield criterion . • Since it is not convenient for numerical treatment in Tresca boundary ， the general finite element software uses Von Mises criterion. • For most metal material ， Von Mises criterion is more suitable to actual circumstances .
Flow rule
Hardening rule
isotropic hardening——homogeneous expansion of initial yield surface （ a ） Von Mises （ b ） Tresca
isotropic hardening——homogeneous expansion of initial yield surface （ a ） Von Mises （ b ） Tresca
Loading and unloading rule
single curve hypothesis • How to determine the constitutive model of material using uniaxial tensile testing ？ • Numerous experiments indicate that ， the equivalent stress-equivalent strain curve obtained by the combination of different stress is basically identical to that obtained under uniaxial tensile testing • Single curve assumption ： for one material ， under the same deformation condition ， equivalent stressequivalent strain curve is unitary . • As per single curve assumption ， equivalent stressequivalent strain curve under complex stress state can be obtained by that under uniaxial tensile testing. • Material constitutive model can be obtained if we get equivalent stress-equivalent strain curve
Verification of yield criterion
Orthotropic yield criterion
The measurement of yield locus for anisotropic material
In order to obtain the true yield locus for metal sheet plate ， the following three experiments shall be carried out ： uniaxial tensile test Pure shear test Equibiaxial tensile test
Pure shear test ASM (American Society for Metals )prepares the sample which can be used approximately to obtain the pure shear stress condition using tensile test ďźŒ and the figure is shown below. There is a groove splited 45 degree in the sample center, and the aligning holes in the two ends guarantee the centering of axial force during tensile test. When the sample is stretched, there will be stress concentration in the root of the groove. Stretching the sample along the two ends of symmetric line, the stress state in the middle area of the material is approximately pure shear state.
Forming limit diagram of plate （ FLD ）
For the laboratory ， in order to obtain the FLD of the plate ， the plate need to be stamped to instable failure under different strain paths. The limit condition of plate will be obtained ， and then the strain under limit condition will be measured. the acquisition of different strain paths The limit strain under different strain path will be obtained by using samples with different width. hemispherical punch test ：
ď Ź the measurement of strain During big plastic deformation ďźŒ the measurement of strain usually uses grid measurement method, namely printing the grid on the sample surface. Calculating the strain in instability area through measuring the grid after deformation.
commonly used grid forms
Circular grid changes into ellipse ďźŒ through measuring the change of its major and minor axises, and the two principal strains of the plate will be obtained.
Johnson-Cook material model
J-C expression under uniaxial stress state
true stress - strain curve of material Typical load - displacement curve of metal material under uniaxial tensile test is shown below. Curve AB is elastic stage ， point B corresponds to
elastic limit load ； point C corresponds to limit load ， which is generally considered as the moment that material starts to neck ； point D corresponds to the point that material starts to crack ， where the load declines dramatically ； point E corresponds to the moment that material completes rupture.
figure ： Typical load - displacement curve of metal material under uniaxial tensile test
1 ） According to uniaxial tensile test of smooth round bar ， the load - displacement curve is obtained 2 ） Obtain the initial true stress and strain curve 3 ） Revise the initial true stress and strain curve with Bridgman method 4 ） With the revised true stress and strain curve ， carry out finite element analysis on the sample ， and extract load displacement curve. Then compared with the load displacement curve obtained from the experiment ， and calculate their deviation. 5 ） According to the relative error calculated ， adjust the stress input value which the strain responses to. And recalculate with the new stress-strain curve. 6 ） Repeat the steps4 ） ~5 ）， till the result of finite element analysis and the load - displacement curve of experiment basically matches.
load - displacement curve of experiment Engineering stress strain curve Initial true stress - strain curve Revised true stress strain curve finite element analysis
comparison
Curve Fitting or not
YES
NO Calculated load -displacement curve
Accurate stress – strain relation
Continue to adjust
Figure: flow chart of true stress - strain curve revised by finite element method
Johnson-cook failure model
The determination of material parameter of fracture strain relevant
when researching the effect of different stress state on fracture ， two directions should be considered . 1 、 Using simple sample with multi-direction loading . advantage ： the specimen preparation is easy ， and it is convenient for dimension measurement. disadvantage ： special loading /controlling and measuring system need to be designed. 2 、 Using simple loading method , but complex stress state is obtained through the changing of sample shape. Using round rod sample with different notches, different stress state can be obtained during uniaxial tensile test （ at the notch area ） .
advantage ： using the common loading and controlling system ， it is convenient to control temperature and loading rate. disadvantage ： In contrast , sample preparation and measurement are difficult. To be synthetically considered ， the second method is adopted . Using the uniaxial tensile test of smooth round rod sample and round rod sample with different notch curvature radius to obtain different stress state ， to calculate the stress triaxiality and fracture strain with the measurement value of sample deformation ， or to obtain the stress triaxiality and fracture strain during tensile process by the method of finite element analysis. To match the relation curve of stress triaxiality and fracture strain ， lots of data will be needed ， and here five sets of data are used to match the curve.
R=8
round rod
R=4
R=2 Figure ďźš the stress triaxility of round rod with different shape
R=1
R=1 the distribution of equivalent stress
R=4 the distribution of equivalent stress
R=1 the distribution of hydrostatic pressure
R=4 the distribution of hydrostatic pressure
Considering the effect of strain rate on facture strain • Through the uniaxial tensile test of smooth round rod at different loading rate ， the effect of strain rate on fracture strain is researched. • Considering the effect of temperature on facture strain • Through the uniaxial tensile test of smooth round rod at different temperature ， the effect of temperature on fracture strain is researched.
test scheme • 1 、 test principle First ， determine the material constant A, B , n in the first bracket. A is the initial yield stress. B ， n are strain hardening parameters. • Secondly ， determine the material constant m in the third bracket. • Last ， determine the material constant C in the second bracket ， which represents the effect of strain rate.
• Johnson-Cook fracture model is relevant to stress state ， strain rate and temperature. • In order to determine the material constant in the first bracket ， smooth round rod samples and round rod samples with notch are designed to obtain different stress state. In the research of the effect of strain rate and temperature on fracture strain in fracture model, the experiment data which is used to determine the effect of the strain rate and temperature in constitutive model can be used .
2 、 specimen preparation With the use of high speed wire-electrode cutting machine, round rods with different size notch are prepared as tensile samples. The detail dimensions are shown below: sampleⅠ
Sample Ⅱ
Sample Ⅲ
Sample Ⅳ
Sample Ⅴ
3 〠testing program With the use of quasi-static tensile test of smooth round rod and round rod with notch , and tensile test of smooth round rod at different loading velocity and different temperature, material mechanical properties at different stress condition and different strain rate are obtained. JOHNSON-COOK constitutive model and fracture model will be used to match the test data, and then the material model parameters will be obtained . 3.1 quasi-static tensile testing for round rod with different notches At room temperature, tensile test is carried out on samples shown in figure 1-5, and the loading velocity is 1mm/min. The longitudinal deformation and transversal deformation (namely diameter change) of sample during loading, will be immediately (real-time) recorded by non contact type double laser extensometer, till the failure of the sample. The diameter of necking location will be measured at the end of the test.
3.2 tensile testing of sample type I at different loading velocity At room temperature, uniaxial tensile testing for sample type I is carried out at universal testing machine, and the loading velocity is 5mm/min 〠10mm/min 〠50mm/min 〠250mm/min 〠500mm/min respectively. Based on the testing data, the yielding stress, tensile strength and fracture strain of material at different strain rate are obtained. Through data fitting, strain rate relevant material constant C in J-C constitutive model and strain rate relevant material constant D4 in fracture model can be obtained.
3.3 tensile testing of sample type I at different temperature At a fixed loading velocity 1mm/min, uniaxial tensile testing for type I sample is carried out at universal testing machine, at different temperature 50℃ 、 100℃ 、 150℃ 、 200℃respectively. Based on the testing data, the yielding stress, tensile strength and fracture strain of material at different temperature are obtained. Through data fitting, temperature relevant material constant m in J-C structure model and temperature relevant material constant D5 in fracture model can be obtained.
Discussion 1 〠fracture strain is equal to the equivalent plastic strain when the crack occurs. Since the fracture strain is very difficult to measure, so that can be calculated approximately through the diameter of fracture section. Sometimes the deviation caused by this kind of approximation can not be neglected. 2. during the deformation of the sample, r and R will change obviously. That causes the stress triaxiality to change. The testing result can be modified by finite element analysis.