Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 177 (2017) 64 – 69
XXI International Polish-Slovak Conference “Machine Modeling and Simulations 2016”
Application of Abaqus software for the modeling of surface progressive hardening Tomasz Domański*a, Alžbeta Sapietováb, Milan Ságab b
a Czestochowa University of Technology, Faculty of Mechanical Engineering and Computer Science, Czestochowa, Poland University of Žilina, Faculty of Mechanical Engineering, Department of Applied Mechanics, Univerzitná 8215/1, 010 26, Žilina, Slovak Republic
Abstract The article concerns numerical modelling of phase transformations in solid state during hardening of steel. Abaqus FEA software is used for numerical analysis of temperature field and phase transformations in analyzed process. Additional author’s numerical subroutines, written in FORTRAN programming language are used in computer simulations where models of the distribution of movable heat source, kinetics of phase transformations in solid state as well as thermal and structural strain are implemented. Thermomechanical properties of hardened material changing with temperature are taken into account in the analysis. Structural transformations are considered as: base structure –austenite, austenite – pearlite, bainite and austenite – martensite. Model for evaluation of fractions of phases and their kinetics is based on continuous heating diagram (CHT) and continuous cooling diagram (CCT). Estimation of phase fractions in hardened element is performed in presented numerical example. Computer simulations allow the determination of the temperature field and the analysis of stress and strain states in hardened element. © Published by Elsevier Ltd. This ©2017 2017The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of MMS 2016. Peer-review under responsibility of the organizing committee of MMS 2016 Keywords: hardening; phase transformations; ABAQUS; numerical modeling;
1. Introduction Surface hardening technology is very often used. There is no comprehensive numerical models which allow provide reliable assessment of phenomena that accompany to such a process of hardening. Progressive hardening is performed for concentrated heat sources with high power. Near to this sources occurs a high temperature and its
* Corresponding author. Tel.: +48-34- 3250-609. E-mail address:
[email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of MMS 2016
doi:10.1016/j.proeng.2017.02.184
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considerable gradients [1,2]. Temperature and phase transformations in hardening steel elements are reasons to generated significant thermal and structural strains and in consequently to generate residual and temporal stresses. 2. Model of thermal phenomena Heat transfer equation with convective unit was used to modeling of thermal phenomena. The arguments of searched temperature fields are spatial coordinates (Lagrange'a coordinates) and time [3-5]: a 2T x, t
wT x, t T x, t v wt
Q C
(1)
where: T=T(x,t) is temperature [K], a is thermal diffusivity, C is specific heat capacity, Q is element of volumetric heat sources that takes into account the heat generation from the source, v is the vector of velocity of the beam, x is the vector of position of considered particle (point), t is time [s]. Equation (1) is solved by methods proposed in [2]. This solution applies to the geometry and configuration of sources which is shown in figure 1a. Y
b)
a) X d Q1 d0
v
Z
Fig. 1. (a) Scheme of considered system, (b) temperature distributions in z=0 the plane of operation of heat source
Two sources: first with a Gaussian distribution was heating source, and second at the distance z = –d from the sprayed water stream. The distribution of second source has selected linear penetration (x=x+d0). Water was sprayed at a distance of 25 mm for the source of a heating source. Width at the sprayed water was 20 mm. In an surface that was sprayed water cooling was carried out by the difference of temperature of the surrounding medium and the surface of the sample (condition Newton).
Q1 x
· § x 2 z 2 P 1 R ¨ exp E y ¸¸ 2 ¨ 2Sr 2 2 r ¹ ©
(2)
where: P is power derived from the heating source, R is the reflectance, E is absorption coefficient. z=z0+v·t, z0 is a central of heat source, v is speed of movement source, t is time. Fig. 1b shows the temperature distributions in time (thermal cycles) on the surface of hardened object along the x coordinate. 3. Phase transformation is the solid state, structural and thermal strains Model the kinetics of phase transformations in the solid state on the base on continuous cooling diagram (CCT) for C60 carbon steel [6,7].
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For determined heating and cooling rates austenitization temperatures (A c1, Ac3) during heating, start (Fs, Ps, Bs, Ms) and final temperatures (Ff, Pf, Bf, Mf) of each phase transformation and final fractions of structure components during cooling are determined in UEXPAN subroutine. In this subroutine interpolated CHT and CCT diagrams are implemented with diagram of fractions of structural components (Fig. 2).
Fig. 2. Scheme of considered CCT diagrams.
Phase fraction of austenite formed during heating is determined using a Machnienko model [8,9]. Phase fractions of the austenite during cooling process are determinated by temperature and cooling rate. Fraction of created phases such as bainite, ferrite and pearlite is also determinated by the model used for transformation of diffusion (Machnienko formula) taking account of the existing fractions of previously formed phases, ie.:
K~A (T , t ) 1 exp( k
TsA T , Ti T ) K (T , t ) K % K~A (1 exp)( k ( si )) TsA T fA Ts T fi
(3)
Transformation of austenite into martensite is defined as follows [8,9]:
K~M (T ) K % K~A (1 exp( k (
MS T m ) )), T [ M s , M f ] Ms M f
(4)
where: Ms and Mf are receptively start and final temperatures of martensite transformation determined for specified cooling time in CCT diagram, while k and m factors are determined experimentally [2]. Isotropic strains from the temperature and phase transformations (structural strains) are determined by formulas [10,11]:
dH TPh
¦
D iKi dT H APh dK A , dH TPh
i 5 i 1
¦
D iKi dT ¦ j 2 H Ph dK j j
i 5 i 1
j 5
(5)
where: i,j = A, B, F, M. and P, D i D i T are thermal expansion coefficients of austenite, bainite, ferrite, martensite
and pearlite, H iPh H iPh T are isotropic deformations of phase transformation: the initial structure in austenite, austenite in bainite, ferrite, martensite or in pearlite.
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4. Mechanical phenomena The information about the stresses associated with hardening are obtained by solving the rate equilibrium equations, complemented by constitutive compounds and appropriate initial and boundary conditions. There are thus the opportunity to reflect changes of material constants of temperature and phase composition in the following increments of load [12-14]:
div σ x,t 0, σT
$ ε e , , σ x, t σ x,T 0, ε x, t ε x,T 0 E $ ε e E 0 kr 0 kr
σ , σ
(6)
where: σ=σ(σσβ) the stress tensor, E is the tensor of material constants, He is the tensor of elastic strains (He=H-Hp-HTPh), H is a total strains tensor, Hp plastic strains, HTPh is isotropic tensor of thermal strains and phase transformations (HTPh=HT+HPh). Young's and tangential modulus was depended on temperature. The yield strength is made dependent on temperature and phase composition. Plastic deformations are determined the right non-isothermal plastic flow [1,6,15]. Huber-Mises yield condition is used, in which the yield stress is determined by relationship:
f
V ef Y T , ¦Kk , H efp 0, Y
Y0 T , ¦Kk YH T , H efp , ¦Kk
(7)
where: V ef is effective stress, Y=Y(T,6Kk, H efp ) is the plasticizing stress of the material phase composition 6Kk in temperature T and effective plasticized H efp , Y0=Y0(T,6Kk) is the yield point, YH is the surplus of the yield points resulting by the strengthening of material [16].
5. Calculation example
0
0 20
10
0
0
20
30
40
0
0
0
0
0 50
60
70
80
00
90
00
10
00
11
00
12
14
y [mm]
13
00
Computer simulation of temperature field, phase fractions of progressive hardening process was performed in Abaqus FEA for dimensions 120x25x10 mm thin plate (Fig. 1a) made of C60 steel with assumed base material structure consist of ferritic – pearlitic structure (60% ferrite and 40% pearlite). Assumed in calculations hardening process parameters were set to: heat source power Q 1=1.5 kW, beam radius r=10 mm, and rate the heating source v=2,5 mm/s. In order to reduce computational time, symmetry of the joint was assumed in calculations. Calculated temperature field in the cross section (plane YZ) of the plate at the time 30s, presented in Fig. 3, and in the cross section (plane XY) of the plate at the time 30 [s], presented in Fig. 4 allows the determination of heat affected zone geometry.
0 5 10
0
10
20
30
40
50
60
70
dimension z [mm]
Fig. 3. Distributions (maps) of temperature (cross section YZ).
80
90
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0
dimension y [mm]
2
4
6
8
10 0
5
10
15
dimension x [mm]
Fig. 4. Distributions (maps) of temperature (cross section YX).
Obtained distributions of phase fractions and kinetics of phase transformations from simulations (in cross section) are presented in figure 5, and kinetics of phase transformations at point x=0.
Fig. 5. Distributions of phase fractions in cross section, kinetics of phase transformations.
6. Conclusions Presented models for determination of movable heat source power distribution, kinetics of phase transformation, CHT and CCT diagrams, implemented in additional subroutines used in Abaqus finite element analysis allowed numerical simulation of progressive hardening process. Obtained temperature field in the cross-section YZ and YX (Fig.3,4) allows the determination of fusion zone and heat affected zone geometry. With so selected parameters hesting source depth of hardened zone reaches about 2 mm. In this zone behind the martensitic structure at about 80 percent which has good mechanical properties. Developed in Abaqus FEA three-dimensional numerical model of hardening allows the prediction of joint structural composition. Estimated properties are helpful in determining a proper set of process parameters necessary to obtain desired geometry of object.
Tomasz Domański et al. / Procedia Engineering 177 (2017) 64 – 69
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