ARCHIVES OF METALLURGY AND MATERIALS Volume 54 2009 Issue 3

A. BOKOTA∗ , T. DOMAŃSKI∗∗

MODELLING AND NUMERICAL ANALYSIS OF HARDENING PHENOMENA OF TOOLS STEEL ELEMENTS

MODELOWANIE I ANALIZA NUMERYCZNA ZJAWISK HARTOWANIA ELEMENTÓW ZE STALI NARZĘDZIOWYCH

This research the complex model of hardening of tool steel was shown. Thermal phenomena, phase transformations and mechanical phenomena were taken into considerations. In the modelling of thermal phenomena the heat transfer equations has been solved by Finite Elements Method by Petrov-Galerkin formulations. The possibility of thermal phenomena analysing of feed hardening has been obtained in this way. The diagrams of continuous heating (CHT) and continuous cooling (CCT) of considered steel are used in the model of phase transformations. Phase altered fractions during the continuous heating (austenite) are obtained in the model by formula Johnson-Mehl and Avrami and modified equation Koistinen and Marburger. The fractions ferrite, pearlite or bainite, in the process of cooling, are marked in the model by formula Johnson-Mehl and Avrami. The forming fraction of martensite is identified by Koistinen and Marburger equation and modified Koistinen and Marburger equation. The stresses and strains fields are obtained from solutions by FEM equilibrium equations in rate form. Thermophysical values in the constitutive relations are depended upon both the temperature and the phase content. The Huber-Misses condition with the isotropic strengthening for the creation of plastic strains is used. However the Leblond model to determine transformations plasticity was applied. The numerical analysis of thermal fields, phase fractions, stresses and strain associated deep hardening and superficial hardening of elements made of tool steel were done.

Praca przedstawia kompleksowy model hartowania stali narzędziowej. W rozważaniach uwzględniono zjawiska termiczne, przemiany fazowej i zjawiska mechaniczne. W modelowaniu zjawisk cieplnych równanie przewodnictwa rozwiązano metodą elementów skończonych w sformułowaniu Petrova-Galerkina. Istnieje zatem możliwość analizowania zjawisk hartowania posuwowego. W modelowaniu przemian fazowych wykorzystano wykresy ciągłego nagrzewania CTPa i ciągłego chłodzenia CTPc rozważanej stali. Ułamek fazy przemienionej podczas ciągłego nagrzewania (austenit) wyznaczono w modelu równaniem Johnsona-Mehla i Avramiego oraz zmodyfikowanym równaniem Koistinena i Marburgera. Ułamek ferrytu, perlitu lub ba- initu, w procesie chłodzenia, wyznacza się równaniem Johnsona-Mehla i Avramiego. Ułamek martenzytu wyznaczany jest równaniem Koistinena i Marburgera oraz zmodyfikowanym równaniem Koistinena i Marburgera. Pola naprężeń i odkształceń otrzymano z rozwiązania metodą elementów skończonych równań równowagi w formie prędkościowej. Wielkości termofizyczne i związki konstytutywne uzależniono zarówno od temperatury jak i od składu fazowego. Do wyznaczenia odkształceń pla- stycznych wykorzystano warunek plastyczności Hubera-Misessa ze wzmocnieniem izotopowym. Do wyznaczenia odkształceń transformacyujnych zastosowano model Leblonda. Przeprowadzono analizę numeryczną pól temperatury, przemian fazowych, naprężeń i odkształceń towarzyszących głębokiemu hartowaniu oraz przypowierzchniowemu elementów wykonanych ze stali narzędziowej.

1. Introduction there are many mathematical and numerical difficulties in its modelling. For this reason there is no model which Heat treatment is a technological process, in which would include phenomenon accompanying heat treat- thermal phenomena, phase transformations and mechan- ment, and therein and hardening. ical phenomena are dominant. Models, which describe The correct prediction of the final proprieties is pos- processes mentioned above, do not take into consider- sible after defining the type and the property of the ation many important aspects [1-6]. As a result of the nascent microstructure of the steel-element in the pro- complexity of phenomenon of heat treatment process, cess of heating, and then the cooling treated thermally.

∗ INSTITUTE OF COMPUTER AND INFORMATION SCIENCES ∗∗ INSTITUTE OF MECHANICS AND MACHINE DESIGN, CZESTOCHOWA UNIVERSITY OF TECHNOLOGY, 42-200 CZĘSTOCHOWA, 73 DĄBROWSKI STR., POLAND 576

To achieve this, it is necessary to establish equations able when refers problems of the heat flow in moving describing temperature fields, phase transformations in objects, or thermal load generated is moving sources of the solid state, as well as strains and stresses generated the warmth [7,11,14]. during the heat treatment [1,6-9]. Fields of the temperature determined from equation Representant of mechanical phenomenon in process solution of the transient heat flow with the convection of heat treatment are mainly stress, and their determina- term: ! tion is depend on accuracy computing temperature fields
∂T and from kinetics of phase transformations in solid state. div λgrad (T) − C + v · grad (T) = −qv (2.1) ∂t The kinetics of phase transformations has significant im- pact on temporary stresses and then on residual stress- where λ = λ(T) is the heat conductivity coefficient, es [1,6,8-11]. Numerical simulations of steel hardening C = C(T) is an effective heat capacity, qv is intensity process need therefore to include thermal strains, plas- of internal sources (one takes into account in him heat tic, and structural strains and transformations plasticity of phase transformations), v is a velocity vector of the in the model of heat treatment [9,12,13]. small parts (points) of the element (object). The last decade showed strong evolution of numer- Superficial warming is realised in the model with ical methods in order to in a greater or smaller extent the boundary condition of the Neumann (heat source design processes of heat-treatment. Every paper dealing q˜n), and superficial heating is modelled with internal with this topic should contain thermal, microstructural sources generating himself eg. during induction heating. and stress analysis. To implement this type of algorithms The cooling is modelled by a Newton boundary condi- one usually applies the FEM, which makes it possible to tion with temperature dependent a convection coefficient: take into account both nonlinearities and inhomogene- ∂ −λ T = = αT | − ity of thermally processed material [2,4,7-9,14]. Special ∂ q˜n (T)(T Γ T∞) (2.2) emphasis put on the development of this branch of nu- n Γ merical methods is inspired by the industry, which de- On the boundary touching with air (the cooling in mands tools improving heat-treatment processes because air) in the Newton condition one founded, that the coef- of modern technologies and costs reduction trends. ficient of taking over of the heat took into account both the radiation as and the convection [1,14]: p α∗ = α 3 | − 2. Temperature felds and phase transformations (T) 0 T Γ T∞ (2.3) where α0 is coefficient of the heat exchange established In the modelling of the heat phenomena, convec- experimentally [1,14], Γ is surface, from which the heat tion effects are in many cases significant and cannot be is taken over, T∞ is temperature of the cooling medium. skipped in the conductivity equation. The arguments of The finite element method in the Petrov-Galerkin the wanted field of the temperature are then space co- formulation is used to solve the problem. The system of ordinates (Euler coordinates). This approach is comfort- equations to numerical solves is in the form [7,14]:        β K +V +BN +M Ts+1 = M −(1−β) K +V T s+ ij ij ij  ij j ij ij ij j (2.4) +β N ∞ s+1 + −β N s −∞ s +β s+1 + −β s −β s+1 + −β s Bij Tj (1 )Bij Tj Tj Qi (1 )Qi q˜i (1 )q˜i where coefficient β represents the schema of the inte- β = PM PM R gration with respect to time, ( 1 – backward schema ∂ φ ( α) K = Ke = λe wi(xα) d j x dΩ Euler, β = 3/4 – schema Petrov-Galerkin, β=2/3 - schema ij ij ∂xα ∂xα e=1 Re=1Ωe Galerkin, β=1/2 - schema Crank Nicolson, β = 0– e = α φ α Ω Qij wi (x ) j (x )d schema forward Euler) [7,14], symbol “s” signed the Ωe time t, however symbol “s+1” – time t + dt. eP=M eP=M = 1 e = 1 e e Matrixes and vectors (2.4) are calculated like: Mij ∆t Mij ∆t Cef Qij e=1 e=1 PM PM R ∂φj(xα) = e = e e ( α) Ω Vij Vij vαCef wi x ∂xα d (2.5) e=1 Re=1 Ωe e = α φ α Γ, Bij wi (x ) j (x )d Γe MPB N = αT e Bij Bij e=1 PM MPB = e v, = e Qi Qijqj q˜i Bijq˜j e=1 e=1 577

e e e where Kij, Qij i Bij are symmetrical matrix of finite el- boundary conditions and his implementation, gives the ements called: conductivity matrix, internal source and possibility of the simulation of the temperature distri- N boundary elements matrix (but in Bij condition boundary bution both in Lagrange and Euler coordinates. In this e α, = Newton considered was), Mij is symmetrical matrix of second case one provide v(x t) 0. e heat capacity (bear also the name – matrix of mass), Vij The implementation the algorithm of presented is convection matrix, M is number of elements, M B is above of solution of the conductivity equation with the number of boundary elements, wi = wi(xα) are weights finite element method lets on the simulation of ther- functions, ϕi = ϕi(xα) are approximations functions [14]. mal loads in axisymmetrical and flat elements (problem Heat of phase transformations is considered in the 2D) and in axisymmetrical disks and spherical elements term source of the conductivity equation (2.1) and is (problem 1D). In problems 2D the bilinear approxima- calculated with the example [7,10,15-17]: tions functions were used for quadrangular elements, X however in problems 1D – linear and square functions. η v = k η In both case, the weight – functions are shift weigh func- q Hk ˙k (2.6) k tions, so-called “upwind function” [14]. η where H k is volumetric heat (enthalpy) k – phase trans- In the model of phase transformations diagrams of k continuous heating (CHT) and cooling (CCT) are used formations,η ˙k is the rate of change k – phase fraction. The algorithm of the solving of the conductivi- (fig. 1) [8,18]. ty equation in the form (2.4) with suitable initial and

Fig. 1. The Time-Temperature-Transformation graphs (CHT) and (CCT) for steel C80U [7,8]

The phase fraction transformed during continuous   heating (austenite) is calculated in the model using the , n(ts tf ) η (T, t) = 1 − exp(−b ts, t f (t (T)) Johnson-Mehl and Avrami formula and modification A   η > η , = − − ln( s) − , > / Koistinen and Marburger formula, for rate heating 100 (T t) 1 exp − (TsA T) T˙ 100 K s A  TsA TfA   K/s [9], fractions pearlite or bainite are determined in − m η = η − − Ms T , = . model by Johnson-Mehl and Avrami formula. The frac- M (T) m 1 exp − m 3 3 Ms M f
tion of the martensite formed, is calculated using the n η(·) (T, t) = ηm 1 − exp (−b (t(T)) ) Koistinen and Marburger formula [1,8,9]: η = η% η η > η% η = η η <η% m . for . and m for . ( ) A A ( ) A A () (2.7) where b(ts, tf )andn(ts,tf) are coefficients calculated as- suming the initial fraction (ηs=0.01) and the final frac- η η% tion ( f =0.99), (·) is maximal phase fraction for estab- lished cooling rate estimated with CCT diagrams, η is A a fractions of the initial austenite, TsA is temperature 578 start austenite transformations, T fA – final temperature was done in the Institute for Ferrous Metallurgy in Gli- of this transformations [9], m is constant from experi- wice by means of a dilatometer DIL805 produced by ment; for considered steel m = 3.3 if for considered steel Bahr¨ Thermoanalyse GmbH. Results of these simula- the temperature start transformations martensite amount tions and appropriate comparisons to the experiment re- Ms=493 K, and final this transformations is in tempera- sults are presented in pictures [7,8]. The example com- ture M f =173 K [7,18]. parisons are presented the figure 2, the kinetic of trans- The purpose of the dilatometric research was to formations established cooling rate are presented on the analyse phase transformations during heating and contin- figure 3 (see [8]). uous cooling of steel considered. Dilatometric research

Fig. 2. Experimental and simulated dilatometric curves (see [8]) 579

Fig. 3. The kinetic of transformations for established cooling rate (see [8])

The coefficient of thermal expand of pearlite struc- into martensite or austenite into pearlite respectively, ture is for considered depended from temperature (see sgn (.) is function of sign. In the group of considered fig. 2), so approximation of this coefficient to square steel (C80U, C90U and C100U) ferrite is not performed, function was applied [7,8]. therefore η3 = dη3 = 0 [8,18]. Based on comparisons of experimental and simu- lator dilatometric curves for the considered steel, values of thermal expansion coefficients and isotropic structural 3. Stresse and strain filds strains of each micro-constituents were determined, they − The equilibrium equation and constitutive relations equal: 22, 10, 10 and 14.5 (× 10 6) [1/K] and 1.0, 4.5, − are used in rate form [7,8,11,14], i.e.: 8.3 and 1.5 (× 10 3) for austenite, bainite, martensite and pearlite, respectively [7,8]. T e e The simulated dilatometric curves were obtained by divσ˙ (xα, t) = 0, σ˙ = σ˙ , σ˙ = D ◦ ε˙ + D˙ ◦ ε (3.1) solving the increment of the isotropic strain in the pro- cesses of heating and cooling using the formula: where σ = σ(σαβ) is stress tensor, D = D(v, E)isthe tensor of material constants (isotropic materials), v is X   Poisson ratio, E = E(T) is the Young’s modulus de- k=5 Tph ph εe dε = αkηkdT − sgn (dT) ε dηk (2.8) pendent on the temperature, however is tensor elastic =1 k k strains. where αk = αk(T) are thermal expansion coefficients Application equilibrium and constitutive equations of: austenite, bainite, ferrite, martensite and pearlite, in rate form permitted change of material property from ε ph = εph k k (T) is an isotropic strain accompanying the temperature and phase compositions. conversion of the initial structure into austenite, austen- Additivity strains was assumed, i.e. total strains in ite into bainite, ferrite or cementite fraction, austenite the around considered points are result of the sum: 580

σ εp where ef is effective stress, ef is effective plastic ε = εe + εTph +εtp +εp (3.2) strains, Y is a plasticized stress of material on the phase p Tph η ε where ε are isotope of temperature and structural fraction in temperature T and effective strains ef : strains (see. (2.8)), εtp are transformations plasticity, and     εp are plastic strains. ,η,εp = ,η + ,εp Y T ef Y0 (T ) YH T ef (3.4) In order to calculate plastic strains a model of non-isothermal plastic flow with the Huber-Misses plas- where Y0 = Y0(T,η) is a yield points of material depen- ticity condition and with the isotropic strengthening is dent on the temperature and the phase fraction, however used, where the actual effective stress depends on phase = ,εp YH YH (T ef ) is a surplus of the stress resulting from composition, temperature and plastic strain [1,7,14]. the material hardening. Flow function (f )havetheform: After considerations (3.4) scalar multiplier of plas-   ticity representation effective plastic strains (εp ), solved =σ − ,η,εp = ef f ef Y T ef 0 (3.3) by formula [7]

     P  · ◦ ε − εTph −εtp + ˙ ◦εe − κT ˙ + κηη 3S D ˙ ˙ ˙ D 2Y T k k ˙k ε˙p = 2Y (3.5) ef 9S · (D ◦ S) + 4κY Y 2 κY = κY ,εp where S is the deviator of stress tensor, (T ef )is microplasticity which appear in weaker austenite phase κT = κT ,η,εp of has caused by difference of volumetric between phas- the hardening modulus, (T ef)isthethermal κη = κη ,εp es. Magee’s interpretation (fundamentally for marten- softening modulus, k k (T ef ) is the modulus of structural softening or hardening [2,3,5]. sitic transformations), this is the result of change in the Can we remark, that rate of effective plastic strain orientation of grain martensite which have been newly εp formed by external load. Priorities of these mechanisms (˙ef ) describe (3.5) is depend from value surplus stress, thermophysical gradient of values and of kind output depend on materials and of kind of transformations. materials structure. Greenwood-Johnson mechanism is priority in diffusiv- Individual modules are determined following: ity transformations, and also in bainitic and martensitic transformations when big difference volumetric between   ∂Y T,η,εp phases is. κY = ∂Y = H ef ∂εp ∂εp ef ef   In this work Leblond’s model has been used to es- ∂ ,η,εp ∂ ∂ ,η YH T timate transformations plasticity [12]. In the literature κT = Y = Y0(T ) + ef (3.6) ∂T ∂T ∂T  there are some other models estimations transformations η ∂ ,η ∂Y T,η,εp κ = ∂Y = Y0(T ) + H ef plasticity (comp. [4,13,19]). However, Leblond’s model k ∂ηk ∂ηk ∂ηk takes into consideration all transformations and it is used The accessible thermophysical values obtained from by authors working on modelling heat treatment the most = ,η experiment are Young’s modulus E E(T ) and tan- frequently. = ,η gential modulus Et Et(T ). Take advantage of unaxial Using the Leblond model, completed by decreasing curves tension or compression, and make use of strains functions (1 - η) which has been proposed by the authors κY additivity, solved hardening modulus ( ), the thermal of the work [12], transformations plasticity are calculated κT softening modulus ( ) and the hardening modulus or as following: κη structural softening ( k )[7].   3.1. Transformations plasticity 0, for η 6 0.03, εtp =  P k ˙  k=5 ph S −3 = (1 − ηk) ε ln (ηk) η˙k, for ηk > 0.03 Numeric simulations of process heat treatment of k 2 1k Y1 steel require taking into consideration in models of (3.7) where 3εph are volumetric structural strains when the transformation plasticity. This phenomenon is a reason 1i of irregular, plastic flow metals, which appears during material is transformed from the initial phase “1” into the phase transformations in solid state, especially dur- the k-phase, Y1 is a actual of phase output (in cooling ing transformation austenit to martensit (A→M) cooling process is austenite). iron. In the literature there two different mechanisms, Current point of plasticity output phase is calculated one which has been proposed by Greenwood and John- by formula son and second by Magge [8,12,13]. In the interpretation = 0 + κY1 εtp of Greenwood-Johnson the transformation plasticity is Y1 Y1 ef (3.8) 581

0 κY1 Tph where Y1 is a yield points of output phase, is the load, t˙ is the vector of nodal forces resulting from hardening modulus of materials about austenite struc- thermal strains and structural strains, t˙e is the vector of εtp ture, and ef is effective transformations plasticity. Be- nodal forces resulting from the value change of Young’s ptp cause no existing suitable date, assumption, that κY1 = modulus dependent on the temperature, t˙ is the vector κY . of nodal forces resulting from plastic strains and trans- The equations (3.1) are solved by means of the FEM formation plasticity. [7,8,14]. The system of equations used for numerical The rate vectors of loads in the brackets are calcu- calculation is: lated only once in the increment of the load, whereas the vector t˙ptp is modified in the iterative process [14]. n o n o n o n o n o Have marked rate of displacement solved rate stress- K U˙ = R˙ + t˙Tph − t˙e + t˙ptp [ ] (3.9) es to result to gradient of displacement rate. The final K U˙ displacements, strains and stress are resulting integration where is the element stiffness matrix, is the vector = of nodal displacement, R˙ is the vector of nodal forces with respect to time, from initial t t0 to actual time t, resulting from the boundary load and the inertial forces i.e.

Zt Zt Zt U (xα, t) = U˙ (xα,τ)dτ, ε(xα,t) = ε˙ (xα,τ)dτ, σ (xα , t) = σ˙ (xα ,τ)dτ (3.10)

t0 t0 t0

Using one step scheme integration’s and marked In interactions process in following “i” steps are by index “s” time t, however by index “s+1” – time solved the system of equations ts+1 +∆ts+1, summation discrete value functions “f” ob- n o n o tain from solutions, in following time steps, carry out [K] δiU˙ = δit˙ptp (3.12) following: and updating successively displacements, strains and stresses   kP=s   +1 f xα, ts = f˙ xα, ∆tk ∆tk+  k=0  (3.11) +1 +1 f˙ xα, ∆ts ∆ts

  Xk=s      X  s+1 k k Tph s+1 i k s+1 f xα, t = f˙ xα, ∆t ∆t + f˙ xα,∆t + δ f˙ ∆t (3.13) k=1 k=0

4. Examples of calculation 2×105 and 4×103 [MPa] (Et =2×10−2E), yield points 150, 480, 1150 and 300 [MPa] for austenite, bainite, Numerical simulations of hardening of the elements martensite and pearlite, respectively, in the temperature 0 made of the carbon tool steel were performed. The ther- 300 K. In the temperature of solidus Young s modulus and tangential modulus equalled 100 and 10 [MPa], re- mophysical coefficients λ and C were assumed as con- stants: 35 [W/(mK)], and 5.0 ×106 [J/( m3K)]. These are spectively, whereas yield points equalled 5 [MPa]. These the average values calculated on the basis of the data in values were approximated with the use of square func- the work [5]. Because in work was presented feed hard- tions (fig. 4) using the following assumptions based on ening assumed that the heats of phase transformation are the work [2]. Fig. 4 assumed to zero. Heat transfer coefficient assumed con- stant equal αT =4000 [W/(m2K)](coolinginfluidlayer 2 −4/3 [20]) and α0=30 [W/(m K )]. The cooling was mod- elled with the Newton condition. The temperature of the cooling medium equalled T∞ =300 K. Young0s and tangential modulus (E and Et)were dependent on temperature, whereas the yield stress (Y) was dependent on temperature and phase composition. Assumed, that Young0s and tangential modulus are equal 582

Fig. 6. Hardened zones in the cross sections

t Fig. 4. Diagrams of functions E(T), E (T)andY0(T,η)

4.1. Example 1

The axisymmetrical object with the following size φ30×60 mm with mouth φ10×mm (fig. 5) underwent hardening simulation. After heating it had an even tem- perature equalling 1150 K, and the output microstructure was austenite. Fig. 5.

Fig. 7. Kinetic of transformations in the superficial points 1 and 2 of the element (fig. 5)

Fig. 5. Scheme of the system and boundary conditions (example 1)

Exemplary residual stresses distributions and plastic Hardened zones in the cross sections and the kinetic strains after hardening, with and without transformation of transformations in the superficial points 1 and 2 of the plasticity: (εtp = 0)and(εtp , 0) are presented in figures element are presented in figures 6 and 7, respectively. 8 and 9, respectively. 583

sity cooling and boundary conditions, for conductivity equations (2.1), presented figure 10.

Fig. 10. Scheme of the considered system (example 2)

The heating executed by volumetric source (simu- lations inductive heating) operate on length 10 mm to depth 1.5 mm. To assume that is depth and harden- ing superficial. The cooling executed by flux to result from difference of temperature between side surface and Fig. 8. Residual stresses in the cross sections (fig. 5), without and medium of cooling (condition Newton) (fig. 10). The ini- with considering transformation plasticity tial temperature of hardened model and cooling medium amount 300 K. The thermophysical coefficient accruing in conductivity equations assumed same heavy how in example 1. The solutions investigated for rate of travel v=36 m/h. Power of heating source, to assure maximal of temperature on surface ≈ 1500 K (Fig. 11), equal 4.1 kW. Average density of power equal ≈ 3[W/mm3]. The calculations investigate in series system, i.e. the heating simulations to follow to moments of estab- lished temperature distributions in control region, and after calculation phase fractions, thermal and structural strains, and then stresses, plastic strains and transforma- tions plasticity. Because the geometry oh hardened elements is ax- isymmetrical, to move object in model of mechanical phenomenon of equilibrium equations (3.1), expressed Fig. 9. Plastic strains after hardening, with and without considering in polar coordinates, reduced to one equations. Prob- transformation plasticity lem 1D is solving. To assumed strain plane state, but to modify to, so that obtain reset resultant of force normal 4.2. Example 2 (N|Γ = 0) in cross-section (Γ). This condition obtain in interactions process to assure conditions [7,11]: The axisimmetrical object with the following size ZR ZR φ30 mm with mouth φ10 mm, made of carbon tool steel σ = , σ = C80U. The control region (Euler coordinates) is assump- ˙ zrdr 0 zrdr 0 (4.1) tion equal 100 mm. The locations of source, zone inten- 0 0 584

After to use constitutive equations on axial stresses (˙σz) [1,7,14]

σ˙ = (2µ + λ)˙εe +λ(˙εe +ε˙e)+2˙µεe + λ˙ e, e = εe + εe+εe (4.2) z z r θ z e e r θ z to obtain the formula on average rate of total axial strain:

R R   R   − λ εe +εe − µεe + λ˙ e + µ + λ ε p + εTph +εtp (˙r ˙θ)rdr 2˙ z e rdr (2 ) ˙z ˙z ˙z rdr r r R r ε˙z = (4.3) (2µ + λ)rdr r where λ = λ(E, v) µ = µ(E, v) are constants Lame´ [1,14]. sumed the loss of deformations of its lateral surface, for The assumption in stress model modified plane state the reason of thermal load asymmetrical. of strain established, that in freely selected disc about The results of numerical simulations presented in differentiable thickness “δ” (see Fig. 10), it can be as- turn figures.

Fig. 11. Temperature distributions on the side surface of the hardening element

Fig. 12. Temperature distribution in the cross-section of the hardened element 585

Fig. 13. Hardened zone

Fig. 14. Kinetic of austenite decay and martensite formation in the superficial point

Fig. 15. Residual stresses after hardening, with and without considering transformation plasticity 586

Fig. 16. Plastic strains after hardening, with and without considering transformation plasticity

5. Conclusions comparison to stress distributions obtained in example 1. The influence on results of transformations plastici- ty simulation is also observed (figs. 15 and 16) and is After deep hardening (example 1) the hardened zone smaller than in case of deep hardening (comp. figs 15, is diversified. The mixture of bainite, martensite and 16 and 8, 9). pearlite occurs (fig. 6). However nearby the frontal sur- The presented model of phase fractions calculations faces bainite, martensite and retained austenite occurs. in cooling process could by used to simulations of hard- The martensite fraction in frontal layer is very high ening elements made of tool steel C80U, C90U and ∼ ( 90%) (figs 6 and 7). The internal stress distributions C100U, because the diagrams CCT for this steel, in are advantageous (fig. 8). Depositions of negative cir- terms of shape are similar [18]. cumferential and axial stresses is superficial, but deep enough (fig. 8). The meaningful influence on results of Acknowledgements calculations of transformations plasticity is observed. It means that in the model taking into consideration the The research was performed within framework of research phenomenon of transformation plasticity is not to omit. project, financed by KBN for years 2007-2009, no. R15 035 02. The results of stress and strain simulations, after such hardening, with and without taking into account trans- formations plasticity are significantly different (figs. 8 REFERENCES and 9). There is still the problem of reliability of the results, but it could be confirmed with very expensive [1] A. J. F l e t c h e r, Thermal Stress and Strain Generation experiments only. in Heat Treatment, Elsevier, London (1989). The plastic strains are in all volume of hardened ele- [2] M. C o r e t, A. C o m b e s c u r e, A mesomodel for ment, but the obtained distributions are more convincing the numerical simulation of the multiphasic behavior of after taken into account transformations plasticity. The materials under anisothermal loading (application to two larger strains are in superficial external layer, and not low-carbon steels), International Journal of Mechanical inversely (fig. 9). Sciences 44, 1947-1963 (2002). Analysing the results obtained in example 2, it was [3] M. P i e t r z y k, Through-process modelling of mi- found that in case of feed hardening of element made crostructure evolution in hot forming of steels, Journal of tool steel the hardened zone after complex thermal of Materials Technology 125-126, 53-62 (2002). load is comparable to results obtained in the process [4]J. Rońda, G. J. Oliver, Consistent thermo-mechanical-metallurgical model of welded of inductive hardening [1,14]. The hardened zone with steel with unified approach to derivation of phase very large martensite fractions is not deep (figs. 13 and evolution laws and transformation-induced plasticity, 14). The obtaining such hardened zone is the result of Computer Methods Applied mechanics and engineering suitable thermal load and cooling (figs. 11 and 12). The 189, 361-417 (2000). distributions of stresses after such hardening are advan- [5]M. Coret,S. Calloch,A. Combescure,Ex- tageous too (compression in surface layer). They are the perimental study of the phase transformation plasticity of most different on the limit of hardened zone (fig. 15) in 16MND5 low carbon steel induced by proportional and 587

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Received: 20 May 2009.