Estimation

Calculating

of the Effect of Stress/Phase Transformation Interaction

Internal Stress during Martensitic Quenching of Steel*

when

By S. DENIS, * * A. SIMON* * and G. BECK*

Synopsis

Usually in the prediction of internal stresses during quenching, in the case of a steel which undergoes phase transformation, a particular effort is made in order to modelize the progress of the transformations. In this work, we try to show in the case of a tool steel (60 NCD I1), which under-

goes only martensitic transformation during quenching, how the appearance of internal stresses is modified by taking into account a specific mechanical behaviour of the metal during phase transformation.

The procedure consists in using the experimentally determined dimensional, mechanical and thermal behaviour of the steel in a thermo-elastoplastic model of calculating stresses and strains in cylinders.

The results on the example of water quenching at 80 °C show that by considering a stress transformation interaction which induces transformation

plasticity the development of the internal stresses during cooling is quite modified. Indeed, the transformation plasticity phenomenon simulated by a reduc-tion in the yield stress of the steel during its transformation leads to residual stress profiles which are inverse as compared with that of thermal stresses, i.e., the stresses are tensile on the surface and compressive in the centre of the cylinder.

I. Introduction

During the course of heat treatment, it is most

important that the deformations and internal stresses which appear during rapid cooling in various parts of

a steel should be totally mastered. Indeed, as a result of rapid cooling achieved by quenching, it is

possible to obtain the most suitable microstructures for giving the metal the desired mechanical character-

istics. This possibility will be limited by the appear-ance of deformations or internal stresses in the steel

which have adverse effects on its use. This latter aspect of quenching is a problem of the moment, in

that it represents the limiting factor to achieving the more rapid cooling required by the development of

steels containing fewer costly elements. If internal stresses still exist at the end of the heat

treatment, this is due to heterogeneous plastic de-formations occurring in the steel. The plastic de-

formations occur during quenching because at a

given moment the density of the metal varies suffi-ciently notably from one point to another in the part to cause higher deformation than the elasticity of

the metal is capable of sustaining. The physical origins of these deformations are of two types : thermal

stresses due to thermal contraction and which depend solely on thermal gradients in the part, and trans-

formation stresses which are due to changes in density when the metal changes from one crystalline structure to another. A qualitative analysis of these phenome-

na has been undertaken by H. Buhler and A. Rose.'

When these stresses are only due to thermal con-traction (in the case of metal which does not undergo phase transformation), internal stresses calculation can be undertaken correctly if the necessary data and a computer programme for calculating internal stresses taking into account the metal's thermo-elastoplastic behaviour are available. We have been able to achieve this for the quenching of aluminium alloys.2,3) A large number of authors4-12) have applied this

method to the case of steel which undergoes one or more phase transformations during quenching. These different investigations cover a varying degree of sim-

plification and offer greater or lesser accuracy regard-ing the data used for the calculation. Some very complicated cases have been investigated, ranging from quenching with different types of transformation

(pearlite, bainite, martensite), to cases where neither temperature nor structure are homogeneous before quenching (in the case of induction hardening7)), or cases where the chemical composition of the part is not homogeneous (case hardening7"0)). In all these investigations, particular effort has been made to modelize the progress of the transformations, but the more basic question of finding out the " actual" mechanical behaviour of the metal during phase transformation and its consequences on internal stresses remains unanswered. Indeed, in the case of a simple and often used

geometrical shape such as the infinite cylinder, the following data are needed in order to calculate the stresses during cooling:

(1) Temperature distribution at each moment and at every point in the part throughout cooling. This can be achieved with accuracy by measuring the temperature distribution with thermocouples inserted to varying depths in the mid-plane of the cylinder.

(2) The thermal expansion coefficient at each temperature can be measured by means of dilatometric tests and it is possible to determine both the part due to thermal agitation and the part resulting from changes in density due to phase transformations.

(3) The mechanical behaviour of the metal at each temperature and in the various structural states observed during quenching must be known. This can be achieved either by carrying out tensile tests on the metal at various temperatures and with different microstructures, or by applying the law of mixtures using the characteristics of each of the different

phases of the metal.

*

**

Received February 2, 1981. © 1982 ISIJ Laboratoire de Metallurgie Associe au C.N.R.S ., Parc de Saurupt, 54042 Nancy Cedex, France.

(504) Research Article

Transactions ISIJ, Vol. 22, 1982 (505)

Further, this characterization of the metal gives rise to a phenomenon which has not yet received a satisfactory explanation, i.e., the interaction between internal stresses and structural transformation in the metal. When transformation takes place in a metal subjected to mechanical stresses, plastic deformation occurs even though internal stress is lower than the

yield stress of each of the phases in question. This is the transformation plasticity phenomenon.13,14> An-other interaction which should be mentioned is the fact that transformation kinetic is altered by stress.

In this paper we shall attempt to show, using a simple case, how the prediction of internal stresses can be modified depending on whether or not this interaction is taken into consideration.

II. Behaviour of the Alloy during Quenching

We have chosen to carry out our investigation on a 60 NCD 11 steel (0.57 % C, 0.65 % Mn, 0.31 % Si, 2.35 % Ni, 0. 75 % Cr and 0.41 % Mo) and were thus able to obtain exclusively martensitic transforma-tion under standard cooling conditions and for average bulk test pieces.

After austenitization at 900 °C for 15 min, this steel is brought to a temperature MS of 247 °C. In the case of this alloy, martensitic transformation can occur even though there is still a high temperature

gradient in the test piece, and, secondly, the alloy's high carbon content produces significant variations in mechanical properties during martensitic trans-formation. We are thus able to observe the charac-teristic behaviour of a steel which undergoes only martensitic transformation during quenching. In this case, the progression of the transformation will only depend on the temperature in the test piece.

Measurements have been carried out in order to define the dimensional and mechanical behaviour of this steel during quenching. The measurements were taken using a dilatometer capable of generating rapid thermal and mechanical cycles.15) Three types of tests were carried out: 1) A dilatometric test during continuous cooling

with no stress applied to the sample, in order to measure the dimensional variations caused by

thermal shrinkage and martensitic transformation. 2) Tensile tests carried out at various temperatures during cooling.

3) Dilatometric tests during continuous cooling with various stresses applied to the test piece prior to

martensitic transformation. These tests supply data on the stress-transformation interactions.

1. Dimensional Behaviour of the Metal during Cooling

Dimensional behaviour is measured simply by a direct dilatometric test giving dL/Lo as a function of temperature (Fig. 1). On the basis of this curve, the deformation of the metal at any temperature can be described by stating that:

r = 13• T ...........................(1)

where ~3 is the slope of the straight line plotted from the zero point on Fig. 1 to the representative point

on the dilatometric curve at temperature T. The temperature-related coefficient 3 variations are thus observed (Fig. 2). Coefficient j3, which is constant down to temperature MS, is equal to the thermal expansion coefficient determined conven-tionally. This coefficient increases progressively whilst the temperature remains below MS. When calculat-ing stresses, this curve offers a practical means of estimating, at any temperature, the part of defor-mation attributable to thermal expansion and to structural transformation.

2. Mechanical Characteristics o f Metal during Cooling In order to measure the mechanical characteristics

of steel at various temperatures in the condition in which it is to be found during the cooling process, i.e., in the metastable austenitic state and in the combined austenitic-martensitic state, we used a dilatometer15) to carry out rapid tensile tests (~ ~ 10-2 s-1) by inter-rupting cooling at various temperatures.

The curve given in Fig. 3 is an example of the stress-strain curve recorded in the case of a tensile test undertaken at 524 °C.

Fig. 1. Change in length,

perature.

d L/Lo, as a function of tern-

Fig. 2. Change in i as a function of temperature.

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In order to calculate stress it was necessary to define the numerical values which could be used in the model. The plastic behaviour of the metal up to a deformation of 0.5 % is represented by a sloping straight line Ep (" plastic modulus "). The point at which this line intersects a sloping straight line that

gives the Young's modulus defines an " apparent" yield stress aEap. The behaviour of the material is therefore represented by two straight lines with dif-ferent slopes : one describing the elastic field, the other describing the first 0.5 % of plastic deformation (Fig. 3).

The curves in Figs. 4 and 5 show the changes in aEap and Ep respectively as a function of temperature. A rapid increase in the two values is observed during the martensitic transformation interval. Young's modulus was taken to be constant and equal to a mean value of 145 500 M.Pa.

3. Interactions between Stresses and Martensitic Trans-

formation In order to obtain data regarding the behaviour of

the metal under stress during quenching, a dilatometer capable of generating rapid mechanical and thermal cycles was used.l5~ By means of this dilatometer, a 25 mm long test piece, 2.5 mm in diameter, can be subjected to predetermined tensile stresses ranging from 0 to 2 000 MPa and to temperature variations which are also predetermined (maximum cooling rate 100 °C/s). The experimental procedure consisted of applying vacuum austenitization at 900 °C for 20 min followed by cooling under argon jets (mean cooling rate 10 °C/s.) : a temperature plateau was maintained at 350 °C in order to apply the tensile stress, and this

was followed by further argon cooling. During this test, variations in temperature, in applied stress and in the length and resistivity of the test piece are recorded. Figure 6 indicates the dimensional behaviour of the test piece at various applied stresses as a function of temperature. The curves obtained reveal two changes in the behaviour of the metal caused by the applied stress. It is observed that, when the stress increases,

(1) the dilatation associated with martensitic trans-formation is displaced to high temperatures, resulting

Fig. 3. Stress-strain curve in simple tension

524 °C during cooling.

obtained at

Fig. 4. Change in ~Eap as a function of temperature.

Fig. 5. Change in Ep as a function of temperature.

Fig. 6. Evolution of continuous

of steel 60 NCD 11 as

stress applied.

cooling dilatation

a function of the

curves

tensile

Transactions ISIJ, Vol. 22, 1982 (507)

in an increase in the temperature Ms (Fig. 7), and

(2) the range of total dilatation measured during transformation increases (Fig. 8). We thus possess certain elements of the stress-transformation interac-tion in the steel for which we are seeking to determine the onset of the internal stresses during quenching. These results concern only the effect of tensile-stresses, but they do help to reveal the range of behavioural changes which can be induced by internal stresses. As is shown in Fig. 7, the temperature MS in the steel increases from 247 to 265 °C, i.e., a rise of 18 °C, when the applied stress increases from 18 to 285 MPa. This latter stress value is in the region of the yield stress of the austenite before transformation. The increase in elongation during transformation which indicates transformation plasticity becomes important once the applied stress is in excess of 40 MPa (Fig. 8). Total deformation (measured with respect to the austenite at 0 °C) during transformation can reach 2.5 % for an applied stress of 285 MPa. De-formation of this type, which is connected with the

progress of transformation, occurs when stress remains below the yield stress of each of the attendant phases or combination of phases. Modelization of such a

phenomenon is complex. As we shall see, this modeli-zation can be initiated by considering that the moment transformation takes place, a decrease occurs in the metal's yield stress. Although the mechanism which induces transformation plasticity13~ is very different from that which is created by a decrease in the metal's yield stress, we shall see that such a simulation fully privileges plastic deformations during transformation.

III. Data Necessary for Stress Calculations

We have taken as our example a 105 mm long cylinder, 35 mm in diameter, for which we shall calcu-late the variations in time and space of internal stresses during quenching in water at 80 °C. First of all, we must find out the temperature evolution within the cylinder during quenching. This is achieved by measuring the temperature along the median plane of the cylinder at four different points situated 2.5, 7.5, 12.5 and 17.5 mm respectively below the surface. Figure 9 shows the curves obtained, and Fig. 10 shows the radial temperature distribution at differ-ent moments during cooling in the cylinder's median

plane. In order to demonstrate the relative significance

that the various phenomena occurring in the metal during quenching have on the development of internal stresses, we have evolved various modelizations re-presenting metal behaviour

(1) We assume that the metal remains austenitic (the values of coefficient 8, QEap and Ep are extrapo-lated for low temperatures on the basis of the values obtained experimentally with the metastable aus-tenite).

(2) We assume that the metal undergoes mar-tensitic transformation, but that no stress-transforma-tion interaction occurs (The values of jS, aEap and Ep are those which were obtained by the dilatometric and tensile tests).

Fig. 7. Change in temperature MS in steel 60 NCD 11 as

a function of the tensile stress applied.

Fig. 8. Change

in steel

applied.

in the dilatometric range of transformation

60 NCD 11 as a function of the stress

Fig. 9. Time-related temperature variation graphs d

water quenching at 80 °C.

uring

Fig. 10. Temperature distributions along the radius of the

test piece at different moments during cooling.

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(3) We assume that the metal is subjected to martensitic transformation and that the only stress-transformation interaction observed is the variation in temperature MS (the values of a, ~Eap and Ep are the same as in model 2, but their variations according to temperature are displaced by + 15 °C, correspond-ing to the temperature MS displacement for a tensile stress of 270 MPa).

(4) We assume that the metal undergoes mar-tensitic transformation and that the stress-transforma-tion interaction induces transformation plasticity. The S values are those used in model 2. Transforma-tion plasticity is simulated by a decrease in apparent

yield stress QEa p (~3 and Ep are the same as in model 2 during the transformation interval). Figures 11(a) to (c) show the variations in j3, QEap and Ep respectively for the various behaviour models considered for the metal. These different behaviour models will now be used in order to calculate the internal stresses and thus isolate the determining

parameters which are at the origin of internal stresses.

Iv. Internal Stresses Calculation

1. Calculation Method

We have developed and applied a method for calcu-lating internal stresses based on the thermo-elasto-

plastic behaviour of the metal during quenching.16,17) In order to take into account the passage of the

metal into the plastic field, we must assume that

yielding occurs if the stresses conform to a general plasticity criterion F[(a), K] =0, where K is a strain hardening parameter. We have adopted the Von Mises yield criterion which, in the particular case of the cylindrical shape where shear stresses are non-existant, is written as :

F[(ci), K] - (ar-Qa)2-f-(~B-Uz)2-F-(Uz-Ur)2--2UE ...(2)

where, Ur, 6B, 6z : radial, tangential and axial stresses ~E : the material's yield stress which

depends upon K. The metal is in the plastic field when F[(a), K] >0. The flow rule associated with this yield criterion18) relates the plastic strain increments between two moments during cooling with the stress state and with its increase in the case of a material with strain hardening.

The general principal of the calculation model consists of establishing an elastoplastic formulation of the stress and strain relations. In the case of thermo-elastoplasticity, at a given moment and at a given

point characterized by its temperature, each com-ponent at of the total strain is the sum of four terms : 1) An elastic strain re 2) A thermal strain ~th due to thermal shrinkage and

phase transformation, written as ,8 T 3) A plastic strain increment dsp 4) A total accumulated plastic strain 4e which is

the sum of all the plastic strain increments calculated for the preceeding moments during cooling

We have at = Ce+Eth+4Sp+~ dep.

These stress and strain relations are associated with

the equations of equilibrium of the stresses and the relations of compatibility of the strains in the material.

The problem is solved by a method of successive approximations.

Fig. 11. Change in

stress aEap

for the four

coefficient /3 (a), " apparent (b), and " plastic modulus " types of model considered.

" yield

Ep (c)

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At each moment during cooling corresponding to a known temperature distribution in the test piece, the plastic strain increments must be calculated in the three directions, radial d gyp, tangential 44 and axial dsz7 in order to know the stress state (o, 7e, 6z) at different points on the radius of cylinder.

2. Results

Figure 12 shows the changes in internal stresses calculated along the radius, at different moments during cooling, for the four types of models investi-

gated and described in Fig. 11. Internal stress evolution, based on temperature profiles in the test

piece (Fig. 10), is shown after cooling has been in progress for 48, 57, 63, 72 and 81 s and at the end of the cooling process (residual stress profiles).

After 48 s, the surface temperature is still 385 °C, i.e., all four models considered are identical because martensitic transformation has not yet taken place. Figure 12(a) shows the variations in radial (6r), longitudinal (as) and tangential (a0) stresses along the cylinder's radius at this given moment during cooling. Still for the same moment during cooling,

Fig. 12 (a') gives the plastic strain increments (4~p, die and 4~?) which correspond to the stresses shown in Fig. 12(a). Total plastic strain at a given moment is obtained by accumulating the plastic strain incre-ments calculated for the preceeding moments and for the moment under consideration. With these in-crements it is possible to visualize the local plastic strains which occur at a given moment during cooling. In particular, the plasticity due to martensitic trans-formation will be clearly revealed.

In order to explain the evolution of stress and strain throughout the cooling process, we shall confine ourselves to following the evolution of 6z and of 4sp and 4~p. These simplified figures are shown in Figs. l2(al) and (a' 1). The stresses are shown to be purely thermal in origin, and the following general conventional evolution is observed; before stress in-version occurs, stresses are compressive in the centre of the test piece and tensile at the surface. Plastic deformation occurs both in the centre and at the surface of the test piece. In the moments which follow, part of the test

piece will undergo martensitic transformation, and

Fig. 12.

Calculations of the onset of

during water quenching at 80

internal °C

.

stresses

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(c, c')(1#-'4): 63 s after the start of cooling

(temperature in the centre : 405 °C, temperature at the surface: 149 °C)

(d, d')(1'-.'4): 72 s after start of

(temperature in the 290 °C, temperature surface: 106 °C)

cooling

centre :

at the

(e,e')(1N4): 81 s after start of

(temperature in the 230 °C, temperature surface: 83 °C)

cooling

centre :

at the

(f,f')(1-4): The end of quenching process

(temperature in the centre : 80 °C, temperature at the a]c(• irs n °r!1

Fig. 12. Continued. (l..'4: see figure caption on p. 509)

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the four models investigated will all differ. Com-

parison of the results obtained for models 1 and 2 reveals how martensitic transformation alters the onset of the purely thermal stresses. Models 2 and 3 reveal the effect of stress-transformation interaction resulting in a change in temperature MS. Models 2 and 4 reveal the effect of stress-transformation interaction resulting in a transformation plasticity effect. If we examine the evolution of internal stresses

during cooling in the case of model 1, we note the following general conventional evolution : At the start of cooling, compressive stress in the centre of the test

piece and tensile stress at the surface; this profile is later reversed leading to tensile internal stresses in the centre and compressive internal stresses at the surface. It should also be noted that after cooling has been in progress for 57 s (Fig. 12(b'1)) no further

plastic deformation occurs. The residual internal stresses observed at room temperature are the result of this plastic deformation at high temperatures.

If we now examine the case of martensitic trans-formation (model 2), we note that at the stage between 57 s and 72 s after the start of cooling, the superficial area gradually undergoes martensitic transformation whilst the centre remains austenitic. The surface starts to become strongly subjected to compression

(= -600 MPa) and the centre off the test piece to tension (cz = +680 MPa, Fig. 12(d2). The centre then begins to transform leading to an inversion in the direction of stress variation and to the following final profile (Fig. 12 (e2)) : -200 MPa at the surface and -50 MPa in the centre. If we follow plastic strain increments we observe that at the start of transformation (Figs. 12(b'2) and (c'2)) plastic de-formation of the area undergoing transformation is induced. This plastic deformation is of the opposite sign compared with the deformation which occurred at high temperatures. When transformation prog-resses in the superficial area of the test piece (Fig. 12

(d'2)), it induces plastic deformation in the centre of the test piece. When the centre undergoes trans-formation (Figs. 12(e'2) and (f'2)), no plastic deforma-tion occurs. As a result, the residual stress state will depend considerably on the range of plastic deforma-tion induced during martensitic transformation. In the case of model 3 (temperature Ms=262 °C

instead of 247 °C), no significant difference is ap-

parent as compared with the evolution of internal stresses calculated for model 2. At the most, the

phenomena mentioned above occur slightly earlier, and the residual stress profile remains practically the same.

Model 4, which is characterized by easier yielding in the areas which undergo martensitic transformation, shows a very different evolution from that observed in model 2. The compressive stresses which develop during cooling in the areas which undergo transforma-tion (Figs. 12(b4), (c4) and (d4)) are indeed much lower. By contrast, however, plastic deformations are much higher (Figs. 12 (b'4), (c'4) and (d'4)). It should also be mentioned that in this case, plastic

deformation always remains localized in the zone

which is undergoing transformation. The accumu-lation of high plastic strain increments due to trans-formation in the case of model 4 gives a higher value than that obtained by the accumulation of high temperature thermal plastic deformation. As these deformations are of opposite signs, this model leads to a residual stress profile of tension at the surface

(~z=+220 MPa) and compression in the centre (c= -480 MPa, Fig, 12(f4)). Transformation has completely reversed the direction of residual stresses.

3. Discussion of the Results In model 4 it must be noted that the physical

mechanisms that are at the origin of the transforma-tion plasticity are very different from those described by a decrease in apparent yield stress 6Eap.

There exist two types of interpretation off the transformation plasticity phenomenon.13,14) An im-

portant consequence which is common for the both interpretations is that plastic deformation occurs even though the stress is lower than the yield stress of each of the phases in question.

This yielding of the metal takes place only when transformation progresses. In fact, for a given stress value and a given progress of transformation, this plastic deformation has a certain finite value. Our simplified modelization, in which we simulate the transformation plasticity by a decrease in the metal's yield stress at the beginning of the martensitic transformation interval, gives the metal the possibility of undergoing plastic deformations at low internal stresses. On the other hand the magnitude of this deforma-tion is not limited by the progress of transformation but is only due to the fixed yield stress variation. Hence, this modelization would only fulfil the condition that the transformation plasticity has a finite magnitude dependent on the progress of trans-formation if the following assumption can be verified : the calculated plastic deformations occurring in the zones of the test piece which undergo martensitic transformation have the same order of magnitude as those which are allowed by the transformation plas-ticity such that it has been defined on Fig. 6. As an example of this comparison, when the surface of the test piece undergoes martensitic transformation at the stage between 54 s and 57 s after the start of cooling, the temperature of the surface decreases from 253 to 198 °C and an equivalent plastic strain in-crement of 0.67 % is calculated. In the same time range, the equivalent stress increases from 250 to 310 MPa. On the experimental dilatometric curves

(Fig. 6) the value of the total dilatation which is possible during the progress of transformation cor-responding to the variation of temperature from 253 to 198 °C is about 1.4 % for a tensile stress applied of 250 MPa.

This type of comparison has been done at other locations in the cylinder. The results show that the calculated plastic deformations and the plastic de-formations allowed by the transformation plasticity

phenomenon have the same order of magnitude.

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Hence, our simulation is a first approach to take into account the transformation plasticity in the predic-tion of internal stresses during the quenching process.

In order to achieve a more accurate modelization of the transformation plasticity phenomenon, it will be necessary to modify the basic formulation of the thermo-elastoplastic analysis. Especially, we will use a formulation of the following type :

Et - Ee'~ Eth'~ QEp'~"QEpt~ QEp ...............(3)

where Et, Ee, et,, QEp, 4Ep were previously defined, and 4Ept is the plastic strain increment representing the transformation plasticity. This term must depend on the stress state and on the progress of transformation. Nevertheless, a comparison between calculated and measured stresses has been performed.

The calculated residual axial stress profiles in the cases of models 2 and 4 and the measured residual axial stress profile (by Sach's method) are shown in Fig. 13.

We note that the measured residual stresses are of compression in the central zone of the test piece and of tension in the area near the surface, as it has been calculated by taking into account the transformation

plasticity in the prediction of internal stresses (model 4). The measured stresses are very different from those calculated in the case where martensitic trans-formation is treated in the " conventional manner" (model 2). Thus, it may be seen that the only manner for carrying out a realistic prediction of internal stresses is to take into account the specific mechanical behaviour of the steel in its transformation range, especially the transformation plasticity phe-nomenon. In order to lower the discrepancy between the

calculated and measured values of the residual stresses, it would be necessary to achieve a more accurate description of the thermal, dimensional and mechanical behaviour of the steel by cooling.

V. Conclusion

In this paper, we have shown, with the help of an example, factors which are the important metal-lurgical parameters that must be considered when

calculating the internal stresses during quenching of an alloy which is subjected to phase transformation during cooling. A simple case from the analytical viewpoint, that of the martensitic transformation of steel, was taken as the example for this investigation.

We have obtained data for calculating the internal stresses during quenching by taking into account either

just the thermal stresses, or the stresses which are of thermal origin and transformation, and by consider-ing a stress-transformation interaction which results either in a modification of transformation kinetic or in transformation plasticity. In view of the results of the internal stress calcula-

tions on the example of water quenching at 80 °C, we are able to draw the following conclusions:

Martensitic transformation treated in the " con-ventional " manner with an increase in the metal

yield stress as transformation progresses, leads to plastic deformations in the zone undergoing trans-formation, or induces plastic deformation in the neighbouring zones which are still austenitic. The deformations are of the opposite sign to those induced by high temperature thermal stresses. These two types of deformations have a comparable range, a fact which leads to low residual stresses.

The stress--transformation kinetic interaction simu-lated by a variation of + 15 °C in the MS point in-duces no considerable or significant change in internal stresses. The stress-transformation interaction which in-duces transformation plasticity simulated by a reduc-tion in the yield stress of the metal undergoing transformation modifies quite fundamentally the onset of internal stresses. The plastic deformations become very considerable during transformation and take

precedence over thermal plastic deformations. The residual stresses then reveal inverse variation as com-

pared with that of a thermal stress profile, i.e., they reveal tensile stresses on the surface and compressive stresses in the centre of the test piece. This type of variations of the residual stresses actually exists in the test piece at the end of the quenching process.

Fig. 13. Residual axial stresses as calculated in the cases of model 2 (full line) and model 4 (dashed line) versus as measured (*) for a cylinder (c, = 35 mm) of steel 60 NCD 11 quenched in water at 80 °C from 900 °C.

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