Jose Jimenez

Lakhdar Taleb

EXPERIMENTAL ANALYSES OF TRANSFORMATION INDUCED PLASTICITY IN 36NICRMO16 CARBON STEEL SUBJECTED TO VARIOUS THERMOMECHANICAL HISTORIES

Keywords: TRIP (Transformation Induced Plasticity), FD (Free Dilatometric) test, Martensite, Austenite, Elastoplasticity. _________________________________________________________________________________________________________________________________

This study deals with Transformation Induced Plasticity (TRIP) observed in martensitic transformation of 36NiCrMo16 steel. In this study, TRIP tests were carried out for two different cases. First, after only free dilatometric (FD) tests, which is used as the reference test for the considered applied stress. Second, with TRIP tests being performed similarly to the first case (same thermal cycle, same applied stress) but with pre-thermomechanical loading histories applied. Such histories may be FD tests, TRIP tests, elastoplastic history, etc... The comparison between the results of the TRIP test (a) and the TRIP test (b) indicate if TRIP holds the memory of the applied loading histories. Current obtained results tell us that TRIP doesn't hold any significant memory. During the autenite → martensite transformation, the material may present recovery from strain hardening. Waiting for more details about the physical phenomena responsible of the absence of TRIP memory, one can point out the importance of this result as it enables to use the same specimen for several TRIP tests. However, this result must be validated using other combinations of loading histories (such as multiaxial, cyclic, among others…).

Introduction

Transformation induced plasticity (TRIP) refers to an anomalous plastic strain observed when metallurgical transformation occurs under an external stress much lower than the yield limit [START_REF] Taleb | A micromechanical modeling of the Greenwood-Johnson mechanism in transformation induced plasticity[END_REF][START_REF] Taleb | New investigations on transformation induced plasticity and its interaction with classical plasticity[END_REF][START_REF] Tahimi | Plasticité induite par transformation de phase martensitique dans l'acier 35[END_REF][START_REF] Taleb | Experimental analysis of transformation plasticity[END_REF]. One of the most common ways of studying the TRIP phenomena is by performing a TRIP test. However, before defining this, we refer first to the definition of a free dilatometric (FD) test.

FD tests consist of subjecting a test specimen to a thermal cycle which leads to the transformation of 𝛼 (ferritic) phase into 𝛾 (austenitic) phase during heating, followed by transformation of 𝛾 phase into 𝛼 phase during cooling. During FD tests, the test specimen is only subjected to temperature variation and no inelastic strain is observed at the end of the transformation [START_REF] Tahimi | Plasticité induite par transformation de phase martensitique dans l'acier 35[END_REF][START_REF] Taleb | Experimental analysis of transformation plasticity[END_REF][START_REF] Boudiaf | Experimental analysis of the correlation between martensitic transformation plasticity and the austenitic grain size in steels[END_REF].

Similar to FD tests, TRIP tests also consist of subjecting a specimen to a thermal cycle leading to the same transformations just mentioned. However, the key difference between the FD tests and the TRIP tests is that during the TRIP tests, the test specimen is also subjected to the aforementioned external stress, provoking said anomalous plastic strain. The external stress is applied just before the starting temperature (Ms) of the transformation during cooling and maintained constant during the whole transformation [START_REF] Taleb | Experimental analysis of transformation plasticity[END_REF].

There have been many contributions towards the understanding of the TRIP phenomena throughout the years. The focus of this paper is to provide our own small contribution as well to this subject, by carrying out some experiments using TRIP and FD tests and analyzing the behavior of 36NiCrMo16 steel test samples, and seeing if, within the experimental results, there is any observable elastoplastic history effect carried forward after different tests.

Experimental procedure

In addition to the FD and TRIP tests, other tests have been carried out in order to have an idea about the elastoplastic behavior of the austenitic phase at a given temperature before Ms (austenite → martensite transformation). Such elastoplastic characterization is performed through the application of a loading-unloading cycle where both directions are considered: tension and torsion. The FD, TRIP and elastoplastic tests are carried out in different orders (and magnitudes of stress in the case of the TRIP test) and are compared in order to visualize the transformation plasticity phenomenon and see if the effects of loading history could be observed.

As a remark, some details about the parameters are mentioned in Table 1: where:

• VS: Virgin Specimen • 𝑇 𝜎 𝑦 /4 : TRIP test under σ y /4

• 𝑇 σ y /4 𝑟𝑒𝑓 : reference TRIP test under σ y /4 (this test is considered as a reference, as it is preceded only by a free dilatometric cycle and will be used for comparisons throughout the article)

• Elastoplastic cycle: it is a quick loadingunloading tensile stress applied before Ms leading to plasticity of austenite.

The 𝑇 𝜎 𝑦 /4 test results will be compared to 𝑇 σ y /4 𝑟𝑒𝑓 and, more generally, the test results of the 3rd column will be compared to the corresponding test results of the last column.

The tests were carried out following the left to right order. An example for the first line of Table 1 is shown, in which the virgin specimen is subjected to successively: FD test → TRIP test under σ y /4 → FD test → TRIP test under σ y /4

In Figure 1 we can observe an example of how these tests are carried out by presenting a simplified form of the loading vs time plot of the first series of tests previously presented. The top part of Figure 1 shows an example of the temperature cycles through which the test specimen goes through. The Ms line represents the temperature at which the transformations would occur. The captions indicate the test cycles carried out. The applied stress is represented on the bottom part of the figure.

Material and specimen

The chemical composition of the 36NiCrMo16 steel test specimen used in this work is listed in Table 2. It is a low alloyed steel with a carbon content of 0,35% which is mostly used for heavy duty machine parts, construction, and equipment, as well as the aviation and aerospace industry. It (and its substitutes) is characterized by its relatively high nickel content, and thus better tensile strength and yield point, as well as its very good hardenability, impact resistance, load deformation, wear, tension resistance and high toughness. As for the specific test specimen, as seen in Figure 2, the central part, which is a tubular cylinder, allows us to obtain a quasihomogeneous stress field and a negligible temperature gradient across its tube section.

Tests Performed and Obtained Results

References: TRIP tests preceded only by FD Tests

Method of Evaluation of TRIP

In a general assumption, evaluating TRIP consists of a relatively simple formula:

𝜀 𝑡𝑝 = 𝜀 𝑡𝑜𝑡𝑎𝑙 -𝜀 𝑡ℎ𝑚 -𝜀 𝑒 (1)
where 𝜀 𝑡𝑝 is the actual TRIP strain, 𝜀 𝑡𝑜𝑡𝑎𝑙 is the deformation measured during the TRIP test, 𝜀 𝑡ℎ𝑚 is the deformation measured during the FD test, and 𝜀 𝑒 is the measured elastic deformation. It is to be noted that this rule only applies when the stress applied to obtain TRIP is significantly smaller compared to the yield stress [START_REF] Taleb | Experimental analysis of transformation plasticity[END_REF], [START_REF] Barbe | Numerical modelling of the plasticity induced during diffusive transformation. Case of a cubic array of nuclei[END_REF]. By subtracting the measured deformation during the FD test and the measured elastic deformation, we obtain the measured deformation during the TRIP test, as can be seen in Figure 3. Knowing this, before evaluating TRIP, it is important to define the FD test, which will be considered as the reference. Indeed, FD test results obtained after applying TRIP tests will not provide the same results as those that were performed before (FD_1). The graphs in Figure 4 (a) for these two particular FD test cases show that for the first case (FD_1), the strain difference from the origin is of about -7.4 * 10 -4 , but in the second case (FD_2), which is after applying a TRIP test, the difference from the origin is of about -17.4 * 10 4 . The reason for why this happens is still being investigated, as well as the noticeable mismatch between the 2 plots at the end of the austenization. However, taking a look at Figure 4 (b), we see that, despite applying more TRIP tests, even for cases in which the stress applied to the specimen is higher, the following FD results (FD_3, FD_4, and FD_5) are almost superimposed. Therefore, FD_2, FD_3, FD_4, and FD_5 may be taken as references that may be used for the determination of TRIP.

References of the TRIP Tests

Here, we present the reference TRIP tests related to the levels of applied stresses.

Memory Effect for the Tests

To evaluate the memory effect, we compare our reference TRIP test characterized by the fact that the thermomechanical history applied on the specimen before the TRIP test is reduced in free dilatometry. Figures 6 (a),(b) and (c) show these comparisons for the cases presented in Table 1. As seen in these figures, there is no clear trend as to why certain tests provide a change from austenite to martensite faster than others. This matter is still being researched in our current investigations. However, what is clear is that the end strain of TRIP tests performed after being submitted to more TRIP cycles tends to be lower, and in the case of TRIP cycles after going through an elastoplastic provides the lowest end strain for all 3 cases.

Analysis of Other TRIP Characteristics

Effect on the Kinetics Plasticity

The kinetics of TRIP are usually expressed by the function defined ratio vs. the volume proportion (Z) of the martensite phase that is formed during these tests:

(𝜀 𝑡𝑝 (𝑍)/(𝜀 𝑡𝑝 (𝑍 = 1)) (3)
This function is estimated from experimental results,

𝜀 𝑡𝑝 = 𝑓(𝑇) (4)
and the volume proportion of the new formed phase (𝜶) vs. temperature

𝑍 = 𝑓(𝑇). (5)
Equation ( 5) is estimated from thermal strain evolution vs. temperature (Equation (4) deducted from free dilatometry test) by assuming the linear mixture rule to obtain the thermal strain (𝜺 𝒕𝒉𝒎 ) of the phases mixture [START_REF] Taleb | Experimental analysis of transformation plasticity[END_REF]. The thermal strain can be expressed by

𝜀 𝑡ℎ𝑚 = 𝛼 • (𝑇 -𝑇 𝛼 ) + (1 -𝑍) • 𝜀 𝑀𝐴 𝑇 𝛼 (6) with 𝛼 = 𝑍 • 𝛼 𝑀 + (1 -𝑍) • 𝛼 𝐴 (7)
where Z is the volume fraction of the martensitic phase, 𝜶 𝑴 is the dilatation coefficient of the martensitic phase, 𝜶 𝑨 is the dilatation coefficient of the austenitic phase, 𝜶 is the dilatation coefficient of the mixture, 𝑻 𝜶 is the references temperature of 𝜶 phases and ∆𝜺 𝑴𝑨 𝑻 𝜶 is the compactness variation (also known as deformation of transformation) between the martensitic and the austenitic phases at room temperature, which, for our case, varies between 0,63% and 0,88% at room temperature at the beginning of the transformation due to strain hardening. We obtained our values for ∆𝜺 𝑴𝑨 𝑻 𝜶 using a graphic method as shown in Figure 7: To interpret this, we measure the distance at the beginning of transformations on the plot of our referential FD test (as shown with the green arrow) to obtain our first percentage. As for our second percentage, we trace a line (as shown in red) parallel and superimposed to the cooling process and use that to position our arrow (yellow) and to measure that distance until the end point of the plot. This gives us our second percentage. With these 2 values, we now have our range in which our value of ∆𝜺 𝑴𝑨 𝑻 𝜶 is found.

Equation ( 6) and Equation ( 7) lead to

𝑍 = 𝜀 𝑡ℎ𝑚 -𝛼 𝐴 •(𝑇-𝑇 𝛼 )-∆𝜀 𝑀𝐴 𝑇 𝛼 (𝛼 𝑀 -𝛼 𝐴 )•(𝑇-𝑇 𝛼 )-∆𝜀 𝑀𝐴 𝑇 𝛼 (8)
With this expression, we can now estimate Equation ( 3), which will provide us the transformation plasticity evolution vs. the martensitic volume proportion for specific tests.

TRIP Parameter 𝒌

cases where the applied stress is small, uniaxial, and maintained constant during the transformation, certain TRIP as Desalos' model for example, may be expressed by the following equation:

𝜀 𝑡𝑝 = 𝑘𝑍(2 -𝑍)𝜎 (9)
where k is generally known as the TRIP parameter. This equation is valid if the applied load remains constant during the transformation, for which a rate formulation is necessary in general.

This parameter can be defined as a function of the transformation progress [START_REF] Taleb | New investigations on transformation induced plasticity and its interaction with classical plasticity[END_REF]. It allows us to analyze the TRIP rate of a test specimen as a function of the volume fraction of the product phase and the applied stress to it [START_REF] Taleb | A micromechanical modeling of the Greenwood-Johnson mechanism in transformation induced plasticity[END_REF]. This parameter can be estimated experimentally using tests where applied stress does not generate classical plasticity or creep, which is why in most cases this is determined using test results that do not surpass half of the test specimen's yield stress.

Equation ( 9) leads to

𝑘 = 𝜀 𝑡𝑝 𝑍(2-𝑍)𝜎 ; 𝑖𝑓 𝑍 = 1, 𝑘 = 𝜀 𝑓 𝑡𝑝 𝜎 (10)
In these conditions, k is constant. of this, it may be determined after a TRIP test as the ratio between the final TRIP and the applied stress.

Desalos' model [START_REF] Barbe | Numerical modelling of the plasticity induced during diffusive transformation. Case of a cubic array of nuclei[END_REF] uses such a method in order to determine the coefficient k. This character makes this model non-predictive.

One can find in the literature the expression giving the parameter k based on micromechanical considerations, such as in [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF] [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF], who presents the parameter with the following expression:

𝑘 = 2∆𝜀 MA 𝜎 𝑦 (11)
where 𝝈 𝒚 is the yield stress of the austenitic phase.

TRIP Evolution versus Volume Proportion (Z) of the Martensite Phase

As seen in other works such as [START_REF] Taleb | Experimental analysis of transformation plasticity[END_REF] [START_REF] Taleb | Experimental analysis of transformation plasticity[END_REF], estimating the expression in Equation ( 2) is highly influenced by the applied load during tests. For this reason, we analyze this case with a test using only 75% of the austenitic phase yield stress during the transformation.

In Figure 6 we present the TRIP evolution of the case presented in Section 3.1.2 vs. the volume proportion determined from the FD test applied just before it. However, as we saw in the previous section, there are expressions such as Desalos' and Leblond's models that allow us to obtain this relation as well. Therefore, we predict the same relation using Desalos' and Leblond's models and compare it to our results. There is a clear distinction between the three results, most noticeably how for our experiment's results, there is a much faster TRIP evolution at the beginning of the transformation than the other two. This difference may be shown considering constitutive equations of both models.

Desalos' model is given by

𝑑𝜀 𝐷 𝑡𝑝 = 𝑘2(1 -𝑍)𝑑𝑍𝜎 (12)
If the applied stress and k are assumed constants, Equation (12) can be integrated, leading to

𝜀 𝐷 𝑡𝑝 = 𝑘𝑍(2 -𝑍)𝜎 (13)
Therefore,

𝜀 𝐷 𝑡𝑝 (𝑍) 𝜀 𝐷 𝑡𝑝 (𝑍=1) = 𝑍(2 -𝑍) (14)
The same reasoning with Leblond's model leads to Equation ( 18) is verified for any 𝑍 > 0.

We can conclude that the TRIP kinetics predicted by Leblond's model are more important than the Desalos' kinetics.

Concluding Remarks

In this paper, we have investigated the TRIP phenomenon in the martensitic transformation of 36NiCrMo16 steel, with a main focus on the memory of TRIP after the application of a thermoelastoplastic history including phase changes. In addition, the TRIP parameter, the kinetics of TRIP, as well as the relation between the final TRIP and the applied stress are also studied.

The obtained results enable us to make the following concluding remarks:

• After "light" thermomechanical history composed of only Free Dilatometry, no memory is observed: The TRIP response is completely superimposed on the reference test under question. This observation is the same for all stress levels considered here: TRIP(𝜎 𝑦 4 ⁄ ),TRIP(𝜎 𝑦 2 ⁄ ), and TRIP(3 𝜎 𝑦 4 ⁄ ). As a consequence, the specimen becomes reusable even if it is necessary to check for oxidation after a certain number of tests. This "economic" option allows us to perform several "low cost" TRIP tests compared to the classical procedure (one test = one specimen). Finally, note that our result confirms the choice adopted in [START_REF] Coret | Experimental study of the phase transformation plasticity of 16MND5 low carbon steel under multiaxial loading[END_REF] [START_REF] Coret | Experimental study of the phase transformation plasticity of 16MND5 low carbon steel under multiaxial loading[END_REF] and [START_REF] Videau | Experimental Study of the Transformation-Induced Plasticity in a Cr-Ni-Mo-Al-Ti Steel[END_REF] [START_REF] Videau | Experimental Study of the Transformation-Induced Plasticity in a Cr-Ni-Mo-Al-Ti Steel[END_REF] where they use the same specimen for several TRIP tests.

• After thermomechanical histories including Free Dilatometries as well as TRIP tests, the amount of TRIP is a slightly reduced (by about 10%).

• After the elastoplastic cycle, where the austenitic phase undergoes plasticity, the difference is more significant. This result is qualitatively in accordance with the test results given in Taleb (2014) [START_REF] Taleb | Transformation-Induced Plasticity (TRIP)[END_REF] where the interaction between TRIP and Classical plasticity is studied. For a quantitative point of view, the pre-hardening in Taleb (2014) [START_REF] Taleb | Transformation-Induced Plasticity (TRIP)[END_REF] is significantly more important than the one considered in our study, as they reached a plastic strain of the order of 5%. This parameter deserves verification in a future extension of our study. The study of the TRIP parameter and kinetics points out the importance of Leblond's model with its predictive character.

Fig. 1 .

 1 Fig. 1. Simplified look of the loading vs. time plot of the first series of tests.

Fig. 2 .

 2 Fig. 2. Dimensions (in mm) of 36NiCrMo16 test specimen.

Fig. 3 .

 3 Fig. 3. (a) FD and TRIP complete cycle at 𝑇 3σ y /4 𝑟𝑒𝑓 ; (b) TRIP test cycle strain portion after applying Equation 1.

Fig. 4 .

 4 Fig. 4. (a) Free Dilatometric test results: FD_1 is carried out using virgin specimen while FD_2 is obtained using the same specimen subjected to TRIP cycle in addition to FD_1;(b) Free Dilatometric test results post TRIP tests.

Fig. 5 . 4 𝑟𝑒𝑓

 54 Fig. 5. Complete thermomechanical cycle comparison between 𝑇 σ y /4 𝑟𝑒𝑓 , 𝑇 σ y /2 𝑟𝑒𝑓 and 𝑇 3σ y /4 𝑟𝑒𝑓

Fig. 6 .

 6 Fig. 6. (a) Comparison between reference TRIP test and following TRIP tests at σ y /4; (b) Comparison between reference TRIP test and following TRIP tests at σ y /2; (c) Comparison between reference TRIP test and following TRIP tests at 3σ y /4

Figure 7 :

 7 Figure 7: Explanation of obtention of ∆𝜺 𝑴𝑨 𝑻 𝜶

  ICRS11 -11 th International Conference on Residual Stresses -Nancy -France -27-30 th March 2022

Figure 7 .

 7 Figure 7. The orange line represents the evolution of the ratio 𝑇𝑅𝐼𝑃 (𝑍)⁄ 𝑇𝑅𝐼𝑃 (𝑍 = 1) vs. Z during TRIP with 75% of the austenitic phase yield stress during the transformation. The blue line represents the prediction using Leblond's model. The grey line represents the prediction using Desalos' model.

Table

  . Different histories applied in the present work.

				Memory after light thermomechanical history composed of one TRIP test
	VS	FD	𝑻 𝛔 𝐲 /𝟒 𝒓𝒆𝒇	FD	𝑻 𝝈 𝒚 /𝟒
	VS	FD	𝑻 𝛔 𝐲 /𝟐 𝒓𝒆𝒇	FD	𝑻 𝝈 𝒚 /𝟐
	VS	FD 𝑻 𝟑𝝈 𝒚 /𝟒 𝒓𝒆𝒇	FD	𝑻 𝟑𝝈 𝒚 /𝟒
				Memory after medium thermomechanical history composed of 3 TRIP tests
	VS	FD	𝑻 𝛔 𝐲 /𝟒 𝒓𝒆𝒇	FD	𝑻 𝝈 𝒚 /𝟐	FD	𝑻 𝝈 𝒚 /𝟐	FD	𝑻 3𝝈 𝒚 /𝟒	FD	𝑻 𝝈 𝒚 /𝟒
	VS	FD	𝑻 𝛔 𝐲 /𝟐 𝒓𝒆𝒇	FD	𝑻 3𝝈 𝒚 /𝟒	FD	𝑻 3𝝈 𝒚 /𝟒	FD	𝑻 𝝈 𝒚 /𝟒	FD	𝑻 𝝈 𝒚 /𝟐
	VS	FD 𝑻 𝟑𝝈 𝒚 /𝟒 𝒓𝒆𝒇	FD	𝑻 𝝈 𝒚 /𝟒	FD	𝑻 𝝈 𝒚 /𝟒	FD	𝑻 𝝈 𝒚 /𝟐	FD	𝑻 𝟑𝝈 𝒚 /𝟒
				Memory after medium thermomechanical history composed of 2 TRIP tests
	VS	FD	𝑻 𝛔 𝐲 /𝟒 𝒓𝒆𝒇	FD	𝑻 3𝝈 𝒚 /𝟒	FD	𝑻 3𝝈 𝒚 /𝟒	FD	𝑻 𝝈 𝒚 /𝟒
	VS	FD	𝑻 𝛔 𝐲 /𝟐 𝒓𝒆𝒇	FD	𝑻 𝝈 𝒚 /𝟒	FD	𝑻 𝝈 𝒚 /𝟒	FD	𝑻 𝝈 𝒚 /𝟐
	VS	FD 𝑻 𝟑𝝈 𝒚 /𝟒 𝒓𝒆𝒇	FD	𝑻 𝝈 𝒚 /𝟐	FD	𝑻 𝝈 𝒚 /𝟐	FD	𝑻 𝟑𝝈 𝒚 /𝟒
					Memory after severe thermomechanical history in classical elastoplasticity
	VS	FD	𝑻 𝛔 𝐲 /𝟒 𝒓𝒆𝒇	FD	Elastoplastic cycle	FD	𝑻 𝝈 𝒚 /𝟒
	VS	FD	𝑻 𝛔 𝐲 /𝟐 𝒓𝒆𝒇	FD	Elastoplastic cycle	FD	𝑻 𝝈 𝒚 /𝟐
	VS	FD 𝑻 𝟑𝝈 𝒚 /𝟒 𝒓𝒆𝒇	FD	Elastoplastic cycle	FD	𝑻 𝟑𝝈 𝒚 /𝟒

  The reference of the TRIP test under 𝜎 𝑦 4 ⁄ ≈ 30 MPa is denoted as 𝑇 𝜎 𝑦 4

	⁄ 𝑟𝑒𝑓 . Similarly, the reference
	of the TRIP tests performed with 𝜎 𝑦 2 ⁄ ≈ 60 MPa and 3𝜎 𝑦 4 ⁄ ≈ 90 MPa are denoted as 𝑇 𝜎 𝑦 2 ⁄ 𝑟𝑒𝑓
	and 𝑇 3𝜎 𝑦 4

⁄

𝑟𝑒𝑓 respectively. We can see these aforementioned references in Figure

4

.