Martensite’s Logistic Paradigm

Guimarães, José Roberto Costa; Rios, Paulo Rangel; Alves, André Luiz Moraes

doi:10.1590/1980-5373-MR-2020-0370

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Articles • Mat. Res. 24 (3) • 2021 • https://doi.org/10.1590/1980-5373-MR-2020-0370 copy

Martensite’s Logistic Paradigm

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

This work introduces a deterministic approach to the martensite transformation curve. Martensite is a nucleation-controlled transformation that has two characteristics: autocatalysis and auto-accommodation. Only a small number of martensite units initially form owing to primary nucleation. These new units may cause the transformation of other units by autocatalysis. We call this kind of transformation chained autocatalysis. Moreover, as the transformation progresses, the auto-accommodation influences the arrangement of new units. This work assumes that the transformation-saturation relates to the exhaustion of the chained autocatalysis, which underlines the microstructure. To compare, we considered the KJMA’s extended-transformation concept that implies assuming exhaustion by impingement. Data from isothermal martensite transformations and anisothermal martensite transformations are used to validate the model. Those data comprised different grain sizes and carbon contents. The model is based upon Verhulst’s logistic concept. We propose that the model's high fitting-capability stems from its deterministic aspect combined with martensite’s self-similarity. Additionally, we suggest that chained autocatalysis controls the rate of martensite transformation. Therefore, the relaxation of transformation strains by plasticity assisted by mutual accommodation determines the transformation's martensite volume in the absence of post-propagation coarsening/coalescence.

Keywords:
Martensitic transformations; microstructure; logistic equation; analytical methods; Avrami’s equation

	Equation 11 Logistic					Equation 12 KJMA
$T$ , K	$φ (T)$	$V_{V i}$	$τ$ , s	ΣSQ	$δ$ < 5%	$φ (T)$	$V_{V i}$	$τ$ , s	ΣSQ	$δ$ < 5%
77	2.8	2.3x10^-3	2.8x10²	6.6x10^-4	79.8%	1.7	8.1x10^-9	6.9x10²	4.6x10^-3	58.3%
133	1.7	6.4x10^-4	2.4x10²	1.5x10^-2	71.4%	0.9	6.8x10^-5	2.6x10²	4.1x10^-2	53.6%
143	1.5	5.5x10^-3	1.7x10²	1.5x10^-2	75.0%	0.7	2.7x10^-4	1.7x10²	6.8x10^-2	61.9%
163	1.3	1.1x10^-3	7.5x10¹	4.9x10^-2	72.6%	0.5	5.8x10^-3	7.6x10¹	1.0x10^-1	65.5%
173	1.5	3.3x10^-4	7.5x10¹	4.7x10^-2	72.8%	0.7	3.6x10^-4	7.7x10¹	1.0x10^-1	66.7%
193	2.5	2.1x10^-3	1.9x10²	8.5x10^-4	35.7%	2.6	1.5x10^-12	1.9x10³	2.6x10^-3	19.6%
203	2.6	4.9x10^-3	3.4x10²	3.3x10^-5	54.1%	2.8	2.5x10^-12	3.4x10³	3.2x10^-5	54.1%

	Equation 11 Logistic					Equation 12 KJMA
Field (kOe)	$φ (T)$	$V_{V i}$	$τ$ , s	ΣSQ	$δ$ < 5%	$φ (T)$	$V_{V i}$	$τ$ , s	ΣSQ	$δ$ < 5%
0	3.5	2.0x10^-8	1.2x10⁴	9.5x10^-5	75.0	2.8	4.0x10^-7	1.2x10⁴	5.5x10^-5	100.0
20	2.7	3.2x10^-5	9.3x10²	5.7x10^-5	95.5	1.9	1.6x10^-5	1.3x10³	2.7x10^-4	72.7
40	2.1	1.3x10^-3	7.6x10²	1.0x10^-4	100.0	1.4	4.7x10^-4	1.2x10³	2.2x10^-4	72.2
60	2.1	2.5x10^-6	2.8x10²	2.4x10^-4	75.0	1.6	1.1x10^-4	2.7x10²	5.2x10^-5	100.0
90	1.7	2.5x10^-4	3.3x10²	6.7x10^-4	60.9	1.4	1.3x10^-3	3.2x10²	7.2x10^-5	95.7

		Equation 14 Logistic					Equation 15 KJMA
D, mm	$T_{i}$ , K	${\bar{φ}}_{G}$	$V_{V i}$	$T^{*}$ , K	ΣSQ	$δ$ < 5%	${\bar{φ}}_{G}$	$V_{V i}$	$T^{*}$ , K	ΣSQ	$δ$ < 5%
0.185	628	4.0	2.3x10^-3	646.6	3.5x10^-2	83.3%	1.4	2.8x10^-4	628.1	4.9x10^-2	69.7%
0.067	628	3.1	4.5x10^-4	643.5	8.9x10^-3	87.7%	1.3	7.1x10^-4	623.1	8.1x10^-2	58.5%
0.025	638	3.6	4.5x10^-5	643.5	1.8x10^-2	79.4%	1.3	5.3x10^-4	613.1	1.1x10^-1	39.7%
0.015	608	2.7	2.2x10^-3	613.2	2.4x10^-2	88.7%	1.0	7.0x10^-4	607.8	1.2x10^-1	59.7%
0.006	608	3.3	6.3x10^-4	613.2	1.2x10^-2	83.6%	1.4	4.5x10^-5	607.8	1.7x10^-1	47.5%

		Equation 14 Logistic					Equation 15 KJMA
wt%C	$T_{i}$ , K	${\bar{φ}}_{G}$	$V_{V i}$	$T^{*}$ , K	ΣSQ	$δ$ < 5%	${\bar{φ}}_{G}$	$V_{V i}$	$T^{*}$ , K	ΣSQ	$δ$ < 5%
0.002	290	3.7	1.2x10^-1	339.7	7.0x10^-4	88.9	1.3	8.6x10^-2	299.2	8.1x10^-4	93.3
0.020	265	4.0	3.7x10^-2	304.7	3.6x10^-4	96.9	1.5	6.4x10^-3	267.8	8.8x10^-4	96.9
0.050	254	5.5	1.2x10^-2	304.1	2.3x10^-4	88.9	2.4	8.0x10^-3	265.4	4.1x10^-4	77.8
0.100	193	6.1	4.6x10^-2	242.6	1.1x10^-4	90.0	2.8	6.6x10^-2	218.4	3.2x10^-5	90.0

		Equation 14 Logistic					Equation 15 KJMA
wt%C	$T_{i}$ , K	${\bar{φ}}_{G}$	$V_{V i}$	$T^{*}$ , K	ΣSQ	$δ$ < 5%	${\bar{φ}}_{G}$	$V_{V i}$	$T^{*}$ , K	ΣSQ	$δ$ < 5%
0.46	586	1.8	1.7x10^-3	587.5	3.0x10^-2	65.1%	1.0	8.7x10^-4	586.3	8.5x10^-2	49.2%
0.66	535	2.3	1.6x10^-2	546.0	1.6x10^-2	69.7%	1.2	2.9x10^-3	535.9	4.0x10^-2	53.0%
0.80	502	2.2	1.6x10^-2	513.9	6.3x10^-3	84.2%	1.2	7.9x10^-3	503.8	2.3x10^-2	52.6%

$T$ , K	$φ (T)$ Difference	$V_{V i}$ Difference	$τ$ Difference
77	48.9%	200.0%	-85.6%
133	61.5%	161.3%	-5.0%
143	72.7%	181.4%	-2.1%
163	88.9%	-133.9%	-2.0%
173	72.7%	-9.0%	-2.0%
193	-3.9%	200.0%	-163.6%
203	-7.4%	200.0%	-163.7%

D, mm	$T_{i}$ , K	${\bar{φ}}_{G}$ Difference	$V_{V i}$ Difference	$T^{}$ Difference*
0.185	628	96.3%	156.6%	2.9%
0.067	628	81.8%	-44.8%	3.2%
0.025	638	93.9%	-168.7%	4.8%
0.015	608	91.9%	103.4%	0.9%
0.006	608	80.8%	173.3%	0.9%

wt%C	$T_{i}$ , K	${\bar{φ}}_{G}$ Difference	$V_{V i}$ Difference	$T^{}$ Difference*
0.46	586	57.1%	64.6%	0.2%
0.66	535	62.9%	138.6%	1.9%
0.80	502	58.8%	67.8%	2.0%

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