Calibration and evaluation of the Lemaitre damage model using axial- torsion fatigue tests on five engineering alloys

The Lemaitre damage model is evaluated using fatigue test data from five engineering alloys: 1045 steel, 16MnR steel, 7075-T651 Al alloy, extruded AZ61A Mg alloy, and extruded AZ31B Mg alloy. Tension-compression, torsion, proportional axial-torsion, and 90° out-of-phase axial-torsion loadings were investigated. The results show that the overall accuracy of the fatigue life estimates made by using the Lemaitre model is comparable to those obtained by fatigue models that require the definition of a loading cycle. A simple and effective method is described for determining the material constants of the Lemaitre model.

Keywords
Multiaxial fatigue; Life prediction; Damage mechanics; Carbon steel; Aluminum alloy; Magnesium alloy

Authorship

Fábio Castro ^**Corresponding author.

Departamento de Engenharia Mecânica, Universidade de Brasília - UnB, Brasília, DF, Brasil. E-mail: fabiocastro@unb.br, cainabemfica@gmail.comUniversidade de BrasíliaBrazilBrasília, DF, BrazilDepartamento de Engenharia Mecânica, Universidade de Brasília - UnB, Brasília, DF, Brasil. E-mail: fabiocastro@unb.br, cainabemfica@gmail.com

Cainã Bemfica

*Corresponding author.

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables

Figures (5)
Tables (5)

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Figure 1
Loading paths used in the fatigue tests.

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Figure 2
Observed live versus estimated live for 1045 steel (a) and 16MnR steel (b).

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Figure 3
Observed live versus estimated live for AZ61 Mg alloy (a) and AZ31B Mg alloy(b).

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Figure 4
Observed live versus estimated live for 7075-T651 Al alloy.

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Figure 5
Observed live versus estimated live based on the modified Lemaitre model for AZ31B Mg alloy.

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Table 1
Summary of the Jiang‒Sehitoglu cyclic plasticity model.

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Table 2
Material constants used in the Jiang‒Sehitoglu plasticity model.

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Table 3
Material constants used in the Lemaitre damage model.

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Table 4
Difference between stabilized stress ranges obtained from fatigue experiments on 1045 steel and from numerical simulations using the Jiang-Sehitoglu plasticity model.

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Table 5
Difference between stabilized stress ranges obtained from fatigue experiments on 16MnR steel and from numerical simulations using the Jiang-Sehitoglu plasticity model.

Figure 1 Loading paths used in the fatigue tests.

Figure 2 Observed live versus estimated live for 1045 steel (a) and 16MnR steel (b).

Figure 3 Observed live versus estimated live for AZ61 Mg alloy (a) and AZ31B Mg alloy(b).

Figure 4 Observed live versus estimated live for 7075-T651 Al alloy.

Figure 5 Observed live versus estimated live based on the modified Lemaitre model for AZ31B Mg alloy.

Table 1 Summary of the Jiang‒Sehitoglu cyclic plasticity model.

1) Yield function

f = ∥ S - α ∥^{2} - 2 k^{2} = 0

2) Flow rule

{\dot{ε}}^{p} = \frac{1}{h} 〈 \dot{S} \cdot n 〉 n

3) Hardening rule

α = \sum_{i = 1}^{M} α^{(i)}

{\dot{α}}^{(i)} = c^{(i)} r^{(i)} [n - {(\frac{∥ α^{(i)} ∥}{r^{(i)}})}^{χ^{(i)} + 1} \frac{α^{(i)}}{∥ α^{(i)} ∥}] \dot{p} (i = 1, 2, ..., M)

S = deviatoric stress, α = backstress, k = yield stress in shear, n = normal on the yield surface, h = plastic modulus function, εp = plastic strain, α(i) = ith backstress part, M = number of backstress parts, p˙ = equivalent plastic strain rate, and c(i), r(i), χ(i) = material constants.

Table 2 Material constants used in the Jiang‒Sehitoglu plasticity model.

1045 steel

E

= 204 GPa,

ν

= 0.27,

k

= 50 MPa

c^{(1)}

= 12583.2,

c^{(2)}

= 1859.5,

c^{(3)}

= 475.2,

c^{(4)}

= 164.4,

c^{(5)}

= 68.9

r^{(1)}

= 55.8,

r^{(2)}

= 56.0,

r^{(3)}

= 55.0,

r^{(4)}

= 54.5,

r^{(5)}

= 116.3

χ^{(1)}

χ^{(2)}

\dots

χ^{(5)}

= 5

16MnR steel

E

= 212.5 GPa,

ν

= 0.31,

k

= 50 MPa

c^{(1)}

= 18373.8,

c^{(2)}

= 1721.7,

c^{(3)}

= 325.3,

c^{(4)}

= 89.8,

c^{(5)}

= 31.5

r^{(1)}

= 67.8,

r^{(2)}

= 65.9,

r^{(3)}

= 64.4,

r^{(4)}

= 63.7,

r^{(5)}

= 116.5

χ^{(1)}

χ^{(2)}

\dots

χ^{(5)}

= 5

7075-T651 Al alloy

E

= 71.7 GPa,

ν

= 0.3,

k

= 100 MPa

c^{(1)}

= 651313.5,

c^{(2)}

= 121390.2,

c^{(3)}

= 29596.0,

c^{(4)}

= 8764.4,

c^{(5)}

= 3007.4

c^{(6)}

= 1158.2,

c^{(7)}

= 489.5,

c^{(8)}

= 223.3,

c^{(9)}

= 108.6,

c^{(10)}

= 55.8

r^{(1)}

= 28.8,

r^{(2)}

= 33.9,

r^{(3)}

= 33.7,

r^{(4)}

= 33.6,

r^{(5)}

= 33.5

r^{(6)}

= 33.4,

r^{(7)}

= 33.4,

r^{(8)}

= 33.3,

r^{(9)}

= 33.3,

r^{(10)}

= 76.8

χ^{(1)}

χ^{(2)}

\dots

χ^{(10)}

= 10

Table 3 Material constants used in the Lemaitre damage model.

Material	$S$ (MPa)	$s$	$D_{c}$
1045 steel	2.1	2.8	0.4
16MnR steel	2.0	2.7	0.8
AZ61A Mg alloy	2.2	1.8	0.1
AZ31B Mg alloy	1.7	2.1	0.1
7075-T651 Al alloy	14.0	0.8	1.0

Table 4 Difference between stabilized stress ranges obtained from fatigue experiments on 1045 steel and from numerical simulations using the Jiang-Sehitoglu plasticity model.

Loading type	Specimen ID	$\frac{Δ σ^{exp} - Δ σ^{num}}{Δ σ^{exp}}$ (%)	$\frac{Δ τ^{exp} - Δ τ^{num}}{Δ τ^{exp}}$ (%)
In-phase axial-torsion	4524	0	5
	4523	0	6
	4528	-2	3
	4516	0	8
	4519	0	7
	4525	0	-3
	4520	1	-4
	4515	2	-1
	4550	0	4
	4514	2	2
	4554	1	0
	4517	2	3
	4526	3	-3
	4503	7	-3
	4501	6	-3
	4522	3	-2
	4548	3	-2
	4521	0	-1
	4509	9	-5
	4530	4	-3
90º out-of-phase axial-torsion	45A3	8	18
	4583	5	4
	4580	18	18
	45B2	14	18
	4586	13	16
	45A5	8	6
	45D4	21	14
	45D1	13	14
	45B5	20	13
	4588	8	5

Table 5 Difference between stabilized stress ranges obtained from fatigue experiments on 16MnR steel and from numerical simulations using the Jiang-Sehitoglu plasticity model.

Loading type	Specimen ID	$\frac{Δ σ^{exp} - Δ σ^{num}}{Δ σ^{exp}}$ (%)	$\frac{Δ τ^{exp} - Δ τ^{num}}{Δ τ^{exp}}$ (%)
In-phase axial-torsion	TU20N	6	6
	TU21N	4	2
	TU09N	13	19
	TU15N	17	21
	TU06S	22	23
	TU09S	23	23
90º out-of-phase axial-torsion	TU07S	32	32
90º out-of-phase axial-torsion	TU08N	31	29
	TU06N	28	26
	TU03N	30	28
	TU14N	29	32

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Calibration and evaluation of the Lemaitre damage model using axial- torsion fatigue tests on five engineering alloys

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[1] *Corresponding author.