Local Strain Energy Density Applied to Bainitic Functionally Graded Steels Plates Under Mixed-Mode (I + II) Loading

doi:10.1007/s40195-014-0182-8

Abstract

Abstract:

In this paper, the averaged value of the strain energy density (SED) over a control volume is used to predict the critical load of V-notched specimens made of functionally graded steels (FGSs) under mixed-mode loading. The studied FGSs contain ferritic and austenite phases in addition to bainitic layer produced by electroslag remelting. The mechanism-based strain gradient plasticity theory is used to determine the flow stress (yield stress or ultimate stress) of each layer. The Young’s modulus and the Poisson’s ratio have been assumed to be constant, while other mechanical properties vary exponentially along the specimen width. The control volume is centered in relation to the maximum principal stress present on the notch edge and assumes a crescent shape. The points belonging to the volume perimeter are obtained numerically. In the present contribution, the effects of notch radius and notch depth on the SED and the critical load are studied. The notch radius varies from 0.2 to 2.0 mm, and the notch depth varies from 5 to 7 mm. By using the SED approach and finite element simulations, the critical load is determined, and the obtained results show a sound agreement with the experimental results.

Key words: Functionally bainitic graded steel, Strain energy density, Mechanism-based strain gradient plasticity theory, Critical load, Notch radius, Notch depth

H. Salavati, Y. Alizadeh, F. Berto. Local Strain Energy Density Applied to Bainitic Functionally Graded Steels Plates Under Mixed-Mode (I + II) Loading[J]. Acta Metallurgica Sinica (English Letters), 2015, 28(2): 164-172.

Figures/Tables 16

Table 1 Chemical compositions of original ferritic and austenitic steels (wt%)

Steel	C	Ni	Cr	Mo	Cu	Si	Mn	S	P	Fe
316L (α)	0.01	9.58	16.69	1.89	0.43	0.53	1.5	0.04	0.04	Bal.
AISI1020 (γ)	0.11	0.07	0.12	0.02	0.29	0.19	0.63	0.08	0.01	Bal.

Table 2 Mechanical properties of single phase steels present in the considered FGSs [22, 23]

Phase	Yield strength (MPa) ^[22]	Ultimate strength (MPa) ^[22]	KIC [MPa. m^0.5] ^[23]	Poisson’s ratio	Elasticity module (GPa)
Ferrite	245	425	45.72	0.33	207
Austenite	200	480	107.77	0.33	207
Bainite	1025	1125	82.08	0.33	207

Fig. 1 Geometry of the specimen under three-point bending in the crack arrester configuration (S = 72 mm, W = 18 mm, a = 5.5 mm, 2α = 60°, b = 5 mm)

Fig. 2 Control volume for homogenous materials a mode I loading; b mixed-mode loading

Fig. 3 Control volume for FGS materials a mode I loading; b mixed-mode loading

Fig. 4 Notch depth and x-y Cartesian coordinate systems (x 0 is the coordinate of the point that crack initiation occurs)

Fig. 5 Discretization of the control volume boundary

Fig. 6 Principal stress contour lines for the case ρ = 2 mm, a = 6.5 mm, 2α = 60°, b = 12 mm

Fig. 7 SED contour lines in the control volume for the case ρ = 2 mm, a = 6.5 mm, 2α = 60°, b = 12 mm \( \left(r_{0} + R_{\text{C}} (x_{i} ) = r_{0} + \frac{{\left( {1 + \nu } \right)\left( {5 - 8\nu } \right)}}{4\pi }\left( {\frac{{K_{\text{IC}} (x_{i} )}}{{\sigma_{\text{ut}} (x_{i} )}}} \right)^{2} \right)\)

Fig. 8 Comparison between the FEM and experimental values of the critical load for different notch depths and a constant notch radius ρ = 1 mm

Table 3 SED and critical load in V-notched α/β/γ FGS for different notch depths and the constant notch radius under mixed-mode loading (ρ = 2 mm; F app = 5 kN; E = 207 GPa)

a (mm)	σ _UT (MPa)	W _cr (MJ/m³)	W _FEM (MJ/m³)	Critical load (FEM) (kN)	Critical load (Exp) (kN)	Error (%)
5	526.5	0.67	0.1371	11.05	11.2	1.3
5.5	639.66	0.988	0.1651	12.24	12.5	2.1
6	777.14	1.459	0.2017	13.45	13.8	2.5
6.5	944.18	2.153	0.2511	14.64	14.7	0.4

Table 4 SED and critical load in V-notched αβγ FGS for different notch radii and the constant notch depth under mixed-mode loading (a = 5.5 mm; F app = 5 kN; σ UT = 639.66 MPa; E = 207 GPa)

ρ (mm)	Wcr (MJ/m³)	W _FEM (MJ/m³)	Critical load (FEM) (kN)
0.2	0.99	0.1651	12.24
0.4	0.99	0.1635	12.29
0.6	0.99	0.1619	12.35
0.8	0.99	0.1604	12.41
1	0.99	0.1588	12.47
1.2	0.99	0.1573	12.53
1.4	0.99	0.1558	12.59
1.6	0.99	0.1545	12.64
1.8	0.99	0.1534	12.69
2	0.99	0.1524	12.73

Fig. 9 Variation of the strain energy density versus the notch radius for a constant notch depth a = 5.5 mm (Fapp = 5 kN)

Fig. 10 Variation of the critical load versus the notch radius for a constant notch depth a = 5.5 mm

Fig. 11 Variation of the strain energy density versus the notch depth for a constant notch radius ρ = 1 mm (Fapp = 5 kN)

Fig. 12 Variation of the critical load versus the notch depth for a constant notch radius ρ = 1 mm

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