PSI - Issue 42

Vítor M. G. Gomes et al. / Procedia Structural Integrity 42 (2022) 1552–1559 V.M.G. Gomes et al. / Structural Integrity Procedia 00 (2019) 000–000

1556

5

Fig. 3. Finite element model and definition of the boundary conditions: A - Graphical interface of FEM model converted from CAD model. 1 / 2 of the full model; B - Rainflow matrices of vertical and lateral displacements; C - Temporal spectrum of lateral and vertical loadings for two loading scenarios in analysis.

4.3. Cyclic Hardening

The spring steel’s cyclic properties are computed considering estimation methods based on its monotonic mechan ical properties (Table 3). The procedure consists in estimating firstly the coe ffi cients and exponents of strength and ductility of strain-life curve. For high-strength steel, the modified universal slopes (Muralidharan and Manson (1988)) is a suitable method, such that σ ′ f = 0 . 623 σ 0 . 832 uts E 0 . 168 , b = − 0 . 09, ε ′ f = 0 . 0196 ε 0 . 155 f ( σ uts / E ) − 0 . 53 , and c = − 0 . 56. Next, the Ramberg-Osgood’s cyclic hardening coe ffi cient and exponent are computed such that K ′ = σ ′ f / ε n ′ f and n ′ = b / c (Stephens et al. (2001)). Table 4 presents the results of the estimated properties.

Table 4. Parameters for Strain-life and Ramberg-Osgood curves. σ ′ f [MPa] ε ′ f [%] b [MPa]

′ [MPa]

n ′

σ ′

c [MPa]

K

y

1954

30.15

-0.09

-0.56

2369

0.1607

872

The limit elasticity surface for a isotropic and kinematic behaviour is written as σ = α + R + σ y n , where α is the back stress tensor computed from Chaboche’s kinematic hardening law with three back stresses, α i . The back stress tensor, α , is computed by summing n α , number of back stress tensors, such that: α = n α i = 1 α i , where, α i is computed as α i = C i 1 − exp γ i ε p , with C i and γ i denoting the back stress components and the rates of kinematic hardening, respectively. We can assume n α = 3. The other parcel is related to isotropic hardening, such that n is the outward normal vector, and R is the expansion of limit elasticity surface. R is computed from exponential’s isotropic hardening law, such that R = R ∞ 1 − exp ( − b ∞ ε p ) . R ∞ denotes the exponential coe ffi cient and is computed by the di ff erence between the cyclic yielding stress and the monotonic yielding stress, R ∞ = σ ′ y − σ 02 . b ∞ is the hardening parameter that governs the saturation rate and values between 0.05 and 50 can reach a stabilization in 10 to 1000 cycles. In this application b ∞ is assumed to be 2. With respect to determination of C i and γ i , they are computed by fitting the stable