PSI - Issue 8

Matteo Loffredo / Procedia Structural Integrity 8 (2018) 265–275

271

M. Lo ff redo / Structural Integrity Procedia 00 (2017) 000–000

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2. the loading curve, including the tensile yield point σ (L) duced. 3. σ y shall depend on p , in order to account for the reduction of σ (U)

y and the hardening behavior, should be properly repro

y due to tensile plastic strain.

4. σ y , after first yielding, should be frozen. 5. the hardening modulus following reverse yielding (Fig.3) should depend on p . 6. after reverse yielding all stress strain curves should concur at the same stress level. 7. in case of re-loading the stress strain curve should fall on a closed loop.

3.2. Loading phase

Requests 2 and 3 of previous section can be satisfied by combining with non linear kinematic hardening non linear isotropic softening in the loading phase. This combination can take into account, at the same time, the loading-phase hardening law and the evolution of σ y (See Fig.3). Eq.2 shows the kinematic hardening evolution law of the backstress vector α and its specialization to the uniaxial case. p is the equivalent plastic strain while C (L) 1 and C (L) 2 are two constitutive constants to be identified. The isotropic softening law has been likewise expressed as shown in Eq.3 where H (L) 1 and H (L) 2 are two typical constants to be identified.

C (L) 1 C (L) 2

(1 − exp( − C (L)

(L) 1 n d p − C

(L) 2 α d p α ( p ) =

2 p )

(2)

d α = C

H (L) 1 H (L) 2

(L) y +

(1 − exp( − H (L)

σ y ( p ) = σ

2 p )

(3)

The hardening and softening laws expressed above, if needed, can be made more accurate by increasing the number of material parameters to be identified. To this purpose, for the hardening law (Eq.2) it is su ffi cient to superimpose several backstresses vectors, and the softening law (Eq.3) can be straightforward modified. This could be necessary for materials where a purely exponential profile is not su ffi cient (depending on the material or on the required accuracy, a double exponential or an exponential + linear trend could be more advisable).

3.3. Unloading phase

The loading phase ends when p has been reached and the load is reversed. At this point, as anticipated in Sec.3, two abrupt changes shall take place. For this purpose a counter parameter Ω has been defined in Eq.4, where n ∗ is the last plastic iteration, preceding iteration n .

Ω n = Ω n − 1 + 1 IF [[sign( n n · n n ∗ ) ≤ − 1]

(4)

Firstly, when load is reversed (then when Ω ≥ 1), the yield locus is frozen (Eq.5) and an updated kinematic hardening law is introduced (Eq.6). The kinematic hardening law that has been modified by introducing a memory backstress vector β which is updated at each load reversal (Eq.7). Eq.6 shows, in addition, the specialization of the kinematic hardening law for the uniaxial case at first unloading. The introduction of β ensures that stress-strain loops are closed (as requested in Sec.2.3) and that their trend remains, regardless of the cycle, exponential. C (U) 1 and C (U) 2 are the