ABAQUS user subroutines for the simulation of viscoplastic - loicz

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12 0 for x ≤ xm 1 h(x) = 2 1− cos π(x − x ⎧ ⎫ ⎪ ⎛ ⎡ m) ⎤ ⎞ ⎪ ⎨ ⎜ ⎝ ⎣ ⎢ xp − xm ⎦ ⎥ ⎟ for xm < x < xp ⎬ ⎪ ⎠ ⎪ ⎩ 1 for x ≥ xp ⎭ where xm and xp are two material parameters. The positive spectral projection operators (fourth-order tensor) for the elastic and the total strains are defined as Pεe = H εe H εe , Pε = H ε H ε respectively. The positive projection of the elastic and the total strain tensors are then given by E e + = Pee : E e , E + < 4> = Pe (9) (10a, b) : E (11a, b) respectively. By introducing a strain-based positive projection operator < 4> T = I < 4> < 4> − ⎛ I − P ⎞ ⎝ εe⎠ : I ⎛ − P ⎞ ⎝ ε ⎠ a symmetric, so-called active damage tensor can be defined as D = a T : D (13) Thus, the effective stress tensor and the damage-active stress tensor accounting for damage deactivation are defined as ˜ S = (I − D a ) −1 2 ⋅ S ⋅(I − D a ) −1 2 , (14) S ˆ = (I − Da) −q T −q ⋅S ⋅(I − Da ) , (15) respectively. If the effective stress tensor and the damage active stress tensor defined in (14) and (15), respectively, are used instead of those defined in (1) and (2), the damage deactivation can be described. (12)
3. Unified models of BODNER-PARTOM and of CHABOCHE coupled with damage For the description of viscoplastic behaviour a lot of unified models have recently been developed. The main advantage of unified models compared to classical plasticity and creep models is the treatment of all aspects of inelastic deformation behaviour including plastic flow under monotonic and cyclic loading, creep and stress relaxation by a single inelastic strain quantity [OLSCHEWSKI et al., 1990]. The total strain rate is decomposed into an elastic and an inelastic part by E Ý = Ý E e + Ý E i As the model proposed by BODNER & PARTOM [1975] and by CHABOCHE [CHABOCHE & ROUSSELIER, 1983] are two popular models, they are chosen to be combined with the above damage model. The effective stress concept of CDM says that any constitutive equation for the damaged material can be derived in the same way as for a virgin material if the stress tensor is replaced by an adequately defined effective stress tensor. Following this concept, the BODNER-PARTOM model and the CHABOCHE model can simply be extended to include material damage by replacing the stress tensor in the respective constitutive equations by the effective stress tensor defined in the expressions (1) or (14). The resulting models are summarized as follows: (16) 13
Page 1 and 2: Technical Note GKSS/WMS/01/5 intern
Page 4 and 5: 0. Nomenclature 3 1. Introduction 7
Page 6 and 7: R ∞ Wi Z I , Z D Zij 6 saturated
Page 8 and 9: W internal variable, eq. (20b) four
Page 10 and 11: where S is the CAUCHY stress tensor
Page 14 and 15: flow rule: E Ý i = p Ý ˜ S ′
Page 16 and 17: 16 −1 −1 D1 = 0 , L1 = 0, Z i1
Page 18 and 19: parameters of Tables 2 and 3, and t
Page 20 and 21: 20 Fig. 1. Experiments and model pr
Page 22 and 23: 22 Fig. 3. Contour plots of the max
Page 24 and 25: 24 Fig. 5. Contour plots of directi
Page 26 and 27: 26 Fig. 6. FE-mesh and loading cond
Page 28 and 29: 28 Fig. 13. FE-mesh and loading con
Page 30 and 31: proove that the model performs well
Page 33 and 34: 6. References AI, S.H.; LUPINC, V.
Page 35 and 36: 7. Appendices: ABAQUS-Inputfiles 7.
Page 37 and 38: ** auflagerung ** *BOUNDARY, OP=NEW
Page 39 and 40: ** E, nu, Do, n, K1, K2, K3, m1 149
Page 41 and 42: ** ====== 2. load ein ** *STEP, INC
Page 43 and 44: 7.3 Appendix 3: Single crystal plat
Page 45 and 46: *NODE FILE, FREQUENCY=0 *EL FILE, F
Page 47 and 48: 7.4 Appendix 4: TiAl turbine blade
Page 49 and 50: *AMPLITUDE, NAME=CYCLEONE, DEFINITI

ABAQUS user subroutines for the simulation of viscoplastic - loicz

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