ABAQUS user subroutines for the simulation of viscoplastic - loicz

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where S is the CAUCHY stress tensor and I denotes the second-order identity tensor. Similarly, it is supposed that the contribution of the stress and damage on the damage development can be represented by a newly introduced damage-active stress defined analogously to eq. (1) as 10 ˆ S = (I − D) −q ⋅S ⋅(I − D) −q = 3 σ σ ∑ σ ˆ ˆ i n ˆ i n i , (2) i=1 σ where q is a material parameter, σ ˆ and n ˆ i i (i = 1, 2, 3) are the eigen-values and the corresponding eigen-vectors of ˆ S . From the point of view of linear irreversible thermodynamics the evolution law for a second order symmetric damage tensor can be generally expressed as: D Ý = S : YD , (3a) < 4> where YD is the thermodynamic force conjugate to the damage tensor, and S < 4> characteristic tensor of rank four, respectively. If the fourth-order tensor S is the damage is symmetric and positive- definite, the thermodynamic restrictions will be automatically satisfied [KRAJCINOVIC, 1983; GERMAIN, NGUYEN & SUQUET, 1983; YANG, ZHOU & SWOBODA, 1999]. Motivated by the results of experimental investigations, it is assumed that only the tensile principal damage-active stresses are responsible for the damage evolution and that damage grows perpendicularly to the direction of the principal damage-active stresses. Thus, consider that damage may also develop partly isotropically, the damage evolution law is assumed taking the following particular form: < 4> D Ý = ⎛ ⎝ βII + (1 − β) I ⎞ ⎠ : ˆ S < 4> where β, B0, m are material parameters. I B 0 m < 4> = ⎛ ⎝ βII + (1 − β) I ⎞ ⎠ : 3 σ ˆ i σ σ ∑ n ˆ ˆ i n i , (3b) i=1 B 0 m denotes the fourth-order identity tensor, and 〈.〉 is the MCCAULEY bracket, which equals one for positive arguments and zero else. Creep rupture is assumed to take place when the maximum principal damage DI reaches a critical value DR. For single crystal superalloys, the initial anisotropy must be considered. The following particular form of damage law is taken for single crystals with cubic symmetry (F.C.C. and B.C.C. single crystals):
4> with R D Ý = β II + β I 1 2 + β R 3 ⎛ ⎞ ⎝ ⎠ : 3 η ˆ iσ i n ˆ σ ˆ σ ∑ i n i (4) ⎡ ηi = ⎣ ⎢ 3 i=1 B 0 m c c c c = ∑ e ei ei ei , (5) i i=1 3 ∑ j=1 σ c ( n ˆ i ⋅e j) 2n⎤ ⎦ ⎥ p , (6) c where β1, β2, β3=(1−β1−β2), B0, p, n and m are material parameters, e j (j=1,2,3) are the lattice vectors, and η i is an orientation function which satisfies the cubic symmetry. Damage can also be inactive. Let us consider a single micro-crack embedded in an elastic material with a tensile load perpendicular to the crack faces. If the load is reversed the crack will close and in a one-dimensional case the material behaves as uncracked. This phenomenon is called “damage deactivation” (not “healing”) in CDM. The damage still exists but the loading condition can render it inactive. For the representation of this mechanism the phenomenological algorithm proposed by HANSEN & SCHREYER [1995] can be used. In this method the microcrack opening/closing effect is introduced by considering the spectral decomposition of the elastic strain tensor E e and the total strain tensor E 3 E e e εe εe ε ε = ∑ εi ni ni , E = ∑ εi ni ni , (7a, b) i=1 3 i=1 e εe e e where εi and εi are the eigenvalues, ni and ni are the corresponding eigenvectors of E and E, respectively. Let the positive (tensile) spectral tensor corresponding to the elastic and to the total strain be defined as 3 H εe e εe εe = ∑ h(εi )ni ni , H ε ε ε = ∑ h(εi )n ini (8a, b) i=1 respectively, with the modified Heaviside function 3 i=1 11
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 12 and 13: 12 0 for x ≤ xm 1 h(x) = 2 1− c
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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