김경종
454 lines
30 KiB
Markdown
454 lines
30 KiB
Markdown
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where $\dot { \bar { \varepsilon } } ^ { p l }$ is the equivalent plastic strain rate and C is the kinematic hardening modulus. In this model the equivalent stress defining the size of the yield surface, $\sigma ^ { 0 }$ , remains constant, $\sigma ^ { 0 } = \sigma | _ { 0 }$ , where $\sigma | _ { 0 }$ is the equivalent stress defining the size of the yield surface at zero plastic strain.
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# Nonlinear isotropic/kinematic hardening model
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The evolution law of this model consists of two components: a nonlinear kinematic hardening component, which describes the translation of the yield surface in stress space through the backstress, ; and an isotropic hardening component, which describes the change of the equivalent stress defining the size of the yield surface, $\sigma ^ { 0 }$ , as a function of plastic deformation.
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The kinematic hardening component is defined to be an additive combination of a purely kinematic term (linear Ziegler hardening law) and a relaxation term (the recall term), which introduces the nonlinearity. In addition, several kinematic hardening components (backstresses) can be superposed, which may considerably improve results in some cases. When temperature and field variable dependencies are omitted, the hardening laws for each backstress are
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$$
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\dot {\pmb {\alpha}} _ {k} = C _ {k} \frac {1}{\sigma^ {0}} (\pmb {\sigma} - \pmb {\alpha}) \dot {\bar {\varepsilon}} ^ {p l} - \gamma_ {k} \pmb {\alpha} _ {k} \dot {\bar {\varepsilon}} ^ {p l},
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$$
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and the overall backstress is computed from the relation
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$$
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\alpha = \sum_ {k = 1} ^ {N} \alpha_ {k},
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$$
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where is the number of backstresses, and $C _ { k }$ and $\gamma _ { k }$ are material parameters that must be calibrated from cyclic test data. $C _ { k }$ are the initial kinematic hardening moduli, and $\gamma _ { k }$ determine the rate at which the kinematic hardening moduli decrease with increasing plastic deformation. The kinematic hardening law can be separated into a deviatoric part and a hydrostatic part; only the deviatoric part has an effect on the material behavior. When $C _ { k }$ and $\gamma _ { k }$ are zero, the model reduces to an isotropic hardening model. When all $\gamma _ { k }$ equal zero, the linear Ziegler hardening law is recovered. Calibration of the material parameters is discussed in “Usage and calibration of the kinematic hardening models,” below. Figure 23.2.2–1 shows an example of nonlinear kinematic hardening with three backstresses. Each of the backstresses covers a different range of strains, and the linear hardening law is retained for large strains.
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The isotropic hardening behavior of the model defines the evolution of the yield surface size, $\sigma ^ { 0 }$ , as a function of the equivalent plastic strain, $\bar { \varepsilon } ^ { p l }$ . This evolution can be introduced by specifying $\sigma ^ { 0 }$ directly as a function of $\bar { \varepsilon } ^ { p l }$ in tabular form, by specifying $\sigma ^ { 0 }$ in user subroutine UHARD (in Abaqus/Standard only), or by using the simple exponential law
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$$
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\sigma^ {0} = \sigma | _ {0} + Q _ {\infty} (1 - e ^ {- b \bar {\varepsilon} ^ {p l}}),
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$$
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where $\sigma | _ { 0 }$ is the yield stress at zero plastic strain and $Q _ { \infty }$ and b are material parameters. $Q _ { \infty }$ is the maximum change in the size of the yield surface, and b defines the rate at which the size of the yield
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<details>
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<summary>line</summary>
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| equivalent plastic strain | α = α₁ + α₂ + α₃ | α₁ = 4.0×10⁴ (1.0 - e^(-20ε̅^pl)) | α₂ = 2.0×10⁴ (1.0 - e^(-500ε̅^pl)) | α₃ = 4.0×10⁴ ε̅^pl |
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| -------------------------- | ---------------- | ---------------------------------- | ---------------------------------- | ------------------ |
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| 0.00 | 0.0 | 0.0 | 0.0 | 0.0 |
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| 0.05 | ~45 | ~25 | ~10 | ~2 |
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| 0.10 | ~55 | ~35 | ~15 | ~4 |
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| 0.15 | ~60 | ~38 | ~18 | ~6 |
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| 0.20 | ~65 | ~40 | ~20 | ~8 |
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| 0.25 | ~68 | ~41 | ~22 | ~10 |
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| 0.30 | ~70 | ~42 | ~25 | ~12 |
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</details>
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Figure 23.2.2–1 Kinematic hardening model with three backstresses.
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surface changes as plastic straining develops. When the equivalent stress defining the size of the yield surface remains constant $( { \boldsymbol { \sigma } } ^ { 0 } = { \boldsymbol { \sigma } } | _ { 0 } )$ , the model reduces to a nonlinear kinematic hardening model.
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The evolution of the kinematic and the isotropic hardening components is illustrated in Figure 23.2.2–2 for unidirectional loading and in Figure 23.2.2–3 for multiaxial loading. The evolution law for the kinematic hardening component implies that the backstress is contained within a cylinder of radius $\sqrt { 2 / 3 } \alpha ^ { s } = \sqrt { 2 / 3 } \bar { \sum _ { k } ^ { N } } { C _ { k } } \bar { / } \gamma _ { k }$ , where $\alpha ^ { s }$ is the magnitude of at saturation (large plastic strains). It also implies that any stress point must lie within a cylinder of radius $\sqrt { 2 / 3 } \sigma _ { m a x }$ (using the notation of Figure 23.2.2–2) since the yield surface remains bounded. At large plastic strain any stress point is contained within a cylinder of radius $\sqrt { 2 / 3 } \left( \alpha ^ { s } + \sigma ^ { s } \right)$ , where $\sigma ^ { s }$ is the equivalent stress defining the size of the yield surface at large plastic strain. If tabular data are provided for the isotropic component, $\sigma ^ { s }$ is the last value given to define the size of the yield surface. If user subroutine UHARD is used, this value will depend on your implementation; otherwise, $\sigma ^ { s } = \sigma | _ { 0 } + Q _ { \infty }$ .
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# Predicted material behavior
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In the kinematic hardening models the center of the yield surface moves in stress space due to the kinematic hardening component. In addition, when the nonlinear isotropic/kinematic hardening model is used, the yield surface range may expand or contract due to the isotropic component. These features allow modeling of inelastic deformation in metals that are subjected to cycles of load or temperature, resulting in significant inelastic deformation and, possibly, low-cycle fatigue failure. These models account for the following phenomena:
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<details>
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<summary>line</summary>
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| ε^pl | σ_max | σ_0 | σ_10 | α |
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|------|-------|-----|------|---|
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| 0 | 0 | 0 | 0 | 0 |
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| >0.5 | >0.5 | >0.2| >0.1 | >0.1 |
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| >1.0 | >1.0 | >0.4| >0.2 | >0.2 |
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| >1.5 | >1.5 | >0.6| >0.3 | >0.3 |
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| >2.0 | >2.0 | >0.8| >0.4 | >0.4 |
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| >2.5 | >2.5 | >1.0| >0.5 | >0.5 |
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| >3.0 | >3.0 | >1.2| >0.6 | >0.6 |
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| >3.5 | >3.5 | >1.4| >0.7 | >0.7 |
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| >4.0 | >4.0 | >1.6| >0.8 | >0.8 |
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| >4.5 | >4.5 | >1.8| >0.9 | >0.9 |
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| >5.0 | >5.0 | >2.0| >1.0 | >1.0 |
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</details>
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Figure 23.2.2–2 One-dimensional representation of the hardening in the nonlinear isotropic/kinematic model.
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<details>
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<summary>text_image</summary>
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limit surface
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√(2/3)ΣN(k=1)Ck/γk
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0
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αdev
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√(2/3)σ0
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S3
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S
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S1
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yield surface
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∂F/∂σ
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S1
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limiting
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location
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of α
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</details>
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Figure 23.2.2–3 Three-dimensional representation of the hardening in the nonlinear isotropic/kinematic model.
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• Bauschinger effect: This effect is characterized by a reduced yield stress upon load reversal after plastic deformation has occurred during the initial loading. This phenomenon decreases with continued cycling. The linear kinematic hardening component takes this effect into consideration, but a nonlinear component improves the shape of the cycles. Further improvement of the shape of the cycle can be obtained by using a nonlinear model with multiple backstresses.
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• Cyclic hardening with plastic shakedown: This phenomenon is characteristic of symmetric stress- or strain-controlled experiments. Soft or annealed metals tend to harden toward a stable limit, and initially hardened metals tend to soften. Figure 23.2.2–4 illustrates the behavior of a metal that hardens under prescribed symmetric strain cycles.
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<details>
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<summary>line</summary>
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| time | σ |
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|------|-------|
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| 0 | 0 |
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| 1 | 1 |
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| 2 | 0 |
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| 3 | 1 |
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| 4 | 0 |
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| 5 | 1 |
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| 6 | 0 |
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| 7 | 1 |
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| 8 | 0 |
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| 9 | 1 |
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| 10 | 0 |
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| 11 | 1 |
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| 12 | 0 |
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| 13 | 1 |
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| 14 | 0 |
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| 15 | 1 |
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| 16 | 0 |
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| 17 | 1 |
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| 18 | 0 |
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| 19 | 1 |
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| 20 | 0 |
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| 21 | 1 |
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| 22 | 0 |
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| 23 | 1 |
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| 24 | 0 |
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| 25 | 1 |
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| 26 | 0 |
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| 27 | 1 |
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| 28 | 0 |
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| 29 | 1 |
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| 30 | 0 |
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| 31 | 1 |
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| 32 | 0 |
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| 33 | 1 |
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| 34 | 0 |
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| 35 | 1 |
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| 36 | 0 |
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| 37 | 1 |
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| 38 | 0 |
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| 39 | 1 |
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| 40 | 0 |
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| 41 | 1 |
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| 42 | 0 |
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| 43 | 1 |
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| 44 | 0 |
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| 45 | 1 |
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| 46 | 0 |
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| 47 | 1 |
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| 48 | 0 |
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| 49 | 1 |
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| 50 | 0 |
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| 51 | 1 |
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| 52 | 0 |
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| 53 | 1 |
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| 54 | 0 |
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| 55 | 1 |
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| 56 | 0 |
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| 57 | 1 |
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| 58 | 0 |
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| 59 | 1 |
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| 60 | 0 |
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| 61 | 1 |
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| 62 | 0 |
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| 63 | 1 |
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| 64 | 0 |
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| 65 | 1 |
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| 66 | 0 |
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| 67 | 1 |
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| 68 | 0 |
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| 69 | 1 |
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| 70 | 0 |
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| 71 | 1 |
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| 72 | 0 |
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| 73 | 1 |
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| 74 | 0 |
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| 75 | 1 |
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| 76 | 0 |
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| 77 | 1 |
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| 78 | 0 |
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| 79 | 1 |
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| 80 | 0 |
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| 81 | 1 |
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| 82 | 0 |
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| 83 | 1 |
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| 84 | 0 |
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| 85 | 1 |
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| 86 | 0 |
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| 87 | 1 |
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| 88 | 0 |
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| 89 | 1 |
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| 90 | 0 |
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| 91 | 1 |
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| 92 | 0 |
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| 93 | 1 |
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| 94 | 0 |
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| 95 | 1 |
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| 96 | 0 |
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| 97 | 1 |
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| 98 | 0 |
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| 99 | 1 |
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| 100 | 0 |
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</details>
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<details>
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<summary>line</summary>
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| ε | σ (Curve 1) | σ (Curve 2) | σ (Curve 3) |
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|------|-------------|-------------|-------------|
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| 0 | 0 | 0 | 0 |
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| 1 | 1 | 1 | 1 |
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| 2 | 2 | 2 | 2 |
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| 3 | 3 | 3 | 3 |
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</details>
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Figure 23.2.2–4 Plastic shakedown.
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The kinematic hardening component of the models used alone predicts plastic shakedown after one stress cycle. The combination of the isotropic component together with the nonlinear kinematic component predicts shakedown after several cycles.
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• Ratchetting: Unsymmetric cycles of stress between prescribed limits will cause progressive “creep” or “ratchetting” in the direction of the mean stress (Figure 23.2.2–5). Typically, transient ratchetting is followed by stabilization (zero ratchet strain) for low mean stresses, while a constant increase in the accumulated ratchet strain is observed at high mean stresses. The nonlinear kinematic hardening component, used without the isotropic hardening component, predicts constant ratchet strain. The prediction of ratchetting is improved by adding isotropic hardening, in which case the ratchet strain may decrease until it becomes constant. However, in general the nonlinear hardening model with a single backstress predicts a too significant ratchetting effect. A considerable improvement in modeling ratchetting can be achieved by superposing several kinematic hardening models (backstresses) and choosing one of the models to be linear or nearly linear $( \gamma _ { k } \ll C _ { k } )$ , which results in a less pronounced ratchetting effect.
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<details>
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<summary>line</summary>
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| ε | σ (Curve 1) | σ (Curve 2) | σ (Curve 3) | σ (Curve 4) | σ (Curve 5) |
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|------|-------------|-------------|-------------|-------------|-------------|
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| 0 | 0 | 0 | 0 | 0 | 0 |
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| 1 | ~0.5 | ~0.3 | ~0.1 | ~0.05 | ~0.02 |
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| 2 | ~1.0 | ~0.7 | ~0.4 | ~0.2 | ~0.1 |
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| 3 | ~1.5 | ~1.1 | ~0.7 | ~0.4 | ~0.2 |
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| 4 | ~2.0 | ~1.5 | ~1.0 | ~0.6 | ~0.3 |
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| 5 | ~2.5 | ~1.9 | ~1.3 | ~0.8 | ~0.4 |
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</details>
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Figure 23.2.2–5 Ratchetting.
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• Relaxation of the mean stress: This phenomenon is characteristic of an unsymmetric strain experiment, as shown in Figure 23.2.2–6.
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<details>
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<summary>line</summary>
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| ε | σ (Curve 1) | σ (Curve 2) | σ (Curve 3) | σ (Curve 4) |
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|------|-------------|-------------|-------------|-------------|
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| Low | Low | Low | Low | Low |
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| Mid | Mid | Mid | Mid | Mid |
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| High | High | High | High | High |
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</details>
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Figure 23.2.2–6 Relaxation of the mean stress.
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As the number of cycles increases, the mean stress tends to zero. The nonlinear kinematic hardening component of the nonlinear isotropic/kinematic hardening model accounts for this behavior.
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# Limitations
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The linear kinematic model is a simple model that gives only a first approximation of the behavior of metals subjected to cyclic loading, as explained above. The nonlinear isotropic/kinematic hardening model can provide more accurate results in many cases involving cyclic loading, but it still has the following limitations:
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• The isotropic hardening is the same at all strain ranges. Physical observations, however, indicate that the amount of isotropic hardening depends on the magnitude of the strain range. Furthermore, if the specimen is cycled at two different strain ranges, one followed by the other, the deformation in the first cycle affects the isotropic hardening in the second cycle. Thus, the model is only a coarse approximation of actual cyclic behavior. It should be calibrated to the expected size of the strain cycles of importance in the application.
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• The same cyclic hardening behavior is predicted for proportional and nonproportional load cycles. Physical observations indicate that the cyclic hardening behavior of materials subjected to nonproportional loading may be very different from uniaxial behavior at a similar strain amplitude.
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The example problems “Simple proportional and nonproportional cyclic tests,” Section 3.2.8 of the Abaqus Benchmarks Guide, “Notched beam under cyclic loading,” Section 1.1.7 of the Abaqus Example Problems Guide and “Uniaxial ratchetting under tension and compression,” Section 1.1.8 of the Abaqus Example Problems Guide, illustrate the phenomena of cyclic hardening with plastic shakedown, ratchetting, and relaxation of the mean stress for the nonlinear isotropic/kinematic hardening model, as well as its limitations.
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# Usage and calibration of the kinematic hardening models
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The linear kinematic model approximates the hardening behavior with a constant rate of hardening. This hardening rate should be matched to the average hardening rate measured in stabilized cycles over a strain range corresponding to that expected in the application. A stabilized cycle is obtained by cycling over a fixed strain range until a steady-state condition is reached; that is, until the stress-strain curve no longer changes shape from one cycle to the next. The more general nonlinear model will give better predictions but requires more detailed calibration.
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# Linear kinematic hardening model
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The test data obtained from a half cycle of a unidirectional tension or compression experiment must be linearized, since this simple model can predict only linear hardening. The data are usually based on measurements of the stabilized behavior in strain cycles covering a strain range corresponding to the strain range that is anticipated to occur in the application. Abaqus expects you to provide only two data pairs to define this linear behavior: the yield stress, $\sigma | _ { 0 }$ , at zero plastic strain and a yield stress, $\sigma ,$ a t a finite plastic strain value, $\varepsilon ^ { p l }$ . The linear kinematic hardening modulus, $C ,$ is determined from the relation
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$$
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C = \frac {\sigma - \sigma | _ {0}}{\varepsilon^ {p l}}.
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$$
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You can provide several sets of two data pairs as a function of temperature to define the variation of the linear kinematic hardening modulus with respect to temperature. If the Hill yield surface is desired for this model, you must specify a set of yield ratios, $R _ { i j }$ , independently (see “Anisotropic yield/creep,” Section 23.2.6, for information on how to specify the yield ratios).
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This model gives physically reasonable results for only relatively small strains (less than 5%).
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Input File Usage: \*PLASTIC, HARDENING=KINEMATIC
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Plastic: Hardening: Kinematic
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# Nonlinear isotropic/kinematic hardening model
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The evolution of the equivalent stress defining the size of the yield surface, $\sigma ^ { 0 }$ , as a function of the equivalent plastic strain, $\bar { \varepsilon } ^ { p l }$ , defines the isotropic hardening component of the model. You can define this isotropic hardening component through an exponential law or directly in tabular form. It need not be defined if the yield surface remains fixed throughout the loading. In Abaqus/Explicit if the Hill yield surface is desired for this model, you must specify a set of yield ratios, $R _ { i j }$ , independently (see “Anisotropic yield/creep,” Section 23.2.6, for information on how to specify the yield ratios). The Hill yield surface cannot be used with this model in Abaqus/Standard.
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The material parameters $C _ { k }$ and $\gamma _ { k }$ determine the kinematic hardening component of the model. Abaqus offers three different ways of providing data for the kinematic hardening component of the model: the parameters $C _ { k }$ and $\gamma _ { k }$ can be specified directly, half-cycle test data can be given, or test data obtained from a stabilized cycle can be given. The experiments required to calibrate the model are described below.
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Defining the isotropic hardening component by the exponential law
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Specify the material parameters of the exponential law $\sigma | _ { 0 } , \ : Q _ { \infty }$ , and b directly if they are already calibrated from test data. These parameters can be specified as functions of temperature and/or field variables.
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Input File Usage: \*CYCLIC HARDENING, PARAMETERS
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Plastic: Suboptions→Cyclic Hardening: toggle on Use parameters.
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Defining the isotropic hardening component by tabular data
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Isotropic hardening can be introduced by specifying the equivalent stress defining the size of the yield surface, $\sigma ^ { 0 }$ , as a tabular function of the equivalent plastic strain, $\bar { \varepsilon } ^ { p l }$ . The simplest way to obtain these data is to conduct a symmetric strain-controlled cyclic experiment with strain range $\Delta \varepsilon .$ . Since the material’s elastic modulus is large compared to its hardening modulus, this experiment can be interpreted approximately as repeated cycles over the same plastic strain range $\Delta \varepsilon ^ { p l } \approx \Delta \varepsilon - 2 \sigma _ { 1 } ^ { t } / E$ (using the notation of Figure 23.2.2–7, where E is the Young’s modulus of the material). The equivalent stress defining the size of the yield surface is $\sigma | _ { 0 }$ at zero equivalent plastic strain; for the peak tensile stress points it is obtained by isolating the kinematic component from the yield stress (see Figure 23.2.2–2) as
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$$
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\sigma_ {i} ^ {0} = \sigma_ {i} ^ {t} - \alpha_ {i}
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$$
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for each cycle $i ,$ where $\alpha _ { i } = ( \sigma _ { i } ^ { t } + \sigma _ { i } ^ { c } ) / 2$ . Since the model predicts approximately the same backstress value in each cycle at a particular strain level, $\alpha _ { i } ~ \approx ~ ( \sigma _ { 1 } ^ { t } + \sigma _ { 1 } ^ { c } ) / 2$ . The equivalent plastic strain corresponding to $\boldsymbol { \sigma } _ { i } ^ { 0 }$ is
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<details>
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<summary>line</summary>
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| ε^pl | σ | Label |
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|------|-------|-------|
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| ε^pl | σ_t^0 | σ_t^0 |
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| ε^pl | σ_t^1 | σ_t^1 |
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| ε^pl | σ_t^2 | σ_t^2 |
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| ε^pl | σ_t^c | σ_t^c |
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| ε^pl | σ_c^0 | σ_c^0 |
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| ε^pl | σ_c^1 | σ_c^1 |
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| ε^pl | σ_c^2 | σ_c^2 |
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| ε^pl | σ_c^c | σ_c^c |
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| ε^pl | σ_n^0 | σ_n^0 |
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| ε^pl | σ_n^1 | σ_n^1 |
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| ε^pl | σ_n^2 | σ_n^2 |
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| ε^pl | σ_n^c | σ_n^c |
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| Δε^pl = ε^pl_t - ε^pl_c | σ_n^t | σ_n^t |
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| Δε^pl = ε^pl_t - ε^pl_c | σ_t^0 | σ_t^0 |
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| Δε^pl = ε^pl_t - ε^pl_c | σ_t^1 | σ_t^1 |
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| Δε^pl = ε^pl_t - ε^pl_c | σ_t^2 | σ_t^2 |
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| Δε^pl = ε^pl_t - ε^pl_c | σ_t^c | σ_t^c |
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| Δε^pl = ε^pl_t - ε^pl_c | σ_c^0 | σ_c^0 |
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| Δε^pl = ε^pl_t - ε^pl_c | σ_c^1 | σ_c^1 |
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| Δε^pl = ε^pl_t - ε^pl_c | σ_c^2 | σ_c^2 |
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| Δε^pl = ε^pl_t - ε^pl_c | σ_c^c | σ_c^c |
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||
</details>
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||
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Figure 23.2.2–7 Symmetric strain cycle experiment.
|
||
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||
$$
|
||
\bar {\varepsilon} _ {i} ^ {p l} = \frac {1}{2} (4 i - 3) \Delta \varepsilon^ {p l}.
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||
$$
|
||
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Data pairs $( \sigma _ { i } ^ { 0 } , \bar { \varepsilon } _ { i } ^ { p l } )$ , including the value $\sigma | _ { 0 }$ at zero equivalent plastic strain, are specified in tabulated form. The tabulated values defining the size of the yield surface should be provided for the entire equivalent plastic strain range to which the material may be subjected. The data can be provided as functions of temperature and/or field variables.
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||
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||
To obtain accurate cyclic hardening data, such as would be needed for low-cycle fatigue calculations, the calibration experiment should be performed at a strain range, $\Delta \varepsilon$ , that corresponds to the strain range anticipated in the analysis because the material model does not predict different isotropic hardening behavior at different strain ranges. This limitation also implies that, even though a component is made from the same material, it may have to be divided into several regions with different hardening properties corresponding to different anticipated strain ranges. Field variables and field variable dependence of these properties can also be used for this purpose.
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||
|
||
Abaqus allows the specification of strain rate effects in the isotropic component of the nonlinear isotropic/kinematic hardening model. The rate-dependent isotropic hardening data can be defined by specifying the equivalent stress defining the size of the yield surface, $\sigma ^ { 0 }$ , as a tabular function of the equivalent plastic strain, $\bar { \varepsilon } ^ { p l }$ , at different values of the equivalent plastic strain rate, $\dot { \bar { \varepsilon } } ^ { p l }$ .
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Input File Usage: Use the following option to define isotropic hardening with tabular data:
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||
|
||
\*CYCLIC HARDENING
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||
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||
Use the following option to define rate-dependent isotropic hardening with tabular data:
|
||
|
||
\*CYCLIC HARDENING, $\mathrm { R A T E } { = } \dot { \bar { \varepsilon } } ^ { p l }$
|
||
|
||
Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Plastic:
|
||
|
||
Hardening: Combined: Suboptions→Cyclic Hardening
|
||
|
||
<!-- source-page: 259 -->
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||
|
||
Defining the isotropic hardening component in a user subroutine in Abaqus/Standard
|
||
|
||
Specify $\sigma ^ { 0 }$ directly in user subroutine UHARD. $\sigma ^ { 0 }$ may be dependent on equivalent plastic strain and temperature. This method cannot be used if the kinematic hardening component is specified by using half-cycle test data.
|
||
|
||
Input File Usage: \*CYCLIC HARDENING, USER
|
||
|
||
Abaqus/CAE Usage: You cannot define the isotropic hardening component in user subroutine UHARD in Abaqus/CAE.
|
||
|
||
Defining the kinematic hardening component by specifying the material parameters directly
|
||
|
||
The parameters $C _ { k }$ and $\gamma _ { k }$ can be specified directly as a function of temperature and/or field variables if they are already calibrated from test data. When $\gamma _ { k }$ depend on temperature and/or field variables, the response of the model under thermomechanical loading will generally depend on the history of temperature and/or field variables experienced at a material point. This dependency on temperaturehistory is small and fades away with increasing plastic deformation. However, if this effect is not desired, constant values for $\gamma _ { k }$ should be specified to make the material response completely independent of the history of temperature and field variables. The algorithm currently used to integrate the nonlinear isotropic/kinematic hardening model provides accurate solutions if the values of $\gamma _ { k }$ change moderately in an increment due to temperature and/or field variable dependence; however, this algorithm may not yield a solution with sufficient accuracy if the values of $\gamma _ { k }$ change abruptly in an increment.
|
||
|
||
Input File Usage: \*PLASTIC, HARDENING=COMBINED, DATA TYPE=PARAMETERS, NUMBER BACKSTRESSES=n
|
||
|
||
Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Plastic: Hardening: Combined, Data type: Parameters, Number of backstresses: n
|
||
|
||
Defining the kinematic hardening component by specifying half-cycle test data
|
||
|
||
If limited test data are available, $C _ { k }$ and $\gamma _ { k }$ can be based on the stress-strain data obtained from the first half cycle of a unidirectional tension or compression experiment. An example of such test data is shown in Figure 23.2.2–8. This approach is usually adequate when the simulation will involve only a few cycles of loading.
|
||
|
||
For each data point $( \sigma _ { i } , \varepsilon _ { i } ^ { p l } )$ a value of $\alpha _ { i } \left( \alpha _ { i } \right.$ is the overall backstress obtained by summing all the backstresses at this data point) is obtained from the test data as
|
||
|
||
$$
|
||
\alpha_ {i} = \sigma_ {i} - \sigma_ {i} ^ {0},
|
||
$$
|
||
|
||
where $\boldsymbol { \sigma } _ { i } ^ { 0 }$ is the user-defined size of the yield surface at the corresponding plastic strain for the isotropic hardening component or the initial yield stress if the isotropic hardening component is not defined.
|
||
|
||
Integration of the backstress evolution laws over a half cycle yields the expressions
|
||
|
||
<!-- source-page: 260 -->
|
||
|
||
|
||
|
||
<details>
|
||
<summary>line</summary>
|
||
|
||
| Point | ε^pl | σ |
|
||
|-------|------|----|
|
||
| σ₁, ε₁^pl | | |
|
||
| σ₂, ε₂^pl | | |
|
||
| σ₃, ε₃^pl | | |
|
||
</details>
|
||
|
||
Figure 23.2.2–8 Half cycle of stress-strain data.
|
||
|
||
$$
|
||
\alpha_ {k} = \frac {C _ {k}}{\gamma_ {k}} (1 - e ^ {- \gamma_ {k} \varepsilon^ {p l}}),
|
||
$$
|
||
|
||
which are used for calibrating $C _ { k }$ and $\gamma _ { k }$ .
|
||
|
||
When test data are given as functions of temperature and/or field variables, Abaqus determines several sets of material parameters $( C _ { 1 } , \gamma _ { 1 } , . . . , C _ { N } , \gamma _ { N } )$ , each corresponding to a given combination of temperature and/or field variables. Generally, this results in temperature-history (and/or field variablehistory) dependent material behavior because the values of $\gamma _ { k }$ vary with changes in temperature and/or field variables. This dependency on temperature-history is small and fades away with increasing plastic deformation. However, you can make the response of the material completely independent of the history of temperature and field variables by using constant values for the parameters $\gamma _ { k }$ . This can be achieved by running a data check analysis first; an appropriate constant values of $\gamma _ { k }$ can be determined from the information provided in the data file during the data check. The values for the parameters $C _ { k }$ and the constant parameters $\gamma _ { k }$ can then be entered directly as described above.
|
||
|
||
If the model with multiple backstresses is used, Abaqus obtains hardening parameters for different values of initial guesses and chooses the ones that give the best correlation with the experimental data provided. However, you should carefully examine the obtained parameters. In some cases it might be advantageous to obtain hardening parameters for different numbers of backstresses before choosing the set of parameters.
|
||
|
||
Input File Usage: \*PLASTIC, HARDENING=COMBINED, DATA TYPE=HALF CYCLE, NUMBER BACKSTRESSES=n
|
||
|
||
Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Plastic: Hardening: Combined, Data type: Half Cycle, Number of backstresses: n
|
||
|
||
Defining the kinematic hardening component by specifying test data from a stabilized cycle
|
||
|
||
Stress-strain data can be obtained from the stabilized cycle of a specimen that is subjected to symmetric strain cycles. A stabilized cycle is obtained by cycling the specimen over a fixed strain range $\Delta \varepsilon$ until a
|