김경종
283 lines
21 KiB
Markdown
283 lines
21 KiB
Markdown
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a minimum requirement, this method is used to define a target equivalent creep strain rate; however, if required, it can also be used to define the target creep strain rate as a function of equivalent creep strain (measured as log strain), temperature, and other predefined field variables. The creep strain dependency curve at each temperature must always start at zero equivalent creep strain.
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A solution-dependent amplitude is used to define the minimum and maximum limits of the loading (see “Defining a solution-dependent amplitude for superplastic forming analysis” in “Amplitude curves,” Section 34.1.2). Any number or combination of loads can be used. The current value of $r _ { \mathrm { m a x } }$ is available for output as discussed below.
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<table><tr><td>Input File Usage:</td><td>Use all of the following options:*AMPLITUDE, NAME=name, DEFINITION=SOLUTION DEPENDENT*CLOAD, *DLOAD, *DSLOAD, and/or *BOUNDARY with AMPLITUDE=name*CREEP STRAIN RATE CONTROL, AMPLITUDE=name, ELSET=elsetThe *AMPLITUDE option must appear in the model definition portion of an input file, while the loading options (*CLOAD, *DLOAD, *DSLOAD, and *BOUNDARY) and the *CREEP STRAIN RATE CONTROL option should appear in each relevant step definition.</td></tr></table>
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Abaqus/CAE Usage: Creep strain rate control is not supported in Abaqus/CAE.
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# Elements
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Rate-dependent plasticity (creep and swelling behavior) can be used with any continuum, shell, membrane, gasket, and beam element in Abaqus/Standard that has displacement degrees of freedom. Creep (but not swelling) can also be defined in the thickness direction of any gasket element in conjunction with the gasket behavior definition.
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# Output
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In addition to the standard output identifiers available in Abaqus/Standard (“Abaqus/Standard output variable identifiers,” Section 4.2.1), the following variables relate directly to creep and swelling models:
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<table><tr><td>CEEQ</td><td>Equivalent creep strain, $\int_{0}^{t} \sqrt{\frac{2}{3} \dot{\varepsilon}^{cr} : \dot{\varepsilon}^{cr}} \, dt$ .</td></tr><tr><td>CESW</td><td>Magnitude of swelling strain.</td></tr></table>
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The following output, which is relevant only for an analysis with creep strain rate loading control as discussed above, is printed at the beginning of an increment and is written automatically to the results file and output database file when any output to these files is requested:
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RATIO Maximum value of the ratio of the equivalent creep strain rate to the target creep strain rate, . $r _ { m a x }$
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AMPCU Current value of the solution-dependent amplitude.
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# 23.2.5 ANNEALING OR MELTING
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Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE
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# References
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• “Material library: overview,” Section 21.1.1
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• \*ANNEAL TEMPERATURE
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• “Specifying the annealing temperature of an elastic-plastic material” in “Defining plasticity,” Section 12.9.2 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
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# Overview
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# This capability:
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• is intended to model the effects of melting and resolidification in metals subjected to high-temperature processes or the effects of annealing at a material point when its temperature rises above a certain level;
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• is available for only the Mises, Johnson-Cook, and Hill plasticity models;
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• is intended to be used in conjunction with appropriate temperature-dependent material properties (in particular, the model assumes perfectly plastic behavior at or above the annealing or melting temperature); and
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• can be modeled simply by defining an annealing or melting temperature.
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# Effects of annealing or melting
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When the temperature of a material point exceeds a user-specified value called the annealing temperature, Abaqus assumes that the material point loses its hardening memory. The effect of prior work hardening is removed by setting the equivalent plastic strain to zero. For kinematic and combined hardening models the backstress tensor is also reset to zero. If the temperature of the material point falls below the annealing temperature at a subsequent point in time, the material point can work harden again. Depending on the temperature history a material point may lose and accumulate memory several times, which in the context of modeling melting would correspond to repeated melting and resolidification. Any accumulated material damage is not healed when the annealing temperature is reached. Damage will continue to accumulate after annealing according to any damage model in effect (see “Damage and failure for ductile metals: overview,” Section 24.2.1).
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In Abaqus/Explicit an annealing step can be defined to simulate the annealing process for the entire model, independent of temperature; see “Annealing procedure,” Section 6.12.1, for details.
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# Material properties
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The annealing temperature is a material property that can optionally be defined as a function of field variables. This material property must be used in conjunction with an appropriate definition of material
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properties as functions of temperature for the Mises plasticity model. In particular, the hardening behavior must be defined as a function of temperature and zero hardening must be specified at or above the annealing temperature. In general, hardening receives contributions from two sources. The first source of hardening can be classified broadly as static, and its effect is measured by the rate of change of the yield stress with respect to the plastic strain at a fixed strain rate. The second source of hardening can be classified broadly as rate dependent, and its effect is measured by the rate of change of the yield stress with respect to the strain rate at a fixed plastic strain.
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For the Mises plasticity model, if the material data that describe hardening (both static and ratedependent contributions) are completely specified through tabular input of yield stress versus plastic strain at different values of the strain rate (see “Rate-dependent yield,” Section 23.2.3), the (temperaturedependent) static part of the hardening at each strain rate is specified by defining several yield stress versus plastic strain curves (each at a different temperature). For metals the yield stress at a fixed strain rate typically decreases with increasing temperature. Abaqus expects the hardening at each strain rate to vanish at or above the annealing temperature and issues an error message if you specify otherwise in the material definition. Zero (static) hardening can be specified by simply specifying a single data point (at zero plastic strain) in the yield stress versus plastic strain curve at or above the annealing temperature. In addition, you must also ensure that at or above the annealing temperature, the yield stress does not vary with the strain rate. This can be accomplished by specifying the same value of yield stress at all values of strain rate in the single data point approach discussed above.
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Alternatively, the static part of the hardening can be defined at zero strain rate, and the rate-dependent part can be defined utilizing the overstress power law (see “Rate-dependent yield,” Section 23.2.3). In that case, zero static hardening at or above the annealing temperature can be specified by specifying a single data point (at zero plastic strain) in the yield stress versus plastic strain curve at or above the annealing temperature. The overstress power law parameters can also be appropriately selected to ensure that at or above the annealing temperature the yield stress does not vary with strain rate. This can be accomplished by selecting a large value for the parameter (relative to the static yield stress) and setting the parameter .
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For hardening defined in Abaqus/Standard with user subroutine UHARD, Abaqus/Standard checks the hardening slope at or above the annealing temperature during the actual computations and issues an error message if appropriate.
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The Johnson-Cook plasticity model in Abaqus/Explicit requires a separate melting temperature to define the hardening behavior. If the annealing temperature is defined to be less than the melting temperature specified for the metal plasticity model, the hardening memory is removed at the annealing temperature and the melting temperature is used strictly to define the hardening function. Otherwise, the hardening memory is removed automatically at the melting temperature.
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Input File Usage: \*ANNEAL TEMPERATURE
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Plastic: Suboptions→Anneal Temperature
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# Example: Annealing or melting
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The following input is an example of a typical usage of the annealing or melting capability. It is assumed that you have defined the static stress versus plastic strain behavior (see Figure 23.2.5–1) for the isotropic
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hardening model at three different temperatures, including the annealing temperature. It is also assumed that the plastic behavior is rate independent.
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<details>
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<summary>line</summary>
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| Point | ε^pl | σ |
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|-------|------|----|
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| σ₁ | θ₁ | σ₁ |
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| σ₂ | θ₂ | σ₂ |
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| σ₃ | ε₂^pl| σ₃ |
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| σ₄ | ε₁^pl| σ₄ |
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| σ₅ | ε₁^pl| σ₅ |
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</details>
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Figure 23.2.5–1 Stress versus plastic strain behavior.
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The plastic response corresponds to linear hardening below the annealing temperature and perfect plasticity at the annealing temperature. The elastic properties, which may also be temperature dependent, are not shown.
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<table><tr><td colspan="3">Plasticity Data, Isotropic Hardening:</td></tr><tr><td>Yield Stress</td><td>Plastic Strain</td><td>Temperature</td></tr><tr><td> $\sigma_1$ </td><td>0</td><td> $\theta_1$ </td></tr><tr><td> $\sigma_2$ </td><td> $\epsilon_1^{pl}$ </td><td> $\theta_1$ </td></tr><tr><td> $\sigma_3$ </td><td>0</td><td> $\theta_2$ </td></tr><tr><td> $\sigma_4$ </td><td> $\epsilon_2^{pl}$ </td><td> $\theta_2$ </td></tr><tr><td> $\sigma_5$ </td><td>0</td><td> $\theta_3$ </td></tr><tr><td colspan="3">Anneal Temperature: $\theta_3$ </td></tr></table>
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# Elements
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This capability can be used with all elements that include mechanical behavior (elements that have displacement degrees of freedom).
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# Output
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Only the equivalent plastic strain (output variable PEEQ) and the backstress (output variable ALPHA) are reset to zero at the melting temperature. The plastic strain tensor (output variable PE) is not reset to
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zero and provides a measure of the total plastic deformation during the analysis. In Abaqus/Standard the plastic strain tensor also provides a measure of the plastic strain magnitude (output variable PEMAG).
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# 23.2.6 ANISOTROPIC YIELD/CREEP
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Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE
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# References
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• “Material library: overview,” Section 21.1.1
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• “Inelastic behavior,” Section 23.1.1
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• “Classical metal plasticity,” Section 23.2.1
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• “Models for metals subjected to cyclic loading,” Section 23.2.2
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• “Rate-dependent plasticity: creep and swelling,” Section 23.2.4
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• \*POTENTIAL
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• “Defining anisotropic yield and creep” in “Defining plasticity,” Section 12.9.2 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
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# Overview
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Anisotropic yield and/or creep:
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• can be used for materials that exhibit different yield and/or creep behavior in different directions;
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• is introduced through user-defined stress ratios that are applied in Hill’s potential function;
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• can be used only in conjunction with the metal plasticity and, in Abaqus/Standard, the metal creep material models;
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• is available for the nonlinear isotropic/kinematic hardening model in Abaqus/Explicit (“Models for metals subjected to cyclic loading,” Section 23.2.2); and
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• can be used in conjunction with the models of progressive damage and failure in Abaqus/Explicit (“Damage and failure for ductile metals: overview,” Section 24.2.1) to specify different damage initiation criteria and damage evolution laws that allow for the progressive degradation of the material stiffness and the removal of elements from the mesh.
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# Yield and creep stress ratios
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Anisotropic yield or creep behavior is modeled through the use of yield or creep stress ratios, $R _ { i j }$ . In the case of anisotropic yield the yield ratios are defined with respect to a reference yield stress, $\sigma ^ { 0 }$ (given for the metal plasticity definition), such that if $\sigma _ { i j }$ is applied as the only nonzero stress, the corresponding yield stress is $R _ { i j } \sigma ^ { 0 }$ . The plastic flow rule is defined below.
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In the case of anisotropic creep the $R _ { i j }$ are creep ratios used to scale the stress value when the creep strain rate is calculated. Thus, if $\sigma _ { 1 1 }$ is the only nonzero stress, the equivalent stress, ${ \tilde { q } } ,$ used in the user-defined creep law is $\tilde { q } = R _ { 1 1 } | \sigma _ { 1 1 } |$ .
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Yield and creep stress ratios can be defined as constants or as tabular functions of temperature and predefined field variables. A local orientation must be used to define the direction of anisotropy (see “Orientations,” Section 2.2.5).
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Input File Usage: Use the following option to define the yield or creep stress ratios:
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\*POTENTIAL
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This option must appear immediately after the \*PLASTIC or the \*CREEP material option data to which it applies. Thus, if anisotropic metal plasticity and anisotropic creep behavior are both required, the \*POTENTIAL option must appear twice in the material definition, once after the metal plasticity data and again after the creep data.
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Abaqus/CAE Usage: Use one of the following models:
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Property module: material editor:
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Mechanical→Plasticity→Plastic: Suboptions→Potential
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Mechanical→Plasticity→Creep: Suboptions→Potential
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# Anisotropic yield
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Hill’s potential function is a simple extension of the Mises function, which can be expressed in terms of rectangular Cartesian stress components as
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$$
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f (\pmb {\sigma}) = \sqrt {F (\sigma_ {2 2} - \sigma_ {3 3}) ^ {2} + G (\sigma_ {3 3} - \sigma_ {1 1}) ^ {2} + H (\sigma_ {1 1} - \sigma_ {2 2}) ^ {2} + 2 L \sigma_ {2 3} ^ {2} + 2 M \sigma_ {3 1} ^ {2} + 2 N \sigma_ {1 2} ^ {2}},
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$$
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where and N are constants obtained by tests of the material in different orientations. They are defined as
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$$
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F = \frac {(\sigma^ {0}) ^ {2}}{2} \left(\frac {1}{\bar {\sigma} _ {2 2} ^ {2}} + \frac {1}{\bar {\sigma} _ {3 3} ^ {2}} - \frac {1}{\bar {\sigma} _ {1 1} ^ {2}}\right) = \frac {1}{2} \left(\frac {1}{R _ {2 2} ^ {2}} + \frac {1}{R _ {3 3} ^ {2}} - \frac {1}{R _ {1 1} ^ {2}}\right),
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$$
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$$
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G = \frac {(\sigma^ {0}) ^ {2}}{2} \left(\frac {1}{\bar {\sigma} _ {3 3} ^ {2}} + \frac {1}{\bar {\sigma} _ {1 1} ^ {2}} - \frac {1}{\bar {\sigma} _ {2 2} ^ {2}}\right) = \frac {1}{2} \left(\frac {1}{R _ {3 3} ^ {2}} + \frac {1}{R _ {1 1} ^ {2}} - \frac {1}{R _ {2 2} ^ {2}}\right),
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$$
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$$
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H = \frac {(\sigma^ {0}) ^ {2}}{2} \left(\frac {1}{\bar {\sigma} _ {1 1} ^ {2}} + \frac {1}{\bar {\sigma} _ {2 2} ^ {2}} - \frac {1}{\bar {\sigma} _ {3 3} ^ {2}}\right) = \frac {1}{2} \left(\frac {1}{R _ {1 1} ^ {2}} + \frac {1}{R _ {2 2} ^ {2}} - \frac {1}{R _ {3 3} ^ {2}}\right),
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$$
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$$
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L = \frac {3}{2} (\frac {\tau^ {0}}{\bar {\sigma} _ {2 3}}) ^ {2} = \frac {3}{2 R _ {2 3} ^ {2}},
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$$
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$$
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M = \frac {3}{2} (\frac {\tau^ {0}}{\bar {\sigma} _ {1 3}}) ^ {2} = \frac {3}{2 R _ {1 3} ^ {2}},
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$$
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<!-- source-page: 289 -->
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$$
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N = \frac {3}{2} (\frac {\tau^ {0}}{\bar {\sigma} _ {1 2}}) ^ {2} = \frac {3}{2 R _ {1 2} ^ {2}},
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$$
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where each $\bar { \sigma } _ { i j }$ is the measured yield stress value when $\sigma _ { i j }$ is applied as the only nonzero stress component; $\sigma ^ { 0 }$ is the user-defined reference yield stress specified for the metal plasticity definition; $R _ { 1 1 } , R _ { 2 2 } , R _ { 3 3 } , R _ { 1 2 } , R _ { 1 3 }$ , and $R _ { 2 3 }$ are anisotropic yield stress ratios; and $\tau ^ { 0 } = \sigma ^ { 0 } / \sqrt { 3 } .$ . The six yield stress ratios are, therefore, defined as follows (in the order in which you must provide them):
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$$
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\frac {\bar {\sigma} _ {1 1}}{\sigma^ {0}}, \quad \frac {\bar {\sigma} _ {2 2}}{\sigma^ {0}}, \quad \frac {\bar {\sigma} _ {3 3}}{\sigma^ {0}}, \quad \frac {\bar {\sigma} _ {1 2}}{\tau^ {0}}, \quad \frac {\bar {\sigma} _ {1 3}}{\tau^ {0}}, \quad \frac {\bar {\sigma} _ {2 3}}{\tau^ {0}}.
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$$
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Because of the form of the yield function, all of these ratios must be positive. If the constants $F , G ,$ and H are positive, the yield function is always well-defined. However, if one or more of these constants is negative, the yield function may be undefined for some stress states because the quantity under the square root is negative.
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The flow rule is
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$$
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d \pmb {\varepsilon} ^ {p l} = d \lambda \frac {\partial f}{\partial \pmb {\sigma}} = \frac {d \lambda}{f} \mathbf {b},
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$$
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where, from the definition of f above,
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$$
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\mathbf {b} = \left[ \begin{array}{c} - G (\sigma_ {3 3} - \sigma_ {1 1}) + H (\sigma_ {1 1} - \sigma_ {2 2}) \\ F (\sigma_ {2 2} - \sigma_ {3 3}) - H (\sigma_ {1 1} - \sigma_ {2 2}) \\ - F (\sigma_ {2 2} - \sigma_ {3 3}) + G (\sigma_ {3 3} - \sigma_ {1 1}) \\ 2 N \sigma_ {1 2} \\ 2 M \sigma_ {3 1} \\ 2 L \sigma_ {2 3} \end{array} \right].
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$$
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Input File Usage: Use both of the following options:
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\*PLASTIC
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\*POTENTIAL
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Plastic: Suboptions→Potential
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# Anisotropic creep
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For anisotropic creep in Abaqus/Standard Hill’s function can be expressed as
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$$
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\tilde {q} (\pmb {\sigma}) = \sqrt {F (\sigma_ {2 2} - \sigma_ {3 3}) ^ {2} + G (\sigma_ {3 3} - \sigma_ {1 1}) ^ {2} + H (\sigma_ {1 1} - \sigma_ {2 2}) ^ {2} + 2 L \sigma_ {2 3} ^ {2} + 2 M \sigma_ {3 1} ^ {2} + 2 N \sigma_ {1 2} ^ {2}},
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$$
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where $\tilde { q } ( \sigma )$ is the equivalent stress and F, G, H, L, M, and N are constants obtained by tests of the material in different orientations. The constants are defined with the same general relations as those used for anisotropic yield (above); however, the reference yield stress, $\sigma ^ { 0 }$ , is replaced by the uniaxial equivalent deviatoric stress, (found in the creep law), and $R _ { 1 1 } , R _ { 2 2 } , R _ { 3 3 } , R _ { 1 2 } , R _ { 1 3 }$ , and $R _ { 2 3 }$ are referred
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<!-- source-page: 290 -->
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to as “anisotropic creep stress ratios.” The six creep stress ratios are, therefore, defined as follows (in the order in which they must be provided):
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$$
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\frac {\sigma_ {1 1}}{\tilde {q}}, \quad \frac {\sigma_ {2 2}}{\tilde {q}}, \quad \frac {\sigma_ {3 3}}{\tilde {q}}, \quad \frac {\sigma_ {1 2}}{\tilde {q} / \sqrt {3}}, \quad \frac {\sigma_ {1 3}}{\tilde {q} / \sqrt {3}}, \quad \frac {\sigma_ {2 3}}{\tilde {q} / \sqrt {3}}.
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$$
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You must define the ratios $R _ { i j }$ in each direction that will be used to scale the stress value when the creep strain rate is calculated. If all six $R _ { i j }$ values are set to unity, isotropic creep is obtained.
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Input File Usage: Use both of the following options:
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$$
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\begin{array}{l} * \text { CREEP } \\ * \text { POTENTIAL } \end{array}
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$$
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Creep: Suboptions→Potential
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# Defining anisotropic yield behavior on the basis of strain ratios (Lankford’s r-values)
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As discussed above, Hill’s anisotropic plasticity potential is defined in Abaqus from user input consisting of ratios of yield stress in different directions with respect to a reference stress. However, in some cases, such as sheet metal forming applications, it is common to find the anisotropic material data given in terms of ratios of width strain to thickness strain. Mathematical relationships are then necessary to convert the strain ratios to stress ratios that can be input into Abaqus.
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In sheet metal forming applications we are generally concerned with plane stress conditions. Consider $x , y$ to be the “rolling” and “cross” directions in the plane of the sheet; z is the thickness direction. From a design viewpoint, the type of anisotropy usually desired is that in which the sheet is isotropic in the plane and has an increased strength in the thickness direction, which is normally referred to as transverse anisotropy. Another type of anisotropy is characterized by different strengths in different directions in the plane of the sheet, which is called planar anisotropy.
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In a simple tension test performed in the x-direction in the plane of the sheet, the flow rule for this potential (given above) defines the incremental strain ratios (assuming small elastic strains) as
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$$
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d \varepsilon_ {1 1}: d \varepsilon_ {2 2}: d \varepsilon_ {3 3} = G + H: - H: - G.
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$$
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Therefore, the ratio of width to thickness strain, often referred to as Lankford’s r-value, is
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$$
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r _ {x} = \frac {d \varepsilon_ {2 2}}{d \varepsilon_ {3 3}} = \frac {H}{G}.
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$$
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Similarly, for a simple tension test performed in the y-direction in the plane of the sheet, the incremental strain ratios are
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$$
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d \varepsilon_ {1 1}: d \varepsilon_ {2 2}: d \varepsilon_ {3 3} = - H: F + H: - F,
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$$
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and
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