김경종
360 lines
22 KiB
Markdown
360 lines
22 KiB
Markdown
<!-- source-page: 421 -->
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<details>
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<summary>text_image</summary>
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σ
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A
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B
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D
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primary loading curve
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exponential/quadratic
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unloading
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C
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0
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ε_max_B
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ε
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</details>
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Figure 23.4.1–4 Exponential/quadratic unloading.
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The unloading response follows the loading curve when the calculated unloading curve lies above the loading curve to prevent energy generation and follows a zero stress response when the unloading curve yields a negative response. In such cases the dissipated energy will be less than the value specified by the energy dissipation factor.
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# Specifying interpolated curve unloading
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The damage model in Figure 23.4.1–5 illustrates an interpolated unloading response based on multiple unloading curves that intersect the primary loading curve at increasing values of stress/strains. You can specify as many unloading curves as are necessary to define the unloading response. Each unloading curve always starts at point O, the point of zero stress and zero strains, since the damage models do not allow any permanent deformation. The unloading curves are stored in normalized form so that they intersect the loading curve at a unit stress for a unit strain, and the interpolation occurs between these normalized curves. If unloading occurs from a maximum strain for which an unloading curve is not specified, the unloading is interpolated from neighboring unloading curves. As the fabric component is loaded, the stress follows the path given by the loading curve. If the fabric is unloaded (for example, at point B), the stress follows the unloading curve . Reloading after unloading follows the unloading path until the loading is such that the strain becomes greater than $\varepsilon _ { B } ^ { m a x }$ , after which the loading path follows the loading curve.
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The unloading curve also has the same temperature and field variable dependencies as the loading curve.
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# Specifying combined unloading
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As illustrated in Figure 23.4.1–6, you can specify an unloading curve in addition to the loading curve as well as a constant transition slope that connects the loading curve to the unloading
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<details>
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<summary>text_image</summary>
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σ
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primary loading curve
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D
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A
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B
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C
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unloading curves
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0
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ε_max_B
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ε
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</details>
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Figure 23.4.1–5 Interpolated curve unloading.
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curve. As the fabric is loaded, the stress follows the path given by the loading curve. If the fabric is unloaded (for example at point B) the stress follows the unloading curve . The path is defined by the constant transition slope, and lies on the specified unloading curve. Reloading after unloading follows the unloading path until the loading is such that the strain becomes greater than $\varepsilon _ { B } ^ { m a x }$ , after which the loading path follows the loading curve.
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<details>
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<summary>line</summary>
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| Curve Type | Point Label | ε (max) | σ (min) |
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| ------------------ | ----------- | ------- | ------- |
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| primary loading | A | ~0.5 | ~0.2 |
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| primary loading | B | ~1.0 | ~0.5 |
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| primary loading | D | ~1.5 | ~0.8 |
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| transition curve | C | ~0.8 | ~0.3 |
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| unloading curve | E | ~1.2 | ~0.4 |
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</details>
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Figure 23.4.1–6 Combined unloading.
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The unloading curve also has the same temperature and field variable dependencies as the loading curve.
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# Defining models with permanent deformation
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These models dissipate energy upon unloading and exhibit permanent deformation upon complete unloading. You can specify the onset of permanent deformation by defining the strain below which unloading occurs along the loading curve.
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The unloading behavior controls the amount of energy dissipated as well as the amount of permanent deformation. The unloading behavior can be specified in one of the following ways:
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• an analytical unloading curve (exponential/quadratic);
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• an unloading curve interpolated from multiple user-specified unloading curves; or
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• an unloading curve obtained by shifting the user-specified unloading curve to the point of unloading.
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Input File Usage: Use the following options to define permanent deformation with quadratic unloading behavior:
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```txt
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*LOADING DATA, TYPE=PERMANENT DEFORMATION, DIRECTION
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*UNLOADING DATA, DEFINITION=QUADRATIC
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```
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Use the following options to define permanent damage with exponential unloading behavior:
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```txt
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*LOADING DATA, TYPE=PERMANENT DEFORMATION, DIRECTION
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*UNLOADING DATA, DEFINITION=EXPONENTIAL
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```
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Use the following options to define permanent damage with an interpolated unloading curve:
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```txt
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*LOADING DATA, TYPE=PERMANENT DEFORMATION, DIRECTION
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*UNLOADING DATA, DEFINITION=INTERPOLATED CURVE
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```
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Use the following options to specify permanent damage with a shifted unloading curve:
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```txt
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*LOADING DATA, TYPE=PERMANENT DEFORMATION, DIRECTION
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*UNLOADING DATA, DEFINITION=SHIFTED CURVE
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```
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# Defining the onset of permanent deformation
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By default, the onset of yield will be obtained as soon as the slope of the loading curve decreases by 10% from the maximum slope recorded up to that point while traversing along the loading curve. To override the default method of determining the onset of yield, you can specify either a value for the decrease in slope of the loading curve other than the default value of 10% (slope drop = 0.1) or by defining the strain below which unloading occurs along the loading curve. If a slope drop is specified, the onset of yield will be obtained as soon as the slope of the loading curve decreases by the specified factor from the maximum slope recorded up to that point.
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<!-- source-page: 424 -->
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# Input File Usage:
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Use the following options to specify the onset of yield by defining the strain below which unloading occurs along the loading curve:
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\*LOADING DATA, TYPE=PERMANENT DEFORMATION, YIELD ONSET=value
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Use the following options to specify the onset of yield by defining a slope drop for the loading curve:
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\*LOADING DATA, TYPE=PERMANENT DEFORMATION, SLOPE DROP=value
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# Specifying exponential/quadratic unloading
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The model in Figure 23.4.1–7 illustrates an analytical unloading curve that is derived from an energy dissipation factor, (fraction of energy that is dissipated at any strain level), and a permanent deformation factor, $D _ { p }$ . As the fabric component is loaded, the fabric stress follows the path given by the loading curve. If the component is unloaded (for example, at point B), the stress follows the unloading curve . The point D corresponds to the permanent deformation, $D _ { p } \varepsilon _ { B } ^ { m a x }$ . Reloading after unloading follows the unloading curve until the loading is such that the strain becomes greater than $\varepsilon _ { B } ^ { m a x }$ , after which the loading path follows the loading curve. The arrows shown in Figure 23.4.1–7 illustrate the loading/unloading paths of this model.
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<details>
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<summary>text_image</summary>
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σ
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B
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E
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primary loading curve
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A
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exponential/quadratic
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unloading
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C
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0
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D
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ε_max_B
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ε_max_B
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ε
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</details>
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Figure 23.4.1–7 Exponential/quadratic unloading.
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The unloading response follows the loading curve when the calculated unloading curve lies above the loading curve to prevent energy generation and follows a zero stress response when the unloading curve yields a negative response. In such cases the dissipated energy will be less than the value specified by the energy dissipation factor.
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<!-- source-page: 425 -->
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# Specifying interpolated curve unloading
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The model in Figure 23.4.1–8 illustrates an interpolated unloading response based on multiple unloading curves that intersect the primary loading curve at increasing values of stresses/strains.
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<details>
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<summary>text_image</summary>
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σ
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0
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D
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ε_max_B
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E
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B
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A
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C
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primary loading curve
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unloading curves
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</details>
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Figure 23.4.1–8 Interpolated curve unloading.
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You can specify as many unloading curves as are necessary to define the unloading response. The first point of each unloading curve defines the permanent deformation if the fabric component is completely unloaded. The unloading curves are stored in normalized form so that they intersect the loading curve at a unit stress for a unit strain, and the interpolation occurs between these normalized curves. If unloading occurs from a maximum strain for which an unloading curve is not specified, the unloading is interpolated from neighboring unloading curves. As the fabric is loaded, the stress follows the path given by the loading curve. If the fabric is unloaded (for example, at point B), the stress follows the unloading curve . Reloading after unloading follows the unloading path until the loading is such that the strain becomes greater than $\varepsilon _ { B } ^ { m a x }$ , after which the loading path follows the loading curve.
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The unloading curve also has the same temperature and field variable dependencies as the loading curve.
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# Specifying shifted curve unloading
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You can specify an unloading curve passing through the origin in addition to the loading curve. The actual unloading curve is obtained by horizontally shifting the user-specified unloading curve to pass through the point of unloading as shown in Figure 23.4.1–9. The permanent deformation upon complete unloading is the horizontal shift applied to the unloading curve.
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The unloading curve also has the same temperature and field variable dependencies as the loading curve.
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<details>
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<summary>line</summary>
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| Point | Curve Type | ε (max) | σ (min) |
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|-------|-------------------------|---------|---------|
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| A | primary loading curve | D | Low |
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| B | primary loading curve | D | High |
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| C | shifted unloading curve | D | Low |
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| E | primary loading curve | D | High |
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</details>
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Figure 23.4.1–9 Shifted curve unloading.
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# Using different uniaxial models in tension and compression
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When appropriate, different uniaxial behavior models can be used in tension and compression. For example, response under tension can be plastic with exponential unloading, while the response in compression can be nonlinear elastic (see Figure 23.4.1–10).
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# User-defined fabric materials
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The mechanical response of a fabric material depends on a number of micro and meso-scale parameters covering the fabric construction and that of the individual yarns as a bundle of fibers. Often a multi-scale model becomes necessary to track the state of the fabric and its response to loading. Abaqus provides a specialized user subroutine, VFABRIC, to capture the complex fabric response given the deformed yarn directions and the strains along these directions.
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The density (“Density,” Section 21.2.1) is required when using a fabric material.
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Input File Usage: Use the following options to define a fabric material through user subroutine VFABRIC:
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\*MATERIAL, NAME=name
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\*FABRIC, USER
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\*DENSITY
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# Properties for a user-defined fabric material
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Any material constants that are needed in user subroutine VFABRIC must be specified as part of a user-defined fabric material definition. Abaqus can be used to compute the isotropic thermal expansion response under thermal loading, even as the remaining mechanical response is defined by the user
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<details>
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<summary>text_image</summary>
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σ
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A
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primary loading curve
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unloading
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nonlinear elastic
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ε
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</details>
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Figure 23.4.1–10 Different uniaxial models in tension and compression.
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subroutine. Alternatively, you can include the thermal expansion within the definition of the mechanical response in user subroutine VFABRIC.
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Input File Usage: Use the following option to define properties for a user-defined fabric material behavior:
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$$
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* \text { FABRIC, USER, PROPERTIES } = \text { number\_of\_constants }
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$$
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# Material state
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Many mechanical constitutive models require the storage of solution-dependent state variables (plastic strains, “back stress,” saturation values, etc. in rate constitutive forms or historical data for theories written in integral form). You should allocate storage for these variables in the associated material definition (see “Allocating space” in “User subroutines: overview,” Section 18.1.1). There is no restriction on the number of state variables associated with a user-defined fabric material.
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State variables associated with VFABRIC can be output to the output database (.odb) file and results (.fil) file using output identifiers SDV and SDVn (see “Abaqus/Explicit output variable identifiers,” Section 4.2.2).
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User subroutine VFABRIC is called for blocks of material points at each increment. When the subroutine is called, it is provided with the state at the start of the increment (fabric stress in the local system, solution-dependent state variables). It is also provided with the nominal fabric strains at the end of the increment and the incremental nominal fabric strains over the increment, both in the local system. The VFABRIC user material interface passes a block of material points to the subroutine on each call, which allows vectorization of the material subroutine.
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The temperature is provided to user subroutine VFABRIC at the start and the end of the increment. The temperature is passed in as information only and cannot be modified, even in a fully coupled thermalstress analysis. However, if the inelastic heat fraction is defined in conjunction with the specific heat and conductivity in a fully coupled thermal-stress analysis, the heat flux due to inelastic energy dissipation is calculated automatically. If user subroutine VFABRIC is used to define an adiabatic material behavior (conversion of plastic work to heat) in an explicit dynamic procedure, the temperatures must be stored and integrated as user-defined state variables. Most often the temperatures are provided by specifying initial conditions (“Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1) and are constant throughout the analysis.
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# Deleting elements from an Abaqus/Explicit mesh using state variables
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Element deletion in a mesh can be controlled during the course of an Abaqus/Explicit analysis through user subroutine VFABRIC. Deleted elements have no ability to carry stresses and, therefore, have no contribution to the stiffness of the model. You specify the state variable number controlling the element deletion flag. For example, specifying a state variable number of 4 indicates that the fourth state variable is the deletion flag in VFABRIC. The deletion state variable should be set to a value of one or zero in VFABRIC. A value of one indicates that the material point is active, while a value of zero indicates that Abaqus/Explicit should delete the material point from the model by setting the stresses to zero. The structure of the block of material points passed to user subroutine VFABRIC remains unchanged during the analysis; deleted material points are not removed from the block. Abaqus/Explicit will pass zero stresses and strain increments for all deleted material points. Once a material point has been flagged as deleted, it cannot be reactivated. An element will be deleted from the mesh only after all of the material points in the element are deleted. The status of an element can be determined by requesting output of the variable STATUS. This variable is equal to 1 if the element is active and equal to 0 if the element is deleted.
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Input File Usage: \*DEPVAR, DELETE=variable number
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# Thermal expansion
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You can define isotropic thermal expansion to specify the same coefficient of thermal expansion for the membrane and thickness-direction behaviors.
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The membrane thermal strains, $\varepsilon ^ { t h }$ , are obtained as explained in “Thermal expansion,” Section 26.1.2.
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The elastic stretch in a given direction, $\lambda ^ { e \ell }$ , relates the total stretch, , and the thermal stretch, $\lambda ^ { t h }$ :
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$$
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\lambda^ {e \ell} = \frac {\lambda}{\lambda^ {t h}}.
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$$
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$\lambda ^ { t h }$ is given by
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$$
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\lambda^ {t h} = 1 + \varepsilon^ {t h},
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$$
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where $\varepsilon ^ { t h }$ is the linear thermal expansion strain in that direction.
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# Fabric thickness
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The thickness of a fabric is difficult to measure experimentally. Fortunately, an accurate value for thickness is not always required due to the fact that a nominal stress measure, defined as force per unit area in the reference configuration, is used to characterize the in-plane response. An initial thickness can be specified on the section definition. Accurate tracking of the thickness with deformation is necessary only if the material is used with shell elements and the bending response needs to be captured accurately. You can compute the thickness direction strain increment when the fabric is defined through user subroutine VFABRIC. For test data–based fabric materials the thickness is assumed to remain constant with deformation. For a test data–based fabric definition, you must use the thickness value specified on the section definition for converting the experimental load data (which are typically available as force applied per unit width of the fabric) to stress quantities.
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# Defining a reference mesh (initial metric)
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Abaqus/Explicit allows the specification of a reference mesh (initial metric) for fabrics modeled with membrane elements. For example, this is useful in airbag simulations to model wrinkles and changes in yarn orientations that arise from the airbag folding process. A flat mesh may be suitable for the unstressed reference configuration, but the initial state may require a corresponding folded mesh defining the folded state. The angular orientation of the yarn in the reference configuration is updated to obtain the new orientation in the initial configuration.
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# Input File Usage:
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Use the following option to define the reference configuration giving the element number and its nodal coordinates in the reference configuration:
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\*INITIAL CONDITIONS, TYPE=REF COORDINATE
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Use the following option to define the reference configuration giving the node number and its coordinates in the reference configuration:
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\*INITIAL CONDITIONS, TYPE=NODE REF COORDINATE
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# Yarn behavior under initial compressive strains
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Defining a reference configuration that is different from the initial configuration generally results in nonzero stresses and strains in the initial configuration based on the material definition. By default, compressive initial strains in the yarn directions generate zero stresses. The stress remains zero as the strain is continuously recovered from the initial compressive values toward the strain-free state. Once the initial slack is recovered, any subsequent compressive/tensile strains generate stresses as per the material definition.
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# Input File Usage:
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Use the following option to specify that initial compressive strains are recovered stress free (default):
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\*FABRIC, STRESS FREE INITIAL SLACK=YES
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Use the following option to specify that initial compressive strains generate nonzero initial stresses:
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\*FABRIC, STRESS FREE INITIAL SLACK=NO
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<!-- source-page: 430 -->
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# Defining yarn directions in the reference configuration
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In general, the yarn directions may not be orthogonal to each other in the reference configuration. You can specify these local directions with respect to the in-plane axes of an orthogonal orientation system at a material point. Both the local directions and the orthogonal system are defined together as a single orientation definition. See “Orientations,” Section 2.2.5, for more information.
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If the local directions are not specified, these directions are assumed to match the in-plane axes of the orthogonal system defined. The local direction may not remain orthogonal with deformation. Abaqus updates the local directions with deformation and computes the nominal strains along these directions and the angle between them (fabric shear strain). The constitutive behavior for the fabric defines the nominal stresses in the local system in terms of the fabric strain.
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Local yarn directions can be output to the output database as described in “Output,” below.
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# Picture-frame shear fabric test
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The shear response of the fabric is typically studied using a picture-frame shear test. The reference and the deformed configuration for a picture-frame shear test under force is illustrated in Figure 23.4.1–11, where $L _ { 0 }$ is the size of the picture-frame, and $\psi _ { 1 2 } ^ { 0 }$ is the initial angle between the yarn directions. The four sides of the specimen are constrained not to change in their length even as the frame elongates and the angle between the yarn directions $\psi _ { 1 2 }$ decreases with deformation.The relationship between the nominal shear stress, $T _ { 1 2 }$ , and the applied force, , is
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$$
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T _ {1 2} = \left(\frac {F L _ {0}}{v _ {0}}\right) \sin \left(\frac {\psi_ {1 2}}{2}\right),
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$$
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where $v _ { 0 }$ is the initial volume of the specimen. The fabric engineering shear strain, $\gamma _ { 1 2 } ,$ , is related to the change in the angle between the yarn directions as
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$$
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\gamma_ {1 2} = \psi_ {1 2} ^ {0} - \psi_ {1 2}.
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$$
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# Use with other material models
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The fabric material model can be used by itself, or it can be combined with isotropic thermal expansion to introduce thermal volume changes (“Thermal expansion,” Section 26.1.2). See “Combining material behaviors,” Section 21.1.3, for more details. Thermal expansion can alternatively be an integral part of the constitutive model implemented in VFABRIC for user-defined fabric materials.
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For a test-data based fabric material, both the mass proportional and the stiffness proportional damping can be specified (see “Material damping,” Section 26.1.1). If stiffness proportional damping is specified, Abaqus calculates the damping stress based on the current elastic stiffness of the material and the resulting damping stress is included in the reported stress output at the integration points.
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For a fabric material defined by user subroutine VFABRIC, mass proportional damping can be specified, but stiffness proportional damping must be defined within the user subroutine.
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