김경종
297 lines
26 KiB
Markdown
297 lines
26 KiB
Markdown
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The dashed curves represent the asymptotic response at two different values of $\beta ( \beta _ { 1 }$ and $\beta _ { 2 } )$ . For fixed values of r and $m , \eta _ { m }$ is a function of $\beta .$ . In particular, if $m = 0$ ,
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$$
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\eta_ {m} = 1 - \frac {1}{r} \mathrm{erf} \biggl (\frac {1}{\beta} \biggr).
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$$
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The above relation is approximately true if $U _ { d e v } ^ { m }$ deu is much greater than m.
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# Specifying the Mullins effect material model in Abaqus
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The primary hyperelastic behavior is defined by using the hyperelastic material model (see “Hyperelastic behavior of rubberlike materials,” Section 22.5.1). The Mullins effect model can be defined by specifying the Mullins effect parameters directly or by using test data to calibrate the parameters. Alternatively, you can define the Mullins effect model with user subroutine UMULLINS in Abaqus/Standard and VUMULLINS in Abaqus/Explicit.
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# Specifying the parameters directly
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The parameters r, m, and $\beta$ of the Mullins effect can be given directly as functions of temperature and/or field variables.
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Input File Usage: \*MULLINS EFFECT
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Abaqus/CAE Usage: Property module: material editor:
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Mechanical→Damage for Elastomers→Mullins Effect:
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Definition: Constants
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# Using test data to calibrate the parameters
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Experimental unloading-reloading data from different strain levels can be specified for up to three simple tests: uniaxial, biaxial, and planar. Abaqus will then compute the material parameters using a nonlinear least-squares curve fitting algorithm. It is generally best to obtain data from several experiments involving different kinds of deformation over the range of strains of interest in the actual application and to use all these data to determine the parameters. It is also important to obtain a good curve-fit for the primary hyperelastic behavior if the primary behavior is defined using test data.
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By default, Abaqus attempts to fit all three parameters to the given data. This is possible in general, except in the situation when the test data correspond to unloading-reloading from only a single value of $U _ { d e v } ^ { m }$ . In this case the parameters m and $\beta$ cannot be determined independently; one of them must be specified. If you specify neither m nor $\beta ,$ Abaqus needs to assume a default value for one of these parameters. In light of the potential problems discussed earlier with $\beta = 0$ , Abaqus assumes that $m = 0$ in the above situation. The curve-fitting may also be carried out by specifying any one or two of the material parameters to be fixed, predetermined values.
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As many data points as required can be entered from each test. It is recommended that data from all three tests (on samples taken from the same piece of material) be included and that the data points cover unloading/reloading from/to the range of nominal strain expected to arise in the actual loading.
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The strain data should be given as nominal strain values (change in length per unit of original length). The stress data should be given as nominal stress values (force per unit of original cross-sectional area).
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These tests allow for entering both compression and tension data. Compressive stresses and strains are entered as negative values.
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For each set of test input, the data point with the maximum nominal strain identifies the point of unloading. This point is used by the curve-fitting algorithm to compute $U _ { d e v } ^ { m }$ for that curve.
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Figure 22.6.1–4 shows some typical unloading-reloading data from three different strain levels.
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<details>
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<summary>line</summary>
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| Nominal Strain | Nominal Stress (Red) | Nominal Stress (Blue) | Nominal Stress (Maroon) | Nominal Stress (Black Dashed) |
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| -------------- | --------------------- | ---------------------- | ------------------------ | ------------------------------ |
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| 0 | 0 | 0 | 0 | 0 |
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| 1 | 10 | 15 | 5 | 10 |
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| 2 | 20 | 25 | 10 | 15 |
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| 3 | 30 | 35 | 15 | 20 |
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| 4 | 40 | 45 | 20 | 25 |
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| 5 | 50 | 55 | 25 | 30 |
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| 6 | 60 | 65 | 30 | 35 |
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| 7 | 70 | 75 | 35 | 40 |
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| 8 | 80 | 85 | 40 | 45 |
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| 9 | 90 | 95 | 45 | 50 |
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| 10 | 100 | 105 | 50 | 55 |
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</details>
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Figure 22.6.1–4 Typical available test data for Mullins effect.
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The data include multiple loading and unloading cycles from each strain level. As Figure 22.6.1–4 indicates, the loading/unloading cycles from any given strain level do not occur along a single curve, and there is some amount of hysteresis. There is also some amount of permanent set upon removal of the applied load. The data also show evidence of progressive damage with repeated cycling at any given maximum strain level. The response appears to stabilize after a number of cycles. When such data are used to calibrate the Mullins effect model, the resulting response will capture the overall stiffness characteristics, while ignoring effects such as hysteresis, permanent set, or progressive damage. The above data can be provided to Abaqus in the following manner:
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• The primary curve can be made up of the data points indicated by the dashed curve in Figure 22.6.1–4. Essentially, this consists of an envelope of the first loading curves to the different strain levels.
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• The unloading-reloading curves from the three different strain levels can be specified by providing the data points as is; i.e., as the repeated unloading-reloading cycles shown in Figure 22.6.1–4. As discussed earlier, the data from the different strain levels need to be distinguished by providing them as different tables. For example, assuming that the test data correspond to the uniaxial tension state, three tables of uniaxial test data would have to be defined for the three different strain levels shown in Figure 22.6.1–4. In this case Abaqus will provide a best fit using all the data points (from all strain levels). The resulting fit would result in a response that is an average of all the test data
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at any given strain level. While permanent set may be modeled (see “Permanent set in rubberlike materials,” Section 23.7.1), hysteresis will be lost in the process.
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• Alternatively, you may provide any one unloading-reloading cycle from each different strain level. If the component is expected to undergo repeated cyclic loading, the latter may be, for example, the stabilized cycle at each strain level. On the other hand, if the component is expected to undergo predominantly monotonic loading with perhaps small amounts of unloading, the very first unloading curve at each strain level may be the appropriate input data for calibrating the Mullins coefficients.
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Once the Mullins effect constants are determined, the behavior of the Mullins effect model in Abaqus is established. However, the quality of this behavior must be assessed: the prediction of material behavior under different deformation modes must be compared against the experimental data. You must judge whether the Mullins effect constants determined by Abaqus are acceptable, based on the correlation between the Abaqus predictions and the experimental data. Single-element test cases can be used to derive the nominal stress–nominal strain response of the material model.
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The steps that can be taken for improving the quality of the fit for the Mullins effect parameters are similar in essence to the guidelines provided for curve fitting the primary hyperelastic behavior (see “Hyperelastic behavior of rubberlike materials,” Section 22.5.1, for details). In addition, the quality of the fit for the Mullins effect parameters depends on a good fit for the primary hyperelastic behavior, if the primary behavior is defined using test data.
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The quality of the fit can be evaluated by carrying out a numerical experiment with a single element that is loaded in the same mode for which test data has been provided. Alternatively, the numerical response for both the primary and the softening behavior can be obtained by requesting model definition data output (see “Output,” Section 4.1.1) and carrying out a data check analysis. The response computed by Abaqus is printed in the data (.dat) file along with the experimental data. This tabular data can be plotted in Abaqus/CAE for comparison and evaluation purposes. The primary hyperelastic behavior can also be evaluated with the automated material evaluation tools in Abaqus/CAE.
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# Input File Usage:
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\*MULLINS EFFECT, TEST DATA INPUT, BETA and/or M and/or R
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In addition, use at least one and up to three of the following options to give the unloading-reloading test data (see “Experimental tests” in the section describing hyperelastic test data input, “Hyperelastic behavior of rubberlike materials,” Section 22.5.1):
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\*UNIAXIAL TEST DATA
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\*BIAXIAL TEST DATA
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\*PLANAR TEST DATA
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Multiple unloading-reloading curves from different strain levels for any given test type can be entered by repeated specification of the appropriate test data option.
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# Abaqus/CAE Usage:
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Property module: material editor:
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Mechanical→Damage for Elastomers→Mullins Effect: Definition:
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Test Data Input: enter the values for up to two of the values r, m, and
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beta. In addition, select and enter data for at least one of the following: Add Test→Biaxial Test, Planar Test, or Uniaxial Test
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# User subroutine specification
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An alternative method for defining the Mullins effect involves defining the damage variable in user subroutine UMULLINS in Abaqus/Standard and VUMULLINS in Abaqus/Explicit. Optionally, you can specify the number of property values needed as data in the user subroutine. You must provide the damage variable, , and its derivative, $\frac { d \eta } { d \tilde { U } _ { d e v } }$ dUdev . The latter contributes to the Jacobian of the overall system of equations and is necessary to ensure good convergence characteristics in Abaqus/Standard. If needed, you can specify the number of solution-dependent variables (“User subroutines: overview,” Section 18.1.1). These solution-dependent variables can be updated in the user subroutine. The damage dissipation energy and the recoverable part of the energy may also be defined for output purposes.
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User subroutines UMULLINS and VUMULLINS can be used in combination with all hyperelastic potentials in Abaqus, including user-defined potentials (via user subroutines UHYPER, UANISOHYPER\_INV, and UANISOHYPER\_STRAIN Abaqus/Standard, and VUANISOHYPER\_INV and VUANISOHYPER\_STRAIN in Abaqus/Explicit).
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Input File Usage: \*MULLINS EFFECT, USER, PROPERTIES=constants
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Abaqus/CAE Usage: Property module: material editor:
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Mechanical→Damage for Elastomers→Mullins Effect:
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Definition: User Defined
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# Viscoelasticity
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When viscoelasticity is used in combination with Mullins effect, stress softening is applied to the longterm behavior.
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In this case specification of the parameter (which has units of energy) should be done carefully. If the underlying hyperelastic behavior is defined with an instantaneous modulus, will be interpreted to be instantaneous. Otherwise, is considered to be long term.
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# Elements
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The Mullins effect material model can be used with all element types that support the use of the hyperelastic material model.
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# Procedures
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The Mullins effect material model can be used in all procedure types that support the use of the hyperelastic material model. In linear perturbation steps in Abaqus/Standard the current material tangent stiffness is used to determine the response. Specifically, when a linear perturbation is carried out about a base state that is on the primary curve, the unloading tangent stiffness will be used.
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In Abaqus/Explicit the unloading tangent stiffness is always used to compute the stable time increment. As a result, the inclusion of Mullins effect leads to more increments in the analysis, even when no unloading actually takes place.
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The Mullins effect material model can also be used in a steady-state transport analysis in Abaqus/Standard to obtain steady-state rolling solutions. Issues related to the use of the Mullins effect in a steady-state transport analysis can be found in “Steady-state transport analysis,” Section 6.4.1, and “Analysis of a solid disc with Mullins effect and permanent set,” Section 3.1.7 of the Abaqus Example Problems Guide.
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# Output
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In addition to the standard output identifiers available in Abaqus (“Abaqus/Standard output variable identifiers,” Section 4.2.1, and “Abaqus/Explicit output variable identifiers,” Section 4.2.2), the following variables have special meaning for the Mullins effect material model:
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<table><tr><td>DMENER</td><td>Energy dissipated per unit volume by damage.</td></tr><tr><td>ELDMD</td><td>Total energy dissipated in element by damage.</td></tr><tr><td>ALLDMD</td><td>Energy dissipated in whole (or partial) model by damage. The contribution from ALLDMD is included in the total strain energy ALLIE.</td></tr><tr><td>EDMDDEN</td><td>Energy dissipated per unit volume in the element by damage.</td></tr><tr><td>SENER</td><td>The recoverable part of the energy per unit volume.</td></tr><tr><td>ELSE</td><td>The recoverable part of the energy in the element.</td></tr><tr><td>ALLSE</td><td>The recoverable part of the energy in the whole (partial) model.</td></tr><tr><td>ESEDEN</td><td>The recoverable part of the energy per unit volume in the element.</td></tr></table>
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The damage energy dissipation, represented by the shaded area in Figure 22.6.1–1 for deformation until $c ^ { ' }$ , is computed as follows. When the damaged material is in a fully unloaded state, the augmented energy function has the residual value $U ( \mathbf { I } , \eta _ { m } ) = \phi ( \eta _ { m } )$ . The residual value of the energy function upon complete unloading represents the energy dissipated due to damage in the material. The recoverable part of the energy is obtained by subtracting the dissipated energy from the augmented energy as $\eta \tilde { U } _ { d e v } ( \overline { { \lambda } } _ { i } ) +$ $\phi ( \eta ) + \tilde { U } _ { v o l } - \phi ( \eta _ { m } )$ .
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The damage energy accumulates with progressive deformation along the primary curve and remains constant during unloading. During unloading, the recoverable part of the strain energy is released. The latter becomes zero when the material point is completely unloaded. Upon further reloading from a completely unloaded state, the recoverable part of the strain energy increases from zero. When the maximum strain that was attained earlier is exceeded upon reloading, further accumulation of damage energy occurs.
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# Additional reference
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• Ogden, R. W., and D. G. Roxburgh, “A Pseudo-Elastic Model for the Mullins Effect in Filled Rubber,” Proceedings of the Royal Society of London, Series A, vol. 455, p. 2861–2877, 1999.
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# 22.6.2 ENERGY DISSIPATION IN ELASTOMERIC FOAMS
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Products: Abaqus/Standard Abaqus/Explicit
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# References
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• “Material library: overview,” Section 21.1.1
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• “Combining material behaviors,” Section 21.1.3
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• “Elastic behavior: overview,” Section 22.1.1
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• “Hyperelastic behavior in elastomeric foams,” Section 22.5.2
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• “Mullins effect,” Section 22.6.1
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• \*HYPERFOAM
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• \*MULLINS EFFECT
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• \*UNIAXIAL TEST DATA
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• \*BIAXIAL TEST DATA
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• \*PLANAR TEST DATA
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# Overview
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Energy dissipation in elastomeric foams in Abaqus:
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• allows the modeling of permanent energy dissipation and stress softening effects in elastomeric foams;
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• uses an approach based on the Mullins effect for elastomeric rubbers (“Mullins effect,” Section 22.6.1);
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• provides an extension to the isotropic elastomeric foam model (“Hyperelastic behavior in elastomeric foams,” Section 22.5.2);
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• is intended for modeling energy absorption in foam components subjected to dynamic loading under deformation rates that are high compared to the characteristic relaxation time of the foam; and
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• cannot be used with viscoelasticity in Abaqus/Standard.
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# Energy dissipation in elastomeric foams
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Abaqus provides a mechanism to include permanent energy dissipation and stress softening effects in elastomeric foams. The approach is similar to that used to model the Mullins effect in elastomeric rubbers, described in “Mullins effect,” Section 22.6.1. The functionality is primarily intended for modeling energy absorption in foam components subjected to dynamic loading under deformation rates that are high compared to the characteristic relaxation time of the foam; in such cases it is acceptable to assume that the foam material is damaged permanently.
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The material response is depicted qualitatively in Figure 22.6.2–1.
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<details>
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<summary>text_image</summary>
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stress
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a
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stretch
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b
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B
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C
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c'
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d
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b'
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c
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</details>
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Figure 22.6.2–1 Typical stress-stretch response of an elastomeric foam material with energy dissipation.
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Consider the primary loading path $a b { b ^ { ' } }$ of a previously unstressed foam, with loading to an arbitrary point $b ^ { ' }$ . On unloading from $\bar { \boldsymbol { b } } ^ { \prime }$ , the path $b ^ { ' } B a$ is followed. When the material is loaded again, the softened path is retraced as $a B b ^ { ' }$ . If further loading is then applied, the path $b ^ { ' } c$ is followed, where $b ^ { ' } c$ is a continuation of the primary loading path (which is the path that would be followed if there were no unloading). If loading is now stopped at $c ^ { ' }$ , the path $c ^ { ' } C a$ is followed on unloading and then retraced back to $c$ on reloading. If no further loading beyond $c ^ { ' }$ is applied, the curve $a C c ^ { ' }$ represents the subsequent material response, which is then elastic. For loading beyond $c ^ { ' }$ , the primary path is again followed and the pattern described is repeated. The shaded area in Figure 22.6.2–1 represents the energy dissipated by damage in the material for deformation until $c ^ { ' }$ .
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# Modified strain energy density function
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Energy dissipation effects are accounted for by introducing an augmented strain energy density function of the form
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$$
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U (\hat {\lambda} _ {i}, \eta) = \eta \tilde {U} (\hat {\lambda} _ {i}) + \phi (\eta),
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$$
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where $\hat { \lambda } _ { i } ( i = 1 , 2 , 3 )$ represent the principal mechanical stretches and $\tilde { U } ( \hat { \lambda } _ { i } )$ is the strain energy potential for the primary foam behavior described in “Hyperelastic behavior in elastomeric foams,” Section 22.5.2, defined by the polynomial strain energy function
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<!-- source-page: 169 -->
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$$
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\tilde {U} (\hat {\lambda} _ {i}) = \sum_ {i = 1} ^ {N} \frac {2 \mu_ {i}}{\alpha_ {i} ^ {2}} \left[ \hat {\lambda} _ {1} ^ {\alpha_ {i}} + \hat {\lambda} _ {2} ^ {\alpha_ {i}} + \hat {\lambda} _ {3} ^ {\alpha_ {i}} - 3 + \frac {1}{\beta_ {i}} \left((J ^ {e \ell}) ^ {- \alpha_ {i} \beta_ {i}} - 1\right) \right].
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$$
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The function $\phi ( \eta )$ is a continuous function of the damage variable, $\eta ,$ and is referred to as the “damage function.” The damage variable varies continuously during the course of the deformation and always satisfies $0 < \eta \leq 1$ , with $\eta = 1$ on the points of the primary curve. The damage function $\phi ( \eta )$ satisfies the condition $\phi ( 1 ) = 0 ;$ thus, when the deformation state of the material is on a point on the curve that represents the primary foam behavior, $U ( \hat { \lambda } _ { i } , 1 ) = \tilde { U } ( \hat { \lambda } _ { i } )$ and the augmented energy function reduces to the strain energy potential for the primary foam behavior.
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The above expression of the augmented strain energy density function is similar to the form proposed by Ogden and Roxburgh to model the Mullins effect in filled rubber elastomers (see “Mullins effect,” Section 22.6.1), with the difference that in the case of elastomeric foams an augmentation of the total strain energy (including the volumetric part) is considered. This modification is required for the model to predict energy absorption under pure hydrostatic loading of the foam.
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# Stress computation
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With the above modification to the energy function, the stresses are given by
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$$
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\boldsymbol {\sigma} (\eta , \hat {\lambda} _ {i}) = \eta \tilde {\boldsymbol {\sigma}} (\hat {\lambda} _ {i}),
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$$
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where $\tilde { \sigma }$ is the stress corresponding to the primary foam behavior at the current deformation level $\hat { \lambda } _ { i }$ . Thus, the stress is obtained by simply scaling the stress of the primary foam behavior by the damage variable, . From any given strain level the model predicts unloading/reloading along a single curve (that is different, in general, from the primary foam behavior) that passes through the origin of the stress-strain plot. The model also predicts energy dissipation under purely volumetric deformation.
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# Damage variable
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The damage variable, , varies with the deformation according to
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$$
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\eta = 1 - \frac {1}{r} \mathrm{erf} \left(\frac {U ^ {m} - \tilde {U}}{m + \beta U ^ {m}}\right),
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$$
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where $U ^ { m }$ is the maximum value of $\tilde { U }$ at a material point during its deformation history; $r , \beta _ { : }$ , and m are material parameters; and $\operatorname { f } ( x )$ is the error function. When $\tilde { U } = U ^ { m }$ , corresponding to a point on the primary curve, $\eta = 1 . 0$ . On the other hand, upon removal of deformation, when $\tilde { U } = 0$ , the damage variable, , attains its minimum value, $\eta _ { m }$ , given by
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$$
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\eta_ {m} = 1 - \frac {1}{r} \mathrm{erf} \left(\frac {U ^ {m}}{m + \beta U ^ {m}}\right).
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$$
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For all intermediate values of $\tilde { U }$ , varies monotonically between and $\eta _ { m }$ . While the parameters r and $\beta$ are dimensionless, the parameter m has the dimensions of energy. The material parameters can
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<!-- source-page: 170 -->
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be specified directly or can be computed by Abaqus based on curve fitting of unloading-reloading test data. These parameters are subject to the restrictions $r > 1 , \beta \geq 0$ , and $m \geq 0$ (the parameters $\beta$ and m cannot both be zero). Alternatively, the damage variable $\eta$ can be defined through user subroutine UMULLINS in Abaqus/Standard and VUMULLINS in Abaqus/Explicit.
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If the parameter $\beta = 0$ and the parameter m has a value that is small compared to $U ^ { m }$ , the slope of the stress-strain curve at the initiation of unloading from relatively large strain levels may become very high. As a result, the response may become discontinuous. This kind of behavior may lead to convergence problems in Abaqus/Standard. In Abaqus/Explicit the high stiffness will lead to very small stable time increments, thereby leading to a degradation in performance. This problem can be avoided by choosing a small value for $\beta .$ . In Abaqus/Standard the default value of $\beta$ is 0. In Abaqus/Explicit, however, the default value of $\beta$ is 0.1. Thus, if you do not specify a value for $\beta ,$ , it is assumed to be 0 in Abaqus/Standard and 0.1 in Abaqus/Explicit.
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The parameters $r , \beta ,$ , and m do not have direct physical interpretations in general. The parameter m controls whether damage occurs at low strain levels. If $m = 0$ , there is a significant amount of damage at low strain levels. On the other hand, a nonzero m leads to little or no damage at low strain levels. For further discussion regarding the implications of this model on the energy dissipation, see “Mullins effect,” Section 4.7.1 of the Abaqus Theory Guide.
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# Specifying properties for energy dissipation in elastomeric foams
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The primary elastomeric foam behavior is defined by using the hyperfoam material model. Energy dissipation can be defined by specifying the parameters in the expression of the damage variable directly or by using test data to calibrate the parameters. Alternatively, you can define the Mullins effect model with user subroutine UMULLINS in Abaqus/Standard and VUMULLINS in Abaqus/Explicit.
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# Specifying the parameters directly
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The parameters $r , m ,$ and $\beta$ in the expression of the damage variable can be given directly as functions of temperature and/or field variables.
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Input File Usage: \*MULLINS EFFECT
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Abaqus/CAE Usage: Property module: material editor:
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Mechanical→Damage for Elastomers→Mullins Effect:
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Definition: Constants
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Using test data to calibrate the parameters
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Experimental unloading-reloading data from different strain levels can be specified for up to three simple tests: uniaxial, biaxial, and planar. Abaqus will then compute the material parameters using a nonlinear least-squares curve fitting algorithm. See “Mullins effect,” Section 22.6.1, for a detailed discussion of this approach.
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Input File Usage: \*MULLINS EFFECT, TEST DATA INPUT, BETA and/or M and/or R
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In addition, use at least one and up to three of the following options to give the unloading-reloading test data:
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\*UNIAXIAL TEST DATA
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