김경종
255 lines
17 KiB
Markdown
255 lines
17 KiB
Markdown
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This flow potential, which is continuous and smooth, ensures that the flow direction is always uniquely defined. The function approaches the linear Drucker-Prager flow potential asymptotically at high confining pressure stress and intersects the hydrostatic pressure axis at 90°. See “Models for granular or polymer behavior,” Section 4.4.2 of the Abaqus Theory Guide, for further discussion of this potential.
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The default flow potential eccentricity is $\epsilon = 0 . 1$ , which implies that the material has almost the same dilation angle over a wide range of confining pressure stress values. Increasing the value of provides more curvature to the flow potential, implying that the dilation angle increases more rapidly as the confining pressure decreases. Values of that are significantly less than the default value may lead to convergence problems if the material is subjected to low confining pressures because of the very tight curvature of the flow potential locally where it intersects the p-axis.
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# Yield function
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The model makes use of the yield function of Lubliner et. al. (1989), with the modifications proposed by Lee and Fenves (1998) to account for different evolution of strength under tension and compression. The evolution of the yield surface is controlled by the hardening variables, $\tilde { \varepsilon } _ { t } ^ { p l }$ and $\tilde { \varepsilon } _ { c } ^ { p l }$ . In terms of effective stresses, the yield function takes the form
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$$
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F = \frac {1}{1 - \alpha} \left(\bar {q} - 3 \alpha \bar {p} + \beta (\tilde {\varepsilon} ^ {p l}) \langle \hat {\bar {\sigma}} _ {\mathrm{max}} \rangle - \gamma \langle - \hat {\bar {\sigma}} _ {\mathrm{max}} \rangle\right) - \bar {\sigma} _ {c} (\tilde {\varepsilon} _ {c} ^ {p l}) = 0,
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$$
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with
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$$
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\alpha = \frac {(\sigma_ {b 0} / \sigma_ {c 0}) - 1}{2 (\sigma_ {b 0} / \sigma_ {c 0}) - 1}; 0 \leq \alpha \leq 0. 5,
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$$
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$$
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\beta = \frac {\bar {\sigma} _ {c} (\tilde {\varepsilon} _ {c} ^ {p l})}{\bar {\sigma} _ {t} (\tilde {\varepsilon} _ {t} ^ {p l})} (1 - \alpha) - (1 + \alpha),
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$$
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$$
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\gamma = \frac {3 (1 - K _ {c})}{2 K _ {c} - 1}.
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$$
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Here,
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$$
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\hat {\bar {\sigma}} _ {\mathrm{max}}
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$$
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$$
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\sigma_ {b 0} / \sigma_ {c 0}
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$$
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$$
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K _ {c}
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$$
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is the maximum principal effective stress;
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is the ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress (the default value is );
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is the ratio of the second stress invariant on the tensile meridian, $q _ { \mathrm { ( T M ) } }$ , to that on the compressive meridian, $q _ { \mathrm { ( C M ) } }$ , at initial yield for any given value of the pressure invariant p such that the maximum principal stress is negative, $\hat { \sigma } _ { \mathrm { m a x } } < 0$ (see Figure 23.6.3–8); it must satisfy the condition $0 . 5 < K _ { c } \leq 1 . 0$ (the default value is );
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$$
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\bar {\sigma} _ {t} \big (\tilde {\varepsilon} _ {t} ^ {p l} \big)
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$$
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$$
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\bar {\sigma} _ {c} \big (\tilde {\varepsilon} _ {c} ^ {p l} \big)
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$$
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is the effective tensile cohesion stress; and
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is the effective compressive cohesion stress.
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<details>
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<summary>text_image</summary>
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-S₂
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Kc = 2/3
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-S₁
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Kc = 1
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(T.M.)
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(C.M.)
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-S₃
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</details>
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Figure 23.6.3–8 Yield surfaces in the deviatoric plane, corresponding to different values of $K _ { c }$ .
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Typical yield surfaces are shown in Figure 23.6.3–8 on the deviatoric plane and in Figure 23.6.3–9 for plane stress conditions.
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# Nonassociated flow
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Because plastic flow is nonassociated, the use of concrete damaged plasticity results in a nonsymmetric material stiffness matrix. Therefore, to obtain an acceptable rate of convergence in Abaqus/Standard, the unsymmetric matrix storage and solution scheme should be used. Abaqus/Standard will automatically activate the unsymmetric solution scheme if concrete damaged plasticity is used in the analysis. If desired, you can turn off the unsymmetric solution scheme for a particular step (see “Defining an analysis,” Section 6.1.2).
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# Viscoplastic regularization
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Material models exhibiting softening behavior and stiffness degradation often lead to severe convergence difficulties in implicit analysis programs, such as Abaqus/Standard. A common technique to overcome some of these convergence difficulties is the use of a viscoplastic regularization of the constitutive equations, which causes the consistent tangent stiffness of the softening material to become positive for sufficiently small time increments.
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The concrete damaged plasticity model can be regularized in Abaqus/Standard using viscoplasticity by permitting stresses to be outside of the yield surface. We use a generalization of the Duvaut-Lions regularization, according to which the viscoplastic strain rate tensor, $\dot { \varepsilon } _ { v } ^ { p l }$ , is defined as
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<details>
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<summary>text_image</summary>
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uniaxial tension
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1/1-α (q̅ - 3α p̅ + βσ̂₂) = σc₀
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uniaxial compression
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σ̂₂
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σ₁₀
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σ̂₁
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biaxial tension
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1/1-α (q̅ - 3α p̅ + βσ̂₁) = σc₀
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σc₀
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1/1-α (q̅ - 3α p̅) = σc₀
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(biaxial compression
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(biaxial tension)
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</details>
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Figure 23.6.3–9 Yield surface in plane stress.
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$$
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\dot {\varepsilon} _ {v} ^ {p l} = \frac {1}{\mu} (\varepsilon^ {p l} - \varepsilon_ {v} ^ {p l}).
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$$
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Here $\mu$ is the viscosity parameter representing the relaxation time of the viscoplastic system, and $\varepsilon ^ { p l }$ is the plastic strain evaluated in the inviscid backbone model.
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Similarly, a viscous stiffness degradation variable, $d _ { v }$ , for the viscoplastic system is defined as
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$$
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\dot {d} _ {v} = \frac {1}{\mu} (d - d _ {v}),
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$$
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where d is the degradation variable evaluated in the inviscid backbone model. The stress-strain relation of the viscoplastic model is given as
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$$
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\pmb {\sigma} = (1 - d _ {v}) \mathbf {D} _ {0} ^ {e l}: (\pmb {\varepsilon} - \pmb {\varepsilon} _ {v} ^ {p l}).
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$$
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Using the viscoplastic regularization with a small value for the viscosity parameter (small compared to the characteristic time increment) usually helps improve the rate of convergence of the
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model in the softening regime, without compromising results. The basic idea is that the solution of the viscoplastic system relaxes to that of the inviscid case as $t / \mu \to \infty$ , where t represents time. You can specify the value of the viscosity parameter as part of the concrete damaged plasticity material behavior definition. If the viscosity parameter is different from zero, output results of the plastic strain and stiffness degradation refer to the viscoplastic values, $\varepsilon _ { v } ^ { p l }$ and $d _ { v }$ . In Abaqus/Standard the default value of the viscosity parameter is zero, so that no viscoplastic regularization is performed.
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# Material damping
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The concrete damaged plasticity model can be used in combination with material damping (see “Material damping,” Section 26.1.1). If stiffness proportional damping is specified, Abaqus calculates the damping stress based on the undamaged elastic stiffness. This may introduce large artificial damping forces on elements undergoing severe damage at high strain rates.
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# Visualization of “crack directions”
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Unlike concrete models based on the smeared crack approach, the concrete damaged plasticity model does not have the notion of cracks developing at the material integration point. However, it is possible to introduce the concept of an effective crack direction with the purpose of obtaining a graphical visualization of the cracking patterns in the concrete structure. Different criteria can be adopted within the framework of scalar-damage plasticity for the definition of the direction of cracking. Following Lubliner et. al. (1989), we can assume that cracking initiates at points where the tensile equivalent plastic strain is greater than zero, $\tilde { \varepsilon } _ { t } ^ { p l } > 0$ , and the maximum principal plastic strain is positive. The direction of the vector normal to the crack plane is assumed to be parallel to the direction of the maximum principal plastic strain. This direction can be viewed in the Visualization module of Abaqus/CAE.
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Abaqus/CAE Usage: Visualization module:
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Result→Field Output: PE, Max. Principal
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Plot→Symbols
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# Elements
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Abaqus offers a variety of elements for use with the concrete damaged plasticity model: truss, shell, plane stress, plane strain, generalized plane strain, axisymmetric, and three-dimensional elements. Most beam elements can be used; however, beam elements in space that include shear stress caused by torsion and do not include hoop stress (such as B31, B31H, B32, B32H, B33, and B33H) cannot be used. Thin-walled, open-section beam elements and PIPE elements can be used with the concrete damaged plasticity model in Abaqus/Standard.
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For general shell analysis more than the default number of five integration points through the thickness of the shell should be used; nine thickness integration points are commonly used to model progressive failure of the concrete through the thickness with acceptable accuracy.
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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 relate specifically to material points in the concrete damaged plasticity model:
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<table><tr><td>DAMAGEC</td><td>Compressive damage variable, $d_{c}$ .</td></tr><tr><td>DAMAGET</td><td>Tensile damage variable, $d_{t}$ .</td></tr><tr><td>PEEQ</td><td>Compressive equivalent plastic strain, $\tilde{\varepsilon}_{c}^{pl}$ .</td></tr><tr><td>PEEQT</td><td>Tensile equivalent plastic strain, $\tilde{\varepsilon}_{t}^{pl}$ .</td></tr><tr><td>SDEG</td><td>Stiffness degradation variable, $d$ .</td></tr><tr><td>DMENER</td><td>Energy dissipated per unit volume by damage.</td></tr><tr><td>ELDMD</td><td>Total energy dissipated in the element by damage.</td></tr><tr><td>ALLDMD</td><td>Energy dissipated in the 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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# Additional references
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• Hillerborg, A., M. Modeer, and P. E. Petersson, “Analysis of Crack Formation and Crack Growth in Concrete by Means of Fracture Mechanics and Finite Elements,” Cement and Concrete Research, vol. 6, pp. 773–782, 1976.
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• Lee, J., and G. L. Fenves, “Plastic-Damage Model for Cyclic Loading of Concrete Structures,” Journal of Engineering Mechanics, vol. 124, no. 8, pp. 892–900, 1998.
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• Lubliner, J., J. Oliver, S. Oller, and E. Oñate, “A Plastic-Damage Model for Concrete,” International Journal of Solids and Structures, vol. 25, pp. 299–329, 1989.
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# 23.7 Permanent set in rubberlike materials
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• “Permanent set in rubberlike materials,” Section 23.7.1
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# 23.7.1 PERMANENT SET IN RUBBERLIKE MATERIALS
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Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE
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# References
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• “Combining material behaviors,” Section 21.1.3
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• “Hyperelastic behavior of rubberlike materials,” Section 22.5.1
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• “Classical metal plasticity,” Section 23.2.1
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• \*HYPERELASTIC
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• \*MULLINS EFFECT
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• \*PLASTIC
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# Overview
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# This feature:
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• is intended for modeling permanent set observed in filled elastomers and thermoplastics;
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• is based on multiplicative split of the deformation gradient;
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• is based on the theory of incompressible isotropic hardening plasticity;
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• can be used with any isotropic hyperelasticity model;
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• can be combined with Mullins effects; and
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• cannot be used to model viscoelastic or hysteresis effects or with the steady-state transport procedure.
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# Material behavior
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The real behavior of filled rubber elastomers under cyclic loading conditions is quite complex as shown in Figure 23.7.1–1. The observed mechanical behaviors are progressive damage resulting in a reduction of load carrying capacity with each cycle, stress softening (also known as Mullins effect) upon reloading after the first unloading from a previously attained maximum strain level, hysteretic dissipation of energy, and permanent set. This section is concerned with modeling permanent set; therefore, the idealized representation of permanent set is described below.
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# Idealized material behavior
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From Figure 23.7.1–1 it is clear that the observed permanent set is different for each cycle, but the material has a tendency to stabilize after a number of cycles of loading between zero stress and a given level of strain. For a given load level along the primary loading path shown with the dashed line in Figure 23.7.1–1, the idealized representation of permanent set will be a single strain value after unloading has taken place. Since rate and time effects are ignored in this model, idealized loading and unloading take place along the same path, whether Mullins effect is included or not.
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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 23.7.1–1 Typical behavior of a filled elastomer.
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The permanent set behavior is captured by isotropic hardening Mises plasticity with an associated flow rule. In the context of finite elastic strains associated with the underlying rubberlike material, plasticity is modeled using a multiplicative split of the deformation gradient into elastic and plastic components:
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$$
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\mathbf {F} = \mathbf {F} ^ {e} \cdot \mathbf {F} ^ {p},
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$$
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where $\mathbf { F } ^ { e }$ is the elastic part of the deformation gradient (representing the hyperelastic behavior) and $\mathbf { F } ^ { p }$ is the plastic part of the deformation gradient (representing the stress-free intermediate configuration).
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An example of modeling permanent set along with Mullins effect for a rubberlike material can be found in “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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# Specifying permanent set
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The primary hyperelastic behavior can be defined by using any of the hyperelastic material models (see “Hyperelastic behavior of rubberlike materials,” Section 22.5.1). If test data input is used to define the hyperelastic response of the material, the data must be specified with respect to the stress-free intermediate configuration after unloading has taken place.
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Permanent set can be defined through an isotropic hardening function in terms of the yield stress and the equivalent plastic strain. In this case the yield stress is the (effective) Kirchoff stress on the primary loading path from which unloading takes place, and the equivalent plastic strain is the corresponding logarithmic permanent set observed in the material. If is the true (Cauchy) stress, Kirchoff stress is defined as $J { \boldsymbol { \sigma } } _ { ; }$ , where is the determinant of .
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