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269 lines
16 KiB
Markdown
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# 22.8 Nonlinear viscoelasticity
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• “Hysteresis in elastomers,” Section 22.8.1
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• “Parallel rheological framework,” Section 22.8.2
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# 22.8.1 HYSTERESIS IN ELASTOMERS
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Products: Abaqus/Standard Abaqus/CAE
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# References
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• “Elastic behavior: overview,” Section 22.1.1
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• \*HYSTERESIS
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• “Defining hysteretic behavior for an isotropic hyperelastic material model” in “Defining elasticity,” Section 12.9.1 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
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# Overview
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The hysteresis material model:
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• defines strain-rate-dependent, hysteretic behavior of materials that undergo comparable elastic and inelastic strains;
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• provides inelastic response only for shear distortional behavior—the response to volumetric deformations is purely elastic;
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• can be used only in conjunction with “Hyperelastic behavior of rubberlike materials,” Section 22.5.1, to define the elastic response of the material—the elasticity can be defined either in terms of the instantaneous moduli or the long-term moduli;
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• is active during a static analysis (“Static stress analysis,” Section 6.2.2), a quasi-static analysis (“Quasi-static analysis,” Section 6.2.5), or a transient dynamic analysis using direct integration (“Implicit dynamic analysis using direct integration,” Section 6.3.2)—it cannot be used in fully coupled temperature-displacement analysis (“Fully coupled thermal-stress analysis,” Section 6.5.3), fully coupled thermal-electrical-structural analysis (“Fully coupled thermal-electrical-structural analysis,” Section 6.7.4), or steady-state transport analysis (“Steady-state transport analysis,” Section 6.4.1);
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• cannot be used to model temperature-dependent creep material properties—however, the elastic material properties can be temperature dependent; and
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• uses unsymmetric matrix storage and solution by default.
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# Strain-rate-dependent material behavior for elastomers
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Nonlinear strain-rate dependence of elastomers is modeled by decomposing the mechanical response into that of an equilibrium network (A) corresponding to the state that is approached in long-time stress relaxation tests and that of a time-dependent network (B) that captures the nonlinear rate-dependent deviation from the equilibrium state. The total stress is assumed to be the sum of the stresses in the two networks. The deformation gradient, , is assumed to act on both networks and is decomposed into elastic and inelastic parts in network B according to the multiplicative decomposition $\mathbf { F } = \mathbf { F } _ { B } ^ { e } \cdot \mathbf { F } _ { B } ^ { c r }$ The nonlinear rate-dependent material model is capable of reproducing the hysteretic behavior of elastomers
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subjected to repeated cyclic loading. It does not model “Mullins effect”—the initial softening of an elastomer when it is first subjected to a load.
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The material model is defined completely by:
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• a hyperelastic material model that characterizes the elastic response of the model;
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• a stress scaling factor, $\begin{array} { r } { \pmb { S } , } \end{array}$ that defines the ratio of the stress carried by network B to the stress carried by network A under instantaneous loading; i.e., identical elastic stretching in both networks;
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• a positive exponent, $m ,$ generally greater than 1, characterizing the effective stress dependence of the effective creep strain rate in network B;
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• an exponent, C, restricted to lie in , characterizing the creep strain dependence of the effective creep strain rate in network B;
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• a nonnegative constant, A, in the expression for the effective creep strain rate—this constant also maintains dimensional consistency in the equation; and
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• a constant, $E ,$ in the expression for the effective creep strain rate—this constant regularizes the creep strain rate near the undeformed state.
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The effective creep strain rate in network B is given by the expression
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$$
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\dot {\epsilon} _ {B} ^ {c r} = A [ \lambda_ {B} ^ {c r} - 1 + E ] ^ {C} (\sigma_ {B}) ^ {m},
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$$
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where $\dot { \epsilon } _ { B } ^ { c r }$ is the effective creep strain rate in network $\mathrm { B } , \lambda _ { B } ^ { c r } - 1$ is the nominal creep strain in network $\mathrm { B } ,$ and $\sigma _ { B }$ is the effective stress in network B. The chain stretch in network $\mathbf { B } , \lambda _ { B } ^ { c r }$ , is defined as
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$$
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\lambda_ {B} ^ {c r} = \sqrt {\frac {1}{3} \mathbf {I} : \mathbf {C} _ {B} ^ {c r}},
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$$
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where ${ \bf C } _ { B } ^ { c r } = { \bf F } _ { B } ^ { c r } \cdot { \bf F } _ { B } ^ { c r }$ . The effective stress in network B is defined as $\sigma _ { B } = \sqrt { \frac { 3 } { 2 } { \bf S } _ { B } : { \bf S } _ { B } }$ , where $\mathbf { S } _ { B }$ is the deviatoric Cauchy stress tensor.
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# Defining strain-rate-dependent material behavior for elastomers
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The elasticity of the model is defined by a hyperelastic material model. You input the stress scaling factor and the creep parameters for network B directly when you define the hysteresis material model. Typical values of the material parameters for a common elastomer are $\begin{array} { r } { S = \mathrm { { 1 . 6 , ~ } } A = \frac { 5 } { ( \sqrt { 3 } ) ^ { m } } ( \sec ) ^ { - 1 } ( \mathrm { { M P a } ) ^ { - m } } } \end{array}$ (sec)−1 (MPa)−m , $m = 4 , C = - 1 . 0$ , and $E = 0 . 0 1$ (Bergstrom and Boyce, 1998; 2001).
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Input File Usage: Use both of the following options within the same material data block:
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\*HYSTERESIS
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\*HYPERELASTIC
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Hyperelastic: Suboptions→Hysteresis
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The input of the parameter is not supported in Abaqus/CAE.
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# Elements
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The use of the hysteresis material model is restricted to elements that can be used with hyperelastic materials (“Hyperelastic behavior of rubberlike materials,” Section 22.5.1). In addition, this model cannot be used with elements based on the plane stress assumption (shell, membrane, and continuum plane stress elements). Hybrid elements can be used with this model only when the accompanying hyperelasticity definition is completely incompressible. When this model is used with reduced-integration elements, the instantaneous elastic moduli are used to calculate the default hourglass stiffness.
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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 have special meaning if hysteretic behavior is defined:
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EE Elastic strain corresponding to the stress state at time t and the instantaneous elastic material properties.
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CE Equivalent creep strain defined as the difference between the total strain and the elastic strain.
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These strain measures are used to approximate the strain energy, SENER, and the viscous dissipation, CENER. These approximations may lead to underestimation of the strain energy and overestimation of the viscous dissipation since the effects of internal stresses on these energy quantities are neglected. This inaccuracies may be particularly noticeable in the case of nonmonotonic loading.
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# Additional references
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• Bergstrom, J. S., and M. C. Boyce, “Constitutive Modeling of the Large Strain Time-Dependent Behavior of Elastomers,” Journal of the Mechanics and Physics of Solids, vol. 46, no. 5, pp. 931–954, May 1998.
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• Bergstrom, J. S., and M. C. Boyce, “Constitutive Modeling of the Time-Dependent and Cyclic Loading of Elastomers and Application to Soft Biological Tissues,” Mechanics of Materials, vol. 33, no. 9, pp. 523–530, 2001.
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# 22.8.2 PARALLEL RHEOLOGICAL FRAMEWORK
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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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• “UCREEPNETWORK,” Section 1.1.23 of the Abaqus User Subroutines Reference Guide
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• “UTRSNETWORK,” Section 1.1.57 of the Abaqus User Subroutines Reference Guide
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• “VUCREEPNETWORK,” Section 1.2.13 of the Abaqus User Subroutines Reference Guide
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• \*HYPERELASTIC
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• \*MULLINS EFFECT
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• \*PLASTIC
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• \*TRS
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• \*VISCOELASTIC
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# Overview
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The parallel rheological framework:
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• is intended for modeling polymers and elastomeric materials that exhibit permanent set and nonlinear viscous behavior and undergo large deformations;
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• consists of multiple viscoelastic networks and, optionally, an elastoplastic network in parallel;
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• uses a hyperelastic material model to specify the elastic response;
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• can be combined with Mullins effect;
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• bases the elastoplastic response on multiplicative split of the deformation gradient and the theory of incompressible isotropic hardening plasticity;
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• can include nonlinear kinematic hardening with multiple backstresses in the elastoplastic response in Abaqus/Standard; and
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• uses multiplicative split of the deformation gradient and a flow rule derived from a creep potential to specify the viscous behavior.
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# Material behavior
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The parallel rheological framework allows definition of a nonlinear viscoelastic-elastoplastic model consisting of multiple networks connected in parallel, as shown in Figure 22.8.2–1. The number of viscoelastic networks, N, can be arbitrary; however, at most one equilibrium network (network in Figure 22.8.2–1) is allowed in the model. The equilibrium network response might be purely elastic or
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<details>
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<summary>text_image</summary>
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0
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1
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2
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</details>
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Figure 22.8.2–1 Nonlinear viscoelastic-elastoplastic model with multiple parallel networks.
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elastoplastic. In addition, it might include Mullins effect to predict material softening. The definition of the equilibrium network is optional. If it is not defined, the stress in the material will relax completely over time.
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The model can be used to predict complex behavior of materials subjected to finite strains, which cannot be modeled accurately using other models available in Abaqus. An example of such complex behavior is depicted in Figure 22.8.2–2, which shows normalized stress relaxation curves for three different strain levels. This behavior can be modeled accurately using the nonlinear viscoelastic model depicted in Figure 22.8.2–3, which can be defined within the framework; but it cannot be captured with the linear viscoelastic model (see “Time domain viscoelasticity,” Section 22.7.1). In the latter case, the three curves would coincide.
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# Elastic behavior
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The elastic part of the response for all the networks is specified using the hyperelastic material model. Any of the hyperelastic models available in Abaqus can be used (see “Hyperelastic behavior of rubberlike materials,” Section 22.5.1). The same hyperelastic material definition is used for all the networks, scaled by a stiffness ratio specific to each network. Consequently, only one hyperelastic material definition is required by the model along with the stiffness ratio for each network. The elastic response can be specified by defining either the instantaneous response or the long-term response.
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# Equilibrium network behavior
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In addition to the elastic response described above, the response of the equilibrium network can include plasticity and Mullins effect to predict material softening. If the plastic response is defined using
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<details>
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<summary>line</summary>
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| Time | sigma1 | sigma2 | sigma3 |
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|------|--------|--------|--------|
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| 1.0 | 1.0000 | 1.0000 | 1.0000 |
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| 1.1 | 0.9500 | 0.9200 | 0.9400 |
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| 1.2 | 0.8800 | 0.8600 | 0.9000 |
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| 1.3 | 0.8400 | 0.8200 | 0.8700 |
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| 1.4 | 0.8200 | 0.8000 | 0.8500 |
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| 1.5 | 0.8100 | 0.7900 | 0.8400 |
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| 1.6 | 0.8050 | 0.7850 | 0.8350 |
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| 1.7 | 0.8030 | 0.7800 | 0.8300 |
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| 1.8 | 0.8020 | 0.7750 | 0.8250 |
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| 1.9 | 0.8010 | 0.7700 | 0.8200 |
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| 2.0 | 0.8005 | 0.7650 | 0.8150 |
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| 2.1 | 0.8003 | 0.7600 | 0.8100 |
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| 2.2 | 0.8002 | 0.7550 | 0.8080 |
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| 2.3 | 0.8001 | 0.7500 | 0.8060 |
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| 2.4 | 0.8000 | 0.7450 | 0.8040 |
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| 2.5 | 0.8000 | 0.7400 | 0.8020 |
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| 2.6 | 0.8000 | 0.7350 | 0.8010 |
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| 2.7 | 0.8000 | 0.7300 | 0.8005 |
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| 2.8 | 0.8000 | 0.7250 | 0.8003 |
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| 2.9 | 0.8000 | 0.7200 | 0.8002 |
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| 3.0 | 0.8000 | 0.7150 | 0.8001 |
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| 3.1 | 0.8000 | 0.7100 | 0.8000 |
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| 3.2 | 0.8000 | 0.7050 | 0.8000 |
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| 3.3 | 0.8000 | 0.7000 | 0.8000 |
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| 3.4 | 0.8000 | 0.6950 | 0.8000 |
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| 3.5 | 0.8000 | 0.6900 | 0.8000 |
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| 3.6 | 0.8000 | 0.6850 | 0.8000 |
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| 3.7 | 0.8000 | 0.6800 | 0.8000 |
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| 3.8 | 0.8000 | 0.6750 | 0.8000 |
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| 3.9 | 0.8000 | 0.6700 | 0.8000 |
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| 4.0 | 0.8000 | 0.6650 | 0.8000 |
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</details>
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Figure 22.8.2–2 Normalized stress relaxation curves for three different strain levels.
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<details>
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<summary>text_image</summary>
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0
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</details>
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Figure 22.8.2–3 Nonlinear viscoelastic model with multiple parallel networks.
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isotropic hardening, the response in the equilibrium network is equivalent to that of the permanent set model available in Abaqus (see “Permanent set in rubberlike materials,” Section 23.7.1, for a detailed description of the model). In Abaqus/Standard the nonlinear kinematic hardening model with multiple backstresses can be specified in addition to isotropic plastic hardening. The nonlinear kinematic hardening model is a generalization of the model used for metal plasticity. See “Models for metals
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subjected to cyclic loading,” Section 23.2.2, for a detailed description of the model, with the difference that the Cauchy stress is replaced with the Kirchhoff stress in the current formulation.
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# Viscous behavior
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Viscous behavior must be defined for each viscoelastic network. It is modeled by assuming the multiplicative split of the deformation gradient and the existence of the creep potential, $G ^ { c r }$ , from which the flow rule is derived. In the multiplicative split the deformation gradient is expressed as
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$$
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\mathbf {F} = \mathbf {F} ^ {e} \cdot \mathbf {F} ^ {c r},
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$$
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where $\mathbf { F } ^ { e }$ is the elastic part of the deformation gradient (representing the hyperelastic behavior) and $\mathbf { F } ^ { c r }$ is the creep part of the deformation gradient (representing the stress-free intermediate configuration). The creep potential is assumed to have the general form
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$$
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G ^ {c r} = G ^ {c r} (\pmb {\sigma}),
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$$
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where $\sigma$ is the Cauchy stress. If the potential is specified, the flow rule can be obtained from
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$$
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\mathbf {D} ^ {c r} = \dot {\lambda} \frac {\partial G ^ {c r} (\pmb {\sigma})}{\partial \pmb {\sigma}},
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$$
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where $\mathbf { D } ^ { c r }$ is the symmetric part of the velocity gradient, ${ \bf L } ^ { c r }$ , expressed in the current configuration and $\dot { \lambda }$ is the proportionality factor. In this model the creep potential is given by
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$$
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G ^ {c r} = \bar {q},
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$$
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and the proportionality factor is taken as $\dot { \lambda } = \dot { \bar { \varepsilon } } ^ { c r }$ , where $\bar { q }$ is the equivalent deviatoric Cauchy stress and $\dot { \bar { \varepsilon } } ^ { c r }$ is the equivalent creep stain rate. In this case the flow rule has the form
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$$
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\mathbf {D} ^ {c r} = \frac {3}{2 \bar {q}} \dot {\bar {\varepsilon}} ^ {c r} \bar {\pmb {\sigma}},
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$$
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or, equivalently
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$$
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\mathbf {D} ^ {c r} = \frac {3}{2 \tilde {q}} \dot {\bar {\varepsilon}} ^ {c r} \bar {\boldsymbol {\tau}},
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$$
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where $\tau = J \sigma$ is the Kirchhoff stress, is the determinant of , is the deviatoric Cauchy stress, is the deviatoric Kirchhoff stress, and $\tilde { q } = J \bar { q } .$ . To complete the derivation, the evolution law for $\dot { \bar { \varepsilon } } ^ { c r }$ must be provided. In this model $\dot { \bar { \varepsilon } } ^ { c r }$ can be determined from either a power-law strain hardening model or a hyperborlic-sine model.
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# Power law model
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The power law model is available in the form
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