ProductsAbaqus/StandardAbaqus/Explicit Material behaviorThe parallel rheological framework allows definition of a nonlinear viscoelasticelastoplastic model consisting of multiple networks connected in parallel, as shown in Figure 1. Figure 1. Nonlinear viscoelasticelastoplastic model with multiple parallel
networks.
The number of viscoelastic networks, N, can be arbitrary; however, at most one equilibrium network (network $0$ in Figure 1) is allowed in the model. The equilibrium network response might be purely elastic or 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. 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 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 3, which can be defined within the framework; but it cannot be captured with the linear viscoelastic model (see Time domain viscoelasticity). In the latter case, the three curves would coincide. Figure 2. Normalized stress relaxation curves for three different strain
levels.
Figure 3. Nonlinear viscoelastic model with multiple parallel networks.
Elastic behaviorThe 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). 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 longterm response. Equilibrium network behaviorIn 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 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 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 subjected to cyclic loading for a detailed description of the model, with the difference that the Cauchy stress is replaced with the Kirchhoff stress in the current formulation. Viscous behaviorViscous 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, ${G}^{cr}$, from which the flow rule is derived. In the multiplicative split the deformation gradient is expressed as $$\mathbf{F}={\mathbf{F}}^{e}\cdot {\mathbf{F}}^{cr},$$
where ${\mathbf{F}}^{e}$ is the elastic part of the deformation gradient (representing the hyperelastic behavior) and ${\mathbf{F}}^{cr}$is the creep part of the deformation gradient (representing the stressfree intermediate configuration). The creep potential is assumed to have the general form $${G}^{cr}={G}^{cr}\left(\mathit{\sigma}\right),$$
where $\mathit{\sigma}$ is the Cauchy stress. If the potential is specified, the flow rule can be obtained from $${\mathbf{D}}^{cr}=\dot{\lambda}\frac{\partial {G}^{cr}\left(\mathit{\sigma}\right)}{\partial \mathit{\sigma}},$$
where ${\mathbf{D}}^{cr}$ is the symmetric part of the velocity gradient, ${\mathbf{L}}^{cr}$, expressed in the current configuration and $\dot{\lambda}$ is the proportionality factor. In this model the creep potential is given by $${G}^{cr}=\overline{q},$$
and the proportionality factor is taken as $\dot{\lambda}={\dot{\overline{\epsilon}}}^{cr}$, where $\overline{q}$ is the equivalent deviatoric Cauchy stress and ${\dot{\overline{\epsilon}}}^{cr}$ is the equivalent creep stain rate. In this case the flow rule has the form $${\mathbf{D}}^{cr}=\frac{3}{2\overline{q}}{\dot{\overline{\epsilon}}}^{cr}\overline{\mathit{\sigma}},$$
or, equivalently $${\mathbf{D}}^{cr}=\frac{3}{2\stackrel{~}{q}}{\dot{\overline{\epsilon}}}^{cr}\overline{\mathit{\tau}},$$
where $\mathit{\tau}=J\mathit{\sigma}$ is the Kirchhoff stress, $J$ is the determinant of $\mathbf{F}$, $\overline{\mathit{\sigma}}$ is the deviatoric Cauchy stress, $\stackrel{}{\mathbf{\tau}}$ is the deviatoric Kirchhoff stress, and $\stackrel{~}{q}=J\overline{q}$. To complete the derivation, the evolution law for ${\dot{\overline{\epsilon}}}^{cr}$ must be provided. In this model ${\dot{\overline{\epsilon}}}^{cr}$ can be determined from either a powerlaw strain hardening model or a hyperbolicsine model. Power law modelThe power law model is available in the form $${\dot{\overline{\epsilon}}}^{cr}={\dot{\epsilon}}_{0}{\left({\left(\frac{\stackrel{~}{q}}{{q}_{0}+a<p>}\right)}^{n}{\left[\left(m+1\right){\overline{\epsilon}}^{cr}\right]}^{m}\right)}^{\frac{1}{m+1}},$$
where
Powerlaw strain hardening modelThe powerlaw strain hardening model is available in the form $${\dot{\overline{\epsilon}}}^{cr}={\left(A{\stackrel{~}{q}}^{n}{\left[\left(m+1\right){\overline{\epsilon}}^{cr}\right]}^{m}\right)}^{\frac{1}{m+1}},$$
where
Hyperbolicsine law modelThe hyperbolicsine law is available in the form $${\dot{\overline{\epsilon}}}^{cr}=A{\left(\mathrm{sinh}B\stackrel{~}{q}\right)}^{n},$$
where
BergstromBoyce modelThe BergstromBoyce model is available in the form $${\dot{\overline{\epsilon}}}^{cr}=A{\left({\lambda}^{cr}1+E\right)}^{C}{\left(\stackrel{~}{q}\right)}^{m},$$
where $${\lambda}^{cr}=\sqrt{\frac{1}{3}\mathbf{I}:{\mathbf{C}}^{cr}},$$
and
The response of the network defined by the BergstromBoyce model is very similar to the response of the timedependent network in the hysteresis model (see Hysteresis in elastomers). However, there are also important differences between the models. In the BergstromBoyce model the equivalent Kirchhoff stress is used instead of the equivalent Cauchy stress, which is used in the hysteresis model. (The two stress measures become equivalent for the case of incompressible materials.) In addition, the material parameters, A, in the models differ by a factor of $3/2$. The parameter in the hysteresis model must be multiplied by $3/2$ to make the parameters equivalent. Userdefined model in Abaqus/StandardA userdefined creep model is available of the following general form: $${\dot{\overline{\epsilon}}}^{cr}={g}^{cr}\left({\overline{\epsilon}}^{cr},{I}_{1}^{cr},{\overline{I}}_{1},{\overline{I}}_{2},J,p,\stackrel{~}{q},t,\theta ,FV\right),$$
where $${I}_{1}^{cr}=\mathbf{I}:{\mathbf{C}}^{cr},$$
and
The tensor $\overline{\mathbf{B}}$ is defined in UCREEPNETWORK. Thermal expansionOnly isotropic thermal expansion is permitted with nonlinear viscoelastic materials (Thermal expansion). Defining viscoelastic responseThe nonlinear viscoelastic response is defined by specifying the identifier, stiffness ratio, and creep law for each viscoelastic network. Specifying network identifierEach viscoelastic network in the material model must be assigned a unique network identifier or network id. The network identifiers must be consecutive integers starting with 1. The order in which they are specified is not important. Input File Usage Use the following option to specify the network identifier: VISCOELASTIC, NONLINEAR, NETWORKID=networkId Defining the stiffness ratioThe contribution of each network to the overall response of the material is determined by the value of the stiffness ratio, $r$, which is used to scale the elastic response of the network material. The sum of the stiffness ratios of the viscoelastic networks must be smaller than or equal to 1. If the sum of the ratios is equal to 1, the purely elastic equilibrium network is not created. If the sum of the ratios is smaller than 1, the equilibrium network is created with a stiffness ratio, ${r}_{0}$, equal to $${r}_{0}=1\stackrel{N}{\sum _{k=1}}{r}_{k},$$
where $N$ denotes the number of viscoelastic networks and ${r}_{k}$ is the stiffness ratio of network $k$. You can specify the stiffness ratio to remain constant during the analysis or to vary as a function of temperature and predefined field variables. Defining a constant stiffness ratioYou can specify that the stiffness ratio remains constant during the analysis: Input File Usage VISCOELASTIC, NONLINEAR, SRATIO=ratio Defining a temperature and fieldvariable dependent stiffness ratioAlternatively, you can define the stiffness ratio as a function of temperature and predefined field variables. Input File Usage NETWORK STIFFNESS RATIO, N=nNetworks, DEPENDENCIES=n Specifying the creep lawThe definition of creep behavior in Abaqus/Standard is completed by specifying the creep law. Power law creep modelThe power law model is defined by specifying five material parameters: ${q}_{0}$, n, m, a, and ${\dot{\epsilon}}_{0}$. The parameter ${\dot{\epsilon}}_{0}$ must be positive. It is introduced for dimensional consistency, and its default value is 1.0. For physically reasonable behavior ${q}_{0}$ and n must be positive, a must be nonnegative (the default is 0.0), and −1 < m ≤ 0. Input File Usage VISCOELASTIC, NONLINEAR, LAW=POWER LAW Strain hardening power law creep modelThe strain hardening law is defined by specifying three material parameters: A, n, and m. For physically reasonable behavior A and n must be positive and −1 < m ≤ 0. Input File Usage VISCOELASTIC, NONLINEAR, LAW=STRAIN Hyperbolic sine creep modelThe hyperbolic sine creep law is specified by providing three nonnegative parameters: A, B, and n. Input File Usage VISCOELASTIC, NONLINEAR, LAW=HYPERB BergstromBoyce creep modelThe BergstromBoyce creep law is specified by providing four parameters: A, m, C, and E. The parameters A and E must be nonnegative, the parameter m must be positive, and the parameter C must lie in $\left[1,0\right]$. Input File Usage VISCOELASTIC, NONLINEAR, LAW=BERGSTROMBOYCE Userdefined creep modelAn alternative method for defining the creep law involves using user subroutine UCREEPNETWORK in Abaqus/Standard or VUCREEPNETWORK in Abaqus/Explicit. Optionally, you can specify the number of property values needed as data in the user subroutine. Input File Usage VISCOELASTIC, NONLINEAR, LAW=USER, PROPERTIES=n Numerical difficultiesDepending on the choice of units, the value of A in the strain powerlaw, hyperbolicsine, and BergstromBoyce models may be very small for typical creep strain rates. If A is less than 10^{−27}, numerical difficulties can cause errors in the material calculations; therefore, a different system of units should be used to avoid such difficulties in the calculation of creep strain increments. In such cases it is recommended to replace the strain powerlaw model with the power law model, which does not have the limitation described above. The strain powerlaw model is a special case of the power law model obtained by setting ${\dot{\epsilon}}_{0}=1$, $a=0$, and ${q}_{0}={A}^{1/n}$. Thermorheologically simple temperature effectsThermorheologically simple temperature effects can be included for each viscoelastic network. In this case the creep law is modified and takes the following form: $$\frac{d{\overline{\epsilon}}^{cr}}{d\tau}=\frac{1}{{a}_{T}\left(\theta \right)}{g}^{cr}\left({\overline{\epsilon}}^{cr},{I}_{1}^{cr},{\overline{I}}_{1},{\overline{I}}_{2},J,p,\stackrel{~}{q},\tau \right),$$
where $\tau $ and ${a}_{T}\left(\theta \right)$ denote the reduced time and the shift function, respectively. The reduced time is related to the actual time through the integral differential equation $$\tau ={\int}_{0}^{t}\frac{d{t}^{\prime}}{{a}_{T}\left(\theta \right)},\frac{d\tau}{dt}=\frac{1}{{a}_{T}\left(\theta \right)}.$$
Abaqus supports the following forms of the shift function: the WilliamsLandelFerry (WLF) form and the Arrhenius form (see Thermorheologically simple temperature effects). In addition, userdefined forms can be specified in Abaqus/Standard. Userdefined form in Abaqus/StandardAn alternative method for specifying the shift function involves using user subroutine UTRSNETWORK. Optionally, you can specify the number of property values needed as data in the user subroutine. Input File Usage TRS, DEFINITION=USER, PROPERTIES=n Material response in different analysis stepsIn Abaqus/Standard the material is active during all stress/displacement procedure types. However, the creep effects are taken into account only in quasistatic (Quasistatic analysis), coupled temperaturedisplacement (Fully coupled thermalstress analysis), directintegration implicit dynamic (Implicit dynamic analysis using direct integration), and steadystate transport (Steadystate transport analysis) analyses. If the material is used in a steadystate transport analysis, it cannot include plasticity. In other stress/displacement procedures the evolution of the state variables is suppressed and the creep strain remains unchanged. In Abaqus/Explicit the creep effects are always active. ElementsThe parallel rheological framework is available with continuum elements that include mechanical behavior (elements that have displacement degrees of freedom), except for onedimensional elements. The parallel rheological framework is also supported with elements that use the plane stress formulation such as solid plane stress elements, membranes, and shells. However, those elements are not supported with compressible materials. If a compressible material is specified with plane stress elements, Abaqus will modify the material to make it incompressible and issue an informational message. OutputIn addition to the standard output identifiers available in Abaqus (Abaqus/Standard output variable identifiers and Abaqus/Explicit output variable identifiers), the following variables have special meaning for the nonlinear viscoelastic material model:
In the above definitions ${r}_{k}$ denotes the stiffness ratio for network $k$, $N$ denotes the number of viscoelastic networks, the subscript or superscript $k$ is used to denote network quantities, and the network $0$ is assumed to be the purely elastic network. If plasticity is specified in the equilibrium network, the standard output identifiers available in Abaqus corresponding to other isotropic and kinematic hardening plasticity models can be obtained for this model as well. In addition, if the Mullins effect is used in the model, the output variables available for the Mullins effect model (see Mullins effect) can be requested. References
