PolyUMod Three Network (TN) Model

As specified by its name, the kinematics of the three-network model consists of three parts, or molecular networks, acting in parallel, see the rheological representation figure to the right.

The total deformation gradient \(\mathbf{F}^{\mathit{appl}}\) contains both a thermal expansion part \(\mathbf{F}^{\mathit{th}} = \left[1 + \alpha (\theta – \theta_0) \right] \mathbf{I}\), and a mechanical deformation part \(\mathbf{F}\):
\[
\mathbf{F}^{\mathit{appl}} = \mathbf{F}\, \mathbf{F}^{\mathit{th}}.
\]
The deformation gradient acting on network A is multiplicatively decomposed into elastic and viscoplastic components:
\[
\mathbf{F} = \mathbf{F}_A^e \mathbf{F}_A^v.
\]
The Cauchy stress acting on network A is given by a temperature-dependent version of the eight-chain representation:
\[
\boldsymbol{\sigma}_A = \frac{\mu_A} {J_A^e \overline{\lambda_A^{e*}}}
\left[ 1 + \frac{\theta – \theta_0}{\hat{\theta}} \right]
\frac{\mathcal{L}^{-1}\! \left( \overline{\lambda_A^{e*}} / \lambda_L \right) }
{\mathcal{L}^{-1}\! \left( 1 / \lambda_L \right) }
dev \left[ \mathbf{b}_A^{e*} \right] +
\kappa (J_A^e – 1) \mathbf{I},
\] where \(J_A^{e}=\det[\mathbf{F}_A^e]\), \(\mu_A\) is the initial shear modulus, \(\lambda_L\) is the chain locking stretch, \(\theta\) is the current temperature, \(\theta_0\) is a reference temperature, \(\hat{\theta}\) is a material parameter specifying the temperature response of the stiffness, \(\mathbf{b}_A^{e*} = (J_A^e)^{-2/3} \mathbf{F}_A^e (\mathbf{F}_A^e)^\top\) is the Cauchy-Green deformation tensor, \(\overline{\lambda_A^{e*}} = \left(tr[\mathbf{b}_A^{e*}] / 3\right)^{1/2}\) is the effective chain stretch based on the eight-chain topology assumption, \(\mathcal{L}^{-1}\!(x)\) is the inverse Langevin function, where\(\mathcal{L}(x) = \coth(x)-1/x\), and \(\kappa\) is the bulk modulus. By explicitly incorporating the temperature dependence of the shear modulus it is possible to capture the stiffness variation of the material over a wide range of temperatures.

The viscoelastic deformation gradient acting on network B is decomposed into elastic and viscoplastic parts:
\[
\mathbf{F} = \mathbf{F}_B^e \mathbf{F}_B^v.
\] The Cauchy stress acting on network B is obtained from the same eight-chain network representation that was used for network A.
\[
\boldsymbol{\sigma}_B = \frac{\mu_B} {J_B^e \overline{\lambda_B^{e*}}}
\left[ 1 + \frac{\theta – \theta_0}{\hat{\theta}} \right]
\frac{\mathcal{L}^{-1}\! \left( \overline{\lambda_B^{e*}} / \lambda_L \right) }
{\mathcal{L}^{-1}\! \left( 1 / \lambda_L \right) }
dev \left[ \mathbf{b}_B^{e*} \right] +
\kappa (J_B^e – 1) \mathbf{1},
\] where \(J_B^{e}=\det[\mathbf{F}_B^e]\), \(\mu_B\) is the initial shear modulus, \(\mathbf{b}_B^{e*} = (J_B^e)^{-2/3} \mathbf{F}_B^e (\mathbf{F}_B^e)^\top\) is the Cauchy-Green deformation tensor, and \(\overline{\lambda_B^{e*}} = \left(tr[\mathbf{b}_B^{e*}] / 3\right)^{1/2}\) is the effective chain stretch based on the eight-chain topology assumption.

The effective shear modulus is taken to evolve with plastic strain from an initial value of\(\mu_{Bi}\) according to:
\[
\dot{\mu}_B = -\beta \left[ \mu_B – \mu_{\mathit{Bf}} \right] \cdot \dot{\gamma}_A,
\] where\(\dot{\gamma}_A\) is the viscoplastic flow rate. This equation enables the model to better capture the distributed yielding that is observed in many thermoplastics.

The Cauchy stress acting on network C is given by the eight-chain model with first order \(I_2\) dependence:
\[
\boldsymbol{\sigma}_C =
\frac{1}{1+q}
\left\{
\frac{\mu_C} {J \overline{\lambda^*}}
\left[ 1 + \frac{\theta – \theta_0}{\hat{\theta}} \right]
\frac{\mathcal{L}^{-1}\! \left( \frac{\overline{\lambda^*}} { \lambda_L} \right) }
{\mathcal{L}^{-1}\! \left( \frac{1} {\lambda_L} \right) }
dev \left[ \mathbf{b}^* \right] +
\kappa (J – 1) \mathbf{I}
+
q \frac{\mu_c}{J}
\left[
I_1^* \mathbf{b}^* – \frac{2I_2^*}{3} \mathbf{I} – (\mathbf{b}^*)^2
\right]
\right\},
\] where \(J=\det[\mathbf{F}]\), \(\mu_C\) is the initial shear modulus, \(\mathbf{b}^* = J^{-2/3} \mathbf{F} (\mathbf{F})^\top\) is the Cauchy-Green deformation tensor, and \(\overline{\lambda^*} = \left(tr[\mathbf{b}^*] / 3\right)^{1/2}\) is the effective chain stretch based on the eight-chain topology assumption [Arruda:1993].

Using this framework, the total Cauchy stress in the system is given by \(\boldsymbol{\sigma} = \boldsymbol{\sigma}_A + \boldsymbol{\sigma}_B + \boldsymbol{\sigma}_C\).

The total velocity gradient of network A,  \(\mathbf{L} = \dot{\mathbf{F}} \mathbf{F}^{-1}\), can be decomposed into elastic and viscous components: \(\mathbf{L} = \mathbf{L}_A^e + \mathbf{F}_A^e \mathbf{L}_A^v \mathbf{F}_A^{e-1} = \mathbf{L}_A^e + \tilde{\mathbf{L}}_A^v\), where \(\mathbf{L}_A^v = \dot{\mathbf{F}}_A^v \mathbf{F}_A^{v-1} = \mathbf{D}_A^v + \mathbf{W}_A^v\) and \(\tilde{\mathbf{L}}_A^v = \tilde{\mathbf{D}}_A^v + \tilde{\mathbf{W}}_A^v\). The unloading process relating the deformed state with the intermediate state is not uniquely defined since an arbitrary rigid body rotation of the intermediate state still leaves the state stress free. The intermediate state can be made unique in different ways, one particularly convenient way that is used here is to prescribe \(\tilde{\mathbf{W}}_A^v = \mathbf{0}\). This will, in general, result in elastic and inelastic deformation gradients both containing rotations. The rate of viscoplastic flow of network A is constitutively prescribed by \(\tilde{\mathbf{D}}_A^v = \dot{\gamma}_A \mathbf{N}_A\). The tensor \(\mathbf{N}_A\) specifies the direction of the driving deviatoric stress of the relaxed configuration convected to the current configuration, and the term \(\dot{\gamma}_A\) specifies the effective deviatoric flow rate. Noting that \(\boldsymbol{\sigma}_A\) is computed in the loaded configuration, the driving deviatoric stress on the relaxed configuration convected to the current configuration is given by \(\boldsymbol{\sigma}_A’ = dev[\boldsymbol{\sigma}_A]\), and by defining an effective stress by the Frobenius norm \(\mathbf{\tau}_A = || \boldsymbol{\sigma}_A’ ||_F \equiv \left( tr[\boldsymbol{\sigma}_A’ \boldsymbol{\sigma}_A’] \right)^{1/2}\), the direction of the driving deviatoric stress becomes \(\mathbf{N}_A = \boldsymbol{\sigma}_A’ / \tau_A\).
The effective deviatoric flow rate is given by the reptation-inspired equation [Bergstrom:2000]:
\[
\dot{\gamma}_A = \dot{\gamma}_0 \cdot
\left(\frac{\tau_A}{\hat{\tau}_A + a R(p_A)} \right)^{m_A} \cdot
\left(\frac{\theta}{\theta_0} \right)^n,
\] where \(\dot{\gamma}_0 \equiv 1\)/s is a constant introduced for dimensional consistency, \(p_A = – [(\boldsymbol{\sigma}_A)_{11} + (\boldsymbol{\sigma}_A)_{22} + (\boldsymbol{\sigma}_A)_{33}]/3\) is the hydrostatic pressure, \(R(x) = (x + |x|) / 2\) is the ramp function, and \(\hat{\tau}_A\),\(a\),\(m_A\), and \(n\) are specified material parameters. In this framework, the temperature dependence of the flow rate is taken to follow a power law form. In summary, the velocity gradient of the viscoelastic flow of network A can be written
\[
\dot{\mathbf{F}}_A^v = \dot{\gamma}_A \mathbf{F}_A^{e-1} \frac{dev[\boldsymbol{\sigma}_A]}{\tau_A} \mathbf{F}.
\]

The total velocity gradient of network B can be obtained similarly to network A. Specifically, \(\mathbf{L} = \dot{\mathbf{F}} \mathbf{F}^{-1}\) can be decomposed into elastic and viscous components: \(\mathbf{L} = \mathbf{L}_B^e + \mathbf{F}_B^e \mathbf{L}_B^v \mathbf{F}_B^{e-1} = \mathbf{L}_B^e + \tilde{\mathbf{L}}_B^v\), where \(\mathbf{L}_B^v = \dot{\mathbf{F}}_B^v \mathbf{F}_B^{v-1} = \mathbf{D}_B^v + \mathbf{W}_B^v\) and \(\tilde{\mathbf{L}}_B^v = \tilde{\mathbf{D}}_B^v + \tilde{\mathbf{W}}_B^v\).

The unloading process relating the deformed state with the intermediate state is not uniquely defined since an arbitrary rigid body rotation of the intermediate state still leaves the state stress free. The intermediate state can be made unique in different ways [Boyce:1989], one particularly convenient way that is used here is to prescribe\(\tilde{\mathbf{W}}_B^v = \mathbf{0}\). This will, in general, result in elastic and inelastic deformation gradients both containing rotations. The rate of viscoplastic flow of network B is constitutively prescribed by \(\tilde{\mathbf{D}}_B^v = \dot{\gamma}_B \mathbf{N}_B\). The tensor \(\mathbf{N}_B\) specifies the direction of the driving deviatoric stress of the relaxed configuration convected to the current configuration, and the term \(\dot{\gamma}_B\) specifies the effective deviatoric flow rate. Noting that \(\boldsymbol{\sigma}_B\) is computed in the loaded configuration, the driving deviatoric stress on the relaxed configuration convected to the current configuration is given by \(\boldsymbol{\sigma}_B’ = dev[\boldsymbol{\sigma}_B]\), and by defining an effective stress by the Frobenius norm \(\mathbf{\tau}_B = || \boldsymbol{\sigma}_B’ ||_F \equiv \left( tr[\boldsymbol{\sigma}_B’ \boldsymbol{\sigma}_B’] \right)^{1/2}\), the direction of the driving deviatoric stress becomes\(\mathbf{N}_B = \boldsymbol{\sigma}_B’ / \tau_B\). The effective deviatoric flow rate is given by the reptation-inspired equation [Bergstrom:2000]:
\[
\dot{\gamma}_B = \dot{\gamma}_0 \cdot
\left(\frac{\tau_B}{\hat{\tau}_B + a R(p_B)} \right)^{m_B} \cdot
\left(\frac{\theta}{\theta_0} \right)^n,
\] where \(\dot{\gamma}_0 \equiv 1\)/s is a constant introduced for dimensional consistency, \(p_B = – [(\boldsymbol{\sigma}_B)_{11} + (\boldsymbol{\sigma}_B)_{22} + (\boldsymbol{\sigma}_B)_{33}]/3\) is the hydrostatic pressure, and \(\hat{\tau}_B\),\(a\),\(m_B\), and \(n\) are specified material parameters. In this framework, the temperature dependence of the flow rate is taken to follow a power law form. In summary, the velocity gradient of the viscoelastic flow of network B can be written
\[
\dot{\mathbf{F}}_B^v = \dot{\gamma}_B \mathbf{F}_B^{e-1} \frac{dev[\boldsymbol{\sigma}_B]}{\tau_B} \mathbf{F}.
\]