The unknowns of the mixture can be determined when the mass balance (
14), (
15) and the momentum balance (
19) are solved. The momentum balance for the mixture is solved instead of the individual phases as the relative momentum change between the phases in negligible. Moreover, the mass of the solid remains constant. Therefore, the constitutive quantities
$$\begin{aligned} \mathcal {R} = \lbrace \mathbf{T}_s, \mathbf{T}_l, \varvec{j} \rbrace \end{aligned}$$
(32)
need to be chosen in a thermodynamically consistent way to solve the balance equations. To this end, the entropy balances of the individual constituents are transformed using the Legendre transformation
$$\begin{aligned} \uppsi _\upalpha = e_\upalpha - \upeta _\upalpha \uptheta \end{aligned}$$
(33)
where
\(\uppsi _\upalpha \) denotes the Helmholtz energy for phase
\(\upalpha = s,\;l\). For an isothermal process where the change of temperature w.r.t time is zero, the transformation of Eqs. (
29) and (
31) yields
$$\begin{aligned} -\uprho _s\displaystyle {\left( \uppsi _s\right) ^{\prime }_{s}} + \mathbf{T}_s:\mathbf{D}_s-\varvec{v}_s\cdot \hat{\mathbf{P}}_s +\hat{e}_s = {\hat{\upeta }}_s \end{aligned}$$
(34)
for the polyamide and
$$\begin{aligned} -\uprho _l\displaystyle {\left( \uppsi _l\right) ^{\prime }_{s}} + \mathbf{T}_l:\mathbf{D}_l -\varvec{v}_l\cdot \hat{\mathbf{P}}_l - \varvec{j}\cdot \hbox {grad}\,\uppsi _l + \hat{e}_l = {\hat{\upeta }}_l \end{aligned}$$
(35)
for the moisture. According to the second law of thermodynamics in a viable process, the entropy of the system must not decrease. Hence,
\({\hat{\upeta }} = {\hat{\upeta }}_s + {\hat{\upeta }}_l\) should always be non-negative. By the addition of Eq. (
34) and Eq. (
35), and by employing the inequality
\({\hat{\upeta }}_s+ {\hat{\upeta }}_l \ge 0\), the Clausius–Planck inequality
$$\begin{aligned} -\uprho _s\displaystyle {\left( \uppsi _s\right) ^{\prime }_{s}}-\uprho _l\displaystyle {\left( \uppsi _l\right) ^{\prime }_{s}} +\mathbf{T}_s:\mathbf{D}_s + \mathbf{T}_l:\mathbf{D}_l - \varvec{w}\cdot \hat{\mathbf{P}}_l-\varvec{j}\cdot \hbox {grad}\,\uppsi _l \ge 0 \end{aligned}$$
(36)
is obtained. The relation (
36) is evaluated for the thermodynamic consistency using the Liu–Müller method [
23,
24]. The mass balance and the momentum balance for the moisture are added to the entropy Eq. (
36) with the help of Lagrangian parameters
\(\Lambda _1\) and
\(\varvec{\Lambda }_2\), respectively,
$$\begin{aligned} \begin{aligned}&-\uprho _s\displaystyle {\left( \uppsi _s\right) ^{\prime }_{s}}-\uprho _l\displaystyle {\left( \uppsi _l\right) ^{\prime }_{s}} +\mathbf{T}_s:\mathbf{D}_s + \mathbf{T}_l:\mathbf{D}_l - \varvec{w}\cdot \hat{\mathbf{P}}_l-\varvec{j}\cdot \hbox {grad}\,\uppsi _l\\&\quad +\Lambda _1(\displaystyle {\left( \uprho _l\right) ^{\prime }_{s}} + {\mathrm{div}}\,\varvec{j} + \uprho _l\,{\mathrm{div}}\,\varvec{v}_s)\\&\quad +\varvec{\Lambda }_2\cdot (\uprho _l\displaystyle {\left( \varvec{v}_l\right) ^{\prime }_{s}} - \uprho _l\varvec{b}_l -\varvec{v}_l{\mathrm{div}}\,\varvec{j} + {\mathrm{div}}\,(\varvec{j}\otimes \varvec{v}_s) -{\mathrm{div}}\,\mathbf{T}_l -\hat{\mathbf{P}}_l)\ge 0. \end{aligned} \end{aligned}$$
(37)
As the process is quasi-static both in terms of the mechanical deformation and the moisture transport, the inertial terms in the momentum balance equation are neglected. Thus, the entropy inequality with the modified momentum balance equation can be rewritten as:
$$\begin{aligned} \begin{aligned}&-\uprho _s\displaystyle {\left( \uppsi _s\right) ^{\prime }_{s}}-\uprho _l\displaystyle {\left( \uppsi _l\right) ^{\prime }_{s}} +\mathbf{T}_s:\displaystyle {\left( \varvec{\upvarepsilon }\right) ^{\prime }_{s}} + \mathbf{T}_l:\mathbf{D}_l - \varvec{w}\cdot \hat{\mathbf{P}}_l-\varvec{j}\cdot \hbox {grad}\,\uppsi _l\\&\quad +\Lambda _1(\displaystyle {\left( \uprho _l\right) ^{\prime }_{s}} + \varvec{w}\cdot \hbox {grad}\,\uprho _l + \uprho _l{\mathrm{div}}\,\varvec{v}_l)\\&\quad +\varvec{\Lambda }_2\cdot (-{\mathrm{div}}\,\mathbf{T}_l -\hat{\mathbf{P}}_l)\ge 0. \end{aligned} \end{aligned}$$
(38)
For simplicity, the model is developed for small deformations, and a geometrically linear deformation is assumed. To this end, the strain is formulated as the symmetric part of the displacement gradient
$$\begin{aligned} \varvec{\upvarepsilon }_s = \dfrac{\hbox {grad}\,\varvec{u}_s + \hbox {grad}^{T}\, \varvec{u}_s}{2} \end{aligned}$$
(39)
and
\(\mathbf{D}_s = \displaystyle {\left( \varvec{\upvarepsilon }_s\right) ^{\prime }_{s}}\) is assumed. Furthermore, the strain is split additively into an elastic and an inelastic part
$$\begin{aligned} \varvec{\upvarepsilon }_s = \varvec{\upvarepsilon }_e + \varvec{\upvarepsilon }_i \end{aligned}$$
(40)
to account for the viscoelastic behaviour of the polymer [
20,
28]. The deformation of the solid part is thus related to the variables
\(\varvec{\upvarepsilon }_{s}\),
\(\varvec{\upvarepsilon }_i\). The fluid deformation is defined by the deformation velocity given by
\(\mathbf{D}_l\). The mass flow of the phases is modelled with the variables
\(\uprho _s\),
\(\uprho _l\) and the gradient of the densities
\(\hbox {grad}\,\uprho _s\) and
\(\hbox {grad}\,\uprho _l\). Thus, the set of process variables for the linear viscoelastic solid interacting with the diffusion process at a constant temperature is chosen as [
11,
20]
$$\begin{aligned} \mathcal {S}=(\varvec{\upvarepsilon }_s,\varvec{\upvarepsilon }_i, \mathbf{D}_l,\uprho _s,\uprho _l,\hbox {grad}\,\uprho _s,\hbox {grad}\,\uprho _l). \end{aligned}$$
(41)
The free energy functions depend only on the process variables
$$\begin{aligned} \uppsi _{s,l} = \uppsi _{s,l}(\mathcal {S}). \end{aligned}$$
(42)
The principle of equipresence is assumed, and therefore, all material time derivatives can be written as:
$$\begin{aligned} \displaystyle {\left( \uppsi _{s,l}(\mathcal {S})\right) ^{\prime }_{s}} = \displaystyle \frac{\partial \uppsi _{s,l}}{\partial \mathcal {S}} \displaystyle {\left( \mathcal {S}\right) ^{\prime }_{s}} \end{aligned}$$
(43)
using the chain rule of differentiation. Substituting the derivative in the inequality (
38) and rearranging the terms lead to
$$\begin{aligned} \begin{aligned}&\displaystyle {\left( \varvec{\upvarepsilon }_s\right) ^{\prime }_{s}}:\left[ -\uprho _s \displaystyle \frac{\partial \uppsi _s}{\partial \varvec{\upvarepsilon }_s} -\uprho _l \displaystyle \frac{\partial \uppsi _l}{\partial \varvec{\upvarepsilon }_s} + \mathbf{T}_s\right] +\displaystyle {\left( \mathbf{D}_l\right) ^{\prime }_{s}}:\left[ -\uprho _s \displaystyle \frac{\partial \uppsi _s}{\partial \mathbf{D}_l} -\uprho _l \displaystyle \frac{\partial \uppsi _s}{\partial \mathbf{D}_l}\right] \\&\quad +\displaystyle {\left( \uprho _s\right) ^{\prime }_{s}}\left[ -\uprho _s \displaystyle \frac{\partial \uppsi _s}{\partial \uprho _s} -\uprho _l \displaystyle \frac{\partial \uppsi _l}{\partial \uprho _s}\right] +\displaystyle {\left( \uprho _l\right) ^{\prime }_{s}}\left[ -\uprho _s \displaystyle \frac{\partial \uppsi _s}{\partial \uprho _l} -\uprho _l \displaystyle \frac{\partial \uppsi _l}{\partial \uprho _l} + \Lambda _1\right] \\&\quad +\displaystyle {\left( \hbox {grad}\,\uprho _s\right) ^{\prime }_{s}}\cdot \left[ -\uprho _s \displaystyle \frac{\partial \uppsi _s}{\partial \hbox {grad}\,\uprho _s} -\uprho _l \displaystyle \frac{\partial \uppsi _l}{\partial \hbox {grad}\,\uprho _s}\right] \\&\quad +\displaystyle {\left( \hbox {grad}\,\uprho _l\right) ^{\prime }_{s}}\cdot \left[ -\uprho _s \displaystyle \frac{\partial \uppsi _s}{\partial \hbox {grad}\,\uprho _l} -\uprho _l \displaystyle \frac{\partial \uppsi _l}{\partial \hbox {grad}\,\uprho _l}\right] \\&\quad +\mathcal {D} \ge 0 \end{aligned} \end{aligned}$$
(44)
with the residual inequality
$$\begin{aligned} \begin{aligned} \mathcal {D}&= \displaystyle {\left( \varvec{\upvarepsilon }_i\right) ^{\prime }_{s}}:\left[ -\uprho _s \displaystyle \frac{\partial \uppsi _s}{\partial \varvec{\upvarepsilon }_i} -\uprho _l \displaystyle \frac{\partial \uppsi _l}{\partial \varvec{\upvarepsilon }_i}\right] -\hat{\mathbf{P}}_l\cdot (\varvec{w} + \varvec{\Lambda }_2) - \varvec{j}\cdot \hbox {grad}\,\uppsi _l \\&\quad - \varvec{\Lambda }_2\cdot \hbox {grad}\,p_l+ \mathbf{D}_l:(-p_l\mathbf{I}+ \uprho _l\Lambda _1\mathbf{I}) + \Lambda _1\varvec{w}\cdot \hbox {grad}\,\uprho _l. \end{aligned} \end{aligned}$$
(45)
While frictional effects in the fluid mainly govern the momentum exchange
\({\hat{\mathbf{P}}}_s = -{\hat{\mathbf{P}}}_l\) [
15], the stress in the moisture can be assumed to be hydrostatic and hence has been transformed to
\(\mathbf{T}_l = -p_l\mathbf{I}\) with
\(p_l\) being the hydrostatic pore pressure. Following the arguments of entropy evaluation and using the fact that the rate of change of the process variables can be arbitrary, the following relationships are obtained as necessary and sufficient conditions [
23,
24]:
$$\begin{aligned}&-\uprho _s \displaystyle \frac{\partial \uppsi _s}{\partial \mathbf{D}_l} -\uprho _l \displaystyle \frac{\partial \uppsi _l}{\partial \mathbf{D}_l} = 0, \end{aligned}$$
(46)
$$\begin{aligned}&-\uprho _s \displaystyle \frac{\partial \uppsi _s}{\partial \uprho _s} -\uprho _l \displaystyle \frac{\partial \uppsi _l}{\partial \uprho _s} = 0, \end{aligned}$$
(47)
$$\begin{aligned}&-\uprho _s \displaystyle \frac{\partial \uppsi _s}{\partial \hbox {grad}\,\uprho _s} -\uprho _l \displaystyle \frac{\partial \uppsi _l}{\partial \hbox {grad}\,\uprho _s}=0, \end{aligned}$$
(48)
$$\begin{aligned}&-\uprho _s \displaystyle \frac{\partial \uppsi _s}{\partial \hbox {grad}\,\uprho _l} -\uprho _l \displaystyle \frac{\partial \uppsi _l}{\partial \hbox {grad}\,\uprho _l}=0, \end{aligned}$$
(49)
$$\begin{aligned}&\Lambda _1 = \uprho _s \displaystyle \frac{\partial \uppsi _s}{\partial \uprho _l} +\uprho _l \displaystyle \frac{\partial \uppsi _l}{\partial \uprho _l}, \end{aligned}$$
(50)
$$\begin{aligned}&\mathbf{T}_s = \uprho _s \displaystyle \frac{\partial \uppsi _s}{\partial \varvec{\upvarepsilon }_s} +\uprho _l \displaystyle \frac{\partial \uppsi _l}{\partial \varvec{\upvarepsilon }_s}. \end{aligned}$$
(51)
The first term of the residual inequality (
45) contains the derivative of the inelastic strain. The evolution of the inelastic strain
\(\displaystyle {\left( \varvec{\upvarepsilon }_i\right) ^{\prime }_{s}}\) should not be arbitrary as it is an internal variable that depends on the relaxation time of the polyamide. Thus, an evolution equation is employed which ensures that
$$\begin{aligned} \displaystyle {\left( \varvec{\upvarepsilon }_i\right) ^{\prime }_{s}}:\left[ -\uprho _s \displaystyle \frac{\partial \uppsi _s}{\partial \varvec{\upvarepsilon }_i} -\uprho _l \displaystyle \frac{\partial \uppsi _l}{\partial \varvec{\upvarepsilon }_i}\right] \ge 0 \end{aligned}$$
(52)
is satisfied which is sufficient to fulfil the respective part of the residual inequality. The interaction force
\({\hat{\mathbf{P}}}_l\) between the polyamide and the moisture is generating a flow. The momentum balance of the liquid phase is not solved explicitly to determine
\(\varvec{j}\), rather
\(\varvec{j}\) is defined as a constitutive quantity. Hence, the interaction force is not included in the set of constitutive variables (
32) and any value of
\({\hat{\mathbf{P}}}_l\) should fulfil the residual inequality, which can be ensured by the choice
$$\begin{aligned} \varvec{\Lambda }_2 = -\varvec{w}, \end{aligned}$$
(53)
making the term disappear in Eq. (
45). The remaining inequality becomes
$$\begin{aligned} \mathcal {D} = - \varvec{j}\cdot \hbox {grad}\,\uppsi _l - \varvec{w}\cdot \hbox {grad}\,p_l + \mathbf{D}_l:(-p_l\mathbf{I}+ \uprho _l\Lambda _1\mathbf{I}) + \Lambda _1\varvec{w}\cdot \hbox {grad}\,\uprho _l \ge 0. \end{aligned}$$
(54)
Here, the chemical potential for the liquid [
8,
14,
22,
32]
$$\begin{aligned} \upkappa _l = \uppsi _l -\dfrac{p_l}{\uprho _l} \end{aligned}$$
(55)
is introduced and substituted in
\(\mathcal {D}\). In this way, the Helmholtz free energy for the solid and the chemical potential for the liquid define the two potentials for the coupled problem. Equation (
54) is transformed with the help of these potentials to
$$\begin{aligned} -\varvec{j}\cdot \hbox {grad}\,{\upkappa _l} + \mathbf{D}_l:(-p_l\mathbf{I}+\uprho _l\Lambda _1\mathbf{I}) - \left( -\dfrac{p_l\varvec{j}}{\uprho _l^2} +\Lambda _1\varvec{w}\right) \cdot {\hbox {grad}\,\uprho _l} \ge 0. \end{aligned}$$
(56)
The inequality should not be harmed for any deformation velocity of the liquid, therefore
$$\begin{aligned} p_l = \uprho _l\Lambda _1, \end{aligned}$$
(57)
and by using Eq. (
50)
$$\begin{aligned} p_{l} = \uprho _l\uprho _s \displaystyle \frac{\partial \uppsi _s}{\partial \uprho _s}+\uprho _l^2 \displaystyle \frac{\partial \uppsi _l}{\partial \uprho _l} \end{aligned}$$
(58)
is obtained. The residual inequality reduces to
$$\begin{aligned} -\varvec{j}\cdot \hbox {grad}\,\upkappa _l \ge 0. \end{aligned}$$
(59)
The inequality can be satisfied if
$$\begin{aligned} \varvec{j} = -\text {k}\left( \hbox {grad}\,{\upkappa }_l\right) , \end{aligned}$$
(60)
where
\(\text {k}\ge 0\) is a material parameter which has the character of a diffusion constant. Thus, the conditions for the constitutive quantities are obtained, which are exploited to get the equations of these quantities in the next section.