A numerical model for chemo-thermo-mechanical coupling at large strains with an application to thermoresponsive hydrogels

Abstract

The aim of this work is the derivation and examination of a material model, accounting for large elastic deformations, coupled with species diffusion and thermal effects. This chemo-thermo-mechanical material model shows three key aspects regarding its numerical formulation. Firstly, a multiplicative split of the deformation gradient into a mechanical, a swelling and a thermal part. Secondly, temperature-scaled gradients for a numerical design comprising symmetric tangents and, thirdly, dissipation potentials for the modelling of dissipative effects. Additionally, the derived general material model is specialised to thermoresponsive hydrogels to study its predictive capabilities for a relevant example material class. An appropriate finite element formulation is established and its implementation discussed. Numerical examples are investigated, including phase transition and stability phenomena, to verify the ability of the derived chemo-thermo-mechanical material model to predict relevant physical effects properly. We compare our results to established models in the literature and discuss emerging deviations.

1 Introduction

Material models for the description of deforming bodies that exchange chemical substances (liquids, gases, etc.) with their environment under varying thermal conditions are referred to as chemo-thermo-mechanical material models. Such models are of great scientific as well as economic relevance since they are applicable to a broad range of materials including but not limited to metals and metal oxides [1,2,3,4], wood [5,6,7], concrete [8,9,10,11], and hydrogels [12,13,14,15,16,17,18,19]. Especially hydrogels have attracted considerable attention in recent years since they represent a comparatively new material class with fascinating properties.

Hydrogels have due to their exceptional deformation-diffusion behaviour a vast variety of application fields, such as drug delivery, tissue engineering, biosensors, artificial organs, food processing, and cosmetics, which makes them a relevant and active field of research [20, 21]. Thermoresponsive hydrogels are a sub-class of environmentally responsive or stimuli-responsive hydrogels [22,23,24] that recently received notable attention due to their ability to undergo large reversible volume changes that can be controlled through variations in the ambient temperature. This feature makes them an attractive material for e.g. sensors [23, 25, 26], actuators [23, 27, 28] and valves [29,30,31] for microfluidic devices as well as for drug delivery [24, 32,33,34] and tissue engineering [32, 34] applications.

The theoretical and numerical description of thermoresponsive hydrogels is challenging since they exhibit not only large volume changes in response to varying temperatures, which entails solute diffusion within the body and solute exchange with the environment. These hydrogels can also feature a discontinuous phase transition from a hydrophilic swollen to a hydrophobic (nearly) dry state when heated above a specific transition temperature, making such models a topic of ongoing research [12,13,14,15,16,17,18,19].

Lastly, hydrogels are frequently discussed for their applicability to model pattern formation and evolution in biological tissue and materials [14, 35,36,37]. For this, their strong reversible volume expansion is combined with suitable geometries (e.g. strips or rings) to study a rich variety of buckling and wrinkling patterns. However, such mechanical instabilities add additional difficulties to the already intricate numerical modelling of these materials.

In this work, we derive an up-to-date and thermodynamically-consistent chemo-thermo-mechanical material model for the treatment of large elastic deformations, coupled with species and heat diffusion. The model includes (i) a multiplicative split of the deformation gradient in the three-field problem. This approach grants the established model considerable generality and makes it applicable to a broad range of material classes and geometries. Furthermore, we incorporate (ii) dissipation potentials and temperature-scaled gradient quantities. With the introduced temperature-scaled gradients, we have been able to obtain a symmetric tangent stiffness matrix that fully facilitates the use of the monolithic finite element scheme. We discuss (iii) pertinent implementation-related details, including the application of a pre-swollen reference configuration. We report (iv) on an error in the analytical description of a widely used buckling example and on its correction. Finally, we conduct (v) an original numerical study on the thermally-triggered swelling-induced buckling of hydrogels, which has received little attention so far, despite its potential for industrial applications.

To demonstrate the capabilities of the established model, we specialise it to thermoresponsive poly(N-isopropylacrylamide) (PNIPAM) hydrogels. To this end, a consistent constitutive approach for the modelling of thermoresponsive hydrogels is employed and discussed. Further, we implement a monolithic finite element formulation of the derived model, discuss critical points of the implementation and conduct a series of numerical studies. These studies show the ability of the established model to simulate the discontinuous phase transition of PNIPAM hydrogels and the even more demanding buckling of hydrogel coronae. In the course of one such study, we have found the mentioned error in an established model on buckling of hydrogel coronae [37]. We discuss these issues in detail and propose corrections. Lastly, we investigate the thermally-induced buckling of thermoresponsive hydrogels as an interesting but yet unexplored direction for further research.

The remainder of this paper is organised as follows. The general model is derived in Sect. 2 and specialised to thermoresponsive hydrogels in Sect. 3. The finite element implementation is discussed in Sect. 4. Next, the conducted numerical studies are presented in Sect. 5. Finally, the obtained results are summarized and discussed in Sect. 6. In the included appendices, further details regarding the derived model and its implementation, as well as on discussed issues are provided.

2 General theory

2.1 Basic quantities

2.1.1 Kinematics

We assume a reference configuration $\,\mathcal {B}_{0}$, in which the body is undeformed and stress-free, and parametrise this configuration with the coordinates $\varvec{X}$ [38]. At the instant of time $t\in \,\mathcal {T}\subset {\mathbb {R}}^+$, the body occupies the current configuration $\,\mathcal {B}$, which is obtained from the reference configuration via the mapping $\varvec{\varphi }: \,\mathcal {B}_0 \rightarrow \,\mathcal {B}$, and parametrise it with the coordinates $\varvec{x}$. The motion of the body, which can be defined as a time-dependent family of configurations, is written as $\varvec{\varphi }(\varvec{X},t)$:

$$\begin{aligned} \varvec{\varphi }: {\left\{ \begin{array}{ll} {\mathcal {B}}_0 \times {\mathcal {T}} \rightarrow {\mathcal {B}}\\ (\varvec{X}, t) \mapsto \varvec{x} = \varvec{\varphi }(\varvec{X}, t) \end{array}\right. } . \end{aligned}$$

(1)

The deformation gradient $\varvec{F}$ maps vectors at point $P \in \,\mathcal {B}_0$ to vectors at $\varvec{\varphi }(P) \in \,\mathcal {B}$ and can be calculated as

$$\begin{aligned} \varvec{F} = \frac{\partial \varvec{\varphi }}{\partial \varvec{X}} = \frac{\partial \varvec{x}}{\partial \varvec{X}}. \end{aligned}$$

(2)

The inverse, the transpose and the adjoint of the deformation gradient will be denoted as $\varvec{F}^{ -1}$, $\varvec{F}^{ \top }$, and $\varvec{F}^{ *}$ respectively. The latter two are linked through $\varvec{F}^{\top } = \varvec{G}^{-1} \varvec{F}^{*} \varvec{g}$. These definitions are in accordance with [38, 39]. At this point, we introduce the metric tensors $\varvec{G}$ and $\varvec{g}$ with respect to the reference and the current configuration, respectively. The Jacobian determinant J of the mapping $\varvec{\varphi }$ is defined in the current context as $J = \text {det}(\varvec{F}) {\sqrt{\text {det}(\varvec{g})}}/{\sqrt{\text {det}(\varvec{G})}}$.

As the measure of deformation we employ the (contravariant-covariant) right Cauchy-Green deformation tensor $\varvec{C}$ and its covariant form $\varvec{C}^{ \flat }$, both of which are defined on the reference configuration:

$$\begin{aligned} \varvec{C} = \varvec{F}^\top \varvec{F}, \quad \quad \varvec{C}^{ \flat } = \varvec{G} \varvec{C} = \varvec{F}^{*} \varvec{g} \varvec{F}. \end{aligned}$$

(3)

2.1.2 Measures of stress

We introduce the Cauchy stress tensor $\varvec{\sigma }$ defined with respect to the current configuration. It is related to the first Piola-Kirchhoff stress tensor $\varvec{P}$ through a Piola transformation: $\varvec{P} = J \varvec{\sigma } \varvec{F}^{ -*}$. The first Piola-Kirchhoff stress tensor establishes a relation between the material surface normal vector $\varvec{N}$ and the stress vector $\varvec{T}$ (force measured per unit surface area of the reference configuration):

$$\begin{aligned} \varvec{T}(\varvec{X},t,\varvec{N}) = \varvec{P}(\varvec{X},t) \varvec{N}(\varvec{X},t). \end{aligned}$$

(4)

2.1.3 Diffusion-specific quantities

In the context of continuum mechanics of deformable bodies, the species concentration $c_{0}(\varvec{X},t)$ with respect to the reference configuration is defined as a scalar field quantity that assigns every point $P\in \,\mathcal {B}_{0}$ at time t a concentration value:

$$\begin{aligned} c_0: {\left\{ \begin{array}{ll} {\mathcal {B}}_0 \times {\mathcal {T}} \rightarrow {\mathbb {R}}^{+}\\ (\varvec{X}, t) \mapsto c_0(\varvec{X}, t) \end{array}\right. }, \end{aligned}$$

(5)

measured in terms of the amount of substance per unit volume of the reference configuration, i.e. $[\hbox {mol/m}^{3}]$. Furthermore, the concentration gradient is defined as

$$\begin{aligned} \nabla _{\!0} c_{ 0} = \varvec{G}^{ -1} \frac{\partial c_{ 0} ({\textbf {X}},t)}{\partial \varvec{X}}. \end{aligned}$$

(6)

We define the material species flux vector $\varvec{H}(\varvec{X},t)$ and the (scalar) material species flux per unit area $H(\varvec{X},t,\varvec{N})$ such that the following relation holds true:

$$\begin{aligned} H(\varvec{X},t,\varvec{N}) = \varvec{H}(\varvec{X},t) \cdot \varvec{N}(\varvec{X},t). \end{aligned}$$

(7)

Furthermore, we introduce the chemical potential $\mu (\varvec{X}, t)$ $[\hbox {J/mol}]$, as another scalar field quantity that resembles the driving force for diffusion processes:

$$\begin{aligned} \mu : {\left\{ \begin{array}{ll} {\mathcal {B}}_0 \times {\mathcal {T}} \rightarrow {\mathbb {R}}\\ (\varvec{X}, t) \mapsto \mu (\varvec{X}, t) \end{array}\right. }. \end{aligned}$$

(8)

2.1.4 Thermodynamic quantities

From the thermodynamic point of view, the body under consideration will be treated as a partially open system. This means that the amount of particles comprising the solid body is assumed as constant over time. However, the body may exchange work and heat with its surrounding as well as amounts of the diffusive substance under consideration.

The absolute temperature $\theta$ is introduced as a scalar field quantity:

$$\begin{aligned} \theta : {\left\{ \begin{array}{ll} {\mathcal {B}}_0 \times {\mathcal {T}} \rightarrow {\mathbb {R}}^{+}\\ (\varvec{X}, t) \mapsto \theta (\varvec{X}, t) \end{array}\right. } \end{aligned}$$

(9)

and represents the driving force for heat conduction. We introduce the material heat flux vector $\varvec{Q}(\varvec{X},t)$ and the material (scalar) heat flux per unit area $Q(\varvec{X},t)$. These two quantities are related via:

$$\begin{aligned} Q(\varvec{X},t,\varvec{N}) = \varvec{Q}(\varvec{X},t) \cdot \varvec{N}(\varvec{X},t). \end{aligned}$$

(10)

2.2 Balance equations

2.2.1 Linear and angular momentum

The balance of linear momentum may be derived in the common manner, neglecting inertial effects, and stated in its local form as:

$$\begin{aligned} \text {DIV}(\varvec{P}) + \varvec{B} = 0, \end{aligned}$$

(11)

with the material body force vector field $\varvec{B}(\varvec{X},t)$.

The balance of angular momentum may be used, in conjunction with the balance of linear momentum, to derive the symmetry property of the Cauchy stress tensor: $\varvec{\sigma }^{ \top } = \varvec{\sigma }$, or the first Piola-Kirchhoff stress tensor: $\varvec{F}^{\,-1}\,\varvec{P} \,=\, \varvec{P}^{\,*}\,\varvec{F}^{\,-*}$.

2.2.2 Species balance

By employing the defined species-related quantities to formulate a transport equation for the concentration and by the use of the localisation theorem the material species balance can be established in its local form as

$$\begin{aligned} {\dot{c}}_{ 0} = - \text {DIV}(\varvec{H}), \end{aligned}$$

(12)

whereat $\dot{(\bullet )}$ denotes the material time derivative.

2.2.3 Energy: the first law of thermodynamics

To this point, there has been no connection made between the mechanical, chemical and thermal evolution of the material under consideration. To establish such a connection we introduce here and in the next section the first and second law of thermodynamics.

The first law can be stated in its material local form [40, Section 64] as

$$\begin{aligned} {\dot{u}}_{0} = \varvec{g}\varvec{P}\!:\!\dot{\varvec{F}} - \nabla _{\!0} \mu \! \cdot \! \varvec{H} + {\dot{c}}_0 \mu - \text {DIV}(\varvec{Q}) + R_{q} \end{aligned}$$

(13)

with the (volume-specific) internal energy $u_{0}(\varvec{X},t)$ and the (scalar) material heat source field $R_{q}(\varvec{X},t)$ of the system with respect to the reference configuration.

2.2.4 Entropy: the second law of thermodynamics

The second law of thermodynamics can be incorporated into non-equilibrium thermodynamics and continuum mechanics frameworks through an entropy balance equation. Such a balance equation includes the (net) internal entropy $\eta _{0}(\varvec{X},t)$ and the (net) entropy production $\Gamma _{0}(\varvec{X},t)$ per unit volume of the reference configuration, as well as the material heat source $R_{q}(\varvec{X},t)$. The local form of the material entropy balance can be found as [40, Sections 27 & 31]:

$$\begin{aligned} {\dot{\eta }}_{0} = \frac{R_{q}}{\theta } - \frac{1}{\theta } \text {DIV}(\varvec{Q}) + \frac{1}{\theta ^{ 2}} \nabla _{\!0} \theta \cdot \varvec{Q} + \Gamma _{0}. \end{aligned}$$

(14)

Combining this entropy balance with the energy balance (Eq. 13) and demanding that the entropy production is non-negative for arbitrary parts of the system: $\Gamma _{0}(\varvec{X},t) \, \ge \, 0$ [40, Section 27], yields the so-called material free energy imbalance [40, Section 64]:

$$\begin{aligned} \varvec{g}\varvec{P}\!:\!\dot{\varvec{F}} - \nabla _{\!0} \mu \! \cdot \! \varvec{H} + {\dot{c}}_0 \mu - {\dot{\theta }} \eta _{0} - \frac{1}{\theta } \nabla _{\!0} \theta \cdot \varvec{Q} - {\dot{\psi }}_{0} = \theta \Gamma _{0} \ge 0, \end{aligned}$$

(15)

whereat the internal energy $u_{0}$ was replaced by the volume-specific (Helmholtz) free energy $\psi _{0}(\varvec{X},t)$ using the Legendre transformation $\psi _{0} = u_{0} - \theta \eta _{0}$.

2.3 Specific assumptions and concepts

2.3.1 Coleman–Noll procedure

Demanding invariance under superimposed rigid body motion, we obtain the following dependencies of the free energy:

$$\begin{aligned} \psi _{0}(\varvec{X},t) = {\widetilde{\psi }}_{0}(\varvec{C}, c_{0}, \theta ). \end{aligned}$$

(16)

By differentiating the free energy with respect to time, taking into account the implicit time dependencies and inserting this expression in Eq. 15, the desired constitutive relations can be found through standard arguments of the Coleman–Noll procedure [41] as:

$$\begin{aligned} \varvec{P}&= 2 \varvec{F} \frac{\partial {\widetilde{\psi }}_{0}}{\partial \varvec{C}^{ \flat }}, \end{aligned}$$

(17)

$$\begin{aligned} \mu&= \frac{\partial {\widetilde{\psi }}_{0}}{\partial c_{0}}, \end{aligned}$$

(18)

$$\begin{aligned} \eta _{0}&= - \frac{\partial {\widetilde{\psi }}_{0}}{\partial \theta }, \end{aligned}$$

(19)

$$\begin{aligned} \theta \Gamma _{0}&= - \nabla _{\!0} \mu \! \cdot \! \varvec{H} - \frac{1}{\theta } \nabla _{\!0} \theta \cdot \varvec{Q} \ge 0. \end{aligned}$$

(20)

The reduction of the free energy imbalance from inequality 15 to the form of 20 implies that the only two possible reasons for dissipation within the described material are the transfer of heat and species across the system boundary. By resorting once again to the Coleman–Noll procedure it can also be found that

$$\begin{aligned}&- \nabla _{\!0} \mu \!\cdot \! \varvec{H} \ge 0 , \end{aligned}$$

(21)

$$\begin{aligned}&- \frac{1}{\theta } \nabla _{\!0} \theta \!\cdot \! \varvec{Q} \ge 0 \end{aligned}$$

(22)

must hold true. This encapsulates two basic physical principles: heat flows from hot to cold regions and substances are transported from regions with higher chemical potential to those with lower chemical potential.

2.3.2 Dissipation potentials

We link the flux quantities to temperature-scaled gradient quantities. This approach yields some desirable formal advantages, as discussed in detail in [42]. Therefore, we define on the basis of [42, 43] the temperature-scaled gradient of the chemical potential ${\overline{\nabla }}_{\!0} \mu$ and the temperature-scaled temperature gradient ${\overline{\nabla }}_{\!0} \theta$ as:

$$\begin{aligned}&{\overline{\nabla }}_{\!0} \mu := \frac{1}{\theta } \nabla _{\!0} \mu , \end{aligned}$$

(23)

$$\begin{aligned}&{\overline{\nabla }}_{\!0} \theta := \frac{1}{\theta } \nabla _{\!0} \theta . \end{aligned}$$

(24)

With these gradients, Eqs. 21 and 22 can be rewritten as:

$$\begin{aligned}&- \theta {\overline{\nabla }}_{\!0} \mu \!\cdot \! \varvec{H} \ge 0, \end{aligned}$$

(25)

$$\begin{aligned}&- {\overline{\nabla }}_{\!0} \theta \!\cdot \! \varvec{Q} \ge 0. \end{aligned}$$

(26)

If we consider first the species flux vector $\varvec{H}$ as an example, it can be shown [44] that Eq. 25 is satisfied, if a chemical dissipation potential is introduced that has units of power per unit reference volume. This potential depends on ${\overline{\nabla }}_{\!0} \mu$ at a given state¹, which is characterised by $\{\varvec{C}, c_{0}, \nabla _{\!0} c_{0}, \mu , \theta , \nabla _{\!0} \theta \}$, such that [42, 44]:

(27)

The factor $1/\theta$ thereby ensures compatibility of the above constitutive relation with the chemical dissipation inequality 25. Furthermore, for inequality 25 to be satisfied, the defined chemical dissipation potential must fulfill further criteria [44, 45], namely: has to be normalised (i.e. for ${\overline{\nabla }}_{\!0} \mu = \varvec{0}$), positive (i.e. ), and a convex function with respect to ${\overline{\nabla }}_{\!0} \mu$.

In the same manner, we also introduce a thermal dissipation potential , which depends on at a given state $\{\varvec{C}, c_{0}, \nabla _{\!0} c_{0}, \mu , \nabla _{\!0} \mu , \theta \}$ and also fulfils the above-stated criteria such that the heat flux vector can be determined as:

(28)

With the introduction of these dissipation potentials, all material response functions can now be derived from the free energy $\psi _{0}$ and the dissipation potentials and .

2.3.3 Entropy evolution equation

The second law of thermodynamics shall be employed as one of the basic equations of the material model, governing implicitly the evolution of the temperature field $\theta$. Therefore, we express Eq. 14 in an alternative form, which we will refer to as entropy evolution equation. To this end we express the entropy production $\Gamma _{0}$ through its form given by Eq. 20 and obtain with the help of Eq. 23 [42]:

$$\begin{aligned} {\dot{\eta }}_{0} = \frac{R_{q}}{\theta } - \frac{1}{\theta } \text {DIV}(\varvec{Q}) - {\overline{\nabla }}_{\!0} \mu \cdot \varvec{H}. \end{aligned}$$

(29)

2.3.4 Multiplicative decomposition of the deformation gradient

The deformation state of a body is determined by the primary variable $\varvec{\varphi }$ and, as a further consequence, by the deformation gradient $\varvec{F}$. The considered deformation mechanisms in this work are elastic mechanical deformation, swelling and thermal expansion. A common approach to take these different mechanisms into account in a unified manner is the multiplicative decomposition or multiplicative split of the deformation gradient [46,47,48].

By using such a split, we write the deformation gradient as [49]:

$$\begin{aligned} \varvec{F} = \varvec{F}_{m} \varvec{F}_{s} \varvec{F}_{\theta }. \end{aligned}$$

(30)

In the above equation $\varvec{F}_{m}$ is the mechanical part of the deformation gradient, accounting for elastic deformation due to the action of external force fields, $\varvec{F}_{s}$ is the swelling part, accounting for swelling-induced deformation due to the absorption of solvent molecules and $\varvec{F}_{\theta }$ denotes the thermal part that accounts for thermal expansion due to temperature changes. Figure 1 gives an illustration.

Fig. 1

The employed multiplicative split of the deformation gradient and its mapping behaviour with respect to the reference and the current configuration as well as the two intermediate configurations $\,\mathcal {{\overline{B}}}_1$ and $\,\mathcal {{\overline{B}}}_2$

3 Specialisation to thermoresponsive hydrogels

We specialise the presented general theoretical principles to the behaviour of thermoresponsive hydrogels at finite strains. Such elastomers are capable of absorbing fluid molecules of low molecular weight and thereby undergo large reversible deformation due to swelling. The swelling characteristics are in general a function of temperature. The mechanical response of such a hydrogel and both its solvent diffusion and heat conduction shall be examined by modelling this strongly coupled saddle point problem.

3.1 Constitutive form of the deformation gradient’s decomposition

In the context of thermoresponsive hydrogels, the deformation gradient might be initially decomposed into an isothermal elastic part $\varvec{F}_{e}$ and a thermal part $\varvec{F}_{\theta }$. The elastic part can then be further divided into a mechanical and a swelling part, yielding a refinement of the employed general multiplicative split (Eq. 30) as:

$$\begin{aligned} \varvec{F} = \varvec{F}_{e} \varvec{F}_{\theta } = \varvec{F}_{m} \varvec{F}_{s} \varvec{F}_{\theta }. \end{aligned}$$

(31)

This form of the multiplicative split is motivated by the fact that the network of polymeric chains, comprising the hydrogel, is elastically stretched due to solvent uptake and when external mechanical forces are applied to it [50, 51]. Additionally, in a non-isothermal setting, further deformation due to thermal expansion will take place.

For the swelling part of the deformation gradient $\varvec{F}_{s}$, we assume the swelling deformation of the body to be isotropic and volumetric.² This assumption is formalised in the literature employing a multiplicative split for material models of polymeric gels—see e.g. [13, 53, 54]—as:

$$\begin{aligned} \varvec{F}_{s} = (1 + \Omega c_{0} )^{\frac{1}{3}} \varvec{I}, \end{aligned}$$

(32)

where $\varvec{I}$ denotes the corresponding mixed identity tensor. The constant factor $\Omega$ denotes the volume of a mole of fluid molecules.

We will resort for the specification of the functional form of $\varvec{F}_{\theta }$ to an approach introduced by [55] for general thermoelastic solids. This approach has successfully been used for other material models—see e.g. [43, 49, 56]—as well. Therefore, the thermally induced deformations, characterised by $\varvec{F}_{\theta }$, are assumed to be also isotropic and volumetric and are expressed as:

$$\begin{aligned} \varvec{F}_{\theta } = e^{ \alpha (\theta - \theta _{\textrm{ref}})} \varvec{I}. \end{aligned}$$

(33)

The constant factor $\alpha$ denotes the thermal expansion coefficient.

Lastly, we introduce, in analogy to the right Cauchy-Green deformation tensor $\varvec{C}$ given in Eq. 3, the elastic right Cauchy-Green tensor $\varvec{C}_{e}$ as:

$$\begin{aligned} \varvec{C}_{e} = \varvec{F}_{e}^\top \varvec{F}_{e} = (\varvec{F} \varvec{F}_{\theta }^{-1})^\top \varvec{F} \varvec{F}_{\theta }^{-1} = J_{\theta }^{\,-2/3}\varvec{C}. \end{aligned}$$

(34)

In the above equation, $J_{\theta } = \sqrt{\text {det}(\varvec{F}_{\theta }^\top \varvec{F}_{\theta })}$ which results with Eq. 33 in $J_{\theta } = e^{ 3 \alpha (\theta - \theta _{\textrm{ref}})}$.

3.2 Free energy function

The free energy function ${\widetilde{\psi }}_{0}(\varvec{C}, c_{0}, \theta )$ is assumed to be additively composed of an elastic part ${\widetilde{\psi }}_{0,e}({\varvec{C}}, \theta )$, accounting for free energy changes due to elastic deformation, a chemical or mixing part ${\widetilde{\psi }}_{0,s}(c_{0}, \theta )$, which accounts for contributions due to species diffusion, a thermal part ${\widetilde{\psi }}_{0,\theta }(\theta )$, accounting for free energy changes due to alterations in the temperature field, and a penalty term ${\widetilde{\psi }}_{0,p}({\varvec{C}}, c_{0}, \theta )$, which enforces volumetric constraints [57]:

$$\begin{aligned}{} & {} {\widetilde{\psi }}_{0}(\varvec{C}, c_{0}, \theta ) = {\widetilde{\psi }}_{0,e}({\varvec{C}},\theta ) + {\widetilde{\psi }}_{0,s}(c_{0}, \theta ) + {\widetilde{\psi }}_{0,\theta }(\theta )\nonumber \\ {}{} & {} + {\widetilde{\psi }}_{0,p}(\varvec{C}, c_{0}, \theta ). \end{aligned}$$

(35)

For the elastic part ${\widetilde{\psi }}_{0,e}$, a neo-Hookean-type material is assumed and formulated with respect to the elastic right Cauchy-Green tensor $\varvec{C}_{e}$ to model the elastic stretching of the polymer network [58, Chapter IX]:

$$\begin{aligned} \begin{aligned}&{\widehat{\psi }}_{0,e}(\varvec{C}_e,\theta ) = {\widetilde{\psi }}_{0,e}(J_{\theta }^{\,-2/3}\varvec{C},\theta ) := \\&{\widetilde{\psi }}_{0,e}(\varvec{C},\theta ) = \frac{N k_B \theta }{2} \left( J_{\theta }^{\,-2/3} I_{\varvec{C}} - 3 - 2\ln (J_{\theta }^{\,-1}J) \right) , \end{aligned} \end{aligned}$$

(36)

whereat the factor $N k_B \theta$ resembles the shear modulus with N being the number of monomers of a prototype polymer chain and $k_B$ being the Boltzmann constant. Furthermore, $I_{\varvec{C}} = \text {tr}(\varvec{C})$ denotes the trace of the right Cauchy-Green deformation tensor.

We use the Flory-Rehner-approach [50, 51] for the mixing part, which is based on the Flory-Huggins theory for polymer solutions. This mixing part ${\widetilde{\psi }}_{0,s}$ accounts for energetic and entropic changes in the polymeric gel due to the mixing of the solvent molecules with the network of polymer chains. This results in the functional form [12]:

$$\begin{aligned} {\widetilde{\psi }}_{0,s}(c_{0}, \theta ) = \mu _{f} c_{0} + R_{m} \theta c_{0}\left[ \ln \!\left( \frac{\Omega c_{0}}{1 + \Omega c_{0}}\right) + \frac{\chi }{1 + \Omega c_{0}} \right] . \end{aligned}$$

(37)

The quantity $\mu _{f}$ denotes the chemical potential of the pure solvent and $R_{m}$ is the (universal) gas constant. The so-called Flory-Huggins interaction parameter $\chi$ characterises the dis-affinity between the polymer and the fluid. An increase of $\chi$ causes solvent molecules to be expelled from the gel and causes the gel to shrink, while a decrease of $\chi$ causes swelling of the polymeric gel due to absorption of solvent molecules [53]. Furthermore, the interaction parameter $\chi$ is assumed to be a linear function of temperature and concentration in the following form [59]:

$$\begin{aligned} \chi (c_{0}, \theta ) = \chi _0 + \frac{\chi _1}{1 + \Omega c_{0}} = A_0 + B_0 \theta + \frac{A_1 + B_1 \theta }{1 + \Omega c_{0}}. \end{aligned}$$

(38)

This allows for the analysis of diverse phenomena associated with a phase transition in hydrogels [15, 60, 61].³ The parameters $A_0$, $B_0$, $A_1$, and $B_1$ vary for different monomers and can be fitted to experimental data.⁴ Alternative functions for $\chi$ are available as well in the literature. See e.g. [62] for a general review, or [12], in which an advanced $\tanh$-approach is used.

For the thermal part of the free energy function we resort to an approach used for the description of general thermo-mechanical materials, pioneered by [55]. Hence, the thermal part ${\widetilde{\psi }}_{0,\theta }$ of the free energy can be stated as [63, Section 7.6]:

$$\begin{aligned} {\widetilde{\psi }}_{0,\theta }(\theta ) = c_{\theta }\left[ \theta - \theta _{\textrm{ref}} - \theta \ln \!\left( \frac{\theta }{\theta _{\textrm{ref}}}\right) \right] , \end{aligned}$$

(39)

whereat $c_{\theta }$ is the specific heat (capacity) and $\theta _{\textrm{ref}}$ is a reference temperature.

Finally, the penalty term is taken as the quadratic form

$$\begin{aligned}{} & {} {\widehat{\psi }}_{0,p}(\varvec{C}_e, c_{0}) = {\widetilde{\psi }}_{0,p}(J_{\theta }^{\,-2/3}\varvec{C}, c_{0}):= {\widetilde{\psi }}_{0,p}(\varvec{C}, c_{0}, \theta )\nonumber \\{} & {} \quad = \frac{\kappa }{2} \left( J_{\theta }^{\,-1}J - J_{s} \right) ^2, \end{aligned}$$

(40)

which includes the penalty parameter $\kappa$ and $J_{s} = \sqrt{\text {det}(\varvec{F}_{s}^\top \varvec{F}_{s})}$ which results with Eq. 32 in $J_{s} = 1 + \Omega c_{0}$. Therefore, this penalty term enforces the incompressibility of both the polymer network and the solvent by further accounting for thermal expansion.

Constitutive models for chemo-thermo-mechanical coupling for hydrogels are not standard. The presented form of the free energy function is chosen in a way such that it resembles a widely accepted chemo-mechanical constitutive model for hydrogels in the isothermal limit case, i.e. for $\theta = \theta _{\textrm{ref}}$. Further, it reduces to an established general thermo-mechanical material model in the dry limit case, i.e. for $c_{0} = 0$. These limit case considerations are further detailed in Appendix A.

Lastly, it has to be noted that for most elastomeric materials the thermal expansion coefficient takes a value in the order of $10^{-4}\hbox {K}^{-1}$ in a usual range of the temperature change considered ($\Delta \theta = \pm 25\textrm{K}$) [12, Footnote 13]. Therefore, it is the variation of the parameter $\chi$ in Eq. 38 that constitutes the main contribution to the thermally-induced volume changes. However, we explicitly account for thermal expansion through $\varvec{F}_{\theta }$ in our model for the sake of generality and to enable dedicated follow-up studies in this regard.

3.3 Pre-swollen reference configuration

If a (nearly) dry state of the body is to be described, the species concentration $c_{0}$ approaches zero. In this case, special care must be taken: Although this scenario represents a physically meaningful state, numerical difficulties arise as the mixing part in Eq. 37 yields ${\widetilde{\psi }}_{0,s} \rightarrow 0$ and the chemical potential in Eq. 18 results in $\mu \rightarrow - \infty$. To prevent these numerical difficulties, an initial concentration [52, 64, 65] or a pre-swollen reference configuration [57, 66] can be employed. In what follows, we introduce such a pre-swollen reference configuration for $\,\mathcal {{B}}_{0}$ as pictured in Fig. 2.

Starting from a dry (i.e. solvent-free) configuration $\,\mathcal {{B}}_d$, the body is assumed to swell homogeneously to the configuration $\,\mathcal {{B}}_{0}$, due to the deformation gradient $\varvec{F}_{0} = J_{0}^{ 1/3} \varvec{I}$. Since in this way, a new configuration is introduced, the species concentration as a volume-specific quantity can now be expressed with respect to the dry configuration, denoted as $c_d$,⁵ and with respect to the pre-swollen reference configuration, denoted as $c_0$, which are related through the identity $c_d = c_0 J_0$ [67]. Both configurations are at the reference temperature $\theta _{\textrm{ref}}$ and stress-free. The imposed requirement that $\,\mathcal {{B}}_{0}$ is stress-free, formalized as $\varvec{P}_d(\varvec{C}_0, c_{d,{\textrm{ref}}}, \theta _{\textrm{ref}}) = \varvec{0}$, where $\varvec{C}_{0} = \varvec{F}_{0}^\top \varvec{F}_{0}$, can be used to establish an explicit relation between $J_{0}$ and the corresponding pre-swelling concentration $c_{0,{\textrm{ref}}}$ with respect to $\,\mathcal {{B}}_{0}$, which reads:

$$\begin{aligned} c_{0,{\textrm{ref}}} = \frac{1}{\Omega }\left( \frac{N k_B \theta }{\kappa } \, \frac{J_0^{2/3} - 1}{J_0^2} + \frac{J_0 - 1}{J_0}\right) . \end{aligned}$$

(41)

Fig. 2

Stress-free dry configuration $\,\mathcal {{B}}_d$ with $c_{d} = 0$ and $\theta = \theta _{\textrm{ref}}$. Stress-free pre-swollen configuration $\,\mathcal {{B}}_{0}$ with $c_{0} = c_{0,{\textrm{ref}}}$ and $\theta = \theta _{\textrm{ref}}$. Both may act as the reference configuration for the deformed configuration $\,\mathcal {{B}}$

We further demand identical energetic states of the stress-free dry and the stress-free pre-swollen configuration, which is formalized as $\int _{\,\mathcal {B}_{d}} {\widetilde{\psi }}_{d}(\varvec{C}_{d}, c_{d}, \theta ) dV$ $= \int _{\,\mathcal {B}_{0}} {\widetilde{\psi }}_{0}(\varvec{C}, c_{0}, \theta ) dV$. This reflects the necessity that either choice of reference configuration should yield the same deformed state of the system. Figure 2 elucidates the mapping

$$\begin{aligned} \varvec{F}_{d} = \varvec{F} \varvec{F}_{0}. \end{aligned}$$

(42)

Furthermore, $\varvec{C}_{d} = \varvec{F}_d^\top \varvec{F}_d = \varvec{F}_0^\top \varvec{C} \varvec{F}_0$ is introduced at this stage. The energetic postulate can then be used to establish a connection between the free energy functions as:

$$\begin{aligned} {\widetilde{\psi }}_{0}(\varvec{C}, c_{0}, \theta ) \,=\, \frac{{\widetilde{\psi }}_{d}(\varvec{C}_{d}, c_{d}, \theta )}{J_{0}}\Bigg |_{\varvec{F}_{d}\,=\, \varvec{F} \varvec{F}_{0}, \; c_d \,=\, c_0 J_0} \end{aligned}$$

(43)

With the help of Eqs. 35, 36, 37, 39, 40, and the definition

$$\begin{aligned} \omega := J_0 J_{\theta }^{-1} \end{aligned}$$

(44)

we reshape this connection to obtain the complete form of the free energy function of the pre-swollen configuration:

$$\begin{aligned} \begin{aligned} {\widetilde{\psi }}_{0}(\varvec{C}, c_{0}, \theta ) = \,&\frac{N k_B \theta }{2 J_{0}} \left[ \omega ^{2/3} I_{\varvec{C}} - 3 - 2\ln (\omega J) \right] \\&+ \mu _{f} c_{0} + R_{m} \theta c_{0}\left[ \ln \!\left( \frac{\Omega J_0 c_{0}}{1 + \Omega J_0 c_{0}}\right) \right. \\&\left. + \frac{ A_0 + B_0 \theta }{1 + \Omega J_0 c_{0}} + \frac{A_1 + B_1 \theta }{(1 + \Omega J_0 c_{0})^2} \right] \\&+ \frac{c_{\theta }}{J_0}\left[ \theta - \theta _{\textrm{ref}} - \theta \ln \!\left( \frac{\theta }{\theta _{\textrm{ref}}}\right) \right] \\&+ \frac{\kappa }{2 J_0} \left( \omega J - J_s \right) ^2. \end{aligned} \end{aligned}$$

(45)

A note for practical implementation: The initial chemical potential of the pre-swollen configuration can be found by $\mu _{\textrm{ref}} = \partial _{c_{0}} {\widetilde{\psi }}_{0} |_{(\varvec{I}, c_{0,{\textrm{ref}}}, \theta _{\textrm{ref}})}$ with the help of Eqs. 18 and 45. After the specification of the constitutive function of the free energy, the quantities that are constitutively linked to the free energy can be derived. The first Piola–Kirchhoff stress tensor can be written by evaluating Eq. 17 as:

$$\begin{aligned} \varvec{P} = J_{0}^{-1} \varvec{F} \left[ N k_B \theta \left( \omega ^{ 2/3} \varvec{G} - (\varvec{C}^{ -1})^{\,\sharp } \right) + \kappa \omega J \left( \omega J - J_{s} \right) (\varvec{C}^{ -1})^{\,\sharp } \right] ,\nonumber \\ \end{aligned}$$

(46)

where $(\varvec{C}^{ -1})^{\,\sharp } \,=\, \varvec{C}^{ -1}\varvec{G}^{ -1}$. The chemical potential can be calculated from Eq. 18 and stated explicitly as:

$$\begin{aligned} \begin{aligned} \mu =&~\mu _{f} + R_{m} \theta \Bigg [ \ln \!\left( \frac{\Omega J_0 c_{0}}{1 + \Omega J_0 c_{0}}\right) + \frac{1}{1 + \Omega J_0 c_{0}} \\&+ \frac{A_0 + B_0 \theta }{(1 + \Omega J_0 c_{0})^2}\\&+ \frac{A_1 + B_1 \theta }{(1 + \Omega J_0 c_{0})^3} (1 - \Omega J_0 c_{0}) \Bigg ] - \kappa \,\Omega \left( \omega J - J_{s} \right) . \end{aligned} \end{aligned}$$

(47)

And finally, the entropy can be calculated by evaluating Eq. 19, which leads to:

$$\begin{aligned} \begin{aligned} \eta _{0} =&~\frac{c_{\theta }}{J_{0}} \ln \!\left( \frac{\theta }{\theta _{\textrm{ref}}}\right) - R_{m} \theta c_{0} \Bigg [ \frac{B_0}{1 + \Omega J_0 c_{0}} + \frac{B_1}{(1 + \Omega J_0 c_{0})^2} \Bigg ]\\&- R_{m} c_{0} \Bigg [ \ln \left( \frac{\Omega J_0 c_{0}}{1 + \Omega J_0 c_{0}}\right) + \frac{A_0 + B_0 \theta }{1 + \Omega J_0 c_{0}} + \frac{A_1 + B_1 \theta }{(1 + \Omega J_0 c_{0})^2} \Bigg ]\\&- \frac{N k_B}{2 J_{0}} \left( \omega ^{ 2/3} I_{\varvec{C}} - 3 - 2\ln (\omega J) \right) \\&+ \frac{\alpha N k_B \theta }{J_0}\left( \omega ^{2/3} I_{\varvec{C}} - 3\right) + \frac{3 \alpha \kappa }{J_0} \omega J \left( \omega J - J_{s} \right) . \end{aligned} \end{aligned}$$

(48)

3.4 Dissipation potential functions

The last step of the specialisation process is the specification of appropriate functional forms for the dissipation potentials. The approach, to derive the species and heat fluxes from dissipation potentials, allows in principle the use of general functional relationships; however, we limit ourselves to linear approaches (as done in e.g. [12, 21, 53, 54, 68] for polymeric gels and [1, 5, 69, 70] for thermo-mechanical materials). Further, we neglect for now the Soret effect [71], i.e. species diffusion caused by temperature gradients, and the Dufour effect [71], i.e. a heat flow caused by concentration gradients, in accordance with existing literature on chemo-thermo-mechanical models for hydrogels (see e.g. [12]). We, however, note that such effects can be included for more specialised treatments [72, 73] and represent an interesting direction for future work.

Fickian diffusion is employed, which relates species flux vectors linearly to gradients of the chemical potential [40, Section 66.4]:

$$\begin{aligned} \varvec{H} = - \frac{1}{\theta } \,\overline{\!{\varvec{M}}}\, {\overline{\nabla }}_{\!0} \mu \end{aligned}$$

(49)

with the temperature-scaled mobility tensor:

$$\begin{aligned} \,\overline{\!{\varvec{M}}}(\varvec{X},t) = \widetilde{\,\overline{\!{\varvec{M}}}}({\overline{\nabla }}_{\!0} \mu ; \varvec{C}, c_{0}, \nabla _{\!0} c_{0}, \mu , \theta , \nabla _{\!0} \theta ). \end{aligned}$$

(50)

Further, Fourier-type heat conduction is assumed, which relates heat fluxes linearly to temperature gradients [40, Section 42]:

$$\begin{aligned} \varvec{Q} = - \,\overline{\!{\varvec{K}}}\, {\overline{\nabla }}_{\!0} \theta \end{aligned}$$

(51)

with the temperature-scaled conductivity tensor:

$$\begin{aligned} \,\overline{\!{\varvec{K}}}(\varvec{X},t) = \widetilde{\,\overline{\!{\varvec{K}}}}({\overline{\nabla }}_{\!0} \theta ; \varvec{C}, c_{0}, \nabla _{\!0} c_{0}, \mu , \nabla _{\!0} \mu , \theta ). \end{aligned}$$

(52)

The temperature scaling is in accordance with the approach outlined in Sect. 2.3.2.

The material under consideration is assumed to be isotropic with respect to species diffusion and heat conduction. Mathematically, this assumption is expressed as:

$$\begin{aligned} \,\overline{\!{\varvec{M}}}&= {\overline{m}} \varvec{I}, \quad \quad&{\overline{m}}&= {\overline{m}}({\overline{\nabla }}_{\!0} \mu ; \varvec{C}, c_{0}, \nabla _{\!0} c_{0}, \mu , \theta , \nabla _{\!0} \theta ), \end{aligned}$$

(53)

$$\begin{aligned} \,\overline{\!{\varvec{K}}}&= {\overline{k}} \varvec{I}, \quad \quad&{\overline{k}}&= {\overline{k}}({\overline{\nabla }}_{\!0} \theta ; \varvec{C}, c_{0}, \nabla _{\!0} c_{0}, \mu , \nabla _{\!0} \mu , \theta ), \end{aligned}$$

(54)

with ${\overline{m}}$ and ${\overline{k}}$ being the scalar, temperature-scaled mobility and conductivity, respectively.

Equation 53 is usually defined by symmetry consideration with respect to the reference configuration. However, [54, 68] argued that it is more appropriate to formulate the demand of isotropy with respect to diffusion in terms of the current configuration,⁶ i.e.:

$$\begin{aligned} \varvec{h} = - {\overline{m}} \; {\overline{\nabla }}_{\!s} \mu , \end{aligned}$$

(55)

and afterwards to express this spatial relation in terms of the reference configuration by the use of the respective transformation laws. Since this approach is advantageous [13, 21, 69], it will also be adopted for the diffusion and thermal dissipation potential here. Therefore, the heat conduction law is assumed to be of the form:

$$\begin{aligned} \varvec{q} = - {\overline{k}} \; {\overline{\nabla }}_{\!s} \theta . \end{aligned}$$

(56)

For the mobility ${\overline{m}}$, we choose an approach that is widely used in the literature concerning polymeric gels [13, 54, 68] and adapt it to the present context:

$$\begin{aligned} {\overline{m}} = \frac{D' c_{0} \theta }{J R_{m}}, \end{aligned}$$

(57)

whereat $D'$ is the (constant) permeability coefficient or diffusivity and the spatial concentration $c = c_0 / J$ is used in this definition since the employed diffusion law is given with respect to the current configuration.

For the conductivity ${\overline{k}}$ a linear interpolation approach [12] is chosen⁷:

$$\begin{aligned} {\overline{k}} = \frac{\theta (k_{p} + \Omega c_{0} k_{f})}{1 + \Omega c_{0}}, \end{aligned}$$

(58)

whereat $k_{p}$ and $k_{f}$ are the (constant) conductivity of the polymer and the solvent (fluid), respectively.

Finally, the desired functional forms of the dissipation potentials can be stated with these considerations as [43, 64]:

(59)

The constitutive functions for the species and the heat flux vector can be determined from these dissipation potentials by the use of Eqs. 27 and 28 and follow as:

$$\begin{aligned} \varvec{H} = - \frac{D' c_{0}}{R_{m}} \varvec{C}^{ -1}\cdot {\overline{\nabla }}_{\!0} \mu , \end{aligned}$$

(60)

and

$$\begin{aligned} \varvec{Q} = - \frac{J\theta (k_{p} + \Omega c_{0} k_{f})}{1 + \Omega c_{0}} \varvec{C}^{ -1}\cdot {\overline{\nabla }}_{\!0} \theta . \end{aligned}$$

(61)

4 Finite element formulation

4.1 Preliminary remarks

With the specification of the constitutive functions for all quantities involved in the model, we have obtained a set of partial differential equations in the variables $\varvec{\varphi }$, $c_{0}$, $\mu$, $\eta _{0}$ and $\theta$. We restate this set of equations as:

$$\begin{aligned}&\text {DIV}(\varvec{P}) + \varvec{B} = \varvec{0} \end{aligned}$$

(62)

$$\begin{aligned}&{\dot{c}}_{ 0} + \text {DIV}(\varvec{H}) = 0 \end{aligned}$$

(63)

$$\begin{aligned}&{\dot{\eta }}_{0} = \frac{R_{q}}{\theta } - \frac{1}{\theta } \text {DIV}(\varvec{Q}) - {\overline{\nabla }}_{\!0} \mu \cdot \varvec{H} \end{aligned}$$

(64)

$$\begin{aligned}&\eta _{0} = - \frac{\partial {\widetilde{\psi }}_{0}}{\partial \theta } \end{aligned}$$

(65)

$$\begin{aligned}&\mu = \frac{\partial {\widetilde{\psi }}_{0}}{\partial c_{0}} \end{aligned}$$

(66)

where $\varvec{P}$, $\varvec{H}$, and $\varvec{Q}$ are related to the stated variables through Eqs. 17, 27, and 28 respectively. By examination of Eqs. 62–66, it can be seen that these equations involve the gradients of $\varvec{\varphi }$, $\mu$ and $\theta$ but not of $c_{0}$ and $\eta _{0}$. This implies that the fields $\varvec{\varphi }$, $\mu$ and $\theta$ are global fields in the sense that their value at a given point of the body depends also on the values of neighbouring points. The quantities $c_{0}$ and $\eta _{0}$, on the other hand, are so-called local fields since their values do not depend on the values of surrounding points.⁸ In the preceding sections we formulated our material model with respect to $\varvec{\varphi }$, $c_0$ and $\theta$ since these quantities are the most tangible and easiest to measure. For the following finite element implementation, however, the species concentration is re-interpreted as a function of the global field variables, i.e. $c_{0}(\varvec{\varphi }, \mu , \theta )$,⁹ and its values are obtained by local condensation [57]. In this way, all field variables can be determined consistently. To do so, we use Eq. 66 as an implicit equation for $c_{0}$ and determine the local values of the concentration for given $\varvec{\varphi }$, $\mu$ and $\theta$ by the residual:

$$\begin{aligned} R_c&= \frac{\partial {\widetilde{\psi }}_{0}}{\partial c_{0}} - \mu , \end{aligned}$$

(67)

which has to vanish at the solution point. Applying Newton’s method (linearisation) with respect to the concentration, yields an iteration procedure

$$\begin{aligned} {R_{c}^{k+1}}&= R_{c}^{k} + \frac{\partial {R}_{c}}{\partial c_{0}} \Delta {c}_0 \; := 0 \end{aligned}$$

(68)

to solve for the increments $\Delta c_0$ until convergence is reached. After the species concentration ${c}_0$ is locally determined, the global fields $\varvec{\varphi }$, $\mu$ and $\theta$ can be determined by the Eqs. 62–65.

From the mathematical perspective, the set of Eqs. (62–65) represents a coupled initial boundary-value problem in the three primary variables $\varvec{\varphi }$, $\mu$ and $\theta$. We specify initial conditions for the deformation field ($\varvec{\varphi }_{in}$), the chemical potential field ($\mu _{in}$) and the temperature field ($\theta _{in}$) at each point $\varvec{X} \in \,\mathcal {B}_{0}$ of the body in the reference configuration. Mechanical, chemical and thermal boundary conditions may be imposed at each point $\varvec{X} \in \partial \!\,\mathcal {B}_0$ of the body’s surface in the reference configuration. The mechanical boundary conditions are either deformation boundary conditions or traction boundary conditions:

$$\begin{aligned} \qquad \qquad \partial \!\,\mathcal {B}_0 = \partial \!\,\mathcal {B}_{0,\varvec{\varphi }} \cup \partial \!\,\mathcal {B}_{0,\varvec{T}}{} & {} {}&\partial \!\,\mathcal {B}_{0,\varvec{\varphi }} \cap \partial \!\,\mathcal {B}_{0,\varvec{T}} = \emptyset \qquad \qquad \end{aligned}$$

(69)

$$\begin{aligned} \varvec{\varphi } = \breve{\varvec{\varphi }}(\varvec{X},t){} & {} {}&\varvec{X} \in \partial \!\,\mathcal {B}_{0,\varvec{\varphi }} \end{aligned}$$

(70)

$$\begin{aligned} \varvec{P}\varvec{N} = \breve{\varvec{T}}(\varvec{X},t){} & {} {}&\varvec{X} \in \partial \!\,\mathcal {B}_{0,\varvec{T}} \end{aligned}$$

(71)

The chemical boundary conditions are either chemical potential boundary conditions or species flux boundary conditions:

$$\begin{aligned} \qquad \qquad \partial \!\,\mathcal {B}_0 = \partial \!\,\mathcal {B}_{0,\mu } \cup \partial \!\,\mathcal {B}_{0,\varvec{H}}{} & {} {}&\partial \!\,\mathcal {B}_{0,\mu } \cap \partial \!\,\mathcal {B}_{0,\varvec{H}} = \emptyset \qquad \qquad \end{aligned}$$

(72)

$$\begin{aligned} \mu = \breve{\mu }(\varvec{X},t){} & {} {}&\varvec{X} \in \partial \!\,\mathcal {B}_{0,\mu } \end{aligned}$$

(73)

$$\begin{aligned} \varvec{H}\varvec{N} = \breve{H}(\varvec{X},t){} & {} {}&\varvec{X} \in \partial \!\,\mathcal {B}_{0,\varvec{H}} \end{aligned}$$

(74)

Lastly, the thermal boundary conditions are either temperature boundary conditions or heat flux boundary conditions:

$$\begin{aligned} \qquad \qquad \partial \!\,\mathcal {B}_0 = \partial \!\,\mathcal {B}_{0,\theta } \cup \partial \!\,\mathcal {B}_{0,\varvec{Q}}{} & {} {}&\partial \!\,\mathcal {B}_{0,\theta } \cap \partial \!\,\mathcal {B}_{0,\varvec{Q}} = \emptyset \qquad \qquad \end{aligned}$$

(75)

$$\begin{aligned} \theta = \breve{\theta }(\varvec{X},t){} & {} {}&\varvec{X} \in \partial \!\,\mathcal {B}_{0,\theta } \end{aligned}$$

(76)

$$\begin{aligned} \varvec{Q}\varvec{N} = \breve{Q}(\varvec{X},t){} & {} {}&\varvec{X} \in \partial \!\,\mathcal {B}_{0,\varvec{Q}} \end{aligned}$$

(77)

Fig. 3 summarises the model.

Fig. 3

Three-field initial boundary-value problem for deformation-diffusion-conduction coupling. The deformation map $\varvec{\varphi }$, the chemical potential $\mu$ and the temperature $\theta$ are global fields, while the fluid concentration $c_{0}$ is a local field for Fickian-type diffusion. The boundary of the solid is decomposed into Dirichlet and Neumann-type parts for the global fields

4.2 Method of weighted residuals and weak form

Employing the method of weighted residuals, we derive the weak form of the set of partial differential equations (62–64). Therefore, Eqs. 62–64—the strong form of the coupled partial differential equations—are weighted with appropriate test functions $\delta v$ and integrated over the domain under consideration. We introduce a vectorial deformation test function field $\delta \varvec{v}_{\varvec{\varphi }}(\varvec{X})$, a scalar chemical potential test function field $\delta v_{\mu }(\varvec{X})$ and a scalar temperature test function field $\delta v_{\theta }(\varvec{X})$, which are defined as [5]:

$$\begin{aligned} \delta \varvec{v}_{\varvec{\varphi }}(\varvec{X})&= \{ \delta \varvec{v}_{\varvec{\varphi }}(\varvec{X}) ~|~ \delta \varvec{v}_{\varvec{\varphi }}(\varvec{X}) = \varvec{0} \quad \forall \varvec{X} \in \partial \!\,\mathcal {B}_{0,\varvec{\varphi }} \}, \end{aligned}$$

(78)

$$\begin{aligned} \delta v_{\mu }(\varvec{X})&= \{ \delta v_{\mu }(\varvec{X}) ~|~ \delta v_{\mu }(\varvec{X}) = 0 \quad \forall \varvec{X} \in \partial \!\,\mathcal {B}_{0,\mu } \} , \end{aligned}$$

(79)

$$\begin{aligned} \delta v_{\theta }(\varvec{X})&= \{ \delta v_{\theta }(\varvec{X}) ~|~ \delta v_{\theta }(\varvec{X}) = 0 \quad \forall \varvec{X} \in \partial \!\,\mathcal {B}_{0,\theta } \}. \end{aligned}$$

(80)

These test functions are then used to derive the weak form of the initial boundary value problem. For the balance of linear momentum, we obtain:

$$\begin{aligned} \begin{aligned} \int _{\,\mathcal {B}_0} \varvec{P}: \nabla _{\!0} \delta \varvec{v}_{\varvec{\varphi }} dV = \int _{\,\mathcal {B}_0} \varvec{B}\cdot \delta \varvec{v}_{\varvec{\varphi }} dV + \int _{\partial \,\mathcal {B}_{0,\varvec{T}}} \breve{\varvec{T}} \cdot \delta \varvec{v}_{\varvec{\varphi }} dA \\ \varvec{\varphi }(\varvec{X},t) = \breve{\varvec{\varphi }}(\varvec{X}) \;\; \text {for} \; \varvec{X} \! \in \! \partial \!\,\mathcal {B}_{0,\varvec{\varphi }} \qquad \quad \varvec{\varphi }(\varvec{X},0) = \varvec{\varphi }_{in}(\varvec{X}) \;\; \text {for} \; \varvec{X} \!\in \! \,\mathcal {B}_{0} \end{aligned} \end{aligned}$$

(81)

The weak form of the material species balance follows as:

$$\begin{aligned} \begin{aligned}&\int _{\,\mathcal {B}_0} {\dot{c}}_{0} \delta v_{\mu } dV + \int _{\,\mathcal {B}_0} {\overline{m}} J {\overline{\nabla }}_{\!0} \delta v_{\mu } \cdot (\varvec{C}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \mu dV \\&= - \int _{\partial \,\mathcal {B}_{0,H}} \breve{H} \delta v_{\mu } dA \\&\mu (\varvec{X},t) = \breve{\mu }(\varvec{X}) \;\; \text {for} \; \varvec{X} \! \in \! \partial \!\,\mathcal {B}_{0,\mu } \qquad \quad \mu (\varvec{X},0) \\&= \mu _{in}(\varvec{X}) \;\; \text {for} \; \varvec{X} \! \in \! \,\mathcal {B}_{0} \end{aligned} \end{aligned}$$

(82)

Finally, the weak form of the entropy balance is written as:

$$\begin{aligned} \begin{aligned}&\int _{\,\mathcal {B}_0} {\dot{\eta }}_{0} \delta v_{\theta } dV - \int _{\,\mathcal {B}_0} \frac{1}{\theta } {\overline{k}} J {\overline{\nabla }}_{\!0} \theta \cdot (\varvec{C}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \theta \delta v_{\theta } dV \\ {}&+ \int _{\,\mathcal {B}_0} {\overline{k}} J {\overline{\nabla }}_{\!0} \delta v_{\theta } \cdot (\varvec{C}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \theta dV \\&- \int _{\,\mathcal {B}_0} \frac{1}{\theta } {\overline{m}} J {\overline{\nabla }}_{\!0} \mu \cdot (\varvec{C}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \mu \delta v_{\theta } dV = \int _{\,\mathcal {B}_0} \frac{R_{q}}{\theta } \delta v_{\theta } dV \\ {}&- \int _{\partial \,\mathcal {B}_{0,Q}} \frac{\breve{Q}}{\theta } \delta v_{\theta } dA \\&\theta (\varvec{X},t) = \breve{\theta }(\varvec{X}) \;\; \text {for} \; \varvec{X} \! \in \! \partial \!\,\mathcal {B}_{0,\theta } \qquad \quad \theta (\varvec{X},0) = \theta _{in}(\varvec{X}) \;\; \text {for} \; \varvec{X} \! \in \! \,\mathcal {B}_{0} \\ \end{aligned} \end{aligned}$$

(83)

Starting from the stated weak form expressions, the set of equations is first discretised with respect to time, then linearised and finally spatially discretised. Since these procedures are standard, they are not reproduced here. However, a detailed derivation is provided from Appendix B to Appendix D for the convenience of the reader.

4.3 LBB condition and Taylor-Hood element

The derived finite element formulation of the material model represents a coupled three-field problem (deformation, chemical potential and temperature), also referred to as mixed method/problem. Such mixed problems have to satisfy the so-called Ladyzhenskaya-Babuška-Brezzi (LBB) condition [75], which is also labelled inf-sup condition [76, Section 4.5], to avoid oscillations of the respective scalar field quantities and therefore to be numerically stable. One way to satisfy the LBB condition is the use of so-called Taylor-Hood elements [77].

In the context of hydrogels, [74] employed Taylor-Hood elements for a chemo-mechanical two-field problem and studied its numerical stability in-depth. A Taylor-Hood element in this context employs an interpolation scheme for the deformation field of one order higher than the one used for the chemical potential field. Since the mathematical structure of the equations is similar for both the evolution of the chemical potential field and the temperature field, we also use this element type for the present problem. However, we leave the numerical proof of meeting the LBB condition to future work, see for instance [78]. Therefore, a quadratic ansatz is used for the displacements and a linear ansatz for the field of the chemical potential as well as the temperature field, which is also illustrated in Fig. 4.

Fig. 4

A Taylor-Hood element (2D for clarity): circles mark the deformation nodes (serendipity element type), squares the chemical potential and temperature nodes and crosses mark the integration points

5 Numerical examples

After the theoretical derivation of the presented material model and the corresponding finite element formulation, the capabilities of this model shall be verified in a series of numerical examples. Four examples¹⁰ are discussed: the free swelling of a PNIPAM hydrogel block, the swelling dynamics of a PNIPAM hydrogel block, the isothermal buckling of a hydrogel corona and the thermally-triggered swelling-induced buckling of a hydrogel corona. For all conducted studies, an in-house Python 3 finite element solver was employed.

5.1 Free swelling of a PNIPAM block

In this first example, we study the swelling behaviour of a thermally responsive poly(N-isopropylacrylamide)-based (PNIPAM) hydrogel. It is well known that such PNIPAM gels exhibit a characteristic phase transition temperature (PTT) or lower critical solution temperature (LCST) [79] at which they undergo an abrupt change from a hydrophilic swollen state (below the PTT) to a hydrophobic (nearly) dry state (above the PTT). This characteristic behaviour has been investigated well experimentally by measuring the volume of the hydrogel as a function of temperature [80, 81]. On the other hand, this discontinuous swelling poses a considerable challenge regarding the numerical modelling of such gels that was addressed repeatedly in the literature [14,15,16,17, 30, 60, 82,83,84].

We consider a block of PNIPAM gel, which is fully immersed in a water bath, i.e. $\breve{\mu } = 0\mathrm{Nmm/nmol}$, and which can swell without mechanical constraints. For this, a cube with an edge length of $10\textrm{mm}$ is used. We conduct quasi-static simulations to exclude dynamic effects. Therefore, a discretisation of the cube with a single 3D Taylor-Hood element was found to be sufficient. If heated/cooled sufficiently slowly, the cubic hydrogel will shrink/swell by the same stretch in all directions. In this case, the ratio between the current volume of the gel and the volume of the dry gel is given by the Jacobian determinant $J_{d}$, which is found by invoking Eq. 42 as $J_{d} = J J_{0}$.

In the vicinity of the PTT an abrupt volume change takes place. For the chosen material data this phase transition occurs in the temperature range $\theta =304.7-305.8\textrm{K}$, see also [14, 15]. In this temperature range, the free energy function exhibits two local minima and the cube would “jump” instantly from one state to the other in nature. However, the finite element simulation only obtains one stable state at a time and modelling the jump would require the implementation of an involved path-tracking method. Therefore, we choose to approach the PTT from a lower and a higher temperature level with two separate setups. Firstly, at temperature $\theta _{\textrm{ref}}=290\textrm{K}$ with a pre-swelling ratio of $J_{0}(\breve{\mu },\theta _{\textrm{ref}})=23.728$ we heat the system up to $\theta =305\textrm{K}$ with a temperature increase of $\Delta \theta = 0.015\textrm{K}$ per $\Delta t = 0.01\textrm{s}$. Secondly, at temperature $\theta _{\textrm{ref}}=340\textrm{K}$ with a pre-swelling ratio of $J_{0}(\breve{\mu },\theta _{\textrm{ref}})=1.197$ we cool the system down to $\theta =305\textrm{K}$ with a temperature decrease of $\Delta \theta = 0.035\textrm{K}$ per $\Delta t = 0.01\textrm{s}$. The material constants are detailed in Table 1.

Table 1 Material properties. The values for $A_0, B_0, A_1$ and $B_1$ are in accordance with [15, 85]. The transport coefficients ($k_{p}, k_{f}, D'$) are set to arbitrary high values in order to immediately achieve the state of equilibrium and therewith mimic a quasi-static setting

Fig. 5

The volume ratio of the free swelling hydrogel as a function of temperature. The pluses ($+$) and crosses ($\times$) give the numerical predictions of the proposed model by heating or cooling the cube, respectively. The circles ($\circ$) present the experimental data by [80, Fig. 8a]. The solid curves give the analytical solution by [15, Eq. 55]

To verify the predictions of the two simulations, the numerical results are compared with available experimental data and analytical calculations in Fig. 5. The work [80] has provided experimental results by preparing PNIPAM gel particles with various C%-values.¹¹ We compare our results with the data obtained for C%=2.50 in [80, Fig. 8a]. Furthermore, an analytical solution of a hydrogel cube under free swelling is presented in [15, Eq. 55]. This work investigated a free energy function, which is similar to the one employed in this work (Eq. 45), and minimised this free energy at a certain temperature to obtain the corresponding volume ratio. As can be seen in Fig. 5, the numerical prediction of our model corresponds very well with both the experimental data and the analytical solution.

5.2 Swelling dynamics of a PNIPAM block

After assessing the capabilities of our model to simulate the phase transition of PNIPAM hydrogels correctly and in accordance with the literature in a quasi-static setting, we now want to study the previously excluded dynamic effects. This seems reasonable, both to test the model’s capabilities in dynamic situations, and to support our earlier findings by showing that the results obtained in the previous example are indeed the limit case for $t \rightarrow \infty$.

We want to study the thermally induced solvent uptake and inhomogeneous swelling behaviour of hydrogels and consider a PNIPAM block with an edge length of 20 mm. This block is, as in Ex. 5.1, fully immersed in a water bath, i.e. $\breve{\mu } = 0\mathrm{Nmm/nmol}$, and can swell without mechanical constraints. Due to the symmetry of the problem, we discretise only an eighth of the cube with $5\times 5\times 5$ Taylor-Hood elements. This time, we use finite values for the thermal conductivity of the gel and the solvent as well as a finite diffusion coefficient. The values are listed in Table 2, the remaining model parameters are the same as in Ex. 5.1. While the thermal conductivity values are chosen to be realistic, we have chosen a slightly exaggerated diffusion coefficient as we are interested in studying the dynamic effects only qualitatively and also in keeping the required simulation time within reasonable limits.

We assume that the block has been equilibrated before the start of the simulation at the temperature $\theta _{\textrm{ref}}=300\textrm{K}$ and at the chemical potential $\breve{\mu }$, giving a pre-swelling ratio of $J_{0}(\breve{\mu },\theta _{\textrm{ref}})=14.482$ (cf. Figure 5). At the beginning of the simulation, the temperature at the faces of the cube is “abruptly” reduced to $\theta =290\textrm{K}$, while the chemical potential is kept at $\breve{\mu }$. This temperature decrease is implemented by a linear ramping, as is common practice in such simulations [57], by a decrease of $\Delta \theta = 0.1\textrm{K}$ per $\Delta t = 0.2\textrm{s}$. In the following time intervals, the faces of the cube are kept at $\theta =290\textrm{K}$ and $\breve{\mu }$, while the bulk material is allowed to adapt to these boundary conditions: we first advance the system by 50 time steps with $\Delta t = 0.5\textrm{s}$, then by 50 time steps with $\Delta t = 1\textrm{s}$, then by 50 time steps with $\Delta t = 10\textrm{s}$, and finally by 25 time steps with $\Delta t = 100\textrm{s}$. This gives a total simulated time of 3095 s.

The temperature $\theta$ and the chemical potential $\mu$ versus the simulation time t are given in Fig. 6. In particular, we analyse two observation points: the centre C of the block (i.e. at $x=y=z=0\textrm{mm}$) and a point O that is located on the space diagonal closer to the cube’s corner (i.e. at $x=y=z=6\textrm{mm}$) and which we will refer to as “outer point” in the following.

The externally applied temperature change causes a rapid decrease in the temperature field within the cube. As can be seen in Fig. 6, the outer and centre point are cooled down to $\theta =290\textrm{K}$ within about 87 s and 175 s respectively. The cooling thereby takes place exponentially as is typical for the assumed Fourier-type heat conduction. While these time values should be treated with caution since they are influenced by the ramping scheme chosen, the general behaviour and the different delays in the onset of the cooling process are physically reasonable.

Table 2 Altered material properties for swelling dynamics of a PNIPAM block

Fig. 6

The temperature $\theta$ and the chemical potential $\mu$ as a function of simulation time t. The solid and dashed lines give the temperatures of outer O and centre C points, respectively. The dash-dotted and dotted lines give the chemical potential of outer and centre points, respectively. The crosses ($\times$) give the prescribed temperature at the outer surfaces of the cube. The curves remain on the same level until the end of the simulation time of $3095\textrm{s}$

The evolution of the chemical potential in the two chosen observation points, as shown in Fig. 6, is less intuitive than the temperature evolution. It can be seen that starting from its initial value $\breve{\mu } = 0\mathrm{Nmm/nmol}$, the chemical potential in both points drops and reaches a minimum value of $-9.717\times 10^{-6}\mathrm{Nmm/nmol}$ after 24.5s and $-13.171\times 10^{-6}\mathrm{Nmm/nmol}$ after 59 s in the outer and centre point respectively. Then, it rises again and approaches the applied boundary value $\breve{\mu }$ for both points. The observed minima in the evolution of the chemical potential are a result of the two different time scales involved in this simulation: On the one hand, heat conduction is a relatively fast process, due to which the chemical potential in the block is lowered in a first phase without notable volume expansion. Thereafter, a slower species diffusion process takes place, in which the chemical potential is equilibrated again due to the uptake of additional solvent, which also causes the observed swelling of the block. The position (in time) and depth of the observed minima in both points are thereby governed by this competition as well.

Fig. 7

The volume ratio as a function of simulation time t. The solid and dashed lines give the volume ratios of outer O and centre C points, respectively. The dash-dotted line gives the radius of the circumscribed sphere to the cube, while the dotted line gives the radius of the inscribed sphere to the cube. The curves remain on the same level until the end of the simulation time of $3095\textrm{s}$

The volume expansion is discussed next. The evolution of the volume ratio for the outer and centre point is depicted in Fig. 7. As can be seen, also the volume ratio curves exhibit a decrease and minimum values before approaching asymptotically the expected final ratio of $J_d(\breve{\mu },\theta =290\textrm{K})=23.728$ (cf. Figure 5). In this case, however, the identified minima in the simulation arise from numerical oscillations in the initial stages, lacking physical significance. Despite efforts to address these oscillations by employing measures like Taylor-Hood elements and a carefully selected ramping scheme for boundary conditions, complete elimination proved challenging. This challenge persists due to the inherent complexities introduced by the two mentioned time scales: the gradual temperature boundary condition ramping, aligned with the heat conduction time scale, successfully eliminates observable numerical oscillations in the hydrogel block’s temperature field (see Fig. 6); however, the gradual application still resembles a “sharp drop”, interpreted in terms of the species diffusion time scale, and therefore introduces numerical oscillations in the field of the species concentration, which governs the swelling process and ultimately the volume ratio field. Moreover, the corners of the cube experience more pronounced swelling compared to the centres of the faces. This phenomenon arises because, at the corners, diffusion occurs from three orthogonal directions. Figure 7 illustrates the radius of the circumscribed sphere to the cube (half of the space diagonal) and the radius of the inscribed sphere to the cube (half of the distance between opposite faces) for reference. The different evolution of these two radii over time is particularly evident in the first 200 s. The observed inhomogeneous swelling behaviour is in good agreement with the literature on transient swelling simulations [12, 15, 21, 57].

In this example, we have examined the ability of our model to capture the temporal evolution of thermally induced swelling processes. We have confirmed that the relevant physical phenomena and the temporal evolution of the pertinent field quantities are qualitatively reproduced in a reliable manner, as well as the correct limit values of all field quantities for $t \rightarrow \infty$. This is not apparent due to the two different time scales involved in this kind of simulation, which also give rise to the observed persistent numerical oscillations.

5.3 Isothermal buckling of a hydrogel corona

We study next a boundary value problem based on the experimental and analytical treatment in [37]. This work investigated coronae (rings) made of a soft hydrogel that had been fixed to a disk of stiff hydrogel at the inner rim. The whole structure was immersed in water and the soft gel swelled and increased its volume, while the stiff gel was of non-swelling nature. At a certain swelling level, a bifurcation point was reached and the soft corona buckled out-of-plane with a wave-like pattern towards the outer rim. The larger the ratio of inner to outer radii $r_i/r_o$, the higher the observed wave number m, which is tantamount to the number of complete deformation waves appearing in the soft corona. These experiments have subsequently also been treated numerically in a finite element context [14, 15, 36, 57].

In the course of our numerical investigations, we obtained significantly different results than the analytical/numerical treatment presented in [37]. A detailed examination of this issue brought us to the following conclusion: In [37], a differential equation and boundary conditions suitable to obtain a numerical solution to the wave number were presented. Most likely, a minor miscalculation of the differential equation happened that yields the observed deviations. This view is supported by our numerical study presented in the following and other results obtained in the literature. We, therefore, propose a corrected version of the analytical theory established in [37] in Appendix E.

Fig. 8

Geometry and boundary conditions of the hydrogel corona. The inner rim is mechanically clamped. Points $P=(r_o, 0, 0)$ and $Q=(0, r_o, 0)$ are defined for observation purposes

We consider a hydrogel corona with the geometry and boundary conditions as given in Fig. 8. The outer radius $r_o$ and the thickness h are kept constant at $50\textrm{mm}$ and $1\textrm{mm}$ respectively. We investigate the ratios $r_i/r_o$: 0.200, 0.228, 0.289, 0.366, 0.410, 0.475, 0.534, 0.623, 0.684, 0.728, 0.761, 0.787, 0.808, 0.826 and 0.840. We use again the material parameters as given in Table 1, since for the considered setting PNIPAM gels and the polyacrylamide gels, studied in [37], have comparable elastic properties (cf. [37] and [86]). Further, for buckling studies geometric parameters are of superior importance compared to material properties. However, the parameters for conductivity and diffusivity are adapted to realistic values in this example and are listed in Table 3. We model only half of the corona since this was found to be sufficient to capture all relevant buckling modes (even and odd) and discretise it by $6\times 60\times 1$ Taylor-Hood elements in radial, azimuthal and thickness direction, respectively. The stabilising pre-swelling parameter $J_{0}$ is set to 2.0 as suggested by [57].¹²

Table 3 Altered material properties for isothermal buckling of a hydrogel corona

The swelling causes buckling of the corona, which is a structural instability arising at bifurcation points. However, the perfect plane corona will not buckle in silico without proper numerical treatment. We have therefore chosen to apply an out-of-plane force F at point Q of Fig. 8 to disturb the system in an initial phase and to aid the corona to take the secondary (post-buckling) path.¹³ Although this strategy is simple, it promotes that no specific post-buckling mode is preferred, but the system will take its first mode with the largest deflection. Such a disturbance is in alignment with the literature of similar numerical treatments, see e.g. [69, 87] and the supplemental material of [88].

The temperature is kept constant at $\theta _{\textrm{ref}}=300\textrm{K}$ throughout the simulation. Influx of solvent molecules is enabled at the corona’s top and bottom faces by appropriate Dirichlet boundary conditions. In doing so we increase the chemical potential at these faces from the initialised system in three load increments. Firstly, we increase in 25 time steps with $\Delta t = 0.1\textrm{s}$ the chemical potential from $\mu _{in}(J_{0},\theta _{\textrm{ref}}) = -1.37\times 10^{-4}\mathrm{Nmm/nmol}$ to $\breve{\mu }_1=-1.32\times 10^{-4}\mathrm{Nmm/nmol}$ and apply the out-of-plane force F in a ramp function. Secondly, we remove the force F and keep the faces at $\breve{\mu }_1$ for a single time step with $\Delta t = 0.1\textrm{s}$. Thirdly, we increase in 25 time steps with $\Delta t = 0.02\textrm{s}$ the chemical potential to $\breve{\mu }_2=-1.28\times 10^{-4}\mathrm{Nmm/nmol}$. This value was chosen since it was found that already at this stage the wave number could be safely determined for all cases.

Fig. 9

Comparison of experimental and numerical results. The dashed line gives the numerical solution to the differential equation as proposed by [37, Eq. 15]. The asterisks ($*$) show the numerical values of this solution given in [37, Fig. 7]. The solid line presents the numerical solution to our corrected differential equation in D20. The open circles ($\circ$) give the experimental data provided by [37, Fig. 7]. The crosses ($\times$) and the upward-pointing triangles ($\triangle$) correspond to numerical data from [57] by employing a minimisation and a saddle-point problem, respectively. The left-pointing triangles ($\triangleleft$) and the right-pointing triangles ($\triangleright$) correspond to numerical data from [14] and [36], respectively. Finally, the squares ($\square$) give the numerical predictions of the proposed model in this work

The predicted buckling patterns of the proposed model are compared with available experimental and numerical data in Fig. 9 as well as with the numerical solution of the differential equation in Appendix E. The wave numbers predicted by our proposed model, given with squares ($\square$) in the figure, match very well with the solution to the corrected form of the differential equation, given as solid line. Furthermore, the numerical data obtained from similar finite element treatments of the corona problem presented by other works match very well with the solution to our form of the differential equation. The solution to the original shape of the differential equation presented by [37] is given as a dashed line and shows significantly different regions for a steady wave number, in particular for small ratios of the radii. The deformation patterns of the investigated ratios are presented in Fig. 10.

Fig. 10

Out-of-plane displacement $\zeta$ as shown in the colour bar [mm] at $\breve{\mu }_2=-1.28\times 10^{-4}\mathrm{Nmm/nmol}$. The ratio $r_i/r_o$ is given to the left of each image, which controls the number of swelling-induced buckles. Furthermore, the obtained wave number m is given in brackets

Fig. 11

The ratio $r_i/r_o=0.808$ is chosen as a representative example. a Out-of-plane displacement $\zeta$ as shown in the colour bar [mm] at $\breve{\mu }_3=-4\times 10^{-5}\mathrm{Nmm/nmol}$. b Chemical potential $\mu$ versus displacement $\zeta$ at point P throughout the computation. The values $\breve{\mu }_i$ of the prescribed chemical potential are indicated by vertical, dash-dotted lines

Next, we take the ratio $r_i/r_o=0.808$ as a representative example to present some additional numerical results. The expected wave number for this corona geometry is 10. Following the previously defined three load increments, we add a fourth to increase the swelling. In the fourth increment, we increase in 125 time steps with $\Delta t = 0.02\textrm{s}$ the chemical potential to $\breve{\mu }_3=-4\times 10^{-5}\mathrm{Nmm/nmol}$ at the bottom and top faces. Figure 11 shows the out-of-plane displacement of the whole structure and of point P of Fig. 8. It can be clearly seen that the displacement initially remains zero until the out-of-plane perturbation causes the structure to buckle. From this point on, the displacement increases steadily.

As a final remark, we want to discuss the terms secondary bifurcation or mode jumping. Both are used to describe a sudden dynamic change in the mode shape, i.e. wave number m in this example, of the buckled state of a structure during a quasi-static loading process. We observe such mode jumping for small wave numbers in our simulation at high levels of swelling. Although mode jumping has received considerable attention in the literature [89, 90], we do not investigate this behaviour in more detail and leave this phenomenon to future investigations.

5.4 Thermally-triggered swelling-induced buckling of a hydrogel corona

In this last example, we extend the study of hydrogel coronae to their thermoresponsive behaviour. The possibility of obtaining defined large deformation patterns that can be controlled by changes in the environmental temperature represents an attractive feature of thermoresponsive hydrogels like PNIPAM gels. However, a systematic investigation of the thermally-triggered swelling-induced buckling of hydrogel coronae was not given in the literature so far to the best of the authors’ knowledge.

We refer to thermally-triggered swelling-induced buckling in the context of thermoresponsive hydrogels as a form of buckling that is caused by a decrease in ambient temperature. This temperature change results in a decrease in chemical potential within the hydrogel and therefore in additional solvent uptake. This, in turn, causes further swelling and if such swelling of a gel with a suitable geometry (e.g. a corona) exceeds a bifurcation point the hydrogel will buckle. The arising buckling patterns are thereby again primarily governed by geometric parameters. The theoretical treatment of such thermally-triggered buckling problems employed in this work is straightforward: Since the analytical theory proposed in [37] and discussed in Sect. 5.3 describes swelling-induced instability formation and buckling without further assumptions on the cause of the necessary volume increase, it should be also applicable in the case of thermally-triggered swelling-induced buckling. To validate this hypothesis, we carried out a number of numerical simulations, described in the following.

The geometry and boundary conditions of the coronae used in this case are the same as already given in Fig. 8. The outer radius and thickness are again kept constant at $50\textrm{mm}$ and $1\textrm{mm}$ respectively. We investigate $r_i/r_o$ ratios of 0.808, 0.826 and 0.840, which are tantamount to wave numbers of $m=10,11$ and 12, respectively. The material parameters are again taken from Table 1 with the adapted parameters listed in Table 3. As done before we discretise only one half of the corona by $6\times 60\times 1$ Taylor-Hood elements in radial, azimuthal and thickness direction, respectively. The pre-swelling parameter is determined as $J_{0}(\theta _{\textrm{ref}},\mu _{\textrm{ref}}) = 10.432$.

We cool down the system by $10\textrm{K}$ in three load increments, starting at an initial temperature of $\theta _{\textrm{ref}}=303.15\textrm{K}$. Firstly, we decrease in 80 time steps with $\Delta t = 0.025\textrm{s}$ the temperature to $\breve{\theta }_{1}=302.15\textrm{K}$ and apply the out-of-plane force F at point Q of Fig. 8 in a ramp function.¹⁴ Again, without out-of-plane disturbance or the implementation of special numerical treatments, like the arc-length method, a deformed but plane corona would be obtained in silico. Secondly, we remove the force F and keep the corona at $\breve{\theta }_{1}=302.15\textrm{K}$ for a single time step with $\Delta t = 0.025\textrm{s}$. Finally, we decrease in 150 time steps with $\Delta t = 0.04\textrm{s}$ the temperature to $\breve{\theta }_{2}=293.15\textrm{K}$. The chemical potential is kept constant at $\mu _{\textrm{ref}}=0\mathrm{Nmm/nmol}$ throughout the simulation.

The buckling patterns obtained from the simulations are presented in Fig. 12. As expected, the wave numbers in this case coincide with the ones obtained in the previous example respectively and the corrected analytical theory. These results, therefore, validate our hypothesis that the analytical theory proposed in [37] and discussed in Sect. 5.3 is equally applicable for the modelling of thermally-triggered swelling-induced buckling of hydrogel coronae. Figure 13 further shows the out-of-plane displacement of the three investigated geometries at point P of Fig. 8. We observe again the displacement initially to remain zero until the out-of-plane perturbation causes the structure to buckle. The higher the resulting wave number, the later the onset of buckling and the smaller the deflection of the wave pattern.

Fig. 12

Out-of-plane displacement $\zeta$ as shown in the colour bar [mm] at $\breve{\theta }_{2}=293.15\textrm{K}$. The ratio $r_i/r_o$ and the wave number m equals a 0.808 and 10, b 0.826 and 11, and c 0.840 and 12, respectively

Fig. 13

Temperature decrease $\Delta \theta$ versus magnitude of displacement $|\zeta |$ at point P throughout the computation for the three investigated ratios $r_i/r_o$ in the legend. The relative position of the prescribed temperature $\breve{\theta }_1$ is indicated by a vertical, dash-dotted line

6 Conclusion

The aim of this work was the establishment and examination of an up-to-date thermodynamically consistent material model for the description of large elastic deformation, coupled with species diffusion and heat conduction that is applicable to a broad range of materials. For this, we derived a base model in a differential-geometric setting, which employs the balance of linear momentum, the balance of species concentration and the entropy evolution equation as governing equations of the global fields, namely deformation, chemical potential and temperature. The species concentration in this setting is a local field variable. This base model was then extended by the inclusion of a multiplicative split of the deformation gradient, dissipation potentials and temperature-scaled gradients, whereby the first two aid the desired degree of generality. Next, we implemented the theoretical model in a finite element code. With the introduced temperature-scaled gradients, we obtained a symmetric tangent matrix that enables the full exploitation of the monolithic finite element scheme since it allows for the usage of symmetric solvers for the Newton-type iterative update algorithm. This offers a computational speedup and a data reduction when compared with non-symmetric solvers. We further discussed relevant implementation-related details such as the use of a pre-swollen reference configuration and the numerical stability of the scheme.

With the obtained simulation framework, we conducted a series of numerical examples to probe the capabilities and limitations of our approach. For these examples, thermoresponsive hydrogels were chosen as a relevant example material class and a consistent constitutive approach was presented and discussed. The examples were thereby selected to not only investigate the model’s ability to capture the physical material behaviour correctly but also its capacity in numerically challenging setups (discontinuities, instabilities). We studied first the thermal swelling behaviour of PNIPAM hydrogels and found very good agreement with analytical as well as experimental results in the literature. In the second example, we examined the swelling dynamics of a PNIPAM hydrogel and confirmed that the relevant physical phenomena and the temporal evolution of the pertinent field quantities are qualitatively reproduced in a reliable manner. In the third example, the swelling-induced buckling of hydrogel coronae was considered. Here, it was found that the model predictions deviated significantly from established results in the literature. Further investigation of this issue revealed that these deviations can be traced back to an error in the considered literature and we proposed corrections in this regard. These corrections are not only of academic value but of actual relevance as shown in the fourth example. In the same, we investigated, based on the corrected theory, thermally-triggered buckling of PNIPAM gels as an interesting but still unexplored direction for further research. Also in this case, very good agreement between the predictions of our model and the analytical theory was found.

As already mentioned, several chemo-thermo-mechanical material models have been established in the literature by now. However, we are confident that the presented model adds value to the field due to the additional modelling options it provides, its advantageous numerical properties and the demonstrated robustness of its implementation, also in challenging numerical settings.

In this work, we limited the discussion to hydrogels as an example material class, entailing certain simplifying specialisations. Further, neither the differential-geometric formulation nor the more general gradient-flux relations that dissipation potentials would allow for were given the space they would deserve. This, however, seemed necessary to centre attention first on the physical validity and numerical stability of our model and a detailed treatise of these aspects is postponed to future work.

Lastly, also the incorporation of plasticity, anisotropy and/or viscous effects in the presented model represents an interesting direction for future work since this would extend the range of applicability of the model to e.g. other polymer-like or bio-materials, such as paper.

Appendices

Constitutive Ansatz for the Free Energy Function

We state the proposed free energy function from Sect. 3.2 again, i.e. Equation 45 for $J_0 = 1$:

$$\begin{aligned} \begin{aligned} {\widetilde{\psi }}_{0}(\varvec{C}, c_{0}, \theta ) = \,&\frac{N k_B \theta }{2} \left[ J_{\theta }^{-2/3} I_{\varvec{C}} - 3 - 2\ln (J_{\theta }^{-1} J) \right] \\&+ \mu _{f} c_{0} + R_{m} \theta c_{0}\left[ \ln \!\left( \frac{\Omega c_{0}}{1 + \Omega c_{0}}\right) + \frac{\chi (c_{0},\theta )}{1 + \Omega c_{0}} \right] \\&+ c_{\theta }\left[ \theta - \theta _{\textrm{ref}} - \theta \ln \!\left( \frac{\theta }{\theta _{\textrm{ref}}}\right) \right] + \frac{\kappa }{2} \left( J_{\theta }^{-1} J - J_s \right) ^2. \end{aligned} \end{aligned}$$

(A1)

and the employed split of the deformation gradient in Eq. 31

$$\begin{aligned} \varvec{F} = \varvec{F}_{e} \varvec{F}_{\theta } = \varvec{F}_{m} \varvec{F}_{s} \varvec{F}_{\theta } \end{aligned}$$

(A2)

as pictured in Fig. 1.

The intermediate configuration $\,\mathcal {{\overline{B}}}_2$ at a (nonuniform) temperature $\theta$ and a (nonuniform) concentration value $c_0$ is obtained from the deformed configuration $\,\mathcal {B}$ by an isothermal, isohygric removal of all applied external forces. Additionally, the intermediate configuration $\,\mathcal {{\overline{B}}}_1$ at a (nonuniform) temperature $\theta$ is obtained from the intermediate configuration $\,\mathcal {{\overline{B}}}_2$ by an isothermal diffusion process to $c_{0,{\textrm{ref}}}$. Both mappings in combination, from $\,\mathcal {B}$ to $\,\mathcal {{\overline{B}}}_1$, give an isothermal destressing to zero stress, which is governed by $\varvec{F}_{e} = \varvec{F}_{m} \varvec{F}_{s}$. Finally, the reference configuration $\,\mathcal {B}_0$ is obtained from the intermediate configuration $\,\mathcal {{\overline{B}}}_1$ by a heat conduction process to $\theta _{\textrm{ref}}$ and is governed by $\varvec{F}_{\theta }$.

Firstly, we consider the model in the isothermal limit case, i.e. $\theta = \theta _{\textrm{ref}}$. This results with help of Eq. 33 in $\varvec{F}_{\theta } = \varvec{I}$, which is tantamount to $\varvec{F}_{e} = \varvec{F}$ and $I\!I\!I_{\varvec{C}_{e}} = I\!I\!I_{\varvec{C}}$. Furthermore, we reduce the interaction parameter in Eq. 38 to a constant: $\chi (c_{0}, \theta ) = \chi$, recall that $J_s= 1 + \Omega c_0$, as found below Eq. 40 in the main text and set $\mu _f = 0$. With this, we rewrite Eq. A1 in the isothermal limit as:

$$\begin{aligned} \begin{aligned} {\widetilde{\psi }}_{0}(\varvec{C}, c_{0}) = \,&\frac{N k_B \theta _{\textrm{ref}}}{2} \left[ I_{\varvec{C}} - 3 - 2\ln (J) \right] + \frac{\kappa }{2} \left( J - 1 - \Omega c_0 \right) ^2 \\&+ R_{m} \theta _{\textrm{ref}} c_{0}\left[ \ln \!\left( \frac{\Omega c_{0}}{1 + \Omega c_{0}}\right) + \frac{\chi }{1 + \Omega c_{0}} \right] . \end{aligned} \end{aligned}$$

(A3)

This is the same shape of the free energy as classically used in the relevant literature [14, 57, 60, 74].

Secondly, we consider the chosen constitutive model in the dry limit case, i.e. $c_{0}=0$. This results with help of Eq. 32 in $\varvec{F}_{s} = \varvec{I}$ and $J_s= 1$, which is tantamount to $\varvec{F}_{e} = \varvec{F}_{m}$ or $\varvec{C}_{e} = \varvec{C}_{m}$. Furthermore, it can be shown with the application of Bernoulli-de l’Hôpital’s rule that ${\widetilde{\psi }}_{0,s}(c_{0}=0, \theta )$ of Eq. 37 equals 0. Next, we introduce $J_m = \sqrt{\text {det}(\varvec{C}_{m})}$ and $I_{\varvec{C}_m} = \text {tr}(\varvec{C}_m)$. With these steps, Eq. A1 can be rewritten in the dry limit as:

$$\begin{aligned} \begin{aligned} {\widetilde{\psi }}_{0}(\varvec{C}_{m}, \theta ) = \,&\frac{N k_B \theta }{2} \left[ I_{\varvec{C}_{m}} - 3 - 2\ln (J_m) \right] + \frac{\kappa }{2} \left( J_m - 1 \right) ^2 \\&+ c_{\theta }\left[ \theta - \theta _{\textrm{ref}} - \theta \ln \!\left( \frac{\theta }{\theta _{\textrm{ref}}}\right) \right] . \end{aligned} \end{aligned}$$

(A4)

This formulation of a nearly incompressible neo-Hookean-type material is in close accordance with the general constitutive approach for the free energy of thermo-mechanical materials as discussed e.g. in [55, 91].

Time-Discretisation

In general, the time parameter t can be defined as $t\in \,\mathcal {T}\subset {\mathbb {R}}^+$. We replace $\,\mathcal {T}$ by a set of discrete time steps $t_{i}$, which leads to a division of the time interval under consideration into finite time increments $[t_{n}, t_{n+1}]$. We define a corresponding time step as $\Delta t_{n+1}:= t_{n+1} - t_{n}$. For notational convenience, we will drop the subscript ${n+1}$ in the following such that all quantities without subscript are understood to be evaluated at time $t_{n+1}$. With this, the rate terms in Eqs. 82 and 83 can be expressed with the finite difference scheme as:

$$\begin{aligned} {\dot{c}}_{0} \approx \frac{c_{0} - c_{0,n}}{\Delta t}, \quad \quad {\dot{\eta }}_{0} \approx \frac{\eta _{0} - \eta _{0,n}}{\Delta t}. \end{aligned}$$

(B1)

Linearisation

We first rewrite the weak forms of the governing equations 81–83 to be implicit functions of their arguments, i.e. functions of the form $\,\mathcal {F}_{i}(\varvec{\varphi }, \mu , \theta ) = 0$ for $i=1,2,3$. Thereafter, we interpret the implicit versions of these equations as a set of nonlinear equations $\varvec{\,\mathcal {F}}({\textbf {x}}) = [\,\mathcal {F}_{1}, \,\mathcal {F}_{2}, \,\mathcal {F}_{3}]^T$ in the set of variables ${\textbf {x}} = [\varvec{\varphi }^T, \mu , \theta ]^T$. Hence¹⁵

$$\begin{aligned} \begin{aligned} \,\mathcal {F}_{1}({\textbf {x}}) :=&\int _{\,\mathcal {B}_0} \varvec{P} : \nabla _{\!0} \delta \varvec{v}_{\varvec{\varphi }} dV \\&- \int _{\,\mathcal {B}_0} \varvec{B}\cdot \delta \varvec{v}_{\varvec{\varphi }} dV - \int _{\partial \,\mathcal {B}_{0,\varvec{T}}} \breve{\varvec{T}} \cdot \delta \varvec{v}_{\varvec{\varphi }} dA = 0, \! \end{aligned} \end{aligned}$$

(C1)

$$\begin{aligned} \begin{aligned} \,\mathcal {F}_{2}({\textbf {x}}) :=&\int _{\,\mathcal {B}_0} (c_{0} - c_{0,n}) \delta v_{\mu } dV \\&+ \int _{\,\mathcal {B}_0} \Delta t {\overline{m}}_{n} J_{n} {\overline{\nabla }}_{\!0} \delta v_{\mu } \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \mu dV \\&+ \int _{\partial \,\mathcal {B}_{0,H}} \Delta t \breve{H} \delta v_{\mu } dA = 0, \end{aligned} \end{aligned}$$

(C2)

$$\begin{aligned} \begin{aligned} \,\mathcal {F}_{3}({\textbf {x}}) :=&\int _{\,\mathcal {B}_0} (\eta _{0} - \eta _{0,n})\delta v_{\theta } dV \\&- \int _{\,\mathcal {B}_0} \frac{\Delta t}{\theta } {\overline{k}}_{n} J_{n} {\overline{\nabla }}_{\!0} \theta \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \theta \delta v_{\theta } dV \\&+ \int _{\,\mathcal {B}_0} \Delta t {\overline{k}}_{n} J_{n} {\overline{\nabla }}_{\!0} \delta v_{\theta } \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \theta dV \\&- \int _{\,\mathcal {B}_0} \frac{\Delta t}{\theta } {\overline{m}}_{n} J_{n} {\overline{\nabla }}_{\!0} \mu \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \mu \delta v_{\theta } dV \\&- \int _{\,\mathcal {B}_0} \frac{\Delta t R_{q}}{\theta } \delta v_{\theta } dV + \int _{\partial \,\mathcal {B}_{0,Q}} \frac{\Delta t \breve{Q}}{\theta } \delta v_{\theta } dA = 0. \end{aligned} \end{aligned}$$

(C3)

In the next step, the obtained set of equations $\varvec{\,\mathcal {F}}$ is expanded in a Taylor series. Therefore, the directional derivative or Gateaux derivative $D\varvec{\,\mathcal {F}}({\textbf {x}})[{\textbf {u}}]$ is employed as an appropriate notion of differentiating the set of function $\varvec{\,\mathcal {F}}$ at ${\textbf {x}}$ in the direction ${\textbf {u}}$ [92, Section 2.3.2]. With this directional derivative, $\varvec{\,\mathcal {F}}$ can be expanded with respect to an initial state ${\textbf {x}}_{0}$ in the direction of an increment $\Delta {\textbf {x}}$ of its argument. Truncating this expansion after the linear term to obtain a linear approximation leads to $\varvec{\,\mathcal {F}}({\textbf {x}}) \approx \varvec{\,\mathcal {F}}({\textbf {x}}_{0}) + D\varvec{\,\mathcal {F}}({\textbf {x}}_{0})[\Delta {\textbf {x}}] = \varvec{0}$. The increment $\Delta {\textbf {x}}$ is a set of increments of the primary variables, namely a displacement increment (vector) field $\varvec{U}(\varvec{X},t)$, a scalar chemical potential increment field $\varrho (\varvec{X},t)$ and a scalar temperature increment field $\vartheta (\varvec{X},t)$. The linearised equation can then be restated as a Newton–Raphson procedure to iteratively determine a solution to the initial nonlinear problem as:

$$\begin{aligned} D\varvec{\,\mathcal {F}}({\textbf {x}}_{k})[\Delta {\textbf {x}}] = - \varvec{\,\mathcal {F}}({\textbf {x}}_{k}). \end{aligned}$$

(C4)

Starting from a given initial state ${\textbf {x}}_{0}$, the increment $\Delta {\textbf {x}}$ is calculated from this equation. Then, a new state is obtained according to ${\textbf {x}}_{k+1} = {\textbf {x}}_{k} + \Delta {\textbf {x}}$ until a specified convergence criterion is satisfied.

The evaluation of the directional derivative $D\!\varvec{\,\mathcal {F}}$ required in Eq. C4 is now possible by applying the rules for directional differentiation to Eqs. C1–C3:

$$\begin{aligned} \begin{aligned} D\!\,\mathcal {F}_{1} =&\int _{\,\mathcal {B}_0} D\varvec{P}: \nabla _{\!0} \delta \varvec{v}_{\varvec{\varphi }} dV - \int _{\,\mathcal {B}_0} D\varvec{B}\cdot \delta \varvec{v}_{\varvec{\varphi }} dV \\&- \int _{\partial \,\mathcal {B}_{0,\varvec{T}}} D \breve{\varvec{T}} \cdot \delta \varvec{v}_{\varvec{\varphi }} dA, \end{aligned} \end{aligned}$$

(C5)

$$\begin{aligned} \begin{aligned} D\!\,\mathcal {F}_{2} =&\int _{\,\mathcal {B}_0} Dc_{0} \delta v_{\mu } dV \\&- \int _{\,\mathcal {B}_0} \frac{2 \Delta t}{\theta } {\overline{m}}_{n} J_{n} {\overline{\nabla }}_{\!0} \delta v_{\mu } \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \mu \vartheta dV \\&+ \int _{\,\mathcal {B}_0} \Delta t {\overline{m}}_{n} J_{n} {\overline{\nabla }}_{\!0} \delta v_{\mu } \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \varrho dV \\&+ \int _{\partial \,\mathcal {B}_{0,H}} \Delta t D\breve{H} \delta v_{\mu } dA, \end{aligned} \end{aligned}$$

(C6)

$$\begin{aligned} \begin{aligned} D\!\,\mathcal {F}_{3} =&\int _{\,\mathcal {B}_0} D\eta _{0} \delta v_{\theta } dV \\&+ \int _{\,\mathcal {B}_0} \frac{3 \Delta t}{\theta ^{ 2}} {\overline{k}}_{n} J_{n} {\overline{\nabla }}_{\!0} \theta \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \theta \vartheta \delta v_{\theta } dV \\&- \int _{\,\mathcal {B}_0} \frac{2 \Delta t}{\theta } {\overline{k}}_{n} J_{n}\left[ {\overline{\nabla }}_{\!0} \theta \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \vartheta \delta v_{\theta } \right. \\&\left. + {\overline{\nabla }}_{\!0} \delta v_{\theta } \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \theta \vartheta \right] dV \\&+ \int _{\,\mathcal {B}_0} \Delta t {\overline{k}}_{n} J_{n} {\overline{\nabla }}_{\!0} \delta v_{\theta } \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \vartheta dV \\&+ \int _{\,\mathcal {B}_0} \frac{3 \Delta t}{\theta ^{ 2}} {\overline{m}}_{n} J_{n} {\overline{\nabla }}_{\!0} \mu \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \mu \vartheta \delta v_{\theta } dV \\&- \int _{\,\mathcal {B}_0} \Delta t D\left( \frac{R_{q}}{\theta }\right) \delta v_{\theta } dV \\&- \int _{\,\mathcal {B}_0} \frac{2 \Delta t}{\theta } {\overline{m}}_{n} J_{n} {\overline{\nabla }}_{\!0} \mu \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \varrho \delta v_{\theta } dV \\&+ \int _{\partial \,\mathcal {B}_{0,Q}} \Delta t D\left( \frac{\breve{Q}}{\theta }\right) \delta v_{\theta } dA. \end{aligned} \end{aligned}$$

(C7)

These equations include directional derivatives that need further specification. We begin with the directional derivative of the concentration field that can be written as:

$$\begin{aligned} \begin{aligned} Dc_{0}({\textbf {x}}_{0})[\Delta {\textbf {x}}] = \frac{\partial c_{0}}{\partial \varvec{F}}\!:\!\nabla _{\!0} \varvec{U} + \frac{\partial c_{0}}{\partial \mu } \varrho + \frac{\partial c_{0}}{\partial \theta } \vartheta . \end{aligned} \end{aligned}$$

(C8)

As laid out in Sect. 4.1, the species concentration $c_{0}$ is a dependent variable of the material model, while the chemical potential $\mu$ is one of its primary variables. Needless to say, this also implies that the chemical potential is not dependent on the other two primary variables, namely the temperature $\theta$ and the deformation in the shape of the deformation gradient $\varvec{F}$. Keeping this in mind, differentiating Eq. 66 with respect to $c_{0}$, $\theta$ and $\varvec{F}$, respectively, and using Eqs. 17 and 65, the following relations can be found:

$$\begin{aligned}{} & {} \left( \frac{\partial c_{0}}{\partial \mu }\right) ^{-1} = \frac{\partial ^{ 2} {\widetilde{\psi }}_{0}}{\partial c_{0}^{ 2}} =: \kappa _{1}^{-1}, \end{aligned}$$

(C9)

$$\begin{aligned}{} & {} \begin{aligned} \frac{\partial c_{0}}{\partial \varvec{F}}&= - \kappa _{1} \varvec{g} \frac{\partial \varvec{P}}{\partial c_{0}}, \end{aligned} \end{aligned}$$

(C10)

$$\begin{aligned}{} & {} \frac{\partial c_{0}}{\partial \theta } = \kappa _{1} \frac{\partial \eta _{0}}{\partial c_{0}}. \end{aligned}$$

(C11)

These definitions require the quantity $\kappa _{1}$, which is derived using Eq. 45:

$$\begin{aligned} \begin{aligned} \kappa _{1}^{ -1} = \frac{\partial ^{ 2} {\widetilde{\psi }}_{0}}{\partial c_{0}^{ 2}} = \;&R_{m} \theta \Omega J_{0} \Bigg [ \frac{1}{\Omega J_0 c_{0}} - \frac{1}{1 + \Omega J_0 c_{0}} \\&- \frac{1}{(1 + \Omega J_0 c_{0})^{ 2}} - \frac{2 (A_0 + B_0 \theta )}{(1 + \Omega J_0 c_{0})^{ 3}}\\&- \frac{2 (A_1 + B_1 \theta )}{(1 + \Omega J_0 c_{0})^{ 4}} \left( 2 - \Omega J_0 c_{0}\right) \Bigg ] + \kappa \Omega ^{2} J_0. \end{aligned} \end{aligned}$$

(C12)

The directional derivative of the first Piola-Kirchhoff stress tensor equals

$$\begin{aligned} \begin{aligned} D\varvec{P} = {\mathbb {A}}:\!\nabla _{\!0} \varvec{U} + \frac{\partial \varvec{P}}{\partial c_{0}} Dc_{0} + \frac{\partial \varvec{P}}{\partial \theta } \vartheta , \end{aligned} \end{aligned}$$

(C13)

with the (first) elasticity tensor

$$\begin{aligned} \begin{aligned} {\mathbb {A}}^{ iJ}{}_{k}{}^{L} = \frac{\partial P^{ iJ}}{\partial F^{ k}{}_{L}}= \;&J_0^{-1}\Big \{N k_B \theta \, \omega ^{2/3} \delta ^{ i}{}_{k} G^{ LJ} \\&+ \kappa \omega J \left( 2 \omega J - J_{s} \right) (F^{ -1})^{ J}{}_{m} g^{ mi} (F^{ -1})^{ L}{}_{k}\\&+ \Big [N k_B \theta - \kappa \omega J \left( \omega J - J_{s} \right) \Big ]\\&\quad (F^{ -1})^{ L}{}_{m} g^{ mi} (F^{ -1})^{ J}{}_{k}\Big \}. \end{aligned} \end{aligned}$$

(C14)

Furthermore

$$\begin{aligned} \frac{\partial \varvec{P}}{\partial c_{0}}= & {} - \kappa \Omega \, \omega J \, \varvec{F}^{ -T} \varvec{G}^{ -1}, \end{aligned}$$

(C15)

$$\begin{aligned} \frac{\partial \varvec{P}}{\partial \theta }= & {} - J_0^{-1} \Big [3 \alpha \kappa \, \omega J (2 \omega J - J_s) \varvec{F}^{ -T} - N k_B \big (\omega ^{2/3}\varvec{F} - \varvec{F}^{ -T} \, \big ) \nonumber \\{} & {} + \; 2 \alpha N k_B \theta \, \omega ^{2/3} \varvec{F} \Big ] \varvec{G}^{-1}. \end{aligned}$$

(C16)

The directional derivative of the entropy can be determined with the help of Eq. 65:

$$\begin{aligned} \begin{aligned} D\eta _{0} = - \varvec{g} \frac{\partial \varvec{P}}{\partial \theta }\!:\!\nabla _{\!0} \varvec{U} + \frac{\partial \eta _{0}}{\partial c_{0}} Dc_{0} + \frac{c_{\theta } + c_{\text {corr}}}{J_{0}\theta } \, \vartheta , \end{aligned} \end{aligned}$$

(C17)

whereat the specific heat (capacity) $c_{\theta }$ is employed [12]. However, we deviate from the commonly used definition $c_{\theta } = - \theta \frac{\partial ^{ 2} {\widetilde{\psi }}_{0}}{\partial \theta ^{ 2}}$, introduce an additional correction term $c_{\text {corr}}$ and define $c_{\theta } + c_{\text {corr}} = -J_{0} \theta \frac{\partial ^{ 2} {\widetilde{\psi }}_{0}}{\partial \theta ^{ 2}}$. This is necessary due to the pre-swollen configuration and the temperature dependence of not only the thermal part of the free energy but also of the other parts (cf. Equations 36 and 37). The explicit functional form of the correction term $c_{\text {corr}}$ is found by differentiating Eq. 45 twice with respect to $\theta$ and determine:

$$\begin{aligned} \begin{aligned} c_{corr}&\,=\, -J_{0} \theta \frac{\partial ^{ 2} {\widetilde{\psi }}_{0}}{\partial \theta ^{ 2}} - c_{\theta } \,= \\&- 2 R_{m} J_0 c_0 \theta \left( \frac{B_0}{1 + \Omega J_0 c_{0}} + \frac{B_1}{(1 + \Omega J_0 c_{0})^{ 2}} \right) \\&+ 2 \alpha N k_B \theta (\omega ^{2/3} I_{\varvec{C}} - 3) \\&- 2 \alpha ^2 N k_B \theta ^2 \, \omega ^{2/3} I_{\varvec{C}} - 9 \alpha ^2 \kappa \theta \, \omega J \left( 2 \omega J - J_s \right) . \end{aligned} \end{aligned}$$

(C18)

Furthermore, the term $\partial \eta _{0} / \partial c_{0}$ is obtained from Eq. 48 and reads:

$$\begin{aligned} \begin{aligned} \frac{\partial \eta _{0}}{\partial c_{0}} =&- R_{m} \Bigg [\ln \left( \frac{\Omega J_0 c_{0}}{1 + \Omega J_0 c_{0}}\right) + \frac{A_0 + B_0 \theta }{(1 + \Omega J_0 c_{0})^2} \\&\qquad \qquad +\frac{(A_1 + B_1 \theta )}{(1 + \Omega J_0 c_{0})^3}(1 - \Omega J_0 c_{0}) + \frac{1}{1 + \Omega J_0 c_{0}} \Bigg ]\\&- R_{m} \theta \Bigg [ \frac{B_0}{(1 + \Omega J_0 c_{0})^2} + \frac{B_1 (1 - \Omega J_0 c_{0})}{(1 + \Omega J_0 c_{0})^3} \Bigg ] - 3 \alpha \kappa \Omega \omega J. \end{aligned} \end{aligned}$$

(C19)

The evaluation of the directional derivatives for the source terms $\varvec{B}$ and $R_{q}$ as well as for the boundary conditions $\breve{\varvec{T}}$, $\breve{H}$ and $\breve{Q}$ strongly depends on their actual functional form. In this work we will restrict the discussion to source terms and boundary conditions, which do not depend on the primary variables and can therefore be treated as constants with respect to directional differentiation.

Spatial Discretisation and Finite Element Method

The basic idea of the finite element method is the division of the domain under consideration, i.e. the body in the reference configuration $\,\mathcal {B}_{0}$, in a number of distinct subdomains, so-called finite elements. Within the domain of a finite element, the global field quantities are approximated by the use of scalar shape functions $N(\varvec{X})$ as follows [93, Section 31.5]:

$$\begin{aligned} \begin{aligned} \varvec{U}(\varvec{X},t) = \sum _{a=1}^{n_{\varvec{\varphi }}} [\check{\varvec{U}}(t)]^{ a} [N_{\varvec{\varphi }}(\varvec{X})]_{ a}, \\ \varrho (\varvec{X},t) = \sum _{a=1}^{n_{\mu }} [{\check{\varrho }}(t)]^{ a} [N_{\mu }(\varvec{X})]_{ a},\\ \vartheta (\varvec{X},t) = \sum _{a=1}^{n_{\theta }} [{\check{\vartheta }}(t)]^{ a} [N_{\theta }(\varvec{X})]_{ a}. \end{aligned} \end{aligned}$$

(D1)

The quantities $N_{\varvec{\varphi }}$, $N_{\mu }$ and $N_{\theta }$ are the shape functions used for the interpolation of $\varvec{U}$, $\varrho$ and $\vartheta$ within the element, respectively. The nodal values, i.e. ${\check{\varvec{U}}}^{ a}$, ${\check{\varrho }}^{ a}$ and ${\check{\vartheta }}^{ a}$, are the values of the a-th node of the element, where $n_{\varvec{\varphi }}$, $n_{\mu }$ and $n_{\theta }$ are the number of nodes per element used for the discretization of $\varvec{U}$, $\varrho$ and $\vartheta$, respectively.

Since also the gradients of the global fields are present in the set of governing equations, we also introduce their discrete counterparts. Hence, $(\varvec{B}_{\varvec{\varphi }})_{ a}:= \nabla _{\!0}(N_{\varvec{\varphi }})_{ a}$, $(\varvec{B}_{\mu })_{ a}:= \nabla _{\!0}(N_{\mu })_{ a}$ and $(\varvec{B}_{\theta })_{ a}:= \nabla _{\!0}(N_{\theta })_{ a}$. With this and by recalling Eq. D1 the needed gradients can be found as:

$$\begin{aligned} \begin{aligned}&\nabla _{\!0} \varvec{U} = \sum _{a=1}^{n_{\varvec{\varphi }}} \check{\varvec{U}}^{ a} (\varvec{B}_{\varvec{\varphi }})_{ a}, \qquad \nabla _{\!0} \varrho = \sum _{a=1}^{n_{\mu }} {\check{\varrho }}^{ a} (\varvec{B}_{\mu })_{ a},\qquad \\ {}&\nabla _{\!0} \vartheta = \sum _{a=1}^{n_{\theta }} {\check{\vartheta }}^{ a} (\varvec{B}_{\theta })_{ a}. \end{aligned} \end{aligned}$$

(D2)

We employ a standard Galerkin ansatz by selecting the same approximation for both the test function fields and the trial solutions for the displacements, the chemical potential and the temperature, respectively.

The fully discretised linear element-level set of equations can now be worked out, starting from the Newton–Raphson scheme introduced in Eq. C4. Following the standard procedure in this context a set of linear equations with respect to the increments of the primary field quantities can be found as $\varvec{\textsf{K}}^{ (e)} \Delta \varvec{\textsf{x}}^{(e)} = \varvec{\textsf{R}}^{ (e)}$:

$$\begin{aligned} \begin{aligned} \left[ \begin{array}{rrr} (\textsf{K}_{\varvec{\varphi } \varvec{\varphi }}^{(e)})_{ ba} &{}(\textsf{K}_{\varvec{\varphi } \mu }^{(e)})_{ ba} &{}(\textsf{K}_{\varvec{\varphi } \theta }^{(e)})_{ ba} \\ (\textsf{K}_{\mu \varvec{\varphi }}^{(e)})_{ ba} &{}(\textsf{K}_{\mu \mu }^{(e)})_{ ba} &{}(\textsf{K}_{\mu \theta }^{(e)})_{ ba} \\ (\textsf{K}_{\theta \varvec{\varphi }}^{(e)})_{ ba} &{}(\textsf{K}_{\theta \mu }^{(e)})_{ ba} &{}(\textsf{K}_{\theta \theta }^{(e)})_{ ba} \\ \end{array} \right] \left[ \begin{array}{r} (\check{\varvec{U}}^{(e)})^{ a} \\ ({\check{\varrho }}^{ (e)})^{ a} \\ ({\check{\vartheta }}^{ (e)})^{ a} \end{array} \right] = \left[ \begin{array}{r} (\textsf{R}_{\varvec{\varphi }}^{(e)})_{ b} \\ (\textsf{R}_{\mu }^{(e)})_{ b} \\ (\textsf{R}_{\theta }^{(e)})_{ b} \end{array} \right] . \end{aligned} \end{aligned}$$

(D3)

Here, the superscript (e) highlights the affiliation of these quantities to a single element. For notational convenience, however, this superscript is suppressed in the following. The vector $\varvec{\textsf{R}}$ denotes the residual vector. Its components can be explicitly stated as (cf. Equations C1–C3):

$$\begin{aligned}{} & {} \begin{aligned} (\textsf{R}_{\varvec{\varphi }})_{ b} =&-\int _{\,\mathcal {B}_0}\! \varvec{g} \varvec{P}\cdot (\varvec{B}_{\varvec{\varphi }})_{ b} \, dV + \int _{\,\mathcal {B}_0}\! \varvec{B}^{ \flat } (N_{\varvec{\varphi }})_{ b} \, dV \\&+ \int _{\partial \,\mathcal {B}_{0,\varvec{T}}}\! \breve{\varvec{T}}^{ \flat } (N_{\varvec{\varphi }})_{ b} \, dA, \end{aligned} \end{aligned}$$

(D4)

$$\begin{aligned}{} & {} \begin{aligned} (\textsf{R}_{\mu })_{ b} =&\int _{\,\mathcal {B}_0} (c_{0} - c_{0,n}) (N_{\mu })_{ b} \, dV \\&+ \int _{\,\mathcal {B}_0} \frac{\Delta t}{\theta } {\overline{m}}_{n} J_{n} (\varvec{B}_{\mu })_{ b} \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \mu \, dV \\&+ \int _{\partial \,\mathcal {B}_{0,H}} \Delta t \breve{H} (N_{\mu })_{ b} \, dA, \end{aligned} \end{aligned}$$

(D5)

$$\begin{aligned}{} & {} \begin{aligned} (\textsf{R}_{\theta })_{ b} =&\int _{\,\mathcal {B}_0} (\eta _{0} - \eta _{0,n}) (N_{\theta })_{ b} \, dV \\&- \int _{\,\mathcal {B}_0} \frac{\Delta t}{\theta } {\overline{k}}_{n} J_{n} {\overline{\nabla }}_{\!0} \theta \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \theta (N_{\theta })_{ b} \, dV \\&+ \int _{\,\mathcal {B}_0} \frac{\Delta t}{\theta } {\overline{k}}_{n} J_{n} (\varvec{B}_{\theta })_{ b} \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \theta \, dV \\&- \int _{\,\mathcal {B}_0} \frac{\Delta t R_{q}}{\theta } (N_{\theta })_{ b} \, dV \\&- \int _{\,\mathcal {B}_0} \frac{\Delta t}{\theta } {\overline{m}}_{n} J_{n} {\overline{\nabla }}_{\!0} \mu \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \mu (N_{\theta })_{ b} \, dV \\&+ \int _{\partial \,\mathcal {B}_{0,Q}} \frac{\Delta t \breve{Q}}{\theta } (N_{\theta })_{ b} \, dA. \end{aligned} \end{aligned}$$

(D6)

The block matrix $\varvec{\textsf{K}}$ denotes the tangent matrix. Its components can be explicitly stated as (cf. Equations C5–C7):

$$\begin{aligned}{} & {} \begin{aligned} (\textsf{K}_{\varvec{\varphi } \varvec{\varphi }})_{ ba}&= \int _{\,\mathcal {B}_0} (\varvec{B}_{\varvec{\varphi }})_{ b} \cdot \left[ \varvec{g} {\mathbb {A}} \right. \\&\left. - \kappa _{1} \left( \varvec{g} \frac{\partial \varvec{P}}{\partial c_{0}}\right) \otimes \left( \varvec{g} \frac{\partial \varvec{P}}{\partial c_{0}}\right) \right] \cdot (\varvec{B}_{\varvec{\varphi }})_{ a} dV, \\ (\textsf{K}_{\varvec{\varphi } \varvec{\varphi }})_{ jbia}&= \int _{\,\mathcal {B}_0} (B_{\varvec{\varphi }})_{ bJ} \left( g_{ jn} {\mathbb {A}}^{ nJ}{}_{i}{}^{L} \right. \\&\left. - \kappa _{1} g_{ jn} \frac{\partial P^{ nJ}}{\partial c_{0}} g_{ ik} \frac{\partial P^{ kL}}{\partial c_{0}} \right) (B_{\varvec{\varphi }})_{ aL} dV, \end{aligned} \end{aligned}$$

(D7)

$$\begin{aligned}{} & {} \begin{aligned} (\textsf{K}_{\varvec{\varphi } \mu })_{ ba}&= \int _{\,\mathcal {B}_0} \kappa _{1} \varvec{g} \frac{\partial \varvec{P}}{\partial c_{0}} \cdot (\varvec{B}_{\varvec{\varphi }})_{ b} (N_{\mu })_{ a} dV, \end{aligned} \end{aligned}$$

(D8)

$$\begin{aligned}{} & {} \begin{aligned} (\textsf{K}_{\varvec{\varphi } \theta })_{ ba}&= \int _{\,\mathcal {B}_0} \varvec{g}\left[ \kappa _{1} \frac{\partial \varvec{P}}{\partial c_{0}} \frac{\partial \eta _{0}}{\partial c_{0}} \right. \\&\left. + \frac{\partial \varvec{P}}{\partial \theta } \right] \cdot (\varvec{B}_{\varvec{\varphi }})_{ b} (N_{\theta })_{ a} dV, \end{aligned} \end{aligned}$$

(D9)

$$\begin{aligned}{} & {} \begin{aligned} (\textsf{K}_{\mu \varvec{\varphi }})_{ ba}&= \int _{\,\mathcal {B}_0} (N_{\mu })_{ b} \kappa _{1} \varvec{g} \frac{\partial \varvec{P}}{\partial c_{0}} \cdot (\varvec{B}_{\varvec{\varphi }})_{ a} dV, \end{aligned} \end{aligned}$$

(D10)

$$\begin{aligned}{} & {} \begin{aligned} (\textsf{K}_{\mu \mu })_{ ba}&= -\int _{\,\mathcal {B}_0} \left[ \kappa _{1} (N_{\mu })_{ b} (N_{\mu })_{ a} \right. \\&\left. + \frac{\Delta t}{\theta ^{ 2}} {\overline{m}}_{n} J_{n} (\varvec{B}_{\mu })_{ b} \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot (\varvec{B}_{\mu })_{ a} \right] dV, \end{aligned} \end{aligned}$$

(D11)

$$\begin{aligned}{} & {} \begin{aligned} (\textsf{K}_{\mu \theta })_{ ba}&= \int _{\,\mathcal {B}_0} \left[ \frac{2 \Delta t}{\theta ^{ 2}} {\overline{m}}_{n} J_{n} (\varvec{B}_{\mu })_{ b} \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \mu (N_{\theta })_{ a} \right. \\&\left. - \kappa _{1} \frac{\partial \eta _{0}}{\partial c_{0}} (N_{\mu })_{ b} (N_{\theta })_{ a} \right] dV, \end{aligned} \end{aligned}$$

(D12)

$$\begin{aligned}{} & {} \begin{aligned} (\textsf{K}_{\theta \varvec{\varphi }})_{ ba}&= \int _{\,\mathcal {B}_0} (N_{\theta })_{b} \; \varvec{g}\left[ \kappa _{1} \frac{\partial \varvec{P}}{\partial c_{0}} \frac{\partial \eta _{0}}{\partial c_{0}} \right. \\&\left. + \frac{\partial \varvec{P}}{\partial \theta } \right] \cdot (\varvec{B}_{\varvec{\varphi }})_{ a} dV, \end{aligned} \end{aligned}$$

(D13)

$$\begin{aligned}{} & {} \begin{aligned} (\textsf{K}_{\theta \mu })_{ ba}&= \int _{\,\mathcal {B}_0} \left[ \frac{2 \Delta t}{\theta ^{ 2}} {\overline{m}}_{n} J_{n} (N_{\theta })_{ b} {\overline{\nabla }}_{\!0} \mu \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot (\varvec{B}_{\mu })_{ a} \right. \\&\left. - \kappa _{1} \frac{\partial \eta _{0}}{\partial c_{0}} (N_{\theta })_{ b} (N_{\mu })_{ a} \right] dV, \end{aligned} \end{aligned}$$

(D14)

$$\begin{aligned}{} & {} \begin{aligned} (\textsf{K}_{\theta \theta })_{ ba} =&\int _{\,\mathcal {B}_0} \left\{ -\left[ \kappa _{1} \left( \frac{\partial \eta _{0}}{\partial c_{0}}\right) ^{ 2} + \frac{c_{\theta } + c_{\text {corr}}}{\theta } \right. \right. \\&+ \frac{3 \Delta t}{\theta ^{ 2}} {\overline{k}}_{n} J_{n} {\overline{\nabla }}_{\!0} \theta \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \theta \\&+ \frac{3 \Delta t}{\theta ^{ 2}} {\overline{m}}_{n} J_{n} {\overline{\nabla }}_{\!0} \mu \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \mu \\&\left. \left. + \frac{\Delta t}{\theta ^{ 2}} R_{q} \right] (N_{\theta })_{ b} (N_{\theta })_{ a} \right. \\&+ \frac{2 \Delta t}{\theta ^2} {\overline{k}}_{n} J_{n}\left[ (N_{\theta })_{ b} {\overline{\nabla }}_{\!0} \theta \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot (\varvec{B}_{\theta })_{ a} \right. \\&\left. + (\varvec{B}_{\theta })_{ b} \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot {\overline{\nabla }}_{\!0} \theta (N_{\theta })_{ a} \right] \\&\left. - \frac{\Delta t}{\theta ^{ 2}} {\overline{k}}_{n} J_{n} (\varvec{B}_{\theta })_{ b} \cdot (\varvec{C}_{n}^{ -1})^{ \flat } \cdot (\varvec{B}_{\theta })_{ b} \right\} dV \\&+ \int _{\partial \,\mathcal {B}_{0,Q}} \frac{\Delta t \breve{Q}}{\theta ^{ 2}} (N_{\theta })_{ a} (N_{\theta })_{ b} dA. \end{aligned} \end{aligned}$$

(D15)

It is desirable that the tangent matrix is symmetric, i.e. that $\varvec{\textsf{K}} = \varvec{\textsf{K}}^T$ holds true. A symmetric tangent matrix allows for a more efficient numerical implementation of the material model, as discussed in depth by [42]. It is far from obvious, though, that a set of governing equations, when linearised and fully discretised, results in a symmetric tangent matrix, especially in the context of multi-field models. In this work, however, the resulting tangent matrix is symmetric as can be verified from Eqs. D7–D15. To ensure this, two measures are taken, which were proposed by [42] for the same purpose. Firstly, the entropy evolution equation is used as the equation governing implicitly the evolution of the temperature field instead of an explicit temperature evolution equation as commonly used in the literature [5, 12, 43]. Secondly, temperature-scaled gradient quantities are introduced and the employed dissipation potentials are defined with respect to them rather than to the regular gradient quantities.

Another fact of interest about the derived system of equations is the following: A close look on Eqs. D4–D15 reveals that almost all quantities in these equations are defined with respect to the current time step $t_{n+1}$ and are, therefore, treated as unknowns. The only exceptions to this are the quantities involved in the definition of the dissipation potentials and the discretisation of the rate terms. Hence, the derived system of equations might be viewed as an “almost fully implicit” numerical scheme for the determination of the increments of the field quantities. Implicit numerical schemes are known to have in general good stability characteristics (see e.g. [76, Section 9.6.1] or [94, Section 9.8]); to be more specific, they are, in principle, unconditionally stable. Therefore, the almost fully implicit scheme, employed in this work, can be expected to possess similar advantageous stability characteristics.

Buckling of a Corona Geometry

In [37] the structure of a thin corona made of a soft gel is experimentally and theoretically described. The soft corona is clamped to a disk of stiff gel at the inner rim, while the outer rim is free. When the corona is homogeneously immersed in water, the gel swells and bends out-of-plane leading to a wavy periodic pattern.

The theoretical description is based on the Föppl-von Kármán equations for thin elastic plates [95] and uses the framework of linear elasticity. The corona has an inner radius of $r_i$, an outer radius of $r_o$ and is described in polar coordinates ($r,\theta$). The already swollen corona is supposed to be pulled at the inner rim inwards so that the inner radius shrinks to the radius of the stiff disk.

The described corona necessarily shows a stress distribution symmetric about the corona’s centre perpendicular to the r-$\theta$-plane, which includes that the stress components do not depend on $\theta$. Additionally, the shear stresses must vanish due to symmetry. The work [37] searched for a solution of the equation of forces in radial direction [96, Eq. 40],

$$\begin{aligned} \frac{\partial \sigma _{rr}}{\partial r} + \frac{ \sigma _{rr} - \sigma _{\theta \theta } }{r} = 0, \end{aligned}$$

(D16)

by using Hooke’s law and the strains $\varepsilon _{rr}=\partial _r u_r$ and $\varepsilon _{\theta \theta }=u_r/r + (1/r) \partial _{\theta } u_{\theta }$ in the form [97, Chapter 3.6]

$$\begin{aligned} u_r = a r + \frac{b}{r}, \quad \quad u_{\theta } = 0. \end{aligned}$$

(D17)

The inward drawing of the inner rim shall be given by $\beta$, which allows to write the boundary conditions as: $u_r(r=r_i)=-\beta$ and $\sigma _{rr}(r=r_o)=0$. Furthermore, if the Poisson’s ratio of the gel is taken to be 0.5 and if we follow [37] by using the outer radius $r_o$ as the unit of length, we obtain

$$\begin{aligned} \sigma _{rr} = - \alpha \left( 1 - \frac{1}{r^2} \right) , \quad \quad \sigma _{\theta \theta } = - \alpha \left( 1 + \frac{1}{r^2} \right) , \end{aligned}$$

(D18)

with $\alpha = \beta E (2r_i)/(3r_o^2+r_i^2)$ and E being the Young’s modulus.

The Föppl–von Kármán equation is given in rectangular coordinates in [98, Eq. 246]. If, temporarily, we change from standard polar coordinates $r, \theta$ to a corresponding radial s—tangential t coordinate system, i.e. with t being tangent to the radian measure $r\theta$, then we find for this particular example

$$\begin{aligned} D \Delta \Delta \zeta - h \left( \sigma _{ss} \frac{\partial ^2 \zeta }{\partial s^2} + \sigma _{tt} \frac{\partial ^2 \zeta }{\partial t^2} \right) = 0, \end{aligned}$$

(D19)

for a corona of thickness h, the bending stiffness D and the corona’s out-of-plane deflection $\zeta$. Equation D19 is tantamount to the corresponding one of [37, Eq. 5].

If we turn back to standard polar coordinates $r, \theta$ and acknowledge the identities $\partial ^2_{ss} = \partial ^2_{rr}$, $\partial ^2_{tt} = (1/r^2) \partial ^2_{\theta \theta } + (1/r) \partial _{r}$ and $\partial ^2_{st} = (1/r) \partial ^2_{r\theta } - (1/r^2) \partial _{\theta }$, then Eq. D19 is reshaped with the help of Eqs. D18$_{1,2}$ to

$$\begin{aligned} \Delta \Delta \zeta + \alpha \Delta \zeta + \frac{\alpha }{r^2} \left( -\frac{\partial ^2 \zeta }{\partial r^2} + \frac{1}{r}\frac{\partial \zeta }{\partial r} + \frac{1}{r^2}\frac{\partial ^2 \zeta }{\partial \theta ^2} \right) = 0. \end{aligned}$$

(D20)

Eq. D20 is tantamount to the corresponding one of [37, Eq. 15]. However, in Eq. 15 of [37], the penultimate term of the LHS contains the factor 3/r instead of the correct factor 1/r. We believe the factor of 3/r to be a miscalculation as this result was used to plot instability regions in [37, Fig. 7].

The boundary conditions suitable for the corona are at $r=r_i$:

$$\begin{aligned} \zeta = 0, \quad \quad \frac{\partial \zeta }{\partial r} = 0; \end{aligned}$$

(D21)

and at $r=r_o=1$ [95, Eqs. 12.6 & 12.7]:

$$\begin{aligned}{} & {} - \frac{\partial (\Delta \zeta )}{\partial r} + 0.5 \frac{1}{r^3} \left( \frac{\partial ^2 \zeta }{\partial \theta ^2} - r \frac{\partial ^3 \zeta }{\partial r \partial \theta ^2} \right) = 0, \nonumber \\{} & {} \Delta \zeta - 0.5 \frac{1}{r^2} \left( \frac{\partial ^2 \zeta }{\partial \theta ^2} + r \frac{\partial \zeta }{\partial r} \right) = 0. \end{aligned}$$

(D22)

The solution to Eq. D20 satisfying the stated boundary conditions can be found numerically and is exemplarily described in [37]. The solution to $\zeta$ must satisfy the periodic form

$$\begin{aligned} \zeta (r, \theta ) = \xi (r) \cos (m \theta ), \end{aligned}$$

(D23)

which introduces the wave number m.

Table 4 shows the obtained wave number in terms of the ratio $r_i/r_o$ of 0.20 to 0.87 since the numerical computations are tedious. We state the numerical results to the correct differential equation in D20 with the factor 1/r in the left column, while the results for the original differential equation with the factor 3/r are given in the right column. As can be observed the difference in the obtained instability regions is significant.

Table 4 Aspect ratio $r_i/r_o$ to wave number m. Instability regions in the left column according to Eq. D20, in the right column according to [37, Eq. 15]