Thermodynamic formulations of continuum growth of solid bodies

Abstract

The thermodynamics of open systems exchanging mass, heat, energy, and entropy with their environment is examined as a convenient unifying framework to describe the evolution of growing solid bodies in the context of volumetric growth. Following the theory of non-equilibrium thermodynamics (NET) introduced by De Donder and followers from the Brussels School of Thermodynamics, the formulation of the NET of irreversible processes for multicomponent solid bodies is shortly reviewed. In the second part, extending the framework of NET to open thermodynamic systems, the balance laws for continuum solid bodies undergoing growth phenomena incorporating mass sources and mass fluxes are expressed, leading to a formulation of the second principle highlighting the duality between irreversible fluxes and conjugated driving forces. A connection between NET and the open system thermodynamic formulation for growing continuum solid bodies is obtained by interpreting the balance laws with source terms as contributions from an external reservoir of nutrients.

Keywords

Volumetric growth non-equilibrium thermodynamics mixture theory thermodynamics of open systems reservoir of nutrients

1. Introduction

The effect of growth in biological materials has been modeled in the literature since the late 1960s by considering that the deformations are due to mass change and to elastic deformation induced by growth [1 –4]. These lines of thought were brought to a more achieved model of the kinematics by Rodriguez et al. [5], who introduced the multiplicative split of the total deformation gradient into a growth deformation tensor describing the local addition of mass and an elastic tensor accounting for the local reorganization of the body required to restore the kinematic compatibility. Without going into a detailed description of the flourishing activity devoted to morphogenesis, growth and remodeling, one may classify growth models into three main families, as described in Ganghoffer and Sokolowski [6]. The respective advantages and drawbacks of those models are discussed in a recent review article [7].

A clear distinction is further made between volumetric growth referring to processes taking place in the bulk of the material, and surface growth, describing mechanisms of mass accretion at a surface [8 –13]. Volumetric growth models are essentially based on the idea of a local change of density or volume as a way to model the ‘squeezing’ of new particles coming from some external reservoir of nutrients. The distinction between surface growth and volumetric growth may be artificial according to certain authors [14], but could be attributed to specific physical mechanisms occurring at different scales [6].

Cells respond to various mechanical and biochemical stimuli in a cascade of events, which are reflected by the coinage mechanotransduction’; the nature of the exact stimuli and the mechanisms leading to the production of an extracellular matrix (ECM) and the growth of new tissue are still poorly understood. From a pure mechanical standpoint, the pathways through which cells sense signals mechanical signals and convert them into chemical signals inducing ultimately a biological response such as growth or remodeling still remain elusive [15]. They involve a signal transduction occurring on the cell surface, which is then transmitted to components such as integrin and growth factors acting as mechanosensory entities [16], leading to activate growth factors (including proteins acting as regulators for the transport of the nutrients within the cell).

From a macroscopic and continuum point of view, the analysis of growing solid bodies require modifications of the classical framework of continuum mechanics; especially, the fundamental assumptions introduced by Truesdell and Noll [17,18] have to be relaxed:

Classical continuum mechanics considers a unique and fixed material body that is a fixed set of material points with a given connectivity and fixed mass;

The reference configuration is given and fixed to define the geometrical properties of the body (shape) once and for all;

The material behavior is defined and fixed throughout the deformation;

In addition to those kinematic assumptions, the mass of the material body is fixed and constant, i.e. only closed systems are considered.

Growth usually involves several constituents (soft or hard tissues, fluid carrying growth factors, nutrients and various cell populations), which the theory of mixtures is able to incorporate. Biological growth has been modeled in this context by various authors [19 –24], allowing the consideration of a solid mixture and multiple fluid constituents. This framework is suitable for the description of complex responses of biological tissues, encompassing both volumetric and surface growth within a common umbrella.

Growth theories assume classically a constant number of particles to be able to map the actual and reference configurations, but they try to introduce the impact of the changing number of particles by either a local density change or a local change of specific volume; these two strategies have given rise to two alternative classes of models in the literature, involving respectively density based or volume based energies [25]. In continuum approaches, the growth is assumed to occur at the scale of tissue elements, defined as small regions of space (representative volume elements) located at a so-called mesoscopic level of description, that receive nutrients and chemical species via diffusion processes, from an externally assumed existing reservoir [26,27]. Accordingly, the sources and fluxes of mass are incorporated into the various balance laws, adopting the thermodynamics of irreversible and open systems as a natural framework that allows a physically consistent treatment of the coupling between mechanics and mass transport. In those works, biochemical and mechanical stimuli for growth are involved in the balance laws that finally lead to the expression of the local dissipation, from which kinetic laws for the identified irreversible fluxes can be formulated.

Since growth involves coupling between mechanics (cells and tissues respond to mechanical signals such as stresses and strains) and biochemistry (diffusion and reaction phenomena occurring at cellular level), the thermodynamics of irreversible processes is a natural framework to incorporate these phenomena of different physical origins under a common and unifying umbrella. A clear exposition of the thermodynamic formulation for such open systems undergoing growth phenomena is in our opinion still lacking in the literature; especially, the connections between different possible thermodynamic approaches is a topic of interest that has not been treated so far in the literature. Two alternative viewpoints can indeed be considered to treat the thermodynamics of growing continuum solid bodies: the non-equilibrium thermodynamic which basically considers closed systems (there is not net source of mass), or the more recent thermodynamic formulation for open systems including mass sources and fluxes. In the present work, we advocate as the principal contribution two main aspects: new developments leading to the derivation of the residual dissipation for growing solid bodies highlighting the density of the grand potential, and a connection between non-equilibrium thermodynamics and open system thermodynamics by closing the system in the last formulation.

The present contribution is organized as follows. In order to set the stage, we recall in Section 2 in a synthetic manner the formulation of non-equilibrium thermodynamics, recalling the principal balance laws to put them later on in perspective with the more modern treatment of growth. Driven by the mass balance including source and flux terms, the balance laws for tissue elements under growth are expressed in Section 3, adopting here the framework of the thermodynamics of open systems. These balance laws are given a different interpretation (Section 4), considering explicitly the reservoir of nutrients feeding the growing solid body as an additional system leading to a closed system formulation. As a consequence, the balance of mass and energy can be written as conservation laws when the reservoir is introduced to close the overall thermodynamic system. The residual dissipation and kinetic laws for the irreversible fluxes are expressed in Section 4. A summary of the main developments and perspectives for future work are given in the conclusive Section 5.

A few words regarding notations are in order. Vectors and higher order tensors are denoted with boldface symbols. The summation convention on repeated indices is presently adopted, otherwise explicitly stated. The partial derivative of a scalar function $f (x)$ is denoted $\partial_{x} f = \frac{\partial f}{\partial x}$ ; the time derivative of a function $a (t)$ is represented by a superposed dot, viz. $\overset{\cdot}{a} (t) = \frac{d a (t))}{dt}$ . The second order identity tensor is denoted $I$ . The short cuts l.h.s. and r.h.s. stand for left-hand side and right-hand side respectively.

2. An overview of non-equilibrium thermodynamics (NET)

From a historical perspective, the first occurrence of thermodynamic considerations for the treatment of irreversible process trace back to W. Thompson in 1854 [8], who analyzed at that time thermo-electric phenomena and established two famous relations bearing his name. In the same period, Clausius (in 1850) introduced the concept of ‘non-compensated heat’ to quantify irreversibility. The first one is a consequence of the conservation of energy, while the second one relating the thermoelectric potential of a thermocouple to the Peltier heat, was obtained by combining both laws of thermodynamics and making an additional assumption about the ‘reversible’ contributions to the powers. Boltzmann attempted (in 1909) unsuccessfully to justify Thompson’s assumption, but it was Onsager in 1931 who proved the second Thompson relation as a consequence of the invariance of the microscopic equations of motion under time reversal. In the same period, Clausius (in 1850) introduced the concept of ‘non-compensated heat’ to quantify irreversibility.

Later on (beginning of the twentieth century), the foundations of the thermodynamics of multiphysical coupled systems subjected to exchange of mass, heat, and work with their surrounding were laid down by a panel of thermodynamicists of the Brussels School of Thermodynamics, including amongst others De Donder, Munster, De Groot, Mazur, Prigogine, to name the principal investigators in the field. They altogether contributed to the development of non-equilibrium thermodynamics, abbreviated NET in the remainder of this contribution.

In order to set the stage, we expose in this section an overview of the thermodynamic modeling of coupled phenomena in open systems (despite the lack of unity in the field, if we follow the belief “as many thermodynamicists, as many thermodynamics” [28]).

This background will be extended in the forthcoming sections to treat the specific situation of growing solid bodies. We consider a multicomponent system with k = 1…N constituents, amongst which chemical reactions occur amongst r of the constituents (with r less than N).

2.1. Mass balance

The rate of change of mass for any constituent k is given for a certain volume V as the sum of material flow of the same component through the boundary and the production of k due to chemical reactions occurring inside V:

\frac{d}{dt} \int_{V} ρ_{k} dV = \int_{V} \frac{\partial ρ_{k}}{\partial t} dV + - \int_{\partial V} ρ_{k} v_{k} . n dS + \sum_{p = 1}^{r} \int_{V} ν_{kp} R_{p} dV

(1)

with $υ_{kp}$ stoichiometric coefficients and $R_{p}$ the rate of chemical reaction p. The scalar quantity $ν_{kp} R_{p}$ is the production of k per unit volume in the p^th chemical reaction. Application of Gauss’ theorem to the surface integral delivers the local mass balance

\frac{\partial ρ_{k}}{\partial t} = - div ρ_{k} v_{k} + \sum_{p = 1}^{r} ν_{kp} R_{p}

(2)

Since mass is conserved in each separate chemical reaction, the stoichiometric coefficients shall satisfy the relation

\sum_{k = 1}^{n} υ_{kp} = 0

(3)

The mass flux $J_{k}$ expresses vs. the local velocity $v_{k}$ , the barycentric velocity $v$ and the mass density $ρ_{k}$ as

v = \sum_{k = 1}^{n} ρ_{k} v_{k} / ρ \to J_{k} = ρ_{k} (v_{k} - v)

(4)

An alternative formulation of the local mass balance writes, involving the concentrations or mass fractions $c_{k} = \frac{ρ_{k}}{ρ}$ defined as the ratio of mass densities of the k^th constituent to the total density,

ρ \frac{{dc}_{k}}{dt} = - div J_{k} + \sum_{p = 1}^{r} ν_{kp} R_{p}

(5)

Summation of all partial mass balance equations (2) and consideration of the definition of the mass fluxes in equation (4) deliver the strong form of the conservation of mass

\frac{\partial ρ}{\partial t} + div ρ v = 0

(6)

with the total density defined as the sum of partial densities

ρ = \sum_{k = 1}^{n} ρ_{k}

(7)

Equation (5) traduces the conservation of the total mass, despite each constituent does not satisfy mass conservation due to mass sources and fluxes within the thermodynamic system, which overall mutually compensate.

The partial densities satisfy the balance law

\frac{\partial ρ_{k}}{\partial t} = - \nabla . (ρ_{k} v) - \nabla . (J_{k}) = - \nabla . (Φ_{m_{k}})

(8)

The flux $Φ_{m_{k}}$ introduced therein decomposes into a convection flux $ρ_{k} v$ and a diffusion flux $J_{k}$ . The definition of the barycentric velocity and diffusion flow with respect to the barycentric motion that entails the sum of fluxes vanishes

\sum_{k = 1}^{n} J_{k} = 0

(9)

As a matter of fact, the present description implicitly relies on the assumption of a closed representative volume element (exchange and production of matter occur within this RVE, with however not net effect), not exchanging mass with its surrounding. This viewpoint is at variant with the more modern approach of growth that will be treated later on, which incorporates both a mass source and mass flux in the overall mass balance.

2.2. Balance of momentum

The dynamical equations of motion write [29]

ρ \frac{d v}{dt} = - Div P + \sum_{k = 1}^{n} ρ_{k} F_{k}

(10)

with $P$ the nominal (the first Piola–Kirchhoff stress tensor), coined the pressure tensor by De Groot and Mazur [29]. The Cauchy stress tensor is symmetrical in the present picture, which will no more be the case when considering an open RVE exchanging mass with its surrounding. The stress tensor accounts for short range interactions between material points, whereas the force per unit mass $F_{k}$ incorporates both external contributions and long-range interactions within the body. Previous balance of linear momentum can be rewritten using the following relation

\frac{da ρ}{dt} = \frac{\partial a ρ}{\partial t} + Diva ρ v

(11)

valid for any property (a consequence of the mass balance equation and the definition of the material derivative) as the following equality

\frac{\partial ρ v}{\partial t} = - Div (P + ρ v \otimes v) + \sum_{k = 1}^{n} ρ_{k} F_{k}

(12)

The second order tensor $P + ρ v \otimes v$ therein deserves the interpretation of a momentum flow, including the convective part $ρ v \otimes v$ , and vector $\sum_{k = 1}^{n} ρ_{k} F_{k}$ is a source of momentum. As a matter of completeness and for the purpose of comparing with the subsequent thermodynamic formulation of growth, we express the balance of kinetic energy

\begin{matrix} ρ \frac{d (v^{2} / 2)}{dt} = - \nabla . (P . v) + P : \nabla v + \sum_{k = 1}^{n} ρ_{k} F_{k} . v \\ \to ρ \frac{\partial (v^{2} / 2)}{\partial t} = - \nabla . (P . v + ρ v (v^{2} / 2)) + P : \nabla v + \sum_{k = 1}^{n} ρ_{k} F_{k} . v \end{matrix}

(13)

2.3. Energy and entropy balance

The total specific energy incorporates in an additive manner the specific kinetic energy $k : v^{2} / 2$ , the potential energy $ψ$ and internal energy u:

e = k + ψ + u

(14)

The principle of conservation of energy writes in global and local forms successively over a time dependent domain $Ω_{t}$

\frac{d}{dt} \int_{Ω_{t}} ρ edv = \int_{Ω_{t}} \frac{\partial (ρ e)}{\partial t} dv = - \int_{\partial Ω_{t}} J_{e} . n ds \Rightarrow \frac{\partial (ρ e)}{\partial t} + \nabla . J_{e} = 0

(15)

with $J_{e}$ the energy flux per unit surface and unit time, expressed as

J_{e} = ρ e v + P . v + \sum_{k} ψ_{k} J_{k} + J_{q}

(16)

This relation in fact defines the heat flux $J_{q}$ ; accordingly, the total energy flux is the sum of the convective term $ρ e v$ , the energy flux $P . v$ due to mechanical work, a potential energy flux due to diffusion of components in the force due to potential $ψ_{k}$ , and the heat flux $J_{q}$ . Using the equation for the rate of total potential energy

ρ ψ : \sum_{k} ρ_{k} ψ_{k} \Rightarrow \frac{\partial ρ ψ}{\partial t} = - \nabla . (ρ ψ v + \sum_{k} ψ_{k} J_{k}) - \sum_{k} ρ_{k} F_{k} . v - \sum_{k} J_{k} . F_{k}

(17)

and adding the balance of kinetic energy, equation (13), we finally obtain the balance of internal energy

\frac{du}{dt} = \nabla . (J_{q}) + ρ^{- 1} p_{i} + ρ^{- 1} J_{k} . F_{k}

(18)

with $p_{i} = - σ : D (v)$ the volumetric density of the power of internal forces, sum of the work done by the pressure and the viscous stress $σ^{visc}$ , viz.

p_{i} = - p \nabla . v - σ^{visc} : \nabla v

(19)

The general form of the entropy balance (s denotes the density of entropy per unit mass) writes as a decomposition of the rate of total entropy into an exchange term $\frac{d_{e} S}{dt}$ and a production term $\frac{d_{i} S}{dt}$ :

\begin{matrix} S : \int_{V} ρ sdV \Rightarrow \frac{dS}{dt} = \frac{d_{e} S}{dt} + \frac{d_{i} S}{dt} \\ \frac{d_{e} S}{dt} = - \int_{V} J_{s, tot} . n ds, \frac{d_{i} S}{dt} = \int_{V} σ_{s} dV \end{matrix}

(20)

in which $J_{s, tot}$ is the total entropy flux, and $σ_{s}$ the entropy production per unit volume and time.

Localization of previous entropy balance then delivers

ρ \overset{\cdot}{s} = - \nabla . J s / T + σ_{s}

(21)

At equilibrium, the total differential of the entropy density is given by Gibbs relation

T \overset{\cdot}{s} = \overset{\cdot}{u} - σ : D^{rev} - μ_{k} {\overset{\cdot}{c}}_{k}

(22)

for $k = 1 . N$ chemical species (otherwise called constituents) having each the chemical potential $μ_{k}$ . The symmetric part of the velocity gradient $D$ is being additively decomposed into a reversible contribution $D^{rev}$ and an irreversible contribution $D^{irr}$ .

The barycentric balance of mass for the k^th constituent writes as in non-equilibrium thermodynamics (accounting for the multicomponent nature of the growing solid body)

ρ {\overset{\cdot}{c}}_{k} = - \nabla . (J_{k}) + υ_{k m} R_{m}

(23)

with the scalar $R_{j}$ denoting the velocity of the chemical reaction (for the $j^{th}$ constituent). Introducing equation (23) and the material derivative, equation (18), into equation (22) leads after a few rearrangements to

ρ \overset{\cdot}{s} = - \nabla . (\frac{J_{s}}{T}) - \frac{1}{T^{2}} J_{q} . \nabla T - p_{i} - \frac{1}{T} \sum_{k} J_{k} . (T \nabla \frac{μ_{k}}{T} - F_{k}) - \frac{1}{T} σ^{visc} : \nabla v - \frac{1}{T} A_{k} R_{k}

(24)

with $A_{k} : υ_{ik} μ_{i}$ the chemical affinities, introduced by De Donder (Munster, 1966), and $J_{s} : \frac{1}{T} (J_{q} - μ_{i} J_{i})$ the total entropy flux, incorporating the heat flux and the mass flux weighted by the chemical potential. As a matter of fact, previous equality introduces a decomposition of the rate of entropy into exchange contributions expressed mathematically as divergence terms (entropy fluxes through the system boundary, compensable), and a remaining part that will be identified as the internal entropy produced (non-compensable). The entropy production is then easily isolated as [29]

σ_{s} = - \frac{1}{T^{2}} J_{q} . \nabla T - \frac{1}{T} J_{k} . (T \nabla (\frac{μ_{k}}{T}) - F_{k}) - \frac{1}{T} σ^{visc} : D (v) - \frac{1}{T} A_{j} J_{j}

(25)

Alternative expressions of the internal entropy production based on different definitions of the heat flux have been formulated in De Groot and Mazur [29]; we will, however, not dwell on these details. It can also be added that, when concerning biological applications, particular formulations of the balance laws could be required, for instance accounting for situations in which non-fixed interfaces exist (see [30,31]); a context in which such kinds of problems naturally arise and must be dealt with is that of phase transitions (see [32 –35]).

We close this section by commenting the assumptions inherent to the NET that has been exposed along this main streamline: as a main aspect, the mass balance equation (6) does not introduce any source term, thus it does not incorporate mass production due e.g. to volumetric growth. One accordingly expects that the consideration of mass production and mass fluxes at the level of the mass balance will have consequences for all other balance laws.

It is the object of the next section to formulate such a true thermodynamic formulation for growing solid bodies, which aims to be more general compared to NET.

3. Balance laws for growing solid bodies in the framework of open systems mechanics

In order to set the stage, we emphasize a few points of vocabulary. One basic notion in thermodynamics is the system, defined as a collection of material particles enclosed by a boundary; the definition of the thermodynamic system and its boundary is completely arbitrary and depends on the viewpoint of the observer. Material points not included into the system boundary constitute its surrounding. Interactions between the system and its surrounding play a prominent role in the analysis of growing solid bodies. The system is closed when no matter crosses its boundary; we observe that NET exposed in previous section deals with closed systems, with mass being exchanged amongst chemical species within the system, but not generating any transfer of mass across the boundary. Whereas a closed system has a constant mass, the conserve does not hold: systems may be open with constant mass (if mass outflow on part of the boundary is compensated by mass inflow on another subpart of the system boundary). Mass may also vary for open systems due to internal sources of mass (mass production), in addition to mass flow across the boundary. NET allows mass production through chemical reactions, but there is no overall mass production according to Lavoisier law since all mass sources are explicitly accounted for within the same system. Observe lastly that closed and open systems are the thermodynamic analogs of the Lagrangian and Eulerian frame of reference: the Lagrangian description deals with the analysis of the motion of a fixed small element of particles with constant mass, whereas the Eulerian observer concentrates on a fixed volume in space witnessing different material particles during time due to a constant flow of mass through its boundary. In order to get rid of this observer dependency, one shall presently assume that growth can be characterized by mass variations occurring for any reference (= Lagrangian) volume.

With these remarks in mind, we shall next proceed with the description of open systems subjected to mass flow and mass production. We consider a volume element with domain $Ω_{t}$ as an open system exchanging mass with its surrounding and witnessing mass production due to internal sources. From a biological viewpoint, the mass flux is typically attributed to cell migration, while the mass sources originate from cell growth or shrinkage, cell division or cell death. This viewpoint differs from that of NET which works on closed systems with no internal mass production. The mass balance for such open systems is first written and its impact on all other balance laws is next examined.

3.1. Mass balance

The material derivative of the mass functional $M [Ω_{t}] : (\int_{Ω_{t}} ρ dV)$ elaborated over an open system with domain $Ω_{t}$ and internal density $ρ = ρ (x, t)$ successively writes in global and local format as

\begin{matrix} \frac{D}{Dt} (\int_{Ω_{t}} ρ dx) = \int_{Ω_{t}} (\frac{d ρ}{dt} + ρ \nabla . v) dx = \int_{Ω_{t}} (π - \nabla . J) dx \\ \Rightarrow \frac{d ρ}{dt} = π - \nabla . J - ρ \nabla . v \end{matrix}

(26)

introducing therein and in a formal manner mass sources $π$ and mass fluxes $J$ . The mass production term $π$ accounts for new particles entering the system, which are quizzed in-between the existing particles, leading locally to a change of internal density.

The general balance law for a physical quantity defined by a specific density writes

\frac{D}{Dt} (\int_{Ω_{t}} ρ adx) = \int_{Ω_{t}} (\frac{d (ρ a)}{dt} + ρ a \nabla . v) dx = \int_{Ω_{t}} (ρ \frac{da}{dt} + a \frac{d ρ}{dt} + ρ a \nabla . v) dx,

(27)

Inserting the continuity equation (26) into previous general equality, equation (27), delivers the general identity

\frac{D}{Dt} (\int_{Ω_{t}} ρ adx) = \int_{Ω_{t}} (ρ \frac{da}{dt} + a (π - \nabla . J)) dx = \int_{Ω_{t}} (ρ \frac{da}{dt} + a (\sum_{k} ρ_{k} {\overset{\cdot}{n}}_{k} - \nabla . J)) dx

(28)

The mass balance can be related to the nutrients transport responsible for the variation of density: the time variation of the chemical concentration of nutrients $n (x, t)$ is due to exchange through the boundary, accounted for by a mass flux $\int_{\partial Ω_{t}} J . n ds$ , and to the source term due to growth expressed in (26), the global quantity; hence, one may write the following physical balance law successively in weak and strong form:

\frac{D}{Dt} \int_{Ω_{t}} ρ dx = \int_{Ω_{t}} ρ \overset{\cdot}{n} dx + \int_{\partial Ω_{t}} J . N ds \Rightarrow ρ \overset{\cdot}{n} + div J = ρ Γ

(29)

The last equality is nothing else than $Γ = (π - \nabla . J) / ρ$ - the strong form of the mass balance, thus equality (29) rewrites

\frac{D}{Dt} \int_{Ω_{t}} ρ dx = \int_{Ω_{t}} {π - div J (ρ)} dx

(30)

with the identification of the source terms $π = ρ \overset{\cdot}{n}$ .

3.2. Balance of linear and angular momentum

The balance of linear momentum writes in global format as

\frac{d}{dt} \int_{Ω_{t}} ρ v dx = \int_{Ω_{t}} f dx + \int_{\partial Ω_{t}} σ . n ds + \int_{Ω_{t}} π v dx - \int_{\partial Ω_{t}} (J \otimes v) . n ds

(31)

with $σ$ Cauchy stress and $f$ body forces per unit physical volume. Localization of this balance law is obtained using the mass balance as

ρ \frac{D v}{Dt} = f + div σ - (J . \nabla) v

(32)

The Cauchy stress appearing there is not symmetrical in general due to the mass flux; the balance of angular momentum leads to the symmetry of the modified Eulerian stress [13,36], incorporating the mass flux

\tilde{σ} : σ - J \otimes v

(33)

Previous balance of momentum entails the following modified form of the principle of virtual work (for virtual velocity fields $\tilde{v}$ )

ρ \tilde{v} . \frac{D v}{Dt} = f . \tilde{v} + \tilde{v} . \nabla . σ - (J . \nabla) v . \tilde{v}

(34)

3.3. Physical picture of a composite system including the reservoir of nutrients

We advocate in this section a different point of view of a composite closed system including both the growing solid body $Ω_{t}$ and the reservoir of nutrients, occupying as the domain $R_{t}$ : by construction, the composite system $Ω_{t} \cup R_{t}$ is closed, and this viewpoint leads to a different interpretation of the mass sources and fluxes that would otherwise vanish for a non-growing body. It will also clearly set a connection with the point of view advocated in NET.

The additive nature of the mass and energy functionals with respect to the domain leads to the following decompositions of the overall mass and energy functionals

\begin{matrix} M_{tot} = M [Ω_{t}] + M [R_{t}] \\ E_{tot} = E [Ω_{t}] + E [R_{t}] \end{matrix}

(35)

This entails the following expression of the variation of the overall mass of the growing body

δ M [Ω_{t}] \equiv δ (\int_{Ω_{t}} ρ dx) = lim_{δ t \to 0} {\frac{M [Ω_{t} + R_{t}] - M [Ω_{t}]}{δ t}} δ t \equiv - δ M [R_{t}] \to δ M [Ω_{t} \cup R_{t}] = 0

(36)

Furthermore, a fraction $δ R_{t}$ of the reservoir continuously feeds the growing body $Ω_{t}$ during the infinitesimal duration $δ t$ , thus it holds that

\begin{matrix} δ M [R_{t}] : \overset{\cdot}{M} [R_{t}] δ t \\ \Rightarrow Ω_{t + δ t} = Ω_{t} + δ R_{t} \Rightarrow M [Ω_{t + δ t}] = M [Ω_{t}] + δ M [R_{t}] \end{matrix}

(37)

in which a superposed dot denotes the material derivative. The overall mass is preserved (this view point is somewhat equivalent to that of NET which implicitly incorporates the nutrients inside the system, which by construction and as a corollary remains overall closed), thus it holds the following equality

\frac{D}{Dt} {M [Ω_{t}] + M [R_{t}]} = 0

(38)

Furthermore, the previous mass balance, equation (26) for the sole open growing system $Ω_{t}$ leads to

\begin{matrix} \int_{Ω_{t}} (\frac{d ρ}{dt} + ρ \nabla . v) dx & = \int_{Ω_{t}} (π - \nabla . J) dx \\ \Rightarrow \frac{DM [Ω_{t}]}{Dt} = \int_{Ω_{t}} \frac{\partial ρ}{\partial t} dx + \int_{\partial Ω_{t}} ρ v . n dx = \int_{Ω_{t}} (π - \nabla . J) dx \end{matrix}

(39)

In equation (39), the contribution $(π - \nabla . J)$ represents the density of the source of mass.

With these elements at hand, we can now bring a new interpretation of the mass balance equation (38): the variation of the mass being exchanged writes

δ M_{exchange} [Ω_{t}] = \int_{\partial Ω_{t}} (ρ v + J) . n ds \to δ M_{exchange} [Ω_{t}] + δ M_{exchange} [R_{t}] = 0

(40)

The last equality constitutes a reciprocity principle for the exchange of mass between $Ω_{t}$ and the reservoir of nutrients. The surface integrals $\int_{\partial Ω_{t}} ρ v . n dx, \int_{\partial Ω_{t}} J . n dx$ therein represent exchange of mass with the reservoir by convection and conduction respectively. Using equation (39), the local variation of mass of the growing body then expresses as

\begin{matrix} δ M_{local} [Ω_{t}] : \int_{Ω_{t}} \frac{\partial ρ}{\partial t} dx = - \int_{\partial Ω_{t}} (ρ v + J) . n dx + \int_{Ω_{t}} π dx \\ \Rightarrow δ M_{local} [Ω_{t}] + δ M_{exchange} [Ω_{t}] = \int_{Ω_{t}} π dx \equiv - δ M [R_{t}] \\ δ M_{exchange} [Ω_{t}] = \int_{\partial Ω_{t}} Tr (Σ_{m}) dx, Σ_{m} : (ρ v + J) \otimes n \end{matrix}

(41)

Convection and conduction fluxes of mass have been condensed into the second order tensor $Σ_{m}$ , the trace of which provides the overall density of mass flux.

This writing leads to a new interpretation of the mass balance when introducing a non-compensated exchange term: the sum of the local mass variation (due to the local change of density) with the mass production of the reservoir and exchange terms vanishes,

δ M_{tot} = δ M_{local} [Ω_{t}] + δ M [R_{t}] + δ M_{exchange} [Ω_{t}] \equiv 0

(42)

Extending this writing, we can further express the material derivative of any global extensive quantity $A [Ω_{t}]$ elaborated over a time dependent domain $Ω_{t}$ as the following expression

\begin{matrix} \frac{DA [Ω_{t}]}{Dt} = \frac{D}{Dt} (\int_{Ω_{t}} ρ adx) = \int_{Ω_{t}} (ρ \frac{da}{dt} + a (π - \nabla . J - ρ \nabla . v) + ρ a \nabla . v) dx = \\ = \int_{Ω_{t}} (ρ \frac{\partial a}{\partial t}) dx + \int_{\partial Ω_{t}} ρ a v . n ds + \int_{Ω_{t}} a (π - \nabla . J) dx \equiv δ A_{local} [Ω_{t}] + δ A^{ex} - δ A [R_{t}] \end{matrix}

(43)

In previous identity, we have decomposed the total derivative of the integral $A [Ω_{t}]$ successively into a term traducing the local variations of the density $a (x, t)$ , an exchange terms with the reservoir $δ A^{ex}$ , and a contribution specific to the reservoir $δ A [R_{t}]$ , all three terms identified to

\begin{matrix} δ A_{local} [Ω_{t}] = \int_{Ω_{t}} (ρ \frac{\partial a}{\partial t}) dx, \\ δ A^{ex} = \int_{\partial Ω_{t}} ρ a v . n ds, \\ δ A [R_{t}] = - \int_{Ω_{t}} a (π - \nabla . J) dx \end{matrix}

(44)

with

\frac{DA [Ω_{t}]}{Dt} = δ A_{local} [Ω_{t}] - δ A [R_{t}] + δ A^{ex}

(45)

The balance of $A [Ω_{t} \cup R_{t}]$ over the whole domain including the reservoir thus writes

\frac{DA [Ω_{t} \cup R_{t}]}{Dt} = \frac{DA [Ω_{t}]}{Dt} + \frac{DA [R_{t}]}{Dt} = δ A_{local} [Ω_{t}] - δ A [R_{t}] + \frac{DA [R_{t}]}{Dt} + δ A^{ex}

(46)

with the material derivative over the reservoir due to the source term (vanishing in the absence of growth) therein given by

\frac{DA [R_{t}]}{Dt} = \int_{R_{t}} a (π - \nabla . J) dx

(47)

Equations (45) and (46) entail that $A [Ω_{t} \cup R_{t}]$ is a conserved quantity if and only if

δ A_{local} [Ω_{t}] + δ A^{ex} = 0

(48)

so that the local variation is compensated by an exchange term with the reservoir; the strong form of previous condition is simply the local conservation law

ρ \frac{\partial a}{\partial t} + \nabla . (ρ a v) = 0

(49)

which holds in the absence of growth for the composite system, including the reservoir. Especially, the balance of energy is obtained from previous general writing by the identification $a = (e + k)$ , with the material derivative of the overall energy given by

\begin{matrix} \frac{D}{Dt} (\int_{Ω_{t}} ρ (e + k) dx) = \int_{Ω_{t}} (ρ \frac{d (e + k)}{dt} + (e + k) (π - \nabla . J)) dx = \\ = δ E_{local} [Ω_{t}] - δ E [R_{t}] + δ E^{ex} \end{matrix}

(50)

Since mass and energy are conserved for the composite system $Ω_{t} \cup R_{t}$ , one obtains the same conservation laws as for NET, since the overall system is closed.

It is next instructive to have a look at the theory of mixtures, which bears strong similarities in its spirit and formulation with NET.

3.4. Insight into the theory of mixtures

Mixture theory combines basically the framework of continuum mechanics for the description of the motion and deformation of solids with chemistry principles; it has been successfully applied to the analysis of tissue growth and remodeling for reactive mixtures consisting of a solid mixture with multiple fluid components, see the review article by Ateshian [19] and references therein; a general theoretical formulation accounting for grain growth (and also for phenomena such as grain boundary migration, grain rotation, and evolution of dislocation density) is provided by Placidi and Hutter [37]. The framework of mixture theories proves interesting as a specific pathway to account for the generation of residual stresses arising from osmotic effects; it accordingly provides a more detailed analysis of the growth mechanisms in comparison to single constituent theories.

The theoretical foundations of the theory of mixtures were laid down more than 50 years ago by Truesdell and Toupin [38]; we shall rely on their presentation to provide a summary of the conservations laws for reactive mixtures, and highlight similarities with the equations written for NET in section 2.

The underlying principle is that the continuum is occupied at each point by all constituents of the mixture, with a motion given by the point mapping

x = φ^{α} (X^{α}, t)

(51)

with $X^{α}$ the reference position of constituent $α$ . The mass balance for constituent $α$ writes

\frac{D^{α} ρ^{α}}{Dt} + ρ^{α} v^{α} = {\hat{ρ}}^{α}

(52)

introducing therein the material derivative of a function (the velocity) following the motion of constituent $α$ , elaborated as

\frac{D^{α} f}{Dt} = \frac{\partial f}{\partial t} + \nabla f . v^{α}

(53)

The source term ${\hat{ρ}}^{α}$ on the r.h.s. of equation (52) represents the mass supply to constituent $α$ due to chemical reactions amongst other constituents of the mixture. The density of the mixture is the sum of the partial densities (as in equation (7))

ρ = \sum_{α} ρ^{α}

(54)

The mixture velocity is the velocity of the center of mass analogous to the barycentric velocity introduced in equation (4). The sum of the densities of all supply terms vanishes

\sum_{α} {\hat{ρ}}^{α} = 0

(55)

Thus, mixture theory is a closed system approach, in line with the strategy of NET. The conservation of linear momentum for the constituent $α$ including inertia terms writes

ρ^{α} \frac{D^{α} v^{α}}{Dt} = \nabla . T^{α} + ρ^{α} b^{α} + {\hat{p}}^{α}

(56)

Here, $T^{α}$ is the partial Cauchy stress, $b^{α}$ represents body forces acting on constituent $α$ , and ${\hat{p}}^{α}$ is the momentum supply to constituent $α$ from the rest of the mixture, deemed as an irreversible contribution. This last term is not present in the balance law of growing solid bodies (see equation (32)), but can be added, as done in Epstein and Maugin [13]. The mixture satisfies as a whole the overall balance of linear momentum

ρ \frac{D v}{Dt} = \nabla . T + ρ b

(57)

This entails by comparing with the summation of the partial balance laws, equation (56), the mixture stress and body force given successively by

T = \sum_{α} T^{α} - ρ^{α} u^{α} \otimes u^{α}, b = \frac{1}{ρ} \sum_{α} b^{α}

(58)

with the relative velocity $u^{α} : v^{α} - v$ therein; note that term $ρ^{α} u^{α}$ is identical to the flux given in equation (4). The partial momentum supply must satisfy the equality

\sum_{α} {\hat{p}}^{α} + {\hat{ρ}}^{α} u^{α} = 0

(59)

The conservation of angular momentum for each constituent writes

(T^{α})^{T} - T^{α} = {\hat{M}}^{α}

(60)

with ${\hat{M}}^{α}$ an antisymmetric tensor the dual vector of which is the angular momentum supply to constituent $α$ from the rest of the mixture. The sum of these angular momenta vanishes for a nonpolar mixture. The overall stress is thus not the simple sum of the partial stresses, which renders the interpretation of such partial stresses in mixture theories complicated.

The conservation of the internal energy $ε^{α} = ε^{α} (x, t)$ for any partial constituent writes

ρ^{α} \frac{D^{α} ε^{α}}{Dt} = T^{α} : L^{α} - \nabla . q^{α} + ρ^{α} r^{α} + {\hat{ε}}^{α}

(61)

with the partial velocity gradient $L^{α} : \nabla v^{α}$ , $q^{α}$ the heat flux, $r^{α}$ the heat supply from external sources, and ${\hat{ε}}^{α}$ the energy supply from the rest of the mixture. Comparing to the overall mixture energy balance law

ρ \frac{D ε}{Dt} = T : L - \nabla . q + ρ r

(62)

this leads to’ to the identification of the internal energy, heat flux, and heat supply of the mixture

\begin{matrix} ε = \frac{1}{ρ} \sum_{α} ρ^{α} (ε^{α} + \frac{1}{2} u^{α} . u^{α}) \\ q = \sum_{α} q^{α} - {{(T^{α})}^{T} - ρ^{α} (ε^{α} + \frac{1}{2} u^{α} . u^{α})} u^{α}, r = \frac{1}{ρ} \sum_{α} ρ^{α} (r^{α} + b^{α} . u^{α}) \end{matrix}

(63)

These complicated relations show that the heat flux is not the mere summation of the partial heat fluxes of the constituents, since it also includes diffusion terms.

The energy supply terms satisfy the equations

\sum_{α} {\hat{ε}}^{α} + {\hat{p}}^{α} . u^{α} + {\hat{ρ}}^{α} (ε^{α} + \frac{1}{2} u^{α} . u^{α}) = 0

(64)

The entropy inequality holds for the whole mixture – and not for each constituent separately – in terms of the partial entropy $η^{α} = η^{α} (x, t)$ of constituent $α$ per unit mass

\sum_{α} ρ^{α} \frac{D^{α} η^{α}}{Dt} + \nabla . (\frac{q^{α}}{θ}) - \frac{ρ^{α} r^{α}}{θ} + {\hat{ρ}}^{α} η^{α} \geq 0

(65)

The entropy inequality of the whole mixture per unit mass writes

\frac{D η}{Dt} + \nabla . (\frac{h}{θ}) - \frac{ρ s}{θ} \geq 0

(66)

with the entropy, heat flux and entropy supply of the mixture given successively by

η = \frac{1}{ρ} \sum_{α} ρ^{α} η^{α}, h = \sum_{α} q^{α} + ρ^{α} θ η^{α} u^{α}, s = \frac{1}{ρ} \sum_{α} ρ^{α} r^{α}

(67)

The entropy inequality is finally written in terms of the Helmholtz specific free energy density for each constituent,

\begin{matrix} ψ^{α} = ε^{α} - θ η^{α} \to \\ - \sum_{α} (\frac{D^{α} ψ^{α}}{Dt} + η^{α} \frac{D^{α} ψ^{α}}{Dt}) + T^{α} : L^{α} - {\hat{p}}^{α} . u^{α} - \frac{1}{θ} q^{α} . \nabla θ - {\hat{ρ}}^{α} (ψ^{α} + \frac{1}{2} u^{α} . u^{α}) \geq 0 \end{matrix}

(68)

This formulation of the residual dissipation is suitable to formulate constitutive relations.

4. Energy and entropy balance laws for open systems

4.1. Balance of energy

The first law of thermodynamics for an open system has to account for the contributions to kinetic and internal energies due to the incoming material. The energy balance in the actual configuration expresses for a material domain as

\begin{matrix} E + K & = (\int_{Ω_{t}} ρ (e + k) dx) \\ \Rightarrow \frac{D}{Dt} (E + K) = \int_{Ω_{t}} ρ \frac{D}{Dt} (e + \frac{1}{2} v^{2}) dx + \int_{Ω_{t}} (e + \frac{1}{2} v^{2}) (π + \nabla . m) dx = \\ = \int_{Ω_{t}} {(f . v + r) + π (e + \frac{1}{2} v^{2})} dx + \int_{\partial Ω_{t}} n . {σ . v + m (e + \frac{1}{2} v^{2}) - q} ds \end{matrix}

(69)

with E the total internal energy with specific density e, $K : \int_{Ω_{t}} ρ \frac{v^{2}}{2} dx$ the kinetic energy with specific density $k : ρ \frac{v^{2}}{2}$ , r the volumetric heat supply (generated by growth), and $q$ the heat flux across $\partial Ω_{t}$ . The material derivative of the energy functional writes

\frac{D}{Dt} (\int_{Ω_{t}} ρ (e + k) dx) = \int_{Ω_{t}} (ρ \frac{d (e + k)}{dt} + (e + k) (π - \nabla . J)) dx

(70)

The rate form of the global energy balance is thus obtained as

\overset{\cdot}{E} + \overset{\cdot}{K} = P_{e} + P_{m} + Q \circ, \to {\begin{matrix} P_{e} = \int_{Ω_{t}} f . v dx + \int_{\partial Ω_{t}} (n . σ . v) ds \\ Q \circ = \int_{Ω_{t}} rdx - \int_{\partial Ω_{t}} q . n ds \\ P_{m} = \int_{Ω_{t}} A_{j} R_{j} dx + {- \int_{\partial Ω_{t}} J . n (e + k) ds + \int_{Ω_{t}} {π (e + k) - \nabla . (μ_{k} J_{k})} dx} \end{matrix}

(71)

The quantities $P_{e}$ and $Q \circ$ denote the power of external forces and quantity of heat respectively. In comparison to the classical formulation for bodies with constant mass, it involves an additional power due to mass production and exchanges, quantity $P_{m}$ .

The overall mass flux is the sum of the partial fluxes $J_{k}$ of the diffusing chemical species, viz.

J = \sum_{k} J_{k}

(72)

Previous global writing of the energy balance can be rewritten

\begin{matrix} \int_{Ω_{t}} (ρ \frac{d (e + k)}{dt} + (e + k) (π - \nabla . J)) dx = \\ = \int_{Ω_{t}} (f . v + r) dx + \int_{\partial Ω_{t}} (- μ_{k} J_{k} . n - q . n + n . σ . v) ds + \int_{Ω_{t}} A_{j} R_{j} dx + \int_{Ω_{t}} π (e + k) dx - \int_{\partial Ω_{t}} J . n (e + k) ds \end{matrix}

(73)

with A_j a thermodynamic affinity conjugated to the chemical reaction rate R_j. Localization of this balance law delivers its strong form

ρ \frac{d (e + k)}{dt} = (f . v + r) - \nabla . {q - σ . v} - \sum_{k} J_{k} . \nabla (e + k) + A_{j} R_{j} - \nabla . (μ_{k} J_{k})

(74)

The balance of momentum, equation (34), written in virtual power form gives in the specific case of real fields the balance of kinetic energy

ρ v . \frac{d v}{dt} \equiv ρ \frac{dk}{dt} = f . v + v . \nabla . σ - (J . \nabla) v . v = f . v + \nabla . (σ . v) - σ : \nabla v - J . \nabla k

(75)

Using this expression of the material derivative of the kinetic energy delivers the balance of energy

ρ \frac{d (e + k)}{dt} = (f . v + r) - \nabla . q - σ : \nabla v - \nabla . σ . v - J . \nabla (e + k) + A_{j} R_{j} - \nabla . (μ_{k} J_{k})

(76)

The kinetic energy therein is given as the sum of the partial contributions

k \equiv \sum_{k} M_{k} n_{k} {v_{k}}^{2}

(77)

A simplified form of the balance of energy is then obtained using the previous balance of kinetic energy as

ρ \frac{de}{dt} = {r - \nabla . q} + σ : \nabla v + {A_{j} R_{j} - \nabla . (μ_{k} J_{k}) - J . \nabla e}

(78)

Coming back to the interpretation based on the existence of an external reservoir of nutrients, the general writing of equations (46) and (47) entails as a specific case the global form of the energy balance

\int_{Ω_{t}} (ρ \frac{\partial E}{\partial t}) dx + \frac{DE [R_{t}]}{Dt} + δ E^{ex} = 0

(79)

with $E^{ex}$ the energy exchanged with the reservoir; accordingly, the sum of the local variation of energy of the growing system, the variation of the energy of the reservoir and the energy fluxes vanishes. This interpretation leads to the following writing of the first principle of thermodynamics:

\overset{\cdot}{E} [Ω_{t} + R_{t}] + \overset{\cdot}{K} [Ω_{t} + R_{t}] = P_{e} + P_{ϕ} + Q \circ

(80)

wherein the dependency on the domain has been indicated under brackets. The power due to mass production and exchanges is incorporated on the left-hand side, whereas the right hand side remains classical (the same as for a non-growing solid body):

\begin{matrix} P [R_{t}] = - \int_{Ω_{t}} ((- v + J) . \nabla e + e π) dx + \int_{Ω_{t}} A_{j} R_{j} dx + \int_{Ω_{t}} {π (e + k) - \nabla . (μ_{k} J_{k})} dx \\ J = \sum_{k} J_{k} \\ P^{ex} = \int_{\partial Ω_{t}} J . n (e + k) ds \\ \overset{\cdot}{E} [Ω_{t} + R_{t}] \equiv \int_{Ω_{t}} (ρ \frac{\partial E}{\partial t}) dx - P [R_{t}] + P^{ex} \end{matrix}

(81)

with $P^{ex}$ the power of exchange terms. The contributions $P [R_{t}], P^{ex}$ are viewed as powers of internal forces.

4.2. Entropy balance

The balance of total entropy $S : \int_{Ω_{t}} ρ sdx$ writes

\frac{d}{dt} \int_{Ω_{t}} ρ sdx + \int_{\partial Ω_{t}} n . J sd σ + \int_{\partial Ω_{t}} n . \frac{q}{T} d σ \geq \int_{Ω_{t}} \frac{r}{T} dx + \int_{Ω_{t}} π sdx

(82)

Introducing therein the specific entropy density $s = s (x, t)$ . Inequality (82) traduces the fact that the difference between the material derivative of the total entropy and the reversible entropy fluxes due to heat and mass exchanges is not compensated, thus it represents an irreversible contribution being fully dissipated. Previous inequality rewrites after few transformations

\frac{d}{dt} \int_{Ω_{t}} ρ sdx \geq {\int_{Ω_{t}} (- \nabla . (q / T) + r / T) dx} - \int_{Ω_{t}} \nabla . (J s) d σ + \int_{Ω_{t}} π sdx

(83)

We further express the left hand side relying on the derivative of an integral based on a specific density using equation (27),

\begin{matrix} \frac{d}{dt} \int_{Ω_{t}} ρ sdx = \int_{Ω_{t}} (ρ \frac{ds}{dt} + s (π + \nabla . m)) dx \equiv \int_{Ω_{t}} (ρ \frac{ds}{dt} + s (π - \nabla . J)) dx \\ \Rightarrow \int_{Ω_{t}} (ρ \frac{ds}{dt} + s (π - \nabla . J) + \nabla . (J s)) dx - {\int_{Ω_{t}} (- \nabla . (q / T) + r / T) dx} \geq \int_{Ω_{t}} π sdx \end{matrix}

(84)

This entails the following expression of the local dissipation:

ρ \frac{ds}{dt} + \nabla . (q / T) - r / T + J . \nabla s \geq 0

(85)

The heat source is next extracted from the energy balance and inserted into previous expression of the local dissipation, using the free energy density $ψ : e - Ts$ , thus yielding

\begin{matrix} r = ρ \frac{de}{dt} + \nabla . q - σ : \nabla v + J . \nabla e - A_{j} R_{j} + \nabla . (μ_{k} J_{k}) \equiv \\ ρ (\overset{\cdot}{ψ} + \overset{\cdot}{T} s + T \overset{\cdot}{s}) + \nabla . q - σ : \nabla v + J . \nabla e - A_{j} R_{j} + \nabla . (μ_{k} J_{k}) \end{matrix}

(86)

This entails after straightforward calculations the expression of the residual dissipation based on the free energy density

- (q / T) . \nabla T - J . \nabla ψ - s J . \nabla T - {ρ (\overset{\cdot}{ψ} + \overset{\cdot}{T} s) - σ : \nabla v - A_{j} R_{j} + \nabla . (μ_{k} J_{k})} \geq 0

(87)

Isolating the thermal dissipation gives

- (q / T) . \nabla T - (s J) . \nabla T \geq 0

(88)

Thus the residual dissipation becomes

- J . \nabla ψ - {ρ (\overset{\cdot}{ψ} + \overset{\cdot}{T} s) - σ : \nabla v - A_{j} R_{j} + \nabla . (μ_{k} J_{k})} \geq 0

(89)

The comparison with the entropy inequality obtained in Section 2 evidences the additional contributions $ρ s \overset{\cdot}{T}$ (which vanishes since we considered isothermal processes) and the irreversible flux of the potential (first term on the l.h.s.).

We next introduce the mass balance for each chemical constituent, distinguishing the transfer of species across the boundary ${\overset{\cdot}{n}}_{k^{ex}}$ from the species produced within the volume by chemical reactions, denoted by the rates ${\overset{\cdot}{n}}_{k^{ch}}$ :

\begin{matrix} R_{j} = {\overset{\cdot}{ξ}}_{j} defined as the rate of advancement of the k - chemical reaction \\ {\overset{\cdot}{n}}_{k}^{ch} = υ_{km} {\overset{\cdot}{ξ}}_{m}, {\overset{\cdot}{n}}_{k}^{ex} = \nabla . J_{k} \end{matrix}

(90)

Thereby, the rate of moles being exchanged obeys a stoichiometry described by the coefficients $υ_{k m}$ and further satisfies a conservation law written as Fick’s law (last equality in equation (90)). The thermodynamic affinity, quantity

- A^{i} : \frac{\partial μ}{\partial ξ^{i}} = \frac{\partial ψ}{\partial ξ^{i}}

(91)

represents a non-equilibrium force driving the i^th chemical reaction; it vanishes at equilibrium.

The free energy density incorporates the temperature, the elastic part of the total strain, and the exchange mole number as arguments; its material derivative writes

ψ \equiv ψ (T, ε^{e}, {\overset{\cdot}{n}}_{k}^{ex}) \to \overset{\cdot}{ψ} = \partial_{T} ψ . \overset{\cdot}{T} + \partial_{ε^{e}} ψ : {\overset{\cdot}{ε}}^{e} + \partial_{n_{k}^{ex}} ψ . {\overset{\cdot}{n}}_{k}^{ex}

(92)

Observe the consistency of the arguments of the free energy density, which depends on the elastic (reversible part) of the total strain and the exchange part of the total mole number. Inserting previous expression into the dissipation delivers

\begin{matrix} - \sum_{k} J_{k} . \nabla ψ - \sum_{k} J_{k} . \nabla μ_{k} - ρ (\partial_{T} ψ + s) \overset{\cdot}{T} - {ρ \partial_{n_{k}^{ech}} ψ . {\overset{\cdot}{n}}_{k}^{ex} + \sum_{k} μ_{k} \nabla . (J_{k}) - A_{j} {\overset{\cdot}{ξ}}_{m}} + \\ + {σ : D (v) - ρ \partial_{ε^{e}} ψ : {\overset{\cdot}{ε}}^{e}_{i}} \geq 0 \end{matrix}

(93)

Equalities (90) allow isolating the following contribution from the dissipation in equation (93):

ρ \partial_{n_{k}^{ech}} ψ . {\overset{\cdot}{n}}_{k}^{ex} + \sum_{k} μ_{k} \nabla . (J_{k}) - A_{j} {\overset{\cdot}{n}}_{k}^{ch} = ρ \partial_{n_{k}^{ech}} ψ . {\overset{\cdot}{n}}_{k}^{ex} + μ_{k} {\overset{\cdot}{n}}_{k}^{ex} - A_{j} υ_{k m} {\overset{\cdot}{ξ}}_{m}

(94)

We thus obtain the local dissipation

\begin{matrix} - \sum_{k} J_{k} . \nabla ψ - \sum_{k} J_{k} . \nabla μ_{k} - ρ (\partial_{T} ψ + s) \overset{\cdot}{T} - {ρ \partial_{n_{k}^{ex}} ψ . {\overset{\cdot}{n}}_{k}^{ex} + μ_{k} {\overset{\cdot}{n}}_{k}^{ex} - A_{j} {\overset{\cdot}{ξ}}_{m}} + \\ + {σ : D (v) - ρ \partial_{ε^{e}} ψ : {\overset{\cdot}{ε}}^{e}_{i}} \geq 0 \end{matrix}

(95)

The Coleman–Noll procedure then leads in a straightforward manner to the state laws (completed by the definition of the thermodynamic affinity, equation (91))

| \begin{matrix} \partial_{T} ψ + s = 0 \\ D = D^{e} + D^{irr} \to σ : D^{e} - ρ \partial_{ε_{i}} ψ . {\overset{\cdot}{ε}}_{i} = 0 \Rightarrow σ : {\overset{\cdot}{ε}}^{e} - ρ \partial_{ε_{i}} ψ . {\overset{\cdot}{ε}}_{i} = 0 \\ ρ \partial_{n_{k}^{ex}} ψ + μ_{k} = 0, \end{matrix}

(96)

This entails the residual dissipation given by

σ : D^{irr} - J . \nabla ψ - \sum_{k} J_{k} . \nabla μ_{k} + A_{j} {\overset{\cdot}{ξ}}_{m} \geq 0

(97)

The second and third terms on the left hand side of previous inequality combine accounting for the state laws into a single contribution, thus

σ : D^{irr} - J . \nabla (ψ - ρ \partial_{n_{k}^{ex}} ψ) + A_{j} {\overset{\cdot}{ξ}}_{m} \geq 0

(98)

The contribution $\sum_{k} J_{k} . \nabla (ψ - ρ \partial_{n_{k}^{ex}} ψ)$ to the local dissipation expresses the convection of the gradient of the potential $ψ - ρ \partial_{n_{k}^{ex}} ψ$ , built as the Legendre transform of the free energy density, and identified to the density of the grand potential [36]

ω [θ, F, μ] : e (F, s, n) - θ (F, s, n) s - μ_{k} (F, s, n) . n_{k}

(99)

The last term in the residual dissipation, equation (98), the scalar $A_{j} {\overset{\cdot}{ξ}}_{m}$ , represents the dissipation due to chemical reactions within the growing body. The grand potential is the full Legendre transform of the internal energy density, and it leads to the zero potential which has been shown to be the adequate thermodynamic potential for growing solid bodies, leading to a universal extremum principle [36].

From the Gibbs and Gibbs–Duhem relation [39], successively the equalities

\begin{array}{l} e (F, s, n_{k}) = Τ (F, s, n_{k}) : F + θ (F, s, n_{k}) s - μ_{k} (F, s, n_{k}) n_{k}, \\ F : d T + s d θ + n_{k} d μ_{k} = 0 \end{array}

(100)

the differential of the grand potential is obtained from a straightforward calculation as

d ω = - sd θ + T : d F - n_{k} d μ_{k} \equiv T : d F + F : d T \to ω = ω (θ, F, μ_{k}),

(101)

resulting in the thermodynamic relations corresponding to the state laws

\frac{\partial ω}{\partial θ} = - s; \frac{\partial ω}{\partial F} = T; \frac{\partial ω}{\partial μ_{k}} = - n_{k}

(102)

One may then rewrite the residual dissipation, inequality (98), as the inequality

σ : D^{irr} - J . \nabla ψ + A_{j} {\overset{\cdot}{ξ}}_{m} \geq 0

(103)

In comparison to NET, the residual dissipation appearing in the thermodynamics of growing solid bodies involves an additional contribution $- \sum_{k} J_{k} . \nabla ω$ traducing the irreversible transport of the gradient of the free energy density by the mass diffusion flux. This contribution reflects the fact that the representative volume element is truly open, which entails a mass balance which is not a conservation law due to the occurrence of mass sources and fluxes. These terms are not present in NET, since mass fluxes occur within the RVE (and mutually compensate), and not at the level of inter-RVE interactions.

Evolution laws for the internal variables can further be obtained relying on a dissipation potential

Ω \equiv Ω (σ, \nabla ω, A_{j}) \to D^{irr} = \frac{\partial Ω}{\partial σ}, - J = \frac{\partial Ω}{\partial \nabla ω}, {\overset{\cdot}{ξ}}^{k} = \frac{\partial Ω}{\partial A^{k}}

(104)

Together with the state laws in equation (96), they build a closed system of differential equations in view of numerical simulations of growing solid bodies.

5. Conclusion

In the present contribution, we have exposed the balance laws of solid continuum bodies subjected to volumetric growth, up to the local dissipation. We have compared two possible thermodynamic frameworks for the description of growth phenomena: non-equilibrium thermodynamics (NET) and the mechanics and thermodynamics of open systems. While both frameworks have the capability of including multiphysical couplings such as interactions between transport of heat and mass (by diffusion, chemical reactions) and mechanics, NET is basically a closed system approach describing interactions between multiple components, with however no net mass production and diffusion (all individual contributions mutually compensate when summed up) at the level of a representative volume element. The theory of mixtures which has been shortly reviewed fits within this framework and has been applied over the last three decades to the analysis of tissue growth and remodeling.

On the contrary, open system thermodynamics relies on a mass balance incorporating mass sources and fluxes viewed as hidden contributions, which entails specific balance laws incorporating these additional terms. As a consequence, we have obtained a dissipation inequality incorporating the gradient of the density of the grand potential carried by the irreversible mass flux through the system boundary; this additional term is not present in the residual dissipation evaluated in NET.

The profound reason of the differences between open system thermodynamics and NET lies in the fact that the source of mass is explicitly taken into account in NET, whereas it is postulated in continuum growth theories in an ad hoc manner as descriptive of new nutrients feeding the system at each material point. As the present contribution shows, both approaches can nevertheless be reconciled by interpreting the balance laws in open systems thermodynamics in terms of an external reservoir of nutriments, so that the overall thermodynamic system becomes closed.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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