Thermomechanics of material growth and remodeling in uniform bodies based on the micromorphic theory

Thermomechanics of material growth and remodeling in uniform bodies based on the micromorphic theory

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Thermomechanics of material growth and remodeling in uniform bodies based on the micromorphic theory Mohammadjavad Javadi, Marcelo Epstein, Mohsen Asghari PII: DOI: Reference:

S0022-5096(20)30140-X https://doi.org/10.1016/j.jmps.2020.103904 MPS 103904

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Journal of the Mechanics and Physics of Solids

Received date: Revised date: Accepted date:

4 August 2019 3 January 2020 11 February 2020

Please cite this article as: Mohammadjavad Javadi, Marcelo Epstein, Mohsen Asghari, Thermomechanics of material growth and remodeling in uniform bodies based on the micromorphic theory, Journal of the Mechanics and Physics of Solids (2020), doi: https://doi.org/10.1016/j.jmps.2020.103904

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Thermomechanics of material growth and remodeling in uniform bodies based on the micromorphic theory Mohammadjavad Javadia,∗, Marcelo Epsteinb , Mohsen Asgharia a

b

Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran Department of Mechanical and Manufacturing Engineering, University of Calgary, Canada

Abstract Based on the micromorphic theory, a novel mathematical formulation for the mechanical modeling of material growth and remodeling processes in finite deformation is developed. These two processes have an important significance in evolution of living tissues. The presented formulation incorporates both the volumetric growth and mass flux phenomena into the modeling with the aid of the micromorphic theory’s capability to include internal structures in materials. The balance equation of microinertia is presented which reveals the importance of rearrangement and alteration of microstructure in the micromorphic material growth. Within the framework of material uniformity, the evolution laws are derived in terms of first-order differential equations for a set of material transplants which satisfy the formal restrictions arising from micromorphic material symmetries, and are consistent with the second principle of thermodynamic. The set of the micromorphic Eshelby and Mandel stress tensors as driving forces for the local rearrangement of material inhomogeneities is also determined. Keywords: the non-classical continuum mechanics; the micromorphic theory; microstructure; growth; remodelling; the Mandel stress; material uniformity

∗

Corresponding author Email address: [email protected] (Mohammadjavad Javadi )

Preprint submitted to Journal of the Mechanics and Physics of Solids

February 13, 2020

1. Introduction The concept of material evolution laws, as manifested by plasticity, material growth and remodeling theories, is fraught with problems, even in the case of first-order materials. This concept is further complicated in the framework of the non-classical continuum theories, especially by the micromorphic theory (Eringen and Suhubi, 1964; Suhubl and Eringen, 1964; Eringen, 1999, 1966; Kafadar and Eringen, 1971). This is because of the incorporation of the internal characteristic lengths of the microstructure into the mechanical analysis, which are of high importance in biological tissues (Eringen and Kafadar, 1976; Eringen, 1999; Mindlin, 1964). On the other hand, the second-order effects of the non-classical theories can be significant when diffusive phenomena are allowed. The origin of the theories of materials with internal structure is the work of Cosserat and Cosserat (1909). In this theory, the basic physical idea of a continuum is endowed with the extra kinematic degrees of freedom afforded by a triad of rigid vectors or directors attached at each material point. This medium is called “micropolar” in Eringen’s terminology. Upon extending the theory in various ways to deformable vectors, this medium with more degrees of freedom is called “micromorphic”. A geometrical description of micromorphic media in terms of non-holonomic frame bundles of second order has been discussed in Epstein and de Leon (1998). The theory of continuous distributions of inhomogeneities was also developed for micromorphic media by Epstein and De Le´on (1996). The geometric description of the generalized continuum theories such as the micromorphic theory is vital in order to obtain the material evolution laws. On the other hand, the mechanical analysis of biological tissues can be much better accomplished by utilizing non-classical continuum theories, especially by the micromorphic theory (Eringen,

2

1999; Eringen and Suhubi, 1964; Suhubl and Eringen, 1964). This is due to the fact that the micromorphic theory within the non-classical framework incorporates the internal traits of lengths and microstructure into the mechanical analysis. These are of a great significance in biological tissues (Goda et al., 2016; Akhtar et al., 2011; Sack et al., 2016). The physical anelastic phenomena of growth, remodeling, aging and morphogenesis are accompanied in biological tissues by a change in internal structure subjected to configurational forces which affect their mechanical behavior. The geometrical description of such physical phenomena plays a remarkable role in their mechanical understanding. One of the exact mathematical classifications of these processes for first order materials can be found in Epstein (2009, 2015). In biological tissue the mass variation is either due to the internal mass generation, or mass flux at the surface, or both. Some remarkable studies on the volumetric growth process have been accomplished with only consideration of mass generation, including studies by Lubarda and Hoger (2002); Rodriguez et al. (1994); Epstein and Maugin (2000a); Li et al. (2011). Epstein and Maugin (2000b) showed that classical continuum mechanics is incapable of predicting and illuminating the mass-diffusive effects in the growth process. They were also successful in taking into consideration the mass diffusive effects into the growth formulation by utilizing the non classical continuum theory of the second order gradient theory. Ciarletta et al. (2012) presented a similar formulation that additionally takes into account the species transportation. The important of growth and remodeling processes in mechanical modelling of biological tissues have attracted many researchers in recent years to predict the mechanical behaviour of tissues (Goriely et al., 2019; Grillo et al., 2019; Tartibi et al., 2019; Grillo et al., 2018; Oomen et al., 2018; Du et al., 2018; Wang et al., 2018; Weickenmeier et al., 2017; Kuhl, 2014; Eriksson et al., 2014; Ambrosi et al., 2011). 3

Garikipati et al. (2004) discussed the mass flux phenomenon within the framework of the classical continuum mechanics utilizing the mixture theory of Truesdell and Toupin (1960). However, it leads to some drawbacks such as the appearance of partial stresses or mass exchanges between the single phases (Ambrosi et al., 2010; Ciarletta et al., 2012). One of the non-classical continuum theories which is capable of considering the mass flux constitutive law is the micromorphic theory. The mechanism of mass transfer in micromorphic media ˙ is more complicated than the second gradient theory (ElZanowski and Epstein, 1992; Epstein, 1999; Epstein and Maugin, 2000b; Ciarletta et al., 2012) due to its internal structures. Both micro and macro elements are affected under the mechanism of mass transfer in which microstructure should also satisfy the balance of microinertia in the micromorphic theory. The aim of this study is to present a formulation for growth and remodeling in biological tissues based on the micromorphic theory to consider the effects of mass flux and the microstructure of biological tissues. The relative merit of the micromorphic theory with respect to other non-classical continuum theories is its degree of freedom and the role of microinertia for the rearrangement of the micro structure. To this end, we first discuss the geometrical description of the finite micromorphic theory. Secondly, the balance equations of mass and microinertia, linear and angular momenta, and internal energy together with the entropy inequality in the presence of growth and mass diffusion are derived from a principle of invariance under general observer transformation. Thirdly, within the framework of material uniformity, by applying the derived entropy inequality, the hyper-elastic constitutive equations with respect to both reference and current configurations are derived. We then develop the evolution laws for the micromorphic materials by enforcing the Principle of Maximum Dissipation on the micromorphic dissipation equation. Next, we address how 4

the evolution laws are a function of the Eshelby and Mandel stress tensors as driving forces for the local rearrangement of the micromorphic material inhomogeneities. We also develop a mathematical restriction on the growth and remodeling evolution laws which include the change of the reference configuration and material symmetry consistency.

2. Micromorphic media In Eringen’s terminology (Eringen, 1999) a micromorphic continuum consists of an ordinary body B to each of whose points a triad of deformable linearly independent directors is attached. This concept generalizes the pioneering idea of Cosserat and Cosserat (1909), who considered the triad to be orthonormal and should remain as such throughout the process of deformation. In order to model materials with internal structure, granular materials and biological tissues in which cells are embedded within an extra-cellular matrix, the application of these sophisticated models of generalized continua is inevitable. In this treatment, the body B is referred to as the macromedium or matrix and the collection of deformable triads as the micromedium, each triad being the representative of an affinely deformable grain. A deformation of a micromorphic medium is described by a deformation of the macromedium as well as a concomitant field of linear transformations of the local triads. From another viewpoint, the micromorphic medium is represented as an ordinary body B encompassing a continuous collection of particles undergoing strictly affine deformations only (Slawianowski, 1974). In the special case when the particles behave rigidly, the mentioned theory collapses to a Cosserat medium. In this case each material point is named as a pseudo-rigid body (Cohen and Muncaster, 1988) or Cosserat point (Rubin, 1985).

5

2.1. Kinematics Following the usual notational conventions, an element in a reference configuration of a micromorphic medium is identified by 12 parameters, namely, the coordinates X I (I = 1, 2, 3) of a point in the macromedium and the three referential directors Dα (α = 1, 2, 3) attached thereat. A motion of the micromorphic medium is, accordingly, represented by 12 (smooth) functions κi , KIi as follows (Epstein and Elzanowski, 2007) xi = κi (X 1 , X 2 , X 3 , t)

i = 1, 2, 3

(1)

α, i, I = 1, 2, 3

(2)

and diα = KIi (X 1 , X 2 , X 3 , t)DαI

where dα are the current directors and the summation convention is enforced. If EI and ei denote, respectively, the referential and spatial covariant coordinate bases, the functions KIi represent the components of a (two-point) tensor K as K = KIi ei ⊗ EI .

(3)

The linear operator K is precisely the affine deformation associated with the directors attached to the macromedium point X = (X 1 , X 2 , X 3 ). The deformation gradient in the case of a micromorphic medium, that is, the derivative of the micromorphic deformation, is a rather more complicated entity than its counterpart for an ordinary medium. In terms of components in a coordinate chart, the deformation gradient i consists of the deformation itself κi , KIi supplemented with the derivatives κi,I , KI,J , where

commas indicate partial differentiation. In other words, at each point X the deformation

6

i gradient consists of the quantities κi , KIi , FIi , KI,J , where FIi = κi,I can be recognized as the

ordinary deformation gradient of the macromedium. i Let us consider the effect brought about on the quantities κi , KIi , FIi , KI,J by a change of

reference configuration. Using a circumflex accent to denote quantities in the new reference configuration, the change of variables is given in terms of some functions ˆA = X ˆ A (X I ), X

(4)

ˆA = K ˆ A (X I ). K I I

(5)

and

The referential directors are related by ˆA = K ˆ A DI . D α I α

(6)

Consequently, taking into account Equation (2), the present configuration component functions with respect to the two given reference configurations are related by 
 i ˆ ˆ i (X(X), ˆ ˆ A (X) . κi (X, t), KIi (X, t) = κ ˆ (X(X), t), K t) K A I

(7)

The deformation gradients relative to the two reference configurations are related by   i ˆ i Fˆ B K ˆA + K ˆi K ˆA , ˆ Ai K ˆ IA , Fˆ i Fˆ A , K )= κ ˆi, K (κi , KIi , FIi , KI,J A I A,B J I A I,J

(8)

where an even more compact notation has been used for clarity. The matrix with components FˆIA is given by the derivatives ˆ A. FˆIA = X ,I 7

(9)

Thus, while both the macroscopic deformation gradient and the microdeformation themselves are composed as ordinary tensors, this is not the case for the gradient of the microdeformation, whose composition law under a change of reference configuration is more involved and will play an important role later in the construction of possible evolution laws of growth and remodeling in micromorphic media.

3. Balance equations 3.1. The master energy balance In deriving the balance equations for a micromorphic medium, we will follow the general line of thought of Green (1965). On the other hand, we incorporate the additional contributions associated with mass growth into the formulation. In general, there are two types of the growth contributions. First, the newly added material of growth phenomenon may have the same properties as the material at the same point. This type of sources is called compliant, according to the terminology of (Epstein and Maugin, 2000b). Normally, this type of contributions balances out and does not appear in the local form of the balance equations. Second, the new material may have sources that are different from the pre-grown material. This type of sources is called non-compliant and we denote the corresponding terms in the formulation with an overbar. They appear as extra contributions in the local form of balance equations. This line of thought, as originally portrayed in Noll (1974) and Green and Rivlin (1964), is based on the postulate of invariance of the energy balance under arbitrary changes of observer. Therefore, the total kinetic energy of the micromorphic medium occupying a

8

spatial volume ω is proposed as follows Z   1 1 T = ρv.v + ρνiν T + pd .v dω, 2 ω 2

(10)

where v and ρ are the velocity and the instantaneous material spatial density, respectively. The symmetric second-order tensor i is the spatial microinertia tensor, ν is the micro gyration tensor and pd is the diffusive momentum per unit spatial volume (Javadi and Epstein, 2018). The superscript T denotes the transpose of a tensor. In equation (10), ν denotes the spatial director velocity tensor with the following components νji = K˙ Ii (K)Ij ,

(11)

where a superposed dot indicates the material derivative. Under suitable continuity conditions, the internal energy content U in the same spatial volume ω can be expressed in terms of energy per unit mass  as follows (Epstein and Maugin, 2000b) U=

Z

ρ dω.

(12)

ω

The entrant mass into the fixed spatial volume ω is of three kinds. The first kind is brought about by the mere fixity of this volume, which implies that there are material particles flowing into ω with the consequent addition of kinetic and internal energy. The second kind of entrant mass is that entailed by the possible creation or destruction of mass per unit volume, such as in processes of biological growth and resorption. The third kind is the exterior mass flux through the boundary. Under such conditions, the total contribution of the entrant masses to the rate of increase of energy in the volume ω with boundary ∂ω and

9

exterior unit normal n is Wentrant

Z    1  1 1 T T ¯ = π  + v.v + νiν + ν iν dω 2 2 2 ω Z  1    1 1 1 − (v.n) ρv.v + ρνiν T + pd .v + m  + v.v + νiν T da. (13) + 2 2 2 2 ∂ω

In equation (13), π is the volumetric mass sources per unit time associated with the micromorphic materials. Moreover, the volumetric non-compliant microinertia ¯i per unit time and per unit volume contributes to the total energy. In this same way we consider a vector field of mass flux based on the Cauchy tetrahedron argument as m = −m.n.

(14)

Additionally, we are not assuming any extra contributions associated to the rotational aspects of the entering mass, as would be legitimate to assume in the context of mixture theory. The remaining contributions to the energy balance equation are provided by the power of the non-mechanical (thermal) power sources as well as the external forces. The former is expressed as Wthermal =

Z

ρrdω +

Z

hda,

(15)

∂ω

ω

where r and h are, respectively, the radiation heat source and the conductive heat flux. The heat flux vector q can be obtained as h = −q.n. As far as the mechanical power is concerned, we assume the existence of body force b per unit mass, as well as micromedium body force (body couple) which, when integrated on each putative pseudo-rigid grain, results in a director-related (’couple-like’) tensor c, assumed to have been smeared over a unit mass of the macromedium. The power associated with diffusive momentum pd is added in the ¯ .v. form of b.pd and some other extra energy parts are contained in non-compliant term p 10

˜ Hence, the power of Corresponding to these body forces, we have surface tractions t, λ. these mechanical forces can be proposed as follow

Wmech =

Z

¯ .v)dω + (b.(ρv + pd ) + ρtr(cν) + p

ω

Z

˜ (t.v + λ.ν)da.

(16)

∂ω

Collecting all the above contributions, the energy balance stipulates that ∂ (T + U ) = Wentrant + Wthermal + Wmech . ∂t

(17)

3.2. Translation invariance Under a translational change of observer with relative velocity a, all quantities in the master energy balance equation remain unchanged except the velocity field itself which transforms according to v → v + a.

(18)

Collecting all the terms affecting the coefficient of a.a, we obtain the equation of balance of mass as ρ˙ + ρ∇.v = π − ∇.m.

(19)

The linear terms in a yield the balance of linear momentum equation in the following form ¯ + divσ − pd divv − m.∇v. ρv˙ + p˙ d = ρb + p

(20)

In the process of derivation of the above equation, we exploited the balance of mass (19) and invoked Cauchy’s tetrahedron argument σn = t, 11

(21)

in which σ represents Cauchy stress tensor. 3.3. Rotation invariance Under a rotational change of observer with uniform rigid body angular velocity Ω, all quantities in the master energy balance equation remain unchanged except the velocity field, the microgyration tensor, microinertia tensor, the body force, the body couple, the diffusive momentum, and the non-compliant momentum which transform according to v → v + Ωx,

(22a)

ν → ν + Ω,

(22b)

i˙ → i˙ + 2Ωi,

(22c)

b → b + 2Ωv,

(22d)

c → c + i(νΩ + Ων + ΩΩ),

(22e)

p˙ d → pd + Ωpd ,

(22f)

¯→p ¯ + 2Ωpd . p

(22g)

Collecting all terms affecting the coefficient of ΩT Ωx, we obtain pd = m. The linear terms in ν T Ω and

(23)

1 T Ω Ω yield the balance of microinertia as 2 ρ

Di − ρiν T − ρνiT = −i + ¯i, Dt 12

(24)

k where inl = inl ,k m and commas indicate partial differentiation with respect to the spatial

configuration. Similarly, equating the coefficients of linear terms in Ω leads to the balance of momentum moments respectively in the following form −∇λ + ˜i − ρc + ρω + s − σ = 0,

(25)

k n where, ω kl = iml (ν˙ m + νnk νm ) is the spin inertia tensor per unit of mass, s is the symmetric m . Furthermore, λ is the higher order second order microstress tensor and ˜iml = mk inl νn,k

micromorphic stress and is obtained by Cauchy’s tetrahedron argument as ˜ λn = λ

(26)

By substitution of equations (19), (20), (24) and (25) into (17), the energy balance equation can be rewritten as ρ˙ + pd .v˙ = ∇λ : ν + σ : (∇v)T + (s − σ) : (ν)T − divq − m.∇ + ρr + b.pd .

(27)

It should be noted that the symmetric microstress tensor s dose not produce any contribution on the energy balance equation (17), since Ω : s = 0 (Eringen, 1999). 3.4. The Clausius-Duhem inequality In the micromorphic framework, by defining the entropy content per unit mass as sρ , the global Clausius-Duhem inequality in the Eulerian veiwpoint can be obtained as follows D Dt

Z

Z  Z    ρr h ρsρ dω ≥ + πsρ dω + + msρ da, θ ω ω ∂ω θ

(28)

where θ is the absolute temperature. It is worth noting that the effects of non-compliant terms have been neglected in this equation (Epstein and Goriely, 2012). Hence, the local 13

form can be deduced as ρs˙ρ ≥

q ρr + m.∇sρ − div . θ θ

(29)

Considering the Helmholtz free energy per unit mass as ψρ =  − θsρ , the combination of Eqs. (27) and (29) yields an equivalent form of entropy inequality with diffusive effects in the micromorphic theory as below ˙ ≤ λ : ∇ν + σ : (∇v)T + (s − σ) : (ν)T − (v˙ + ∇ψρ + sρ ∇θ − b).m − 1 q.∇θ. ρ(ψ˙ ρ + sρ θ) θ (30) Equation (30) expresses the thermodynamical consistency for the energy dissipation within a growing micromorphic hyperelastic continuum which results in mass transport. Similarly, the Lagrangian version of the second law of thermodynamics can be presented as ˙ ≤ T I F˙ i + S I K˙ i + ΛIJ K˙ i − (v˙ i F i + ψρ,I + θ,I sρ − bi F i )M I − 1 Q ¯ I θ,I , (31) ρR (ψ˙ ρ + sρ θ) i I i I i I,J I I θ where ρR denotes the possibly time-varying mass density in the reference configuration, T is the macromedium Piola stress tensor, S is the micromedium Piola stress, Λ is the micromedium Piola hyperstress tensor, M is the mass flux vector in the reference configuration ¯ is the referential heat flux vector. The relations between the Lagrangian and Eulerian and Q quantities can be obtained as TiI = JF σil Fl−I ,

JF = detF,

(32a)

a −L −I −B , SiI = JF (sli − σil )Kl−I − JF λml i KB,L Fm Ka Kl

(32b)

−I ml −J ΛIJ i = JF λi Fm Kl ,

(32c)

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ρ = JF−1 ρR ,

(32d)

M I = JF Fi−I mi ,

(32e)

¯ I = JF F −I q i . Q i

(32f)

4. A micromorphic theory for growth and remodeling processes in uniform media 4.1. The theory of uniformity and the micromorphic archetype According to the seminal work of Noll (1967) a simple body is called materially uniform when at any of its points, the constitutent material is the same. Moreover, the notion of uniformity was cast into the framework of the micromorphic media by Epstein and De Le´on (1996); Epstein and de Leon (1998); Epstein and Elzanowski (2007). Considering the Helmholtz energy content at a specific point of a micromorphic medium, for any point throughout the body, the energy content per unit volume can be written as ψR (X, t) = ψR (F(X, t), K(X, t), ∇K(X, t); X, t),

(33)

where the energy content depends explicitly on point X. More specifically, if the micromorphic body is uniform, then the elastic potential energy depends on point X only through the uniformity fields P(X, t), Q(X, t) and R(X, t). These fields represent the implants from the micromorphic archetype to point X (see Figure 1). According to this ansatz, it can be concluded that i ψR (X, t) =ψR (FIi (X, t), KIi (X, t), KI,J (X, t); X, t) = JP−1 (X, t)ψ¯R (FIi (X, t)PαI (X, t), i I (X, t)), KIi (X, t)QIα (X, t), KI,J (X, t)PβJ (X, t)QIα (X, t) + KIi (X, t)Rαβ

15

(34)

Figure 1: Uniformity fields associated with reference configuration

where ψ¯R is the energy potential in the micromorphic archetype, ψR is the energy potential per unit reference volume and JP = det(P). 4.2. Thermodynamics consideration of the micromorphic theory We assume that the Helmholtz energy function per unit volume in the micromorphic archetype, ψ¯R and in the reference configuration, ψR are presented as i ψR (X, t) =ψR (FIi , KIi , KI,J , θ, θ,I ; X, t) = JP−1 ψ¯R (FIi PαI , i I KIi QIα , KI,J PβJ QIα + KIi Rαβ , θ, θ,I PαI ).

(35)

By abuse of notation, we do not indicate the arguments (X, t) of the tensor fields. Consequently, the material time derivative of equation (35) can be written as ψ˙ R = − JP−1 PI−α P˙αI ψ¯R + JP−1

¯ ∂ ψ¯R ˙ i P I + F i P˙ I ) + +J −1 ∂ ψR (K˙ i QI + K i Q˙ I ) ( F α I α I I α I α P i i ∂(FK PαK ) ∂(KK QK α)

∂ ψ¯ i i i I I P˙βJ QIα + KI,J PβJ Q˙ Iα + K˙ Ii Rαβ + KIi R˙ αβ ) (K˙ I,J PβJ QIα + KI,J i i K ∂(KM,N PβN QM + K R ) α K αβ ∂ ψ¯R ∂ψR ˙ (θ˙,I PαI + θ,I P˙αI ) + JP−1 θ. (36) + JP−1 K (∂θ,K Pα ) ∂θ

+ JP−1

16

Let us define the Helmholtz free energy per unit mass, ψρ as ∂  ψR  ρ˙ R ˙ ¯ P )ψR , ρR ψρ = ρR = ψ˙ R − ψR = ψ˙ R + (trL ∂t ρR ρR

(37)

¯ P is the “inhomogeneity velocity gradient” with respect to the micromorphic archetype. where L By utilizing equations (36) and (37) and then substituting into (31), we obtain    ∂ ψ¯R ∂ ψ¯R ∂ ψ¯R −1 −1 I I I i I I ˙i ˙ P − T Q + J − S F + J R i KI i α αβ I P P i i N M i K ∂(FKi PαK ) α ∂(KK QK ) ∂(K P Q + K R ) α α M,N β K αβ    ¯ ¯ ∂ ψR ∂ ψR i + JP−1 PβJ QIα − ΛIJ K˙ I,J + JP−1 Fi i i K i N M ∂(FKi PαK ) I ∂(KM,N Pβ Qα + KK Rαβ )   ∂ ψ¯R ∂ ψ¯R ∂ ψ¯R −1 i S I ˙ + JP−1 K Q KIi + J θ P + JP−1 ,I S,I γ α P i i i K N M K QK ) ∂(KM,N Pα Qγ + KK Rγα ) ∂(θ,K Pα ) ∂(KK α    ∂ ψ¯R ∂ ψ¯R i J i ˙ I + J −1 ˙I K P Q K + JP−1 I,J β α I Rαβ P i i K i N M i K ∂(KM,N PβN QM + K R ) ∂(K P Q + K R ) α α K αβ M,N β K αβ     ¯ ∂ ψ¯R 1 ¯I −1 ∂ ψ ˙ I ˙ i i I + JP−1 P θ + ρ s + J θ + ( v ˙ F Q θ,I ≤ 0. + ψ + θ s − b F )M + ,I R ρ i ρ,I ,I ρ i α I I P ∂(θ,K PαK ) ∂θ θ



JP−1

(38) i , θ˙ and θ˙,I . It follows from this linearity It is clear that equation (38) is linear in F˙ Ii , K˙ Ii , F˙ I,J

that the conjugate terms of these quantities within parenthesis must vanish identically, viz. ∂ ψ¯R ∂ψR PαI = , i K ∂(FK Pα ) ∂FIi

(39a)

∂ ψ¯R ∂ ψ¯R ∂ψR I QIα + JP−1 Rαβ = , i N M i K i K ∂(KK Qα ) ∂KIi ∂(KM,N Pβ Qα + KK Rαβ )

(39b)

∂ ψ¯R ∂ψR PβJ QIα = , i i N M i K ∂KI,J ∂(KM,N Pβ Qα + KK Rαβ )

(39c)

TiI = JP−1

SiI = JP−1

−1 ΛIJ i = JP

∂ ψ¯ ∂ψR =− , ∂θ ∂θ

(39d)

∂ ψ¯R ∂ψR PαI = − = 0. K ∂(θ,K Pα ) ∂θ,I

(39e)

sR = ρR sρ = −JP−1 −JP−1

17

These restrictions are the same as the constitutive restrictions of thermoelasticity. The residual inequality obtained by the remaining terms can be considered as the dissipation D per unit reference volume  ∂ ψ¯R ∂ ψ¯R ∂ ψ¯R −1 −1 i i S F + J K Q + J θ P˙αI P P i i K ) S,I γ K ) ,I ∂(FKi PαK ) I ∂(KM,N PαN QM + K R ∂(θ P ,K α γ K γα   ∂ ψ¯R ∂ ψ¯R −1 i K i P J Q˙ Iα K + J + JP−1 I P i i N M + K i RK ) I,J β ∂(KK QK ) Q ∂(K P α α K αβ M,N β   ∂ ψ¯R 1 ¯I θ,I ≤ 0. K i R˙ I + (v˙ i FIi + ψρ,I + θ,I sρ − bi FIi )M I + Q + JP−1 i N M i K θ ∂(KM,N Pβ Qα + KK Rαβ ) I αβ

 −D = JP−1

(40) By substitution of equations (39a)-(39e) into the equation (40), the micromorphic dissipation can be simplified as −α −β M ˙ K −α NL ˙ K M ˙ K −γ ˙ L −α D = − mM L Pα PM − nK Qα QM − nK (Rαβ − Rαβ (Qγ QM ))QN PL

1 ¯I − (v˙ i FIi + ψρ,I + θ,I sρ − bi FIi )M I − Q θ,I ≥ 0, θ

(41)

−α in which the component form P˙αL PM is the macro inhomogeneity velocity gradient at the −α reference configuration, Q˙ K α QM are the components of the micro inhomogeneity gyration K at the reference configuration and R˙ αβ represents the time rate of the third-order material

transplant that is related to the micro inhomogeneity gyration gradient. The latter relation expresses the condition for the thermodynamical compatibility of the growth and remodeling processes for the micromorphic theory. In particular, the Mandel macrostress, the micromedium Mandel stress and the micromedium Mandel hyperstress in general form can be obtained for the micromorphic theory as M i KM i mM KK,L , L = Ti FL + Λi

18

(42a)

M i MN i nM KK,N , K = S i KK + Λ i

(42b)

NL i L nN K = Λi KK .

(42c)

Equations (42a)-(42c) are identical to the presented Mandel stress tensor in the work of Epstein and Elzanowski (2007). In this study, we confine our attention to the rate-dependent type of growth and remodeling. Clearly the dissipation function can be expressed as D = D(F, K, ∇K, P, Q, R, m, n, n, θ, ∇θ) = z : ς,

(43)

˙ Q, ˙ R, ˙ Q, ¯ M} where ς is the collection of the thermodynamic fluxes formally in the vector ς = {P, and the vector z contains their conjugate thermodynamic forces. In this step, we utilize the Principle of Maximum Dissipation (Hackl and Fischer, 2007) on equation (43) and maximize the dissipation D, i.e. max{D|z; D = z : ς}.

(44)

Suppose that the Lagrangian Lz is defined by Lz = D + ζ(D − z : ς).

(45)

In the above relation ζ indicates the Lagrange multiplier. In order to achieve the maximum dissipation, we should solve ∂Lz /∂z = 0 leading to (1 + ζ)

∂D − ζς = 0. ∂z

(46)

Multipling equation (46) by z and solving for ς yields ς=

D ∂D . ∂D ∂z :z ∂z 19

(47)

By using equation (47) it can be inferred that ˙ = F(P, Q, R, m, n, n, θ, ∇θ), P

(48a)

˙ = G(P, Q, R, m, n, n, θ, ∇θ), Q

(48b)

˙ = H(P, Q, R, m, n, n, θ, ∇θ), R

(48c)

¯ = Q(P, Q, R, m, n, n, θ, ∇θ), Q

(48d)

M = M(P, Q, R, m, n, n, θ, ∇θ),

(48e)

where equations (48a)-(48e) are the evolution law for the micromorphic material which have the general form of a coupled system of first-order differential equations. Additionally Equation (48d) and (48e) indicate the heat flux and the mass flux constitutive relations, respectively. 4.3. Micromorphic evolution laws In this section, we are interested in determining general guidelines of evolution laws for micromorphic materials. According to the work by Epstein and Maugin (2000b), the material evolution laws should satisfy some general formal restrictions. The first restriction implies that the evolution laws do not explicitly depend on the body point X. This condition was already implemented in the equations (48a)-(48c) and was given by uniformity condition. Secondly, the chosen particular reference configuration does not affect the evolution law. Lastly, the material evolution laws must be independent of the change of material archetype which demonstrates the symmetry of the evolution laws. By ignoring the thermal effects, the 20

micromorphic evolution laws (48a)-(48c) do not alter by variation of θ and ∇θ. Hence, the only variables are confined to the transplant maps and the Mandel stresses. In an alternative form, one can replace the Mandel stress tensors by the Eshelby stress tensors as follows ˙ = f (P, Q, R, b, c, c), P

(49a)

˙ = g(P, Q, R, b, c, c), Q

(49b)

˙ = h(P, Q, R, b, c, c), R

(49c)

where f , g, and h are tensor-valued functions. The Eshelby stresses b, c , c and the Mandel stresses m, n, n are related through equations b = ψR I − m, c = −n, c = −n. 4.3.1. Reduction to the archetype As mentioned above, the particular mathematical expressions of the evolution laws (49a)(49c) still depend on the choice of a micromorphic archetype as well as the choice of reference configuration. By taking a fixed archetype, we focus on investigating the effect of the change of reference configuration on the expression of the evolution laws. Figure (2) shows the ˆ to identify quantities pertaining to the second reference basic scheme, where we use “” configuration. The evolution laws in the second reference configuration are given by ˆ cˆ, cˆ), ˆ˙ = ˆ ˆ Q, ˆ R, ˆ b, P f (P,

(50a)

ˆ cˆ, cˆ), ˆ˙ = g ˆ Q, ˆ R, ˆ b, Q ˆ(P,

(50b)

ˆ cˆ, cˆ). ˆ P, ˆ˙ = h( ˆ Q, ˆ R, ˆ b, R

(50c)

21

Figure 2: Uniformity fields associated with two different reference configurations

In the framework of the micromorphic theory the change of reference configuration is completely expressed by the following mapings (Epstein and De Le´on, 1996)

Y M = Y M (X I ),

(51a)

HIM = HIM (X J ).

(51b)

We also introduce the derivative of equation (51a) with respect to the reference configuration as EIM =

∂Y M . ∂X I

(52)

ˆ Q, ˆ R} ˆ in two reference Now, the relation between the material implants {P, Q, R} and {P, 22

configurations can be shown as PˆαM = EIM PαI ,

(53a)

ˆ M = H M QI , Q α I α

(53b)

M I M ˆ αβ R = Rαβ HIM + PβJ QIα HI,J .

(53c)

On the other hand, the transformation of the macromedium Eshelby stress, the micromedium Eshelby stress and the micromedium Eshelby hyperstress can be presented as ˆbN = J −1 [bB E −J E N + cAB E N H L (H −I ),M ], M J M B I B A L E

(54a)

−1 A −I N −I AB −I N AB L N cˆN M = JE [cI HM HA + cI HM HA,B + cI (HM ),L EB HA ],

(54b)

−1 AB −I S S N cˆN M = JE [cI HM EB HA ],

(54c)

where JE = det(E). Obviously, the similar transformation forms can be extracted for the micromorphic Mandel stresses. The macromedium Piola stress, the micromedium Piola stress and the micromedium Piola hyperstress are transformed respectively, according to the following formulae TˆiM = JE−1 [TiI EIM ],

(55a)

M SˆiM = JE−1 [SiI HIM + SiIJ HI,J ],

(55b)

SˆiM N = JE−1 [SiIJ EJN HIM ].

(55c)

23

Now, by considering a given micromorphic point and instant of time, if the reference configuration changes such that the point X is brought into coincidence with the archetype, we can conclude that EαJ = Pα−J ,

(56a)

HαJ = Q−J α ,

(56b)

J −J Hα,σ = Rασ .

(56c)

Additionally, by substituting equations (56a)-(56c) into equations (54a)-(54c), we also have the following values for the pull-back of the macromedium Eshelby stress, the micromedium Eshelby stress and the micromedium Eshelby hyperstress to the archetype as ¯bβ = JP [bB P J P −β + cAB RI Q−σ P −β ], α J α B I σα A B

(57a)

−ρ −σ −β −σ −β I −β AB I L AB c¯βα = JP [cA I Qα QA − cI Qα Rρσ QA PB QL + cI Rασ PB QA ],

(57b)

AB I −β −γ c¯βγ α = JP [cI Qα QA PB ].

(57c)

¯ over all new quantities according To make a difference between the quantities, we insert “” to this transformation. At the same time, under such a reference configuration change, the time rates of the implants in the new reference configuration become P¯˙αβ = PI−β P˙αI ,

(58a)

¯˙ β = Q−β Q˙ I , Q α α I

(58b)

24

¯˙ γ =Q−γ (R˙ K − P˙ J QI RK Q−ρ P −σ − P J Q˙ I RK Q−ρ P −σ ) R αβ β α ρσ I β α ρσ I K J J αβ ˙K ¯˙ σ K ¯˙ ρ K = Q−γ K (Rαβ − Pβ Rασ − Qα Rρβ ).

(58c)

These quantities should be termed as the “inhomogeneity velocity gradient” for the micromorphic materials at the archetype level. To this end, the evolution laws can be obtained as

P¯˙αβ = f¯αβ (¯bσρ , c¯σρ , c¯στ ρ ),

(59a)

¯˙ β = g¯β (¯bσ , c¯σ , c¯στ ), Q α α ρ ρ ρ

(59b)

¯ γ (¯bσ , c¯σ , c¯στ ). ¯˙ γ = h R αβ αβ ρ ρ ρ

(59c)

With the aid of equations (58a)-(58c), the micromorphic evolution laws can be rewritten as P˙αI = PβI f¯αβ (¯bσρ , c¯σρ , c¯στ ρ ),

(60a)

Q˙ Iα = QIβ g¯αβ (¯bσρ , c¯σρ , c¯στ ρ ),

(60b)

K ¯ γ ¯σ ¯σ , c¯στ ) + f¯σ (¯bσ , c¯σ , c¯στ )RK + g¯ρ (¯bσ , c¯σ , c¯στ )RK . R˙ αβ = QK γ hαβ (bρ , c ασ α ρ ρ ρ ρβ ρ ρ β ρ ρ ρ

(60c)

It can be observed that the micromorphic evolution laws depend on material transplants linearly. Equations (59a)-(59c) represent the evolutions equations reduced to the archetype. One of the important points of this case is the careful calculation of the contributed arguments in the change of the reference configuration (the pullback of the Eshelby or Mandel stresses to the archetype). This is clearly a constitutive property which, once known, authorizes us to calculate how any point in the reference configuration responds to the Eshelby stresses. 25

Therefore, the evaluation of evolution laws reduces to the determination of the functions f¯αβ , ¯ γ illustrate how the archetype would ¯ γ . Note that the functions f¯β , g¯β , and h g¯αβ and h α α αβ αβ respond to the application of the micromorphic Eshelby stresses measured therein. 4.3.2. Material symmetry consistency and actual evolution in a micromorphic evolution law In this section, the consistency of the material symmetry during the material evolution is studied. We show that the presented evolution laws (60a)-(60c) need to fulfill some extra conditions in order to achieve the consistency of the micromorphic material symmetry. We should bear in mind that in the micromorphic medium, the evolution laws at each point remain invariant after applying the material symmetry group. A similar notion was implemented in the first- and second-grade material (Epstein and Elzanowski, 2007; Epstein, 1999). For incorporation of the material symmetry entering the treatment of evolution laws two approaches are taken into account. First, the material symmetry is considered as an element of the symmetry group of the reference crystal which is fixed during the material evolution. Second, the material symmetry is adopted as a time dependent element. In the first case, the following laws of transformation for the implant maps under a change of archetype {A, B, D} can be presented as (see Figure 3) PˆαI = PρI Aρα ,

(61a)

ˆ I = QI B ρ , Q α ρ α

(61b)

ˆ I = RI B ρ Aσ + QI Dρ , R αβ ρσ α β ρ αβ

(61c)

where {A, B, D} are the elements of the micromorphic symmetry group. 26

Figure 3: Uniformity fields associated with two different micromorphic archtype

Moreover, the pull-backs of the macromedium Eshelby stress, the micromedium Eshelby stress, and the micromedium Eshelby hyperstress in the new archetype are expressed in terms of the old pull-back Eshelby stresses (57a)-(57c) by ˆbβ = JA [¯bρ Aσ A−β + c¯λτ Dµ A−β B −ρ ], α σ α ρ µ ρα τ λ

(62a)

−ρ −σ −β ν µ ν −σ −β cˆβα = JA [¯ cρσ Bασ Bρ−β − c¯λτ ¯λτ ν Bα Dρσ Bλ Aτ Bµ + c ν Dασ Aτ Bλ ],

(62b)

ν −ν −β cˆβν cλτ α = JA [¯ ν Bα Aτ Bλ ].

(62c)

Additionally, the evolution laws in the new archetype can be written as ˙ PˆαI = PˆβI fˆαβ (ˆbσρ , cˆσρ , cˆστ ρ ), 27

(63a)

ˆ˙ I = Q ˆ I gˆβ (ˆbσ , cˆσ , cˆστ ), Q α β α ρ ρ ρ

(63b)

K K ˆ γ ˆσ ˆσ , cˆστ ) + fˆσ (ˆbσ , cˆσ , cˆστ )R ˆK ˆ˙ αβ ˆK ˆ ασ + gˆαρ (ˆbσρ , cˆσρ , cˆστ R =Q ρ )Rρβ , γ hαβ (bρ , c ρ ρ β ρ ρ ρ

(63c)

ˆ γ are the tensor-value functions in the new archetype. Since the change where fˆβα , gˆβα and h αβ of archetype (material symmetry groups) is considered to be independent of time, the time rates of the new implants in terms of their old counterparts are extracted from equations (61a)-(61c) as ˙ PˆαI = P˙ρI Aρα ,

(64a)

ˆ˙ Iα = Q˙ Iρ Bαρ , Q

(64b)

ˆ˙ I = R˙ I B ρ Aσ + Q˙ I Dρ . R αβ ρσ α β ρ αβ

(64c)

¯ γ ) and the In order to find the relations between the old evolution functions (f¯βα , g¯βα and h αβ new ones, the equations (60a)-(60c) and (63a)-(63c) are substituted in equations (64a)-(64c), and we get fˆαβ = f¯ρν Aρα A−β ν ,

(65a)

gˆαβ = g¯ρν Bαρ Bν−β ,

(65b)

¯  B ρ Aσ + g¯ Dρ − f¯ν Aρ A−σ D − g¯ν B γ B −ρ D ]. ˆ γ = B −γ [h h  ρ β ν ασ γ α ν ρβ ρσ α β ρ αβ αβ

(65c)

Equations (65a)-(65c) illustrate the principle of material symmetry consistency for micromorphic materials. Consequently, by applying the transformation {A, B, D} which belong 28

ˆ γ are to the symmetry group of the original archetype, the evolution functions fˆβα , gˆβα and h αβ ¯γ . related to their respective counterparts f¯βα , g¯βα and h αβ As mentioned previously, the material symmetry group in the first case can not be evolved in time during the material evolution. While, in the second case, two evolutions ˆ I (t)} and {PαI (t), QIα (t), RI (t)} at a point are related by the time deˆ Iα (t), R {PˆαI (t), Q αβ αβ pendent element of the symmetry group {A(t), B(t), D(t)}. In other words, the material symmetries can evolve in time during the material evolution. Based on these conditions, differentiating equations (61a)-(61c) with respect to time, evaluating the result at an instant wherein A and B are the unit tensor and D = 0 yield ˙ PˆαI = P˙αI + PρI A˙ ρα ,

(66a)

ˆ˙ Iα = Q˙ Iα + QIρ B˙ αρ , Q

(66b)

ρ I I I ˙ρ I ˙σ ˆ˙ αβ R = R˙ αβ + Rρβ Bα + Rασ Aβ + QIρ D˙ αβ .

(66c)

Comparison of relations above with those obtained from (60a)-(60c), implies that the relation ¯ ρ are as below between evolution laws fαρ , gαρ , hραβ and their counterparts f¯αρ , g¯αρ , h αβ f¯αρ = fαρ + A˙ ρα ,

(67a)

g¯αρ = gαρ + B˙ αρ ,

(67b)

¯ ρ = hρ + D˙ ρ . h αβ αβ αβ

(67c)

ρ In these equations, the quantities A˙ ρα , B˙ αρ , and D˙ αβ are arbitrary infinitesimal generators

of the micromorphic symmetry group. These formulations represent the principle of actual 29

evolution. These equations result in the extra constraint on the material evolution laws in which the transformation {A(t), B(t), D(t)} can evolve in time. 4.4. Pure remodeling and growth in micromorphic theory In order to assess the growth and remodeling processes in the framework of the micromorphic theory, the potential energy function (35) is employed at the fixed material point X0 . The existence of a time variable t in the energy function enables us to investigate the material evolution phenomena such as growth, remodeling, aging and morphogenesis. During a remodeling process, the material point remains materially isomorphic which is equal to the uniformity condition (35). Additionally, without loss of generality we consider the initial condition as P(0) = I,

(68a)

Q(0) = I,

(68b)

R(0) = 0,

(68c)

where I is the second-order identity tensor and 0 is the third order zero tensor. Let ρ¯R be the constant mass density and Iαβ 0 be the constant microinertia of the micromorphic archetype. The density and microinertia at the reference configuration are given by ρR (t) =

ρ¯R , JP

I J IIJ (t) = Iαβ 0 Qα (t)Qβ (t).

30

(69a)

(69b)

In equation (69b), both microinertia tensors in the reference configuration and material archetype are symmetric. It is obvious that JP > 1 implies the reduction of the mass in the reference configuration. On the contrary, for JP < 1 the mass in the reference configuration increases (Epstein, 2015). Moreover, the time evolution of the material transplant Q leads to the alteration of the microinertia tensor in the reference configuration which is an important process in materials with microstructure. This effect can not be captured in the classical continuum theory. Now, differentiating equations (69a) and (69b) with respect to time yields

ρ˙ R = −ρR PI−α P˙αI ,

(70a)

IN ˙ J −α I˙ IJ = IM J Q˙ Iα Q−α Qα QN . M +I

(70b)

On the other hand, from equations (19) and (24) the mass balance and the balance of microinertia in the reference configuration can be represented respectively as K ρ˙ R = Π − M,K

(71a)

n n n KLl ),S + ¯IKL KK KLl , ρR I˙ KL KK KLl = −M S (IKL KK

(71b)

and

n where Π = JF π (JF = det(F)), M S = JF Fn−S mn and ¯IKL KK KLl = JF ¯inl . By simplifying

equation (71b), we obtain n ρR I˙ KL = −M S (Kn−K Kl−L )(IM N KM KNl ),S + ¯IKL .

31

(72)

By substituting equations (70a) and (70b) into (71a) and (72), one can get Π and ¯I explicitly in the following form K Π = −ρR PI−α P˙αI + M,K ,

¯IKL = ρR (IM L Q˙ K Q−α + IKN Q˙ L Q−α ) + M S (K −K K −L )(IM N K n K l ),S . α α N n M N M l

(73a)

(73b)

These equations imply that (assuming that the mass and microinertia consistency conditions are satisfied) no separate constitutive law needs to be given for the volumetric mass source Π and volumetric microinertia ¯I in the absence of the mass flux, since they will be dictated by the evolution of the micromorphic material implants. Based on Epstein and Elzanowski (2007); Epstein (2015), the time evolution of the determinant of P in the micromorphic theory determines the process of instantaneous growth or resorption on the macroelements. Alternatively in the above mathematical description, growth is taking place ˙ is negative, while the reduced value of the determinant of P demonstrates when tr(P−1 P) the accumulated growth from a fixed time origin. In the special case when P is spherical, namely when P = P˜ (t)I, this process is called pure growth. However, if P is unimodular so that there may be pure distortions without any growth or resorption this is a process known as pure remodeling. In the first-order theory, the trace of the inhomogeneity velocity gradient will carry the information about volumetric growth. In the context of the micromorphic theory, the definitions of growth and remodeling change considerably compared to ˙ −1 at the refclassical continuum mechanics. The micro inhomogeneity gyration tensor QQ erence configuration plays an important role in the arrangement and re-accomodation of the microstructure which can be expressed in equation (73b). Let us consider the special case when the mass production and mass flux are absent. Based on (73a) and (73b) this condition 32

˙ The pure remodeling on this level can be characterized ˙ = 0 and ¯I = ρR I. leads to tr(P−1 P) ˙ −1 )IT which by the relation ρR I˙ = 0, and this relation leads to the second-order tensor (QQ is a skew-symmetric tensor. As a result we obtain the following relation ˙ −1 = −I(QQ ˙ −1 )T I−1 . QQ

(74)

˙ −1 ) = 0. If we have ρR I˙ 6= 0 After some algebraic manipulations it can be shown that tr(QQ then we can call this process an evolution that leads to the change of material microinertia. This process can only happen in materials with a microstructure such as micromorphic materials. In this paper, for micromorphic materials this process is referred to as growth in inertia. Unlike the process of growth and remodeling in the classical continuum mechanics, there are two levels of growth and remodeling in the micromorphic theory. The first level is similar to the classical one (Epstein and Maugin, 2000b; G¨oktepe et al., 2010; Papastavrou et al., 2013; Buskohl et al., 2014; Budday et al., 2014; Rausch and Kuhl, 2014; Rejovitzky et al., 2015; Balbi et al., 2015; Epstein, 2015; Xue et al., 2016) indicating the variation of density for growth reaccomodation of the same material point within the same neighbourhood for remodeling. On the other hand the second level is associated with microelements inside each material point, in which the growth and remodeling processes can be considered as an alteration of microinertia tensor. We refer to the first level as macro-scale growth and remodeling and the second level as micro-scale growth and remodeling.

5. Conclusions The focus of this paper is on growth and remodeling processes in a uniform micromorphic media. Firstly, we derived a number of material evolution laws such as those found in material 33

growth, remodeling and plasticity for a set of material transplant maps within the context of material uniformity. It was found that the final form of the material evolution law is invariant with respect to the change of the reference configuration as well as the change of the micromorphic archetype. Secondly, we demonstrated how the micromorphic theory as a non-classical continuum theory is able to capture the influence of the internal structures of the biological materials within the growth and remodeling processes in living tissues. Next, we derived the micromorphic Eshelby and Mandel stress tensors by implementing the second principle of thermodynamics. We then introduced a new definition of growth and remodeling processes in the microstructure via the variation of the microinertia. Lastly, we identified the fact that the growth and remodeling processes in the context of the micromorphic theory are dependent on the evolution of the micromorphic material transplants which leads to changes in density and microinertia. As the essential outcome of the current study, it was deduced that the growth process in the macrostructure results in modifications in density while the growth process in microstructure causes changes in microinertia.

Author statement Mohammadjavad Javadi: Conceptualization, Methodology, Writing- Original draft preparation. Marcelo Epstein: Conceptualization, Methodology, Writing - Review and Editing. Mohsen Asghari: Conceptualization, Methodology.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 34

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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