Material growth and remodeling formulation based on the finite couple stress theory

Journal Pre-proof Material growth and remodeling formulation based on the finite couple stress theory M. Javadi, M. Asghari, S. Sohrabpour

PII: DOI: Reference:

S0020-7462(19)30246-X https://doi.org/10.1016/j.ijnonlinmec.2020.103413 NLM 103413

To appear in:

International Journal of Non-Linear Mechanics

Received date : 9 April 2019 Revised date : 19 November 2019 Accepted date : 6 January 2020 Please cite this article as: M. Javadi, M. Asghari and S. Sohrabpour, Material growth and remodeling formulation based on the finite couple stress theory, International Journal of Non-Linear Mechanics (2020), doi: https://doi.org/10.1016/j.ijnonlinmec.2020.103413. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.

Journal Pre-proof

Material Growth and Remodeling Formulation Based on the Finite Couple Stress Theory M. Javadia,b, M. Asgharia, S. Sohrabpoura a

Department of Mechanical and Manufacturing Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4

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b

Mechanical Engineering Department, Sharif University of Technology, Tehran, Iran

Corresponding author’s Email: [email protected]

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Abstract

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The mathematical formulation for material growth and remodeling processes in finite deformation is developed based on the couple stress theory. The generalized continuum mechanics of couple stress theory is capable of capturing small-scale cellular effects and of modeling mass flux in these processes. The frame-indifferent balance equations of mass, linear and angular momentums, as well as internal energy together with the entropy inequality are first introduced in the presence of the mass flux based on the finite couplestress theory. Then, within the framework of material uniformity the Eshelby and Mandel stress tensors as driving or configurational forces for local rearrangement of the first- and second-order material inhomogeneities are determined for the Cauchy stress tensor as well as the couple stress tensor. In the next step, the basic kinematic tensors are multiplicatively decomposed into elastic and anelastic parts, and by utilizing the derived entropy inequality, the hyper-elastic constitutive equations with respect to both reference and current configurations are obtained. Additionally, an admissible form for each of the two evolution laws of classical and higher-order material transplant tensors of material growth which satisfy the general formal restrictions are developed as a function of classical and hyper versions of the Mandel stress. Moreover, in a numerical study the effects of presented evolution laws on the growth of a cubic materially isotropic object under a specific oscillating external loading, corresponding to some diagonal classical stress and skewsymmetric couple stress tensors in the reference configuration, are investigated. Keywords: the finite couple stress theory, length scale, growth, remodeling, the Mandel stress, the Eshelby stress

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Nomenclature

t n the stress vector

u internal energy per unit mass

s n the couple stress vector

φ ( X ,t) the configuration map

c body couple per unit mass

F deformation gradient tensor

r position vector in the current configuration

F e elastic part of deformation gradient

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Journal Pre-proof F g growth part of deformation gradient

p d extra diffusive momentum

χ motion of a continuous medium

p non-compliant momentum

Y mapping for change of reference configuration

v velocity vector in current configuration

V inverse velocity vector

E Lagrange strain tensor

q heat flux vector in the current configuration

C the right Cauchy-Green deformation tensor

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Ee elastic part of Lagrange strain tensor

K curvature tensor

η entropy density per unit mass

K e elastic part of curvature tensor

h bulk heat supply per unit mass

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E g growth part of Lagrange strain tensor

! heat flux vector in the reference Q configuration

θ absolute temperature

K g growth part of curvature tensor

P the stress power per unit reference volume

H g third-order growth symmetric tensor

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H e third-order elastic symmetric tensor

Ψ ρ Helmholtz free energy per unit mass

ρ0 mass density in the reference configuration

ρ g mass density in the intermediate configuration

Ψ R Helmholtz free energy per unit reference volume

Γ rate of mass growth in the reference configuration

! Helmholtz free energy per unit volume Ψ R in the intermediate configuration

γ rate of mass growth in the current configuration

ε permutation tensor

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ρ mass density in the current configuration

Ψ R Helmholtz free energy per unit volume in the archetype

T s Lagrangian stress tensor

M mass flux vector in the reference configuration

ΛD Lagrangian hyperstress tensor

b the Eshelby stress tensor

m mass flux vector in the current configuration

d the Eshelby hyperstress tensor

M! positive material constant

β! the Mandel stress tensor

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τ Cauchy stress tensor

α! the Mandel hyperstress tensor

µ couple stress tensor

β the Mandel stress tensor in the intermediate configuration

f body force per unit mass

P(X,t) first-grade uniformity field

α the Mandel hyperstress tensor in the intermediate configuration

Q(X,t) second-grade uniformity field

k1 classical remodeling stiffness

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Journal Pre-proof k2 couple-stress remodeling stiffness

P XY first-order material implant between X and Y

D s the mechanical dissipation

Q XY second-order material implant between X and Y

D m the mass dissipation

a magnitude of normal traction

J determinant of F

b magnitude of couple traction

J g determinant of F g

a0 ,b0 constant amplitude

J e determinant of F e

ω angular frequency

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Lg remodeling velocity gradient

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µ shear modulus

κ , κ ′ higher order material constant

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J P determinant of P

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Journal Pre-proof 1. Introduction The physical phenomena of growth, remodeling, aging and morphogenesis may occur in biological tissues subjected to environmental conditions which affect their mechanical behavior. The importance of growth and remodeling in mechanical analysis of biological

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tissues has led many researchers to focus on these concepts to a greater degree in recent years in order to predict the behavior of tissues with more accuracy (Di Stefano et al., 2018, Grillo

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et al., 2012, Ciarletta and Amar, 2012, Zollner et al., 2013, 2012; Rausch et al., 2013; Rausch and Kuhl, 2014; Holland et al., 2013, 2017, 2018; Wong and Kuhl, 2014, Dervaux and Amar, 2008, Amar and Goriely, 2005, Hamedzadeh et al., 2019, Grillo et al., 2015). Changes in appearance, volume, mass and the internal structure of tissues accompanied by material

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evolution illustrate the occurrence of growth, remodeling, aging and morphogenesis. The mathematical description of these four physical phenomena has a vital role in their mechanical

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understanding. Differentiating these processes from each other and predicting what may happen under various conditions is complicated. In this work, we only focus on the growth and remodeling processes, which respectively correspond to added mass and to re-accommodations of the same material within the same material neighborhood (Epstein and de Le´on, 2000; Epstein, 2015).

An important paper done on growth and added mass was done by Segev and Epstein (1996). They proposed the permissible geometrical framework for growing bodies and investigated two general kinds of growing bodies. The first kind considers growing bodies

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whose particles are identifiable throughout the various growth stages, while in the second one the various growing body points are assumed not to be identifiable. Moreover, DiCarlo (2005) presented a thermodynamical evolution law for the remodeling process in the bone model. He illustrated that during the remodeling process the bone orthotropy axes are reoriantated. Furthermore, Epstein (2009) presented the distinction between the remodeling and aging

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Journal Pre-proof processes based on the implication of material symmetry groups. He concluded that in the case of triclinic solids there is no canonical way to obtain a remodeling component as distinguishable from an aging component of an evolution process. The mass variation in biological tissues is either due to the mass generation in the tissue,

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mass flux at the surface, or both. The mass generation and mass fluxes are very important factors in the development and evolution of the biological tissues. Many remarkable studies on

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the volumetric growth process have been accomplished by only considering mass generation, including studies by Lubarda and Hoger, (2002); Rodriguez et al. (1994); Epstein, and Maugin, (2000a). Epstein and Maugin (2000b) showed that classical continuum mechanics is incapable

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of predicting and illuminating the mass-diffusive effects in the growth process. They successfully took the mass-diffusive effects into the growth formulation by utilizing the nonclassical or generalized continuum mechanics of the second order gradient theory. Ciarletta et

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al. (2012) presented a similar formulation that additionally took into account species transportation. Generalized continuum theories can be better applied to the mechanical analysis of biological tissues from the perspective of micropolar theory (Eringen, 1999, 1966; Kafadar and Eringen, 1971). The reason for this is because the generalized continuum theories such as micropolar theory, the couple stress theory, and the second gradient theory each incorporate the internal characteristic lengths or microstructure into the mechanical analysis, which are of high importance in biological tissues (Eringen and Kafadar, 1976; Eringen, 1999; Mindlin, 1964). A paper by Garikipati et. al. (2004), considered the mass flux phenomenon within the

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framework of classical continuum mechanics, as a result, they utilized the mixture theory of Truesdell and Toupin (1960). However, this utilization led to some drawbacks such as appearance of partial stresses or mass exchanges between the single phases (Ambrosi et al., 2010; Ciarletta et al., 2012, Ciarletta and Maugin, 2011).

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Journal Pre-proof The aim of this study is to present a formulation for growth and remodeling in biological tissues based on the couple stress theory. The couple stress theory allows incorporation of mass flux into the formulation and provides us with the opportunity to include the microstructure of biological tissues. One of the advantages of the couple stress theory is that it is less complex

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than other generalized theories, because it only considers the rotational motion of microstructures. It is unlike the micromorphic theory in which deformation of microstructures

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is also included, and the microstructure’s rotation is identical with that of the macro-element, and unlike the micropolar theory in which the microstructural rotation is independent (Eringen, 1999; Eringen, 1966; Mindlin and Tiersten, 1962).

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In the sections below, a number of concepts will be addressed. We will first discuss the balance equations of mass, linear and angular momentums, and internal energy together with the entropy inequality in the presence of mass flux based on the finite couple-stress theory.

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Secondly, the basic kinematic tensors are multiplicatively decomposed into elastic and anelastic parts within the framework of material uniformity. Thirdly, by applying the derived entropy inequality, the hyper-elastic constitutive equations with respect to both reference and current configurations are derived. We will next address how Eshelby and Mendel stress tensors, which are related to the configurational forces, are extracted for the couple stress theory and how these stress tensors can be the driving force for the local rearrangement of the first- and second-order material inhomogeneities. However, it should be noted that some theories do not agree with the idea that configurational forces drive material inhomogeneities

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of growth or remodeling, but those theories consider that these forces only modulate them (Taber and Eggers, 1996; Ambrosi and Guana, 2007; Ambrosi et al., 2008). In the next step, we develop a thermodynamically admissible form for both growth and remodeling evolution laws which satisfy the limiting factor of evolution as a function of classical and hyper versions of the Mandel stress. Lastly, we design a numerical study in which we investigate the effects

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Journal Pre-proof of presented evolution laws on the growth of a cubic isotropic object under a specific oscillating external loading, corresponding to some diagonal classical stress and skew-symmetric couple stress tensors in the reference configuration. 2. Preliminaries

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A configuration of a simple body B is identified by a mapping

(1)

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φ : B → S : X → x = φ ( X ,t),

where S is the physical space which here is considered to coincide with the affine space E 3 (the 3-D Euclidean space) modeled on ! 3 . The map φ ( X ,t) , called the configuration map,

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maps the material points X = ( X 1 , X 2 , X 3 ) in the body B into spatial points x = (x1 , x2 , x3 ) in S . The tangent of φ ( X ,t) , denoted F and called the deformation gradient of φ ( X ,t) , i.e.

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F( X ,t) :TX B → Tφ ( X ,t ) S,

(2)

maps material vectors in the tangent space TX B into spatial vectors in the tangent space

Tφ ( X ,t ) S .

In coordinate charts {X K } in B and {xk } in S (Alhasadi et al., 2019), the deformation gradient is represented by

FkK = ∂φk ( X K ,t) / ∂ X K = φk ,K ( X K ,t),

(3)

where the uppercase and lowercase indices respectively are related to the reference and current

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configurations. Moreover, (•),i and (•),I imply ¶ (•) / ¶x i and ¶ (•) / ¶X I , respectively. The Lagrange strain tensor E is related to the deformation gradient tensor as (Truesdell and Toupin, 1960)

E=

(

(4)

)

1 T F F−I , 2

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Journal Pre-proof or in the component form

E MK =

(F 2 1

kM

(5)

FkK − δ MK ) ,

where d MK is the Kronecker delta in reference configuration.

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The Couple stress theory is one of the theories in the category of generalized theory of continuum mechanics which was initially presented by Toupin (1962) and Mindlin and

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Tiersten (1962) in which each material element interacts with neighboring elements by exerting couple stresses along with classical or force stresses. In classical continuum mechanics, it is well known that the internal energy per unit mass u at a specific point of an elastic body is a

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function of the Lagrange strain tensor E . In the couple stress theory, the specific internal energy u is also dependent on the traceless second-order tensor K = ∇ × E (Toupin, 1962,

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Mindlin and Tiersten, 1962), whose components are written as follows 1

K AM = − E KM ,N ε KNA = − ε KNA FkM ,N FkK , 2

with K II = 0,

(6)

where e KNA are the components of the permutation tensor ε in the reference configuration, and

(•),N stands for the components of the material gradient. 3. The couple-stress growth and remodeling thermomechanical theory Consider a continuous volume body V bounded by a surface S in the reference configuration. After deformation, the volume changes to v and the bounding surface to s in

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the current configuration. The vector n is unit outward normal to the bounding surface in the current configuration. In this section, the balance principles, including balance equations of mass, linear momentum, moment of momentum and energy, as well as the entropy inequality for a growing body are developed in the framework of the couple stress theory under the condition of mass fluxes and volumetric mass sources.

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Journal Pre-proof 3.1. Balance equation of mass The mass density in the material and spatial configurations are here indicated by

r0 (X K , t ) and r (x k , t ) , respectively, where r0 = J r , and J = det(F) . Internal mass creation/annihilation and mass fluxes on the boundaries cause the variation of mass in the body.

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The rate of mass growth in the material and spatial configurations are here denoted by Γ and

γ , respectively. Moreover, the mass flux vector in the material and spatial coordinates through

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the boundary with exterior unit normals N and n , respectively, are indicated by M and m . The following relations between the material and spatial coordinate parameters hold (Epstein

Γ = Jγ ,

M = J F −1m.

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and Maugin, 2000b; Alhasadi et al., 2019)

(7)

The local mass balance equation in the material configuration as well as its spatial form can

¶r0 ¶t

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also be written as (Epstein and Elzanowski, 2007) = G r0 - ÑR . M, X

(8)

and its spatial form in the local form is as Dr + rÑ.v = g r - Ñ. m, Dt

(9)

where D(i) / Dt denotes the material time derivative, i.e. a partial time derivative holding the X fixed, by following the gross motion of the medium, whose velocity field is v (Epstein and

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Goriely, 2012). The gradient operators ∇(i) and ∇ R (i) also represent the gradient with respect to the current and reference configurations, respectively. It would be useful to note that in view of Cauchy’s theorem, the incoming mass fluxes (surface) in Lagrangian and Eulerian form are calculated by m = -m. n and M = -M. N , respectively (Ciarletta et al., 2012).

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Journal Pre-proof 3.2. Balance of linear momentum and moment of momentum In the classical continuum theory, the effect of a surface element on an adjacent one is representable by a surface force-traction vector per unit area (or force-stress vector) t n . In the couple stress theory, the interaction between two neighboring surface elements is additionally

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measured via a couple-stress vector (or surface couple-traction vector per unit area) s n . The force-stress vector and the couple-stress vector on the surface give rise, respectively, to the

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force-stress or Cauchy stress tensor t and the couple-stress tensor µ in the body. It should be noted that these two stress tensors are generally asymmetric.

The addition of mass involved in the growth implies the presence of new sources of

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momentum, angular momentum, energy and entropy. In the particular case when the velocity, angular velocity, specific energy and specific entropy of the entering mass are the same as the

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local base material, the case is ''reversible'' growth; otherwise, we have the general case of ''irreversible'' growth (Epstein and Maugin, 2000b), which is the case under consideration in this study. In other words, due to the fluctuations of incoming mass the effects of extra diffusive momentum as well as the contribution of the non-compliant momentum representing any desired deviation due to mass flux (Epstein and Goriely, 2012; Javadi and Epstein, 2018) are present in the irreversible growth. The effects of extra diffusive momentum and non-compliant momentum will be considered in this study. The balance of the momentum for the body in the

d

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current configuration is as follows (Epstein and Goriely, 2012) ( ρv + p dt ∫ v

d

) dv = ∫ ( ρ f + p) dv + ∫ t n ds + ∫ ργ v dv + ∫ n.(m ⊗ v) ds, v

s

v

(10)

s

where f is the body force per unit mass, t n = n.τ is the stress vector or surface force-traction per unit spatial area acting on points of the boundary surface, p d is the extra diffusive

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Journal Pre-proof momentum per unit spatial volume and p is the non-compliant per unit spatial volume. It should be noted that the integral is carried out on the volume body convecting with the motion of the medium; that is an important point for the way in which the time derivative of the volume integral is taken.

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The balance of the moment of momentum for the body in the current configuration can

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be written as (Epstein and Maugin, 2000b; Malvern, 1969; Alhasadi et al., 2019) d

r × ( ρ v) dv = ∫ (r × ( ρ f + p) + ρc + r × ργ v) dv + ∫ (r × t dt ∫ v

v

n

+ s n + r × n.(m ⊗ v)) ds,

(11)

s

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where s n = n.µ

(12)

represents the couple-stress vector acting on the points of the boundary surface, c is the body

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couple per unit mass in the current configuration and r is the position vector of material particles relative to a fixed origin. As stated by Epstein and Goriely (2012), the problem of inertial-observer dependence is associated with Eq. (11). With consideration of extra diffusive momentum p d , we offer the following form for the balance of the moment of momentum d

r × ( ρv + p dt ∫ v

d

) dv = ∫ (r × ( ρ f + p) + ρc + r × ργ v) dv + ∫ (r × t n + s n + r × n.(m ⊗ v)) ds. v

(13)

s

It can be shown that this equation possesses the property of being the material frame indifferent.

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Using Green-Gauss theorem and substituting Eq. (9) into Eqs. (10) and (13), the local form of the momentum and the angular momentum equations in components form are obtained as (Epstein and Goriely, 2012; Javadi and Epstein, 2018; Toupin, 1962; Mindlin and Tiersten, 1962)

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Journal Pre-proof ρ

(14)

Dvl Dpld + = ρ f l + pl + τ kl ,k − pld vk ,k − mk vl ,k , Dt Dt

µkl ,k + r c l + t mn e mnl = 0.

(15)

where v!l are the components of the acceleration vector v! . From Eq. (15), the components of

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the skew-symmetric part of the Cauchy stress tensor can be expressed as (Mindlin and Tiersten,

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1962) 1

τ [ pq] = − (ε pql µ kl ,k + ρ cl ε pql ), 2

(16)

where τ [ pq] represents the components of tensor t A as the skew-symmetric part of the Cauchy

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stress tensor τ , i.e. τ [ pq] = (τ pq − τ qp ) / 2 . If the couple stress tensor µ, body couple c and mass flux vector m were absent, t A would disappear, and the Cauchy stress t would be symmetric

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as in classical continuum mechanics. According to the results of (Epstein and Goriely, 2012; Javadi and Epstein, 2018), we have p d = m . This equation shows that the diffusive momentum is a direct result of mass flux; in other words, once we have mass flux we also have diffusive momentum, and one cannot exist without the other. Moreover, they are identical in magnitude and orientation. Substituting Eq. (16) into (14), the local balance equation of the momentum is rewritten as

⎛ Dv ⎞ Dml 1 τ ( kl ),k − ε klm ( µ nm,nk + ρ cm,k ) + ρ ⎜ f l − l ⎟ + pl − mk vl ,k − ml vk ,k − = 0, 2 Dt ⎠ Dt ⎝

(17)

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where t ( kl ) are the components of the symmetric part of t . 3.3. Balance of energy and the entropy inequality In view of the first law of thermodynamics, the rate of change of the sum of the internal and kinetic energies of the body in the framework of the couple stress theory with the

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Journal Pre-proof volumetric mass sources and mass fluxes can be written as follows (Mindlin and Tiersten, 1962; Epstein and Goriely, 2012; Javadi and Epstein, 2018; Epstein and Maugin, 2000b; Epstein and Elzanowski, 2007) ⎛1

d

⎛

⎞

⎛

1

⎞ ⎞ ⎛ v.v ⎞ + mu ⎟ − n.q⎟ ds ⎟ 2 ⎠ ⎠ ⎠

ρ v.v + ρu + p .v ⎟ dv = ∫ ⎜ t .v + s .∇ × v − n.⎜ m ⎜ dt ∫ ⎜⎝ 2 2 ⎠ ⎝ ⎝ ⎝ d

n

v

n

s

(18)

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⎛ ⎛ v.v ⎞ ⎞ 1 + ∫ ⎜ f.( ρ v + p d ) + p.v + ρ (c).∇ × v + ργ u + ρ h + ργ ⎜ dv , 2 ⎝ 2 ⎟⎠ ⎟⎠ v⎝

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where u is the internal energy per unit mass, q is the heat flux vector and h is the bulk heat supply per unit mass. However, we have considered for a potential non-compliant momentum input p , contributing to the energy rate in the amount p.v . The velocity field receives

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mechanical power from the body force f per unit mass. In addition, however, since we have the small diffusive momentum pd per unit volume, we have added the corresponding power

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f . pd (Epstein and Goriely, 2012; Javadi and Epstein, 2018). Applying the Green-Gauss theorem to surface integrals in Eq. (18) and substituting Eq. (15) into the result, and in view of the fact that the gradient of the curl of any vector field is a traceless second-order tensor (and accordingly ∇(∇ × v) with the components vn,mi ε mnj is a traceless tensor which implies that the spherical part of µ , i.e. µ s , does not contribute to the energy equation), we arrive at the following equation (Mindlin and Tiersten, 1962; Epstein and Goriely, 2012; Javadi and Epstein, 2018; Epstein and Maugin, 2000b; Epstein and Elzanowski, 2007) ⎛

Du

k

v

Dvk ⎞ ⎛ ⎞ 1 d v = ∫ ⎜ τ ( kl ) vl ,k + µ klD vn,mk ε mnl − qk ,k + ρ h + f k mk − mk u,k ⎟ dv, ⎟ Dt ⎠ 2 ⎝ ⎠ v

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∫ ⎜⎝ ρ Dt + m

(19)

where µklD = µkl - (1/ 3) µii d kl are the components of µ D , the deviatoric part of the couple stress tensor. The local balance of energy from Eq. (19) becomes (Mindlin and Tiersten, 1962; Epstein and Goriely, 2012; Javadi and Epstein, 2018; Epstein and Maugin, 2000b; Epstein and Elzanowski, 2007)

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Journal Pre-proof ρ

Dvk Du 1 + mk = τ ( kl ) vl ,k + µ klD vn,mk ε mnl − qk ,k + ρ h + f k mk − mk u,k . Dt Dt 2

(20)

The second law of thermodynamics or Clausius-Duhem inequality with mass sources and mass fluxes is given by the following inequality (Epstein and Maugin, 2000b)

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⎛ ρ h ργ η ⎞ ⎛q ⎞ d ρη dv ≥ ∫ ⎜ + dv − ∫ n.⎜ + mη ⎟ ds , ∫ ⎟ dt v θ ⎠ ⎝ θ ⎝θ ⎠ v s

ρ

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or, in local form

⎛q ⎞ Dη ρ h ≥ − mk η,k − ⎜ k ⎟ , Dt θ ⎝ θ ⎠ ,k

(21)

(22)

where h is the entropy density per unit mass, q k are the components of the heat flux vector

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and q is the absolute temperature in the macro element. The contribution of the non-compliant internal energy and non-compliant entropy is neglected in the presented energy balance

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equations and the entropy inequality. According to Epstein and Goriely (2012), a noncompliant property (such as velocity, internal energy, or entropy) is related to the differential values of that property for the entering mass (or volumetric mass source) and for the local substratum. Therefore, when we assume that the new mass enters with the same specific energy and specific entropy as the local substratum, the non-compliant internal energy and entropy are indeed neglected (Epstein and Maugin, 2000b; Alhasadi et al., 2019). Adopting the Helmholtz free energy per unit mass, Ψ ρ = u − θ η , then eliminating h between Eqs. (20) and (22) yields

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the following entropy inequality for the couple stress theory ⎛ DΨ ρ Dθ ⎞ ⎛ ⎞ qθ Dv 1 ρ⎜ + η ⎟ ≤ τ ( kl ) v(l ,k ) + µ klD vn,mk ε mnl + mk ⎜ f k − k − Ψ ρ ,k − θ ,k η ⎟ − k ,k . Dt ⎠ 2 Dt θ ⎝ ⎠ ⎝ Dt

In writing this result, the relation

t ( kl )v l ,k = t ( kl )v ( l ,k )

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was used.

(23)

Journal Pre-proof 3.4. Constitutive modeling and dissipation inequality In view of the fact that in finite deformation couple stress theory the internal energy u per unit mass for a homogeneous material is a function of E and K , the Helmholtz free energy Ψ ρ = u − θ η is considered as a function of E , K and q , i.e. Ψ ρ = Ψ ρ ( E,K,θ ;X) . With the

DΨ ρ Dt

=

∂Ψ ρ DE MK ∂E MK

Dt

+

∂Ψ ρ DK AM ∂K AM

Dt

+

∂Ψ ρ Dθ ∂θ Dt

.

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components of these two tensors as well as Dθ / Dt , i.e.

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aid of the chain rule, the material time derivative can be written in terms of the rate of the

(24)

The rate of the components of the Lagrange strain tensor E is related to the components of the

DE MK = Dt

1 2

( vk ,l

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spatial velocity gradient as (Eringen, 1980)

+ vl ,k ) FlM FkK = v( k ,l ) FlM FkK .

(25)

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Imposing the material time derivative on Eq. (6) leads to

(26)

DK AM DE MK ,N = ε ANK . Dt Dt

Substituting Eq. (25) into (26) yields

(

)

DK AM DE MK ,N = ε ANK = ε ANK v( k ,l ) N FlM FkK + v( k ,l ) FkK FlM ,N + v( k ,l ) FkK ,N FlM . Dt Dt

(27)

The last term on the right hand side is negligent because of the symmetry of FkK ,N with respect

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to its indices K and N coupled with skew-symmetry of e ANK with respect to the same two indices. Moreover, the first term in the parenthesis on the right-hand side can be rewritten as

v ( k ,l ) N =

¶v ( k ,l ) ¶X N

=

¶v ( k ,l ) ¶ x n = v ( k ,l ) n FnN , ¶ x n ¶X N

(28)

hence, we rewrite Eq. (27) as

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Journal Pre-proof

(

)

DK AM DE MK ,N = ε ANK = ε ANK v( k ,l )n FnN FlM FkK + v( k ,l ) FkK FlM ,N . Dt Dt

(29)

The determinant of a two-point tensor, e.g. A , is det (A) = (1/ 6) ε qnk ε ANK AqA AnN AkK . With −1 some algebraic manipulations, it reads as ε ANK AnN AkK = det (A) ε qnk AAq for non-singular A .

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Now, the two-point deformation gradient tensor F is as follows

ε ANK FnN FkK = J ε qnk FAq−1.

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Combining Eqs. (30) and (29) results in

DK AM 1 −1 −1 −1 = vk ,ln ε qnk J FAq FlM + v( k ,l ) J ε qnk FNn FAq FlM ,N . Dt 2

(30)

(31)

re-

In order to obtain the above formula, the relation v ( k ,l ) n e qnk = (1 / 2)v k ,ln e qnk was also used. Here, substituting Eqs. (31) and (25) into (24) produces

⎡ ∂Ψ ρ ⎤ ∂Ψ ρ ∂Ψ ρ Dθ 1 ∂Ψ ρ ⎡ε qnk J FAq−1 FlM ⎤ vk ,ln + =⎢ FlM FkK + J ε qnk FNn−1 FAq−1 FlM ,N ⎥ v( k ,l ) + . ⎦ Dt ∂K AM 2 ∂K AM ⎣ ∂θ Dt (32 ⎢⎣ ∂E MK ⎥⎦

urn al P

DΨ ρ

)

Now, substituting Eq. (32) into (23) and following the methodology of Coleman and Noll (1963) leads to the constitutive equations for the couple stress hyperelastic theory as

τ ( kl ) = ρ

∂Ψ ρ ∂E MK

η=−

∂Ψ ρ ∂K AM

FAl−1 FkM ,

Jo

µ klD = ρ J

⎤ ∂Ψ ρ 1 ⎡ ∂Ψ ρ FlM FkK + ρ J ⎢ ε qnk FNn−1 FAq−1 FlM ,N + ε qnl FNn−1 FAq−1 FkM ,N ⎥ , 2 ⎢⎣ ∂K AM ∂K AM ⎥⎦

∂Ψ ρ ∂θ

(33)

(34)

(35)

.

16

Journal Pre-proof Eqs. (33)-(35) present the thermoelastic constitutive equations for the couple stress theory with volumetric mass sources and mass fluxes in the current configuration. With such constitutive identification, the Clausius-Duhem inequality reduces to (36)

of

⎛ ⎞ qθ Dvk mk ⎜ f k − − Ψ ρ ,k − θ ,k η ⎟ − k ,k ≥ 0. Dt θ ⎝ ⎠

The left-hand side of this equation indeed represents the dissipation in the Eulerian (the spatial)

pro

form. Later in this section, its Lagrangian counterpart will also be provided. The dissipation inequality will be utilized in further sections in order to develop equations for the mass flux. The first two terms on the right-hand side of Eq. (20) represent the stress power per unit

reference volume as 1

re-

current volume. Multiplying those by J = det(F) results in the stress power P per unit

1

P = J t ( kl ) v ( l ,k ) + J µklD v n ,mk e mnl = J t s : sym(Ñv) + J µ D : ÑÑ ´ v. 2

urn al P

2

(37)

This stress power can also be written in terms of the Lagrangian parameters as (Mindlin and Tiersten, 1962)

S P = TMK

DE MK DK AM DE DK + Λ DAM = TS : + ΛD : , Dt Dt Dt Dt

(38)

where TS is the work conjugate to E , i.e. the symmetrical part of the well-known second Piola-Kirchhoff stress tensor, and the traceless tensor LD is the work conjugate to K . The

Jo

Eulerian stress tensors tS and µ D are related to the Lagrangian stress tensors TS and LD as follows (Mindlin and Tiersten, 1962)

τ ( kl ) =

1 S −1 −1 TMK FlM FkK + ⎡⎣ Λ DAM ε qnk FNn FAq−1 FlM ,N + Λ DAM ε qnl FNn FAq−1 FkM ,N ⎤⎦ , J 2 1

17

(39)

Journal Pre-proof (40)

µklD = Λ DAM FAl−1 FkM .

Now, the Eulerian form of the second law of thermodynamics presented in Eq. (23) can be transformed into the Lagrangian form as follows DΨ ρ Dt

− ρ0

⎛ ⎞ Q! DK AM D(FiJVJ ) Dθ S DE MK η + TMK + Λ DAM + M K ⎜ f i FiK + FiK − Ψ ρ ,K − θ ,K η ⎟ − K θ ,K ≥ 0, Dt Dt Dt Dt ⎝ ⎠ θ

(41)

of

− ρ0

where V is the inverse velocity vector. Moreover, the relations of other terms appeared in this

pro

equation with their Eularian counterparts as follows

Q! K = J qk FKk−1 ,

re-

M K = J mk FKk−1 ,

VK = − vk FKk−1 ,

(42)

(43)

(44)

have

− ρ0

DΨ ρ Dt

urn al P

In the isothermal case, the terms in which the temperature is present are omitted in order to

S + TMK

⎛ ⎞ DE MK DK AM Dvi + Λ DAM + M K ⎜ f i FiK − FiK − Ψ ρ ,K ⎟ ≥ 0. Dt Dt Dt ⎝ ⎠

(45)

4. Theory of uniformity

Jo

4.1. The couple stress material uniformity

According to the definition presented by (Epstein and Elzanowski, 2007; Noll 1967), a material body is generally said to be materially uniform if all of its points are materially isomorphic to one another. On the other hand, two material points are said to be materially isomorphic to each other when they are made of the same material, i.e. they have the same

18

Journal Pre-proof chemical identity. Henceforth, we confine our study only to local, elastic materials, where only the present state of deformation in an arbitrarily small neighborhood determines the point-wise constitutive response of any material point. In this way, the constitutive response is independent of the history of deformation and also independent of deformation in areas out of that small

of

neighborhood of the material point. Now we consider a material element of very small size for any material point of the

pro

body as the material surrounding the point in an arbitrarily small neighborhood. For a uniform body, the pictures of the two different material elements at a fixed time under a microscope are not necessarily the same, and maybe the atomic arrangement of one point is seen rotated and/or

re-

distorted with respect to the other. This possibly observable difference in the two pictures can be fixed by imposing a simple appropriate deformation to one of the material elements, without any detriment to its chemical identity. Therefore, at a fixed time, the constitutive response of

urn al P

any of the two material points, say X1 and X 2 , of a uniform body, which are materially isomorphic, can be mathematically related by pre-imposing some appropriate extra deformation to the original deformation of one of the two material elements with respect to its local reference configuration. The inverse of that pre-imposed deformation should be applied to the second material element in order for the constitutive responses to be mathematically identical. Therefore, the described mathematical relation of “being materially isomorphic” is symmetric (Epstein and Elzanowski, 2007).

Jo

The mathematical relation of being materially isomorphic is also reflexive, because any material point is trivially materially isomorphic to itself. Moreover, that relation is transitive, meaning that if X 2 and X 3 are also materially isomorphic, then X1 and X3 are materially isomorphic as well.

19

Journal Pre-proof In view of the mentioned three properties of symmetry, reflexivity and transitivity, it is deduced that the material isomorphism is of equivalence relation. As an important consequence of this equivalence relation, we are allowed to consider an arbitrary fixed material point, say

X 0 , on the uniform body, or not on the body but made of the same material of the uniform

of

body, to say every material point X of our uniform body is isomorphic to X 0 . The fixed material point X 0 is called “the archetypal material point” or “the crystal of reference”

pro

(Epstein and Elzanowski, 2007; Epstein and Maugin 1990, 1995).

For the sake of convenience, in the following we imagine the archetypal material point

X 0 is being placed outside of the body. The material archetype (or, the archetype, for short)

re-

is the cubic material element containing X 0 which is in a natural state when it is under no deformation. To achieve the constitutive response of any material point X in our uniform body

urn al P

with its own deformation state, we can pre-impose an appropriate extra deformation to its original deformation with respect to the local reference configuration. Now, we consider that total resulted deformation for the archetype. Then, we determine the constitutive response of the archetypal material point (please see Fig. 1)

For the hyper-elastic local bodies made of simple materials in the framework of the classical continuum theory, the mathematical formulation relating to the constitutive response of the material points of a uniform body to one another has been represented by (Epstein and

Jo

Elzanowski, 2007; Epstein and Maugin, 2000b). In this section, this mathematical formulation is extended to the framework of the couple stress theory. In the couple stress theory, the internal energy content per unit mass can be written in general as u = uˆ ( E, K; X ) ,

(46)

in the absence of thermal effects. The dependence of uˆ on the material point X may be because of a situation in which the different points of the body B are made of different materials, i.e. 20

Journal Pre-proof when we have a non-uniform body. On the other hand, the body may be still materially uniform, which is the case considered in this section. More specifically, the body is materially uniform if, and only if, there exists two uniformity fields described by P( X) and Q ( X) from the archetypal material point X 0 to each material point X Î B to allow us to write the constitutive

of

equation (46) for any material point of the body in a way that is not explicitly dependent on X . The fields described by P( X) and Q ( X) are discussed and explained in the following. Now,

pro

we consider the following definition for the uniform body in the framework of the couple stress

urn al P

re-

theory.

Jo

Fig. 1. The relation between archetypal, reference and current configurations

Definition 4.1. A couple stress-based body is materially uniform if there exists two tensor fields: one a second-order tensor field P(X) mapping the archetype in ! 3 onto the tangent space of material point X in the reference configuration, i.e.

21

Journal Pre-proof (47)

P(X) : ! 3 → TX B,

and the other a third-order tensor field Q(X) as (48)

Q(X) : ! 3 × ! 3 → TX B,

of

with QIαβ = QI βα , such that the general form of the constitutive equation (46) can be recast in

pro

the following form

ˆ IJ , K LI ; X A ) = u (Eαβ ( X A ), K ρα ( X A )), u(E where

re-

1 1 E ab (X A ) = E IJ PI a (X A )PJ b (X A ) + PI a (X A )PI b (X A ) - dab 2 2

urn al P

and

(49)

(50)

K ra (X A ) = J P K LI PI a (X A ) Pr-L1 (X A ) + E IJ Q I ag (X A ) PJ b (X A ) e rg b 1 + Q I ag (X A ) PI b (X A ) e rg b , 2

(51)

with the upper case, and Greek indices referring, respectively, to the reference configuration and the archetype. In fact, Eαβ is the pull-back of the Lagrange strain tensor

E IJ = (1/ 2)(FT F - I)IJ = (1/ 2) ( FkI FkJ - d IJ ) from the reference configuration to the

(

)

Jo

archetype, i.e. E ab = (1/ 2) Fk a Fk b - dab where Fk a = (FP ) k a = FkM PM a . Similarly, K ρα is the pull back of K IJ = (Ñ ´ E) IJ = -(1/ 2) e KNI FkJ ,N FkK from the reference configuration to the archetype, i.e. K ra = -(1/ 2) e gbr Fk g Fk a ,b = -(1/ 2) e gbr Fk g (FkM PM a ),b , which leads to

K ra = -(1/ 2) e gbr Fk g FkM ,b PM a - (1/ 2) e gbr Fk g FkM Q M a b with Q M a b as equivalent to PM a , b . Inspired by de León and Epstein, (1993), de Leon and Epstein, (1996) and Epstein and

22

Journal Pre-proof Elzanowski, (2007) who provided the terminology for uniformity fields for their own case, we call the two fields P( X) and Q ( X) , respectively, the first-order implants and the second-order implants from the archetypal material point to the body points. It should be noted that the inverse of the mapping imposed by the pair {P(X), Q(X)} on

of

the archetype to give the reference configuration for the material point X is not obtained by

pro

simply inverting each component of the pair and considering P -1 (X), Q-1 (X) (with the note that the inverse of a third-order tensor such as Q is not relevant). Indeed, the inverse of

! P(X) :TX B → " 3 , ! Q(X) :TX B × TX B → " 3 ,

−1

{

}

! ! = P(X), Q(X) , where

re-

{P(X), Q(X)} , denoted by {P(X), Q(X)}

(52) (53)

is a pair that when its composition with the pair itself is considered, it maps the archetypal

urn al P

configuration to itself. In fact, the archetypal configuration is the natural state with no rotation and no stretching, and also with no gradient of stretching and rotation. Therefore,

{P, Q} ! {P," Q" } = {I, 0}.

On the other hand, according to the compositional law given in (Epstein and Elzanowski, 2007), the component form for such a compositional transformation is

{P

}{

} {

}

, QIαβ ! P"α J , Q" ρ JK = PIα P"α J , QIαβ P"α J P"β K + Q" ρ JK PI ρ .

Jo

Iα

{

} {

(54)

}

In view of PIα P!α J , QIαβ P!α J P!β K + Q! ρ JK PI ρ = δ IJ , 0 , we can conclude that

P!α I ( X ) = Pα−1I ( X ),

(55)

Q!γ AD ( X ) = − QIαβ ( X ) Pγ−1I ( X ) Pα−1A ( X ) Pβ−1D ( X ).

(56)

23

Journal Pre-proof Now, it would be worth writing the transformations needed for relating the constitutive responses of two different material points X and Y of the uniform body. If we want to go from the tangent space TX B of the point X to the tangent space TY B of the point Y via the following transformations (de León and Epstein, 1993) (57)

of

P XY :TX B → TY B,

pro

and

Q X Y :T X B ´T X B ® TY B ,

(58)

{

}

−1 ! ! Q(X) It is an option to first go from X to the archetype via the pair {P(X), Q(X)} = P(X),

{P

XY IJ

} {

}{

re-

, then from the archetype to Y via the pair {P(Y), Q(Y)} . Hence, we can write

}

XY , QIAD = PIγ (Y ), QIαβ (Y ) ! P"α M ( X ), Q"γ AD ( X ) ,

(59)

urn al P

which according to the result of compositional transformation law given in (Epstein and Elzanowski, 2007) results in

PIJXY = PIα (Y )Pα−1J ( X ), and,

XY QIAD = PIγ (Y )Q!γ AD ( X ) + QIαβ (Y )Pα−1A ( X )Pβ−1D ( X ).

(60)

(61)

4.2. The Eshelby stress and hyperstress in the couple stress-based bodies

Jo

The Helmholtz free energy function Ψ R per unit reference volume, and the uniformity condition (49) can be written as Ψ R (E IJ , K LI ; X A ) = J P−1 Ψ R (Eαβ ( X A ), K ρα ( X A )),

24

(62)

Journal Pre-proof where ψ R is the Helmholtz free energy function per unit volume in the archetype. It is clear that if P = I and Q = 0 , then Ψ R expresses the value of the archetype energy function. Components of the symmetric part of the well-known second Piola-Kirchhoff stress tensor T S

TIJS =

of

as the work conjugate to the Lagrangian strain E is defined as

∂Ψ R . ∂E IJ

(63)

Λ DLI =

pro

Similarly, the deviatoric part of the Lagrangian hyperstress tensor ΛD is written as

∂Ψ R . ∂K LI

(64)

re-

Combining Eqs. (62)-(64) and using the chain rule of differentiation, results in

urn al P

∂Ψ R = J P TIJS Pα−1I Pβ−1J − Λ DLI QK µγ ε ργ β PLρ Pα−1K Pµ−1I , ∂Eαβ

∂Ψ R = Λ DLI PLρ Pα−1I . ∂K ρα

(65)

(66)

Subsequently, by using Eqs. (65) and (66), the derivatives of Eq. (62) with respect to P and Q , can be expressed as

∂Ψ R S = −Ψ R Pη−1A + TMC C AC Pη−1M + Λ DRL K LA Pη−1R + Λ DRL ε LKC (E RA,K − E RK ,A )Pη−1 C ∂PAη 1 −1 D J Λ C Q ε P P P −1 P −1 , 2 P ML AJ K µγ ργ β Lρ J β µ M η K

Jo

−

(67)

∂Ψ R 1 = − J P−1Λ DIL C BJ ε ρηβ Pθ−1I PLρ PJ β , ∂QBθη 2

(68)

25

Journal Pre-proof T where, the tensor C = F F is the well-known right Cauchy-Green deformation tensor. By

isolating Eqs (67) and (68) of the material implants P and Q , these equations can be expressed independently of the archetype. Therefore, the Eshelby stress and the Eshelby hyperstress for the couple stress theory are obtained, respectively, as

d NMB = −

of

∂Ψ R ∂Ψ R PNη − Q , ∂PAη ∂QAθη Nθη

pro

bNA = −

∂Ψ R P P . ∂QBθη Nθ Mη

re-

More clearly,

urn al P

S bNA = Ψ Rδ NA − TNC C AC − Λ DNL K LA − Λ DRL ( E RA,K − E RK ,A )ε LKN ,

1 d NMB = − Λ DNL C BJ ε LMJ . 2

(69)

(70)

(71)

(72)

It should be noted that in the absence of couple stress effects, tensor b becomes the common classical Eshelby stress tensor. In addition, according to the relation between the Mandel and Eshelby stress tensors that were investigated in Epstein and Elzanowski (2007), the Mandel stress and hyperstress tensors for the couple stress theory can be written as

Jo

β! = Ψ RI − b,

α! = −d.

(73)

(74)

26

Journal Pre-proof 5. The multiplicative decomposition approach to drive growth and remodeling evolution laws in the framework of the couple stress theory In this section, a widely used multiplicative decomposition approach is extended to the couple stress theory in order to represent growth and remodeling evolution laws. This

of

decomposition is developed based on the uniformity definition of the couple stress theory

pro

5.1. Multiplicative decomposition of the couple stress kinematics terms

The structural evolution of a medium that grows can be addressed by multiplicatively decomposing the involved kinematic tensors. The notion of multiplicative decomposition for the kinematic tensors in the growth process takes inspiration from the conventional

re-

decomposition of elasto-plastic deformation gradient into elastic and plastic parts (Bilby, 1957 Kroner, 1959; Lee, 1969). This idea was first applied to biomechanic cases of growth by

urn al P

(Rodriguez et al., 1994) as g FkK = FkKe FKK or F = F e F g .

(75)

It can be considered that F g ( X ,t) maps vectors of TX B into “relaxed” vectors of another vector space as (Ciancio et al., 2008; Micunovic, 2009; Goriely, 2016; Di Stefano et al., 2018)

F g ( X ,t) :TX B → N t ( X ),

(76)

and, F e ( X ,t) maps vectors of N t ( X ) into vectors of Tx B as (Di Stefano et al., 2018) (77)

Jo

F e ( X ,t) : N t ( X ) → Tx B,

where the vector space N t ( X ) with coordinate chart {X K } is the intermediate configuration with material state of stress-free or of naturality. This intermediate configuration can be achieved by distorting the material elements in TX B or in Tx B , in a generally incompatible way. It ought to be noted that the intermediate configuration is not a true configuration; therefore, a growth mapping cannot be defined from the reference configuration to the 27

Journal Pre-proof intermediate one. The intermediate configuration is just a collection of natural states of the different material elements. Consequently, none of F g ( X ,t) and F e−1 ( X ,t) can be initially considered as the tangent maps of deformations that determine a configuration of the material as a subset of the Euclidean space.

of

In the framework of the couple stress theory, in which the second gradient of the deformation is also involved in the constitutive equation through the tensor K = ∇ × E , in

pro

addition to tensor F g , each point of the body in the reference configuration is also linked by means of a third-order symmetric tensor H g to the intermediate configuration as (78)

re-

H g ( X ,t) :TX B × TX B → N t ( X ),

g g = H IKJ with the symmetry H IJK . Similarly, in order to link each point of the body in the

intermediate configuration to the current configuration, alongside with the tensor F e , a third-

urn al P

order symmetric tensor H e is proposed as

H e ( X ,t) : N t ( X ) × N t ( X ) → Tx B,

(79)

e e = H iKJ with the symmetry H iJK . In this way, the pair {F, ∇F} is multiplicatively decomposed

as

{F, ∇F} = {F , H } ! {F , H } , e

e

g

g

(80)

where the components of the gradient of the deformation gradient tensor ∇F are

Jo

FiI ,J = ∂φi ( X K ,t) / ∂ X I ∂ X J = φi,IJ ( X K ,t).

(81)

In this step, by virtue of the uniformity theory in the framework of the multiplicative decomposition, the first- and second-order growth elements of the deformation should be related to the implant transformations as follows

28

Journal Pre-proof

{F , H } = {P,Q} = {P,! Q! } , g

(82)

−1

g

which gives

FαgI = Pα−1I ,

(83)

H g γ AD = − QIαβ Pγ−1I Pα−1A Pβ−1D .

(84)

of

In this step, by considering the component form of Eq. (80) and the rule for compositional

{F

e kK

}{

pro

transformation in components presented in Eq. (54), we can write

} {

} {

}

e g g g e g g g , H kKN ! FKK , H KKN = FkKe FKK , H kKN FNN FKK + FkKe H KKN = FkK , FkK ,N ,

e g g g H kKN FNN FKK + FkKe H KKN = FkK ,N ,

and consequently

Jo

urn al P

e g g−1 g−1 H iKN = FiI ,J FJNg−1 FIKg−1 − FiBe H BAD FAK FDN .

re-

which gives

29

(85)

(86)

(87)

Journal Pre-proof Fig. 2. Illustration of multiplicative decomposition involving the Lagrangian finite strain tensor E and admissible couple stress tensor K .

B 0 , B and B are the reference configuration, the intermediate

By substituting Eq. (75) into (5) and (6), it can be written

and,

(

pro

e g g g E MK = E AB FAM FBK + E MK ,

of

configuration and the current configuration respectively.

)

e g g g e g g g K AM = ε ANK RABN FNN FAM FBK + E AB H AMN FBK + RMKN ,

where,

(F 2 1

e RABN =

e kB

(H 2 1

FkAe − δ AB e kBN

), E

g MK

e FkAe + FkBe H kAN

=

(F 2 1

), R

g AM

g MKN

g FAK − δ MK

=

(H 2 1

),

re-

e E AB =

g AMN

g g g FAK + FAM H AKN

(88)

(89)

(90)

).

urn al P

In addition, in view of Eq. (30) for the two-point tensor F g , we obtain

g−1 g g J g ε DNB FAD = ε ANK FNN FBK , J g = det (F g ).

(91)

Substitution of Eq. (91) into Eq. (89) leads to

(

)

e g−1 g e g g g K AM = J g K DA FAD FAM + ε ANK E AB H AMN FBK + RMKN ,

where,

e e K DA = RABN ε DNB .

(92)

(93)

Jo

In the condition where there is uniformity in the material body, the Helmholtz free energy per unit volume in the reference configuration, which in general condition is described as Ψ R (E MK , K AM ; X I ) , can be written as a function as Ψ R (E MK , K AM , P!AM ( X I ,t), Q! AMN ( X I ,t)) , g g ( X I ,t), H AMN ( X I ,t)) . Finally, based on or in view of Eq. (82) in the form of Ψ R (E MK , K AM , FAM

the multiplicative decomposition, we can write

30

Journal Pre-proof ! (E e , K e ), Ψ R (E MK , K AM ; X I ) = J g Ψ R AB DA

(94)

! is the Helmholtz free energy functions per unit volume in the intermediate where, Ψ R configuration. According to works by Epstein and Maugin (2000b), and Epstein and Elzanowski (2007), the material evolution laws should satisfy some general formal restrictions.

of

The first restriction implies that the evolution laws do not explicitly depend on the body point X . Now that we have Eq. (94), on the right hand side of which the dependence of the energy

pro

function on X does not explicitly appear, the way has been paved for the first restriction to be met. Here before proceeding to work on Ψ R for later use, we prescribe a general form of the constitutive equation for the vector-type mass flux M as a function of the same variables of

g g M J = M J (E MK , K AM , FAM ( X I ,t), H AMN ( X I ,t)).

re-

the energy function Ψ R in the following form

(95)

urn al P

The material time derivative of Eq. (94) yields

g ! ( E e , K e ) DE e DΨ R ( E MK , K AM ; X I ) ∂Ψ g−1 DFKK ! e e R AB DA AB = J g FKK Ψ R ( E AB , K DA )+ Jg e Dt Dt Dt ∂E AB

+J

g

(96)

! ( E e , K e ) DK e ∂Ψ R AB DA DA e ∂K DA

Dt

.

The rate of the Helmholtz energy per unit mass Ψ ρ is related to its rate per unit reference volume Ψ R as DΨ ρ Dt

= ρ0

⎛ DF g g −1 ⎞ DΨ R D ⎛ Ψ R ⎞ DΨ R 1 D ρ0 = − Ψ = − tr ⎜⎝ Dt F ⎟⎠ Ψ R . Dt ⎜⎝ ρ0 ⎟⎠ Dt ρ0 Dt R Dt

Jo

ρ0

(97)

! , where the To get the relation above, it suffices to see that Ψ R = ρ0 Ψ ρ = ρ g J g Ψ ρ = J g Ψ R

constant ρ g with a vanishing time rate is the mass density in the intermediate configuration,

(

) (

)

(

)

g g g g −1 and (1/ ρ0 ) D ρ0 / Dt = 1/ det(F ) D det (F ) / Dt = tr (DF / Dt)F .

31

Journal Pre-proof By substituting DΨ R / Dt from Eq. (97) into Eq. (96), it can be written

ρ0

DΨ ρ

=J

Dt

g

! DE e ∂Ψ R AB e ∂E AB

Dt

+J

g

! DK e ∂Ψ R DA e ∂K DA

Dt

(98)

.

This result obtained for ρ0 DΨ ρ / Dt will be substituted into Eq. (45). Moreover, in Eq. (45),

of

we want to replace the rates of E MK and K AM with appropriate expression in terms of elastic and growth parts of the relevant kinematic tensors. To do that, in view of Eqs. (88) and (92)

pro

we write

e g g g DE MK DE AB DE MK g g e DFAM g e g DFBK = FAM FBK + E AB FBK + E AB FAM + , Dt Dt Dt Dt Dt

re-

and

e g g ⎛ e DFNN DK AM DK DA g−1 g g g e g DFAM g = Jg FAD FAM + ε ANK ⎜ RABN FAM FBK + RABN FNN FBK Dt Dt Dt Dt ⎝

+

g NN

F

g AM

F

(100)

g e g g DFBK DE AB DFBK g g e DH AMN g e g + H AMN FBK + E AB FBK + E AB H AMN Dt Dt Dt Dt

urn al P

+R

e ABN

(99)

g ⎞ DRMKN . Dt ⎟⎠

By these substitutions, Eq. (45) can now be read as e ! ⎛ g ∂Ψ ⎞ DE AB g g g g R − J + TMK FAM FBK + Λ AM ε ANK H AMN FBK ⎜ ⎟ e Dt ⎝ ∂E AB ⎠

+

(101 )

e ! ⎛ g ∂Ψ ⎞ DK DA g R −J + J g Λ AM ( F g−1 ) AD FAM ⎜ ⎟ e Dt ⎝ ∂K DA ⎠

Jo

DFJIg ⎛ 1 1 e g e g g e g g + TIK E JB FBK + TMI E AJ FAM + TIK FJKg + TMI FJM + Λ AM ε AIK RABJ FAM FBK ⎜ Dt ⎝ 2 2 ⎞ 1 e g g e g g e g g +Λ AI ε ANK RJBN FNN FBK + Λ AM ε ANI RAJN FNN FAM + Λ AM ε ANI E AJ H AMN + Λ AM ε ANI H JMN ⎟⎠ 2 +

g DH AMN Dt

⎛ ⎞ Dvi ⎛ 1 e g g ⎞ ⎜⎝ Λ AM ε ANK E AB FBK + 2 Λ AM ε ANK FAK ⎟⎠ + M K ⎜⎝ f i FiK − Dt FiK − Ψ ρ ,K ⎟⎠ ≥ 0.

In the above equation, the terms on the left hand side, excluding the last one with M K , represent the stress power in the decomposed form into elastic and anelastic parts, with the 32

Journal Pre-proof e e terms containing rates of E AB or K AD corresponding to reversible elastic energy while others g containing rates FJIg or H AMN corresponding to irreversible anelastic energy. Since this

equation must be satisfied in any conditions, the following set of constitutive equations in the

∂E AB e

! 1 ⎡ ∂Ψ g−1 g−1 g g−1 g g−1 FMA FKB − ⎢ eR FDA FOK ε ANK H AON FMA 2 ⎣ ∂K DK

+

! ∂Ψ R ∂K DA e

g g−1 FDA FMA ,

∂K DK e

(102)

⎤ g g−1 g g−1 FDA FOK ε ANM H AON FKA ⎥, ⎦

(103)

re-

Λ DAM =

! ∂Ψ R

pro

! ∂Ψ R

S TMK = Jg

of

framework of the couple stress theory is obtained

Furthermore, by using Eqs. (39) and (40), the Eulerian version of those equations can be

stress tensor as ! ∂Ψ R

τ ( kl ) = J e

µklD =

∂E AB e

! ∂Ψ R ∂K DA e

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derived for the symmetric part of the Cauchy stress tensor and the deviatoric part of the couple

FlAe FkBe +

! ∂Ψ R

∂K DM e

e ε DKN FkKe H lMN ,

FDle−1 FkAe ,

(104)

(105)

where J e = det (F e ) . Therefore, the inequality of Eq. (101) reduces to the following dissipation

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inequality:

! ! ! ⎡ ∂Ψ ⎤ ∂Ψ ∂Ψ e e e e R R LgJA ⎢ J g eR C JB + Jg K JD + Jg ( RCBJ − RCJB ) ε DAB ⎥ e e ∂K DA ∂K DC ⎣ ∂E AB ⎦ g ! ⎡⎛ DH JMN ⎞ g−1 g−1 ⎤ ⎛ ⎞ ∂Ψ Dv 1 e g R + Jg C ε − H CMN LgJC ⎟ FNF FMA ⎥ + M K ⎜ f i FiK − i FiK − Ψ ρ ,K ⎟ ≥ 0, ⎢ JB DFB ⎜ e 2 Dt ∂K DA ⎝ ⎠ ⎠ ⎢⎣⎝ Dt ⎥⎦

33

(106 )

Journal Pre-proof where C e = F eTF e and Lg =

DF g g−1 F is infelicitously called the inhomogeneity or remodeling Dt

velocity gradient in the intermediate configuration. It should be noted that Lg does not indeed represent the gradient of any velocity field, because F g is not the gradient of any vector-like

of

field. In Eq. (106), the energy conjugate to LgJA plays the role of the Mandel stress tensor in the

β JA = J g

! ∂Ψ R e ∂E AB

e C JB + Jg

! ∂Ψ R e ∂K DA

e K DJ + Jg

! ∂Ψ R e ∂K DC

pro

intermediate configuration β , i.e.

e e ( RCBJ − RCJB ) ε DAB ,

(107)

g g g−1 g−1 / Dt − H CMN LgJC )FNF FMA represents the Mandel and the energy conjugate to (DH JMN

1

! ∂Ψ R

2

e ∂K DA

α AJF = J g

re-

hyperstress tensor in the intermediate configuration α , i.e.

e C JB ε DFB .

(108)

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It might be useful to write the Mandel stress tensors in terms of the stress tensors and the

! / ∂E e and ∂Ψ ! / ∂K e from (102) and (103) kinematic tensors. To this end, we extract ∂Ψ R AB R DA as

! ∂Ψ 1 S g g g R = g TMK FAM FBK + Λ DAM K AM δ AB , e ∂E AB J

(109)

! ∂Ψ g g−1 R = Λ DAM FAM FAD . e ∂K DA

(110)

)

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(

So, we can write

S g g e g e g g−1 e g g−1 e e β JA = TMK FAM FBK C JB + Λ DAM K AM C JA + J g Λ DAM FAM FAD K DJ + J g Λ DAM FCM FAD ( RCBJ − RCJB ) ε DAB ,

34

(111 )

Journal Pre-proof (112)

1

g g−1 e α AJF = J g Λ DAM FAM FAD C JB ε DFB .

2

Since the inequality of Eq. (106) should be satisfied at all times, including when the mass flux

M = 0 where the total dissipation corresponds to the stresses, and when the mass flux is the

g ⎡⎛ DH JMN ⎞ g−1 g−1 ⎤ g D s = β JA LgJA + α AJF ⎢⎜ − H CMN LgJC ⎟ FNF FMA ⎥ ≥ 0 ⎠ ⎢⎣⎝ Dt ⎥⎦

pro

and

of

sole source of dissipation, we can conclude

⎛ ⎞ Dv D m = M K ⎜ f i FiK − i FiK − Ψ ρ ,K ⎟ ≥ 0, Dt ⎝ ⎠

(113)

(114)

re-

where D s and D m are respectively the mechanical and mass dissipation. These two results will be used in following sections to present the growth and remodeling evolution laws and mass

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flux constitutive equation.

5.2. The mass transport constitutive law

Eq. (114) enforces suitable thermodynamical constraints for extracting the mass flux constitutive equation. This equation expresses the internal product of two vectors ( M and FT .f − FT .( Dv / Dt) − ∇ R Ψ ρ ) and should be greater than or equal to zero. An admissible form

for the mass flux constitutive equation, which satisfies Eq. (114) can be expressed in a close

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relation to the inverse velocity V

(115)

⎛ ⎞ ! V.⎛ FT .f − FT . Dv − ∇ Ψ ⎞ V, M= M R ρ⎟⎟ ⎜⎝ ⎜⎝ Dt ⎠⎠

where M! is a positive-definite scalar function. The material gradient of the Helmholtz energy g g ( X I ,t), H AMN ( X I ,t)) that appeared in Eq. (115) is function per unit mass Ψ ρ ( E MK , K AM , FAM

now decomposed as

35

Journal Pre-proof ∂Ψ ρ ∂XK

=

∂Ψ ρ ∂E MN ∂E MN ∂ X K

+

∂Ψ ρ ∂K MN ∂K MN ∂ X K

+

g ∂Ψ ρ ∂FAM g ∂XK ∂FAM

+

g ∂Ψ ρ ∂H AMN g ∂XK ∂H AMN

(116)

,

S g / ρ0 and ∂Ψ ρ / ∂K MN = Λ DMN / ρ0 . In order to find ∂Ψ ρ / ∂FAM where ∂Ψ ρ / ∂E MN = TMN and

g ∂Ψ ρ / ∂H AMN , in view of the dependency of the Helmholtz energy function to the variables as

of

g g e e Ψ ρ ( E MK , K AM , FAM ( X I ,t), H AMN ( X I ,t)) or Ψ ρ (E AB , K DA ) , from its material time derivative we

∂Ψ ρ DE AB

∂Ψ ρ DK AD

+

+

g ∂Ψ ρ DFAM

+

pro

can write g ∂Ψ ρ DH AMN

∂E AB Dt ∂K AD Dt Dt ∂FAM Dt ∂H AMN !### #"#### $ !#### #"##### $ g

totalmechanical power

g

mechanicaldissipation power

=

e ∂Ψ ρ DE AB

+

e ∂Ψ ρ DK DA

∂E AB Dt ∂K DA Dt !### #"#### $ e

e

,

(117)

mechanicalreversible power

is

clear

that

the

re-

The total mechanical power has also been presented on the left hand side of Eq. (101), and it mechanical

reversible

power

e e e e (∂Ψ ρ / ∂E AB )( DE AB / Dt) + (∂Ψ ρ / ∂K DA )( DK DA / Dt) has been represented there by all the

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terms which correspond to the elastic deformation. The combination of these facts yields g g g ⎛ DF g ⎞ (118 ⎡⎛ DH JMN ∂Ψ ρ DFAM ∂Ψ ρ DH AMN ⎞ g−1 g−1 ⎤ g−1 g JM + = − ρ0−1 ⎜ FMA [β JA ] + ⎢⎜ − H CMN LgJC ⎟ FNF FMA ⎥ (α AJF )⎟ . g g ) ∂FAM Dt ∂H AMN Dt ⎠ ⎢⎣⎝ Dt ⎥⎦ ⎝ Dt ⎠

From this important result we get ∂Ψ ρ ∂FJIg

(

)

g g−1 g−1 = − ρ0−1 β JA FIAg−1 − α AJF H CMN FICg−1 FNF FMA ,

and

g ∂H JMN

(

)

g−1 g−1 = − ρ0−1 α AJF FNF FMA .

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∂Ψ ρ

(119)

(120)

Substituting Eqs. (119) and (120) into Eq. (116), then putting the result into Eq. (115), we can write

36

Journal Pre-proof ⎛⎛ Dv ⎛1 ⎞ S M L = M! ⎜ ⎜ f i FiK − i FiK − TMN E MN ,K − Λ DAM ⎜ ε ANB FkM ,N Fk ,BK ⎟ + ρ0−1 β JA FIAg−1 Dt ⎝2 ⎠ ⎝⎝

(

)

(121)

) )

g g−1 g−1 g +α AJF H CMN FICg−1 FNF FMA H JIK VK VL .

It should be noted that Eq. (121) is in agreement with the vision of Coleman and Gurtin (1967) in the sense that the terms with orders higher than that of ∇ RF have been discarded. That is

of

because according to Eq. (95) the mass flux should be represented as a function of variables

pro

with an order which is less than ∇ RF . 5.3. The evolution laws for material implants

When we study an initially uniform body undergoing processes of remodeling and

re-

growth, its material points do not change in chemical identity; therefore, the body remains uniform. On the other hand, each material element in the body evolves during the passing of time due to those processes. The difference between the evolution of the material elements

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changes the pattern of inhomogeneity in the body.

To study the material evolution, we can consider one and the same material element at point X at different instants of time, and investigate its constitutive response, because the material evolution is a point-wise phenomenon. The temporal change of the selected material element can be expressed as material evolution laws which identify the governing relation between all quantities that act as driving forces and the time evolution of that point. To present a particular theory of material evolution, we should provide an evolution law describing the

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way in which evolution happens, usually in the form of a system of first-order ordinary differential equations for the implant tensors (in the case of our couple-stress based growth and remodeling processes those are tensors P and Q) as a function of time only. The material evolution laws express the governing relation between the time rate of the material inhomogeneities and all quantities that act as driving forces of the evolution process, which is the temporal change in the material implants. 37

Journal Pre-proof In the following, the evolution laws for the material growth and remodeling processes in the framework of the couple stress theory are presented through the second principle of thermodynamics. In view of Eq. (113), the mechanical dissipation can be described as a function of the form DF g DH g DF g DH g , , α, β) = A1 : + A2 ! , Dt Dt Dt Dt

of

D s = D s (F e ,H e ,F g ,H g ,

(122)

pro

where A1 and A2 are conjugate thermodynamic forces of the thermodynamic fluxes DF g / Dt and DH g / Dt , respectively. In the following, we enforce the Principle of Maximum Dissipation (Hackl and Fischer, 2007) on Eq. (122). To this end, a Lagrangian LA is considered

re-

as

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⎛ ⎛ DF g DH g ⎞ ⎞ LA = D s + ζ ⎜ D s − ⎜ A1 : + A2 ! , Dt Dt ⎟⎠ ⎟⎠ ⎝ ⎝

(123)

where ζ is a Lagrange multiplier. In order to achieve the maximum dissipation, we should solve ∂LA / ∂A1 = 0 and ∂LA / ∂A2 = 0 , which leads to 1

2

∂D s DF g −ζ = 0, ∂A1 Dt

(124)

∂D s DH g (1+ ζ ) −ζ = 0. ∂A2 Dt

(125)

(1+ ζ )

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Multiplicating Eq. (124) by A1 and Eq. (125) by A2 , then solving for DF g / Dt and DH g / Dt , we get the following two equations (126)

D s ∂D s DF g = = f (F g ,H g ,α,β), ∂D s Dt ∂A : A1 1 ∂A1

38

Journal Pre-proof (127)

D s ∂D s DH g = = g(F g ,H g ,α,β). ∂D s Dt ∂A2 ! A2 ∂A2

where f and g are some tensor-valued functions. Eqs. (126) and (127) are the evolution laws for the couple stress theory. According to Epstein and Maugin (2000b), and Epstein and

of

Elzanowski (2007), the chosen particular reference configuration should not affect the evolution laws. In this respect, we also consider the evolution laws in another arbitrarily chosen

pro

reference configuration, governed by the following transformation

YM = YM ( X K ),

(128)

and write the following new form for the evolution equations

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re-

DFˆ g ˆ ˆ g ˆ g ˆ ˆ β), = f (F , H , α, Dt

ˆg DH ˆ ˆ g , α, ˆ β), ˆ Fˆ g , H = g( Dt

where

(129)

(130)

g g ∂XK FˆKM = FKK ,

(131)

∂XK ∂XL ∂2 X K g g g Hˆ KMN = H KKL + FKK ,

(132)

∂YM

∂YM ∂YN

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∂YM ∂YN

g g DFˆKM DFKK ∂XK = , Dt Dt ∂YM

(133)

g g g DHˆ KMN DH KKL ∂ X K ∂ X L DFKK ∂2 X K = + , Dt Dt ∂YM ∂YN Dt ∂YM ∂YN

(134)

39

Journal Pre-proof

It should be noted that through the transformation of the reference configuration the Mandel stress tensor and the Mandel hyperstress tensors do not change, as is the case for F e and H e , thus we have

αˆ = α,

of

(135)

pro

βˆ = β.

(136)

Substituting Eqs. (131), (132), (135) and (136) into Eqs. (129) and (130), then the obtained results into Eqs (133) and (134), we reach

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re-

⎛ g ∂XK ⎞ ∂XK ∂XL ∂2 X K ∂X g g g g fˆKM ⎜ FKK , H KKL + FKK , α AJF , β JA ⎟ = f KK (FKK , H KKL , α AJF , β JA ) K , ∂YM ∂YM ∂YN ∂YM ∂YN ∂YM ⎝ ⎠

(138)

⎛ g ∂XK ⎞ ∂XK ∂XL ∂2 X K g g gˆ KMN ⎜ FKK , H KKL + FKK , α AJF , β JA ⎟ ∂YM ∂YM ∂YN ∂YM ∂YN ⎝ ⎠ g g = g KKL (FKK , H KKL , α AJF , β JA )

∂XK ∂XL ∂YM ∂YN

g g + f KK (FKK , H KKL , α AJF , β JA )

(137)

∂2 X K ∂YM ∂YN

.

As a particular case, we chose the change of the reference configuration to be

∂Y = Fg . ∂X

(139)

Now, from Eqs. (137) and (138) we have

(140)

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g g g−1 f KN (δ KN , 0, α AJF , β JA ) = f KK (FKK , H KKL , α AJF , β JA )FKN ,

(

)

g g g g g−1 g g−1 g−1 g KMN (δ KN , 0, α AJF , β JA ) = g KOS (FKK , H KKL , α AJF , β JA ) − f KK (FKK , H KKL , α AJF , β JA )FKA H AOS FOM FSN .

40

(141 )

Journal Pre-proof Recalling that

g g f KK (FKK , H KKL , α AJF , β JA )

and

g g g KOS (FKK , H KKL , α AJF , β JA )

are indeed

g g presenting the components F!KK and H! KOS , respectively, from Eqs. (140) and (141) we obtain

g g F!KK = f KN (α AJF , β JA )FNK ,

of

(142)

g g g H! KOS = FMO FNSg g KMN (α AJF , β JA ) + H NOS f KN (α AJF , β JA ).

(143)

pro

These two equations demonstrate the evolution equations in the intermediate configuration. The Mandel stress tensors β and α are the driving stresses for the first- and second-order material inhomogeneities. In the simplest case, the second-order tensor valued function

re-

f (α,β) can be assumed to be coaxial with the Mandel stress tensor β , and the third-order tensor valued function g(α,β) can be assumed to be coaxial with the Mandel hyperstress

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tensor α , i.e. g g F!KK = k1 β KN FNK ,

(144)

g g g H! KOS = k2 α KMN FMO FNSg + k1 β KN H NOS ,

(145)

with k1 and k2 as some positive material constants which play the role of classical and couple stress remodeling stiffnesses.

6. An example with numerical illustrations

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6.1. Definition and formulation

In this section, a specific case is numerically investigated in order to help us better understand the growth processes under the framework of the couple stress theory. We consider a very small cubic chunk made of an isotropic material with the following Helmholtz energy per unit volume in the intermediate configuration function as (Ristinmaa and Vecchi, 1996)

41

Journal Pre-proof

(

! = µ tr(C e ) − 2ln Ψ R 2

(

))

det(C e ) +

(146)

κ e K : D : Ke, 2

where the classical material constant µ is the shear modulus, κ is a higher-order material constant which is written as µl 2 with l denoting the material length scale parameter, and D a

of

fourth-order tensor whose components are DIJKL = 2 (δ KIδ LJ + κ ′δ KJ δ LI ) with −1 < κ ′ < 1 . In the example κ ′ is assumed to be zero. With this energy function, we have

pro

! ! ∂Ψ ∂Ψ R R = 2 = µ (I − C e−1 ), e e ∂E ∂C

re-

! ∂Ψ R = 2κ K e . e ∂K

(147)

(148)

Consequently, for the given material, the constitutive Eqs. (102) and (103) result in

(

)

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T s = J g µF g−1 (I − C e−1 ) F g−T − 2κ tr K eT F g K g F g−1 F g−1F g−T ,

ΛD = 2κ F gT K eF g−T .

(149)

(150)

The couple stress tractions on the boundary areas of the cubic element are assumed such that we have the following form of the uniform Lagrangian couple stress distribution in it

⎡ 0 −b(t) b(t) ⎤ ⎢ ⎥ [Λ] = ⎢ b(t) 0 −b(t) ⎥ . ⎢ ⎥ 0 ⎥ ⎢⎣ −b(t) b(t) ⎦

(151)

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Moreover, the gross uniform distribution of the second Piola stress is assumed to be of the form T = a(t)I.

(152)

The values of the stress components are considered to be oscillatory of the form

a (t ) = a0 sin(w t ),

(153)

42

Journal Pre-proof b (t ) = b0 sin(w t ),

(154)

in which a0 and b0 denote the amplitudes and w is the angular frequency. As the initial conditions, we adopt F = I , F g = I , K = 0 and K g = 0 at the reference

elastic and growth parts of the basic kinematic tensors

pro

F = m (t ) I,

of

time t = 0 . Consequently, in view of equations in Section 5 we have the following forms for

re-

F g = h(t)I.

-n (t ) n (t ) ù é 0 ê [K ] = ê n (t ) 0 -n (t ) úú , êë -n (t ) n (t ) 0 úû

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e

⎡ 0 −g(t) g(t) ⎤ ⎢ ⎥ [K g ] = ⎢ g(t) 0 −g(t) ⎥ . ⎢ ⎥ 0 ⎥ ⎢⎣ −g(t) g(t) ⎦

(155)

(156)

(157)

(158)

With the aid of Eqs. (111) and (112), we can now write the following results for the Mandel stresses β and α as:

⎛ m2 (t) ⎞ β=⎜ 2 a(t) h2 (t) + 6 b(t) g(t) + 6 b(t) n(t) h3 (t)⎟ I, ⎝ h (t) ⎠

(159)

1 α 112 = α 113 = α 221 = α 223 = α 331 = α 332 = b(t) m2 (t) h(t), 2 1 α 121 = α 131 = α 212 = α 232 = α 313 = α 323 = − b(t) m2 (t) h(t), 2

(160)

)

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(

43

Journal Pre-proof and other α I J K components vanish. Substituting Eqs. (159), (160), (156) and (158) into the evolution laws (144) and (145), we get (161)

⎛ m2 (t) ⎞ dh(t) = k1 h(t) ⎜ 2 a(t) h2 (t) + 6 b(t) g(t) + 6 b(t) n(t) h3 (t)⎟ , dt ⎝ h (t) ⎠

)

(162)

pro

1 dg(t) 1 1 dh(t) k2 − g(t) 4 = b(t) m2 (t) h(t), 3 2 dt h (t) dt 2 h (t)

of

(

On the other hand, the combination of Eqs. (149)-(150) with (151), (152) and (155)-(158)

⎛ h2 (t) ⎞ µ ⎜ 1− 2 ⎟ = a(t) h2 (t) + 6b(t)g(t), ⎝ m (t) ⎠

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b(t) = 2κ n(t).

(163)

re-

yields

(164)

Substituting Eqs. (163) and (164) into (161) and (162) leads to the following set of coupled first-order differential equations for h(t) and g(t) as the functions governing the evolution of growth in the example under investigation:

⎛ ⎜ dh(t) 1 3 b2 (t)h3 (t) = k1 h(t) ⎜ a(t) h2 (t) + 6 b(t) g(t) + dt κ ⎜ 1− 1 a(t) h2 (t) + 6 b(t) g(t) ⎜⎝ µ

(

)

(

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1 dg(t) 1 1 dh(t) k2 b(t) h3 (t) − g(t) = 2 dt h3 (t) 2 ⎛ h4 (t) dt 1 2 ⎜⎝ 1− µ a(t) h (t) + 6 b(t) g(t)

(

44

)

)⎞⎟⎠

.

⎞ ⎟ ⎟, ⎟ ⎟⎠

(165)

(166)

Journal Pre-proof It is convenient to introduce dimensionless parameters to investigate the problem numerically. To this end, we define the dimensionless time τ = k1 µ t , the dimensionless stress α = a / µ , ! τ ) = l g(τ ) . With this change of the dimensionless couple stress β = b / µl , and γ = k2 l 2 / k1 , g(

⎛ α (τ ) h2 (τ ) + 6 β (τ ) g( ⎞ ! τ) dh(τ ) 2 3 = h(τ ) ⎜ + 3 β ( τ ) h ( τ ) ⎟, ⎜⎝ 1− α (τ ) h2 (τ ) + 6 β (τ ) g( ⎟⎠ dτ ! τ)

)

pro

(

of

variables, Eqs. (165) and (166) read

! τ) 1 ! τ ) dh(τ ) γ 1 dg( g( β (τ ) h3 (τ ) − = . 2 dτ h3 (τ ) h4 (τ ) dτ 2 1− α (τ ) h2 (τ ) + 6 β (τ ) g( ! τ)

6.2. Numerical results and discussion

))

(168)

re-

( (

(167)

The set of equations in (167) and (168) have been solved numerically by assuming

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certain values for the previously defined dimensionless parameters presented in table 1. The ! τ ) have been depicted in Figures 3 and 4 with β0 = 0.06 . results for h(τ ) and g(

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Table 1: Parameters employed in the numerical analysis

Since ρ0 (τ ) = ρ g (0) J g , where J g = h3 (τ ) is the determinant of F g , which is the growth part of the deformation gradient, it is clear that for h(τ ) > 1 we will have an increase in the material element mass density, meaning that there will be absorption of a new mass or in other words a

45

Journal Pre-proof positive growth. With h(τ ) < 1 , we have the opposite case, i.e. mass desorption or loss of mass. Figure 3 indicates a net increase in h(τ ) , i.e. the general tendency is a rise in h(τ ) , while the loading and stresses are purely periodic with zero net variation. This means that with a pure oscillatory loading, overall, we have a positive mass growth along with remodeling of the

of

macro-element, although that mass growth takes place in an oscillatory manner. A similar ! τ ) , which describes the situation is observed in Figure 4 for the growth component g(

pro

remodeling of the microstructure due to its rearrangement and re-accommodation. In other ! τ ) has a net increase. words, while the stresses are purely periodic, g(

To study the effects of the couple stress intensity on the remodeling and growth in the

re-

given problem, in Figure 5 the function h(τ ) has been depicted for different values β0 of the couple traction intensity, including β0 = 0 that represents the results of the classical theory. It

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is clear that the presence of the couple stresses gives more intense remodeling and higher positive growth, and with higher values of β0 , i.e. with higher couple stresses, we have more

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powerful remodeling and growth.

46

re-

pro

of

Journal Pre-proof

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Fig. 3. Remodeling induced in macro-element by oscillatory tractions

47

Journal Pre-proof

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re-

pro

of

Fig. 4. Remodeling induced in microstructure by oscillatory tractions

Fig. 5. Remodeling induced in macro-element by oscillatory tractions

Figure 6 shows the relative density ρ0 (τ ) / ρ g = h3 (τ ) for different values of couple stress intensity β0 . The green curve corresponding to the classical theory with β0 = 0 relatively represents the lowest increase in ρ0 (τ ) / ρ g . Other curves indicate that with higher couple

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stresses we have a greater increase in the relative density.

48

re-

pro

of

Journal Pre-proof

7. Conclusions

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Fig. 6. Variation of the relative mass density versus time for different intensity of the couple stresses

In this study, a theory for growth and remodeling processes in finite deformation has been formulated in the framework of the generalized continuum mechanics of couple stress theory. The developed theory can also be used to capture small-scale effects of the microstructure. The couple stress theory also provides the potential for incorporating mass production and mass

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flux phenomena into the formulation, which is unlike the case in classical continuum mechanics. The main balance equations, including the mass, linear momentum, angular momentum and internal energy balances together with the entropy inequality have been derived. The configurational forces of the Eshelby and Mandel stresses for local rearrangement of material inhomogeneities for both the classical and couple stresses are presented; this is done

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Journal Pre-proof in view of the material uniformity theory. Moreover, with the aid of the Clausius-Duhem inequality the hyperelastic constitutive equations have been obtained by multiplicatively decomposing the Lagrangian strain tensor and its curl into elastic and growth parts. Additionally, the two evolution laws for the classical and second-order tensors F g and H g of

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the material growth process have been presented.

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Acknowledgments

We would like to express our sincere gratitude to Dr Marcelo Epstein for his guidance and

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comments on this study.

References

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Highlights • The mathematical formulation for material growth and remodeling processes in finite deformation is developed based on the couple stress theory. • The balance equations of mass, linear and angular momentums, as well as internal energy together with the entropy inequality are introduced in the presence of the mass flux. • Within the framework of material uniformity, the Eshelby and Mandel stress tensors as driving or configurational forces for local rearrangement of the first- and second-order material inhomogeneities are determined. • The basic kinematic tensors are multiplicatively decomposed into elastic and anelastic parts. • The hyper-elastic constitutive equations with respect to both reference and current configurations are obtained.

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Signature

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1. Mohammadjavad Javadi Sigaroudi __________________

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value 0.1 (Epstein and Elzanowski,2007)

Symbol 𝛼0 = 𝑎0 /𝜇

0-0.06

𝛽0 = 𝑏0 /𝜇𝑙

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20 (Epstein and Elzanowski,2007) 1

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Parameter the dimensionless force traction amplitude the dimensionless couple traction amplitude the dimensionless angular frequency the dimensionless remodeling stiffness

𝜔/𝑘1 𝜇 𝑘2 𝑙2 /𝑘1