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Abstract

Based on micropolar theory, we develop a novel mathematical framework for modeling finite elastoplastic deformations. The proposed formulation accommodates both large strains and finite rotations while exploiting the ability of micropolar theory to represent underlying material microstructure. The balance equations are obtained from an invariance principle with respect to general observer transformations. Within the setting of material uniformity, the plastic evolution laws are expressed as first-order differential equations for a set of material transplants, subject to the formal restrictions dictated by micropolar material symmetries and the constraints of the second law of thermodynamics. In addition, we identify the micropolar Mandel stress tensors as the energetic driving forces governing the local rearrangement of material inhomogeneities.

Keywords

Non-classical continuum mechanics micropolar theory microstructure finite plasticity the Mandel stress material uniformity

1. Introduction

Formulating material evolution laws, whether in plasticity or in growth and remodeling, is challenging even for first-order materials, and becomes considerably more complex in non-classical continua, particularly in the micropolar continuum [1 –3], due to the influence of internal microstructure. The foundations of theories describing materials with internal structure trace back to the seminal work of the Cosserat brothers [4]. In their formulation, a continuum is endowed with additional kinematic degrees of freedom by attaching a triad of rigid directors to each material point. In Eringen’s terminology, such a medium is known as a “micropolar” continuum.

To analyze finite-strain plasticity, the multiplicative decomposition of the deformation gradient into elastic and plastic parts, $F = F^{e} F^{p}$ , is widely used. The idea of decomposing the deformation gradient dates back to the seminal works of Stojanović et al. [5], Stojanović [6]; see also [7 –13] for later developments. A comprehensive geometric treatment of this decomposition is provided in Yavari [14]. These ideas have their roots in the classic works of [15 –20].

Within this framework, it is necessary to introduce an intermediate configuration representing a stress-free or natural state of the material. This intermediate configuration is constructed by distorting material elements, either in the reference or current tangent spaces, in a generally incompatible manner. Importantly, the intermediate configuration is not a genuine configuration of the body; as such, no true plastic mapping can be defined between the reference configuration and this intermediate state. Rather, it should be viewed as a collection of natural states associated with individual material elements. Consequently, neither $F^{p}$ nor $F^{e}$ can, at the outset, be interpreted as tangent maps of deformations that place the material in a well-defined configuration within Euclidean space. The extension of the multiplicative decomposition to micropolar continua was developed in Steinmann [21], where the author introduced a multiplicative split of both the classical deformation gradient and the microrotation tensors, without defining any additional plastic tensor for the analysis of the micropolar curvature measures.

Developing material evolution laws is essential for understanding inelastic behavior, such as plastic deformation. In this regard, a correct geometric description of the underlying theory is crucial for formulating appropriate evolution laws. A comprehensive geometric treatment of both classical and non-classical continuum theories is presented in Epstein and Elzanowski [22]. In this approach, the concept of material isomorphism is extended to the entire body, allowing one to study material inhomogeneities and their evolution within the framework of uniform materials. One of the major advantages of this viewpoint is that it removes the unnecessary reliance on an intermediate configuration. Instead, the notions of a reference crystal or material archetype naturally emerge, objects that do not correspond to actual configuration spaces. In this framework, the uniformity maps are given by the inverse of the plastic deformation tensor [23, 24]. Many studies have employed these approaches to model inelastic deformation within non-classical continuum theories. For example, Javadi et al. [25] developed growth and remodeling evolution laws for uniform bodies within the framework of micromorphic theory and subsequently proposed related evolution laws based on finite couple stress theory [26]. More recently, Steigmann et al. [27] formulated a plastic evolution law for hemitropic Cosserat solids in uniform bodies.

The aim of this study is to develop a formulation for the elastoplastic behavior of micropolar materials that accounts for the effects of microstructure. A key advantage of the micropolar theory, compared with other non-classical continuum frameworks, lies in its additional degrees of freedom and the role of microinertia in describing microstructural rearrangements. To this end, we first present the geometrical foundations of finite micropolar kinematics. We then derive the balance equations of mass, microinertia, linear and angular momenta, and internal energy, together with the entropy inequality, from an invariance principle under general observer transformations. Within the framework of material uniformity, and by enforcing the derived entropy inequality, we obtain the hyperelastic constitutive relations with respect to the reference configuration. Subsequently, we develop evolution laws for micropolar materials by applying the Principle of Maximum Dissipation to the micropolar dissipation equation. We further show that these evolution laws depend on the Mandel stress tensors, which serve as driving forces for local microstructural rearrangements. Finally, we establish a mathematical restriction on the plastic evolution laws that ensures consistency with changes in the reference configuration and preservation of material symmetry.

2. Micropolar media

In Eringen’s terminology [1], a micropolar continuum is defined as an ordinary body $B$ , to each point of which a triad of rigid, linearly independent directors is attached. This concept originates from the pioneering work of Cosserat and Cosserat [4], who introduced an orthonormal triad that is required to remain orthonormal throughout the deformation process. In order to model materials with internal structure, metamaterials 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 framework, the body $B$ is referred to as the macromedium or matrix, while the collection of rigid triads constitutes the micromedium, with each triad representing the internal structure. A deformation of a micropolar medium is therefore characterized by the deformation of the macromedium together with a concomitant field of linear transformations acting on the local triads. In this case, each material point is named as a pseudo-rigid body [28] or Cosserat point [29].

2.1. Kinematics

Following standard notational conventions, an element in the reference configuration of a micropolar medium is characterized by six parameters: the coordinates $X^{I}$ $(I = 1, 2, 3)$ of a point in the macromedium and the three referential rigid directors $D_{α}$ $(α = 1, 2, 3)$ attached to it. A motion of the micropolar medium is therefore described by six smooth functions, $κ^{i}$ and $ω^{i}$ , defined as follows:

\begin{matrix} x^{i} = κ^{i} (X^{1}, X^{2}, X^{3}, t), ω^{i} = ω^{i} (X^{1}, X^{2}, X^{3}, t), i = 1, 2, 3 \end{matrix}

(1)

where $κ^{i}$ represents the coordinates of a material point in the current configuration, and $ω^{i}$ is the rotation vector. The matrix representation of this vector, denoted by the hat operator $\hat{ω}$ , can be expressed as

\begin{matrix} \hat{ω} = (\begin{matrix} 0 & - ω^{3} & ω^{2} \\ ω^{3} & 0 & - ω^{1} \\ - ω^{2} & ω^{1} & 0 \end{matrix}), \end{matrix}

(2)

which belongs to the Lie algebra $s o (3)$ ¹. To obtain the corresponding field of orthogonal transformations $R \in SO (3)$ , the exponential map is used, such that

\begin{matrix} R = \exp (\hat{ω}) = I_{3 \times 3} + \frac{\sin (| ω |)}{| ω |} \hat{ω} + \frac{1 - \cos (| ω |)}{| ω |^{2}} {\hat{ω}}^{2} . \end{matrix}

(3)

Then, it can be written as

\begin{matrix} d_{α}^{i} = R_{I}^{i} (X^{1}, X^{2}, X^{3}, t) D_{α}^{I} α, i, I = 1, 2, 3 \end{matrix}

(4)

where $d_{α}$ are the current directors and the summation convention is enforced. If $E_{I}$ and $e_{i}$ denote the referential and spatial covariant coordinate bases, respectively, then the functions $R_{I}^{i}$ represent the components of the two-point tensor R as

\begin{matrix} R = R_{I}^{i} e_{i} \otimes E^{I} . \end{matrix}

(5)

The linear operator R represents the affine deformation associated with the directors attached to a macromedium point $X = (X^{1}, X^{2}, X^{3})$ . In a micropolar medium, the deformation gradient, i.e., the derivative of the micropolar deformation, is inherently more complex than in a classical continuum. In a coordinate chart, the deformation gradient comprises the deformation itself, represented by κ and R, together with their gradients, $\nabla κ$ and $\nabla R$ . In other words, at each point X, the deformation gradient is characterized by the set ${κ, R, F, \nabla R}$ , where $F = \nabla κ$ corresponds to the conventional deformation gradient of the macromedium.

Accordingly, the energy density per unit volume of a micropolar medium can be expressed as

\begin{matrix} U (F, R, \nabla R; X), \end{matrix}

(6)

where ∇⁡ denotes the gradient operator with respect to the reference configuration. Assuming the strain energy to be Galilean-invariant, we can then write

\begin{matrix} U (F, R, \nabla R; X) = U (Q F, Q R, Q \nabla R; X), \end{matrix}

(7)

where $Q \in SO (3)$ is an arbitrary spatial rotation tensor. A necessary and sufficient condition for this is that [27]

\begin{matrix} U (F, R, \nabla R; X) = U (U, Γ; X), \end{matrix}

(8)

where $U = R^{t} F$ and $Γ = R^{t} \nabla R$ . The superscript t denotes the transpose of a tensor. In Eringen’s terminology, U is referred to as the Cosserat deformation tensor, and Γ as the wryness tensor [2].

Remark 2.1. Since the wryness tensor belongs to the Lie algebra $s o (3)$ , it can be reduced to a second-order tensor as

\begin{matrix} Γ = axl (R^{t} R_{, I}) \otimes E^{I}, \end{matrix}

(9)

where (,) denotes the differential derivative.

Let us now consider the effect of a change in the reference configuration on the quantities κ , U, and Γ. Denoting the corresponding quantities in the new reference configuration with a circumflex accent, the change of variables can be expressed in terms of appropriate transformation functions:

\begin{matrix} \hat{X} = \hat{X} (X), \end{matrix}

(10)

and

\begin{matrix} \hat{Q} = \hat{Q} (X) . \end{matrix}

(11)

The referential directors are related by

\begin{matrix} {\hat{D}}_{α}^{A} = {\hat{R}}_{I}^{A} D_{α}^{I} . \end{matrix}

(12)

Consequently, in view of equation (4), the functions of the present configuration with respect to the two reference configurations are related by

\begin{matrix} (κ (X, t), R (X, t)) = (\hat{κ} (\hat{X} (X), t), \hat{R} (\hat{X} (X), t) \hat{Q} (X)) . \end{matrix}

(13)

The deformation gradients corresponding to the two reference configurations are related through

\begin{matrix} (κ, R, F, \nabla R) = (\hat{κ}, \hat{R} \hat{Q}, \hat{F} \hat{P}, \hat{\nabla} \hat{R} : [\hat{Q}, \hat{P}] + \hat{R} \nabla \hat{Q}), \end{matrix}

(14)

where ${(C : [A, B])}_{JK}^{i} = C_{MN}^{i} A_{J}^{M} B_{K}^{N}$ [30]. The matrix $\hat{P}$ is given by the derivatives

\begin{matrix} \hat{P} = \nabla \hat{X} (X) . \end{matrix}

(15)

Consequently, although both the macroscopic deformation gradient and the microdeformation are conventional tensors, the gradient of the microdeformation transforms more intricately under a change of reference configuration, and this behavior will be crucial in formulating potential evolution laws for plasticity in micropolar media. Following the previous notation, we can establish the relation between the strain tensors $(U, Γ)$ and $(\hat{U}, \hat{Γ})$ in the new reference configuration as

\begin{matrix} U = {\hat{Q}}^{t} \hat{U} \hat{P}, \end{matrix}

(16)

and

\begin{matrix} Γ = {\hat{Q}}^{t} \hat{Γ} \hat{P} + axl ({\hat{Q}}^{t} \nabla \hat{Q}) . \end{matrix}

(17)

3. Balance equations

3.1. The master energy balance

In formulating the balance equations for a micropolar medium, we follow the general approach of Green [31]. This approach, as originally formulated in Noll [32] and Green and Rivlin [33], is based on the postulate that the energy balance remains invariant under arbitrary changes of observer. Consequently, the total kinetic energy of the micropolar medium occupying a spatial volume ω is expressed as

\begin{matrix} T = \int_{ω} (\frac{1}{2} ρ v . v + \frac{1}{2} ρ ν i ν^{t}) d ω, \end{matrix}

(18)

where v and ρ denote the velocity and the instantaneous spatial mass density, respectively. The symmetric second-order tensor i is the spatial microinertia tensor, and ν is the microgyration tensor [1], with the following form:

\begin{matrix} ν = \overset{\cdot}{R} R^{t} \in s o (3), \end{matrix}

(19)

where a superposed dot indicates the material time derivative. The internal energy content U in the same spatial volume ω can be expressed in terms of energy per unit mass ϵ as follows

\begin{matrix} U = \int_{ω} ρϵ dω . \end{matrix}

(20)

The rate of increase of energy in the volume ω with boundary ∂ω and exterior unit normal n is

\begin{matrix} W_{entrant} = \int_{∂ω} - (v . n) (\frac{1}{2} ρ v . v + \frac{1}{2} ρ ν i ν^{t}) da . \end{matrix}

(21)

The remaining contributions to the energy balance stem from non-mechanical (thermal) sources and external forces, with the former expressed as

\begin{matrix} W_{thermal} = \int_{ω} ρrdω + \int_{∂ω} hda, \end{matrix}

(22)

here, r and h denote the radiation heat source and the conductive heat flux, respectively, with the heat flux vector q related to h through $h = - q \cdot n$ . Regarding mechanical power, we assume the presence of a body force b and a body couple l per unit mass. These give rise to corresponding surface tractions t and m. Consequently, the power associated with these mechanical forces can be expressed as

\begin{matrix} W_{mech} = \int_{ω} (b . ρ v + l . ρ ν) dω + \int_{∂ω} (t . v + m . ν) da . \end{matrix}

(23)

Accounting for all of the above contributions, the energy balance requires that

\begin{matrix} \frac{\partial}{∂t} (T + U) = W_{entrant} + W_{thermal} + W_{mech} . \end{matrix}

(24)

3.2. Translation invariance

Under a translational change of observer with relative velocity a, all quantities in the master energy balance equation remain invariant, except for the velocity field, which transforms according to

\begin{matrix} v \to v + a . \end{matrix}

(25)

By collecting all terms multiplying $a \cdot a$ , the mass balance equation is obtained as

\begin{matrix} \overset{\cdot}{ρ} + ρ \nabla . v = 0 . \end{matrix}

(26)

The terms linear in a yield the balance of linear momentum in the form

\begin{matrix} ρ \overset{\cdot}{v} = ρ b + div σ . \end{matrix}

(27)

In deriving the above equation, we employed the mass balance (26) and invoked Cauchy’s tetrahedron argument,

\begin{matrix} σ n = t, \end{matrix}

(28)

where σ denotes the Cauchy stress tensor.

3.3. Rotation invariance

For an observer undergoing a rigid rotation with constant angular velocity $Ω \in s o (3)$ , the master energy balance is preserved in form. Only the velocity field, the microgyration tensor, the microinertia tensor, the body force, and the body couple are affected by this change of frame, and these fields transform as follows:

\begin{matrix} v \to v + Ω x, \end{matrix}

(29)

\begin{matrix} ν \to ν + Ω, \end{matrix}

(30)

\begin{matrix} \overset{\cdot}{i} \to \overset{\cdot}{i} + 2 Ω i, \end{matrix}

(31)

\begin{matrix} b \to b + 2 Ω v, \end{matrix}

(32)

\begin{matrix} l \to l + i (ν Ω + Ω ν + Ω Ω), \end{matrix}

(33)

Collecting all terms that contribute to the coefficients of $ν^{T} Ω$ and $\frac{1}{2} Ω^{T} Ω$ leads to the balance of microinertia expressed as

\begin{matrix} ρ \frac{D i}{Dt} - ρ i ν^{T} - ρ ν i^{T} = 0 . \end{matrix}

(34)

Likewise, by equating the coefficients of the terms linear in Ω , one obtains the corresponding balance of momentum moments in the form

\begin{matrix} \nabla λ + ϵ : σ + ρ (l - ω) = 0, \end{matrix}

(35)

where $ω = axl (i (\overset{\cdot}{ν} + ν ν))$ denotes the spin–inertia tensor per unit mass, and (:) represents the double contraction [34]. In addition, λ represents the couple stress tensor, obtained via Cauchy’s tetrahedron construction as

\begin{matrix} λ n = m . \end{matrix}

(36)

By substituting equations (18), (20)–(23) into equation (24), the energy balance equation takes the form

\begin{matrix} ρ \overset{\cdot}{ϵ} = λ : {(\nabla (axl (ν)))}^{t} + σ : ({(\nabla v)}^{t} + ν) - div q + ρr . \end{matrix}

(37)

3.4. The Clausius–Duhem inequality

Within the micropolar framework, defining the entropy per unit mass as $s_{ρ}$ , the global Clausius–Duhem inequality in the Eulerian description can be expressed as

\begin{matrix} \frac{D}{Dt} \int_{ω} ρ s_{ρ} dω \geq \int_{ω} \frac{ρr}{θ} dω + \int_{∂ω} \frac{h}{θ} da, \end{matrix}

(38)

where θ denotes the absolute temperature. Consequently, the local form of the inequality can be expressed as

\begin{matrix} ρ \overset{\cdot}{s_{ρ}} \geq \frac{ρr}{θ} - div (\frac{q}{θ}) . \end{matrix}

(39)

By defining the Helmholtz free energy per unit mass as $ψ_{ρ} = ϵ - θ s_{ρ}$ , the combination of equations (37) and (39) leads to an equivalent form of the entropy inequality within the micropolar theory, given by

\begin{matrix} ρ ({\overset{\cdot}{ψ}}_{ρ} + s_{ρ} \overset{\cdot}{θ}) \leq λ : \nabla (axl (ν)) + σ : ({(\nabla v)}^{T} + ν) - \frac{1}{θ} q . \nabla θ . \end{matrix}

(40)

Equation (40) represents the thermodynamic consistency condition for energy dissipation in a micropolar hyperelastic continuum. Similarly, the Lagrangian form of the second law of thermodynamics can be written as

\begin{matrix} ρ_{R} ({\overset{\cdot}{ψ}}_{ρ} + s_{ρ} \overset{\cdot}{θ}) \leq T : \overset{\cdot}{U} + Λ : \overset{\cdot}{Γ} - \frac{1}{θ} \bar{Q} \nabla_{R} θ, \end{matrix}

(41)

where $ρ_{R}$ denotes the mass density in the reference configuration, $\nabla_{R}$ is the gradient with respect to the reference configuration, T is the macromedium Piola stress tensor, Λ is the micromedium Piola hyperstress (couple stress) tensor, and $\bar{Q}$ is the referential heat flux vector. The relationships between the Lagrangian and Eulerian quantities are given by

\begin{matrix} T = J_{F} F^{- 1} σ R, J_{F} = \det F, \end{matrix}

(42)

\begin{matrix} Λ = J_{F} F^{- 1} λ R, \end{matrix}

(43)

\begin{matrix} \bar{Q} = J_{F} F^{- 1} q . \end{matrix}

(44)

\begin{matrix} ρ = J_{F}^{- 1} ρ_{R}, \end{matrix}

(45)

4. A micropolar theory for plasticity processes in uniform media

4.1. The theory of uniformity and the micropolar archetype

According to the work of Epstein and Elzanowski [22], Noll [35], a simple body is said to be materially uniform if, at every point within the body, the constituent material is the same. The concept of uniformity is here extended to the framework of micropolar media. Considering the Helmholtz free energy density at a given point in a micropolar medium, for any point throughout the body, the energy content per unit volume can be expressed as

\begin{matrix} ψ_{R} (X, t) = ψ_{R} (U (X, t), Γ (X, t); X, t), \end{matrix}

(46)

where the energy density depends explicitly on the point X. More precisely, if the micropolar body is uniform, the elastic potential energy at X depends on this point only through the uniformity fields $P (X, t)$ , $Q (X, t)$ , and $O (X, t)$ . These fields represent the implants from the micropolar archetype at the point X (see Figure 1). Based on this ansatz, it follows that

\begin{matrix} ψ_{R} (X, t) = & ψ_{R} (U (X, t), Γ (X, t); X, t) = J_{P}^{- 1} ψ_{R} (\bar{U} (X, t), \bar{Γ} (X, t)), \end{matrix}

(47)

where ${\bar{ψ}}_{R}$ is the energy potential in the micropolar archetype, $ψ_{R}$ is the energy potential per unit reference volume, $J_{P} = \det (P)$ , and

\begin{matrix} \bar{U} = Q^{t} U P, \end{matrix}

(48)

\begin{matrix} \bar{Γ} = Q^{t} Γ P + \tilde{Γ}, \end{matrix}

(49)

where $\tilde{Γ} = axl (Q^{t} O)$ .

Figure 1.

Uniformity fields associated with reference configuration.

Remark 4.1. In the context of micropolar media, the corresponding material implant satisfies $Q \in SO (3)$ and $\tilde{Γ} \in s o (3)$ .

Remark 4.2. In comparison with the well-known multiplicative decomposition of the micropolar deformation gradient [21], the elastic Cosserat deformation and wryness tensors are denoted by $\bar{U}$ and $\bar{Γ}$ , respectively. The associated plastic deformation and rotation-related tensors correspond to the inverses of the implant maps ${P, Q, O}$ , namely ${P^{- 1}, Q^{t}, - Q^{t} O : [Q^{t}, P^{- 1}]}$ .

It should also be noted that in Steinmann [21], the inverse of the independent third-order tensor O is interpreted as the gradient of the plastic part of the rotation tensor $R^{P}$ .

4.2. Thermodynamics consideration of the micropolar theory

We assume that the Helmholtz free energy per unit volume is given by ${\bar{ψ}}_{R}$ for the micropolar archetype and by $ψ_{R}$ in the reference configuration, expressed as

\begin{matrix} ψ_{R} (X, t) = ψ_{R} (U (X, t), Γ (X, t), θ, \nabla_{R} θ; X, t) = J_{P}^{- 1} ψ_{R} (\bar{U} (X, t), \bar{Γ} (X, t), θ, \nabla_{R} θ P) . \end{matrix}

(50)

For notational simplicity, the explicit dependence of the tensor fields on $(X, t)$ is omitted. Accordingly, the material time derivative of equation (50) can be expressed as

\begin{matrix} {\overset{\cdot}{ψ}}_{R} = - J_{P}^{- 1} tr (P^{- 1} \overset{\cdot}{P}) {\bar{ψ}}_{R} + J_{P}^{- 1} \frac{\partial {\bar{ψ}}_{R}}{\partial \bar{U}} \overset{\cdot}{\bar{U}} + J_{P}^{- 1} \frac{\partial {\bar{ψ}}_{R}}{\partial \bar{Γ}} \overset{\cdot}{\bar{Γ}} + J_{P}^{- 1} \frac{\partial {\bar{ψ}}_{R}}{\partial (\nabla_{R} θ P)} (\nabla_{R} \overset{\cdot}{θ} P + \nabla_{R} θ \overset{\cdot}{P}) + J_{P}^{- 1} \frac{\partial ψ_{R}}{∂θ} \overset{\cdot}{θ} . \end{matrix}

(51)

where the stress tensor at the level of the micropolar archetype is defined as $\bar{T} = \frac{\partial {\bar{ψ}}_{R}}{\partial \bar{U}}$ , and the couple-stress tensor as $\bar{Λ} = \frac{\partial {\bar{ψ}}_{R}}{\partial \bar{Γ}}$ . We now introduce the Helmholtz free energy per unit mass, denoted by $ψ_{ρ}$ , as

\begin{matrix} ρ_{R} {\overset{\cdot}{ψ}}_{ρ} = ρ_{R} \frac{\partial}{∂t} (\frac{ψ_{R}}{ρ_{R}}) = {\overset{\cdot}{ψ}}_{R} - \frac{{\overset{\cdot}{ρ}}_{R}}{ρ_{R}} ψ_{R} = {\overset{\cdot}{ψ}}_{R} + (tr {\bar{L}}_{P}) ψ_{R}, \end{matrix}

(52)

Here, ${\bar{L}}_{P} = tr (P^{- 1} \overset{\cdot}{P})$ defines the “inhomogeneity velocity gradient” relative to the micropolar archetype. By combining equations (51) and (52) and inserting the results into equation (41), we arrive at

\begin{matrix} (J_{P}^{- 1} Q \bar{T} P^{t} - T) : \overset{\cdot}{U} + (J_{P}^{- 1} Q \bar{Λ} P^{t} - Λ) : \overset{\cdot}{Γ} + (U^{t} T + Γ^{t} Λ + J_{p}^{- 1} \frac{\partial {\bar{ψ}}_{R}}{\partial \nabla_{R} θ P} \nabla_{R} θ P^{t}) : (\overset{\cdot}{P} P^{- 1}) \\ - (T U^{t} + Λ Γ^{t}) : (\overset{\cdot}{Q} Q^{t}) + (Q^{t} Λ P^{- t}) : axl ({\overset{\cdot}{Q}}^{t} O + Q^{t} \overset{\cdot}{O}) \\ + (J_{P}^{- 1} \frac{\partial {\bar{ψ}}_{R}}{\partial (\nabla_{R} θ P)} P^{t}) \nabla_{R} \overset{\cdot}{θ} + (ρ_{R} s_{ρ} + J_{P}^{- 1} \frac{\partial \bar{ψ}}{∂θ}) \overset{\cdot}{θ} + \frac{1}{θ} \bar{Q} \nabla_{R} θ \leq 0 . \end{matrix}

(53)

It is apparent that equation (53) is linear in $\overset{\cdot}{U}$ , $\overset{\cdot}{Γ}$ , $\overset{\cdot}{θ}$ , and $\nabla_{R} \overset{\cdot}{θ}$ . Therefore, the corresponding conjugate terms within the parentheses must vanish identically, i.e.,

\begin{matrix} T = J_{P}^{- 1} Q \bar{T} P^{t}, \end{matrix}

(54)

\begin{matrix} Λ = J_{P}^{- 1} Q \bar{Λ} P^{t}, \end{matrix}

(55)

\begin{matrix} s_{R} = ρ_{R} s_{ρ} = - J_{P}^{- 1} \frac{\partial \bar{ψ}}{∂θ} = - \frac{\partial ψ_{R}}{∂θ}, \end{matrix}

(56)

\begin{matrix} - J_{P}^{- 1} \frac{\partial {\bar{ψ}}_{R}}{\partial (\nabla_{R} θ P)} P^{t} = - \frac{\partial ψ_{R}}{\partial \nabla_{R} θ} = 0 . \end{matrix}

(57)

These conditions coincide with the constitutive restrictions of classical thermoelasticity. The remaining terms yield a residual inequality, which can be interpreted as the dissipation $D$ per unit reference volume.

\begin{matrix} - D = & (U^{t} T + Γ^{t} Λ) : (\overset{\cdot}{P} P^{- 1}) - (T U^{t} + Λ Γ^{t}) : (\overset{\cdot}{Q} Q^{t}) + (Q^{t} Λ P^{- t}) : axl ({\overset{\cdot}{Q}}^{t} O + Q^{t} \overset{\cdot}{O}) \\ + \frac{1}{θ} \bar{Q} \nabla_{R} θ \leq 0 . \end{matrix}

(58)

where we define $B = - U^{t} T - Γ^{t} Λ$ , $C = T U^{t} + Λ Γ^{t}$ , and $D = - Λ$ as the Mandel stress, Mandel couple stress, and Mandel hyperstress tensors, respectively, within the framework of micropolar theory. The component form $\overset{\cdot}{P} P^{- 1}$ is the macro inhomogeneity velocity gradient at the reference configuration, $\overset{\cdot}{Q} Q^{t}$ are the components of the microinhomogeneity gyration at the reference configuration and $axl (Q^{t} \overset{\cdot}{O})$ represents the time rate of the third-order material transplant that is related to the microinhomogeneity gyration gradient. In equation (58), assuming the conduction term $\frac{1}{θ} \bar{Q} \nabla_{R} θ$ is non-positive, the inhomogeneous contribution satisfies the following sufficient condition.

\begin{matrix} - D = & (U^{t} T + Γ^{t} Λ) : (\overset{\cdot}{P} P^{- 1}) - (T U^{t} + Λ Γ^{t}) : (\overset{\cdot}{Q} Q^{t}) + (Q^{t} Λ P^{- t}) : axl ({\overset{\cdot}{Q}}^{t} O + Q^{t} \overset{\cdot}{O}) \leq 0 . \end{matrix}

(59)

4.3. The principle of maximum plastic dissipation

To derive the plastic evolution law compatible with the uniform micropolar media, we use the principle of maximum plastic dissipation. In other words, the loading/unloading conditions and the form of associative flow rule is in agreement with principle of maximum plastic dissipation and can be shown as the Kuhn–Tucker optimality conditions [36]. To move further we consider following form of micropolar yield function in stress, couple stress, and hyperstress space by $ϕ (B, C, D) = 0$ . Then, the admissible set of stress states, in convex set is defined as

\begin{matrix} E = {(\tilde{B}, \tilde{C}, \tilde{D}) \in ℝ^{9} \times ℝ^{9} \times ℝ^{9} | ϕ (B, C, D) \leq 0} . \end{matrix}

(60)

Denoting by ${B, C, D}$ the actual states of stresses, the principle of maximum plastic dissipation is equivalent to the statement that, for given ${\overset{\cdot}{P}, \overset{\cdot}{Q}, \overset{\cdot}{O}}$ ,

\begin{matrix} D (B, C, D; \overset{\cdot}{P}, \overset{\cdot}{Q}, \overset{\cdot}{O}) \geq D (\tilde{B}, \tilde{C}, \tilde{D}; \overset{\cdot}{P}, \overset{\cdot}{Q}, \overset{\cdot}{O}) \forall (\tilde{B}, \tilde{C}, \tilde{D}) \in E . \end{matrix}

(61)

The maximum dissipation problem can be recast into a minimum problem as

\begin{matrix} (B, C, D) = \arg [\min {- D (\tilde{B}, \tilde{C}, \tilde{D}; \overset{\cdot}{P}, \overset{\cdot}{Q}, \overset{\cdot}{O})}] \forall (\tilde{B}, \tilde{C}, \tilde{D}) \in E . \end{matrix}

(62)

Following the work done by Simo et al. [37], we define the Lagrangian functional associated with equation (62) as

\begin{matrix} L = - D (\tilde{B}, \tilde{C}, \tilde{D}; \overset{\cdot}{P}, \overset{\cdot}{Q}, \overset{\cdot}{O}) + \overset{\cdot}{\bar{γ}} ϕ (\tilde{B}, \tilde{C}, \tilde{D}), \end{matrix}

(63)

We assume that the plastic consistency parameter $\overset{\cdot}{\bar{γ}}$ belongs to the admissible set

\begin{matrix} \overset{\cdot}{γ} \in K_{p} : = {δγ \in L^{2} (Ω) | δγ \geq 0} . \end{matrix}

(64)

The Kuhn–Tucker optimality conditions and the associated plastic flow rules can then be derived as

\begin{matrix} \frac{∂L}{\partial B} = - \overset{\cdot}{P} P^{- 1} + \overset{\cdot}{γ} \frac{∂ϕ}{\partial B} = 0, \end{matrix}

(65)

\begin{matrix} \frac{∂L}{\partial C} = - \overset{\cdot}{Q} Q^{t} + \overset{\cdot}{γ} \frac{∂ϕ}{\partial C} = 0, \end{matrix}

(66)

\begin{matrix} \frac{∂L}{\partial D} = - Q axl ({\overset{\cdot}{Q}}^{t} O + Q^{t} \overset{\cdot}{O}) P^{- 1} + \overset{\cdot}{γ} \frac{∂ϕ}{\partial D} = 0, \end{matrix}

(67)

\begin{matrix} \overset{\cdot}{γ} \geq 0, ϕ \leq 0, and \overset{\cdot}{γ} ϕ = 0 . \end{matrix}

(68)

By using equations (65)–(67), it can be inferred that

\begin{matrix} \overset{\cdot}{P} = f (P, Q, O, B, C, D), \end{matrix}

(69)

\begin{matrix} \overset{\cdot}{Q} = g (P, Q, O, B, C, D), \end{matrix}

(70)

\begin{matrix} \overset{\cdot}{\tilde{Γ}} = h (P, Q, O, B, C, D), \end{matrix}

(71)

where $\overset{\cdot}{\tilde{Γ}} = axl ({\overset{\cdot}{Q}}^{t} O + Q^{t} \overset{\cdot}{O})$ , and equations (69)–(71) represent the associative flow rules, or evolution laws, for the micropolar material. These equations take the general form of a coupled system of first-order differential equations.

4.4. Micropolar evolution laws

In this section, we focus on establishing general principles governing the evolution laws of micropolar materials. Following the framework presented by Epstein and Maugin [38] and Epstein and Elzanowski [22], these evolution laws are required to satisfy several fundamental formal constraints. First, the evolution laws must not explicitly depend on the material point X; this condition, referred to as the uniformity condition, is already incorporated in equations (69)–(71). Second, the selection of a specific reference configuration should not influence the form of the evolution law. Finally, the evolution laws must remain invariant under changes of the material archetype, reflecting the inherent symmetry of the evolution laws.

4.4.1. Reduction to the archetype

As noted above, the specific mathematical forms of the evolution laws equations (69)–(71) still depend on the choice of a micropolar archetype as well as the reference configuration. By fixing the archetype, we focus on examining how changes in the reference configuration affect the form of the evolution laws. Figure 2 illustrates the basic scheme, where quantities associated with the second reference configuration are denoted by “ $\hat{□}$ ”. The evolution laws in this second reference configuration are expressed as

\begin{matrix} \overset{\cdot}{\hat{P}} = \hat{f} (\hat{P}, \hat{Q}, \hat{O}, \hat{B}, \hat{C}, \hat{D}), \end{matrix}

(72)

\begin{matrix} \overset{\cdot}{\hat{Q}} = \hat{g} (\hat{P}, \hat{Q}, \hat{O}, \hat{B}, \hat{C}, \hat{D}), \end{matrix}

(73)

\begin{matrix} \overset{\cdot}{\hat{Γ}} = \hat{h} (\hat{P}, \hat{Q}, \hat{O}, \hat{B}, \hat{C}, \hat{D}) . \end{matrix}

(74)

Figure 2.

Uniformity fields associated with two different reference configurations.

Within the framework of micropolar theory, a change of reference configuration is fully described by the following mappings:

\begin{matrix} Y = Y (X), \end{matrix}

(75)

\begin{matrix} H = H (X) . \end{matrix}

(76)

We further define the derivative of equation (75) with respect to the first reference configuration as

\begin{matrix} K = \frac{\partial Y}{\partial X} . \end{matrix}

(77)

The relationship between the material implants ${P, Q, O}$ and ${\hat{P}, \hat{Q}, \hat{O}}$ in the two reference configurations can now be expressed as

\begin{matrix} \hat{P} = K P, \end{matrix}

(78)

\begin{matrix} \hat{Q} = H Q, \end{matrix}

(79)

\begin{matrix} \hat{O} = H O + \nabla^{1} H : [Q, P], \end{matrix}

(80)

where $\nabla^{1}$ denotes the gradient with respect to the first reference configuration. The transformations of the Mandel stress, Mandel couple stress, and Mandel hyperstress can be expressed as

\begin{matrix} \hat{B} = J_{K}^{- 1} (K^{- t} B K^{t} - \hat{\tilde{Γ}} H Λ K^{t}), \end{matrix}

(81)

\begin{matrix} \hat{C} = J_{K}^{- 1} (H C H^{t} + H Λ K^{t} {\hat{\tilde{Γ}}}^{t}), \end{matrix}

(82)

\begin{matrix} \hat{D} = - J_{K}^{- 1} (H D K^{t}), \end{matrix}

(83)

Here, $J_{K} = \det (K)$ and $\hat{\tilde{Γ}} = axl (H \nabla^{2} H^{T})$ , with $\nabla^{2}$ denoting the gradient taken with respect to the second reference configuration. Now, for a given micropolar point at a specific instant in time, if the reference configuration is chosen such that the point X coincides with the archetype, it follows that

\begin{matrix} K \to P^{- 1}, \end{matrix}

(84)

\begin{matrix} H \to Q^{t}, \end{matrix}

(85)

\begin{matrix} \nabla^{2} H \to O . \end{matrix}

(86)

By substituting equations (84)–(86) into equations (81)–(83), we obtain the following expressions for the pull-back of the Mandel stress, Mandel couple stress, and Mandel hyperstress to the archetype:

\begin{matrix} \bar{B} = J_{P} P^{t} B P^{- t} + {\tilde{Γ}}^{t} \bar{D}, \end{matrix}

(87)

\begin{matrix} \bar{C} = J_{P} Q^{t} C Q - \bar{D} {\tilde{Γ}}^{t}, \end{matrix}

(88)

\begin{matrix} \bar{D} = - J_{P} (Q^{t} D P^{- t}) . \end{matrix}

(89)

To distinguish the transformed quantities, we place a “ $\overset{ˇ}{□}$ ” over all new quantities resulting from this transformation. Meanwhile, under this change of reference configuration, the time rates of the implants in the new configuration become

\begin{matrix} \overset{\cdot}{\overset{ˇ}{P}} = P^{- 1} \overset{\cdot}{P}, \end{matrix}

(90)

\begin{matrix} \overset{\cdot}{\overset{ˇ}{Q}} = Q^{t} \overset{\cdot}{Q}, \end{matrix}

(91)

\begin{matrix} \overset{\cdot}{\overset{ˇ}{Γ}} = \overset{\cdot}{\tilde{Γ}} - axl ({\overset{\cdot}{Q}}^{t} O) - axl (Q^{t} O : [\overset{\cdot}{\overset{ˇ}{Q}}, I]) - \tilde{Γ} \overset{\cdot}{\overset{ˇ}{P}} . \end{matrix}

(92)

These quantities can be interpreted as the “inhomogeneity velocity gradient” of micropolar materials at the archetype level. Accordingly, the evolution laws can be expressed as

\begin{matrix} \overset{\cdot}{\overset{ˇ}{P}} = \bar{f} (\bar{B}, \bar{C}, \bar{D}) = \overset{\cdot}{γ} \frac{\partial \bar{ϕ}}{\partial \bar{B}}, \end{matrix}

(93)

\begin{matrix} \overset{\cdot}{\overset{ˇ}{Q}} = \bar{g} (\bar{B}, \bar{C}, \bar{D}) = \overset{\cdot}{γ} \frac{\partial \bar{ϕ}}{\partial \bar{C}} \end{matrix}

(94)

\begin{matrix} \overset{\cdot}{\overset{ˇ}{Γ}} = \bar{h} (\bar{B}, \bar{C}, \bar{D}) = \overset{\cdot}{γ} \frac{\partial \bar{ϕ}}{\partial \bar{D}} . \end{matrix}

(95)

Using equations (93)–(95), the micropolar evolution laws can be reformulated as

\begin{matrix} \overset{\cdot}{P} = P \bar{f}, \end{matrix}

(96)

\begin{matrix} \overset{\cdot}{Q} = Q \bar{g} \end{matrix}

(97)

\begin{matrix} \overset{\cdot}{\tilde{Γ}} = \bar{h} + axl ({\bar{g}}^{t} Q^{t} O) + axl (Q^{t} O : [\bar{g}, I]) + \tilde{Γ} \bar{f} . \end{matrix}

(98)

Remark 4.3. The plastic multiplier $\overset{\cdot}{γ}$ can be determined by enforcing the consistency condition $\overset{\cdot}{γ} \overset{\cdot}{\bar{ϕ}} = 0$ [13]. This condition states that, during plastic loading (i.e., when $\overset{\cdot}{γ} > 0$ ), the rate of the yield function must vanish, $\overset{\cdot}{\bar{ϕ}} = 0$ . Consequently, one obtains

\begin{matrix} \overset{\cdot}{\bar{ϕ}} (\bar{B}, \bar{C}, \bar{D}) = \frac{\partial \bar{ϕ}}{\partial \bar{B}} : \overset{\cdot}{\bar{B}} + \frac{\partial \bar{ϕ}}{\partial \bar{C}} : \overset{\cdot}{\bar{C}} + \frac{\partial \bar{ϕ}}{\partial \bar{D}} : \overset{\cdot}{\bar{D}} = 0 . \end{matrix}

(99)

It is evident that the micropolar evolution laws depend linearly on the material implants. Equations (96)–(98) represent the evolution equations reduced to the archetype. A key aspect in this context is the careful evaluation of the contributions arising from changes in the reference configuration, specifically the pullback of the Mandel stresses to the archetype. This is inherently a constitutive property which, once determined, enables the prediction of how any point in the reference configuration responds to the Mandel stresses. Consequently, the problem of evaluating the evolution laws reduces to determining the functions $\bar{f}$ , $\bar{g}$ , and $\bar{h}$ , which characterize the response of the archetype to the applied micropolar Mandel stresses.

4.4.2. Material symmetry consistency and actual evolution in a micropolar evolution law

In this section, we investigate the consistency of material symmetry during material evolution. It is demonstrated that the proposed evolution laws equations (96)–(98) must satisfy additional constraints to ensure the preservation of micropolar material symmetry. Specifically, in a micropolar medium, the evolution laws at each material point remain invariant under the action of the material symmetry group. A similar principle has been applied in the theories of first- and second-grade materials [22, 39] as well as in the micromorphic theory [25].

Two approaches are typically considered for incorporating material symmetry into the treatment of evolution laws. First, the material symmetry is regarded as an element of the symmetry group of the reference crystal, which remains fixed throughout material evolution. Second, the material symmetry is treated as a time-dependent quantity. In the first case, the transformation laws for the implant maps under a change of archetype ${G, \tilde{G}, S}$ can be expressed as shown in Figure 3.

\begin{matrix} \hat{P} = P G, \end{matrix}

(100)

\begin{matrix} \hat{Q} = Q \tilde{G}, \end{matrix}

(101)

\begin{matrix} \hat{O} = O : [\tilde{G}, G] + Q S, \end{matrix}

(102)

where ${G, \tilde{G}} \in SO (3)$ , $axl (G^{t} S) \in s o (3)$ , and $axl ({\tilde{G}}^{t} S) \in s o (3)$ , all belonging to the micropolar symmetry group. Furthermore, the pullbacks of the Mandel stress, Mandel couple stress, and Mandel hyperstress in the new archetype can be expressed in terms of the corresponding pullback Mandel stresses from the previous archetype, as specified in equations (87)–(89).

Figure 3.

Uniformity fields associated with two different micropolar archtype.

\begin{matrix} \hat{\bar{B}} = J_{G} G^{t} \bar{B} G + {\tilde{\tilde{Γ}}}^{t} \hat{\bar{D}}, \end{matrix}

(103)

\begin{matrix} \hat{\bar{C}} = J_{G} G^{t} \bar{C} G - \hat{\bar{D}} {\tilde{\tilde{Γ}}}^{t}, \end{matrix}

(104)

\begin{matrix} \hat{\bar{D}} = J_{G} G^{t} \bar{D} G, \end{matrix}

(105)

where $\tilde{\tilde{Γ}} = axl ({\tilde{G}}^{t} S)$ . In addition, the evolution laws in the new archetype can be written as

\begin{matrix} \overset{\cdot}{\hat{P}} = \hat{P} \hat{\bar{f}} (\hat{\bar{B}}, \hat{\bar{C}}, \hat{\bar{D}}), \end{matrix}

(106)

\begin{matrix} \overset{\cdot}{\hat{Q}} = \hat{Q} \hat{\bar{g}} (\hat{\bar{B}}, \hat{\bar{C}}, \hat{\bar{D}}), \end{matrix}

(107)

\begin{matrix} \overset{\cdot}{\hat{\tilde{Γ}}} = \hat{\bar{h}} (\hat{\bar{B}}, \hat{\bar{C}}, \hat{\bar{D}}) + axl (\hat{\bar{g}} (\hat{\bar{B}}, \hat{\bar{C}}, \hat{\bar{D}}) {\hat{Q}}^{t} \hat{O}) + axl ({\hat{Q}}^{t} \hat{O} : [\hat{\bar{g}} (\hat{\bar{B}}, \hat{\bar{C}}, \hat{\bar{D}}), I]) + \hat{\tilde{Γ}} \hat{\bar{f}} (\hat{\bar{B}}, \hat{\bar{C}}, \hat{\bar{D}}), \end{matrix}

(108)

where $\hat{\bar{f}}$ , $\hat{\bar{g}}$ , and $\hat{\bar{h}}$ are the tensor-valued functions in the new archetype. Since the change of archetype (material symmetry groups) is assumed to be time-independent, the time rates of the new implants in terms of their previous counterparts are obtained from equations (100) to (102) as

\begin{matrix} \overset{\cdot}{\hat{P}} = \overset{\cdot}{P} G, \end{matrix}

(109)

\begin{matrix} \overset{\cdot}{\hat{Q}} = \overset{\cdot}{Q} \tilde{G}, \end{matrix}

(110)

\begin{matrix} \overset{\cdot}{\hat{O}} = \overset{\cdot}{O} : [\tilde{G}, G] + \overset{\cdot}{Q} S . \end{matrix}

(111)

To establish the relationships between the original evolution functions ( $\bar{f}$ , $\bar{g}$ , and $\bar{h}$ ) and the new ones, equations (96)–(98) and equations (106)–(108) are substituted into equations (109)–(111), yielding

\begin{matrix} \hat{\bar{f}} = G^{t} \bar{f} G, \end{matrix}

(112)

\begin{matrix} \hat{\bar{g}} = G^{t} \bar{g} G, \end{matrix}

(113)

\begin{matrix} \hat{\bar{h}} = & {\tilde{G}}^{t} \bar{h} G + {\tilde{G}}^{t} axl ({\bar{g}}^{t} Q^{t} O) G + {\tilde{G}}^{t} axl (Q^{t} O : [\bar{g}, I]) G + {\tilde{G}}^{t} \tilde{Γ} \bar{f} G \\ - axl ({\tilde{G}}^{t} {\bar{g}}^{t} \tilde{G} {\hat{Q}}^{t} \hat{O}) - axl ({\hat{Q}}^{t} \hat{O} : [{\tilde{G}}^{t} \bar{g} \tilde{G}, I]) - \hat{\tilde{Γ}} G^{t} \bar{f} G . \end{matrix}

(114)

Equations (112)–(114) illustrate the principle of material symmetry consistency for micropolar materials. Accordingly, by applying the transformation set ${G, \tilde{G}, S}$ , which belongs to the symmetry group of the original archetype, the evolution functions $\hat{\bar{f}}$ , $\hat{\bar{g}}$ , and $\hat{\bar{h}}$ are related to their corresponding counterparts $\bar{f}$ , $\bar{g}$ , and $\bar{h}$ .

As previously noted, in the first scenario, the material symmetry group is assumed to remain fixed throughout the material evolution. In contrast, in the second scenario, the two sets of evolution functions ${\hat{P} (t), \hat{Q} (t), \hat{\tilde{Γ}} (t)}$ and ${P (t), Q (t), \tilde{Γ} (t)}$ at a given material point are connected through a time-dependent element of the symmetry group ${G (t), \tilde{G} (t), S (t)}$ . In other words, the material symmetries themselves may evolve dynamically over time. Under these conditions, differentiating equations (100)–(102) with respect to time and evaluating the result at an instant where G and $\tilde{G}$ are identity tensors and $S = 0$ , yields

\begin{matrix} \overset{\cdot}{\hat{P}} = \overset{\cdot}{P} + P \overset{\cdot}{G}, \end{matrix}

(115)

\begin{matrix} \overset{\cdot}{\hat{Q}} = \overset{\cdot}{Q} + Q \overset{\cdot}{\tilde{G}}, \end{matrix}

(116)

\begin{matrix} \overset{\cdot}{\hat{\tilde{Γ}}} = {\overset{\cdot}{\tilde{G}}}^{t} \tilde{Γ} + \overset{\cdot}{\tilde{Γ}} + \tilde{Γ} \overset{\cdot}{G} + \overset{\cdot}{\tilde{\tilde{Γ}}}, \end{matrix}

(117)

where $\tilde{\tilde{Γ}} = axl ({\tilde{G}}^{t} S)$ . A comparison of the above relations with equations (96)–(98) indicates that the evolution laws ${f, g, h}$ and their counterparts ${\bar{f}, \bar{g}, \bar{h}}$ are related according to

\begin{matrix} \bar{f} = f + \overset{\cdot}{G}, \end{matrix}

(118)

\begin{matrix} \bar{g} = g + \overset{\cdot}{\tilde{G}}, \end{matrix}

(119)

\begin{matrix} \bar{h} + \hat{\tilde{Γ}} \bar{f} = h + {\overset{\cdot}{\tilde{G}}}^{t} \tilde{Γ} + \tilde{Γ} \overset{\cdot}{G} + \overset{\cdot}{\tilde{\tilde{Γ}}} . \end{matrix}

(120)

In these expressions, the quantities $\overset{\cdot}{G}$ , $\overset{\cdot}{\tilde{G}}$ , and $\overset{\cdot}{\tilde{\tilde{Γ}}}$ represent the arbitrary infinitesimal generators of the micropolar symmetry group. These relations embody the principle of actual evolution and, consequently, impose an additional constraint on the material evolution laws.

The physical meaning of actual evolution is tied to the notion of a material G-structure. Because the material transplants $P (t)$ , $Q (t)$ , and $O (t)$ are not unique in the plastic deformation of a micropolar body, and because at each material point with a continuous symmetry group each fiber of the G-structure is diffeomorphic to the Lie group of symmetries of the archetype, not every change in these transplants represents a true material evolution. For a plastic deformation to correspond to an actual material evolution, the fields $P (t)$ , $Q (t)$ , and $O (t)$ must evolve in such a way that their instantaneous rates leave the material G-structure at that point. Mathematically, as expressed in equations (118)–(120), this requires that the evolution function take values outside the material Lie algebra associated with the symmetry group. For example, according to equation (118), any evolution for which the resulting tensor f is skew-symmetric does not constitute actual evolution; it represents only a change within the material G-structure rather than a genuine modification of the material response.

Remark 4.4. The distinction between fixed and time-dependent material symmetries depends on the structure of the material symmetry group. If the symmetry group is discrete, each fiber of the material G-structure consists of isolated points, so any smooth change of the implant field necessarily leaves the original G-structure; consequently, every non-trivial evolution corresponds to a true change of material structure. For continuous symmetry groups (Lie groups), the fibers are smooth manifolds, and certain changes of the implant field may remain within the same G-structure, representing only internal relabeling. In this case, the principle of actual evolution requires that the evolution law produce values outside the Lie algebra of material symmetries to ensure that the evolution is physically meaningful.

For further discussion on this important point, the reader is referred to Epstein and Elzanowski [22].

5. Conclusion

In this work, we have developed a comprehensive framework for finite elastoplastic deformations within the context of micropolar theory. The formulation consistently accounts for large strains and finite rotations while capturing the influence of material microstructure. By deriving the balance equations from an invariance principle under general observer transformations, we ensure a physically consistent description of the system. The proposed evolution laws, expressed as first-order differential equations for material transplants, respect the constraints imposed by micropolar symmetries and the second law of thermodynamics. Furthermore, the identification of the micropolar Mandel stress tensors as the driving forces for local material rearrangements provides a clear energetic interpretation of plastic flow at the microscale. Overall, this framework offers a robust and thermodynamically consistent approach for modeling complex elastoplastic behavior in micropolar media and lays the foundation for future computational and experimental investigations.

Footnotes

ORCID iD

Mohammadjavad Javadi

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

Notes

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Finite strain elastoplastic behavior of uniform bodies based on micropolar theory

Abstract

Keywords

1. Introduction

2. Micropolar media

2.1. Kinematics

3. Balance equations

3.1. The master energy balance

3.2. Translation invariance

3.3. Rotation invariance

3.4. The Clausius–Duhem inequality

4. A micropolar theory for plasticity processes in uniform media

4.1. The theory of uniformity and the micropolar archetype

4.2. Thermodynamics consideration of the micropolar theory

4.3. The principle of maximum plastic dissipation

4.4. Micropolar evolution laws

4.4.1. Reduction to the archetype

4.4.2. Material symmetry consistency and actual evolution in a micropolar evolution law

5. Conclusion

Footnotes

ORCID iD

Funding

Declaration of conflicting interests

Notes

References